Classification Theory (Studies in Logic and the Foundations of Mathematics)

CLASSIFICATION THEORY AND THE NUMBER OF NON-ISOMORPHIC MODELS STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS VOL...

Author: S. Shelah

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CLASSIFICATION THEORY AND THE NUMBER OF NON-ISOMORPHIC MODELS

STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS VOLUME 9 2

Editors

J. BARWISE, Stanford H. J. KEISLER, Madison P. SUPPES, Stanford A. S. TROELSTRA, Amsterdam

NORTH-HOLLAND AMSTERDAM. NEW YORK * OXFORD .TOKYO

CLASSIFICATION THEORY AND THE NUMBER OF NON-ISOMORPHIC MODELS R E V I S E D EDITION

S. S H E L A H The Hebrew University, Jerusalem, Israel

1990

NORTH-HOLLAND AMSTERDAM'NEW YORK.OXFORD.TOKY0

ELSEVIER SCIENCE PUBLISHERS B.V Sara Burgerhartstraat 25 P.O. Box 211 lo00 AE Amsterdam, The Netherlands Dietributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y.10010, U.S.A.

Library of Congress Cataloging-in-PublicationData Shelah, Saharon. Classification theory and the number of non-isomorphic models/S. Shelah. -- 2nd ed. p. cm. -- (Studies in logic and the foundations of mathematics; v. 92) Includes bibliographical references. ISBN M - 7 0 2 W 1 1. Model theory. I. Title. 11. Series. QA9.7.S53 1990 51 1’.8--dc20

8S29756 CIP

First edition : 1978 Second edition : 1990

ISBN : 0 444 70280 1

@ ELSEVIER SCIENCE PUBLISHERS B.V., 1990

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the written permission of the Publisher, Elsevier Science Publishera B.V., P.O. Box 211, lo00 AE Amsterdam, The Netherlands. Special regulations for readers in the USA: This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the Publisher. No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or idem contained in the material herein.

Printed in Great Britain

CONTENTS

Contents. . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgements . . . . . . . . . . . . . . . . . . . . ix Introduction. . . . . . . . . . . . . . . . . . . . . . . xi Introduction to the revised edition . . . . . . . . . . xv Open problems . . . . . . . . . . . . . . . . . . xvii Added in proof . . . . . . . . . . . . . . . . . . xxiii Notation . . . . . . . . . . . . . . . . . . . . . . . . . xxxl

CHAPTER I. PRELIB~INARZES ....

............ Q 0. Introduction . . . . . . . . . . . . . . . . . . . . . Q 1. Preliminaries and saturation . . . . . . . . . . . . . . 8 2. Order. stability and indiscernibles . . . . . . . . . . . . CHAPTERI1. RANKSAND INCOMPLETE TYPES.

1 1 1 9

........ Q 0. Introduction . . . . . . . . . . . . . . . . . . . . . Q 1. Ranks oftypes . . . . . . . . . . . . . . . . . . . . Q 2. Stability. ranks and definability . . . . . . . . . . . . Q 3. Ranks. degrees and superstability . . . . . . . . . . . . Q P. The f.c.p., the independence property and the strict order property . . . . . . . . . . . . . . . . . . . . . . .

18 21 29 41

CHAPTER I11. GLOBAL THEORY .

82

.............. Q 0. Introduction . . . . . . . . . . . . . . . . . . . . . Q1.Forking . . . . . . . . . . . . . . . . . . . . . . . Q 2 . The finite equivalence relation theorem . . . . . . . . . Q 3. The instability spectrum . . . . . . . . . . . . . . . . Q 4 . Further properties of forking . . . . . . . . . . . . . . V

18

62

82 84 94

101

108

vi

CONTENTS

f 6. The fist stability cardinal f 6. Imaginary elements

............... .................. $ 7. Instability . . . . . . . . . . . . . . . . . . . . . .

122 130 137

CHAPTERIV. ~

150

........... f 0 . Introduction . . . . . . . . . . . . . . . . . . . . .

f f f f f

E

M

1. The set of axioms 2 Examples of F’s

O . .D. .~ .

...................

................... . 3. General properties of F-primary models . . . . . . . . . 4. Prime models for stable theories . . . . . . . . . . . . 6. Various results . . . . . . . . . . . . . . . . . . . .

CHAPTER V . MOBE ON TYPESAND SATURATED MODELS.

... f 0 . Introduction . . . . . . . . . . . . . . . . . . . . . f 1. Orthogonality. regularity and mhimelity of types . . . . . f 2 . Dimensions and orders between indiscernible sets . . . . . f 3. Weighted dimensions and superstability . . . . . . . . . f 4. Semi-regular and semi-minimaltypes . . . . . . . . . . f 6. Multi-dimensional theories . . . . . . . . . . . . . . . $ 6. Cardinality-quantifiers and two-cardinal theorems f 7 Ranksrevisited

.

. . . . .

............ .......

.. CHAPTERVI . SATURATION OF ULTBLPRODUCTS

...... $ 0. Introduction . . . . . . . . . . . . . . . . . . . . . f 1. Reduced products and regular filters . . . . . . . . . . f 2 . Good filters and compactness of reduced products . . . . . f 3. Constructing ultrafilters . . . . . . . . . . . . . . . . f 4 . Keisler’s order . . . . . . . . . . . . . . . . . . . . f 6. Saturation of ultrapowers and wtagoricity of yuuudoelementary claases . . . . . . . . . . . . . . . . . . f 6. Saturation of ultralimits . . . . . . . . . . . . . . . . CHAPTER VII . CONSTRUCTIONOF MODELS .

......... $ 0. Introduction . . . . . . . . . . . . . . . . . . . . . f 1. Skolem functions and generalizations of eaturativity . . . .

150 152 157 174 183 204 223 223 230 240 249 267 284 289 305 321 321 324 333 345 370 379 390 397 397 400

vii

UONTENTS

Q 2. Generalized Ehrenfeucht-Mostowski models . . . . Q 3. On the f.c.p., uniform trees and ID(T)I > IT/= KO . Q 4 . Semi-definability . . . . . . . . . . . . . . . .

.

$ 6 Hanf numbers of omitting types

...

...

... ............

411 419 426 432

.

CHAPTER VIII THE NUMBEROF NON-ISOMORPHIC MODELS IN PSEUDO-ELEMENTARY CUSSES . . . . . . . . . . . . . . 440 $ 0. Introduction . . . . . . . . . . . . . . . . . . . . . $ 1 . Independence of types . . . . . . . . . . . . . . . . . $ 2 . Unsupmtable theories . . . . . . . . . . . . . . . . Q 3. Saturated models and the w e h = lTll . . . . . . . . . $ 4. Categoricity. etlturation and homogeneity up to a cardinality

440 444 455 464 47 1

.

CHAPTER I X CATEUOBJCITYAND THB: NUMBER OB MODELS IN ELEMENTARY CLASSES. . . . . . . . . . . . . . . . . . 479

..................... Q 1. Supratable theories and categoricity . . . . . . . . . . Q 2. On the lower parts of the spectrum . . . . . . . . . . .

479 481 497

CHAPTER X . CLASSIFICATIONFOR F&-SATURATED MODELS

508

Q O . Introduction

$ 0. Introduction . . . . . . . . . . . . . . $ 1. Preliminaries . . . . . . . . . . . . . . $ 2 . The dimensional order property . . . . . . . $ 3. The decomposition lemma . . . . . . . . . $ 4. Deepness . . . . . . . . . . . . . . . . $ 5 . Deep theories have many non-isomorphic models $ 6. Infinite depth . . . . . . . . . . . . . . $ 7 . Trivial types . . . . . . . . . . . . . . CHAPTERX I . THEDECOMPOSITION THEOREM .

.

. . . . . . . . . . . . . . . . . . . . . . . .

. . . .

$0. Introduction . . . . . . . . . . . . . . . . . $ 1. Stationarization . . . . . . . . . . . . . . . . $ 2. The axiomatic treatment . . . . . . . . . . . . $3. Specifying the axiomatic treatment . . . . . . . .

508 509 512 520 527 533 548 550

557 557 557 561 572

viii

CONTENTS

XI1. THE MAIN GAPFOR

THEORIES .

590

$0. Introduction . . . . . . . . . . . . . . . . . $1. On FA"and F{ . . . . . . . . . . . . . . . . . $2. Stable systems . . . . . . . . . . . . . . . . $3. On good sets . . . . . . . . . . . . . . . . . $4. The otop/existence dichotomy . . . . . . . . . . $5. From the (Xo.2)-existence property to the (A. 2)-existence property . . . . . . . . . . . . . . . . . . . $6. The book's main theorem . . . . . . . . . . . .

590 591 598 603 608

XI11. FORTHOMAS THE DOUBTER . . . . . . .

622

$0. Introduction . . . . . . . . . . . . . . . . . $ 1 . Can the models be characterized by invariants ? . . . . $2. On having many models. no one elementarily embeddable into another . . . . . . . . . . . . . . . . . $3. On the Morley conjecture . . . . . . . . . . . . $4. ](Ha. T)for a large enough . . . . . . . . . . . .

622 623

CHAPTER

CHAPTER

COUNTABLE

616 620

627 634 643

........................ 653 $ 0. Introduction . . . . . . . . . . . . . . . . . . . . . 653 5 1. Filters. stationary sets and families of sets . . . . . . . . 653 5 2. Partition theorems . . . . . . . . . . . . . . . . . . 659 5 3. VarioUereeults . . . . . . . . . . . . . . . . . . . . 666 Historicalremazka . . . . . . . . . . . . . . . . . . . . 673 Referen....................... 684 Index of definitions and abbreviations. . . . . . . . . . . . 691 Index of symbols . . . . . . . . . . . . . . . . . . . . 703

~P~PNDIX

ACKNOWLEDGEMENTS

My reseamh in this area started with my thesis, and I thank M. 0. Rabin for his kind guidance. I thank the Israel Academy of Science for partially supporting my research, aa well aa the United StatesIsrael Binational Science Foundation (grant 11 10)and the NSF (grants 14PH747 and MCS-08479). Much of the proofreading and checking waa done industriously by M.Abramsky who waa my research assistant from 1973 to 1976,and for this I am grateful to him, aa well as to L. Marcus who did a similar work previously. They are responsible for the fey paragraphs which are in good English. I owe a special debt of gratitude to D. Ehrman, J. Alon and mainly D. Sharon for typing, correcting and recorrecting the manuscript. Lastly, I thank J. Bddwin, G. Cherlin, S. Koppleberg, D. Monk, M. Rubin and P. Schmidt for detecting various errors and inaccuracies.

ix

This Page Intentionally Left Blank

INTRODUCTION

The aim of this book is to represent works of the author on claesi6ication and related topics. The author, in a moment of insanity, believed this would be the easiest way to represent his work. There is no point in trying to convince you that I think the book is important; since otherwise I would not have spent my time writing it. Anyhow, I have said it in the introduction to [Sh 75~1,which is supposed to be a propaganda for this book. There is also no point in explaining how the subject evolved because I say almost everything in the introduction to [Sh 741. Let us note only that the founding stone is [Mo651 (there are historical notes at the end of the book). So we shall now explain how to read the book. The right way is to put it on your desk in the day, below your pillow at night, devoting ycuueelf to the reading, and solving the exercises till you know it by he&. Unfortunately, I suspect the reader is looking for advice on how n& to read, i.e. what to skip, and even better, how to read only some isolated highlights. If you are generally not interested in model-theory but are interested in ultrafilters and ultraproducts, you can read Chapter IV, Sections 1, 2 and 3. There, many claesical results are represented but also some new ones. We prove that there is a A-good not A+-goodultrafilter iff A is regular (improving Keialer, cf. [Ke 651); show that p < 2” A pwo = p iff there is an ultrafilter D over A and ni < w such that n,/D = p, thus anawering a question of Keisler (cf. [Ke 0781). We also answer one of the questions from [CK 731 (due to Keisler): there is a regular ultran,/D > No* n,/D = 2”. filter D over A > No, D not good, but Also an HI-incomplete D over I is A-good iff for every order J, JI/D is A-saturated. Also Chapter VII is quite isolated. The first section deals with a generalization of A-saturated, so that such models exist in more cardinalities. The second section presents a generalization of Ehren-

n

n

xi

n

xii

MTaODUCTION

feuchtMoetowski models, and w u m e only the knowledge of Skolem functions. The third section uses the second, and some definitions from previous chapters, and pretends to show the generalization of Section 2 is useful, so e.g., we have a model generated by a tree of indiscernibles and this is important for unsuperstable theories. The fourth section again requires no prerequisite, and there we prove that for any theory, and p 2 2ITl, it has p-universal, p-stable models of arbitrarily high cardinalities. In Section 5 we present the theorems on Hanf numbers of omitting types. Chapter VIII is devoted to the number of non-isomorphic models, but for this we use only some definitions (or rather equivalent conditions explicitly stated and used as alternative definitions).Also Chapter VII, Section 3 is used in Chapter VIII, Sections 2 and 3. Most of Chapter VIII, Section 1 is quite easy (and has real results) and there is not much dependence. In Chapter VIII, Section 2 we have somewhat heavier material; you can start by reading Theorem 2.2(1): if T is unsuperstable, h > IT1 is regular, then T has 2”non-isomorphic models of power A. Chapter VIII, Sections 2 and 3 are aimed at proving the same for any h 2 I TI + 24,; but Chapter VIII, Section 3 and the later parts of Section 2 me, in their present form, suitable for your too arrogant students (but no guarantee). Section 7 of Chapter I11 is independent, and contains results on the possible K(h) = sup(lS(A)I: IAI s A} and related problems. Also the combinatonal appendix needs no prerequisite, but it is not so interesting. If you arrive here I hope you understand that you should really start to read from Chapter I. Now Chapter I, Section 1 is a restatement of some classical theorems (compactnese and LowenheimSkolem which are proved later, elementary chains, saturated models, etc.). Now the reader should know what are firat order theories, formulas and satisfaction; and no more; but I would be very interested to meet a successful reader who has not read before a significant part of [CK731,or [Sa 721.We here a h make some conventions. Chapter I, Section 2 contains central definitions (stability, indiscernibility) and some theorems on existence of indiscernibles and the connection between unstability and order. Chapter 11, Sections 1 and 2 are easy and central. Here ranks are introduced and stability of formulas and theories are really introduced, i.e., we give many equivalent conditions. Chapter 11, Sections 3 and 4 are quite peripheral and independent. In Chapter 11, Seotion 3 we investigate more deeply ranks, and connect

INTRODUOTION

...

Xlll

them with superstability. Only Claim 3.12 of Chapter I1 is used in Chapter 11, Section 4. In Chapter 11, Section 4 we investigate some more properties of formulm and theories: the finite cover property, the strict order property and the independence property. Almost the only use later is in the investigation of Keisler orders and s&~rationof ultrapowers, in Chapter VI, Sections 4 and 6. In Chapter 111,Sections 1, 2, 3, 4 and 6 are very necessary for what follows; Sections 1-4 can be read only successively, Section 6 depends mainly on Chapter 11, Sections 1and 2. Unlike Chapter I1we concentrate here on complete types and investigate forking, and in Sections 1-6 deal with stable T only. Forking is introduced in Section 1, investigated in Section 2, used in Section 3 to (almost) give the stability spectrum theorem, and in Section 6 really to give it. In Section 4 we me the connection between ranks and forking, use them to prove results on both sides. I n Section 6 we extend our models so aa to have names for equivalence classes, this is important when we look for canonical forms. In Section 7 we investigate various properties, like in Chapter 11, Section 4, mostly for unstable T. Chapter IV deals with prime models. As we have to deal with five kinds of prime models, we give an axiomatic setting (in Section 1) fmd which examples satisfy which axioms (in Section 2, the results sum up in a table at the end). In Section 3 we prove theorems in this axiomatic setting (prime models exists, they realize only isolated types, etc.). Section 3 uses only Chapter IV, Section 1, but in Chapter IV, Section 2 extensive use of Chapter 111, Section 4 is made. The fourth section is somewhat heavier; we here concentrate on stable theories, and give characterization of prime models (implying uniqueness). Only a few lemmas from it are used later. The fifth section contains scattered results, most of them do not require much knowledge. The main theorem says, e.g., that for countable stable T, if a primary model (i.e. a prime model constructed step by step) exists, the prime model is unique. We also prove a theorem on the existence of a model of T omitting < 2ITI complete types. Chapter V contains some of the deepest results of the book. It uses Chapter I V but usually not in a deep sense, the reader would better go to its detailed introduction. Now in Chapter VI we turn to ultraproducts, Sections 1 , 2 and 3 do not refer to anything. I n Sections 4, 6 and 6 we use Chapter 11. The firat half waa already reviewed. In the second half we investigate

xiv

I"F&ODUOTION

Keisler order on theories ([Ke 671) and get quite a complete picture (to complete the picture we should know more on unstable theories without the strict order property), and find, quite accurately, how saturated am ultraproducts and ultrdimits. Chapters VII and VIII were already reviewed. Chapter VIII, Section 4 is to a large extent a summary of results, and has some not so hard theorems. In Chapter IX,Section 1we prove the categorioity theorems, relying on various previous resulta (including Chapters V and VIII). We also prove some results on the number of models in elementary claases (in Chapter VIII we concentrate on pseudo-elementary claeeee), and when a model can be represented as a union of strictly increasing chain of length 6. In Chapter IX, Section 2 we deal with the number of countable models of a superstable T ;and what can be the number of F&,-models of T in He (it is 1, IaI' (p s 21Tl), or 2 2la1),and just models when T is totally-transcendental. The Appendix contains the combinatorid results we need.

Note. (1) Each chapter hae its introduction, it may be wise to look at it again during the reading. (2) Exercises am scattered randomly among the sections. Some are the result of the public pressure due to the prejudice that examples clarify notions, others me variants of theorems I want to mention, or remnant of obsolete proof. And some were the result of my preferring to give an exercise with a generous hint rather than a theorem with a thin proof. (3) The change in the name of the book is not incidental, but a change in point of view during the yeam in which the book waa written, explained in [Sh 76~1. The reader may ctlso want to know which of my papers becomes obsolete by this book. So this is the fate of [Sh 69a], [Sh 70a], [Sh 7Ob] [Sh 711, [Sh 71b], [Sh 71d], [Sh 72a], [Sh 741 and [Sh 7481. Also [Sh72] except Section 2 which deals with the uncountable caw, [Sh 72b] except the purely combinatorid part and the theorem on L,,,,. But in [Sh 71b], the proof of the last theorem is not covered. We did not deal here with the results on non-elementary claases ([Sh 691, Section 6; [Sh 69~1,[Sh 701, [Sh 76a] and [Sh 7601).

INTRODUCTION TO THE REVISED EDITION

In this edition four new chapters (X, XI, XI1 and XIII) have been added. (In addition, many corrections have been made, including many which previously were in the “Added in Proof’’ part of the First Edition.) The additional chapters present the solution to countable first order T of what the author sees as the main test of the theory (XII, 6.1).

In Chapter X we introduce the dop (dimensional order property) and show that it is a very meaningful dividing line for superstable theories: if it holds, T has many &-saturated ( = Fgo-saturated) models; if it fails, we have a structure theory: every &-saturated model M of T is Fio-prime over a non-forking tree of models (M,,:qeI c u’A) (each M,,F;fo-primeover 9). If the tree I were to have an infinite branch, the theory is deep and still has many models. Otherwise, we can compute the number of #,-saturated models. This chapter is an improved version of [Sh 83a], [Sh 84bl. Mainly the IE case is explicated and the facts on trivial (regular) types are gathered in one section. In Chapter X I we prove the needed decomposition theorems assuming (*)

there are prime atomic models over MI U 4 when &, 4M,, A& {illl,&&} is independent over 4.

Chapter XI1 is the crux of the matter. We prove that the negation of (*) above implies that in models of T we can define a relation ;the definition being aR6 iff a type which order a large subset of p ( z ,a,6) is omitted. Chapter XI11 is intended to exemplify that Theorem XII, 6.1 fulfills its aim. For this we consider several questions which are

‘“w(

xv

xvi

INTRODUCTION TO THE REVISED EDITION

solved using its partition to cases. In Section 1 we deal with “when La,,-equivalent models of cardinality A are isomorphic ”. In Section 2 we deal with computing IE(A,T).Considering what was done in Chapter X, the main problem is t o show IE(h,T)= 2A when T has otop. Usually, the reader thinks this is immediate, forgetting that elementary embedding does not necessarily preserve the omission of types. In Chapter XII, $3,we prove the Morley conjecture : for countable first order T, No< A < y * I ( h , T)< I @ , T) except when T is categorical in N, (so (VA > N,)I(A, T)= 1) but I(No,T)= No. In Chapter XIII, $4, we compute I ( h , T ) for A large enough (though we do not know if a relevant invariant can be >No but

< 2x0). For a discussion on the significance of the new chapters, see [Sh 85bl. We thank Leo Harrington for hearing the proofs in Chapter XI1 while they were generated, and the reader may thank him for persuading me to prove the otop there using finite sequences rather than some considerably more complicated and infinite versions. We thank Saffe and Hrushovski for pointing out the need for corrections, Grossberg for proofreading VIII, and the Israel Journal ofMuthemutics for allowing us to use portions of [Sh 831 and [Sh 83a] which appear there. Last, but not least, we thank Damit Sharon and Alice Leonhardt for typing the new material and corrections. S. SHELAH November 1987

OPEN PROBLEMS

A. Completions Here we deal with problems which just complete theorems in the book. We feel existing methods will suffice. (1) Complete the computation of the possible function I(A,T) ( A 2 No,T countable complete first order theory). See XIII, $93 and 4. Though some think this was “the problem”, I could not make myself excited about it. Still it would be nice to know. The most appealing part is : ( l a ) What can SND(T)be? If it is a limit cardinal > K O ,what can k(T), Z( T)be ? Hopefully SND(T)E (w 1)U {&, (2’~)’} : even so, we shall need another cardinal invariant to compute I(A,T)for N, 6 A < If we are interested in I(&, T), too, we should ask ( l b ) Does the assumption I ( N o ,T)= No,T superstable, bound the depth of T ?[ ( w + 1) is possible.] (2) What can A be from Theorem V, 5.81 (3) Is DP(T)> lTl+possible ? Remember DP(T)< S(lTI), all successor < [TI+are possible; however, cf. IT1 > Noimply S(lT1)> /TI+. (4)Suppose A 2 K,(T)(or even A = K,(T)), M is F,”-atomic over A , and for every I E M indiscernible over A , dim(1,A , M ) < A. Then, is M necessarily F,”-prime over A ? Even various weakenings are open. You can assume A > IT1 [is regular] or it can be to prove for Ceq(this strengthens the hypothesis - we have more I ’ s ) , or try the parallel for Fi. Note that this is an attempt to improve the characterization theorem IV, 4.14 ; note also that for T superstable this is true (see IV, 4.18). ( 5 ) Suppose T is stable (D(T)( > IT(‘“,(Vx < ,u)[x‘~< JD(T)I]K

+

xvii

xviii

OPEN PROBLEMS

regular ~ K ( T Prove ). that for every A = A<“ 2 IT1 T has a t least min(ZA,2.1 non-isomorphic K-compact models of power A. See IX, 1.20. (6) For a shallow superstable without the dop variety the depth is < w. (This means that if every completion of an equational theory is shallow superstable without the dop, then every such completion has finite depth.) See [Sh 861. (7) Is every unidimensional stable T,superstable ? (8) See questions in Baldwin and Shelah [BSh 851 and [Sh 86aI.

B. Classifying further unstable theories Clearly not all unstable theories are equally complicated, and we may want finer divisions : Theorem II,4.7(1) suggest the strict order property and independence property, and IV, $7 the tree property. We shall deal here with some problems that look like reasonable approaches. (1) What can { ( A , p )be: every model of T of power A has a pcompact extension of power A} ? The interesting case is IT1 < p < A, x<’’ < A < 2 X (see Ex. VIII, 4.5, which is solved in [Sh 81, Th 1,18]),and we can assume T i s unstable but simple (i.e. without the tree property, since otherwise we have an answer). By [Sh 801, a positive non-trivial answer is consistent ; by [Sh 811 if we change the problem a little (restricting the pcompactness to a predicate) this cannot be proved in ZFC. If the idea can be transferred to the usual case, we may still hope for w 1 cases (with [Sh 811’s example being for 3 and w’s one are those like Tind; the needed analysis may look like [Sh 83al). (2) Continue the investigation of Keisler order. See VI and [Sh 72, $11. Note that the countable theories left are those unstable T without the strict order property, and that the simple partition with the tree property seems meaningful, so [Sh 801 seems relevant. It seemed that the inner partition of unstable simple theories (if exists) is different than in (1).Also note that this involves building special ultrafilters, so we may choose another variant of the order (e.g. limit ultrapower M i I W with W HI-complete ; see Keisler [Ke 631).

+

OPEN PROBLEMS

xix

(3) The following definition was suggested by Grossberg and the author.

DEFINITION 3a : Let h * (p)‘” mean that if I , A E M , M t T , IAl < x, T..X 14 = h then some J E I is an indiscernible sequence over A of power We can ask,

CONJECTURE 3b : Suppose T is without the independence property, then for some a, IT1 +x) += (p);:’. This is confirmed in [Sh 86a] for theories that have no monadic expansions with the independence property. Note that we get complete negative results if T has the (**)-independence property, where

DEFINITION 3c: (i) T has the *-independence property, if for some q@,, z,,Z,,8) for every k < w and w E k there are 6 and (i < k) such that for every distinct i, < i, < i, < k (ii) T has the (**)-independence property if there are formulas qn(z0, . . . ,z,,g,) ( / ( E t ) constant) such that for every W, E “ w ( n < w ) there are 6,(n < w ) at(i< w ) such that for n < w and i, < .. . < in-,

We may weaken (ii) and still obtain the result. (4)Find what K = { A :T has a universal model of power A} can be. Note that for “saturated” we know the answer (see VIII, 4.8). Also, for a strong limit h > ITI, h E K ,and if T has a saturated model of power A , then A E K .Surprisingly, some independence results were found: for Tord see [Sh 80a], and for Tind [Sh 841.

xx

OPEN PROBLEMS

C. Problems parallel (to the book) (1) Prove the main gap (etc.) for uncountable theories. [Here and elsewhere we concentrate on the main gap, but expect the parallel to XI11 too.] Note that almost everywhere in the book the countability of the theory is not particularly important. I n fact, there is just one point (but a crucial one) in which we use countability: in XII, $4 (the otop/existence dichotomy). It seems we should try to deal with inevitable types (instead of isolated ones) (see proof of IV, 5.9) or even with just prime (but not necessarily primary) models over suitably chosen non-forking trees of models. We may concentrate on unidimensional superstable T (this suffices to prove either I(A,T)< Z2lT1 or T)2 loll). Note that if q ( x ,a) is weakly minimal, i% U p(M,a) E A E M , then M is Fio-atomic over A. (2) Prove the main gap for the class of (TJ+-saturatedmodels. Again only “one point ” is missing : the existence of a regular type ~ Note the analysis of non-multidimensional T (in in X, $ 3 ’ proof. V, $5) and the dimensional continuity property (see X, $2) (which take care of the cases that an increasing union of saturated models do not). We can look also a t other kinds of saturation. (3) Prove the main gap (etc.) for the class

for T countable. See VI, 5.3, 5.4, 5.5 and 5.6. A priori we considered the order of difficulty in increasing order as: Ch. X (i.e. the main gap for Fgo-saturated models), question (2) above, the present question C3 and Theorem XII, 6.1. The fact that XII, 6.1 was solved before C2 and C3 is due to our greater interest, but we may have been wrong. A relevant theorem is [Sh 751. Note that a hidden order property for those classes is: (Pr)

I n CeQ,for some formula p(x,g , @ for every k,n < o,there are at, (t< n)such that for every t,< t,< n,m, c m 2< n,

OPEN PROBLEMS

xxi

I V ( ~ ” “ >at,,@eslk d I v ( ~ ~ “a,m z , @m,)I, d I v ( ~ , ~ ~<, N, o~. ~ ~ I By [Sh 751 if T has (Pr), I(A,K,) = ZA for A > 2*0 (using cardinality quantifiers). (4) Prove the main gap for strongly A-saturated models (i.e. for Fi-saturated models strongly A-homogeneous, i.e. if (ai:i < a < A ) r M ( b , : i < a < A ) s M realizes the same type, then some automorphism of the model takes one to the other). The following problem should be easier to solve. (5) Prove the main gap (etc.), for the class of Nosaturated models (Tcountable). (6) Classify when we replace “isomorphic” by “L,,,uequivalent ”. (6a) Classify the possible Scott heights of models of T. See the works of Nadel and [Sh71a]. This does, of course, have some connection to the Vaught conjecture. (7) We know that { A :T has a rigid model of power A} may be “bad” (see [Sh 76~1).But what for IT(+-saturated models? What about the number of non-isomorphic ones when a t least one exists? See [Sh 83b] on very partial positive results. (8) Vaught conjecture, i.e. (for complete countable T) I ( K o ,T)> No=I ( N o ,T)= 2*0. We can look also at variants of it: the number of minimal, models or rigid models. Morley [Mo] proves I ( N o ,T)> K, *I(No, T)= 2’0, and the proof applies to many other cases (but this really belongs to description set theory, where stronger theorems have since been proven, e.g. that of Burgess; see [Sh 84bl). Some people think this is the most important question in model theory as its solution will give us an understanding of countable models which is the most important kind of models. We disagree with all those three statements.

xxii

OPEN PROBLEMS

D. Further reaching problem (1) Define " T has a structure theory over a class K" and classify accordingly. It seems reasonable to look a t specific K's. A reasonable interpretation is: for some K , A for every model M of T there is a model I E K ,and functionsf:M-t"I, g : M - t A such that if a : ~ M ( = t 1,2,i < n),g(a2)= g(a,2) and ( f ( a 2 ) :i < n), (f(a,"): i < n) realize the same atomic type in I , then (a: : i < n), (a,":i < n ) realizes the same type in M ; moreover, if F,BeKI,h < h and c,J realizes the same atomic type in I , then

(3a c M ) [f(iz)= F

A g(a) = &] iff

(3a EM)[f(a) = Z A g ( a ) = a].

So the main gap says which T has a structure theory over K = { I :I a tree with d o levels and depth
u,

~ ( A , N= ) { ( M ,a ) a e N :/ ~ ~T,b ( 1 ~ 1 1= A,,MrPM = N>. (2b) Replace (M,a)aeNby M in (2a) above. (2c) When over M r P M there is a prime model. (3) Classify for $EL,,,, or $€LA+,,, or for abstract elementary classes see [Sh %a]. For example: is categorical in one u , 2 then it is categorical (3a) If $ E LA+,, in every ,u 2 &(A). (3b) If $€LA+,,,@ as in VII, $2, @ is such that E M ( I , @ ) I= $ for every I , then either for every A 3 a,( I@l)({EM(I,@)/ E : III = A}( = 2A,or (possibly changing @) for every Al{EM(I,@)/ E 111 = A} = 1. See [Sh 751, [Sh 83a], [Sh 85aI.

ADDED IN PROOF

In order to make the process of writing converge, the author “closed” it to innovations in the summer of 1983 (except that Chapter X, Section 5 , was expanded drastically to include the proof for I E ) . So let us comment on some of the relevant improvements and advances. 1. Uniqueness of prime models (IV, 5.6)

This is old, but still note [Sh 79a] while reading Theorem IV, 5.6 (which says that if T is countable and stable, and there is a primary model over A , then the prime model over A is unique). By [Sh 79a] the assumptions “countable ” and “ stable ” cannot be omitted and an alternative proof of the theorem is given. 2. Regular, semi-regular and simple types (Chapter V )

See several works of S. Buechler, the thesis and some works of E. Hrushovski and Buechler and Shelah [Bush] and Hrushovski and Shelah [HSh]. More on the dichotomy of multi-dimensionality for stable theories (strengthening it) will appear, relevant to problem (12).

3. Chapter VI S. Koppleberg continues the work on ultraproducts of finite cardinals (and gets quite general sets). 4. Numbers of non-isomorphiclpairwise non-embeddable models

(Chapter VIII) There are improvements and (more) independence results and xxiii

xxiv

ADDED IN PROOF

counterexamples restricting possible improvements, see [Sh 80a], [Sh 841, [Sh 85e, Ch 1111, [Sh 88b] and [Sh 891. The improvements are : (a) We improve the proof of VIII, 3.2 (give new results for nonfirst order cases). See [Sh 85e, Ch. 1111. (b) Improving VII, 1.8: if T E T , are complete, countable T , KOunstable not stable, then IE(h, T,, T)> a2}for h > KO by [Sh 88b, Theorems 1 and 21. then IE(h,T , ,T ) (c) Improving VIII, 1.2 if T E T , , ID(T)1> IT1l+, > lD(T)INofor h > JT,I+.See [Sh 88b, Fact 31. thenlE(h, T , , T ) = (d)If T is unsuperstable, h > (T,I +KO,T G T,, 2A(and general principles), which implies this is proved (see [Sh 831 for many cases and some principles, and [Sh 891 for all). For independent results : (A) It is consistent that for some unsuperstable countable T , and T , , T G T , , lTll = K, and I ( K , , T , , T ) = 1 (by [Sh 8OaI). (B) It is consistent that for T = Tord,the theory of dense linear order, or TZd, for some T,, A = lTll > KO,I E ( h , T , , T )= 1 (by [Sh 80a] T = Tord,h = K,, by [Sh 841 T = TZd,h = K , , or one of many others). Mekler generalizes the results on graphs to classes of universal theories (C) For a quite comprehensive list of restricting independence results ID(T)I> IT1[,see [Sh 88bI. 5. Decomposition theorem (Chapter X I ) We here prove [by XII, X I 2.4, for (Tt, Ga)] (*)

if T is countable superstable without dop and otop, then for every M there is a non-forking tree ( N , :7 E I ) , M primary over UN,, IIiVJI < 20 ' . Moreover, any suitable tree of this form by (i.e. using (Ti,, E,))can be extended such that the model M is prime over it.

By [Bush 891 this missing axiom for, e.g., (TLo,C,)holds (in the cases we need) so we can apply this pair in Chapter XI11 to improve the cardinals in the results ( 11N711 < KO). Earlier results do this too. Saffe (in Trento 7/1984) suggests using the notion of elementary submodels used in I X , 1 . 1 , 1.5 used also in V, $6, i.e.

A G,*B if for every ~ E Bfinite , d and G E A and p such

ADDED I N PROOF

xxv

that l=cp[b,a]there is F E B , satisfying l=cp[6’,it] and Bytp(6,a),A ) = R”(tp(F,a),A),

) . we can get there IIN711 < No.But he and prove (*) for (Tt,c ~ Hence, does not prove Ax(C1). Somewhat later, Harrington proves it and the author notes that since trivially [A c , B * A c,B] it follows c,)hold for (Ti,, s d )[see that the other axioms which hold for (Ti,, iTI,3.17(3)][only for Ax(A5) use (Cl)],hence X, 2.7, X, 2.15, X, 2.16 ilpply. Thus, showing that very little need to be added to X I to get the above improvement, as well as replacing XIII, 1.1, 1.4, $2 “ A > ITI+2Ko” by “ A > ITI},“(2No)+”by “K1”. Let us turn to the list of open problems. 6. Problem A7 The problem was: “IS every unidimensional, stable T,superstable ? ” Hrushovski answers positively, using the interpretation of groups. 7. Problem C4

The problem was: “Prove the main gap from a Fi-saturated strongly A-homogeneous model”. For A = KO, we compute the possible spectrum functions [Sh 88aI. Also, the related problem of a A-resplendent model, A > IT1,was solved.

8. Problem 0 2 (Classification over a predicate) This is essentially solved (but using independence results for nonstructure, and much is not yet written, see [PSh 851, [Sh 85d] and [Sh 86bl). 9. Problem 0 3

The classification of a universal class (i.e. MEK if and only if every finitely generated substructure of M belongs to K) is essentially solved see [Sh 85e] and [Sh 891, but not completely written. see a work by Grossberg and the On the categoricity of author (on the categoricity of w successive cardinals) in preparation ; and a work by Makkai and the author (and more in preparation).

xxvi

ADDED IN PROOF

10. Chapter III, Section 5 (on instability and D ( T ) )

Newelski had shown that we cannot in general improve the results there. 11. SND

E. Hrushovski has proved that for countable first order T , SND(T) satisfies CH. We report below on other relevant works of the author. 12. Better invariants

We prove that if T is superstable without dop, for characterizing up to isomorphism He-saturated models we need only finitary invariants; more specifically we can use the theory of the model in the logic with quantification over sets which are the algebraic closure of finite sets [in a""]and has quantifiers on dimension. 13. Multidimensionality

As we say previously the results of V, $ 5 can be improved. I n fact we have now a typescript [Sh 891 proving the following. Assume T is stable (first order complete) and it is multidimensional (see V, $ 5 ) , then there is a (stationary) type orthogonal to the empty set [hence : if T is stable in N, and a < p then T has 2 21,-al pairwise non isomorphic F;t -saturated models of cardinality N,]. 14. Isomorphic ultrapowers

We proved that it is consistent with ZFC, that there are countable models M , N which are elementary equivalent, but for every ultrafilter D on w , M"/D and N"/D are not isomorphic, moreover Th(M) = Th(N) does not have the strict order property and even for different ultrafilters we do not get isomorphic ultrapower. This will probably appear in the proceedings of the MSRI 10/89 symposium in set theory. It also seems that, e.g., in some models of set theory, some ultrafilter will not give isomorphic ultrapower in the use of Ax Kochen.

ADDED IN PROOF

xxvii

15. Universal models

In a work in preparation by Kojman and the author more is done on characterizing the possible classes { A : T has a universal model of power A } for T first order, e.g., T is countable with the strict order property or unsuperstable and K, < 2'0, then T has no universal model in K,. 16. Resplendency

We note that the following are equivalent for a first order T : (a) T is stable, (b) if M is a ITI+-saturated model of T satisfying the following then it is saturated: for every c i € M for i < ITI, if q is a consistent theory extending Th(M,ci)i
17. Omitting types We forgot to mention a work of Hrushovski and the author, to appear in the Israel J . of Math., characterizing quite fully the possible sets {h(T,p ) : p a type} where h(T, p ) is sup { (IM11 : M a model of T omitting p } for T (countable first order) stable and for T superstable. 18. Universal classes

The appearance of [Sh 891 has now been delayed for several years. I n addition to the four chapters which appeared (in [Sh 85e]), a reasonable amount of material has been available in the form of a preprint before those four were sent for printing (i.e., Chs. V, VI, and 111, $6). Meanwhile Ch. VII has been lengthened and rewritten and more material has been added (mainly 111, $ 7 ) . We shall summarize this below. Note also that works of Baldwin and Shelah on the primal framework are beginning to appear (it is a more general framework of course). 19. Chapter III

Sections 1-5 have been revised (mainly the later parts of $3, e.g., a complete proof that if in EM(1) we can define (in any logic) order

xxviii

ADDED IN PROOF

on { ( q ( x t ) : i < p ) : t E I } then for any h 2 p+++ 17’1 we have 2A pairwise non isomorphic models of the form EM(I). Section 6, a survey and results on black boxes, is mainly in more finished form (but there is the knowledge that, e.g., a club of 6 < u1 pp(&) = 2’8 helps to get better black boxes on (16+l)f). We add section 7, done later, where it is proved that, e.g., IE(A,T,,T ) = 2A when h > ITl\ (even if A is singular). Some of the cases depend on “guessing clubs in ZFC” which has meanwhile found other applications. 20. Chapter IV

This has been revised but nothing novel added ; here and later we work mainly on Axiom Framework 1. 21. Chapter V

In Section 1 we prove that if K is ( < p , < p)-smooth and ( < p+,p ) based and satisfies LSP(p), then K satisfies smoothness, is ,u*-based if the assumption and satisfies LSP(p*)for every p* 2 p. For p = fails we have strong non structure results, hence we assume the conclusion. Having smoothness, we can allow ourselves to work inside the monster model 6 ; tp (@, A ) is now defined as {F(@): F an p)-homogeneous is the replacement automorphism of 6 over A } , (DK, of y-saturated, NF(M,,M,,M,,M,) is the replacement of: M iare algebraically closed sets (not necessarily universes of models of the fix first order T ) M , c M l c M,, M , C M , EM, and tp(M,,M,) does not fork over M,. Now Sections 2-7 are concerned with “all that we know on stable theories for (TI+-saturated models holds in this context”. Well, some definitions have to be adapted - we deal only with types of the form tp(N,M),where NF(NnM,N,M,a), usually [IN11 < xK.The proof of “ every type p , dom(p ) of power < p has a p+isolated extension ” is somewhat more complicated (we have to add variables to the type during the extensions, in order to continue to deal with types of the allowed form only, i.e., of models).

xK,

22. Chapter VI

Here we generalize the superstable theory to our context. But first we have to be able to deal with types tp(a,M),i.e., types of elements.

ADDED IN PROOF

xxix

We say tp(a,N) does not fork over M < N if for some Nl we have M < Nl and a E Nl and NF(M,N, N , , a). We define stationarization and parallelism too, and prove the expected results (though, e.g., the uniqueness of stationarization is no longer totally trivial). Next K ( K ) is no longer a cardinal but a set of regular cardinals < xK,K E K ( K )iff for some (Mi, i < K ) increasing continuous sequence of Ma < 6,and some a E a we have : tp(a,M,) forks over M ifor every i < K . K is called superstable iff K ( K )is empty. Under those definitions the theory generalizes - e.g., non superstability implies strong non structure theorems, the union of an <.-increasing chain of (DK,p)homogeneous models of length 6, cf(6) not in K ( K ) is (DK,p ) homogeneous, also the theory of regular types. [Also generalizing Chapter X here is straightforward, and we can prove the categoricity theorem, i.e., like Los conjecture.] 23. Chapter V111

The aim of this chapter is as follows. We strengthen the notion of elementary submodels t o : M < * N iff M < N and for every type over M , < M with < K variables with llM,II < h which is realized in N , is realized i n M too. Our aim is to show that all the good properties still hold (i.e., Axiomatic Framework 1 ) . More exactly, we show it except when the class has some non structure property. Unfortunately one of these gives h (usually) A+ non isomorphic models (rather than 2”). Essentially the point is that we do not know to discard classes which behave like the class of linear well ordering. One of our problems is to get non structure from order. The problem is that the relevant order is not preserved when we pass to Ehrenfeucht Mostowski models. The solution is a theorem on the existence on indiscernibles which is novel also in the first order case (and is explained below), and then use universality. Subsequently we prove non structure when unions are not O.K. Maybe it is the right place to say again what the point is of the indiscernibility theorem in Chapter I, last section, of [Sh 85el. For a first order T,to get an indiscernible subset J of I of power A, we need a large cardinal (indiscernible set rather than just n-indiscernible set). Of course we get the theorem assuming stability of T,i.e., no order on n-tuples for every n. There we say that if I consists of ntuples it suffices to assume there is no long order on 2n-tuples (and if we phrased it rightly - on n-tuples). What we have to say on order,

xxx

ADDED IN PROOF

stability and existence of indiscernibles, for any model - not necessarily of a stable theory, is said here in Chapter I, 2.12,2.10 [the first says there are indiscernibles in the presence of stability and lack of long order, the second says that the lack of long orders implies stability; the case of order of length w is spelled out in 2.11, and remember the trivial IS(A)I < &,lSv(A)l (or see 11, 2.15)]. The indiscernibility theorem we use in ([Sh 891) Chapter VII says, is )given, ) IAJEK, e.g. for first order stable T,that if ( A t : t ~ P G z ( A ~ ( 5 "< ) K (where Y s 2 ( h = ) { t : t E A, It1 < 2)) then we can find Y G A, IYI = p + and ( B t : ~ E Y ~ ~ ( such Y ) ) that: (a) ( B t : ~ E Y ' , ~ ( Yis) ) z ( ,Y ) t, is not a subset independent, i.e., for every t , tp(B,, u{B,:~ € 9 of s}) does not fork over {Bs:s a proper subset of t } , (b) A , c B,, lBtl 6 K, provided that A > a 1 ( p K ) . Similar theorems hold for the structures { p : p a finite sequence of ordinals < A ) and for the structures {p : p a strictly decreasing (finite) sequence of ordinals < A}.

NOTATION

We use i,j, a,8, y , 6,5 for ordinals; k, I, m, n for natural numbers (i.e., finite ordinals); K , A, p, x for cardinals (i-e.,initial ordinals, i n h i t e if not stated otherwise). 8 is reserved for limit ordinals. i < a, K < A mean i E a,K E A, respectively. A sequence is a function 3 whose domain, Dom 3, is an ordinal, which is also called the length of B and denoted by I@). B [ i ] denotes the image of i under 8, i.e., the ith member of the sequence. B 1 a (or sometimes I I a) is {(i,8[i]): i E a n I@)}. B Q t means I = t 1 I @ ) ; we write B is an initial segment of t . For any set A, "A = {a: Z(3) = a,Range 3 E A}, "'A = UBtaBA. We sometimes write A", A'" instead of aA,"'A, respectively. Sequences of ordinals are denoted by q, v , p, a (U is also used to denote permutations). The empty sequence is denoted by ( ). Ha is the ath infinite cardinal. 2,(A) = >(A, a) is defined inductively 23(A*B); aa(No)is written by >,(A) = A, and for a > 0, >(A, a) = aa. A+ is the cardinal successor of A; "'A = &s,,
zE
.

xxxi

xxxii

NOTATION

is a non-empty set (the “universe” of the model), and if R is an n-place predicate symbol of L then RM,the interpretation of R,is an n-place relation over 1M(, and if F is an m-place function symbol of L then FM,the interpretation of F,is an m-place function from to !MI. Since lMl is the universe of M , IlMll is its cardinality and it is called the cardinality of M . L(M) is the language such that N is an L-model. Models are denoted by M ,N and occasionally by 9. A, B and C denote sets contained in some [MI ;a, b and c denote elements of some ]MI and a, 6 and 2 finite sequences of elements; 8-6 denotes the concatenation of 7i and 6. We write iZ E A instead of E “’A and sometimes write E M instead of B E IMI. We often write just ti instead of Range 8, e.g., we write A u iZ instead of A U Range and if I is a, set of finite sequences we write U I or sometimes just I for U {Range8:si E I}. If p(Z) is an L(M)-formula E IMI Z(a) = Z(Z) we write M p[a] to mean p[a] is true in M . Th(M) is the set of all sentences true in M. M is elementarily equivalent to N, written M = N, if Th(M) = Th(N). If P is a unary predicate symbol of L(M)we often write P ( M )for PM (note that P ( M ) c ]MI),and if cp(E; 8) is an L(M)-formula, Z E 1M1, Z(B) = Z(p), we write p ( ~a); for (6: 6 E 1M1, Z(6) = Z(Z), M I= p[6; a]}. M is a submodel of N, written M E N, if L = L(M) E L(N), [MI c INI, for every R E L and E IMI, si E RM O E E RN,and for every F E L and 7i E !MI,FM(a)= J”(7i).If M i c M,for all i < j .c a then the union of the M i , Ui
xxxiii

NOTATION

If we are not interested in A, we just write p is a A-m-type in B. Types are denoted by p, Q and T . We write m-type instead of L-m-type, A-type instead of A-1-type, p-m-type instead of {p}-m-type, and A-( < K,)-type instead of A-m-type for some m < w . But from 111, $ 3 on type [A-type] means ( < K,)-type [A-( < KO)-type]. p is a complete A-nz-type over A in M if it is maximd, that is, if z E A , p ( ~8); E A , @) = I @ ) , then p(Z; Z)E p or a) E p. z realizes p in M if V ( X ;a)~p * M l=pl[a; a]. We identify 7-p with pl. p restricted to A, written p A, is {?(it; a) ~ p E: A}; p restricted to A , writtenp A, is {?(it; 8)~ pp(f; : E A or -,p(E, 8)E A } ; ~restricted positively to A, written p t+ A, is {p(p;a) ~ pp (: ~jj;EA}. If A = {p} we write p t p and p p instead of p {p} and p I+ {p} respectively. The domain of p , Dom p , is the smallest set over which p is a type. We sometimes consider types with an infinite number of free variables. Let A be a set of m-formulas, i.e., A = {p@; gf): Z(Z) = m, i c 1 A l } cll(A) denotes the closure of A under negation, i.e.,

a)

r+

cl,(A) = (pi(%gf):i < Id]}u {7p&3; &): i c = {$f(%

pi}

a):i < plwl}.

clf(A) denotes the closure of cl,(A) under conjunction, i.e.,

clr(A) denotes the Boolean closure of A, i.e., clW =

{V A

$dZ;

$&;

#if):

E c l l ( 4 IwI,lvfl

< go}.

fEW jEUt

cl,(A) denotes the closure of A under negation, conjunction and permutation of variables, i.e., cl,(A) = @(A') where

A' = {pf(u(Zf)):

= iiT&, 0 a

permutation of 2, (strictly speaking of Range Zf), i c Id]}.

We use mainly clg(A). We define pf as p if i = 0 and lp if i = 1, pif("')is p if " . . ." is true and lp if " . . ." is false. Sometimes when we do not want to write p(Z;a) out in full too many times we write "let p = p(Z; Z)" in an abuse of notation. When we say a formula" we may mean p, p(E), p ( B ; 8) or p(B; Z). Letfbe a function. For a set A,f(A)= { f ( a ) :a E A}; for a sequence 3, (6

xxxiv

NOTATION

Range I E Domf, f(3) is a sequence t, Z(Z) = I @ ) , t[i] = f(a[i]); for a type p , f(p) = {tp(Z;f(a)): tp(Z; a) ~ p } (In . practice there will be no inconsistencies.) I,J denote orders or index sets. I, J denote sets of finite sequences; this has been reversed in Chapters I-V and IX. Theorems, lemmas, claims, conclusions and corollaries are numbered together and are often referred to just by their number, e.g., 11, 4.16 means Theorem 16 of Section 4 of Chapter 11.Within Chapter I1 itself the theorem would be referred to as 4.16. Exercises, problems, questions and conjectures are numbered together; they and definitions are referred to always with name attached, e.g., Definition VII, 3.2 means Definition 2 of Section 3 of Chapter VII. Within the chapter itself, the chapter number is not given. However in Chapters X-XI11 definitions are numbered together with theorems. When we wish to write two almost identical statements we often write only one and insert the variations from the other statement at the appropriate places in round or square brackets, see e.g., Lemma I, 1.12 and Definition I, 2.2(2).

CHAPTERI

PRELIMINARIES 1.0. Introduction We here introduce our notation, which is quite standard, then, in Section 1, give some of the classical theorems of model theory (compactness, Lowenheim-Skolem, and the existence and uniqueness of saturated models). They are included mainly for the sake of completeness. Some of them are proved later (compactness after VI, 1.3, Lowenheim-Skolem in VII, 1.1-1.3). In Section 1 we also introduce &-a saturated model such that we restrict ourselves to elementary submodelsof it of cardinality < I(all. By this way we do not lose generality, and hopefully improve presentation. In Section 2 we deal with some problems concerning stability of models rather than of theories (on which we concentrate in the rest of @'(A, M), stability, indiscernibility the book). We define tp,(a, A, N), and splitting; and then investigate the connections between stability, order and the existence of indiscernible sequences: By 2.6 if, e.g., pi = tp(a,, A u U {a,:j < i}) does not split over A, j < i => p , c p , for i < a,then {ai:i < a) is an indiscernible sequence over A. (This will be the way by which we construct indiscernible sets.) We prove in 2.8 that if Th(M) is stable in A, I E M , 111 > A, IAI I A, A G N,then there is a J E I indiscernible over A, IJI > A. The method here is to find p E Sm(C,N )and B E C, such that for every C', C 5 C' E IN[, p has a unique extension in Sm(C,N )not splitting over B, and when IC'I s A, this type is realized by some E E I. In 2.9 we prove that unstability implies the existence of ordered I (rememberthat by 11, 2.13, T is unstable if€ it has the order property). 1.1. Preliminaries and saturation

We shall use freely the: COMPACTNESS THEOREM 1.1: If T , is a set ofsentences,und every finite subset of T , hm a d e l , then T , hae a model. 1

Proof. See Chapter VI after Lemma 1.3.

DEFINITION 1.1: (1) M is anelenzentaryszlbW1of N , M < N if EN and for every E 1M1, 'p E L ( M ) , M C 'p[a] e N C 'p[a]. (If not mentioned otherwise, we assume L ( M ) = L ( N ) . ) (2) M,,i < a is an elementary chain if i < j < a => M,< M,. (3) M,,i < a is a wntinwMLB elementary chain if it is an elementary chainandSM,=U,,,M,. (4) If iK is an elementary submodel of N , N is called an elementary extension of M. LEMMA 1.2 (Tarski-Vaught Test) : M ie an elementary submodel of N iff M ie a m.&moukZ of N , and when 8 E 1M1,b E IN[,p E L ( M ) ,N C p[b, a], then there ie a b' E 1M1 for which N C p[b', a]. Proof. The "only if" is clear. So suppose the second condition is satis-

E [MI, 'p(5)E L and we shall prove by induction on 'p that M C p[7i] o N C p[a]. If 'p is atomic, this holds as M is a submodel of N . If we get 'p by

fied. We let

connectives, the result follows by using truth tables. So we are left with the c&88 p = (3y)$(y, 3).If M C I@], then for some b E 1611,M C $[b, a], so by the induction hypothesis N C $[b, a] thus N k (3y)$(y, a), i.e., N C p[a]. If N C 'p[iZ], then for some b E INI, N C $[b, a], so by assumption for some b' E I M I, N C $[b', a], so by the induction hypothesis M k$[b', a] thus M C ' p [ i i ] . We have shown iK C p[a] o N C p[a] and thus have finished the induction and the proof of the theorem.

T H E ELEMENTARY CHAIN LEMMA 1.3: (1) If M,,i < a is an elementary chain, then M,< U,< a M,. (2) < ie a partial order; and if M < N,then p is an m-type in M iff p ie an m-type ouer 1M1 in N . If M,4 N , 1 = 1 , 2 and Ml C Ma,then M I < M a , 80 if M,< N and Mf E M , f o r i < j < a,tlrenU,,,M,
M

=

U,< a M,;so by definition, M,c M

for i < a. We E lM,l, p(3) E L ( M , ) , hi C 'p,P] e

shall prove that for every i < a, 211, C @I; by induction on p. The only non-trivial case is p = (3y)$(y, 3). If M tC p[a], then for

CH.

I, 8 13

PRELIMINARIES AND SATURATION

3

some b E lM,1, M i C #[by a] hence by the induction hypothesis M C #[by a] so M C (p[a].If M C p[Ti], then for some b E 1M1, M C #[by a]; but as lMl = lMil for some j,i Ij < a,b E IM,I. By the induction hypothesis M , C #[b, a], hence M , C (p[zi], so, aa M , < M,, M , k ~ [ a ] . (2) Immediate.

ul
DOWNWARD LOWENHEIM-SKOLEMTHEOREM1.4: I f A then M irag an elementary subntodel N (L(N) = L(M)), A IlNll = I 4 + ILL

E 1M1, E INI,

Prmf.See VII, 1.1, 1.2 and 1.3. LEMMA 1.6: (1) If a, E 1M1 for i < a, then (M,.. . , a , , . . .),<=< ( N , . . .,a,, . .){ iifs iK < N . (2) If M c N , Th(M) G Th(N) and in M every element i8 an individual constant, then M < N .

.

<(1

Proof, Immediate.

LEMMA 1.6: If p i8 a type in M , then in 8 m e elementary extenSgon N of M , i8 T d i Z e d and IlNll I 11M11 -k IL(bl)l.

Proof. By 1.5(1) we can assume every element of M is an individual constant (so we can assume p is over 0 ) . Let c be a new individual constant and

"

Tl = Th (N) { d c ): d4 E PI.

As p is finitely satisfiable in M , every finite subset of T , has a model. Hence by the compactness theorem, T,haa a model N , and by the

downward Lowenheim-Skolem theorem we can assume llNll I I T,I = )1M)1+ IL(M)J.As N is a model of Th(M), if for every individual constant c , E L(M) we identify c y and c:, then M will be a submodel of I?. By 1.6(2), M < N and by the definition of T,,cN realizes p .

DEFINITION 1.2: (1) A model M is A-saturated if every type in M over some A G 1M1, IAI < A, is realized in M . (2) A model M is A-compact if every type in M of cardinality < A is realized in M . (3) A model M is 8uturated (compact)if it is IlMll -saturated (-compact).

4

PREIJXD?ARIES

[OH. I,

81

THEOREM 1.7: (1) 8wppse llMll + JL(M)I I A = A'", lMl infinite, C~ ext& of cardinality A. then M ha a K - C O ~ ~ ) C Gelementary ( 2 ) 8-e lldlll + IL(M)( I A = A<", LoE L ( M ) , M infinite and for every m thme are I A conzlplete L,-m-typea over 0 in M . Then M ha an elementary extension of power A, w h e L,-reduct ie K-saturated.

Proof.(1) We shall define by induction on a s A models Ma such that:

(i) M o = M and for limit 6 s A, = U a < d Ma; (ii) 1Mal # lHa+ll,M a < M a + l , IIMa+,II I A + IIMaII; (iii) every type over Ma of cardinality < K is realized in M a + ,. We can easily prove by induction on a s A, that < a * M b < M a (by 1.3) and that llMall IA; t~ lMal # lMa+,1 clearly lliKAII = A. NOW if p is a type in MA,lpl < K , then for some a < A, p is over lMa( (as lpl < K s of A because A<" = A). As M a < M A , p is in Ma, hence realized in M a + , SO (as M a < M a + , < MA)p is realized in M A .SOM A is K-compact, IIMAII= A and M <MA (as M = N o <MA). SO it Suffices t0 define the &fa's; SO let No= dl and = u # < d &fa. Now suppose M a is defined, and we shall define Ma+,. Let (I)i:i c A} be a list of all types in Ma of cardinality < K (as llMll + IL(M)I I A = A<", their number is s A; by allowing repetitions we get i c A). M7e define an elementary chain Mk, i I A, such that M: = Ma 11Mk11 I A, Mt = u i < d N L (for limit 8 5 A) and Mk" IX3aliZeS pi (possible by 1.6 and 1.3). As 1M1 is infinite, also 1M;I is infinite, hence qa = {x # a: a E [ M i l } is finitely satisfiable in M i ; hence by 1.6, M i has an elementary extension of cardinality I A which realizes qa, and lMa+,l # lMal, and we call it M a + , . So clearly M , < M ; < M , + , , every type in M a of cardinality < K is realized in M a + l . We have finished the definition of the Ma's, thus the proof, (2) A similar proof.

DEFINITION1.3: Let L(M) E L(N). An (M, N)-eZementarry mapping is a (one-to-one)function f,Dom f E IMI, Range f c IN!, such that for every a,, . .,a,, E Dom f and formula tp(xl, . . ., 2), E L(M), M C Cp[a,, . .,a,] o N C tpV(a,), . .,f(a,,)].When no confusion can azise we omit " ( M ,N ) " .

.

.

.

Remark. Note that if M = N,then the empty function is an ( M , N ) elementary mapping. We denote elementary mappings by f, P and g.

OH. I, 8 13

5

PRELJMINARIES AND SA"RA!L'IOW

DEP'INITZON 1.4: (1) M is ieonwrphic to N, Written M z N, if L(M) = L(N) and there is an (M, N)-elementarymapping f with Dom f = 1M1, Range f = IN].f is called an M m p h i m n from M onto N. ( 2 ) f is an auhwrphim of M if f is an isomorphism from M onto M. DEP'INITZON 1.6: (1) M is K - h o n t o g e n e o w , if for every (Af,M)-elementary mapping f , lDom f I < K , and every a E IMI, there is an (bl,M)elementary mapping g which extends f , Dom g = {a} u Dom f. ( 2 ) M is etron~lywhontogeneow if for every ( M M)-elementaxy y mapping f , lDom f I < K , there is an automorphism of M extending f. (3) M iS 8trOnglY K - S d U Y d d if it iS K - & W & b d and SkOIlgly Khomogeneous.

Proof. The proof is, in fact, identical to the proof of 1.7, provided we prove the analog to 1.6 : (*)

If f is an ( M M)-elementary y mapping, a E ]MI,then there is an elementary extension N of M ,and an extension g of f , which is an (N,N)-elementary mapping and Dom g = {a} u Dom f .

Proof of (*). Let Dom f 23 = {d%bl(l,,

* '*

Y

2:

{a::i < a}, and f(at) = b:, and

b:,,,): i(l)Y * * * < a,M =! d a , a,(,,, * * * Y a:(&

We want to prove that p is consistent, i.e., finitely satisfiable. Clearly p is closed under conjunctions, so it sufficesto prove, for ~ ( zbtcl,, , . .) E p that bl C (3z)(p(2,bi(l), . . .). By the definition of p , M C v[a, . .I, hence iK C (3z)v(z,a t ( l ) j ..); . so aa f is an elementary mapping, M C (3z)cp(z, bicl,, . . .). So p is consistent, hence by 1.6, p is realized i,n some elementary extension N of M ;say by b E INI. Let us exfendfto g by letting g(a) = b. It is easy to check that N, g satisfy our demands.

.

.

DEFINITION 1.6 : ( 1) M is h-univereal if for every N elementarily equivalent to M and A c IN1, IAI 5 h there is an (N, M)-elementarymapping f,Dom f = A. M is universal if it is IIMII-UnivemI. (2) M is ( < h)-univerealif M is p-universal for every p < A.

6

PRELIMINARXES

[OH. I,

31

THEOREM 1.9: (1) EVWY A-saturated nzodel A-mw. (2) If A > (L(M)I,then M is A-saturated ifl M is A - m p a c t . (3) Every A-saturated model is A-honzogeneoua and A-universal. (4) Every ( c No)-univer8al, A-homogeneous model is A-saturated. ( 6 ) If p I A, then every A-saturated model is p-saturated, every hcumpact nzodel is p - m p a c t , and every A-universal model is p-universal.

(6) Let A > Nobe a limit cardinal. If M is p-saturated (p-homogeneous) for every p c A, then M is ~-8aturated(A-homogeneous).(Thisdoes not hold for universality but holds for compactness.) (7) If M is A-saturated ( A - m p a c t ) and LoE L(M), then M 1 Lo is

A-saturated (A-compact). ( 8 ) Every strongly A-homogeneous model is A-homogeneow. Proof. (1) Immediate. (2) Immediate. (3) For this it suffices to prove:

CLAIM 1.10: Let A E B E IMI, IAI < A, IBI I A, f an ( M , N ) elementary mapping, Dom f = A. If N is A-saturated OT N = M is A-homogeneoue, t h n we can extend f to an ( M , N)-elementary mapping 9, Domg = B.

A u {bt: i < a} (so lBal c A for a < IBI) and we shall define by induction on a 5 IBI ( M , N ) elementary mappings f a ; fo = f, fd = Ut
Proof of 1.10. Let B = {bt: i c IBI}, B,

=

Proof of 1.9 (continued). (4) Let N be ( < No)-universaland A-homogeneous; and we shall prove it is A-saturated. We prove this by induction on A. So let A E IN I, [ AI c A, p a type over A in N and we shall prove that p is realized in N . By 1.6 there are M and a such that N < M a E [MI and a realizes p. There is an (M, N)-elementary mapping f, Dom f = A u {a} [if A is finite-by the hypothesis; if A is infiniteN is IA1 -saturated by the induction hypothesis, hence N is IA I-universal by 1.9(3)]. Then g = f 1 A is an ( N , N)-elementary mapping, and so g , = g - l is an ( N , N)-elementary mapping. As N is A-homogeneous,

OH. I, 8

13

PRELIMINARXES AND SATURATION

7

IA I < A, some (N, N)-elementarymapping ga extends g1 and Dom ga = Dom g, u { f (a)}.It is easy to check that g2(f (a))realizes p . (6)Immediate. (6) Immediate. (7) Immediate. (8) Immediate. THE UNIQUENESS THEOREM 1.11 : If M = N , MyN are saturated and have the same power, then MyN are isomorphic. Moreover, any ( M , N)-elementarymupping f with domain of curdinulity < 11iW11 can be extended to an isomorphiamfrom M onto N.

Proof. Let A = 11M11 = IlNIl, [MI = {a,:i < A}, IN1 = {b,: i < A}. Now N)-elementary mappingsf,, such we define by induction on a < A, (M, f,, lDom f a + , - Dom f,l that: f o = f (ortheempty function),f d = lJ,
+

LEMMA 1.12: Let M be a A-cumpact [A-eaturated]model. If p is a ( < KO)-type(or even a type with IA free variablu) in M , lpl < A [IDompI < A], t h p is realize& in M .

Proof. Easy, so left to the reader. Remark. We use Lemma 1.12 very frequently but without explicit mention. Conventions.We work within ZFC set theory in the standard way, i.e., all the symbols and formulas of a language L are sets of heredita,ry power < ILI. We assume also that there exists an inaccessible cardinal I? (i.e., a regular cardinal I? such that (VA < 4(2A < I?)) bigger than all the cardinalities we shall deal with. Note that (R(I?),r) is a model of ZFC set theory (R(I?)is the set of all sets of hereditary power < E ) . Every theory T will have a saturated model 6 = 6 ( T )of cardinality I?, (exists by 1.7)and, unless otherwise stated M , N will be elementary submodels of Q of cardinality < I?; A, B, C subsets of 161 of cardinality
8

PREIJMINARIEIS

[CH. I,

81

All our important properties (such as rank, forking eto.) are preserved under automorphisms of B and therefore a h under elementmy mappings as an elementary mapping is a (B, B)-elementary mapping and can be extended to an automorphism of B by 1.11. We do not state and prove these preservation theorems as they are trivial. The assumption on E is justified as (1) It does not, in fact, add any extra axiom of set theory as a hypothesis to our theorems. (2) Any model of T of cardinality < t? is isomorphic to some M < B. (3) I f M < S , llSll < t?,thenSisisomorphicover 1M1 tosomeN
EXERCISE 1.1: Prove that every finite model is A-saturated (for any

A). EXERCISE 1.2: Give an example of a ( < A)-universal model which is not A-universal for every A. CONCLUSION 1.13: I f M i g a saturated model, T = Th(M) G T I ,T, wnktent, ITl! s IlMll = A, then M can be expanded to a nodel of T , . Proof. We can assume T , is complete. If T , has a saturated model of cardinality A then its restriction to L = L(T)is also saturated by 1.9(7) and the theorem follows by 1.11. In generd let N , be a A+ -saturated model of T,,N the L-reduct of N,. Define inductively on a < A, N:, N i such that: N i < N , , N : < N , IlNiII = A, < a implies IN;] c IN;\ G [ N i l , and N: is saturated. Now N o = Ucr
EXERCISE 1.3: Prove 1.13 directly. EXERCI8E 1.4: Call M P-saturated, P a one-place predicate, if every H that ~ P(z) ~ ,€13, is realized in H. typep over A G IHI,IAI < ~ ~ such (1) Prove that if M is saturated, N is P-saturated, L(N) E L(M), llMll = 11N11, M 1 L(N) = N, then there is an (N, M)-elementary mappingf: IN1 + lMl,f(PN) = PM.

CH.

I, 5 21

ORDER, STAB-

AND MDISCERNIBLES

9

(2) Suppose L(T) has the one place predicate P and the two place predicate c, and for every model M of T and a,, . . .,a,, E PMthere is b E PM such that M C ( V ~ ) ( Z Eb = V;=l z = a,). Prove that if for a < S < A+, IIM,II = A, P(M,) = P ( M o ) , M , is P-saturated, then U, IPMI 2 KO A = A < A > IT1 + No,then T has a model N , llNll = A+, IP(N)I = A. (Show first that we can assume w.1.o.g. L(T)has a predicate E aa in (2), and then define inductively saturated models 211, (a < A + ) of T , llMall = A, P(M,) = P(Mo).For a = 0, 1 use the msumption, for limit 6 use (2) and (1) and for a = fi + 1 remember M , z Mo.)

1.2. Order, stability and indiscernibles

DEFINITION 2.1: Let A c !MI. (1) tPA(6,A, M ) = {'p(3;a)t: a E A , t E (0, l}, 'p Ed, M C 'p[6;B13. (2) #?(A, M ) = {tPA(6,A, M ) : 6 E "IMI}. (3) If A = L we omit it, if m = 1 we omit it, if M = B we omit it, and if A = $4, M = 6,we omit them. Remurk. If tp(B, A ) = tp(6, A ) and E E A, then tp@-5, A) = tp(6-E, A).

DEFINITION 2.2: (1) The model M is stable in (A, A ) if for all A c 1611, IAI s A and m < W , IS?(A, M)I s A. (2) The theory T is sktble in (A, A ) if every model of T is. We sometimes say "M[T] is (A, A)-stable", instead of "M[T]is stable in (A, A)". (3) IfA = L we omit it. (4) T is stable if there exists a A in which T is stable. ( 6 ) T is superstable if there exists a A such that T is stable in all p 2 A.

LEMMA 2.1: If A is regular, A c IMI, IAI < A, but ISm(A,M)1 2 A, then there ie aJinite B c 1611,IBI < mysuch that 1S(A u B)I 2 A. Proof. By induction on m. Form = 1, trivial. Now assume the claim for m and look at Sm+'(A,M).If IS"(A, M)I 2 A we are through by the induction hypothesis. Soassume ISm(A,M)1 < A. For allq ES"+,(A,M ) define

..

q* = {(3zm)$(zo, .y

2,;

a): $(so,. . .

y

2,;

a) E q}.

10

[CH. I, 8 2

PREIJMINARIES

Clearly q* is a (consistent) m-type over A and haa a unique extension q+ E P ( A , M).Now I@+:q ~ 8 " + l ( AM)}1 , s I P ( A , M)1 < A, so by the regularity of A, for some p E P ( A ,M) l{q E P + ' ( A , M ) : q+ = p}I 2 A. I n other words p has 2 A extensions in 8"+l(A, M). Thus for any (bo, ,b,,,-,) rea,lizingp in M !&(Au {bo, ,bm-,})I 2 A.

...

.. .

Remurk. If M is (IAI+ + No)-homogeneous we can strengthen the above conclusion to J8(Au B, M)l 2 A.

COROLLARY 2.2: T is stable in s P.

p

iff for every A, IAI s

p

IB(A)I

Proof. p+ is a regular cardinal. DEFINITION 2.3: (1) (2:i < a} is a A-n-indiscernible sequence Over A if for every io < . < in-' < a,jo< - - < jn-l< a,and permutation u of (0, . ,n - l},

--

..

tpd(8(O)-.

. .-af'(m-l), A) = tpd(gjU(O)-.

. .majUCn-l) A ) . 9

If i < j < a * # # a' we call the sequence non-trivial. (2) If A = L we omit it; if the sequence is A-n-indiscernible for all n we say it is A-indiscernible; if A = 0 we omit it; if A = {Q}we write Q instead of A. For simplicity we treat I = {a*: i < a) aa if it were ordered by < . We always aasume Z ( 8 ) = Z(7io) = m. I is A-(
.

A).

If i < j < a 8 # iij we call the set non-trivial. (2) We adopt the same conventions as in Definition 2.3(2).

LEMMA 2.3: (1) If A c A, and I = {#: i < a} is a A,-n-indiscernible set [sequence] over A, then I is a A-n-~ndiscernibleset [sequence] Over A. (2) If I is a ~-n-indiscernibleset [seq-] over A for all Q E A, then I is a A-n-indiscernible set [sequence] over A. (3) Let m < w and for di k < m let A, be aJinite set of form& and nk < w. Let a 2 n = max{nk: k < m} [a > n]. Then there is aJinite A such that for any I = {$: i < a}, I is a A-n-indiscernible set [seqt44?nce]

OH.

I, 5 21

ORDER, STABILITY AND INDIBCERNIBLES

11

iflfor all k < myI is a A,-n,-indiscernible set [sequence].( A &o depends on m = l(@.) (4) If {8:i < a}, a 2 w [a a limit ordinal] i s a A-n-indiscernible set [ s e q ~ over ~ ] A and p > a, then we can de$w 8 for a 5 i < p such tluz.t {#: i < 8) is &o a A-n-indbcernible set [seqwnce]over A. (6)If (it = a) E A , then a trivial indiscernible set or sequence lrae only one element. Note. The requirement in (4) that a be a limit ordinal is n e v : take a = w + 1, A = {(p(z, y)}; and let CE Ccp[at,a'] 8 i < j < w + 1, (E k (+)cp(a", y ) ; then {ai: i < w + 1) is a A-2-indiscernible sequence which cannot be extended. The proof is left to the reader. DEFINITION 2.6: Let I be an W t e set of finite sequences, all of the tame length, (p(Zo,. . . , P - l ; g ) a formula, and E a sequence. cp(ito, . . .,3-l; a) is connected and antisymmetric over I if for m y n distinct sequences sio, . . .,3-l from I there is a permutation u of n such that kcp[7iU(O), , , .,7iU("-'); E] and there is a permutation u of n such .,Zj'(n-l). 51. that k+Ti"(O',

..

Y

THEOREM 2.4: (1)If I is an inJinite set of sequence.3 of the same length, A , A and n arejhite, then I iraS an inJinite subset {8:i < w} which is a A-n-indiscernible sequence over A. (2) IfA,A,narefinite,thenforeveryk < wthereisl = 1(1Al,A,n, k ) < w such t h d every set of sequences (of the same length) of cardinality 2 1 iraS a subset of cardinality 2 k which b a A-n-b&mrnible sequence over A.

Proof. (1)follows from Ramsey's theorem and (2) from the finite analog of Ramsey's theorem. (See 2.1 of the Appendix.) DEFINITION 2.6: (1)The type p ( A l ,A,)-splits over the set A if there me 6, E such that tpAl(6,A ) = tpAl(E,A) but there is Q E A, such that v(Z; 6), +I; E) El). (2) p splits over A if p (L, L)-splits over A'. Remark. Clearly if p splits over A 2 B, then p splits over B. LEMMA 2.6: For 1 5 n < y I w let A,, be sets of formulas closed under permutations of variables. Let I = (8:i < a} and A, = U (8: j < i}

12

PRELIMINARIES

[OH. I,

f2

u A . Assume that for all i < a, 2 I n + 1 < y , p r = tp(tif,A,) does not (A,, A,+,)-split Over A and that p j A, E pi A,, for all j c i < a, 1 I n c y. T h e n I is a A,-n-indiscernible sequence Over A for 1 4 n < y.

Proof. Let i(0) c i(1) c . . c i(n - 1 ) < a, j ( 0 ) c j(1) < . . . c j (n - 1 ) c a, Q = Q ( Z O , . .,Zn- l; g ) E A,, and E E A. We must show I=Q[T~"O), . . .,a'("-l ) ;El iff t ~ [ ? i j ( . .~.),,aj("-I); El. The proof is b y induction on n. For rc = 1 , p, t A, 5 pj(,, t A,, pic,, t A, so the result is obvious. Now assume the result for n. Let = max{i(n),j(n)). By the induction hypothesis $(,In. . .^ a " n - l ) and af(o)^.. .^aj(n-l) realize the same A,-type over A. Now ps does not (A,, A,+,)-split over A so if Q ~ A n + 1then ,

.

(by the remark after Definition 2 . 1 ) . Thus since pjcn, A, + ,, pi(,) A, + E 2% IA,+l we get

LEMMA 2.6: L e t M be stable in h and assume that in M :

(*I

There is n o n c o and increasing sequence and ~ E S " ( A ~ ,such M ) that p r A,+1 splits over A , for all i c A.

If A E !MI and I E "1611, III > h 2 IAI then there is I' II'l > h such that I' is an indiscernible sequence Over A . Proof. First we shall show: (**)

There exist B, C, A E B E C E 1611, ICl I A, and p ES"(C, M ) such that (1) for all C',C G C' s IN[, IC'l I X p has an extension p' ES"(CI)realized in I - "C' which does not split over B (in particular p does not split over B), and (2) for all E' E /MI there is E C such that tp(E, B) = tp@', B).

E

I,

OH. I, 9 21

ORDER, STABILITY AND INDISUERNIBLES

13

By way of contradiction assume -,(**). We define by induotion t m hmwsing aequence {B,},,, suoh that B, s ]MI,lBfl 5 A as follows: Bo = A , Bd = uf
Proof.By way of contradiction assume that {Af}tLA is an increasing i? E A,,, sequence, p ES"(A~),and p 1A,,, splits over A,. Choose @, such that tp(6,, A,) = tp(i?,A,) and v,(Z; @), 7pf(Z; 8)E P for some formula v,. Let 7i realizep. Let p = mi+: 2y > A}. Define by induction on Z(q) elementmy mappings P,,, r ] E u r 2 , Dom P, = Ale,,): If Z(q) = 0 P, is the identity on A,. If 1(q) = 6 then P, = u a < d PVla.If l(r]) = i + 1 let P,,-be an arbitrary extension of P, to A,+,. B',,-<,> will be defined as follows: Let Q be such that 0 1 A, = P,andQ(6,) = P,,-<,>(i?). Q is an elementary mapping and P,,-<,> will be an arbitrary extension {F,(6fni?): 7 ~ f + l 2i ,< p}. Clearly IBI 5 H,. of B t o A,,,. Let B = &<,, Ia21 5 A. For every r ] E '2 extend P,to a mapping Pg with domain A, u {a}. Let p,, = tp(Pg(7i),B). Now assume r ] # Y E u2. Let a = min@:7[8] t@]}. W.l.0.g. q[a]= 0. Thus va(Z; P)EP and C(pa[7i; 5'1.

u

+

14

PRELXMINARIES

[OH. I, 8 2

Thus b,[P;(a);P;(@)], and cp,(Z; P;(&)) ~ p ,Similarly . 7cpa(a(p; Pk(b)) ~ p , , But . Pb(6.) = F,,(F); thus -,q,(f; P ; ( b ) )~ p , ,In . other words p,, # pv. Hence ISm(B)I2 l{p,,: q ~ ’ 2 } 1= 2u > A 2 IBI. So T is not stable in A, a contradiction. THEOREM 2.8: If T ie stable in A, IIl > A L IAI, then there is J IJI > A, which is an indiscernible sequence over A.

c I,

Proof. Immediate by the two preceding lemmaa. Remark 1.J is in fact an indiscernible set over A , see 2.9 and 11,2.13(1) and (9). Remark 2. If K is regular, and for all IAI < K and m < w , ISm(A)I< K, and there is A < K for which 2.6(*) holds, then for all A and I with. IAI < K s IIl, there is J c I, IJI 2 K, J an indiscernible sequence over A. The proof is similar. LEMMA 2.9: If I is a A-indiscernible sequence over A but not a Aindiscernible set over A then there are n < w , cp(Zo, . . .,P - l ; 8) E A, and 8 E A such that cp(fo,. . , P - l ;8) is connected and antisymmetric over I.

.

Proof. Immediate.

THEOREM 2.10: Suppose >

xOsjb
(IA('

+ 2a'),

cp = cp(P; g), l(3) = m, l(g) = n, IS;(A, M)l 2

(1) k t fl(31, 3 2 , z3; ill, ; 1 6 ) = l(iii), g3) = [ ~ ( z ga) i ; ~ ( 3 1?&)I, i = 1, 2, 3. Then there are in 1 M 1 sequences 8' i < h such t h d M C B[Sit; af]o i < j. (2) Swppose h -+ ( X ) % [e.g., h = x = ESo or x = K,+, h = ( P o ) + , see Appendix, DeJinitions 2.1 and 2.51 then there are in 1M1 sequences Si', 8, i < X a n r E t ~ { O , l ) m h t h a t MC c p [ i ? ; i E ' I t o i < j.

Remark. For a finite A , see a stronger result: 11, 4.10(4), Appendix 1.7( 2).

zOSp
Proof. Let K = (IAI” + 2a’) and 8, i < K + be sequences from 1H1 such that the types tpJ8, A , H)are distinct and let y5 = +(p; Z) = cp(3;g). Now define by induction on a IA sets A, c ldll such that

QH.I,

§ 21

15

ORDIOR, S T A B m Y AND INDISOIRNIBLES

A . = A ; for limit 8, Ad = u a < d A,; and for every p ES;(A,, bl) U #$(A,, M ) and B s A,, IBI i:'1.1 + No, p 1 B is realized by a sequence from A,+1 2 A,, and 1A.I 5 IAIla+l1. So a < h =IA,I < XB<,mn(lAI+2)m"lfl hence JA,I < K . Now we s h d prove: (*)

There is i < K + such that for every a < h and every B c A,, IBI < '1. + 8 0 , t p ~ zAa+1, , M ) ($9 tp)-splits over B.

Suppose not, then for every i < K + there are a, < A, B, E A , which contradict (*). Clearly we can replace {8:i < K + } by any subset of the same ctrtrdinality. As h 5 K we can aasume a, = a for every i. Then aa a < h the number of subsets of A, of cardinality <1.1 + No is ~ Z o r r r IAaP < ~ 5 Zorucn ~ A ~ ~ aI +Kl ~so"we can aasume B1 = B for every i. Let IBI" = p < '1.1 + KO s A. By the requirements on A,+1 there is B E C 4 A,+1. ICl 5 (n + 1)2" in which every p ES$(B,M ) is reahed. Clearly IQ(C, M)I 5 21'1" I2((n+1)*a')"5 K so we can assume that for every i < K + , tp,(8, C, M ) = p. Now M tp,(F, A, AZ) # tp,(E1, A, M ) there is @ E A such that M C &'" a] G iv[Z1; a]. Choose 8' E C such that tp& B, M )= tpy(si', B, M) (possible by the definition of C). As tp,(Zl, A,+l, M ) does not (#, 9)split over B, for I = 0, 1, M C tp[8; iz] = ~ [ 3a']. ; So M C tp[F;a'] = - q [ E 1 ; a'] but this contradicts tp,(i?, C,M ) 3: p. So we have proved (*) and now we prove: (1) Define by induction on a < A, sequences a,, 6 ,, E, E Aaa+a aa follows. Suppose we have defined them for every fl < a. Let B, = UB
= tp[Z,; a,], a,] A l@a; 6,].

M I=q[8;a,]

M

C (p[Ef;

hence M C cp[Z,; (ii) If a < 8, then tplr(aB,BB,M ) = tp&, hence

6,] = tp[E,; 6,]; B,, M), but E, E B,

M C +[a,; z,] = $[6,; E,] or iK c VIE,; a,] So if we let = aan6,*a,, we prove (I).

= tp[~,; 6,~.

16

PRELIMINARIES

[CH.

I, f 2

6,, Z, be as in the proof of (1). For the dehition of A -+ (x):; and why No + (No)!;see Appendix, Definitions 2.1 and 2.1. (2) Let a,,

Here we define the colouring f of A: if a < < A, f ({a,18)) = 0 when M C p[E,; a,], and f ({a,F}) = 1 otherwise. As A -+ (x)&f is constant on (allpairs from) a subset of A of cardinality x, so, by renaming, on x. If M C 7p[Eo; a,], then for a, < x, M k p[Z,; a,] e a 2 8, so let = a,, F = E, for a < x and t = 1. If M C p[Zo; a,], then for a, < x, M C p[E,; 5B]o u < fl so let 8, = 6,, b = 5, for a < x and t = 0. CONCLUSION 2.11 :If ]@(A,M )I > IA I + No,then there are sequences an,En in ]MI (n < w ) and t E (0,l} such that M C ?[El; a,]' e 1 < n. THEOREM 2.12: Suppose M is stable in A, and in it there is no ordered (by 8ome formula of L(M)) set of sequences of length p, and p = A, IL(M)l < cf A or p = A+. If I c ]MI, IIl > p, then there is an indiscernible set J G I of length A+.

Proof.Use the ideas of the proof of 2.6 and 2.10, and use 11, 2.16. EXERCISE 2.1: Suppose A. < A, < < A,, I E IMI, IIl > hl: and ill is stable in A, for i s n, then there is an (n + 2)-indiscernible sequence J c I, IJI = &+. EXERCISE 2.2: (1) Find a model of power N,, which is stable in No, but there is no indiscernible sequence of length N, in it. (2) Moreover, for some symmetric tp = p(z, y), there is no p-2indiscernible sequence of power N, in it. (Hint: Use a dense Specker order for ( l ) , and add a well-ordering to it for (2) (see e.g. [GS 73]).) (3) In (1) and (2), show that we can choose a model with'a countable, N,-categorical theory. EXERCISE 2.3: Suppose A c B c C and for every 5 E C there is a 8 E B such that tp(E, A) = tp(6, A). If p ES"'(B)does not split over A, then p has a unique extension in Sm(C)which does not split over A. EXERCISE 2.4: Generalize 2.12 to (A, A)-stability. EXERCISE 2.5: Suppose A is regular, M a model, and for every A c /MI, ]A1 < A implies JS"'(A,M)1 < A, and in M there is no ordered set

OH. I, 8 21

ORDER, STABILITY AND INDISCERNIBLES

17

of sequences of length p for some p < A. Then for every I c 1M1, III 2 A there is an indiscernible set J c I, IJI = A.

EXERCISE 2.6: In 2.12 (cf. Exercise 2.6) add an A E IMI, [ A ]< of p( IA I < A) and demand that J is indiscernible over A. EXERCISE 2.7: (1) SupposeS E S;(A), 9 = cp(3; g), 181 > IAI. Prove that there are li, E A, pk € 8 (n, k < o),and t E (0, l}, such that cp(Z; B,$ €pic iff n < k (iff +Z; $pk,of course). (Hint: Generalize 2.10(1) and 2.11.) (2) Generalize 2.10(2) similarly. EXERCISE 2.8: Let cp = q(Z;g),

h be regular, 5 E A E J M1 , (A1 < A. Show that if b = 5-En. ^E, I{p E Smn(A,M ) : cpn(F;b)e p)I 2 A, then for some B G (MI, ( B J5 m(n - l), I(p E P ( A u B): cp(Z; E) E ~ } 1 J A.

EXERCISE 2.9: (1) Suppose T is stable, A regular, IAI < A =. IS(A)I < A, and p < A =. p" < A. For any lAol < A and elements ah (i < K , a < A) there is a set S c A, IS1 = A, such that for any i(0) < . < i(n) < K , {(af,O),. . .,ap)):a ~S)isindiscernibleover A,. (Hint: You can msume T is stable in some p < A by 11, 2.13 or see 111, 4.23.) (2) Generalize (1) when we replace the condition on T by a condition on M . EXERCISE 2.10: Let M be an infinite K+-saturatedmodel, and define the model M" m follows:

pf"I

=

Ff(a)=

p 1U"pfIY a

a[.]

ifaEM, if a E "IMI,

PM"= IMI and M

E

ME.

Show that if M is A-stable, A = A", then M" is A-stable; and if Th(M) is A-stable, A = A", then T h ( P ) is A-stable; but if A < A" or M is not A-stable, then M" is not A-stable; and if A < A", Th(M") is not A-stable. Also show that Th(M") depends only on Th(M) and K .

CEAPTmR 11

RANKS AND INCOMPLETE TYPES

II.0. Introduotion Section 2 (and Section 1 on which it relies) are the cornemtones of the book, but they a,re not difficult. This chapter has esaentidy two interlocked aims; the first is to investigate the family of formulas cp(Z; a)(aE Q)('p = 'p(Z; g) fixed); or other words the family of the seta 'p(Q, E Q). We try to classify them by complexity, in Sections 2 and 4. Clearly this hae various implications for the theory T.The second aim is to investigate various notions of ranks (e.g., possible values, equality between distinct ranks), we deal with it in Section 1; somewhat in Section 2, and mainly in Section 3. The properties we shall deal with me the order property ( =instability), the f.c.p. (the hite cover property) the independence property and the strict order property. We say T haa such a property if some 'p(z; g ) has the property (or, equivdently, by our theorems, some 'p(Z;g) has the property). Essentially we prove that the order property implies the f.c.p. and the order property is equivalent to the disjunction of the independence property and the strict order property. Other properties of this sort are suggested in Chapter 111, Section 7, (K&T) = a, K ~ J T=) 00, essentially). The most important of these properties is stability (=not the order property) with which we shall deal in Section 2. The main theorem (2.2) lists many propertiea equivalent to unstability of a formula, some of them are helpful in proving assertions about stable formulas, and some are helpful in proving assertions about unstable formulas. For unstable formulaa the important properties are: the order property (i.e. for some a,,, &,(n < W ) t: 'pp,, 6,J if€ 2 > k) (this shows the similarity to the "classical " cam-the theory of dense linear order); and for every X there is an A, IAI I; X < lQ(A)l.Among the properties important for stable 'p m:l&(A)I I; IAI + KO (=stability 18

OH. 11, 8 01

INTRODUOTION

19

in every A); P ( p , v, a)s P ( Z = Z,9 , 2 ) < w ; and every cp-m-type over A is definable by a formula over A, of a fixed form. For theories we get a similar theorem (2.13): T is stable in every A = iff T is stable in some A iff every v(x;8 ) is stable iff every ~ ( 38);is stable iff T does not have the order property (i.e. no formula orders an infinite set of sequences), ifF every infinite indiscernible sequence is an indiscernible set. We also prove that if IIl 2 A > IAl, A regular and A is finite, then some J E I is A-indiscernible over A, IJI = A. In Section 4 we shall deal with the other properties. We say v(Z; 0 ) does not have the f.0.p. if for some k, < KO we can strengthen the compactness properties of EE to: if p is a set of cp-m-formulas,and every subset of p of cardinality sk, is realized, then p is realized. The “classical~yexample of a theory with the f.c.p. is the theory of one equivalence relation such that for every n < w there are (insnitely many) E-equivalence claases of n elements. It appears that for stable T, the possession of the f.c.p. is equivalent to several natural properties (see the f.0.p. Theorem 4.4). In particular if T has the f.c.p., then there are a formula E ( z , y; 2) and sequences Z,, such that E ( z , y, En) is an equivalence relation with k, equivalence classes and n 5 k,, < No(this shows the a 5 i t y of any stable T with the f.c.p. to the classical example, and is helpful in proving assertions about such theories). On the other hand if T does not have the f.c.p., then Rm[8(3;a), rp, A] = k is an elementary property of a; and there is k;,@ < KO such that for each a, for all A 2 k;,, the value of P [ B ( Z ; a), 9,A] is fixed; and the value of Mlt[8(Z;a),q] is bounded by some ki,, < KO; and for each finite A, there is k, < KO such that any set of (A, m)-formulas p is realized iff each q E p , IqI < k, is realized; and every (A, m)-type has a finite subtype of bounded cardinality which has the same rank Rm(- ,A, A). Let =(A) be the first regular cardinality p such that IAl 5 A =]l?t(A)l< p. It appears that the number of possible functions is small: for stable v = rp(Z; 0): n ( 1 < n < KO),and A + . For unstable v = v(Z; 8):Ded, A and (2”+ ;when they are distinct, rp satisfies the second case iff v(Z; ti) has the independence property. The strict order property of T is equivalent to the existence of a formula ~ ( 2g); defining a quasiorder (i.e.,a transitive, reflexive, anti-symmetric relation) with infinite chains. In Section 4 we also st& to investigate dimensions. Ranks were invented and investigated for their use in stability. Several kinds of ranks were used, and most of them are particular caaes of P ( p , A , A), on which we concentrate. We investigate them

20

RANKS A N D INCOMPLETE TYPES

[CH. 11,8 0

alao when there is no apparent application; more information is obtained in Chapter 111, Section 4, and Chapter 5, Section 7. Rm(p,A, A) is interesting mainly for A = 2, No,00 and A = L or A

finite. What is the meaning of the rank Rm(p,A , A)? For finite p , we can say that it measures the complexity of the family of sets {a: a realizes p u { c p ( ~ ; 6))) for v E A , 6 E 6. The basic properties are proved in Section 1. The rank is monotonic in the parameters (decreasing for p and A, increasing in A), and every type has a finite subtype of the same rank. When A 2 No we can extend p while preserving the rank, i.e., if p is over A, there is a complete q 2 p, Rm(p,A , A) = Rm(q,A, A); when A = 2, v E A there is no si such that

Rm[pu {cp(?C;

A , A]

= Rm(p,A ,

A) for t E (0, 1);

so for A = L, p over A , p has at most one extension to a complete type over A of the same rank. For every A, the maximal number of such complete extensions is, essentially denoted by Mlta(p,A, A). In Section 1 we also prove that for limit A, Rm(p,A, A) = Rm(p,A,A+). In Section 2 we prove that for finite A, Rm(p,A, 2) < w or Rm(p,A , 00) = a;and Rm[e(Z;a), A, k] = 1 is an elementary property of a. In Section 3 we complete the investigation on the stability spectrum for countable theories. If IT1 < 2H0, then T is stable in IT1 ifF T is stable in some A < 2H0iff T is stable in every A 2 ITI iff R(x = x,L, 2) < 00. Also T is superstableiff T is stable in every A 2 2ITI iff T is stable in some A < AHo iff R(z = x,L, 00) < 00. But the aim of Section 3 is the investigation of R and D. We prove that Rm(p,A, A) is < 141 + Noor 00 usually (i-e.,A = a, or A 5 No). Also ah = R m (p,A A) , (except for being decreasing) is essentially arbitrary for KO I A I I TI +,has restricted changes for A s No,is fixed for A > (2lT1)+.In fact it is fixed for A > lAl+ + No except when for some p , ! A ] + s p 5 (2A)+, lAl+ < A < p implies a, = CO, and A > p + implies aA = a, < 00; and a,,= 00 except possibly when p is a limit cardinal (rememberthat for limit p , a,,= a,+) so ah (A > Id 1 No)has at most three values (see Theorem 3.13 and Exercise 3.16). Moreover for A = L, A s KO we can compute Rm(p,A, A) from R m ( p , A 2) , (Exercise 3.21); and R m ( p , A ,a),for finite p , is usually directly characterized as the maximal a for which TCl +(Ap, h) is consistent for some (A, a)-functionh. We deal much with Dm(p,A, A), +

+

+

OH.

11,5 13

BANKS OF TYPES

21

which is a version of Rm(p,A , A) (essentially we require that the many contradictory extensions are finite) and we use it to prove assertions about R for A > lAl+ + No and prove that for A > IAl + Nothey am usually equal. We also give a method of constructing counterexamples (Exercise 3.20) and many exercises. +

IK.1. Ranks oftspea DEFINITION 1.1: Let p be an m-type, A a set of m-formulas, A a cardinal (possibly finite),or A = 00. We define the rank Rm(p,A, A) or Rm[p,A , A] by defining inductively when P ( p ,A, A) 2 a, a an ordinal. (1) Rm(p,A,A) 2 0 when p is a (consistent) type. (When p is inconsistent we stipulate P ( p ,A, A) = -1.) (2) Rm(p,A , A) 2 6 for 6 a limit ordinal if Rm(p,d, A) 2 a for all a < 6. (3) P ( p , d , A) 2 a + 1 iffor a l l p < handall finiteq E p thereare types {qi}isrrwhich are d-m-types (i.e., rn-types whose formulas m e all of the form ~ ( 38); or +E; a) where ~ ( 35);is in A) such that: E q, (or vice (i) for i # j there is a formula such that q~ E qi, l~ versa). In this cme we say that qi and q, are explicitly contradictory. (ii) P ( q u q i,A, A) 2 a for all i Ip. (4) Now if Rm(p,A,A) 2 a but not Rm(p,A,A) 2 a + 1 we say Rm(p,A, A) = a. (It is easy to see by induction on a that Rm(p,A, A) 2 a implies P ( p ,A, A) 2 p for all /3 5 a.)If Rm(p,A, A) 2 a for all a we define P ( p , A, A) = 00. DEFINITION1.2: If Rm(p,A, A) = a # 00, define the mu2tipZicity: (1) Mltl(p, A, A) is the first power po such that there is a finite q E p such that q has no more than po A-m-types qi satisfying 3(i), (ii)from Definition 1.1 (po may be finite). (2) Mlta(p,A, A) is the first power po such that there are no more than po A-m-types qi satisfying 3(i), (ii) from Definition 1.1 for p = q. If Rm(p,A,A) = 00, define Mltl(p,A, A) = Mlta(p,d, A) = a. If Mltl(p, A, A) = Mlta(p,A, A) we shall just write Mlt(p, A, A). If a statement is true for both Mltl(p, A, A) and Mlt2(p,A, A) we shall write Mltl(p,A, A). If A = KO we may omit it.

Remark. Clearly if Rm@,A, A) # 00, then 0 < Mlta(p,A, A) 5 Mltl(p, d, A) < A and if p is finite, then Mlta(p,A, A) = Mltl(p, A, A). In fact Mltl(p, A, A)+ < A. See Lemma 1.9.

22

RANKS AND INCOMPLETE TYPES

[m.II,§ 1

Ndation. If p = {8(z; a)} we may write P [ 8 ( z a), ; A, A] instead of P[{B(z; a)},A, A]. The same convention applies to Mlt and D (see Definition 3.2). 8pcial Cases. P ( p ,A , A) 2 1 if for all p < A and all finite q E p there are Am-types {q,},s,, each of which is consistent with q, but every two are explicitly contradictory. P ( p ,A, A) 2 2 if for every p < A and every hite Q E Z, there Am-types {q,},s,, which pairwise explicitly contradictory and P ( q u q,, A , A) 2 1 for all i. Example. Let M be a model with equivalence relations {Ea: a < a,} such that a < /3 =+ E, E En, E, divides every equivalence class of En into infinitely many classes, every equivalence dam of 1, is infinite, and if a, = y + 1, then E, has h h i t e l y many equivalence classes. Let Ta*= Th(iK) and let 6 = 6(Ta')be the saturated model of Ta' of cardinality I?. Then for every a E 1 6 1: (1) Rl[{z = a}, L,a]= 0, (2) R1[{zEoa}, L,001 = 1, (3) R'[{zE@}, L,001 = 1 + a. Remark. Tawadmits elimination of quantifiers and is totally transcendental (see Definition 3.1).

Proof. (1) ClearlyRl [{z = a}, L,a]2 0, and there are no q,, q1 such that qo u {z = a} and q1 u {z = a} are consistent while q,, q1 are explicitly contradictory, otherwise a would realize qo and ql. Thus R1[{z= a}, L,A] 2 1 for A 2 2, in particular for h = ao. (2) First we show R1[{zE,a},L,ao]2 1. Let p be any cardinal. Choose elements a,, i s p, in the 1, equivalence class of a and let r, = {z = a,} u {z # a,: i # j s p}. Clearly they are pairwise explicitly contradictory, but a, realizes {zE,a} u r,. In order that R1[{zEoa},L, a] 2 2 there must be two explicitly contradictory types p,, pl such that for i = 0 , l R1[{zE,a}upi, L,a]2 1. In particular {zE,a} upi = q, is consistent, i = 0, 1. By Theorem 1.6, extend q,, ql, to complete types q:, q: over the same sets and of the same rank. But E, E E, for all a < a, so q:, q: can be explicitly contradictory only if there is a, such that [z = a,] E q:, [z # a,] E q: or vice versa. But then from the proof of (1)it is clear that R1(q,*,L, ao) 2 1. So R1[{zEoa},L,ao] I2. (3) We leave this &B an exercise for the reader.

aII, 8 11

RANKS OF TYPES

23

DEFINITION 1.3: p t- q when every sequence realizing p realizes q. If p t- q and q t-p we write p = q: p is equivalent to q. Remark. For finite types, p I- q is equivalent to (VZ)[A p A q], and for infinite types to the condition that for every finite q‘ E q there is a finite p’ E p such that p’ t- q’. --f

THEOREM 1.1: (1) If p1 t-p2, then Rm(p,,A,A) s Rm(p2,A, A). I n p ~ t k t this h ~ie the m e w h Pa E p,. (2) If e q d i t y hQZd.8 in (l),t h Mlt’(p,,A, A) s Mlt’(p,,A, A). Proqf. (1) We prove by induction on a that Rm(pl,A,A) 2 a =Rm(p2,A , A) 2 a. For a = 0 or limit there is no problem. Assume for a and prove for a + 1. Let R”’(p,,A, A) 2 a + 1 and let qa E pa be h i t e , p < A. We shall h d typea {q,},<,, aa required in the definition. Since p , t- p2, clearly p1 t- qa so there is a finite q1 E p , such that q1 t- pa. Choose pairwise explicitly contradictory A -m-types {qi}, , such that R”’(ql U qf,A , A) 2 a for all i s p. By the induction hypothesis, Rm(qau q,,d, A) 2 a for all i s p. Thus R”’(p2,A, A) 2 a + 1. (2) Similar proof.

THEOREM 1.2: FOTall p, A, A, m there it3 a $nite q E p such that l P ( p , d ,A) = R”’(q,A, A), and Mltl(p, A, A) = Mltl(q,A, A).

Proof.If q s p , then Rm(p,A, A) s Rm(q,A, A) by the previous theorem. So if the left-hand side is co, we may take q arbitrarily, for example q = 0. If the left side is a # 00, then Rm(p,d, A) ;t a + 1 and thus there is a finite q c p and there is p < A such that there me no pairwise explicitly contradictoryd-m-types@(},s,,satisfying Rm(qu q,, A , A) 2 a. But that m e w that Rm(q,A, A) 2 a + 1. By 1.1(2) Mltl(p,A, A) IMltyq, A, A) = pq, say. Among the pairs q,pq satisfying the above mentioned condition, choose one with minimal pq. Thus Rm(q,A,A) = a, Mltl(q,A, A) = Mtl(p, A, A). LEMMA 1.3: (1) If A , c A,, A 2 K , then R”(p,A,, A) s Rm(p,d,,K ) . (2)If there is eqdity in (I),t h Mltz(p,A,, A) I Mlt’(p,A,, K ) . PTOO~. (1) We prove by induction on a that Rm(p,Al,A) 2 a * Rm(p,A,, K ) 2 a. The proof is immediate. (2) Likewise.

24

RANKS AND INCOMPLETE TYPES

[CH. 11,s 1

LEMMA 1.4: (1) If Rm(p,A,2) = a # oc, then there is no q(Z;8) E A and sequence m h that P [ p u ( ~ ( 0a)t}, ; A , 21 = a,for t = 0 , l . (2) If Rm(p,A,A) = a # a and Mltl(p,A, A) = 1 then the parallel w i u s i o n still ho&. ( 3 ) If Rm(p,A , A) = a # a0 and Mlt'(p, A , A) < No,t h there is no q ( Z ; 8) E A and sequence ii m h that Rm[pU {p(Z; a)", A , A] = a and Mlt"p u { q ( Z ; 8)t},A , A] = MltJ(p,A, A) for t = 0 , 1.

Prqof. (1) If there were such a p we would choose qo = {q(Z, a)}, q1 = {+Z, a)} and get Rm(p,A, 2) 2 a + 1. (2), (3) The proof is similar. Remark. If q E p , p A ESF(A), Rm(q,A , 2) = P ( p ,A , 2) = u # 00, then p ! A may be defined in terms of q in the following sense: q ( f ; i Z ) ~t pA o p ( Z ; j i ) ~ o l , ( A ) , a ~and A , Rm[qu{q(Z;iE)},A, 21 = a ; for, if q(3; 5)~p 1A, then q G q u (q(3;a)} E p and the rank of p U (~(3; a)} is between the rank of q and the rank of p , and thus is a. And if p(Z; g) E cl,(A), a E A, q(Z;a) q! p, then +Z; a) E p , since p t A E @(A). Thus the rank of q u {-.q(Z; a)} is a and by the previous lemma the rank of q u {I@; a)} is not a. (Compare Theorem 2.12.) COROLLARY 1.6: If Q c p , p ESF(A),Rm(p,A , 2) = Rm(q,A , 2) = a # a,then p is the unique type in #?(A) extending q with rank a. THEOREM 1.6: If p is an m-type over A , A 2 there is q €Sm(A),p c q, Rm(q,A , A) = a.

Ho,P ( p , A , A)

= a, then

Proof. Denote r = {+Z; a): Rm[{$(Z;a ) } , A , A] < a, E A}. It is sufficient to show that p u is consistent since then we can find qEBm(A)extending p u r. Thus P ( q , A , A) 5 Rm(p,A,A) = a but there is no finite q1 c q such that Rm(ql,A, A) < a (since then we would have P [ { Aq,}, A, A] < a and c7A ql) E T c q, thus making q contradictory). So by Theorem 1.2, Rm(q,A,A) = a. Now we show that p u r is consistent. Otherwise there is a finite r E p of equal rank and there are #:, 8 E A , i = 1,. . ., n, such that Rm[{$& af)},A, A] < a and r u {+(l, at): 1 5 i 5 n} is contradictory. Thus r I- qh1(Z, al) v - - - v #,,(?$, a"). Now Rm(r,A, A) = a > max, P[{$@, $)}, A, A]. On the other hand, Rm(r,A, A) 5

r

OH. II, 5 13

25

RANK8 OF TYPE8

CLAIM 1.7: Tor all r, h 2 No,

Proof.Let #,

= #,(if; 8,); note that for 1 I; i I; n {#,} u r k {Vr-l #,} u r,andsobyTheoreml.l(l)P[{#,}ur,d, A] 5 R”[{V:,,$,}u r,A,A].

Thus maxi.,Sn P[{#,} V r, A , A] 5 For the other direction we prove (*)

=[{y #,}ur, 4 A]

2

B =2.

Rfil[{V?=i#i}U r,

A].

max R”[{#,}u r, A, A1 2 1

8.

The proof is by induction on B. For = 0 (*) says that if {V, #,} u r is consistent, then there is i such that {$,} u r is consistent, and this is correct. When B is a limit ordinal the claim follows directly from the induction hypothesis. We must now show the passage from p to B + 1. Assume by way of contradiction that Rm[{V,#,} u r, A , A] 2 p + 1 but max, P[{#,} u r, A , A] < B + 1. Since P[{#,} u r , A , A] < B + 1 for every i = 1,. . . , n , there me pi < A and finite q, s r such that there are no { Y , } , ~ ~ satisfying , the q,, p = (n + 1) max,p,. demands of Definition 1.1(3). Let q = A is infinite by aaumption, 80 p < A. Thus there is a finite q c r and there me p l , . . .,pn < A such that for all i there do not exist pairwise u q u r,, A , A] explicitly contradictory types {rj},Sr,such that P[{#,} 2 /3 for all j 5 p,. On the other hand, since P[{V,#,} u r , A , A] 2 B + 1, and p < A, for all finite q E r, there are pairwise explicitly contradictory {rjlfS,,such that P[{V$,} u q u r j , A , A] 2 6 for all j 5 p. By the induction hypothesis with q u r, in place of r , we get that u q u rj,A , A] 2 B for all j.Thus for all j 5 p there is max, P[{$,} i(j), 1 5 i(j) 5 n, such that P[{$,tn} u q u r,, A , A] 2 p. There are p + 1j ’ s and n i’s, 80 there is i, equal to i(j) for at least max, plo + 1j ’ s (by the definition of p). Then w.1.o.g. i, = i(j)for all j 5 pto,and 80 u q u r,, A , A] 2 /3 for all j I; pie. This contradicts what we obtained previously by the choice of the p,. Thus we have proved (*), Claim 1.7 and Theorem 1.6.

uXl

COROLLARY 1.8: (1) If A 2 KO,in DeJinitim 1.1 and 1.2 we can re9lace “explicitly contradictory” by “contradictory”; or by “ q u q, u qj

26

RANKS AND INCOMPLETE TYPES

[m. 11, 8 1

implierr a wntradktbn for all i , j" and even by " P ( qU qf u q,, A , A) < a fori < j 5 p". ( 2 ) If A is a set of m-formulas and A , = @(A), t h n for every *type p und infinite cardinality A, Rm(p, A l , A) = P ( p ,A, A). LEMMA 1.9: (1) I f R m ( p , d , A = ) a # 00, h 2 KO,then the value ,uo= Mltl(p,A , A) is achieved, that is, for every finite q E p vor q = p ] thre are Am-typea {qf}f
OH. 11, 8 13

RANKS OF TYPES

27

(4) Mltl(q,A) = Mlta(q,A) (i.e., Mltl(q,A, KO) = Mlta(q,A,No)).

Proof. ( l ) , (2) and (3) are left as an exercise. (4) If P ( q , A , No) = 00 this holds by Definition 1.2. Otherwise let a = R”’(q,A , No), k = Mltl(q, A) so by the remark after Definition 1.2, k < KO and k 2 Mlta(q,A). For proving the other inequality, choose finite p c q such that a = R”(p,A, No), Mlt’(p, A) = Mlt’(q, A) (by Theorem 1.2). By Definition 1.2 there are pairwise explicitly contradictory A-rn-types r, (i < k) such that a = R m ( pu rf,A , No). If for each i < k, a = Rm(qu rf,A , No), then Mlta(q,A ) 2 k so we finish. Otherwise for some j < k, Rm(qu r,, A , No) < a hence for some finite p1, p E pl c q, P ( p 1 u r j , A , KO) < a. As p E p1 E q clearly a = Rm(p,,A , Ho), k = Mltl(pl,A) so as p1 is finite, there are pairwise explicitly contradictoryd-m-typesr: (i < k)such that R”’(plu r:, A , No) = a. Choose A such that rf,r i (i < k) and q are over A , and for i < k, choose rf such that r: c rf E S ~ ( A )Rm(p, , u r f , A , No) = a (by 1.6) and similarly r;, such that r, s E @ ( A ) , a = Rm(pU r;, A, N o ) . The rf (i Ik) are distinct (for i(1) # i(2) < k =+ rill, ri(2) are explicitly then a IRm(plU r;, A, KO)hence contradictory; and if i < k,rf = a IRm(plu r,, A, No)contradicting the choice ofp,, r j ) . As E &?(A), the rf (i Ik) are pairwise explicitly contradictory, and a I Rm(pu r f , A, KO)hence Mltl(p, A ) 2 k + 1, a contradiction.

e e,

LEMMA 1.11: There i8 AT depending 012 T only Rm(p,A , A) = Rm(p,A , CQ) for every p and A.

<

S W ~

t h t for A 2 AT

Proof. For every O(Z; a) and A, Rm[B(Z;a),A , A] is a decreasing function of A, hence there is A = A(B(Z; a ) , A ) such that for all p 2 A R”’[O(Z;a),A , p] = R”’[O(Z;a),A, A]. It is clear that A(B(Z; a),A) depends on O(Z; j j ) , tp(a, 0), A only (as rank is preserved by automorphismsof CE). As the number of such triples is I2ITI, there is & = sup{A(B(Z;a),A): O(Z, a),A } < 00. As for finite p, Rm(p,A , A) = R”’(A p , A, A), and every type has a finite subtype of the same rank (see l . l ( l ) , 1.2), for every p and A, A 2 AT implies R”(p,A, A) = R m ( pA, , AT). NOWwe can easily , 2 a implies prove by induction on a that for A 2 AT, R m ( p , A A) R”’(p,A, 00) 2 a. Hence for A 2 AT, R”’(p,A,00) 2 R”’(p,A, A), and 1.3 implies equality. EXERCISE 1.1: Let f be an automorphism of Q. Show that Rm(p,A, A) = R”’v(p),A, A] for everyp, A, A; wheref(p) = {v(Z; f (a)):~ ( 3a); ~ p } .

28

RANKS AND INCOMPLETE TYPES

[OH. 11, 8 1

EXERCIBE 1.2: Show that for every A, A, that if for no p does R"(p,A, A) = a,, then P ( p ,A, A) 1 a, =- R'"(p,A , A) = 00, for every m-type p. EXERCIBE 1.3: Show that for all A 2 X,, Rm(p,L, A ) = 0 iff p is (consistent and) algebraic (i.e. realized by only a finite number of sequences). EXERCIBE 1.4: Show that if T is the theory of KOindependent unary predicates, then for all p, Rm(p,L, 00) = 1 o p is not algebraic and for p not algebraic Mltl(p, L, 00) = 2% (So T = ((3%)Ate,,,Pt(~)il(iEs): u E w s w , w finite}.) EXERCISE 1.5: (1) Show that if 8 is a limit ordinal (or 0), then for all p,

where k, 1 < w and k* = min{j: n5 2 nk}. (2) Generalize Theorem 1.6 accordingly. (Hint: take 1 = 0.)

EXERCISE 1.6: Let k, 1 be natural numbers, k 2 2 , l 2 1. Show that for all a, p and A ((3) and (4) are rephasings of ( 1) and (2), respectively): (1) R"(p,A, k) 2 1.a * Rm(p,A,k') 2 a, (2) R"(p,A, k ) 2 w - a * Rm(p,A,KO)1 a, ( 3 ) R'"(p,A,k)S l.R'"(p,A,k') (1 - I), (4) R'"(p,A,k ) < w * R m ( p , AXo) , W. (See also Exercise 3.22(1).)

+

+

EXERCISE 1.7: Show that if any one of p , A, or A is finite, then Mlt'(p,d, A) = Mlta(p,A, A). Show that in general equality does not hold. EXERCISE 1.8: Prove that 1.10(3) may fail for A < 8,. EXERCISE 1.9: Prove that in 1.10(3) even for e t e A , 'p E d Mlt'(p, A, A) 1 Mlt"p

u {'p}, A , A1 + Mlt'rp u {+,

A , A].

EXERCISE 1.10: Show that if Rm(p,A, A ) 2 n, then for some finite A l E A, Rm(p,A l , A ) 2 n for p finite, A finite or 00. Show that those restrictions are necessary.

OH. 11, 8 21

29

STABILITY, RANKS AND DEFINABILITY

II.2. Stability, ranks and definsbility Now we show that there is no essential difference between A-m-types for Gnite A and p-m-types.

LEMMA 2.1: Let A = {pk@; #A

gk):

k < 1 < O}, Z(Z)

=

m and

= #,(Z;go,...,gai-i;z,zo,...,zai-i) ai-1

= k=O

(z = zk

--f

A ( z = zk l p k - l ( z ; g k ) ) A V Z = zk A = zk A A k<Sl k
p k ( z ; g k ) ) I\

k-1

~

(

2

Z

=

Zn).

Then: (1) For every type p and power A, Rm(p,A , A) = Rm(p,#A, A). (2) For every fornzula p(Z; a) (where p E A or lcp E A ) and co # c1 a’) (more exactly, there is a‘, c U {c0,cl} m h that cp(Z; a) = k(VZ)(p(Z; T i ) = a’))).Hence for every A-m-type p over A , IAI 2 2, there is a #A-typeq over A equivalent to it (i.e., p = q). Also i f p E rS,m(A), then q E Q A ( A ) . (3) For eoery 6 m h that k(3Z)#A(z; 6) there ie G 6 and p (p E A or - , pE A )for which p(Z; a) = 6). Hence for every #A-m-typeq over A thereisa A-m-typep OverA equivalent toit. Ale0 q E STA(A )implies p E @(A).

Proqf. Let co # cl tpk(Z;

--

E A.

( 2 ) and (3) may be proved as follows:

a) = #,(Z; co, . . .,co, co, . . ., co, . . ., a,. . .,co, . . .,c,; Yo

Yl

c1, co, 2,

20,

. - ,c1, co, .

- * . , Zk,

gk

Y Yat - i

,co).

Z k + l i * . *,z11-1

In other words, in # A we substituted (co, . . .,co) for gn,n # k, for g k , c1 for z and zk, co for zn, n # k. A similar trick works for l p k ( i & a). And on the other hand,

cat-1) pk@, a k ) or l p k - I ( Z , a,) or is false (the c,’e are not necessarily distinct). So in this way q may be defined from p and vice vem. Now for the case where p[q] is complete. If p[q] is a complete A-type (#,-type) over A, IAI 2 2, then (i) for every k, a k E A , #A@;

a08

* -9

a21-1; c k , CO,

q

*

- 9

pk(% a k )

or q

lpk@,

9).

30

[CH.

RANKS AND INCOMPLETE TYPES

11, 8 2

(ii) For every c, c,, 8, E A ,

. . .;c, coy. ..) p I- 7#&I(Z;go, . . . ;c, coy . . .). p I- #&@;a,,

or

The proof is straightforward.

Proof of (1).By the above it is sufEcient to show: Claim 2.1(4): If A,, A ;are sets of m-formulas which are closed under l, and: (iii) For every v(E;g) ELI, and there is a h i t e A,-m-type p z such that v(Z;a) = A p;, and p: and p i @are explicitly contradictory, or C,(3Z)v(3; a). (iv) For every I@; a) E A , and there is a finite A,-m-type such that p(Z; a) 3 A g and g and Q are explicitly contradictory, or b-l(3Z)(p(Z;

a).

Then for every type r and cardinal A, Rm(r,A,, A) = P ( r , A,, A).

Remark. If we amume (iii) alone we get Rm(r,A,, A)

5

Rm(r,A,, A).

Proof.By the symmetry of the tlssumptions it is enough to prove for h i t e r that Rm(r,dl,A) 2 8 * Rm(r,A,,A) 2 8. For 8 = 0 or limit this is immediate. So assume it for a and let 8 = a + 1. Let p < A. Since Rm(r,A,, A) 2 a + 1 there are A,-m-types {rJlr. which are pairwise explicitly contradictory and Rm(rU r,, A,, A) 2 a. By (iii) we can v(Z;a) E r,}, and r f , r; are explicitly contradictory define r: = for i # j.Also r, 3 r:. Thus Rm(ru $ , A , , A) = Rm(ru r,,A,, A) 2 a. So by the induction hypothesis and 1.2, P ( r u r:, A,, A) 2 a. Thus Rm(ryA,,A) 2 a + 1 = 8. This completes the proof of the claim and the lemma.

u {c:

Remark. We w i ll use Lemma 2.1 without explicit mention. In the following theorem we show that the unstability of equivalent to many conditions.

v(E;g)is

THE UNSTABLE FORMULA THEOREM 2.2: Thefollowingpropertiea , = l(Z), are equivalent (relative to a given theory T ) . of v = q ~ ( Z ; g )m (1) cp i8 unstable (in every insnite power); i.e., for every A 2 Mo there is A m h that IS;(A)I > A 2 IAI.

OH. 11, $21

31

STABILITY, RANKS AND DEFINABILPTY

(2) cp ie urntable in at leaet one power h 2 X,. ( 3 ) cp lrae the order property; i.e., t k r e are 6,n < w BucIG thad for every k < o {cp(z; p)’f(ksn): n < w} is consistent. (4) For every n < o,F(cp, m, n) i s &tent, where

r(9, m, a) = {dq;&,P: r) E “2,/3 < a)

( d y we omit the m). ( 6 ) r(9,a) i s am&ent for every o r d i d

a.

(0) P ( Z = Z,cp, 00) = 00. (7) P ( Z = 1 ,Q, 2) 2 0 . ( 8 ) There are A , p , p €@(A),

m h thad p is not A-definable (see Definition 2.1). (9) There ie 7u) 9 euch that for every A, IAI 2 2, and p EAY;(A),p is (9,A)-definable (see Definition 2.1).

DEFINITION 2.1: (1) The cp-m-type p is $@;Z)&fiined C$[Z; Z] and Tcp(Z; a) EP * C+[7i; Z]. (2)

p is

if cp(Z; a) EZ)

-

The cp-m-typep is (+@; Z), A)-defidle if there is a Z E A such that

$(a;+defined.

(3) The cp-m-type p is A&$nuble if there is a $ for which p is ($, A)definable. (4) The m-type p is A-definable if p r cp is A-definable for every cp.

Remark. If p is a complete cp-m-type over A, then the mows in (1)go both ways for CTE A. We sometimes write “definable over A ” instead of “A-definable”. The proof of Theorem 2.2 will be broken down into a series of lemmas and theorems and completed with Theorem 2.12. From now till then all nohtions will be aa in Theorem 2.2. Obviously, (1) (2).

LEMMA 2.3: (2) 3 (3). Procf. Let l&(A)I > h 2 IAI 2 KO. By Conclusion I, 2.11 there are sequences 3, n c o such that either (i) for every k < w , {cp(Z; &)‘(i
32

RANKS AND INCOMPLETE TYPES

[CH.

11, 5 2

LEMMA 2.4: (3) =- (4).

r

r

Proqf. Define an order < on a 2 2 as follows: if 7 k = v k, ~ [ k=] 0, v[k] = 1, then 7 < v ; if 7 t k = v 1 k, Z(7) = k, v[k] = 1 then 7 < v ; if 7 k = v t k, Z(7) = k, v[k] = 1, then v < 7.It is easily seen that < is a total order. So by the compactness theorem and the fact that 'p has the order property we see that the set

r

T u {tp(Z,,; gy)if(n
= n, Z(v)

< n}

is consistent for all n < w. So we get that

F('p,n ) = ('p(8,; is consistent for every n <

g,,lk)W[kl: q E "2,

k < n}

w.

LEMMA 2.6: (4) => (6).

Proof. This is immediate since every finite subset of F('p,a)is equal to a subset of some F('p, n),after changing names of variables. LEMMA 2.6: (6) * (1).

Proqf. Let A be infinite; we must show that 'p is unstable in A. Let M be a model of F(tp, p ) where a,, realizes g,, and E,, realizes 3,. Let A = {a,,: Z(v) < p}. Cleazly IAI 5 (Za<#1a21).K0 I p . A . N , = A (since A < 2*, we have p 5 A). For 7 E '2, let p,, be the tp-m-type E,, realizes over A. If Z(7) = Z(v) = p, 7 # v , then p,, # p,,, since if a is the first ordinal such t h t 7[a]# v[u], then 'p(3;i i , , l a ) ~ [ a El p,, and 'p(3;Siy,a)v[al = -'p(E; B,,lcr)'[al E p,,. Thus p = infb: 2' > A} and let

u

Ifl34I 2 Ib,,: 4 7 ) = d l = l{7: Z(7) = PI1

= 2'

>A 2

14.

This completes the proof of the equivalence of properties ( I ) through (6).

LEMMA 2.7: (6) => (6).

Proof. Let a; realize ijp and Z; realize 3, for all a,p E a>2, 7 E a2. Let A be any cardinal. For all 7 E h'2 let p,, = (tp(3;si~l,#c6': B < Z(q)}; p,, is a (consistent)'p-m-type. Let p v be such a p,, with minimal R"(p,,, 'p, A), say R"(pv, 'p, 4 = a.

CH. II,#21

33

STABILITY, RANKS AND DERDTABILITY

Suppose a # a.Let @, y < A, be a sequence of zeros of length y and py = ~ ) , - i i - ~ ~The > . p% are pairwise explicitly contradictory, p,, c py, and P ( p V , 9,A) 2 a,by the minimality of a. So R"(pv,9,A) 2 a + 1, a contradiction. Hence P ( p v ,9,A) = a, so R"(3 = 8,9,A) = 00. As A was arbitrary, by 1.11 P ( 3 = 3, 9,a)= 00. LEMMA 2.8: (6)

(7).

Proof. Exercise. Notation. From now on in this section, R ( p )or P ( p )denotes P ( p ,9,2).

LEMMA 2.0: (1) Let p be an arbitrary m-type. R ( p ) 2 n iff q 9 ,n) = {#(Zn; q :#(Z;a) Ep, q E n2}

u ( ~ ( 3g n~l r );n [ k l : q E "2, k < n } is &tent. (Note t7& &'I n) = I'(9, n ) = I'{;+(q, n).) (2) For any rn-type p , for all k, n < o,Rm(p,9,k ) 2 n + 1 iff {#(Z,;

a): v E " + l k ,#(Z; a) ~ p }

u {v(Z,,: 2t.f) 3

-,'p(Ep; 2:'):

q, p E n+lk, Y E nzrk,Z(v) = rn < n

+ 1,

v=qtm=prm,i=q[m],

j = p b l , i # j} i8 consistent. P ( p , 9,k) = n, Mlt(p,9,k) 2 k, < k iff the set of fomnulas given above is inconsiStent, but if thefurther wnditbna q[O] < k,, p[O] < ko are added, the set ie d e t e n t . (3) For any m-type p for all n < o,Rm(p,9,No) 2 n + 1 ifl

{$(a,; a):Y €*+I u {p)(%,,;

*j)

w , #(Z;a) El)}

= +zp;

* f ) : q, p E n+lo, vE

Z(v) = m

+ 1,

v = r) t m = p r m , i = q[rn],

j = 4m3, i # j} ie wn&tt?nt. Rm(p,9,Ho) = n, Mlt(p,9,8,) 2 ko < No iff the set of fotmukce given above i8 i?acmaisM,but if thefurther conditbna q[O] < k,, p[O] <

ko are added the 8et is &tent.

Proof. (1) Firat suppose R ( p ) 2 n. By the compaotness theorem it is n ) is consistent when p is finite. Now we sufficient to show that ,&'l

34

RANKS AND INCOMPLETE T Y P E S

[OH. 11, 8 2

define 8, for a E k>2,by induction on k 5 n, such that for all 9,q E k2, R(p,) L n - k where p,, = p u {p(Z; a,,,,)W[fl:j c k}. For k = 0, po is just p; so R(po) 2 n. Suppose the definition for k < n is completed, and look at k + 1. Let Z(q) = k; then R(p,,) 2 n - k. Thus there is a sequence a,, such that if pc(o> = p,, u (~(3; a,,)} and pn-(l> = p,, u {+3; a,,)},then R(pc<,>)2 n - k - 1 for i = 0 , l . And this is the condition for k + 1. Thus when I(?) = n we get R(p,) 2 0. Thus pn is consistent. Let 4 realize p,,. Taking a,, for Z,,and a,, for g,, we see that FJp, n ) is Consistent. The proof of the other direction is similar and is left to the reader (now the induction is downward). (2) and (3) have a similar proof.

COROLLARY 2.10: (1) (7) * (4). (2) R ( p ) 2 w iff R ( p ) = 00 iff P ( p , p , 0 0 ) = 00 iff there are an, w , m h that for every k < w p u {p(Z; an)u(ksn): n < w } is consietent.

n

<

Proof. (1) This is because F{;=;)(p,

n ) is equivalent to F(p,n ) .

This completes the proof of the equivalence of (1) through (7). (2)

R(p) 2

w implies FJp, n ) is consistent for every n, hence TJp, a )

is consistent for every a,so by the same proof aa that of 2.7, P ( p ,p, 00) = 00. Trivially P ( p , p , 0 0 ) = 00 implies P ( p ) = 00 implies P ( p ) 2 w . The Iaat "iff" is proved aa in 2.3 and 2.4 for finitep, and then by compactness.

LEMMA

2.11: (1)

(8).

Proof. There is an A such that l8;(A)I > IT p E @ ( A ) which are A-definable is

s I{$(#; a): $ € L(T), a € A}I

I IT1 *

2

nea,

So some p E #;(A) is not A-definable.

THEOREM 2.12: (1) Assum R(Z = 3) < w . Then there is a formula $(g; Z)euch that forevery A ([A12 2), everyp E#;(A)is (y5, A)-&$dZe. (2) Tor every O(Z; g), Z(3) = m, $nite A , k, and n there is a #(g) euch that: for all 6, P [ O ( Z ; 6 ) , A , k] 2 n if k$[6].

CH.

11,s 21

S T A B a Y , RANKS AND DEFINABfLITY

35

( 3 ) FOYeuery finite A , n, tp(Z; g) E A , and O(Z; 2) there is a #(g; Z) w h that: if O(Z; a) ~ p Rm(p, , A , 2) = P [ O ( Z ; a), A , 21 = n, then p 1 tp is

#(g; a)-&finuble. Remark. Theorem 2.12(1) proves (9) =- (7). It is obvious that (8) + (9) so we have (1) + (8) =+ (9) =- (7) and thus (1) through (9) are equivalent, mmpleting the proof of Theorem 2.2.

Procf. (1) Let A be any set, p E 8;(A). We shall show first that p is (#, A)-defined for some 4. By Theorem 1.2 there is a h i t e q E p such Assume q = {tp(Z; zit)"["]: 1 < n} where 7 that R(q) = R ( p ) = k (
.-

Hence tp(Z-;a) ~p iff k#p;a], E A, andp is (#, A)-definable.Now we show that the choice of # depends only on 9,and not on p. In fact it suffices to prove that there is a finite A such that every p E #;(A) is ($, A)-definable for some # EA. For if A = {#*(#; 2):i < n} then

#(g; Z) = #(g; ZO, . . .

y

=

A 1%z

En-1.9 Z,ZO,

zt +#t(g;

--

- 9

Zn-1)

zt)I

I
is the required single formula. By way of contradiction assume there is no such fhite A. Let P be a new one-place predicate and b,, . . .,b, - new individual constants. Remember Z(Z) = m and let E(#) = n in v(E;8).For any (not necessarily bite) set d of formulas #(g; ZJ,, let I

(where 6 = ( b o , . . .,bm-l)).

36

RANKS AND INCOMPLETE TYPES

[OH.

11,8 2

Now T , is consistent for any finite A, since by assumption there are A, p , p E~S:(A) and p not (#, A)-definable for any # E A. So letting M C T , A E 1M1,and 6 realize p we get (M, A, b,, . . ., bm-l) C T,. Thus T A , is consistent where A,, is the set of all formulae of L of the A, b,, .,bm-l) C T,, where form # ( j j ; 2,) with l(jj) = n. Let (M, A = PM.If p is the 9-type 6 realizes over A, then for no # is p (#, A)dehable, contradiction. This proves (1). (2) is clear by Lemma 2.9(2) using Lemma 2.1. (3) is left as an exercise.

..

THEOREM 2.13: The following propertie8 of T are equivalent: (1) T is unstable. (2) T is unstable in at least one A, hiT' = A. (3) gbme formula Q(X; j j ) is unstable. (4), 8 m formula ~ ( 3j j ; ) is unstable, l(3) = m. (6) There is a formula ~ ( 3jj) ;and sequence8 (3n :< w } , l(3) = @) = l(Z),w h that for all n, k < w , CQ[P; ak]o n < k. (6) There is a formula Q = Q ( ~ O , .. .,P-l;a) and an infinite set I of sequence8 of length l(Zo) = . . . = Z(P-') such that Q is connect& and antisymmetric over I (see Definition I , 2.5). (7) There is an infinite indi8CemLible sequence (of sequences) which is not an indiscernible set. ( 8 ) There is n < w , afinite A , and A m h that for all no < w there is a A-n-indiscernible sequence I over A which is not a A-n-~ndiscernibleset over A , and IIl 2 no. (9) There is a formula Q = v(f0,. . .,P-'; ij) such that for all no < w there is a set I of rn, sequences of length 1(Z0) = . . = l ( 9 - l ) and a sequence ii w h that p(Z0, . . ., 9 - l ; a) is connected and antisymmetric over I .

-

The proof will be delayed until after Lemma 2.16. LEMMA 2.14: For every A, Q, IS(A)I 2 18:(A)I. I f ~ ( 3j j ; ) is umtable in A, then T is unstable in A.

Proof. Exercise.

OH.

11,s 21

STABILITY, 8 -

AND DEFINABILITY

37

Proof. (1) Let A = {cp,(Z;g): i < IAl). The mapping g: @(A, M) -+ < e , ( A , M) defined by g(p) = (. . .,p !9, . . .)f, is one-one, since if p # q €#;(A, M), then there is i < lAJ and E E A such that a) E p CPI(E; 8) 9; thus g(p) z g(q). cpf(~;

nf

-

e

nf<

A,. By the assumption on A we can Let A, = ISgi(A,M)I. So h 5 find an i for which A, r A, and that is exactly what had to be proved. (2) Follows from (1)by taking A = (plAl) . +

LEMMA 2.16: Let I = {Sif:i < i,) be a A-n-indiscernible sequence over A which is not a A-n-indiscernible set over A. Then there is #(E; #,2) E A, P,501 iff o < a < p, where E E A such that for o < a # 8, . . n a w + n - 1 ( a d n(p 1) 5 i,). ga E-pa-pa+l-.

+

Proof. By I , 2.9 there is cp E A and E E A such that cp(E1, . . .,En; E) is connected and antisymmetric over I. So by 3.9 of the Appendix it follows that there is such a #. Proof of Theorem 2.13. By definition, (1) =. (2). By Corollary I, 2.2, (2) implies that there is an A for which IS(A)I > h(=AITl) 2 IAl. So (2) => (3) by Lemma 2.15(2). By adding dummy variables, (3) =- (4),. By Lemma 2.14, (4), =- (1). So (1) through (4), are equivalent. Now we show (3) =- (5): (3) implies that cp has the order property, by Theorem 2.2. Hence there are Tin, n < w, such that for every k < w Pk = {cp(x;a-n )u(ksn):n < w) is consistent. Let bk realize p,, and set E" = (bn)^Tin. Then Cp[bk, Tin] iff k 5 n. Let # = #(xl, 9 ;x,, ga) = p(xl; jja) A x1 # xa. Then k#[Ek; B] iff k < n, and this is (5). (5) (4), (for some m ) : Aasume (6) and let b = Sian+'. Then for every k {cp(E;B)u(ksn):n < w ) is consistent (since it is realized by Siak). So cp has the order property and thus is unstable by Theorem 2.2. (6) =- (6) is clear. (6) =. (7): By Theorem I, 2.4, for every finite A, n(cp E A), the infinite set mentioned in (6) has a subset {af:i < w) which is a A-n-indiscernible sequence over Si. This is of course not a A-n-indiscernible set over a (by (6)). Thus by compactness we get (7). (7) =- (5) by Lemma 2.16.

38

RANKS AND INCOMPLETE TYPES

[CH.

11, 5 2

(6) =- (9) and (7) 3 (8) are obvious and (9) =- (6) and (8) =- (7) follow by compactness. This completes the proof of the theorem.

THEOREM 2.17: Let T be stable and A be$nite. Then there i s afinite A* m h that if for fi < a I), is the A*-m-type that SiS realizes over A, = A u (8,: y < p}, Po = p,, and for every PEA* Rm(pB trp) = Rm(pot v) (m = l(7iB))and pB is (#&j,Z); A)-&efinuble, #,,,from 2.12; t h n {a,: jl < u} is a A-indiscernible sequence over A.

u

Prmf. Since A is bite, there is an n < w such that any sequence (6,: y < /3} is A-indiscernible over a set B iff it is A-nl-indiscernibleover B for all n1 5 n. Now define a sequence of finite sets A,,,A,,-,, .. .,Ao by downward induction such that (1) A E A,, (2) each A, is closed under permutations of variables, and (3) if 'p = p(Eo,, . .,2-l;g) Ed,, l(P)= - - = l(2-l) = m, then every pm-type p such that P ( p ) = R m ( p A ) is (#, A)definable for some # = #@,, . . .,2-l,g; Z) E A,-l. (3) is accomplished by Theorem 2.12(1)-(3). Now choose A* = U,=.,,A,. So in order to prove the conclusion of the theorem it is sufficient to show that {BE: p < a} is A,-i-indiscernible over A for all i s n, since A E A,. Now if y < p < u, then for every T E A * , po 19, c p , IF, p , r ' p and R(Pv t ?J) = R(Po t 'p) = m p , d - BY corollary 1.6, Po t v has a unique extension in #:(A,) of equal rank, so since I), 'p, p , t 'p A, E 4XAY) and RbO t 94 2 RRP, t q-4 t A,] 2 R(PB t tp) = R(Po t 94 we have p , = ( p , 1 tp) A, for every Q €A*, y < /I< a. So by the definition of A*, tp(EY, A,) A, E t p ( 8 , A,) 1A, for all y < p < a, i In. It also follows from (3) that t p ( 8 , A,) does not (A,, A,+,)-split over A for i < n, p < u so the result follows by Lemma I, 2.6.

-

r

r

COROLLARY 2.18: If T is stable and for every p < u 2 w p , is the m-type realizes over A, = A u (3'y:< Is), for every R m ( p , 1tp) = P ( p 0 r v), andpo s p,, then {a,: p < a}is an indiscernible set over A.

u

Remark. We can use many other parallel conditions in place of "for every 9,Rm(% t y) = R'"(po t v)", e-g., ~ ( P B L, , 2) = *(PO, L,2).

Proof. By the previous theorem it is an indiscernible sequence over A. So by Theorem 2.13(7) it is an indiscernible set over A. THEOREM 2.19: Assume T is stable, A j n i t e , A regular, I a set of sequences of length m, IAl < A I 111. Then there is J E I , IJI = A, which is a A-indiscernible set over A.

CH. 11,s 21

STABILITY,RANKS AND DEFINABILITY

39

Proof.Let A* be uin Theorem 2.17.Among the A*-m-typesp, lpl < A, which are realized by at least A sequences from I, choose one p , with minimal ( P ( p , 'pj, 2): j < no) (ordered lexicographically, where A* = {'pj: j < no}).We crtn assume that p , is a tspe over A, 2 A, lAol < A. Now we shall define ?2 E I for i < A. Let 7 i j , j < i < A be defined, A, = A, u (J {aj: j < i}. l&$(A,)l < A since T is stable and lA,l < A. Thus there is p , E ~ * ( A ~p o) E , pi, and p f is realized by A sequenoes from I. Clearly for j < no, R"(p,, 'pj, 2) = P ( p o ,'pj, 2). Choose 8 E I - "(A,) realizing pi. By 2.17 and 2.13(8),J = {?2: i < A} is a A-indiscernible set. If A = No, the claim follows from Theorem I, 2.4(1) and the fact that as A is finite every set of sequences is a A-n-indiscernible set for all sufficiently large n. LEMMA 2.20:Buppose v(J; @)is stable. T h there are jinite A, n such that if I is a A-n-indiscernible set of sequencerr of length I(@) and 7i is any sequence of length 1@), then

I{EE

I : kp[a; ~]}l< n or ~ { E EI : C7'p[zi;

a}]< n.

Proof. By contradiction. Assume that the conclusion is false. Then by and an indiscernible set I such that compactness there are an { E E I: h@; Elt} is infinite for t = 0, 1. Let {tT}n < @E I. For every w E w , {'p(Z; F)ii(new): n < w } is cornistent (since I is indiscernible). Thus

1

I).

B?( (J

2 2h > No =

n<m

This contradicts the stability of 'p.

EXERCIBE 2.1: Show that in Lemma 2.20, for big enough A, (1)we can choose n = R"(E = E , 'p, 2) + 1, (2)we can chooss n = 2"+ where k = Rm(Z= Z, #, 2)and #(Z; 8) = 'p(%

q,

(3)when I is infinite we can choose n = R"(Z = J, 'p, 00).

EXERCIBE 2.2: Show that Exercise 2.1 cannot be essentially improved. That is for some stable (even No-stable) theory T,infinite indiscernible set I, 'p and E,

I@ E I : C'p[E; iz]t})l

2 n for t E (0, I},

40

RANKS AND INCOMPLETE TYPES

[CH.

11, 8 2

but Rm(f = f , I,A) 5 n for every A 2 2 and Rm(Z = 3,$, 2) = pog, A ] , P ( f= Z, $, A) = 1 for A 2 X,. (Hint: Let T = Th(M), 1M1 = w u {w G o:IwI = n}, P = o,RM = {(n, w ) : n E w } and ~ ( sy),= R ( r , y), I = 0.)

EXERCISE 2.3: Let (I, s ) be an order; B L1 model HI as follows:

=

{E,: 8 E I). We define the

E f l = {(f, s>: (Vt 5 4f(t)= s(t)}; TI = Th(H’). (1) Show that TI admits elimination of quantifiers. (2) Let K(I) = inf{K: I contains no increasing sequence of length K } . Show that TI is stable in A if€ A = 111 + A<x. Compare this with the example after Definition 1.2. There, (I, < ) is a*,i.e., (a,3 ) .

IM’I = d f : f : I + u } ;

EXERCISE 2.4: Suppose M is K-compact,p is an m-type in M , 1pl < K , a, E 1M1, and vl E L(H), 1 = 1, 2. If for all E E 1M1 realizingp, i=pl[E;a,] o Cv,p; a,], then h p l [ E ; a,] e ttp,[Z; 4 3 for all E E 6 realizing p . EXERCISE 2.5: Relativize Theorem 2.2 to a formula d(Z; a) (i.e., in (1) and (2), replace “9 stable in A ” by “IAI s A implies l{p € @ ( A ) :p u { d ( f ; 6 ) is consistent}l I A’’, in (3), add { d ( Z ; E ) } ; in (4) and ( 6 ) , add {O(Zn;Z): 9 ~ ~ to 2 ) P,in (6)and (7), replace f = Z by 8(f;a); and in (8) and (9) demand e(z; a) E p . EXERCISE 2.6: Do the same with Theorem 2.13. EXERCISE 2.7: For any finite A, n, and 6(f;5) there is an @)-type q, such that for every E Rm[O(Z;a), A , No] 2 n iff E realizes q. (Hint: Use 2.9(3).)

a)

EXERCISE 2.8: Let ~ ( f ; = +(g;3);and show that v(Z;g) is stable iff +(g;Z) is stable. (Hint: Use 2.2(3).) EXERCISE 2.9: If T is stable, p E S:( IiKI),iK E A, then there is a unique p E q € @ ( A ) such that for every I,R ( p 1 v) = R(q tp), and if p is $(z;a)-defined,E E IHI,then so is q.

OH. 11,s 31

41

RANKS, DEGREES AND SUPERSTABILITY

II.8. Ranks, degrees and superstability THEOREM 3.1 : For a finite rn-type p the following are equivalent: (1) R r n ( p , A , 2 )2 l A l + + So, ( 2 ) R r n ( p , A ,A) = a0 where A = (2"0)+, (3) There are ~"(5; g,,) E A and a,, for q E " > 2 such that for every 7 E "'2, 1, U p,, [p,, = {Q,,l,(Z; a,,ln)R[nl: 72 < $(q)}]i8 COWi8t&.

Remark. From 3.1 it follows that (1) and ( 2 ) are equivalent also for infinite p (by 1.2). If in ( 2 ) we replace A by No,the equivalence is still correct, but the claim is stronger with A. If we replace A by A1 > A, the implication (1) * ( 2 ) is not correct.

DEFINITION 3.1. If Rm(Z= 3, L, 2 ) < 00, T is called totally tranecendental.

Proof of 3.1. ( 2 ) ==

(1) is trivial by 1.3(1).

(3) + ( 2 ) We shdl prove by induction on a that Rm(pup,,, A , A) 2 a for every q E * > 2 . As p o = p , this suffices. For a = 0 it follows by the

consistency of p,, u p ; and for a a limit ordinal from the induction hypothesis. So let a = p + 1 , q E " > 2 ; then { p v :q Q Y E "2) is a family 0f2~0d-m-types which are pairwise explicitly contradictory. By 1.1(1) and the induction hypothesis Rrn(pv up , A , A) 2 hence Rm(p,,up ,A , A) 2

p+

1 = a.

(1) =- (3) We shall define by induction on n < w subsets

U, of lAl+ + No of cardinality lAl+ + KO, formulas Q , , E A for q ~ " ' 2 and sequences for q E ">2, i E U,,such that R'"(p!,,A, 2 ) 2 i for i E U,, 7 ~ where ~ 2p i = p U {pTlr(Z;@$)'['I: k < Z(7)). For n = 0 let U, = {a: a < 141 + No}and p'o = p, so by (1) the induction hypothesis holds. For n + 1, for every i < IA I + Nochoose j = j(i), i < j E U,, so Rrn(rJ,, A , 2 ) > i for q E "2, hence for some Q!, E A and a!,, Rm[& u {tp!,(Z; a!,)'},A , 21 2 i, t E (0, l}. Hence for some Un+lc {a: a < 141' + No}, lUn+lI = l A l + Ho,foreverya,pE Un+l, q E "2, v; = Q:. Let for q E "2, Q,,= Q; (for any a E Un+L) and for when q E , > 2 and $ when q E ,2. Hence is i E Un+l, +

+

+

is consistent so some assignment Z, show that (3) holds.

H

6,,,gv H iiv realizes it. These av

42

RANKS A N D INCOMPLETE T Y P E S

[OH.

11,s 3

Remark. For A = L them conditions are equivalent for difFerent m's by I, 2.1, i.e., (1)1 o (2), o (3), for d 1, myn. Proof. (3)==(2)trivially, by the same A. (2)*( 1) Suppose not ( 1). For every p E @ ( A ) there ie a finite qp G p suoh that P ( p , A , 2) = P ( q p y A 2) y < co (by not (1)). By 1.6,p(1) # ~ ( 2* ) Q P ( ~ )# qfis hence l8?(A)I s I{qp:p

E 8?(A)}l s

+

S

I@:

q a finite A-m-type over A}\

+ KO,

a contradiotion.

(1) * (3) By 3.1 we know that (1)implies condition (3)from 3.1. Letting A = U {a,,: r) E @>2}, and q,, be a type in &(A) exfending p,, [we use the notation of 3.1(3)] we see

I&WI because if r) #

Y

2

E "2, let n =

V q l k ( Z ; $Ik)nrkJ

EI)n~

k,,:r ) E'21

=P o

min{k: r)[k] # v[k]}. Then

l Q , , l k ( % q,k)"rkJ

= q J v l k ( R a"Ik)v[k' E q v .

On the other hand, lAl I; I">21.N0 = No.

CONCLUSION 3.3: (1) If IT1 < 2n0, T ie stable in IT1 (or in some A < 2b), then T ie shbk in every cardinality p 2 ITI. (2) A totally tranrrcencEental T i s stable in every A 2 I TI. DEFINITION 3.2: We define the

degree Dm(p,A, A) ( p ,A , A are as in Definition 1.1) by defining inductively when Dm(p,A, A) 2 a (a an ordinal): (1) D"(p,A, A) 2 0 when p is a (consistent) type (when p is an inconsistent set of m-formulas, we let the degree be -1). (2) D"(p,A, A) 2 8 (8 a limit ordinal) if D"(p,A, A) 2 a for all a < 8.

OH.

11, 8 31

RANKS, DEQREES AND SUPER8TABIIJTY

43

(3) Dm(p,A,A) 2 a + l i f f o r d l p < Aandallfhitarspthereare, a finite q z r , n < w, $(Z;0 ) E A, and sequences a{,i 5 p such that: (i) Dm(qu {$@;at}, 4 A) 2 a, (ii) {$(z;a,):i 5 p} is n-contradictory (or n-inconsistent) over q, i.e., for every w E p, IwI = n, t=-,(3Z)(AiEw $(3;a{)A A q). (4) Exactly as (4) of Definition 1.1.

Remark. We ignore the degree for A < X,, and will be mainly interested in the c&88 A > ITI+.We can always replace q by A q.

U

a,)},A, A)

=

y:

(4) If p is an m-type over A, A 2 No,A is given, then there i s q , p E q E &?m(A) m h thud

W p ,4 4 = D"(q, 4 A). Proof. The proof of (2) is like l . l ( l ) , 1.3(1); the proof of (1) is like 1.2; the proof of (3) is like 1.7 and the proof of (4) like 1.6.

LEMMA 3.5: (1) If X E {R, D},p a$nite type and 00 > Xm(p,A, A) 2 a, then for some A-m-type q, Xm(pU q, A, A) = a. (2) If X E{R, D} a = (21Tl)+,Xrn(p,A, A) 2 a, then Xm(p,A, A) = 00. If X = R we need only a = (21*l+No)+.

Proof.(1) We suppose there is no such q, and prove by induction on

82

a that q = p vq,,

q1 a A-m-type, Xm(q,A,A) 2 a implies

Xrn(q,4 A) 2 8. For 8 = a this is the aasumption, for 8 a limit ordinal it follows by induction. For = y + 1 > a, and e.g., X = D, let rl c q be finite and p < A. Then there are a finite r z rl and formulaa $(Z;a,) (i 5 p, $ E A ) n-contradictory over p v r such that

Dm[pu r u {$(Z;at)},A, A] 2 a

44

[CH.

RANKS AND INCOMPLETE TYPES

11,f 3

(they exist as Dm(pu q , A , A) 2 a implies Dm(pU q,d, A) 2 a So by the induction hypothesis

+ 1).

D"[p u er u {#(% q},4A1 2 y As this holds for any finite erl, Dm(q,A , A) 2 p. Thus Dm(p,A, A) = 00, a contradiction. The proof for

X

=

R is similar.

(2) By 1.2 and 3.4(1) we can assume p is finite; if X = R we can also assume IT1 = lAl 24,. If (2) d m not hold, then by part (1) there are finite p , (i < a) such that X m ( p , , A ,A) = i. By 1.1(1) and 3.4(2) we can assume p , = {Ol(3;q)}.By the definition of a, for some i < j < a, 8, = 8, and tp(q) = tp(z,); but Xm(pr,A,A) = i < j = Xm(p,,A, A), a contradiction.

+

DEFINITION3.3: Let &(a) be the set of strictly descending nonempty sequences of ordinals < a. A (A, a)-functim is a function

A : &(a) --+ ((~(3,z), #(E; g), n ) : Q E L,# E A , n < o,Z(Z) = m}; say h(q) = (v,, #, n,) (for some fixed m). Let 8 = 8(z; a) be an m-formula, p a cardinal, A a (A, a)-functionand U G &(a). Define T,"(8,h)to be the following set of formdm containing free variables of the form &,,

rfce,h) =

{(33)[w ;a)

A

= ( ) or

r)

r)

u {+z)[/\ r)

E

A

0 c ISIt,)

E

~ ~ ~~ , I i~ . v l i )( A 3vqlI(~; ; r,lI,vl~I-l~~~]:

U ;and v E " p where k

#,@; iir,v-dA

=

Z(r))}

'~~(3; %,v)]:

lEW

U,w

= p , IwI

= n,, v E "p where k=Z(q)- 1

If U = &(a), we write F,*(8, h). THEOREM 3.6: Let IT1 + 1. c cfp. Then Dm[8(3;@ , A , p + ] 2 a if there M a ( A , .)-function h euch that I',*(8(3; ti), h) is wn&tent.

Remark. For the if part no restriction on p is needed. Proof. (e) Suppose r,*(O(Z;a), h) is consistent, and let it be realized For r ) E &(a) u {< )} and by the assignment #,,v H 6,,,, Z,,v H v E "p, where k = Z(r)), let pn,v =

a)} u {#qli(z; h 1 1 , v t i ) : tJ

0 <

{vr,d%~ , l I . v l ( I - l J :

5

J(q)}

0 < 1 5 l(r))}.

OH. 11,s 31

BANKS, DEGBEES AND SUPERSTABILITY

45

We oan easily prove by induction on /3 that Dm(pn,v, A, p + ) 2 /3 where p 5 r][l(q)- 13 for 2(q) > 0 and D m ( p < > , < > , A , p2+ a. ) As p 0 , < ) = {O@; a)},this proves one direction. (+) We prove by induction on a, for all O(Z; 6)at once. a = 0 h is the empty function. a = 6 (a limit ordinal). So by the induction hypothesis, for every /3 < a there is a (A, /3)-functionhBfor which I',*(O(Z; a), hB)is consistent. For Y E &(a) define h(v) = h,ml+l(~). It is easy to check that a combination of the assignments realizing the I',*(O(Z; a), hB) realizes

r,*(eg; a),a). a = /3 + 1 By

the definition there are p(Z; a), n < w and Amformulas $@;6,) (i 5 p ) n-contradictory over cp(Z;E) such that Dm[p(Z;a) A $(Z; &),A, p'] 2 /3 and p(E; a) k O(Z; 6). Hence, by the induction hypothesis for every i 5 p there is a ( A , /3)-function h, such that I'y*[p(Z; a) A $(Z; 6,),h,] is consistent. Define ( A , a)-functions hf (for i 5 p ) as follows: If r] E de(/3), hf(r])= h,(r]),if r] = (p), k'(r]) = (p, $, n> and if r] = (/3)^u, u E &( then /I) hf(r]) , = h&) (clearly this exhausts all possibilitiee). If for some i 5 p, Fy*(O(Z;a),h*) is consistent, we me through. Otherwise, for each i some finite subset of it is inconsistent, hence for some finite u(i) c &(a) I';tf)(e(Z; a),hf)is inconsistent. The number of possible u(i)is 5 1.1 + No,and for each finite u the number of possible hf r u i s j I T I . A s c f p > IT1 + IaI,thereisVcp,IVI = p s u c h t h a t for all i E V,u(i) = u,h' 1 u = h* 1u.W.1.o.g. u is closed undertaking non-empty initial segments. Now using the $(Z;6J (i E V ) and p(Z; 8) we can ewily show that I';(O(E; a), h*) is realized, a contradiction.

46

R A N K S A N D INCOMPLETE T Y P E S

[CH.

11, 8 3

Proof. (1) It suilices to prove (1) for finite p, so by the remark to 3.6 it suffices to prove (2).Notice that 3.6 holds when ITI < cf p = a. (2), (3) By 3.6, for some (A, p)-function h, F,*(Ap, h) is consistent. NOWwe define for i < p, 0 < k < W ; v k , #k, n k , 7; such that 7: E &(/A), E(qi) = k, q;[k - 11 > i, h(qi 1 I ) = (vl, n,) for 1 5 E s k. For k = 1 there = (~1,#1,ni, such that .{I < p : h((a)) = < ~ i ,$1, nJ}1 = p, and for i < p choose q: = ( a ) such that a > i, and h((a)) = (ql,#1, nl). If we have defined for k , there am ( P k + l , # k + l , nk+l such that = (P)k+l, # k + l , nk+l>}l = p ; < p: and for i < p choose q i + l = qg-(a) where a > i, h(q:-(a)) = these v k s #k, nk Prove (2), (3)* ( ' P k + l , #k+l, n k + l > *

THEOREM 3.8: If A >

l!Pl+,

then Dm(p,A,A) =

Dm(p,A,ao).

Remark. If you wonder why we do not require only A > (IdI then see Exercises 3.10 and 3.14.

+ KO) , +

Proof. Let p = IT1 +,and w.1.o.g. p is {O(Z; a)}. If Dm(p,A,p + ) 2 a, then by 3.6 (when a < p) and 3.7 (when a 2 p ) there is a (A, a)-function h such that F,*(O(Z;a), h) is consistent. Hence for every A, Ff(O(3;a), h) is consistent; hence P [ O ( Z ; a ) , A , A] 2 a. By the monotonicity of Dm(O(Z;a ) , A , A) in A (by 3.4(2)) it follows that for every A > p, D"[O(Z;a),A , A] = D"[O(Z;a),A, p + ] . NOWwe can e d y finish the proof by proving by induction on a that D"[O(Z;a),A, p + ] 2 u implies Dm[O(Z;a), A, a]2 a using what we have just proved (for every p).

Proof. We shall define, by induction on k c w, formulas tpi and natural numbers rnk such that (1) for every k, there are a,,, q E " > p , such that (i) for every q E "p, p,, = p u {v:(Z; 0 < n < w} is consistent, (ii) for everyq E "p,n < w, w c p, IwI 2 mi+,, {&+l(Z; Gv(i)): i E w} is inconeietent.

CH. 11,s 31

RANKS, DEGIREESAND SUPERSTABILITY

47

Letf(k) = min{n: m: > 2) u {a). (2) For every k, if f ( k ) < w , then f ( k + 1) > f ( k ) or Z = f ( k ) = f(k + l), and mf > m:+l, (3) if n < f ( k ) , then tpl = &+l. If we succeed in defining them, and define tpk as yz for large enough k ( f ( k ) > n), then clearly 3.9 is satisfied by g~, = Alsis, ?@; Yj) A A p. It is also clear that for k = 0 there me such &, mz (by 3.7(3)). So it suffices to prove, that if q!,,m!, are defined for Z I k, 0 < n < w , then we can define &+l, m:+l, 0 < n < w . Now we can assume there are a,, r ) E O>p, such that (1)is satisfied by v;, m:, (0 < n < a),a,; and (iii) ifr)E@>p,Z< w , i l I ia s . - - sil < p,jl s j, I . . - s j, < p (wherei, = i,+l oj, = j,+,fora = 1,. . .,Z - l)andvl,. . . , v , E ” > ~ then the two sequences

h

Tin-
*

-

n-

*

~V
realize the same type over 0, which depend on Z(r)) only and

A,

=

u {a,,:

Z(v)

s

Z(r))

or v Z(q) #

r);

v E O>p} u Domp.

Remark. (1) We can choose the v’s as void sequences. Hence in particular, (aV<,): a < p ) is an indiscernible sequence over A,. (2) This is possible by 2.6 of the Appendix and compactness. See VII, 3.6. So we have q$, m: (0 < TZ < w ) , and 4, r ) E O’p, such that (i) and (ii) from (1) hold, and also (iii). We must define g$+l, .I+:, Iff@) = w , clearly 3.9 holds. So letf(k) < w and r ) E f(k)-lp,and let 1, be a sequence of n ones. Suppose first that

p* = p u {&a; a,,,): n < f(k)} {&~>+tt(’;

is consistent. Then define

‘Vl-)’

<

w,

{&k)@

aV))

48

RANKS AND INCOMPLETE TYPES

[CH.

11, 8 3

Clearly this definition satisfies our demands. So suppose that p* is inconsistent. Hence it has an inconsistent finite subtype, which we can mume is

Let us define

LEMMA 3.10: For A 2 N,, Rm(p,d, A) 2 Dm(p,d, A). Proof. We prove by induction on u that Dm(p,d, A) 2 a implies R"(p,LI, A) 2 u. For a = 0, or u a limit ordinal it is immediate. Let a = + 1, and w.1.o.g.p is finite. So by definition for any p < A there are $(Z;6,)(i d p ) ,n-contradictory over some finite r 2 p, with Dm[ru {$(Z; 6i)}yA , A] 2 @, hence, by the induction hypothesis, R"[r u {$(Z; gi)}, A , A] 2 @. By 1.6 there are pi E Sm(A),r u {$(Z; 6,)) c pi, and Rm(pf,d,A) 2 8, where A = Uis,bi u Domp. As the $ ( f ; gi) are n-contradictory over r , for every q E ST(A),I{i: pi 1LI = 911 < n, hence, as p was arbitrary, Rm(p,d, A) 2 @ + 1 = u. THEOREM3.11: D"(p, d, A) = Rm(p,d,A)providedthat:d = cl;(d,), h = p + , p is regular; p 2 !TI+or p = No > lLI1l; and R"(p,d, A) < 00 or T is stable and pi < p (i < Id1[)* pi .c p .

n,<,,,,

Proof. By 1.2, 1.1(1) and 3.4(1), (2) we can mume p = {8(Z; a)}, and by 3.10 it suffices to prove by induction on a that R"(p,d, A) 2 u =Dm(p,d , A) 2 a. For a = 0 or u a limit ordinal this is immediate, so let u = + 1. If R"(p, d, A) < 00, by 3.6(1) we can assume R m ( p , d A) ,

OH. 11, 5 31

RANKS, DEUREES AND SUPERSTABILITY

49

+

= a. By 1.8(2) a s p is finite and P ( p ,A, A) 2 p 1, there are A,*types pf (i Ip ) pairwise explicitly contradictory, much that Rm(pu pf,A , A) 2 8. As Rm(p,A, A) is not 1, I{i: Rm(pu pf,A , A) 2 l}] < p, so w.1.o.g. Rm(pupf,A, A) = 8 for every i Ip. By 1.6 we can assume that pi EST~(A)(i Ip ) for some A, and of come i # j * pf # p j . Notice that by 1.3, for every Q E A, Rm(p,Q, a)s &"($I, Q, A) < rn hence by 2.10(2) no(tp)=def Rm(p,Q, 2) < w and

+

+

there is nl(tp) <

such that for no 8, (I < nl(cp))does

w

A

(gZ)[

I < nl(cP)

A

Q(Z; 8k)if(kr1) A d(iE;a)]

k
hold. If T is stable, we can choose A and distinct pi E Sfl(A)(i Ip ) such that Rm(pupf,A , A) 2 8. As Pj < P ( j

<

*

1411)

n

Pj

< P,

1<1&1

for some Q € A l there are p distinct pf Q and w.1.o.g. i < j s p * pi Q # p j Q. NOWRm(p,Q, 2) < w by the stability of T (2.13, 2.2) so we can proceed as before so no(p),nl(v) < w exist. Now the proof splits into two cases:

lA1].We can find Q' such that for each n < w, 61)1f(1=n): I I n} E pfCn), where i(n) < p. (If T is stable Q' is Q or

Case I : p = No > {?'(if;

l~ and if Rm(p,A, A) < a,then by 1.6 of the Appendix such a Q', Q' cl,(dl) exists.) Let n1 = nl(cp') and for n I n1 let

E

By Theorem I, 2.4 we can assume that {61:I < w } is a #,-n,-indiscernible sequence over for each n 5 n,. For every I < w let #(Z;

El) =

A

TQ'(Z;

5k)

A

Q'(Z;6 n l + J *

k
It suffices to prove that {8(Z;a) A #(if; E l ) : I < w } is n,-contradictory, 1+(if; ZI),hence Rm[8(Z;a) A +(if; E J , 4, A] because by our choice pi(nl+l) 2 8, hence by the induction hypothesis Dm[8(Z;a) A #(Z;El), A, A] >- p, hence by definition Dm[e(Z;a),A, A] 2 /3 + 1 (as A = p + , # E A ) . Now if I(k) < w (k < n,) me distinct and WZ)

A [e@; a) A +(Z; El(k))],

k
50

[a. 11, Q 3

RANKS A N D INCOMPLETE T Y P E S

hence for every n s nl,

hence by the $,-n,-indiscernibity of 6,( I <

w ) over a,

a contradiction. Case 11: p > No, so we can assume A = A,. If T is stable and 2 IT1 +,then by the induction hypothesis, Dm(p,A, A) 2 /3 so by 3.7 and 3.8, Dm@ , A , A) = 00; so Dm(p,A,A) = R"(p,A, A). Let R"(p, d, A) < GO, then clearly by the above R"(p U p i ,d , A ) = /?< I TI+.Now choose for each i < p a finite q, E p, such that R"(p U q,, A , A ) = p. Hence for each i < p, l{j< p : qs E p,}l < p so we can assume that i < j < p implies qs $ p,. As d is closed under conjunction we can replace qt by A qr = pi@;5,) and as the number of possible pi is I d I < p and p is regular, we can assume that pr= p for every i < p. As p , ~ S , "( 1A ) for i < p, clearly for i <j < p, -p(z;5,)~ p ,so, the p , rp are distinct. So whether R m ( p , A A) , < 00 or T is stable and pf < p (i < lA1l)* < pi < p, we can m u m e fi < I TI and there is p E A such that

p

nf

+

@ = {q Es;(A): Rm(p u q, A , A) 2

p}

has cardinality z p . We now need:

CLAIM 3.12: Suppose @ E { p € @ ( A ) :p v {8(z; a)} is d s t e n t } and R"[8(Z; a), p, 21 = no < w . Let K be when 1@1 W infinite, and is finite. Then we can find r, (i < K ) auch that: log, log,I@I if (1) rf W a p-m-type Over A ,

+ 2, ( 3 ) for each i, there is a unique pf E @ such tlrat rf c p i , (4) i # j implia pf # p,. We shall prove this claim later. Now apply Claim 3.12 to our @, and we get r, (i < p) aa mentioned there and let A r, = p*(Z;Ef). So R"[B(z;a) A p*(Z; E,), A , A] 2 6, but for i # j, (2) lrfl = no

Rm[e(z; a) A

F*(Z;zf) A p*(z; E,),A, A] <

p.

CH. 11, Q 3)

RANKS, DEGREES AND SUPERSTABILITY

For any sequence i, t 721 = n,(cp), let $@;a') =

=

A

(i(O),

. . .,i(nl)),

51

i(0) < . . . < i(nl) < p,

-p*(~; Eia)) A Q*(R

Eitcn,)).

Icnl

Clearly R"[O(Z; a) A #(E; Z), A , A] 2 j3, hence by the induction hypothesis and 3.6, for some (A, j3)-function A,, I',*[O(?z; a) A $(-x ; Z), h'] i s consistent. For each such T define a (A, a)-function I' as follows: If 7 E (wp),then 747) = hl(d, i f 7 = (B), then h'(7) = ( 6 , no), if 7 = (j3)-v, v E &(/I), then hf(q)= h,(v). If for a suitable T, T*,[O(Z;a), h,] is consistent, then by 3.6, D"[O(Z;a),A , A] 2 a, so we finish. Hence assume that for each T, for some finite u(i) c &(a), I';A9[O(Z; a), hf] is inconsistent. Clearly the function P defined by P(S) = (u(t),h' 1 u(i)),has 5 I TI possible values. As p > IT[,p is regular, there is y < p such that for any j, L y, 2, < o,il(0) < ... < il(nl - 1) < y (for I < I,) there is ja< y such that for any Z < I,, iI(nl - 1) < j, and $9

.

F((il(0),. . ,it@,

-

11, jl)) = P(idO),

. . .,i

h

-

I), jd)

(this is a variant of the Downward Lowenheim-Skolem theorem, see VII, 1.4). So we can define y(I) < y (I < w ) such that y(I) < y(2 + l), and if Z(0) < < Z(nl - 1) < k < w , then

---

As in Case I we can assume that the formulas d ( q a) I\ $(z; E<~0)....,7tn1-1).7tn1+I)>1 (2 < 4

.;

+

are n,-contradictory, but of course P((y(O),, , y(nl - l),y(n, I))) = P((y(O), . ,y(n, - l),y ) ) and let it be (u, h t u ) so clearly I'&( O(Z; a),h) is consistent, a contradiction.

..

Proof of Chim 3.12. We define r,,pi by induction on i, so suppose K , and we have defined r j ,p j for every j < i. Define an equivalence relation Ei on the cp-m-formulas over A: cp(iE; Zl)Eicp(Z; 8 2 ) iff for every j < i, cp(Z; al) E p j o cp@; 3,) E p j . If I@]is finite, the number of B,-equivalence classes is 5 2'; hence

i<

1{pE @: ~ ( 3~;~ ) B , v a,) ( z ;* [v(z; a,) EP 0 c~(z; a,) EP]}~5 2,', so there are p o E @, cp(Z; ah) E cp@; a:) are Ei-equivalent.

~ O ,

+Z;

a:) E

~ O ,

such that cp(Z; a:),

52

RANKS AND INCOMPLETE TYPES

[OH.

11, 8 3

We can choose such po, 7i6,$ also for infinite 6.Efhae < K equivalence c l m a e ~ t h e r w i s echoose representatives tp(E; a,) (jc K ) , and let 6, (j< i ) ream p,. Then by Exercise I , 2.7 and compmtness there 7i(I(k)]ii(k*1) for k, 1 4 w ; are a(n) < K , p(n) < i such that ktp[6BB(1); contradicting 2.10(2). I n the same way we ctan prove that

* ~ ( 3G ; )EP]}~< K ; a,) * [tp(z;a,) so necessarily suitable po, at, a: exist-. t(Z)E (0, 1) so that Now we define by induction on 1 s no,p', 1{p E 6:tp(z;

. . .,tp@; 3 + l ) N 1 ) } E p 1E 6, (i) rf = {tp(z;so),tp@; (ii) Rm[{B(B; a)} u r f , tp, 21 s no - E or l{p E @: ri c p } ] = 1. For I = 0, po, 3+ are defined and let t(0) = 1, so clearly (i) and (ii) hold. If we have defined for I, and I{p E 6:r: c p]l > 1, then for some tp = tp(3; rf u {tpt} c pist E 6 for t E {Oj 1). Clearly there is t(E + 1) E {0,1} so that m[{e(z;a)} u r! u { p + i ) } ,tp,21 < B ~ B ( a)} Z ; u r f , tp,21 I no - E and let p' + 1 = p'st('+1). If we have defined for 1 but 1{p E 6 ;r: E p}[ = 1 let = 3, t(E + 1) = t(E). Now let r' = r&+ 1. It is easy to check that all the conditions are satisfied. THEOREM 3.13: (1) (A) P ( p , A , A) = P ( p , A , w), and (B)P ( p , d , A) 2 lAl+ No implies P ( p , A , A) = 00, provided that: (i) A 2 (214)++ + N 0 , or (ii) A > Id(+ KO,Rm@,A , A ) < 00 A not singular nor successor of singular. (2) T h e m 3.11 hkls for all A > p > No( p aa in 3.11).

+

+

Proof. (1) W.1.o.g. we can aesume p is finite. If d is finite, (A) follows from 2.10(2) if Rm(p,A , A) 2 w and from 2.9(3) (using compactness) if P ( p , A , A) < w , and (B) follows from 2.10(2). So w.1.o.g. d is infinite, Id I = ITI, and by 1.8(2) we can m u m e d = Cl$(d). If (ii)holds, let A = p+, p regular (if A is regular but not a successor by 1.9(2)we can replace it by A+). By 3.11, Rm@,A , A ) = D m ( p A , , A) and by 3.8, Dm(p,d,A) = D m ( p , A , c o ) . By 3.10, Rm(p,d,co) 2 Dm(p,A,00) hence Rm(p,d,co) I Rm(p,A,A) but by 1.9(2), Rm(p,A, co) Rm(p,A, A) hence (A) holds. (B) follows similarly from 3.7(1).

OH.

11,s 31

RANKS, DEGREES AND SUPERSTABILITY

53

Let B(Z;a) = A p . We can assume that (i) holds. By 1.3( 1) it suffices t o prove (A) and (B) for A = (2IA')++.If R ( p ,d, A) = co, (B) is trivial, and otherwise 3.11 applies and we can finish as above. So we need only prove (A) on the assumption that R"'(p, d, A) = 00. So Rm(p,d,A) > (21T9+,hence there are A-m-typesp, (i < (21''I)+),pairwise explicitly contradictory such that R m ( pU p , , A , A ) > (2ITI)+,hence R m ( pU p,, A , A ) = 00. By 1.6 we can assume that for some A, p f ~ S y ( Afor ) every i. As (2'4)' = ){p,:i < (2\")+}1 5

s

n

n

r

l{p, cp: i < ( 2 9 + } )

@Ed

I{q E #:(A): Rm(Pu !I,4

4

= m)1,

@€A

for some c p ~ dthere me distinct q,;#:(A) (i < (2IA1)+) such that Rm(pu q,,d, A) = CQ. If Rm(p,cp, 2) 2 W, our conclusion follows by 2.10(2) and 1.3 hence we can assume Rm(p,cp, 2) < w . So we can apply Claim 3.12to 4j = {q €#:(A): Rm(pu q, d, A) = co}, andget cp-m-types r, (i < IdI'), 11. 5 2 Rm(p,'p, 2) =def n<>Suchthat Rm(pU r,, A , A) = CQ, R"'(p u r, u r,, A , A) <: 00 (for i # j < Id1 +). We can apply this construction to p u r,, etc. Hence we can define for r) E "'(Id1 +) d-mtypes r,,, and natural numbers n,, so that: (0) rwllE r,, when 1 < Qq), (1) R"'(pur,,,d,A)= c o f o r e a c h r ) ~ " > ( ( d ( + ) , (2) IY,-<,>l = n,,,r o = 0, (3) R m ( p v r , ~ < , ) u r , , - < , ) , d , A<) CQ (r)E"'(Jdl+),i# j < Id]+. Now for r ) ~ " ( l d l +let ) r,, = Ul<er,,ll, so clearly Rm(pU r,,,d,A) = m. Let p , Y,, be over A, so by 1.6 there are PE#?(A), r,, s r", P ( p u r w , d , A ) = m f o r r ) ~ " ( J d l + ) h e n c e i f rrZ) # v E ' ( l d ) + ) , then Y, $ rm. Let A r,, = #,,(f; a,,),now we can prove that for some #,,(Z; &) andK s ldl+, = {B(z,; a): v E "K} u {#n(zv; &)if(vln=? v E "K, r) E "K, 0 < n < W}

+

r,

is consistent; and that this implies the consistency of I', for each p, and this implies Rm(p,d, 00) = co. As the proofs are similar to 3.6 and 3.8 we leave the details to the reader. Instead of 3.6 we can use 1.6 of the Appendix for m = 1. (2) Easy by 3.11, 3.8 and 3.13(1).

THEOREM 3.14: The following are equivalent.

(1) T i e superstable, i.e., stable in every large enough A (infact A 2 21Tl). (2) T is stable in some A for which A% > A. (3) R"5 = 5, L, (21TI)++] < IT!+.

54

R A N K S AND INCOMPLETE T Y P E S

(4) For ~ o m em < w , Rm(2= 8,L, co) <

[OH. 11, $ 3

00.

( 5 ) T i s s t a b Z e a n d D l ( z = s , L , I T J + +<) I T [ + . T is stable and for some m < w , Dm(2= 8,L, co) < 00.

(6)

Proof. (1)

=- (2) Immediate.

(2) =- (6) By 3.9 (for p = 0) and compactness, if ( 5 ) fails, but T is stable we can find 4, r) E "'A, satisfying (i) and (ii) from 3.9, and if A = lJ{a,,: r) E ">A}, q,, E S ( A )p,, , E q,,, then lS(A)I 2 l{q,,:r) E "A}I = P o , as the q,, axe pairwise contradictory by (ii); but IAI 5 A, a contradiction. (6) =- (6) Follows by 3.4(2) (6) =- (4) By 3.13(2). (4) + (1) For every Q = ~ ( 2g), ; l(2) = m,by 1.3(1), Bm(2= 8,Q, co) I Rm(2= 8,L, co) < 00, hence, by 2.2, ~ ( 8 g);is stable. As this holds for every such Q, T is stable. So by 3.13(1), R*(Z = 2, L, A) < co where A = (2ITI)++. Now for every A, for each p ES*(A), choose finite qp E p , R*(qp,L, A) = Rm(p,L, A) which is < co. By definition of rank, 1{p ESm(A): qp = q}l s 21TI so IS*(A)I 5 I{qP: p E S ~ ( A ) .21r' } I = IAI + IT1 + ZITI. So T is stable in every A 2 2ITl. (3) * (4) Immediate, by 1.3. (6) 3 (3) By 3.11.

LEMMA 3.16: If IST(A)I > IAl + IT/,then for some countable B c A , IS,(A)I 2 IS,(B)I 2 2*0, and there are ~ E AQ ,, , E ~such that for k < w , q Ek2: (3x1

A Qnli(z, anli)vtzl*

l
Proof. Choose, by induction on l < such that a,, E A and for every v E '2,

w,

formulas ~,,(z;a,,) for r) E ' > 2

{ p :p E S , ( A ) ,~ ~ ~Sivlk)v[kl ~ ( 2 for ; every k < l } has cardinality > IAI + ITI. Then we let B = (J {a,,: r) ~ " > 2 } ,so 1B1 5 No, B c A , IS,(A)I 2 IS,(B)I 2 2 b becaum the types {Q~I&; avIk)v[kl:k < w } are consistent and pairwise contradictory for v

E "2.

EXERCISE 3.1: Prove that i f p 2 Id1 + No,p+ 2 A, and Rm(p,A , A) 2 S(p), then R*(p, d, A) = co. [On S(p) see Definition VII, 6.1(3) and Theorem VII, 5.6.1

CH.

11, 8 31

RANKS, DEGREES AND SUPERSTABILITY

55

EXERCISE 3.2: Prove that if p + 2 A, X = R or X = D and T is stable, X m ( pA, , A) L [(No+ I A l ) Y ] + , then X*(p, A, A) = 00 (forX = D use Exercise 3.10).

EXERCISE 3.3: For

a formula B = B(Z;a) define the model Qo.

universe is (6 E Q : 6 C Orb; a]}, and its relations a m

.

Re = {(Zit . . 4): Q C v[Zly y

Its

.. . 4, a]} y

for every q~E L(T). Assume T is stable or at least that for every q ~ , P(e(z;a), v, 00) < a. Prove that P [ B ( Z , a), L, A] computed in Q is equal to R(x = z, Lo,A) computed in Go. Moreover, for each A c L there is do c Lo (and vice versa) such that = Ido)and P [ B ( Z ; a), A, A] computed in Q is equal to Rm(E= Z,Ao, A) computed in Qo. Moreover we can assume that for every v(Z;g) E A there is p o ( f ;go)E do,and vice versa, such that (i) for every 6 E Q there is go E Q0 such that for every 3 E Go, 5 = @OY

* *

.>,

(E

.

k c p [ ~ ~ , .. ; 61

A

A B[Z,; a]

iff Cro C qo[z;6O],

1

(ii) for every 6 O E CoyCro C (W)

6O)

or there is 6 as in (i).

EXERCISE 3.4: Give an example of a theory T suchthat D(x = x , L, 00) = 1, R(x = x , L, 00) = 00. (Hint: See 4.8(2).) EXERCISE 3.5: Give an example of a theory T,, ITaI = 1.1 such that D(x = x , L,00) = a, R ( z = z,L, 00) = 00.

+ KO

EXERCISE 3.6: Give an example of a theory T,1111 = A, such that R(x = z,L, p ) < 00 iff p > (2"+. (Hint: It has A one-place predicates

Onls. 1 EXERCIBE 3.7: Prove that if A is finite, and for every k < o, P ( p ,A , k) 2 n, then P ( p ,A, KO) 2 n.

DEFINITION 3.4: (1) Define CR(p, A) as an ordinal or

as follows: CR(p, A) 2 0 iff p is a complete type, and CR(p, A) 2 a + 1 8 p is a complete tspe and for every p < A and finite B c Domp, there are pairwise contradictory complete types qt (i 5 p), p t B E qi, and CR(qi, A) 2 a(2) T is transcendental if CR(p, 2) < 00 for every type p . 00,

56

RANKS AND INCOMPLETE TYPES

[CH.

11, 8 3

EXERCISE 3.8: Prove for CR(p, A) theorems parallel to l . l ( l ) , 1.2, 1.3(1), 3.1 and 3.2. EXERCISE 3.9: Investigate CR(p, A). (Hint: See V, Section 7.) EXERCIBE 3.10: (1) If T is stable, show that in (3) of the definition of D”(I,,A, A) we can restrict q to the union of r and a finite clg(A)-rn-type. (2) If in addition A = clg(A) we can replace “{y5(f;a,); i s p} is n-contradictory over q” by “ { y 5 ( f ; a,);i s p} is n-contradictory” (i.e., n-contradictory over the empty set) and thus omit q completely and use just r. (3) Hence show that if T is stable and A = clg(A) it is sufficient to define a (A, a)-function as a function

A: &(a) -, {(#(f; g), n ) : y5 E A , n < w , Z ( f )

= m}

and we a n then replace IT1 by (Id1 + No)in 3.6, 3.7, 3.8 and 3.11.

EXERCISE 3.11: Show that in the previous exercise the stability is indeed necessary. EXERCISE 3,12: Let p be a finite rn-type, A c L; prove that the < 2n0, and always following conditions are equivalent when (2) * (1) * (3). (1) R”(p,A , 2) = a, (2) for some A, I{q € @ ( A ) :I, = q}l > IAI lAl No, (3) for some A, IAI s KO, [{q E S ~ ( A )p: c q}l 2 P o .

+

+

EXERCISE 3.13: Relativize 3.3(1) and 3.14 to 6(f;8) (as.in Exercise 2.5) and check the other theorems to see if they can be relativized similarly. EXERCISE 3.14: (1) Show it is possible that Dm(f= Z,$, N,) = 00 but Dm(f= f,y5, A) = 2 for every A 2 8,. (Compare with 3.8.) (2) Show that in (1) we can replace two by any a. (3) Show that in any example of (l), T is necesmrily unstable. [Hint: Let T, be the theory saying: P , Q form a partition of the universe; P,(O < n < w ) a m disjoint subsets of P ; xRy -+ P ( z ) A Q ( y ) and S(z) = { y : z R y } (i.e., we look at P as a family of subsets of Q); the sets S(z) for z E PI are pairwise disjoint; P,(cO), P,(c,) for 1 E “w, and

CH. 11, f 31

RANKS, DEQREES AND SUPERSTABILITY

57

the c,’s are distinct; and let It be a one-to-one function from O’w into w , for every r ) E n < w and distinct

n

21 E Pn+1,

S(cnlk)

kSI(n)

n

S ( z I )= 0. I < htn)

Now T1has a model completion with elimination of quantifiers satisfying (1) for 51, = zRy.1

EXERCISE 3.15: Show that in Theorem 3.11, we can replace “ p is regular ” by ‘‘cf p > I TI ” and if T is stable we can replace “p 2 I TI , pf < p (i < [All) * ~ l < l A l l
EXERCISE 3.16: Show that Exercise 3.16 is best possible, i.e., in 3.11 ”. we cannot replace “ p 2 (2IA1)+” by “cf p 2 [Hint: Assume 2Nn = Et,+,+, for n < w . M is the model with universe K, and the one-place relations Pt,(u < K,, n < w ) such that P:, E {i:K, < i < N,+,}, and {PZ,:u < K,} is independent for each n < w . Let T = Th(M), so IT1 = N,, but +

I

1

R ( z = z,L, A) = 2 a0

ifA > if A = K,,, or No+a+l, ifA < No+,.

For each u < Nu+,, or u = 00 we can expand M to M, by one-place predicates such that 2 is replaced by u, ID(T,)I = Ha+,, lTll = N, where T, = Th(M,).] We can use Exercise 3.20 for more general examples.

EXERCISE 3.17: Prove that in Claim 3.12, when 0 is finite, we can 0 I@[ ” (see 1.7 of the Appendix). replace “ K = log, log,I@I ” by “ ~ ~ “I EXERCISE 3.18: (1) Prove that the following conditions on A 5 x are equivalent: (A),,, For some T, IT1 I A, ID,(T)I = x. (B) For some Boolean algebra B, llBll IA and on B there are x ultrafilters. (2) If A, x satisfy (B) for a 1-homogeneous B, show that in Exercise 3.6 we can replace (2*))+by x+. EXERCISE 3.19: Generalize 3.9 to Dm(p,A, p + ) : (1) for stable T, (2) in general.

58

RANKS AND INCOMPLETE TYPES

[CH.

11, $ 3

EXERCISE 3.20: Let R(A, T ) = R1(z = X, L, A), Mlt(A, T) = Mlt(z = z,L,A). Suppose T,is a complete theory in L, (i < K ) , L, pairwise disjoint, (Vz)P,(z)E T,. Let M , be a model of T,, M = Mi (i.e., IM/= Ui<, 1M,1, L(M) = U,<,L,, and if R c L i is a predicate, RM= RM1;for simplicity only, assume L has no function symbols) and let T = Th(M). Then T, 2 sUPf
z,<,

p =

2 (Mlt(A,Ti):i <

K,

R(A, Ti) = P}.

If R(A, T) = /3 + 1, Mlt(A, T) = 1. If R(A, T) = /3, Mlt(A, T) = 2{MlW, Ti): R(A,T,)= P}. (3) Let A < No,/3 = sup,<, R(A, T,), /3 = S + k, where k < No S is limit or zero. Then (i) B I R(A,T)5 6 w , (ii) if {i < K : R(A, T,) 2 S } is infinite, then R(A, T) = S + w , Mlt(A, T) = 1, (iii) if {i < K : R(A, T,) L S } is finite, then R(A, T) < S + w ; and also if k, = 2 {Mlt(H,, !Pi): R(A, T,)2 S}, 6 + 1 = R(A, T ) ,then k, = max{k: Ak I I } (see the next exercise).

+

EXERCISE 3.21: If A c KO,a = Rm(p,L, A), /3 = Rm(p,L, H,), then w/3 I a < + w , and if kl = a - w/3, ka = Mlt(p, L,A), k = Mlt(p, L, KO),then A k i I k < A k i + l , and k, = max(1: Z.Akl I k}. Also k is the maximal number of pairwise contradictory extensions q of p satisfying P ( q , L, A) L w/3 (see Exercise 1.6). Show that we can replace “L” by “A” when A = clk(A). EXERCISE 3.22: Show that the results of Exercises 1.6 and 1.6 cannot be improved. In particular show that: (i) For every a, and w a I /3 < w a + w there is a totally transcendental theory T, IT1 = la1 + KO, R(2, T ) = ,9, R(K,, T ) = a. (Hint: By induction on /3 using Exercise 3.20.) (ii) Suppose n < w , H, = A, < A, < * - - < A, = 00, a1 > a, > . . > a, where for q c 1 < n, A1 is not the successor of a limit cardinal. Then there is a theory T, IT1 I lal[ p (where p+ L A,,-l) such that R(A,T) = al when A1 I A < A1+l. (Hint: Use N u , lNul = * > p , Pf;’. = {v E N u : r ) Q v } for r) E O’p.)

+

UH. 11,6 31

RANKS, DEGREES AND SUPERBTABILITY

59

QUESTION 3.23: Investigate the connection between R1(x = x, L, A) and Rm(3= 8, L, A). (Hint: See V, Section 7 . ) EXERCIBE 3.24: Let

Q’

be an expansion of

Q

by

K

<

E

individual

constants. Show that

Rm(p,L,A) = P ( p , L’,A),

Dm(p,L’,A) = Dm(p,L’,A).

EXERCISE 3.25: Show that if P ( p , A , X,) 2 /3 + 1, q c p is finite, A = clg(A), then there are A-m-formulas v,,(8;Z,,), pairwise contradictory, such that for n < w . P(qU {v,,(iE;a,,)},A , KO) 2 (Hint: Show there are distinct q,, €Bm(A), P ( q u q,,, A , KO) 2 1.6 and 1.9(2)) and then use 1.6 of the Appendix.)

/3

(by

EXERCISE 3.26: Let p be a strong limit cardinal of cofinality 24, (i.e., (Vx < p)2x < p and cf p = X,). (1) Show that if p is an m-type over 6,

x

= I{qEfJ%4):puqconsistent}l > IAI

+ [dl + p,

then x 2 2u; so, e.g., ISm(A)I> IAI + Id1 + p implies IBm(A)I 2 2 ~ . In particular, there are A-m-formulas~ ~ (a,)3 over ; A for r ] E *>p, such that for every r ] E *p,

P U {vnlr(%anlr) A

l(pV(8;

&): 1 <

W, Y

= (r]

1 J)-(i), i < $11)

is consistent. (2) Prove that for finite p , P [ p ,A, (2”) +I = 00 iff there are cl,(A)-mformulas tpn = ~ “ ( 84) ; ( r ] E “ ’ p ) such that for any r ] E *p

P u { ~ n l lA vnl(I-l)*:0 < < a < dJ- 111 is consistent ifF for some A, G A and A, Idl[ + IAI “9

1{q E BTl(A):p Z n < a pn =

~9

u q is consistent}] > p iff there are p,, (n < and v;(z; 4) for

7 = (
such that for any aI <

p

U

*

- 9

< p I (0 < 1 <

{~v:(z; a:)

a1 <

Bm-J),

A

.

= p and

w ) such that

Y < tcms

w)

e(3; 8:;): 1 < w }

is consistent where qr = ((a,, Po), . ., (aI-1, /3*-,)). (3) If the p,, are as in (2) and for every n < w , for some k < w , A D ( l A l + , I A l , p k , p n )fails (see Definition VII, 1.11; e.g., if = p or p = Xu+,), then in (2) we can add “iff Rm(p,A,p ) 2 lAl+ ”.

60

RANKS AND INCOMPLETE TYPES

[CH.

11, 8 3

EXERCISE 3.27: If T is stable, Dm[O(Z;Ti), A, a303 2 u + 1, then there are n < o,8 formula $(Z; 8) and an indiscernible set I baaed on G (eee Definition 111, 1.8) such that {I,4(Z; 6): 6 E I}is n-contradictory but Dm[O(Z;a) A $(Z;6),A, 001 2 a for 6 E I . EXERCISE 3.28: Dm(p,A, A) = Dm[p,cl,(A), A] for everyp, A; if T is stable and for each $(Z;#) E A, n < o,#,,(Z;go,. . .) = Al<,,$(Z;&) €A. EXERCISE 3.29: Show that in Exercise 3.28 both conditions are necessary.

*.

EXERCISE 3.30: Letp, A, be fixed, and for k < o,R"'(p,A , k ) = uk, R T p u {v*},A, kl = = uk. (1) uk = 6 + nk and t$ < S imply (2) If A = clz(A), then for some t E (0, l}, uk 5 t$ + 1. (3) We cannot improve (2) (to ak 5 G). (4) If u: = 8 + n: for t E (0, l}, then nk 5 nt + nf. [Hint: (1) Use Exercise 1.6: (4) We can msume p is finite and define finite A-rn-types p n (q E k , n = nk) such that (i) pr<,>(i < k ) are explicitly contradictory, (ii) Rm(pn,A, k ) 2 6 for q E n a k , where pw = p u { p n l l 1: 5 l(q)}. We prove by downward induction on l(q) that (u)for some t, Rm[pnu {$},A, k ] = uk, or (/3) let

u

Rm[pnu {cp">, A, k ] = 8

+ n:Sk,

Mlt(pn u {$},A, k ) = 1 : s k

then

QUESTION 3.31: Is Exercise 3.30(4) best possible? (Thisis a problem in finite combinatorics.) EXERCISE 3.32: Suppose Z(Z) = Z(g) = m and O(Z,g; C) k cp(Z; 7i) A $(g; 6), q ( Z ; G) k (3g)O(Z, 5; C), and for every 2, O(Z; 2; C) is algebraic. Prove that for every infinite A, Rm[v(Z;Ti), L,A] 5 R"'[$(Z;6),L,A].

CH.

11, 8 31

61

RANKS, DEGREES AND SUPERSTABILITY

DISCUSSION 3.16 : The following generalizes Exercise 3.3, and shows how the work of Poizat can be put in our context. Let p be a type over a set A. Define 6, = (a, a)asA;Q, is a model with universe {a E C: a realizes p } , and relation R, = {E E Q,: E satiafies tp}, for any tp = tp(3, a), si E A. In 11, Exercise 3.2 we deal with the case p a finite type, then Q,, is a saturated model, and so we can translate results on the stable case to the non-stable case. Here Q,, is not necessarily sctturated, but it is like the Q when we deal with kind I1 in [Sh 76~1.As said there, all results on ranks, degrees and also forking goes through, and some caaes are discussed in detail. Let me elaborate a little more: CLAIM 1: Q, is a i - h o g e n w w r ; moreover, i f (ar:i < u), (br:i < u) realizes the same quantifier-free formulas in 6, (equivalently, the same form* with parametersfrom A in a), then there is an autommphisnz f of Q, which taka (ar:i < u> to (b,: i < a) (and, in fact, is a ratridion of an automorlphh of Q which is the identity on A).

Proof. Find the required automorphism of Remark. This would hold also for Easily:

IQ’I

Q.

r E &A).

= {a: tp(a, A ) E F},

CLAIM 2: (1) Q, is i-compact for quantifier free form&, ( 2 ) Q, satisfies the assumption of kind 11, hence all recrults on forking and ranke generalized.

in

But we may still be interested in the connection between what occurs Q and in 6 ,.

ASSERTION 3: (1) A set I E IQ,I is discernible in Qp iff it is indiscernible in Q over A. (2) Two such injinite sets are equivalent in 6, iff they are equivalent in Q. ( 3 ) For B E C E IQ,l, and IQpl, tp(si,C, Q,) fork over B iff tp(Si, C u A, Q ) fork over B u A. (4) Similar equivalence holds for stability in a power. Procf. E.g. (3) p = tp(8, C,6,) does not fork over B iff some infinite indiscernible set I over C of sequences realizing p has, under auto-

62

RANKS AND INCOMPLETE TYPES

[CH.

11, 5 4

morphism of Q, which are the identity over B, a bound number of images up to equivalence, so we finish. For usual rank there is no nice translation, but if in the definition we do not take a h i t e subtype but deal with the same type, or deal with C P (and A = 0 for simplicity) the situation will be like 11, Exercise 3.3.

II.4. The f.c.p., the independence property and the strict order property DEBTNITION 4.1: (1) ~ ( 3g ; ) has the finite cover property (f.c.p.) if for such that arbitrarily large natural numbers n there are go,. . ., CT(~Z)

A c~(z;a') k-zn

but for every 1 < n, C(~Z)

A

cp(z;ak).

k
(2)

T has the f.c.p. if there exists a formula ~ ( xj i );which has the

f.c.p.

Note. Here x is a single variable, see also Theorem 4.4. LEMMA 4.1: The formula v(Z; g ) h not have the f.c.p. iff there is a natural number n Buch that: If I' is a set of ym-formulas and every subset of r of cardinality < n is &tent, then r is consistent.

Proof. Immediate, by the definition. THEOREM 4.2: (1) If T does not have the f.c.p., then T is stable. (In other words, every umtable theory has the f.c.p.) (2) If cp(Z; g ) ie umtable, then the formula

#(Z;2) = #(Z; g1, p,83, g4) =

[Cp(z;gl)= 7(p(~;ga)lA [v(z;g3) = v(~;g4)]

has the f.0.p.

Proof. (1) By Theorem 2.13, as T is unstable, some y(x;g ) is unstable, hence (1) follows from (2). (2) As v(Z;j i ) is unstable, by Theorem 2.2(1), (3) it haa the order property, i.e., there are Tio, . . .,a', . . .,1 < w such that for every k < w , p k = {v(-; --I i i ( k S l ) : 1 < 5 a ) 4 is consistent. Let n be any natural number. For any k < n define Ek = lioniPn7ikn i i k + l .We claim that q = {#(Z; Zk): k < n} is inconsistent, but for every

OH.

11, 8 41

F.O.P., INDEPENDENCE AND OBDEE

63

1 < n, ql = {$@;ak): k < n, k # 1) is consistent. As n is arbitrary, by Lemma 4.1, this will prove the theorem. Let us fist prove that q is inconsistent. Suppose 6 realizes q. Then as C$[6; ao], also Cq[6;aO] E 7q[6; F]. On the other hand for every k < n, t~ +[6; ~ k ] ,clearly b[6; ak] = q[6; ak+1]. _ . Hence

kv[6; it01 0 kq[& ]'ti

0 Cq[5;

as] 0. ' * 0 C q [ E ;].ti

0 C--lp[6; aO1,

a contradiction. Therefore q is inconsistent. Now if 1 < n, then clearly a sequence realizing p r+ realizes q,; hence q1 is consistent. THEOREM 4.3: (1) There is a stable theory with the f.c.p.; and there is a stable theory without the f.0.p. ( 2 ) There are stable t k i e s which are (i) superstable and without the f.c.p., which are also stable in KO, (ii) superstable with the f.c.p., which are also stable in No, (iii) unsuperstable without the f.c.p., (iv) uneujperstable with the f.c.p.

Proof. Clearly (2)implies ( l ) ,so we prove (2) only. (i) The theory of a model whose only relation is equality. (ii) Let M be a model with the equality relation, and an equivalence relation, such that for every n there is an equivalence class of cardinality n. Clearly its theory satisfies our demands. (iii) Let M be a model such that IMI = Oo;the relations of M are equality and for every n the equivalence relation En defined as

En = ((7, v): r), v E %, r) n = v 1 n}. The theory of M is the required theory; see Exercise 2.3. (iv) By combining the two previous examples we can easily construct such a theory. THE f.c.p. THEOREM 4.4:Let T be stable. Then thefollowing assertiw are equivalent (where 1 5 m < w , 2 I A I m): ( 1 ) T h a the f.c.p., i.e., some q ( x ;8) hccs the f.0.p. (2), 8ome q(3;g) has the f.c.p., l(3) = m. (3), Not for every Enite A , is there a k < o such that a set p of A-mformulas is realized iff every q E p , 141 < k is realized. (P), For somejnite A , k, and formula O(3; g), Rm[B(?Z; it), A , No] = k

64

[CH.

RANKS AND INCOMPLETE TYPES

11, $ 4

is not an elementary property of ii (that is, there is no formula $(g) such that Rm[B(Z;a), A , KO] = k o h,h[Z]). (5), There are finite A, k and a formula 0 such that for no 1 < w Rm[B(Z;a),A, HO] = k * Mlt[B(Z; a), A] I 1. (6)A,mFor some finite A,for no k < w does the following hZd: for every set p of A-m-formulas, there &a q E p , IqI < k, auch that Rm(p,A, A) = R"(q,A, 8. (7), For some finite A, for no k < w does the following h Zd:for every A-m-type p , Rm(p,A, k ) = Rm(p,A, 00). (€9, There is a fornula; y(Z, g; 2) Buch that Z(Z) = Z(g) = m and (A) for every Z, y(Z, g; 8 ) is an equivalence relation. (B) for every n and for some En,y(Z, g; Z,,) has 2 n but onlyfinitely many equident chases.

Proof. We shall prove (1) * (2), * (l), (2), * (3), =s (€9, + (4), =(8)m * (2)m and (3)m * (6)A,m, (5)m * (7)m, (vm)[not (3)m and not (s)ml * [not (6)A,m and not (7),]. Clearly this is sufficient (as(1) does not depend on m and A). (1) * (2), Trivial. (1) We shall prove it by induction on m = 1(@. For m = 1 (2), this is true by definition. Hence assume we have proved it for m and we shall prove it for m + 1. Suppose it is not true. Then there is y = y(Z; g) = y(zo,. . .,x,; 9) and for every n < w a set r,,of formulas of the form y(Z;a) such that rnis inconsistent but every subset of r,,of cardinality < n is consistent. By Lemma 4.1 there is 1 < w such that if F is a set of formulas of the form y(zo;cl, . , c,, a), and every subset of F of cardinality < I is consistent, then r is consistent. Let

..

+(xi,

---

9

;2 ,

go,

--

~1-1)=

and let for every n

r,*= {+(zl,. . .,x,;

(3zo)[

A q(zo, z1, . k
9

zm; g k

)I

ao,. . .,

for y ( 3 ;Tio) E F,,, . . .; y(Z; 8 - l ) E F,,}. Clearly every subset of r,*of cardinality
r = {y(xo;c1,. ..,c,, a): y(z0, . . .,x,; a) E Fn}. Clearly every subset of r of cardinality I Z is consistent, .and so by the definition of 1, F is consistent; hence there is co which realizes it. So (co,cl, . . .,c,) realizes r,,,a contradiction. So the implication is proved. (2),

(3),

Trivial, A = { y ( Z ; g)}. (€9, Suppose A exemplifies (3),;

* (3),

so for every k < w there is a

CH.

11, § 41

F.C.P., INDEPENDENCE AND ORDER

65

set p k of A-m-formulas,which is not realized, but every q c pky IqI < k, is realized. W.1.o.g. l ) k has minimal cardinality; sop, is finite, lpkl 2 k, and every proper subset of l ) k is realized. Let n(0) = Rm(Z= B, A, 2), so Rm(p,A,A) I n(0) for everyp and A. Now for each k, we define by induction on Z s n(0) + 1 sets pfc c p k . Let p! = 0, and pfc' be a subset of pk of minimal cardinality such that pfc c pi+' and Rm(pfc+l,d, NO)I n(0) - 1 - 1 (as R"(pk,dy 80)= -1, pf, exists). There is a maximal Z(0) such that for some n < w, W = {k < w: Ipfco)l I n} is infinite. So for some n(l), 4 2 ) < w, 8, and infinite Wl,for each k E W,,Ipf0)I = n(1) and A yfco)= 8(Z; 4) and P [ p $ O ) ,A, No]= n(2); necessarily n(2) = n(0) - Z(0). (n(2) I n(0) - Z(0) by the definition of thepf, andn(2) 2 n(0) - Z(0) by the choice of Z(0)) and n(2) > 0 (n(2) 2 0 for we can choose k > n(1) and n(2) # 0 by the choice of Z(0)). Clearly JpfZo)+ll tends to infinity with k; and for proving (5), it suffices to prove that Mlf[B(Z; z k ) , A] = Mlt[p$O),A] tends to infinity with k. We prove that, more specifically, Mlt(p$O),d)2 Ip$O)+lI IpjZ0)I.Let A be such that all p k are over A. For each Q E pfZ0)+l- pfco), by the choice of pZo)+l , Rm[p$O)+ - { Q } , A , No] 2 n(0) - Z(O), so there is q,,, ES"(A), pZ0)+l - {tp} c q,,, and Rm(q,,A,No) 2 n(2). But Rm(pfc'o)+l, A, No) I n(0) - Z(0) - 1 so 1~ E q. Clearly the qVys show that Mlt(pfc",A) 2 lp$o)+ll- Ip$O'I. (5), + (4)m Suppose A, k, 8 exemplify (6),,,;so there are a,(Z < w) such that Rm[8(Z;a,),A, KO] = k and Mlt[O(z; a,),d] 2 1. So for some A, I{q E S ~ ( A , )P: [ { 8 ( z a,)} ; u q, A , KO] = k}l 2 1, SO if I 2 24n,by 1.6 of the Appendix there are tp;, 8; (i < n ) such that R"(pf, A, No) = k, p ; = ttp;(z;b y ( * = j ) : j5 i} u p(z; a,)} andyfEcll(A).SowecanfindZ(n+ 1) < wandtp, suchthatfori < n < w = (p,, and let for i < n, A pi(,,, = 8,(Z; EF). If (4), is not true, there and are k,bt, such that: for every E, P [ d t ( Z ; a), A , No] = k 4 C@] similarly for 8 , $ . By a compactness argument we can find a, 8, (j< w) such that: @)

P [ 8 ( Z ;a),A,

KO] = k,

Rm[{8(Z;a)} U {Cp,(Z; 6j)u(j=f):j I i}, A,

KO] = k

contradicting the definition of the rank. (a),,, => (8),,, We choose A, k, 8 exemplifying (4),,,, so that k is minimal. Clearlv for everv 8,. the property P [ 8 , ( S ; Zl),A, KO] 2 k is elemen-

66

[CH.

RANKS AND INCOMPLETE TYPES

11, 8 4

tary (as it is equivalent to -,V14kR”[e,(Z;a,), A , KO] = I, and by the minimality of k). Moreover also the property “Rm[8(Z;a),A, KO] > k or Rm[B(Z;@ , A , No] 2 k and Mlt[O(Z; a ) , A ] 2 1” is elementary (it is equivalent to the existence of Qt,, E A (i # j c 1) and 6‘, such that

A R”[{e(Z;a)} u {Ti,,@; &,,):

i < E},d, KO] 2 k

j
where Q‘.,(% h,,) = l Q , , i ( R 6,,‘)). If R”[e(Z; a),A, KO] = k S- Mlt[O(Z; a),A] II for some I (not depending on a) it will follow that also Rm[O(Z; a),A, KO] = k is elementary, contradicting our hypothesis ((4),). So for every I < w there is 8, such that R”[e(Z; a,),A, No] = k, Mlt[O(Z; a,),A, KO] 2 I. So for some A, (by 1.10(1) and l.lO(4)).

q

= {q E ,Y:(A~): R*[{e(z; a,)} u q, A ,

K,]

=

k}

has cardinality Mlt[B(Z; a,),A] and q E @, implies Mlt[{O(Z; a)} u q, A ] = 1. By Claim 3.12, for each j c w , there is I ( j ) c w (e.g., 2,’ + 1) and distinct q{ E Qltf)(i < j)and 4 E q{, Ir{l 5 Rm(Z = Z, A , 2) + 2, and 4 c q E @,cn =- 9 = qi. We can find 8* and b: (i c j,j c w ) such that for every i c j,j c 0, O(Z; a,(,)) A A TI = 8*(Z; 6{, a,(,)),hence

Rye*(%;6{, a,,,,), A , N,] = k, mt[e*(2; 6{, a,(,)),41 = 1. Clearly for every cp E cl,(A) there is a formula +p, depending on A , m, k and 8* only, such that: Rm[e*(Z; E{, ai(j))A ~ ( 2a),;4, No] 2 k * W,JC Z{, 5i,,,,I, and for each F(Z; 5 ) exactly one of &,[E, 6{, 6{, si,,)] holds. We define an equivalence relation E(ji,, jia; a): b,, b, are equivalent i f (i) Rm[8*(2;6, a),A , No] = k A Mlt[B*(Z; 6,,a),A] = 1 iff

iX,(n],97@,

m[e*(z;6,

a),A , KO] = k

A

mt[e*(z;6,

a), A ]

= 1.

(ii) if the condition from (i) holds, then for each Q E A

w)[+,Jz,

61,

a)

#cp(g,

62,

a)].

It is easy to check that E(&,ji2;a) is an equivalence relation, expressible by a first order formula (for (i)look at the beginning of the proof of (4), =- (8)”), and E(Jl, fla; a,,,)) has c KO but 2j equivalence classes. (8), =- (2), Let 7 ~ ( Z , ji; Z) be as in (8),,,, and we shall prove that -,Q(Z; ji, Z) has the f.c.p. For each n let Zn be such that Q(Z,ji; En) has ~n but c K O equivalence classes; let a,, . . ., a, (I 2 n ) be a set of

CH.

11, $ 4 1

F.C.P., INDEPENDENCE AND ORDER

67

representatives, so p = {+Z; a,, 4): 1 Ii IZ} is inconsistent but every p' E p , p' # p is realized. (3), =- (0)A,mTrivial (asp is realized iff Rm(p,A , A) # -1). (5), * (7), Immediate (use a minimal k). We finish by:

CLAIM 4.5: Suppoee not (3), and not (ti),,,for every m. Then (A) not (0)A,m, (B) not (7)m* Proof. (A) By (B) it suffices to prove (A) for any fixed A < KO.By , I Theorem 2.2(7) and Theorems 1.1(1)and 1.3(1),for allp, R m ( p , d A) Rm(Z= Z,A , 2) = no < 0. By 2.1, w.1.o.g. A = {cp}, so it suffices to prove that for every k I no there is n,(k, A) < w such that every cp-mtype of rank < k has a subtype q of cardinality In,(k, A) such that Rrn(q,v, 4 < k. Define for t E (0,I}, +t(. . ., Z, . . ;g) = cp(Zq; and let

.

+(.

. ., P,, . .. ; . . .,z-t;

=

A {cp(~,;

z$')

k >n

G

j 9

* *

*)nskh.vsk>A;f,l
7cp(fp; 2)'):

v E k > A ; q,p E 'A;

Z(U), v = 7 1 n = p n, i = q[n],j = p[nl}. For notational simplicity let +t = f ( t ( Z * ; g), = +(Z*; Z). By Lemma 2.9(2) it is clear that for every cp-m-type q Rm(q,cp,A) 2 k iff {+(Z*; Z)} u {Q(Z*; a): cp(Z; a)t E q} is consistent. Taking in Theorem 4.4(3) A = {+, +O, +l}, m' = Z(Z*^Z) the result follows. (B) Since for every A 2 2, and p : (0 Rrn(p,cp, 4 I no -def - Rm(- = 3,cp, 2) < w , (ii) Rm(p,cp, n ) = R m ( p cp, , a) ( n IA) implies Rm(p,cp, n ) = R Y P , A); it suffices to prove: for every n there are a(n),p(n) < w such that for any set (*I of cp-m-formulas, p , if for every q c p , IqI I a@), RYq, cp, R n ) l 2 n , then R"(p, q,00) 2 n. (Take n2 = max{p(n): n s n , } , so for (A) k = m&x{a(n):n Ino}.) We prove it by induction on n. For n = 0, Rm(p,cp, A ) 2 0 iff p is consistent; hence by "not (3),,," our assertion holds. Suppose we have proved for n, and we shall prove for n + 1. By Theorem 2.12(2) for every q E O'2, k, Z < w there is a formula +[J such that =

+

cp9

Rm[{cp@;Si,)R['': i < Z(q)},cp, k ] 2 Z iff kgJISi.o .,.,82(q)-1]. Let m, = Z(1)where cp = cp(Z; g); and A, = {I#")*":q E a(n)r2}. Let k(n) < w be such that

68

[OH.

RANKS AND INCOMPLETE TYPES

11, 8 4

(i) every inconsistent set of A,-m,-formulas F has an inconsistent subset of power < k(n),and (ii) there are no a,, Sl such that C&; a,] = 7p[6,; at] for i # j < k(n) [k(n)exists by "not (3)," and Theorem 2.2 and 4.31. Now let P(n + 1) = 24k(n)+ /3(n),and a(n + 1) be such that if r i a a of power
P[r,

+

r

+ +

THEOREM 4.6: (1) If T does not have the f.c.p. and A hJinite,then there is n,(A) < w such that: if {ay:y < a 2 n,(A)} iS a A-n-indhcernible set over A, B > a, t h n we can deftneZ f o r a I; y < fi so that {a: y < /?}is a A-n-indiscernible set Over A. ( 2 ) I f A c l M l , V ~ 1 M fory 1 < aandMis(IAl+ lAl+ lPl 1.1 +)-cornpact,then we can choose E [MI for a I y < 9, (asin (1)). ( 3 ) If A, A are Jinite a < P = w, A c !MI, E IMI for y < a then we can choose cUE(MIfor a < y < /3 (under the assumption of ( 1 ) ) .

+

+

+

Proof. ( 3 ) follows from (2), and (2) will be clear from the proof of (1). So we shall prove (1) only. If n is too large, every set of distinct 8 s is a A-n-indiscernible set. Similarly if m = Z(ao)is too large. So we can prove the theorem for fixed n, m, and still get an nl(A) which is good for all n, m. Let A* = {p(Zo,.. .,i f " - ' ; 8):p(P, .,it"-'*, 8) = +(F(O), . . .,zo(n - 1)* 8) for some #(Zo,. .,if"-1*,8) E A and permutation 0 of n}.

.. .

J

CH. 11, 4 41

F.C.P., INDEPENDENCE AND ORDER

69

As T does not have the f.c.p., by Theorem 4.4(3), there is a natural number n3 = nl(A) such that: If I'is a set of formulae of the form ~ ( Zil, 9 .; ,Z-l, a), cp E cl,(A*) and every subset of I' of cardinality
..

pa = {cp(~; W-',

.. .,avo,E)*: E E A , yo < - - - <

yn-2

cp ELI* and Ccp[Z-',

< a,t E {0, I},

. . .,Tio, a*}.

Z-2,

For this, it is clearly sufficient fo prove that pa is consistent. By the definition of nl, it suffices to prove that every subset of p a of Casdindity < nl is consistent. Let q be a subset of pa, 1q1 < nl. Clearly in q apptw ~ ( -n l)(nl - 1) P a . So if a > (n - l)(nl - l ) , then there is 7iv, which does not appear in any of the formulas of q, hence i 2 realizes q, so q is consistent. So p a is consistent; hence we can define P,hence we mn define iir for a 5 y < 8, by induction, as required. 4.2: (1) A formula cp@; g) has the independence property if DEFINITION for every n < w there are sequences aI (I < n ) such that for every w E n, r

(2) T has the independence property if some formula cp(x;a) has the independence property.

DEFINITION 4.3: (1) A formula cp(Z; g) has the strict order property if for every n there are (I < n ) such that for any k, I < n C ( 3 Z ) [ i c p ( E ; a,) A cp(E; a,)] 0k < I . (2) T has the strict order property if some cp(x; g) has the strict order property.

Remark. ( 1 ) As in Definition 4.1(2), we can also in Dehitions 4.2(2) and 4.3(2) replace cp(z; by cp(Z; 3); in this case we use Theorems 4.11 and 4.16. (2) We shall show that a theory is unstable iff it has the independence

a)

property or the strict order property; but neither of these properties implies the other (see 4.7 and 4.8).

70

RANKS AND INCOMPLETE TYPES

[CH.

11, 8 4

Proof.(1) This follows from (2). It T is unstable, some (p(z;j j ) is unstable (by 2.13); hence by (2) (p(z; i j ) has the independence property or #,,(zPo,. ; . .,ijn-l) has the strict order property (for some n, q ) . Thus in the first case T has the independence property, and in the second c&88 the strict order property. Suppose on the other hand that T,and thus some (p(z;i j ) , has one of these properties. If (p(z;j j ) has the independence property, then by (2), (p(z;j j ) is unstable and by Theorem 2.13, T is unstable. If (p(z; 5) has the strict order property, then so does #,, for q = (0) so by (2) again, (p(z;j j ) is unstable and 80 is T. (2) Suppose (p@; i j ) has the independence property. Then clearly (p(3;j j ) has the order property. So by Theorem 2.2 (p(Z; i j ) is unstable. Now assume that for some q #,, has the strict order property. Then #,, has the order property and thus #,, is unstable. So there is an A such that ISFq(A)I > IAI 2 KO, where Z(Z) = m. For every qES$&4) let tZq realize q, and define p* = {(p(Z; a)t: ii E A , t E (0, l}, kgl[iiq;a]'},

i.e., the (p-m-typeiiq realizes over A. Clearly p* E S ~ ( A ) and ; for p, q E STw(A),p* = p* implies q = p; hence p # p implies q* # p*, so

This means that (p(Z; j j ) is unstable. So the strict order property of #,, implies the unstability of (p; and also the independence property of (p(Z; j j ) implies (p(Z; i j ) is unstable. So it remains fa be proved that if (p = (p(3;j j ) is unstable, then it has the independence property or for some q , & has the strict order property. By Theorem 2.2, (p(3;j j ) has the order property, so there are sequences lie, . . . Z7. . . such that for every k < o there exists Ek such an]iff k s n. By the compactness theorem and Theorem I, that kv[ek;

OH.

11, 8 41

71

F.C.P., INDEPENDENCE AND ORDER

2.4, we can assume that (3: n < w ) is an indiscernible sequence. If for every n < w , w c n,

k(32)

I

A ~ ( 3~;k ) M ( k e w ),

[k
then clearly ~ ( 3j j ; ) has the independence property and so are finished. So we assume that there are n < w , w E n such that k7(38)[

A Cp(3; ak)if(kew)]. k
Let IwI = no. There is an u < such that:

w

such that we can define w,for I Iu

( 1 ) ~ ~ = ~ , ~ , = { n - n ~ , n 1-,..., n ,n +- I } , ( 2 ) for every I I a, lwll= no,w 1E n, ( 3 ) for every 1 < u there is k, < n such that wl+ 1 = w,v

{h}(and 80 k, E ~

1

k, , 4 wI+1, k, + 1 $ w I , k, +

step raise w = wo to w,). We are assuming i=1(33)[Ak
{k, + 1) -

1 E W I +(we ~ step by

On the other hand by

A ~ ( za*)if(kewa)]. ; k
Hence there is 1 < u such that

A ~(3;

kl(3Z)L

(kea)],

where 8 = w,,

k
and k(31)[

1

A ~ ( 3ak)if(kE” ; , where t = w ~ + ~ . kcn

Let

=

k,,and

*

= *(3;j j , =

PO,.

A

. ., g - 1 , p + a , . . ., p-1)

~ ( 2jjk)if(kes) ; A ~ ( 93j j. )-

ken

k+8.8+1

We shall prove that $ has the strict order property. Let y < w , and define 6 = @On. . .-a8 - 1^@B + 2 + Y” . . ,^an - 1 + 7, By the indiscefibfiity of ( 2 :i < w ) , and the definition of $, if p I k < 1 < /I+ 2 + y , then (as this holds for k = 8, I = /3 + 1, y = 0 ) , k(3z)[$(P;

a’, 6)

ak)];

A --1cp(3;

72

RANKS AND INCOMPLETE TYPES

[CH. 11,5 4

but (by the same argument)

Hence, observing again the definition of #,

As this is true for every y < w , clearly # haa the strict order property, and it is also clear that # is of the required form. As this waa proved under the assumption that cp(S;g) does not have the independence property, we have proved the theorem.

THEOREM 4.8: (1) There is a theory To,, with the 8 t k t order property and without the. independence property. (2) There w a theory Find with the independem property and hthout

the strict order property. Moreover, there is no form& (p(xl,. . .,x,,;a) which M connected and adqpnmetrh over an infinae 8et. Remark. Cf. Definition I, 2.5. Of course by 2.13 there are m < w and a formula y3 = #(.", *, P - 1 ; 6), Z(P)= * * = Z(P-l)= m such that # is connected and antisymmetric over en infinite set of sequences of length m.

..

Proof.(1) Let Tordbe the theory of the rational order, i.e., dense order without first or laat element. Clearly (p(s;y ) = [a: < y ] haa the strict order property. The checking that it does not have the independence property is left to the reader aa an exercise. (2) In the language of T , there will be only the equality sign, a one place predicate P(x),and a two place predicate %By.Its axioms will be: (1) x E y implies T P ( z ) ,P ( y ) ;that is, ( V q ) [ x B y+ ,P(z) A P ( y ) ] . (2) if P ( y ) ,then y is uniquely determined by {x:xEy}, and conversely; i.e.,

OH.

11, 8 41

73

F.C.P., INDEPENDENCE AND ORDER

(3), For every 2n different elements in l P , xl,. . .,q,,zl, . ..,xn there is a y such that zlEy,. . .,xnEy, 7x1Ey,. , lxnEy. That is,

..

(Vq,.

. .,xn)(Vxl,.. .,xn) [0
#

d

A

7p(xk) A ~ p ( x ' ) ]

0 < 1Sn -+

( 3 ~ )A (%&Y

A

0
~~EY)].

(4), The mme as (3)n, interchanging P and l P ,

xEy and yEx.

That is, (VYl,

-

* *,

Yn)(vY1,

*

9

Y")

[A

[Yk

# Y' A

p(Yk)

A

0 .c k s n 0
+(3Z) (5) All models of

A

0
A

1

(%By,A 7ZEYk) *

Tinaare infinite, i.e.,

(3z0,.. ., z,)

p(Y')l

zi # z, for every n > 0.

t<jSn

It is not hard to prove that Tinais consistent, by building a model for it. It is also easy and standard to prove it has elimination of quantifiers, and is complete. By this it can be shown that no formula 'p = 'p(Z; 9) has the strict order property. On the other hand, clearly y(x; y) = zEy has the independence property (hence the order ProPY)*

a)

THEOREM 4.9: If 'p@; i8 urntable, A < K < Ded A, then there is an A such that IAI 5 A, ISg(A)I 2 K . (See Definition 1.4 of the Appendix.)

Proof. By Exercise 1.4 of the Appendix, there is an ordered setJ, IJI 2 K , with a dense subset I, IZl = A. As 'p(Z; 9 ) is unstable, by Theorem 2.2 'p has the order property. Hence by the compactness theorem there are BE, 8 E J such that: for every t E J, {'p(Z; B8)if(tsE):8 E J)is consistent. Let A = (J{aE:8 E I}. Clearly IAI s I Z l - Ho = A. For every t E J let p t = {'p(Z; aE)if(t<8): 8 E I}. Clearly p t is a consistent 'p-m-type over A. Let pt c qt E S t ( A ) . Now if al, 8 2 E J , a1 < 82, then there is t E I , 81 < t < a2, so y(Z; a,) E qa1, -I~(Z; a,) E qsa. Hence a1 # 8 2 implies q8,

f

l#t(A)l 1 1{q8:8 EJ}I = IJI =

a)

K

> A = IAl.

THEOREM 4.10: (1) If 'p = 'p(Z; hag the independence property, then for every A there is an A m h that IAI = A, ISg(A)I = 9'.

74

RANKS AND INCOMPLETE TYPES

[OH. 11, 5 4

(2) If f o r some infinite A , IS;(A)I 2 Ded,lAI, then Q ( Z ; ~ )has the independence property. (Ded,+ee Definition 1.4 of the Appendix.) ( 3 ) If cp(Z;g) has the independence property and Z(g) = k, then f o r every n c w there is an A , IAI s nk, I8;(A)I 2 2". (4) If f o r every n c w there is afinite A , IAI 2 2, euch tlrat I&(A)I 2 IAI", then ~ ( 3g);has the independence property.

Proof. (1) Let

r = { (W[A QP;SJ A A l ' p ( Z ; W)]: LEW

w G A, u

LEU

c A; w n u = 0; w, u are finite.

r

As Q has the independence property, u T is consistent, hence has a model N.Let 6, realize gt and A = Uf

r,

IS;(A)I 2 I{qw: w E A}1 = l{w: w E A}l = 2 A thus proving (1). (2) Let U = {Q@; 6):6 E A}. The mapping p + p n U from S;(A) to subsets of U is one-to-one. Hence {pn U :p ES;(A)}is a family of power 2Ded,lAI = DedJUI. So by Theorem 1.7(1) of the Appendix )E U,i < n, such that for every w E n for every n c w there are ~ ( 36'; there is p w E S;(A) such that for i c n ~ ( 2at) ; epWo i E w. This implies that Q(Z; g) has the independence property. (3) The proof is similar to that of (1). (4) The proof is similar to that of (2), this time using 1.7(2) of the Appendix. THEOREM 4.11 : The following statements about a theory T are equivalent. (1) Some formula p(x; g) has the independence property, i.e., T hm the independence property. (2), Some formula Q(Z; ji) ( l ( Z ) = m ) hm the independence property. ( 3 ) There are Q ( X ; g) and n < w such that f o r every k c w there is an A , IAI = k , IS,(A)I 2 2k'n. (4), There is ~ ( 3g), ; m = I @ ) , 8uch that f o r every n there is afinite A , 2 2, p;(A,I 2 pq".

CH.

11, f 41

F.C.P., INDEPENDENCE AND ORDER

75

If for ~0112e A, (vp)b < Ded, A -+plTl < Ded, A], the follo2oing statements are equivalent a8 well. ( 6 ) For every p there is an A , IAI = p, IS(A)I 2 2u. (6), For s m A, ISm(A)I 2 Ded,lAl, and (Vp)[tc < Ded,lAl < D4lAll. Procf. Clearly (1)implies (2), (by adding dummy variables to p). By Theorem 4.10(3), (2), implies (4), and by 4.10(1),(2), implies (6),, when assuming the existence of the suitable A. Suppose (6), holds, and let A = 1.41; x = Ded, A; so by Lemma I, 2.1 for some By 1B1 = IAI, IS(B)I 2 x. As x is regular, clearly by Lemma 2.16(1)for some 'p, 1SJB)I 2 x. Hence by 4.10(2),'p = p(z; #) has the independence property. So (6), implies (1).By 4.10(4) (a), implies (2),. By 4.10(1), (1) implies ( 6 ) , and by 4.10(3), (1) implies (3),and trivially (3)implies (4),, (6) implies (6),. So we have (1)=+ (2), + (4), =. (2),, (1) =+ (3)+ (4),; and when suitable A exists (1)=- (2), =+ (6), * (l), (1)* (6)3 (6),. So the theorem is proved, by the consistency results of Baumgartner (see [Ba 761). DEFINITION4.4:(1)Kz(A, T)is the first cardinal p such that IAI s A implies lS;(A)I < p. (2)Kr;(A, T)is the first regular or finite cardinal which is 2 KT(A, T). (3)Usually we omit T. If d = L we omit it, ifd = {p} we write just 'p and if m = 1 we omit it. THEOREM 4.12:Kr;(A) can be only one of the following functions: constantn (1 < n <

w),

A+,

Ded, A,

(2"+.

Moreover, emh of these functions is realized (by 8ome p, T ) .I n fact, p is unstable ifl Kr;(A) = Ded, A for all A or Kr;(A) = (2")+for all A. Proof. First suppose that p(Z;#) is unstable. Then by Theorem 4.9, A < K < Ded A implies there is an A, IAI s A, IS;(A)I 2 K. Hence K;(A) 2 Ded A. If for every A, Kr;(A) = Ded, A, the conclusion of the theorem holds. So suppose for at lectst one A, Kr;(A) > Ded, A. So by Definition 4.4,there is A, IAI s A, 15;(A)I r: Ded, A. As Ded, A is regular, by Theorem 4.10(2)' p ( f ; #) has the independence property. So by Theorem 4.10(1)for every A there is an A, IAI s A, ISz(A)I 2 2A,

[OH. 11,8 4

RANKS AND INOOMPLETE TYPES

76

hence Kr;(A) > 2A. But always IS;(A)I 5 2 1 4 + N o , hence Kc(A) I; (2"+. So if p is unstable, Kr;(A) = Ded, A (for every A) or Kt$!(A)= ( 2A) (for every A). So suppose p(5; 8) is stable. For every n, if for one A, Kc(A) > n, then there is A, IAl I; A, ISg(A)I 2 n. It is easy to find B E A, IBI 5 KO such that ISg(B)J2 n; hence for every p, aa p 2 K, 2 IBI, Kr;(p) > n. Hence if for some A, Kr$(A) = n, then for every p, Kr;(p) = n. As for every A, and a, ti realizes a type in 8; (A), clearly IS;(A)I 2 1, son > 1. So suppose Kr;(A) 2 No for some A. Then for any n there is A,, l@(An)l 2 2". We can define by induction on k < n, sik, ~ [ ksuch ] that +

1{p ES;(A,):p(5;sil)n[ll~p for every 1 < k}l 2 2n-k and {p@;al)n[lI: 1 < k - 1) u {p(z;a k - 1 ) 1 - n t k - 1 1

1

is consistent. For simplicity suppose every $11 is zero. Hence {p(?; jjl)if(l
IAI

=

A,

pg(A)I 2 l{pa:u < A}l

=

A.

Hence for every A, Kg(A) 2 A + . As p(5; g ) is stable, IAI 5 A implies ISg(A)I s A, so K;(A) 5 A + . So K;(A) = A + . Clearly in the c m s K;(A) is (2"+, A + , n ; Kr;(A) = K;(A). Let us prove that each of the mentioned functions is Kri(A)for some p and T. (1) By Theorem 4.8(2) there are T and p(z; y ) such that p(z;y) has the independence property. Hence Kt(A) = (2A)+. (2) Let T be the theory of the order of the rationals, and p(z;y ) = z < y . Clearly K$(A)= Ded A. (3) If T is the theory of equality, ~ ( zy;) = [z = y ] ,then K$(A) = A+. (4) Let T be a theory with an equivalence relation E with n 2 1 equivalence classes, and p(z; y) = [ z E y ] .Clearly Ki(A) = n + 1. THEOREM 4.13: Swppocle #(g; 5) = p(3; jj) and #(jj; 3) does nd have the independence property. Let

OH.

11, 5 41

77

F.C.P., INDEPENDENCE AND ORDER

Then there is n = n(cp) < w such that {a*: p < a} is a A,-n-indiscernible set, E a 8equence then either

(1) if

< a: hp[7iB; a]}l < n or

I@ < a: C-,cp[iZ*;E]}I

< n;

(2) if {a*: p < a} b a A,-n-indbmible sequence and E b a sequence, then there are 0 = a. 5 a, 5 5 an = a such that: if i < n, a, 5 jl, < a,+,, then ~tp[iP'; E ] iff ~tp[7iB'; E].

---

Proof. (1) As $@;3)does not have the independence property there is no < w such that

r = { ( 3 p ) A +(y";

w

~I)U(~€ W ):

I<no

= no .

1

is inconsistent. Let n = no. Now suppose our conclusion is incorrect. Then there are distinct ordinals pol . ..,fin- yo, . . .,yn- < a such that

,,

,

E l , . . ., hp[iZ*n-1; El, C - l c p [ Z i Y o ; E l , . . ., t7cp[ZiYn-1; 51. Remembering that {a*: p < a} is a An-n-indiscernibleset, we can see hp[$o;

that taking 8 for f' for I < n, I' is satisfied. Hence contradiction. (2) The proof is essentially the same.

r is consistent,

DEFINITION 4.6: (1) Let I be a A-n-indiscernible set in M (i.e., 8 E I * A, n, M )is the first cardinal p such that there exists a maximal A-n-indiscernible set I,, in M , I c I , , II,I = p (I, being maximal means that there is no A-n-indiscernible set I, in M , I , c I,, Ii Z Id. (2) Similarly we define dim(I, A, M), dim(1, n, M),with A-indiscernibility, n-indiscernibility instead of A-n-indiscernibility; and the same with dim(I, A,
THEOREM 4.14: Suppose T a h x not have the independenceproperty, A is$nite. Then there are a natural number n* = n*(A)and a finiteA* such that: for any n; if I, is a A-n-indiscernible set in M , I, is a A*-n*hdiscernibk Set in M , 11, n Is1 2 n*, then n*[dim(I,, A, n, M)]"-l 2 dim(I,, A*, n*, M).

[OH. 11, 0 4

RANKS AND INCJOYPLETE TYPES

78

So if one of those dimensions is M

t e , then

dim(I,, A, n, M )2 &(I,,

A*, n*, M).

Prmf.Remember that by Lemma I, 2.3, for any finite k; A,, . . .,Ak; n', , ,nk there are finite A, n such that I is A-n-indiscernible if€ it is Ai-nf-indiscernible for every i, 1 s i s k. Remember also that for sufficiently large n, every set of sequences (of length m ) is A-nindiscernible. Thus it suffices to prove the theorem for a fixed n. Let

..

A

= {qk(Zo,.

. .,3-l): k < k,

<

w}

where &to)= - - - = Z(3-l) = m. Define, for k < E,, u a permutation of n $&?, . .,3 - 1 -,5--O) = Cpk(?iYO', * . .,2 9 - 1 ) 1 and let zk = fin.. . - 3 - 1 9 80 4 k . u = $k,u(Zk; ZO). Now no $k,u(Zk; Zo) hae the independence property (by Theorem 4.11, ae T does not have the independence property). So there is i = i(k, U) < w such that

.

is inconsistent. Define n* = max{i(k, u ) + n: k < k,, u a permutation of n}

u is a permutation of n

A'

= {qk,s(fo; P):k <

k,,

'pk,@O;

u a permutation of n,

P)= #k,u(Zk;

ZO)}

i..

u {Zo # g}, I ( @

= m.

We shall now show: If I, is ad-n-indiscernibleset in M ,I, ad1-(< n*)indiscernible eet in H,11, n I,[ 2 n*, then n*[dim(I,, A, n, M)]"-'2 dim(Ia,A',
u

OH.

11, 8 41

F.C.P., INDEPENDENCE AND ORDER

79

sequences of I,. So every formula ~ ( 36); in q is realized by every i5 E I, n I , except at most n - 1. Hence I{E E I,: C&; 6]}1 2 II, n I,I - (n - I) 2 n* - (n - I) > i ( k , U) where Q is tpk,o or - , Q ~ , ~ By . the definition of dl and i(k, u), clearly I{Z E I,: C - p [ E ; 6]}1 < i ( k , o) < n* (by 4.13). We may aasume the number of formulas 8(3;g) appeasing in q is IIdall.

Henceif Ira] > n*~Iln-l~Aal = n**)I,)n-l,thenthereisaZ €1, - I, which satisfies every formula in q, hence I, u {Z} is a A-n-indiscernible This contradicts the choice of I, as a maximal set, and aa i5 E I,, Z E 1". A-n-indiscernible set in H.So [ I a [I n**II,l"-l so by replacing n* by n** we finish the proof. THEOREM 4.16: If T &ea not have the independence property and 111n IpI 2 H,; I , , I , are indiscernible sets, t h n dim(I,,A, M ) IT1 = dim(I,,A, Y) IT].

+

+

Proof. The same as the previous theorem. THEOREM 4.16: If s m e ~ ( 3Z);has the strict order property then T has the strict order property.

Proof. We prove it by induction on m = l(3).If m = 1, there is nothing to prove. Suppose we have proved for m and we shell prove for rn + 1 ; for notational simplicity let Q = ~ ( z8; , Z), 1(g) = m. Let {az:1 < w } be an indiscernible sequence such that

C(32, g)[lQ(z,8; 8,)A

Q(z,8; %)I

iff 1 < k (exists by compactness and Theorem I, 2.4). If there is b such that C ( 3 g ) [ i Q ( bg, ; %z) A Q(b,8; %+I)] for every 1 < w , then 8(g ; z,Z) = ~ ( z8; , Z) has the strict order property (this is exemplified by (b)n6&, 1 < w ) hence we finish by the induction hypothesis. Otherwise, by compactness there is a minimal k < w such for every 1 < k + 1. that for no b, C(3g)[-,tp(b, g ; Zai) A v(b,g ; 82z+l)] Clearly k 2 1. Let

80

RANKS AND INOOYPLETE TYPES

[CH. 11, $ 4

(by the minimality of k and the indiscernibility of {a,: 1 < w } ) and the same x shows that (ii)holds. By this and compactness, it follows that $(x; Lo,. . .,Zak-1) haa the strict order property, hence we have finished the proof.

QUESTION 4.1: Suppose ISg(A)I 2 DedJAI; does cp necessarily have the independence property? (See 4.10(2).) EXERCISE 4.2: Show that 4.6 may fail when T is stable but does not have the f.c.p. EXERCISE 4.3: Show that we can add to 4.4 the conditions: (9)" There is a finite A such that for every 1 < w there is a Aindiscernible set I, of sequences of length m, such that 1 S dim(I,,A, 6 ) < KO. (lo), For some finite A,, A, for no k < w does the following hold: for every A,-m-type p , Rm(p,A,, k ) = Rm(p,A,, a). (ll)*,,, For some finite A,, A, for no k < w does the following hold: for every set p of A,-m-formulas, there is q E p , 1q1 < k such that A) = R"(q, A,, A). R"(p, EXERCISE 4.4: Show that T has the strict order property iff some cp(Z; g) is a partial non-trivial order, i.e., it is transitive [ ( V Z , j j , Z)(cp(Z; g ) A cp(g; Z) +cp(Z; Z))], irreflexive [(V3)+3; Z)], asymmetric [(VZ, jj)(cp(3;j j ) -+ +g; Z)] and with arbitrarily large finite .. .,Zn) A,<,,cp(Z,; ?El+,)]. chains [for every n, (3Z0, EXERCISE 4.5: Let T& be the theory with the following axioms: ( V X Y ) ( X R ~~ R x ) , ( V X ) i XRX,

(VX,,...,X~;Y~,...,&)[/\X~ 1. j

# y j + ( 3 z ) ( A1 z t R z A

(1)

AJ l ~ j R z ) ] . (2)n

Show that also TZd satisfies the demands in Theorem 4.8(2).

OH.

11, 5 41

81

F.O.P., INDEPENDENCE AND ORDER

EXERCIBE 4.6: For stable T,regular A, and a model M the following conditions are equivdent. (1) There are finite A, m and a A-m-type p over IM), 1p1 = A, such that every q E p, 1q1 c A, is realized in M , but p is not realized in M . (2) There are finite d, m, A-m-types r , p over ]MI, 1p1 = A, Irl IA, such that for every q c p, Iql < A, r u q is realized in M, but r u p is not realized in M . (3) There are finite A, I , n such that A = dim(I, A, n, M). (4) There rn A, m a n d p E AT( IMI) such that M realizes every q c p, IqI < A, and some r c p, 1.1 = A, but not p itself. [Hint: For (2) + (3) let p = {pi(Z; Zi): i < A}, and let 6, realize Y u {p@; Zj): j c i}. Now use 2.18 and 4.13(1). For (3) + (4) use p' = {p@;a): Z E M, p E cll(A) and I{E

E J:

Cp[E; st]}l 2

KO}

for any J, I c J' E 1611, IJI = A, J G J' enough indiscernible.]

EXERCIAE 4.7: Suppose T does not have the f.c.p. and M omits some A-m-type of cardinality A, and A is finite. Then for some E(3, g ; 2) E L and B E !MI, E(Z, g; E ) is an equivalence relation, and in M it has IA but 2 Noequivalence clams. [Hint: By 2.1 we can assume p = p 1' p and by 4.2(1) T is stable, hence by 2.13 and 2.2 n1 = Rm(p,p, No) c o.We can aasume p haa a unique extension q in 8;(1611).We prove by induction on n,. Choose a finitep, c p, n1 = Rm(p,,p, KO),let $($; 2) be aa in 2.12(1) and let n* be maximal such that {r EA;((M():p1 E r , n* = Rm(r,p,

No)}

is infinite, so n* < nl. Let pa be finite, p1 G c q, such that p1 s r E 5:(1M1),Rm(r,p, KO) > n* and r # q implies pa $ r . Let d(2; E) (a E IMI) say: {p(Z; go): $(go;Z)} is consistent, has rank (Rm(-, 9,KO)) = n*, and extends pa (0, E exist by Theorem 4.4). We could have chosen p1 such that n* is minimal. Then

E(Zl, Za; Z) = 8(2,; Z)

A 0(Za; E ) A

(Vg)[$(g; 21)

$(g; Z,)]

is as required.]

QUEATION 4.8: Suppose the condition of Exercise 4.6 holds. Is there an equivalence relation over M with exactly A equivalence classes ?

CHAPTERI11

GLOBAL THEORY

III.0.Introduction The main interest and motivation in this chapter is in uncountable theories. The results of Chapter I1 are sufficient to find the stability spectrum of T (i.e., the class of cardinals in which T is stable) when T is countable. We shall prove: THE STABILITY SPECTRUM THEOREM: For any stable T,there is a cardinal K ( T ) I I TI such that T is stable in A iff A = A
The main idea behind the theorem, is that if T is unstable in A, A 2 2ITI, then there is a sequence of equivalence relations (in T ) E, (i c 6) each E, partitions many equivalence class of A,<, E,, into infinitely many parts, and AI61 > A. (This is almost true, as by 6.13 if No < K < K ( T )K, is regular, then there are such Ei(i < K ) . ) For proving the stability spectrum theorem, we seek a notion, which we call " p forks over A " and which satisfies: (0) K ( T ) is the minimal cardinality K ; such that for every p E Sm(A) there is B G A, IBI < K ( T )such that p does not fork over B (in fact, K ( T )I ITI+). (1) ForeveryB G A, IBI < ~ ( T ) , { p ~ S ~ ( A ) : p d o e s n o t f o r k o v e r B } has small cardinality (in fact I 2 9 (2) I f p E Sm(A)forks over B, then for some finite C c A, p (B u C) forks over B. (3) If p E &"(A,),A , (i I 6 ) is increasing, and p A , + 1forks over A, then T is unstable in A whenever Alal > A. Clearly properties (0), (1) enable us to prove stability results, whereas (2), (3) enable us to prove instability results. The notion of splitting is almost suitable, but it satisfies (3)only for 82

CH.

111, 01

INTRODUCTION

83

A < 2161.This was the motivation for defining: p strongly splits over B if ~ ( 3a), , -,T@, 6) ~p where a, 6 belong to an infinite set indiscernible over B. This is sufficient to prove the stability spectrum theorem for A 2 2ITI. However, for finer considerations it is not good enough, as there is an unjustified dependence on the parameters appearing in the type, and as it does not necessarily satisfy: (4) If p does not fork over B, Domp E A then there is q, p E q E Bm(A),which does not fork over B. All this leads to defining “ p forks over B” by: for someA, Domp E A , each extension of p in Bm(A)splits strongly over B. Notice that for B c A, p E #“‘(A),“ p does not fork over B” is a good substitute for “p and p 1 B have the same rank” and it looks more natural (see Section 4 for the connection). In Section 1 we define forking, and prove its simplest properties. It is interesting to note that “ p E @“(A)does not fork over A” is relatively problematic, and in its proof we assume T is stable and use the ranks Rm(p,A, KO)for finite A’s. Hence in Sections 1-6 we deal with stable T only. Note that some of the lemmas in Section 1 are needed only to get better results in Section 2. We say p is stationary over B, if for every A, Dom p E A , there is one and only one q E Sm(A)extending p which does not fork over B. It is interesting to know how much information p should contain in order to be stationary over B (assuming p does not fork over B); and the finite equivalence relation theorem answers it. A formula v(Z;a) is almost over B if some $(Z, p, 6) (6 E B) is an equivalence relation with a) = ~ ( pa). ; So finitely many equivalence classes, and $(E; p; 6) t- (~(5; p E Bm(A)(B s A) is stationary over B iff it does not fork over B, and it “decides” all formulas almost over B. Hence each p E B ( ~ M )is) stationary over [MIand we can prove property (1) of forking. In Section 3 we prove the stability spectrum theorem for A 2 2ITI. We also define the dimension dim(1, A, M) (Definition 3.3) and show that it is well behaved when it is ~ K , ( TWe ) . prove that a K,(T)saturated model is not A+-saturated iff some I has dimension s A. We also prove that A-saturation is preserved under unions of length 6 if cf S 2 K ( T),and that a A-stable T has a saturated model of cardinality

A. In Section 4 we give a quite complete picture of the connection between ranks and forking, being stationary and the multiplicity; and show that complete types over 1M1 have usually multiplicity 1. But more important are the results:

OLOBALTHEORY

84

[CH. 111, 5 1

THEOREM 0.1: (1) (symmetry) tp@, A u 6) does not fork over A iff tp(6, A U a) doee not fork over A (8633 4.13). (2) (tradtivity) if A E B E C, and tp(a,C) dim not fork over B, tp(a, B ) does nd fork over A then tp@, C) doee not fork over A (see 4.4). (3) Let B c A , then tp(6, A u a), tp(iZ, A) donot forkover B u a, B reap. iff tp(7i-6, A) do@ not fork over B (4.14, 4.16). (4) T h type p does not fork over lbll iff p i8 finitely sati8jiuble in bl (8W

4.10).

This theorem will enable us to introduce in Chapter IV F',-constructions. In Section 6 we complete the stability spectrum theorem for h I2ITI, by proving in the appropriate caaes,that there we many independent formulas (over F)using very specific dependence relations. The idea of Section 6 is that sometimes equivdence claaaee are not leas real than elements, hence when we seek a canonical form, sometimes it is an equivalence class. So we introduce P,which is just like Q except that we have elements for equivalence classes. We get, e.g., that for stable T , for eachp E Lgm( 1N1)there is a canonical'B G IMI,over which p does not fork, B is the algebraic closure of some C, ICl < K ( T ) . In Section 7 we are interested mainly in unstable theories, and get various results. Some theorems are generalizations of the stability spectrum theorem. We find q r a ( T )s /TI+ such that K < K,,(T) iff in T there are K independent orders iff for some A, IS"(A)I > Ax for every h < DedlAI. (Thus we get a complete characterization of Kr(h) for countable T,and a better picture in general). We also find when there are > h pairwise contradictory types of cardinality < K over a set A, IAI = A. Those proofs indicate an alternative proof to the stability spectrum theorem for h 2 2ITI without using forking or ranks (see Exercise 7 . 8 ) . In Sections 1-5, T will be stable and A finite in all unstarred theorems and l e m m . No unstated assumptions are made on T or A in starred theorems and lemmm.

III.1. Forking Notation.For an index set U let 2, = {xu:u E U}and for a sequence Ti = ( u ~ ) ~let + ,Pa = A type in 2 , over B is a consistent set of formulas using only variables from P, and parameters from B. The

type is complete if it is maximal among the types in Z, over B. For an m-type p and a set I of sequences of length m let p ( I ) = { Z E I : a realizes p}. Similarly we define cp(I;6) for a formula ~ ( 26). ; Insted of %Awe sometimeswrite just A.

DEFINITION 1.1: tp,(B, A) = {cp(Z6; a): 6 E B,a E A, Ccp[6; a]},tp*(6, A) = {cp(Ze; 8):si E A , E E 6, C cp[E; a]}.

DEFINITION 1.2: p split8 strongly over A if there is an indiscernible sequence I = {Z:n < w } over A and a formula cp such that cp(Z; ao), -,cp(Z; al) ~ p (Compare . Definition I, 2.6.) Remark. Clearly ifp splits strongly over A then p splits over A.

DEFINITION 1.3: The formula cp(Z; ti) &ideaover A if there we n < and sequenoes a’, 1 < w, such that (1) @(Ti, A = tp(8, A), (2) {cp(Z; #); 1 < w} is n-inconsistent (see Definition 11,3.2(3)(E)).

w

DEFINITION 1.4: The type p in I , forb over A if there are formdaa cpo(Zo;So), . .., cpn-l (3”- l; Z-l),with Z c 3, for k < n, such that ( l ) p V k < n vk(*;a k ) , (2) cpk(P;ak)divides over A for dl k < n. Instead of writing “{cpO(z; a)} forks” we write “cp(2; a) forks”. Note. p is not in general a type over A ; see Corollary 1.3.

LEMMA 1.1*: (1) If

g, k$(g; E) --+ cp@; 13)and cp(1; E) divides over A , then #(g; E) d i v k h Over A. (Inparticular this holds when $ is obtained by adding dummy variublea to cp.) ( 2 ) Ifp i s a type in Za, @finite) then p forks over A iff there are formula ZE

.

. .,cp’-’(Za; 3 - l ) smh thUtp k V k < n cpk(Za; ak),andcpk(Za; 7ik) divides over A for k c n. (3) cp(Z; 8) d i v k h Over A iff there are n c w and a sequence I = {a‘: 1 c w } indiscernible Over A such that ti = iio and {cp(Z; 8): 1 < w} is (po(Za; go),

n-ietent. (4) If cp(E; a) d i v k h over A then cp(I;a)forbover A. (6)p f o r b over A iff some finite &type q of p forb Over A type in Iafor some finite a). ( 6 ) tp*(C, B) fwks ovw A ifffor tp(8, B))f o r b Over A.

8Ome

(so q is a

E E C, tp*@,B) (OT qUiVdently

QLOBAL THEORY

86

[CH. 111, 8 1

(7) I f p , c p , f o r i < j < Sa~nop,forksoverA,tlaenU,,,p,doea not fork over A. (8) If A c B, q and p are types in the same variables, q t- p [ p E q] and q doecr not fork [aplit ~trongly]over A then p doea not fork [aplit strongly] over B. (9) I,fpu{$'(Z; a')}forksOverAforl < n < w,thenp~{V,,,,#(Z;ti~)} forks over A.

Proof. ( 1 ) is immediate and (2) follows from the parenthetical remark in (1).

(3) If the stated condition holds then clearly cp(Z; a) divides over A. Now suppose cp(Z; a) divides over A. Then there are n < w and sequences a', 1 < w such that tp(si, A ) = tp(2, A), and {cp@; 8 ) :1 < w} is ninconsistent. By Ramsey's theorem and compactness there are E l , 1 < w such that (6': 1 < w} is an indiscernible sequence over A , tp(6', A) = tp(a, A) and {cp(Z; 6l):1 < w } is n-inconsistent. So there is an automorphism P of Q such that F 1 A is the identity and F(6O) = a. So (P(6'):1 < w ) is the required sequence. (4) Immediate. (A converse is Exercise 4.15.) (5) Since i f p t- V,,,, qi then for some finite q c p , q t- V,<,, 'pi. The rest are immediate.

LEMMA 1.2: Lei p , q be m-types, p is Over A . (1) If q ( Z ; g ) E A , l(3) = m, cp(3;a) divides Over A , then Rm(p,A , No) > m p u (94%a,}, A , No]. (2) If q 1 p , q forks over A , then there is afinate A , m h that for every Pnite A 1A,, Rm(q,A , X,) < Rm(p,A , 24,). (3)* If cp@; g) E A , q(3;a) divides over A , A 2 2, then

-

R"(P, 4, A) < 00 Rmrpu {q(% a)},A, A1 < m p , 4, A). Dm(p,A , A) < co + Dm[pu {q(Z; a)},A , A] < Dm(p; A , A). (4)* If q 2 p , q f o r b over A , A 1 X,, then there i8 a jhite A , such that for every A 2 A,,

Rrn(;p,4, A) < + Rrn(q,4, 4 < R Y p , A, 4 D"(p,A, A) < a0 =+ P ( q , A , A) < P ( p , A , A). (5)* If in (4) we mame only that q split8 over A , then the conclusion is valid for A = 2. Proof. (1) By Lemma 1.1(3) there is I = {a': 1 < w} which is an indiscernible sequence over A, a = sio, and {cp(Z;8): 1 < w} is ninconsistent. Let p i = p u (~(3; 8 ) } for 1 < w, a = Rm(p,A , X,).

OH.

m, 8 11

FORKING

87

Clearly a 2 P ( p l , A ,No) = P ( p o , A ,No),by Theorem 11, 1.1(1) and Exercise 11, 1.1 and a < m as T is stable. Suppose a = P ( p l ,A, No) by way of contradiction. Let B = A u U I . By Theorem 11,1.6there are ql E 8T(B)such that Rm(plu ql,A , No) = a. Notice that p u ql = p l u ql. Now 1{2 < w : p(Z; 8)E qk}l < n 80 l{ql:ql = qk}1 < n. Thus there are infinitely many distinct q k , which are clearly pairwiee explicitly contradictory. Since P ( p u qky A , KO)= a, Rm(p,A , KO)> a, contradiation. (2) Since q forks over A, let q k V k < n q k ( Z ; ak)as in Lemma 1.1(2) where ~ ~ (Z k3) divides ; over A. Take A , = {vk(Z; gk): k < n} and let d 2 dobe finite. clearly q k p U {Vk<,,vk(f; ak)}k p . Thus

R"(q, A, No) 5 P[ pu =

{ v pk@;ak)},4 NoI k
max P [ p u {$(it; a")}, A , No], ken

this by Claim 11,1.7.By part (1)just proved, P [ p u {cpk(Z; Gk)}, A , < P ( p ,A, No).Since n < w we get

No]

Rm(q,A , No) 5 max Rm[pu {vk(E; 7ik)}, A , No] < Rm(p,d, No). ken

(3), (4),(6)are proved similarly.

COROLLARY 1.3: If p

$8 a type

in it, Over A thenp does not fork Over A .

Proof. Suppose p does fork over A. By 1.1(ti), ( 1) we can aasume p is an m-type. Taking q = p in part (2) of the above lemma P ( p ,A, KO)< R"(p, A , No): contradiction. Remark. Cleasly if p is a type over A then p does not split strongly over A. This also follows from Theorem 1.6. THEOREM 1.4*:If p is a type in f , over B which does not fork over A , then there ie a complete type q z p in it, Over B which does not fork over A .

Proof. Let r = {#(ita; 8):+ E U, i Ti E B, #(Za; 6 ) forks over A} and define a): #(Za; a) E r}.We now show that q' is consistent. If not, then it has a finite inconsistent subset

q' = p u {-,#(Za;

ak):k < n < w } , k Vk
p' u (-,yP(Zaa,; where p' V k < n #k

forks over A so yP E r. Thus p forks over A by Lemma 1.1(9)so p forks over A by Lemma E p,

88

QLOBALTHEORY

[OH.

111, $ 1

1.1(8), contradiction. Thus q' is consistent, q' is a type in itu over B. Let q 2 q' be any complete type in ?Zu over B. We now show that q does not fork over A. If it does, then q k V k < n ~ k ( Z a ,ak), ; .Ti, E U,where 'pk divides over A . Let q* E q be a finite subtype such that q* V k < n qk and let O(Za; 6) = A q* (aE u, 6 E B).q is complete so that O(Za; 6) E q. Clearly O(Za; 6) forks over A so (I(%;6,J E r.So E q' E q, contradiction. Thus q does not fork over A . COROLLARY 1.6*: Let A , E Bl,A , E B,,C,,Ca be any sets, F an elementary mapping from B, onto B,; F(A,) = A,, tp*(C,, B,)does not fork over A,. Then we can extend F to an elementary mapping F', Dom F' = B, v C,m h that tp*(F'(C,), B, V C,)do@ not fork over A,.

Proof. Directly from the theorem. THEOREM 1.6: (1)If p strongly splits over A , then p f o r b over A . (2) p f o r b over A iff there is a set B such that p is a type over B and for every q 2 p (in the same variables) which is mnplete over B,q strongly splits over A .

Remark. B may be chosen such that B

- Domp is finite.

Proof. Without loss of generality we may assume that p is an m-type. (1) By definition there are a set I = (3"'n < w } indiscernible over A and a formula Q such that Q(Z; a0),7 ~ ( Z ; a') ~ pNow . by Lemma 11, 2.20 there is n(y) < w such that for all 8, I{n < w: CQ[E; a"]>l < n(q) or I{n < w : Clp[8;7in]}I < n(q). Define 6" = aannSian+land $(Z; 6") = Q(Z; aan)A l ~ ( Z ; Zan+l). Clearly (6": n < w } is indiscernible over A . If w E w , IwI = n(tp), then {#(Z;6"): n E w} is inconsistent; otherwise, letting E realize this set, I{n < w , Ccp[E; Z]}l 2 1{2n:n E w}l = n(rp) and I{n < w : Cl~[8; a"]}l 2 1{2n + 1: n E w}1 = n(tp),Contradiction. Thus by Lemma 1.1(3)#(Z; 6O) divides over A. In addition p I- $(Z; 6O) since tp(Z; ao),+%; a,) ~ pHence . p forks over A. (2) * By definition, p k V k < n rpk(Z; ak)where tpk(8;Z k ) divides over A. So by Lemma 1.1(3)there are sequences (or sets; it is the same by = iZk, and there are n(k) stability) {a:: i < w } indiscernible over A, such that { q k ( Z ; a:): i < w } is n(k)-inconsistent. Let B = Dom p v {@:k < n , i < n(k)}andletqE8m(B)extendp.SincepI-Vk
m, § 11

89

FORKINQ

c Now assume p is a type over

B and every q, such that p c q E

Sm(B),strongly splits over A. By Theorem 1.4 if p does not fork over A there is q E Bw(B ) extendingp which doea not fork over A ,contradiction by (1).

DEFINITION 1.6: Let I be an infinite indiscernible set and A any set. Av,(I, A) = {cp(Z; a)': cp(Z; 9) E A , t E (0, l}, E A, and for all but finitely many E E I, hp[E; a]'}. If A = L we omit it. DEFINITION 1.6: The infinite indiscernible sets I , , I , equivalent if there is an infinite J such that I, u J and I , U J m e still indiscernible. LEMMA 1.7: (1) For every infinite indiscernible set I, Av(I, A ) is a complete type over A. ( 2 ) If I is an infinite indbcemible set over A then B realizee Av(I, A u U I ) ifl I u {E} is indiscernible Over A.

Proof. (1) By Lemma 11, 2.20 for every E A and formula cp Icp(I;a)! < Noor(lcp(I;7i)1 < Ko;thuslcp(Z;a)~Av(I, A)orcp(Z;7i)~Av(I, A). In other words, Av(I, A) is complete. In order to see that Av(1, A) is consistent, notice that every finite subset of Av(I, A) is realized by dl but finitely many members of I. (2) Left rn an exercise. LEMMA 1.8: The following d i t i m on the infinite indidcernible seta I , , I , are equivalent. (1) I , , I , are equivalent. ( 2 ) FOTany set A Av(I,, A ) = Av(I,, A). (3) There is a &el M y1M1 z I , , I , such that Av(Ii,

= Av(Ia, 1611).

Proof. (1) ==( 2 ) Let J be an infinite set such that I, uJ , I, u J are indiscernible. By definition Av(I,, A) z Av(I, uJ, A), but since both are complete types over A, they are equal. Likewise, Av(I,, A) = Av(I, u Jy A) = Av(J, A ) = AV(I1 U J y A). SOAv(I1, A) = Av(I,, A). (2) * (3) Trivial. (3) * (2) If (2) does not hold then there are A and a formula p(Z; a) such that cp(Z; 7i) E Av(I,, A ) and cl p(Z; a) E Av(I,, A). By Lemma 11, 2.20 there is n(cp) < o such that for any indiscernibleset I*and sequence a*, Icp(I*; &*)I < n(cp)or I-&*; a*)l < n(cp).Choose 6', . . ., 6n(@)E I, such that b[6*; a] (1 s i s n(cp)) and E l , . ., F(@) E I , such that

.

90

t-p[i?;

GLOBAL THEORY

a] (1

5

i

5 n(T)).Thus t(3Z)

[CH.

AYL$ (~(6';5) A

M is a model, 6', i? E lMl there is a* E lMl such that nW

'A (&';

a*] A

-q@;

+8;

m, 0 1

Z)). Since

a*]).

1=1

Thus (cp(Il; a*)],llT(I2; a*)] 2 n(T).Hence ~ ( 2a*) ; E Av(I,, 1M1)and +Z; a*) E Av(I2, IMI), contradiction. (2) * (1) D e h e by induction on n < o sequences c7L which realize Av(I1, C,,) = Av(I2, C,) where C, = U I, u U I 2 U U {i?: i < n}. The sets I, U {b:n < o}and I, U {b:n < w } are indiscernible by Lemma 1.7(2) and so I, and I, are equivalent. DEFINITION 1.7: p is stationary over A if (1)p does not fork over A and (2) p has no two contradictory extensions which do not fork over A.

Remark. By Theorem 1.4 an equivalent condition is that for all B such that p is a type over B, p has a unique extension to a complete type (in the same variables) over B which does not fork over A. LEMMA 1.9*: If A E B, p does not fork over A , p A , then p does not split over B.

r B is stationary over

Proof. By way of contradiction assume tp(6, B) = tp(Z, B) and ~ ( 36), ; ltp(5; a) ~ pLet . P be an elementary mapping such that P B is the identity and P(E) = 6. ( p B ) u {tp(Z; 6)) does not fork over A, since it is a subtype of p. Likewise (p t B) u {-,cp(t; a)} does not fork over A. Thus P [ ( p B) u {+if; a)}] = p 1 B u {-,cp(f;6)) does not fork over A (since P 1 A is also the identity). But then p t B has two contradictory extensions which do not fork over A , contradiction.

r

DEFINITION1.8: The infinite indiscernibleset I is based on A if for every set B, Av(I, B) does not fork over A. (The interesting cam is when B 2 A, and this gives an equivalent definition.)

Remark. I is based on U I. LEMMA 1.10: (1) If p is an m-type over B z A which is stationary over A , B, = B U Uf< 8,p E p , = tp(Zj, Bj),and p j does not fork over A , then I = {ii': i < i,, 2 o}is an indiscernible set over B based on A. (2) I n the notation of ( l ) ,for all C z B, Av(I, C) is the unique extension of p in P ( C ) which does not fork Over A .

,

Proqf. (1) By Lemma 1.9 p , does not split over B and for all i < j < i,, p i c p j because p is stationary over A. So by Lemma I, 2.6, stability, and Theorem 11, 2.13(7) I is an indiscernible set over B. Now we show that I is b a d on A. Let B be any set, and define C = B u B’ u Uief0 3.Inductively define ?if, for io s j < io w , such that p , = tp(?ij,C u Ute8 )extends p and does not fork over A. This is achieved by 1.4. Clearly p , t B, also eatisfies the above conditions. Thus, aa above, {sij: j < i, + w } is an indiscernible set over B. Now

+

Av(I, B’) c Av(I, C) = Av({Si’: j < i,}, C) = Av({?i’: j < i, = Av({iZ’:

i,

since pi0 = tp(8, C) for i , I j < i, so neither does Av(1, B‘). (2) Follows from (1).

Ij c

+

W.

i,

+ w }, C )

+ w } , C ) = pfo,

But pf0does not fork over A

LEMMA 1.11: (1) If p E S ~ ( ~does M ~not) fork over A c \MI, and M M (( IA I + 2)ITl) -saturated, then p is stutionary over A . (2) If A c B and r, = { p E Sm(B): p doe0 not fork over A}, then IFl[ I (IAI + 2)lT1for any B. (3) If A c B and I’a= { p E Sm(B): p doe0 not ~tronglyelit over A}, t h n Ira[ I (IAI + 2)ITI. (4) There are at nmt (IAI + 2)ITI non-equivalent infinite indiscernible set8 based O n A . +

Proof. (1) Assume p is not stationary over A. Then there is B 3 and p c ql, qa ES”(B),ql # qa, and q,, qa do not fork over A. Let 6 E B be such that ~ ( 36);€ q l , l@; 6) € q a . Now we define an increasing sequence A, c I”, lA,l I (IAI + 2)1TI, sequences 6,E IM( and typesp, E S ~ ( ~ M for~i )<, ((IAI + 2)lTl)+.Let A, = A, and if 6 is a limit ordinal A, = U { A j :j < S}. Assume A, is defined; we shall be such that tp(6,, A,) = tp(6, A,); define 6,,p , and A , + l . Let 6,E there is such a 6, by _saturativity.By the assumption on q,, qa we have that p t A, u ( ~ ( 2b,)}, ; p A, u {ltp(P; 6,)} do not fork over A. Without loss of generality ~ ( 26,); ~ p Let . p , ‘be such that p 1 A, u {-vp(Z; 6,)) E p , E Sm( / M I ) ,and p , does not fork over A (by Theorem 1.4). Take A , + , = A, u 6,. So we have defined A,, p , and 6,. Now from {6,:i < ((]A1+ 2)IT9+} choose an (infinite)indiscernible sequence over A. If i, < il and gf0,6,, are in the indiscernible sequence over A then p(3; gf0),- , ~ ( f6,1) ; ~ p , ,

QLOBAL THEORY

92

[OH.

UI, 8 2

pf, splits strongly over A and thus forks over A (by Theorem 1.6); contradiction. (2) Follows from (3) since strong splitting implies forking. (3) Define an equivalence relation between h i t e sequences from B as follows: 6 E if there are a,, . . .,a,,, a, = 6,a,, = E, and infinite indiscernible sets I,, . . .,I,,-1over A such that a,, ar+1E I , for all 0 I1 < n. Let B G B be such that every h i t e sequence in B has an --equivalent sequence in B . The mapping p HI, 1 B' is a one-one (Assumep1 # pa E Fa,p1 B' = pa 1 B'. mapping from Fainto Sm(B'). Let v(Z;6) €pl, +Z; 6) and 6 E E B'. If p(Z; E) E p1 then q ( Z ; E) E 212 andpa splits strongly over A, contradiction;if -vp(Z; 2) E pl, then p1 splits strongly over A, contradiction.) By Theorem 11, 2.13 T is stable in (IAI + 2)lTl so (3) will follow if has 5 ( [ A [+ 2)ITI equivalence classes (since then B' we show that may be chosen such that ISm(B')Is (IAI + 2 ) 9 . If not then there is I = {&: i < ((IAI 2)lTl)+},6f 3, 6, for all i # j,and we may assume I(&) = 2(Ej),for all i,j. By Theorem I, 2.8 I haa a subsequence of the same power which is indiscernible over A. This is a contradiction to the inequivalence of the 6:. This proves (3). (4) Assume there are { I j : j < ( ( [ A + ] 2)ITI)+} pairwise nonequivalent indiscernible sets baaed on A. Choose M such that IMI 2 A v I,: j < ((IAI + 2)lTl)+} and M is ((IAI + 2)lT1)+saturated. Let p j = Av(I,, 1M1). By Lemma 1.8 the inequivalence of the I, implies that pf # p j for all i # j.But by definition the fact that I, is based on A implies that p j does not fork over A. So there are > (IAI + 2)lTl complete types over which do not fork over A, in contradiction to (2). This completes the proof of Lemma 1.11. 80

-

-

-

-

+

u {u

CONCLUSION 1.12: For every sequence 6 and set A there is an indiscernible set I over A and based on A , such that 6 E I . If M is a ( [ A [+ KO)+saturated mode2 such that A v 6 c IMI then we may take I c 1M1. If tp(6, A ) doe%not fork over B G A then we can c h e I 8 0 that I is based on B.

Proof.Let Ml be a (( [ AI + 2 ) 9 -saturated model such that lMl I z A v 6. tp(6, A) does not fork over A by Corollary 1.3. Thus by Theorem +

1.4 there is a complete type r z tp(6, A) over lMll which does not fork over A. By Lemma 1.11(1) r is stationary over A. Now by repeated use of Theorem 1.4 and by Lemma 1.10(1) we may build an indiscernible

CH. In, 5 11

FORKING

93

set (over lM1l) of sequences realizing r based on A. Call this set {E,: i < w}. Now tp(6, A) = tp(Eo,A) so there is an automorphism P of Q which is the identity on A and such that P(Eo)= 6. Without loss of generality P(E,)E [MI for all i < w , so we may take I = {P(E,):i < w}. If tp(6, A) does not fork over B G A then we can choose r so that r does not fork over B and then I is baaed on B. Remark. See 6.3(6). EXERCISE 1.1: Show that in Lemma 1.7(1) it is possible to replace the assumption that T is stable by the (weaker)rtssumption that T does not have the independence property. Exercise I.IA.Another&leis: T = Th(N),N= (IN,E , C , P ~ ) ~ , ~ where E is an equivalence relation over N with infinitely many equivalence classes, each infinite, PI5 c/E, and every Boolean combination of the P,has infinite intersection with c/E. Let A = 0, p = 0, q = {xEc} and A = {P,(x):1 < n} u {E(x; y)}. Clearly p u q fork over A, but R(p, A, 2) = R(p u q, A , 2) = n when n > 0. For 2 I h < KO,we use N = ("1, E, e, Pt,a)l
[OH. 111,8 2 w } is

an indis-

m.2. The finife equivalence relation theorem Notation. FEm(A)will denote the set of formulas 'p = 'p(Z; 3; a) such that I(%) = l(8) = m and ti E A, and 'p is an equivalence relation with a finite number of equivalence classes, FE(A) = Um
DEFINITION 2.1: (1) The formula 'p(Z; 6) is almost over A if there is E(Z;8) E FE(A) auch that (V?@j)[E(S;9) ('p(Z; 6)= 'p(8;6))]. In this case 'p is said to depend on E. Notice that if 6 E A then 'p@; 6) is almost --f

over A. (2) The type p is almost over A if every formula in p is almost over A. (3) stp(a, A) = the strong type of 7i over A = {E(Z;a): E(Z;8) E FEm(A)}.Similarly define stp*(C, B) for a set C.

Remark. realizes stp(si, A ) and stp(i5,A ) I- tp(a, A). Notice also that stp(a, A ) is almost over A, and if atp(a, A ) = stp(6, A ) and 'p(Z; a) is almost over A then k'p[a; a] = 'p[6; a]. If realizesp, p is almost over A, then stp(a, A ) t- p. LEMMA 2.1*: (1) FEm(A)is closed under conjunction, i.e., if 'pi@; 5; a,) E FEm(A),i = 1, . . .,n then 'p(Z; 8; a) =def ATzl 'p,(Z; 8; a,) E FEm(A). Hence stp(8, A) is closed under conjwnctiow. (2) If E(Z;8) E FEm(A) and E ' ( P 3 ' ; 8-8') =def E(Z;g), then E' E FE"(A) (where n = m + l(x'), l(Z') = Z(8')). (3) If S = (xo,. . .,x,,,-~),8 = (yo,. . .,Y,,,-~), 0 is a permutation of 0, . . .,m - 1 and E(Z;8) E FEm(A)then

Proof. Immediate. LEMMA 2 . P : (1) The set of formulas which are almost over A is closed under all the connectives and quuntiifiers, under the addition of dummy variables, and under permutation of variablea. (2) The number of formulas which are almost over A is, q to logical

OH.

In, 8 21

THE FINITE EQUTVALENCE RELATION

95

equivalence, at most I A \ + I TI. Thus the number of non-equivalent types which are almost over A is 5 2IAI ITI. (3) If stp(Z, A ) = stp(6, A ) then there is an a u t m p h i a m F of Q such that F(8) = 6, F t A is the identity, and F preserves the formulas which are almost over A . We can replace a, 6 by sets. +

Proof. (1) The cases A and are easy to see ( A by the preceding lemma), as are dummy variables and permutations of variables. Now let tp(zo,Z; 6) be almost over A ; i.e., there is E(zo,3;yo, ij) E FE(A) such that E(zo,Z;yo, jj) k (cp(zo,f ; 6 ) = tp(yo,8; 6)). Let tp*(Z; 6) = (3z0)tp(z0, Z;6). Define

E*(% Y) = ( V ~ O ) ( ~ Y O ) W 3; ~ OYo, , 8) A (VYo)(~zo)E(zo, 3;Yo, g).

It is not hard to see that E* E FE(A) and that tp* depends on 2*; in fact n(E*) I2n(B).Note that the above proof works even when Z = g is the empty sequence. (2) and (3) immediate. LEMMA 2.3*: (1) Theformula tp(Z; a) is almost over A iff r = {tp(Z; F(a)): F is an aubnwrphiam of Q which is the identity on A} contains only a jnite number of nonequivalent formuha iff r contains < IlQll nonequivalent formulas. (2) tp@; a) is equivalent to a formula over A iff dl the formulas of r above are equivalent. Proof. (1) The third statement implies the second by compactness and the inverse implication is immediate. Also the direction * in the first "iff" is clear by definition. Now for the other direction, if r has only a finite number of non-equivalent formulas, then there is n such that the ; set Fl = {O(g,;a): O(&; a) E tp(a), A, i < n}u {l(VZ)[tp(Z;jji) = ~ ( 3g,)]: i < j < n}has a contradiction. Let n be minimal. Then we can assume that there is a contradiction in

96

GLOBALTHEORY

[CH.111, g 2

J%; %) = (Vg)[O(g; 2) --+ ( q ( Z l ; g) = v(Z2;@)I E FE(A), n(E) I 2"-l, and ~ ( 2a); depends on 1. ( 2 ) Similar.

NOW

LEMMA 2.4*: If E(Z;8) b an equivalence re2atio-n almoflt Over A , then there i8 an equivalence relation E' Over A which refim E ; Le., (Vq)(E'(Z; -+ E(Z;g)), 8wh that if E ?uufinitely many equivalence c l ~ s s e sthen , 80 does E'.

a)

Remark. This lemma says it is pointless to replace " E over A" by " E almost over A" in Definition 2.1(1).See Exercise 2.3.

Proof. By Lemma 2.3(1) there are E,, i < n, equivalence relations such that every automorphism P of 6, such that P A is the identity, takes E to one of the E,, and for every Ef there is such an P. Thus every automorphism of 6 which is the identity on A yields a permutation of Ef(Z;g) is preserved (up to equivalence) under all the E,. Thus automorphisms of 6 which are the identity on A . So by Lemma 2.3(2) E'(Z; g) = A,<,,Ef(Z;g) is over A , and certainly refines 1.I f E has k equivalence classes so does each E,, hence E' has at most kn < No equivdence classes. LEMMA 2.6: Let I = {a,: 1 < w } be an indbcernible 8et which b based on A . Let v ( Z ; g ) be a formula and f o r all n < w define q,,(~;a") = large ~ h q n i8 , a l m t Vwcan-l.lw1-n A l c w ~ ( 3a,)* ; T h for all Over A . (The size of n depends on tp only.) Proof. By Lemma I I , 2 . 2 0 there is n such that for any indiscernible set I, and for all of length 1(Z), Iv(a,I ) ! < n or 17(p(a, I)I < n. Thus for any 6, bn[6; a"] * there is w c 2n - 1 , 1.11 = n such that bq[6; si,] for a l l i ~ w *lq(6,I)I >n=+llq(6,1)l < n=+q(6;8)EAv(IY6)=+lq(6,I)I 2 KO * 17q(6,1)( < n *there is w G 2n - 1, IwI = n such that Cq[6;Zf] for all i E w * bvn[6; an]. So bn@; a")o ~ ( 6g;) E Av(1, 5). Now by Lemma 2.3(1) if qn(Z; an)is not almost over A ,then there are {P,: i < ((]A1+ 2)lT1)+},Ff 1 A is the identity, F , ( P ) = 4, tp(q, A) = tp(P, A), and such that q n ( E ; q )are pairwise non-equivalent. Thus for all i < j < (([A1+ 2)IT1)+there is gf,, such that hp,,[6f,,; q]= l p n [ 6 i , j ; a71- Thus, by the above, v(&,j;8)E Av(P,(I)y & , j ) 0T T ( & , j ;5) E Av(P,(I), 6,,,).Thus Y , ( I ) ,P,(I) are not equivalent (see Lemma 1.8). Since I is based on A and E; A is the identity we have that each F f ( I ) is also based on A .

OH.

111, $21

THE FINITE EQUIVALENCE RELATION

97

Thus we have ((IA1 + 2)lTl) non-equivalentindiscernible sets based on A , in contradiction to Lemma l . l l ( 4 ) . +

LEMMA 2.6: (1)* If p €Sm(B) does not fork over A c B and q z p is a type (in the same wa?$abka) which i8 dm8t over Bythen q does not fork over A . Also Rm(q,A , No) = Rm(p,A , KO)for all jinite A . (2)* If p E Sm(B)does not fork over A G B, E ( f ;g) E FEm(B),and q = p u {E(Z;a)}is umaistent, then q does not fork over A . Also R”(q,A , No) = R’”(p,A , No)for all (cjinite)A . (3) For all A , B, C (with A c B ) there is an elementary mapping 4, Dom F = A u C, F 1 A is the identity, such that for all 8 E C, tp(F(8), B) does not fork over A , and stp(8, A ) = stp(F(8),A ) .

Proof. (1) Assume that q does fork over A ; then by Lemma 1.1(6) q has a finite subtype q’ which forks over A. Every formula in q’ is almost over B y so by Lemma 2.2(1) ~ ( 36);= A q‘ is almost over B. Thus Q depends on some E ( f ;ti) €FEm(B). Let realize q. Clearly k(E)[E(Z; a) + Q(Z; 6)],so if tp(f;6) forks over A so does p u {E(Z;a)}. Thus the firat part of (1) follows from the first part of (2). The second clause (about Rm(q,A, No)) similarly follows from the proof of Rm[p u {E(Z;a)},A , No] = Rm(p,A, No). (2) Notice 8 does not necessarily realize p. Choose a maximal set al,. . ,,an such that tp(iZk,B) = tp(8, B) but k7E(Z;ak), for all 1 s 1 # k I n. (n < w since n(E) < No.) Thus tp@, B) u {-,E(f; a”): 1 I k I n} is inconsistent. tp(7i, B) is closed under conjunctions so there is tp(f;8 ) E tp(7i, B) such that tp(f;8 ) t VFX1E(Z;a k ) . Assume p u { E ( f ;a)} forks over A . Then for all k , p~ {E(Z;ak)}forks over A . So by Lemma 1.1(9) p u {VL,l E ( f ;a“)} forks over A . p u {E(Z;ti)} is consistent so ( 3 g ) [ ~ ( J8 ); A E(E;g)] ~ p Thus . if 6 realizes p there is 7i* satisfying tp(p; 8 ) A E(6;I) hence , V1=l E(7i*;8”). Thus also VLs1E(6;a”). In other wordsp t VzSl E ( f ;Ek),or p t p u {VpPIE(Z;tik)}. Thus p also forks over A , in contradiction to the hypothesis. The second clause is proved using Claim 11, 1.7 and Exercise 11, 1.1. (3) Clear by Corollary 1.3, Theorem 1.4 and (1). COROLLARY 2.7: If p is alm8t over A then p does not fork over A .

Proof. Let q 2 p be a complete m-type over A u Dom p . q 1 A E @ ( A ) and so does not fork over A by Corollary 1.3, and clearly q 1A up is a

(consistent) type almost over A. Thus q 1A up does not fork over A by Lemma 2.6(l ) , and so neither does p . THEOREM 2.8 (The Finite Equivalence Relation Theorem): Let p, q E S ~ ( ~ MpI )#, q, A E 1611,p , q do not fork over A , and M is (([A1+ 2)1TI)+-suturated. Then there is E ( f ;j i ) E FEm(A) such that P ( f ) u q(ji) k 1 E ( f ;9).

Prmf.Since p # q and both are complete over 1611 there is c p ( f ; 6), 6 E 1611,such that cp(Z; 6) ~ pl c ,p ( f ; 6) E q. By Conclusion 1 . 1 2 there is a set I = {6*:i < W } c 1611 indiscernible over A and based on A such that 6, = 6. p, q do not fork over A, thus they do not strongly split over A, so c p ( f ; 6,) ~p and lcp(f; 6,) E q for all i. By Lemma 2.6 (and using its notation) there is n <

w

such that

cpn(P; 6") is almost over A. Let E ( f ;j i ) E FE(A) be such that cp,, depends

+.

on 4:(Eji)[E(E;j i ) (cp,,(f; 6") = cp,(ji; En))]. Notice cp,,(f; 6") E p , lcpn(f;6")E q. Thus p ( f )u q(#) t- (cp,,(f; 6") h Tcp,,(ji; 6"))I- l E ( Z ;ji).

COROLLARY 2.9: ( 1 ) For dl and A , stp(a, A ) b s W k r y over A . ( 2 ) If p , q eSm(B) do not fork over A c B, p # q, then there is E ( f ;j i ) E FEm(A)swh that p(Z) U q(ji) t- i E ( f ;j i ) .

Remark. This improves Lemma l . l l ( l ) ,2.8. Proof. ( 1 ) Let p = stp(si, A). p is almost over A, so by Corollary 2.7 p does not fork over A. Now if p were not stationary over A, then p would have two contradictory extensions pl, pa which do not fork over A. Let M be a (([A1+ 2)ITI)+-satwatedmodel such that p1,pa are over [ M I . So A c [ M I . Let p', p a be extensions of pl, pa, respectively, in Sm(1611)which do not fork over A. Let E ( f ;j i ) E FEm(A)be such that pl(Z) U pa@)k i E ( f ;ji). But p c pl, pa, E ( f ;a) E P , SO that ~ ~ (U2 pa(y) k l E ( f ; j i ) A E ( f ;a) A E(@;a), a contradiction. (2) Assume that there is no such E. Let = p ( f )u q(ji) u { E ( f ;8): E E FEm(A)},r is consistent (since FEm(A)is closed under conjunctions). Let f I+ a, ji H 6 realize I'. Thus stp(8, A ) = stp(6, A), a realizes p and 6 realizes q. So stp(a, A ) is not stationary over A, in contradiction to ( 1 ) .

r

)

100

QLOBAL THEORY

[CH.

111, $ 2

(2) If J M an in$nite ~ndiscernibleset based on A , t h there an equivalent indiecern~leset I c 1M1 (I ie also indiscernible Over A ) . 8ee De$nition 1.6.

Remark. Notice that when 12'1 is small there is a simple direct proof. The interesting case is when I2'1 is very large in relation to A. Prmf. Let p be an m-type almost over A . By Corollary 2.7, p does not fork over A , so there is a complete m-type q z p, over 12111 u Domp E Sm( 1M1).Clearly q 1 1MI also which does not fork over A ; so q I i < w + w} in M such does not fork over A . Now define a sequence (8,: that a, realizes q 1 ( A u U {a,: j < i}) for all i < w + W. This is possible by saturativity. So by Lemma 2.12 q 1 ( A u {a,: j < w } is stationary over A. By Lemma 1.10 I = {a,: w s i < w + w } is an indiscernible aef over A , whose average is q. And so by Lemma 2.10 we see that &, realizes p. This proves (1). For part (2), choose distinct 7 i , ~ J and , 6,oIM1 realizing stp(i%,,A ) and 6,,~lMl such that tp(?ion. . .ha,,, A ) = tp(6,^- - .^6,,,A ) and let I = {b,,: n < a}. LEMMA 2.14*: If p = tp(iz, A ) ie 8&Li%Onary over B c A , t h p k stp(a, A), and thw, for every 'p which is d w t over A , p k 'p or p t- -,'p.

Proof. Let E E FEm(A)where m = I@). We must show that p t- E(Z;a). If not, then there is 6, not E-equivalentto 3,such that q = p u {E(Z;6)) is consistent. Then by Lemma 2.6(2) q and p u {&'(it; a)} are two contradictory extensions o f p which do not fork over B, contradiction to the fact that p is stationary over B.

Proof. (1) Let 'p(Z;6) depend on EtzFE(lMl) and let n(E) = n < W. Since M is a model there are representativea a,, ..., E 1M1 of all the equivdence claases of E. We have C ( E ) Vr<,,B(Z;$) and t(VZ)(E(Z; a,) +'p(Z; 6))or C(E)(E(Z;a,) + -,'p(Z; 6)) for i < n. Without loss of generality we can assume c(vz)(E(z; a,) + ' p ( ~ ; 6)) for

a.m, § 31

THE INSTABILITY SPEOTRUM

101

i < m I n and k(Vz)(E(Z; a,) + +it, 6)) for m 5 i < n. So k'p(Z; 6 ) E V,<,,, E(3;a,) if m > 0 and k+Z; 6) if m = 0. (2) Let 8 realize p. By Corollary 2.9(1) it is sufficient to prove that p t- stp(a, A), i.e., for all E E FEm(A),p k E(Z;a). So let E E FEm(A). By definition n = n(E) < No, and so k(33,- .Z,,)(Vg) V!-l E(g;3,). Now since the parameters of E me from A E 1 x1,and M is a model, there are a,, . . .,a,, E ]MIsuch that C(VQ) VTI1E(g;a,). Let CE@;a,). Thus p k E(3;a,) and so p t- E(3;a). The second clause follows h m the first clause by Lemma 1.1(8) and the fact that stp(8, lHl)k tP@i,1x1,.

-

EXERCISE 2.1: Is Lemma 2.16(2) still true when T is not stable?

EXERCISE 2.2: Show that ifA G B, p = tp(@,B) does not fork over A then p is stationary over A iff p I- stp(a, A ) iff p t- stp(a, B). EXERCISE 2.3: The E' we get in 2.4 is the coarsest equivalence relation over A refining E . EXERCISE 2.4: If I , , I , me (inhib)indiscernible sets b w d on B, and for some a, E I,, stp(a,, B ) = stp(ai,, B) then I,, I , are equivalent. EXERCISE 2.5: If stp(B, A) = stp(6, A), then there is an indiscernible set I based on A such that I u {a}, I u (6) me indiscernible over A. EXERCISE 2.6: If p does not fork over A, p E Sm(B),stp(a, A) stp(6, A), Ti, 6 E B then for every 'p, 'p(3; a) EP o 'p(Z; 6)~ p .

=

EXERCISE 2.7: If T is stable in A, IAI 5 A, then the number of nonequivalent types almost over A of the form stp(a, A) is 5 A. EXERCISE 2.8: Show that ifp is stationary over A but p u q forks over A then there is 'p E q such that p u {'p} forks over A.

III.8. The instability spectrum THEOREM 3.1: The following conditions on K , T , m are equivalent: (1) There is an bm?&ng sequence A,, i 5 K , and p E S ~ ( A , J such that for all i < ~ , 1p f o r k over A,. ( 2 ) Like (1) but now p 1 A, + q l i t s strongly over A,.

,

QLOBAL THEORY

102

[OH. 111,f 3

(3)A There exists an increasing sequence A,, i IK , a type q E Sm(A,), sets I f = {a!: j < A} E A , + indiscernible over A,, and tp,(B; a&)E q such

that j > 0 =r 7tpf(3; 7i;) E q, and there are m, < w m h that {tp,(z;a;): j < A} is m,-i&stent. (Soq 1 A , + splits strongly over A,for all i < K . ) (4)A fib ( 3 ) but ~ Where A, = INiI, H 1 . 1 being (IN,I1+-8&urated. DEFINITION 3.1: K"(T)= the first infinite K for which (1) above fails; K ( T )= sup,,, K"(T).The first regular K 2 K ( T )is denoted by K,(T),or m(T)(so it is K ( T )or K ( T ) + )For . unstable T we stipulate K"(T)= K,(T)

= 00.

Theorem 3.1. Trivially (4)A + (3)A and by definition (3)A 3 (2) (for any A). By Theorem 1.6(1), (2) * (1). So it remains to prove PVOOf Of

(1)

* (4)**

Since p / A , + , forks over A, we have p t A , + l I- v k < , , , & ? 8 ; @ j ) where each tpk(B; aL), k < n,, divides over A,. So by Lemma 1.1(3) there m {aL,[:1 < w} indiscernible over A,, a3, = a ,; and {tpk(B; i&): 1 E o} is mL-inconsistent. Without loss of generality we may w u m e A d = u f < d A,, S a limit ordinal (otherwise define A; = A,, for i a succesmr, and A; = u f < d A,; then A; G A d and p A d + l = p 1 forks Over A d , so it forks over A;, and thus the A;'s satisfy (1)). Now let B, = U k , l g , [ . Define by induction elementary mappings P, and models Mi such that Dom P d = A d , = u ( < d M,, Dom P,+ = A,+1 u B,, M, is ( A + IA,l)+-saturated, is 11M,11+-sa,turated, i < j + P, A, = f, A,, i < j * M i E M,,Range P,E lM,l and for any 7i u B,, tp(Pf+l(a),IM,I) does not fork over P,(A,). For i = 0 take Po as the identity on A,, and take any appropriate Mo. For i = 8 let F d = u , < d A,, M6 = u i < d Mi.BY (h'0lhy 1.5 we may define A,,, LJ B, + !Mil u Pf+l(A,+l u B,) with the desired property, and then extend ]Mil u EI,+l(A,+lu B,) to an 11Mi (1 -saturated model Mi + 1. This completes the definition. Now choose Q ~~"(IM,I)su~hth&tP,(1]) E q.ForeveVi,p 1 I- V k < q &(B;ai) 80 P,+,(P t CP;(Z; P,+,cm. E', t A , + , = A , + , 80 pc(p A t + l ) v k < n C 't+l(%))* so q v k < m dC@; Pf+l(8k))* q is complete so for each i < K there is k(i) < n, such that tpfccr,(l; F,+ l(%k(,,)) ~ q { ~. o , , I 1: < o} is indiscernible over A, so { y i + l ( z k ( f ) , [ ) : 1 < W} is indiscernibleoverP,+l(A,)= EI,(A,),andifSiEB, uAi+l,tp(F,+l(7i),1Mi1) does not fork over P,(A,). Hence by Corollary 2.11(1) {P,+l(a3,,i,,,): 1 < w} is indiscernible over ]Mil.Define 6; = F,+ l(?&,),t). Since M , + is IIM,((+-saturated, and IIM,(( > X we can define 6;E JM, for

r

+

vk<,,,

+

r

OH.

111, Q 31

THE INSTABILITY SPECJTRWM

103

A such that {6j: j < A} is indiscernible over IM,I. Now 6j): jE w } is mi,,,-inconsistent. Hence {j < A: tpis(f;6:) E q} is

w I;j <

{tpfc,(f;

finite. We may in fact aasume (by omitting if necessary) that only I f = {6$:j < A}. The conclusion holds with this If,&,), M,, q.

y&,)(f; 6;) E q, and j > 0 implies 7&,)(f; 6;) E q. Let

COROLLARY 3.2: For all p E S"(B) there is A that p does not fork Over A .

c B, IAI

< K"(T) such

Proof. By way of contradiction, aasume there is no such A . We shall define A,: i I; K"(T) such that A, c B and 1Ai1 < lil+ + H, for i < K"(T).Take A , = 8, Ad = U , < d A , , AKyT)= B. Assume A, is defined. By rtssumption p forks over A,. By Lemma 1.1(6) p has a finite subtype p' which forks over A,; let A, + = A , u Dom p'. So by Theorem 3.1(1) K"(T)< K"'(T), contradiction.

Proqf. If not, then by Theorem 3.1(1) there are an increasing sequence A,, i I; IT1 +,m < w , and p E B " ( A , ~ ,such + ) that for all i < IT1 p t A,+l forks over A,. Also, by Theorem 11, 1.2, for every A there is a(A) < !TI+such that P ( p , A , No) = P [ p &,),A, No].The number of possible finited is IT1 so a = supdrutea ( A ) < IT1 +.But p forks over A , and so by Lemma 1.2(2) there is A , such that P ( p ,A,, No) < P ( p 1A,, A,, H,), contradiction. +

r

LEMMA 3.4: If J iS an indi8CemGible set Over A , 6 any sequence, l(6) = m, t k n there iS I c J with I I I < K"( T )m h that J - I i8 indi8CemGible Over

Au6uUI. Proqf. By Corollary 3.2 there is B c A U UJ , IBI < K"(T),such that tp(6, A U UJ ) does not fork over B. Let I c J, 111 < K"(T),be such that B c A U U I . Clearly J - I is indiscernible over A u U I , and thus by Corollary 2.11(2), 1.6(1) J - I is indiscernible over A u 6 u 1.

u

Remurk. If T does not have the independence property (but is not T),see Definineceswrily stable) a similar conclusion holds, using qnp( tion 7.3.

104

@LOB&

THEOBY

[OH.

III,5 3

COROLLARY 3.6: If J i8 an indbcernibk 8d Over A , B i8 any S d , then there i8 I E J such that J - I i8 indiscernible over A V B V U I , and (1) IIl 5 K ( T ) + IBI. (2)If IBI < Cf(K(T))then III < K ( T ) (The . interuting m e is where I JI is hrge enough in r e W m to I BJ.)

Proof. By way of contradiction msume there are A , m such that IAI I; p < ISm(A)I.By Lemma I, 2.1 we can wsume m = 1. By Corollary 3.2 for every p € # ( A ) there is B, E A , lBpl < ~~(2') such that p does not fork over Bp.The number of possible Bp's is so there is a B for which 1{p E ~ ( A B,, ) : = B}I > p. By Corollary 3.3 K ( T )s ITI' and 80 IBI 5 ]TI.Let B E ]MI,11iK11 = ITI. Then p < I{~EB(A):Bp = B}I iI { ~ E B ( Ap) :does not fork over B}I s l{p E 8(1M(U A ) :p does not fork over B}I = I{p E 8(lM1): p does not fork over B}I (this by Lemma 2.16(2)) s ISr(lNl)l. But clearly IS(1MI)l s 2ITl, contradicting p 2 2ITl.And if T is stable in po and 11M11 = IT1 s popthen IS(lMI)I 5 po contradiofing po s p.

THEOREM 3.7: If p < p X d n then T is n d 8tabk in p.

Proof. Since p < p e M nthere is K < K(T)such that p" > p. Let K be the minimal one. Thus p e E = p. Now since K < K( T), for some m K < ."( 2') 80 by Theorem 3.1, (3), we have an inoreeeing sequence A,, i s K , a type p E Sm(An),sets I' = {q:j < p} indiscernible over A,,I' c A,, md cpi(Z; $) EZ) such that j > 0 =+ yv@; 4)EZ). Now for r ) E " p define an elementary mapping F,,suoh that (1) Dom F,,= U {$: j < p, i < Z(r))}, (2) 7 = p i Fnextends F,,, (3) if Z(q) = i then for every j:

(4&-<,>(a = &r(al), (B)q-<,>($) = FV(ad),

(C) for everg a # 0,j = &-(o>(G). The definition prom& by induction on Z(r)). For Z(7) = 0, F,,is the

CH.

111,s 31

THE INSTABILITY SPECTRUM

105

uy
empty mapping, and for Z(7) limit take P, = P,ly. If Z(q) = a + 1, P,ts(o) is any extension of P,ra with the right domain and P,rc-
P.

For 7 E "p, P,(p 1 Dom F,)is consistent since P, is elemenfary. Let P,(p 1 Dom P,)s q, ~8"m(B).We shall show that for 7 # p E "p, Qn # Qo.

Let a be the firat ordinal such that ~ [ a#] p [ a ] ; let q 1 a = p 1 a = v. Assume q (a 1) = u-(jl), p (a 1) = v-(j,), jl # j,. We may assume without loas of generality that j, # 0. If also j, # 0, then .Fp@yl) = Pv-(t,)(iZ~l) = li'v-(o)(a~l); P,(a;) = Pv-(jl)(iig)= Py-(o)(ay1).Thus Pp(ayl)= P,(G$), ~p.(iE; at) E p SO q a ( t ; F , ( a ~ )E)q,. T T a ( t ;BYl) €2) since j, # 0 SO i ' p a ( Z ; Pp(Zyl)) E q,,, thus 1vcr(E; E q p , and Q, # Qe. Similarly if j, = 0, Pp(a:)= F,(ay,) but tpa(t; Pp(iit))E qp and l T a @ ; P,(a?,))E ~ l r *Again qn f q p Thus IlIm(B)I2 I{q,: 7 E "p} = pn > p and T is not stable in p.

+

+

P,(m)

COROLLARY 3.8: (1) If po ie th first cardinal 2 IT1 in which T is etable, t h po 5 2ITl, T is stable in p iff p = p, + P < " ( ~ .I n fact po = sup(l8(A)1:A of cardinality 5 IT[}. ( 2 ) T ia awperstable iff K ( T )= No. ( 3 ) Eithr (i) K"(T)= K ( T for ) every m or (ii) for some m, m < m, iff K"(T)= K,( T ) if K"( T ) # K ( T ) .In t7re latter m e K,( T ) is singular and K ( T )= ( K ~ ( T ) ) + . Remark. See Exercise 3.3. Proof. (1) Follows from Lemma 3.6 and Theorem 3.7, and the remark to 3.6. (2) If K ( T )= Nothen p = p<'c(T) for all (infinite) p. So T is stable in p for p 2 2ITI,thus T is superstable. On the other hand if K ( T )> No then for every cardinal N,, cf(N,+,) = w so N>+a, > Ha+,. Thus K,="(,) > N,,, and T is not stable in Ha+, so T is not superstable. (3) Clearly K"(T)5 K"+'(T)(by extending the m-type i n Theorem 3.1 to an (m + 1)-type). For every regular K there is a strong limit cardinal A > 2ITI of cofinality K , so A'" = A, A" > A. If K < K ( Tthen )

106

GLOBAL THEORY

T is not stable in h by Theorem 3.7, and if K h by Lemma 3.6. SOfor regular K ,

K

[OH.

2 ~~(2') then

III,f)3

T is stable in

< K ( T iff ) K < K,(T),SO we finish.

DEFINITION 3.2: I is a maximal indiscernible set over A in M if I is indiscernible over A , I E ]MI,and there is no J 3 I, J E IMI,J indiscernible over A. DEFINITION 3.3: dim(I, A, M) = min{l JI: J is equivalent to I , and J is a maximal indiscernible set over A in M}.If A = 0 we just write dim(I, M).If for all J as above dim(I, A, M )= IJI, we say that the dimension is true.

Remark. This definition differs technically from Definition 11, 4.5, but no confusion will arise.

LEMMA 3.9: I f I is a d m a l indhcernible set over A in M ,then 111 K ( T )= dim(1, A , M ) ~ ( 5 "and ) ifdim(I, A , M ) 2 ~ ( 5 " )then ~ the dimension is true.

+

+

Proof.Let J be a maximal indiscernible set over A in M ,equivalent to I, and for which IJI = dim(I, A, M). Clearly IIl 2 IJI, so that IIl + K ( T )2 IJI + K ( T ) .Thus it is sufficient to prove that III + K ( T )I I JI + K ( T or ) in fact IIl I IJI + K ( T ) . By way of contradiction assume IIl > IJI + ~(2'). From Corollary 3.6 we have I, E I, !IlI 5 IJI + ~ ( 2 "such ) that I - I, is indiscernible over A u U I , u UJ. From the equivalence of I and J we can find an infinite I* such that (I- 11)u I*is indiscernible over A u UJ and J u I*is indiscernible over A (take I*= (8,: n < w } where the type 8, realizes over A u I U J u U {al:1 < n} is the average of I over the same set). Now with the aid of I*we shall show that (I - I,)u J is indiscernible over A. Let a,,. . .,8, be distinct sequences in (I - 11)u J , E E A , r p ( f , , . . .,3,; E) a formula. Let 6,, .. ., 6, be any distinct sequencesin I*. Define 8; aa follows: for 8, E J, 6, for 8, E I - I , .

u u

Now Crp[zi1, . . ., 8,; EJ o Crp[8;, . . .,8:; El since (I - I,)u I* is indis. . ., 6,; a] since cernible over A U UJ, and hp[8;, . . .,8:; El o C&, J u I*is indiscernible over A. But then the truth of rp(8,, . . .,8,; a) is

CH.

111, Q 31

THE INSTABILITY SPECTRUM

107

not dependent on the particular a,. Thus (I - I,) u J is indiscernible over A, in contradiction to J’s maximality in M. LEMMA 3.10: (1) Let M be an ( K , + K ( 2’))-saturatedmodel. Then M is h-saturated iff for every infinite indiscernible I c 1M1,dim(I, M ) 2 A. (2) We can replace the msumption “ (N, + K ( T))-saturated” by “every type which is almst Over some A c IMI,IA I < K ( T ) ,is realized in M ” (i.e.,by “F~tn-satu*ated”. Bee Definition%I V Y1.1 and IV, 2.1). Proof. (1) If M is h-saturated then clearly there is no indiscernible set of power < h which is maximal in M.So one direction is clear. Now assume M is not h-saturated, h > K, + K ( T), and let p E #“(A), A c IMI,IAI < A, be a type omitted in M. Extend p to p ~ P ( l M l ) and choose B c “1, IBI < K ( T )so , that q does not fork over B (by Corollary 3.2). By 1.12 and 2.13(2) since M is (N, + K(T))-saturated there is I c ldll which is an infinite indiscernible set such that p = Av(I, IMI).Let J z I be an indiscernible set maximal in M. If dim(I, M ) < A, we are through. Otherwise IJI 2 &(I, M ) 2 A. By Corollary 3.6 there is J’ c J , IJ‘I s lAl + K ( T )such that J - J’ is indiscernible over A ( K ( T< ) A, IAI < h so IJ’I < hand thus J - J ‘ # 0). But then since Av(J, A) = Av(I, A) = p 1 A = p we have that all the sequences in J - J’ realize p, contradiction. (2) Similar proof. THEOREM 3.11: If { d f ( } { < h iS an ~nCreaeing8Of modele, and K ( T ) Cf 6, then = ui<6 Mi i8 h - 8 d U T d e d .

h-8&U&ed

Proof. Clearly if h s cf(6) then M is h-saturated. So we may assume h > cf(S), and in partioular h > K(T).First assume cf S > No so we can use 3.10(1). Let I c IiKl be an infinite indiscernible set. Let B c lH), IBI < K(T),be such that Av(I, ]MI) does not fork over B. Since K( T ) s of S and cf 6 is regular, there isj < 6 such that B E 1M,1. By Corollary 2.13(2) there is J E lM,1 equivalent to I and since M,is h-saturated there is J’ z J , J ‘ c IM,I, indiscernible and IJ’I 2 A. Thus by Lemma 3.9 dim(I,M) = dim(J’,M) 2 h and by Lemma 3.10(1) M is h-saturated. If cf S = 8, we can use 3.10(2) instead of 3.10(1) (by 2.13(1)) and easily reprove a suitable version of 2.13(2) (by Exercise 2.4). THEOREM 3.12: If T is A-stable then T h a h-saturated nzodel of 7A.

108

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[OH. 111,8 4

Proof. We define an increasing sequence {&f,};s), such that IIM,II = A, M,,= U,< M,and every p E B( IM,I) is realized in M,+ It is easy to see that M,,is cf(b)-saturated. Thus if A is regular then MAis Asaturated.

If A is &gulax then A = x;<,,f(A) A,, cf(A) < A, < A,’for all i < j c cf(h). SO for all i < cf(A) MA = U j < A M j + A : ; cf(j + A,+) > A, SO M,+,:is A,-eaturated. Now if K ( T ) cf(A) then by Theorem 3.11 we have that MAis A,-saturated, for all i. Thus M Ais A-&urated. If ~(2’)> cf(A) then A A. So by Theorem 3.7 T is not stable in A, contradiction.

EXERCIBE 3.1: In Lemma 3.4 prove that we have 111 < ~ ~ ( 2 ” ) . EXERCISE 3.2: Give an example in which K ( T )is singular. EXERCISE 3.3: Prove ~ ~ ( 2= ’ )~(2’).[Hint: Suppose A, (i < K ) is increasing, tp(an6, A,+l) forks over A, for i < K . By 4.16 tp(iZ, A, +1) forks over A, or tp(6, A, + u a) forks over A, u a.3 EXERCIBE 3.4: Prove that in Lemma 3.4, when J is infbite, there is a minimal I = I , among the possible I’s, i.e., I , is contained in every possible I . EXERCISE 3.5: Prove the paxallel of 3.11 for A-aomp&ctnesa.

III.4. Further properties of forking THEOREM 4.1 : Let p E Bm(B), A E By then p doe0 not fork Over A iff for allJinite A , Rm(p,A , No)= Rm(pt A , A , KO). The analogous theorem for stationary types is: THEOREM 4.2: Let p E Bm(B),A E Bythen (1) p M etatimry Over A i+@for all Jinite A , P ( p , A , KO) = P ( p r A , A, KO)and Mlt(p, A, No) = 1. If p M etationary wer A and the rn-type q 2 p , then q doe% not fork Over A iff for all Jinite A and all A, 2 s A s KO,R”(p,A , A) = Rm(q,A , A). (2) p it3 e t a t i m r y over A iff p is etation4zry Over B and &oes not fork over A . ( 3 )In (1) we can restrict ourselves to all large enough A .

OH.

m, 8 41

FURTHER PROPERTIES OF FORKING

109

Proof of Theorem 4.1. The direction e follows from 1.2(2). Now w u m e p does not fork over A. Let 8 realizep. Let q = stp(8, A), r = stp(8, B). By 2.6(1) (and since q t - p t A, r t-p),for every finite d, R"(r, d, No) = Rm(pu r , d , No) = Rm(p,d,No); Rm(q,d,80)= W q U P t A , d , NO) = R"(p A , d, No).So it is sufficient to prove Lemma 4.3.

LEMMA 4.3: Let q = stp(8, A), p a type realized by 8 which does not fork Over A , and q E p . Then for everyfinite A , 2 5 A s No, (1) Rrn(q,d,4 = Rrn(p,d, 4. (2) Mlt(q, A, A) = Mt(p,4 A). Proof. First we prove (1) for A < KO. Let p be over B, where A G B. By 11, 1.1 Rm(q,d,A) 2 R m ( p , d A). , By 11, 2.13 and 11, 2.2 Rm(q,d, A) = n < w for some n. By 11, 2.9(2) there are cpf;j(E A ) , iZ$j for q E ">A, v E "
q,{)t: l < n, i i

< A, j < A, i # j;

= q[l] and t = 0 or j = q[l] and t = 1)

is consistent; let a, realize this set. Define 6"Jq E "A)':6 (q E ">A, i < A, j < A) by 2.6(3) such that (i) stp(b,, A) = stp(7i,,A), q E "A i < A, j < A. atp(6f;j,_A) = stp(7it;', A), q E ,A'" (ii) t p * ( u {b,,: q E "A} u U {&j: q E n > A , i < A, j < A}, B) does not fork over A, and extend the corresponding type of the 6;i,lat when 1 < n, By 2.9(1) 6,,(7E "A) realizes p . Clearly hpnll[6n; 9 E "A, i # j < A; and i = q[l],t = 0 or j = q[Z],t = 1. So by 11, 2.9(2) (now using the other direction) Rm(p,d,A) 2 n = Rm(q,d,A). So we prove the equality. The proof of (1) for A = KO,and of (2) is similar using suitable parts of 11, 2.9. Now we return to

Proof of Themem 4.2. (1) Suppose p is stationary over A. So it does not fork over A, hence by 4.1 for every finite d, Rm(p,A , No) = R"(p A , 4 No). Let M be a (I BI + ITI +)-saturatedmodel, B c "1. Let r E rSm( 1M1) p E T, T does not fork over A , and let 8 realize r . By 2.14 p k q =deI stp(si,A), hence by 4.3 Mlt(p,d, A) = Mlt(q,d, A) = Mlt(r,d, A) for

r

+

110

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[CH. 111, 9 4

every finite A, and 2 IA I8,. But by 11, 1.10(1) Mlt(r, A, A) = 1 so Mlt(p,A, A) = 1 and we finish. Now suppose p is not stationary over A. If it forks over A, then by 4.1 for some finite A, Rm(p,A,8,) # Rm(p 1 A , A, 8,). So assume p does not fork over A. By 2.9(1), if p = tp(8,, B), q = stp(8,, A) then p If q. So for some E E FEm(A)p u {-,E(Z; a,)} is consistent; let si, rertlizeit.By2.6(1)Rm[p u {E(Z;aI)},A,KO] = Rm(p,A,N,)forZ = 1, 2 (where A = {A')).But E(Z;8,) k -,E(Z; a,). Hence Mlt(p, A, KO) 2: 2. The second sentence is emy, so we have finished the proof of 4.2(1). (2) The proof is immediate by part (1). From 4.1 we see COROLLARY 4.4: Let A E B E C and p E Sm(C).p does not fork over A i$p does not fork over B and p 1 B does not fork over A . THEOREM 4.6: Letp be an m-type over A , q a A-m-type,A not necessarily finite, swh t?ut p U q (bconeistent and) f o r b over A . Then for A 1 8,: * ( p , A , A) < 00 * Rm(pu q, A , A) < Rm(p,A , A).

Remark. (1) This improves 1.2. Note that in 1.2(4) we have some unspecified A,, and in 1.2(3) we have a formula which divides. (2) In the c o m e of proving this theorem we shall show: (*)

If p forks over A, A c B, then there is an elementary mapping F, F 1 A = the identity and P ( p ) forks over B.

Proof. Let M be a p+-saturated model ( p = IAI + A + I TI)such that A c [MIand q is a type over [MI. Assume that a = Rm(p,A,A) = Rm(pu q, A , A) < 00 and we will get a contradiction. ( ) that Since A 2 No,by 11, 1.6 and 11, 1.1 there is q' E S T ( ~ M such Rm(pu q', A , A) = a. By the definition of rank there axe .c A such q'. Let them be {q,: j < j , < A} and let 8,realize p u q,. Choose A, c 1M1, IA,I 5 IT1 such that tp(a,, [MI) does not fork over A, and let B = U,
CR.

111, $41

FURTHEBPROPERTIESOFFORKINQ

111

contradictory, $ A 8 t VzvZ(Z;El,,). Since M is a model we can choose (by 2.6(3)) 6',E;,, E IMI, I < Z,, i < w, such that

U{El,,:

I < I,, i < w } , A ) = stp*(6' u (J {Ei,,: < ,z i < w } , A )

stp,(6 u

I?

and tp*(6' u U {Ei,,: I < I,, i < w}, B) does not fork over A. Thus also = ~ m [u p {$(z; a) A e(z; 6')}, d, A] $(z; a) A e(z; 6')k vl(z; E:,,), and {pz(5;Ei,,): i < w} is m,-contradictory, I < I,. But stp(E,,,, A ) = stp(E,,,,A), I < I,, i < o (by the indiscernibility of {Ez,,: i < w } over A ) so by 2.9(1) tp(Ei,,, B) = tp(E;,,, B), I < Z,, i < w . Hence $(if; a) A O(Z; 6') forks over B. By 11, 1.6 we may extend p 1 = pu { $ ( ~ ; aA) 8(Z; 6')}toatypepuq',q'ES~(IMl)suchthatRm(puq',d,A) = u (because Rm(pl,A,A) = a by Exercise 11, 1.1). Thus there is a j < j, such that O(2; 6') E q, (since 8(Z; 6') is a Aformula). Since @, realizes q, u p and tp(Z,, IM[)does not fork over B, the formula $(Z; a) A B(3; 6') does not fork over B. Contradiction.

vz

EXERCISE 4.1: Show that in 4.6 the condition A 2 omitted.

KO cannot be

Remark. See Exercise 4.18. Hint: Let Af = (B, R)be a finite model, R a two place relation, such that for any a, b E B there is an automorphism f of M ,f ( a ) = b. Let us define a model N:

IN1 = w x B x{O, l}, RN = {<<% a, 1)s (n, b, 1)): M t= R[a,bl} BN = {((n,a, 0 ) , (m, b, 1)): n = m or a # b} PN = {(n, a, 1): n < w , a E A } . Let T = Th(N); it is KO-stable; let A = 0, p = { P ( z ) } , q = ((0, a, 0)Sz: a E B}, d = (8, It}.

COROLLARY 4.6: Let p = stp(3, A), A 2 KO,Z(a) = m and d is not neceesarily finite. (1) There do not exist two contradictory A-m-types q,, qz such that R"(p,d, A) = R y p u q1, A , A) = R"(p u qa, A , A) < 00. (2) If Rm(p,A , A) < co then Mlta(p,A , A) = 1. (3) If Q E S ~ ( ~ M IRm(q,d, ), A) < 00 then Mlta(q,d, A) = 1 (see 2.16(2)).

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[CH.111, 8 4

LEMMA 4.7: Let q = stp(& A ) , l(a) = m , p be a A-m-type such that p u q does not fork over A (A not necessarily Jinite). Then for A 2 KO, R"(q, A, A) = a < 00 Rm(g U p , A , A) = a. Proof. Let B 2 A u ?j u Dom p. Choose r E Sm(B) extending p u q such that r does not fork over A. Then r k q u p t- q and thus R"(r,A, A) 4 R"(p U q, A, A) 4 Rm(q,A , A) = a. On the other hand by 11, 1.6 there is p' E S t ( B ) ,such that Rm(qup', A , A) = Rm(q,A , A). By 4.6 q u p' does not fork over A and thus by 2.9(1) p' c_ r hence p E p'. LEMMA 4.8: (1) If p does not fork over A then p is "de$nuble almost over A". This meuna thad for all 'p there is a formula t,hcp(Z; Ecp) which U almost E,,,); i.e.,~ ( za) ; E p * I=+&; Ecp] over A, swh thad p t 'p is de$nuble by t,hcp(z; and -,'p(E; a) ~p k + J a ; Eel. (2) If A c B E C,p E Sm(C),p 1 B is stationary over A andp does not fork over A, then p is deJintabk over B. Proof. (1) Immediate by 4.3 (for A = 2) and 11, 2.12(3). (2) As of (l),noticing that by 2.14, if ti realizes p , p t B k stp(zi, A). LEMMA 4.9: Letp ES"((M(),A$nite.Foreueytypeq €Srn(B) extending p , which does nd fork over IMI,q t A = Avd(l, B) for some in$nite A - i d i s m i b k set I E IMI. Proof. By Lemma 11,2.20 there is a finiteA, such that for all 'p(Z; 8) E A, a, and any infinite A,-indiscernible set J, I'p(J,a)] < Noor 17'p(J, ti)( < No. Using Theorem 11, 2.17 we can find a hi& A* such that if pk = tpd.(7ik,Ak),Ak = A u {al: I < k}, and for all 'p E A* and k < w 'p, ' p a 2, (where l @ k ) = m), then { a k : k < w) Rm(pk I'p,'p, 2) = *(PO is a 4,-indiscernible set over A, i f p k satisfies the condition from 11, 2.17. suoh that p 1'p is (+@,A)-dehable, for every Choose a finite A c 'p € A * ('pa from 11, 2.12) and define tik (k < w ) accordingly. As in 2.13 the result follows (by defining {a,:w s i < w + w } and Exercise 11,

u

2.9).

EXERCISE 4.2: Show that I does not depend on q but only on p and A.

UH.

111, 8 41

FURTHER PROPERTIES OF FORKING

113

COROLLARY 4.10: (1) The formula ~ ( 3a); does not fork over IN1 iff there is 6 E m h that C Q [ ~ ;a]. (2)* If there is 6 E A m h that k q [ 6 ; a] then ~ ( 3a);does not fork over A .

Proof. (1) Assume ~ ( 3a); does not fork over IMI. Then there is q E Sm(lN1 u a) which does not fork over JNl, ~ ( 3a);E q (by Theorem 1.4). Let p = q 1 1 Y1.By the previous lemma there is I c I Y I such that q 1Q = Av,(l, u a). Thus ~ ( 3a);is realized in N.This proves one direction, and the other follows from (2). (2) If ~ ( 3a); forks over A then ~ ( 3a); k V,<,,~ ~ (a,) 3; where ~ ~ (a,) 2 divides ; over A. But if ~ ( 3a);is realized in A then there is i such that ~ ~ (a,)3is;realized in A. By the definition of "divides" there are n, c w and sequences a { , j c w, such that tp(a{, A ) = tp(a,, A) and for every i c n, (~'(3;a{):j c w} is n,-contradictory. But the sequence 3; contradiction. in A which realizes Q,(P; a,) realizes also all the ~ ~ (a{), An alternative proof of Corollary 4.10. Choose q E S"'(liKl)such that q u { ~ ( 3a)} ; does not fork over 1Nl,and n(tp) c w such that for any indiscernible set I , lrp(I;a)l < n(9) or ) 7 ~ ( I , s i ) lc n(cp). Let I = {6,,: n c w} be an indiscernible set based on q, so ~ ( 3a); E Av(I; T i ) hence

LEMMA 4.11 : If p E ST(/ M I ) ( A is not necessarily finite) is definable Y l (see 4.8(1)) then p doef, not fork over A . almost over A where A G l

Proof. Without loss of generality Y is IAI +-saturated, since we may extend the given N to a IAI +-saturated model and then extend p to a complete A-m-type over the new model which is still definable almost

114

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[OH. 111, 8 4

over A. The implication about forking will then hold for p. Also we may assume that A is closed under conjunction and negation. By way of contradiction assume p forks over A. Thus by Lemma 1.1(6) we get a ~ ( 3Z);~p which forks over A, and by definition there is #@; ii) almost over A which defines p t tp. Without loss of generality ii E 1M1.So if Z' E lMl and stp(Z', A) = stp(Z, A ) then C#[Z; a] = #[Z'; ii] and thus v(3;a') E p . Now define sequences En E lMl,n < w such that tp(Zn,A u U,<,a,) does not fork over A, and each En realizes stp(Z, A). By Lemma'l.lO(1) {En: n < w } is an indiscernible set bawd on A. Clearly C#[Z,; ii]; thus ~ p @ ; an) E P ; thus for every n, 4n = V w s a n - l , l r u l - n A 1 . W ~ ( 3 a,);EZJ and [by l.l(9)] it forks over A. This contradicts 2.6 (because of 2.7).

EXERCISE 4.3: (1) Find an example of p E S,"(B),p is definable over A c B, but p forks over A. (2) In 4.11 let q be an m-type over A, p u q consistent; show that p u q does not fork over A. COROLLARY 4.12: Let p ES"(IMI),A c ] M I ,p does not fork over A iffpi8deJinablealmostoverAiffforall~, R m ( p , q No) , = Rm(p A , v , KO).

Proof. By 4.8, 4.11 and 4.1. THEOREM 4.13 (The symmetry theorem): If A c C , B c C, tp(ii, C) does not fork over A , tp(6, C ) does not fork over B then tp(ii, C u 6) forks over A iff tp(6, C u a) forks Over B.

Proof. Suppose not; so by the symmetry of the hypothesis on ii, 6; we can assume tp(ii, C u 6) forks over A, but tp(6, C u 7i) does not fork over B. Hence there is Z E C and a formula ~ ( 36,;Z) which forks over A , and Cv[ii,6, Z]. Now define by induction on n c w sequences a, 6, such that (i) stp(ii,,C) = stp(ii,C) (ii) stp(6,, C ) = stp(6, C) (iii) p , = tp(ii,, si u 6 u C u U {ai: i c n} u U,{6,:i c n})does not fork over A. (iv) qn = tp(En,7i u 6 u C u U {7ii: i 5 n} u U {6{:i c n})does not fork over B (this is possible by 2.6(3)). Now t+p[ii, 6,,,Z] [aa by 2.9(1) stp(6, A ) is stationary over A, we have assumed tp(6, C U ii) does not fork over B, and by (iv), (ii)] hence

OH. 111, 8 41

6,, 81 for k

FURTHER PROPERTIES OF FORKING

115

does not fork over B, hence, by 4.8(1) is definable almost over B; but by (i) stp(a,, C) = stp(7i,C)]. On the other hand if k > n, as q(E;6,E) forks over A, by (ii) also q(3; 6,, E) forks over A, but by (iii) p , does not fork over A , hence hp[Z,,

In [q,,

-lq(f; En, a) Ep,. so C-lCp[iz,, 6,, El. So bq&, 6,, El o k In, hence by 11, 2.13(5)

T is unstable. Contra-

diction.

THEOREM 4.14: If A c B, p = tp(Tt-6, B) does not fork over A , then q = tp(& B u 6) does not fork over A u 6. Procf. Suppose the hypothesis holds, but the conclusion fails. So for some 5 E B, !=tp[Z, 6, but v(Z;6, E) forks over A u 6. Now define by induction on n sequences En, a,, such that (i) stp(Zi,, A u 6) E stp(zi, A u 6). (ii) stp(En,A) = stp(E, A). (iii) pn = tp(a,,, A u 7i u 6 U E U U {tii: i < n} u U {Ei: i < n})doe8 not fork over A u 6. (iv) r , = tp(E,,, A U a u 6 U E u U {a,: i In} u U (5,: i < n})does not fork over A (this is possible by 2.6(3)). As E E B, tp(a-6, A U E) does not fork over A ; hence by the previous theorem 4.13 (and by 1.3), tp(E, A U @ u 6) does not fork over A, hence by (ii), (iv) and 2.9(1), tp(E,, A u 7i u 6) = tp(E, A u si u 6), hence

!=tpFi,6,znI*

But by (i) stp(zi,, A u 6) = stp(Z, A u 6), so by (iv), ~(a,,, 6; Z) = 6;3) E r,, when n Ik. Combining we get Cq,P,, 6, a,] when n s k. Suppose now n > k, by (ii), (iv) tp(EnE, A ) = tp(6-& A) hence q(3; 6, a,) forks over A , hence by (iii) -,q(Z; 6 , E J ~ p , ; hence b+Z, 6, E,]. Combining we get kq[a,,,6,4] iff n Ik, so by 11, 2.13(5) T is unstable, contradiction. q@,

LEMMA 4.15: (1) If tp(Z, B) does not fork over A E B, and tp(6, B u a) does not fork Over A u a, then tp(a-6, B ) does not fork over A . (2) If thejrst two types are stutimry over A , the third is also. ( 3 ) In ( 1 ) we can use an infinite a.

Proof.(1) Let {gi: i < i,} be a maximal sequence with the property that for all i # j, stp(&,,A u 7i) f stp(6,, A u a). By Lemma 2.2(2) i, is certainly s ~ ~ ~ I + Now I ~ I . by Lemma 2.6(3) there is an elementary mapping F with Dom F = A u 3 u U (6,:i < i,} such that F 1 A is

116

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[CH. 111, § 4

the identity, and for all E E Dom P,tp(P(E), B) does not fork over A, and stp(5, A ) = stp(P(Z),A). Thus, in particular, stp(8, A ) = stp(F(Si),A) and tp(7i, B),tp(P(a), B) ES~(')(B) do not fork over A. But by Corollary 2.9(1) stp(G, A) is stationary over A , and thus tp(a, B) = tp(F(Ti),B). Thus we can find an elementary mapping G with Dom G = B u Range F such that G B is the identity and C(P(7i))= 3. Now let bf = G(P(gi)).By the maximality of the 6, there is no 6' such that stp(6', A u a) $ stp(67, A u a) for all i < i,. Thus there is an i < i,, such that stp(6, A u a) = stp(67, A u a) (6 is from the statement of the lemma). Now tp(67X, B) does not fork over A since an@' E G(Range P). Thus tp(6f, B u T i ) does not fork over A u a, by Theorem 4.14. But we have also assumed that tp(6, B u 7i) does not fork over A u a, so by Corollary 2.9(2) tp(67, B u a) = tp(b, B u a). Thus tp(a-6, B) = tp(7in6:, B), and tp(Si-6, B) does not fork over A. (2) Immediate by 2.14. (3) Same proof as (1).

EXERCISE 4.4: Prove that if p # q E Sm( B ) p A = q 1 A = r and Rm(p,L, A) = Rm(q,L, A) = Rm(r,L, A) < co,then there is E E FEm(A) such that p(Z) U q(#) k ,E(Z; 5). EXERCISE 4.5: Prove that if p €Sm(B), A c B and R m ( p1 A, L, A) = Rm(p,L, A) < 00 then p is definable almost over A. EXERCISE 4.6: If in addition to the assumptions of Exercise 4.5 Mlt(p A , L, A) = 1, then p is definable over A. EXERCISE 4.7; !Cry to generalize Exercise 4.4 and Exercise 4.5 to B). other infinite A's, where Sy(B)would replace Sm( LEMMA 4.16: Let p E Sm( IM I), w k r e A c IM) and M is IA1 -saturated. (1) p f w k g over A iff p strongly splits over A. (2) [ p A is stationary over A a n d p does not fork over A ] iffpdoes not split over A . +

Proof. ( I ) One direction is just Theorem 1.6(1), and the other direction is left as an exercise. (2) One direction is Lemma 1.9, and the other direction is left as an exercise.

OH.

111, 8 41

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LEMMA 4.17: Let I,J be inJinite indiscernible sets. (1) If Av(I, U I ) = Av(J , U I)and Av(I, UJ) = Av(J, UJ)t h I and J are equivalent. (2) If for all distinct a,, . . . ,a,, E I and distinct 6,, . . ., 6,,E J tp(a,^ - - .^a,,) = tp(6,^. '^En), then Av(I, U I ) = Av(J, U I) =- I,J are equivalent. (3) AV(I, U I)is St&WMW'Y Over U I . (4) Av(I, U I,) is stationary Over U I , for some I , E I , 1111 < min{N,, K(T)}, and Av(I, U I ) does not fork over U I , .

Proof. Let M be a (IIl+ + IJI+)-saturated model such that U I u UJ E [MI. Let p = Av(I, [ M I ) , q = Av(J, [ M I ) . Clearly p and q do not split over U I and UJ respectively. So by Lemma 4.16, p U I is stationary over U I and p does not fork over U I; a similar claim holds for q. This proves ( 3 ) . Now from Av(I, U I) = Av(J, U I)we get that p U I = q t U I and thus (by Theorem 4.2(1)) for all finite A, Rm(p,A,2) = Rm(p1 U I , A , 2) = Rm(q1 U I , A , 2) 2 Rm(q,A , 2 ) . If we assume that tp(a,^- - -^a,,) = tp(6,". - .^En) as in (2), equality holds; thus p = q so (2) holds. Also in case (l),by the symmetry in the assumptions, again equality holds, thus (1) holds. Let us prove (4). If I, s I, 11,1 = No then by (3) Av(I,, U I,) = Av(I, U I,) stationary over U I,, and as Av(I, 1M1) does not fork over U I, (as it does not split over U I,)Av(I, U I)does not fork over U I,. If KO < K(T),we finish. If K ( T )= No, for some finite I, E I, Av(I, U I)does not fork over U I,. Choose a E I - I,; let I, = I, u {a}. As Av(I, U I,)k stp(8, U I,) we finish. 4.1 : An m-type p is stationary if for every A, Dom p DEFINITION there is a (unique) type q €&"(A)such that (i) P c q. (ii) For every finite A Rm(p,A , 2) = P ( q ,A , 2 ) .

E

A,

Remark. The uniqueness follows by 11, 1.4(1). DEFINITION4.2: (1) The stationary m-types p , , pa are called parallel if for every A , Dom p1 u Dom pa E A, the unique types q,, q2 from the previous definition are equal. (2) We call p a stationarization of q if q is stationary p parallel to q OT q is complete over some A, and for some E realizing q, p is parallel to stp(E, A). A stationarization of I, over A is any stationarization q E Sm(A)of p.

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[CH.111, 5 4

Remark. I n (2), in any case p is stationary, and the two cams are wnsistent .

LEMMA 4.18: (1) Let p sSm(B),t h n p is stationary over B iiff p is 8kl4%Onary. (2)* I n Dejinition 4.2(1) it suflces to say that this ?wLoldsf o r some A. (3)* Parallelism is an equivalence relation. (4)* If p , q are stationary and f o r every jinite A , R m ( p ,A , 2) = R m ( pu q, A , 2) = Rm(q,A , 2) then p , q are parallel. (6)* If p is a stationary m-type, Domp E A , there is a unkpe

q E &'(A) parallel to p . Proof. ( 1 ) By 4.2. (2) Immediate. (3) Immediate, by the definition and (2). (4) Immediate. (6) Immediate by the definitions.

DEFINITION 4.3: An (infinite)indiscernible set I is based on a s t a t k r y

type p if Av(I, U I ) is parallel to p.

DEFINITION 4.4: (1) A set I of sequences is independent over (A, B ) where A E B if for every B E I, tp(B, B u U ( I - a)) does not fork over A. When A = B we write A instead of (A, A). (2) We say I is independent over p, if p E Sm(A),I is independent over A, and each a E I realizes p . LEMMA 4.19: (1) If I is independent over p , p E Sm(A),p is stationary, then I is an indiscernible set over A based on p . (2) If I is an indiscernible set over A , based, on p €Sm(A),then I is independent over A and over p . If C E A , p does not fork over C , then I is independent over (C, A). ( 3 ) If I is independent over ( A ,B), J independent over ( A ,B u U I ) then I u J is independent over ( A ,B). (4) If I is independent over ( A ,B), A E C E B, E B, J E I , then I - J is independent over (C, B, u UJ ) . (6)If I = {ai: i < a}, tp(B,, B U U {B,: j < i})do@ not fork over A E B then I is independent over (A,B).

CH. I I I , § 41

FURTHER PROPERTIES OF FORKINQ

119

Proof. (1) By l.lO(1). (2) E a y by 1.7(2). (3), (6) Immediate, by 4.13,4.14 and 4.16. (4)Immediate. DEFINITION 4.6: Let A c B, J a set of sequences of length m. (1) If p ES"(A), dim(p, B, J) is min{)Il: I c J, I a maximal independent set over (A, B) of sequences realizing p}. (2) If p is a stationary type, and there is q ES*(B)parallel to it, dim(p, B, J) = min(lI1: I a maximal set indiscernible over B, of sequences in J realizing q}. See Lemma 4.20. (3)The dimensions in (l), (2) are called true if for every such maximal I, IIl is equal to the dimension. (4)If J = "A, we write A instead of J. If B = Domp, we omit it. LEMMA 4.20: (1) Definitions 4.5(1)and 4.5(2)are Consistent, i.e., when both apply the resulting dimemione are equal. (2) If I is based on p ( p stationary, I an (infinite) indiscernible set) then dim(I, A , M ) = dim(p, A , M).

Proof. Immediate (use 4.19,see Definition 3.3). THEOREM 4.21: (1) If I is idpendent over ( A , B), K 2 K ( T ) , cfK > ICl, then for some J c I , IJI < K , I - J is independent over ( A , B u C u J ) . Moreover there is a minimal such J . (2) If dim(p, A , J)is 2 K ( Tthen ) it is true; and if the dimemion is not true, for every suitable maximal J (from Definition 4.5) IJI < K,(T).

Proof. Similar to 3.6,3.9,Exercise 3.4. EXERCISE 4.8: Show that for some unstable T , there is p ~S"(141), A c ICl, such that p does not split over A but is not definable over A. EXERCISE 4.9: Show no stable T satisfies Exercise 4.8. EXERCISE 4.10: Let A c B, 7i algebraic over A. Then tp(8, B) does not fork over A (see Definition 6.1). CLAIM 4.22: (1)Let A c B E C be such that for all ~ E there C is ZEC such that tp(Z, B ) does not fork over A and tp(Z, A ) = tp(6, A). If ii is any sequence such that tp(8, A ) is stationary and tp(8, C ) does not fork over A then: tp(8, B ) k tp(8, C) iff tp(7i, A) t- tp(7i, C).

120

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( 2 ) We can waive tire statiOnarity of tp(8, A), for every E E C ; and then:

if tp(E, A) is stationary

stp(i3, B) I- tp@,C) if tp(& A) F tp(8, C).

Proof. ( 1 ) -= is immediate. =- Let q ( Z ; 6) E tp(& C), 6 E C; we shall show that tp(8, A) I- q ( Z ; 6). Let a be aa above. By Lemma 1.9 tp(8,C) does not split over A, and q(Z;6) E tp(8, C), so q(3;a) E tp(8, C). Since tp(8, B) t- tp(8, C) we have tp(8, B) k q ( Z ; a). Thus there is a E B and #(Z; a) such that k#[8; 83 and I-#(Z;a)+q(Z;5). Clearly tp(8, A u z) does not fork over A so (by 4.13) tp(& A u a) does not fork over A. So as before, we can define {a,,: n < w } to be an indiscernible set over A u Z based on A such that stp(a,,, A) = stp(a, A). By Lemma 2.6 there is n for which #n =

V

wS2n-1

Iwl = n

few

4)

is almost over A. Of course, #(Z; a,,) I- q ( Z ; a) for all n and so #,, I- q ( f ;a). So by Lemma 2.14, since #,, is dmost over A, tp(8, A) I-#,, (or a,,) be a stationmization of tp(& A) I- -I#~, let q E Sm(Cu Un<@ tp(8, C), so #(Z,a,,) E q hence #,, E q, hence tp(8, A) u {#,,} is consistent; contradiction). Thus tp(8, A ) I- q ( Z ; a); but tp@,A) = tp(6, A) 80 @(a, A ) I- q ( E ; 6). (2) Similar proof.

EXERCISE 4.11: Suppose A is finite, ~ E S ~ (and B )rl, i < A, are A-m-types such that the types p u r, are rn,-contradictory, rn, < w , i < A, and IBI + IT1 < A. (1) Show that for some i, p u ri forks over B. ( 2 ) Show that if 1231 + I TIK< A we can relax the finiteness condition on d to [dl < K . In fact, A 2 No,A > 2141 suffioe. QUESTION 4.12: Generalize the properties of the rank D such aa symmetry, monotonicity, etc.) to the c&88 where T is unstable but D ( p , L, a)< a for every type p. THEOREM 4.23:Suppose A > Nois regular and (Vp < A)(pCKcT) < A). If IIl L A > IAI then there are J , I , c I , IJI = A, II,l < K(T),such that J i s independent over A U U I , (hence if stp(8, A u U I,), 8 E J , is $xed, J i8 indbcernible over A u I,,).

u

Remark. We give this theorem partly to show an alternative proof of Theorem I, 2.8: the existence of large indiscernible sets.

CH. 111, Q 41

FURTHER PROPERTIES OF FORKING

121

Proof.Let a,, i < A, be distinct members of I , and let A , = A u U,<,a,. For each i let B, c A,, 1B,1 < K ( T )be such that tp(a,, A,) does not fork over B, and let f(i) = min{j: B, c A,} so cf 6 2 K ( T )=- f(i) < i. As (Vp c A)(p K,(T)hence by 1.3(3) of the Appendix, So = (6 < A: cf 6 2 ~ ( 2 ’ ) is ) stationary. Hence by 1.3(1) of the Appendix, there is a stationary set S, E So on which f is constant, i(0) say. As by 1.3(1) of the Appendix there are a stationary set S, c S, and B c A,,,, such that i ES, 5 B, = B. Let I , c I be minimal such that B c A u U I , and let J = {a,: i E S,}. Clearly the conclusion holds.

EXERCISE 4.13: Given an increasing sequence of sets A, (a < <), for every [ we can find elementary mappings F,,q E c>[, such that: (i) q + v implies F, extends F,. (ii) Dom P, = Al(,). (iii) tp,(Range F,,U {Range F,:Y E [>t, not q v}) does not fork over Ua< I(,) Range Fnlw If tp,(A,, U,<, A,) is stationary, this construction is unique and for any 8, E A,+,,we call {FJa,):q E ,+l<, i < [}an indiscernible tree, as it satisfies Definition VII, 2.4, and VII, 3.1.

+

EXERCISE 4.14: Suppose K < K ( T and ) q~,, a < K , are as in Theorem 3.1. Find sequences a,,, q E ” A such that: (i) {a,: q E “‘A) is an indiscernible tree. *TI) where q E “A, v E A. (ii) hp,[E,; ?iv]ii(v (iii) {rp,(Z; a,,-( ,)): i < A} is m,-contradictoryfor v E ,A, some m, < w. +

EXERCISE 4.15: Show that if T is stable then we can take n Definition 1.4 of forking. Hence a formula divides iff it forks.

= 1

in

EXERCISE 4.16: Show that in 4.23, if we weaken the conclusion “lIol < x(T)” to “(Iol< A ” we can weaken the hypothesis on A to “ A > K,( T) is regular”. EXERCISE 4.17: (1) Suppose M is K-compact, p an m-type, 1p1 < K . Then p does not fork over [MI iff p is finitely satisfiable in M iff p is realized in M. (2) Suppose M is K-SatUrated, p €Srn(A),K,(T)+ [ A [ +I K . Then p does not fork over [HIiff p is finitely satisfiable in M iff p is realized in M.

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[ax.III,§ 6

EXERCIBE 4.18: Show that in 4.5 we can omit A 2 No if we add A = clz(A) or p is stationary over A. [Hint: We can aasume that q is finite, so in either case that q = { q ( Z ; a)}, so we can msume p is finite. Let M be a ( A + IT1 + IAl)+-saturated model, A u si s 1611.Let for (a < A) be elementary maps, fa 1 A = the id., fo 1 1M( = the id., tp*(f,(lMI), U B , , f s ( l M l ) )does not fork over A. Then note: (A) q, = { ‘ p ( f ; f , ( s i ) ) u ( 4 = j ) : a < A} are pairwise explicitly contradictory, and of equal rank, hence Rm(p,A, A) > Rm(pu qo, A , A). . (B) N o element of 1M1 realizes ‘p(Z;f,(si)) A A p (a > 0) (by 4.10, Exercise 4.19), hence R m ( pu q, A , A) = R m ( pu qo, A , A) (by Exercise 4.20).]

EXERCISE 4.19: Suppose A E B, C,tp,(C, B) does not fork over A, and q is a type over B forking over A. Then q forks over C. [Hint: Otherwise let tp(& B u C) extend q and not fork over C. By the Symmetry Theorem 4.13 tp*(B, C u a) does not fork over C. But (by symmetry) also tp*(B, C) does not fork over A , hence by transitivity (4.4) t p , ( B , C ~ d ) does not fork over A, so by monotonicity tp*(B, A u 2) does not fork over A. So by symmetry, tp(& B) does not fork over A, but q c tp(& B) and q forks over A, contradiction.] EXERCISE 4.20: Suppose p is an m-type over l M ) , p 2 1131’ + A, M is p-saturated, p ~ { q ( Z ; s i ) }is not realized in ill; and pl = p u {-,c@; a)}. Then P ( p , A, A) = P ( p l ,A, A). [Hint: Trivially P ( p , A , A) 2 R m ( p l , A A), , so we prove by induction on a that R”’(p,A,A) 2 a =- R m ( p l , A A) , 2 a. The point is that the types “witnessing” for the rank can be chosen over !MI and if r is an m-type Irl < p, then p u r is consistent iff p1 u r is consistent. Use over lM(, Exercise 4.17(1).] In fact, M p+-compact is sufficient.

I

111.5. The first stability cardinal

From now on, until after Lemma 6.11 in this section let B c IMI, p €Sm(B), and B = {q ~S~(lill1): p c q and q does not fork over B}. By Lemma 2.16(2) every q E B is stationary over Byand by Corollary 2.9(1) for every si realizing p there is a unique q,, E B such that q,, k stp(si, B) so if si realizes q E B then q,, = q.

CH. 111, 8 61

THE FIRST STABILITY CARDINAL

123

LEMMA 5.1 : Choose a jixed a* which reulim p . Then there are A* and Er(3;#) E FE”(B)for i < A* m h that: (1) p(Z) u {Er(Z;a*): i < A*} is stationary over B . ( 2 ) For all j < A*, p(Z) U {Ef(Z;a*): i < j}U {-$?(Z; a*)} i s ~ 0 1 ~ eistent.

Prooj. By Corollary 2.9(1) p(Z) u {E(Z;a*):E E FE*(B)} is stationary over B. So there is a set Qil of minimal power A* such that pgl = p ( 3 )U {E(Z;B*):EE @,}isstationaryover B.Let Qi, = {I#,@;@: i < A*}. Let U be the set of all j < A* such that p ( Z ) u {E,(Z;a*): i < j}u {-,E,(Z; a*)} is consistent, and let Qi = {E,(Z;j j ) : i E u). We claim that p z I- pgl. We shall prove this by proving by induction on i that p$ F E,(Z;a*). Assume this is true for i < j.Now if j E U the result is immediate. If j4 U then by the induction hypothesis p: I-p(Z)u {E,(Z;a*): i < j} and aa j 4 U p(Z) u {E@; a*): i < j} IE,(Z; a*), 80 pg F E,(Z; a*). This proves the claim, and in particular p z is stationary over B. Thus lQil = A*. Sincep(3) u {E,(Z;a*): i < j, i E U}U {-,E,(Z; a*)} is consistent for each j E U,by changing the indices we get exactly (2). LEMMA 6.2: (1) For all a,, i < A*, if p‘ = p ( Z ) u {Er(Z;a,): i < A*} is &tent (A* and E: are from the previ0u-s lemma), then p’ W 8tatimry over B . (2) For all ai, i < A*, if p’ = p ( ~u) { E ~ ( a,): z ; i < j } is coneistent then pf u {-,I#?@; a,)} is codstent.

Proof. (1) Let a’ realize p‘ and set r = p(Z) u {Ef(Z;a’): i < A*}. Now by Lemma 5.1(1) p(Z) u {Ef(Z;a*) :i < A*} is stationary over B, and a’ realizes the same type over B ~ J JZ*(p); so r is also stationary over B . It is not hard to see that p‘ I- r I- p’; thus p’ is also stationary over B. ( 2 ) Now let a‘ realizep’ and take r = p(Z) u {Er(Z;a’): i < j}.Since Z* and a’ realize the same type over B, r u {,E,*(Z; a’)} is consistent (by Lemma 5.1(2)), and of course r u {E?(Z;a’)} is consistent. As above p’ I- r I- p j , so that p’ u {E?(Z;a’)}and$ u {-,E;(Z; a‘)}are consistent. If t=E?[aj;a‘] then p’ u { 4 ’ ? ( 2 ;a’)} I-pj u {-,E;(Z; a,)} and if C-,E,*[aj;a‘] then p j u {E,*(Z;a’)} I- p’ u {,E;(Z; a,)}. In any caae (2) is proved. Remark. This shows that A* is in fact not dependent on the choice of a* as assumed in Lemma 5.1. Clearly it does not depend on M either, 80

124

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[a. III,§6

DEFINITION 6.1: A*(p) is the A* defined in 6.1.

Proof. (1) For all q E U ~ < ~define * ~a,,+ by~ induction ~ on I(q)

such that p , = p u {El* (z; - a,lcr+l)): i < I(q)} will be consistent and C7E:[+<,,; a,,-<1>]for i = Z(7). Assuming p , is defined for I(q) = i , choose such that p , u {E:(Z; 7i,-
DEFINITION 6.2: Let Eo EFE"(B), @ E FE"(B). ECN(Eo,@) is the maximal n such that for some E realizing p , r(@, E, n ) = UfSl p(Zf) u {E(zf;a): E E 0, 1 s i 5 n} u {-,EO(Zf; z,): 1 5 i < j 5 n} is consistent. (ECN stands for class equivalence number.)

Remark. Clearly ECN(Eo,@) I n(Eo). LEMMA 6.4*: (1) In the above &$nit& ''80me 15'' may be replaced by any E". (2) If Q1E @a then ECN(Eo,@a) 5 ECN(Eo, Q1). 4'

Proof. (1) is true because p is a complete type over B so for every E, 8' realizing p there is an elementary mapping P such that F 1 B is the identity and P(E) = 8'. (2) is immediate.

DEFINITION 6.3: Let {EO}, @, 'YE FE"(B). Then Eo depends on @ (mod Y)if ECN(Eo,'Y) > ECN(Eo,@ u Y).

CH.

111, 5 61

THE FIRST STABILITY CARDINAL

125

DEFINITION 6.4: Let l7,@, Y E FEm(B). l7 is

independent over Qi (mod Y )iffor all E E 27, E does not depend on @ u (27 - {E))(mod Y ) .

LEMMA 5.6*: (1) For every 0,E there is a flnite Y E Qi such tluct ECN(E, @) = ECN(E, Y). (2) If E depend%on @ (mod Y) then for someflnite l7 E @, E depends on 27 (mod Y ) . Proof. (1) Let n = ECN(E, @). Then F(@,8, n + 1) is inconsistent. Let F' be a finite inconsistent subset and let Y be the set of E E FEm(B ) appearing in I", I" E F(Y,8, n + 1 ) so ECN(E, Y ) I n. But by Lemma 6.4(2) ECN(E, u') 2 n, so we have equality. (2) Immediate by part ( 1).

LEIKMA 6.6*: If El depends on @ u {Ea} (mod Y ) ,but El doe% not depend on @ (mod Y ) ,then Ea depencls on Qi u { E l } (mod Y) and even on { E l } (mod Y u @). Proof. Let a realize p and denote pg = p u {E(Z;a): E E l7}. Let a:, . . ., ail (a!, . . . , a:,) be representatives of the equivalence classes of

El ( E a )which contain sequences realizing p. Let S f = { k : 1 I k Inl, p& u { E l @ ;a;)} is consistent}. Since El depends on @ u {Ea} (mod Y ) but not on @ (mod Y ) there is k E S1 such that p&y u {Ea(Z;a), El(%;ah)} is inconsistent. Let a* realize u { E ~ ( za;)}. ; Thus for E E @ u Y , I. E[a*;a] A E1[a*;ail. Also for some m 5 na k Ea[a;a%]so pgUyu {Ea(Z;a;)} is consistent (it is realized by 3).But p&,u~Bi) U {Ea(Z;a:)} is inconsistent as every sequence which realizes it also realizes p&, u {Ea(Z;a), El(%;a;)}. Thus ECN(Ea,Qi u Y U { E l } ) < ECN(Ea,Qi U Y ) IECN(Ea, Y ) , and the conclusion follows. LEMMA 6.7*: If for all i < a E' does not depend on Qi u {Ej: j < i) (mod Y ) ,then { E L i: < a} is independent over Qi (mod Y ) . Proof. By Lemma 5.6 it suffices to prove this for finite a. For a = 0 or a = 1 it is trivial. Now suppose the claim is true for a but fails for a 1. For some i < a E* depends on l7 = @ u { E f :j < a 1 , j # i}

+

+

(mod Y ) ,but ELdoes not depend on l7 - {E"} (mod Y ) .So by Lemma 6.6, with E' for E l , E" for Ea, l7 - {E"} for @, and Y for Y , we get a contradiction.

126

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[CH. 111, f 6

LEMMA 6.8*: Let l7 be independent over @ (mod Y). Then: (1) for every E there is a Jinite II* c l7u @ a c h that l7 - 17* is independent over @ u {E}(mod l7* u Y). (2) For every Q1 there is l7, E nu@, < 1a11++ No, such that n - n1is independent over @ U (mod U n1). Proof. (1) By Lemma 6.6(1) there is a finite l7* G l7 u @ such that ECN(E, Y U @ u n) = ECN(E, Y u n*). Clearly by 6.4(2) l7 - l7* is independent over @ (mod Y u n*). Suppose that - 17* is not independent over @ u {-E> (mod Y u n*). Then for some E* E l7 - l7*, E* depends on @ u { E } u (l7 - 17* - (E*}) (mod Y u n*). But E* does not depend on @ u (l7 - l7* - {E*}) (mod Y u l7*) (by 5.4(2): the monotonicity properties of dependency). So by Lemma 6.6 E depends on @ u (I7- I!*) (mod Y u n*), hence ECN(E, Y u l7*) > ECN(Z, 17* U Y U @ u (l7 - l7*)),contradiction to the definition of l7*. Now the claim follows by Lemma 6.4(2). (2) Similar.

LEMMA 6.9*: If No < A I A* (see Lemma 5.1)and A is regular, then there isaJiniteYc{E::i< A}andSsA,lSl =A,suchthatl7={E::i~S} is independent over the empty set (mod Y) and ECN(Er, Y u (l7 - {E:})) 2 Bfori€S.

Proof.Let l7, = {E:: i < j};by Lemma 6.1 ECN(E,*,nj)2 2. If 6 is a limit ordinal less than A then by Lemma 6.6(1) there is f ( 6 ) < 6 such that ECN(E,*,n d ) = ECN(E,*,nf(d)). By the regularity of A and 1.3( 1)of the Appendix there is i, < A such that So = (6 < A : f (6) = i,} is of cardinality A. For each 6 E Sothere is afinite ITdE l7,,such that ECN(E,*,n d ) = ECN(E,*,n,,)= ECN(E,*,nd). As I17i,l < A, IS,l = A, A is regular, and l7*Jhas c A finite subsets, for some finite l7,S = (6: f (6) = i,, n6= l7} is of cardinality A. The result now follows by 6.7. LEMMA 6.10: Let A, I A* be singular. Then there is Y G { E r : i c A,}, IYI = cf A,, and S E A,, IS1 = A,, such that {E:: i ES}is independent over the empty set (mod Y )and ECN(E:, Y)2 2 fbr i ES. Proof. Let A, = ~ o < c r < c f h Aa o where Aa is regular and for 0 < u < < cf A,, AB > A, > cf Ao. Let l7, be defined aa in the previous lemma.

CH. 111, 6 61

THE FfR8T STABILITY CARDINAL

127

Now define by induction finite Y, E nAa+, and S, c lSal = such that {Ef : i ELI,} is independent over UB
V;, then (by 6.8(2)). Let V , = fin<, {El:i E V,, a < cf A,} is independent over

0 mod

[u

Y,u

IS,

- Val

I cf A,

and

U { E , :~ E S , - V,, a < cf A,}].

a
LEMMA 6.1 1* :Lei l7 be independent Over @ (mod Y)and ECN(E, Y)2 2 for E E l7.Then for every nl E l7, Fnt = p(P) U p(@ U {E(P;8): E E @ U Y U nl} U { T E ( E ;8):E E l7 - nl} is cm&tent.

Proof. By the compactness theorem it suffices to prove for finite l7. We - nll = 0 there is shall prove it by induction on Il7 - nll. If nothing to prove. So assume it is proven for n, and let Il7 - rill = n + 1 and E* E l7 - nl. Thus Fn,u{E., is consistent; so let Z Ha, ij H 6 realize it: so CE*[8;61. Now we know that

ECN(E*, Y U @ U (l7 - {I#*))) = ECN(E*, Y ) 2 2 .

r

s o = p ( z ) u {E(z;6): E ( Z ;8) E Y u 0 u (n- {E*))}u { 7 ~ * ( 6~)); is consistent. Let E H E realize it. Then P H 8, jj i+ Z realize Fnt. QUESTION 5.1: There is a countable C E IMI such that for q # r E 9' q 1 C # r 1 C (assume M is IBI +-saturated).See Lemma 6.3.

Now we abandon our fixed ByM yp .

DEFINITION 5.6: D,(T)

= LIm(0),D ( T ) =

U, D,(T).

LEMMA 5.12*: There is a countable A such that 1Sm(A)I2 ID(T)I for 8 m e m < w.

GLOBAL THEORY

128

[CH. 111,8 6

Remurk. (1) We can take m = 1 unless I (D(T)I is singular of cofinality > w . (2) If cflD(T)I > KO we can choose an empty A.

Proof. Let p = ID(T)I. For all m < w and A 5 IDrn(T)I,A regular, there is a set AA,, of power < m such that lS1(AA,,,)1 2 A (this by Lemma I, 2.1). Case 0. If p

5

KO there is nothing to prove.

Case 1. If p > No is regulax then p = IU,,,D,(T)I 5 2 ID,,,(T)I, so there exists m for which p = ID,(T)I and thus lS1(Afi,,,)12 p and lAfi,rnl < m < No* Case 2. If p > Nois singular of cofinality > No,then again there is m for which IDrn(T)I= p and so ISm(0)l2 p. Case 3. If p > No is singular of cofinality No, let p = Zn<@pn and A = Urn,, Afi+,m.Then IAI s K O and IS(A)I 2 ID(T)I. LEMMA 6.13: Bwppme IS(A)I 2 A > IAl
Proof. For every p € # ( A ) there is B, c A, [BPI< K ( T ) such , that p does not fork over B,. IS(A)I 2 A > IAI<"(" so for some Bo l{p E S ( A ) :B, = Bo}I 2 A. Now if IS(BO)l2 A then (1) holds. Otherwise IS(Bo)l < A. Thus for some p o , 9' = { p ES(A): p 1 Bo = pot B, = Bo) is of cardinality > A . If A*(po) 5 KO (from 6.1) then by Lemma 6.3, 2w0 2 2A*(P0)2 191 ' 2 A. Thus (2) holds. 2 If A*(po)> No, by Lemmas 6.9, 6.10, 6.11, ISa(Bo)l 2 2A*(po) I{p E S ( A ) p: Bo = p o , B, = Bo}I 2 A. So for some B', IS(B')I 2 A and IB'I = 1BI + 1 < K ( T ) . THEOREM 6.14*: If X = ID(T)l < IT1 then L has a sublangwcge L* of power A such that for every predicate R of L there is a formula (infact a predicate) R* of L* such that (VZ) (R(Z)= R * ( Z ) ) E T . (We say in this w e that T is a dejinitionul extension of T n L*.)

Proof.For all m and all p # q E D"(T) choose a formula v = rp,,q(Z) such that tp ~ pl ,p E q. Let Qrn= {v,,&Z): p # q E Dm( T)},and let Y,

m. ID,§61

T E E FIR8T STABILITY UARDINAL

129

be the closure of Qm under the connectives. Claim: For every formula $(E) (I(%) = m) there is an equivalent formula in Ym.

Prwf. By the definition of the ( P ~ , ~the , set r = {'p(Z) = ' p ( i j ) : 'p E Gm}u {$@)= +(@} is inconsistent. Thus there is a finite @ c Qm such that {'p(if) E ' p ( i j ) : ip E @} u {$(it) E +(jj)} is inconsistent. Let @ = { ' p l : I < n}. Thus for every 7 E "2 there is t(7) such that ~ ~ ( Z ) nk[ ~ I@)~(*). l Thus

$(z) E

V A 'pl(~)n[lI,

nEna l < n ttn) = 0

This proves the above claim. Now we define L*. For every m and ' p ( E ) E Ym,if there is a predicate R such that (VZ)('p(Z) = R(Z))E T let R, be any such R. If there is no Ym} such R, let R, be an arbitrary predicate. Then L* = {R,: 'p E Urn is the required sublanguage.

Remark. Since lD(T)Jis always 2 K O we can assume by the above theorem that ID(T)I 2 ITI. Now we come to the main theorem of this section. THEOREM 5.15: Let A designate an in$nite cardinal. Then every stable theory T eatie@9 one of the following: (1) T ie stable in A iff h = ID(T)I + hex(*). (2) T is etuble in X iff A = 2"o + Acre(*); and also ID(T)I = \TI < 2%

Proof. If T is stable in A then lSm(0)I IA and so ID(T)I IA. Also by If there is a countable Theorem 3.7, A = so A = ID(T)1 + A such that [#(A)[2 2No then we also get A = 2Ko + A A 2 IAI. Thus the assumptions of Lemma 5.13 hold for A+ in place of A, but 5.13(1) does not hold (otherwise ID(T)I<"'*)2 A + ; by our hypothesis ID(T)I IA and thus > A, a contradiction). Thus 5.13(2) holds, i.e., 2No 2 A + . I n other words A < 2Ho; so ID(T)I < 2No and thus IT1 = ID(T)I by Lemma 11, 3.15. Of course there is a countable A for which IS(A)I 2 2% (by Theorem 11, 3.2).

Thus T is not stable in any power < 2% But by our hypothesis if A 2 2n0, A = ACE(*) then T is stable in A. Thus (2) holds.

130

OLOBALTHEORY

[CH.

111, Q 6

COROLLARY 6.16: If T is superstable then ID(T)I or ID(T)( + 2 b is thefivst power in which T i s stable and it i a I8(A)Jfor some countable A.

LEMMA 6.17: If T is countable and superstable, but not stable in So,then there is m such that IDm(T)I 2 No.(Thus T is not N,-categorical by I X , 1.6.)

Proof. Similar to the proof of Lemma 6.13.

III.6. Imaginary elements

DEFINITION 6.1 : (1) The sequence 6 is defined by the formula;@I a) if ~(6 a);= {6}. It is defined by the type p if 6 is the unique sequence which realizes p . It is definable over A if tp(6, A) defines it. (2) The formula v(Z;a) is algebraic if v(Q; 3)is finite. The type p is algebraic if it is realized by finitely many sequences only. The sequence 6 is algebraic over A if tp(6, A ) is algebraic. (3) The definable closure of A, dcl A is {b: b definable over A}. (4) The algebraic closure of A, rtcl A is {b: b algebraic over A}. If A = acl A we say A is algebraically closed. (6) When dcl A = acl A, cl A will be their common value.

LEMMA 6.1 : (1) The type p defines a sequence iff for some finite q c p , A q defines that sequence. (2) The type p is algebraic i;fsfor some Jinite q c_ p , A q is algebraic (moreover, we can choose a q such that p and q are realized by exactly the same sequences). Proof. Immediate.

LEMMA 6.2: (1) A c dclA c acl A. (2) When A G By dcl A c dcl B and acl A c acl B. (3) If B = dclA tiLen B = dcl B. (4) If B = acl A then B = acl B. (6) 6 &a definable [azgebraic]over A if 6 E dcl A [6E acl A]. Proof. Easy, e.g., (4) If c E acl B then for some b,, . . .,bk E By n < w and formula 9, b(3 '"z)~(z,b,, . . .,bk) A ~ [ cb,,, . . .,b,]. AS b, E B =

OH.

111, 8 61

acl A there are #,[b,; a,]. Define

O(z;

*

.

131

IMdGlINARY ELEMENT8 EA,

n(i)<

w

and

#' such that

k(3s%t$4f(z; $)

k

* Y

7ik)

=

* *

A #i(Yi; %) A

Yk)[ 1-1

(3 '"z)Cp(z~y1,

A d z , Yl,. *

*Y

-

* * 9

A

yk)

n)];

so l=O[c; a,,. . .,ak](where yf H bi). But the number of possible (y,, . . .,y k ) is Inf=, n(i),and for each such (y,, . . .,yk) we can find In 2's. Hence IO(C; a,, . . ., ak)l 5 n n(i)< KO.

n:=,

LEMMA 6.3 : (1) If p E Sm(A)then p 7uz.~a unique exteneion q E Sm(dclA ) , infactp = q and everyformula Over dcl A is equivalent to aform& over A. (2) If a E acl A , then v(Z; 7i) is almost Over A. HoreOver if O(Z; 6 ) is almost Over acl A then it is almost Over A. ( 3 ) If q E Sm(aclA ) , B E A , then q forks Over B i f l q / A forks over B. (4) stp(a, A ) = stp(a, acl A), tp@, acl A ) k tp(7i, A). ( 5 ) Let T be stable, I a n (injinite)indiscernible set based on A. Then I is trivial (all of I's elements are e q d ) iff Av(I, A ) W algebraic.

Proof.(1) is trivial, and (3), (4) follow from (2) by 2.6(1) (and 1.1(8) of course). So let us prove (2); so by 6.2(5) msume #@;a) is algebraic, Z E A, k#[iz; Z]. Define E(Z;jj; a) = (VZ)[#(Z; Z) --f cp(Z; 2 ) = p(g; Z)]. Clearly E = E(Z;9) is an equivalence relation, and if I#(Cr; E)I = n, it has I2" equivalence classes. So E E FE(A). As !$[a; Z], E(Z;8) --+ q ( Z ; 2 ) = c p ( j j ; a), i.e., v(Z;7i) is almost over A. The assertion about O(Z; 6)follows by 2.4. Now ( 5 ) is easy.

Now we define the model C q .

DEFINITION 6.2: (1) The universe of C q is {a/E: m < w , aim16QI, E = E(Z, jj) E L is an equivalence relation (without parameters)} (we mean that ZJE, = aa/Ea iff El = E,, l=El(Zl,B2)). We identify a/= with a. (2) The relations of QeQ will be equality and (i) P, = {a/E:6 s 161), (ii) a function P, from P, onto P, defined by P,@) = 8 / E (so F, is a partial function, i.e., a relation), (iii) for every formula p(Z) E L R, = {(a,/=, . . .): Crp[a,, . .I}.

.

132

[a111, . 86

QLOBAL THEORY

(3) The language o f Cq is Leq, and its theory Teq. (4) 6:a w i l l denote a submodel of Cq whose universe is P, u u u Pgnfor some P, and some n < w . So V$ is not uniquely defined.

ITs,

LEMMA 6.4: For every f m & c p ( f ) in Lea there is an equivaknt form& which ie a Boolean cmnbinath of formzclae of the following forma: (i) a pmpoaitiOrca2 txn&znt (true or f a k e ) , (ii) x = y , (%)

pE(Z),

(iv)

AT- 1 pEI(xi:I)

W i ,

-+

---

9

&)[AT- 1 FE,(&)= xt W L * - - &)I3

-+

Proof, Left as an exercise for the reader (equivalenae is relative to T). CONCLUSION 6.5: Qbq w not saturated; but if we a&d to it elenzents, witextending the r e l a t h (80 a 1 2 a ~element 4 get a eaturated model.

llCll

mure

usP p )we

DEFINITION 6.3: (1) If p is a set of m-formulas in Q, then pel = {B&; T i ) : c p ( f ; a) €I)} u {P=(xt):i < I ( f ) } . (2) If M < Q we let Meq be the submodel o f Cq, whose universe is {a/,?#:a E IiKI}. See Lemma 6.6. (3) In attributing properties we shall not distinguish strictly between iK and MeQ,p and pm (this is justified by 6.6). LEMMA 6.6: (1) p is &tent iff pw is Coneietent. If p is an m-type in W, p k A,<,,P - ( x , )then for some m-type q in 6, p q =p. (2) When M < Q, MeQ< C a . ( 3 ) For every W < Cq,there is a un&w Jf < Q such that Meq = H'. Proqf. Easy.

DEFINITION 6.4: (1) The kind 0, o f Zi = (ao, . .., is the unique formula O(xo,.. .,xrn-1) = A,<,,,PEI(xt) such that CQC (2) By a ( < Ko)-typein Cq we shall mean a type p such that 0, E p for any 7i realizing p.

LEMMA 6.7: (1) For h 2 ILI,T is stable in h iff Teq is stable in A. Also K ( F )= .(TOq). ( 2 ) If p is a type in Q, A c 161 t h : p f o r b Over A (in6)iff peaf o r b over A (in Cq) and p is statbnary iff peq i s etatbnary. ( 3 ) If 7i E Q, A c 1Q1 then [tp(7i, A , Q)rq = tp@, A, Cq). not fork over B. (4) If B c A, a E acl B, then tp(a, A )

OH.

111, 5 61

IMAGINARY ELIEMENTS

133

( 6 ) If B c A, 6 E acl(B u a), 6 # acl A, and tp(6, A) f o r b Over B, then tp@, A ) f o r h over B (hot?& in Q and in P a , in particular for b = FE(W

Proof. (1) By 6.6; and the second phrme by (2), (6). (2) By 6.4. (3) Easy. (4) mvial. (6) Easy by (4) and 4.16.

Following Definition 6.4(2),the theory we have developed for Q bidefor Pa.We shall we this freely.

LEMMA 6.8: For any equivalence relation in Qeq, E = E(Z,g) = E(Z,jj;E ) ( l ( Z ) = I @ ) = m ) between 8eQuences of k h d 8 (80 E(Z, g) k 8(Z) A tJ(g)),we can c h e a relation P,, and partial function F , (both definable by f m u l u 8 in Leq with E a8 parameter; but the f m u l u 8 themselves do not depend on E ) which satisfy: FE is a function from sequences of length m of kind 8 to P,, Buch that t e [ q A epa], implies FB(E1) = F&) 0 ~ E [ E ,cia]. , Proof. Technical, so we leave it to the reader. Remark. We shall not distinguish strictly between the FE, P, from 6.8 and those from Definition 6.2. So we write ii/E instead of YE(@. DEFINITION 6.6: For a formula ~ ( 3jj); let EJjj, 3) = (VZ)[cp(Z; 8) E ~ ( 3Z)]; (in this is done for fixed kinds of jj,5.) Clearly E, is an equivalence relation, and ?(Z; a) is equivalent to a formula over 7i/E,.

LEMMA 6.9: I n gW,for algebraically c h e d A: (1) Every p E Sm(A)is stationary over A (for stable T). (2) Every formula cp(Z; 3) which is almost over A , is equivalent to a formula Over A. Procf. (1) By 2.9(1), (1) follows from (2). - (2) As cp(Z; a) is almost over A, it depends on some E E FEm(A).Let b,, . . ., 6, be representatives of the equivalence classes of E. Clearly for ell i FE(6,)is algebraic over A [as P E ( P E ( 6 , ) ) and I-P,(s) = VYI x = p E ( 6 i ) ] , So FE(6,)E A . As a(Z; a) depends on E , for some w, I-cp(Z; a) = V,Ew E(Z; 6,). But kE(Z; 6,) = [FE(6,)= FE(Z)] hence kcp(Z; 7i) = [V,Ew FE(Z) = FE(6,)].The second formula is over A, 80 we f i s h .

134

GLOBAL THEORY

[OH. 111, f 6

THEOREM 6.10 (The Canonicity Theorem for Types): Let T be stable. for e v e y statiomy m-typep there is a 8et A = Cb(p) (the canonical In CQ, h e of p ) and a s t a t k r y type q E P ( A ) ,q = Ctp(p) ( = tlre canonical type of p ) euch t W : (1) p and q are pralkl. (2) Any automorlphh F of Cq t a b p to a parallel type iff F f A = identity. ( 3 ) Tor any a E C q , a E A iff every autommphism. F of C q which take8 p to a parallel type necessarily satisfie F(a) = a. (4) There is B E A , IBI < K ( TOver ) which no type parallel to p forb. ( 6 ) If a type in Sm(C)is parallel to p and CEO@ not fork over B E C then A E ad B. ( 6 ) A E dcl(Domp ) .

Proqf. Let r E Sm(QBO) be parallel to p. By 11, 2.2 for every 'p(Z; 8) there is a formula &(g; 5,) which defines r f 9,i.e., C+,[Z;

E,]

o 'p(3;3)E r .

Let

A = dcl{e,/E*.: 9 E Lea}, q=rfA. Clearly r is definable over A , hence does not fork over A by 4.11. So q is stationary and q, r are parallel, hence p , q are parallel (this is (1)). Clearly an automorphism F of Cq takes p to a parallel type iff F(r) = r if€ for every 'p, k+&; a,) = +,(g; F(E,))iff for every 'p, E,IE,. = F(E,J/E*eifF F f A = identity (this is (2)). If a $ A then as A = dcl A there is a' # a, tp(a, A ) = tp(a', A ) hence there is an automorphism P of QBO, P r A = identity and P ( a ) = a'; so P(r) = r, F(a) # a' (this is the missing part in (3)). F'rom (3) it is clear that A does not depend on the particular choice of the $ :,I Also A depends only on r , i.e., on the equivalence class of p under parallelism. The same is true for q, hence the notation A = Cb(p), q = Ctp(p) is justified. As we could have chosen E, E Domp (by 11, 2.12) and as E,/E*. E dcl(5,) {E,/E**: p€Leq} E dcl(Domp). Hence by 6.2(2), (3) A E dcl(Dom p), so (6) holds. As for (4), for some B s A, 1B1 < K ( T )q, does not fork over B, but r does not fork over A, hence by 4.4 r does not fork over B. As any type parallel to p is c r also (4) holds. So we &118left with (6). By 4.4; r does not fork over B, hence is definable almost over B, hence by 6.9(2) definable over a d B, so we could have chosen Ee E acl B hence A E acl B.

OH.

111,Q 61

135

IMAQINABY ELEMENTS

CONCLUSION 6.11 : When T ie etuble, for euey etutionary type I, in Q there is a p r d W type q, euch t7td for every atdmnorphh F of B:

F ( p ) i e p r d k d t o p iff F(q) = q.

LEMMA 6.12: In P,if I is an in&iecernib& eet baaed on the etcrtiOnary type p (i.e., Av(I, U I ) , and p are p r d W ) then I ie i n d h i b l e over Cb(p).

Proof. Any permutation of I is induced by some automorphism F of which maps Av(I, U I ) onto itself, hence p to a parallel type hence F 1 Cb(p) = identity. So the result is clear by 6.10(2). The following theorem shows that if T is stable but K ( T )is large, then T contains an underlying structure of equivalenoe relations. This corroboratea our intuition that the natural example is in fact quite general. V q

THEOREM 6.13: In Q, if No < K < K(T)< 00, K ie regular, then there are m < w and fomnukza cpt(E, #) E L i < K , I @ ) =. I(@) = m euch t h d : (i) cpt(3, #) &fine0 an equivalence relation. (ii) For every h there are an(7 E "A) such t7td ktp,[a,,, a,] i f l v 1 (i + 1) = rl t (i + 1). Proof, Let M,(i I; K ) be an increaaing sequence of models, 42, < 6, = UL
a,, *

* *

9

W s'c")z)$:(~, a,, -

a,:)],

and

*

9

am(,))

$&; am * - * am(,)) 1~P(c:,U I ) (see Exercise 6.1) end w.1.o.g. $,(lQml; a,, .,a,,,(,)) n 1M,1 c lMacL+l)l. Clearly if tp(cl, U I ) = tp(c2, U I ) then cl E Cb(p) o c2 E Cb(g) hence Ili
..

..

136

I#,(

[CH.

111, 8 6

> KO we can assume m(i) = m, k(i) = k, 1, = l,(i) = am)l (for eve^ i). lpq1 ;a09 .,am)l, 1%= ldi) = l#;(Ma(f);

As

Let

GLOBAL THEORY K

.

.

+

3

.

Trivially cp, ie an equivalence relation and as #,(ICaI ;a,, . . ,am)G 1M;?, + ,)I, $ 1 ME?,)]we can find 7i, satisfying (ii) (as in Theorem 3.1). By 6.4 we can go back to cE. EXERCISE 6.1: Prove that if 6 E acl A, then for some Si E A and tp, cp(3; Si) = tp(6, A). EXERCISE 6.2: Suppose c, E acl(A u 8) for some Si but c, acl(A u (c,: j < i)), for i < K. Prove K < K(T). EXERCISE 6.3: Prove that in 6.13 the restriction on general be weakened. QUESTION 6.4: Can we in 6.13 replace "of K > Ho"?

"K

K

+

cannot in

> H, regular" by

CLAIM 6.14: Let T be stable. (1) (in CQ) If i3 E ad(A u B), tp,(A, B u C) does not fork over B, then tp(a, A u acl B) k tp(a, A u B u C). B E C and (2) 8e(i) stp(6, A u B) I- tp(6, A u C), (ii) tp,(6 u A, 0)does not fork over B, (iii) tp,(6 u A, B) i s stationary, then tp(6, A u B) k tp(6, A u C). Proof. (1) As tp,(A, B u C) does not fork over B, by 6.7(5), tp,(A u a, B u C)does not fork over B; so by 6.3(3)tp,(A u5, acl B u C) does not fork over B, hence over acl B. So tp,(C, acl B u A u i3) does not fork over acl B. But, by 6.9 tp,(C, acl B) is stationary hence by Exercise 2.2 and Lemma 6.3(4), tp,(C, acl B u A) t- tp,(C, acl(B u A)) t- tp*(C, ac1 B u A u i3) hence tp(a, A u ac1 B) k tp@,A u ac1 B u C) (see IV, Ax(V.1) for

F", IV, 2.8).

OR. I n ,

8 71

INSTABILITY

137

(2) We work in Qeq, 80 by 0.3(4), Exercise 2.2, tp(6, acl(A u B)) k stp(6, A u B) so by (i) (a)tp(6, acl(A u B)) k tp(6, A u C). By 6.14(1) and (ii)foreveryFEacl(AU B),tp(c,A U aclB) I- tp(6,A U C),hence: (8)tp,(acl(A U B), A U acl B) 1tp,(acl(A u B), A u C). Now (a)and

imply: ( y ) tp(6, A u acl B ) t- tp(6, A u C). [If 6, realizes tp(6, A u acl B) let f, be an elementary mapping, f, (A u acl B) is the identity, f(6) = 6,. Now let f extend f,, Dom f = 6 u acl(A u B), hence Rangef = 6, u wl(A u B). Let g be the identity mapping on C. Clearly (f (A u acl B)) u g is an elementary mapping. Hence by @) (f acl(A u B)) u g is an elementary mapping, hence by (a)f u g is an elementmy mapping, 80 6, realizes tp(6, A u C).] As tp(6 u A, B)is stationary, by 6.3(4) tpJ6 uA, B) 1tp(6 u A , acl B) hence (8) tp(6, A u B) k tp(6, A u acl B).By ( y ) and (8) tp(6, A u B) k tp(6, A u C), which is our desired conclusion.

(8)

r

III.7. Instability DEFINITION 7.1 : (1) Let K s d ( T ) be the first infinite cardinal K for which T doea not have K independent orders by m-formulas: i.e., there are no ; < K ) l(3) = m and 4 (j< w) such that for every formulas ~ ~ (&)3 (i r) E

{Q'(z; $)U'"["r":j

<

w,

i<

K}

is consistent. (2)

=

Kkl(T).

Remark. The expression "ird" is an abbreviation for "independent orders".

Proof.Immediate. THEOREM 7.2: P ( h ) 2 sup{(pX)+:p < Ded A, K < K&(T)}(we m8ume A 2 Kird(T), of course) (see DeJinition I I , 4.4).

[CH. 111, 6 7

GLOBAL THEORY

138

Proof.Let p < Ded A, and Z an order 111 = p, with a dense suborder J, 131 = A. Let K < t&(T) and let tpr (i< K ) be the formulaa from Defini(8 E J ) so

tion 7.1. By compactneae define

is consistent. Let A IAI s h and

=

U {g:i

<

K,

8E

that for every r) E "I,

J}, and p,, E q,, E Sm(A).Then

THEOREM 7.3: Led A and A be infinite set8 such that: (i) J8r(A)I 2 A > 214 IAI, where h i e regu2ar. (ii) J8Tl(A)J< h for uZZA, E A , !All < Id!. (C) There i8 no S c &'(A) such that IS( 2 h and I{p IAI for every 'p E A . Thf?7& < K z d ( T ) .

+

tp:

p

s

Proof.Let A = {tp,(Z; 8):i < ldl}, and A, = {tp,: j < i}. We d e h e by induction on a a function F, from Q ( A )to subseta of I A I :

In the definition a appears only once: in (Vfl < a);hence for y < a PAP) 2 PAP). Now for some p E SZ(A), P,(p)l = Id I (in fact this holds for a's much higher than w). For suppose there is no such p. As ISZ(A)I 2 A, h > 214, h regular, for some w, E ldl, n < w , S , = { p E S ~ ( A )P,(p) : = w,foreveryn < w}hascardinality Th.Letd, = { t p , : i E ~ , < , w , } , so by assumption lA*I < ( A1, hence by (ii) lST*(A)l < A. As h is regular, for some p o E ST.(A),

Inn<(u

has c d i n d i t y z h . Now for every t; < 141, 1{p 1 tp,: p ESa}l 5 IAI. We prove this by w,, p ES,, then tp, ELI* and p 'pt = induction on t;; if t; E

nneop

CH. III, 5 71

139

INSTABILITY

p o rcpt. If for some n [$w,then by the definition of F,, and as (Vp,q E B,)(VI < w ) [ F I ( p= ) FI(q)]we get (lettinggA.= { p A*: p EBS}):

r

I{P 'Pr: P

s

E&}l

1u

{p

vc:p E Sa and the number of q vr,for q E 8 2 ,

A*SA( lA*I<&

'

q Id* = p r d * i s I

2 2

A*EA(

lA*I

<no

1{p 1tpr: p ES,, p

s1~1}1 d*

=

r and the number of

TESd.

q r~r,forq~& q ,rd* = r i a ~1.41}1

I Idl.lSd*l.I4I

I 4

as

IBAOI s

n

18@,1 5 IAI'"'' = IAl

cpiEA*

by the induction hypothesis, and Id1 < IAI as

2141 < h I ISd"(A)I 5 214+14.

So by S , we get a contradiction to hypothesis (iii) of the theorem. Fn(p*). Hence for somep* €&(A), IwI = Id1 where w = By compactness it suffices to prove that for any i(0) < i(1) < i(2) < ... < i ( n - 1) < l d I , i ( I ) E w , Z < n a n d k < o t h e r e a r e ~ ~ ( ( I < n ,k)j < such that for any 7 E "k, {tprcI,(P;Bj)ift*f112j):2 < n, j < k} is consistent. Denote d1= {'pica,: u < 2). Now we define by induction on I In for every 7 E I ( IA1+) a type q,, E ST(A) such that (i)i(O), . . ., i(n - 1) E P,+ -l(q,,);(ii) when v is an initial segment of 7,qv r At(")G q,,; (iii) i # j => qV-<() dl # qv-<,) d1 when I = I(v) + 1. We let qo = p * ; and if q,, is defined, I(q) = I we can define q,,-
on<,

[CH. 111, 8 7

OLOBALTHEORY

140

CONCLUSION 7.4: If for 8ome A , IS(A)I 2 A, A regular, A > 2ITI JAJITI and for p. < Ded,JAJ,p e K < A then ~i,.d(T) > K.

+

Proof. By 7.1 we can assume T does not have the independence property Choose a d with minimal cardinality for which ISd(A)I 2 A. By 11, 2 K. 4.10(2) when Idl! < K , IS,,(A)I 5 ISo(A)I < A hence So our conclusion follows by 7.3 as (i), (ii)hold by the choice of d and the existence of S as mentioned in (iii)contradicts (AIITI < A.

naed,

THEOREM 7.6: Suppose T does not have the independence property, A c B, p ES"(A).Then there is C E B, ICl I IT1 and q ES"(B),8wh that p c q, and q does not split over A u C . Proof. For every C

E

B let

r(C) = {rp(Z; a) = v(Z;6 ) :a, 6 E B, tp@, A u C)

=

tp(6, A u C)}.

It is clear that tp(E, B) does not split over A u C iff i3 realizes r(C). So it suffices to prove that for some C, C E B, ICl 5 IT1 and p u r(C) is consistent. Suppose there is no such C, and define by induction on a < IT!' sets C,E B , IC,l I IT].Let C, = 0, and C, = Ut
< lTl+,%(a) = n, vf = v1}1 =

ITI'.

So assume that n(k) = n, q$ = tpf for every k < w (by renaming). Let A = {vI: 1 < n}, and define p,, ES"(A u CI(,,J for q E to>2 by induction on l(q).Let p0 = p, and if p,, is defined 1(q) = k,let p,,-(,) be any extension of it in Sm(Au Ck+l)'So by (ii) for some 1 < n, yI(Z;a:) = +pI(Z; 6:) ~ p , , - ( , )Let . ~ ~ ( 3 ;~ p , , - ( , )then , p,, u {TrpI(Z; a:)t} is ES"(A U Ck+l),p,, _c p,,-(l), consistent (by (i)) hence there is lvI(Z; ~ p , , - ( Now ~ ) . for every k < w let Ak = U { E { : j < k, I < n}; for q # v e k 2 , by the definition (p,, I d ) r A k # ( p v I d ) A,, hence IflT(Ak)IL 2k, and lAkl 5 ksm,, where m, = ZI<,,@!). so by 11, 2.1

OH. 1x1, 8 71

INSTABILITY

141

and 11, 4.11 some q(3;j i ) E A has the independence property, a contradiction.

DEFINITION 7.2: (1) Kedt(T)is the first regular cardinal K such that: there are no formulas q,(3; 8,) ( l ( Z ) = my i < K , i successor) numbers n, < w and sequences sin, r) E Iczw, for which (i) p,, = {q@; a,,l,): i su.ccessor, i < K } is consistent for r) E Icw, a < w, r) E "'w, i = l(7) + 1) is n,-contradictory, (ii) {q@, i.e., for w c w , IwI = n,, C7(3P)[l\,,, q,(Z, (2)

KrOdt(T)

=

edt(T).

Remark. The expression "cdt " stands for "contradictory types".

Proof. (1) Let 'p(z; j i ) have the strict order property. Choose q, = ~ ( z ; j iA~lq(z;jids ) ni = 2. (2) Because we can choose all the (p,% equal. (3) By adding dummy variables. (4) By 11, 3.6. (5) Clearly Krgdt(T) 5 K,(T);the converse holds by (4) and by 6.13. THEOREM 7.7: The following conditions on T , m < w and a regular cardinal K are equivalent. (1) K < K e d t ( T ) . (2) K < K&(T), and moreover, in DeJinition 7.2(1)we can have n, = 2 for every i. (3) For every A, A'" = A, there is a set A , IAI = A, and a set S of m-types over A which are contradictory in pairs, IS1 = A", where p E S *

14 =

K.

(4) There is a set A ,a set S of m-types over A and a cardinal x such that:

(i) 181 > IAI'" + 21Tl+X, (ii) no sequence realizee > x types from 8, (iii) for every p E 8,1231 5 x.

[m. m, § 7

GLOBAL THEORY

142

Proof. ( 2 ) s- (1) Immediate. (1) s- (4), ( 2 ) =. ( 3 ) Easy, by compactness. (3) s- (4) Choose in (3) a A which is a strong limit cardinal > 21Tl+", which hes oofinality K . Then the A and S from (3) will satisfy (4), for X = K. (4) =. ( 2 ) Let w.1.o.g. IS1 = A+, where A = IAl'" 21Tl+X,andB = {pi:i < A+}. We can mume the types in S am pairwise contradictory, for define by induction ui E A+ aa follows: u(is a maximal subset of A+ , for whioh j < i * zc, n uj= 0, and U {pa: a E u,} is consistent. By (4)(ii) 5 x ; so letting pi = U {pa: a E u,}we get what we want. So now let p , = {q',@;g): a < x}. As the number of sequences (9;:a < x ) is 5 ITIx 5 2ITI+X I A we can assume tp; = rpa; and For i < A+ we define similarly that tp*(A') is fixed where A' =

+

luil

ua

finite subseta u', of x by induction on a. If u; is defined for fl < a and there ia a h i t e u E x and there am distinct [(n) < A+(n < a) such u;, = uj; and the formulaa that qQ)= 4 for jE UBea Armcp,(Z; a:((A)) am pairwise contradictory then we let u; = u. The first a for whioh we cannot define uI, will be denoted by a(i).Clearly a(i) < x ; and it suffices to prove that for some f < A+, a ( [ ) 2 K . (Then the rplof Definition 7.2 will be A {tpj:jE w f } remembering that tp,(A,) is fixed.) So suppose for every f < A + , a(6) < K , and we shall get a contradiction. As there are xen I 2% < A possible sequences (&: a < a ( i ) ) (as IAI'" I A < IS1 5 lAlx, K 5 x ) for some S1s 8,ISll = A+, and p , E S implies ~ a(;) = a*, and u', = w, for a < a*. As IAI'" I, A, there implies % = aj for is Sa s 8, suoh that IS21 = A+, and pi E S ~ $((A)

jE U a < A Now let S, = {[(i): i < A+}. For i < j < A+, pcCO U pCw is contradidory (we nofed it at the beginning of the proof), so there is a finite subset h ( i , j )of x suoh that

is contradictory. But A+ 2 (2X)+, hence by 2.6 of the Appendix for some S3 E SayIS3] 3: x , h has a fixed value, u. This shows that for any pi E fI3 we could have defined .;u = u; a contradiction.

THEOREM 7.8: T h e k a themy T , without the Btrict oraZerpoperty,for whkh K&(T) = a.

CH. 111, Q 71

143

INSTABILITY

Proof.The set of axioms of T will be as follows: (1) E is an equivalence relation with infinitely many equivalence alaeees

W w E 4 ( V q ) ( z E y= YE%), ( V q z ) ( d yA yEz + xi%),

@x0, . . .,z,,)A l z i ~ z ,for every n > 0. i<jsn

(2) For two distinct B-equivalence classes x,/E, x2/E, the family of

sets

{z: z E q , zRy} for y E x,/E forms a partition of x l / E

(VX?.l)(ZBY -BY), +

( V q ) ( i ~ E-+ y ~ Z ( Z EA Z~ R z ) , (vZyZ)(Z&/

A ZR2 A

2 E y j - y = 2).

(3) These partitions are independent, moreover, for m (vzi,. *

*

n < w,

(i\T z ~ y i i\ l z ~ z i

,Xn)(vyl,- - yn)(vz)[ 8

s

A

1-1

111

A -@IEYk A 1x1 = x k )

isi
A

Iernsk

z*Ez

A

A

z*Ry,

kasn m-1

A

A

x~Rz*

1-1

A

1-m

7xlRz*)].

We leave to the reader to prove that T admits elimination of quantifiers is complete and satisfies our requirements.

THEOREM 7.9: There ie a theory T , and a type p ES(O),Such that for every cardinal A there M a set A , IAI = A, which sa.tisjEes: for every q E S ( A ) ,p E q, and B E A , IRI < IAI, q aplita strongly over R.

144

GLOBALTHEORY

[CH. 111,f 7

Remark. This shows the limitations on generalizing 1.3 to unstable theories.

Proof.A set of axioms for T will be: (1) E is an equivalence relation, on {z:P(z)};each equivalence c l w has exactly two elements

(Vz)[zEz = P(z)],

(vq)[zEY = yEz1, (Vzyz)[~EyA yEz

--f

zEz],

(Vz)[P(s)+ (31YM # Y (2) When -4'(z), P(y) then

A XjJY)l.

R chooses an element from y/B,

(Vz?dCP(z)v lP(Y) lZRY1, (vZy)[iP(S) A P(y) + (3!Z)bEZA ZRZ]]. ---+

(3) The choices in (2) are independent, i.e., for every w ( l ) ,. . .,w(n) E (1,. . ., n}

It is emy to see that T has elimination of quantifiers, and when M is a model of T, A = P ( M ) ,p = {lP(z)} satisfies our demands.

Proof.(1) We can use 7.4 and the independence proofs of set theory or prove like 11,4.16.

CH.

111, $71

INSTABILITY

145

As K&t( T)s

K c d t ( T), it suffices to prove by induction on m that For m = 1 it is trivial, so suppose we have proved it for m and K < KrtL l ( T)is regular; we shall prove K < Kr,’dt( T). So there are formulas tp,,(y, Z; 2,) (Z(2) = m ) (0 < a < K , a successor) and a,, ( r ) E KA) as in Definition 7.2; and n, = 2, and {a,,: r ) e K’A} is an indiscernible tree (see Definitions VII, 2.4, VII, 3.1, VII, 3.2, Theorems VII, 3.6, VII, 3.6) and A > I TI is a strong limit cardinal of cofinality K . Choose a maximal set w c K such that Fw is consistent, where rw= {tpa(yn,2;ii,,,,,) 0 < a < K a successor, r) E %, and y + 1 # w 3 r)[y] = 0 ) (note that if w(i)(i < 6) is increasing, then F,,,, is increasing and Fv,w(,)is consistent iff U, F,,,, is). If IwI = K clearly {tp,,(y; Z, 2,,): a E w } proves our assertion. Otherwise let Po < K , ( V a E w ) ( a < Po), and for every r) E KA let (8 = (0, . . . <),

(2)

Kr,”at(T)

s

K&t(T).

),

F“ = {tpa(yv-,,, 2;a(,,-,,)lu): 0 < [Y

. . .) A p : p

rn =

a

<

K,

a

successor, v

+ 1#w md y G

<

&I

5

E flu,

4yl = O},

F nis finite}.

Now no sequence Z realizes > 2IT1 of the types 9. [Otherwise by 2.8 of the Appendix there are q, (2 < w) and y < K such that r), y = qo 1 y , and the q,[y] are distinct. Then w u {y + 1) contradicts the maximality of w.] DEFINITION7.3: (1) &,(T)is the first cardinal K , such that there are no formulas tp,(Z;&) (i < K ) (Z(2) = m ) and natural numbers n,, and sequences iii(a < w) such that i < K } is consistent, (i) for any r ) E IEw, {tp,(f;iikLt1): (ii) for all i < K {tp,(it; a:): a < w} is n,-contradictory. (2) Kinp(T) = SUP, .Ep(T)* (3) KrEp( T) [Krinp( T)]is the first regular cardinal 2 K E ~T) ( [Kin,( T)]. Remark. The notation “inp ” stands for “independent partitions ”.

DEFINITION 7.4: (1) Krrd( T )is the first cardinal K such that there are no formulas tp,(Z; &) (i < K ) (Z(x) = m ) and sequences iii(a < w) such that (i) for r ) E Icw {tpl(2; iii)if(n[f12u): i < K , a < w } is consistent, (ii) for i < K , a < w , k(V2)[tpi(2; iii)+-tpl(%; a:,,)]. (2)

Ksrd(T)

=

Krrd(T)*

Remark. The notation “srd ” means “strict independent order ”.

146

[CH. I n ,

QLOBALTHEORY

57

THEOREM 7.11: 8-e Kcat(T) = 00, K ~ , , ( T <)00 then there ie a fomzukc (~(2; #) and 8equenCes a,, (q E ,>w) such that (i)(~(2; a,,,,,): n < w } i8 d S t e n t for each q E Ow (ii) if 7,v E are 4-~nconzparuble,then k(3z)[q(z; 4) A p(z;

Proof. By the assumption K&t(T)= 00, the proof of 7.6(2), 7.10(2), and 7.7(2), there are a formula cp(z; fj)and sequences a,, (7E ,'w) such that consistent for each 7 E ,w, for k # I < w , 7 E " > o . By ~(2, Definitions VII, 2.4, VII, 3.1 and Theorems VII, 3.6, VII, 3.6,we can assume ( y ) {an: q E O > W } is an indiscernible tree, i.e., tp(Znln. . 0) depends on the lexicographic order among the q,'s, and Z(qf), h(qf,7,)= max{Z: qr I = q j t I). By Ramsey's theorem, the fact that we can lake any subset of the levels and compactness we can assume ( 8 ) in ( y ) only the order relations among the Z(qf), h(qf,7,) matter. For 0 , . .., 0 (ktimes) we write 0,. Now { ~ ( zlicI,ol+l)): ; 2 < w } cannot k < w } : 2 < w } proves K,,,(T) be consistent (as then {{(~(z;7i<~,~,,~>): = 00). Hence there is a maximal ra for which (a) { ~ ( za,,,,,): ; n

<

w } is

W 3 z ) [ d z ,%-u)) A

{ d z ;a < t , o l + 1 ) ) : < n} u { d z ; a): k > n}

is consistent. Hence there is m < {(P(z; ~ < I , 0 , + , ~ ) < :

w

such that

4 u {Pk, a)} u{ d ~ a < n;+ l , o k > ) :n + 2 5 k < n + m + 2)

is incorisistent. Now define, for every q E O>w h(q) = (n, O n + l , q [OI, Om, ~ [ l ]Om, ,

v*@;

-* an 1 - A (P("; I
---

9

7 [ J ( 7) 1],Om>,

A

~ < l , o ~ +A~ > ) ~ ( 2 ah
It is easy to check that: (1) For every q E ,'w, {v*(z;CZJ:2 < w } is consistent (by the choice of n). (2) If q, V E ' O ' W , l(q) < Z(v), q < I r v (lexicographic order) but not q Q v , then Cl(3z)[q*(x; a:) A 'p*(z$71 ; (remember that the indiscernibility applies to levels too, and the definitions of m and n). So the 'p*(z; a:) satisfy "half" of our demand (ii)(and (i),of course). By compactness we can define ( q E @ > w )which satisfy (a),@), ( y ) , (6) and when q, v E @>a,Z(q) < Z(v), v < I Z q but not q 4 v then C1(3z)[v*(z; at) A v*(z;a:)] (by reversing order).

CH.

111, 8 71

INSTABILITY

Now repeat the process and we get tp**

THEOREM 7.12: If T does K,",d(T)= Kh(T)* Proof. By definition, always

147 as required.

not have the independence property, t k n

KTrd(T)

5

KZd(T),

so it suffices to prove

K < K!&(T) assuming K < K$(T), and let tpi(Z; $) (i < K) exemplify the latter. We define by induction on 5 formdm #& gf) (i < 5 ) such that, letting

(*)

(i)

there are LZ; (a < w , i < K) such that: for every q E ICw,UieK pici,is consistent where

p i = {fit(%;ak)uUra): a < (ii) (iii)

t,hf(%;

G)I- #@;

for i <

w},

5,

{? :a I
U, # U B< a$.

Clearly (*),, holds, and ( * ) r suffice and (*)a follows from (*)E( 5 < 8). So suppose (*kyand we shall prove (*)t+l. As T does not have the independence property, for some n < w and finite w c n, and q E 5 ,

(J picf,u {vf(it;G ) u ( a e w ) :a < n}

% is inconsistent. So as in the proof of 11, 4.7(2), there are u,v c n, k c n such that (k + 1) E U - v , k E V - u, u - {k + 1) = w - {k}, and for every q E Kw

u {tpf(Z; L Z ~ ) ~ ( ~ E aV ) <: n} f
+c

is consistent but for some p E ICw,

u

pLCau {vi(z; ak)if(aeu): a <

w}

l
1#c

u,

is inconsistent and we can replace < K , , + pLrilby a finite subset q. We continue as in 11-4.7(2), conjuncting t,h with A q. We let t,hE = t,hy and is satisfied because by the indiscernibility assumption for any k < w , we can assume qG

{tpi(z, GL)u(D(')"a):

i # 5, u <

w;

but not Z(i)

s

a

< I($)

+ k}.

148

QLOBAL THEORY

[m. m, § 7

EXERCISE 7.1: Prove that K&,(T)5 K'"(T),K&,(T)s e d t ( T ) Show . that even for stable T equality need not hold. EXERCISE 7.2: Prove that when T does not have the independence 2 e-a(T). property,

em

EXERCISE 7.3: Suppose T does not have the independence property; and there are K cp,, $a, i < K , a < w such that (where Z(Z) = m) for any r) E =w, {cp,(Z; G)K(a*"[fl): i < K , a < w } is consistent. Prove K < K&,(T). EXERCISE 7.4: Prove that in Defhition 7.3 we can w.1.o.g. take n, = 2.

QUESTION 7.5: Prove that q,,(T) = KE,(T)= KLJT). QUESTION 7.6: ISKm(A)= K1(A)? EXERCISE 7.7: Suppose A E By p €Sm(A). T does not have the independence property. Then there are q E Sm(B),q 2 p and C G By ICl < K&,(T)such that: if {a,,:n < w } is an indiscernible sequence over A u C,and a,, E Bythen cp(3;a,) E q e cp(3;a1)E q. Remurk. Instead of "T does not have the independence property" we n < w } is k-contradictory for can add "where {cp(Z; a,) = +Z; some k < w y y .

EXERCISE 7.8: Prove Theorem 3.8 by the,method of the proof of 7.3 (changing the definition of Pa). QUESTION 7.9: Investigate K F ~ ( TIf) . T does not have the indepen( TK )~ ~ ( at T )least , when both are regular dence property is K ~ ~ = cardinals? PROBLEM 7.10: Characterize the possible functions R(A), and K(A, K ) = sup{ISI + : S is a set of 1-types, contradictory in pairs; p E S == 1p1 < K ; and the types in S are over some A, IAI 5 A}. (Possibility: Maybe the function A(A,@) should be used, where @ = (w,: i < K ) , w, a subset of {j:j < i} and k(A, a) = sup{IZ n +: Z E rc2A; Z closed under initial segments; and for i < K , I{7 w,: r) E Z } ~IA} (remember r) is a function with domain {j:j< i}).}

CH. 111, f 71

149

INSTABILITY

EXERCISE 7.11: Suppose in P,in acl(A v a) there is I independent ~ (IIl.T ) over A, assuming T is stable (see Definition 4.4). Prove K ~ ~ > EXERCISE 7.12:There is a theory T without the strict order property, Kcdt(T)= 00,K ~ ~ (
--+a4

--+

QUESTION 7.13: Can we in 7.10(1)replace K r by K ? DEFINITION 7.6: Let K2ct(T)be the first cardinal K for which we cannot find an m-formula cp,(Z; ti,) (a < K ) and sequences a, (q E ='u) such that: {va(Z; a,,,): 0 < a < K , a a successor} is con(i) For every q E sistent. (ii) If TI%, V E ~ a,/3 , < K are successors, and q, v are 4 incomparable then t7(3Z)(v,(Z; 8,) A v8(Z;By)). We let ~ ~ ~= sup,,, ~ ( Krct(T), t ) KeCt(T)[ K ~ ~ ~be~ the ( T first ) ] regular oardinal 2 K,",t( T ) [KSct( T ) ] .

QUESTION 7.14: Investigate .Fct( T ) , e.g., is KrCdt(T) = -iIlp(T)

Kict(

T ) = KsCt( T ) is

+ Kract(T)?

EXERCISE 7.15: Show that if Kctd(T)<

00,

then 1.3 holds.

CEUTEB IV

PRIME MODELS

IV.0. Introduction ‘Wegive here an axiomatic way to construct prime models (Sections 1 and 3) and prove that some sakisfy the axioms (Section 2) and then give more informetion (mainly characterization (Section 4)) under stability assumptions. We first examine here the most common case. So let T be a countable theory. We call M prime = (FMo-prime) over A if A c 1M1 and whenever A c INI,there is an elementmy mapping f:M +N,f A = the identity. So, in a sense, a prime model “contains” only what every model should contain. It is easy to see that the sequences algebraic over A should be in the prime model, and no other elements should be in it. But, aa we are considering elementary mappings, it is natural to look at the complete types over A which sequences from M realize. Clearly if M is prime over A, p E Sm(A)is realized in M ifF it is realized in every model N,A c 1471. Which me those types? Clearly. they include all the isolated types (p is isolated if for some tp(Z; a) ~ ptp(Z; , a) k p ) , and we prove (6.3) that for countable A the converse holds too.On the other hand ifa E A, C(3z)tp(z,a),there should be an element c realizing tp@, 7i) in M.So we prove (6.10) that a necessary condition for the existence of prime models over every A, is that for any A, E A, and consistent tp(z, a) there is an isolated q, tp(z, a) E q E &A). On the other hand this is sufficient, for then we can define inductively cI (i c ao)such that tp(c,, A u {c,: j < i}) is isolated, and A u {q:i < ao}is the universe of a model M (such a model is called primary and the sequence a construction). If A c INI,we can easily define f : M +N,by defining f(ci) inductively. Naturally after proving existence, we want to prove uniqueness. For the primary model it is not too difficult ; but for the prime model it fails in general (see [Sh 79al). Here we come again to stability. If T is N,-stable, we can prove that over every A a prime model exists (by 2.16(4), 2.17, 3.12) and charac150

OH. Ivy5 01

INTRODUCTION

151

terize the prime model by cclocal” conditions, and prove its uniqueness ( M is prime over A iff A s [MI, M is atomic over A (i.e., for each E E ]MI, tp(E, A ) is isolated) and there is no uncountable set z M indiscernible over A). For superstable T,we cannot prove the existence of prime models, but if one always exists we can characterizeit similarly (see 4.18). For stable T we cannot prove existence nor (natural) characterization, but when prime models always exist (or even if over A there is a primary model) the prime model is unique (6.6). Those uniqueness theorems reprove the uniqueness of the algebraic closure of a field (which is a trivial application as this was the classical example model theorists had in mind), but also prove the uniqueness of the differential closure of a differential field (for characteristic zero we have an KO-stabletheory, for other characteristics we have a stable theory). However the algebraic closure has an additional property: minimdity. We call M minimal over A if A c 1M1 and A c IN1 G implies N = M . I n general the prime model need not be minimal (e.g., when T is the theory of infinite models, with the equality sign only). However for KO-stableT,a minimal model N over A is necessarily prime over A ; and a prime model over A is minimal iff in M there is no infinite indiscernible set over A. Now a natural generalization of “prime model” is “model prime among the A-saturated or A-compact models ”, with suitable generalizations of isolated types. However for stable T,the notion E-saturated ( =every type almost over a set of cardinality < A is realized) is better than A-saturated. More exactly: for A > KOwe get the same saturation, but the definition of isolated satisfies more axioms, insuring existence when A 2 K ( T)and helping to prove uniqueness. A fourth example is the following construction, analogous to the construction of primary models: if T is stable and countable, for every A there is MyA G ]MI, such that for every Z E IMI,Q E L there is #(it; 6) E tp(Z, A), #(E; 6) 1 tp,(E, A ) (this is helpful for two cardinal theorems for stable theories, see V, Section 6). Trying to develop each of these cases separately will cause much repetition, so some general setting is needed. We suggest here one,letting F be the set of pairs ( p , B) where p E S ~ ( A )B, E A, p is "isolated" over B in some proper sense (the main example was F& where FA= { ( p , B): there is q E p 1 B, IpI + IBI < A, q t-p}).So an F-construction over A is (A, (Zf: i c a), ( B , :i < a)), where (tp(Zi,A u {E,:j < i}), Bf) E F; and we define naturally F-saturated, -primary, -atomic. In Section 1 we list the axioms, and give some basic definitions and

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trivial lemmas. We do not claim this set to be dehitive, it is enough for us if it is quite useful; and it covers more cases than it was originally tailored for. More exactly, though we have “ p is isolated over B y ’in mind, what we get is applicable to “p is definable over B ”. So in Section 2 we prove that certain F’s satisfy most axioms (sometimes under additional hypotheses). In addition to the four previous cmes, we deal with F{ = {(p,B): p does not fork over B} for stable T used in 6.15 and VIII, 2.7. In general, we prove that existence axioms hold using various stability assumptions. We sum up our results in a table. In Section 3, we investigate the conclusions we can draw from the axioms, mentioning each time exactly on which axioms we rely (this is important aa some of the examples fail to satisfy some of the axioms). We shall mention here only the uniqueness of the F-primary model and its F-atomicity. In Section 4 we concentrate on stable theories and prove characterization and uniqueness theorems. The way is to find some properties of prime models, proved via their being included in primary models; and show that any two models having these properties, are isomorphic, e.g., for Fff-primenesswe prove it when cf h r: K ( T);partitioning it to two cases (cf h > KO,cf h = KO) covered by two %heoreme with quite different proofs. (For cf h = KO,K ( T)= No,so we oan use induction on ranks; for cf h > KO we use a different argument which fails when cf h = KO.)An important technical lemma is that if p E #“(A) is FAisolated T stable, p does not fork over B G A then p B is Fi-isolated too. The case h 2 K ( 2’) > of h remains obscure (we do not know even if Fi-primary models are Fi-atomic).The last section, 6, deals with various theorems. The most important theorem is the uniqueness theorem. In the characterization theorems we demand mainly cf h 2 K(T),in 6.6 for regular h this can be replaced by A+ 2 K ( T).We use F{ to construct models with an absolute indiscernible set (if A 2 p, T stable in p, then there is I G !MI, 111 = 11M11 = A, I absolutely indiscernible) and investigate the possibilities of omitting types (e.g., if ITI = h is regular, p , E D(T)are not Fi-isolated, (i < i(0) < 2A) then T has a model omitting every p i (see 6.17)). IV.1. The set of axioms

We give here a list of axioms for the concept of primeness, and mention some obvious connections. We will not msume all the axioms hold

OH.

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THE SET OF AXIOMS

153

all the time, but specify each time the axioms used in proving a certain daim . F will denote a family of pairs (p,A) where p ES"(A') (m < w ) for some A', A E A'. We assume that F is closed under changing the names of the variables and their order, that is, if p ES"(B),A E B, and q = {v(yo,. ,y m - 1; a): P ) ( z ~ .( .~.),, Ti) E p } (u a permutation of (0, . . .,m - 1)) then (p, A) E F o (q, A) E F. We let A(F) [&(F)]be the first [regular] cardinal A (if it exists), such that (p, A) E F =- IAI < A. For any A let FA= {(p,A ) : IAI < A, ( p ,A) E F}. We call a type p F-isolated over A if (p,A) E F; p is F-isolated if there is A such that ( p ,A ) E F. Let p E F(B) mean (p, B) E F. In order to get an intuitive feeling for the following axioms think of (p, B) E F as meaning that the type p is isolated (in'the usual sense) over B.

..

The A x i m Ismnorphiern. Ax(1):F is mapped onto itself by any automorphism of Q, i.e., for any pair (p,A) and automorphism F of (E, (p,A) E F e ( P ( P ) ,W )E ) F. Trivially Ieolated Type0 Ax(11.1): If si E B E A, IBI < A(F),p = tp(Z, A), then (p, B)E F. Ax(II.2): If B G A, Ti is definable over B, IBI < A(F), p = tp(si, A), then (p, B) E F. Ax(II.3):If si is algebraic over A , p = tp(Ti, A), then (p, B) E F for some WteBcA. Ax(II.4): If E E B C A , q(Z;8) E p = tp(G, A), IBI < A(F), then ( p , B ) E F. Monotonicity

Ax(III.1): If B c A E Dom p, (p, B) E F, then p A E F(B). Ax(III.2): If B E A E Domp, IAI < A(F), (p, B) E F then (p, A) EF. Ax(1V): If Q = tp(si-6, A), p = tp(si, A), B c A and (q, B) E F then ( p , B ) E F. Notation. If p = tp*(C, A), B c A, then we say (p, B) E F when for every E E C, tp(8, A ) E F(B). We say tp*(C, A) is F-isolated if tp(E, A) is F-isolatedfor every E E C. Note that when Ax(1V) holds, tp*(E, A) E F(B) iff tp(E, A) E F(B).

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[OH.

Iv, 5 1

8y?ndry and Transitivity

Ax(V.1): If q = tp(6-6, A), p = tp(6, A u 6) and B 5 A, and (q, B) E F then (p, B u 6) E F. Ax(V.2): If ICl < h(F), q = tp*(6 u C, A), p = tp(6, A u C), B c A and (q, B) E F then (p, B u C) E F. Ax(V1): If ByC c A, tp(6, A u a) E F(C) and tp(a, A) E F(B) then tp(a, A u 6) E F(B). Ax(VI1): If B E A, tp(6, A u C) E F(B u C), and tp*(C, A) E F(B), then tp& u C, A) E F(B). Continwity Ax(VII1): If A, (i < 8 ) is increasing with i, ~ E S ~ ( U , < , , and A,) p 1A, E F(B) for i < 6,then p E F(B). Ax(1X): If A,, B, (i < 6) are increasing with i, 6 < cf A(F), p~ sm(utcd and p 1 E F(Bf),then ( p , ul
Ax(X.1): If 6 E A, k(3z)(p(z; a), then for some pair ( p , B) E F, B c A, (p(z;6) E p E S(A). Ax(X.2): If A, c A, for i < j 5 a,6 E A,, k(3z)(p(z; a), then for some p ES(A,),(p(z;6) ~ pand , p A, is F-isolated for every i s a. Ax(XI.1): If (p, B) E F, p E @(A), A c C , then there are B c C, and q E Sm(C)such that p E q, (9, B') E F. Ax(XI.2): If A, E A, for i < j 5 a and p E S ~ ( A is , ) F-isolated, then for some q EB"(A,) p c q, and q A, is F-isolated for every i I a.

r

r

Generalized Transitivity Ax(XI1): If C,, Ca E B E A , p E Srn(A),( p B, C , ) E F, and ( p ,C,) E F, then (p, C,) E F. Remurk. We make no use of Ax(XI1) and use Ax(X.2) and (XI.2) rarely.

LEMMA

1.1: (1) Ax(II.4) implies Ax(II.3) and Ax(II.2); Ax(II.2) implies Ax(11.1). If A(F) = No,Ax(X.1) impliefl Ax(II.3). (2) Ax(V.2) and (IV) imply Ax(V.1). (3) Ax(II.1)and Ax(VI1) imply Ax(1V). (4) If aX(V.l), (IX) Itold, then Ax(V.2) lsoIde for ICl < cf h(F) 80 it h o d ls when h(F) is regular.

m. IV, 8 13

THE SET OF AXIOMS

155

( 6 ) Suppose Ax(X.1) h0ld.a f o r jnite A. Then Ax(XI.l) implies Ax(X.l),and Ax(XI.2) implies Ax(X.2). ( 6 ) Ax(X.2)implies Ax(X.1); and Ax(XI.2) implie8 Ax(XI.1). (7) If Ax(1X) hot%, then Ax(XI.l) implies that Ax(XI.2) h o b f o r a < cf h(F).

Remark. Except for trivial cases Ax(VI1) implies Ax(IV) (as we may take C G B assuming Ax(II.1)).

Proof. (3) Apply Ax(VI1)with B, A, an6,0 for B, A , a, C. DEFINITION1.1: (1) A set A is called ( F , p)-saturated if for every (p, B ) E F , Dom p c A , lDom pI < p, p is realized by some sequence from A . (2) p ( F ) is the first cardinal p such that for every h 2 p and A ; A is (F,p)-saturated iff A is ( F , A) saturated (if there is no such cardinal, p ( F ) = coO f course). (3) A is F-semi-8c~tUTatdif it is ( F , lAl)-saturated. (4) A is F-saturated if it is ( F , IAI +)-saturated (or equivalently, ( F , p(F))-saturated;or (F,p)-saturated for every p). LEMMA 1.2: (1) Assume Ax(1V) and (V.1). A is (F,p)-saturate&if wknpES(A'),A' E A,IA'I < p , B s A',@, B)~F,pierecclizedinA. (2) Assume Ax(VIII),(111.1).( p , B ) E Fi,fforevery$niteC G Dom p , P t ( B u C) E F(B). ( 3 ) When p < h I00, ( F , &saturation impliea ( F , p)-ttaturati~n. (4) When p 2 p ( F ) , ( F , p)-saturationie equivalent to F-saturation. ( 6 ) Every F-saturated set is F-semi-saturated. ( 6 ) Assume Axiom (X.1). Then every ( F , X,)-saturated set is a model.

Proof. Immediate. The difference between (1) and the definition is that here p is a one-type, and not an arbitrary m-type. This is where Ax(1V) comes in. (2) is proved by induction on lDom pI . DEFINITION1.2: (1) An F-mtruction is a triple .d = ( A , {a,: i < a}, (B,: i < a))

such that pi(&) = tp(a,, U {a,: j < i} U A ) E F(B,). (2) U,= U,(.d) = {j < i: (B,n a,) - ( A U U {ZB: fl < j}) # 0).

PRIME MODELS

156

[CH.

Ivy5 1

A set U E u is close& (i.e., &-closed) if (Vj E U ) ( U , E V ) . U: = U~(d)isthesmallestclosedsetcontainingUf-B,* = B,u(B,uZ,:j c U,"}. A set B of elements is closed if for some closed U c u, UjSu 7i;, E B E UjSu a, u A , and j E U implies B, E B. (3)ForUc(j:j
d(4. (4) When it is clear what d is, we omit it. We denote 4 0 ) by A ; we also sometimes ignore (B,: i < u) in defining d. (5) Instead of a,we can use any well ordered set I,and in this case al(1) ( = after last) will be an imaginary element of I which comes after all its (real) elements. In this caae d ( s ) will mean &((t: t < 8, t E I));so Dom d = d(al(1)).

DEFINITION 1.3: C , is F-constructible over A , if for some F-construction d,d ( 0 ) = A,, Dom d = Co. DEFINITION 1.4: (1) If C is F-constructible over A , and C is F-saturated [(F, p)-saturated] then we say C is F-primary [(F, p)-primary] over A. (2) C is F-primitive [(F, p)-primitive] over A if A E C and for every C', such that A E C' and C' is F-saturated [(F, p)-saturated] there is an elementary mapping f,from C into C', where f 1A is the identity. (3) C is F-prime over A if it is F-primitive over A and F-saturated, C is (F,p)-prime over A if it is (F, p)-primitive over A and (F, p)saturated. (4) If in any of the above C is the universe of a model M ,we say M is an F-primary [ - . -1model. The same holds for Definition 1.1. LEMMA 1.3: Assume Ax(IV), (V.1). (1) If C is F-constructible over A, then C has a F-construction over A in which l ( 4 ) = 1. (2) If for j < a, A j + l is F-mtrmtible [pimitive] over A,, and for limit 6 I u,A , = A,, then A, is F-constructible [primitive]over A,. (3) If for 0 < j Iu,A is F-prime [primary] over < A,, then A, is

ul<, ,

u,,

F-prime [primary] over A,. (4) A is F-primitive over A, and F-constructible over A. If A is (F, p)saturated, it is also (F,p)-primary and (F, &prime over A.

Proof. Immediate.

a. IVY5 21

EXAMPLES OF F

LEMMA 1.4: (1) If A(F) is regular, then in De$ndtdon 1.2 A(F) > IB:I

157

+

IUfl. So if in addition Ax(III.2) 7rOl&B; then w.1.o.g. we can a88ume U: = U,,B: = B,. (2) Asclume Ax(III.l). If B ie F - m t m t i b l e over A , C E B, then there ie C' c By C c c' euck that A V c' is F-conetructible over A , and l q < ICl' + w 9 . DEFINITION 1.6: We say A is F-atomic over B if B c A, and for every

a E A, tp(8, B) is F-isolated.

QUESTION 1.1: Are there any more connections between the axioms ? DEFINIRON 1.6: For a class K of models, we say "M is prime over A for K (or among the members of 9) if M E K, and A c N E K implies there is an elementary embedding of bl into N, f A = the identity; let K-prime stand for "prime among the tc-compact models of T.Similarly for "primitive for K".

IV.2. Examples of F's Here we present some examples of F's, and show which axioms they satisfy. For F = Fy,, we get the usual concept of primeness. DERTNIITON2.1 : (1) Let p be an m-type, or even a type with infinitely many free variables. (i) p is a E"t,-tYpe if 1p1 < A. (ii) p is a PA-typeif lDomp1 < A. (iii) p is a Fi-type ifp is almost over some B of cardinality < A. (2) For 2 E (8, t} = { ( p ,B ) : for some A, myq; B c A , p ES"'(A), I BI < A, q an Fg-type over Byq E p , q k p ] . FR is defined similarly, but with q almost over B. (3) M is Fg-compact if every c-type over ldll is realized in M. Remark. For understanding Ff, let us for a moment work in aeq,and define card* A = min{l BI : B E acl A = acl B}. Then Fi is defined as F; replacing IAI c A by card* A c A.

DEFINITION 2.2: Fj; = {(p,B):for some A, m , p E S ~ ( A )B, G A , IBI < A and p does not fork over B}. DEFINITION 2.3: F i = { ( p ,B ) :for every 'p there is q* E p IByIqo(pI

A,

158

PRIME MODELS

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Ivy8 2

such that qq t - p I+ cp =def{tp(Z;a): p ( Z ; a ) ~ p and } IBI 5 h + IT]; c f h > IT1 * IBI < A} Note that p 1 +p differs from p p in that the latter also contains formulas of the form +Z; a). In the above deftnitions the letters t , a, a, f,and 1 stand for type, set, almost, forking, and local, respectively. To be precise we should say “Ff in CE” or write Fg[Q] for all X E { a , f ,1, a, t}, that is the definitions depend on the model Q. We use the square brackets to differentiate this from F:(IQI) = {p: (p, IQI) E F ~ } . We use Ff only when T is assumed to be stable. We use the following lemma freely in dealing with Ft.

LEMMA 2.1: Let B

G A , p = tp(Z, A). Then (4)* (1) o (2) o (3). If T is stable all the conditions are equivalent. (1) ( p , B ) EF:, i.e., there is q E p , q t-p, q almost Over B. (2) For a m type q almat Over B, q t- p . (3) StP@,B ) t - P . (4) stp(Z, B) k stp(Z, A).

Proof.Trivially (1) =r (2). By theremark to Definition III,2.1(2) =- (3). Suppose T is stable, (3) holds and F’ realizes stp(F,B),by 111,2.6(1) stp(F,B)does not fork over B, so by (3)(and 111,1.18)also p does not fork over B, but F’ realizes p (by (3));hence, by 11,2.6(1) stp(?,A) does not fork over B, so by 111, 2.9 stp(F,A) = stp(C’,A). By the choice of F’, (4)holds. Hence by 111,2.9 stp(F,A) = stp(F’,A);so (4) holds. Letusassurne(4)or(3)andweprove(l).Letq = {F(Z, G):rp(Z, a) ~ p , p(Z, a) is almost over B}. It suffices to prove q t-p. For any O(Z; a) ~ p , by (4) or (3)there is E E FEm(B)such that E(Z, a) t- O(Z; a) (remember that by 111, 2.2(1) FEm(B)is closed under conjunctions). Let #(Z) = (Vjj)[E(Z,fj)--f O(g; a)]. Clearly the parameters of # are from A ; also E(Z;2) t- #(Z) E #(a) hence # is almost over B. As E(Z, Z) k O(5;a), C#[Z]. As #(Z)is over A, #(Z)~p and as #(Z)is almost over By#(Z)E q. From #’a definition it follows that #(Z) lo(?.; a) (as E ( 3 , Z)) hence q I- O(3; a). So q t p and we finish. In the following lemma we indicate many trivial facts concerning the Fg, their relation to previous notions, and the connections among the various Fg.

m. IV, 5 21

EXAMPLES OF F

159

LEMMA 2.2: (1) A is F&-sdurded if it is the universe of a model; if Ax(X.1)hot%, ale0 the converse is true. (2) We get equivalent definitim of Fa,, FL, if we assume q consists of one formula and in all me8 (in Definition 2.1(2), 2.2 and 2.3) we can a%&um thd q i 8 c h e d under conjunction8 (for FE, by 111, 2.2(1)). (3) For x = t, 8, a, f, h(Fg) = A, and h(Fi) = A+ + IT1 when of h I ITI,and h(Fi) = h othemuiee. (4) AqJ s A, A%) p(F3 + XI, p ( F 0 = a, p(Fi) 5 A+ + ]TI+and when of h > IT[, p ( F i ) s h (for F! w e theproof of (7); for FL we 111, 1.3,III, 1.4).(Bee Exercise 2.2.) (6) A is F ~ - 8 d U T d e d if for 80112e h - m w t Af, lMl = A. FW x = 8, a, [t], &J Fz-8durated ifl &f is Ff-compact. (6) A i8 F1,-8dUrdediff for some h-SdUrded Af, 1M1 = A. (7) FOPh > No, A is Fi-saturated ifffOr 80me h-8dUrded Af, lMl = A. (8) If h > I TI,Ff, = q,and if cf h > I TI , Ff, = FA. If h > I TI,p is F,0-iSO&C?d iffit i 8 pA-lkOlated. (9) If A s K , x = 8 , t , a, f, 1 then Ff E F. (10) Fi 2 FAE Fi c Fi E FL (the last in~luSimfor 8tUbk T, Of +

x

[v]

course).

For x = t, 8 , a , f , 1, ( p , B ) E Ff in (E iff (p", B ) E Fg in CQ(where A s lal, p E Bm(A,(E), B c A). (12) For x = t, 8, a, f, 1, if a?is an E-comtructim in Q, t h it is an (11)

l+mn&wtion in QBq. Hence,for A, C c 161: (i) If C ie e-constructible (primitive)over A in Q, then this hold%also in geq,also for ecl C, (ii) For x = s,a , if C is F;-saturated then in aeqacl(C) is F;-saturated. (iii) 8imilarly for Ff-primary (prime). (iv) If A E 6 and in aeq acl(A) is Fi-saturated and A = acl A n 6 , then in 6 , A is F:-saturated (z = t , s, a). (v) If B E A E a, A is F;-atomic over B in 6 iff this holds in tieq,i f this holds when we replace A andlor B by their algebraic closure. (vi) Let x = s,a. If B E A E 6, and in 6""A is F:-primitive over B, then in a A is F,"-primitive over B. (vii) If B E M E 6 , and in 6""M"q is F:-prime over B, then in 6 M is F;-prime over B. (viii) A E 6 A = acl A in a. Then A is Fi-compact i# acl A (in6"") is Fi-compact. ) (13) Ifp€E"'(A),pEF,O(B),I, r B i s s t a t i o n a r y t h e n p € F ~ ( B (for 8tuble 2"). ( B y 111, 2.14.)

Proof.(7) One direction is trivial. Now assume M is A-saturated, p is almost over B c 1M1, IBI c A. We must show that p is realized in M . Let a realize p. Define IN1 (i 5 w ) such that at realizes a,). Then Bo realizes p . (The proof is like the proof for tp@, B u U,<# stable T, in III,2.13( 1). For y E p let E be the equivalence relation over B that p depends on. Now since E has finitely many classes there is i such that Z,and are in the same class. Thus at satisfies B(Z;7i) and so does ao). (12) Notice that all the F’s satisfy Ax(II.3), hence acl(A) is Fconstructible over A.

Remark. Notice that although F&-saturation and F;,-saturation are the same, not necessarily F;, = F,: (see Exercise 2.2(4)).

Now we check the axioms one by one. In first reading you can omit F;, F:. LEMMA 2.3: A d m (I)i8 8di8jitd by Ff for x = t , 8 , a, f,2.

Proof.Immediate. LEBfMA 2.4: (1) A & m (1I.j)j = 1, 2, 3,4 are 8db@d by for x = 8 , t , a,f,1. (2) Moreover, if a ie algebraic over B, B c A , IBI < h thm tP@,A ) E W B ) .

Proof.(1) Immediate. (2) By 2.1. fOr LEMMA 2.6: A d o m (111.1)i8 8diSF by q,

5=

t , 8, a , f , 1.

Proof. For x = t , 8,2 this iS immediate. For x = a this follows from 2.1(3), and for x = f this is immediate (see 111, lJ(6)). LEMMA 2.6: AXim (111.2) i8 Sat&@& by Pf, fOr

X =

t , 8 , a,f,1.

Proof.Immediate.

Proof. So let p = p(B) = tp(7i, A), q = q(3, g) = tp(7in6, A), B c A, and (q, B ) E R, and we should prove (p, B) E Ff. For x = f this holds

CH.

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EXAMPLES OF F

161

by 111, l.l(f3). If x = t , 8, there is an Fz-type r = r(3, fj) over Byr I- q. By 2.2(2) we can m8ume r is closed under conjunctions. So r' = {(3g)(p(B,fj; 8): (p(% fj; 8) E r } is an Fi-type over Bylr'l = Irl and r' I- p . Hence (p, B) E FZ. The cases x = a, 1 are treated similarly (for x = a notice 111,2 4 1)). LEMMA 2.8: (1) Axiom (V.1) ie eatisfied by F;, for z = 8, t , a, f, 1. (2) Axiom (V.2) is satisfied by Fz for x = 8, a, f; and also for x = t , when A is regular. If we assume cf A > ICl it holds for x = t , 8 , a,f, 1. (See Exercise 2.3,4.)

Proof.(1) Follows from (2), by 1.1(2) and 2.7. (2) Weassumep = tp(i3, A u C), q = tp,(si U C , A ) ,B G A , ICl < A and (q, B) E F;. We must prove ( p , B u C ) E FZ. If x = f this follows from 111, 1.1(5), 4.14. If x = s,t[x = a] then clearly there is a type r [almost] over B, such that r -I q, r = U { r F : B ~U a0, where rcF tp(c,,A)is an F$-type. Let r' = {cp(E; E, 6):cp(B,,, Z6; 6)E r, 6 E B, 8 E C).

Now r' is a type [almost] over B U C , and r' I- p. If x = 8, a ( B u C( < h hence r' is an Fi-type, so we finish. If x = t , ICl < cf A, then lr'l = 1.1 < A and we finish. So let x = I , ICl < cf A, and let (p = cp(Z; Z). Assume Z = (zo, . . ,zk - ,), then there are formulas (pl = (pl(Z;Zll; El,,) for 1 < n( = 2k) such that for all partitions {i,, . . .,i,} u {j,,. . .,j t } = k there is I < n such that ( p l ( f ; zil, . . .,z,,; z,,, . ..,z,J = cp(Z; zo, ..., zk-,). By assumption for every 1 < n and Z1 E C there is a type r&) c tp(anEl, B) such that r&) k tp(SinEl, A) t + pl. Define r@,) = {8(z;E,, E ~ ) :8(g8,B ~ , z2) , E r1(El)},

.

r=

U {r;(8,):El EC,1

< n}.

It is easy to check that Irl < A, r I- p I+ (p, x = 1.

so we finish the last case

LEMMA 2.9:Axiom ( V I )is satisjied by F$for x = t , s,f,if h is regular or c f h > I TI,also by Fi,and if T is stable, by Fftoo. Procf. We should prove that if B , C E A, (ql,C)EF; where q1 = tp(6, A U a), and (p, B) E F : where p = tp(8, A ) then (p,, B) E F;, wherep, = tp(zi, A u 6). Let q = q1 r A .

162

[OH. Ivy8 2

PRIME MODBL8

For x = f this is 111,4.13. For x = s, t , we first claim that if Ti, realizes p , 6, realizes q, then 8,^6, realizes tp(an6, A). Because tp(Zl, A ) = tp(8, A ) for some elementary mapping f,f 1 A = the identity and f (a,) = a; so w.1.o.g. Ti = 6,.AS C E A , (q, C) E F;, tp(6, A ) I- tp(6, A u a),SO tp(iZ-6, A ) = tp(Tin6,, A), so the claim is proved. Hence if 8, realizes p necesaarily tp(aln6, A ) = tp(a-6, A), SO tp(iZl, A u 6)= tp(& A u 6) = p,. SO p I- p,. But for some F;-type p' over B, p' E p , p' I- p , hence p' I- p1 hence ( p , , B) E q. For 1: = a the proof is similar. We first show that if a,, 6, realize stp@, B), stp(6, C) respectively then tp(Ti-6, A ) = tp(Siln6,, A). Hence stp(Z, B) I- tp(Ti, A U 6) so we finish by 2.1. We are left with x = 1. Let ~ ( 5 3 Z)~ ; be a formula. As in the proof of 2.8 we can find n < w and, for 1 < n, 'pI = cpI(Za; ?&,a,) such that for every E E A u 6, there are I < n and E' E 8, 8' E A such that hp(Za; 8) = (pI(Za,6,8'). As (ql,C)E F there ~ are types p i c q1 YC, p i = qi(zh), where $1 = #I(%; zap 21) = FI(Za; 536, zI)*Let (I\ = p1 1' $1, [dl < {0&3, $): i < (p:(} be closed under conjunctions (by 2.2(2)) where i$E C. For every 1 < n, i < lp:l let 0&(Za,

z<,i, 2,)

=

(vZ6)[ef,1(z5~

Zf,J

dzao,

-+

QI.

As (p,B) ~ F f hthere , are types pf,l= pf,1(3a)c p 1 B such that pf,rk p f O&; and let p* = Uf,I pr,[.So as in the case x = 2 we1assume hisregulasorcfh > ITI,wehaveIp*l < h ; a n d o f c o u r s e p * s p r B . Now it is easy to check p* I- p1 v. +

r'

THEOREM 2.10: A d m (VII) is satisfied by Fg when a,f, T is stable, or x = t , I , h is regular.

2 = s

or x =

Prmf. Let B c A , p = tp(si, A U C), p = tp*(C, A), r = tp& u C, A) and (p, B u C) E FRY(p, B) E F;. We must prove B) E F; for 8 E C where = tp(a-8, A). CLAIM 2.11: For sets A , C, suppose for every 8 E C pi I- qe = tp(8, A), and p' I- p = tp(& A u C), p' is over A u C, (p', pi are closed under conjunctions).Define ?-&=

{(3%d)[#(3a, E l

E y

5,%d; 6')

a E cy #(za;

Then ri,c I- ra,e = tp(ii-E, A ) .

4,E2)]: a, 6,) Ep', 0@CyEd; )'6

A t)(?t&, z y

E pied}.

OH. IVY8 21

EXdlldPLJES OB

163

F

Proof of 2.11. Let 'p = 'p(ZayZ, 6,) E T,,,, and we shall prove r;,E I- (p, and tGs will suffice. By the hypothesis p' I-p and t'p[7i, 8,6,] hence p' I- 'p(Zdd,a, 6,) ~ p Hence . (as p' is closed under conjunctions) there exists # = 2, a, 6,) ~ p (we ' add dummy variables if necessary to make room for 2) such that a E C,6, E A and #(ZayE, a, 6,) I- (p(ZaYE, 60). Hence b * [ Z , a, 6,] where 'p*(Za, 4, 6,) = (E6)[#(Za, Z, Zd,6,) + (p(Zad, $, 6011. I- 'p*(ga, Za;6,) hence for some Hence (p*(Zay%a; 6,) E qa-d, SO @ a ~ %; 6 8 ) E d - d , F(Ed)(vZd)[O(Z&Z d ; 63)+(p*(Ze~Z ~6 Y a)Ia

It is easy to check that (33d)[$@tw

%Y 4, 61)A

@ a ~ Zdp

6311 I- CP(%F,, %Y

60)

and that the antecedent of this implication belongs to &. So we finish. Continuation of the poqf of 2.10. Clearly if p', qi are over B u C, B respectively (and w.1.o.g. C n A E A), then T $ , is ~ over B ; so the caae x = s is immediate. If x = t , then I C 1 < A as (p,B U C) E Fi and there are p', qi satisfying 2.11 such that 1 p' 1, I < A. As h is regular in that case it follows that IrL,,l < h so we finish. The C= 2 = f follows by 111, 4.16. So now we deal with the m e 2 = I ; again we assume h is regular. Let 2 E C, and 'p(Za, Za; E ) be a formula. By the hypothesis there is p' c p closed under conjunctions, lp'1 < A, p' is over B u C and p' I- p 'p. Let p' = {#l(Z,,, E, a(i),6 :): i < a < A}, and 'pr(ZayZd(:);6*,6:) = (VZ,J[#1(Z4yZ,,Zao, b:) -+ 'p(Z4, ZEy6*)], for a(;)E C, 6 : E B, 6* E A . By the hypothesis (q,-dcf),B) E Fi so for some &acn c qpdct, 1 B y IQ&l < A d-aco I- Qa-aco t' 9:. Let +

q;-&l) =

@*,I: j < .(i) < A}

{4.,(5,%(1);

(again w u m e it is closed under conjunctions), 6;,,E B. Let e&(Z8~

za; gtj)

= (3Zd(l))[#l(% %Y

ZrZ(f)>

6:)

A

ei,j(3t~Zd(f); 6 : , j ) 1 .

Let r& = {O&(z8,Z,; 6f.J :j c a(i),i c a}.It is easy to check r;,, E r,,a is over B and of cardinality < A . It suffices to show T L , ~k r5, I+ Q. For let 'p(?i?,,, Za; 6*) E r,,,, (8* E A ) so k'p[iZ, E, 6*]. Hence 'p(Z,,, 2, b*) ~ p , hence for some i < a, #&,, E, a(i),6:) I- 'p(Zay8, 6*)hence b(fia)[${(Z8Y

SO

t'p:[~, a(i),6*,671, hence

q)

'p@aY

EY

6*)].

'pr(~~, it&:);6*,6:)

E q ~ - d ( { ) hence for

164

[CH. Iv,6 2

PXME MODELS

some j < a(i) of d

6:,,)

~f,j(% gd(f); @,j)

A

I- &Zd, Zd(,); 6*, 6:). So by the definition

(3zd(f>)t#f(Zd,

hence

% zd(f;, 6:)

%Y

zd(O;

ef,j(Zdt

%d(f),

#f(%

hence A

'

qP(zdY

6f,j)1 I- d z d Y

%; 6*)

6*)1

d&(za,zd;67,J I- d z d , zd;g*)

so we finish also the case x = 1. We are left with the case x = a. First we prove

LEMMA 2.12: Let T be atable (1) If B E A, tp(Ti, A) ~ q ( B ) , E ~ a c l U l ( Ti)thtp(Ti^c', B A)EF$(B). (2) Let B c A, tp(Ti, A) E Fa@) iff tp(Ti, ad A) E F"(clc1 B).

Proof.(1) Let O(z; 8, 6)be algebraic, 6 E B and C@, 8,6]. By 2.4( 1) for ,$I we can assume s(z; a, 6) I- tp@, B u a). Let e ( l q a, 6) =

. .,B,,},

Z0 = 5. By 111, 2.6(3), (1) there are Ti', 5; such that stp(TinZon ^Z,, B) = stp(Si'^ELn. .^$, B) and stp(Ti'nZLn- ^-' cn, A ) does not fork over B. So stp(Ti', A) = stp(8, A), (by 111, 2.6 and 111, 2.9) hence we can msume Ti' = Ti. So (EL,. ., 5;) is a permutation of (EOy ,En), hence we ctln w u m e Ei = $. So tp(Si^Z,, A) does not fork {Eo,.

- ..

--

.

...

.

over B. By 2.1 it suffices to prove stp(8"Zo, B) I- tp(Ti-Zo, A). If this is not true there is Ti*^8*, stp(Si*^Z*, B) = stp(Ti^ZoyB), tp(Ti*^Z*, A) # tp(Ti^E,, A). So necessarily tp(Si*^E*, A) forks over B. Again we can assume Ti* = Ti, hence for some i E* = Ef, contradiction. (2) Immediate, by 2.1 part (4), and 111, 6.3(4). Conclusion of thproof of 2.10. We have to prove it for x = a. By 2.2(11), it suffices to prove it in P. Let C' = acl(B u C), so by 2.12(1) tp*(C', A) EF$(B)and by 2.12(2) tp(Si, A u C') E F"(B U C') = F"(C'). SO tp@, cl) I- tp(Ti, A u C'), stp*(C', B) I- tp(C', A). Now use 2.11 with p' = tp*(Ti, B u C'), qi a type almost over B, which is over A, and such that qi I- tp(E, A) (qi exists by 2.1). This proves the assertion.

LEMMA 2.13: A d m (VIII) i8 8athfied by

for x = t, 8, a, f.

Proof of 2.13. Let A, (i < 6) be increasing, p E ~ ~ ( UAf), , , ,pf = p 1 A, E F$(B).We must prove ( p , B) E F;. For x = f,it follows by 111, 1.l(7).

OH.

Iv, 9 21

EXAMPLES OF

F

165

For x = 8, t let q, be an Fg-type over B yq, E p,, q, t p,. I f x = s , p rBisanFi-type,p t B t q , t p , h e n c e p t B t U , , , p , = p . If x = t , qo is an Fi-type, qo t p o ;and clearly q, G p o ,hencep, t q, t p,, hence !lo I- Po Uf < d Pr = P. For x = a, by 2.1, if li realizes p, stp(8, B) t p , hence stp(li, B) I- p hence (p, B) E F:.

LEMMA 2.14: Adom (IX) is satisfied by F : for x 6 < Cf A, it8 ConclUsiOn hkh.

=

t , 8, U , f; for 1, if

Proof. Let 6 < cf A,. A,, B, are increasing with i, p ES"'(U,<~ A,), and

( p I A,, B,) E F$. w e must prove (PY Ul
DEFINITION 2.4: (1) Al(T) is the first cardinal A such that there are no A, (i < A) and p ES"'(U,<~ A,) such that i < j =-A, E A,, and p t A, + splits over A, (if there is such A, and a0 otherwise). (2) A,(T) is the first cardinal A such that there are no A, and p,, E #"'(A,,)for q E h 2 2 such that (i) Pr- z P,,-<1>3A,- = A,,,> for rl E (ii) if q is an initial segment of Y then p,, c p v . (3) A3(T)is the first cardinal A such that there are no m-types p,,, q E h > 2such that (i) p,,-v P,,-<~>is inconsistent for q E h>2, (ii) if q is an initial segment of Y then p,, c p v .

DEFINITION 2.6: (1) A1(T)[Aa(T)]i s the first cardinal A such that for every B c A, p ES"'(B)there is C c A, ICl < h and q ES*(A), such that p E q, and q does not split over B u C [and q ( B U C ) t q]. (2) A3(T)is the first cardinal A such that for every m-type p over A, there is an m-type q over A, 1q1 < A, and p U q t p' E #"'(A)for some p'.

THEOREM

2.16: (1) K ( T )5 Al(T) IAa(T) < h,(T). (2) T is unstable in every p < 2"scT). (3) If T is stable then A,( T)5 I TI +

.

166

PILflldE MODBLS

[aIV, . 82

(4) h3(T)= KO iff T is totally trammndental (see Exercise 2.23). (5) h’(T) 2 P ( T ) 2 h3(T).

Remark. If 2ITI* > 2ITl, then (2) implies (3) trivially.

Proof.(1) If T is stable K ( T )5 X,(T) by 111, 3.1, as strong splitting implies splitting. If T is unstable, it is an easy exercise. W e have, in fact, proved that h,(T) 2 ha(T) in I, 2.7. Now ha( T) 5 h3(T)is trivial. (2) Let p < 2<WT);for some h < h3(T),p < 2A; taking the first such A, we have 2 < A 2 p < 2”. Hence there are p,,, q E h>2, as in Definition 2.4(3). For every q ~ ” > 2choose finite B, so that (P,,-<~>1 B,,) u (p,,n<1)1 B,,) is inconsistent, and let B = (Jve*>S B,,. For Y E ”2, pv = U,<,, pvr, is consistent, so for some qv E Sm(B),p v u qv is consistent. Clearly Y # q ==qv # q,,, hence ISm(B)I1 2” > p 2 2<” 1

pi.

(3) Suppose T ie stable, &(T)> lTl+,so for h = lTl+,there am I), (q E h>2) aa in Definition 2.4(3). Let {A,: i < ITI} be a list of all

finite sets of formulas in L. We define by induction on i 5 ITI sequences a ( i ) < IT1 and i < j * a(i) < a ( j ) ,qi a(i) = 9,. For i = 0, 7, E or i limit ohoose in any acceptable way. For i + 1 choose q,+’ = q , h ~ ~ A ’ 2so that P ( p , , , + l , d ,2) , is minimal. For Y = qTI,pvn
(1) #(T)5 h,(T)for 1 = 1, 2, 3. (2) Swppae A, E A, for i < j 5 a and p is an m-type over A,:

THEOREM 2.16:

(i) There ie q ES~(A,), p E q and B C A,, IBI < h,(T) 8wh t7mt P ” q t @ n A d t-q IA,. (ii) There is qESm(A,) p E q and r E q, Irl < h3(T) swh t h t puriFqtA,wherer;=r la,. (iii) There are q E S ~ ( A , ) ,p E q and B c A,, IBI < K ( T ) ,and c c q t B, Irl < K(T), m h that p u (r A,) hm no extdm in Sm(A,) whichforks over B n A,. (iv) There ia qE@(A,), p c q and B E A,, IBI < hl(T)m h that: p u q 1 ( B n A,) hm no extewion in P ( A , ) which qlits over B n A,; for euch i.

Proof. (1) For 1 = 1 it is easy, and for 1 = 2, 3 it follows from (2) for a = 0.

OH.

I v , 8 21

167

IOXAMPLEB OF F

(2) As the proofs are similar, we prove only (ii). Suppose the con-

olusion fails, and we get a contradiction. We define for every sequence q of ones and zeroes, Z(q) < A3(T) by induction on I(q), cm ordinal P(q) 5 a and an m-type q, over ABtn,,such that Iq,l < Il(q)l + KO and p u q, is consietent, and for i < p(q),p U (q, A,) has a unique extension in Bm(A,),and v 4 q =+ q, s q,. For the empty q, q, = 0, B(q) = 0 and when l(q) is a limit ordinal 8 let p;, = U,
LEMMA 2.17: (1) If A r A3(T), t 7 m ediofi a&ma (X.l) and (XI.l), and if A r &(T), then e a t i e h &o azbm (X.2) and (XI.2). (2) If A r Aa(T), t h PAeatis@ axioms (X.l) and (XI.l) and if A r Aa(T),then PA satiojk4 ale0 &om9 (X.1) and (XI.2). Proof. Immediate, by 2.16(2)(i),(ii) (for A r A,(T)) and Definition 2.6 (when A 2 A1(T)). LEMMA 2.18: (1)Adoma (X.l), (XI.l), (X.2) and (XI.2) are eatkclie& by F(,,PAifA > I T I , T ~ z e m T G e t a b k i n 8 0 m e p< 2A. (2) Adoma (X.l), (XI.l), (X.2) and (XI.2) are satiejkd by if A r K(T). (3) A d m (X.l), (XI.I), (X.2) and (XI.2) are eatiefied by F{ if T io etable. (4) Ax(X.1) ie eatisjied by Fi if T ie stable, A r (TI (e.g., by every countable stable T). (6) A d m (X.l), (XI. l), (X.2) an& (XI.2) are eatiefid by FW, if T k totally tranecendmtal. Proof. Part (1) follows by 2.15(2)(3), 2.16(1) and 2.17(1), (2). Part (2) follows by 2.16(2)(iii);part (3) by 111, 1.4. Part (6) follows by 2.15(4), 2.16 and 2.17(1). So we are left with (4). Let p be an m-type over A, and {cp,(P; g,): i < ITI) a list of the formulas in L. Define by induotion on i < I TI formulaa $,(1;a,) (a, E A)

PRIME MODELS

168

[OH. I v ,

52

such that p , = p u {$,(3; Z,): j < i} is consistent. If we have defined $,@,a,) for i < a < ITI,clearly p , is consistent also when a is a limit ordinal. Choose $,@; a,) so that Bm(p,+,,tp,, 2) is minimal. It is easy to check that for some q a ~ S ; ( A )P , ~ + ~Hence I-~~ plT, . has a unique extension in Sm(A):U,< I TIq,. So Ax(X.1) (and a little more) holds.

LEMMA 2.19: When x = t , 8, I , f,F = Fsf,F 8ati8&? Ax(XI1). Also for x = U, T 8table t h b holds. Proof. For x = t , 8, for some Fg-type q E p C,, q I-p 1 B (as (P t B, Ci)E Fg) hence q I- P t Ca; but as ( p ,Ca) E R,p 1Ca 1p , hence q b p , SO (p,Ca) ~ F s fFor . x = u the proof is similar using 2.1(4). For x = f Clearly p does not fork over B (asC, E B, (p,0 2 ) E Pi) and p t B does not fork over C,, hence by the transitivity of forking (111, 4.4) p does not fork over C,. We are left with x = I , so for every 'p, for some q E p 1 Ca, 191 < A, Q I-p 1' tp, it is also clew that p t C, kp IC,, hence p 1C, 1 q, so for every $ E q there is a finite re E p 1C,,re 1 $. Now r = {rs: 41, E q} E p C,, and r I- q I- p 1 'p, so (p,C,) E Fi.

u

+

DEEWITION 2.8: Ff:= {(p,A): p E#"(B),where m < h and p does not split over A}.

w, A

E B, IAI <

LEMMA 2.20: (1) Ff:sat&$es Axbna (I), (11.1, 2, 3, 4), (111.1, 2), (IV), (VII), (VIII) urvd (IX). = A, p(FR) = a). (2) (3) If h 2 hl(!P),then Ff:8 d i 8 f i 8 &onZ (X.l), and if h 2 &(T)then Ff satisfies axioms (X.2), (XLl), (XII.2) and every p € S " ( A ) is Ffisolated. (4) If T doee not have the independence property t7m P ( T ) 5 I TI . +

Proof. (l),(2) are immediate, and (4) is a restatement of 111, 7.6. (3) is similar to 2.16.

DEFINITION 2.7: Fg

= {(a,B): psSm(A)for some m < w , B E A , IBI < A, and there is q E p t B, 1q1 < X such that q has no extension in Bm(A)which forks over B}.

Remark. We can wume q is closed under conjunctions.

LEMMA 2.21: (1) (V.1) an& (IX). (2) If i8 regdur

8di8fies uxbm8 (I), (11.1, 2, 3, 4), (111.1, 2), (IV),

F{ 8di83U d80 (v.2).

CH. I v ,

I

II.1 II.2 II.3

II.4 III. 1 III.2

Iv

v.1 v.2

vl

VII VIII Ix

f 21

+ + + + + + + + + Aregulfbr + Areguler + +

x .1

A2

x.2

A 2

XI. 1 XI.2

A 2 A 2

w

EXAMPLES OF F

+ + + + + + + +

+

+ + + +

+

T stable T attable T stable T St8ble T St8ble

+

+ + + +

+ +

169

+ + +

+ +

+ + + + +

+ + + +

+

+ + + +

+ + +

Skolem funations Skolem funations

+ + + +

T stable T stable T 8t8bh + Table + T stable x Am& x X X + T stable T stable A regular x + Tatable TstableAmgular + ? + T Stable x ? + T stable x Aa(T) A 2 P ( T ) A 2 K(T)T a b l e T stable A 2 A1(T) A 2 K(T)Skolem A 2 IT1 funations x A,(T) A 2 Aa(T) A 2 K(T)T stable A 2 Al(T) A 2 K(T)Skolem funations x X ? P ( T ) A 2 P ( T ) A 2 K(T)T stable x X ? A,(T) A 2 A,(T) A 2 K(T)T stable X t X + T stable T stable +

+ +

+ +

+ + + +

+ +

+

+ + +

+

+

+

( 3 ) Let ~ ( 5 "<) A , F i satisjies Ax(X.l) and (X.2). Then A is Fi-saturated ifs A is the universe of a A-compact model.

Proof.Easy. We can sum up (most of) our results (see Table 1) which includes also results on F s which will be defined later. In each place we write a plus if the axiom is true for that F and a cross if there is a counter-example. I n the remaining cases we give a sufficient condition for the axiom to hold or just put a question mark. On Fi see VII, 4.4, Exercise VII, 4.2. See also 4.4-0. The exercises, usually, show that appropriate lemmas cannot be improved, and are ordered approximately in the same way. Whenever h is singular h = A, where A,, K < A, K = cf A.

zfeX

EXERCILSIF 2.1: Show that in Lemma 2.1, for unstable T, not necessarily (3) e (4). [Hint: Let M be a model of Find(see 11, 4.8(2)). R ( z , y, z) = P ( z ) A P(y)A -,P(z) A [zEz = zEg], N = (IHlP, , R),

170

PRIMlO MODELS

[OH. I v ,

82

T = Th(N) (so N < Q ) , and choose a, c such that N k P(c) A ,P(a). Then B = 0, A = {a}, ;E = (c) form a counterexample.] Deduce in (Ieq a counterexample to Ax(IV), Fi,. EXERCISE 2.2: (1) Show that for every A for some T , p(FA)= p(F1) = 80.

FA satisfies “every FA-type over A has a FA-iSOht0d complete extension over A” then p(Fi) -< A. (3) Show ,u(F:) = A , ,u(F:) 2 A and for A > KO,p(F,”) = A. (4) Show that for some stable T, p(F&) = 8,. (5) For every K , show that there is a F,-saturated, not 8,-compact (2) Show that if

model.

Remark. See Lemma 2.2(4). [Hint: (1) Take Tin* (see 11, 4.8(2)) choose A E I“(, ( An P(q1 = ( A- P(Q)I = A,andlet T = Th(C,. ..,a,...)aeA. (2) (3) By 4.10 and VIII, 4.8 for each regular p < A, T has a psaturated model which is not A+ -saturated. (4) Let T = Th(’2,. . ., E n , . .) when qEnv i f f q n = v n. (6)like (l).]

.

EXERCISE 2.3: For h singular, Ax(V.2) may fail for FA even for stable T (by 2.2(8), 2.8(2), A 5 IT1 of course). [Hint: Let Q, Pt (i < K ) be disjoint one-place predicates, Pf a (partial) one-place function, from Q onto P,.E t (a < A,) are independent equivalence relations on Pf (i < K ) . For every G,EP,(Q), {Pi(%) = cf A &@): i < K } is realized in Q. Let C, E P,(Cr),a E Q(Q),P&) # ct, A = B = 0, C = {cf:i < K } , CZ = u.] EXERCISE 2.4: Show that for A satisfying cf A 5 ITI, Ax(V.2) fail for Ff,(when of A > IT1 it holds by 2.2(8), 2.8(2).) [Hint: Let I be indismrnible, IIl = A, ~ E IC, = I - {a}, and B = A = 0. Remember h ( q ) = A+ + IT1 by 2.2(3).] +

QUESTION 2.5: Show that Ax(V1) may fail for Fi,when A is singulax CfA 5

pi.

EXERCISE 2.6: Show that for A singular Ax(VI1) may fail for FA, (notioethat by 2.2(8), h 5 12’1). [Hint:As in Exercise 2.3 but P,(a) = c,, B = 0 , A = {at: a < &, i < K } , afrE Pi(&)and Efr(c,,a;) iff a = 8.1 QUESTION 2.7’: Show that for A singulax Ax(VI1) may fail for F:, with IT1 = K (=of A),

OH.

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171

EXERCIBE 2.8: Show that Ax(VI1) may fail for Fa when T is unstable. [Hint: As in Exercise 2.1, and let b E R(Q),Q C - , E ( b ,c, a). Let B = 0, A = {b}, c = {a},B = (c}.] EXERCILYE 2.9: Show that Ax(1X) may fail for Fi,when cf A 5 I TI, and we can choose I TI = cf A. [Hint: Let L(T)consist of equality only; af (i 5 A) be distinct elements. B, = A, = {a,: j < i} for i < h and

P

= tP(aA,

Ul
EXERCIBE 2.10: Show that Ax(VII1) may fail for F!, when cf A I IT I, and we may choose 11 ' 11 = cf A. [Hint: Let P,(i < K ) be infinite pairwise lBfl = A, 6 = K , A , = U,<,B,, disjoint one-place predicates B, E P,(@), a 4 u1 PdQ),P = tP@, UI W I EXERCILYB 2.11: Show that Ax(X.1) (hence (X.2)) may fail for FZ x = t , 8 , a even for stable T,A = ITI.[Hint: (1) z = t : T will have independent one-place predicates P, (i < A). A = 0, Q = (a: = x) contradicfs Ax(X.1). (2) z = 8 : P, (i < A) and Q will be disjoint one-place predicates, IP,(Q))= 2, P,(i < A) one-place partial function from Q onto Pi, such that for every cf E P,(Q),{P,(x) = c, A &(%): i < A} is realized. Then A = ut
C

= {c,: i

< A},

EXERCISE 2.13: Show that Ax(X.l) may fail for Fi (IT1 = H,, OT T stable, but IT1 > A; in fact we can .choose a superstable T).[Hint: (1) Tin*,(2) Let T be T A +but the counterexample is for Teq(for TA*see Exercise 11, 2.3, for "eq" see 111, Section 6). Clearly ~ ( 2 1 ~ 9 .= ) A++, ITe'] = A+. Let I, (a < A + ) be the set of one-to-one functions from u into A, Z = U,Z,, and choose a, E Q for S E Z such that a$,at iff

PRIME MODELS

172 8

1i

=t

1i; then A

=

[OH. I v ,

42

{a8/Ei:8 E I , i < A+} c P, 'p = P = ( z )form a

counterexample, (3) Let A be regular

T

= Th(N),

M

=

(AA x

0,.

. ., P,,, . . .,. ..,Ev,. . . ) , , s ~ > h , v s ~ A

where P, = {(p, n ) E 1611: r) Q p} and ( p , n)E,(r), m ) if€ for some a p t a = 7 1 a = v 1 a, p(a) = r)(a)# v(a) or p = r). The counterexample is for TW (assuming N < a): A = {(p, n)/Ev:v E AA, ( p , n ) E 1N1,v # p}, and 'p = P=WI

EXERCISE 2.14: Show that Ax(X.2)may fail for Fi, T stable, ITI = A. [Hint: Let T = TA,A regular (see Exercise 11, 2.3), choose u,E(L: for 7E + 1) such that a,,Eiaviff r ) i = v i. The counterexample is for Teq:A, = 0, A , = (a,,/#,:i < A, r ) E "}, A , = A , u (a,,:r) E A(A + l), (A + 1)E Range(r))}and 'p = PI(z).]

r

EXERCISE 2.15: Show that Ax(XI.l) (hence (XI.2)) may fail for FL, T stable, A = ITI.[Hint: Let P,(i < A) be pairwise disjoint, infinite one-place predicates c E 1C1 - U, Pi(a), ci E Pi( a), A = 0, C = {q: i < A}. SO tp(c,A) is F&-isolated, it has a unique extension in S(C):tp(c, C), which is not Fi-isolated.] EXERCISE 2.16: Show that Ax(XI1) may fail for (Tunstable). [Hint: Use T,a, b, c from Exercise 2.8: we let C1 = 0, B = C, = {a}, A = {a,c}, P = tp(b, A).] EXERCISE 2.17: Show that (VI) may fail for Fr. [Hint: Use the theory of the rationals as an ordered set, B = C = 9, A = {l}, 6 = (3), a = (2).] EXERCISE 2.18: Show that Ax(XI.1) (and XI.2)) may fail for FX. mint: Let T be as in Exercise 2.11, hint 2. We let A = 0, B = 0, C = Pi(C), choose a EQ(&)and let p = tp(a, A). (We use the notation of Ax(XI.l).)

u

EXERCISE 2.19: Show that Ax(XI1) may fail for Fhp. [Hint: Use the previous example. C, = 0, A = Cp = {P,(a)},B = Po@),p = tp(a, B).] EXERCISE 2.20: Prove 2.12 for F( instead @.

CH.

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EXAMPLES OW F

173

EXERCISE 2.21: Assume Ax(III.1)and (VII).If Ax(X.1)[or Ax(X.2)] holds, then it holds for v = p(P; a) too. EXERCISE 2.22: Show that A3(T)= KO iff A3(T)= KO. [Hint: Suppose A3(T)> No,then T is not totally transcendental (by 2.15(4)); let M'be a IT1 +-saturatedmodel of T . Then A = [MI,q = {7v(z; 8): R(cp(P;a), L, 2) < 00) exemplifies A3(T)> KO.Now A3(T)> Noimplies A3(T)> Noby 2.16(2), so as A3(T),A3(T) 2 24, we finish.]

QUESTIONZ.Z3:Showthatbetween~~(T), IT[,X(T)A,(T)(1 = 1 , 2 , 3 ) there are no connections except those from 2.15(1), (6), 2.16(1) Exercise 2.22. [Hint: (1) T , : Let E, (i < A,) beindependent equivalencerelations, each with two equivalence clwes. Then K J T , ) = KO, ITl/ = A,, A,(T,) = A1(T1)= AS(T1) = Aa(T1)= KO. A3(T1)= A3(T1) = A:. (2) T,: Let AI = (A(a)3,. . .,P,,. . .) where, for r) E A(2)>2Pn -{v E *ca)3:7 Q v}, and A, = 2
QUELYTION 2.24: What happens if we change the definition of A3(T) to: A3(T)is the first cardinal A such that if r is a set of m-formulas, p G r a n m-type, then there is q E r, 1q1 c A, p u q is consistent but for every tp E r e i t h e r p u q I-v o r p u q I- -p? (i) Does the value of A3(T)change? (ii) Do we have nicer or better results with this definition? QUESTION 2.25: What happens if in the definition of ha(T )(Definition 2.5(1))we replace q 1 (B U C) I- q by p U q 1 C I- q ? (i) Does the value of ha(T ) change ? (ii) Do we have nicer results with this definition ? EXERCISE 2.26: Prove that T is transcendental iff A,(T) = KO (see Definition 11, 3.4). EXERCISE 2.27: Suppose T is stable, M is F&,-saturated,p E Sm( ]MI). Prove that there is a B E 1611, IBI < K ( T )such that p does not fork over B and p 1 B is stationary.

174

P R W E MODlPLS

[OH. Iv,

83

EXERCISE 2.28: Show that if F = {(p,B):p is definable over B} then Ax(XI.l) holds for F if for some M ,B c IMI c A . EXERCISE 2.29: Let F = R, (1) If cf h 2 K ( T ) F , satisfies Ax(XI.l),then F satisfies Ax(XI.2). (2) If h 2 h3(T)+ K ( T )and cfh > K ( T )or h > A3(T), then F satisfies Ax(XI.2). Remark. (1) For (X.2)see 1.1(6). (2) For x = 8 we can get a result using h3(T), but with ha(T)it is not clear. [Hint: Use 4.3(1).] EXERCISE 2.30: Suppose p = tp(ii, A) is Fi-isolated. (1) If cf A > X,, and FZ satisfies Ax(XI.l), then p is Fi-isolated. (2) If cf h = X, < h but P ( T ) < A (or Fi satisfies Ax(XI.1) for arbitrarily large p < A), then p is FA-iSOlated.[Hint: see 3.2.1 EXERCISE 2.31: Show that Ax(VI1) holds for Fi when (C(< cf A.

EXERCISE 2.32: F; may fail to satisfy Ax(V1). Proof. Let P, (i < A), P , Q be one-place predicates, P,(E) pairwise disjoint, P,(6) E P(&),P(%)nQ(6)= 0, P,a function from Q onto P,, and for every c, E P,(G),{Q(z) A P,(z) = c,: i < A} is realized. Let B = C = A = 0, b E Q ( ~ ) a, E P ( 6 ) - UfeA P,((E).If we add a,: Po --t Pi one-to-one, onto, Q(x) --f U,P,(z) = F,(x)we get T superstable.

IV.3. General properties of F-primary models We use the notation of Definition 1.2 and Section 1 in general. Before eaoh theorem the axioms used are listed and also assumptions on h(F). THEOREM 3.1: (1) [Ax(XI.l)] Over every A , for any p there is an (F, p)-p'mry set (so by 1.2(6), if Ax(X.1)holds it is an (F, p)-primry d l ) . (2) [Ax(XI.l)'JIf p(F) < m, then over every A there is an F-p*mary set . (3) [Ax(XL1)]8uppme IBI < p => ISm(B)II poratleaet IBI < p 3 I{p: p E 8m(B), p is an F-bolated tgpe}l I; p.

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PROPERTIES OF F-PRIMARY MODELS

175

If p is regulccr then over every A , IAI I; p there is an (F,p)-primary set of cardinality p, and so it is F-semi-saturated. (4) [Ax(III.l)] In (3), imtead of requiring p to be regular we can aaeume T stable and; (i) c f p r: A(F) + K ( T )or (ii) c f p 2 h(F), and also p E F(B) implia p does not fork over B. (6) [Ax(X.l)] Over every A, there is an F-mtructible model. Proof. (1) We define by induction on i, sequences a,, and sets B,, such that pi = tp@, A,) E F(B,) where A, = A u U {Ti,: j < i}. Suppose we have defined Zf for i < u,and A, is not (F, p)-saturated. Then there is a pair (q, C) E F , C c Dom q E A,, lDomq( < p, such that q is not realized in A,. Among these pairs choose one, (q,,C,) with minimal j ( q ) = min{j: j I u, Dom q E A,}. By Ax(XI.l) we can find a pair - . _ _ _ ( p a ,B,) E F , q c pa, pa E Sm(A,), and choose a sequence a, realizing Pa-

If for some a A, is (F, p)-saturated then clearly we are through. We shall show this holds for some a < ((IAI + 2)fl+ITI)+.The number of pairs (p,C) E F , Domq E A,, lDom qI < p is I IA,lfl+lTI. Hence [{u:j(q,) 5 j}l I IA,Ifl+ITI,so for some p, < (IA,I”+ITI)+,(Vu)(u 2 j + p, *j( q , ) > j). We can easily prove by induction on j < ((IAI + 2)’+ITI)+ that /3, < (IAI + 2)”+ITI)+.Hence there is p* < ((IAI + 2)”+ITI)+,cfp* = p + , and j < p* - p , < p*. Clearly A,* is (F,p)-saturated, so we finish. (2) Immediate, by (1) for p = p(F). (3), (4) We shall define Tif for i < p2 (ordinal exponentiation) such that: (a) d = (A, {ai: i < pa}) is an F-con8truction. (b) For i < p write & ( p s i ) as U,<”Cj where 1Cj1 < p and u < * Cf, c Ci. We then require that for each i each F-isolated type in Sm(C:) ( j < p) is realized by some sequence in d ( p m ( + i 1)). If p is regular clearly &(pa) is (F, p)-saturated aa desired. So we have to prove (4) only. If (p,C) E F , lDom pI < p there am q, p E q E Sm(,c9(pa)) and C‘, such that (q, C‘) E F.If cf p 2 h(F) + K ( T), then for some u < p, j < p, C‘ c Cf and q does not fork over Cy.Now we can findforeachp,u < p < p , Z B ~ d ( p . (+p 1))realizingq t&(pp).Aswe can m u m e p > X,, {a,: u + w I/3 < p} is an indiscernible set, whose average is q. Hence by 111,3.5 some ZB realizes p . The other case ((ii))is handled similarly. (6) Easy, like (l),using the Tarski-Vaught test.

176

PRIME MODELS

[OH.

Iv, 8 3

THEOREM 3.2 [Ax(II.l), (III.l), (111.2) and (VII); A(F)regular]: (1) If C is F-cowtrudible over A t h C is F-atormic over A. (2) A is F-atornic over A. (3) If tp(6, A) ia F-iaolated then A u 6 is F-atomic Over A. (4) If A , E A , c A,, A , is F-atorni~over A,, A, F-atornk Over A,, then A , ia F-atomic over A,. (6)If for i < 6 A, ia F-atomic Over A,, i < j < 6 A, c A,, t h A, is F-atomic over A,,.

u,<6

-

Remurk. For (5) we need no axiom at all, and for (2) we need Ax(II.1) only. Proof. (1)Let & be an F-construction of C over A. It suffices to prove by induction on a that &(a) is F-atomic over A = d ( 0 ) . For a = 0 this follows by (2), and for a a limit ordinal, by (6).For a = /3 + 1, d @is) F-atomic over A by the induction hypothesis and &(a)[ =&@) u is F-atomic over &(/3) by the construction, and (3). Hence by part (4) &(a) is F-atomic over A. (2) Trivial. (3) If E E A u 6, then E = 7in6,, where 6, E 6 and i%E A. Let tp(6, A) E F(B). So by Axiom (111.2) tp(6, A) E F(B u 8). By Axiom (11.1) tp&, A) E F(a u B), hence by Axiom (VII) (as A u a = A ) tp,(Bn6,, A) E F ( a U B). SO tp(E, A) = tp(8-61, A) E F(a u B). (4) Let E E A,, hence for some B G A,, tp(8, A,) E F(B). For every 6 E B tp(6, A,) is F-isolated, hence tp(6, A,) E F(c6)for some 0 6 c A,. By Axiom (111.2) and the regularity of h(F) tp,(B,A,)EF(C) (C = U~EB Cb).By Ax(III.l), (111.2), tp(E, A, u B) E F(C u B) hence by Ax(VII),tp,(B U a, A,) E F(C), in particular tp(E, A,) E F(C), so we finish. (6)Trivial. THEOREM 3.3 [Axioms (III.l), (VI) and (VIII)]:8 q p e I
Proof. It suffices ,toprove by induction on a E J (on the order J) that (pa(&*), BE)E F. Suppose we have proved it for every a’ < a and we shall prove it for a. For t E I,let C, = B, u [&(t) n &*(a)]. We prove by induction on t E I (in fact t E I u {a1I})that (p,(&*) 1C,, B,) E F.

CH. IV,

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PROPERTIES OF F-PRIMARY MODELS

177

Notice that C, is increasing with t, and Cal(r)= d * ( s ) so p s ( d * ) 1 C,,,,, = p S ( d * ) so , this will suffice. For the &st t in I , and in fact when I C t I 8 , A u B, E C, E d ( s ) . As by definition ( p , ( d ) ,B,) E F , this holds by Axiom (111.1).If t is limit, it follows by Axiom (VIII). So suppose t is the successor of t(1) I C t > 8. If J C s I t(1) then C, = C,(,), so there is nothing to prove. Otherwise C, = C,(,, U &), and the result is an application of Axiom VI * So we finish the induction on t , hence on 8 , hence finish the theorem. CONCLUSION 3.4 [Ax(III.l), (VI) and (VIII)]: If d = ( A , {a,:i < a}) is an F-construction, B E &(a), I BI 2 AJF), then for s m e well ordering I of a, d*= ( A , {a,: i E I } ) is an F-construction, and B is included in d * ( s I B IsiB1 ) , the IBlth element of I .

Proof. Immediate by 3.3. LEMMA 3.6 [Ax(II.l)]: If d = ( A , {a,: i c a}) is an F-construction, Ia then d' = ( d ( p ) , {a,: i c a}) is also an F-construction.

p

Proof. Immediate. LEMMA 3.6 [Ax(II.l), (III.l), (111.2), (V.2) and (VII); A(F) regular]: If d = ( A , {a,:i < a}) isanF-construction,C G &(a), ICl c A(F)then d'= ( A U C,{a,:i ia}) is also an F-construction.

Proof. By 3.2 and 3.6, for every i c a, and E E &(a) pj = tp(E, d ( i ) )is F-isolated, hence for some Bj c d ( i ) , Bjl c A(F), (pj, BE) E F . Let B* = {Bj: E E C u a,} so IBil c X(F) by the regularity of A(F). By Ax(III.2) (p:, B,) E F for every i3 E C u a,. So tp,(C u a,, d ( i ) )E F(BL), hence by Ax(V.2) tp(a,, d ( i )u C) E F( B' u C ) . So we finish.

u

I

THEOREM 3.7 [Ax(I), (II.l), (IILl), (111.2), (V.2) and (VII);h(F) regular]: (1) If B is a n F-primury set over A then ( B ,a)oeAis A(F)homogeneous: i.e., i f a,, b, E B for i c a < A(F) and tp*((a,: i < a), A ) =tp*((b,: i < a), A ) then for every a, E B there is b, E B such that tp*((a,: i I a), A) = tp,((b,: i s a), A). (2) #uppose B, is F-primary over A, (1 = 1, 2), f an elementary nwpping, A, c Dom f E B,, A, c Range f E B,, lDom f - All < A(F), !Rangef - Aal c A(F). Then for every 7i E B, there is f ' extending f, satisfying all the abovementioned conditions and 7i E Dom f '.

178

[OH. Ivy5 3

PRIBfE MODELS

( 3 )In ( 2 ) f o r every C, C B,, IC,I < A(F) we canJind an elementaryf’; Domf’ 2 A , U C,, Rangef’ 2 A , U C,. (4) Instead of “F-primary”, we can in (2), (3) assume only “(F,p)primary” where p 2 lA1l+ + A(F). ( 6 ) I n (1) it su.ces to assume B i s (F, p)-saturated, p 2 IAI + A(F), and every sequence from B realizes over A an F-isolated type (i.e., B is F-atomic over A). +

Remark. Concerning Theorem 3.7, note that in (2), (3) it suffices to assume B, is ( F , p)-saturated, (F, p)-atomic over A,, p 2 lAll + A(F). Also in (3), for any C, E B, (1 = 1 , 2) ICJ IC,I < A(F) there is an elementary f’, Dom f’ 2 A , U C,, Range f ’ z A , U C,. +

+

Prmf. (1) Follows from (2). (2) By 3.6 B, is F-primary over Dom f, so by 3 . 2 = ~ tp(a, Dom f ) is F-isolated, so for some C E Domf, ( p ,C) E F. Hence by Axiom (I), (f( p ) ,f (C)) E F. As B, is F-saturated some 6 E B, realizes f (p).Extend f to f’ by defining f ‘(a) = 6. Clearly f ‘ satisfies our requirements. (3) Prove by repeating (2) A(F) times. (4), (6) The same proof, in fact, as of (l),(2). THEOREM 3.8 [Ax(I), (II.l), (III.l), (111.2), (V.2), (VI), (VII) and (VIII);A(F) regular]: (1) Suppose B, is F-primary over A, (1 = 1, 2) and f is an elementary mapping from A , onto A, then we can extend f to an elementary mapping from B, onto B,. (2) The assertion is true also for B, (F, p)-primary over A,, when IAZl < lBll 5 P-

CONCLUSION 3.9 [The same assumptions as in 3.81: The F-primary set over A is unique u p to isomorphism over A (i.e., u p to isommphitwm which are the identity on A). Proof of 3.8. (1) Let d, = (A,,{Gf: i < a,},{Bf: i < a,}) be an Fconstruction of B, over A,. We define by induction on p < a* elementary mappings fs where a* = max{a,, a,} such that: (1) f o = f. (2) For /? < a,fa extends fs, and for limit 6,fd = (3) Dom fs E B,, Range fs E .Bas

u B < d fs.

(4) Dom fs is closed, i.e., for some V;, DomfB = A U U {at: i E 7;) and i E V i =c B: E Dom fs.

CH.

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179

(6) Range f8 is closed (the same as in (4) for V;). (6) E Dom faa+,, E Range faa+, (when is defined). For @ = 0, @ a limit ordinal, there is no problem. So suppose f6 is defined. By 3.3, 3.6

df= (Dom f6, {af:i <

a,}),

d!j = (Range f8, {af: i < a,})

are F-constructions. Now we define by induction on y c h(F) elementary mappings f] such that: (i) f] is increasing in y , fj = fE, fj = f;, (ii) Domf] c B,, Range fJ c B,, (iii) lDom f] - Dom f 6 1 c h(F). Let Dom fJ = Dom f6 U U {a:: j E V j S y } ,Range fj' = Rangets U {a;: jE VjjY}.For y = 0, y a limit ordinal-the definition is by (i). By 3.6, B,, B, are F-primary over Dom f], Range f] respectively. Hence by 3.7(2)we can extend f; tof;+ satisfying (i),(ii),(iii)and such that for y even:

u

VL+l =

G , Y u u {U,(SB,):jE V $ J

"

{a>

({a}appears only when @ = 2a + 1, a < a,). For y odd, replace 2 by 1, and {a} appears only when @ = 2a, a < a,. Clearly f8+, = U y efSyA (A = h(F)) satisfies our requirements. Now fa. is the required mapping. (2) The proof is similar.

Remurk. In fact the regularity of h(F) is needed for 3.3, 3.6 but not in this proof. THEOREM 3.10 [Ax(I)]: (1) If B is F-constructible over A, then B is F-primitive over A. (2) If B i s Fqrimury over A, then B is F-prime over A.

Proof. (1) Let d = (A, {ai: i < a}) be an F-construction of B over A and let B* be an F-saturated set, A c B*. We should find an elementary mapping f from B into B*, f t A = the identity. Define by induction on /3 I CL elementary mappings f8 from .at@) into B* so that f, extends fE for /3 c y. Let fo be the identity over A, and for limit 8, then by fd = U f l < a f s * I f f 8 is defined, ( ~ E B4) Y E F (PE = tp(aE, B*, so f8(236) is f6(B8)) E F, and DomLfs(pfl)] Axiom (I), (fj(p8)~ realized by some i;B E B* (as B* is F-saturated). Extend f8 to f8 + by defining f8 + 1(7iB) = 68. Clearly f a is the required mapping. (2) Immediate.

,

180

PRIME MODELS

[CH.

Iv, 0 3

LEMMA 3.11: (1) If A G B c C, C is F-primitive over A then B is F primitive over A. (2) If A c B E C, C is F-prime over A, B F-saturated then B is Fprime over A. (3) The same for (F,p) instead of F.

Proof. (1) Let B' be an F-saturated set, A G B'. Then there is an elementary mapping f from C into B', f 1 A = the identity. Let g = f 1 B, so g is an elementary mapping from B into B', g 1 A = the identity. Hence B is F-primitive over A. (2) Immediate from (1) and the definition. (3) Proved similarly. CONCLUSION 3.12: Let p(F) < a0 then: (1) [Ax(I), (X.l) and (XI.l)]Over every A there is an F-prime model. (2) [Ax(II.l), (III.l),(111.2), (VII) and (XI.1);h(F) regular] I n every F-prime set B ouer A , every sequence realizes over A an F-isolated type. (3) [Ax(I), (ILI), (III.l),(111.21, (V.2), (VII) and (XI.1);A(F)regular] If B is F-prim over A, C E B, ICI < cf h(F) then B is F-prime over A U C . The Same hoZds.for F-atomic.

Proof. (1) Immediate, by 3.1(2), 3.10 and Exercise 3.1. (2) By 3. I( 2) there is an F-primary set B* over A. So by the definition of F-primeness there is an elementary mapping f from B into B*. f 1 A = the identity. By 3.2 for every 6 E B* tp(6, A ) is F-isolated. Hence for every 6 E B, tp(f (6),A) is F-isolated. But as f 1 A = the identity, 6, f (6) realize the same type over A, so we finish. (3) By 3.1(2) over A there is an F-primary set B*; and by the definition of F-primeness there is an elementary mapping f from B into B*, f F A = the identity. By 3.6 B* is F-primary over A uf ( C ) , and by 3.10 B* is F-prime over A uf (C). As A uf(C) c f (B) G B* by 3.11(2) f (B) is F-prime over A u f (C). (By Ax(1) the property of being Fsaturated hence F-prime over a set is preserved by elementary mapping.) Hence B is F-prime over A u C. The proof for F-atomic is similar.

LEMMA 3.13 [Ax(III.I), (VI) and (VIII)]: Suppose A,C,, (i < a) are given sets, and for every 5 EC,, for some BE E A, tp(E, A u U j < C , j )E F(Bi). Then for every i < a, 5 E C,, tp(E, A u U j + C,) , E F(BE).

Proof.By 1.2(2) (and Ax(III.1) and (VIII))it suffices to prove that for

CH. I v ,

8 31

181

PROPERTIES OF F-PRIMARY MODELS

u,*,

i < a, E EC,,a E C,, tp(E, A LJ 2) E F(@). Let a = where 2, E C j ( l ) 0 s 1 s k, andj(0) < . . . < j ( n ) < i < j ( n + 1) < < j (k).We prove by induction on 1 I k + 1that tp(E,A u a,-. .-a, EF(&). For 1 s n + 1 this holds by Axiom (111.1).If it is true for 1, 1 s k,it is true for 1 + 1 by Axiom (VI). For 1 = k + 1 we get what we want. Z o n e .

-

---

THEOREM 3.16 [Ax(I), (III.l), (111.2), (V.2), (VIII) and (XI.1); h(F) regular] : (1) 8uppose A c [MI, M is F-saturated, and F-atomic over A. Letp, ~ S ( A ) f oi r< a,and A* = A u {c: c E IMI, c realizes a p f for an i < a}. Then hf is F-atomic over A*. (2) For F = q,we can assume thcct pf's are types almost over A , when T is etable. Proof. (1) Let 7i E 1M1, by Ax(XI.1) a s p = tp(Z, A ) is F-isolated, p has *) is also F-isolated. So for some C G A*, an extension q E ~ ' " ( A which ( Q , C ) E F(so ICl < X(F)) and some 6~ ]MI realizes q. Let C = {c,: i < ICl}. By the hypothesis for every E E C u 6, tp(E, A ) is Fisolated, say tp(E, A ) E F(B,). By Axiom (111.2) and the regularity of X(F), tp,(C u 6, A ) E F(B), where B = {Be: E E C u 6}. Hence by Ax(V.2), for every a < ICl,

u

p a = tp(c,, A u 6 u {cf:i < a})E F ( Bu 6 u {cf:i < a}). Now we define, by induction on i, c; E 1M( such that the mapping

fa(a s ' ICI), f,(a) = a for a E A, f,(6) = 7i and f&) = c;, for i < a, is

elementary. It is trivial to definef, and fa (6 a limit ordinal). For note that by Ax(I), f,(p,) is F-isolated, (wa < h(F)) hence some ci E 1M1 realizes it. Let C' = {c;: i < ICl}; as tp(c;, A) = tp(c,, A), c, E A* clearly C' c A*, and IC'I < X(F).We shall provep* = tp(B, A*) E F(C'), and thus finish. By Ax(VII1)it sufficesto prove tp(& A u C' u E) E F(C') for any E E A*. As before, we can extend ficl to an elementary mappingf', whose domain is c [MI,and whose range is C u Rangef,,,. By the choice of C, and Ax(III.I), tp(6, Domf') E F(C), hence by Ax(1) tp(7i, Rangef') E F(C') so tp(Z, A u C' u E) E F(C'). (2) Work in Ceq.By 2.12(2)Meqis F-saturated and also F-atomic over A , hence it is F-atomic over acl A (see 2.1). Now w.1.o.g. each pt is over acl A ; hence, by 3.14(1) Meqis F-atomic over (acl A ) U A*, and by 2.1 Meq is F-atomic over A * so M is F-atomic over A*. LEMMA 3.16 [Ax(III.2) and (V.2); X(F) regular or [Ax(III.2), (V.1) and

PRIME MODELS

182

[OH.

Iv,

3

(IX)]]: If C is F-atomic over A , and IC - A1 s h(F) [or IC - A1 I cf h(F) when A(F) is Singular], then C is F-constructible over A.

Proof. Let C - A = {c,: i < a = ICl}, and let A, = A u {c,: i < j}. It suffioes to prove tp(c,, A,) is F-isolated. As h(F) is regular [or as IC - A1 5 cf h(F)] and Ax(III.2) holds, tp,({c,: j Ii},A ) E F(B) for some B c A, IBI c h(F). Hence, by Ax(V.2) [or by l.l(4)] tp(c,, A,) E F(B u {c,: j c i}).

THEOREM 3.17: (1) [Ax(XI.2) or Ax(XI.l), a c cf h(F),and Ax(IX)] If A, E A, f o r i c j Ia then there are sets B, (i s a)Buch t7t.d A, E B,, i < j =- B, c B,, B, U A , is F-constructible over A , U U,<,B, when i I/3 s a,and B, is (F, p)-saturated. (80B, u A , is F-constructible over A , when i I/3 Ia.) (2) [Ax(X.2) or Ax(X.l), a < cf h(F), and Ax(IX)] If A, c A , for i c j Ia, then there are models M,,A, E !Mil, i < j * [Mil c IM,l, [MilU A , $8 F-constructible over A , U lM,l when i IB Ia.

u,<,

Proof. Similar to 3.1 (remember 1.1(7))#

CLAIM 3.18: [Ax(I), (IV), (V.1) and (VII)]If Ax(XI.l) [or (XI.2)] are satisfied for l-types then it is satisfied. Proof. Easy.

THEOREM 3.19: ( 1) [Ax(I), (X.l), (XI.l)]Let M be F-saturated. If M is F-minimal over A (seeDefinition 4.4), then in M there i s no in$nite indiscernible set over A. (2) We can replace the three azim by the acreunzlption “over any A there ie an F-p~imed”. Proof. Just the proof of one implication in 4.21. EXERCISE 3.1 [Ax(X.l)]: If A is F-primary over B, then A is an F-primary model over B. EXERCIBE 3.2: In 3.16(1) let p , be an m,-type over A, and A* = A u I,, I , = {E E M :E realizes p,}. Show the conclusion still follows. Similarly 3.16(2).

u,

EXERCISE 3.3: Show that 3.6 holds for any closed C (assuming the appropriate axioms),

OH.

PELIlldE MODELS BOB STABLE THEORIES

I v , $41

183

QUESTION 3.4: Is there in any of the lemmas of Section 3 an unnecessary axiom ? EXERCISE 3.5 [Ax(I)]: Suppose B is F-primitive over A, f an elementary mapping, Dom f = A, and Range f C _ C,C is F-aahrated,then we can extend f to an elementary mapping 9, Dom g = B, Range g E C. EXERCISE 3.6: Suppose A is singular, and Ax(VI1) holds when ICl < cf A, and Ax(II.l), (111.1) and (111.2).Then: If ICl < cf A, C is F-atomic over A, tp(ii, A u C) is F-isolated, then C v ii is F-atomic over A.

IV.4. Prime models for atable theories In this section, we w u m e T is stable, except in the stafied theorems.

LEMMA 4.1*: If (p(Z; a) i8 almost over B, B +(Z;7;) such that (i) #(Z;6 ) i8 over A and dmost over B; (ii) k(p(~;a) --f #(z; 6); (iii) if E E A then k(p(~;a) -+ ? EA i we can take # = (p.)

E

A , then there is a fomnula

e(z; a) i# I#(z; 6)-+ e(3; a): (Note that if

.

Procf. Let a,, . .,a,, be a maximal set, such that Z0 = a, the (p(Z; a,) (I s n) are not equivalent, and tp(i&,A) = tp@, A) (n < No by- 111, 2.3(1), using the fact that (p(Z; a) is also almost over A). Let f(Z; a’) .= Vlsn(p(Z; at).Clearly by 111, 2.3(2), f(Z; 6‘) = #(Z;6) for some # and 6 E A. It is easy to check (i), (ii), (iii).

LEMMA 4.2: flwppose cp(Z; 8 ) does not fork over C , C G A . Then there is a f m u l a #(Z; &) such ticat; (1) #(Z; 6) ia over A , but do@ not fork over C . (2) {(p(E; a), +Z; 6))f o r b ower A (hence over C ) . (3) For every ~ E aAd 8, I(p(~;a)-+0(3;a)implies k#@;6)+

eg; a).

(4) If C E B E A, tp(& A) &OM not fork over B, t h we can c h e +(it;6 ) which ie alnzost over B.

Proof. Let B be as in (4); for (l), (2) and (3), B = A satisfies that tp@,A ) does not fork over B. By 111, 1.12, there is an indiscernible set

PRIME MODELS

184

[CH.

Iv, 5 4

I = {a,,: n < w } over A, based on B, 3 = a,,. Hence by III,2.5, for some n,p,,(Z; a*) = Vros2,,-l,lrl=,, Atpro p(E; a,) is almost over B. Cleady by 111, 1.4, there is q E Sm(U1 U A ) , p(3; a,) E q, q does not fork over C; for every I , q ( Z ; a,) E q (as q does not split strongly over C by 111, 1.6(1)) hence. p,, E q so pa@;a*) does not fork over C. Clearly for every 1 > 0, {p@;a), +E, a,)} splits strongly over A, hence forks over A. Hence a), Vlsw - q ( E ; a,)} forks over A (by 111, for every w E 2n - 1 (~(3; 1.1(9)), hence {q@; a), --,p,,(Z; a*)} forks over A. Also, if E E A, t - q ( ~a); 3 d(Z;17)then t-p(Z;a,)3 e ( E , E) hence t-p,,(~; a*) --f d(Z; a). Now apply 4.1 to p,,(Z; a*), B and A. We get #(Z; 6) satisfying (i), (ii) and (iii) from 4.1. It is easy to check that it satisfies the conditions (1)-(4) from 4.2.

THEOREM 4.3: Swppocre p cBrn(A),C E B c A , p doee not fork over C .

Ey,-b~Wed, t h p 1 B b FA-boWed. (2) If p i8 q-i80kZ&?d,A 2 K=( T ) ,t h p r B b ~ - h o k d . (3) If p 1C i8 8tdbWY9ICl < A, p b F”~-bOlate&, then p PA-i801ated, provided that A 2 K,(T). (4) If p is Fi-isolated A 2 K,(T), then p rB is Fi-isolated. ( 5 ) If p is F!,-isolated, then so is p r B . (1) If p

i8

rB

b

Proof. (1) As p is ~A-isolated, there is q c p , q I-p, Iql < A, and as in 2.2(2) aasume q is closed under conjunotions, and let q = {p,(Z; a,): i < a < A}. As p does not fork over C and pi@;4 )~ pby, Lemma 4.2 there is #@; 6,),5, E B satiaf(1)-(3) from 4.2 (B here and (by (2) from 4.2) corresponds to A there). As p,(Z; a,) {p,(Z; a,), -,#,(Z; 6,))forks over 0,but p does not fork over C,we have that #,(Z; 6,)~ p SO . r = {#,(Z; 6,): i < a} E p , SO r E p B; and clearly 1.1 < A. If QZ;5 ) By then for some i < a, t-p,(f;a,) + B(3; a), hence by (3) from 4.2, t-#,(Z; 6,)+ O(Z;a). Hence r t- p r B. (2) As p is FR-isolated, for some C1s A, IC,l < A, there is a type q E p almost over C,, q t- p . For every E E C, there is BEc By IBEI < K ( T )such that tp(E, B) does not fork over BE(by 111, 3.2). Let B* = u,,l B, U C; then as A 2 K,(T),p*l< A . Now for every q = q(3; E q there is +JZ; satisfying 4.2(4) (A, B there correspond to B,B* here). Let T = {+p:rp~q}.Clearly T is almost over B*, and as in part (l),T E p , T I-p rB. As p*l< A we finish the proof. (3) By (2), for some B* c B, p BEF;(B*). As ICl < A we can assume C G B*. By 2.2(13) p 1 B E Fi(B*).

a

4)

m. IV, f 41

PRIME MODEM FOR STABLE THEORXES

185

Proof. (4) So suppose q = {tpr(E;st):i < u} E p , u < A, U,< a si, E C* E A , IC*l < A, and q has no extension in Sm(A)forking over C*.

W.1.o.g. q is closed under finite conjunctions. We produce #,(3;6,)

(6, E B) as in the proof of 4.3( l),r = {#,(Z; 6,):i < a} c p, and w.1.o.g. p , tp*(C*, B) does not fork over C* n B. Now suppose for some 7i E B and 8, r u {8(E, 5 ) ) is consistent. For each i, not #,(if, Ei) Ia) hence tpr(E; b$ I- -d(?Z;a) fail, so as q was closed under conjunction, q1 = q u {8(E; si)} is Consistent. By the choice of q, q1 does not fork over C*, so let Erealize ql, tpiE, A ) does not fork over C*. As {rp,(iE; $), -,I,$,(?& 6,)} fork over C*, $,@, b,) E tp(E, A), hence r u {8@; a} c tp(8, A). As p’ = tp@,A) does not fork over C*, also tp(E, B u C*) does not fork over C*, and as in addition tp*(C*, B) does not fork over B n C*; tp,(C* u E, B) does not fork over B n C*, hence tp(Z, B) does not fork over B n C*. So also r u {@, a)} E tp(Z, B) does not fork over B n C*. S o r ~p r B , C * n B c B , I r l , I C * n B I < AandforeveryO,aiB, r U { O ( Z ; h ) } consistent implies r U { O ( Z ; ~ ) } does not fork over A ; clearly p I B is Fg-isolated. ( 5 ) Just like 4.3(2).

EXERCISE 4.1: Does a similar assertion hold for Fi (i.e., an assertion like 4.3( 1) or at least 4.3(2))1 Prove that the answer is no. Remark. The question is interesting mainly for h = H,, since if h > KO then Fk-saturation is equivalent to Fi-saturation (by 2.2(6), (7)). Similarly for other concepta (primeness, etc.).

EXERCISE 4.2: Show that in 4.3(2) we cannot omit the assumption that h 2 K ~ ( T ) . DEFINITION 4.1: For x = t, 8, a, h 2

K,

let

F;,= = {(p,B): for some A, m; p ES”(A), B c A , IBI < fork over B and p is an Fg-isolated type}.

K,P

does not

Note that (p, B) E Ff,= does not imply (p, B) E Ff.

LEMMA 4.4: (1) For x p

i8

=

t , 8 , a, K 2 K(IT),p

E @ ( A ) ,p

is e-i8olaterl i#

F,,-GO&ed.

(2) I n tlce following conceptsfor x = t , 8 , a; it does not matter w&kr

F = V, or F = Ff,%:F-construction
186

PRIME MODELS

[m. IVY5 4

Proof. Immediate. Remark. ButforF-construction(A,(a,:i

< a ) , (Bt:i < a))itmatters.

THEOREM 4.6: If x = t , a , K = K,(T)< d~ then E,ls aatiaJie~the foZlozoing aziomCr; (I),(11.1) [Z = 1, 2, 3, 41, (III.l),(111.2), (IV), (V.1) a d (v.2)[When Cf h 2 K(T)], (W), (VII) [When Cf h 2 K(T)], h(VII1) for 6, cf 6 < cf h and (IX) [when cf h 2 K] and (XII). Also h(Ff,,) = K and P(XrC) = P ( F 3 .

Proof. Most of them follow by combining the results on Fg, and FL. Ax(III.1) follows from 4.3. For Ax(V.2) notice Lemma 2.8(2), &B ICl < K 5 cfh. For Ax(VI1) note that ICl < K 5 cf h so there is a B' c A such that tp*(C, A) E E ( E )and the rest follows by Ax(III.2). For Ax(VI), use Ax(IV) and (VII) to show tp(sin6, A) is Fz-isolated, and (V.1) to show tp(si, A u 6) is E-isolated. THEOREM 4.6: If x = t , a , A 2 K = K,(T),Fg 8ati8jEeS Ax(X.l) [Ax(XI.l)] then also Fgels8atiafiecl Ax(X.l) [Ax(XI.l)]. The same holds for Ax(X.2) and (XI.2).

Proof. Immediate. CONCLUSION 4.7: If K ( T )5 cf A, A is F$-conductible over x = t , a, then A is Ff-atomic over B.

By

Proof. Immediate, by 4.4 and 4.6, applying 3.2(1) to Fg,E ( K = K r ( T ) ) . CLAIM 4.8: Let F%8atisfy Ax(X.l), (XI.1); x = t , a, cf h 2 K ( T ) . (1) If M is Fg-prime over A , C E IMI,ICl < cf A then M is Fg-pime over A u C. (2) If .d = ( A , {al: i < a}) i s an Ff-MWition, C E &(a), ICl < cf A, then .d = ( A u C,{a,: i < a } ) is an ~ - m t r r u c t i o n .

Proof. (1)Like 3.12(3), using (2) instead of 3.6; (2) like the proof of 3.6, using 4.7 instead of 3.2, and 2.8(2) instead of Ax(V.2). THEOREM 4.9: Suppose z = t , a, F = Fg satie$ea Ax(X.1) and (XI.1). M is an F-pime model over A, and I c IMI b an inJinite indiscernible set over A . (So w.1.o.g. I is m u x i d in M.)

OH.

Ivyf 41

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187

(1) If cf h 2 K ( T )then , dim(I, A , M ) 2 h ezcept, poseibly, when: (i) cf h = No < A, and (ii) dim(I, A , M) = No,and (iii) for no B c A u I and J E I, IJI = No, is Av(J, A u U J)E F(B), a d (iv) if x = a, for no B -c A, tp*(U I, A) E F(B). (2) dim(I, A, M ) 5 h when cf h 2 K,(T). (3) If cf h 2 K ( T ) N, OT h = KO = K ( T ) ,then Av(I, 1M1) is

+

Ff+-iadated. (4) If cf h 2 K ( T )+ N,, then M is h-hmmgeneoua over A (as in 3.7(1)). ( 5 ) dim (I,A,M) < A when h is regular, A+ 2 K ( T ) .

Proof. (1) Assume dim ( I , A , M )< A and we shall get a contradiction. Clearly, h > KO. We can assume 111 = dim(I,A,M). Choose distinct ~ , E I(n< w ) . By 3.12(2) (using 4.7 instead of 3.2) and Section 2, there are B, c A such that tp(gon... ^ ~ , , , A ) E F(B,). So when cfh >KO, B = U,,,B, has cardinality < h hence, tp,(UI, A ) E F(B) (by the indiscernibility), so when x = a, we get a contradiction to (iv). Similarly, when x = t let J = {G,,; 0 < n < ~},~obyAx(V.1)tp(G~,A~Uln_~Z~)~F(BuU~~~ hence tp(lio,A u UJ) = Av(J, A u U J)E F(B), contradiction to (iii). So (iii) [(iv)] implies cf h = No (i.e., (i)); but they follow from 4.10(1) below. So we can assume cf h = Nohence K ( T )= No,and have 1 1 > KO.Choose disjointJoyJ , s I , lJol < X,, to prove (ii);so suppose 1 I Jll = KO,such that Av(I, U I)does not fork over UJoyAv(I, UJ,) is stationary. By 4.8 M is F-prime over A u UJoy so w.1.o.g. J, = 0. Let N be F-prime over A u J , , ao there is an elementary embedding f of M into N,f 1 A = the identity. As 1 1 1 > Nofor some li E I f (a) realizes Av(J, A u UJ , ) , so by 4.7 Av(J, A u U J , ) is F-isolated, hence by 4.10(1) below dim(I, A, bl) 2 A.

u

CLAIM 4.10: (1) Under the assumption of 4.9(1) if 2 = a, tp*(U I,A) E F(B) or for s m injnite J s I , Av(J, A u U J) is F-isolated, then Av(I, A u (J I)is Fz-isolated where p = h + 1 1 1+; so 1 1 1 2 A. Proof. For simplicity we concentrate on 2 = a,tp*(U I , A) E F(B). By the indiscernibility of I , Q = Av(I, B u U I) k Av(I, A u U I). So it suffices to prove that Av(l, B u U I) is FE-isolated. By the indiscernibility of I and Ax(V.l), for every finite J E I there is an

188

[OH.

PRI116E MODELS

Ivy8 4

F-type qJ c q almost over B u UJ, q, t- Av(I, B u UJ). Clearly q = U {qJ: J c I, J finite} is an FZ-type almost over B u U I, q t- Av(I, B U U I),so we finish. For 4.9(2) we prove: CLAIM 4.10: (2) If UJ c B E C, J an injinite i?zdiecernibie set, IJI 2 A 2 K,(T),c i8 F-constructible ower B, F = F;, x = t , 8, a then Av(J, B) t- Av(J, C). (3) The claim holds for FX,h >, K , ( T )too. ,

Proof. Let S = ( B y{6;: i < u},{B;:i < a}) be an F-construction of C over B. We prove by induction on i that Av(J , B) k Av(J , B(i)).For i = 0 it is trivial, for i = 6 a limit ordinal it is immediate. For i = j + 1, B, c S(j),IBjI < A, so by 111, 3.6,for every a E S(j)there is JdE J. IJ,J < lB,l+ + KJT)IA s I JI,such that J - J,is indiscernible over u B, u UJd.Let B realize Av(J, B(i)).In all cases there is an F-type p c p , = tp(6,, S(j))[almost] over B,,p t-p,. Nowp t- tp(6,, S(j)u a) (for otherwise for some E S(j)C q[6,; By Z] but p u {-,p(E; E, a)} is consistent. So for any 5’ E J - Jay p u {-,q(E; E’, a)} is consistent, and, of course Cq[6,; E’, a]. Contradiction to p t-p,). By Ax(V1) for FL Av(J, a(j)) = tp(8, a(j)) I- tp(E, S(j)U 6,) = Av(J, B(i)). By the induction hypothesis Av(J , B) t. Av(J , a(j)). Hence Av(J, B) t- Av(J, B(i)).So we finish the induction, hence for i = u we get our result. (3) Because an Fi-construction is an F;-construction. EXERCISE 4.3: I n 4.10(2) for x = t show that A 2 K ( T is ) sufficient. Continuation of 4.9. (2) Suppose not, so we can assume IIl = A+. By 111, 4.17(4) for some J E I, I J1 < min{K(T), X,}, p = Av(1, 1611)does not fork over UJ a n d p (UJ)is stationary. By 4.8 M is F-prime over A u UJ. So we can replace A by A u UJ , I by I - J , retaining all our previous hypotheses and in addition p A is stationary, p does not fork over A. Let I = {gf: i < A+}, M* be an F-primary model over A u U I, where I, = {al: i < A}, H* exists by 3.1; and by 4.10, Av(1, A u U I , ) t. Av(I, 1M*I). As A c 1M*1, H*is F-saturated, there is an elementary mapping f:M --+ M*, f t A = the identity. So {f(af): i < A+} is an indiscernible set based on p A, hence equivalent t o I. As K,(T)5 A, for some if(af)realizes Av(1, A u I,)contradiction to Av(I, A u U I,) t- Av(I, lM*I).

r

u

CIE.

Iv, 5 41

PRIME MODELS FOR STABLE THEORIES

189

(3) Here the hypotheses of (1) and of (2) hold. We can assume Av(I, IMI) does not fork over A, Av(1, A) is stationary, as in (2) [by (1) we can assume A I111,hence I JI < 8, Icf A s A when cf A 2 K ( 2') + K,, and IJI < KO = 1 1 1 when A = No = K(T)].We can have I = {af:i < A} (by (1) and (2)), and define M*, f as in (2). Let the image of M by f be N. So it suffices t o prove Av(I, INI) is Fg+-isolated. By Claim 4.10 Av(I, A u U I) is Fg+-isolated, but Av(I, A u (JI)1 Av(I, 1M*1) hence Av(I, IN*!) is Fg+-isolated. Now as Av(I, IM*I) does not fork over A c IN!,by 4.3, Av(I, INI) is Fg+-isolated. (4) Let f be an elementary mapping, A c Dom f c IMI, Range f E [MI,f 1 A is the identity and JDomf - A1 < A; and a EM. We have to find an elementary mapping fl extending f, Dom fl = Dom f u {a}, fl(a) E IN). Let M l be FZ-primary over Domf, so we can assume IM'I E 1M1.. As M is Fg-saturated, by 3.10(2), Exercise 3.6 we can extendftoanelementarymappingfl, Domfl = lM1l, Rangefl c !MI. Choose C c lMll, C < ~(2') such that tp(a, lM1l) does not fork over C. We now define by induction on u < w + w , a, E IMII realizing tp(a, A u C u {af:i < u)). (Remember M is Fz-atomio over A by 4.7, so the above mentioned type is Fg-isolated by 2.8(2), as ICl + KO < K ( T )+ Kl Icf A, hence it is realized in M1.)So let I = (a,;w 5 a < w + w } ; clearly I is indiscernible over A u C, Av(I, lM1l) = tp(a, IM'I). By 4.9(1) A Idim(I, A , M), so let J 2 I be indiscernible, I JI = A. Now some b E J should realize tp(a, Domf); so we can definefl by: fl(a) = fl(b), fl 2 f. ( 5 )Since F satisfies Ax(X.l), (XI.i), there is over A an F-primary model N . Since M is F-prime over A , w.1.o.g. M < N . Suppose I C M is indiscernible over A , 111 = A+. As in the proof of 4.9(2), w.1.o.g. Av(I,A U I ) does not fork over A,Av(I,A) is stationary. As A+ 2 K ( T )clearly

(*)

For every El all but d A of the members of I realize Av(I,A U C).

By 3.4, since I C N , there is an F-construction over A , &' = (Al(ai:i < A + ) ) , such t h a t 1 S &'(A+). Choose a < A+, c f a = A, such that )I n &'(a)l= A and for every /3 < a some CEI n &'(a) realizes (this is possible by (*)). Again by (*) some J E I realizes Av(I, d(/3)) Av(I, &'(a))butby3.2,&'(A+)isF-atomicover&'(a) (and2EI c &(A+)) so tp(d, &'(a))is F-isolated. So for some D c &'(a),tp(d,&'(a))E F(D), so I D1 < A. Since cfa = A for some /3 < a,D G &'(/3). But by the choice of a some J ' E I belong to &'(a)and realizes Av(I, &'(/3)), i.e. it

190

PRRtfB MODELS

[a. w ,8 4

realizes tp(d, d(/3)). This contradicts tp(d,&(a))EF(D),DG &(p) (for x = a remember Av(I,A) is stationary, hence tp(d,d ( p ) ) is stationary). We now prove some preliminaxies to the characterization theorems.

LEMMA 4.11: h 4 x = t , a . Assume 8&i8@3 Ax(X.1), ( x I . l ) , h 2 K ( T and ) cf h > No. If M is e-cumpact and Fz-atomic over A , I E ]MI an infinite indiscernible set over A and p = Av(I, IMI) is c+-ieolated then there is an indiscernible set J ouer A , equivalent to I , J c 12111, IJI = h and Av(J, A u UJ) k Av(J, 1611).

Proof.By 111, 4.17(4) there is a C E IM], such that p = Av(I, 12111) does not fork over C and p t C is stationary, and ICl < 8,. By a hypothesis there is B E IN], IBI = A, ( p , B) E Ff+. We can aasume C E B [by Ax(III.2)] and let E realize p , So tp(E, B) k stp(E, B) (by 111, 2.14) hence tp(E, B) k tp(E, lM1) = p (see 2.1 for x = a). Let {6,: i < A} be a list of all finite sequences from B, each sequence appearing A times. Now define by induction on i < A, a, E 12111 such that a, realizes Av(I, A,) where A, = C u A u {a,: j < i}. If at is defined for i < a then as in the proof of 4.9 above, there is an F-type q a c p overC u A u U {a,: j < a},q, k p t A,. If q, k p 6=, 8, will be any element of which realizes q, (there is one, aa M is ; F-compact, q, an F-type).Otherwise there is a formula ~ ~ (6,)3 such that qk = q, u {tp,(Z; 6,)3 is consistent for t E (0, 1). As p 1 C E q, p t C is stationary, for one such t, q: forks over C. W.1.o.g. it occurs for t = 0, and let G, E [MIrealize q:. As q, k p 1A, and p 1C is stationary, J = {a,: i < A} is an indiscernible set over A equivalent to I (as both are based on p C) by 111, 1.10. As h 2 K ( T )by , 111, 2.4 for every 6, there is J’ c J, IJ’I < K ( T )5 IJI such that J - J’ is indiscernible over 6, u C u A u UJ’. Hence for every 6 E B the set {i < A: 6 = 6, there is tp@; 6,) aa mentioned above} haa cardinality < A. Hence for some a, 6, = 6 and q, k p t 6,, hence p (A u UJ) k p 6, (q, c p (A u UJ) of course). Aa this holds for any a,p t (A u UJ) k p B, but p 1 B k p by the choice of B. Hence p (A u UJ) k p so we finish.

THEOREM 4.12: (1)* Swppose tp@,, A ) tp(a,, A,) where for 8 E p, A , = A U U{it,:j~S}(notice i = {j:j< i}). (The iti’$ m y be inJinite, in fact).

OH.

I v , 8 41

191

PIlfME MODELS FOB STABLE THEORIES

Assurne case (i): for no i < p does tp(6, A ) k tp(6, A u a:), or case (ii): No < pandfor nos E p, IS1 < pdoes tp(6, A,) t- tp(6, A#). Then p < ha(T)(see Definition 2.6). (2)* Suppose p 2 h and that in the hypothesis of (1) we udd to: case (i): tp(6, A ) is Fg-isolated (z= t , s), case'(ii):for every finite S G p, tp(6, A,) is F$-isoZuted; and (A) p > A, or (B) p = h is regular ( > No). Then Ff (z= t, s) does not satisfy Ax(XI.1). (3) The same a8 (1) replacing tp by stp and the conclzcsion is p < K ( T ) . (4) If in (3) p 2 h and in the hypothesis we add to: case (i): tp(6, A) is Ff-isolated, case (ii): for every finite S E p, tp(6, A,) is B'g-isolated; and (A) p > h or (B) p = h is regular > KO. Then Ff does not satisfy Ax(XI.1). Proof. (1) Suppose case (i)holds but p L ha(T).Then there is q E &'"'(A,), p = tp(6, A ) E q and a < p such that q 1 A, k q (this can be done by the definition of P ( T ) and by renaming the ail's which is possible by 3.13 applied to F&). By renaming, we can assume 6 realizes q, without changing the hypothesis. w tp(6,A) Vtp(6, A u a , ) a h tp(iZ,, A) Y By Ax(V1) for Fm, tp(iZ,, A u 6). So there are E E A and 'p such that b[6; a,, i5J but i i realize it. By assumptp(iZ,, A) u {-,'p(6, f,E)} is coneistent, and let Z tion, there is an elementary mappingf,f 1A, = the identity,f (a;) = a,. We can extend it to an elementary mapping g whose domain includes 6. Then g(6) realizes q A, but not q 1A,+1, a contradiction. So we are left with case (ii),and we reduce it to case (i) as follows: Let So = {a < p: tp(6, A,) If tp(6, A,+l)}. Now w.1.o.g. So = p. [ w e define by induction on a 5 p sets &a) E p, S(a) increasing and continuous, lS(a + 1) - S(a)l < KO.If S ( a )is defined tp(6, As(,))Y tp(6, A,) 80 for someS(a + I), #(a) E S(a + I), IS(a + 1) - S(a)l < No,a €#(a + I), tp(6, A,(,)) br tp(6, As(,+1,), and S(a + 1) is minimal under those conditions. Let S(a + 1) - &a) = {~,(,.o), - . ,%&k(a))}, a,* = %a,o)na,(=.l) -. .nnow the az's satisfies all the hypotheses on the iZ,'s, and for them So = p.] So for every a there a m n(a) < o,i(Z,a) < a (for I < n(a)),formulas 'pa,and 5, E A such that C'p,[6, a,; 7il(o,,), ;E,] but ...,EJ} is consistent. If KO < x 5 p, x tp(6, A,) u {-,'pa(%; a,, regular, by 1.3(1) of the Appendix there is a stationary Sx E x, such that a € S X* n(a) = n, i(l,a ) = i(Z) (for I < n). Then let A' =

.

.

...

192

PRIME MODELS

[OH.

Iv,0 4

A u U1<,, a,iI,and is a#,where fl is the ath element in dx.Clearly our hypothesis holds for a:, A', 6 (a < x) hence by (i) x < ha(T). So we are left with the c&88 p = ha(T)> No is singular. Let p = 2idId X W , c f p < x ( 4 < t c 9 x ( i ) Wular. For every regular x < p, find Is,, n = n,, i(Z) = i,(Z) aa above. We can aasume that for 5 # < cf p, ix(& 4 dHc).We let A' = A U (a,: for some 5 < cf p. Z < nx(o, i = iXto(Z)},and {a:: a < p) be {a,: i Ed,(o, 5 < cf p}. (2) The same proof. (3) The same proof as (1). (4) The same proof as (2).

THEOREM 4.13: Suppose Ff

= F, z = t , s , a , F

satisJies Ax(XI.l),

and

(A) h > 8, is regular, or (B)z = t , 8; cf h 2 ha(T)+ HI, OT (C) z = a, cf h z K ( T )+ N,. If for every i < 6, E E C,,for .9ome @ E A tp(E,A,) E FS(&), where A, = A U c,,and for i < 8, tp(6, A,) i8 F-i80latfd,then tp(6, A,) i8 F-?kO&e&.

Prmf.W e prove it by induction on cf 8. If cf 8 < cf A, thia is immediate by Axiom (IX). Suppose the conclusion fails. Then we can eaaily define by induction on i < cf 8, a(i) < 6, and a, E C,,, such that (1) j < i 3 a ( j ) < a(i), (2) tP(6, A,{)) LJ tP(6,AM:) u 4 ) for 5 = t, 8, (3) stp(6, A,,)) Y stp(6,A,,, u a,) for x = a. W.1.o.g. 6 = cf6 (by renaming), so we already know 6 2 cfh. If A is regular, z = t , s we get by 4.12(2) case (ii) (for p s f6)that Ax(XI.l) fails for Ff contradicting an assumption. If h is regular z = a, we get a similar contradiction by 4.12(2) case (ii). So we have proved case (A) of 4.13. In case (B) of 4.13 apply 4.12(1) case (ii) and in case (C) of 4.13 apply 4.12(3) case (ii).

Remark. I n 4.13 for x = t , s we need not assume T is stable.

DE~MTION 4.2: Let cfh 2 K ( T )+ N,, z = t , 8 , ( t . Then M is d e d Ff-admissible over A if: (1) bd is q-saturated. (2) A c IM(, and M is F$-atodc over A.

OH.

Iv, 8 41

PRJME MODEL9 FOR STABLE THEORIES

(3) For any infinite indiscernible set I (of elements) over A ,I G

Av(I, 1iK1)is F,"+-isolated.

193

]MI,

Note. If there is no I aa in (3) then of c o m e (3) holds vacuously. THE FIRST CHARACTERIZATION THEOREM 4.14: Suppose cf h 2 K ( T )+ K,, x = t,8,a, F = FZ, F 8Udi8fies A X ~ (x.1) M and (xI.1). If h i8 8i747UhrY2 = t, U%8U??M 8Udi8$&9 Ax(XI.1)fc" arbitrarily large regular p < h. For x = t, h 2 h3(t)or at leaet Pi-saturated implie8 h - v t . Then: (1) i8 Fprime over A M i8 F-admissible over A . (2) The Fprime model over A is unique up to i s m p h i m over A . ( 3 ) If M I is F-adrnissible over A,, 1 = 1,2; f an elementary mapping fr.m A , onto A , then we extend f to an i 8 0 t ~ ~ p hfrom ~ s n M1 ~ onto Ma. Remark. In the proof, if x = a, you should replace t p by stp in many cmes. Proqf. (1) If M is F-prime over A , then M is F-saturated, by definition, iK is F-atomic over A by 3.12(2) (using 4.7 instead of 3.2), and if I c ldll is an indiscernible set of elements over A then Av(I, !MI) is Ff+isolated by 4.9(3). Hence M is F-admissible over A . As there is an F-prime model over A by 3.12(1), the other direction follows by (3). (2) Follows by ( l ) , (3) and the existence of an F-prime model over A by 3.12(1). ( 3 ) Assume x = t or x = a (the result for x = 8 follows by x = a as h > KO (see Exercise 2.30, 2.2(6), (7))). Let h = & A, be such that if A is regular, = K,(T) K,,A, = A ; and if A ,is singular then A, is regsatisfies Ax(XI.l) ;and ular, = cf A ;and if x = t , A singular then Fi(,) h(i) = A, > lil, A, 2 A, + K ( T )+ K,. For every C E lMIl 1 = 1, 2, there is A, = h(E) such that tp(E, A,) is F&-isolated (we can assume 1M,1 n IiKaI = 0 ) . We shall define elementary mappings fa (a < x ) such that (i) ,!?< EL implies fa extends fs, (ii) A, c Domf, = A: c lM1l, A , E Rangef, = A: c IMaI, (iii) M,, M , are F-admissible over Dom fa, Range fa respectively, (iv) if 2 = 1, 2 E E lMI1, then tp(E, A f ) is FE-isolated, where p = A, h@). For the checking of F-admissibility in (iii) notice that by (3)conditions (1) and (3) &om Definition 4.2, are immediate; hence only (2) (on being F-atomic) needs proof, and it follows from (iv).

x

+

x

+

z,<,

194

PRIME MODELS

[CH.IV,

Q4

Case 1. a = 0.

Let fo = f; trivially all conditions hold. Case 2. a = 8, a limit ordinal.

Let fd = U,
+ 2n, 8 is 0 or limit, fa is defined; we shall define

Choose by induction on i elements a: = a:,, E lM,l - At, such that for some B:,, E A:, tp(a:,,, A: u {&:j < i}) E F;(B:,,) where p = A,. The first i for which we cannot find such a:,, is i(a). Now we define j < i } ) ]by induction a:,, E 1bl,1 which will realizef,[tp(a:,,, A: u on i < i(a) (this is possible as bl, is F-saturated).We extendf, to is an elementary mapping and by letting (a:,,)= a:,,. Clearly (i)and (ii)am clear. As for (iv), if E E 1bl,1,by the induction hypothesis tp(Z, A;) is FE-isolated, p = A, + h(E),and the h ( j ) are regular > KO. So (iv) follows by 4.13(A),by induction on i(a). Case 4. a = 8 define f a + 1'

+ 2n + 1, ( 8 = 0 or 8 limit) fa

is defined and we shall

We define by induction on iinfinite non-trivial indiscernible sets (of elements) over A:, Ji,, such that: (A)For some infinite I:,, E A:, I:,, u J:,, is indiscernible. (B) For any n, a,, ,a,, E J;,,, for some Q,, c A:,

. ..

u

(C) Av(J:*,Y A: u JL) AV(Ji,,Y 1~11). Let i(a) be the first i for which there is no such J:,,. By 3.13 clearly J:,, is indiscernible over U U,+,Jk,, v A. Now we prove by induction on i that when p = A, + h(E), then tp@,A u U U,<,J:,,) is Fg-isolated. For i = 0 this is by the induction hypothesis on a, for i limit, by 4.13,and i successor by 4.16 (see below) (as A, 2 K ( T ) + K,, Aj is regular). So first we prove CLAIM 4.16:Swppme cf A 2 K ( T ) A ,

u U J v 5 i s F:-atomic

Over A

OH.

Iv, 8 41

PRIME MODELS FOR STABLE THEORIES

195

(z= t , 8 , a ) and Av(J, A u UJ) t- Av(J, A u UJ u E), and J ie indiscernible over A. (1)tp(8, A U J)i8 Fg-iSOlated, Fd&dt h d cf r\ 2 K(T). (2)If J is bmed on A, Av(J, A) i s dationary, tp(E, A u UJ)doee not fork over A , then tp(E, A) k tp@,A u UJ ) ; stp(E, A) k stp(E, A u UJ).

u

CLAIM 4.16: SuryHEse B s A , tp(?i,, A) dim nd fork over B, and tp(sil, B ) i8 StatiOnaY, 1 = 1, 2. (1)I f tp(7i1, A ) k tp(81, A u a,) t h tp(a1, B) k tp(ai,,B u a,). (2) If stp(a,, A ) k stp(lZl, A u a,) then stp(a,, B) k stp(a,, B u Z,). Remark. As Ax(V1)is satisfied by FO,,the assumption and conclusion of (1) and (2) are symmetric for al, a2.See also Ex. 4.10. Proof of 4.16. (1)Suppose that the conclusion fails, so there are 8 E B and a formula v, such that tp(Si,, B) u (cp(Z; a,, Z)t} is consistent for t E (0, 1). So there are $, t E (0,l}, such that tp($, B) = tp(a,, B), hp[ii;, Zi, 8It. By 111, 2.6(3) there me sequences 6p, 6!, 6? such that tp(6pn6:n6,, A) does not fork over B, and stp(6pnb:^6,, B) = stp(Bpnti:nB,, B). Hence stp(6,, B) = stp(a,, B). But the last type is stationary over B, by 111, 2.9, hence tp(6,, A) = tp(a,, A). Also tp(6:, B) = t p ( 4 , B) = tp(a,, B) (i = 0, 1) and the last type is stationary, by hypothesis. So tp(6i9A) = tp(7il, A). But clearly b[@, 62, Elt (t E (0, 1)). SO tp(7il, A) = tp(@:,A) It tp(6!, A u 6 ,). AS tp(B,, A) = tp(6,, A), tp(8,, A) If tp(a,, A u a,), so the assumption fds. (2) If stp(Z,, A) k stp(a,, A V a,) then stp(a,, A) k tp(a,, A u a,), Hence tp(8,, A U a,) does not fork over A, and even over B. As tp(al, B) is stationary,by 111, 2.14, tp(8,, A) k stp(a,, A u a,) hence tp(G,, A) k tp(a1, A u a,). So by (l),tp(al, B) t- tp(a,, B u a,). Again by 111, 2.14,tp(a,, B) k tp(Zi,, B U a,) t- stp(8,, B u a,), so we finish. Proof of Claim 4.15. (1)ChooseI G J, II1 < K ( Tsuch ) that tp(8,A u UJ) does not fork over A u U I , Av(J, A u UJ) does not fork over A U UI,and Av(J,A U UI)is stationary (by Theorem 111, 3.2 and 111, 4.17(4)). So 11 U 4 < K,(T)hence by Ax(V.2) for F:,x(T,),it suffices to prove (2). (2) For simplioity, let z = 8, t . Let J = {ai: i < p}, a,, realizes Av(J, A u U J u a), and A, = A u U {a1: i < a}. By assumption tp(iZ,, A,) k tp(a,,, A, u a) hence by Ax(VI) (for P",) tp(Z, A,) k

196

P&IME MODELS

[OH.

m,5 4

tp(Z,A uSi,).ByClaim4.16(l)foreverya < p,tp(Z,A,) ktp(Z,A,,ua,,). As {B1:i s p} is indiscernible over A u B (by 111,2.10)tp(iZ,, A, u a) = tp(Z,,, A, U a) hence tp(Z, A,) k tp(Z, A, u a,). So by induction tp(Z, A ) k tp@,A#).(For x = a, use 4.16(2), and 111, 2.14.) Continuadim of the p m f of 4.14. AS in Cam 3, we can define J:,{

c lMal

[i < i(a)]such that Jielis indiscernible over A:, and for every distinct a:, ...,U",J:,~ a;^...^a% realizes f,[tp(a: ^...^a:, B:,,)] (if 2 = a write stp instead of tp) and Av(Ji,,,A: u U J i , { )kAv(Ji,,, lM1l). Such J;,{ exist by 4.11. So IJ:,{l = A = IJ:,,], so we can extend fa to fa+,, with domain A: U U U1 J:,{, which maps J:,{ onto J:,{. By 3.13 fa+, is elementary. We have proved that for B E lMll, p = A, + h(Z), tp(Z, A u U J:,J is FE-isolated. A similar claim can be proved on any Z E 1M,1.So (iv) holds.

u1

So we fmish the definition. Notice that if we interchange bl, with Ma, A, with 8 2 , a:,{ with at,{,J:,{ with J:,{ fa with f l l all conditions still hold. Also f, = U,<, fa is an elementary mapping, Dom f, c IM,l, Range f, c IMSl, f, extends f. So by symmetry it suffices to prove lM,l = Dom f,. Let c E IMII - Dom f,, and we shall get a contradiction. Now we prove (*)

If a < x, p E #(A:) is realized in MI, then it is realized in A;.

As M, is F-atomic over A:, p is F-isolated, so by Ax(XI.1) there are

q E #(A:),p E q such that q is F-isolated. As of x 2 K ( T)q does not fork over A: for some fl < x. Clearly for some j,q is e(,-isolated. If A is singular choose y = h(i) < A, such that a,b < i, A ( j ) < y . So by 4.3 q' = q 1A: is q(,,-isolated, so for some B G A:, (q', B) E PgCn, and aa A(i) is regular > A(j),B E A: for some f < y. We can aasume an element of M, mlizing q', is even so we could have chosen as except when q' is realized in A:, so we get the conclusion of (*). If A is

regular the proof is similar but easier. As c E lMll - Dom f,, for some a(0) < x, tp(c, A:) d m not fork over A:(o).By (*) we can define by induction on i < w + w + w, a, E A:,,,, a(i) < x incmaaing, a(i + 1) 2 a(i) + w (remember cf x > Ho),so that a, realizes tp(c, A:(#,),a, E A,(, +,). By 111, 2.12 and 111, 1.10, I = {a1: w s i < w + w} is an indiscernible set, tp(c, A&@)) is stationary, a(i)(eoflisalimit ordinal). andAv(1, A:) = tp(c, A:). Letfl = U1
OH.

IV, 8 41

197

PRIME MODELS FOR STABLE THEORIES

tp(c, A:(,)), Av(J, A; +, u U J) k Av(J, 1MII). (J is non-trivial as c $ A:.) As J is not J j,),,+(,, for some distinct b,, . . ., b, E J tp(6, A; + 1) Y tp(6, A; +,) where b = {b,, .. .,b,), w.1.o.g. n is minimal. Jj+l,j, thereisaminimalt a i@ + 1) h A j + , = A;+, u U Uj
,,.

- 9

kt~((aa+a+~,.--,aa+a+n-l),BuJ~+I,,)

by the minimality of n, and

by Claim 4.15(2), a contradiction by Ax(VI1) (for A singular reduce it to some regular A' < A).

DEFINITION 4.3: M is F-*admissible over A (x = t, 8, a, F

=

Fg) if:

(1) M is F-saturated.

(2) A s 1MI and M is F-atomic over A. (3) If I c ]MI is an infinite indiscernible set (of elements) over A, then dim(1, A, M) 5 A. (4) If I = {a,: n a w) c lMl is indiscernible' over A, but tp(ao,A u {a,: 0 < ?a < w)) is not F-isolated then dim(1, A, M )5 No (this can happen only when cf A = No, and is not needed for A = No). We demand (4) only when KO = c f A < A.

CLAIM 4.17 :Let F = F,Z,x = t, s, a and M be F-*admissible [F-atomic, F-saturated] over A, and pi (i < a)types over A (if x = a almost over A), and A* = A U {G:ti€ JMJ, @ realizes eome pi, i < a) U B, where B E W I , IBI < c f h and for some regular p < A, tp,(B, A) E F,"(C), (CI < p. Then M is F-*admissible [F-atomic, F-saturated] over A*, provided at least one of the following conditions holds : (1) A ie regular and F eatis* Ax(X.1) and (XI.1). (2) A is sing&?', for arbitrarily hrge regular p s A, % satie$a Ax(X.l), (XI.1) and x = a a cf A r K(T),x = B =- cf A 1 Aa(T) and x = t a cf A r Aa(T). satiefi (3) A is eingular, for arbitrarily large regular p 5 A, Ax(X;l), (XI.l), and x = t, and for some p a A each p, is a FL-type (or an FL-isolated type).

198

PIlIME MODELS

[OH. Iv,5 4

Proof.Clearly it suffices to prove the version with “F-atomic”, and WSuming B = 0.

(1) Immediate, by 3.15. (2)Similar proof, e.g., if z = a, and 5 E 1611,then for some B, c A, lBll < A, stp(Z, B,) C stp(E,A). By 2.16(2),(iii),we can find B, c A*, lBal < ~(2’) and E’ such that E’ realizes stp(Z, A), and stp(E’, B, u B,) Istp(E’, A*). As lBal < cf h we can continue = in the proof of 3.16. (3) Let E E 1611,and we must prove that tp(E, A*) is F-isolated. By the hypothesis, there is a regular p < h such that FL satisfies Ax(X.1) and (XI.l),tp(E, A), and p , (i < a) are FE-isolated.So tp(E, A ) has a FL-isolahd extension q E #“‘(A*)realized by 5’ E !MI.So for some r c q, Irl < .p, r C q, and let r = {q,(Z, a,): i < /3 < p}. For each i, there are sit, a?, p6, . . .,phcr,such that a, = at%?, at E A , si; E A, = U {a E M: a realized pi, 1 I n(i)} and p{ E {p,: j c a}. By 11, 2.2(8) for every i there is 8 , A,~ and 6, such that for every A,, Cq,[E’,iZt, a] E O,[si, at, 6,].As M is Fi-saturated, pi is FL-type, there are I,!If(#, 2:) ~ p { , such that the above mentioned fact holds for every 7i E A; = U {a E M: 7i realizes I,!Iifor some 1 I n(i)}. So we can define by induction on i < /3, 6; E A, such that tp(6, A u E’ u {6,:j < i}) is q-isolated, and for every si E A;, Cq,[E’; sit, a] EE O,[si; a:, 6iJ.So clearly we can assume r is over A u (6:: i < /I}and the rest is as in 3.16.

THE SECOND CHARACTERIZATION THEOREM 4.18: Suppose K ( T )= KO,F = F;, z = t, 8, a and F eatisfies Aziome (X.1)and (XI.l) and when A is singular, for arbitrarily large regular p < A, F; satiefiee h(x.1) and (xI.1). Aeeume (i) z = a; or (ii) x = s , t and h > No;or (3) z = 8, t, T totaliy tramcendentul. Then (1)M is F-prime over A iff M $8 F-~admis~ble over A. (2) The F w m e &l Over A is unique up to ~eornorlph~sm, Over A. (3)If M , ie F-*admissible over A,, 1 = 1, 2, and f is an elementary mapping from A, onto A, then we can extend f to an i8omOr.ph~87?~ from Mi onto Ma. Proof. As in 4.14,it suffices to prove (3).For simplicity restrict ourselves to thecase 5 = a. Clearly it suffices to prove Claim 4.19(2)below, by applying it to qo = {z = z},y = 1.

CH.

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PRJME MODELS FOR STABLE THEORIES

199

CLAIM 4.19:8uppoee M , , A , (1 = 1, 2) and f are as in 4.18(3). (1)If a: E lMll, tp(a:, A,) = f[tp(ar, A , )] then we can extend f to an elementary m y p i n g from A , u {a E lMll: a realize0 stp(a:, A,)} onto A , u {a E 1M,1: realize0 stp(af, A,)}. (2) If qf (i < y ) are l-typeaOver A,, then we can extend f to an elentent a y mapping from A , u {a E lMll: a realize0 8ome qi} onto A , u {a E 1M,1:a realiz~f(qf)}.

Proof of 4.19. We prove by induction on a that: 4.19(2) holds, when D(q,, L, 00) < a and 4.19(1)holds, when D[stp(a:, A,), L, 003 s a. Caae I. a = 0. Then (2) is empty, and in case (1) by definition of degree stp(ar, A,) is algebraic, hence is realized by a* only, so extend f by lettingf(a:) = a:. Case I I . a > 0, we have proved (1)and (2)for all fi < a, and we now prove (2) for a.

c,

Choose UZEIM,I (i such that for i < j < stp ( a : , A l )f stp(a:,A,), a: realizes some q, and [ is maximal. Let A; = A, U (U:UEIM,I,a realizes some stp(a:,A,), i <j} (so for 1 = 2 it is defined only when (a::i < j )is defined). We define by induction o n j elements a: E 1M,1 (i < j)and an elementary mapping f, from A{ onto Ah, such that fo = f, i < jimplies f, extendsf, and f,+,(a,’) = a;. So clearly fc is the mapping we need. Notice that by Claim 4.1.7,M , is F-*admissible over A{. For j = 0, j limit ordinal, there is no problem, so let j = i + 1. Choose a: E IMsl, which realizes ff(tp(at,A t ) ) (possible aa M,,Ma me F-*admissible over Dom fi, Range ff, respectively). By the induction hypothesis we can extend fi to an elementary mapping f,+,from A\+, onto A:+’ (first, since M , , M 2 is F-admissible over Dom f,,Range fi, respectively, we can extend fi t o an elementary mapping f: with domain A: U a:, qgff:(a:)~M2;second, by 4.7 and (2) for D(stp(@,Ai),L,0 0 ) we can find!,,,). CaseI1I.a > O.Wehaveproved(1)foreveryp < a,and(2)forfi s a, and we shall prove (1)for a.

200

PRIME MODEL8

[CH.

m,8 4

Let I, E lM,l be a maximal independent set over A, whose elements realize stp(a:, lM1l). Let I, = { a ; : j < a,}and a; = a: so if a, 2 w, I,is indiscernible over A, (by 111, 1.10 and 111, 2.9(1)). By Definition 4.3, 4.9 and 4.10, we can Mume that (A) w1 I a, so, for some regular p s A, for every i < A tp(ai, A, u {a;:j < i}) is FUO+lil +-isolatedhence w.1.o.g. a, = A, or (B) a, I w. By F-homogeneity of (Y,,a)croAl it is easy to see that the same case occurs for I = 1, 2. If (B) holds, define by induction elementary mappings f,,,n < w, such that: (i) fo = f, f,,+l extends f,,, M,[M,] is F-admissible over Domf,, [Rangef n l . (ii) ut, i < n belong to the domain 0ff3,,+,. (iii) at,i < n, belong to the range 0ff3,,+~. (iv) Domf,,,+, = Domf3,,+, u {a E lYll:a realizes stp(af, A,), and tp(a, Dom fsn + forks over A l}. This is easily done. Iff3,, is defined, tp(a:, Domf,,,) is F-isolated (by Definition 4.3, part (2)), hence f3,,[tp(4, Domf,,)] is F-isolated, so by by some bi, and letf3,,+,(u;) Definition 4.3, part (2), it is realized in Y,, = bi, D0mf3,+1 = Domf,,, u {a:}. If n = 0 we can let bi = a% = a:. The definition of f3,, + 2 is similar by Definition 4.3( 2) and that of f3,, + is by the induction hypothesis on 3n + 2, by (1) [notice that if a realizes stp(a:, A,) then tp(a, Domfl) k stp(a:, Al); and by 111, 1.2(4), forking implies decreasing in degree]. Now U,,<@ f,,is the desired mapping (by the maximality of I,). So w u m e (A) holds, i.e., a, = A. So I, is indiscernible over A,, and, using 4.17, for every i < cf A, M , is F-*admissible over Bf= A, u {a;:j < i}. Let Ci = A, u {a E Y,: a realizes stp(af', A,), tp(a, Bf) forks over A,}. Clearly for every i I A, we can extend f toft by letting &(a;) = a;(j < i), and if i < of h then extend it by 4.18(1) to an elementary mapping fr from 0: onto q. If tp(ai, 0;)forks over A,, then by F-homogeneity it is eaey to prove that if a mdizea stp(a:, A,), a E M,then stp(a, CL) forks over A,. Hence we can prove (1) as in (B),(with C5,u {a E liK,l:tp(a, C5,) forks over A,} included in Domf3,,, &ngef3,, for I = 1,2 reap. So assume tp(ai,Ck) does not fork over A. Then, again, tp(at,Cf) does not fork over A, for i < A. Also if F E Cf (i < A) then stp(c,B;) Istp(c,B;+J (by the non-forking assumption, i.e. otherwise, sinceM, is F-saturated, we can find Z E M , realizing stp(F,B;), but not stp(c,

OH.

Iv, 8 41

PRIME YODELS FOR STABLE THEORIES

20 1

B:+Jbut necessarily C’E C: and tp(af,C:) fork over A;,a contradiction) hence stp(C,B:)-I stp(C,Bi). In fact for every CE W J ,for some finite J E I,,stp(C,A, U J)t- stp(C,A, U I).So Ml is F-atomic over Bi, and we can easily show that M , is F-*admissible over Bi (notice we can reorder the ai’s). So by 4.18(2) (for a)we can extendf, to the desired mapping.

8 k h of the proof for other ma. Let x = 8; by 2.2 FA-primenesa (seturation) is equivalent to R-primeness (saturation). By Exercise 4.8 FA-*admiasibilityimplies F+dmLsibility. So as we have proved 4.18 for x = a; it suEces to prove that over any A there is en FA-*admisaible model, but this follows by Ax(XI.l) and Exercise 4.14 and 4.16. If x = 8, t T totally transcendental, use R ( p , L, 2) instead of D ( p , L, m), and the proof is similar. If x = t, h > KO,in proving 4.19, case 11,we first, as in the proof of 4.14, extend f to f‘, such that (bl,,Dom f‘), (Ma, Range f’) are F-*admissible, and Dom f’ is the universe of e model; so each p, is stationary. If x = 8 , t h > of h = KO, in proving 4.19, case III(B) let h = z n < a p n , p n < pn+1, F ,. satisfies Ax(X.1), (XI.l), pn regular, and tp((a&. . .,a;>, A ) is F,I-isolated, and e.g., (iv)’ Dom f3n+3 = Domf,,+a U {a EM,: a realizes stp(a:, A,) and tp(a, Dorn fsn+a) forks over A, and is F,,n+,-isolated}. In cases 11, III(A), the change is similar, and when cf h > KO-is easier.

DEFINITION 4.4: bl is F-minimal over A, if M is F-saturated, but there is no F-saturated model N , A G IN1 E IMI,IN1 # ] M I . LEMMA 4.20: If Over A there are an F-prime model and an F-minimal model, tlren, up to ietnnorphim Over A , there i8 a unique F-prim model Over A, and a unique F-ncinimal model over A and they are ietnnorphic Over A .

No F-minimal over A. By the d e h i tion of F-prime there is an elementmy mapping f:M,--t N o f 1 A = the identity. By the F-minimality of No, f ( M o ) = No, so they are isomorphic over A. Hence if MI, bl, am F-prime over A, they me isomorphic to N o over A, hence isomorphic over A. Similarly any two F-minimal models over A am isomorphic over A. Proqf. Let blobe F-prime over A,

202

PRIME MODELS

[CE. I v ,

84

THEOREM 4.21: Let F = Ff, x = t, a, a, of h 2 K ( T ) ,F satisfies Axioms (X.1) and (XI.l),and: T is totally transcendental or x = a =h > KO,x = t =- of h > No.Suppoae M is F-aaturated and F-atomic over A. Then il4 i a F-minimal over A if in M there &3 no in$nite non-trid indkcemible aet over A. Proof. suppose I C 1M1 is an infinite non-trivial indiscernible set over A. We can assume I is a maximal one, and let I = {gi: i < a} (a 2 w ) . Let N be any F-prime model over A u U I (exists by 3.1). We can assume N < il4, hence N omits Av(I, A u U I),by the maximality of I . Hence also any F-prime model over B = A u {a,: 0 < i < a} omits Av(I, I?); and let N , E il4 be one such model. So N, is F-saturated, A c INl/ E 1iK1,but a o $ IN1].We conclude hf is not F-minimal. Now assume M is not F-minimal, so there is an F-saturated N, A E IN1 E ldll IN1 # 1il41. Choose C E 1M1 - INI, and C c IN!, ICl < K ( T )such that tp(c, INI) does not fork over C.For simplicity let x = a, choose c, E IN1 which realizes stp(c, A u C u {c,: i < n}) (exists, &B bl ia F-atomic of A, and by Ax(V.2) (or 2.8), tp(c, A u C u {ci:i < n}) is F-isolated). So clearly {c,,: n < o}c is an infinite non-trivial indiscernible set over A (see 111, 1.10 and 111, 2.9).

EXERCISE 4.4: Prove 4.18, when x dental (see Definition 11, 3.4).

= 8,

h = No,and T is transcen-

QUESTION 4.5: In the characterization theorems, can we weaken the assumption “F satisfies Ax(X.1) and (XI.1)” by “over A (A,) there is an F-primary model”? (see 6.6). EXERCISE 4.6: Let M be F-**admissibleover A, F = Fie, x = a, t, a if iK is F-saturated, F-atomic over A, and for every finite B c 1M1,and p ES(A u B), dim(p, A u ByM ) is s No. Prove a characterization theorem parallel to 4.14, 4.18, for countable superstable T . (Instead of countable, you can assume h*(p) s Nofor every Fio-isolatedcomplete typep; h*(p)is definedin Definition 111,6.1). Generalize similarly 4.21. QUESTION 4.7: In 4.12, try to replace K ( T )by K,,,~(T), at least for stable T . EXERCISE 4.8: Prove that if 211 is Fi-admissible over A, h > X, then

OH.

Iv, 8 41

PRWE MODELS BOB STABLE THEORIES

203

M is F+dmissible over A ;and similarly for -*admissible and -**ahis-

sible. [Hint: Use 2.2; note that Fi-isolation (atomicity) implies Risolation aa for Definition 4.3(4)note that tp(ao,A u {a,,: 0 < n < w } ) Istp(ao,A u {a,,: 0 < n < a}),so if tp(ao,A u {a,,: 0 < n < w } ) is not Pi-isolated, it is not Fi-isolated.]

QUESTION 4.9: In the proof of 4.14, A regular, can we choose x = K,(T)for A singular, can we choose x = cf A ? EXERCISE 4.10: Show 4.16(2) holds also when tp(a,, B ) is not stationary. PROBLEM 4.11: In the characterization theorem 4.14, can we weaken Definition 4.2(3) to “dim(I, A, M ) 5 A ” ? EXERCISE 4.12: Show that 4.18 may fail for x = 8 , t , A = No.[Hint: T = Th(M), M = (m2 x w , . . . , E n , .. .), where
204

PRIME MODELS

[m. IVY6 6

IV.5. VariouS results THEOREM 6.1: Swppwe IT1 = A, p s A, p replur. T h (1) ==(2) (3) + (4) where: (1) ID(T>l < 2'. (2) There arenozand formulascp,(Z)(7 E "'2) (ud?mutpranz&rs)mh with r . that for every 7 E '2, {qnlt(Z)qrt1: i < p} is &t& (3) For every m-type p over 0, IpI < A, there is an m-type q over 0 m h t W Iq1 < 1 1 , a n d p u q ~ r f o r s ~ r ~ D , ( T ) . (4) (A)T hm a model A2 of cardinality A, which is FA-conetructible(over 0 ) hence FA-pirnitiveand FA-at0mic(over 0). In fact there is such a model over any A , IAI < A. (B)I f A<" = A, we can have M ~-COm2)aCt(Irence Fx-eaturated). model of cardinality A, t h M (C) If A = Ash, T hm a Aie F A @ ~ andytherefore FA-p+-i~ and A - p i m e .

Remark. By 111, 6.14 we can assume IT1 s ID(T)I. Proof. We leave it to the reader, as it is similar to previous proofs, except (4) (C), where for h regular it is easy; for h singular, see Ex. 5.17, 5.18.

Remark. Notice that every Fi-saturated model has cardinality 2 A. A converse to 5.1 is : THEOREM 6.2: Suppose IT1 = A, and T hae a A-compact m d e l of cardidity A (e.g., when A = A
Proof. ( 1) By 6.1,if 6.1(3) holds, T has a A-prime model of cardinality A. Now suppoae M is a. A-prime model, but the m-typep is a. counterexample to 6.1(3). As 1p1 < A, M A-compact, some Z E 1 x1realizes it, thus po = tp@,0) is not FA-isolafed.The following theorem, 6.3, shows

OH.

IV,5 63

205

VlCIlIOUS RESULTS

that there is a A-compact model which omits p o , thus M cannot be elementarily embedded into it, contradiction. The rest of the proof will follow Theorem 6.3. THEOREM 5.3: 8uppose pi ie an m,-type mer A for i < a s A and for no mi-type q Over A , q kpi, 1q1 < A. Assume IAI IT1 = A. (1) T hua a model M ,( A s 1M1)of cardidity A omitting each pi (i.e., tb model ~ f X d ~ Z eit). 0 7u) 8epWW (2) If A = A < A ,or T has a A-compact model of power A , A = 8, then in (1) we can choose the model to be Fi-saturated, and even A-compact.

+

Proof of 5.3. (1) For simplicity let A = 0. Let {&: i < A} be a list of all finite sequences from {z,:i < A}, each sequence appearing A times and i < A} be a list of the formulas of the form ~(z, g,) let {tpi(z, fjMi)): (9 E L , j < A) such that &, gd0 E {z,:j < i}. We now define by induction on i < A a consistent set I', of formulas (with no parameters) such that in I', appear only the variables {z,:j < 2i}, and lI',l < lil+ + No. Let I', = 0, I', = Ufcd I', clearly the induction assumptions hold. So suppose I',is defined, and let

I':

=

I'f

u "z)Pi(z,

37(1(f,)l

+ P?f(%:arY

& f d

u { % i + l z x,:

5 2;).

Trivially I': is consistent, and the variables that appear in it are only from {zi: i < 2(i + 1)) and its cardinality is < lil+ + No.Let I'F be the closure of I': under conjunctions, and let r, = {(3)v(&,Z): 2 disjoint to &, but ~{z,:j < 2(i and v(&,Z)

E

+ l)},

ria}.

So lril < A, r, an Z(&)-type. Hence when Z(BJ = m,, j < a, r, Vp, (changing the names of variables accordingly). Suppose & is the jth appearance of ji, in {gB:fl < A}: i f j 2 a or Z(&) + m,, let Ti+, = ria; otherwise for some $&) ~ p , ri, u {--&(&)} is consistent, so let I',,, = I': u {+,(&)}. Clearly the induction assumptions are satisfied, and

I',

U i < A pi. As

FAis consistent there is an aasignment xi H a, realizing it. By the Tarski-Vaught test {ai: i < A} is a universe of a model M , which by the construction has cardinality A and omits the =

Pi's. (2) Similar proof.

Remark.For oountable T the condition on the p,'s is also neceseary for (1). Similarly if A =

A
it is ne(feBB&Fyfor (2).

PRUKE MODELS

206

[a. IV, 8 6

Continuatiotc of the pmf of 5.2. (2) If there is an F,-prime model M1, by 6.2(1) condition 6.1(3) holds, hence by 6.1 there is an FA-primary model AM,.As Ml can be elernenfitwilyembedded into M,, and by 3.2( 1) M , is E*,-atomic,dso Ml is F',-atomic. So by 3.16 Ml is Fi-construotible (over 0), hence Ml is F , - p h q too. So by 3.9 MI, M, are isomorphic. As we could have replaced Ml by any F,-prime model, the F,-prime

model is unique. Ite homogeneity follows by 3.7. (3) Follows from the proof of (2). (Note the remark before 6.2.) CONCLUSION 6.4: Let T bs countabZe. T ?msan F&Wm7nodeZ iflfor every formulcc cp(Z), (3Z)cp(Z) E T there is a form& #(Z),(3Z)[cp(Z) A #@)I E T , and cp(Z) A #(Z) k r E D,(T) (Z(Z) 3: m) for 8ome r .

THEOREM 6.6: 8wppose A+ = 12'1 s ID(T)I, A regular. Then (1) * (2) * (3) where: (1) lD(nl < (2) For every m < O, and m-type p , 1p1 < A, there i s an m-type q, 1q1 < Asp u q i e &tent, andp u q k r E D,(T) for some r . (3) T hae an F$,-atonCic h l . Remark. In (3) we cannot demand that the model has cardinality A+ even if A < (D(T)I,see Exercise 5.13. If h = A
Proof. (1) =r (2) similar to 2.16. (3) It is easy to see that it sufiices to prove: (2) If A is Pr,-atomic, IAI < A+, a0E A, C(3z)p(z,a0),(where L(T)),then there is b, such that A u {b} is E*,-atomic and bP[b,aol. For E A , let q4 be an @)-type over 0, q,, k tp(7i, 0), lq61 < A, q,, closed under conjunctions. Let us prove (*). Let {ai: i < a s A} be the set of all finite sequences from A. We now define by induction on i,a l-type pi over A, lptl < A. Let po = {cp(z; a)}, = Ui
(*I

'p E

Qt

= {(3@[&,

%s

z)

A

#(Bs z)]:

~p(z;8,s E ) E pis C disjoint to ?it, #(#, E ) E q,,,-e, CE

Clearly lqil s Ip;I

+ z{lqh-el: C

C_

Dom p i } .

Dompi} < A and qi is consistent.

UH. IV,

8 61

VARIOUS RESULTS

207

So by (2) there is r, = r,(x, g), q, c r l , lrll < A, r , consistent, and r, I- r' E D,(T) for some r:, m = l(zi,) + 1. Let Pf + 1 = P: u {1G(x,a): 1G@,

It is easy to check hence the theorem.

@ E4

is consistent, so if b realizes pa, we finish (*)

THEOREM 6.6 (The Uniqueness Theorem): (1) Let T be stable and h regular, A+ 2 K ( T ) .If T hua an PAprimary model over A then all the EY,...prinze models over A are &~nmphk Over A . ( 2 ) Let T be stable, p + 2 K ( T ) Let . M be an Fg-pimary model Over A

(x = t, a ) for which there are p r e l u t i m R, on m h that C E B, IBI s p , (B,R,),.,, < (JMl,R,),<,,=r C u A is Ft,-constructible over A . Then all the F A - pmodels i ~ over A are isomorphic Over A . Remark. Notice we do not demand that our F aatisfiesAx(XI.1)or (X.1). In [Sh 79a] a simpler alternative proof appears. Theorem 5.6 is a generalization of:

THEOREM 5.6': If IT ie stuble and m d u b l e and T hua a primary d over A , then dl the prime models over A are i s m p h i c over A .

l

QUE8TION 5.1: What about x = 82

Proof.In (2) we can w u m e p 5 A. Let p be h and x be t in (l),p be p and z be t or a in (2). In (1) let M be an Fi-primary model over A. (1) Let N be q-prime over A; we shall prove it is isomorphio (over A) to dl. We can assume, by the definition of F-primeness, that A E IN1 c IMI.AaMisFg-prirnary,letd = (A,{a,:i < cc},{Bi:i < a}) be a Fg-construction of [MI.We can assume a, # A u {a,: j < i}. Clearly tp(a,, d(i))does not fork over B: (by 111, 4.4). So IB:I 5 p. For B, C s IMI, call the pair (B, C) good if (i) a,E B [a, E C] implies B: G C[B E B:]. (ii) for every 6 E B, tp(b, I NI) does not fork over B n I NI , (iii) for every 6 E B nC, tp(6, IN I) does not fork over B n C n IN I. If a set D E dl satisfies (i) we call it closed (as we have a fixed construction at). Now we prove CLAIM 6.7: Suppose (B, C ) is a good pair, I BI > p. Then we can $nd B,, i < IBI m h that B = UfB,, lB,l s p and former y i -= IBI,

sets

208

PRIME MODEL8

[m.Ivyf 6

Proof of 5.7. We prove it by induction on IBI > p. Suppose we have proved the assertion for every cardinality < IBI (and >p). Let B = {b,: i < IBI}, and define by induction on i, sets B, s Bysuch that (1) j < i =s. b, E B,, and Bj E B,; Bd = uj
,

(@I.

CH. I v ,

8 61

VARIOUS RESULTS

209

(i) tp[& (B n INI)u (B n C)] does not fork over (B, n INI)u (Bi n C)[by (611(j) tp[z, (B n INI)u (B, n C) u a] does not fork over (B n IN]) u (B, nC) p y (i) and the monotonicity of forking]. (k) tp[E, (B n INI)u (B,n C) u a] does not fork over (A* n INI)u (B, n C), hence over (A* n INI)u (B, n C) u a [by (g) and (j), using the transitivity of forking 111,4.41. (1) tp& u a u (B, n C),B n IN!]does not fork over A* n IN1 [by (h) and (k), using Axiom VII for F',]. (m)tp(6,B n INI) does not fork over A* n IN1 [by (l), aa 6 =

E a u a u (Bfn C)]. (n) tp(6, IN]) does not fork over A* n IN1 [by (m) and (ii), using

zniz

III, 4.41. Thus (B,+,,C,)is good. If IBI = p+, then lB,l c- p and we finish. Otherwise we can a,ssume IB,I > y, and let a(0) = 0, a(< + 1) = a(i) + lB,l (ordinal addition), a(6) = Ulc6a(i), (so a(i) < IBI for i < 1231) and use the induction hypothesis on (B,+,,C,)to define B;, a(i) I; j < a(i + l), and the Bj show the claim holds. Continuation of the proof of 5.6. Now clearly (1M1,0) is a good pair. Assume 11M11 > p, as llMll s p is a trivial case; so by 6.7 there are B,i < 11M11,suchthat lB,l s p,andlettingC, = U,<, Bj,C,,M,, = 1iK1, B d c c d (for limit 6) and (B,+,, C,) is a good pair. By 1.3(2) it suffices to show that IN1 n [A u C,,,] = IN1 n [ A u C,u B,,,] is Fg-constructible over IN1 n [A u C,]. Now does not fork over B,+, n C,n IN1 [by the (p) tp*(B,+, n C,,"1) goodness of (Bf+lYC,)I. (9) tp,(B,+, n C,,IN1 n [A n &+,I) does not fork over IN1 n [A u (B,,, n C,)] (by (p) and monotonicity of forking]. (r) tp,((NI n B,+,, [A u (B,,, nC,)]) does not fork over IN1 n [ A u (I?,+,n C,)] [by (9)and Ax(V1) for F{ with A* = B* = C* = IN1n [ A u V4+, nC,)l, and any a E IN1 n B,+,, 6 E (B,+,n Cdl. (a) d is also an F',-construction and B,+,,C, are closed for this construction my Bt's definition and (i)]. (t) tp*(B,+,, A u C,)does not fork over A u (B,,, n C,)[by (a) 3.3 and Exercise 3.61. (u) tp,(lNI n Bi+l,A u C,) does not fork over IN1 n [A u (I?,+,n C,)][by (r),(t)and transitivity of non-forking 111, 4.41. (v) tp,(lNI n B , + , , AuC,) does not fork over IN1 n (A uC,)[by monotonicity of forking].

210

PRXME MODELS

[OH.

my 6

(w)By (i), B,, ,, C,a m dosed in the F;-construction d,hence [by 3.31 B,,, is K-oonstruotible over A u C,,henoe (3.2) Ff-atomic over A u C,[remember we are in 6.6(1)]. (x) IN1 n B,,, is Fg-atomic over IN1 n (A u 0,) [by (v), (w) and 4.31. (9) IN1 n B,,, is Ff-constructible over IN1 u A u C,)[by (x) and 3.16, h i s regulas]. So we f i s h the proof of 6.6(1). (2) Left to the reader. CONCLUSION 5.8: If T is stuble, IT1 is regula;r and Over A there is an Flt,,-primary model, then the FrT,-primemodel over A is unique. THEOREM 5.9 : Let Sf = { A : IAl = h imply over A there is a prime model among the Ff-compact ones}. (1) p 2 h = A<, 3 (TI,AES: implies (2) &o=minSt,, < a0imlplie8A&~< ~(~T~,l)(seeVII,DeJinition5.1), (3) For x = a, a, eirnitar results hol& with 2ITI i w t d of ITI.

Proof. ( l ) ,(2). 5.9A DEFINITION: (i) A type pc=S(A)is Ff-inevitable if every F f compact model JM, A c 1N1,realizes it. For F: = F&, we omit it. (5)@(A) is the assertion that for every A of cardinality A, and Fftype q over A, there is an Fg-inevitable p E 8 ( A ) ,q E p . 5.9B CLAIM: (1)Ifh = h'"+lTI, then he&': @&:(A). (2) If &t(p) hob% whenever A I p < A, = A,'* + IT1

A ESL.

+ A+,

then

Proof.(2) Given A, IAl = A, we define by induction on i elements a, such that tp(a,, A,) is inevitable where A, = A u {aj:j < i},and for each i, and each 1-type q over A,, lql < K,for somej < A,, q is realized by a,. It is easy to see A,, is the universe of a K-compaot model bl, A E IMl, M is F,"-primeover A . (1) The "if" part follows from (2) (for A, = A+). For the "only if" part, let IAI = A, q a 1-type over A, 1q1 < K. Let iU be a K-prime model over A (exist by an assumption), and U E M realizes q, then p = tp(a, A ) E S ( A )extend q and is Fk-inevitable (because if A -c W ,H' E"L,-compact,then M can be elementarily embedded into bl' over A , hence bl' realizes p ) .

m. IVY5 61

VARIOUB a E B l n T s

211

5.9C CLAIM: The following are equivalent (i) q ia a 1-type over A , 1q1 < K , q has no Fi-inevitable completion in 4A), (ii) there are F i - m p a c t d l . 9 M yM a (a E N )sucla that: A E M , A c Ma, and f o r every a E N ,if a realim q, then M a omit tp(a, A ) .

Proof. (i) * (ii): Let M be any Fi-compact model, A E M. If a E M , a realizes q, then by (i), tp(a, A) is not inevitable, so there is a E",compact model Ma, A E Ma, Ma omit tp(a, A). If a E M does not realize q, let M a = H. (ii) == (i): If q has an Fi-inevitable extension p € & A ) , then p is realized in M (by the definition of Fi-inevitable), so let a realize it, but then M aomit p , contradiction. 5.9 D COROLLARY : Suppose &:(A) fail, or even A $S:, then for every p = peK TI s A, QL(p)fail and 80 p $8;.

+I

Proof. If A $ 8;,then by Claim B, &:(A') fail for some A' 2 A, so w.1.o.g. &;(A) fail. So for some A, IAI = A and 1-type q over A, 1q1 < K , q has no inevitable extension in #(A). So by Claim C, there are M , M a (a E A ) as mentioned there. By the downward LowenheimSkolem theorem to the appropriate model, we get that Claim C(ii) holds for some A' E A, lA'l = p, so Claim C(i) holds so Qbb)fail, henoe by Claim Byp $8:.

5.9 E COROLLARY : There is a sentence t,h E LIT,+," which has a model of power A i# not Qbo(A).

Pro@ Formalize Claim B(ii).

PROBLEM 5.2: What is the &st cardinal A, such that if F& satisfies Ax(X.l), IT1 = No, and over some A the F&,-prirne model is not unique, then there is such A, IAI < A (similar problems exist for other Fs, and for uncountable T). THEOBEM 6.10: Let A > IT],A = AeA. Then the following conditim are equivalent. (1) et, 8ath$a Axiom (xI.l), (x.1)a d F~-saturatiOnimqdies Ampa~tne88. ( 2 ) Over any A there is a A-prime model. ( 3 ) Over any A , IAl < A , there is a A-prime model.

(4) If 1p1 < A, p an m-type over A , IAI 5 A, p hua an Fi-ieolated & d o n in #“‘(A). ( 6 ) T ~ 8am M a8 (4) for arbitrary A .

Proof. (1) * (2)By 3.1(2), 3.10(2).

(2)* (3) Trivial. (4)=- (6) Suppose A is a counter-example to (6). As A = Aeh, A is regular. Defbe by induction on i 5 A, A, E A , lA,l 5 A. Let A, = Domp, and A, = UICd A,. If A, is defined, for every m-type q over A,, p s q, 141 < A, q haa at leaat two extensions in P ( A ) , so for some Q,(J;B,) (8, E A ) . q u {Y,(E; is consistent for t E {o,~}. Let 8, = A,+l u {a,: p E q, 141 < A, q an m-type over A,}. Clearly lA,+lls X a8 I TI < A = Aeh. Nowp and Ahprovide a counterexample to (4). (6)=t- ( 1 ) Clear. (3) (4)Letp be an m-type over A , IAI s A, 1p1 < A. By (3)there is an A-prime model M over A. So some E lM[realizes p. If q = tp(8, A) is Fi-isolated we finish. Otherwise by 6.3 there is an Acompact model N,A E INI,N omits q. But aa M is A-prime there is an elementary mapping f from M into N, f t A = the identity, 80 f(a) E IN1 realizes q, contradiotion.

-

PROBLEM 5.3: Find a similar theorem when A < ACh. PROBLEM 5.4: What can {A: Ff satisfies Ax(XI.1)) be, z = t , 8 , a ? THEOREM 6.11: If T is countable F& does not sati8fy Ax(X.l), them FZ does not satiefy Ax(x.1)for x = t , 8 , a and any p and Ft does not 8atiefy Ax(XI.1)for z = 8, a and any p or z = t , p > 12‘1.

Proof. So there a m A, a E A and $(x, a), p such that (3x)$(z;a), but , there are no p €&A), q E p , such that 1q1 < A, $(z;a) ~ p and q kp I+ 9. Let P = A, and extend (Q, P) to a E-saturated model (Sl,pl). We can assume Q, = Q, and let P, = A,. As (Q, P,), (Q, P ) are elementarily equivalent, for every h i t e 1-type q, over A, such t h t (3z)[$(x,a) A A q] there is 5 E A,, such that {$(E; a)} u q u {p(x, 5)t} is consistent for t E (0, 1). By the p-saturation of (SlyP1), this holds for any q, lDom qI < p. The conclusions me immediate.

DEFINITION 6.1: The set I E 1M1 is absoluteiy indi8cernible in M yif every permutation of I is induced by an automorphism of M.

a.IV, 5 61

VABIOUS RESULTS

213

LEMMA 6.12 : ( 1 ) An absolutely indiscernible set I in M , is indiscernible. (2) If M i s strongly (I(+-saturated,(see Definition I, 1.4(2)) I c lbl( I an indiscernible set, then I is a n absolutely indiscernible set in M . Proof. Immediate. Remark. So the real problem is to find M and I absolutely indiscernible in M ,such that ( I ( = (IM((. THEOREM 6.1 3: Let I be an indiscernible set, T stable in ( I ] .Phen I is absolutely indiiwrnible in s m M , ((M(( = ( I ) .Moreover, I is a d d indiscernible set in M.

Proof. Let u = ur(T);so as T is stable in ( I ( ,l I ( < s= 11)(by 111, 3.7) so K s I I I and even K s of 1 I ( . Let bl be q-primary over (JI . M exists by 3.1, 2.18(2); clearly (lMl( = ] I ((as T is stable in II)).By 4.10 I is a maximal indiscernible set in M , and by 3.8(1) I is absolutely indiscernible in M .

THEOREM 5.14 :Suppose Tisstable, h = A<",and Tisstableinsomep, u s p s h . ( 1 ) Over any A, lAl s h there M a Fi-cawtr~c4ibl.emodel M over A, 11M11 = A m h that: if B s J M I , IBI c A, p e s r n ( Au B ) doe8 not fork over C c_ A u B, IC( < u then p is realized in M . ( 2 ) The rrwdel M in ( 1 ) i s unique u p to isomwphim over A. Proof. ( 1 ) The proof is similar to the proof of 3.1(3), (4). ( 2 ) The proof is similar to the proof of 3.9 (in fact, by 3.1, 3.8(2) reap. applied to T h ( M ,a),,,). CONCLUSION 6.16: If I i s a (non-trivial) indbcemrible set of s e q w , and T i s stable in some p s I I1 then ) I [i s an absolutely indiscernible set in a nzodel M, 11M11 = III. Proof. B y 6.14(1) and 6.14(2). THEOREM 5.16: Suppose (TI = A < X , A G L = L ( T ) , ( A ( = A, and one of the following coltditions holde. ( A ) A is regular, x = 2". ( B ) There i s a tree I c A22 (i.e., 7 Q V E I 4 7 E I ) , so that II n a2( < h fora < h but ( I n%2( = x (for x = 2", I n A > 2i a a Kurepatree).

214

PRIME MODELS

[CH. IV,

55

(C) x = A+, A regular. (D) There i8 a tree Z c hr2, x = IZ n "21 > IZ n A > 2 ) + . Then (1) there are models Mi (i < x) of T , each of cardinalityh, such that: i(1) # i(2) < x, a, E IMi(DI,tp&, 0 ) = tp&, 8 ) implies that there are pt c tp(a,, O ) , lpll < A, pl k tpd(iji,, 0) (1 = 1,2). (2) If A'" = A, we can aseume each Mi it? ~ - w m p m t . Remark. Case B applies, e.g., for singular A when (Vp < A ) @<'IA
x

We get X = ANQ. We can conclude:

CONCLUSION 5.17:If IT1 = A < x, A, x SatiSfy (A) OT (B) OT (C) OT (D) from 6.16each?, E D( T ) ( i < io < x ) is not F,iaoZated then T ?maa d e l of cardidity X which Omit8 every pi. Proof of 5.16. We can w u m e w.1.o.g. that d is closed under Boolean combinations. We leave (2) to the reader and concentrate on (1).

Case C . We define the models M , (i < x = A + ) by induction on i. If we have defined M, for j < i let dji = {tp,(a, 0): si E IN,!, j < i} (so lQi1 S A) and it suffices to find a model Hi of T of cardinality A, 8UCh that if tpd(Ti,O)E Qi, IN,(then for some q E tp,(a, 0), Iql < A, q I- tp& 0) and a little more. This follows by

CLAIM 5.18: If IT1 = A, pi (i < io I A) types (in L ( T ) ,over 0 ) then T ha% a. nzodel M of cardinality A, such that: (i) 7i E M , realimp, implies the &stence of q E tp(si, O ) , lql < A fur which q kpi. (ii) If si E "1M1 a d fur 8 m e m-type q (over 8),Iql < A, q I- tp,(si, 8 ) then this hl&S fur 8Orne q E tp(8, 0). Proof. The proof is like that of 5.3(2), but we also do an assignment like (iii) of the proof of Case A. If A = No the set of types forbidden by (ii) of 5.18 is countable, 5.3 applies directly. Caee B. Let {p?,(z,&): i < A} be a list of all formulas cp(z, 8) E L p E {z,:j < A} and wsume jji E {z,:j < 2i}. We define by induction on u < A, for each r) E Z n "2 a consistent set of formulasin the variables {z,: i < 2a}, for A = KO, r,,is finite, and for A > KO, s No +

r,,

CH. IV,

5 61

215

VARJOUS RESULTS

[ I n '21 (note that w.1.o.g. IZ n '21 is non-decreaeing). For

a = 0 let

r,, = 0, and for limit a, q E ' 2 let r,,= UB<= r,,,. Suppose we have defined for a, and let us prove for a + 1. Let {(q:, qa, Z:, Z;): i < i(O)} be a list of all quadrupals (ql, q,, Z,, Z,) where ql,q, E ' 2 n I are distinct, Z,, 2, are finite sequences of variables from {zj:j < 2a} of equal length. So for A = No, i(0) < No, and for h > No, i(0) < No + IZ n a 2 ) + .We now define by induction on i s i(O), consistent set of formulm (9 E ' 2 ) in the variables {zj:j < 2a} such that II'il < No + II n '21+, and j < i implies G r,l. Let O r , = r,,,and for limit ri = U f CPi. , If we have defined for i, and there is a formula Q@) E A such that ri,u { Q P ( Z ~ ) ~ ( ~ ~1 ~=) } 1 , 2 , are consistent, choose = r:,U {tp(E)}JZ1 = ripU {lp(z)}.For all other r]'s, and if there is not = r;.Now let, for q E ' 2 , l E (0, 1 ) such Q for 7 = ql, q, too,

r!

rn- = ricO) u {@z)qa(~ iia) u {Zj

#

z,a+1:j

QtAzsa, ga)}

< 2a).

q u, qla

For each r] E A2 n I , = < A is consistent so some assignment qwuaf,eatisfiw it. By Tasaki-Vaught test there is a model iWn with universe {ut: i < A}. It is easy to check that the models me as required.

Owe A. As Case B wae proved, we can assume h > No, and ae Caee C was proved we can assume 2" > A + . Let {Q,(z, &): i < A, i a successor}be a list of all formulae ~ ( zg, ) E L, g E { z j : j < A}sothat&tz{zj:j< 2i}.Let{Z,:a< A}bealistofall 5 E { z j : j < A}, such that for each such 8, {a:Za = Z } is a stationary subset of h (possible by 1.3 of the Appendix). We aesume Za E {zj:j < 1 + a}, and let L = Uuch La, 1L.I < A, La (a < A) is increasing and continuous. We shall define by induction on a < A, for each r) E ' 2 a consistent set of formulas I',, in the variables {z,:i < 2a) such that Ir,,l< A. 8&e

(i) (a = 0). r<, = 0.

A d w e (ii) (alimit). For each q

E '2,

F,,= UB
8 d w e (iii) (a = + 1, successor). Let (~~(2): i < i(O)} be a list of the formulas from LBwith 2 E {z,:i < 2a}. For each q E B2we define by induction on i < i(0) formulae q(2) such that F,,U { 8 { ( 9 ) :j 5 i} is consistent, and 8; E A. If we have defined for j < i, and there is 8 E A such that r,,u {O{(Zj): j < i} u {8(2)}is consistent, and T, ~ ~ (k 2 ) -,8(2), choose such 8 = ef,, otherwise choose 6; E A such that

PRIBXE MODELS

216

[m. IV, 8 6

r, u {%(2):j Ii}is consistent (note that consistency is preserved for limit i’s). NOWlet for q E 82, r,l be r, u {8#9): j < i(0)) u {-,(3z)tpB(z,g8)} if consistent and U { q ( I j ) : j < i(0)) U {tp&ag, g8)} otherwise. For I = 0 , 1 let = r; u {z,# za8+1: i < 2/31.

r, r,-(,)

(iv) (a = /3 + 1, /3 limit: and for no y < 8, I , = 28). For q E 82 letp, = {(38)’p(2~4,8):cpisafiniteconjunctionofformultlsof r,,;z,n8=0}. If there is a type q = q(Ifi),19.1 < A, such that p , U q is consistent, and for every 8(z) E A, Z(Z) = @), for some t E (0, 1) p,, u q k O(Qt, then choose such q,; and if there is no such p, let q, = 0. Let for 1 = 0, 1

&&a&e

~ V < O=

r, U qn-

S*e (v) (u = /3 + 1, /3 limit, and for some y < /3, I, = I,& Let us define p n (q E 82) as in the previous case, and let Ls, = {q e 62: there is 8 E A such that p , u {0(28)t} is consistent for each t E (0, 1)). By the previous case, m Ipnl < A, for each q = q(Ifl), 1q1 < A, q E S ~if, I), u q is consistent, then for some 8 E A , p n u q u {0(18)t} are consistent (t = 0, 1). So we can find 0; E A such that for every v €82, I),, u {8i918)””9:y < /3} is consistent. We can assume, that i f q l , 11 E S and ~ for every 8(z) E A , pv1 k 8(Q e p , , I- 8(zB), then for every Y E 8’2, Oil = 0;, (remember that we assume A is closed under Boolean combintttions). Now for every q E 5,l = 0 , l let

r,-(l)= r, u {qY(I~)“”; < pi. For E 82 - S, let r,,..,,,= r,. :. Let for each q E ”, r, = Uaeh 80 clearly it is consistent

and let ztH a!,be an msignment satisfying it. So by the Tmki-Vaught test (and Subcase (iii))there is a model M, of mrdinality A whose universe is {a:: i < A}. We now prove (*)

.. .

Suppose Z = ( x ~ ( ~ ),,ztCk)), ql # qa E ’2, and let Ti, = (a::o),. . .,Ti$:)). If tpd(Til,tp) = tp&, tp) then for at least one 1, there ie q E tp(Til, ‘p) 1q1 < A, such that Q I- tPd@l,t p h

For suppose (*) fails, choose a. such that q1 1 a. # qa 1 ao,and for some y < ao, I , = 2. So if a. < /3 < A, /3 limit = I , then qll/3, 7alP

OH. Ivy9 51

217

VARIOUS RESULTS

Clearly there is a closed unbounded set W E A, such that for p E W, I = 1, 2, P,,,l@ is s L, and, for it, it is complete (La, for each formula #(8), z E {q:i < 2p}, # E Lg, for some t E (0, l}, I',,,,, t- #(Z)t)). We can also assume that (for each E W ) is limit, b > ao. As {b: 28 = Z} is a stationary subset of A there is a p E W , 28 = 8. So p,,rlBE flfl (as a,, Ti3 contradicts (*)) and for every 8 E A , t- 8(Z8) o pnal,k 8(Z,). @?or otherwise we can suppose l)nl18 k 8(Z,) but p m l , It 8(Z8), hence for some #(zJ EI)nllB, #(Z8) k 8(Z8) but 1)va18 u {10(8B)} is consistent. For some y , a. < y < 8, # E L,; so by subcase (iii)for some 81 $(ZB) 81(2fl), 1 e 1 ( 8 8 ) E p " l l ( y + l ) hence l O l ( Z 8 ) Epn118. But as #(z,) Epnllfl, e(E,)E Z ) ~ l l B contradiction (to tPd(al, = tpd@2~ v))]. So, by Subcase (v) Oil = Oi2 for every Y € 6 ' 2 , and $:IY(Z8)"W E r,,rl(B+l) E r,,,where y = min{y: q l [ y ] # qa[y]}, and we get a contradiction (to tpd(Til,0) = tpd(Sia,0)). SO we proved (*). NOWfor each q E "2 we define g(q) as the set of v E " 2 such that for some Ti &Ifv, 8 E H,,, tpd(ij;,0) = tpd(6, 0), but for no q s tp(Tiv,O), 1q1 < h doea Q k tpA(Ti9 0). NOW,

(**I

Is(.l>l

Otherwise there are distinct v, E g(q), (a < A + ) and for each of them The number there are witnessee Tia, 6,, and let a, = (?i$y*o), ,aty*k(a))). of possible k(a),i(a,0), .,i(a, k(a)),6, is A, hence there are a # /3 for 0) and we get e d y which they are equal. Clemly tpA(a,, 0) = tpA(aB, a contradiction to (*). Now as we have assumed 2" > A+ (by c w (C))by 2.8 of the Appendix there is U G "2, IUl = 2" such that q # V E U implies q $ g ( v ) . It is easy to check that {Ma: a E U}is a family of models exemplifsing our assertion.

..

...

Case D.Use the proof of Theorem 5.19, Case I (with I instead "2,

omitting condition (4) there). In the end use A P 2.8. We can improve Theorem 5.16 (hence 5.18) by:

THEOREM 5.19: Suppose [TI= A,

= 2A,d E L = L(T),Id1 = A,

then: (1) There are models Mi (i < X ) each of cardinality A, m h that: i(1) > i(2),al€A&), tp,(&,, $3) = tp,(a,,$3) implies there i s p , 5 tp(a,, p)), lpll < A, pl t- tpd(iil,0). If A is regular or 2 < A < 2 A there is also Pa E t~&2,0), l ~ 2 l< A, Pa t- t p ( G 0). (2) If A<" = A, we can waume each Mi is K-OMn'paCt.

218

[ax. IV, 8 6

PRIlKE MODELS

Remurk. So if IT1 = A, Pi E 8$(0) (i < a < 2 9 , then for some M , llJfll = A, and 8 E M ,pi = tpd (a, 0) implies q p , for some q C tp (a,g), IqI < A. We can have A pairs ( A , m)’s.

Proof. (1) By Theorem 6.16 we can assume w.1.o.g. A is singular. Caee I: 2‘A < 2A.Clearly (2”: p < A) is not eventually constant so we can choose A, (a < K = cf A), a < s < SAD, A = A,. Let {vf(z;Bf): i < A} be a list of all formulas ~ ( zjj) ; E L, jj E {z,:j < A}, 80 that g1E {z,:j < Pi}. We define by hduction on i < A, for each q E a set r,,of formulas in the variables {z,:j < 2i}, consistent with T, such that (1) lrnls lZ(q)l + KO,and when 6 = I(q) is limit, r,,= Il,,,,, (2) Y 4 q implies r,,c r,,, (3) if i 1 < Z(q), then (3%)vf(z; &) -+ v,(za,;&) E r,,and

z,,,

ua
+

{za,+l # z,:j s 2i}

c r,,,

5 i < A,+, 7 E f2, Y E (&a)2, Z E {z,:j < A,}, and rv1 r(Z) (4) if T(Z)ES~~(PJ), but not rJ-r(Z), then for some O ( z ) E r , ~ , E + ( z ) , for any O E ~ ’ ~ ) . If such r exists, we denote it by r,(Z). (6) If q E f2, Z E {z,:j < 2i}, Z E (0, l}, r,-o>(Z) does not exist, then for some 0 €4, e(z)t E r,,-&). (6) IfqEf2,ZE{z,:j < i},andforsomeq(Z),r~Sfi(~)(O), Iq(Z)I s lil, q ( ~t-)r , r,,u q ( ~is) consistent, then r,,I- r. Let for q E ”, zf I+ af,be an aasignment satisfying r,,and M,, a model with universe {a:: i < A}, and let g ( q ) = { v c A 2 :For some ~ E E , , , 6 EM,,, tpd(a,0) = tpd(6,0) but for no q -c tp(iiv,0), 19.1 <. A, does q t- tpd(a,0)). Now Ig(q)l s 2<” by (6) above. So there are qCE ”2 (5 < 2A) such that f < 6 =rlr;$ S(TJ. But by (4)above, [Ta E S(rlS)l* 1716 E dra)l. Henoe {iK,,<: 5 < 2”) ie as required. Case 11:2 c A= 2A.Like Case I, omitting condition (4),and try to define inductively qe E A2 (5 < 2”), < 6 =- qa 4 g(qC). We call A s A2 rare iffor some y = y ( A ) < A

(vq,Y)(qEA A

YEA

A

7 17 = Y

rr

+ q = v),

and clearly g(q) is the union of A ram sets. It ia enough to prove A2is not the union of <2” rare sets; and by assumption for some p < A, 2’ 2”, and w.1.o.g.p < A,,. If A, E A2is rare, i < io < 2Aylet y1 = y(Af),and choose induotively q, E <*J2 (a < of A) increasing by 4 ,such # q,+, (for a zero or limit that i < io, ‘yc < Aa, Y E A, implies Y 1 A,,

-

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219

there is no problem, for successor-by cardinality consideration). The limit of the qcris q E ", q $ Uiei0 A,.

QUEBTION 5.5: Can we assume in 6.16 only that T is stable I TI s II I ? EXERCIBE 5.6: Let A s ICI. Define L(A) aa L u A, i.e., we add the elements of A to L aa individual constanta, T ( A ) = T U { ~ ( i z )iz: E A , Crp[iz]}. Investigate the connections between the fulfillment of the axioms by Fg for T and T ( A ) . (2) Use (1) to generalize the uniquenew Theorem 6.6. (1)

EXERCIBE 5.7: Let T be a theory in L, ILI = A, L = U, No. (2) In 6.6(3) we can assume the model haa cardinality rA and is K-compact, when K < A, A'" 5 A+. [Hint: Let T = Th(M), M = (m'2,. ,P,,,..)*a1 where P,,= {q 1n: n < w}.]

..

.

220

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[m. IV,8 6

DEFINITION 5.2: M is a full model if for every formula V(Z) in L ( M ) I{iz E IMI:M C cp[7i]}l < KO or = 11M11.

EXERCIBE 5.14: Let T be a counfable complete theory. Give a necleesary and sufticient condition for the existence of a full model of T cardinality K, which omits the types p , i < w. Generalize 6.17. [Hint: , Z0,. . .,y,, 9)the The condition ie that for no p@) and cp = ~ ( 3yo, following hold : (i) TI- (3""yO)(3~,)(3""y,)(3q),. . . , (3"3,)(3~,)(3~)p, for every n < w. (ii) For every $(Z)EP,(Z), for some n < w

TI- ~ ( ~ " " y O ) ( ~ ~ O ) ( ~ ~ *"* *~ l ) ( ~ ~ l )

EXERCIBE 5.15: Show that if 6.1(3) holds, any EY,-atomic, FAsaturated model is A-compact.

EXERCIBE 5.16: Concerning Theorem 6.2(1), show that for some T, it has a A-compact model, but 5.1(3) fails. [Hint: See Exercise 2.2, hint ( l ) ,with A the universe of a non h-compact model.] EXERCIBE 5.17: Suppose IT1 = A, singular but not strong limit, and T has a A-compact model M, of cardinality A. Choose p < A, 2' > A.

(1) *(T)s cf A. (2) There are no formulaa cp,(Z, a,) (q E "'2, iZq E lMol) such that for any q E fl2 {cp,,&, i~,)*t"l: a < p} is consistent. (3) If A c IMol, m < w , q an m-type over A, then for some m-type p over A, lpl < p and q u p has a unique extension in Hm(A) (so a condition stronger than 6.1(3) holds). (4) T has a A-prime,FA-~tOmio model Af < N o (over 0) of cardinality A. (6) If cf A > KO, any Py,-atOmi~ (over 0) A-compact model is Ahomogeneous and the conclusion of 5.2(2), (3) holds. (6) The A-prime model of T is unique. Hints: (1) See VIII, 4.7 for suitable reducts of M,. (2) As llAfoII = A, M o is A-compact. (3) By (2) (aa in the proof of 2.16(1). (4) By (3) (Mis PY,-constructible) and 4.7. (6) By (l),(3) and &B 4.9(4), any FtA-atomicover 0, A-compact model

m. IV, 4 61

22 1

VARIOUS RESULTS

is A-homogeneous. The uniqueness and characterization of the A-prime model, are trivial now. (6) Proved as 4.18.

EXERCIBE 5.18: Suppose IT1 = A, A singular and strong limit and T has a A-compact model of cardinality A, M,. Then (1) ~ ( 5 "< ) cfh, and even A
T is stable in A, (3)The statement Exercise 6.17(3)holds for p = A, (4)Exercise 6.17(4);( 6 ) , (6)holds. [Hint: (1) See VIII, 4.7, for suitable reducts ofM,. (2) If not, by 111, 6.16, by (l), ID(T)I > A, hence for some m, ID,(T)I > A. Let M be a model of T of cardinality A, so IMI= U f < c l h A ( iA ) ,( i ) increasing IA(i)l < A. For each i , define an equivalence relation E, on the set of m-formulas: cpl(z)Eicp2(Z)iff for any li E A,, M C cpl[li] = 9&]. So E, has s 2lA({)lequivalence classes; so the ~ number of p E D,(T) such that cpl(Z)E,cp2(Z) implies cpl(Z) E I o cp2(Z) E P , is ~ 2 ~ l ' ' ' "< l A (as IA(i)l < A, A strong limit). Hence for some p E D,( T), for every i , there are m-formula cp:(Z), d ( Z ) such that +(Z) ~p o cpi(Z) EP and cp:(Z)Eicp&(Z). Hence #,(Z) = cp:(Z) = - p , " z ( z ) ~but p no n ~ A ( isatisfies ) @@). So M omits {@'(~):i < cfA} E p, hence is not A-compact. (3) We first prove: (2)

(*) There am no formulee PO,^,^ = cp,,v,a(Z,li,,,v,(r)over iKo for i < cf A,

7 E nj<, v E nj
x{
X,), such that:

if then {cpnll.vl,.rtc,

A --rvnl,,vl,,v[,J:

i < of A}

is consistent.

This is proved using Exercise 6.18(2),with similar technique. Next we prove: (**) If p = p(Z) is an m-type over A s IiK,l, 1p1 < A, there is a q(Z), 1q(Z)1 < cf A, such that p(Z) u q(Z) is an m-type over A, and: r € S m ( A )r, z p ( z ) U q(z)implies r is Fi-isolated. If (**) fail we can construct a counterexample to (*), and using (**), Exercise 5.18(3) is easy. (4)As in Exercise 6.17.1

222

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[CH.

Iv, 9 5

EXERCIBE 5.19: Show that Theorem 6.2 holde also for singular A, when cf A > KO,but when cf A = No some parts may fail. [Hint: See Exercises 6.17, 6.18.1 EXERCIBE 5.20: Generalize Theorem 6.2(2) and (3) and the previous exercises to the case A < ITI.

CHAPTERV

MORE ON TYPES AND SATURATED MODELS

V.O. Introduction In this chapter we shall deal with stable theories only; note that for the concepts we investigate here, there is no need to distinguish between parallel types, nor between equivalent indiscernible seta. In Sections 1-6 we investigate the notion “dimensions”, and also regular, orthogonal and minimal types for stable and superstable theories. In Sections 6 and 2 we use this to investigate theories with few quite saturated models. In Section 6 we deal with cmdinality quantifiers and strong transfer theorems. At last, in Section 7, we deal with a generalization of “algebraic”, to “of small cardinality by a cardinality quantifier”, with appropriate rank; and deal again with ranks. The “classical ” example of those who dealt with categoricitywas the algebraically closed fields. There what is the (transcendence)dimension is well understood. The natural first try (see Definition 111, 4.6) is aa follows: let p E S ~ ( A )and , we define on {a E B: p = tp@,A)} a dependence relation: depends on I if t p (a, A u I) forks over A. Unfortunately not all axioms of dependence relations which enable one to define dimension are satisfied (transitivity is lacking). When there are (in a given model or set) lmge independent sets we still get a true dimension (see 111, 4.21(2)), but when, e.g., A E M yand every independent set ( ~ p ( d l is ) )finite, this does not help. So it is natural to try to deal with the types p for which transitivity is satisfied, and we shall call them regular (see Definition 1.2 and 1.9 for equivalent definitions, including the one above). For developing this we define when p i E #‘“(A) are weakly orthogonal (pl(Z) u pa(#) is complete) and when p , , pa are orthogonal (every stationarizations me orthogonal). The main result is contained in 1.14 and 1.16 : among the sequences realizing over A, a regular type, the axioms of dependence (needed to define dimension) are satisfied. 223

w

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[CH.

v, 8 0

Naturally we want to have existence theorems for regular types. For stable T,we can prove that every non-algebraic type a n be extended for every p for some t, p U {pt} is algebraic). to a minimal type p (h., Minimal types are regular and we a n understand them better (see 1.17 and 1.18). But we want to know that there are really many regular types. For superstable T , M , < M,, M , # M , H I F;,-saturated; for some a E lMll - IM,l, tp(a, lM1]) is regular; and for T totally transcendental, F(t,-saturativity is not needed (see 3.6 and 3.19). So regular types are important mainly for superstable T.In this context it is interesting that if M is E-saturated, K 2 K J T ) ,p I = tp(si,, 1611) is regular, then p , is realized in the Ff-prime model over U iff p , is not orthogonal to pa. From this we can prove non-orthogonalityis an equivalence relation among stationary regular types, and it is a meaningful equivalence relation. After this theorem it is natural to define the following orders among indiscernible sets: I s o J if dim(I, M ) s dim(J, M ) (for every M ) and I s 8J if when I u J s M I < M, are Fg-saturated ( K = K,(T)) Av(I, lN1l) is realized in Ma, then Av(J, lMll) is realized in M, (but so this implies Av(I, lM1l)is more complicated than Av(J, lM1l), maybe it would have been better to reverse the orders). Those (quasi-) orders are investigated in Section 2. Our intuition is that for “simple” theories (in an appropriate sense) the F&(,,-saturated models are characterized (up to isomorphism)by some dimensions. So the simplest J for any indiscernible theories will be the unidimensional, (i.e., I sro sets I,J), and we would like some dicotomy, saying meaningful aasertions on unidimensional and non unidimensional theories. So the climax of Section 2 is Theorem 2.10, which says: (1) T is unidimensional, and every Fgr(,,-saturated model is a Ert,-prime model over U I, for some A, where I, is an indiscernible set of cardinality A and has a fixed type, and I, is based on 0. Or (2) T is not undimensional and for every p 2 K,(T),T has a Fi+ -saturated models of arbitrarily large cardinals, saturated, not which me not even p -universal. In Section 3 we shall show that for those orders, for superstable T , above each element there are only finitely many elements up to equivalence by the natural equivalence. Though we get true dimension for regular types, we still want to get somehow the true dimension for the other types; and this is the aim of Section 3. Maybe the following example will clarify it. SupposeMis an

e

+

OH.

v, § 01

INTRODUCTION

225

algebraically closed field, {a,,a,} is transcendentally independent, and iz = (a,, a,) and p = tp(iz, 0). Now this type is not regular; but it is clearly a matter of “notation” essentially. We should define a weighted dimension; i.e., let w(iz, A) be the transcendence dimension of acl(A u T i ) over acl(A), and the parallel to the dimension being true is

(*I

for I = {4:i < a},

(We can define naturally w for infinite sets or sequences.) The first’questionis how to define the weight w(iz, A). Our suggestion is as follows: Assume A = 1H1,H Ero,-saturated, N is F&(n-prime u a. If we can find a set J E IN1 independent over IMI,of over ( M ( sequences realizing over Idl( regular types such that N is Qrtn-prime over IdlI u UJ we define w(iz, IMI)= IJI (and for any p, w(p) is w(a, lUl)where tp(zi, lMl) is a stationarkation of p, w(zi, A) = w(tp(zi,A))). Now the main properties we succeed to prove are: (A) If T is superstable, w(a, A) is always well defined end finite(see 3.9).

(B) If I is independent over A, for every there is J E I, 1JI 5 w(a, A) such that I - J is independent over (A u a, A ) (see 3.16). Note that previously we could get IJI < K(T); however we then get I - J is independent over (A u zi u UJ, A), and this cannot be done here (even the vector space over a bite field c m serve as a counterexample). Now by (B)we can show that dimensions are “almost” true, i.e.: (C)If p E Brn(A),J E p ( Q ) I, l s J is a maximal set independent over A ( I = 1, 2) then 1111I; w(p)JIal. However, the weight defined in Section 3 does not satisfy (2); so we continue our sectrch in Section 4. First let us contemplate an example. Suppose H is the disjoint union of two algebraically closed fields: P, and P,, and a l ~ P INow . clearly tp((al,a,),O) has two dimensions, and the natural way is to try to decompose it. So (already in 3) for a stationary regular typep, we define lowp(zi,A) just as w(zi, A), however it is not I JI but E J: tp(8, 1HJ)is not orthogonal to p}. Note that then w(8, A) = lown(zi,A), where among any family @ :pregular, stationary not orthogonal top,,} just one appears. We call

zn

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226

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q = tp(Z, A) unidimensional if q swp * p sWq; for supentable T this means that for some p , p swq, p regular. We may now ask: (a) For a and A, and stationary regulmp not orthogonal to tp(si, A), can we find equivalence relations B over A, such that tp(a/E,A) is

unidimenaional not orthogonal to p ? (b) Does the parallel of (*) to low, hold? i.e., let p be stationary and regular: I = {&: i < a},then (b') Does (b) hold at least when U I E q(Q q EB'"(A),q unidimensional not orthogonal to p 1 Unfortunately the answer to all those questions is negative (see Exercise 4.11).However some weakening has positive answers. The moat important notion of Section 4 is a semi-regular type (me Definition 4.1).Its important properties are: (i) If p ia semi-regular, it is unidimensional, 80 all the regular types not orthogonal to it are pairwise not orthogonal. (ii) (b') holds when q is semi-regular. (iii) (P) If a$acl A = A, tp(Ti, A ) not orthogonal to a minimal type q, then for some equivalence relationE over A , tp(iz/E, A ) is semiregular (semi-minimal,in fact) not orthogonal to q. (iv) (Pa)If T is superstable, ti $ A = acl A, then for some equivalence relation E over A, tp(a/B, A) is semi-regular (hence not algebraic) (see 4:ll). (v) If bl is Fg-saturated, K 2 lA I -t K,(T),p = tp(7i, A ) semiregular, then for some indiscernible set I E M over A, based on p , Av(1, A u I) I- Av(I, 1N1)(see 4.22). Note that (v) would not hold for a non-unidimensional type. However if we me ready to wave (i), (v), we can replace semi-regulm by cl~{p}-simple( p some regular (and stationary) m-type) and (ii), (iv) holds, the family of such types is closed under supertypes, and a stre@he+ of (iv) holds. If B $ A = aclA, tp(a,A) is not orthogonal to the (iv') (P) stationary regular type p , then for some equivalence relation B over A, tp(Z/B, A) is clj{p}-simple and low,(?i/E, A) > 0. Note that the family of regular types fail to satisfy (iv), and it should be easy to find families satisfying (iv) (or (iv')) but not (ii).The distinct roles of regular, semi-regular and simple will be exemplified in the proof of IX, 2.3 and 2.4.(Simplicity is defined in fact for a regdm family of types, see Definition 4.3.) +

u

UH.

v, 6 01

227

INTRODUOTION

Sections 3 and 4 are relevant mainly to superstable theories (as for stable theories we do not have existence theorems for regular types). So in Section 5 we return to stable theories. In Section 2 we have dealt with unidimensional theories, and in Section 5 we continue this by dealing with theories with a bound number of dimensions called non-multidimensional (this means, e.g., that {dim(I,M): I E ilf) 64 F&(nsaturated, has a bound AT). The main result is again a dichotomy (see 5.8 and 5.9): (a)If T is multidimensional, H, 1 N# 2 K,(T),T stable in K,, then T has at lead 2 I K - B + non-isomorphism F&-saturated models of cardinality N,. (8) If T is not multidimensional, for every F&(,-saturated model indethere is N < M, llNll I 2ITl, N F&,-saturated, and J G pendent over INI, such that 64 is F&,-prime over IN1 u (JJ . By (fl) we get the number of Q,,-saturated model of T of cardinality N, quite accurately (when a - 8 is big enough, T stable in N,accurately). I n IX, 2.3 we get accurate results for superstable T. The main lemma in the proof is as follows. We ask for an indiscernible set I over A , what can be {dim(J,M):AE MF:r(,,-saturated, stp,( J, A ) E stp,(I, A)}. So we choose I, realizing

m,

does not fork over A. Maybe we can choose the dimensions dim(I,, M ) at random. Otherwise we prove that for some a < K,(T), {a,:i < a}is independent over M , a, realizes Av(I,,M) and N is F:r(,,-prime over M U {ai:i< a>then N realizes Av(l,,M) (see 5.3). In Section 6 we try to deal with cardinality quantifiers, and twocardinals theorems. Think of a complete (stable)theory T , in a language L(3<X),x regular; and we assume that every formula is equivalent to a predicate (in T).So we can replace T, by its fbst order part, T, together with the following requirement on the model M: if (V@[(3<Xz)tp(z, 8) = #(@I E T,, tp, # first order then M C #[a] e Itp(M, 7i)l < x for each 7i E !MI. We generalize this somewhat in order to include more cases (e.g., we want just P ( z )to have a fixed cardinality). So T will be a (complete, stable) first order theory, R = (F, W, A, p ) where: F is as in Chapter IV; W is a set of triples (p(x, ij),&8),x ) which means that for good M, M I.#[6] * Iv(M, 7i)l < x (note we demand only e-and not 0).

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We do not require anything from W , but for any formula tpfz, a) we define C(cp(z, a)), which is the minimal x (or co) such t b t our previous requirements imply Iq+lf,ti)l < x (for good bl; we just notice algebraic formulas are < and the union of less than sets each of cardinality < x is < x ; we use just the rules corresponding to this). We formulate six conditions C1-6 from which we prove various results. For a function h, h(x) < x, we define an h-good model, assuming C3 it says that for zi E 11111, h(C(tp(z,a))) IItp(H,a)l < C(tp(z, a)). The main result is 6.7 which says that: any good model can be extended to an F-saturated 8-good model. Note that the question whether ltp(Y,a)[ < x in any good Y,is not an elementary property of 8,but is equivalent to “ 6 does not satisfy a type pX”. Now if F = R,K 2 K J T ) ,p. = K,(T),h = (2IT1)+ and h 5 x, T stable in x for each x appearing in W, then all the conditions are satisfied. But there are other cmes which interest us, where this is not the case. Hence we suggest another set of conditions C*1-8, which maybe me less nice looking. Their consequences are somewhat weaker, but they are satisfied in more cams. In 6.12 we prove the existence of k-good models. Note that in treating C*1-8, we use more good sets and less good models. In 6.14 we find when the various conditions me satisfied, and we go on to “applications” to twocardinality quantifiers, which should exemplify the flexibility of our treatment. Those assertions fail in general for unstable theories.

x,

x

THEOREM: (A) If T is (stable and) countable, M < N , M # N , P ( M ) = P ( N ) there is a model N* of any cardidity > lIN/l, such that N < N*, P ( N ) = P(N*) (see 6.14, the countability is essential; it is generalized to F-saturated:Tnodels). (B) If IPdWl = X I , x o < * * * < x,,, and IT1 s xo I- - Ix” then T b a model N , (P,(N)I= x1 (see 6.16). ( C ) Suppose xo < .. < x,, are regular, for any ~ ( z5) , and 1 for some &@), for every E M ybl C $:[a] o lv(bl,T i ) [ < xI. Then for any xi, IT1 < xo < . . < xn T has a model N , such that for any ‘p, 2 and 8 E IMI, Itp(M, a)] < x1 o bl C @ (See $ 6.16). i] (D) If T is 8wpeT8kZht!eyIP(&f)I < h = 1 1 N 1 1 , h regular for SiW&@iCitY, then there is N < My IlNll = A, IP(M)l 5 2’*’ (see 6.16, in fact we can get IP(M)I 5 I TI ;for T stable this fails).

-

In Section 7 we return to ranks. First, we note that in Section 6, the is quite similar to “tp(x,a) property “Itp(M,“)I < x for each good M y ’

OH.

v, 8 01

229

INTRODUOTION

is algebraic”. Part of Section 7 is devoted to this generalization, so we have K-minimal, and other concepts in the same vein, and a rank D ( p , L, 9) where now D ( p , L,K) 2 a + 1, for finite p , means that for any x, for some good M , p has > x pairwise contradictory extensions q over M , D(q, L, 9)2 a. Some of the theorems are meaningful for the usual notation. In 7.3 we show that when tp s K# (i.e., for good M , Itp(M)l s I#(M)l + xOIx), then we can partition tp to “small” parts, indexed by #, and when tp (or #) is (weakIy) K-minimal, we get better results. So in 7.4 we can characterize when tp s K#. I n 7.6 (and 7.7), we deal with tp(z, fj)and q such that

for any good M , {(tp(H,a ) ] : realizes q, tp(z, a) not K algebraic} has cardinality < &, where A. < A=.

(*4)

We show that then we can replace ho by some k < w , q by a h i t e subtype. When tp(z, 8)is weakly K-minimal for a realizing q then also L< ltp(M, Z)l = 1tp(M, 6)1 for any good M , or tp(z, a), tp(z, 6) are K algebraic” is a first-order property of a-6 with finitely many equivalence classes. It follows that I{Itp(M, a)l: tp(z, a) not K-algebraic}l < A. < AK for good M ,implies that “tp(z, a) is not K-algebraic” is a first order property of a. We can get (still assuming (**)) also that “tp(z, a) is weakly K-minimal” is first-order property of for the trivial R; and if ~ ( za), is minimal, T totally transcendental, we can get in fact a similar result (see 7.7(2)). Later we try to prove theorems of the form: if q(Q)is divided, the parts indexed by #(a), each part has a bounded rank, then we can bound tp’s rank using #’a rank and the bound on the ranks of the parts. More accurately, suppose

m,Y) 1

#(Yb

( W ( z ,Y). By 7.8, if /3 = R[#(z),L,so], R[B(z,b), L, KO] s a for every b E #(Qh then RCtpk), L, solI4 8 + 1) or a = 0, R[tp(z), L, sols B. tp(4 A

tp(4

For the rank R(-, L, 00) we get a similaz result when /3 < o,and a better bound (a + 8) when a,B < o.It follows (see 7.10) that when

T 5

$9

* R(% L,K O )

R(*, L, No) < R(#, L, 00) <

w

< a,

* R(tp, L,CO) < o.

At last we deal with another rank, L, defhed for complete types, without taking finite subtypes (so we lose the finite characteristic). We let L ( p ) 2 a + 1 if L(q) 2 a for some extension q of p which forks

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[OH. v , $ 1

over Domp, and let L(a,A ) = L(tp(B,A)). This ranks satisfies more eht@ -( “natural than the above: L(a-6, A) s L(a, A u 6) @ L(6, A) sum” of ordinals, see Definition 7.6). The rank has other natural properties, e.g., T is superstable iff (Vp)L(p)< 00, if tp(a, A) does not fork over B c A then L(G,A) = L(6,A ) ; L ( p ) s R ( p , L,00). In this chapter T will be stable.

V.l. Orthogonality, regularity and minimslity of types DEFINITION 1.1: (1) If p(Z1), @a) are complete types over A, p an m-type, q an n-type, we call p weakly orthogonal to q i f fp(Zl) u @a) is complete (over A). (2) Let p1 be complete or stationary and p a complete or stationary. Then p1 is orthogonal to payif for every A, Dom p1 u Dompa E A, A the universe of a F$,,-saturated model, and any stationarizations q1 of p l y1 = 1, 2 over A ; q1 is weakly Orthogonal to qa (see Definition 111, 4.2(2)).

(3) The infinite indiscernible sets I , J are orthogonal if Av(I, U I), Av(J, UJ ) are orthogonal (see Lemma 1.1(3)). (4) The type p is orthogonal to the set A if p is orthogonal to every complete type over A.

LEMMA 1.1: (1) If p is algebraic then p is orthogonal to any type q (when both are stationary or complete). If p €Sm(A),p is realized by jzcst one sequence, then p is weakly orthogonal to any complete type over A. (2) If p 1 = pI(EI)= tp(7il,A ) then p1 is weakly orthogonal to pa iff @(a,, A) 1tp(7iay A u 7il) iff tp(7i1, A ) F tp(si1, A u as). ( 3 ) Weak orthogonality and orthogonality are symmetric relations. (4) I f tp(Sil-&,A) is [weakly] orthogonal to tp(61^6aYA) then tp(7il, A) is [weakly]orthogonal to tp(6,, A ) (foralE1(7il),Z(aa),Z(&), Z(Ea)). ( 6 ) If p I (1 = 1, 2) are complete types over A , and they are not orthogonal, thenfor some jinite B there are statimrizations q1of p I over A u By such that ql, qa are not weakly orthogonal. ( 6 ) When p I = tp(7i1,A,), p1 is orthogonal to pa iff for any a; realizing P I , stp(a;y A,) is orthogolzal to Stp(a;, Aa).

Proof. Clearly (2) is a restatement of Ax(V1) for FL (see Lemma IVY 2.9). Then (3) follows; and (4) follows from Ax(1V) for FL. Now ( l ) , (6) are trivial and (6) is easy.

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23 1

THEOREM 1.2: (1)Let B c A , and for 1 = 1, 2, p , i8 a complete type over A , p , does not fork over B. If Pa 1 B i8 8tU4%mtZ?’y, plypa are weakly orthogonal,then p1 1 Bypa 1 B are weakly orthogonal. (2) If plyp a are complete types over A , and they are orthogonal, and one of them is stationary then they are weakly orthogonal. (3) If bl i8 F;,,-saturated, p , are wrnplete types over hf then plypa are orthogonal iff plypa are weakly orthogonal. (4) I f p , i8 parallel to qr for 1 = 1, 2 ( h e w they are stat~onay)then plypa are orthogonal iff ql, qa are orthogonal.

Proof.(1) This is a restatement of Claim IVY4.16. (2) Follows by the definition and (1). (3) By 111, 2.15(2) plypa are stationary, so if they are orthogonal,

they are weakly orthogonal by (2). So suppose they are not orthogonal. Then there is a l!‘&,-saturated N , M c N , and stationarbations q, of p , over N such that qly are not weakly orthogonal. By 111, 3.2 and 111, 2.9 there is C E M , ICl < K ( T )such , that p , doea not fork over C,p , C is stationary, hence q1 does not fork over C , hence q1 is definable over C , by 111,4.8(2). There is 6 E N such that ql(Z)U qa(g) U {cp(Z, g ; 6)’) is consistent, for t E (0, l}. Choose 6, E Y,tp(6,, C) = tp(6, C) (since Y is F&+&urated). By the F&-saturation of N and the definability of q, over C, ql(Z) u qa(g) u {cp(?Z,g; 6,)’) is consistent (for t E (0, 1)) hence pX(Z)u pa(@u {cp(Z, 8; 6,)’} is consistent, hence plypa are not weakly orthogonal. (4) Choose a F&,-saturated model My Domp,, Domq, c Y for 1 = 1, 2. Let r, be the stationarization of p , (and q,) over Y.By the symmetry it suffices to prove rl, ra are orthogonal iff plypa are orthogonal. The “only if” can be proved by (l), (3) and the “if” by the definition.

r

LEMMA 1.3: The types p , q are orthogonal (in Q ) iff pea, 40 are orthogonal in Qeq.

Proof.Use 1.2(3) for Ceq, remembering lMeql

= dcllMl in V a .

LEMMA 1.4: (1) fluppwe {at: i < n} is intiependent over A (see 111, Definition 4.4(1)). Then tp(6, A ) is orthogonal to tp(B,; A ) for i < n iff tp(6, A ) i8 orthogonal to tp(BonB1n- * *-Bn-1, A). (2) If p , = tp(BtyA) are stationary and p i n o i s e orthogonal (i< a) then Ut
232

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[CE.

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Prmf. (1) If tp(6, A) ia orthogonal to tp(Sion-..n7i,-,, A) then by 1.1(4) tp(6, A) is orthogonal to tp(Zt,A). For the other direction, by 1.1(6), 1.2(4) we can assume A = M is F;(,,-saturated. Now mume tp(Si,, A), tp(6, A) are orthogonal for 1 < n, and prove by induction on k < m that tp(6, A) k tp(&A u Ulcka,). For k = 0 it is immediate; for k 2 1 it follows by 1.2(2),(4). So tp(6, A ) and tp(Sion. -hSisil)_l, A) are weakly orthogonal, hence by 1.2(3) they am orthogonal. (2) The proof ie immediate by (1).

w,

-

CONCLUSION 1.6: (1) Let I be an indiemnible eet over A , based on A , and tp(Z,A), Av(I,A) are o r t h u g d . Then tp(Z,A), tp,(I, A) are orthogonal (i.e.,for Si, E I , tp(Z, A), tp(SiOn. A) are o r t h u g d for all n < w ) . (2) If I , J are indiscernible eets over A , baaed on A , and Av(I, A ) ie orthogonal to Av(J,A ) then tp,(l, A ) ie o r t 7 w g d to tp,( J , A ) .

-

Remark. For a strong converse to 1.6(2) see 2.1. Proof. (1) By 1.4(1). (2) By a double use of 1.6(1).

LEMMA 1.6: (1) If for i < a, tp(6, A ) ie o r t h o g d to

and tp(6, A ) ie e t a t i m y then tp(6, A ) k tp(6, A u Ut at).f?otp(6, A) o r t h O g d t0 tP(Z0”- .^a,,, A ) (We (1.2(3)). (2) We can r e p h in (1) t p by stp and m i t the aBmmpth that tp(6, A ) i 8 8kctiOnary. i8

Proof. (1) Immediate by induction oq a,using 1.2(2). The eecond part by 1.2(3). (2) Can be

easily proved in a similar way, or through W.

LEMMA 1.7: If pt, (i < a ) are p i r w h e o r t h o g d , but q ie not orthogmucltop,, qieBtationarythena < ~ ( T ) , a n d e v e < nK~,,~(T). Proof. By 1.2(4),1.1(6)we can assume q, p , are complete types over a Ft(,,-saturated model M. Then use IV, 4.12(3).

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233

DEFINITION 1.2: (1) A non-algebraic type p €&"'(A)is regular if for every B 2 A and r such that p E r E Bm(B); if r forks over A then p , r a m orthogonal. (2) A stationary type p is regular if its stationarization over Dom p is regular.

LEMMA 1.8: (1) I f p , q are parallel (and stationary)then p is regular iff q M regular. (2) If tp(Ti, A ) is regular, then stp(Ti, A ) is rqular. ( 3 ) I f tp(B, A ) = tp(6, A), then stp(B, A ) is regular iff stp(6, A ) M regular. (4) tp(Ti, A) M regular in (E iff it is regular in P. Proof. (1) By the definition, we can assume p , q are complete. Let C = Dom p u Dom q, and let Y be the stationarization of p (and q) over C. By the symmetry it suffices to prove that r is regular 8 p is regular. If p is regular, and r -c r1 EB"'(A), r1 f o r b over C c A, then r1 forks over Dom p, hence is orthogonal to r . So suppose p is not regular; then there am pl E P ( B ) ,Dom p E By p E p , , p , forks over Dom p but p , is not orthogonal to p. We now know p and r are stationary and parallel. By 111, 2.6(3) there is an elementary mapping f of (E such that f 1 Domp = the identity, and for every 6 E B stp(6, Dom p ) = stp(f (6),Dom p), and tp(f (6),C) does not fork over Domp. Hence pa = f ( p l ) is not orthogonal to p, so p 2 is not orthogonal to r. Let p 3 be the statiomrization of pa over C u Dom pa; p 3 1C does not fork over Dom p, by Ax(VI1) for F/, But p C p z E p,, hence, as p is parallel to r, r E p , . Now p , , being parallel to pa, is not orthogonal to r. This shows r is not regular. (2), (3) Immediate. (4) By 1.3.

.

QUEBTION 1.1: Does the converse of (2) hold? THEOREM 1.9: Let p eBm(A).The following conditim are equivalent: (1) p is regular. (2) If A s By I v {E} a set of sequences realizing p and f o r every Ti E I , tp(Ti, B) f o r b over A , and tp(Z, B v U I ) f o r b over A , t h n tp(E, B )f o r k s over A . (3) If B = A u UJ, J v {Ti, Z} i s a set of s e q u e m realizing p , and tp(Ti, B ) f o r b over A and tp(E, B v Ti) f o r b over A, then tp(Z, B ) f o r b over A .

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(4)I f . 2 K,(T),K> IAI,A E M , M b e - 8 d U r d d , p E q=tp(ii,M), q does not fork over A , and N i s @-prime over M u ii; then for no 6 E IN1 - lMl, 6 realizes p and tp(6, M) forb over A . (Ifp is stationary, then for all 6 E IN1 - 1M1,6 realizes p * 6 r e d i m q). ( 6 ) The hypo&& and con~luSionof (4) b o d l for 8 m M , 6 , N , K .

Remark.. If p is stationary, we can in (4) omit the condition

QUEBTION 1.2: Can we in (4)replace by

K(

K

> IAl.

e by Fx?Can we replace K,(T)

T )2

Proof. (1)* (2)Supposep is regular, A E B, tp(?i, B) forks over A for every 7i E I, and tp(8, B) does not fork over A, and we shall prove tp(i5, B u U I) does not fork over A (clearly this is sufficient). By 111,2.6(1) and 111,1.4stp(8, B) has an extension q, q z T E Bm(Bu UI), which does not fork over A. Hence it suffices to prove stp(E, B) k stp(8, B u U I), so it suffices to prove this for finite I. Let I = {a1:i < n}; so aa tp(Z1,B U {a,:j < i}) forks over A and extends p, it is orthogonal to p, hence to stp(E, B). Hence by 1.6(2), stp(E, B) k stp(i5, B u U I ) . (2)* (3) Immediate, as (3)is a particular case of (2);taking I = {a}. Before proving (3) + (4) we need:

CLAIM 1.10: (1)If M is F:-saturded, K 2 K,(T),p = tp(& 1M1) i8 orthogod to r ~B"(11111),T is non-algebraic and N i s Cqrime over ]MI U zi, then T ha% a unique extenaim in Sm(INI), hence i s not realized in N. (2) Moreover,for every 6 E INI, 6 $1M1, tp(6, IMI u a)forks over [MI. Proof of 1.10. (1)As M is @-saturated, there is an infinite indiscernible set I, 111 = K , I c 1611,Av(I, [ M I ) = r. Hence by the hypothesis and IV,4.10(2)(remembering K 2 K,(T))Av(I, ]MI)= T k Av(I, IMI u a) 1 AVV, pq). (2)The same proof. Continuation of the proof of 1.9. ( 3 ) =. (4)Suppose (4)fails, and let T = tp(6, IM]), which is stationary; hence by the claim tp(6, lMl u a) forks over ]MI,hence tp(zi, lMl u 6) forks over lMl (by Ax(V1)for FL). Let J = {Z: 8 E IMI,8 realizes p}, B = A u UJ ; we shall show that A, B, a (for 8) and 6 (for a) form a counter-example to (3). Clearly

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ORTHOQONALITY, REGULARITY AND Y -

tp(si, B) does not fork over A, and tp(si, IMl u 6) forks over A , and tp(6, !MI)forks over A. We need only to replace M by B in the last two types. By the following claim, tp(si-6, [ M I )does not fork over B, thus by 111, 4.14, the result follows.

CLAIM 1.11: If x = 8, t, a,M a q-compact maEel, p , (i < n ) Fg-type8 over ] M I , and si, realizes p,, and B, = U {a: 8 E M, E realize8 p,}, then Ute,, Domp, u U,<,,B,) k tp(sion. - --sin, M ) . tp(aon. snsi,,,

-

Proof. W.1.o.g. p , is a l-type (if it is an m type, let

-

p ! = ((32,. ~ x i - l q + l .-x , , , - ~ )A q: q a finite subset of p,} and replace the p,’s by the pi’s and the sit’s by si,[l]).We can also aasume p , = p (replacep , by p = {V, <, tp,: tp, E p,}, clearly an element realizes p iff it realizes some p,). Let B = B,, si, = (a,), si = ( a o , .. .,a,,-l). Suppose Ctp[si, E l , E E M ; then by 11, 2.2 and 11, 2.13 for some 6 E B and #, Ctp[si‘, E] o !+[a’, 61 for any 8’ E B. As M < 6,M F:-compact, CJI[si, 61. Supposetp(si,B UDomp) lftp(Z, E), so somesi*realizestp(8, B uDomp) (hence a*[l]realizes p) but C-,tp(si*, 17).So, q = {tp(xi;6):Z < n, tp@;

6)~ p } u {#(xo,. . .,%t-1; 61, 1 d x 0 , * *

- 9

a,}

%-l,

is realized by a*. Clearly q is a Fz-type, hence q is realized in M, by a**. So a** E B, t#[a**, 61, C-,tp[a**, E ] contradiction to the definition of #.

-

Continuation of the proof of

1.9. (4) 3 (6) Trivial. (6) (1) Suppose p is not regular (i.e., (1)fails), K 2 K,(T) IAI +, A G 1M1, M q-saturated, and tp(si, IMI) is a stationarization of p and

+

N a FE-prime model over 1M1 u si. As p is not regular there is B, A E B, and q E P ( B ) ,p E q, q forks over A, q is not orthogonal to p . We can assume that IB - A1 < min{x(T), Hl},and q is stationary. As M is Fl-saturated, we can msume B c M , and let q1 E P ( ~ M I ) be a stationarization of q. So ql, tp(& IMI) are not orthogonal, (by 1.2(4)).By 1.2(3)there is a 6realizing q1 such that tp(6, U a)forks over A (as q1 is stationary, being over a model). So (by 111, 1.1(5)), for some $ , c E M , l=$(6,6,@ and $(Z,6,@forks over Clearly q U {$@, aLq} is a consistent F,“-typeover N , hence realized by some EN. This b contradicts ( 5 ) ,so not (1)implies not ( 5 ) .This completes the proof of Theorem 1.9.

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THEOREM 1.12 : 8 u p e p , q are complete typea Over IM I, 6 reulizes q, M ia F39aturatedYK 2 K,(T),N is Fg-prim over lMl U 6. If p , q are not orthogonal and p i.9 regukcr, then p is realized in N .

Proof. By 1.2(3), asp, q am not orthogonal, they am not weakly orthogonal. Hence p has an extension p1E P ( ~ M u 6), ~ which forks over 1611; let Ti realize p1 (Ti is not n e d y in N ) . Choose C s lMl, ICl < K ( T such ) that p = tp(& IMJ)does not fork over C and tp(Ti, C) is stationary (by the saturation of bl). Let B = {Z: Z E N , E realizes p 1C). Then tp(Ti, C u (B n M))does not fork over C.If tp(a, C u B) does not fork over C then by 1.11 also tp(Ti, N ) does not fork over C , contradiction as 6 E N . So tp(Ti, C u B) forks over C,hence over C u (B n M ) (by 111, 4.4). So for some soy.. .,a,, E N realizing p , and E E C, and 9, cp(E, a,, . . ., d,, Z) E tp(si, N ) forks over C u (B n M).If for 3,) fork over C then it every i the type tp(&, C u (B n M )u would be orthogonal to p (as p is regular) hence by 1.6

u

u,<,

tp(Ti, C u ( B n M))k tp a, C u ( B n H )u

U2, 198

But the second type forks over C u (B n M )contradiction. So the a, for which the above tspe does not fork over C realizes p and is, of course, in N .

CONCLUSION 1.13: (1) The r&ion of non-orthogdity among tire stationary regular typea, i.9 an equivalence relation. ( 2 ) .If p , q are ~tationary,regular, and not orthogonal,then r i.9orthogonul t o p iff r ia orthogonal to q. Proof. (1) Reflexivity is trivial (using non-algebraity). Symmetry was proved in 1.1; and transitivity is immediate by 1.12 (going to parallel types) (2) Also easy. Remember that Ti depends on I over A if tp(Ti, A u I ) forks over A .

-

u

THEOREM 1.14: FW a set A tlte relation of &pendew Over A among sequences F for which stp(F,A) is regular satis$es the axioms of dependency relation (Def. AP 3.3). If p , is regular, on {F:stp(c,A) regular not orthogonal to p , ) it is a nice dependence relation.

Procf. (1) The exchange principle. It suffices to show that if {a,:j < a} is not independent over A, then some Ti, depends on {al: i < j}over A. This is 111,4.19(6).

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(2) The hite character. That is, if a depends on I over A then iz depends on some finite J c I over A. Easy, see 111, 1.1(6). (3) Transitivity: Here we use regularity. Suppose iz depends on I over A ,each 6 E I depends on J over A and I,J mindependent over A. We mn replace I by a finite subset, and then J by a finite subset, without changing our assumptions. W e can work in F,and replace A by acl A, (see 1.8(4) and 111, 6.3(3)), so every complete type over A is stationary. Let I , = (6: 6 E I, tp(6, A) is not orthogonal to tp(iz, A)}, J , = {Z: 3 E J, tp(E, A) is not orthogonal to tp(iz, A)}. Now iz depends on I,, for otherwise let I I, = {st: 1 < n}, and as tp(iz, A u I , ) does not fork over A and is stationary 1.6 shows tp(iz, A u U I,)k tp(8, A u U I , U {6[:1 < n}).Hence the latter does not fork over A, contradiction. Similarly, every 6 E I , depends on J1 over A. Let I, = (6C i < k), and assume iz does not depend on J , over A. As tp(st, A u UJ,) forks over A, tp(6{,A u J , u (6': j < i}) forks over A, hence (by the regulazity of tp(8, A)) is orthogonal to tp(ZiyA), hence to tp(iz, A), hence to tp(8, A u UJ,)(using 1.13, 1.8(1) and 1.2(4)). So by 1.6 tp(iz, A u J , ) t- tp(iz, A u J , u I,),so the latter does not fork over A, so tp(7iyA u U I , ) does not fork over A, contradiction. (4) Full transitivity (for {F:stp(c,A ) regular not orthogonal to pol):Repeat the last part of the proof of (3).

u

-

u

u

u

u

u

u

CONCLUSION 1.15: (1) If p E Sm(A),A E M , I, regular (or euery e t a t i o n a ~ z a t hof 7i ie regular) then dim@, M ) is true. (2) If i5 E I * atp(c',A ) is regular, J,, J , are mu&mal subsets of I independen4 over A , then I J,I = I J,I. Proof. By the previous theorem and AP 3.10.

LEMMA 1.16: Let A G IMl,p E S ~ ( A I) ,r lMl a d d i n r E e p e n d e n t 8ei over A of 8t?quences realizing p in M ,p regular, M ie c-saturated,

IAl+ + K ( T )s K , K regular. (1) For every ~tatiorucrizution q = tp(i3, 1M1) of p over M y stp(B, A u UI ) t- q, anal if p i s stationary, q ( A u U I ) t- q. ( 2 ) For euery i5 E M there is J E I , IJI < K ( T )8w:h that tp(c',A u U J ) k tp(c',A u U I ) . (3) Let p be r@r, Mi (i 5 a) c - s a t u r d modeb, i < j ==M , (M,; Md ie F:-pimuy over UiSdlMil, M, =; M , and I , G lMi+,l ie a mmxthal independent set over IMiI of sequence8 realizing 8ome stationof M , arization of p over (Mil. Then U,,,I, u I i s a d d &ei idover A of sequence%realizing p.

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Proof. Easy, using 1.11 and 1.14. DEFINITION1.3: (1) An m-type p is minimal ifp is not algebraic but for t = 0, 1, both non-algebraic. no cp(Z, 8 ) are p u {cp(E, (2) An infinite indiscernible set I is minimal if Av(I, U I ) is minimal. (3) A formula cp = 'p(Z, a) is weakly minimal if 'p is not algebraic but it has a bounded ( < R ) number of non-algebraic pairwise contradictory extensions. THEOREM 1.17: (1) The m-type p ie minimal iffthere is aJinite A , euch that for every Jinite A 2 A,, P ( p ,A , No) = 1, Mlt(p, A, KO) = 1 ifffor every cp, P ( p , 'p, No) s 1, and Bm(p,cp, KO) = 1 * l W p , 9, No) = 1, andfor some 9,R"@, 9,KO)= 1. I f p is complete, MIt(p,q,KO)= 1 for every p. ( 2 ) The m-type p ie algebraic ifffor every A , Bm(py A , KO) = 0 ifffor every 'p, lP(p,cp, KO) = 0 iff P ( p , L , A) = 0 (any h L KO) iff

Dm(p,L,m) = 0. (3) The fornzuZa cp(Z; 8 ) ie weakly minimal iff Dm['p(Z;a), L, a33 = 1. If p ie etatimucry, 'p(Z, 8 ) E p ie weukly minimal, p i s not algebraic, then p ie minimal. (4) If Bm(p, L, KO) = 1, Mlt(p, L,KO) = 1, then p w minimal; and the converse holde for finite p.

Proof.(1) Suppose p is minimal, then let A, = {E = PI. As p is not algebraic, there are distinct 8, (n < w ) which realizes it, hence p u {Z = a,} is consistent and pairwise contradictory. So for every A 2 A,, R"\(p, A , KO) 2 1. Now it is easy to prove (by (2)) that if d is finite, Rm(p,d,KO) > 1 or Rm(p,A,8,) = 1, Mlt(p,A, KO) > 1 then for some cp(2, @)E A , and 8, p u {cp(f,a)'} is not algebraic for t = 0, 1, contradicting the minimality of p. This proves that the first condition implies the semnd and third. The other direction is easy. (2) If p is algebraic, some finite q c p is realized only by n < No sequences; hence q does not have n + 1 pairwise contradictory ex1, k-contradictory extensions. On the other tensions, and no kn hand if p is not algebraic, for every h there are distinct (i < A ) which realizes p , so p u {Z = 8{}are consistent and the formulas Z = are pairwise contradictory. Observing this, (2) is easy. (3), (4) Easy too.

+

THEOREM 1.18: (1) if p is a minimud m-type over A , t h it unique nowalgebraic & d o n q E Sm(A).

hae

a

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(2)If p is a m i n i d type, then there is a unique (up to equivalence) non-trivial injnite indiscernible set of sequences realizing p . ( 3 ) If p is minimal, p c q, q forks over Dom p then q is algebraic. (4)If p E Sm(A)is minimal, then p is stationary and regular. ( 6 ) A complete type parallel to a m~nimaltype is minimal. ( 6 ) If tp((al, . . .,a,,), A) is minimal, then for some 1 ii is algebraic over A u {a,} (where ti =
.

Remark. I n view of (l), (4)and (€9,we shall call minimal types stationary, and extend the definition of parallel accordingly; (and also accompanying concepts, “ I based on p ”).

Proof. (1)A s p is not algebraic, we can find > 21-4 elements realizing it, so infinitely many of them realizes some q E Sm(A),which is the right extension. Clearly it is unique. (2)As p is not algebraic, there is at least one such I . If I , J are two such sets, let Domp u I u J E M ; then Av(I, M ) , Av(J, M ) are nonalgebraic extensions of p in S*(M), hence they are equal, hence (by 111, 4,17(1)),I , J are equivalent. (3) Follows by 1.17(1) and 111, 1.2(4)and 1.17(2). (4)By 111, 4.2, and 1.17(1) p is stationary. By 1.1 every type is orthogonal to an algebraic type hence p is regular. (5) Easy, using 111, 4.2. (6) By 111, 6.2(6) for some 1, a, # acl A . If a, 4 acl ( A U {al}),then tp(6, A u {a,})forks over A (as a, # acl A , q = a, forks over A ) but it is not algebraic, contradicting (3). (7) Easy, like IV, 2.16(2),remembering the finite character of being an algebraic type. +

EXERCISE 1.3: In 1.16(1), we can get q ( A u U I)I- q when some Si, E I realizes stp(8, A). EXERCISE 1.4: Suppose tp(6-6, A ) is regular, ti 4 acl A. Prove tp(6, A)isregular.[Hint:LetA c IMI,MFR-saturated,A > IT1 + IAI, and w.1.o.g. tp(6-6, [ M I ) does not fork over A. Suppose tp(sil, 1611) extend tp(si, A) and fork over A; and it suffices to prove that tp(si, IMI), tp&, 1611) are weakly orthogonal. Choose 6, such that tp(ii-6, A) = tp(iilnb,, A). Clearly also tp(Si1”gl, 1M1) forks over A. Hence

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tp(ZlnEl, [ M I ) is weakly orthogonal to tp(a-6, 1M1). So by 1.1(4) tp(Z,, IMI), tp(?Z,1611) are weakly orthogonal as desired.]

EXERCI8E 1.5: Suppose M is Fg-aaturated, K 2 K,(T),tp(4, IMI)are orthogonal and tp(6,, 1 x1u a,) are e-isolated (for 1 = 1,2). Prove tp(iT1-6,, 1M1)( I = 1,2) are Orthogonal; and generalize by replacing the sequences by sets. (Hint: Use 3.2.) V.2. Dimensions and orders between indiscernible sets

DEFINITION 2.1:

(1) Let I, J be infinite indiscernible sets. I I , J if for every K 2 K,(T)and Fg-saturated model M and I,, J, E M (infinite) indiscernible sefs equivalent to I, J respectively, dim(I, M ) s dim(J,, M )(w stands for “weakly”). (2) Let I, J be infinite indiscernible sets, I S , J when for every K 2 K,( T) and e-saturated model M and I,,J1 s M equivalent to I,J respectively and Fg-saturated model N, M < N if Av(I,, M ) is realized in N then also Av(J,, M )is realized in N (8 stands for “strongly”). (3)For stationary types p, q, p I , q (p saq) if the corresponding relations hold between indiscernible sets based on them.

LEMMA 2.1: (1) FOPany in,nite indiscernible eet I , I I, I , I I , I . (2) I ,, saare tradtiwe. (3) I I,J*I s,J. (4) If I,, I , are equivalent and J,, J, are equivalent then for x = 8 , w, I , 5, J, 0I , 5, J,.

Proof.(1) Immediate. (2) Suppose I,, 1 2 , I, are infinite indiscernible sets, M a Fg-saturated model, 1; c M, Ij c M and I ; , Ij are equivalent to I,,I,, respectively (where K 2 K J T ) ) w.1.o.g. , I, = I ; , I, = Ij. We can find, for 1 = 1, 3 sets A, E U I,,such that I, is based on A,, Av(I,, A,) is stationary and lAll < min{K(T), 24,) (see 111, 4.17(4)). So we can find an elementary mapping f, f 1 ( A , u A,) = the identity, and f maps A , into M ( B exisfs as M is R-saturated). So we can find in M an infhite indiscernible set I; based on f(Av(I2, As)). Clearly, I ; is not necessarily equivalent to I,. But for x = W, 8 , I , 5,. I, iff I ; 5, I s ; and I, I , I, iff I , sZ I;. So if I, I , I , s, I,, dim(I,, M ) I dim(&, M) and dim(I;, M) I dim(I,, M).As M ,I ; , I ; were arbitrary, it follows that I, I , I,.

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24 1

Also if I, I, I , 5 , I,, zi, realizes Av(I,, M), N is FE-saturated, E 1 x1then Av(l;, M) is realized by some E N (aaI, 5 , I ; ) , hence some EN realizes Av(I,,M) (as 1 ~ 1 , )Again . I, s , I , follows. ( 3 ) Suppose I SWJ, so for some Fi-saturated model N,K 2 K J T ) end I,, J , E M equivalent to I, J respectively, but dim(I,, M ) > dim(J,, M ) 2 K ; w.1.o.g. I,, J , are maximal indiscernible sets EM, lJll = dim(J,, M),II,l = dim(I,, M ) . Hence ]Ill > lJll 2 K . So by 111, 3.6 there is I , E I,, ] I a [= I Jll such that I, - I , is indiscernible over lJ I, u lJ J,. By IV, 3.1 there is a F,4-prime model N over UZ, U UJ,, N G M . By IV, 4.10 N omits p = Av(J,,UZ, U UJ,). Choose si E I, - I,, so si realizesp; but M omits (by the definition of J , ) Av (J,, N). This situation shows I J. (4) Trivial.

IN1 u zi,

LEMMA 2.2: Assume K 2 K , ( T ) , M is F3saturated I, J c M inJinite indi8~et~ibk 8d8, and zi realizes Av(I, M ) . T h I I, J iff in the’F,Op i m e model N over N u a, Av(J, M ) is.realized.

Proof. The “only if” part follows by the definition. So suppose Av(J, N )is realized by 6 E N , and we shall prove I I, J. So let K( 1 ) 2 K,( T ) , M,is F&saturated, I,,J, E H,are equivalent to I,J respectively, zi, realizes Av(I,, M,), N , be a F,”-saturatedmodel, lM1lU si E N,. We must prove N, realizes Av(J,, M ) . As there is a Fi,,,-prime model N , c N over 1M,1 u a,, we can assume N, is F,O,,,-prime over lMll u a,. Choose A 2 ~ ( 1 + ) K , A regular A > IT1 + llMll + l[Mlll, = A, and let M* be a saturatad model of power A, 1M1 U 1M11E 1M*1. We can assume that a, = si realizes Av(I, M*) = Av(I,, M*). Let N* be e F:-prime model over IM*I u si. By the symmetry in assumptions, it suffices to prove that Av(J, M ) is realized in N iff Av(J, M*) is realized in N*. Suppose Av(J , M ) is realized in N by 8 E N. Let d = (IMI

U a, {bf: i

< a},{Bf:i < a})

be a Fi-construction of N. We prove by induction on p that dz = (IM*I uzi,{b,:i< /3})isa~-construction,andtp,[si~{b,:i < p}, 1M*1] does not fork over IMI. So d; is a FE-construction, hence a Ficonstruction, hence tp(8, 1M*1 u zi) is F;f-isolated,hence tp(E, 1M*1) is realized in N*, does not fork over M , and extends Av(J, M), so

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tp(E,IM*I) = Av(J, M*) so this will finish one direction. We prove by induction on 8; for fl = 0 and @ a limit ordinal it is trivial. For + 1 we know q = stp(b8, BB) stp(b,, 1M1 u a u {b{: i < p}). It suffices to show q I- tp(b,, 1M*1 u iz u {bi: i < 18)) (the non-forking of tp*@ u {b,: i s

p}, p*1>

over u iz will then follow). So let ZlE a U {b,: i < p}, Za E lM*l, and ~ ( z8,,Z) be a formula; and a,, Za)t for some t. Choose C E 1M1 such that we shall prove q I- ~(z, tp(B8 U 17, U iz, 1M*1)does not fork over 0,ICl < K (C exists by the induction hypothesis, and aa K J T )s K ) . Choose EL E 1M1 such that stp(E;, C) = Stp(E2,C) (possible as M is q-saturated). By the definition , Ea) e Q t- ~ ( zZ,, $). As for some t, ~ ( z El, ,EL)t E of C, q I- ~ ( zEl, tp(b8, u a u {biz i < P}), Q I- ~ ( zEi, , EdtNow we prove the other direction, i.e., suppose that 5 E N* realizes Av(J, 1M1*). As h is regular ~ K J T tp(E, ) , IM*I u a) is q-isolated. So there are C E B E lM*l, ICl < K ( T ) ,IBI < A, tp(E, B u a) ttp(i3, IN*\ u iz), tp(3-5, 1M*1) does not fork over C, and tp(anE, C) is stationary. As M* is a saturated model, there is an automorphism f of M , f 1 ( I V J) = the identify, f (C) c M , (w.1.o.g. I, J are countable) f (J)are equivalent to and for K = No,we can demand just that f (I), I, J resp. ; so we can assume C G M. Now claim III,4.22(2) shows tp(a,Cu iz) I- tp(E, lM*l) (1ettingC u = A, B u a = B, 1M*1 ua = C).

lJfl

THEOREM 2.3: Thefollowing crmditiona on the infinite indi8cernible set8 I , J , and infinite cardinals p > A 2 K J T )are equivalent: (1) I s,J. ( 2 ) If M is Fi-saturated,I , J E M , then dim(I, M ) I dim(J, M ) . ( 3 ) There i s ru) q-saturated ncodel M , with I , , J1c M equivalent to I , J rapectively, dim(I,, M) = p, dim(J,, M ) = A. (4) Por every I , , J1 equivalent to,I, J reqectively, 1111,lJll 2 K,(T), I1(J1)indiscernible ower U J , (UI,), Av(J,, U J , ) has a F&,-isoZated extension in P ( U I , u UJ,). Remark. So if there is a ql-saturated model M , with I,, J, c M , equivalent to I, J respectively such that dim(I,,iK,) = p1 > dim(J,,M,) 2 A , 2 K J T ) then for all ,uLz > A, >, K,(T) there is a F:,-saturated Ma with I,, J, E Ma equivalent to I,J respectively such that dim(I,, Ma) = pa, dim(J,, Ma) = ha.

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Proof of Theorem 2.3: (1) * (2) by the definition. (1) =F (3) by the definition. (2) or (3) =- (4) Suppose I,,J1contradict (4). Without changing the hypothesis on I,, J , , we can assume 1111=A!, IJ,1 = A (by 111, 4.22). Let N be Fi-primary over U I, u UJ,. By IV, 4.10 N omits Av(J,, U I, u UJ , ) and Av(I,, U I, u UJ,). From the first omitting it follows that dim(J,, N) I II,l lJll = p; but dim(J,, N) 2 lJll = p, so dim(J,, N) = p. (See 111,3.9).On the other hand dim(I,, N) 2 II,l = A; suppose dim(I,, N) > A, then there is I*E N equivalent to I,, II*l = A + . As A 2 K,(T),some E € I * realizes Av(I,, U I,).Let d = (I, u J,, {af:i < ao}) be a Fi-construction of N. Going through q,n,(T) (see IV, 4.6-4.7) we can aasume E E {af: i < a}, a < K,(T)(see IVY1.4(2)) so there are I, s I,, J , E J,, 11, u Jal < KJT)such that tp*({af:i < a}, U I, u UJ , ) does not fork over U (I,u J,). We can also aasume Av(I,, U I,) does not fork over U I,, Av(I,, U I,) is stationary, and similarly for J,, J, (see 111, 4.17(4)) and for every E E U (I,u J , ) , tp(E, U (I,u J,) is stationary. Now we prove by induction on i that

+

91 =

tP(.fJ

u (Inu ” b,:j Ja)

< i}) tP(%

u

u J , ) u {a,:j < i}).

(I1

This holds by 111, 4.22(2) as p1is Ff-isolated. So clearly

u

u

e: < 4 ( I , u J , ) ) I- t p * ( h : i < 4, (I1u J , ) ) tp*(E, U (I, U J , ) ) k tp(8, U (I,U J , ) ) . So we could choose

tp*({.,:

hence a < w , so we could choose I,, J, such that 1 1,u Jal < K ( T )and , remember tp(E, I, u J,) I- tp(E, I, u J,). So tp(Z, I, u J , ) is F&)isolated, and it extends Av(I,, U I,),contradiction to the assumption on I,, J,. So dim(I,, N ) = A, dim(J,, N) = p, and (3) fails. If we choose A > III + IJI, we can assume I , J E [MI, hence getting a contradiction to (2). So we proved that (2), or (3) implies (4). (4) =- (1) Suppose K 2 K,(T),M is F:-saturated, I , , J , E M are equivalent to I,J respectively; we shall prove dim(I,, N )s dim(J,, N). Suppose not, and w.1.o.g. I,, J, are maximal indiscernible sets in N. By 111, 3.9 !Ill= dim(I,,M) > dim(J,, M) = lJ1l,and of course lJll 2 KJT).By 111, 3.5 there is I, E I,, II,l 5 lJll K,(T)= lJ1l, such that I, - I, is indiscernible over J,. By 111, 4.17(4) there is J , E J , , lJ81 < K,(T), Av(J,, UJ,)does not fork over UJ,,

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u

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Av(J,, J,) is stationary; and similarly I, E I , - I,, and we can J,)does not fork over J,. Then tp*(I, - 'I, - 12, assume tp,(I,, 1, U J1)does not fork over I, U J,, hence tp*(J, - J,, (I, - I,) u J,) does not fork over I, U J,. So I ; = I , - I, - I,, and J; = J, - 5, scltisfies the hypothesis of (4) and = 1111, IJ;I = IJll. Now we define by induction on p < IJ, I sequences E,, E IblI and setsI: E I ; such that J: = J; u {Zt: i < p}isindiscernible, II:l I IJll and I ; - I;, J Z satisfy the hypothesis of (4). For p = 0, p limit this is J; u (I;- I:)) clear. For p + 1, by (4) there is a Fk,,,-isolated type q EP( which extend Av(J:, JZ), so some E,, E M realizes it, and for some 11,1I;+11 I; IJ11, I ; E I:+, and tp(E,,, (I:+, - I:) u J;) k tp(E,,, ( I ; - I:) U J*). Clearly the induction hypothesis are satisfied, and for /3 = IJll ,JZ contradicts dim(J,, M) = IJll.

u

u

+

u

=

+

LEMMA 2.4 : (1)If p , q are stationary, not orthogonal, and q is regular then p G 8 q (hence p <,a). (2) If p , q are stationary and orthogonal then p $, q (hence p & q). Proof. (1) Immediate by 1.12. (2) Immediate, by IV, 4.10, and 1.4, 1.10. THEOREM 2.6: Let I be a (non-trivial)indiscernible set, II I = p > A 2 K,(T),M a Ft-prime model over I , J E M a (non-trivial) injnite indiscernible set. 'Thendim(J, M ) = A iff dim(J, M ) # p iff I $, J.

Proof; As bl is R-saturated, dim(J, M) 2 A. If J' E M is equivalent to J , IJ'I > p, then for some J " E J', IJ"I Ip, J' - J " is indiscernible over (JI, and its cardinality is, of course > p , contradiction to IV, 4.9. Hence A 5 dim(J, bl) Ip, dim(1, M ) = p. If I I,J then dim(J, M) 2 dim(1, M ) = p hence dim(J, M) = p. If dim(J, M) = p clearly dim(J, M ) # A. To complete the proof we assume I $ w J and prove dim(J, f l ) = A. By 2.3(3) there is a Fisaturated model N, and I,,J, E N equivalent to I, J respectively, dim(I,, N) = p, dim(J,, N) = A. Choose J p E J , lJal < K(T),such J, that Av(J, Ja)is stationary, Av(J, J) does not fork over (by 111, 4.17(4)). By IV, 3.12(3) M is Ft-prime over I u J,. W.1.o.g. we can find an elementary mapping f of I u J, into N, such that f (I) is equivalent to I,, and f ( J a ) c J,. By the primeness we can extend f to an elementary mapping f ' of M into N. So A Idim(J, M) I dimlf'(J),f'(M)] Idim(J,, N) = A. So dim(J, M) = A, and we finish.

u

u

u

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THEOREM 2.6: 8uppoee I is a non-trivial indiscernible set, IIl z p > A z +(T),M &I q+me over I, and J c M an infinite ind&Imible set. Then I is orthogonal to J iff Av(J, M) i s F,”+-isolatediff Av(J, M) i s q-h&t?d fOr 8 x 5 p.

Proof. We can w u m e I JI = A, I, E I , II,l = A, I - I, is indiscernible over I , U J, I, U J c N c M ,N is Fi-primary over I, u J, and is ~-conetrudibleover I u J, and M is 5-constructible over I u N (see IV, 3.17 and IV,3.9). Clearly I - I, is indiscernible over N. If I, J ctre orthogonal then by 1.4 Av(J, N)I- Av(J, N u I) and by IV, 4.10 Av(J, N u I) t- Av(J, M ) and Av(J, J u I , ) F Av(J, N).As (J( (I,(= A, it follows that Av(J, N ) is F;+-isolated, hence Av(J, M) is F;+-isolated, hence it is Fi-isolated for some x < p. Now w u m e Av(J, M )is q-isolated, x s p, and we prove I, J am orthogonal, and so we shall finish. Let A c M ,IAI < x 5 p be such that Av(J, A) I- Av(J, bl). By IV, 3.17 and IV,3.9 there is a model N, A E IN1 E 1611, N Fi-prime over I, for some I, c 1(1111< p ) and Fi-constructible over I, bl is R-constructible over I u N, IN n I1 2 No,IN n JI 2 X,. So Av(I, N) is realized in bl (by any c‘ E I - I , ) and Av(J, N)I- Av(J, H).Hence Av(J , N),Av(I, N) are weakly orthogonal, hence by 1.2(3)they are orthogonal.

+

CONCLUSION 2.7: If 1,J are (non-trivial, inJinite) indiscernible sets A, I, J are not orthogonal, t k n tp,(J, A), tp,(I, A) are not weakly

Over

orthogonal.

Remark. See 1.5(2) for a converse.

Proof.Suppose tp*(I, A), tp*(J, A) are weakly orthogonal. As I,J are infinite, we can w u m e IIl > IJI > K = ~ ~ ( 2 + 1 )IAI +,I indiscernible over A U J, and J indiscernible over A u I. By Ax(V.2) for Fb, for any c’ E J (letting J‘ = J - {E}) tp(E, A u J’)F tp(E, A uJ’ u I).So if N is F:-primary model over A u J’ u I, then Av(J’, A u J’)= tp(E, A U J’)I- Av(J’, N).So Av(J’, N)is q-isolated, ( A = IJI) hence by the previous theorem I,J me orthogonal. , E A,porthogonaltoBortotp,(I,A). CONCLUSION2.8:I f p E P ( A ) B I an (infinite)hdi8Cernibk.set over A, tp,(I, A ) does not fork Over B, then p is ort7cogOnalto AV(I, U I).

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Proof. We can assume A = "1, bl F&,,,-Saturated. Let J be an indiscernible set baaed on p. By 1.4 tp*(I, M),tp,(J, M ) are weakly orthogonal, hence by the previous conclusion p, Av(l, Ul)are orthogonal.

THEOREM 2.9: Let I be a (non-trivial)indiscernible set, IIl = p > A 2 K J T ) ,M a q-prime &l over U I , A c M , IAI < A, p E P ( A ) , A1 the $rst cardinal 2 A in which T is stable, p > hl, and J = (6 E M :6 realizee p}. Then IJI 2 p i f IJI > A1 iff p has an d o n q such t7rat Av(I, I ) I t 0 Q.

u

Proof. If p has an extension q, Av(I, U I ) 5 , q; then let q E Sm(Au B ) IBI < K ( T )so , we can assume B c M , (of course, q is stationary). So p = dim(I, M ) 5 dim(q, M ) 5 IJI. If I JI 2 p then I JI > Al, and if I JI > A, then by I, 2.8 there is an indiscernible J, s J , IJ,I = A:. So dim(J,, M ) > A1 2 A, hence by 2.5 I 5 , J, so letting q = Av(J,, A u J,), Av(I, U I) I, q and q extends p (of course J, is not trivial, q not algebraic).

u

Notation. (for Theorem 2.10). (1) Let IN, be a non-trivial indiscernible set, IIKol = KO.For every A let IKo E I,, I , indiscernible, lIAl = A, and let M(I,) be a F&,- prime model over I,. (2) Let x 2 K J T ) ,T stable in x.

u

THEOREM 2.10: The following 8tatenwnt.3on T are equivalent (remember T i8 Stable). (1) For every A 2 K ~ ( T T ) , has nzaximally-~-saturated rnodels of arbitrarily high cardinality. (2) As (l), but the models are in addition A-universal but not A+universal. (3),,, Not every F & ( ~ ) - 8 d U T d e dmodel i8 i8onzOrphiC to M(Ij,)for 8Ome h. (4), Not every two F&(~)-8aturated. d e l e of T of cardinulity x+ are iscnniwph&% ( 6 ) For some (non-trivial infinite) indiscernible sets I , J, I $, J. (6) There are (non-trivial infinite) indiscernible sets I , J such that J is minimal and I is orthogonal to J. (7) There are (non-trivial infinite) indiscernible sets I , J , I $, J, J is based on 0 such thd (A) I is minimal, or (B)J * I, J s, J * for some minimal J*, and I , J are orthogonal.

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DEFINITION 2.2: If T fails to satisfy the above conditions, it is called uni-dimensional. Proof of 2.10. Trivially (2) =- (1); (6) + (6) (by 2.4(2)). (6) * (1) By 2.3. (1) (a), By 111, 3.12 T hae a saturated model of cardinality x + ; and by (1) and w T is stable in x , T has a maximally-~ro,-satturated model of cardinality x+ . As A L K,( T)they are not isomorphic. (4), =- (3)rHO Clearly p 5 II,l I llM(I,,)\l;and if p I x, then as T is stable in x, IlH(I,)I( s x. So IIM(I,)II = x+ iff p = x+ ;hence one of the F&,,,-saturated models of T of cardinality x+ , is not isomorphic to any

-

+

W,).

(3)IH0 3 (6) Let J be a countable (non-trivial) minimal indiscernible set (existsby 1.18(7)). Let M be a F;=,,,-saturated model not isomorphic to any M(I,). We can wsume J E M and let J , E H be a maximal indiscernible set in M extendingJ. By 1.16 and 1.18(4) for every E E I M I tp(8, J , ) is l?&,-isolated. If I*E 1M1 is a (non-trivial)infinite indiscernible set over lJJ,, then we can assume that for some J , E J,, IJ21, IJ , - J21 2 KO, IJ2I < Kr(T) + HI,tP*(I*, U J 2 ) t. fp*(l*,J 1 ) SO by 2.7 I*,J , are orthogonal, so (6) holds. So suppose there is no such I*, then by IV, 4.14 (the characterization theorem) M is FEr,,,-prime over lJJ,. Let p = IJ , I, then in M(1,) we can find a maximal indiscernible set J,, such that for distinct a!, . . .,a; E J , , 1 = 1, 2 , tp(7iin. - .^7i;) = tp(sifn. . .-a:). As before we can assume M(1,) is F&,-prime over J,. If IJal = p, there is an elementary mapping f from J, onto J 2 . By IV, 4.14 we can extend it to an isomorphism from M onto M J I ) , contradicting the choice of M. If I J21 # p then as dim ( J 2 ,M) = IJ,I, by 2.6 J, SwI,, hence J,, I , are orthogonal (by 1.12) so (6) holds. (6) 5 (7) Let I, J be as in (6), and J, be a non-trivial infinite indiscernible set, based o n 9 (exists by 111,1.12).If J $,J1 then taking J, J , for I, J shows that (7)(A) holds. So assume J surJ,. If J, is orthogonal to I then I $, J , so taking I,J,, J for I, J, J* shows that (7)(B) holds. ( J , swJ holds as they are not orthogonal, and J is minimal hence regular). But J, .is orthogonal to I by the following claim :

u

u

u

CLAIM 2.11: If J sWJ , , I is orthogonal to J then I is orthogonal to J , .

Proof of 2.11. Let M be A-saturated, IlMll = A > K ~ ( T(by ) stability.) and w.1.o.g. J,, J, I be countable and c M .Define J* so that IJ*l = A+,

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J* U J is indiacernible and J* is indiscernible over 1M1;and let N be E-prime over IMI u UJ*. Clearly dim(J, N ) 2 IJ*I = A + , hence dim(J,, N) 2 A+ > A, so some 2 E N realizes Av(J , , 1Ml).On the other hand, as I is orthogonal to J , Av(I, IMI)F Av(I, 1M1 u UJ*) and k by IV, 4.10(2) Av(I, lMl u UJ * ) FAv(I, IN]), hence Av(I, ]MI) Av(I, [MIU a), so by 1.2(3) J , , I am orthogonal.

-

Continuation of the proof. ( 7 ) ( 2 ) Let I,J be as in (7) and if poaaible, as in (7)(A); and ,u > h 2 K,(T). We can assume J,I are indiscernible over U I, UJ reap. and 111 = p, IJI = A. Let N be a F:-primary model over U I u UJ ; so by 2.3 we know N is maximally FZ-saturated, and of come lldlll 2 111 = p . As Af is E-saturated, it is A-saturated, hence A-universal. Choose an indiscernible set J 1 over U I , J G J1, I JII = A + , and 1 = A+. Suppoae f is an elementary mapping from chooae If E I, 1' U I' u UJ1 into N; we try to get a contradiction,and thus show that N is not A+-universrtl. Clearly dim(f(Il), N)) is > A thus aa in 2.6 I s , f ( I 1 ) ; and similmly I 5 , f ( J 1 ) .If J* 5, J I , J*, J* minimal then for some minimal J** E IMI, f ( J 1 ) 5, J**, hence I 2, J**. As dim(J , H)= A, dim(J**, M ) = p, clearly J** sWJ , so we could have started with J , J**instead of J,I, and then (7)(A)holds, so I is minimal. If I is minimal, thenf(ll) is minimal too, hence by 2.4, 1.18(4)f(P) I , I, thus (asI 5, f ( J 1 )and by 2.1(2))f(11) I , f ( J 1 )hence I' I , J 1 contradiction. As we proved (6) =- (6) =$ (1) =- (4), * (3),,0 =- (6) =- (7) (2) =- ( l ) , we finish.

-

PROBLEM 2.1: Is there a uni-dimensional, stable, but not superstable theory? (see Exercise 6.4). QUESTION 2.2: In which places in this section can we replace K J T ) by K ( T)? Or should we find a natural K"( T), K( T)I K"( T)5 KJ T)for those theorems ? PROBLEM 2.3: (1)For every stable IT and A < K J T )does T have a A-saturated model which is not A+ -universal? (2) For every atable theory T with ID(T)I > No does T have a model which is not N,-universal, or at least not I TI -universal? EXERCISE 2.4: (1) Prove (1) above when A 2 KJT).

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(2) Show that if universal.

K,(

249

T ) < I TI then T has a model which is not I TI-

EXERCISE 2.5: Suppose p 5 , q, r I,q, and show that p , r axe not orthogonal, except when they are algebraic. V.3. Weighted dimensions and superstability THEOREM 3.1: Suppose {a, 6} is independent over A and stp(C’,A) is regular. Then it is impossible that both tp@, A u E), and tp(6, A u 5 ) fork over A.

Proof. W.1.o.g. we work in Beq, A = acl A. Let M be X-saturated, IAl + 12’1 < X = llMll,A U a u E u E E M,andJ = {a:aEM,$realizes stp(C,A)}. By 11, 2.2, 13 there is J , c J , lJll < 121’ such that tp(& A U U J ) is (A u J,)-definable (see Definition 11, 2.1(4)). Clearly if tp,(Ji, A U 7i) = tp,(J,, A u a), J ; E J (with suitable indexing) then there is an automorphism f of N ,f A = the identity, f (a) = a, f maps Jl onto J ; ; hence we can replace J , by J;. So we can assume that tp(J,,A u Z U 6) does not fork over A u a . Hence tp(6, A u u UJ , ) does not fork over A u a (by Ax(V1)for FI,)so it does not fork over A (by 111, 4.4). Similarly we can define Ja E J so that tp(6, A u U J) is (A u UJ,)-definable, and

u

tp*(J,,Auau6uUJ,) does not fork over A

U 6.

tp,(6

Hence (by Ax(VI1) for FI,))

u J,, A u a u

u J,)

does not fork over A, so tp,(J2, A u J , ) does not fork over A, in particular J , n Ja = 0. Choose maximal I, c J , independent over A. So I, u I , is independent over A, I, n I , = 0. We can find & E M , i? realizes stp(c,A),but tp(E’, A u

UJ , u UJ , u ti u 6)

does not fork over A. So (3,a’} is independent over A. But {a, E} is not independent over A , hence tp(anE, A) # tp(a-E’, A). As Z, E’ e J, and tp(8, A u U J ) is (A u UJ,)-definable, it must be that tp(E, A u J , ) # tp(E‘, A u J1).As stp(Z,A) = stp(E’,A ) is s t a t i o n q tp(8, A u J , ) forks over A. So E depends on I , over A (by 1.9). Similarly E depends on I, over A . Contradiction by 1.14.

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LEMMA 3.2: Suppose M is Fi-saturated, A r K ~ ( Ttp,(Al, ), [MI u A,) doesnotforkover lM[ andN,ie Fiqrimeover [MI u A, (1 = 1,2). Then: (1) (i) tp*([N1[,INaI) does not fork over [MI. (ii) stp*(IN,I, IMl u A,) btp*(lN,I, (MI u A1 u IN,[). (2) If tp(6, )MI u A,) ie Fi-ieokted then: (i) tp(6,lMI u A,) does not fork: over IMI. (ii) stp(6, I MI u A,) k tp(6, [ MI u A, u A,). Proof. (1)We can in the lemma replace [N,1by sets B, Fg-constructible over [MI U A,, and then prove i t by induction on the length of the constructions. So it suffices to prove (2). (2) If this is true for every finite subset of A,, it is true for A,; so we can msume A, = 8. Similarly we can assume IA,I < A. AS Kr(T) S A, there is B G ]MI, JBI < A such that tp,(A, u 8, ]MI) does not fork over B, tp,(A, u 8, B) is stationary and stp(6, B u A,) k stp(6, [MI u A,). As tp(8, 1M1) is stationary by 111, 2.15(2) and tp(8, [ M I U A,) does not fork over [ M [ (by assumption)tp(8, 1 M 1 u A,) is stationary. If (ii)fails, then there is Si E (M1 such that stp(6, B u A,) br tp(6, B U A, u Ti u 8) and w.1.o.g. Ti E B. Now as M is Fi-saturated there is 8' E I MI, stp(8', B) 9 stp(8, B), so atp(a', B u A,)

E

stp(8, B u A,)

(remember tp,(A,, lM[ u 8) does not fork over 1M1, hence over B (by 111, 4.4)). Hence stp@, B u A,) br tp(6, B u A, u 8'), contradicting stp(6, B u A,) b stp(& [ M 1 u A,). Remark. Really 3.2(2)(ii)implies tp(6, M U Al) I- tp(6,M U A, U As), as tp,(Al U 6, M), tp,(A,, M) are stationary. CLAIM 3.3: If M ie Fi-euturated, A r K,(T) regular, tp(ii, [MI)regular, N is FR-primary over [ M1 u Ti, and 6 E IN[, 6 $ 1 M 1 then N is F",primary over IMI u 6. Proof. Choose B G [ M1, I BI < A so that tp(5-6, M) does not fork over B, tp(~i-6, B) is stationary, and stp(5, B u 5) k stp(6, (MI u a) (such B exists by III,3.2,111,2.9(1)and IV, 3.2).By III,6.14 tp(6,B u a)-I tp(6, lMl u a)(B, IM[, a correspond to B, C, A.) If stp(z, B u 6) br tp(Si, 1M 1 u 6) then some 3' E N realizes stp(Si, B u 6), but tp(zn5, I M I) # tp(a'- 6, As tp(anb, B) = tp(n ^'6, B), tp(6,B u a') forks

m).

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251

wl)

over B, so & # M , hence by 1.9 tp(&, = tp(a, WI); tp(an&,B)= tp(a’^b,B), tp(a^b,M) # tp(n’^ b,M), tp(6,B u a) -I tp(6,M u a), a contradiction. Hence tp(&, U 6) is Fi-isolated, and the rest is easy by IV, 3.16 and IV, 3.6.

wI

THEOREM 3.4: Suppose A G B, f an elementary mzpping whse domain is €3, f A is the identity, and p cSm(B)is stationary, and stp,(B, A ) E stp*(f (B),A), stp*(f (B), B) does not fork over A . Then p is orthogonal to A iff p is Orthogonal to f ( p ) .

Proof. Define elementary mappings fi(i < IT1 +) so that stp,(B, A ) EE stp,(f,(B), A), and tp*(f,(B), f,(B))does not fork over A, and we let fo = the identity over B, fi = f. Suppose p is orthogonal to f ( p ) ,so clearly the f i ( p ) (i < I T ] + )are pairwise orthogonal. So if p is not orthogonal to A, for some a, p is not orthQgona1to stp(& A), hence stp(a, A) is not orthogonal to every ff(p), contradicting 1.7. So we have proved: if p is orthogonal to f (p)then p is orthogonal to A. Now suppose p is orthogonal to A. W.1.o.g. there is I E B, p = Av(I, B), I indiscernible over A. So p is orthogonal to tp,(f(l), A), hence by 2.8 p is orthogonal to Av(f(I), f ( I ) ) ,which is parallel to f (PI = Av(f (0,f(B)).

u,
u

DEFINITION3.1: Then pair (p,tp) is called regular, if tp = p(E, a) E p and (clearly the definition does not depend on M)

(*I

Let p be over [ M I ,M Fg-saturated ( K = K ~T)), ( tp(Si, 1M1) a stationarization ofp, N F;-primary over u si. Then for every 6~ INI, 6 4 IN], such that 6 realizes tp; tp(6, [ M I )does not fork over E.

Remark. Clearly if ( p , tp) is regular, then p is regular. Also if weakly minimal, p ~ pp is , minimal then (p,p) is regular.

tp

is

CLAIM 3.6: Suppose M, N are Fgr,,,-saturated, IMI c IN], si E IN], c m and u = R[tp(iE,E ) , L,001 where (i) for every 6 E INI, 8 # /HI, u I R[tp(6, / M I ) ,L, 001, or (ii)for every 6 E tp(N, a), 6 # !MI;

si 4 [MI, u = R[tp(Z, \MI), L, m] p@,5 ) E tp(Z, INI). Suppose also

a I R[tp(8,

pq), L, ml.

Then p is regular and even (tp(t3, ]MI),tp(3, E)) is regular.

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Proof. Choose A s ]MI, IAl < K ( T )so that tp(ii, IMI) does not fork over.A, tp(a, A) is stationary and E E A and a = R[tp(ii, A), L, a33 (A exists by 111, 3.2,111, 2.9 and 11, 1.2). If q E S” (IMI)is not algebraic (where rn = I@)), tp(& A ) s q, and q # tp(a, M ) then p forks over A, hence by 111, 1.2(4) R(q, L, oo) < a,so N omits q by hypothesis. Thus we can conclude, by 1.9, that tp(si, 1M1) is regular. Similarly ‘p) is regular. (tp(ii, IMI),

DEFINITION 3.2: The weight of p, w(p) ( p stationary or complete) is the minimal cardinal p 2 0 (possibly finite) such that for some regular h 2 K ~ ( T )some , Flf-saturated model M , and a stationarization sif E N tp(8, IMI) of p , there is an independent set {af:i < p} over IN[, where N is q-primary over 1M1 u si, such that N is Flf-primary over 1M1 v {a1: i < p} and tp(sif,[ M I ) is regular. If there are no such B, ii, iil, N w(p) is said to be a3 or not defined. We let w(a, A ) = w(tp(3,A)). LEMMA 3.6: (1) If w(p) is defined, (i.e.,
Proof. (1) By 111, 4.21 it suffices to prove the second part. If tp(ai,WI U a) does not fork over M , N omits it by IV, 4.10(2). (2) Left to the reader as an exercise. (3) Easy. LEMMA 3.7: When T is superstable, h 2 KO regular, M is Fg-saturated, ii a sequence then we can Jind n < o,a, (1 < n), M , (1 In ) such that No= M , Mz+l is F,”-prinzaryover lMzlV a,, tp(Z,, lMzl) is regular; and M,, is Flf-primaryover 1MI V 3.

Proof.We define by induction on 1 < o,3, and M , , M , being Fgsaturated. Let M o = M, and if M , is defined let N , be K-primary over lM,l v a. If a E M,, we finish, otherwise among a, E lN,l - lM,l there is one with minimal CL = R[tp(Ti,, IM,l), L, a].Thus by 3.6 tp(a,, lM,l)

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is regular. By 1.10(2) tp(Ti,, lM,l u a) forks over IM,1, hence (by Ax(V1) for FL) tp(Ti, IM,I u Ti,) forks over IM,I. Let M I + ,be Fa-primary over 1M,1 u Ti,, F,Q-constructible over IM,I u a (exists by IV, 3.17). So clearly tp(zi, IMz+ll)forks over IM,I. Clearly M , c M , E M , c - As T is superstable, K ( T )= X, so for some n < w , M , + , is not defined, so Ti E M,,. As M , + , is Fa-constructible over 1M,( u Ti, M , is F:-primary over WI U a (by IV, 1.3(2)). +

LEMMA 3.8: &uppose that A 2 K,(T) is regular, M , (i s a ) is FZsaturated, Mf+l is Fg-prime over lMrl U Tif (i < a),Ma is Fg-prime over Ui
IN,],

Proof. We first prove that we can assume that for every i, tp(Ti,, lM,l) either is Orthogonal to 1M,I or does not fork over IMol. For each i ,

choose B c / M i l , IBI < K ( T ) ,and A c IMol, IAl < K,(T)such that: tp(Ti,, ] M i l )does not fork over B, tp,(B u Z,, IM,l) does not fork over A , tp(ai,B),tp,(B U ai,A)are stationary, and w.1.o.g. A c B: and if tp(Tif, / M i l )is not orthogonal to lMol, it is not orthogonal to A. As M , is F,Q-saturated,we can choose B‘ G lMol, 7i; E lMol tp,(B u Z,, A ) = tp,(B’ U a:, A). By 3.4 tp(Ti,, B) is orthogonal t o A iff it is orthogonal to tp(Ti;, B’). So if tp(Ti,, IM,l) is not orthogonal to lMol, it is not orthogonal to A, hence not orthogonal to tp(Ti;, I?’).As tp(Ti,, B) is regular, also tp(Tii, B’) is regular, hence for for some Ti: E M i + , , tp@, lMil) extends tp(Tii, B‘) and it does not fork over B‘ (by 1.12). By Claim 3.3 M i + , is Fi-primary over ]Mil u T,:i and clearly tp(si:, IM,])does not fork over lMol. So we can replace Ti, by T.:i So we can assume that tp(Z,, / M i l )is either orthogonal t o lM,l or does not fork over 1M01. So [ M a [is lMol u {E,: i < p} where for each i one of the following cases occurs: (i) tp(E,, A,) is Fa-isolated, where A, = lMol U {E,: j < i}. (ii) tp(E,, A,) is regular, orthogonal to lMol and stationary; moreover, it does not fork over some B, c A,, lBil < K,(T),where tp(Et, Bi) is stationary. (iii) tp(E,, A,) is regular and does not fork over IMol. As in IV, 1.4(2),4.4-7 we can assume that for some y -cK , ( T )y, < p ~ E A , By , . 111, 3.2 and 111,2.9, we can choose B C W,J,IB( K,(T)

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254

[CH.

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suchthattp*[{Ej:j < y}, IMoI]doesnotforkoverB,andtp,[{Ej:j < y } , B] is stationary and if case (ii) occurs for E,, tp(E,, B u {Z,: j < i}) is stationary. Let C = U {Z,: tp(Z,, A,) does not fork over lMol, and is regular}. We now prove that for i < y , tp(E,, A, u C) is F;f-isolated.If c u e (iii) holds, E, E C E A, u C, so this is trivial. If case (i) holds, then tp,[C - A,, A,] does not fork over IMol, so our conclusion follows by IV,4.10(2). If case (ii) holds, as tp(Zf,A,) isorthogonal to liKol,also tp(Ef, B u {Zj: j < a}) is orthogonal to B. As tp*[{Zj: j < i}, IMol] does not fork over B, for every 6 E lMol tp(6, B u {Ej: j < i }) does not fork over B, but the latter is hence it is orthogonal to tp(Ef,B u {Ej: j < i}); s h t i o n q , so they are weakly orthogonal by 1.2(2). Hence tp(Z,, B U {Z,:j < i})k tp(E,, B u {Ej:j < i} u 6).

As 6 E lMol was arbitrary, tp(Zf,B U { E j : j < i)}I- tp(Ef,lMol u {Ej:j < i}) = tp(E,, A,).

As i < y < K,(T), IBI < K J T )I A, tp(E,, A,) is F:-isol&d

hence q-isolated; so u before tp(Z,, A, u C) is q-isolated. So we finish the three cases hence A, is R-constructible over lMol u C, so by IV, 3.2 tp(8, lMol u C) is F!-isolated so we finish. THEOREM 3.9: (1) If T is euperstable,for every a, A w(a, A) is welld e j i d and jinite. (2) Assumi?>he conditions of Lemma 3.8, w(8, IMol) exist8 and is

4,el

14.

Procf. (1)By 3.7 it suffices to prove part (2). Before proceeding to (2) we make two observations: Observation 3.9A: Suppose IAl + IBI + ICl < K , C x = t , 8, a and K 2 K,(T)is regular. If (i) tp,(A, C* U B) is F-isolated over C u B, (ii) tp*(B, C U A ) I- tp*(B, C*), (iii) tp,(B, C u A ) is F-isolated, then tp,(B, C* u A) is F-isolated over C u A.

c C*, F

= FE for

Proof of 3.9A. The idea appears in the proof of 3.3, so we leave it to the reader. Observation3.9B: Suppose q E Sm(INI)is regular; I,J are sets of sequen-

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255

ces realizing q, independent over [MI, 111 + I JI < K , M is FE-saturated ) regular, and A c liKl where K 2 K ( T is u I) forks over A then there is If for every E E J , tp(E, B c IMI, IBI < K such that tp,(U J, B U U I) t- tp*(U J, ] M I ) .

u

Proof of 3.9B. Choose B c 1M1, 1B1 < K such that t p * ( u (Iu J),IMI) does not fork over B, and q B is stationary, and let I* E ]MI be a maximal independent set over B of sequences realizing q B. Clearly t p * ( u J, B u I)is orthogonal to q, hence to tp*(U I*,B) hence 6) tp*(U J , B u 4 t tP*(U J , u I*). Now for any ~ , E I = + ( 6 ~ 1 M ) 6: realizes q B} tp(60n..-n6,,, B u U I*)is orthogonal to q, hence (ii) t p ( u J, B u I*)I- tp(U J, B u IJ I+). By claim 1.11 (W tP(U J , B u I + ) tP(U J , 1 q. Combining (i), (ii),and (iii) we are finiahed.

u

u

u

u u

Proof of 3.9(2).We can easily add to the conclusion of 3.8 that (letting IMol), tp(6,, IMol) are orthogonal or equal. So let = No) I = {6,: i < rs) = U,
w21)

u

u

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standing for C, C * , B , A , respectively) tp,(Ul, /MI U u I * ) is Fiisolated. By IV, 3.6 as N is FT-prime over WI U UI,it is also F;-prime over 1M1 U I U I*,but as t p ( u I, u (J I*)is F~-isolated,N is Pi-prime over [MI u U I*,hence tp(a, !MI u I*)is FR-isolated. By 1.4(1)I* is independent over WI, so as N , is F,”-prime over 1M1 uJ, by 3.2, I* - J is independent over (INal,!MI).As tp@, lN21) is orthogonal to each p j , it is orthogonal to tp*(U (I*- J), IN,[), hence tp*(U (I* - J), lN,l U a) does not fork over IN,l, hence over IMl, hence tp(7i, 12111 u U I*)does not fork over IMI u U J. So by IV, 4.3, as tp@, 1M1 u I*) is l$-isolllted also tp@,IMl u U J ) is qisolated. On the other hand, as J E N,, N , c-prime over 1H1 u a, also U a) is F$-isolated,so by 3.6(3) w@, is well-defined and tp,(J, equalto I J I s IIl = 1/31 s 1.

u u

u

1x1)

LEMMA 3.10: Let p be etationary or complete. (1) The type p is algebraic iff w(p) = 0. (2) If p is regular then w(p) = 1.

EXERCAYE 3.1: Prove that the converse of (2) fails. (Hint: Exeroise 4.11.)

Proof of 3.10. (1) Trivial by Definition 3.1. (2) Immediate by 3.3 and Definition 3.1.

LEMMA 3.11: (1) If {it, 6} ie independent over A then w@-6, A ) = w(a, A ) w(6, A ) (i.e.,if the right side is de$ned, then 80 is the left side and they are e q d . ) (2) w(a, A ) I; W(a-6, A ) s w(E, A ) + w(6,A u a).

+

Remurk. Notice that if {a, 6} is independent over A then w(6,A ) =

w(6,A u a).

Proqf. (1) We can msume A = IMl,Y is q-mturated, A = x,(T). Let N , be q-primary over [MI U and also over IMl u {Ei: i < w(a, A)}, and N , be E-primary over 1M1 u 6 and also over

lMl u {E*: w(a, A ) s i < w(a, A ) + w(6, A)}, where tp(E,, M ) is regulax and {Et: i < w(8, A)}, {Et: w(3, A ) I; i < w(a, A ) + w(6,A)} are independent sets over A. By 3.2 tp,(lN,I, IN,l) does not fork over IMI, so {Et: i < w(a, A ) + w(6, A)} is independent

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257

over ]MI. By 3.2 INl] U IN2[ is Q-constructible over IM1 u B u 6, so tp*[{ci: i < w(Z, A ) w(6, A ) } , 1M1 u ii; U 63 is F,Q-isolated. In a similar way tp(Z-6, [MI U { E f : i < w(ii;, A ) + w(6, A)} is F!-isolated. So we finish. (2) Left to the reader.

+

LEMMA 3.12: (1) If { E f : i < a} is independent over A , (and Ef 4 acl A ) tp(Ef,A u a)fork8 mw A (for i < a) then1.1 s w(a, A ) . (2) If N i 8 CF'??W over IMI U a, M it? F;-8durded, h = K ~ ( T ) , {El: i < a} independent over 1611, E, E IN1 - 1611 t 7 m1.1 5 w(a, A ) .

Proof. (1) We can clearly assume w(a, A) < m, i.e., defined and that A = 1611, M is R-saturated, A = ~~(2').Let N be Fff-pfimctryover U a, and {a,: i < p = w(a, A ) } be aa in Definition 3.1. Now dehe by induction on i < p sets u, E a such that tp*[{Z,: jE a{},1611 u {aj:j < i}] does not fork over 1611,and la - ufl s li l,j < i u, c u,. For i = 0, u, = a, and for fi&t 6, = nf
lUfl},

(MIu {E,:j < 4) u {a,:j s i}]

does not fork over 1H1 U {E,: j < t} U {Z,:j < i}, hence it does not fork over 1611.So clearly tp,[{Z,:j < lufl,j # 5}, IMI u {7ij:j I i}] doesnot fork over 1M1. So let uf+l= uf - (5). For any jE u,, we get tp[E,, ldll u {ii;,:j < p}] does not fork over ]MI.So by 3.2, tp(E,, ]MI u a) does not fork over ]HI, contradiction. So u,,= 0 hence 1.1 s p. (2) Follows by (1) and l.lO(2).

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CONCLUSION 3.13: Let I , A be given. ( 1 ) If J,, J , are muximal subsets of I independent over A , then I J,I s 2 {w(7i,A ) : 7i E J,}. ( 2 ) If p E Sm(A),J a maximal subset of (5: 5 E I , 5 realizes p} independent over A then dim(p, A , I ) s I JI s w ( p )dim@, A , I ) . (3) If p E Sm(A),w ( p ) = 1 then dim(p, A , I ) h true. Remark. See Definition 111, 4.5 for dim(p, A , I ) . Proof. ( 1 ) C l w l y i f for some B E J,, w(a, A ) = ao, the conclusion is trivial, eo we can asaume w(Zi,A ) < ao. E'iret asaume J , ia finite and let del J = { c l : l < n ) . Let F C = E , , " . . . - - cn-,, so by 3.11(1) k = w(F*,A)= 2 1 , n ~ ( F l , ~SO ) .if k < IJ.1, by 3.12(1) for some FEJ., tp(c,A U F*) does not fork over A, hence {Zi: i < n} u {Z} is independent over A , contradiction to the maximality of J,. Henoe So we finish for finite J , and for the U t e J,, i f A s 1 J,I is regular, IJ,I > IJ,I, then for some finite J ; _c J,, for A many ZEJ,, tp(5,A u UJ i ) forks over A , and we get a contradiction as above. ( 2 ) is immediate from (1)and the definition o f dim, (3) follows from (2). LEMMA 3.14: If 1M1 c IN], M , N are F&,,-durated then there is a cardinal p > 0 8w:h that for every d m a l set {a,: i < a} independent over lbl), of 8 ~ ' l M W e 8at € A??, w(%, 161))< 00, this e q ' l d h hQEds; p = 2ica 1H1). ~

(

$

9

Proof. Through the regular types and is left to the reader. THEOREM 3.16: ( 1 ) If T is muperstable, I independent over A , a 8q'&WK% t h for 80TPl.4 J E I , IJI 5 w(G, A ) , ( I - J ) u {Ti) i 8 independent over A . ( 2 ) POTeulperetablt?T , dim(p, A u Zi, I ) s dim(p, A , I ) I; dim(p, A u a, I ) + w(Z, A ) . Remark. Work for stable T too. Proof. ( 1 ) W e can assume w.1.o.g. that A = 1611, M is E-saturated A = K ~ ( TLet ) . I = {d: O < i < a),n = d ' ; by 3.6(2) there is a family

OH.

v, f 31

259

WEIGHTED DIMENSIONS AND SUPEBBTABILPTY

{Z,!,: k < w(Z,,A)} independent over A , tp(Z,!,,A) regular, and tp[Z,, 1M1 u {Z,k:k < w(2,A)}], tp,[{a:: k < w(a, A ) } ,A u @]are Fffisolated. By 3.2 {Z,!,:O < i < a, k < w ( 2 , A ) }is independent over A. By 1.14 there is u c {i:0 < i < a},1 . 1 s w(a0,A ) ,such that {ak:i $ u,k < w ( Z , A ) }is independent over (A u {a:: i = 0 and k < w(Z,,A)},A). By 3.2 the results &re immediate. (2) Left to the reader. Now we define upper and lower weights.

DEFINITION 3.3:Letp be stationary or complete.Define the upper weight p , upw(p), aa the supremum of the cardinals p such that there exists MyN , I satisfying the following: (1)M is q-saturated, K = K ~ T). ( (2) p has a stationmization which is a complete type over realized in N by a. x1u a. (3) N is @-primary model over 1 (4)I is a set of sequences realizing non-algebraic types over and I is independent over 1M1. (5) Ill = p. Define U P W ( ~ )= upw(8, A) = Upw(tp(tZ,A)).

1x1

1x1

DEFINITION 3.4:For p as above define the lower weight, low (p),as the maximal cardinal p such that there me M yN , I , satisfying: (1)M is E-saturated, K = K,(T). (2) tp@, 1M1) is a stationarization of p . (3)N is @-primary over [MI u 8. (4) I E IN1 is a set of sequences realizing regular types over 1H1,I independent over ]MI, and is a maximal such set. (6) III = p. If in addition, q is regular and stationary definelow&) to be the maximal p satisfying the above conditions plus B E 1 tp(c', lMl)is not orthogonal to q.

LEMMA 3.16: (1)In Dejnitiona 3.3 and 3.4, if th appropriate weigh4 is p then for any MyN , I , satisfying (1)-(4), aho 111 = p. (2)upw(p) = upw(r), low(p) = low@), low&) = low&) if p = tp(a, A), r = stp(a, A), or p , q are parallel. ( 3 ) If w ( p )is dejned, then upw(p) = w(p) = low(p) (so this alway8 blh for superstable T ) .Also low@) s upw(p) < K ( T ) . (4)low&) = 0 iff p is orthogonal to q (q reguhr, of course).

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( 6 ) If q, r are non-orthogonal,and regular then low&) = low,@). ( 6 ) If {q,: i < a} is a maximal family of stationary, regular,p a i r h e o r t h q d types not o r t h d to p thm Z,low,,(p) = low(p).-

( 7 ) upw(7in6, A ) 2 upw(7iyA) + upw(6, A u ii) when {ii, b} is independent over A . (8) lowq(Gn6,A) s lowq(7i,A) + lowq(6,A u 8). (9) low(Gn6, A ) 5 low(7iyA ) + low(6, A u 3). (10) If {a, 6) is independent over A , thm equality lrolds in ( 8 )and (9). (11) lowq(8,A) s lowq(Gn6,A) and similarly for low and upw.

Proof. We leave to the reader (1)-(7) and (lo),(11). (8) W.1.o.g. A = 1611,M is F,O-saturated, K = KJT),q is over 1611. Let N , be q-primary over IMI u ii, let 6' be such that tp(iZn6,1M1) = tp(an6', 1M1)and tp(6', N,) does not fork over 1M1 U 8. Let Na be F,O-primary over lNll u 6'; and let q,, Pa be the stationarization of q over IMI, lNll respectively (SO clearly q E q1 c qa). Let I , be B maximal set of sequences from INl] realizing qI which is independent over 1M1 when 1 = 1, and over lNll when 1 = 2. Clearly I , U I , is a set of sequences from IN21 realizing q1 which is independent over 1611.It is a maximal such set. For suppose E E "21, E realizes 42. By the maximality of I,, -tp(c', - lNll u l a ) forks over lNll hence over 1M1. By 1.11 letting J = {d: d EN,, realizes q,} tp(i3, IMI u J u l a ) forks over IMI. But for every a E J , tp(& IMI u I , ) forks over IMI, hence by 1.9 tp@,1M1 u I , u l a ) forks over 1M1. If E realizes Q, but not qa this holds trivially so I , U I , is indeed maximal. Now let N* E N , be F,Oprimary over 1611 u u 6'; and I be a maximal set of sequences from N* realizing qr which is independent over Then by 1.14 111 = 10~,,@~6, A), and by 1.14, [ I ] I; ] I , U l a 1 = Il,l + IIal; and this proves our atmrtion. (9) Follows by (6) and (8).

a

w.

CLAIM 3.16A : ( I ) In DeJinition 3.4 there is a maximum. (2) Assume M is Ff-saturated, K = cf(K) 2 K,(T),IAI < K , p is regular (and over M ) I is a maximal family satisfying: (i) for each @ € I ,tp(6,M) is regular not orthogonal to p , (ii) for each @EI , tp(& M U A ) fork over M, (iii) I is independent over M. Then (a) 14 = low,(fp*(A,M)), (b) tp(U I,M U A ) is Ff-isolated, (c) if N is Ff-constructible over M U A Ff-primary over M U uI , then tp*(A,N) is orthogonal to p .

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WEIUHTED DIMENSIONS AND SUPEBEITABILITY

261

( 3 ) Suppose for p ~ { l , 2 } K, , M , p , A,, I,, N, are as in ( 2 ) and {A,,A,} is independent over M and N is Ff-primary over Nl U N,. Then K , M , p , A , U A,, I , U I,, N are as in (2). Remark. We can replace p by a regular family 9 (see Definition 4.5: Proof. Straightforward.

QUESTION 3.2: Is the independence aasumption necessary in 3.16(7) ? When equality holds ?

QUESTION 3.3: In Definition 3.3 is the mpremum obtained? QUESTION 3.4; In Question 3.3, prove the answer is positive when upw(8,A) =

KO.

PROBLEM 3.5: Try to generalize the results of this section on superstable theories to stable T for which K ~T )~ = (KO. DEFINITION 3.5: (A) Define lgw(p)just as low ( p )but p is minimal and (4) for B E I , in the F,”-primary model over ]MI u B there is no independent set with more than one element, and (6) I is a maximal set satisfying (a), of sequences 5 E IN I, 5 # 1M1. (B) Define ugw(p) just as lgw(p), omitting (6), and (3) N is c-primary over uI . If there are no such M , N , I , ugw(p) = 00.

LEMMA 3.l7: (1) If tchd(T)= No,t h Igw(p) < .KO. ( 2 ) We get tlce aame IIl for eveq M ,8, N in De$nitim ( 3 ) If {a, 9 ie independent over A then: ugw(8-6, A ) s ugw(B, A) (4) If

T ie -~&bZe

3.6.

+ ugw(6, A).

t h ~ ( p=) ugw(p) = Igw(p).

(6) lgW(P) 5 Ugw(P).

Prmf. We leave it as an exerck to the reader. QUESTION 3.6: Try to &pt the theorems and definitions of Seotion 2 (e.g., s ~sJ, , Section 3 (e.g., w ( p ) )and also Section 1 to other kinds of F, mainly F& (replace, if necessary “ T superstable” by “ T totally transcendental” or add it) and also FL,,(for countable T ) . (Seebelow.)

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MORE ON TYPES AND SATURATED MODELS

[OH.

v, f 3

THEOREM 3.18: Suppose

tp = tp(Z,E) E p E Bm(A);then the following 8di8fy: ( 1 ) * ( 2 )0 ( 3 ) 0 ( 4 )0 ( 6 ) and if IAI + IT1 5 so then &o ( 6 )0 (6);and if T is totdly tramthen also ( 7 )o (6). ( 1 ) The pair ( p , t p ) is not regular. ( 2 ) There are tpl(E; 4),#,(Z; at) (for i < k ) , B,(Z; go,. . .,&i), a,) (i < 1 5 k ) and$nite A, (i < k ) , such that, letting a, = Rm(pl,A,, 2 ) (i < k ) for any stationarimtion p1 of p, the fol2owing conditions hold: (i) tpl(Z; El) E p, n ( Z , E l ) I- ~ ( 3El.; at) I- tp(Z; a) for i < k. (ii) (iii) tpl(Z; a,) 1Vi< I (3g0, .,gn(i,)CAjsn(i) #i.,(gj;iZ,) A e:(z;go, . ., &(ip 411 v V l r t e k $i(Z; Zi(). (iv) P [ $ , ( Z ; Ti,), A,, 21 < a, when i < k. (v) For any i < I , and Eo, . . .,

&$~~rta

.

( 3 ) The same aa (2) but n(i) = 0, #i = $, 8, = 8, A, = A for i < 1; and for 1 5 i < k , A, = AO,$, = $; and 8, i s a amcatenation of sequence8 satisfying t p ( f ; E ) V3 = E for i < k. ( 4 ) I f A G !MI, tp(6, 1 ~ 1 i)s a statiolucm'mtion of p, 1M1 u 6 c IN1 ~ E ) but tp(6', 1M 1 ) i s not a then for some 6' E INI, 6' 4 I M I , 6' ~ a t i s $ etp(3; stationarization of p. (6) There are K > IAl + ITI, A c IMI, M a c-saturated model, N E-primary over IM) u 6, tp(6, ) M I ) is a stationarization of p, and 6'E INI, 6' $1 M I , 6' satisfies t p ( ~ z), ; but tp(6', ! M I ) i s not a stationarization of p. (6) There are a countabie M and 6 such tluct tp(6, I M I ) is a idatimarization of p, A r IMI, and for every N , for w h i d lMl u 6 c INl,'smne 61 E IN(, 25' $ lMl eatis* q ( Z ; a) but tp(6', ( M I )i e nd a stationarizcction of P. (7)like (6),for not nmeawri2y countable M . Remark. By III,4.2, for all stationarizationsp, ofp, *(ply A,, 2) is the same number.

Procf. (2) ==(4) Let A , M ,N , b be aa in (4). By 11,2.12(2)the properties of zi, (i c k) me expresaable by hitely many formulas, with parameters from El. Hence we oan assume a, E IMl;El E IMI as A c 1M1. As q = tp(6, 1M1)is a stationariaation of p , P ( q , d , , 2) = a, (for

OH.

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WEIGHTED DIMENSIONS AND SUPERSTABILITY

263

j c k), hence by (2)(iv) +(Z; 4) E p when I I i < k; and clearly by (2)(i)cpl(Z; a,) E q. So by (2)(iii)necessarily for some i < I,

- .-

N C (380,

gn(d [A $&& a,)

A

jsn

el@; go, . . .,tin(,),al)].

Hence there are a,, . . .,an(,)E IN1 such that N t $@,; a,] (forj 5 n ( i ) ) and N C 4(6; a,, . . .,a,,(,);%). By (2)(v)Rm[UZ; a,,. . ., a,), A,, 21 c a,, hence fI,(s; a,, .. .,d,,(,),a,) 4 p; but p = tp(6, 1611) so for some j I n ( i ) 4 [MI. But as N C $,[a,; a,], by (2)(ii) N C cp[Z,; a]. So a, satisfiescp(3,a), and as $,@, 4) E tp(Z,, [ M I ) ,clearly Rm[tp(Zj, [ M I ) A,, , 21 I Rm[$l(Z; a,),A,, 21 < al hence tp(z,, IMI) is not a stationarization of p. So a, satisfies the requirements on ii', hence (4) holds. (4) + (6) Trivial. (6) + (3) Let {p,: i c I 2ITI}be a list of the stationarizations of p inSm(lYI)andp, = tp(6, 1611).Choose $,@; a,) E tp(61, 1611)such that +,(E, Ti,) E p o and we can assume that Go E u I u a (by 1.11) where I = {a E !MI:Ccp[a;Z]}, and $,(E, a,) I- cp(Z; a). SO by 1.11tp(6', A u U I U 6) is F$-isolated, hence it forks over A u I hence tp(6, A u U I u 61) forks over A u I. So we can find Zo E A u U I c IMI and 8 such that C8[6, 6', a,] and 8(3; 61, a,) forks over A u U I. Hence by 111,4.2 we can find a finite d such that

zj

u

u

m p , u {em ii', go)}, A , 21 < RW,, A , 2). As by 1.11tp(6,A u UI)= p,, we can replacep, by tp(6, A u U I). By replacing 8(E; 61,ao) by 8(Z; ii', a,) A $ l ( ~al) , where $ l ( E ; al) E tP@, A u I),

u

~ y p u, {e(Z; 61,a o ) } ~21, =

iq,e(Z; P,a)),A , 21

~ m [ { q ( ~ ;

we can msume that Rm[8(Z;6', a,), A , 21 < Rm(po,A, 2). Similarly by 11,2.12,3we can assume that for every 6",a",R"[B(z;6",&), A , 21 < Rrn(p,,d, 2). So clearly for every 6* realizing p , there is 6** satisfying $,@; a,) such that !+9[6*;6**,a,]. Clearly for every i < there are a,, 7i' E [MI such that p,, $,(Z; a,), 8(Z; 9, a') satisfy the parallel assertion. Hence for every 6* satisfying cp(Z; 13) 6* satisfies some formula $, -+h E n,<,p,, or for some i c a C (3g)[$,(g;8,)A 8(6*; 8, a,)]. By a compactness argument we get i(n) < a, (n c I ) and -,$,,(P;@)E 0,< 8 p , ( I s n c k) such that: Q(Z, a) I-

V

tSn
$n(Z,

a:)

A

V (~Y)[$O(@,zf(n)) A n
W ;8, @?I*

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[CH.

V, 5 3

Clearly for each n, 1 s n < k, p u {#,,(Z;a:)) forks over A, hence there is rp1(3;El) ~p such that pl(Z; c,) t p(Z; 8) k d pl(Z, El) A Sn(Z;a:) forks over A. By 111, 4.2 there is a finite AO such t h t for every 6" realizing p, and n < 1 P[stp(6", A) u {pl(%,4) A #,,(f; a:)}, AO, 21 < ~ [ s t ~ ( 6A), " ,AO,21. By 111, Definition 2.1 there is E(z, #, a*) E PE(A) such that for any 6" realizing p Rm[stp(6",A), do, 21 = P[E(3, p,a*), do, 21. Now by notational changes we can assume B, = Gi, and

Now (3) is clear (by 1.11 it is easy to get iZi E

(3) 3 (2) Trivial.

U I u E).

(1) 3 (6) Immediate by the definition of a regular pair. (4) 3 (6) (when !A( + ( T ( IKO) Immediate. (6) 3 (5) (when I A I + I TI I No) For each +b = #(Z, 3) such that E JM1 (e.g., is inconsistent) and p u {#(Z;a)} forks over A, k(33)(rp A #) let r* = {rp(3;E ) , #(Z; a)}u {Z # a: a E I M I} and r be the set of such r,'s. So r is countable and there is no model N, IMI u 6 c IN1 omitting all types in r. Hence by IVY6.3(1) there is r, E r,8, E 1MI and a formula rpl(Z; a,, 6) such that C(~Z)~,(Z; a,, 6) and p , ( ~a,, ; 6) t- r,. So choose a I TI +-saturated model M,, M E M,, such that tp(6, 1M, 1) does not fork over (M1. Ae clearly pl(Z; iZl, 6) I- I # J for every Z E IM,I, in the FAT)-prime model over I MII u 6, some b' 1Mll will satisfs p(l, 5) but tp(6, '1 MI]) will not be a stcttionmhation of p. Hence (6) holds. (4) =- (7), (7) e=(6) (for T totally traneoendenfal) Left to the reader. CONCLUSION 3.19: (1) If T is totally-tramoendental,M E N, M # N then for 80W b E IN1 - 1611, tp(b, 1611)is regular. (2) If M N, b E IN1 IMI, V(X,5) ~ t p ( bIMI), , R[97(~,E), L,KO1 = R[tp(b, I MI), L, Ho] < ao and for no b, E IN1 - (MI which adis* F ( x , ~ ) R[tp(bl, lMI), No1 < R[tp(b, IMI), Ly KO] t h [tp(byIMl), cp(x, E)] is regular.

=

-

Proof. (1) By (2). (2) Use 3.18 for A = IMl, p = tp(b, ]MI). Clearly condition (4) fails, by our hypothesis, hence (1) fails, but not (1) is our conclusion. EXERCIBE 3.7: Suppose a,, ti, are sequences, M is K-saturated.

UE.

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265

(1) If tp(7il, 1M1 U Zz) is c-isolated then tp(Z1, IN1 u a,) is F&isolated [i.e., for every p there is &(Z; EJ E tp(zi,, Ill21 u 4 )such that #*(% C@) tP@@lY1 4 u aZ)l* (2) The converse of (1) holds when K 2 K ( T). (3) Suppose there are N ,a,, a,, #@(Z;ZQ) as above, and tp(ai;,INI) is parallel to tp(Z,, [MI),N is F&,-saturated. Then we can find a:, E i such that tp(Zi;, I N [ ) = tp(si;, I N ] ) , and #@@, z;) E tP(Zi;, IN1 u a;), #@(% )Z; I- tP&;, 1" u a;).

EXERCIf3E 3.8: (1) Suppose I soJ ; then

u I))

w(Av(&

2 w(Av(J,

u

J)),

and for superstable T if they are finite and equal then J < , I . (Hint: See 3.9(2),and its proof.) (2) For superstable T I I J iff for every regular q, low,(Av(I, I ) ) 2low,(Av(J, J ) ) ,and I S wJ iff for every regular q,

u

u

low,(Av(I, UI))< 1

=- low,(Av( J, UJ ) ) < 1. (3) The relation I = J = d d I s8J A J s o I is an equivalence relation, and for superstable T, for each I, the number of J/= , I s8J is finite (in faot s 2 w ( A v ( ' 9 . A similar result holds for sw.

In (I series of exercises, we now check whether Section 3 Exercise 3.8(2) generalizes to P"?-primeneee.

EXERCIf3E 3.9: If M is K-compact, 1M1 5 A, B is Ff,-constructible over A, 6 E B then tp(6, A) forks over ]MI. (Compare with l.lO(2). What about P i ? ) [Hint: It sufficesto prove that if tp(6, A) does not fork over IN], tp(c, A ) is Ft,-isolated, then tp(6, A u {c}) does not fork for this it suffices to prove tp(c, A) k tp(c, A u 6). Suppose over ]MI; . not, then for some p c tp(c, A) lpl < K , p k tp(c, A), p is closed under conjunotions and for some 8 and Z E A, p u {O(z, 6, a)'} is consistent for t = 0, 1. Let Y = {(3z)(p A e(z, g, a)) A (3z)(p A 4 ( z , g, a): 'PEP} so Y E tp(6, A), 1.1 .c K . As tp(6, A) does not fork over IMI,it is finitely satisfiable in it. Hence by the definability lemma and the Kcompactness of My some 6' E IN]realizes it. So p u {e(z, F,7i)t} is consistent for t = 0, 1, oontradiction.] DEFINITION 3.6:The pair (p, p) (p ~p €Brn(A)for some A) is strongly regular if it fails to satisfy conditions (2)-(6) of 3.18. We d l the complete type p strongly regular if for some Q (p, p) is strongly regular.

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v, 9 3

EXERCIBE 3.10: Show that the pair (p,’p) is strongly regular, iff q~ ~p €Brn(A)for some m, A, and every stationary and complete q containing ‘p is orthogonal to p or is a stationarization of p . EXERCIBE 3.11: Suppose T is totally transcendental M c N, M # N. Show that for some c E IN1 - IMI, tp(c, 1M1) is strongly regular. . EXERCIBE 3.12: If (~(33;a), p) ( p E Sm(A)) is strongly regular, the dependence relation on ‘p(C, a) “tp(ii, A u iil u . - u an) does fork over A, or do not extend p ” satisfies the axioms of a nice dependency E relation (Def. A P 3.4). Prove a similar assertion for several (cpa,p,)p, Sm(a)(A) strongly regular, pairwise not orthogonal ; together.

-

EXERCIBE 3.13: Suppose N is F&,-primaryover [MI U ii, tp(iZ, lMl) is strongly regular, 6 E INl, 6 4 1M1. Then N is F&o-primaryover u 6. (Hint: See 3.3 and 3.18(7).) EXERCIBE 3.14: For totally transcendental T, show that in 3.1 we can assume tp(iZi,,lMll) is strongly regular; and does not fork over or is orthogonal to ] M I .We can replace Fft by Fco. (Hint: Use Exercise 3.11 and Exercise 3.9. Or see XI, $3.) EXERCIBE 3.15: Suppose T is totally transcendental, tp(iZ, 1M1) is x1u iz) is orthogonal to lMl. Show that not strongly regular, and tp(6, 1 necessarily tp(6, u 7i) is FM,-isolated. Remark. It is Fi,-isolated by the proof of 3.9. In the example one equivdenoe relations and Nodisjoint one place predicates suffice. EXERCIBE 3.16: Suppose M c N, ii E IN!,iz $Ibil, p E B ~ ( ~ M is ~ ) strongly regular and not orthogonal to tp(ii, 1M1).Then p is realized in N . (Hint: Use Exercise 3.12. This is similar to 1.12 and see.) EXERCI8E 3.17: Show that in 3.15(1) we cannot demand I - J is independent over (A u UJ u a, A). (Hint: Use vector spaces over a finite field, I a linearly independent set, A = 0, J E I finite, a = ZmJ x-) EXERCI8E 3.18: If 6 E acl(A u ii) then stp(iz, A) s8stp(6, A) hence w(6, A) s w(iz, A). Check the parallel wertions for other weights.

OH.

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267

EXERCI8E 3.19: Suppose K 2 K J T ) ,M is e-saturated, N e-prime over IMI u U I , I a set of sequences realizing over IMI regular types, and J E IN1 a maximal set independent over !MI,of sequencesrealizing regular types. Then N is FE-prime over 1611 u U J . (Hint: over See the proof of 3.9(2).)

THEOREM 4.1 : Assunte: (i) K 2 K ( T )is re@hr. (ii) M E N are F,O-saturatecE,A c 1611,IAl < K . (iii) p ie a stationary regular type. (iv) El E N (1 < k < No)are ewh tlrat p l = tp(El, 1MI) is regular and not orthogonal to p . (v) p , does not fork over A (1 < k). (Vi) p l r A &? stationary (I < k). (vii) E v e y 6~ IN1 - IMI which r d i z e a p , r A r d i z e a p , (1 < k). (viii) I G is a maximal independent (over !MI) set of sequences which realize &er [MI regular t y p a not ortirogrmal to p . (ix) E ie algebraic over IMI u U l < k 8,. Then tp@,1M1 U I)is F$isolated, and even Q-bolated.

u

DEFINITION 4.1 : (1) A stationary type q is sem-regular if it is not algebraic and there me K , M , N , A , k, 5, 5, &B in Theorem 4.1 and tp(i5, lMl) is parallel to q. Clearly we can aasume there is an n s k such that El depends on {Z,: i < I ) over [MI iff 1 2 n ; and 1 < n =. p , = po, and I = {E,: i < n}. (2) If in addition the types tp(E,, IMI) are minimal, q is called semiminimal, and then we can wsume n = k. Clearly every stationary regular (minimal) type is semi-regular (minimal). Proof of Theorem 4.1. By 111, 3.6 there is J E I, I JI < K ( T )such that tp(E’,IMI UUI)doesnotforkoverlNl u UJ,whereE’ =Eon...-- c k - l ; let J = {a,: i < 18). Choose B E 1611, IBI < K , A c B such that tp,(E U U l c kEl u UJ, IMI) does not fork over B and its restriction to B is stationary (possible by 111, 2.9(1) and 111, 3.2), and E is algebraic over B u 5’. By the maximality of I , for every I < k , tp(El, ]MI u I u U,<,E,) forks over 1611,hence tp(El, lMl u U J u UfC1 a,) forks over 1611,hence

268

MOEE ON TYPES AND SATITBATED MOD=

[OH.

v, 8 4

it is orthogonal to p t (as pt is regular). As non-orthogonality is an equivalence relation among the stationary regular types (by 1.13(1)) it is orthogonal top. So by 1.6(1) tp(Eon. IMI u UJ) is orthogonal to p , so tp(E‘, M u UJ) is orthogonal to p, hence by 1.4(1) to tp,(I - J, u J).SOby 1.2(1): (1) tp(8‘, IMI u U J) t tp(a‘, ]MI u U I).Let I, = {&: i < y } be a maximal set of sequences in 1M1, each realizing some p t t B, which is independent over B. By the Definition of B, tp(Z’, ]MI u UJ)does not fork over B u UJ,I, is independent over B u UJ and tp(8’, B u UJ) is also orthogonal to p t 1 B. As the latter is stationary: (2) tp(E’, B u U J) I- tp(6, B u UJ u U I,). Now suppose a E C = U {a: B E 1M1, si realizes p t B, 1 < k} and we shall prove that tp(E’, B u UJ u U I,), tp(& B U UJ u U I,) are weakly orthog) such that tp(& B u U I,) does not onal. Let J , c I,, IJ,I < K ( T be fork over B u UJ,, and we can assume a = Jon,. . .,-a,, a, E [ M I , a, realizes pipts 1 B. If the above mentioned types are not weakly orthogonal there are JI,, J , E JI, c I,, IJI,I < K ( T )and a sequence a‘ = . in N such that

u

1x1

tp(a’, B u

u u Ju

JI,)= tp(Z, B U

u u J

U

JI,)

but tp(c’-d,B u U J U UJ;) # tp(c’-d’,B U U J U UJ;) (because K > IBI + IJI + IJ,I and N is FE-saturated). By the maximality of I,, tp(& B u UI,) forks over B, hence tp(& B u UJ,) = tp(4, B u UJ,) forks over B, hence tp(&, B u J & )forks over B, hence tp(&, 12111) # pi(1,.By hypothesis this implies a; E [MI, hence 2’ E 1M1. Now by the choice of 3, for some 6 E u J, tp(6-i?, B U (J I, u a u a) forks over B U J,, remember tp(6^c’, B)is stationary, hence tp(b^E’, WI) forks over B. But by the choice of B, tp(6“ F’, WI) does not fork over B, contradiction. So for every 2~C,

u

u

tp(Z’, B u

u

I, u

U J ) t tp(Z’, B u u I, u uJ u 2)

hence (3) tp(E‘, B u U I , u UJ) t tp(8, B U UJ u C). Now let C, = realizes p t 1 B, 1 < k), so by 1.11 for every 6 E tp(b, B u C) k tp(6, B u C,) hence (noting 8’ u J E C,)

1x1,

U (7i: B E N , ii

tp&’ u J, B u C) t tp*(d u J, 1H1) (as this holds for every 6). Thus (4) tp($, B u UJ u C) t tp(Z’, [MI u UJ ) .

CH.

v, Q 41

SEMI-REGULAR AND SEMI-MINIMAL TYPES

Combining (2), (3), (4) and (1) we see that tp(", B u (J J) I- tp(Z',

1"

As E is algebraic over B u Z, and IBI, IJI <

u K,

u

269

I).

we finish.

LEMMA 4.2: For every semi-regular type q, there is a stationary regular type p such thal: (1) p , q are not orthogonal. (2) For any type r, r is orthogonal to p iff r is orthogonal to q. (3) If I , J are in$nite indiscernible sets based on q, p r a p . , then I s,J & I . Proof. Easy. (For (3) use Theorem 4.1.) CONCLUSION 4.3: The relation of being non-orthogonal, b an equivalence reWion among the semi-regular types. Proof. By the preceding lemma and the fact that non-orthogonality among the stationary regular types is an equivalence relation. Notation. Hence if p is stationary and regular, q is semi-regular and they axe not orthogonal, we shall define low&) as low#).

LEMMA 4.4: If tp(Z,, A) are minim1 and pipwise nd orthogod, then tp(Eon. . ^Zk- A) is semi-minimal. Prmf. Easy (as here Condition (vii) in Theorem 4.1 holds trivially.)

Proof. We use the notation of 4.1, end assume in addition that N ie q-primary over IMI U U J . Then by 1.14, IJI 5 k < No. Hence by 3.9 w(Z, lMl)s k < X,.

LEMMA 4.6 : (1)Let 6~acl(A U c),6$ acl.4. Ifstp(r,A) is semi-regular [semi-minimal] then stp(6,A ) is semi-regular [semi-minimal] too, etp(6, A ) ie not ort?mgonal to stp(5, A ) ; and w(6,A ) s W ( Z , A ) , and lowJ6, A ) s low,@, A) for any p .

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MORE ON TYPES AND SATURATED MODELLI

(2) If {a,

[OH.

v, 5 4

6) ie indepdd over A

and stp(Z, A), stp(8, A) are eemiregular and not orthogonal then stp(6-Z, A) ie semi-regular, not orthogonal to either of them, and w(6-Z, A ) = w(E,A ) w(6, A).Bimilarly for eemi-minirnal.

+

Proof.W.1.o.g. we can assume, in both parts, that A = IMl,Y is Esaturated, where K = K,(T)and then the proof is easy (using Exercise 4.8.).

THEOREM 4.7: Buppoee p = stp(Z, A) i s eemi-regular and A E B. If stp@,B) f o r b over A then IowJa, B ) < w(Z, A ) = w ( p ) .

Proof.W.1.o.g. we can assume A = 1M(, Y is R-saturated where K = T).By 4.6, as p is semi-regular there am El (1 < k) and N such that: for no 1 < k, there is 5; E INI, Z; + IMI,realizing tp(Z,, Cb(tp(Zon.- .n Zk-,, /MI)) but not tp(Z,, IMI); tp(El, IMl)is regular, not orthogonal to p , {El: 1 < ko = w(ii, 12111)) is independent over 1Y1,N is %-primary over 1M1 u ii and also over 12111 u Ulcko Z,; E IN1 for 1 < k, and ii is K,(

algebraic over 12111 u Ulek Z,. We can assume that tp(Z,, B u Ute, Zt) forks over IMI iff 1 2 k,; so 0 I k, 5 k, I k. Notice that for 1 2 k,, tp(Z,, B u U1<,4) is orthogonal to p , as tp(Zl, 12111) is regular, not orthogonal top. As ii is algebraic over B u c k Z,, clearly low,(i~,A) = w(ii,A) = ko and low,,(Z, B) I low,(Eon- - .nEk-l, B); and by 3.16(8) it is easy to check that the latter is k,. Hence it suffices to prove k, < k,; soasaumek, = ko.By4.1forsomeCE 1M1,Icl < K,Z~E~C~(CUU,,,Z,), stp(80n' "hZkk_ly u U { < k o 4) btp(Zon** ' h Z k - l , u u { < k o Z{). w e can assume that tp(Z,O^. ~ ~ n E k - , y 12111) does not fork over C. As tp(Z,O^. 'hZko-l, B ) does not fork over 1M1,it does not fork over C. Thus clearly (by the q-eaturativity of Y)stp(Zon. -hZkk-l, C uUi
u,

-

-

-

anZk-,,

CONCLUSION 4.8: A type p ie regular iff it ia eemi-replar, and

w(a)

1.

THEOREM 4.9: (1) 8-e for e v q b E By stp(b, A) ie eemireplar, not 0rthogon.d to the &a&nmy regular type p . Then there ie a cardinal p = low,,(B, A) euCi, th& for every I = {5{: i < a}, U I = B, the foUozoins hda8.- I( = G low,(h, A u U,<' 5,).

OH.

v, 5 41

SEMI-REUULILB AND SEMI--

TYPlDS

271

Remark. We define low,(B, A) by the natural extension of Definition 3.4. (2) If p i 8 r@r a d 8 ~ ~ tp*(B, r y A) h orthogod to p , t h for every ii, lowp(si,A) s low,(ii, B). (3) If for every 6 E I, stp(6, A) i8 regular, not orthogonal to p , I = {6,: i < a} then z,
ujei

Prmf. (1)Choosearegular~2 ~,(Z’),letM be F:-etlturated, tp,(B, IMI) does not fork over A E 1611, and let N be e-prime over !MIu B. W.1.o.g. p ~ f l ~ ( I M and l ) , let J E IN1 be a maximal set of sequences realizing p , independent over 1M1.So IJI = lowp(B,A) by the natural extension of Definition 3.4, and it clearly depends on tp,(B, A) alone lowP(6{,A u Uj<,Ej). (and not on MyN, J).Let p = Now we prove IJI s p (this does not depend on the assumption We define by induction “stp(b, A) is semi-regular for every b E By’). on i s a models N,: Nd = M , N , is e-prime over (INjI U 6j),but 6j. we choose it such that tp,(B, N , ) does not fork over A u Ujcf Now let J , c lN,+ll be a maximal set, independent over INil, of sequences realizing the stationwization of p over N,. Clearly IJ,I = lowp(6f,A u Uj<,5j), and U j e u J jE 1N.I is a maximal set, independent over lNol = 1611, of sequences realizing p (see 1.16(3)). Clearly there is an elementary embedding f of N into Nu over M yhence

z,
ujcL

IJI

5

Z < U

IJd

= P.

Now let us prove p s IJI, thus finishing. As for each b E B, tp(b ,]MI) is semi-regular, there is a finite set Jb c N of elements realizing over lMl regular (complete) types not orthogonal to p , such that b~ ad(lbll U Jb). SO B C aCl(lbll U J*) where J* = U k B J b . Now we define by induction on i < a, J: c Jb,, such that J: is a minimal set (by E), independent over (1M1 u UJP, 1M1) satisfying 6,E acl((M1U UJ:+J where Jf = {E E J * :tp(E, ldll u J:) fork over IMI}, J: = u c e j J : E J * . Now IJI 2 IJtI = IJ:l, so it suffices to prove

u

u

u

zl
272

MORE ON TYPES AND SATURATED MODELS

[CH.

v, 8 4

third relation by (2) below, aa for eachj, and c E Jf+l - Jf,tp(c, u uJi U UEG1bt)forks over WI, in fact over /MI U uJi U (for c E J r as Jf E Jb,(and we could choose such J,,) otherwise by 1.9, hence is orthogonal to p, and use 1.6.) (2) We can mume A = lMl,M F$saturatd, K = K,(T),and B s N, N F:-prime over M U B and tp(n, does not fork over B. Let Nl be Fg-saturated, Q-constructible over 1M1 u 8 and also over IN1 u 7i (see IV, 3.17) and N2Q-prime over INl] u INI,hence also over IN1 u 8. We can msume p ~ f P ( l M land ) J s INl! is a maximal set, independent over 1M1 of sequences realizing p. Clearly 1JI = low,(Zi, A). Now by 1.4 tp*(U J , IMI),tp,(B, IMI)are orthogonal and so by 3.2 tp,(u J, ]MI),tp,(lNI, 1M1) are orthogonal. Hence J is independent over (INI, IMI) hence low,(8, B) = low,(7i, N) 2 IJI = low,(8, A). Note also Av(J, U J ) 1Av(J, IN1 u J), and Av(J, IMl u J ) tAv(J, IN&. (3) Eaey by 1.14.

m)

LEMMA 4.10: If p = stp(8, A) i8 semi-minimal, A s B, t h stp(8, B ) is either algebraic, or semi-minimal and not orthqonal to p .

Proof. W.1.o.g. A

=

1M1,M is Q-saturated,

K

=

K J T ) .So there are

Zl (1 < k ) such that tp(Zl, IMI) is minimal not orthogonal to p, and 8 E

acl(lM1 u u I < k 4). By renaming we can aasume that tp(zl, B u U1
-

-

THEOREM 4.11 (in P): (1) If 8 $ ad A, T is superstable t h n there is an e q u i w reladion 1 which is almost Over A such thud stp(8/E, A ) is semi-regular (so 7i/E $ acl A). (2) If tp@,A ) is not o r t h g d to some minimal type q, then for 8 m equialence relation E which is almoet Over A , stp(a/E,A) is semiminimal, and not orthogonal to q.

Remark. E is a formula E ( f ,@,

6),1(f)= I(@ ) , 6 E A .

Proof. Choose a regular K > IAI + K(T),and a E-saturated model M such that A E M , tp(8, lM() does not fork over A, and in part (2) the

CH.

v, 6 41

SEMI-REGULAR AND SEMI-MINIMILL TYPES

273

minimal type q is over lN1.Lk N be q-primary over IN1 u a. For part (1) choose c E IN1 - IMI such that R(p, L, 00) is minimal where p = tp(c, IMI)and for part (2) so that p is a minimal type not orthogonal to stp(i3,A). In both cases p is regular (by 3.6, 1.18(4)). Now choose B E 1M1,A c B, 1B1 < K J T )+ IAI such that tp(7in(c), lMl) does not fork over B, tp(Si-(c), B) is stationary, R[tp(c, B), L, co] = R[tp(c, p q , L, 001. Now we can find elementary mappingsf,(i < K ) such that Domf, = B, Range(f,) IMI and StP,(h(B), A ) = stp*(B, A ) and StP,Lfi(B), U ,< ,f,(B )]does not fork over A, andf, = the identity. Let B, = f r ( B ) andp, be the stationarization off,(p 1 B) over 1M1, sopo = p . Clearly aa stp(8, A) is not orthogonal to p 1 B, it is not orthogonal to each pi, hence by 3.4 the p,’s are pairwise not orthogonal. If b E IN1 - IN1 realizes p , B,, it necessarily realizes p , (in part (1) because of the minimality of the rank, in part (2) by the minimality of p , B,). Clearly it suffices to prove: +

=

r

CLAIM 4.12 (in Eeq):Let a , c , A , M , N , f i , B , , p , be , ~ as above (the minimality and superstability conditions are not necessary). Then there is an equivalence relation E almost over A so that s t p ( a / E , A ) is semi-regular and not orthogonal to p , and if the p0s are minimal, it is semi-minimal.

Proof. By the choice of B = B,, tp(c, B, u a) forks over B, so there is a 6 E B,, such that btp[c, a, 61,and {tp(z, a, 6)) forks over B. Let C, = {d: d E N,d realizes p , 1 B,};clearly Cb[stp(Si, B u C,)] is algebraic over B U zi (see 1.11, Cb-canonical base, see 111, 6) and there are 0 and B E Cb[stp(a, B, u C,)] E acl(B, u C,) n acl(B, u a) such that for d E C,, btp[d, a, 61 iff be[& d, 61. Let I@?,if, 6,) (6, E B,) be an algebraic formula such that C+[E, a, 6,] and $(Z, a, 6,) k tp(5, B, u a). Now let E(Z, 8) say that for infinitely many i < K, (e(E)[$(%, Z,f,(6,)) = $(Z, g,f,(E,))]. By 111, 2.6 B(Z,g) is almost over A. If #(Z, a,f,(g0))is realizedbyE!(Z < k < Ko),then8/1iaalgebraicoverU,,,(B,u ( J I < k B ! ) , because E(z, a) iffforinfinitelymanyi’s(W)[$(Z,~,f,(6,)) = V,J = B!]. Hence tp(a/E, IMI)is semi-regular (and if the p,’s are minimal, semiminimal) provided we show it is not algebraic (as clearly 13:E IN1 as it is algebraic over B, u C,).But for big enough n, Z/E is algebraic over U,<,,(B, u {a!: Z < k}). Clearly tp,({Zi: 1 < k, i < n}, 1M1)doe8 not fork over U,<,,B, (as 17:is algebraic over B, u a), and it is not algebraic by the choice of ‘p. But clearly tp({i$: Z < k,i < n}, IMl u a / q )forks over

MORE ON TYPES AND SATUBATED MODELS

274

[CH.

v, 3 4

IMI,thus 8/E $ M.Clearly 8/E E dcl[(clcl A) u7i] hence 8/E E acl(A u8). Of course, when p is minimal, stp(@/E,A ) is semi-minimal. More < z, < I$) E fully, tp(a/E, M) is semi-regular and @ / Ec dcl (M U dcl(M U C,. Hence, tp(a/E, A ) is semi-regular as tp(a/E, M ) does not fork over A and tp(@/E,M) is semi-regular.

ui

ul
CONCLUSION 4.13 (P): 8wppo8e F eatieJka Axifma (X.l) a d (XI.19. (1) If T ie &wpr8tabk,for every COnetmCCtiOll d there ie a conetruetion d' such thd d, d' have the earn dumain and range, and tp(a;, di)ie algebraic or eemi-regular. ( 2 ) Forunidimensionul T ,thesameconclusionwithtp(a,',A ; )algebraic or eemi-minimal.

Proqf. Immediate.

DEFINITION4.2: The m-formula ~ ( b7i), is semi-weakly-minimal if for some weakly-minimal +(q,,?), every 6~cp(6, a) belongs to acl(F U

+VL f3). THEOREM 4.14 (in P): 8uppose tp(8, A) ie not o r t h a g d to e r n e minimd type q to which e m weakly-minimdform& Q b e 4 8 . Thenfor e m equ~valencereh4bn E which ie over A ,7i/E8ati8Jka 8ome semi-weaklyminimal formukt # which is over A ,but 8 / E $ acl A. Also every etationav type which conhim # i8 nd orthogod to e m etationary type to which Q b-8.

Procf. The proof is the same as that of 4.11(2) with the following additions. Ueing the notation of 4.11(2), let Q(Z; 8:) E tp(c, B,) be weakly minimal, and let ~ ~ ("say" 3 ) w.1.o.g. that for inhitely many i < K, (3 < '"Z)#(z; % h(60))A

(vZ)[$(z; f~h(bo))

* QO@Yfi(6:))l-

Let Qa(y) "sayyythat y = Z/E for some f satisfying ~~(3). Then Qa(y) is eemi-weakly minimal, but it and J me only almost over A. This can be oorrected (see 111, 2.4). For the last phrase, see 4.6. THEOREM 4.16:8-e p , q are wnaplete type%over A, and are stationM?/. If p , q are eemi-minid and not o r t h a g d then there are 1, < w , and BI (1 < 1,) rdizing p such thud every eequi?m?rdizing p i8 algebraic over UlCl,, 8, u A u U (6: 6 realizes q}.

OH. V, 9 41

SE~I-BE~ULAR AND SEBCI-MINIMALTYPES

275

Pro$ Let h > IAI + IT1 be regular, and it2 a R-saturated model, A c 1611,and r ~ B ~ ( l Ma1minimal ) type not orthogonal top (nor q). Let I = {ii E Q: 7i realizes p } , J = (6E Q : 6 realizes q}. Let pl, q1 be Clearly for every ii realizing stationarizatione of p, q (reap.) over IN). p , there are c2(Z< n) such that r2 = tp(c2,M) is minimal, not orthogonal t o r, @Eacl(wIU UlGncl). Also for any F realizing a realizing p, such that cEacl(m U I%). Similar assertions holds for q,; so for every @ realizing p , there are 6,, . . , 6l rea,lizing q1 such that ii E ad(]MIu Ufdl&); hence for some 5 E lit21 we can have ii E acl(5 u Uiel6{). Clearly there are C1c (UI ) n !MI,Cg E (UJ) n lit21 where ICll ICgl s IT1 such that tp(5, (A u U I u UJ))is definable over ClU Ca (as M is Fi-eclturated, 1.41+ IT1 < A). By 1.11for each nrealizingp, there are 6,,,. . . ,62realizing q1 such that 7i E ad(UlsI6 U Cl U C, U A). We can replace Cl,Ca by finite subsets without changing the last conclusion, by a compaotneae argument. SO let Cl = U(
r2 there is an

.

+

stp*({$: i < Z(O)}

u {q:i < 2(1)}, A ) = Stp*({a::

i < Z(O)}

u {6f: i < Z(l)},A ) ,

does not fork over A. Then for every 7i realizing p there is j < K(T), such that tp(ii, A U Ut7i{ u U1@) does not fork over A. As p is stationary for every a' realizing ply and ii' realize the same type over A u U17i{ u U16{; hence for every 7i realizing g there me EOy .,6l realizing q such that ii E aol[(A u Ui,, a{) u (U:,, 6{ u U1&)I; so we finish.

..

QUEBTION 4.1: Try to replace in 4.9 semi-regular by a weaker conmpt, and investigate this concept. (See Definition 4.2 and 4.4 for a suggestion). EXERCHE 4.2: Generalize 4.9 to semi-weakly-minimalformulas. QUEBTION 4.3: It is true that in 4.11 we amnot replace semi-regular (semi-minimal) by regular (minimal) even if we replace ii/E by any b E ad(A u a) - acl A ? What if T is N,-ategoriWl ?

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MORE ON !l'YPHS AND SATUBATED MODELB

[av, .8 4

EXERCISE 4.4 (Cq): Suppose p E S(A)is minimal, I and J am sets of elemente realizing p , which are independent over A, but I uJ is not independent over A. Show acl(A u I),acl(A uJ)not nectxsady have an element in common, which is not in aclA [e.g. T = theory of the complex field]. QUESTION 4.5: Clemeralize 4.16 to semi-regular types. EXERCISE 4.6: Prove that if &?,a) is semi-weakly minimal, then D"[cp(Z;a),L, co] < w . [Hint: See Section 7.1 QUESTION 4.7: Prove 4.11 does not hold for stable T in general; and try to find a substitute. Remark. Let T be defined by the following axioms: (i) P, (r] E @'2) a m pajrwise disjoint, (ii) P; (1 = 0,1) are partial one-place functions, from P, onto pn-, (iii) U, is a partial two-place function,

T exemplifies the non-exiatence of regular types. EXERCISE 4.8: SUPP Mi E Ha E M3,p ES"(A), A E MI,and for I = 1, 2, every 6~ lMt+ll, 6 # 1M,1 which mlizes p, realizes a stationariaation of p over M,.Show that every 6 E 1M31 which realizes p, realizes a stationarimtion of p over lMlI. EXERCISE 4.9: Show that 4.4 does not hold for regularity, even for Ko-stable T . [Hint: Let M = (1611,P , Q , E , P , +), is the disjoint union of P and Q which me W t e , E an equivalence relation on Q, + is a two place function from P to P , ( P , + ) is an abelian group, each element of exponent two and P is a two-place (partial) function, P(a,b) is defined iff aEb, a, b E Q , and P(a, b) + P(b, c) = P(a,c). Choose a,, a, €&(a)- 1611, t ~ B ( a 1C), A E(a1, a,) A P(a1, as) # c for every c E M.Then tp(a,, ]MI) is regular but tp((al, ag), IM1)is not semi-regulw.]

DEFINITION 4.3: (1) A formula (~(3, a) is (strongly) semi-minimal if for every Z satisfying it, stp(Z, a) is semi-minimal not orthogonal to some (fixed) minimal type oontaining a minimal formula, or is algebraic. (2) A formula ( ~ ( 5a), is almost minimal iff it is equivalent to a disjunction of minimal formulas. EXERCI8E 4.10: (1) (P@, a) is semi-minimal iff there is an almost minimal I,$@, 6) and Z suoh that p(C, a) E ad(E u $(C, 6)). (2) The formula p(Z;a) is almost-minimal ifF R”’(p,L,No) = 1. (3) In 4.14, if in the hypothesis we replaoe weakly minimal by minimal, then in the oonolusion we get a semi-minimal formula. For A = a01A, even strongly semi-minimal. (4) The following families of formulas me olosed under disjunotion: almost minimal, semi-minimal, weakly minimal and semi-weakly minimall.

EXERCI8E 4.11: Let V be an infinite veotor w o e over the field with : two-elements. Let us define for each n, iK( =M,,)

lMl P

4

=

=

---

V uV x V x x V (n times), V (a one plaoe predioate),

= (<(%’

-

.s

ai, a1+1, *

* *,

a,,), (a13 *

*

R = ((a, b, c): a,b, c E V,a + b = c}, RI

- 9

.

= (~~,~4,...,~l,CCI+1,...,CCn~,~~,,...,~i,~l+l C, 01,

80

.,4)): ..,4 E V},

ai, 4 + 1 , * * a1,. .,a,, 4 + 1 # .

-

+ c#~;+2,-#4): an, 4 + a s

-. 4E v), - 9

m<,,

= (pfl, P,El, R, T = ThW). Choose N X1-satmated. (A) Prove T is totally transoendental and unidimensional, and oategorical in every A. (B) Show that here it is possible that (when n 2 3) w (6,A ) = w(a-6,A) = w(ii,A u 6) = 1. (C) Show that 4.16 (UuLllot be substasltially improved.

DEFINITION 4.4: (1) Let B denote a subset of B* (stp(Z, A): A, a}. B is oalled regular if: (i) Whenever A E B, stp(si, A) E 9, stp(ii, B) fork over A, then stp(a, B) is orthogonal to B (see 2). (ii) If A G B, stp(zi, B) does not fork over A, stp(a, A) E B then Stp(a, B) E 9. 5

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MORE ON TYPES AND SATURATED MODELS

CUE.

v, 6 4

(2) A type q is orthogonal to B if it is Orthogonal to every p E d? If 9 = {p}, we write p instead of { p } . (3) An m-type q is strongly Orthogonal to B if q E ql, ql complete implieg q1 orthogonal to 9. (4) For a regular set B lowg(q) = lowp(q),where B(1)E 9

zpOBc1)

is a maximal subset of pairwise orthogonal types (it is well defined) (see 4.16( 13)). Similarly low,(a, A) etc. (6) For a set of types 9 let olO(9) = { p :p = stp(ii, B) is a sbtionariea;tion of q E # q E p } n B*, oll(9) = { p :p pardel to some q E LP) n 9*, olj(9) = { P ( p ) :p E P an automorphism of CE and Pw 1 a01A = the identity} n 9*, 0l3(9) = { p :p regular, stationary not orthogonal to LP) n 9*. Now oPJ(9) = 01‘ ol’(9) and we let ol’(9) = oli(9).

DEKUCITION 4.6: b t 9 be (L r e g ” f 8 d y . (1) An m-type q is B-simple when for some A, every ii realizing q is algebraio over A U U{b:stp(b,A) is 9-regular (see (4) below)} U {b:stp(6,A) is strongly orthogonal to B}. (2) A stationary nz-type q is strongly 9-simple if there am Emturated mod& bl, N ( K = K~(!P)),M < N,and 8,B, E N(Z < n ) suoh that (i) tp(ii, [MI) is a stationarization of q, (ii) tp(c,, WI) is 8-regular (see (4)below), (iii) si € aol(lMl u B*). (3) An nc-type q is 9%emi-regular, if (2) holds with the addition B lM(, then B realizes (iv) if B E IN1 realizes tp(Bl, Cb(tp(B,, [MI)), tP(% “1). (4) An m-type q is 8-regular if every extension of q which forks over Domq is strongly orthogonal to 9, but q is not strongly orthogonal to 9. ( 6 ) W e say si is 9-simple over A if stp(ii, A) is 9-simple. Simil&ly for 9-semi-regular,strongly @-simple and 9-regulm.

u,
+

LEMMA 4.16: (1) I f p i8 a regular etdionary type tlren: q ie 8e~ni-regukcr not orthogonal to p iJq ie oP{p)-senti-rq&r,

not algebraic (eo oP{p} ie regular). (2) If q is @-semi-regular, then it is strongly 8-simple, and if 9 = c12(9)any complete strongly 9-simple q, is 9’-simple. Also if q is

H.

v, 8 41

SEMI-REQULLSB AND SEMI-MINIMAL TYPES

279

9-regular, it is 9-simple and i f all stationarizations of a complete p are in 9,p is 9-regular. A n y stationary 9-regular type is 9-semi-regular (9is a regular family, of course). (3) If p = tp(si, A ) ie r@r, p E q, q forbover A , p E B regular, then q is strongly orthogonal to 9.Also p i s 9-regular. (4) If r k q, q $8 9%mple, ie regular then r is B-&mple. 80if A E B, si 9-eimple Over A then li is B-eimlple over B. If r k q, q ie @-regular then r is 9-regular or is strongly orthogonal to 9. If a i s 9-simple, then w,(E,A) < w . (6) If q ie strongly orthogonu1 to r k q then r ie etrongly orthogonal to 2 ( 6 ) If si ie 93-eemi-regular over A thn low&, A ) = w(si, A ) . (7) If q ie B-regular, q over A , q E ql, q1 forks over A then. q1 ie etrongly orthogonu1 to B. ( 8 ) 8uppxe q, r are parallel, then q ie etrongly B-eivle if r ie etrongly P-eimple, a d q ie B-eemi-regular i# r ie B-eemi-regular. (9) If B e d i e f i (i) of Definition 4 4 1 ) then cl0(B) is regulur. If 9 c then: P1is regular, implies clO(9) is regular. (10) If p i s minimal, 9 = el3@},q complete and stationary then :q is semi-minimal not orthogonal to p iff q is strongly 9-simple not algebraic. (1 1) If p , q are parallel and complete (9= cI2(9),9 regular), then p is 9-simple iff q is 9-simple, and p is 9-regular iff q is 9-regular. (12) Suppose 9, = el2(g1)c g2are regular families: i f q i s g1regular and complete [91-semi-regular][strongly P1-simple],then q i s g2-regular [92-semi-regular][strongly P2-sirnple].

e

e

Proof. Easy. I n (2), note that even a stationary q may have a stationarization in 9 but not be in 9 (instead 9 = c12(9) also 9 = cll(9) suffice). In (4)) use (13) below.

(13) Let T be superstable. Every complete stationary 9-regular type p is regular.

Remark. Part (13) sounds, a t first, a truism, but an extension may be orthogonal to 9 while it is not orthogonal to the type. Proof of 4.16(13). W.1.o.g. p e S m ( M ) , M is Fgo-saturated, C E Mr, is over M and r e 9 is regular, not orthogonal to p and d realizes p . If p has weight > 1, there are n < w , n > 1, and d$(i< n ) such that tp(di,M) is regular, { d i : i< n} is independent over M and tp(d,,^dln*m.dnE,_l,M U d), tp(d,M U u,,,di) are Fgo-isolated(by V,

280

MORE ON TYPES AND SATTJBATED MODELS

[UH.

v, 8 4

3.9(2) and see Def. 3.2). Since we have assumed p is not orthogonal to r , w.1.o.g. (by 3.9(1)'sproof) tp(do,M) is not orthogonal to r . Now p , = tp(d,M U dl) forks over M, whereas tp(do,M U dl) does not fork over M, so tp(do,M U z U dl) forks overM, and hence should be orthogonal to r . But tp(80,M U d u dl) forks over M, whereas tp(zo,MU dl) does not fork over M so tp(d,,,M U d U dl) forks over M U dl: hence, tp(d,M U do u dl) forks over M u dl. So stp(d,M U d l ) ,stp(do,M U dl) are not orthogonal. But the latter is regular and parallel to tp(&,M) which is not orthogonal to r , so stp(d, M U dl) is not orthogonal to r, contradicting the 9-regularity of p . So p should have weight 1. If it is not regular it has an extension q, forking over M , not orthogonal to a regular type r' iff r' is not orthogonal to r (as w 0 @ ) = 1 , p not orthogonal to r ) . We conclude that q is not orthogonal to r , but that q forks over M and hence over C, a contradiction. So p is regular, so we have proved 4.16(13).

LEMMA 4.17: (1) If zi E d ( A u 6), 6 ia 9dmpZe over A then zi ie Beimpk over A, and low& A) s low& A ) . 8im;hrly for B-eemiregular, 9-regular and (except BE acl A ) strongly 9-simple. (2) If q ie B-eirnpk, q ia over A, ti r d i w q, then low&, A ) iajinite. (3) B = { p :p a minimal type} is a r@r famiZy. Only aebraic type8 are e t r q l y o r t h o g d to 9. (4) tp(zi-6, A) ie &&n& iff tp(zi, A), tp(6, A) are 9-eimpk. ( 5 ) For any ordinal u , 9 : is a regular family, where 8:g' @ : p a regular type, R @ , L , a )= 01, and p is orthogonal to every stationary q for which R(q,L , 00) < CO} n 8*. (6) 8: = clo(9:) = cl'(8:) = c12(9:), and any formula $(x, C) with R($(x,c),L , m ) = u is Pz-regular or is strongly orthogonal to 8:.Also, every regular p E 9: contains ( i f it is closed under finite conjunctions) a 9:-regular formula. ( 7 ) For superstable T ,for any stationary regular r , for a unique u the type r is not orthogonal to S:,so some S:-regular formula ~ ( xF) ,is not strongly orthogonal to r . Proof. Easy.

LEMMA 4.17(8): (aq)Suppose r is regular and stp(B,A) is not orthogonal to r , r not orthogonal to 9:. Then

OII.

v, 8 41

SEMI-RE(XULAR AND SEMI-MINIMAL TYPES

281

(i) There is a formula E = E ( Z , y) an equivalence relation over A such that tp(a/E,A) is @,*-simplebut not orthogonal to r . (ii) Moreover, tp(a/E,A) contains a formula O(y,b) that is @',*-simple. (iii) Also, for every 6' realizing tp(5, $),O(y,6') is P,*-simple.

4.17 A remark : Instead of 9': we can use any regular 9'= el; '(9'if ) A = aclA and any ~ € 9has ' a 9'-regular formula in it, e.g. any ~ 1 ~ ~ ~9 ( 9E' 9 )': ,, is O.K. and el3@),p stationary regular. Proof. Choose cp(x,C) as in Lemma 4.17(7). Clearly, {cp(x,C)}has a stationary completion p which is regular, not orthogonal to r , and w.1.o.g. p(x, ES(C)(as we can increase C). So q(x,@, stp(@,A) are not orthogonal, hence w.1.o.g. their stationarization over A U F are not weakly orthogonal. Let {c, :n < o}be an indiscernible set based on A,c0 = F. So each p, = &,I?,) is regular, not orthogonal, to A (as po is). Hence, by 3.4 the qn's are pairwise, not orthogonal ; hence, by 1.13(1) (as qo,r are not orthogonal) each q, is not orthogonal to r . Also for each n, stp(@,A) has non-orthogonal stationarizations over A u cn. Now repeating the proof of 4.11 we get E , an equivalence relation over acl A as required in (i). So let E l , ... ,E , be the images of E under the automorphisms over A. As @',* is preserved by the automorphism, for every d and 1 the ~ , then type stp(d/E,,A ) is @',*-simple.Let E*(Z,y) 'ZpA f - l E z ( y), @/EEdcl(A U U,a/E,) so tp(a/E,A) is S',*-simple. Now tp(@/E,A) contains a formula O(y,b) such that for some n, ~ ~<( n )1 realizing tp(c,A ) r

1

As each q(Z,4) is @',*-regular(by Lemma 4.17(6))clearly O(y,6) is @,*-simple.So (ii) holds. Also for every automorphism F of aeq,F(Fl) realizes tp@,g), hence each cp(x,F(~?J) is 9':-regular, hence O(x,F(6)) is @',*-simple.This proves (iii).

THEOREM

4.18: 8uppOcre

B is regular, B

= ol%(B) and

{6{:i < a}. 2 " h lowB(U I , A ) = Zlsrrlow&,, A u U,
I

=

every

282

MORE OW TYPE8 AND SA!lWBATmD MOD=

[OH.

v, 8 4,

Remurk. This really generalizes 4.9( 1). LEMMA 4.19: (1) If A c B, 9 regular, B = c1:(9), ii 9-simple Over A then low&, B ) s low& A ) . (2) If in addition nis 9-semi-regular over A , tp(n, B )fork over A then low,(E, B ) < low,(& A ) . Proof. We prove (2) like 4.7, and (1) like 4.9(2).

THEOREM 4.20 (in P): S p e p is stationary and regular, and stp(ii,A ) is not orthogonal to p . Then for some equivalence relation E almost over A , stp(ir/E, A ) is strongly c13@}-simple, low,(a/E, A ) > 0. It is also cP@}-simple. Proof. Like 4.11. Similarly

CLAIM 4.21: flwppo8e I c ~ ( a6), , I an infinite i n d h r n i b l e set, tp,(I', A ) = tp,(I, A ) implie%I', I are not orthogrmal. Then there are n, 6,(l < n) i2 E A , and # m h that: ( A ) tP(61, A ) = tp(6, A ) . (B)For s m c', #(a, a) c acl(a u Ulenq(&, 6J), in fact there ie a set {6fni5y: 1 < n, a < K } (any K ) independent over A stp(6?,A ) = stp(6f, A ) m h that for every a(1) < K , #(a, a) G ac1(Ul<,,Ef(I) Ul<,,#(a, 5?(p"))). (C) There is an (infinite)indiemible set J E #(a, a) baaed on A m h that tp#, A ) = tp*(I,A ) implies I', J are not orthogonal. (D) If J , c #(a,@b an infinite indiscernible set, then for some 1, and I , c ~ ( 68,), , J,, I , are not orthqtmul. (E) If Av(I, U I ) is regular, then Av(J , lJJ ) is semi-regular. ( F ) If Q(Z; a) is @-regular, #(z; a) is @-simple. LEMMA 4.22: 8 ~ p p K~ = e [A]' + K,(T),bl is F,O-sdurded, p = stp(a, A ) is semi-rephr, then there ie an indiscernible set I E ldll based on p , Av(I, A U U I ) I- Av(I, M). Proof. First choose a maximal set I of sequences from M realizing p, which is independent over A . Let B be a maximal subset of ill satisfying (i) A u U I E B, (ii) B is q-atomic over A u U I (exists aa A U U I satisfies those conditions, and they are preserved by unions of increasing chains). By IV, 3.2, for no b ~ w 1 - B is

OH.

V, $41

SEIUI-REGULAR AND SEMI-BEINRUL TYPES

283

tp(b, B) E-isolated, hence B = INI,N F,O-saturated. We can find a complete regular type q over !.Al not orthogonal to p, and let J, = (4:i < a} c be a maximal set of sequences realizing q independent over Let p , be the stationarization of p over clearly M omits p,, hence a < KO,and even a < w(p). Choose B c N, IBI < K ( T )such that q does not fork over B, q 1 B is stationary, A c B, tp(B,A U UI)does not fork over A U (B n I), and p, rB, q rB are not weakly orthogonal. Let J c !.Al be a maximal set of sequences realizing q rB and independent over B. Now J U J, is a maximal set of sequences from PI realizing q 1 B and independent over B (1.16(3)). Next choose I, c I, J , c J , lIol = lJol = K such that I - I, is independent over (A u I, u B u J,, A ) and J - J , is independent over (A u I, u B uJ,, B). Let N* G M be Q-prime over A U I, U J , u J,. By IV, 4.11 there is a set I, G IN*I of sequencesrealizingp,independent over A, AV(I2,A u Ul a ) l-Av(l,, p*l)(the hypothesis of IV, 4.11 holds as iF* is F:-prime over 0 ,by IV, 4.9(3)). Now 1-1, is independent also over ( p * I , A )and , by IV, 4.15N*is Ff-atomic over A U UI,,hence N* U UI is F:-atomic over A U UI,, where I, = ( I - I o ) U I,.Clearly for every F E J,tp(4 B U UI)fork over B hence tp@, p*IU I,)fork over B, hence is orthogonal to p so also IN*]u I u 5 is E-atomic over A u U I,. Repeating (and noting Jo c lN*l) we get that 1N*I u I , u ( J u J,) is E-atomio over A u 13.So we can find a maximal set B, s liKl E-atomic over A u I,, IN*] U ( 1 3 U J U J O ) c B, and @n B1 = 1N11,N, E-&watd, and let p’, q’ be the stationarizations of p, q, resp., over lN1l. But J u Jo c lNll was a maximal set of sequences in M realizing q B independent over B, hence iK omits q’, hence p’ has a unique extension over WI. As by 4.10(2) Av(I,,A U I , ) t- Av(I,, WJ) we finish.

m.

u

m,

u

u

u

EXERCI8E 4.12: Show that in 4.18 and 4.19(1) the demand P = c l i ( 9 ) is necessary. [Hint: Let T be a theory with only one equivalence relation, with infinitely many equivalence c l m s , each of them infinite, and choose distinct a, E B, B C alEaa A -,alEa3. - For 4.18 let 9 = c13{tp(a,, a,), tp(aa, a,)}, A = 0, a = 2, b, = (a,), b, = (a2): and for 4.19(1) B = c13(tp(aa,a,)}, A = 0, B = {a,}, 8 =

(aa>.l EXERCIBE 4.13: Show the if in Definition 4.3( 1) (i) we demand only “stp(7i, B) is Orthogonal to every p E B which does not fork over A,” then 4.18 may fail. (Hint: Use the example from Exercise 4.12.)

284

MORE ON TYPES AND SATURATED YODELS

[a. V,f 6

EXERCI8E 4.14: Look through Sections 1-4 and generalize the relevant theorems to the notions from Definition 4.5 and 4.4 (in addition to Theorems 4 . 1 M . 2 2 ) . EXERCI8E 4.15: Suppose q, A are as in Definition 4.4(1) A , Dom q E B G C, 7i realizes q, Then low&&C) s lowg(& B). V.5. Mdti-dimensional theories

DEFINITION 6.1: Let { J t :i E 8) swI if for every %-saturated model (where K = K J T )+ 181+), such that I , Ji E IMI. min dim(J,, M ) s dim(I, M ) t€S

LEMMA 6.1: The following conditions on I , J , (i€8)are equivalent (letting K = K,(T)+ I8l+): (1) {Jt:i € 8 ) s, I . (2) If dl is F:-8aturated, J* = {a;: i €8, j < K } d8 independent over 1M1, I , J , E M and izf realizes Av(Ji,] M I )and N $8 Fg-prime over 1M1 U UJ*, then Av(I , IMI) is realized in N . ( 3 ) There are a FE(m-8aturated nzodel MyI' c M equduab4 to I , and 8' c 8, 181 ' < K ( T )and J; E M(i€8')equivalent to Ji, and J* = (8;: i EB', j < K(T)} idpendent over 1M1, a: realizes Av(J;, 1611)such tirat in the F&,-ppime model over 1M1 u UJ*, Av(I', lMl) is realid. (4) There are JI equivalent to Ji (i E B) and I' equivalent to I such that (i) p ' l Y IJII = 4 T ) , (ii) I' is indi8cernible over Utes Ji and J ; is indiscernible over I' u U {Ji: i # j~ 8 ) , (iii) Av(I',1') has a F:(,-isolated complete extmion over I' u

u

{ e l t : i€S}. (6) For 8 m e h 2 p 2 K , there i8 ru) %-saturated MyI , Jt E M such that dim(I, M ) = p, dim(J,, M ) = h (i €8).

Prmf. As the ideas essentially appear in Section 2, we leave it to the reader aa an exercise.

CONCLUSION6.2:(1){Ji:i~rS) 1,Ii,fJfor8melsll E 8,lS'I < K ( T ) , {Ji:i €8')I , I . (2) The holding of { J i :i €8) I,I depend on I , J,only up to equivalence.

Proof.Immediate by 6.1.

OH.

v, 8 61

MULTI-DIMENSIONAL THEOFLIES

285

THEOREM 6.3:LYqposeI , Jfare indiscernibleover A (i < K ( T), a < K ( T)) a d I I l , IJfI 2 Kr(T),IAl c ~ r ( T ) a n d t p , ( J ? , A u U ( I u U { J < ~ :aB or = a,j c i})does not fork over A , and stp,(Jf, A ) = stp,(Jf, A). Let J * = Ur,cr Jr, B = A U U (IU J*). Assume {Jf: a, i c K ( T ) &,I. } (1) IfJ = {ti?: a,i < K ( T ) )is i-endent Over By: 3 realizes Av(Jr, B), then Av(I, A U I)hae an l&&solated mp l e t e extenaim q Over B u UJ. (2) If id additiort LIZ F&)-sdurated, B c (MI,{a?:a, i < K(T)) i8 independent over (1611,B ) , then q I- Av(I, 1M1). Proof. (1) W.1.o.g. we assume 111 = lJfl = A = (2ITI)+,and let p = Av(I, B). Clearly there are J' E J , IJ'I < K ( T )and ql, p G q1 E Sm(Bu J'), such that ql has a unique extension Pa E P ( Bu U J) and let 6 realize ql. By notational changes we can assume that J' = {a?: a,i < a, < K(T)},and that we can choose C c B, ICl < K ( T ) C c B, = A u U (J'U I U U { J f : a,i < a,}) such that q1 does not does not fork over C. Choose JT = fork over C. Clearly also a, s a < K ( T )i, < K ( T )such } that J += U { J T : j c A + } is independent over B u lJJ and ?&realizes Av(Jf, B u lJJ).We shall show that q1 k tp(6, B u U (Ju J + ) ) ,for this it suffices to prove that for every n < o,q1 I- qn = tp(6, B U (Ju U,<,,J?)),and qn does not fork over C and this we shall prove by induction on n. For n = 0 there is nothing to prove, For n + 1 choose an elementary mapping f such f maps Jf that f is the identity over B,, and for i , a, a, 5 a c K(T), onto Jf u {at,:j < n) and f (a?) = a&. As q1 k qa clearly f (ql) kf(qa), so iff (ql) = q",f(q,) = qn+ we finish. But this holde as q1 does not fork over C, C c Bo,f 1 B, is the identity, clearly f (ql) 2 q1 does not fork over C, hence doe8 not split strongly over C, but also qn satisfies this and q1 E q" ; hence qn = f ( q J ; similarly f(q,) = qn+'. We c m conclude that ql I-tp(& B u (JuJ + ) ) .Let M be R-prime over B and Flf-constructible over B u J+; and N be Flf-prime over IMI u J + .As for a 2 a,, dim(Jf, N) 2 A+, dim(I, N) 2 A+ [clearly there is an elementary mapping f, f 1 A u (JI is the identity, and f maps Jf onto (a,i < K ( T ) hence ) {Jf:a,i < K ( T ) }sWI implies (JP: a. s a < K(T),i < K(!P)} sWI]. So there is I+c IN1 indiscernible over lMl u 6, 1 ' 1 = A+, I u I+is indiscernible over A. By IV, 4.10(2) tp(6, B u UJ + )I- tp(5, INI); hence

u

{k,:

ql I- tp(5,

u (Iu I + ) u A ) .

We can aasume that if 5 E I is not disjoint to C

-A

then 5 E C. By

MORE ON TYPES AND SATURATED MODELS

286

[UE.

v,8 6

the proof of IVY4.3 we o m deduoe that stp(6,A u C) k tp(6, A u U ( I , u I + ) ) (, I , = {E E I : E E c))hence stp(6, A uC) k tp(6, A u UI ) . (2) The proof follows eesily now.

DEFZNITION6.2: The (infinite) indisoernible set I is oalled multidimensional over A if whenever I j (js K( T))are suoh that tp,(Ij, A u Ut<j I j ) does not fork over A and stp,(Ij, A) = stp,(I, A) then { 4 : j < 4T))$ w 4 r 4 n . DEPTNITION6.3: The theory T is multi-dimensional if Borne I is multidimensional over some A. LEMMA 6.4: I f I is orthogonal to A then I is multidimen&onal over A .

Proof.Immediate by

1.6 and 2.7.

THEOREM 6.6: 8wppo8e T is not multidimnakmal , and M is F&nsaturated,and A,, s 2ITI i8 the 3 r d cardinal in which T i a stable. Then t h e i s N -c M , [IN11 s A, and a set J E M in&pover M such that M is a e(n-prim over e IN1 u J .

u

We first prove a olaim: CLAIM 6.6: 8 u p p e A,, is aa in 5.6, N o E bl, c My and they are q,,-saturated and llNoII s A,,, and for every 2 E Mytp(8, Jbl,l)for over h N o or E E INol. There are N,, N o c N1 c M,, llNLIIs A, and J c M idpendent over (IMol, lNll) m h that if E E ( M I , 8 $ lblol, i? ~ e a l i z ~ i Av(I*, ~ M , ~I* ) ,any countable set indiacernible over No, I* not multiINo!and Cb(Av(I*,(JI*)) c lMol ,then tp(Z,M , u J ) di&alover fork Over lMol or tp*(I*, IN,l)forh over IN,]. Also N , is F&-saturaterl.

u

Proof of 5.6. Let {(I,,J,): u < a,} be a maximal family such that: (i) I, is countable, and it is indiscernible over INol, and I, is not multi-dimensional over lNol and Cb(Av(I,, (J I,))c IMol. (ii) J, E bl is a maximal set independent over (lMol u UB<, JB, lMol)of sequenoes realizing Av(I,, JM,I),and J , is not empty. (iii) tp,(l,,Up,Jp U pol) does not fork over POI. The number of possible tp,(Ia, IN,!) is sA,,. [If A20 = A, by the A,-stability of T.If A$ > A,, then necessarily K ( T )= No,so ohoose I: s I,, I: finite so that Av(l,, I,) does not fork over I:, and let I: = {a;, . . .,z&-,}, a& E I, - I;; now tp(a,an(,), I N O l ) determine tP*(Ia, INol)*I

u

- - - --

Similarly, for each i there is C, E INol, IC,l a K J T )and I: 5 lNol such that tp*(I,, INoI) does not fork over C, and stp,(I:,C,) = stp*(l,, Cf). So if a. 2 &+ then w.1.o.g. for every a < &+, tp*(I,, INo!)= tp,(I,,, ",I). As I , is not multi-dimensional over INol, we get a contradiction by 5.3. (As U,:j < K ( T )I } wI,*,Av(l,*,POI) is realized in M . ) So uo < &+ and we choose N,, N , E M , llNIII 5 Ao, I N 0 1

u

u

{
Cb(Av(Ia,

u I,))

E

and our conclusion follows.

Proof of 5.5. We define by induction on i s K,(T),N,, M,, J , 80 that:

(i) Ni _C dl, llNfll 5 A. and N , is F:r,,-saturated. (ii) J, E M, J , is a maximal set independent over (IN,I u Uf<, lMfI, INfI). (iii) M , is E(,-prime over (JJ , u INfI, and M , E M . (iv) N , + , c M,, Nd 2 Uffork6 over We choose N o to satisfy (i),and M , by (iii).J , and N , + are defined by Claim 6.6,a d Nd by (iv). If M,(, # M choose E E M ,E $ and a countable indiscernible Set I over NXr(T), Cb(Av(1,U I))E M,,,(,, such that Av(I, MXdn) = tp(Z, ME,(,). By (v) we get a contradiction (when ~~(2') = No, we have to work a little more).

U,<,

Remark. In 6.6 we can msume only M is E(,,-satur&ted. 6.8: For every nun-multidimensional T,there is a cardinal h 5 (2IT')+m h that: (1) A the $r8t c u r d i d for which there are no I,(a < A) m?& that { I E :a < p < A} $ I,. (2) Let K,( T ) 5 Ha < HE, T stable in HE, and p i8 the number of nun~8onwrphkIQm-8aturatedd l a of T of cardinality &. Then

THEOREM

288

MOBE ON TYPES AND SATURATED MODEL8

[OH.

v, 5 6

Proof. Left to the reader as an exercise. PROBLEM 5.1: Make 6.8(2) more precise. In particular try to prove that A is always a successor. (See IX,Section 2.) THEOREM 6.9: If T i s multi-dimeneional, T stable in 8 8 , KJT),< Nay a < t h T has 1 216-al n o n - i s w p h i c Ga-8aturated d e l s of cardinality N, provided that p < N,.

Proof. Let I be (an infinite indiscernible set over A which is) multidimensional over A ; and we can assume IAI < KJT),I countable. Choose I $ ( N a 5 p 5 N,, f < NB)y /I$ = p , such that Stp*(I$,A) E stp,(l,A), and tp,(lf,A U U r<Elt) does not fork over A. For any set 8 E 8, = {A: N, 5 A < h regular or A = Ha}, let p = min(S U {K,}) and M , be a Fz-prime model over

1

uA
A U U ( I : : r \ E S U { N B } y [ < NB}. Clearly M , is a F(fa-saturatedmodel (of T)of cardinality 8 8 , and it is not hard to prove that B ( Z , ) n8, = S where 8 ( M ) = {dim(J, M): J an infinite indiscernible set, J s M } . So we finish.

PROBLEM 5.2: In Definition 6.2, can we replace “over some A” by “over 0”. PROBLEM 5.3: Suppose A , I, J? are aa in Theorem 6.3. Is it necessarily true that for some a,{J; :i < K ( 2’)) <EIl. EXERCIBE 5.4: construct a two-dimensional (b.,A = 3 in Theorem 6.8) oounstable, stable but not superstable theory. EXERCIBE 5.5: In 6.3 replace K J T )by any now).

K

2

K,(T) (so lAl <

K

EXERCI8E 5.6: Suppose A -c By tp(si, A) has no extension over B which fork over A, and tp*(B, A) is stationary. Prove tp(si, A) k tP@, B). QUEBTION 5.7: Can we in 6.3 get F*,,,-ieOlation?

UE.

v, 8 61

QUU"IEE8

AND TWO-0ARDINA.L THEOREMS

289

EXERCIHE 5.8: Suppose I is indiscernible over A. Prove that for some B c A, IBI < K,(T),tp*(UI,A) does not fork over B: provided that I is based on A. V.6. Csrdinalify-quantifiersand two-csrdinal theorems We first define a general context, and then get conclusions on the logics with cardinality quantifiers and two-cardinal theorems. Let K = (F, W,A, p) be 8 quadruple (we write A = hK, p = FKs etc*) satisfying the following conditions: Condition 1. F is aa in IV, Section 1, satisfying Axioms (I), (III.l), (IV), (VI), (VIII) and (Xl).

Condition 2. W is a set of triples (cp(z, g), $(g), x) where x is a regular cardinal 2 Ax and IAI < x implies that for any F-maximal F-comtructible modelM over A , llMll < x,and ifM is F-maximal, A E M , IAl < x, then there is an F-maximal N , A C N C M , llNll < x. Remark. A model M is called F-maximal, if for no a $ bl, is tp(E, l M ) ) F-isolated; similarly for a set. Notation. Let CarK be the set of cardinale appearing in W and 00. We say C"(3'xz)cp(z, a)" (or C, or C, instead C) if for some triple ((~(5 $), , #(Z), x ) E W, C$p].Clemly this is an elementary property of a. Define induotively C,,(cp(Z; a)) = min{~:Icp(Q, B)I = K < No or there me $(y; 6), O(E; y; c') such that: O(Z; y; 5) k cp(Z; a) A $(y; 6), cp(3, a) k (3y)O(Z; y; C'), C"(3'ly)#(y, 6)" for some x s K and for each b,: n > 0, C,,-,(O(Z; bl, a)) s K or O(Z; bly a) is algebraic}. Let C(C~(Z; a)) = min,C,,(~(Z:a)),C(p)= min{C(cp(~,a)):p kcp(~;a)},C(a,A)= C(tp(zi,A)) and let for a4 (infinite)indiscernibleset (of sequences)I,H ( I ) = SUP(K: if J is an infinite indiscernible set and cp(z, a) E Av(J , a), I="(3 <xz)cp(z,a)", x < K then I, J are orthogonal}, For a complete or stationary type p , let H ( p ) be H ( I ) ,for any indiscernible set based on some stationarization of p , let H(a, A ) = H(tp(ii, A)). Let C*(p)(z,a)) = min{K: there is a F-maximal model M , E bl, such that for every p , cp(z,a)~p E 8(1611) implies H ( p ) s K } (clectrly C*(cp(z, a)) s C(cp(z, a)). Let B 4 A (or B 4 A) means that for any a E A, H(a, B ) 2 h (B,A a n be replaced by models.) The sup in the definition of H ( I ) is taken on K E CK;~ ~ similarly elsewhere.

290

MOBE ON W E 8 AND SATURATED MODELS

[OH. V, 8 6

Condition 3. For any formula p(x, 7i) C*(p(x,a)) = C((p(x,a)), i.e., for any F-maximal model M , 7i E 1 M 1, there is a type r such that p(x, 7i) E r E 8(lM 1 ) and H(r) = C(p(x,a)). Condition 4. h 1 p+

+ IT1 + h(F). +

Condition 5. I f M is F-maximal IMI c B , N is a F-constructible over B, c E IN1 - IM1 then tp(c, B ) forks over ]MI. Condition 6. I f M, (i < p) is increaeing, and each M, is F-maximal, then Ut,,M, is maximal. I n each theorem (Cn,, n,, . . .) will denote the conditions on which the proof o f the theorem depends:

CLAIM 6.1: (1) For formulas #(x, a),p(x, 8) the following conditione are equivalent: (i)If 3, 8 E 1 MI then 1#(M, a)l < Ip(M, 8)l + KO (this we denote by $ J ( x ¶ ~s p(x, 8)). (ii) There are a model N,and k, 1 < o,and finite A such thcct in # ( M ya) thre is no ( A , 1)-indiscermible set over Q ( M ,Z ) of cardinality k. (a,8 E 1 M 1, of couree.) (iii)Ira #(a,7i) there is no infinite set indiscernible over p(Q, 8). (iv) There are no lnodele M , N , 7iu8 s 1 x1 c INI, such that Q(M,a) = p(N, 81, $(MY a) # $J(N,a). (v) There are M , u 8 E M and finite k, A such that there are no 6, E #(M; 7i) (i < k ) for which tp,(6t, Uj, Ej u v ( M , E ) ) increaee with i and 6, # 6i+1and ( E = g ) E A . (2) For every c a r d i d K 1 KO and formula p = p(Z,g) t h r e ie a set p; such that for every 9, C(p(1;6 ) ) 5 K iff 6 eatisfie8 sorne #(g) ~ p ; . (3) I n (2)P: = U n c a P:", where (i)Cn(p(l;6 ) ) 5 K iff 6 satisfies some #(g) E p:", (ii) p;en is the set of the following formulas: for each #(1;g,), O ( 1 ; y; I ) a d f i n i t e q p;(:;,',,, tznd k < o and <#(x;gl), $l(gl), x ) E W ,X 5 K the formula saying: there are g,, I such that (VZg)(O(Z,y; 2) -t p(8; g ) A +(Y; and (=)(~(z;-+( 3 ~ ) 8 Y( ;~2 ;) ) and and (VY)(V q ( ~2), v (3*k1)O(8,y, 2 ) ) (for n = 0, q = 0). ( 4 ) p; is closed under disjunctionrr ( m e @Iy: finite disjunction is equivalent to another formula) and contains all formulas ( 3 < k ~ ) c py). (~, Clearly C(V:, pi(% 7i,)) = maxi C(pf(3;%)). C ( A pi) 5 mini C(pf). +

,

=

a,))

,

OH.

v, 8 61

QUANTIFIER8 AND TWO-UARDINAL THEOREMS

291

If e(a;g, a) F V(Z; a) A #(g; 6), V ( E ;a) F (3g)e(z, g; a), C(+(g; 6)) and for every d C(e(Z, d, a)) I K , then C(tp(Z; li)) 5 K . ( 6 ) If p E P ( B ) A , E B, p does not fork Over A then C ( p ) = C ( p 1 A ) and H ( p ) = Wz,t A ) . (7) For every p , for somejinite q E p , C ( p ) = C(A q). (5)

I

K

Proof. (1) Left as an exercise to the reader. (Hint: prove +ii) + ,(iii) * ,(i) e= -,(iv) +ii) and, using 11, 2.17 (ii) 9 (v).) (2) Follows by (3). (3) Easy, by induction and compactness. (4) Trivial. (5) By induction on the first n for which Ca((+(y,6))5 K and for every d C,(O(z, d, c)) IK (this suffices by (2) and compactness). (6) Left to the reader. (7) Left to the reader. =j

LEMMA 6.2. For any stationary type p E Sm(A)and formula #(it; a) the following condetiortf,are equivalent: (i) For every model M , i f A u 8 c ]MI then dim(p, M ) < 1#(M; li)I +

+ x,.

(ii) There is an equivalence relation E = E(Z,g, d ) , d E A u 3 such that: (a) i f A U a c B, tp(6, B) i8 any etationarization of p , then 6/E q! acl B (in Qeq, of course), @) for any 6, 6/E E dcl [#(a; a) u 23 (and by a $zed formula). (iii) There ia a model bl, A u ii E 1611,61 i s F3mturded, K 2 K J T ) , and tp(6,IMl) is a e t a t h d & i o n of p , and N is c-pdmeOver 1611 u 6, and $(My a) # +(N, a). (iv)If I is an indis~rnibleset b a e d on p , and is i d h e r n i b l e Over a, then I is not indiscernible over u +(C, a). ( v )not every J c #(a, a) is orth0gona;l t o p . (vi) There 6 an equivalence relation E = E(Z, @, a), d E A , such that: ( a ) FIE 4 acl A, where b realizes p , @) C((3Q)S= 5 Q(+(%m, ( y ) 8Ome 8 t d h r y type q, +(Z, a) E q, i8 not orthogorutl to tp(6/E, A).

@/m

Proof. (ii) s (i). Let I be an indiscernible set baaed on p , I E (611, IIl = dim(p, M ) . By (ii),if I = {6:: i < III} then the &/E’s (i < IIl) are pairwise distinct, hence 1{6/E: 6 E 1611}1 2 IIl. By (ii)@) 1{6/E: 6 E 1611}1 < 8, + l+(M, ti)[ ; so (i)follows. -(iv)*-(i) If (iv) fails, let M be F:-saturated (choose K = +

((Al++lTI)+), I U {a} E WI. So in the F,"-primary model over UI U @ U @(M,81, no non-algebraic q, +(z,a)E q ~ S ( Ua +(M, a)), is realized. Choose a set J, indiscernible, over a U @(M, a), 1 E J, IJI > I@(M,@)I, and then the F,"-primemodel over U J U @ U @(M,a) contradicts (i) (mapping A to M). ~ ( i i i= ), ~ ( i v Let ) M, 6, N be as in (iii), except that $(M, 8) = $(N, 8). Clearly tp(6, A u 8 u $(N, 8))is (A u 8)-definable, hence we choose &,,EM, such that 6, 6, realizes the same strong type over A u 8 u U,,, 6{, then they realize the same type over A u 8 u Ute, 6t u #(N,B). So (6,: n < 0 ) proves +iv) (we wsume w.1.0.g. lAl < K(T)). (iii)+ (ii) Choose z E $(N, a) - $(M, 8) SO tp(z, !MI u 6) forks over lMl hence tp(&A u ii u #(My8) u 6) forks over A u 8 u $(Mya), and let 9(1,6, a,, forks over 1611, where aaE A u li, dl E +(M, a) and l=p[~,6,dl,d,]. Let dl = d!^- qndf, l=@[q, a],

aa)

-

J

We leave the checking to the reader. (v) o (iii) Immediate. (vi) (v) Immediate (we do not use (vi) (p)). (ii)+ (vi) We work in Cq. Let a = ~ l ^ l i ,a, E A, I = {a,: n < U) an indiscernible set over A, based on A, Go = li. Clearly C($(x, a,)) = C($(x, 8))so for every n C(V,,, $(x; q))= C($(x; 8)); and for every n, if En = E(z, g, a,, a,,), then for every 6, 6/E, E dcl[a,, u $(a, a,,)]. For each n let IP = V,wI,n,w,,n-, /\,,Z(~,ji, a,, at); so for every 6, 6/EnE d~l[Ul< an - 1 (7iI U #(a, %))I heno0 C((3g)(x= 81E")) C($(X,a)) (by 6.2). By 111, 2.6 for some la, En is dmost over A. By a cosmetic treatment we get an E* which is over A and C((3ji)(x= g/E)) s C($(X,a)). Remark 6.2A. In (vi) ( y ) we can have q regular and if tp(a,A) is stationary we can have tp(b/E,A) semi-regular (see 4.1).

LEMMA 6.3: (C2) (1) Tirefouowing cimditione on M, A, where IMI c A, are equivalent: (9 M<"A, (ii) for no E A, 6 E I M l , ~$, d t m C(cp(x,6)) c A, t(3x)#(x, 8, 6) and $(x, li, 6) I- { ~ ( x6), A x # a: a E IMI),

OH.

v, 3 61

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AND TWO-CARDINAL THEOREMS

293

(iii) for no a E A , 6 E IMI,'p, $ and x < A doee 1="(3<x)'p(z, 6)", C(~Z)$(Z, a, 6 ) and $(z,a, 6)I- {'p(z,6) A z # d : d E 1611). (2) Working in C o , if H @ , A ) < A, then for a m formulas 1 = E ( f ,g; 6),cp(x;6),E is an equitnzlenm re2ation, 6 E A, tp(a', A) I- 'p(f/E;6), 4 a01 A. C('p(z;6)) < A and ( 3 ) H ( I ) = SUP{K:ijJisindiscemzibZe,q(z,&)EAv(J,a), C(q(~;a))+ < K t h I , J are orthogonal).

Proof. ( 3 ) Let K be the supremum mentioned in (3). Trivially K 5 B ( I ) , so suppose 'p(1; a) E Av(J, a), C('p(S;a) < H ( 1 ) ) . By 6.2(vi) and the definition of C, the rest is easy. (2) !hivial by 6.2(vi). (1) By (3), (i) (ii), and trivially (ii)* (iii). +i) * -@) So let 6 E A, H(6, l M ( ) < A, so by the definition there a)) < A, and some J s $(a, a) is not orthogis $(z,a),such that a($(%, gonal to any I b d on tp(6,1M1). Using 6.2(ii) to see what is the requirement on a, we see that we oan wume a E 1611. By the proof of 6.2(ii), for some and F E WI, $(s,a,c)l-q(x, a) is consistent, but not realized in -,(ii) * +i) So let 'p(z; 61,$(z,a, 6)oontrctdiots (ii). As C('p(z; a)) c A, for some n C,,('p(z, a)) < A, and we can eaeily prove by induotion on n. that the existenoe of suoh cp, $ oontradiots (iiil.

wl.

COROLLARY 6.4: (C2)If M < A iwlies 'p(N, a) c 1M1,if 'p(N,@ c 1MI.

G

N , then M

E M ,I = " ( ~ < ~ X ) V ( X ,

N ifl 3 E M ,C('p(z, a)) a))',x < A i w l i ~

LEMMA 6.5: (1) [Cl, 2, 61. If bl 4" A , bl W F - m i ~ ~ t ' N d , k Foorcetrzcctible over A , thm ikf 4"N. (2)[C2]If A, (i < a) W increming, B
4.

(6) [C2] If A

294

[m. V, 4 6

YORE ON TYPES AND SATURATED MODEU

( 8 ) Condition (6) lrolcEs for p = K J T )if condition 5 hO&, wave condition 6, and replace condition 4 by h 2 !TI+

+

we can K,(T)++ so

W).

(9) [ c l , 41 Any regular p’ 1 h, sat~s$e% cond&%m6. (10)The condition on in C2 is equivalent to :if B is F-constructible

x,

x

x.

over A and IAl < then IBI < (11) [Cl]Any F - d d set ie (the universe of) a d e l .

Proof. (1) If the conclusion fails, then there are a E 1M1, Q and c E IN1 - lMl such that ~.“(~‘*z)Q(z, a)”, krp[c, a],x < h (by 6.4). By the definition of <”, tp,(A, IMI), tp(c, 1M1) me orthogonal, hence weakly orthogonal contradicting (C5). (2) Trivial, by the definition. (3) Immediate: first part by 6.4,second part by 6.5(1) and 6.5(2). (4) Trivial, by the definition. ( 5 ) Define inductively BIfsuch that: M , is ITI -compact, B E 1M,1 , a, E ~iK,+l~, tp*(M,, B u U j < a Z j ) does not fork over B u U,<,iZj and BI, + is FpT,+ -constructable over IM, I u 4, Ma is PpT,+ -constructible IBIfI) 2 A, hence M , <”Mi ua,,soas 6.5(1) over Uf Mf.By 6.5(4)a@,, ill, <”Mi+1, 80 by 6.5(3) ill, <” M a SO H(Z,o>^. * *-8f(,,), 1Mo1) 2 A for every n c w, i(O), .. .,i(n) c a. Hence we finish. (6) Left to the reader. (7) Left to the reader. (8)-( 11) Immediate. +

DEBTNITION 6.1: A model M will be called good (or K-good) if a E 1N1, l = “ ( 3 ~ h ~ ) ~impliea ( ~ , ~ Iy(iK,Zi)l )” < A; and it is called h-good if in a))] s ~Q(BI, a)l. (If condition 3 holds addition for 8 E lMl ~[C*(Q(Z, this is equivalent to h [ C ( ~ ( a))] z , s Icp(M, a)l.) LEMMA 6.6: [C2] M &s g o d ifa E M implie%IQ(N,8)1 < C(Q(Z,a)). Prmf. See 6.4.

THEOREM 6.7:[Cl,2,4,5,6]If M is F-rrwxid and good, 7G a function from Car K to cardinals, h(h) < A, t k n thre i8 a nzodel N , BI < N , N i.9 h-good, F-maximal. We first prove:

LEMMA 6.8: (1) [Cl, 2, 4, 5, 61 8wppme M is g d , IAI < hK, N i s Fconstructible mer lMl v A . Then N is good.

OH.

v, 8 61

QUANTIR'IEaS AND TWO-OARDINAL THEOREM8

295

(2) [Cl, 2, 4, 6, 61 8wppoee M , (i < 6) ie good and F - d m l , i < j < 6 * Mi < M,; cf 6 < AK and M* ia F-constructible over uf
Proof. (1) Suppose N is not good, so that is cpo(s,a)@E INI) such that k"(3%)pO(z,a)" but jcpo(N,a)[ 2 x. Choose N* c N such that IIN*II < x (see Cl), A u a E IN*I, tp(IN*I, IMI) does not fork over IN*[ where IM*I = n IN*I, and M* 41' bl (see 6.4) and N* is F-constructible over PIUA, p*luA, and N is F-constructible over (MI U (see IV, 3.3) and N* is F-maximal (use C6). By 6.5(7) N* <x+N, but Icpo(N*,ao),)l < x < (cp,(N,a)l, contradiction. (2) A similar proof. (3) By (C6) tp(co, IMI u a) forks over IN], so let CB[co, a,&], e(z,a, 6,) forks over [MI, and B(z, y, 6,) k cp(z, 6) where 6, E M , a 5 6,. So let E(z,y, 6,) = (Vz)[#(z, a) -P e(z,z, 6,) = B(y, z, 6,)], Then clearly co/E$Meq,and I{c/E: c ~ p ( M , a ) } IT1 [(C4)] there is a < x such that for each equivalence 6) u 6 either I{c/B:c E p(dA,, 6))( < x relation 4 = E ( z , y, a) a E ?(Ma, and for each c E (p(Md, 6) there is c' E tp(Ma, 6), kE[c, c', a] or cE cp(M,, 6))1 = x, I{c/E: c E 9 o f a , 6)}1 2 N x ) . so by 6.8(2), (3) cp(Jf,, 6)is constant for a < i < 8. In both cases we get that is good. Let a.

4))

4)

m,

296

MOBE ON TYPE8 AND SATURATED MODELS

[UE.

v, 4 6

be the first a for which we cannot define Ma + (clearly if A 2 11M11 ,and h 2 x for every x E Car K,x < a,T stable in x then a. < A+). Clearly Ma, is aa desired. If h is a function into h:x regular, and IAI < x , N F-constructible over A implies llNll < x}; let Kh = (F, Wh,h(h),p), where Wh = ( ( 9 9 h(x)):(cp, x> E w). D~~lwrrrow 6.2:

#Y

$9

+

+

LEMMA 6.9: If h is imrm'ng h(hK) 2 ITI p+ h(F) and ae above; then Kh 8&i8@ c n if K 8&&@ c n for n = 1, 6 and h[CK(v(x,ti))] = Cg+p(z, a))and eimilarly for C*(tp(x,a)). +

Proof. Trivial. CONCLUSION 6.10: [Cl-61 If h i8 cc8 in Dejnition 6.2, A(A) = hl(h)+, h ( a ) < 00, then there i 8 a F - d m u l model M such that for every a E ]MI, h"d~9 a111 = IdW all.

Proof. Apply 6.7 to Kh. Unfortunately not all the conditions necessary for 6.10 are satisfied in all interesting cases. E.g., F; satisfies C1-6 for K 2 K,(T),but for E", we have difficulties in C1, 3, 5, and for FfTlwe have difficulties in C1, 3. Hence we suggest here an alternative set of conditions, denoted by C*i (if it is the same as the previous condition, we do not redefine it). Then we prove a parallel to 6.7 and 6.8 this time ensuring only the existence of a suitable model. We work in P a . Condition* 1.F is as in IV, Section 1, satisfying Ax(III.l) and Ax(X.l).

Condition* 2. (a) W is a set of triples (p(z,tj),@(tj),x). (b) If x appears in W ,IAI < x, B is F-constructible over A then IBI < and > IT1 +A, 2 A, and x is regular also.

x;

x

x

We define C(p(x,a)),H(I), C@), < A as before; and Condition* 6. At lest one of the following holds: (a) If A, E B (i < p ) is increasing and eaoh A, is F-maximal in B, then &,, A, is F-maximal in B. (A is F-maximal in B if i! E B, 5 $ A implies tp(5, A) is not F-isolahd.) (b) If A, (i ?j x E C d K)is increasing and contimoue then for some i < x, A, is F-maximal in A,, provided that lA,l < x . Note that we weaken C1, change C6, and add:

m. v, 8 61

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AND TWO-OABDINAL THEOREMS

297

Condition* 7. If p eBrn(A),a E A, A = dcl A, tp(z, a) ~ p C , = A n dcl@u p((Eeq,a)) and p is F-isolated then p C is F-isolated.

Condition* 8. If c E dcl(A u a) - dcl A, tp(8, A ) is F-isolated then tp(c, A ) is F-isolated.

DEFINITION 6.3: A = dcl A is called good if for every tp and Itp(A, 7i)l < C(tp(Z, a) (remember we are working in F).

E A,

LEMMA 6.11: (1) (C*2, 68) If A = {af:i < x}, x appears in R, then there is a closed unbounded 8 c x such that 6 €8,cf 6 = p, impliee {aj:j < 6) is F-?ruz&nd in A; 80 C*6(b) holds. ( 2 ) We can replace in 6.3, 6.4, 6.5 and 6.9 (C2) by (C*2a),(C,) by (C*n) (for n # 2 ) (and get true lemmas). (3) (C*2) If A is good, IBI < A=, then dcl(A u B) is good. (4) (c*2) I f A, (i < 6) ie increasing, each A, is good and of 6 < hK A, is good. then UfCd (6) (C*l, 2, 6, 7 , 8 ) If A is good, B is F-cawtructible Over A then B ia good.

Proof. (1) For every 4 < x we define inductively on a s p i, < x, Aa c A, [A"] < x. We let i, = 6 and if A" is define, i,,, is minimal i > i, such that A" E { a j : j < i,+,}, and for limit a i, = Uflcplifl. Now Aa is a maximal set E A which is F-constructible over {uj:j < ia}, so by C*2 ]Aa] < x. Clearly i, satisfies the demand on 6; so i, ~ 8 =, (6 < x : cf 6 = p, {a,: i < 6) is F-maximal in A}. Hence 8, is an unbounded subset of x. Clearly the closure of 8,prove the lemma. (2) Easy.

(3) Suppose not, then for some E C = dcl(A u B), h and tp, C(tp(z, 8)) s h s Itp(C, a)], so w.1.o.g. we can wume [ A ]= A, and we find A, c A, lAll < h such that 8 E A,, C(&c, a)) s himplies #(A, a) E A,. We can wume that tp,(B, A) does not fork over A,. By Exercise 6.7 we get A , be a F-construction of B over A, s.t. A U { a j : j < i} # dcl(A U { a j : j < i}) implies a,Edcl(A U { a j :

j < i}), and dcl(B) is not good, and a is minimal under those

conditions, so by 6.11(3) a is limit. Let Bo = dcl(B). Let ~ E B and , 6)) = < Iq(B0,b)l,so by the be such that C(q-~(z, minimality of a, Iq-~(B',,b)l = x, and for every < a, lq-~(B~,g))( <

x

x

a

x,

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[OK.

v, 8 6

where B, = dcl d,.Let C = dcl(6 u rp(CQ, 6)),C,,= B, n C; again by the minimality of a fl < a implies IC,l < x , and clearly for limit 6 C, = Us<(lC,. Hence by 6.11(1) (or C*6(b)), as cfa = x , for some 6(0) < a, Cd(0)is F-maximal relative to C,. Let y < a be the la& ordinal such that C, = C,,) (there is one, as for limit 6, c d = U, < d C, and lCol > IC,o)l) and let c E C , + ~- C,. By C*8 tp(c, By) is F-isolated, . hence by C*7 (after a cosmetic treatment of rp) tp(c, C,) is F-isolated; so tp(c, CN0))is F-isolated, c E C, - cd(o), contradiction to the choice of

w.

THEOREM 6.12: The parallels of 6.8(2), 6.8(3) and 6.7 (even if we repkcce M by a good 8d) b&, hence abo of 6.10. Proqf. Ewy, by checking the proofs.

THEOREM 6.13: Let K = (F, W,A, p), K 8dbfi C2, and h > IT]. (1) If F = E,K ~ ( T S )K < h, p = K,(T)< h t h (a) C1-6 are satie$ed, and ale0 C*1-8, C*6(a), (b) in C2, the C o n d d t h on x is equivalent to:

x ie regular, and T ia &able in arbitrarily large x1 < x. (0) F - d m Z &I equivalent to F-8atUTated. (2) If F = q,&(T) Kr(T) s K < A, p = K J T ) < A, then (a), (b), (c) of (1) holds, except possibly (C*7). ( 3 ) If F = pK,ha(T) K ~ ( T S )K < h, p = ~~('ll) < h, t h (a) C1-6 b&, and C*1-8 b&, (b) in C2 the condition on x i s equkxdent to: x > IT1 iS regular and for

+

+

x1 < x,x : " ( ~<) x,and AA' < x when A'

c A,(T), F - d d is equivalent to F-eaturded. (4) If F = F b l , h > IT/, p = ITI,and c 2 kg eatiecfied, then

every (0)

(a) C*l,2,4,5,7,8aresatisJied(andC2,4,5)and whenJTIisreguZar, (or h > I TI +,and we cirange p to I TI +) C6, C*6 too is 8dis@?d, (b) in C2(b) the condition on x is equivalent to: x 2 h is rqu.lar, min Car K satisfi the condition ae etated and for e u e y xI < x, x;lT1 < x orx;=(m < X , (0)

F--'ml

i s equivalent to ITl-compccct.

Proqf. (1) Parts (0) and (b) are quite immediate, so let us prove (a) remembering C1* C*l, C2 =$ C*2, C3 = C*3, C4 = C*4, C5 = C*5, and C*6(a) * C6, C*6(a) A C*2 C*6(b). Condition 1. Immediate (see Table 1 in IV,Section 2).

OH.

v, 8 61

QUANTIFIERS AND TWO-CARDINAL THEOREMS

299

Condition 2. Easy by (b).

Condition 3.Let Af be F-maximal, a E IMI, t(3z)tp(z,a). We can clearly , r € # ( A ) such that find A E [MI,a E A , IAI < K ( T )and (a)C ( r ) = C(tp(z, a)),i.e., r k #(z,6) implies C(tp(z, 8 ) ) 5 C(#(z, 611, (p) no extension of r which is a type over IMI, satisfies (a)and forks over A. Choose any T,, r E rl E .S1(IMI), which satisfies (a).Suppose H(r,) < C(tp(z, a))and let c realize r,, so for some E = E(z,y, 6)(6 E 1Af1) c/E $ lAfeql and for some #(z,61) (in iKeq), C(#(z, 6')) < C(tp(z, a)), C#[c/E, 611. Choose c' E !MIsuch that tp(c, A u 6 u 6') = tp(c', A u 6 u 6') and then r u { E ( z ,c', 6)) will contradict (a)or (8).

Condition 4. Immediate. Condition 5. See IV, 4.10(2). Condition* 6(a). Easy, by IV, 4.3(2). Condition* 7. Let ( p , B) E F, IBI < K , and choose B, E C, lBll < K such that tp,(B, C) does not fork over B,. By the definability Lemma 11, 2.12 and IV, 4.3(2), it is clear that ( P ~ C , B , ) E F .

Condition*8. If ~(z, a, 6) (6 E A ) define c, and r I- tp(si, A), r closed under conjunctions then r' I- tp(c, A) where r' = {(3g)(#(g,6')A ~ ( za,,6)): #(ij;6') E r} so the result is clear.

(2) The proof is like the proof of (1) only in the proof of C1, Ax(X,l) we need ha(T)4 K ; and in the proof of C3 we need ha(T) 5 K (remember that by IV.2,16(1) A2(T)IAa(T)).Those are the only places we use

&(T)5

K.

(3) The proof is like the proof of (l),only in the proof of C1, Ax(X.l) we need A3(5!') 5 K (remember P ( T ) 5 A3(T)by IV, 2.16( 1)in the proof of C3 we need A3(T)5 K . We use h3(T)5 K in no other place. (4) Parts (b), (c) are easy.

Condition* 1.Immediate (me Table 1 in IV, Section 2; for C1 Ax(VII1) is missing and Ax(V1) for singular K ) .

Condition 2. Holds by an assumption.

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MORE ON TYPES AND SATURATED MODELB

[OH.

V,3 6

Condition 4. Immediate. Condition* 5 . By IV, 3.2, N is F-atomic over B, hence tp(c,B) is Fisolated. By IV, 4.3(5), tp(c, M) is F-isolated, contradicting M being F-maximal. Note that IV, 4.3(5), implies C*6a holds when p > 12'1.

Condition* 6. Part (b) holds, 80 suppose A, (i Ix ) is aa mentioned there, and for each i there is Z, E A,, 5, q! A,; and tp(Z,, A,) is F-isolated. Then for each i there is p , c tp(Z,, A,), 1p,1 < I TI such that p , k {Z # E: 5 E m(oA,} where m(i) = @,). If cf i = IT1 there is h(i) < i such that p , is over Ah(,),hence by 1.3 of the appendix there are myj, so that So = {i < x : cf i = 12'1, h(i) = j,,,m ( i ) = m} is stationary. By C2 there is a F-maximal bl, A,, c ]MI, lldlll < x. As x 2 A, > 12'1, and IT( are regular, there is L' E L ( T ) ,IL'I < 12'1, and a stationary 8, G So such that p , is a L'-type for each i €S1. By part b of 0.13(4) it is eaay to check that IAI < x implies 18; ( A ) [ < x, hence by Exercise I, 2.6 there is an unbounded 82 c 8, such that {Z,: i ~ 8 is ~E-indiscernible ) over Are, an easy contradiction.

x

Condition* 7. Eaay by the definability Lemma 11, 2.12. Condition* 8. As in 0.13(1).

THEOREM 6.13(5). If A > K 2 K,(T),F = F:,p = K,(T),then (a) C*l, 2a, 4, 6, 7, 8 hold and, in fact,C*6a holds. (b) In C2 for regular x > 2ITI, the condition on x is equivalent to (VX1 < x) (x:""' < x), and for regular x <2ITI the condition on x is satisjied if (c) F maximal is equivalent to K-compact. (d) Though C5 is missing, the parallel of 6.7 holds. Proof. ( 5 ) (a), (b), (c) No new point (we use IV, 4.3(4)). (d) As in the proof of 6.7 we define by induction on i < a. M,,Cr; such that N , is increasing, M , = M , M t + l is F-constructible over M , U a,, for i limit M , is F-constructible over u,,,M,. By the correct choice of the at's we can ensure M., is h-good provided we have proved each M iis good. We prove this by induction on i , so by

OH.

V, 8 61

Q U ~ T ~ ~ I EAND B S TWO-OILBDW~ZI THEOREME

30 1

Lemma 6.11 the only problematic case is i = S, S limit, cf S 2 A,. Clearly, U,M, is K-compact (by 6.13(5)(c)),henceMd = uj
CONCLUSION 6.14: Suppose M E N , M # N , P ( M ) = P ( N ) (Y-a one place ~ e d i d eand ) P(x)is not algebraic. (1) If T ie countable,for every h 2 IlNll there is a mode2 N , , N c N,, llNlII = A PW,)= P ( W .

( 2 ) Poreueryh 2 p 2 ITI,tlrereieanzodelN,,llNIII = A, IP(Nl)I = p. (3) If M,or N,is ITI-mwpct, then for every h 2 IlNll there i s N , ,

N c N,, P(N,)= P ( M )and llNIII 2 A. (4) If M or N is F - d d , and F satis$m C 6 , l thenfor every A 2 IlNll there is N , , N s N,,P(N,)= P ( M )and llNIII 2 A.

Remark. Note that C5 deals with F only so it is meaningful to say "F satisfies (26".

m,

Prmf.(1) Let K = (Fko, K,,KO>, wy,= { ( P ( Z ) , (Vy)(y = y), x)). C h m c E IN1 - 1611,A c IMl,IAI < K ( T such ) that tp(c, 1M1)does not fork over A ;now add the elements of A aa individual oonstmts of Q, and get a,. In Q, for every model X,, A E 1M,1 henoe there is Y E S(lM,l) p d e l to tp(c, 1M1)so H(r) = 00 henoe C*(z = x) = 00. So apply 6.13(4), and 6.12. (2) For oountable T it follows d y by (l),for unoounthble T apply 6.7 for R = (F&(m, Wp,ITI + K ~T( ) ,KJ T ) )and get the result when T is stable in h and y . For the general case apply Exercise VII, 5.1 to the ( ~ ~ just 8 8 mentioned. (3) Apply 6.14(4) to F = FfrI (by 6.13 the h p t h e e i s of 6.14(4) holds). +

+

MORn ON TYPE8 AND SA!CURATED MODELS

302

[OH.

v, 8 6

(4) Define WX,as in (1) and let K = (Fly W$, x, x ) where x = (21Tl)+. Choose c E 1M1 - IN1 and define c,, Ni (i s A) inductively: N o = N, Nd is F-constructible and F-maximal model over UjCdNj; tp(ci, Ni) is a stationmization of tp(c, IMI),and N,,,is F-constructible and F-maximal over Ni u {c}. We should prove by induction on i that N N,and N ( x IN,I u {c,} which is easy by 6.6.

+

CONCLUSION 6.16:Suppme h ( i ) (i < a ) i s an increasing sequenoe of rqular cccrdinala, M a rn.o&l and for every cp(x, and i < a there i8 I&, m h that for every B E IMI M C $!!.1,[7i] if Itp(M, B)l z h(i) (and of couree T = Th(M) i s etable).Thenfor any non-decreaSing sequence p ( i ) (i < a ) ofcardinale ZITI, Tlrasanzode1N;mhthatforanya~INIandtp(x, y ) Itp(N, ~ )=l min(tc(i):N C It",[i%]}.

a)

Proof Left to the reader (use VII, Section 5 ) . Note that w.1.o.g.

I4 G ITI. THEOREM 6.16: suppo8e T i 8 8't+Y8tabkS 11M(1= A, IP(H)I = p, h > p. suppose ah0 that (i) A* 2 p* 2 2ITI, (ii) h 2 A*, p 2 p*, (iii) A > h * o r h = h * , c f h > p o r h = h * , c f h s p * . Then H hae an (elementary)8ubmOcEel N , IlNll = A*, IP(N)I = p*. Remark. See 6.17.

Proof.Suppose first that h > A* or h

= A* = of A, so by Exercise I, 2.6

there is I c 1M1, 111 = A*, I indiscernible over P ( M ) . Hence I is orthogonal to any J c P(a),so if N 1is the &,-prime model over I then IP(N,)I 1; 2ITI. Now over I there is a Fg,-constructible model N, E M , so there is a F;to-primemodel N , over I, N, c N,.Hence IP(N,)I s IP(N,)I < 2ITt,and llN,II 2 111 = A*. If we choose A c P ( M ) , IAl = p*, and N &B Fh-constructible over A u I, N c M,then llNll = A*,

IP")I

= P*.

NOWsuppose h = A*, then let h = Z:(
CH.

V, 5 61

QUANTIFIERB AND TWO-CARDINAL THEOREMS

303

where A is m y subset of P ( M )of cardinality p* (we leave the cheoking that N eatiefies our conclusion to the reader). We are left with the case cfh > IP(M)I;and then we can choose JI G J,, 1511 < No and J, is based on JI and Av(J,, JI) is stationary we can assume that IJ;I does not depend on i, and that for some finite A c M , tp,(JI,P(M) U A U Ul<*Jy)does not fork over A and stp(J;,A) is constant. Then we proceed as in the previous case.

u

THEOREM 6.17: I n 6.16 we can replace (i) A*

u

2 p* 2

(i’) A* 2 p* 2 ITI.

Pro($. Let F = F&Y Tv

=

K

2ITI by

{
(F,

w,I TI +,HO),

so the conclusion of 6.13(6) holds and let us work in CQ.Now we can find I E M y 111 = A* and Z E M such that I is independent over (P(M) U Z, Z) (for A > A* or A regular use a variant of 111, 4.23, in the other cases work analogously to the proof of 6.16 and replace Z by a set). Choose M, < M y IIMoII = p, Z u P ( M ) E IMol, and w.1.o.g. I is independent over (lMol,a), so it is easy to show a E I 3 H(a, Z) = co. Choose Ml < M,,C E M,,IlM,II = p* = ~ ~ P ( M land ) ~ ~I yis independent over (lM1l u P ( M )u Z, a). Let A = dcl(lMll u U I), so A c H,A = dcl(A), we shall prove that A is good, and this suffice (take N s M F-constructible over A, so by 6.11(6) N is good, so 80 IP(N)I 2 IP(NlI 2 p*, IP(NI p*, by P ( N ) P ( A ) 2 wfl), hence IP(N)I = p*,andA E N * 2 IAI = A*,andclearly IlNll s IAI IT1 = A*). Suppose A is not good, so for some si E lMll u U I , C(&, a)) s p* but Itp(A,si)l > p* (see 111, 6.3(1)), and choose a finite I, E I, s i ~ 1M11u U I,. As a)l > p* 2 u U I,)] IT!,there is c ~ q ( A , a ) cEdcl((M1l , U UI)-dcl(lM,I U UI,). By changing if necessary I,, we can assume cEdcl(lM1l U I, U {6})-dcl(lMll U UI,) for some ~ E I SO . clearly tp(6, IM, I u U I ~ )tp(c, , u UI,) are not orthogonal (as they are not weakly orthogonal and the first, not forking over M I ,is stationary). But the first, being a stationarization of stp(6, IMll) E stp(6,M0) satisfies H(6,1MlI U UI,)= 00, whereas by the choice of c ,

=

+

l(lJfl

+

wlI

u

H(c, IMll u 1 0 ) C(P)(2,a) 5 ( p * ) + contradiction. Note that instead of 6.13(6) we can use IX, 1.2, 3.

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[OH.

v, $ 6

Remark 6.17.4. The superstability assumption in 6.16 is necessary, counterexamples can be built as in IX, Exercise 1.2. QUESTION 6.1: Try to eliminate the stability assumption from the general presentation. EXERCISE 6.2: Generalize the theorems in this section to the c&88 which deals with dimensions of indiscernible sets instead of cardinalities of sets v ( M ,Si). That is suppose not W is given, but a function C, whose domain is a family of indiscernible sets, and C ( I )is a infinite cardinality aa in Condition 2. Assume that if C ( I ) ,C(J)are defined and distinct then I , J are orthogonal. Let H ( I )be the supremum of A such that H ( J ) < A implies I , J, am orthogonal; and define H ( p ) , H(a, A) and B IPMI 2 KO. (ii) T has a model M , such that for no N M c N , M # N , P ( M ) = P(hT). [Hint : InL(T)there are only an equivalence relation E and a one place prediccLte P.] EXERCISE 6.7: Prove 6.2 for not necteeearily stationary p, when SiEA.

EXERCISE 6.8: Prove that in 6.14(1) the countability of T is necessary even when it is superstable.

OH.

v, o 71

305

RANKS REVISITED

R" = w + l , = P" U Q" U R", Pr = {w:v Q 7 or 7 4 U E " ' ~ } , E is a three-place relation E RN x RN x Q", EN = { ( n , B , ( p , { n , P } ) ) : n < B G w , p E n 2 , 1 < n , k : < n } ,

9,"= { ( p , In, PI> :n < P G w , 7 Q p E "21, Ff is a partial one-place function,

r

F,N((p,{n,B}))= p k when k is maximal such that p r k

4

v.

Using the automorphism of saturated models we can see that T is superstable. Let M be the submodel of N with universe

PN u w

u U "2x{{n,/3)}. n
Then M,N is a counterexample.]

Remark. On categoricity for logics with generalized quantifiers, see the end of VI, $6,

We use the convention of Section 6, and A = A;, be a function aa in Definition 6.2, K = K".For simplicity we deal with p(z;jj) rather than

94% !i)*

DEFINITION 7.1 : We call q(z;8) K-algebraic if C(p(z;Ti)) < 00; we call p(x; a) K-minimal if it ie not K-algebraic but it hae no two contradictory extensions which are not K-algebraic; we call p(z;8) weakly K-minimal if it is not K-algebraic but whenever #(z,6) forks over 8, p(z;a) A #(z;6) is K-algebraic; and we say p(z;a) sK#(z;6) if for some x for every K-goodmodel bl 3 8 u 6,Ip(bl; a)( 5 ($(bl;611+ x.

306

MOEE ON TYPES AND SATUBATED MODELS

[av, .5 7

CLAIM 7.1: (1) The only information f r m K needed to Dejinition 7.1 is

wg = {(dx; 9)s $ @ ) ) : f o r some x ( d x ; 91, $(@,x )

wK}

(provided that K satisfies C1, 2, 4, 5, 6) for 5 , being aa we want. Alternatively we can define ~ ( xa); I , $(x; 6) by: in (Q, 6 ) i f we add to W, ($(x; g), 9 = 6, Xo), a d get K 1 , ~ ( xa), is K,+ebraic.) (2) For every set Wa = {(tp,(x; &), $,(&)): i < a} we can jind a K satisfying conditions 1-6, such that Wg = Wa. So henceforth we aamme K Satisfies C1-6, h k > (2'IT')+. (3) I n particular, for the empty W",in (2) we get the trivial K,which we denote by KB.So K,-algebraic, KB-minimal,weakly KB-minimal,~ ( xa); $(x; 6), means algebraic, minimal, weakly minimal, p(x; 3) I $(x; 6) respectively; and for KB, pgcs,o,= {(Fkx)p(x;9):k < KO} (see 6.1(2)). (4) If p(x; a) is weakly K-minimal, the dependence relation on V(Q,a) " b depends on {bl, b,, , b,} iff H(b, {b,, ,b,) u 3) < 00 (or equivalently, C(b, a U { b l , . .,b,}) < 00) satirrfies the axbm.9 of dependence relation mentimed in 1.14 with transitivity ale0for non-inde(pendentsets. (6) If p(x; a) ie weakly K-minima2 and hp[c;a] and H(c, 8) = 00, then stp(c; iZ) M regular. Tire family B = { p :p a K-minimal type} is regular and ~ ( xa), is @-regular (see DeJinition 4.5). (6) For every Q = Q ( x ; % ) : p(x;a) is not K-algebraic iff ii realizes {-y~;p~% where } p t = pv =del{rp:( 3 ~ ) q ~ psee ; }6.1(2)). (7) The formula p(x; ii) is not weakly K-minimal nor K-algebraic iff

... .

. ..

there are f m u h #(x; 6,) (i < l T l + )such that: (i) $<x;6,) Q(X;a), (ii) #(x; 6,) is not K-algebraic, (%) for i < j < l!Z'l+, $(x; 6,) A #(x; 6,) is K-algebraic. ( 8 ) ~ ( z ; aG) K $ ( z ; 6 )iff for any F:r,,,-saturated modeZM<,"N, if ii u b E 1611,$ ( N , b) c 1M1 t k n p(N, a) c 1611. Proof. Easy, hence we leave it to the reader. (For (7) use 11, 3.12.)

DEFXNITION7.2: (1) We define Dm(p,d,K ) as Definition 11, 3.2, but Dm(p,A , K ) 2 a + 1 iff for every A and finite q E p , there me mcontradictory #,@; a,) for i < A such that: (9 D"[q u {$@;a,)},L,KI 2 a, (i) I = {a,:i < A} is an indiscernible set, (iii) H ( I ) = 00, (iv) #,@; 9) E d.

Equivalently, there are a K-good M and pairwise explicitly contradictory r, (i < A) such that q, r, are over IMI and Dm(qu r,, L, K ) 2 a. W e define Mltl(p, L , K ) similarly. ( 2 ) For a stationary type p, let w K ( p )= 2 flow&): q a regular stationasy complete type, H(q) = GO) and wK(a,A ) = wK(stp(a,A ) ) . (3) W e call cp(x;8 ) K-weakly-minimal i f D[cp(x;a), L, K ] = 1. The following claim sum up the trivial facts concerning Definition 7.2. CLAIM 7.2 : ( 1 ) The two ver&olze of DejEnition 7.2(1)are really equivalent and the parallels of 11, 3.4 and 11, 3.5 holds. If M is K-eaturated,K ( T ) 5 K , A c 1M1, IAl c K , p is over A not K-abebraic, then for some q E Sm(M)extending p, H(q) = a,D(q, L, K ) = D(p, L, K). ( 2 ) If T is &uperetable, wK(p)b dway8 defined, Jinite, and S w ( p ) ; and it eatieJies the parallels of 3.10, 3.1 1, 3.13 and 3.16 when " I independent over A" is replaced by "I K-independent over A" which ~ n z e a n Is ie independent over A , and t p ( 4 A ) ie not K-a€gebraicfor any 5 E I . (3) If K ie trivial, w K ( p )= w(p),and D(p, L , K ) = D(p, L , a). (4) If ~ ( x6 ;)ie K-weakly-minimal, b [ b ; then w,(b, 6 ) = 1. Ah.xzye

a,

(6) Any weakly Kg-minimal fonnula ie K+ueakly-minid, but nd wnverady. However for trivial K , a fornzula ie K-weakly minimal iff it ie weakly K - m i n i d iff it w weakly m i n d . (6) f l w e (i) O(X, 9; 2) t- cp(z;6 ) A $(S; 61, (ii) c p b ; 6 ) (3%)8(x,g; a), (iii)for euey O(x, a) ie Kdqebraic. Then D[cp(x;a), L , K'j 5 D[$(Z;6 ) , L , K].

a,

a,

Proof. Easy, so we prove only (6). (6)Now we prove b y induction on a that (forevery suitable cp, $, 8, Z?)

(*I

Dm[+;

a), L, K ] r a implies D"'[$@;b), L , K'j r a.

For a = 0 or a limit there is no problem, so suppose a = p (*) holds for 8.

+ 1, and

Let p be any cardinality, and we can assume it is regular and bigger than 2", x = X, (the supremum o f all cardinalities x appearing in W,). So there are explicitly contradictory types p, (i s p) over a K-good

308

MORE ON TYPES AND SATURATED MODELS

model M, euoh that cp(x; a) E pi, and D(pi, L, K) r pi is closed under conjunotions, and let

[OH.

V, 5 7

8. We osn assume

(eo olearly qi(@k #(g; 6)). It sufiioes to prove: (a) Drn(qt9L Y 2 p. (b) The union of any p qiYeis oontradiotory, be&uee then by 7.2(1) we om find A E 1M 1 and q:, q, c q: E @(A), D(q:, L, K) = D(qi, L, K), hence I{q: i s Cc)l = p. A8 this holds for every suitable p, this implies Dm(+(g,b),L, K) 2 p+ 1 = a by the parallel t o Th. 11, 3.11, Ex. 11, 3.27.

Proof of (a).It euffioee to prove that every finite subtype of q, haa degree r p , eo aa pi is o l d under conjunations, it eutlFioee to prove that for every TI(%;a,) E pi -[(35)[e(x; a; a) A y1(x; a,)], L, gl r p. w e ~ u m cpl(x; e a,) cp(x; a). Now cpl(x; a,), ( 3 ~ ) [ 9 (#;~ ;5) A a,)], O(X, g ;a) A cpl(x; a,) satbii8~ the hypotheeia (for V(X;a), #(g; 6), e.(g;a) map.). Henm by the induotion hypotheeis

Proof of (b).If not let a* recllizes p q,'e, no by notational ohange we osn aaeume it E M (6.8(1)) and satisfies all of them. So for wah i f i u {8(x,d*; E)) is finitely satiefhble in M; hence it definea a filter D,on 8(M, a*, a), generated by {a E B(M, a*; a): hpl[a, a,]) for cp,(x, a,) €pi. p, 80 for some i # j D, = D,, but then Let le(M, a*, E)( = X , eo 2-' p, u p, is finitely satidlableyctontradiating their choice.

LEMMA 7.3: (1) Bwppoue p(x; a) s, #(x; 61, then there w a fopmula

e(x, 9; 5) BW:A thad: (i) e(x, Y; 5) k cp(x; a) A #(y; 61, (ii) v(x; a) k (330e(x, y; i), (iii) for each a ~ c p ( 6a), , except < IT1 of them, there is b euch that C8[a, b, 07 and 8(x, b, c') forb over a. (2) If in addition cp(x; a) ie toeakiy K - m i n i d , then for come el(y, Z) E PO(,;,,^, t= ( V Y ) ~ , ( Y , Q . (see 7.1(6)). (3) If in (I), in addition #(y, 6) b weakly K-minimal, t7m for eonte ea(Y, Z) E Pe,,cy;z,#pC(VY )ea(Y, a), ~ o ( Y X ; , 2) = e(x, y, f 1. ( 4 ) There ia a set p(Z; 8w:h thad cp(x; a), #(x; 6) eatbJies the con-

a)

OH.

v,S 71

RANKS BEVISITED

309

c l d o n of (l)(i),(3)iff iin6 satiejkz at leuat one formula of p(Z, g). 8imihrly for (2), and for (3), and for their combination. Proof. There is no new point (see 7.1(8) and the proof o f 6.2). DBIFXNITION 7.3: An n-witness (or just a witness) for ~ ( xii); sK#(x; 6) is a sequence 8 = (O,(x;ji,, gl, a): 1 s n ) such that (i) PI = (yo,. . ., Y , - ~ )2,, = (zO,.. .,z ~ - ~E )=, iihb, (ii) ke0(x,go, ZO, a) m ~ ( xa), ; (iii) e,+,(x, g,+,, z,,,, a) t- e,(x, g,, $, z) A #(Y,, 8 ) A el(zl,g,, 81,a), (iv) 8,(x, g,, Z l , Z ) k ( ~ Y , ) ( V ~ I ) [#I,%, ~ I ( ~5) I ,+ e,+l(x,jit,,, Z,+l, a)], (v)for some e*(gn,2,s 2 ) '=(V&,Zn)e*(gn,Zn, a). LEMMA 7.4: ( 1 ) If 8 = (8,: 1 s n ) is an n-witness for ~ ( xii); s, #(x;6), then for any a, b, (8,: 1 r n - 1 ) is a witneas for ~ ( xii), A el(%,b, a, a) sK#(x; 6),where (so yt take the role of yt + ,). (2) ~ ( xii); sK#(x; 5) iff there is a witness for it. Proof. ( 1 ) Trivial. (2) Eesy, b y 6.l(v)and the dehability lemma. LEMMA 7.5: (1) 8wppocre q(g) is an rn-type, ~ ( xji); a f m u l a and for every ii realizing q, ~ ( xii); 6 ,not K-algebraic; and for some &, for every h,-goo& M {lv(M,Z ) l : a E A2 realizes q}, has cardinality c A, < Ag. Then there are #(ji) E q (msuming q i s closed h for every good M , {Ip(M,ii)l: u d r c o n j u n c t h ) and k < KO 8 ~ &that M C #[ii]}hm cardidity s k, and C#[ii]implies ~ ( xii); G not K-algebraic. ( 2 ) I f i n ( I ) , whenever ii realizes q, ~ ( xii); is weakly K-minimal then we can assume: (i) E(Z, ji) ia an equivalence relation on #(Q), E ( f ,y) k # ( f ), (ii) E has finitely many equivalence claeses, and i n each of them there is a sequence a realizing q, (iii) i f k#(a) A $(6), then CE(2iY6)i$ i n any hK-good M 2 ii u 6, Iv(M9 811 = I d M , 6)1, (iv) there are 8(x,y, Z ) , 81, 8, such that if E(8, 6) then for some 5, 7.3(1)(i),(ii),(21, (3)bid8 (with41, = p), (v) C#[ii]implies y(x; ii) is K-weakly-minimal,

310

MORE OW TYPES AND SATURATED MODEL8

(vi) if K is trivial, R[q(x;si), L,A] c#[a'] impziea R[q(z,a'), L, A] = 1.

= 1 for every

[OH.

v, 8 7

a realizing q,

then

r,

Proof. (1) Let for every a, be the union of the following sets of formulaa: (a) { q m ,3,):n < w, i < a}, (b) the set of formulas saying: {z?: n < w } is a non-trivial indiscerndQ,@,) u g,, ible set over (c) the set of formulas saying H({z;: n < o})= m, (i.e., for each formula # = #(qg ) and 8(@) cplrand i < a let n(0)be as n in 11, 2.20 and we have the formula

u,<,

(VP)[

u,<,

e(g)

-+

V + A +(ZT,

w E anco, 1 nEw Iwi > n(0)

g)],

(dl q(!i,f)Clearly if bl is an h,-good model, a, (i < n ) realizes p, and Iq(M, ao)l < Iq(M, sil)l < - . . < Iq(M, Z,,-l)l, then r,,is consistent. It is also clear that if each F,, is consistent then each Fais consistent. Lastly, it is clear that if is consistent, A < A,, we can find a model M and a,, I, (i < A) such that si, realizes q, I, E q ( M , si,) is indiscernible over U,<,q ( M , a,) U U,<,a,, H(I,) = 00. Then we can contradict the hypothesis by A = A,,, and we find # easily. (2) Let us defhe 6 if for every Is,-good M 2 a u 6, Ip(M, a)l = lq(M, a)].First we prove {a/-: iz realizes p} is bounded, in fact < Ao = (2a1Ti)+ < A,, otherwise there are a, (i < ( 2 1 9 + ) realizing q, pairwise } an non -equivalent. By I, 2.8 we can assume {a,: i < ( 2 2 i T 1 ) +is indiscernible set. As not sio a,, and tp(sio-sil, 0) = tp(silnBo, 0) for no i # j does q(s,a,) S , q(s,a,), hence there is an indiscernible set I{ E q(Q,a,), H ( I { ) = 00, I{ indiscernible over a, u si, u q(Q,a,).By Erdos-Rado theorem (2.6 of the Appendix) (as the number of J E p(Q,a,), H ( J ) = 00, up to equivalence, is 5 21Tl) we can aasume that for a < fl < y < ITI ,I: I$ are equivalent. Using again the symmetry, for any i there is an indiscernible I, E p(Q,a,), H ( I , ) = co,I, indisoernible over U,#,si, u U,#,q(Q,a,) (let I, = I ; for i > 0), and by this we get a contradiction to the hypothesis by 7.6(1). Let si, (i < a < A O ) be a maximal family of pairwise non-- -equivalent sequences realizing it. Let r = r(Zl, 4)be such that p(s, El), p(s; 6a) satisfies the conclusions of 7.3 (1) (i), (ii), (21, (3) iff satisfies at lemt one formula of r. Let

r,

-

-

+

ri(%,

{$I(%;

2,) A #a(%,

31):#i@i,

%a),

#dzi, %) E r};

V,§ 71

OH.

-

RANKS REVISI!CED

31 1

6,, 6, realizes q, 6, 6, then 6,^6, satisfies at least one formula Z), E r,} is contradictory, so of rl. Hence q(Z) u {-,$(Z, af): i < a,$(Zl, there is a finite, contradictory subset. Clearly it involves only finitely many at’s so a < o and let k = a,and we can replace q by some $(if)E q (or $ = A ql, q1 c q, q1 finite). Also only finitely many formulas #:(a, it,) E rl(ZlyZ), me involved. Let so if

E(%, 32) = $@I)

A

A

(%)[v n

$%I,

@

A

Now if CE[6,,6J,then for some n(l),n(2),8,C #$&,

v $%%Yg ) ] . n

4 A $n(aJ6a, 81,

henpe : (i) cp(z, 6JY cp(z, a) are both K-algebraic, or for every A,-good M 2

6, u a, Icp(M, 6,)l

= lcp(Jf,

q,

(ii) similarly for cp(z, 6,), cp(z, 3). Hence a similar conclusion holds for ~ ( z6,,), cp(z, 6,). Now if k#[6,] there me ai and n such that k#:[6,, 4 3 , hence cp(z, 6,) is not K-algebraic and for every ir,-good iK, Iq(M, &)I = Icp(iK, S)l. Hence if#:[6,, at] A #@, 43, i # j then k,E[6,, 6,]. So clearly 1 is an equivalent relation on #(a), and a,, . ,a,,-,am representatives, and k8[6,, 621 iff C$[6J A $[6,] and for every hK-good ,iK 2 6, U 82, lcp(iK; 6,)l = Icp(iK; 6.JI. So 7.6(i), (ii), (iii) holds and (iv) is proved similarly. For (v) use 7.2(6) and for (vi) Exercise 11, 3.32.

..

LEMMA 7.6: (1) 8 q g w e that for every realizing q cp(x;a) is weakly minimal,and R[cp(s,a),L,A] = 1, Mlt[cp(s,a),L,A] = p and l{ltp(iK, a)l:a realizes q, a E Af)1 s A, for 80me &,. Then there are #, E a8 in 7.6(2) (for K = K,, A, > A,,) and k WA that in addition: C#@] i m p l i ~R[cp(z,a), L,A] = 1, Mlt[cp(s,a), L,A] 5 pk.

( 2 ) Suppose in addition that (i) for a realizing q, ~ ( xa); is minimal, (ii) (a)!=+[ti] and rpl(x,al)l-rp(x; a) implies {lrpl(M,&)I : & E M realizes tp(al,pl)}is bounded or @) T doee nd have t k f.c.p. Then there are formulas cpl(x, 9,) ( I < n) m h that: (a) !=t,hl[6]impZies vl(z, 6 ) i8 m i n i d , (b) the fornculcG (3~09 * * .s

R-1)[A #I(R)A ( v ~ ) ( c Pg) (~, I

i.9

w n a h t d with #(@.

VI ‘~1(zR))] ,

312

MORE ON TYPES AND SATURATED MODELS

[CE.

v, 8 7

Proof. Left to the reader.

THEOREM 7.7: (1) Buppose for a m A, < Ax for every hx-good nzodel M {Irp(M,a)l: 6 E iK,rp(x, a) not K-algebraic}lras cardinality < A,. Thenfor a m formula #:,for every 6, cp(x, a) iS K-algebraic iff I=#@]. ( 2 ) 8-e the hypothe& of 1 ?told%for every 9.8 u m e rp(x, a*) iS K-minimal, T totally tranecerrdenta. Then in addition to the conclueion of 7.6(2) (for q = 0 ) ) there are rpl(x;g1), El, #ly &o eatiafving the wncluSione of 7.5(2) and (v') h,h1[8] imp&ee pl(x,8) ia K-minimal, (vii)if 1=#~[6] thenfor 8 m 6, C#[G]and Q ( X ; a) sKrpl(z;6) sKrp(z;a); and conversely for every such a there iS euch a 6. Proof. (1) Let q(z,g)be a formula, and q be a type such that q ( x ;a) is not algebraic iff 6 recllizes q (see 7.1(6)). Now apply 7.6(1), 80 we get +(g) = A ql, q1 c q finite,and aatiafiea our demands. ( 2 ) Let $ be as in 7 . 5 ( 2 ) .We now define by induction on n < w , formulas rp,,(z,g) for q E ">2, such that, letting Cp"(z;y") = rpv = rp(z;g) A At
+

(*)

-

~(3g)(3.. .g,,. -),,sn>ap(gya*)

A $WV]

A ttcnza

which by (1) is equivalent fo: (*')

There are 6, a,, (q E ">2)such that CE(&a*) and for each q E "2 r p ( q 6) A AIet(,,) rpnll(z, iZ,,ll)*[ll is not K-algebraic.

If we can d e h e them for every n, we get a contradiction to the total transcendency of T.So for some n Q,, is defined for q E " > 2 but we cannot choose for 7 E "2. Choose a maximal set S E n2, such that (**I)

there a m 8,a,, (q E "'2 u B) such that E(6, a*) and for iE (0, l}}, every 7 E "2 u {qn
is not K-algebraic.

A

1c Kn)

~

~

~ l ~il i ) (w c~i l ~

B exists aa "2 is f i f e by the choice of n, 8 # "2, so we can choose V E " ~ - Sand , let e*(g",g*) be ( 3 . . . ~ ~ . . . ) , J [ E (Ag ,&EJ7&g)], ~*) whereJ= " < 2 u #-{vrZ;Z < n}. NOWlet #l(gv) = (3g*)[e*(r,g*) A #@*)I, rpl = pv; we leave the rest to the reader.

m. v, B 71

313

RANKS REVISITED

THEOREM 7.8:8tqpose that (i) e(z,y, a) I- d z , a) A #(?I, 61, (ii) ~ ( 2a) , I- (3y)8(~, y, a), (iii) B = R[#(z,61,L,KO], (iv) for every b, R[8(z,b, a), Then R[cp(z,a), L, No] I; a(@

L,No] I; a.

+ 1) or a = 0, R[cp(z,a), L,No] I; 8.

Proof.We can aasume a > 0 as for a = 0 the conolusion follows by Exercise 11,3.32(or 7.2(6)forKg).We prove the assertion by induction on fl for all cp(z; a), #(y, 6),8(z, y; a) and a. For B = 0 #(y; 6) is algebraic (by Exercise 11, 1.3), so let #(a, 6) = {bo, . .,bn}, hence cp(z; a) I- Vlsn8(z, b,, a), hence by 11, 1.7

.

R[cp(z,a), L,KO]s max R[8(z,b,; a), L, KO] ?j a s a(0 + 1). ISn

So suppose p > 0, and we have proved for every 8' < 8. We can assume Mlt[#(y; 6),L,No] = 1 (otherwise we can find n c No and #dy; 6,)(I< n ) such that M Y ;6) = v , < n M y ; Q, R[#,(?/; 611, L,No1 = 8, Mlt[#,(y; &), L] = 1, we use the assertion for cp(z; a) A (3y) [e(% Y; a) A #i(Y; FJI, #dY; 61), Y, 5) A #i(Y; 6,) instead V(z; a), #(y; 6), 8(z,y; c') resp., and use 11, 1.7:

R[cp,(z,a), L,No1 = max R[v(z;a) I
A (3y)(o(z, Y; a) A

#&, h)),L,KO]

By Exercise 11, 3.21 it is easy to prove: CLAIM: Rm[cp(z;a), L,No] 2 y iff there are formulas cp,,(z; a,,) for 7 E ds(oy) (see Def. 11, 3.3) such that (i) Q()(z; a()) = d z ; a), (ii)for v 4 7,cp,,(z;a") I- cpv("; a"), (iii) if 7,u ~ d s ( w yare ) incomparable by Q thenq,,(x;a,,),q,(Z; a+)are contradictory, (iv) for 7 E h(coy)VWcp,(z; , a,,). Suppose R[q(z,a), L, No] > a(p + 1) = a/3 + a, and we shall get a contradiction. Let y = ap + a + 1, and choose q,,(z, a,,) for 7 E &(coy) as in the claim, and let

#,,(Y, 6,) = #(Y, 6) A (W(Q2,Y,a) A P"(", a,,)). Notice that R[cp,,(z,a,,), L, No] 2 y,, where Z(7) > 0, coy,, I; q(Z(7) - 1) < w ~ , , + w , or l(7)= 0 , y,, = y = ap+a+ 1. Hence y,, > ap

implies R[#,,(z,&,), L, KO] = p (if it is < 8, by the induction hypothesis yn s R[cp,(z,a,,), L,No] 4 a@' + 1) I, ap contradiction). Now as Mlt[#(z, 6),L,No] = 1, there is a unique q €#(au 6 u U {a,,: 7 coy)}) -

314

MORE ON TYPES AND SATURATED MODELS

[OH.

v, 8 7

such that #(z,6)E q, R(q, L, KO) = 8, and let b realize q. So if 7 E &(coy), y, > a8 then #,(z, 6,) E q hence k#,[b, 6,], therefore v,(z, a,) A e(z, b, 8) is consistent. If y, > a@ implies v,(z, a,) A O(z, by 2) is not algebraic, then by the "if" part of the claim,

R(v(>(z,a( >) A e(z, bya), L, KO) 2 a

+1

contradiction. Hence there are Y E &(wy), k < W ; yv > ap, k ( 3 s k z ) [vv(zy a,) A e(z, by a)]; hence +*(z,a*) = (3 ~ [ v ~ (av) y A, e(y, z,a)] E q. By Exercise 11, 3.32 and monotonicity,

RCvv(z, a,)

A

(32)[8(z,2, 8)

a*), L,KO1

5 "*(%

5

A #*@,2*)1, L,K O 1 R[@(x,6),L,KO1 5 /9 s a@.

As p' =defR[@(x,6) A i @ * ( x ,e), L , KO]< /3, hypothesis R[vv(x;7iv)

A (32)[8(2,

2,

a)

A +*(z,

So by 11, 1.7, R[qV(z,aV), L,KO]

by the induction

a*)], L,KO] s

a@'

+ 1) s 4.

a8 < yv, a contradiction.

LEMMA 7.9: Let cp(x;Z),@(x;6),O(x,y , q be as in 7.3(1) and assume Y€d. ( 1 ) If D[@(x,6), L , K ] = B, and for every b D[O(x,b, c), d , K ] < n then D[v(z,a),d, K] s n(/3 + 1) except when n = 0, and t h 5=

(2)

D[v(z, a),4 RI 5 8. If in (1) /3 = k c w, then D[q(z;a),A , R] I n

+ k.

Prmf. (1) Using 7.2(1), (6) we can w u m e n > 0 , 8 > 0, and prove by induction on 8; suppose we have proved for every g < 8, but D[&; a),d, R] > n(/3 + 1) = n/3 + n, and we shall get a contradiction. So we can find an hK-good F&,-saturated model My V 6 u 2 E 1M1, mi < W , p i ~ c l , d Z, , , E ] M ~for ~ E I x , 0 < 2 s n, x = xg, such that : (i < x ) are m,-contradictory for 2 < n, 7 E 'x. (i) pi+l(zy (ii) D[v*(z,a"),A , R] > np where r) E "x, Q*(G

w = ~ ( za), A
By 7.2(1) we can h d a model N , M nfl, Cv*[a,,an], and b, E INI, Ce[a,, b,, a]. If for some 7, D[tp(b,, (MI),L,K ] < 8, we get a contradiction by the in-

OH.

v, f 71

RANKS REVISITED

315

duction hypothesis. Hence the number of possible tp(b,, IN]) is <x, and x is regular, hence by 2.6 of the Appendix we can assume that for hence we can msume b, = b. If for each every r) E "x, q = tp(b,, IN[); r) E "x, v*(z,a") A 8(z, by a) is not K-algebraic, then D[q(z,a) A O(z, b, a), A, K] > n contradiction, hence for some r) it is K-algebraic, so there is 0*, CB*[b,@, a] and CO*[b',a',Z] implies v*(z; a') A B(z, b', E') is K-algebraic. Let

v+(z;a+) = v*(z, an) A (3y)[e(z, y, a) A e*(y, a",a)], #+@; ) '6 = I/@, 6) A 8*@, F,a), e+(z,y,a*) = ~ ( z , y , a )A p+(z,a+)A #+(Y, 6 + ) . Clearly v+(z,a+)E tp(a,, 1611) hence D [ v + ( za, + ) , A ,K] > n/3, and clearly D[#+(z;6+),L,K] 5 D[#(x, 6),L,K] = 8, and for every b' D[O+(z,b', a+), A, K] = 0 ; and thus our hypotheses me satisfied; contradiction. (2) It is easy to prove that if D[v(z,a),A, K] 2 1 c w , then for some fhite A' E d, D[rp(z,a),A', K] 2 1. Hence we can assume A is finite; so for every formula vl(zs&) and 1 c w there is a type qZ(yl)such that D[vl(z, Bl),d,K] 2 Z if€ iZl realizes &(cpl). We now prove the assertion by induction on k; and the cams rn = 0 or k = 0 are easy. So suppose D[v(z,a),A , K] > n + k, hence there are rn < w , c p l ~ o l l Aand (i c x = x:) and an +good model 111, a i U u E , Uf c 1N1,suoh that: (a) D[&, a) A vi(2, at), A , KI 2 n + k, (b) the vl(z, 4) are rn-contradictory. We oan now continue as in (1) (for n = 1). The only difference is that instead of m r t i n g "for some r), p*(z, a") A O(z, b, a) is K-algebraic" we aasert "for some i,B(po(z,b, 8,a), A, 9)< n," where

v ~ zb,,s,a) = v(z,a) A cpl(s, a{) A e(z, b, a), and we replace cck8*[b',iP, i5J implies cp*(z, a') A O(z, b', E'] is Kalgebraic" by " CB*[b', a', a'] implies D[tpo(z,b', a', E'), A, rcJ < n" (SO

-,e*

E qvp)).

CONCLUSION 7.10: (1) If v ( ~ a), S K #(z, 6), #(Z,6) is K-weaklyminimal or even D[#(x;6), L,Kl = k < w , then D[p(z;a), L,K] is finite too. If there is an n-witness for v(z, a) sK#(z,61,then D[v(z;a),

L,K] < kn. (2) 8-e

316

MORE ON TYPES AND SATURATED MODELS

[CH.V , 5

7

(i) $(x, 6) ie K-weaUy-minimal, (ii) rp(xs 8 ) S K $(x, 6)s (iii) for m y B(x, 8) there h e*(g, 6 ) euch that $(x, 6 ) A 8(x,~ ) * h K-algebraic iff t=O*p,61. Then tirere h a forrnUla rp*(i~,6) euch tW +*p, 61 and +*[a1, 61implies D[rp(x,a'), L , Kl = D[rp(x,a), L , K]. (3) If for mne $*, ~$*[6*]impliee $(x; 6*) s rp(x; 6*) I#(x; 6.) then we can omit 6 from rp*. Proof. Left to the reader.

LEMMA 7.11: ( 1 ) 8-e

(i) r p ( 8~ ) s $(x, 61, cp(x, a) not dgebraic, (ii) R[#(x, 6), L,a]= R[$(x,a), L , A] = 1. Then R[&, a), L , a]= R[rp(x,a), L , A]. (2) If v ( x ,a) < $(x, b),then R(p(x,a),L, K O )< [R($(x,6), L, K,)lw.

Proof. ( 1 ) B y induction on the length of the witness, refining the proof o f 7.10 to deale with multiplicity. ( 2 ) By 7.8 and witnesses. DEFINITION 7.5: (1)For every complete type p E rSm(A)we define a rank L ( p ) as an OW or a by:

L@) L(p)

2 0

i f p is a type

2 8

if L ( p ) r a for eeah a < 8,

2

a

+1

if there is p,, p G p,, p, forks over A and L(p1) 2 a.

( 2 ) L(a, A ) is L(tp(8,A)).

DIFINITION 7.6: W e define a Q fl (the natural sum) b y double induction on a and 8:

EXERCIBE 7.1: Show that Q is commutative and aseociative.

LEMMA 7.12: Here the t y p wiW be mpZete.

( 1 ) p E q impliee L ( p ) r L(q); and if for nop 1 po, L ( p ) = a, then p 2 po, L ( p ) > a impliee L ( p ) = m; L ( p ) = 0 iff p w algebraic.

(2) If p E #"(A) does not fork over B E A t h L(p) = L(p 1 B). (3) L(p) R(PY L Y (4) T ie euperekzble i$ for every p, L(p) < oo. (5) If R[v(x, a), L, GO] 2 n, then for eome 6, b[6; a] and L(6; a) 2 n. (6) &(a-6, A) s L(a, A u 6) 0 ~ ( 6A). , Proof. (1)Trivial. (2) By (1) it s d c e s to prove by induction on a that L(p 1 B) r a implies L(p) r a; and for a = 0, a = S this is trivial so let a = /3 + 1. As L(p B) athereareZ,C, B E C,c'realizesp 1 Band L(4C) 2 By tp(Z,C)forksover B. Let 7i realizes p ; so we can assume firet that stp(a, B) = stp(Z, B) and then also tp,(C u 5, A) does not fork over B (by 111, 2.6(3)). Then by 111, 4.14 tp(Z, A u C) does not fork over C = B u C henoe by the induation hypotheah, L(Z, A u C) r: 8. Now tp(a, A) E tp(Z, A u C) and clearly tp(B, A u C) forks over B; hence it forks over A (by 111, 4.4, aa tp(5, A) = tp(a, A) does not fork over B). So by definition L(a, A) > L(E, A u C) r 8, so L(Z, A) r a. (3) See 111, 1.2(4). (4) The only if follows by (3) (and 11, 3.14). The if pcL1.t is left to the reader. (6) We prove by induction on n: for n = 0 it is trivial, for n + 1 we cutn find, n-contradictory formulaa v*(x, q) (i < (2ITI)+)such that R[tp(E. ii) A tp*(Z, a,), L, co] 2 n. So for some i q*(E, at) forks over 8. By the induction hypothesis for some 6 L(6, u ii;,) r n, kF[6, a] A tp*[6, 41. Clearly tp(6, a u ~ i ,forks ) over a, hence L(6, a) r n + 1. (6) It suffices to prove by induction on y that L(an6, A) = y implies L(a, A u 6) @ L(6, A) r y. For y = 0, y = 8 there is no problem; so let y = fi + 1. As L(B-6, A) > /3, there are Byaf, 6: such that A c B, tp(t~,^6~, B) extend tp(7in6, A) and forks over A, and L(aln6,, B) 2 y.ByIII,4.15tp(al,BU ~~)forksoverAortp(61,B)forksover~.As clearly L(B,, A u 61) = L(a, A u 6); this implies L(7il, B u b,) < L(ti;,fl U K1) = L(a1,AU 6) or L(6,B)< L(b,,A)or L(a,A U 6) = co or L(b, A) = a.Hence in all caees

L(a,, B u 6,) 0 ~ ( 6 B) , ~< L(B,A u 6) But by the induction hypothesis

318

MORB ON TYPES AND SATURATED MODEL8

DEFINITION 7.7 : Let acl,(A)

[a. v, 8 7

= {b : tp(b,A) is K-algebraic}.

THEOREM 7.14 (Qeq): (1) Suppose K satisfie0 C1-6, p = K,(T),Card K = {(2IT1)+}, T superstable non-multi-dimensional and ~ ( x=)(x= x) is not K-algebraic. Then there is a formula O(x) (with no parameters), types q t ~ S ( a c l p ) , c q@), and types r l , qi c rrES(ci)such that (i) If M +N , N, N are F&,-saturated, thew O(N)= O(N) iJ q , W ) = q,(N)for every i. (ii) r, is K-minimal, q, is o13{r,}-Bimpleand q , ( P q ) s acl,(r,(QeQ)u Z,). (2) Suppose in addition T is totally tranacendental, then we can aasume there are K,-minimal p,(x, 8,) E r,, and #i(x,Zi) E q,, Z, E a01 8, w h that "cl,(q),(ceq, Gc) u Qr(% a,) I- *&, a,! b w q , a,)

a.

Proof. We hereby prove Theorem 7.14(1) by a series of claims. Of course T is stable, but the use of stability is mentioned explicitly. We , K = {(2ITl)+}, and works shall assume K satisfies C1-6, p = K , ( ~ ) Card in FACT A :B = ac1,A implies B = acl B = acl, B and 6~ B if tp(6,A ) is K-algebraic.

FACT B : If 6~ acl,(A U 6)- acl, A, then tp(a,A U 6) forks over A . Proof. Suppose tp(a,A u 6) does not fork over A. Let C, = acl(A U a) n acl,A, C , = acl(A U 6) n ac1,A. As we are working in Ceq, tp(a,acl,A) does not fork over C,, and similarly tp(6,aclKA) does not fork over C,. As we assume tp(a,A u 6) does not fork over A, also tp,(acl(A u a), acl(A U 6)) does not fork over A , hence tp,(a U C , , A U b U C,) does not fork over A, hence tp(a,A U 6 U C, U C,) does not fork over A U C,. By 6.3(2)A U C, < A U C, U a, and H(a,A U C,) = 00. Similarly, H(6,A U C,) = 00. Let M be h,-good, {a$:i < be an indiscernible < be an set over A U C, U C, based on stp(8,A U C,) and {6,;i indiscernible set over A U C, U C, based on stp(6,A U C,), where is large enough, w.1.o.g. A U a U 6 U C, U C, U {a6,6$:i< G M . Then M n acl,(A) has power < hence for some i < tp(at, acl,(A)) does not fork over A U C,. Now M n acl,(A U at)has power < hence for some j,tp(b,,M n acl,(A U at))does not fork over A U C,. Hence b,#acl,(A U at), but clearly stp(ainb,,A U C, U C,) = stp(a-6,A U C , U C,), hence tp(ai"b,,A) = tp(an6,A), contradicting an assumption of Fact B.

x}

x,

x,

x} x} x

x

m. v, Q 71

319

RANKS RBVISITED

FACT C:If T is superstable there is a weakly K-minimal ~ ( xa). ,

FACT E: Suppose 1{1tp(M,6): 6 E M)1 < KOfor any M ,~ ( x6,) is weakly Kminimal, then there are n < w , a, realizing tp(a, 0 ) (1 < n) and non K-algebraicf m u l a B(z)(with no parameters) B(Qeq) E acl(rp*(Ceq,a*) u a*) where ‘p*(x,a*) = V,<, y(x, a,) is weakly K-minimal too.

Proof Like 4.11. FACT F: If ~ ( x6 ,) is weakly K-minimal, B(x) not K-algebraic, T superstable, B(Qeq) E acl,(cp(Qeq,a) u a), then there are complete types pi (i < io), e(x)€pi,piK-minimal such that: if M <; N are Fio-saturated, Dom pi u 6 s 1x1,then B(M) # B(N)iflsome c E IN1 realizes some pi.

u1

Proof of Fact F. Let {pi: i < io}be a maximal list of K-minimal types to which B(x) belongs and are pairwise orthogonal. The “if ” part is clear, so let us prove the “only if”. Let c E B(N) - B(H), be with minimal R(tp(c, IMI), L, ao), so tp(c, 1611)is regular. By a hypothesis a) such that c E acl,({b,, . . ., b,} u / M I ) , there are b,, . . .,b, E ‘p(Qeq, and w.1.o.g. we choose n, c, bl, . .., b, such that n is minimal. As c E N - M ,M N necessarily n 2 1. I f n = 1, then clearly tp(c, ]MI) ie H-minimal, hence not orthogonal to some pi (i < io), so some C’ E IN1 realizes pi, SO we finish. Now we suppose n > 1, and we shall get a contradiction; let 6 = .,b?-1), and choose finite A E 1x1 such that zi E A, c E acl,(B u b u {b,,}), tp(6 ( 0 , b,), IMI) does not fork over A and tp(6 (c, b,), A ) is stationary. As n is minimal c, b, 4 acl,(lMl u 6), so (as~ ( z8,) is weakly H-minimal) {b,, 6} is independent over (IMl, A). Choose 6’ E !MI, 6’ mlizing stp(6, A), then

-

tp(6, A u {b,,}) = tp(6’, A

u {b,}),

and we can find c’ E N,tp(6’ (c’, b,?, A ) = tp(6
u 6’ u {c’})

320

[OH.

MORE ON TYPES AND SATURATED MODELS

v, $ 7

fork over A u 6', hence tp(b,, IMl U {c'}) fork over A u 6', but tp(b,, IN)) does not fork over A henoe over A U 6',so tp(b,, 1M1 u {c'}) fork over IMI,so c' $ IMI.So c', b, contradiot the minimality of n.

FACT G : 8wppose p = p ( x , a) is K-minimal, and stp(8,0) = stp(a', 0) implies p ( x , a), p ( x , Z') are not orthogonal. Then there is q E 8(acl0), Z E acl(0) u q ( C Q ) ,q -c r E ~ ( Z )such that: r is K-minimal, q is clj{p}eimple, and q ( 4 O q ) c acl,(r(CQ)u a); and there are n < w , Zl (I < n) realizing stp(a, 0) such that

So q is c13{r}-simple.(We wume T is superstable.) The proof is like 4.1 1, except r . As q is not K-algebraic, there is a Kminimal r E S ( q , E G q(aeq) extending q (we can have F G q(EeQ)by the Definability lemma). Necessarily T , p ( z ,a) are not orthogonal (as q(M) # q(N), M C N are FGo-saturated, al E M implies p ( N ,a) = p ( M , i ~ ) )As . they are K-minimal necessarily for some finite B, (a U E E B)p(a"'J,a) E acl,(B U r(aeQ), henceq(aeq) G acl,(B U r(aeq). Let E' E q(aeq) be such that tp*(B,q(aeq)) does not fork over I?', E G C'. So tp,(B, acl(q(aeQ))is definable over acl(F'), hence q(aeq) E acl,(c' U r(aeQ)).

Remark. ( 1 ) We could, by choosing the p , looking at tp(z,al),have that t3E qr. (2) The proof of 7.14(2) is similar, remembering 7.7(2).

T H E O R E M 7.14(3): We can add to 7.14(1) and 7.14(2).

(iii) q,(aeq) c dcl O(a). (iv) For any model M , and c:, G E M realizing tp&, aclq) letting Br = c13{rr}, we have : low,(q,(M), aclq) = low,(@, aclq) low9(rr(M, G), $ U aclq). (v) For i Zj,qj i s strongly orthogonal to B,, so qj is 9',-simple and for d € q j ( a ) 10w,~(d) , = 0.

+

CHAPTERVI

SATUaATION OF ULTRAPRODUCTS

VI.

0. Introduction

The ultraproduct construction is one of the most important methods in model theory, and unlike the primary model, the ultraproduct does not depend on the specific theories we deal with. The ultraproduct n , , , M , / D , for D an ultrafilter, M , L-models, is the product when we identify any two elements of n , , , M , (i.e., functions from I)which m e equal for “almost ” every i. The important properties of the ultraproduct me: (1) Log’s Theorem: a (first-order) sentence is satisfied by the ultraproduct 8 it is satisfied by almost every M,. (2) The L,-reduct of the ultraproduct is the ultraproduct of the L,reducts. It is also important to know: (3) The ultraproduct is A-compact if the ultrafilter is A-good and N,-incomplete (thus we can construct A-mturated models). (4) Two L-models are elementarily equivalent iff they have isomorphic ultrapowers. So (4) gives an “algebraic ” characterization of elementary equivalence. It is also important that by ultraproducts, we can get an ccalgebraic’yproof of the compactness theorem. The ultraproduct is not central in this book (and outside this chapter, we shall use it only once or twice), and the first three sections of this chapter have nothing to do with stability. However we have developed here a classification of theories using stability and associated notions, We different from the one Keisler [x67] gives (using his ordering show that our classification is helpful in the analysis of Keisler’s order, thus giving more evidence of the naturalness of the classification. We also deal with the categoricity of pseudo-elementary classes, and saturation of ultraproducts. I n Section 1 we give the basic properties of ultraproducts and reduced products and regular filters. In Section 2 we deal mainly with A-good filters. This property is 32 1

a).

defined combinatorically, but its main property is: an u l t r d t e r D is A-good iff for every family of A-saturated L-models M,, i E I, M,/D is A-compaot iff for every A-saturated atomic Boolean algebra M y M r / D is A-saturated. If we require D to be N,-incomplete aa well, we .can omit the condition that the models am A-eaturated. The Boolean algebra comes quite naturally, but we can replace them by dense order. This is done aa follows: suppose M1/D is A+-compacf for every dense order M.For any model N,we expand it by encoding the set of finite sequences of formulae (with parameters)which are consistent, and some natural relations and functions and get Nl. Note they form naturally a tree and by it we can “ s p k ” on sequences of formula in N’,/D which are not n e d y “standard”. We look in N’,/D,and any 1-typep in W / D , 1p1 < A, and define by induction on i s lpl elements a,of the tree, increasing with i , and “consistent ” with p (in the inner sense we demand this for every finite subset of p separately and this is expressible): and letting p , = {v,: i < IpI}, we want that 9, w i l l “appear” in the “sequence” a,, the element “realizing” alpl is as required. The translation of the tree to the dense (linear) order is by intervals. Now from this we can deduce for singular A that A-goodness implies A -goodness. The importance of constructing ultrafilters is clear, and this is the subjeot of Section 3. Some of those problems are connected to problems on large oardinale, and consistency results and are outside our scope (e.g., N,-complete ultrafilters, non-regular ultrafilters). It is easy to construct regular ultrafilters as they sped on the existence of a family of sets, and this is done in 1.3(4). Good ultrafilters posed a more difficult problem ae their definition says: for every function into D there is a function into D . . . Keisler [K 641, constructs a good ultrafilter on A when A+ = 2” in A+ steps; in each step we have a uniform filtergenerated by A subsets of A. When 2” > A+ ,this seems to fail, but Kunen [Ku 721 suggests an alternative proof he takes a family of 2” independent functions from A onto A; and construct the ultrafilter in 2* steps, after the ith step, we have to delete s 8, + li I of them only. Now it is natural to ask where there is a (regular)p-good u l t r d t e r over h whioh is not p+-good. But p s A + , and by 2.10 p is regular. For p a successor we use the product of ultrafilters, whose most important property is NIX*/Dl x D, = (Mr/Dl)J/Dl, i.e., if we take ultrapower twice, it is like taking it once for the product. By this we get h+good, not A +-good ultrafilters over p 2 A. We use independent functions to construct ultrafilters with various properties; e.g., +

.

+

m.

0 01

INTRODUaTION

323

(1) For p 2 A, h regular, over p them is a regular ultrafilter which is A-good but not h -good. (2) If 2” 2 h = A%, then there is a regular ultrafilter D over p and At < w (t E I ) suoh that nt/D = h (the conditions h = P o s 2Y are neoeeeary (for an infinite A), see Exercise 2.10). (3) If 2” 2 A, p 2 K ; h > K regular, then there is a regular ultrafilter D over p suoh that h = lcf(K, D ) =d& mink:in {a E #ID: # / D t a < a for every a < K } there is a set unbounded from below of cardinality x}. We prove also that Thd is essentially “simpler” than the theory of linear order. In the exeroises, we indicate why any two elementarily equivalent models have isomorphic ultmproducts, and other results. In Section 4 we define Keisler’s order and prove some theorems on it. We oould change the definition slightly, e.g., by omitting the requirement that D is an ultrafilter over A, and &ill get the same results. We also show that some ultraproducts are not A-compact. Section 6 is the heavy section, utilizing the results from I1 on d-nindiscernible sets and their dimemione, to determine how saturated are ultraproducts. We find which elementary olasses (of countable type) contain categorical pseudo-elementary claases. E.g. ,we prove +

ntsI

THEOREM 0.1: If bf W a model of T , T countable and without the f.c.p., D an K , - i m p l e t e UUraJilter Over I , then M1/DiS ~~/D-eaturated. THEOREM 0.2: If T i e countable, euperdable and without the f.c.p., then t h e i e a theory T I ,T E TI,lTIl = 2u0,8Wh that any uncountable model of T , h a a eaturated L(T)-red&. Other theorems (6.1, VIII, 2.1) show this result is the best possible. Summing up our results we get the following picture for Keisler’s order on countable theories. Let H- = { T : T countable, without the f.c.p.}, Kscp= (5”: T#K,,,,, is countable and stable with the f.c.p.},

{ T : T countable, unstable K ~ ~ < ( TOO},) Kcdt= { T : T countable, Kodt(T)= GO, T without the strict order property}, =

H,,

= { T : T countable and with the strict order property}.

then they are THEOREM 0.3: (1) If T,,TI are both Hmin[Kaop][Hmu] @‘-equi~alent, i.e., T I @ Ta @ T l .

In Section 6 we find how compaot rn ultralimite.

PROBLEM 0.1: It would be very desirable to prove that (1A) T,, Ta E Ktndimplies TI, Ta are @-equivalent, (1B) T , , T , E Kdt (or we should aek &o whether K,,(T) = 00, K,,,(T)= 00) implies T,, Ta rn @equhalent. This will complete the model-theoretiosham of investigating Keisler’s order for countable theories. For this it seems reasonable to try to find for T E KiDda theory parallel to 11, Section 2 for stable theories. PROBLEM 0.2: It would be desirable to replace in 0.3(2) the “consistent with ,’ by “provable form ,’. This is a problem in constructing ultrafilters. Another problem on ultrafilters is 6.1. PROBLEM 0.3: On Keisler’s order for uncountable T see Shelah [Sh 721. We conjecture that for every T there are a countable T,and @ T , and T’@ Ta]. simple Ta (Definition 6.4) such that T’@ T o [T’ PROBLEM 0.4: We conjecture: if T is not @-minimal then T is not @,,,-minimal, and if M1/D is (21’I)+-compact,M is A-compact, then M’/D is A-compact (see 6.7). In this chapter D will always denote a filter (or ultrafilter) over I (or sometimes J ) (For the definition and some theorems see Section 1 of the Appendix.)

VI.1. Reduced praduots and regular filters

DEWINITION 1.1: Let D be ti filter over I and for each iE I let Mt be an L-model. (1) We define an equivalence relation zD (or z for short) over

I I

f z g iff {i E I : f ( i ) = g(i)} E D . (2) We define the L-model n , , M , / D (the reduced product of the

ni,1 Jft

Mi’s):

*

OH.

VI, $ 1 3

325

RIDDUOED PRODUUTS AND REUULAR FILTER8

n,,,

(A) Its universe is {f/k :f E [Mil}. (B) If R is en n-place predicate symbol in L then

.

RM = {(f1/#, . .,fn/z): {i €1:(fl(i),. . .,f,,(i))E RMd}E D}. (C) If P is an n-place function symbol in L then PM(fl,. . .,fn) = f where for every i E I , f ( i ) = PMi(f,(i), ...,fn(i))and PM(f1/#, . . .,fn/z) = PM(f,, .. .,fn)/#. (3) If D is an ultrafilter, the redud product is called an ultraproduct. If M , = N for every i E I, then we write N1/Dfor M,/D and call it a power instead of a product.

nt,g

Remurk.As D is a filter, z really is an equivalence relation andf, z gl for 1 5 1 s n implies {i:( f , ( i ) , ., f , , ( i ) ) E R'c} E D o {i: (gl(i), . . ., gn(i)) E RMg}E D a d PM(fly* * * ,fn) z P"(g1, * * gn) 80 ntsl Mt/D is a well-defined L-model. Notice that for a term T , 7(f1,. . .,fn)(i)3: 4fl(i), ' * Y f n ( i )I*

..

-

8

-

nrSI

Notation. ( 1 ) If M , is an L-model for i E I, P # L, N = M t / D and (a,,P,)is an L u {P}-modelfor every i E I, then in an abuse of notaPN)= (X;, P,)/D. tion (N, (2) We do not strictly distinguish between a E M,/DI and a representative of a in lM,l. We write ( a [ i ] :i E I ) for the representative of a. Also (in this chapter) if ii = (aoy...,a,,-l) E M,/DIythen 7i[iI = (ao[i],...,a,". (3) We identify a and
n,,,

n,,,

In,,,

In,,

Remark. Note that if La E L,, M t is an Ll-model and Mf is the L,M f / D is the La-reduct of Mt/D. reduct of Mt (for i E I),then

n,,,

n;,g

THEOREM 1.1 (EoiYs Theorem): Let D be an dtrajlter over I . ( 1 ) M r / D is elementarily equivalent to M . (2) If M,is an L-model (for each i E I ) and cp(Z) E L, ii E M,/DI then n M t / D C cp[ii] i,# {i E I : M,C cp[ii[i]]}E D.

In,,,

ICl

( 3 ) The natural embedding of M into M 1 / D(defined by a H ( a : i E I ) ) is elementary, so we can say M < W / D .

Proof.( 1 ) Follows by (2), for sentences. (2) Easy by induction on formulas (only for negation does the proof

fail for filters). (3) Follows by (2).

DEFINITION 1.2: (1) A basic Horn formula is a formula of the form A,<,,'pi +'p where p,, cp am atomic, n c w , or ~ ' p . (2) The family of Horn formulae is the cloeure of the family of baaio Horn formulas under conjunction, exietenticll quantiihtion and universal quantification (but not disjunction or negation).

Remark. In Definition 2.1(1) we o m take n = 0, so every atomio formula is a basic Horn formula. Also, every negation of an atomic formula is a baeio Horn formula. THEOREM 1.2: If p ( l ) i8 u H m forncukc,

M

=

n,,MJD, iz

E

1M1

and

{i E I : Mi C tp,P[i]]}E D,

t h iK b p[iz]. Proof. By induction on 'p.

DERTNITION1.3: (1) The family {Xi: i c A} of subsets of I is regular if for w E A, x,z 0 1 ~ <1 N,.

n f6Y

(2) The above-mentioned family regularizes D (which is a filter over I)if it is regular and X,E D for i < A.

(3) The filter D over I is A-regular if some {X,: i < A} regularizes it.

(4) (6) (6)

D is regular if it ia 111-regular. D is A-incomplete if there are X,E D( i < a c A), X,= 0. D is A-complete if for X,E D (i c a c A), X,E D.

nlSa

LEMMA 1.3: (1) If D is A-regular, p 5 A, then D ii3 p-regular. (2) T k $ &r D i8 N,-btiX~m~Zete iff D i 8 K,-reg&r. (3) D is not 1 1 1 +-regular. (4) Over every infinite I there is a IIl-regular filter, hence a regular ultrafilter. Proof. (1) Immediate. (2) Immediate. (3) If {X,: i < 1 11+} c D is regular, each X,,being a member of D, is non-empty, so choosej, E X,.Clearly for somej*, I{i c 1 11 :j,= j*}l = 1 1 1 but {X,: j, = j*} 2 {j*}# 0, contradiction. (4) Clearly it suffices to find a set J, I JI II] over which there is a regular filter. So let: J = &,(I) = {w: w E I , lw] c No}, for W E J , +

+

n

3 :

Oa.

VI, 0 13

327

REDUOED PRODUaTB AND REGULAR FILTERS

X,

= {w: W E Jw, c w} c J. The filter D generated by {X,: WEJ) (see Definition 1.1 and Theorem 1.1(1), (2) of the Appendix) is regularized by {X{f,:i E 4. The second phrase follows by 1.1(4) of the

Appendix.

Proof of the C e n e 8 8 Th-mrem (I, 1.1): From 1.1 and 1.3 we get a proof of the Compactness Theorem. Let T be a set of sentences, such that every finite t c T has a model M t . Let J = BN0(T)and let D be the filter over J defined in the proof of 1.3(4), and D* an ultrafilter over J,extending D (existsby 1.1(4) of the Appendix). Let M M,/D*, thenforevery#ET,{wEJ:M, C#} 2 { w E J : # E w }= X , , , E D E D*. Hence, by h g ’ s Theorem, M C #, so M is a model of T. 2 :

n,,

DEFINITION1.4: A type is atomic if all its formulas are atomic. THEOREM 1.4: (1) Let D be afilter over I . Then the following condition% are q u i d & . (A) D i8 A+*. ( B ) For ewery L and every family Mt (i E I ) of L-models, every atomic MJD. type p aver 0 in each Mi, 1p1 5 A, i8 reat%zed in ( C ) For every M every atonzic type g over in W / D , lpl 5 A, is realized in M’/D (notice: p is in M iff it is in MIID). (D) Condition (C) holds for M = (rSNo(A), r). (2) If in (1) D is an ultrafilter we can replace “atomic type” by “type” in the umditifma.

nisr

nfsr

Proof. (1) (A) * (B) Let N = M i / D , {X,: a < A} regularize D , p = {v,(Z): a c A} be the type. For m y finite w E A let a,[<] E 1M,1 realize {v,(Z): a E w}. Now as {X,: a c A} is regular, for every i E I, w(i) = {a < A: i e X,} is finite. Define E IN1 by 8[i] = GwiJi]. Hence for a < A {i E I : Mi C rp,p[i]}2 {i E I:a E w(i)} = {i E I : i E X,} = X , E D . Hence N C tpa[i3], so a realizes p. (B)* (C)Use (B) for M i = ( M a)oslarl, y for every i E I. (Clearly p is a type in M.) (C) * (D) Holds by their definition. (D) * (A) Let N = M1/Dand let p = {{a} E 2: a < A}, so clearly p is finitely satisfiable in M y(hence in N ) . Suppose a E IN1 realizes p , and let for a < h

X, = {i E I : M

C {a} E a[i]}.

[am, . 81

SATURATION OR’ULTRAPRODUCTS

328

nUeu, Xu= { i ~ w l :c a[i]},

Then X, E D aa N C {a} c a, and for w c A, so clearly it is empty iff w is irhite. (2) The same proof using LOPSTheorem.

DEFINITION 1.5: M is (A, d, m)-compact if every ( A , m)-typep over 1M1, in M , 1p1 < A, is realized in M . For d the set of atomic formulaa of L(M), we write at. Similarly the sets of quantifier free formulas, conjunctions of atomic formulaa, and formulaa of quantifier depth s n are denoted by qf, cnat, and qd,, respectively; i f m = 1 we omit it. DEFINITION1.6: (1) For models M,, Ml and an ordinal a, (L(M,) n L(Ml) 2 L) we define a game Gt(Ho,Ml) between the players I and I1 aa follows: A t the @h move player I chooses I E {0,1} and afi E lMil, and then player I1 chooses a;-i E ~ M l - i ~ . The play ends after a moves and then player I1 wins if for every atomic formula cp(z,,. . .,2), E L and ordinals p(O), . .,@(n)< a, Jfo C ~ [ a & ) , $(,)I * C ~ [ a j ( o* ) ,* a;,,)], and player 1 wins otherwise. (2) A strategy (of a player) is a sequence of functions f B @ < a ) which “tells” him what to do ( f Bfor the /?th move) depending only on the previous choices in the play. A winning strategy is a strategy such that in any play in which the player chooses according to it, he wins. A player wine in the game if he has a winning strategy. (3) We omit L if L = L(M,) = L(Ml).

---

9

-

.

9

LEMMA 1.5: (1) In the game Gz(Mo,Ml) nd bothpZaymu win. ( 2 ) The relation “player I1 wine in Gz(M, N)” is an equivalence relation among models. ( 3 ) If player 11 [I] wine in Gi(MoyM , ) , L(1) E L, /3 5 a [a 5 8, L E L(l)] then he m’nein G&l,(M,, Ml). (4) If M,, M , are ~sonwrph~c, then player I1 Wine in Ua(M,, Ml). ( 5 ) If M,, M , are elementarily equivalent, and A-eaturated,then player I1 wine in G”M,, M I ) . Proof. (1) Suppose both players win, then in some play, both use their winning strategy, but one of them must lose this play, contradiction. (2) Combine the strategies to prove transitivity. Symmetry is trivial, and reflexivity follows by (4). (3) Trivial.

OH.

m,8 11

329

REDUCED PRODUCTS AND REUULAR BILTEFtS

(4) If 8 : IM,l+ lMll is the isomorphism then player I1 chooses a;-1 so that g(a;) = a;. (5) Player I1 has to choose at-' such that tp,({az: a zs; p), 0, 22,) =

tP*({d: a

PI, 0, Ml).

Proof. Immediate.

THEOREM 1.7: The L*-mud.ele M,, M,are elementarily equivalent ifffor every jinite L E L*, n < w , player II wine in G;(M,, Ml). Proof. Let us define by induction on n, the sets F;,,,(L), F;,JL) of , .,z,,,-, E)L. formulas ~(z,, (i) F;,,,(L) is the set of atomic formulas y(s,, . . .,z,,,-,E)L. (ii) If F;,,,(L) is defined, F;,,,(L) is the set of formulas of the form A {cp(Z)u(@'6r): cp(@ E F;,,,,(L)} for F c F;,,,(L). (iii) If F;,,,+,(L) is defined

.

-

e + l , m ( L )= { ( ~ ~ r n ) F ( ~* o ,- 2,): dzo, * * * z,) E % n + l ( L ) I * Clearly any sentence in L is equivalent to a disjunction of some of the sentences in F;,,for every sufficiently large n (in fact for its quantifier depth), and this holds for any formula. If M,, M, are elementarily equivalent, L c L* finite, n < W, a winning strategy for player I1 in Gft(M,, M,)is to preserve the satisfaction of

(*I

9

M , C ?[at,.. .,a!-,] iff M, C cp[ua, F(zO,

* *

- 9

zk-l)

E

9

. . .,a:-,]

for every

F;-k.k(L)-

For k = 0 this holds as M,,M, me elementarily equivalent; and by the definition of F;-k,k(L) it is easy to play so that (*) is preserved. For the other direction, it suffices to prove that if ut, . . ., a!-,, a;,. . .,a;- have been chosen in a play in which player I1 uses a winning strategy for Gft(M,, M , ) then (*) holds (because then for any

SATURATION OF ULT'RDRODUOTS

330

[CH. VI, 5 1

$ E (JI'l,o(L), L c_ L*, L finite,Mo C $ e Ml C $, and aa any sentence $ of L* is a sentenceof some finite sublenguage, bl,, M,are elementarily equivalent). The proof is may, by induction on n - k.

If & Mi I$ are elementarily ', equivalent L-modeze for i E I, ILI S A, D a A-regeclarfilter over I , a < A+, thenplayer I1wiw in Ga(Mo,M1), wlrere N o = Mf'/D, M 1 = M i / D . (80 Mo, M 1 are elementarily equivalent.)

THEOREM

1.8:

no

n16x

. ..

Proof. The number of formulaa ' p ( ~ ~ < ~ , ,, z4(,,)), E L,p(1) < a,is 5 A, let {'pj(~sco,n,.. .,z,,~,,~,.,): j < A} be an enumeration of the set of thew formulaa (possibly with repetitions). Let {Xj:j < A} E D regularize D , and for t E I, let w(t) = {j:t E Xj}, (which is finite), let 80

hO.t),

---

Zv(Mt).t)} =

u

{%#(o.n,...} where y(0, t ) <

- - - < y(n(t),t).

j€ur(t)

Let Lt be the minimal sublanguage of L such that vj E & for jE w(t). So by the previous theorem, player I1wins in P&(t)+l(Mf, Mi)for every t. Now we shall describe the winning strategy of player I1in GC(M0,W ) : when he haa to choose E lM1-il he chooses each aj-I[t] separately. If /? $ {y(k,t ) : k I n(t)},he chooses a;-I[t] arbitrarily. If fl = y(k, t ) he imagines he is playing qt)+'(Yp,M i ) , that a~(,,,,[t], , . ., a:(&-l,o[t], a k - la[t] and a:(k,t,[t]have been chosen, and he chooses ai,;;.',,[t]by his winning strategy in GZt)+l(Mf,M;).It is eaay to check that this is a winning strategy for Gg(M0,Ml). CONCLUSION 1.9: If M yN are elementarily equivalent, D ie a A-regular Jilter over I then M x / Di s (A, A , m)-wmpad iff N x / Di8 (A, A , m)-wmpa&.

LEMMA 1.10: If for each t E I player I1 win8 in Ga(Mf,Mt),D a jElter over I , then player I1 wins in G a ( n , , M f / D , Mt/D).

n,,,

Proof. In view of the proofs of 1.8, 1.9, we leave it to the reader.

CONCLUSION 1.11: If M,, N , are elementarily equivalent and hsaturated, D a N e r , then M = M t / D ie (A, A , m)-wmpact ifl N = N t / D i 8 (A, 4 m)*pa&.

n,,,

n,,,

Proof. By 1.6(6) player I1 wins in Gh(Mt,N,). By 1.10 player I1 wins in GA(M,N ) . Henoe, if A > No,by l.6(4) M is ( h , A , m)-compact iff f l is (A, A, m)-compact. If h = KO,every model is (A, A , m)-compact; so we finish.

CH.

m,8 11

33 1

REDUCED PRODUCTS AND REGULAR FILTERS

Notation. If A,, i E I is a family of non-empty sets and D is a filter over I we define A,/D = (flz :f E A,} where z is the equivalence relation from Definition 1.1. If A, is a ctlrdind A, for every i E I , we write n ter &/D for IrIteI WDI and nier A, for 4-$0 llrI:er WDII

nier

nt,,

Inw

ll~:ll/D.

= ntsr

If D i s afilter over I, J c I then D 1J = { X n J : X ED } s o when I - J $ D , D tJisnottrivid.

nt,

EXERCIBE 1.2: (1) If D is a filter over I, J E D then M,/D z M t / D J (the isomorphism maps (art]:t E I ) / D to (a[t]:t E J ) / D ) ; 80 Titer &/D = nteJ &/D IJ . (2) If J c I, I - J $D, D a p-regulrtr filter over I then D J ie

nteJ

p-regular.

EXERCIBE 1.3: (1) Show that if (M, P) = nteI(Mt, P J / D then

IPI = nt,r IPtIID.

n,,,

nlol

(2) Show that A,/D < A, ; and {t : A, < p,}E D implies n t e r &/D s n t e r pt/D. ( 3 ) Show that if Q E 1A,/D > 0 then A,/D 2 A,/D, J , for any filter D , 1 D,J E I.

n,,,

n,,,

r

THEOREM 1.12: (1) If D i s a p-reguZurjZter over I , & (t E I ) (injnite) cardinale, and h = nter&/D > 0, x = mink: {t E I: & s x} E D} then h 1 x". (2) If in (1) p = IIl then h = f. (3) Moreover tkre are natural numbers nt mlr that nt,l n t / D 2 2", and if D i s r e g h r ntsrn t / D = 21'1.

Proqf. (1) By Exercise 1.3(3) we can assume D is an ultrafilter (by extending the original filter to an ultrafilter containing {t: & > x,} for each x1 < x (by 1.1(2), (3) of the Appendix). Case p 5 &

I: x > p. By Exercise 1.2(1) we can w u m e that for every t , s x. Now we want to define models Mt (t E I)so that llMt 11 = At

rm. w, 8 1

SATURATION OF ULTRAPRODUUTB

332

nt,,

and f pairwise contradictory types will be realized in dd,/D. Let L consist of the one-place predicates Pa (i < x, a < p). We define M t such that for each a, for i < & the Pi(Mt) are pairwise disjoint, for & 5 i < x Pk(iWt) = lM,l and for every finite w E p and function f :w+ X M ,I= (3x)[/\,,,P{(")(x)].It is quite easy to construct such a model :

lMtl = {cy: w c p, IwI < No,f:w + & a function}, Pk(Mt) = {c/" E lMtl: a E w, f (a)= i or i 2 &}. By the definition of x, for every i < j < x, a < p {t E I:M: I=-1(3%)(pk(%) A Pi(%))} 1 {t E I: li

1, ljI < &} E D,

hence M = M J D C 7(3s)[Pk(x)A P i ( z ) ] . Let for each r) E ' x , p,, = {e[al(z):a < p}, 80 the p , , ' ~are pairwise contradictory; 60 it suffices to prove they are realized in M . But clearly each p,, is finitely satisfiable in each M t , 80 by 1.4 it is realized in M.

Cme 11: x 5 p. By Exercise 1.3(2) it follows from 1.12(3). (2) h 2 xfi by (1) and h Ixu as we can assume (again by Exercise 1/41)) A, 5 x for every t , so by Exercise 1.3(2), &/D5 h: S

n:pr

x ~ r=~f . (3) We use a method similar to the one used in (1). Let L consist of j < p} regularize D, the one place predicates Pf, (1 = 0 , 1 , a < p), {X,: and w ( t ) = { j : t E X,}which is finite. We define Mt such that llMtll = 2Iw(:)l, and denote this number by nt. We define it such that f o r j E w ( t ) , P:, Pf are complementary and for any finite w E p

Mt I=

A Po,(%) A A -w Pi(%)).

crew

crew(:)

For this let

]M*l = {cw: w G w ( t ) } ,

Pf,(M,)= {cw: 1 = 0 and u E w or I = 1 and a E w ( t ) - w or cc $ w ( t ) } . Let M = n f s , M t / D ,then P : ( M ) , P i ( M ) are a partition of IMI, and for every w E p, p , = {Pf,(x):a < p and a E w e 1 = 0) is realized in M, 80 llMll 2 2", but 11M11 = ntelllMtll/D = n:EIn,/D. When D is regular we get equality as in (2). CONCLUSION 1.13: If D is a regular filter over I , t 7 m hr/D = hir!.

OH.

VI, 8 21

FILTERS AND COMPACTNESS OF PRODUCTS

333

EXERCI8E 1.4: Suppose the filter D is p-regular for every p < A, and Xis singular. Prove that D is A-regular. [Hint: If {Xf : i < p} regularizes D for each p < A, K = cf A < A, A=

2 p(a), p(a) < A,

then

{Xf@) n Xi:a < K , i < p(a)}

a<x

proves our aesertion.]

EXERCISE 1.5: Suppose A is a regular cardinal, prove there is a filter D which is p-regulw for each p < A, but is not A-regular. [Hint: D will be generated by {Xf : i < p < A}, for each p, {Xf :i < p} is regular, but for every a < w , and distinct pn < h (n < a), and finite sets w, c {i: i < p,} nlEtu. Xtm # 0. Use 1.4 of the Appendix to prow this

nnsa

is possible.]

QUESTION 1.6: In Exercise 1.6,if K

2

AWo

clearly we can find such a

D over K . Can we always find such a D over A ? (For A a successor it is trivial.)

EXERCISE 1.7: Prove that in 1.4(1) (and (2)) to the four equivalent conditions we can add: (6) If F is a set of IA atomic formulas (in 5 A variables) which is finitely satisfiable in each Mt, then F is satisfiable in MJD. (6) In (6) instead of atomic formulas we can allow Horn formulas.

n,,,

EXERCIBE 1.8: Prove that for any ultrafilter D end cardinal A, D is A-incomplete iff D is not A-complete. EXERCIAE 1.9: If M = (A, R,, . . .), M 1 / D = N = (A', Ri, . . .) and A E lMll, then N ; = ( M I ,A , R,, .. .),/D = ( M { / D ,A', R;, . . .). (Itis E, M), where H ( Kis ) the family convenient to use this for MI = (H(K), of sets hereditarily of power < K , M E B(K)). VI.2. Goad Biters and compactness of reduced products

DEFINITION 2.1: (1) A filter D (over I) is A-good if for every p < A, every f:A&) + D which is monotonic [i.e., w c u f(u)c f ( w ) ]has a refinement g : flH0(p)--t D [i.e., g(w) E f ( w ) ] which is multiplicative [i.e., g(w u u) = g(w)n ~ ( u ) ] . (2) D is good if it is 1 11+-good.

Remark. (1)Clearly we can add “g is monotonic”, because multiplicity impliea monotonicity (2) We can delete the ctssumption “f is m ~ n ~ t o n i cbecauee ” for any f : 8&) + D, let f * be defined by f * ( w ) = f (u).AS l{u: u E w}l = 21wl c KO,and D, being a filter, is closed under finite intersections, f *(w) E D, and any multiplicative refinement off *, is a multiplicative refinement of fi and f * is monotonic.

.

nUsw

C

W

2.1 : Euery j&r

is Et,-good.

Prmf.Let f : Bb(w) --+ D. Define g(w) = n{f(u):mas u s max w}. As for emh w E ~ ~ ( max w ) w < u,and the number of u c {n:n 5 max w} is finite, clearly g(w) E D. The function g is multiplicative aa max(u U 20) = max{max v, max w}.

nter

Proof. (1) * (2) Let N = Y t / D ,p be an (at,m)-type over IN1 in N , 1131 < Asp = {v,(Z; a,): a < p < A} (v, atomic). Let for w f ( w ) = {t E I : Mt t (33) A,o,,, tpa(P; BJt])}. As p is finitely satisfiable in N,some 8, E IN1 realims {cp,@; aa):a E w}, so X& = {t: Mt t cpa[Zw[t], a,$)]} E D, (for a E w ) hence X;w E D, but clearly it is a subset of f ( w ) , henm f ( w ) E D. As D is A-good, there is a multiplicative function g :El,&) + D refiningf. Let for t E I , ~ ( t=) {a < p : t E g({a})}. NOW,for every finite w E w(t), a E w t E g({a}), hence by the multiplicativity of g, t E g(w); and aa g refinesf, t E f (w). Hence Mt t (32) A,, q~,(3,7i,[t]),and &B this } finitely holds for any finite w c w(t), pt = {v,(Z; Za[t]): a ~ w ( t ) is satisfiable in Ht. As Mt is h-compact, there is 8[t] E lMtl which rsalizes pt, and let 8 = (. ,8[t], . . .)ter/D.Let us prove that 8 realizes p; for every a

naew

..

{t: Mt C v,@[t], 7i,[t]]} 2 {t: a E w(t)} = {t: t E g({a})} = g({a}) E D

hence N t v,[& a,].

OH.

w, 8 21

FILTERS AND OOMPAOTNESS OF PRODUCTS

335

(2) + (3) Trivial.

(3) + (1) Let p < h andf :LYNo(p)3 D be monotonic. For each t E I we define a&] E ]Mi for a < p such that: (*)

for every finite w rzLYN0(p),

bl c (3z)( A z c a&] aEw

A

P ( z ) e t Ef (w).

)

[If we want to define a,[t] for finitely many a's only, we can easily By the h-saturation of M we can choose all a,[t], choose them in M,,. a < PI. So let a, = (. . .,a,[t], . .)ter/D and p = {z E a,: a < p} u {P(z)}. If p c p is finite, then for some finite w c p p E {z c a , : a E w } U {P(z)},and, by (*) {t: M t ( 3 z ) [ / \ a e w 2 %[tI A P(z)I}= {t: t Ef(W)} = f ( w ) E D. Hence M t (3%)A p, so p is snitely satisfiable in N = Mf/D and by (3) it is realized in it, so let c E IN1 realize p. c[t] c a&] A P(c[t])},and it is easy to Define g(w) = {t €1: Aasw check that g: LYwo(p) + D is multiplicative and refines f.

.

THEOREM 2.3: Let D be a$lter over I , h a cardinal > K O then tk follm'ng conditions are equivdent : (1) D is &good and N , - i m p l e t e . (2) Por every family of L-models M t (t E I ) , M J D is (A, at, m)-

ntpf

co?wt.

(3) N = Mi/D ie ( p + , at)-compactfor every p < h (M,,--asde$nedin 2.2( 3)).

Proof. (2)

=r

(3) Trivial.

(3) * (1) The set {{a} c z:a < p} is a set of formulas over lM,,l, and it is snitely satisfiable in N,,, hence some c E IN1 realizes it. Let for

a
X, = {t E I : M,, t {a} c c[t]} so clearly X, E D, and if w c p is infinite, t E naEW X,,then M,, t {a} E c[t]for each a E w, hence w s c[t] E (M,,I contradiction. So {X,:a < p}

regularizes D; hence, for any h-saturated MyM,, < M , by 1.8 player IIwin~inaY+~(Mi/D, M'/D);and, by 1.6(4),asN~/Dis(p,a&)-compact also M x / Dis (p, at)-compact. So by 2.2 D is h-good. As D is Ko-regular, it is 8,-incomplete, by 1.3(2). (1) + (2) As in (3) * (1) it suffices to prove that D is p-regular for every p c A.

SATURATION OF ULTRAPRODUCTS

336

[OH. VI, 8 2

CLAIM 2.4: If D ie pf-goOd and N,-incomplete then D is p-regular.

Proof. As D is X,-incomplete, there are X, E D for n < w , X, = 0. Let f :S,(p) + D be defined by f (w)= X l w l ,and let g :BN0(p)+ D be multiplicative refinement off. Then for a < p, g({a}) E D,and for g({a}), then for each n < w , choose any infinite w E p, if t E w(n) E w, lw(n)l = 12, and then t E g({a}) = g(w(n))c.X,, hence t E X, = 0, so necessarily g({a}) = 0, hence {g({a}): a < p} regularizes D .

naew

naew

n,,<,

THEOREM 2.5: Let D be a A-good, K,-incomplete Jilter over I , D, a Jilter over I , D E D,. Let ILI < A, L = L(MP) = L(Mi) for t E I and M' = rite, MflD,. If NP,M i are elementarily equivalent for every t E I , then player I1 wine in GA(M0,M1).

Remark. Compare with 1.8, 1.10. We assume more and prove more. See Exercise 2.3.

Proof. We use the notation of Definition 1.6. Players 11's strategy is to play so that for every formula ~ ( .q. .,z,) E L, and a, > - . - > a, {t E I : MP C ~ ( a : ~ .[ .t.],aEn[t])} , = {t E I : M i C ~ ( a t , [ t. ]. ., , ain[t])} mod D and therefore mod D, since D E D, (where X = Y mod D means I - (X - Y ) - ( Y - X ) E D ) . Suppose a:, a: have been chosen for i < a < A, and player I has chosen 1 E (0, 11, af E IM'I, and player I1 has to choose at-'. By the symmetry we can assume 1 = 0. Let @ be the set of formulas Q = Q(Z, ql,. . ., xi*), Q E L , i,, . . ., in < a. Let @ = { Q ~ :fi < p = I@]}; clearly p < A, hence by 2.4 D is p-regular, so some family {XB: fi < p} regularizes D . Let

. . .I>.

h(t) = @ < p : MP c ~ ~ ( a E aPo&I, [tl, For w E SHo(p)we define

f

(W)=

{t E I : fi E W

* t E x,9,and M i

b (3%)

A

By the induction assumption {t

E

1: M: b (3%)

A

. . .)>

'p8(Z; a;l,8)[t1,

Eewnh(t)

= { t E I : M i k (32)

A

. . .)}.

' p , 9 ( X ; ai,,8))[t],

E€wnh(t)

...

~ ~ (ai,,,,[t], 2 ; t?ewnh(t)

CH.

m,8 21

FILTERS AND COMPACTNESS OF PRODUCTS

337

hence it is eaay to check that f ( w )E D. So some g :S,,(p) + D is multiplicative and refines f . Notice that @: t E g({/3})} c (Is: t EX,} which is finite. So for each t E I, define a:[t] E lM1l so that it realizes ( ~ ( 2a:(1,8,[t], ; . . .): /3 E g(t)} and a: = (. ..,a;[t],...),,I/D. For the induction hypothesis, for each Q, we get only an inclusion ; by applying it also for -Q, we get equality mod D. THEOREM 2.6: The following d i t i o n e on theJilter D over I , and the cardinal A > Noare equivalent: ( 1 ) D iS A-good. (2) For every A-saturated nmkl M of Tora[the theory of deme linear order I with no Jinite or last element], M I / D is (A, at)-mpact (where M = (pq, I)). ( 3 ) For every A-saturated d l M of Tor,, and for every set A s IMI/DI which is linearly ordered by s , every atomic 1-type p over A in MIID, IpI < A, is realized in MIID.

Proof. The implication ( 1 ) =. (2) follows by 2.2 and (2) =- (3) is trivial; so assume (3), and we shall prove ( 1 ) .

CLAIM 2.7: Assume (3) from 2.6, and let N = ( I N [ , I ) be a A-saturated rooted tree (i.e., I is a partial order, and for every a E IN I, {b E IN I : b I a} is linearly ordered, and 0 I a for every a E I N / ) ;8uch that each a E IN1 hag 1 NII = 11 NI( immediate successors, and for every a, b E IN I there is is a nmxiwl c, c I a and c I b. Then ( 1 ) If N'/D C cf I c, for i < j < a, where u < A, then for some c, NI/D C c, I c for i < a. (2) I f " / D ca1, I 4 1 , A C h , I CXa, A C X a , I c L , f o r i ( l ) < $1) < u,i(2) < j(2) < p, where u,p < A, thenfor some c, NI/D C c: I c I cs for i < a,j < p. Proof. We first note that if A is a linearly ordered subset of IM'/DI, M a model of Tor&, then if every (at, 1)-type over A in M 1 / Dis realized in MIID, then for every m, 1 I m < w , every (cnat, m)-type over A in MI/D is realized in MIID. Now let us choose for each a E IN I ; a A-saturated linear order
SATURATION OF ULTRAPRODUCTS

(2) u < by 1 = 0, (3) b < u,E = 1, (4) let c be the maximal element for which c s a,c s b; c < a, c < by and a' cCb' where a', b' ES'~,a' 5 a,b' s b. It is easy to check that N, = (A*, s*)is B dense h e m order, with firat and last elements and it is A-saturated. Hence it follows 888iIy by our hypothesisthat if A E IN',/DI is linemly ordered by s*, then every atomic (n 1)-typep over A in N i / D , 1p1 < A, n < o is realized in

+

N',/D. Now define a function P. (1) For a E IN],F(a) = (Po(a),F l ( a ) )where Po(a)= (a, 0), Fl(a) = (a, 1) so Po(a),Fl(a) E lNll and for a E INr/DI,

--.

<<

< -

- -

-

~ ( a=) . . .. . ~ o ( a [ t ~>tJD, ) P iWI) * > t d D ) E IWDI (2) For atomic formulas 'p = p(z; y), define m'p) =

P('p)(%B,,

=

P ( z = Y) = 1% = Yo

m

5 9 ) = [zo

F(y 5

2)

=

[Yo

s Yo S

6 , B =
A 21 =

A

5A

YiI,

Yo

5 Y1 A Y1 5 5 1 1 ,

20

5 21 A z 1 5

YJ.

(3) For an atomic formula 'p(z; a) define

W z ;4 ) = P('p)(zo:O, z1;p ( a . (4)

For a set p of atomic formulas 'p(z; a) let

P ( p ) = {F('p@;4 ) :' p k 4 EP}. Now it is eaay to check that for every such type p in N, p is realized in N iff F ( p )is realized in N , . Hence by 2.8 below for each atomic type p in Nr/D,p is realized in N r / Diff P(p)is realized in N',/D. For each p as in Z.Q(A)or (B), F ( p ) is an atomic 2-type over some linearly in N',/D, so F ( p ) is realized in ",/D, hence ordered subset of JN',/DJ p is realized in Nr/D, hence we get our conclusion.

i.e.,

P(F~(Z',

=

#I($,

61) (I

= 1,2) implies 9 1 = pa

(3) For every p E Dom F , p ia r e d i d in M

* #1

=

$2.

ifl

P ( p ) = {P[p(Zl, a)]:y(Z1; a) E p } ie r e d i d in N. Then if p = {pt(Z1;at):i < a} i s a set of atormi0 fotrnulae in Mr/D, u n d f m w y i < a{tEI:p t ( P ; ~ i [ t ] ) ~ D o m E 3 ~ D t ~ p i a r ~ i d i M r / Diff P ( p ) = {P[yf(Z1;Q ] : i < a} ia r e d i d in Nr/D.

Prmf. Suppose E realizes ~ ( E lMI/Dl); E for every t E I let w(t) = {i < a:M C pt(8[t]; a#]) and pi(%';a#]) E Dom E") E D and define P[t] E IN1 so that it realizes {P[pl(Z1; qt])]: i E w(t)}.

It is easy to oheok that Z1 realizes F ( p ) .The other direotion is e a q too.

M be any h-saturated model, IL(M)l s IlMll, and we shall prove that iKz/Dis (A, at)-compaot. This is suffioient by 2.2. Let Continuation of the pmf of 2.6. Let

r ={(po(2,a~),...,pm-l(2,a~-l)): m < o , a t ~ l J f l r ( ~ r ~ L ( J f ) , pf atomio, and Af C (32)

A pt@,af)}. iem

Clearly II'J = J1MJ1,so there is a one-to-onefunction g from ]MIonto I'. Let us define some new relations and functions over lMl (aeauming that they 4 L(N)). (0) c* w i l l be g'l(( )). (1) s : a s b o g(a) is an initial segment of g(b). ( 2 ) Fl: P ( a ) is an element of M realizing g(a). (3) Pe (for eaoh conjunotion p(z; @)of atomio formulas): PJb, a) o -oh oonjunot of p(z;a) appears in g(b). (4) Qe (for eaoh oonjunotion p(z;j i ) of atomic formulas): Q,(b, a) o cp(x;a)is consistent withg(b),i.e., g(b)^ (+o(x;d'),...,pnpl(x; a"-') ~r where p(z; a) = At<,,#{(2;#). ( 6 ) FZ (for eaoh p(z;1)&B in (4): PZ(b,a) = c ifQ,(b, a) and g(c) = g(b)^(#o(s; a'), ,#,,-l(s;a"-')) or +,(b, a) and c = b.

. ..

340

SA"RA!FIOW

[m.

OR'ULTRAPRODUOTS

52

(notice that in (6) if M C (34942;a), then M C (3z)[Af., qr(z;at) A

d x ;a)l).

Let us define N = (Myc*, s , PlyP,,&, P:, P:),, and let N1be a A-saturated elementary extension of N , [INl1 = [INl1 , and Ml be the L(M)-reduct of N , . Clearly Ml is A-saturated, hence by 1.11 M*/D is (A, at)-compaet iff M , = M i / D is (A, at)-compact, so it suffices to prove that M i / D is (A, at)-compact. Clearly Ma is the L(M)-reduct of N , = N',/D. It is easy to check that (lN1l, 5%)satisfies the requirement of claim 2.7, hence also the conclusion of that claim. Let p = {tpf(z;af):i < a. < A} be a cnat-type in M , which is w.1.o.g. closed under conjunctions, and we will prove that it is realized in M,. We define by induction on i Ia. elements cf E such that: (A) j < i * N , C C$ 5 cf. (B) For every fl < a o , Na C &(pa(cfs3fi)(C) If i = j + 1, N , C P,,(Cf,a,). Case I: i = 0. Let co = c* (the root of the tree). Conditions (A), (C) are vacuous, as for condition (B), for every /3 N C (Vij)(Vz)[rps(z, 8 ) + &,&*, jj)] and this is a Horn sentence, so N , also satisfies it. As p is a type in Ma, N , C (32)9~&, Zfi) hence Na C QCp4(c*,a#).

+

Case 11: i 1. Let cf+l = P:,(cfyZf). For condition (A) notice that (i) N t (Vzyz)(x 5 y A y ls; z +z 5 z), (ii) N I=(VzP)(z s P:JXY P)),

and both are Horn sentences, hence Na satisfies them. , by (i) and the induction hypothesisj < i By (ii) N , C cf s c , + ~and * N , Ccj 5 c:+1. As for Condition (B), for each fl < a. there is y = y(fl) < a. such that p,(z; PI = 'pv(z;Pi,Sa) = vfi(z;Pi) A vi(%9,) and 8, = afi-4, (remember p is closed under conjunctions). Clearly N C (VZ, zl,gl,~,)[&,,(z; j j 1 , P,) A 21 = P:Jz9 Pa) &cpp(zly8113 and as this is a Horn sentence Na satisfiesit. As N , C &,&; a, si,) A cf+1 = qc,Y clearly N2 c &,a[c:+l,%I. +

a

OH.

VI, 8 21

341

FILTERS AND COMPACTNESS OF PRODUCTS

As for Condition (C),N C (Vg, zl,z)[&,,(z, 8) A z, = P;Jz, 8)+P,Jzl,g)] so by condition (B) clearly N , C P,,(ct+,,at). Case 111: i = 6 is a limit ordinal. By Claim 2.7( 1) and Condition (A) for j < 8, clearly there is do E lN,l such that j < 6 =r N , C c, 4 do. So do satisfies Condition (A) on c6, but not necessarily Condition (B). We now define by induction of j 4 a. elements d, E lN,l such that (*)

jl < j, 4

ao, y

<6

*N,

C cy 4 dj2 A d5* I

We have already defined do; for limit j, the existence of d, follows by part (2) of Claim 2.7 (using (*) and Condition (A)). For j -+- 1 let a?,+, = Pz,(d,, a,), and we shall prove that (*) holds. Notice that

a,

N C(vW~)(q,(z9 2) and this is a Horn sentence, hence N , satisfies it, hence N, C d,,, Id,, hence (as &'a is transitive)j, < j + 1 5 N , C d,+, Id,l. On the other hand, N C ( ~ 8 N V Z l 9Z,"l

I

2,

A &a&

8)

--f

21

I F:,(z,,

a11

and this is a Horn sentence, hence N, satisfies it. As for y < 6 N , C cy 5 d, (by the induction hypothesis on j ) and N , C &,,(cy, 8,) (by the induction hypothesis on i = 6) clearly N , C cy I P&(d,, a,), but P:,(d,, 8,) = d,+,. So we prove (*) f o r j + 1. So we finish defining d,,j I ao,and let c6 = dao. Condition (A) holds by (*), Condition (C) is vacuous and Condition (B) holds because for eachp < ao,

N c (vg)(vzl,z)r&,,(z, 8) /t 21 5 z QQJq(Z1r ?a1 and this is a Horn sentence, hence N , satisfies it, and N , I= duo s dB+,. Now N , C &e,,(dB + i s 3,) because N C (vz, 8, z)[~g(z,8)+Q,p,(P:,(z, ji), 811; this is a Horn sentence, and as p is a type in M,, for some b E ]N,l N , t ag[b, @gI hence N , I=&,p(p:,(d,, %), ad. So we finish the definition of ct, i Iao.We shall prove that P1(cao) realizes p and thus finish the proof. Now for each /I< a. N C (Vg, zl,z2)[Pe,(z1,8) A z1 S 22 + P,,(z,, and N C (Vz)(Vjj)[P&, 8)+tpB(P1(z), g)] and they are Horn sentences Gg) A cg+, I c,, by Condition hence N , satisfies it, and N , C Pmp(cB+,, (C). Hence N , C rpg[P1(c,,), BB] for every /3 < ao. +

a)]

THEOREM 2.9: Let h > No, D a Jilter over I . Then the following conditions are equivalent.

342

[CH.

SATURATION OF ULTRAPRODUCTS

82

(1) D is A-good and K,-inmmplete. (2) For every model H of Tor,, M'/D is (A, at)-mpact.

(3) NLlD is (A, at)-compact, where N , order, for euch p < A.

= <"'p,

<),

<--whit

Proqf. (1) + (2), (1) =+- (3) Hold by 2.3. (2) * (1) By 2.6 it s d c e s to prove that D is 8,-incomplete. As (&, 5 )I/Dis (A, at)-compact (&-therationals) for some c E I(&, s ) I / D ~ , n 5 c for each n < w , and let

X,

= {t E I : n

< ~ [ t ] Clearly }. X, E D, (IX, = 0. n

(3)

* (1) The proof is similar to that of 2.6, hence we leave it to the

reader.

CONCLUSION 2.10: If A i.9 a singular Cardinal and thefilter D is A-good, t h D i8 A+-good. Proqf. We use 2.6(3), so let A€ be a A+-saturatedmodel of Tor,, A s IN1 = IM'/DI be linearly ordered by s , and p be an atomic 1-type over A in N , 1p1 s A, and we shall prove that p is realized in N , thus finishing. If for some a, (z= a)EP, a realizes p (because p is finitely satisfiable in N).SO let p = {U 5 Z: u E A,} U {X < a: u E As}. As A , c_ A, A, is linearly ordered by 5 , so it hm a cofinal sequence {a,:i < pl}, pl a regular cardinal (so ( V u ~ A , ) ( 3< i p,)(a s a,) and a,E Al). As p1 is regular, p1 < A. Similarly there are a' E Aa (i < pa) such that ( V a ~ A , ) ( 3
As lpll c A, and D is A-good, by 2.2 N realizes plyso we finish. Rernurk. See Exercise 2.7.

Up to now we have dealt with filters and atomic compactness. By the following lemma, we do not need to deal separately with ultrafilters and compactness. LEMMA 2.11 : Let D be an ultrafilter over I , A > Noa cardinal. (1) For every M yMI/D is (A, &)-cumpact iff for every A€, W / D is A--pact.

OH.

5 21

FILTERS AND UOMPAUl"Es8 OW PRODUaTS

( 2 ) For evey L, and L d l s Mty

343

n:,, M J D is (A, ati)-wmpwt iff for

every L and L-nwaeb Mt, ntsIMt/D $8 h-wm?mct. ( 3 ) A8 ( I ) [2] W for A d W t Z & & M[MJ

Proof.In all parts the if part is trivial. For the other direction, for each model iK define M* aa follows:

L W * )= {%a:

(%3)

9 m E L(M)}Y

w Places)

1M*1 = 1M1,

R$*) = {a E IN!: M k tp[a]}.

By Logs Theorem it is easy to check that (ntsIM,/D)* = nt,I M r / D hence M r / Dis A-compact iff AZ*'/D is (A, at)-compact and nsIMt/D is A-compact ii€ nt61M:/D is (A, at)-compact, and iK is A-saturated iff M* is A-saturated. Hence the "only if" part follows easily. THEOREM 2.12: 8uppo8e 2" = A+, IIl = A, D a good, K,incumplete, UltraCjElter over I ; M yN are elementarily equidnt nzodeb of m r a i d i t y
(and saturated).

Proof. By I, 1.11 every two elementarily equivalent A+-satursted models of power A+ are isomorphic. By Log's Theorem Mr/D,N r / D am elementdy equivalent. So, by the symmetry, it suffices to prove that Mr/D haa cardinality s h and is A -saturated. Now +

llMr/Dll < ~

+

~

< M(A+ )A~< 2"'" ~ =~ A+ r

~

and by 2.3 and 2.11 M r / D is A+-compact. As JL(M)Is A, HI/D is A+saturated, so we finish. THEOREM 2.13: If D & a good ultrafler over I , nteIn J D = 2Irl.

nter nt/D 2 KO,then

Proof. I f D is 8,-complete, clearly for some n {t: n, = n} E D hence nt/D = n; hence D is XI-incomplete (by Exercise 1.8) so by 2.4 D is regular. For each n let M , be the following model: M,, = (n + 2", P,,Bn>, where P,,,B,, am one-place and two place relations respectively, P,,= (0, ...,n- l } and R, is such that for every w E P, for some i,n < i < n+2", and for every k < n , k E w o R , ( k , i ) . Let for t ~ 1m(t) , = [log, ntl - 1, 80 ll~m(:)ll5 %,but for every n {t: lJfln(:)ll 5 }. E

344

8A"RATION OF ULTRAPRODUCTS

[OH. VI, 5 2

s 2"+l}$ D. Hence M = (A, P , R ) = nteIM,,,,,)/D is an infinite model and is 111 +-saturated. So clearly JPI> IIJ (otherwise { P ( z )A x # a : a ~ Pis} omitted by M) and ]A1 2 211' (for let C E P, ICI = 111, then for everyB s C, p , = { R ( a ;x)if(aEB): ~ E Cis }realized by some b,eA, so IAl 2 211' = 21'1. We can conclude {t: nt

and the other inequality is Exercise 1.3(2).

EXERCISE 2.1: Show that to the list of equivalent conditions in 2.3 we can add: (4) If p is an atomic 1-type in M = niEIM f / D of cardinality
EXERClSE 2.2: Show that in 2.5 instead of assuming q , Mi are elementarily equivalent for every t E I it suffices to assume n t e r M!/D and rite, MilD are elementarily equivalent. EXERCISE 2.3: Show that in 2.6 we can omit " D is 8,-incomplete" but must then demand that for each t E I, M;, M: are A-saturated and elementarily equivalent. EXERCISE 2.4: Show that to the list of equivalent conditions in 2.9 we can add. (4) (i) N : / D is (A, at)-compact for some p. (ii) D is p-regular for each p < A. EXERCILYE 2.5: Let E be a family of subsets of I closed under finite intersection, and let E generate the filter D . Every monotonic f :SNo(p) -+ E has a refinement g :SNo(p)+ E which is multiplicative iff D is p+-good. EXERCISE 2.6: (1) Every A-complete filter is A+-good.(Hint: See 2.1.) (2) If D is p-complete for each p < A, A singular, then D is A+complete. EXERCISE 2.7: I f A is regular 5 Ill, then over I there is a non-A+complete filter D which is A-complete. Moreover D is p-good iff p S, A+ (Hint: D is generated by {X,: i < A}, X , decreasing, 0, X,= 0.)

CH.

M,5 31

345

CONSTRUOTINQ ULTRAFILTERS

EXERCISE 2.8: Suppose the filter D over I is p+-incomplete and A+-good. Show there am X,E D (i < A) such that the intersection of any p X1)s is empty. EXERCISE 2.9: In Definition 2.1(1), when A = p $ , it suffices to take I.1 = Po.

EXERCISE 2.10: Suppose D is an 24,-incomplete filter over I, and A = ntErnt/D2 24,. Prove that P o = A. [Hint: Let M = (w, <, +, x ,P). where P(x) = [GI,N = Mr/D,and n* = (. . .,nt,. . . ) / D . and for a E INI,1. = l{b E INI:N t a < b}l. Then In*] = A, 1. 2 KO =- 1. = IP(a)l,and let nt = P(n*), nf+, = P ( n f ) ,and for every sequenoe (b,,; n < w ) , bk < n f , let

=

+ n:b, + - .. + nzbk < x < ntob0+ - - . + nfb, + nz: k < w}

1); = (n,*bo

clearly p; is realized by some ai;, thus providing n elements of {b: b < n*}.]

k

I

Inf = AHo distinct

EXERCISE 2.11: In Theorem 2.12 instead “ D good” we can assume only “there is a good K,-incomplete filter D, G D.” EXERCISE 2.12: Show that no 24,-incomplete filter over I is 1I1+

+-

good. VI.3. Constructing ultrafilters We shall show that over every cardinality there is a good ultrafilter. We define the product of ultrafilters, and find how regular and good the product is. Then we assume 2Wo > N, and MA (Martin’s axiom) to prove the existence of an ultrafilter D over w , such that for every N,saturated model M of Find,M a / D is N,-saturated. THEOREM 3.1 : Over any A there ia a good K,-inunnplete ultrajlter.

Proof, We first give a definition and prove some claims. Meanwhile let D denote a filter over A, and 59’ denote a family of functions from A onto A.

DEFINITION 3.1 : 59’ is called independent mod D if for every n < . . .,j,-, < h distinct go, . . .,qn-, E 59’ and every jo, < A: go(a) = jo, . . .,g,,-l(a) = jn-,}# 0 mod D (and D is non-trivial). {a

w

and

CLAIM 3.2: Let Do be the@er over A generded by {A}. T h t h e ia a d mod Do. famay 8 of eardidity 2" which b i

Procf.This is a reetatement of 1.6 of the Appendix for a p d c u l m c&88.

CLAIM 3.3: Let Q be h u i ? e p e mod D, and 8 E A. Then for 81nne jinite W C 0,5% - 8' ia i d mod Dl or mod Dayw h e DI[Da] ia thejElter generated by D U {IS)[DU {A - IS)].

Proof.If 8 is not independent mod D,, then for some n < w there are ..,j,,-,< A such that: distinct go, . . ,g,,- E Q and there are joy. W, = {a < A:go(a) = j o y .,g,,-,(a) .. =j,,-,}= 0 mod D,, hence W,c A - 8 mod D (notice that n = 0 mems D , is trivial). Let 5%' = 8{go, . .,g,, - and assume 8' is not independent mod Day80 there are m < w and ju, . . .,jm-l< A and distinct go,. . .,gm-' E Q' such that Wa = {a < A: go(.) = jO,.)a(., ' ' '"'# = jm-'} = 0 mod Day hence Wa c 8 mod D. So W,n Wa = 0 mod D, contradicting the independence of 8 mod D.

.

,

.

.

CLAIM 3.4: 8-e Q is independent mod D, g E Q, 9' = B-{g}, and f :BNo(A)+ D W momtonic. Then there is a f l e r D , D c D , and a mdiplicdve function f' ;tlN0(A) --t D' refining f m h that Q' W i d pended mod D'. Procf. Let {w,: a < A} be an enumeration of Lgn,,(A), let f'(w) = @: g@) = a, 8 E f (w,), w C w,} and D the filter generated by D u i < A}. Clearly D' is non-trivial and f ' is multiplicative, into D', and a refinement off. Let us prove 3'is independent mod D , so let jo, .. .,j,,- < A, and go, . . .,gn-, E Q', the glYsbeing distinct and

v({i}):

W

=

(8: go(8) = j o , . .

. Y

=

it-,}.

!&I-,@)

We must prove W # 0 mod D . For this it suilhes to prove that if w E&~(A), W nf'(w) # 0 mod D. Let w = w,, 80

so we

finish, &B 8 is independent mod D and f (w) E D.

OH. VI, 8 31

347

CONSTaUCTINa ULTRAFILTERS

Proof of 3.1. Let (8,:a < 2”, a > 0 even} be an enumeratiiga of (8:8 G A}, and dfor: a < 2A, a odd} an enumeration of the functions f : &,(A) +- {S:8 G A}, each appearing 2A times. Let Do be the filter over A generated by {A}, and Sobe independent mod Do and of cardinality 2 A . Now we define by induction on a D,, g, such that: (i) ’3, is independent mod D,, 5 K O + lal, (5)1’3 (iii) for /3 < a, DB G D,, ’38 2 3 ’, (iv) if a is a limit ordinal D, = U,<, D,, ’3, = g, (v) if a is even, a > 0 , 8, E D,+, or A - 8, E D,+l, (vi) if a is odd, f , :&,(A) + D,, f a monotonic then there is a multiplicative refinement f: of f a , f: : &,(A) + I),+,. The induction is easy: for a = 0 we have defined, for a limit see (iv) (using iii), for a + 1, a even, a > 0 use Claim 3.3 and for a + 1, a odd use Claim 3.4, and for a + 1, a = 0 let D , = {{a:n < g(a) < w } : n < w}, ’3, = ’3,,-{g], where g E go. Now D,A is an ultrafilter by (v). I f f : &,(A) + DaA, then for some a < 2A,f :&,(A) +- D,, and for some /3, a < /3 < 2”, f,3 = f , /3 is odd, so (when f is monotonic) there is a multiplicative refinement f ’ off, f ’ : &,(A) + DB+1 E D+ SO D,A is a A+ -good ultraater over h. AS D, DaA, D,A is N,-incomplete.

n8<,

CONCLUSION 3.6: If My N are elementarily equivalent models of T , of cardinality 5 A + , ITI 5 A, 2A = A+, then for some ultrajlter D over A, M A / Dg N A / D .

Proof. Immediate by 2.12 and 3.1. DEFINITION 3.2: If D, is a filter over I, (1 = 1, 2) then D1 x D, is the family of sets 8 E I , z I , such that: {i E I,: {jE I , : (j,i ) € 8 )E Dl} E D,.

LEMMA 3.6: (1) D1 x D, is ajlter over I , x I,. (2) If D, is an ultrafilter over I , ( I = 1, 2) then D , x Da ie an dtrajlter over I , x I,. (3) I%i,j>er1 Jfi,j/Di x Da I L r P ( n r e r , Jfi,j/Di)/Da. (4) Mrixra/Dlx D, 2 (Mrl/DJa/Da.

Proof. Immediate.

SATURATION OF ULTRAPRODUCTS

348

[OH. VI, 8 3

LEMMA 3.7: (1) If D , OT Da i8 h-regular then D , x Da is h-regular. (2) D , x D, i8 h - g d ifl D , iS hyood; PO& that D , i8 8 1 incomplete.

Prmf. (1) Immediate by 1.4(1)(A), (C) and 3.6(4). (2) Immediate by 2.3 and 3.6(4). If D, is h-good, for any h-saturated model M , Mri ra/Dl x D, (M'l/Dl)'a/D, (by 3.6) is (A, at)-compact (by 2.3). Hence by 2.3 D , x D, is h-good. If D1 x Da is h-good we get the result similarly. CONCLUSION 3.8: POTevery h Ip, there is a regular ultrafilter D Over h+-good but not A+ +-good. If A+ < 2", D i8 not A+ +-goodbut n,/D 2 Noimpliea n,/D 2 2A > A+.

p Whkh is

n,<,

n,
Prmf. Let D , be a regular ultrafilter over p, and D, a A+-good K,incomplete ultrafilter over h (exists by 3.1) D, is not A+ +-goodby 2.4 and 1.3(3). So D , x D, is an ultrafilter over p x h which is h+-good but not A + +-good. As IX x pI = p, we finish. The second part follows by 2.13. LEMMA 3.9 (MA): There i8 an ultrajlter D Over w such that: If IP"nI I No,and M = < Ir) M , / D and p is an m-type in M of cardinality < and P(xo) A - A P(x,,,-,)~ pthen , p i8 realized in M .

n,

Remark. On Martin's Axiom the rertder can consult, e.g., Jech [Je 741. It is consistent with ZFC + 2N0 = K, for any regular Eta, if ZFC is consistent and it implies h < 2No + 2" = ~ H o .As we use it only rarely, we do not elaborate.

Proof. It is sufficientto prove the lemma just for models M , such that llJfnII < 2'0, IUMn)J < 2'0. So let {S,: a < a odd} be an enumeration of {S: S E w } and {(p,, ( M t : n < w ) ) : a < 2N0, a even} be an enumeration of all pairs ( p , (M,: n < w ) ) where L = L(M,) has cardinality < 2N0, IIM,II < 2N0, and p is a set of cardinality < 2No of ~(3,7i)( I @ ) = m, E E Ma), and lPMnl < No;each pair appearing 20' times. We define by indaction on a < 2n0, families E, of subsets of w , such that (1) E, generates a non-trivial filter [E,] over w which is regular, (2) /3 < a + E6 E E, and for limit a,E , = u s <, E , and IE,l < 2H0, (3) for a odd, S, E E,,, or w - S, E E,+,,

n,,
CH. VI, 8 31

a,),

(4) for a even if for every tpl(Z,

{i < W: Zt b (%)(p(Zo) A tpl(Zs

then for some 6 E

{i <

w:

349

COWSTRUCTINQ ULTRAFJLTEE8

%[iI) A

*

+

*

. . .,tpn(P, a,,) €pa, A

* * .

A p(Zm-l)

A Ts(z, an[iI))}E [#a]

nieO M f , 6 = (bo, . ..,b, -

Mf k P(bo[i]) A

* * *

A

P(b,-,[i])

I),

for every tpl(Z; 8,) E pa

A p1(6[i], 81[i])} E [Ba+l].

Clearly if we succeed then [,?#,NO] is the ultrafilter we want: and there is no problem for a = 0 or a limit, or a odd (by 1.1 of the Appendix). So aasume a is even and pa,M: (i < w) satisfy the hypothesis of (4) (the other cases are trivial). Let us define a set V of “forcing conditions” a forcing condition is a finite set w of equations xl[i] = a where I < m, i < w, a E P I such ; that q[i] = a E w and zr[i] = a’ E w implies a = a’. V is a partially ordered by inclusion, and it is countable, hence there are no N, pairwise incompatible conditions. For eachq(E;G) = {A q : q C pa,q finite} and finite interseation S of members of Ea let

w,

tpm

=

a))

{w E V : for some i E 8 and bo, . . .,b, - E P ( M f )

Mp c cp[bo,. . .,bm-,, tqi]], {ZO[i] = bo,.

. .,X,-l[i]

=

b,-,}

E

u}.

Clearly each V(8,tp(Z; a)) is dense in V (by our hypothesis from (4)) and their number is <2% Hence by MA there is a generic V* E V , i.e., every two members of V* have a common upper bound, and V* n V(S,tp@; a)) # 0 for every tp(Z; a) EI);, S a finite intersection of M f , b,[i] = a iff q[i] = a E U {w: w E V*} members of Ea.Let bl E ( I < m). 6 = ( b o , . . .,b,-l) is well defined aa V* is generic. Let

ni
E,,,

=

Ea U{{i < o:Mf C tp(6[i], a[i])}:cp(Z; a) €pa}.

[Ea+,]is a proper Glter as V* intersects each V(S,~ ( fa)) , end the conclusion of (4) clearly holds.

THEOREM 3.10 ( M A ) : (1) There is a regular ultrafilter D Over w, 8uch that if M is a A-snturated &el of Ti,,(see 11, 4.8), A 5 2H0, then M m / D ie A-saturated too. ( 2 ) If h < 2’0, there is a regular ultraJilter D over h which is not N,good such that for any model M of Tind,MAID is A+-saturated.

350

SATURATION OF ULTRAPBODUOTS

[OH.

VI, 0 3

Proof. (1) Let D be the ultrafilter from 3.9. If p is any 1-type over P / D , 1pI = p < A, let p = {q&; a=):a < p}, let A, = U {aa[i]:a < p} by 1.6 of the Appendix, and the elimination of quantifiera of Tina,and the p+-eaturation of bl, there is BI E 1611, lBil = KO,such tha&any non-algebraic finite type over A, is realized by an element h m BI. Let us expand M to Mi = ( M , P'a), P a = BI and apply 3.9. (2) Let D, be a good K,-incornplete ultrafilter over A (exista by 3.1)) so D, is regular by 2.4, and Da the ultrafilter from (l),then D = D, x D, satisfies our demands (Dl x Do is regular by 3.7(1) and MAX"/D = (MA/D,)"/D,,iKA/Dlis A+-eaturatedby 2.3 and 2.11 as D, is A+-good and K,-incomplete. So by (1) (MA/D1)"/D,is A+-saturated too). Let 9 denote a family of functions g : A 3 A, g onto A, D a filter over A. Exercises 3.2-3.4 are from [Sh 7 1 ~ 1 .

DEFINITION 3.3: (g,, g2,D) is K-independent if whenever j, < A (5 < to < K) g' E g a (1 < n < o)and f,, f' E g1 me distinct (5 < to, 2 c n) then {a c A: fr(a)= j, for t < to,f'(a) = g'(a) for 1 < n} # 0 mod D.

EXERCISE 3.1:There is a gl of cardinality 2Asuch that is K-independent, Do = {A}, provided that A = A<".

8, Do)

EXERCISE 3.2: If ( g , , a , D ) is K-independent, D = [El (the filter generated by E ) , Yta is a family of functions from A into a, a < K , then 1 1 + I g s l iK , such that ('3;, g a , D) there is '3; G g,,Igl - yP;( s 1 is K-independent.

EXERCISE 3.3: If (g,, 0, D) is K-independent, S c h then for some 9;c g, Ig, - gil < K and (g,,0, D') is K-independent, where D' = [D u {s)]or D' = [ D u {A - A!?}]. EXERCISE 3.4,: Assume (gl,0, D) is K-independent, D = [El. Assume also that M i (i < A) is an L-model, lPMiI s x < K, aB,,,E Mi forB < Po < 2", 1 s rn s n(B).AssurneqBELand{qs(z,YB,,,. . , Y ~ , , , ( ~ ) )/3: c Po)is closed under conjunctions, and for every B < B0

n ~ < ~

.

OH.

VI, 8 33

OONSTRVOTWO WLTRAR'ILTIRS

35 1

EXERCILYE 3.5: (1) There is an ultrafilter D over A suoh that, if h = A<",then: (i) If M, N are elementarily equivalent and of cardinality C K , then MA/Dz NA/D. (5)If IlMll < K, 2Y s 2Athen M A / Dis p+-saturated. ($) If IpMtl s KO < K, 2' s 2A,p a l-tme in H { / D ,P ( X 0 ) lpl s p then p is realized in nfeA M,/D. (iv) If 2" s P, 2%= A, M a model of Thd of mrdinality sA, then M"/D is p+-satuahd. (v) If IlHilJ5 x < K, MI, M ' are elementarily equivalent where M' = ntxA M!/Dthen M', M' isomorphio (Z = 1,2) provided that IL(M1)1 s A.

n{<~

Remark. Notice the following difference between the proof of Exeroise 3.6 aketohed by the preceeding exercises, and the proof of 3.1. In the proof of 3.1 the funations in g are used as partitions of A, whereaa in the proof of Exeroise 3.6(1) they are used as elements in the ultraproduote. EXERCI8E 3.5: (2) In Exeroise 3.6(1) (and Exercise 3.1-4) instead of atesting with the filter Do = {A}, we can stwt with any K-complete filter D, over h provided the oonolusion of Exeroise 3.1 holds. (In Exeroises 3.2 and 3.4 [D, u El replace [El, and in Exeroise 3.6 D, c D.] Of oourse, we c)an start also with [D, u El, provided that IEI < 2", and the ooncluaion of Exeroise 3.1 holds. If there me A, c A, A, # mod D,, A, n A, = 0 (for i < j c A) then the c#>nclusionof Exeroise 3.1 holds. EXERCIHE 3.6: Let f be a 2-place function from 8%(p)to subset of I , let D be a p -good filter over I , and X , (i< p) be subsets of I . Suppose (1) For m y 8, t Ergwo(P), +

te8

x, n n( I - x,)c f(8, t ) mod D.

(2) If 8, t €8&), (3) f(0,O) = 1.

tct

8A

t # 8 thenf(8, t ) = 8.

352 (4)

[am, . 83

SATURAlTON OF ULTRAPRODUOTS I f 8 , t , WE&&), f(8,

t)

u

8 n t (f(81,

= 8, 8 U t C

ti): 8 C

81

C U,t

U,

then

E ti

c U , 81 u

=

U}.

Then there am subsets Y1(i < p ) of I such that: (A) Y1= X 1mod D for each i c p. (B) For every 8, t E &&),

ny1n n( I - yl) E f(8, t ) . 168

1e:

On Exercises 3.6 and 3.7 see [Sh 72~1.

EXERCIBE 3.7: Prove that the following conditions on the filter D over I and the cardinality A > Noare equivalent. (1) For every set of L-models Mt (t E I ) M,/D is A-compact. (2) D is K,-incomplete, A-good and B(D)is A-saturated, where B(D) is the Boolean algebra of the subsets of I mod D. (Hint: Use the Feferman-Vaught theorem; see e.g. [CK 731 and Exercise 3.6.)

n,,

EXERCI8E 3.8: In Exercise 3.7 we can replace A-compact and Asaturated by (A,d)-compact for d the set of Z,, (or n,)formulas (A C,-formula is a formula of the form #(#) = (3Zl)(VZ2). .‘p(Zl, . . .,Z,,,g), ‘p quantifier free; a lI,,-formula is the negation of a C,-formula).

.

EXERCI8E 3.9: Prove that B(D)is isomorphic to M i / D where M, is the Boolean algebra with two elements. EXERCI8E 3.10: Show that in general, in 2.3(3) we cannot replace “for any p < A” by “for some p < A”. (Hint: Use Exercise 3.5.) EXERCIHE 3.11: (1) Show that for any A-good filter D, B(D)is (A, at)compact (SeeExercise 3.12). (2) Suppose M , is a ( A + , at)-compact Boolean algebra, of cardinality A+, and 2A= A+. Prove that there is an K,-incomplete good filter D over A such that B(D)is isomorphic to M,. (3) We can suppose M , is a compact Boolean algebra of cardinality 2A,and (Vp < ZA))(2J’s 2”), and get the same conclusion. Remark. In (3) the existence of M, implies (Vp < 2”(2” s 2A)by VIII, 4.7. Similarly in (2), it implies 2“ = A+.

[Proof: (2), (3). Similar to the proof of 3.1. In the ath step we have a filter D, = [E,], lEal s A + lal, a subalgebra N , of N o , llNIIll s 1. + H,, an embedding Ha of N , to the

OONSTRVOTWO WLTRAR'ILTIRS

Boolean algebra of the subsets of I such that 0 # a E lMol implies &(a) # 0 mod D,, and a family 9, of functions from h to A, such that for every distinct g,EY, and j,< h (I < n < w ) and a € Val,a # 0 {i< h:go(i) = j o , . . . , g , , - l ( i ) =j,,,}nH(a) # OmodDaandlSs h + .1.1 (Clearly Nu,Ha,E, increaae with a, and S, deareaees with it, eaoh of them oontinuously. In the end D = D,n, H = H 2 ~ . ) We have steps of t h kinds: Kind I: We want to include b in the domain of H . Choose g E 9,, and let Nu+,be generated by Nu and b, = {i: g ( i ) = 0}, S,+, = S, - (s) and

B,+,

= E,

u (I

- Ha+,@,) n H,+,(a): a E Nu,a n b,

=0

and b, is b or its complement}. Kind 11: We want to refine the monotonic funofion f : S&) + D,. We do it exactly aa in 3.1. Kind 111:We want to decide the fate of S c h (that is, w e want that for some a E No+,, 8 = Ha+,@) mod Let here A denote a function: A = {(gf,jf): i < io} where io < A+ + 1.1 and gf E S are distinct. For such A let P be the filter generated by Eh9 +

~h

=

B, u {&+(a> = jf:a < A}: i < io}

and S h = g,

- {gf: i < io}.

Cme (i): For some a E Nuand A, H&) = S mod Dh. We then d e h e : Nu+,= Na,H,+l = H,,9,+l = gh,E,+, = Eh(andofcoureeD,+, =

[E, + ,I). It is eaay to check our demands hold. Cme (ii): Not (i), but there are a, b EN,, A such that H,(a) E 8 c H,(b) mod Dh, and b - a is an atom of Nu (i.e., (Vc E N,)(b - a E c v ( b - a ) n c = 0) ) . We then define Nu+,= Nu,So+,= gh, I#,+, = Eh u { A - (IS - Ha@))}.The only way in which our demands may fail is that for some A,, A E A,, and c EN,, c # 0 such that H,(c) G S - Ha@)mod P'. But then Ha@)n Ha@) = 0 mod D', hence c n a = 0 ; but also H,(c) E H J b ) mod P',hence c c b; so as b - a is an atom of Nu,c = b - a. Now it follows that Ha@) = Ha@) U H,(c) = S mod P', so caae (i)holds, contradiction. Cue (iii):Not (ii) or (ii)but for some a E Nuand A, H(a) E S mod Dh, and for every b EN^, h' : h E hl, H,(b) c S mod Dh' implies b E a (in N,, of course). We then choose S,+,, B,+, as in 0&88 (ii). The

only way in which our demands may fail is that for some C E N ~ , c # 0 and hl, h E hl, the following holds:

H(c) s 8 - H J a ) mod P', but then again c n a = 0, and H(c u a) c S mod P', so c u a c a, henoe c = 0, contrediotion. Cme (iv):Not (i),(ii) or (iii), but for some b E Nu, h: S c H(b) mod P, andic h 1 , c ~ N a , 8 c H ( c ) m o d P ' i m p l i e e b c c . The p m f is similar to that of c&88 (iii); this time Ea+, = Eh U {A - (H(b) - S)). Cme (v): None of the previous ones. Choose h such that if c E Nu, c f 0, and 8,E {S,h - 8),B c I , , and Ha@)nS1= 0 mod Dh' then Ha@)n8, = 8 mod P.Now letp = (8 c b: b E N , , S c Ha@)mod P} u {a E x: a E Nu, &(a) E S mod P}. It is easy to check p is a type in s llNall + No s A, p is atomic, and N is (A+,&)Nu, and compect, some c E M,,r d i z e s p. D e h e Nu+, aa the subalgebra of N o generatedbyIN,I u {c},Ha+,eXtendHa,Ha+,(c) = 8,andE,+, = Eh. We leave the reader the checking.] EXERCISE 3.12: Show that the Boolean algebra M o is (A,&)compact ifffor every a,b < A, a,,bt E N osatisfying N o C a,,,, s a,,, s bj(,) s a$(,)for i(1) < i ( 2 ) < a,j(1) < j(2) < there is c E Mo, M o k a, I; c s bjfori < a,j < P. EXERCISE 3.13: Suppoee M,,is a Boolean algebra of cardin&ty 2". Then for some regular, goad filter D over A, M o is elementarily embeddable into B(D),and B ( D ) is p-saturated, where p = mink:2' > 2"). PROBLEN 3.14: For any A, characterize the possible B(D) for D a filter over I. EXERCISE 3.15: Prove that for any n and A, there is a filter D over h such that B(D) is ( A + , C,,)-compact but not (A+, l7,,)-comprtct, and vice vem. (Hint for Exercises 3.13 and 3.15: The proof is like Exercise 3.11, but Nu (a < 2A) is not predetermined.) EXERCIBE 3.16: Define a h-good filter in a Boolean algebra just as in Definition 2.1. Prove the parallel of Exercise 3.6. Show that if B, axe Boolean algebras, D, a filter in B,, Ir : 8, --+ 8, a homomorphism onto h'l(1) = D,, and D,, D, are h-good, then the filter h-l(D,) is &good. [Hint: Let g : S,&) +h-l(DS),p < No;so hg :S&) --+ D2, so it h a

OONSTRVOTWO WLTRAR'ILTIRS

a multiplioative refinement g1 :B&) + D, and choose X , E B,, h(X,) = g,({i}). Hence X , E g(e) mod D,. Define g1 :B,&) --+ D,

bY g1(8) =

u { - (nX , 1

g(t)) : t

t8t

.},

a multiplimtive refinement of gl, and g*: 8,&) p(8) = X , n fl(8) is aa regarded.]

n,.,

S' :~~;EOW -+ D, --t

h-l(D,) defined by

EXERCISE 3.17: Supp~eep s x = x<= < 2n, and ahow thare is a regdar, K - p o d filter over p generated by a frtmily of sx sets. (Hint: Use 3.1, and show that any regular, good ultrafilter over p haa such subfilh, using Exeroise 2.6.) EXERCIBE 3.18: Suppose E is a family of subsets of A, IEI s A dosed under finite intersections, 0 $ E , and D is a A-regular filter over I. (1) There is a function B :[El --+ D suoh that: (i) for A, B E [El, B(A n B ) = G(A)nG(B),A c B +G(A)c U(B), (ii) if A, €[El (i < S),

nA, = 0 l
Q(A,) = 0. f>d

( 2 ) [El is p-good [c1-1~gulm] iff [{Q(A):A E g]is p-good b-regulas3 (for eeoh p). (Hint: (2) ;See Exaroiee 2.8.)

EXERCIBE 3.19: If D is A-regular, A < = = A, then there is a K-good, A-regular filter D, s D .

DEWINITION 3.4: D is a, uniform filter if all members have the same Oardindity (it is IIl when D is a filter over I ) . EXERCIBE 3.20: (1) A ultrafilter D over A is d o r m ii€ where @ = {A E A: I A - 41 < A} which is auniform filter. (2) If D is a uniform ultrafilter D over A, of[(A, < )"D] > A. (3) If D is a regular ultrafilter over A, 6, limit, then

ED

r

EXERCIBE 3.21: Iffor some ultrafilter D over A, Do c D, Do a filter and o ~ [ ( K ,< )"/Dl = p, then there rn functionsfr : A K , (i < p ) suoh that for no g : A - K for every i < p ( K , c ) ~ / D Cfr , < g. --f

356

[a. VI, B 3

SATUI&ILTION OF ULTRAPRODUUTS

EXERCIBE 3.22: If Do is a filter over A and there are functionf, : A 3 K (i < p ) such that for every g : A -+ K for some i < p ( K , < )"Do C g < f,, then for every ultrafilter D over A, extending Do cf[(K, < )"D] s p. If also i < a < p implies ( K , <)"Do Cf, < fa, then cf[(K, <)"/Dl = p. Remark. The family of ft's as in Exercise 3.22 are called a (Do,p)-scale for K . On independence results conoerning scales see, e.g. [He 741. The existence of an 2b-s~aleis weaker than MA. The following exercises a m an improvement of Theorem 3.10.

EXERCISE 3.23: Suppose D is A-regular M = nt,rMt/D, A c ]MI, IAI S A then there are finite Pt c lMtl such that A s PM where W, PM)= I I t a r (Mt, pt)/D. EXERCIBE 3.24: Suppose Dan ultrafilter over I, llMtll s A, A c 1M1, n t e r Jft/D. (1) If IAI < cf[(h, <)'/D],then for some P, s lMtl, lPtl < A, and A c PM where (MyPM)= (Mt,P,)/D. (2) If A = Ha+,, IAI < cf[(N,+,, <)'/Dl for 0 s 2 s n, then for some P, c lM,l, lPtl < K, and A E, 'P where

n,,I

(Jf,PM)=

n

(Mt, P J D .

t€t

Remurk. Now a combination of Exercise 3.6 and 3.24 gives results like 3.10 (for the existence of such filters the existence of scales is sufficient, see Exercise 3.22). EXERCISE 3.25: Suppose A = 2" < 2"+, and some p+-complete filter D, over A satisfies Exercise 3.1 and a ( D l , A+ +)-scale for p+ exists. Then (1) over A there is a regular and good ultrafilter D, such that for every A+ +-saturated model of Thd,MA/Dis A+ +-saturated; (2) over A+ there is a regular not good ultrafilter D, such that for every model M of Tina,M " + / Dis A+ +-saturated. EXERCISE 3.26: (1)If A = KO and MA holds, M , N are elementarily equivalent IL(M)I 5 A, 11M11 + IlNll s A then for some regular ultrafilter D over A, MAID N X / D . (2) If, e.g., IlMll + llNll + IL(M)I s p + , M , N elementarily equivalent and the assumption of Exercise 3.26 holds, then for some regular ultrafilter D over h M A / D= NA/Dprovided that A+ = 2".

=

+

OONSTRVOTWO WLTRAR'ILTIRS

CONJECTURE 3.26: (1) In Exercise 3.26(1) we can eliminate MA. (2) It is consistent with ZFC, that for A = KO for every regular ultrafilter D over A+, of@+, < ) " + / D= A + + , cf(h, <)"+ID> A++ (thus Exerciee 3.6(1) is essentially best possible). We may repltloe A, A+ by A+, 2". DEFINITION 3.6: For an ultrafilter D (over I) and reguIar cadinal 1Cf(K, D ) is the smallest cardinality A such that: there is a subset of {a E d / D :d / D C a < a for each a < K } which is unbounded from below, and has cardinality A.

K;

Remarke. (1) We do not distinguish between K and the model ( K , <) (till the end of this section). (2) Clwly lCf(K , D) is an W t e regular cardinal or 1. (3) Till the end of thh section, an ultrafilteris over I if not mentioned otherwise. CLAIM3.11:(1)IfK > III~~@?',IOf(K, D ) = 1;inf&hf(K, D ) = 1 iff D ie cf ~damndinglycomplete (8ee De$nition 6.1). ( 2 ) d / D k not ( K + lcf(K, D ) ) + - w t . (3) If D ie a A-good N,-inco))zlpleteultrajilterand K < A, thenlcf(K,D ) 2 A. Procf. Immediate. THEOREM 3.12: For every p = pNo 5 2", and regular K, No < K there is a regular ultrajEZter D over A such that lcf(Ko,D ) = K and

sp

-

THEOREM 3.13: (1)Suppose No = A0 < A, < * * < A,, = A+ each 4 i 8 regular and hI+1 s p l s 2", pl regular (for 1 < n). Thenfor 801126 regular A1-good ultra$lter D over A, lcf(K, D) = p l whenever A, 5 K < A1+,. I n fact D i8 not A?-good. (2) 80 in particular for every h and regular p s A, thre i 8 a regular ultraMer over A which ia p-good but not p+-good. THEOREM 3.14: Stqpme in the previowr thorem that A, = K, and < xk.= 2A and xf k < w , xl= xp (1 $ k) are given xo < x1 (i < 2, 1 < k or i = 0, 1 = k) are regular m r d i d >No, X! s xi, X I 5 X I . Let Jl = (a E oA/D:1.1 = xl} <

*

a

-

358

where

1.

=

n a [ i ] / D= l{b E d / D :o A / DC b < a}], f
then we can aizmme in addition, thad (i)for each a E o A / D ,1. ie jinite or E hl:1 s k}, (ii) for I < k cf(J,)= x: (Jl is order as a subset), (iii) for 1 < k cf (J:) = xe (JF the inverse of JJ. DEFINITION 3.6: (1) In Definitions 3.1 and 3.3 we allow the functions in 9, g1resp., not be neclessarily onto A, and then demmdj, E Range@,), j c E RtmgeVt) reSP* (2) For a family 9 with domain A let (FI for finite interseotion): FI(9) = {h: h a funotion, Dom h c 9, lDom hl < No and h ( f ) E Rengedf)}. For h E F I ( 9 ) let A , = {i < A: f ~ D o m h implies f(i) = h(f)}. FI,(Y) = {A,,: h E FI(9)). CLAIM 3.15: (1) All relevant krnmaa remain true ~ dthe new r Dejinition 3.6:3.2,3.3, 3.4,Exerci8es 3.1,2,3and 4.E.g., 3.2 becomee if IJ,l s A for i < 2”, then there ia a f m i l y (fi: i < 2”}, fc fr.mA onto Jt, independent mod{A}. (2) If 9 is-i mod D, for a < 6 , D , (a < 8 ) an ilacrerreing 8 7 Of $&3’8 Over I t h 9 6 ~ d p e n d e n mod t ua<6 D,. (3) If D i8 a@er over A, 9 idpendent mod D, then there i s a d d jEJter D* 2 D, modulo which 9 is i d (i.e., there is no D1 # D*, and D1 2 D* such ticat Q itt idpendent mod Dl). (4)9 is idpendent mod D iff A,, # 0 mod D for every h E FI(9); crnd I1 -C 1, iffAh, E AhI. (We &Card the w e IRange f I = 1.)

DEFTNITION 3.7: (1) In a Boolean algebra, a partifion is a maximal set of pairwise disjoint non-zero elements. An element is baaed on a partition W ,if b E W impties b c a or b n a = 0; the element is supported by a set W of elements if it is baeed on some partition W1E W (noticeif W is a partition, a is baaed on W 8 W supports a).A set W is dense iffor every non-zeroa there is in W b c a. (2) If D is a f l h r over I,instead of speaking on B(D), AID, ,we speak on {A: A s I), A , . (and then everything is mod D.) (3) CC(B),for a Boolean algebra B, is the minimal regular cardinal A 2 Nosuch that every partition of B hae omdimlity
..

...

OONSTRVOTWO WLTRAR'ILTIRS

CLAIM 3.16: I n a Bodean algebra

(1) aiebaaedtm W i f f l -aiebaeedon W , (2) if q ie baaed on Wl (Z = 1,2) then a1 n a,, a, u a,, a, a, are Basedon{cnd:ctsW1,cEc W a , c n & # O}wh&hieapadition, (3) for any eet W of ehnente olosed under finite intereectiorce 8uch that

-

everyaiz WbeJonstoapwtition E W,theeetofelenzenteitswpporbiea a d id&aU e k m m h of W , (4)if a , b are supported by Wand c E W =+ [c E a o c c b ] then a = b .

d+brcz,

CLAIM 3.17: Buppoae D ie a m a x i d j E u e r Over I modulo which 9 ie i-. (1) FI,(S)ie h e mod D; hence euch ehnent ie twppted by it. (2) Fw each f E a, U-l(j):j E Range f } is a pddtbn mod D. (3) If An E FI($), Dom hn(n < w) are pairtoke cEi.$Xnt, A E I, and A n A,, = 0 mod D then A = 0 mod D. (4) Led 9 be the diejoint union of Yr,, 9 a , A E I ie euby FI8(g1),h E FI(EB), A = h1 U ha, h1E FI(gJ. If A, c A mod D then A,, c A mod D. (6) CC(B(D)) = KOifffor dyfinitely many f E B ie IRange(f)l > 1, and for 7u) one IRange(f)j = No. Othertoke CC(D) ie the fir8t regular A > No ezlol, that f E 9 =s IRange(f)l < A. Moreover, if p L h ie regular, A, # Omod D for i < p, then there ie 8 s p, 1 8 1 = p euch t7mt for n < w, and dietilcct i(Z)€8, A,(!, # 0 mod D. (6) IB(D)I s 191tre + 2 ' " ~ hK = CC(B(D)).

nl<,,

Prmf.(l)OtherwieeforsomeA E I , A # €imodDandA, $ A m o d D for eaoh h E F I ( ~hence ) Ah A (I- A) # 0mod D. Hence 9 is independent m o d [ D u { I - A}], oontradioting the maximality of D (aa A # 0 mod D). (2) Clearly the f - l ( j ) ' s are pairwiee disjoint mod D. Suppose A n f-'(j)= 0 mod D for eaoh j E Range(f),but A # 0 mod D. By (1) for some h E FI(O) A , s A mod D, and olearly for some h,, h E hl E FI(Q),f E Dom hl. So also A,, c A mod D; but let h,(f) = jo,so A n A,, E A nf-l(j0)= 0 mod D, mntradiotion. (3) Suppoee A # 8 mod D heme by (1) for some Is E FI(g),Ah E A mod D; for all but hitely many n's, Dom h n Dom hn = 0, so h U hn E FI(9). NOW Ahuh,, # 0 mod D and Aruh, E A mod D, and A n &,ha E A n A,, = 0 mod D mntrdotion. (4) Let A be based on the partition {Ahl: g < g(O)}, h, E FI(9,).Now, A h E A m o d D , hence A n A h , P 0 m o d D ~ 8 , , n 8 , = 0 r n o d D

360

[m.

SATURATION OF ULTBdPtlODUCTS

83

hence (m g1u 9a is independent mod D ) A n Ahr = 0 mod D =Ah1 n A,, = 0 mod D hence (otherwise Ah1 - A contradicts the maximality of the partition) Ah1 E A mad D. (6) Notice t h t if f,,E are distinct, [Range > 1, j: # j: E Rangef,,,and h,, is defied by h,,(j',) = j p for 1 < n and h,,(f,,)= j:, then h,, E FI(9)and A,JD (n < o)are KO pairwise disjoint non-zeroelements of B(D). By 3.17(2) f E 9 3 ]Rangef I < CC(B(D)),hence the first phrase follows, and the second follows from the third. So let p L h be regular, A, # 0 mod D for i < p. By 3.17(1) there are h,EFI(Q) such tht A,,, c A, mod D. By its definition h > No, hence p > 8, and we msume p is regular; and Dom hi is finite. So by 1.4 of the Appendix for some n < w end 8, E p, IS1l = p, for every i # j E S I D o m h i n D o m h , = cfo,...,fn-l}. As (Rangeft\ < h s p, there are 8 c 8, and j, E Range fi (1 < n ) such that 181 = p and for every i €8,hi(fo) = joy h,(fl)= jl, . ., = j,,-,.Clearly 8 is m required. (6) By 3.16(4) each partition W supports s 2IwI elements. By 3.17(1) every element of B(D) is supported by {Ah/D:h E FI(9)).By 3.17(6) IFI(9)I 5 I 91 + K , and every partition of B(D) has cardinality < K . Collecting those faots the result J B ( D ) s J Jgl*x + 2 < = becomes obvious.

f,l

.

LEMMA 3.18: 8 u w e D a maximal filter mod& which g*u g is independent, 9*, 9 are disj~id;f E 9* u 9 implies IRange(f)l < cf a: of a > No, 9 = u B < a g8,g8 increas&ng and let 9 8 = g - g8. Swppo8e ale0 that the $&ere Di(i < a ) satisfy D, s D, for i < j < a anal: (i) Di is generated by D and sets &upported (mod D ) by F I p * u 9,).

(ii) g*V gEis independent mod D,.

(E) Di is rrmxhal with re8peCt to (i) and (G). T b

(1) D* = Ut
OH.

m,8 31

361

OONSTRUOTINQ ULTBAFILTIDRS

Proof. Part (3) is trivial (aa the family of D’s satisfying (i) and (ii) is closed under amending chains), and part (2) first statement follows by (1). The second statement of (2) holds easily. (1) As 9*is independent mod D, for each i,cleady it is independent mod D*. So we have to prove only the maximality. So let A E I , A # 0 mod B*; now for some S+ c 9,19’1 < CC(B(D)),and A is supported by FI,(9+ u 9*) mod D ; 80 for some y < a 9+c g, (as cf a > IRange (f)l for f E 9*U 9,and cf a > No, by 3.17(6) cf a L CC(B(D))hence of a > IS+I). Now [D, u { I - A}] is a filter properly extending D,, and satisfies (i), hence by (iii) should f d to satisfy (ii). That is for 8ome h E FI(9* u 9,) Ah !z A mod D,, hence (by (ii) and 3.16(2)), for some B E D,, BID is supported by FI,(9* u 9,)and Ah n B C_ A mod D (by 3.16(2)) SO Ah S A U (A - B )mod D. h t h. = k1 u W’, W1 E FI(S*), ha E FI(9’). By 3.17 (4) (where D, A U (A - B ) , 9*u 9,,9,stands for D, A , 9,Sl).Ahi s A u (A - B ) mod D hence A,1 E A modD,. As this holds for every A # $4 modD*, D* is a maximal filter modulo which 9*is independent. (4) Suppose feYp is a counterexample. Then for some A E 1, A # F, modDp and for every t~ Range(f) A n f ’ ( t )= pI modDBso for some B, supported by FI,(Y* U Y@),A n f ’ ( t )c B, modD and I - B,€Dp. By 3.17(2) there is a partition ( A , $ / D : i< 5) of B(D) on which A is based; by a hypothesis 5 < cfa and w.1.o.g. for some 5 < f [ , for i < 5, A,* c A modD, and for i 2 5(i < 5) Ah1n A = F, mod D. So for i < 5, A,( n f’(t) E B, modD. Let hi,o = hi (Y* u g’), so as B, is supported by FI,(Y* U Yp)a n d f E W = ~-9?’,by 3.17(4) Aht E B, modD. Define A* = Ui..cAht,or now: h - s t l y , A c A* modD [as for i < 5 A,* G A,i,Oc A* so A -A* is disjoint to Ahi(i< 5) and for i 2 5 (but < E ) ( A-A*) fl A,* E A n A,, = Y, modD, so A - A * = Y, modD would contradict the choice of
r

x,t,o

362

SATURATIOX OB ULTBhPaODUaTS

[=.VI, 9 3

A,,,oc B, modD]. But I - B t ~ D Bhence , I-A* ED!, hence I - A ED@, a contradiction. Remark. We have shown that B(D) is complete (this holds generally when for some 8,D is maximal such that Y is independent modD). C L A I M 3.19: ( 1 ) Let Y be in dependent modD, and (f-'(t)/D: t E Range f) is a partition of B(D)for every f E 8 [which holds if D is m a z i b l over which Y is independent but also if D, Y are Da, YBfrom 3.18 (see 3.18(4))].If g : I + K , and KI/DI=u < g/D for every a < K , and j~$9, then d / D tf/D < g/D (hencefor every ultrafilter D* 2 D, K'/D* I= f/D* < g/D*). (2) 8 w p p e CC(B(D))s K , K , p regular, f, : I --+ K for i < p; and KIIDCa <&ID < f!/D,foreverya < K , j < i < p;butforanyg:I-tK if for every a < K d / D Ca < g/D then for 80112e i , d / D t g / D > &ID. Then for every dtrafier D* 1 D, lof(K, D ) = p (and thi8 iS exempli&?d by the f,/D*'8). ( 3 ) Part o w bZd8 for [ D u {Ah}]when A E FI(B),f $ Dom A (k for [ D u {A}]when A # 0 mod D).

Proof. (1) Suppose not, then A = {t E I :f ( t ) 2 g(t)} # 0 mod D , hence by a hypothesis for some j, < K A n fl(j,,)# $j modD, hence

contradioting KIID k j o < g/D. (2) Otherwise there is g/D*(g: I --+ K ) contradioting it. Let A, = {t: a > g(t)} and define induotively a(i) < ~ ( a ( i ) suoh that A, - ANi) # 0 mod D. If a ( i )is defined for every i < K , - A,)/D: i < K } contradiots CC(B(D))s K , hence for some i, a(;) is defined but not a(i + 1). Define gl E I -t K , by

then gl contradiots the h y p o t h e . (3) Etlay. CLAIM 3.20:8wppwe 9 is a nun-empty family of facnctions olrto w, D a d d j 2 t e r modulo which Y is i n & p e .

OONSTRVOTWO WLTRAR'ILTIRS

nt, +

for any draJilter ( 1 ) I f f E Q, t h ( 1 f ( t ) ) / D s lQINo f(t)/D* = ]{a:o l / D Ca < f/D)] s ]ZSPINo. D* 2 D, (2) I f f e Q,, gl : I +w , wl/D t gl/D s f / D ( I = 0, 1) a d for every n g;l(n) = g; '(n)mod D t h wl/D C go/D = gl/D.

nt61

nt,

(1 + f (t))/D P m f . (1)The phase affer "hence " is trivial, so suppoee > p =da IQINo, so there me g, :I + w, g,(t) s f (t)suoh that {t: g,(t) # g,(t)) # 0 mod D for i < j < p+. The number of possible sequences ( g i l ( n ) / D :n < w ) is s p (it is s IB(D)INo;but by 3.17(6) CC(B(D))= N,, hence by 3.17(6) IB(D)I s Po, but pH0 = p). So w.1.o.g. g i l ( n ) / D = gyli(n/D)for i < j < p+ but we get a contradiction by part ( 2 ) . (2) Suppose not, so A = {t E I : go(t) # gl(t)) # 0 mod D. By 3.17(2) there is n < w suoh that A nf-l(n) # 0 mod D. Let for 1, k s n A,,, = {t E A: f (t) = n, go(t) = k, gl(t) = 1) so clearly A n f -l(n) = U,, r n Ak,,, hence for some k, E A,,, # 0 mod D. As A,,, s A, olearly k # I ; it is also olem that A,,, r gcl(k) and A,,, r g i l ( l ) ; hence g o l(k)n g i l(1) # 0 mod D, but g; ' ( k )is disjoint to gcl(E) and g; '(1) = g; '(1) mod D, contradiotion. CLAIM 3.2 1 : #uppose Q*, 9, g6y W', a, D,, D* are ars in Claim 3.18. l h g e ( f ) I < K ; K > KO or Q* = 0, Buppose f u r t h e m e tlrat f E g* K = N o ; ~ i 8 r e g u l a r ; f 6 ~ ~ 6 + l - ~ 6fOTeVe?'yf< and K,{t:[< f 4 ( t ) E K ) ~ DB+ I' Then for every ultraJlter Dl 2 D*, l c f ( ~Dl) , = of a and thia is exemptifid by f6/D1 ( p < a) and d / D 1 t f6/D1 < fy/D1for y < )8 < a.

Proof. By 3.19(1) d / D * b f4/D* < fy/D* for y < < a, and by 3.19(2) (for K > No) and 3.18(2) (for K = No) it suffices to prove

(*I

If g : I + K , and for every f < K , d / D * t f < g/D* then for some 43 < a d / D S t= g/D* > fs/D*.

Suppose g falsifies (*) then for m y [ < K , B, = {t: [ < g(t)}E D* hence for some a(& < a, Be E DNO. As of a > K , a(*) = supt,, a ( [ ) < a, so Be E DM.) for every f , SO {t: [ < g(t))E D,(.) for every [, henoe by 3.19(1) { t :f,,,,(t) < g(t))= ImodD,,,,, hence { t :f,(,,(t) < g(t))= I mod D*, oontradiotion. CLAIM 3.22: If I J,I s h for j < jo s 2Qhen there are functions f, from h d o J,, and a regular A+ -good@er D, such tlrat Q is independent mod D, and D is a d m a l euchjiaer.

364

[aVI, . 03

SATURATION OF ULTRAPRODUCTS

Proof. By 3.16(1) and 3.2 there are functions f j ( j < j,) from h onto J , and g, from X onto X (for i < 2”) such that Q* u Q is independent mod{A}, where Q* = uj:j< jo}, ’3 = {g,: i < 2”). Let Do be a maximal filter modulo which Q* u 9 is independent {H,: 0 < i < 2“} a list of all functions from Sw0(A) into (8: 8 s A}, each appearing 2” times. We define D,as in 3.18 (letting B, = j < i}).We use f o to define a regular D,. I f Range (H,) c D,, we work aa in Claim 3.3, using f, to get D’ z D, and extend it to D t + , by 3.18(3), then UtD, is the required flter.

uj:

CLAIM 3.23: Let D be a maximal Jilter d&which 3 i a independent, p = CC(B(D)), forinJinitelymanyf E B IRangef I > 1 and D* 2 D an ultraflter. Then D* ie not pi-good. Proof. Choose distinct f , E g and thenj,, E Range f , such that f;l(j,,) 4 D* (possible as IRange fnl > 1). Let A, = - f i l ( j n ) E) D* and define g :SN0(p)--t D* by g ( 4 = A,,,. We suppose 9, :&&) + D* refine g and is multiplicative, and get a contradiction. As gl({a}) E D* (for a < p), g,({a}) f 0 mod D. Hence for some ha E F I ( S ) Ah. E gl({a}) mod D. By 3.17(6) p > No,and it is regular by 8 1 = p and for some k Dom ha n definition; so for some S c p, 1 n < w } -C df,: n < k} for every a €8.By 3.17(6) for some 8, c S, I8,l = p, for any n < w , and distinct a(l)~ 8 , ( <1 n ) Ahacl, 0 mod D. Choose 8 = (a(2): 1 s k}, a(2)€8, distinct so g(8) E I f
n,<,(A

u,:

nl<, +

= nlrk

CLAIM 3.24: 8uppoae D i8 an N,-inmmplete &mal f l e r Over I d u b which g u g*ie independed, 9 u Q* a family of functione from I onto o,B* = df:i < S}, of S > KO. DeJine D1a8 theplter generated by (A) t k d W 8 O f D, (B)A: = {t: g(t) < f,(t)}when g :I --+ w , and for ewery n < w , g-’(n) i8mqpo9de&byFI,(SU{fj:j< i ) ) , a n d f o r a n y h ~ F I ( S u { f ~ < : j i}) for ~cnm? n A , n {t: g(t) 5 n} # 0 mod D. Then (1) g &I independent mod D,. (2) For any UltraJilterD* 2 D,. (i) ox/D*C f,/D* < f,/D* for j < i < 8 ,

OH. VI, 5 31

CONSTRUCTING ULTBAF'ILTEBB

(ii) for any a E w'/D*, for 8ome i < 6, d / D * C a < fJD* or fm 8 +W , d / D * C g/D* = a and o'/D C n < g/Dfor every n < w .

365 m

g :I

Proof.(1)Note that by this we prove D, is non-trivial (take A the empty function). So suppose gl, i(Z) (1 < n) as in (B), A E FI(S), and we have to prove Ah n A;,) # 0 mod D.

n

I
Let {j(k):k < m} = {i(Z): Z < n},j(k) < j ( k + 1) (note an ordinal may appear several times in the list i(Z) (1 < n)).We now define by induction on k 5 m A, E FI(S u S*), such that (letj(m) = 6): (i) Dom A, c 9 u { f j : j < j(k)}, (ii) A, = A; A, (k I; m ) is increasing, (iii) if i(2) s j(k) then for some unique nl,

Ah, E gi'(n1) mod D, (iv) Ak+l(fj,k)) is the minimal natural number n, n > nI whenever i(Z) = j(k),1 < n. For doing this it suffices to prove that: if A1 E FI('3 u {ff:j < j*}), i(Z) = j*,then for some ha, n*, A' E ha, ha EFI(Su {fj: j < j*})and Aha c g;'(n*) mod D. But &8 i(l), g, satisfy (B)for some nl, Ahi n {t E I : gI(t) 5 n'} # 0 mod D, hence for some n*, Ahi n g;'(n*) # 0 mod D. NOWAh', gil(n*) ~ I W supported by FI,(S U {fj: j < j*}), hence (by 3.16(2)) also Ah1 n g i ' ( n * ) is, hence there is ha E FI(S U {tf: j < j*}),Ah* E Ahl n g;-'(n*) mod D. so we can define the hk's. NOW it is trivial to check that AhmE Ah n fllcn A$:) hence Ah n # 0 mod D. (2) (i) Clearly fi : I + w for every n f i ' ( n ) is supported by FI,(Gf,}), hence by FI,(S U dfor: a < j}) and by 3.17(2) A # 0 mod D implies that for Borne n A n f;l(n) # 0 mod D. Hence A:, is one of the generatore of D,, BO {t E I : fj(t) c fr(t)}= A:, E D' c D*. (ii) Suppose a = f/D*, and let {A,: a < a,} be a maximal family of pairwise disjoint subsets of I mod D, such that A, # 0 mod D, but for every n < W ,{t E I : n 2 f ( t ) } n A, = 0 mod D. As CC(B(D)) = N, (by 3.17(6)) a, < wl, 80 w.1.o.g. a, 5 o.If a0 < w , let A = UI<,,, A,; and if a, = o,let A = (A, - {t: f ( t ) 5 2)). I n any case for each n < W , Z < a, A n f - l ( n ) = 0 mod D, and A, c A mod D , hence df-l(n)/D:n < W } u {AID} is a partition of B ( D ) . Clearly there is < 6 such that the f-l(n)'s (n < o) and A are supported by u G.: j < 18)).

n

w9

SATWaTION OF ULTBaPaODUCTS

366 Case (i).A

cm. m,0 3

# D*. Define

Cletlrly A: is one of the generators of D, (by 3.16(2)), hence wr/D* C a = f / D * = g/D* < fs/D*. Case (ii).A E D*. As D is 8,-incomplete, then are B, E D, B, = 0; w.1.o.g. B,+, E B,,. Define

nn<,

Then wr/D* C a = g/D*, and wl/D C n < g/D for every n < W .

Proof of Tlwore~n3.12. Let D, g be such that 4is a family of p functions from h onto w , independent mod D , D a maximal filter over h modulo which '3 is independent, and D is regular (by 3.22). Let '3 = (fi:i < p ~ } (pK-ordinal product) = {A: i < p}. We define Dl (i < p ~ as ) in 3.18 (for g* = 8) such that A," = { t : n
2

Idf,/D*:i(*) < i < p ~ } 1= p

(for i < j o A / DC fj/D* < A/D*, henae fJD* # f,/D*). On the other hand n,<&a)/D* s ph = p by 3.20(1).

Proof of Theorem 3.13. Let D be a maximal filter modulo which 59 is independent, D regular (eee 3.22) suoh that 9 9 UIen9,,9; = cf::A, s K < A, + ,, K regular, i c 2 5 ~ (ordinal ~) multiplication) wheref: is onto K . We define by downward induotion, filters D, (Z 5 n) such that (i) 0, = D , D,,, c 0 , s (ii) D, is a maximal filter modulo which UL<, qLis independent, (iii) for any ultrafilter D* extending D,, A, s K < K regular lcf(K, D*) = pi.

OONSTRVOTWO WLTRAR'ILTIRS

If D1+,is defined, we define DlS1(i < 2 ” ~&B ~) in 3.18 (with U L eyPk, I

ub:A, s

K < At+,, K regular, i < b}, 2“ p, for 9*,yPB, a reap.) such that {t E I: f 5 f i ( t ) }E D1,l+l for each f < K. Let D, = UfD,,,, so (i) and (ii)holds by 3.18, and by 3.21 (iii)holds. For 1 = 0, for even i’e we do &B above, and for odd i’s we immitate the proof of 3.22 (and 3.1) to gat a hl-good u l t r d t e r Do. Now Do is not ht-good by 3.23 and 3.17(6).

Proof of Theorem 3.14. Combine the previous proofs, but use also Claim 3.24.

THEOREM 3.26: For every reguhr A s IIl, there W a regular ultrajilter D Over I such thud (i) D W h-good but not A+-good, (ii) lof(K, D) = Afor every K < A, (iii) every function f :8,&1) + D, IRange (f)l < A, can be r e j d to a multiplicative one. Proof. Let 9 = Ua
+

\

EXERCISE 3.27: Generalize Claims 3.16-3.24, when we replace FI,(S) be any family S of subsets I , ‘Y independent modD” by “S n Dc = 0” (i.e., Dc disjoint toS, Dc = { A E I : A - A E D}) [Hint: (1) 3.17(6), (6), 3.10(1), 3.20 (aasume CC(B(D)))3.21, 3.22, 3.23 and 3.24 (see 3J8) are left to the d e r . (2) 3.16(2), (3), (4), 3.17(1) and 3.19(2) are trivial. (3) 3.17(2) becomes:if B1 E S,A # B E S1 A - A n B E D ; A 3 8 - 8’ * (3B E 8,)(3C E 8)(C E A n B), then 8,is a partition. (4)3.17(3)becornes:ifrS1 E S , ~ ~ ~ ( V A E S ) ( ( ~ B , C E S A )n( BC) and ( V B E S ~(A ) n B = 0 mod D) then A = 0 mod D. 6) 3.17(4) becomes: if D is & maximal filter diejoint to {{B n A,: B E A, h E P I ( 9 ) } ,B,, B, supported by 8,h E F I ( 9 ) and B, n A , E Bo mod D then B, E B2 mod D.

c

\

368

SATURATION OF ULTRAPRODUCTS

[CH. VI, 6 3

In 3.18 replace Yl,, FI,(9f1u 9*))}by B,, {B n A,: B el?’, h E FI(9*)} (9* c 9)mp., and w u m e KO + CC(B(D))< cf a.] (6)

EXERCIBE 3.28: (1) Let B be a Boolean algebra, CC(B) 5 A then B is (A, at)-compaot iff B is complete. (2) if A = CC(B(D)),D a A-good filter, then B(D) is complete. EXERCIBE 3.29: Suppose B is a complete Boolean algebra, CC(B) 5 A, IlBll 5 2A. Then for some regular filter over A, B B(D). [Rint: If llBll 5 A, this is easy by Exercise 3.27 (on 3.22) and Exercise 3.28. Otherwise let f i be from A onto A, D a maximal filter modulo which i < 2A}is independent; let B, be the subalgebra of B(D)generated , a homomorphism from 5, onto B, which we extend step by f i l ( j ) / D h,, by step, and get h* : B(D)+ B, and let D* = {A: I ( A / D )= l).]

ui:

EXERCIBE 3.30: Suppose Di (i < A) filters over I , D a filter over A, and let D* = { A : {i < A: A E D,} E D}. (1) D* is a filter; and if p 2 CC(B(D,)),p*-+ (p)! (e.g., p* = (2 f,/Di. Then for any ultrafilter D’ 2 D*, h f ( K , D) = p. (3) Ifp; = n t e r nt/Di then n i c h 2 ntsr n t P * (4) Suppose each 0, is a maximal filter modulo whioh B is independent, p 2 CC(B(D,))(see 3.17(6)), x regular > 2” and (Vx, x x ) (x:” < x). Then x 2 CC(B(D)).[Hint: (1) If A , # 0 mod D(a < x ) let f (a,8) = min{i: A,, A , # 0 mod Q, A, n A , = 0 mod Di}and see Defnition 2.1 of the Appendix. (2) Use 3.19(2). * (4) Suppose A, # 0 mod D(a < x ) are pairwise disjoint mod D. As x > 2A, w.1.o.g. S = {i: A , # 0 mod 0,)does not depend on a. By 2.8 of the Appendix and the proof of 3.17(6) w.1.o.g. for each i ~ l ? , and distinct ~ ( l (I) c n), A,cl)# 0 mod D,. Hence A, n A , 0 mod D, contradiction. The proof gives more than required.]

J

EXERCISE 3.31: (1) Suppose for each a < A, K,,

pa are regular cardin-

OH.

VI, 5 31

CONSTRUUTINO ULTRAFILTERS

369

zarh

als > A , K~ < pa, and K~ # K~ for a # j?,III 2 K ~ pa , I 21'1. Then for some regular ultrafilter D over I, lcf(Ka,D) = pa. (Hint: Use Exercise 3.30 and the proof of 3.13.) (2) In (1) if ga(9)is a family of 21'1 functions from I onto K,(A), U agau 9 is independent mod Do, then there is a filter D z Do, g is independent mod D, such that for every ultrafilter Do 2 D and a < A, lcf(Ka,Do) = pa. [Hint: (2) By the proof of 3.13, for every finite w E A there is a maximal filter D, modulo which 3 ' is independent, and for a E w f t / D (t < p a ) exemplifies lcf(Ka,D l ) = pa for any ultrafilter D12 D,, and then use Exercise 3.30(2).]

EXERCISE 3.32 (G.C.H.): Suppose K < plC5 21'1, regular for each K ES; K ESimplies K 5 II1 is regular, and for inaccessible A { K ES: K < A} is bounded below A. Then for some regular ultrafilter D over I,lcf(K, D) = plCfor each K E S. [Hint: Let {fg: a < 21'1, K ES} is independent mod Do, Do regular, and Range (f,") = K . For regular x, 8, c { K ES:x s K } , let (*)$,

there is a filter D = D(S,, x ) 2 Do such that (i) {f,":a < 21'1, K < x} is independent mod D, (ii) for every ultrafilter D* 2 D, K ES,, 1Cf(K, D*) = p x and this is exemplified byf,"/D*(a < pa) .

We prove it by induction on the order type of S, (for x a successor of a singular, use x+ and then add x+, if necessary).] DEFINITION3.8: For a model M and a complete Boolean algebra B and ultrafilter D of it, we define the Boolean ultrapower N = M('B)/D as follows (the sup exists by the completeness): (i) On the set {f:Dom f is a partition of B, Range f E M } we define an equivalencerelation x :f, x fa iff sup{a, n a,: a, E Rangef, ( I = 1, 2) f,(a,) = f,(a,)} E D. By 3.16(2) x is an equivalence relation, and for everyf,, . . .,f, there are f: x f, which have the same domain. Now IN1 will be the set of equivalence classes of x and

RN = {(f,/x,...,f,/x):

(similarly for J").

370

SATURATION OF ULTRAPRODUCTS

[CH. VI, $ 4

EXERCISE 3.34: Prove that Boolean ultrapower is a particulaz case of limit-ultrapower from Definition 4.2, hence the parallel of Exercise 4.10 holds. EXERCISE 3.35: Let D be a filter over I , 5 = B(D), K, = CC(B), D , 2 D an ultrafilter, and D* an ultrafilter of 5 , D* = { A / D :A E Dl}. We define a function H:w(m)/D*+ wl/D, as follows: i f f / z E w(S)/D*, let Domf = {a,: n c a,, I; w } (as C C ( 5 ) = Kl), let a, = A , / D and w.1.o.g. I = Un
ateI

Remark. By this, the problem on “what can be n,/D: n, c can be reduced to a problem of Boolean ultrapowers. EXERCISE 3.36: I n 3.19(2) replace the a <

K

byf’ (jc

w}”

K).

EXERCISE 3.37: Suppose g is a family of functions from I into the ordinals, and D is a maximal filter over I modulo which Q is independent. Let p = CC(B(D)),and suppose D is A-good. Then for any ’1 and: ultrafilter D, 2 D, and regular K , x c A satisfying K , x 2 p + 13 K > 2 X or x > 2’F, or K = x are weakly compact wI/D, has no ( K , x)Dedekind cut (see VII, Definition l.lO(B)). +

EXERCI8E 3.38: Use Exercise 3.37, Theorem 3.25 and Exercise 3.30 to investigate the possible { ( K , x): d / D has a ( K , X)-Dedekind cut}. Phrase the open problems. QUESTION 3.39: Investigate the problems from Exercise 3.38. VI.4. Keisler’s order

DEFM~TION 4.1: (1) For every A Keisler’s QA-order on theories is defined aa follows: T , @A T,if when M ,is a model of T,(1 = 1 , 2 ) and D a regular ultrafilter over A, the A+-compactnessof HG/D implies the A+-compactnessof M:/D. (2) Keisler’s order @ on theories is defined aa follows: TI@ Ta if Ta. for every A T ,

OH. VI,

8 41

EEISLER'S ORDER

37 1

T,and T,are @.,-equivalent (@-equivalent)if T,@., Ta @., T1 (Ti@ Ta @ Ti). (3)

LEMMA 4.1: (1) T7w relations @., @ are transitive and re$hAve. (2) @.,- and @-equivalence are equivalence relations. Proof. (1) Easy to check the transitivity by the definition. The reflexivity follows by 1.9. (2) Follows from (1). Remark. So naturally, @ (@J is an order on its equivalence classes.

LEMMA 4.2: (1) T ie @.,-&TTuz~ (i.e., TI@A T for every T , ) if fw every model M of T and regular ultraJilter D over A,

M A / Di8 A+-COm$MGt+ D is A+-g&. (2) T is @.,-minimal (Le., T @., TIfor every T,)iff for every model M of T and regular ultra$lter D over A, M A / Die A+-mpact. (3) T is @-maximal [-minimal] iff it is @.,-WWX~TTUZ~ [-minimal] for every A.

Proof. (1) Let M , be a model of T,, M a model of T, and D a regular ultrafilter over A. If T satisfies the condition mentioned in (l),and M A / Dis A+-compact,then D is A+-good,hence by 2.3 and 2.11 M t / D is A+-compact. As this holds for every M,, D; TI@., T, so T is @.,-maximal. Suppose now T is @.,-maximal, D a regular ultrafilter over A and M a model of T and assume M A / Dis A+-compact. By the @,-maximality of T,Th(Mh)@.,T [ M Afrom 2.2(3)] hence M i / D is A+-compact. By 2.3 this implies that D is A+-good. (2) If T satisfies the condition, clearly it is @.,-minimal. If T is @.,-minimal, then T @., T, [the theory of (A, =)I. But for every regular ultrafilter D over A, II(A, =)h/DII = AA/D= 2A > A, hence (A, =)"D is A+-compact.So clearly T satisfies the condition. (3) Immediate.

THEOREM 4.3: Any theory with the strict-order property is @-mtwiml. Proof. Just like 2.6 and then use 2.11 and 4.2. (Alternatively use Exercise 4.6.)

372

SATURATION OF ULTRAPRODUCTS

[CH.

VI, 8 4

THEOREM 4.4: Let M be a moclel of an urntable theory T , D an ultrafilter over I and suppose that tkre are finite cardinah m,, i E I , a h that No s m,/D < 2A. Then N = M'/D is not A+-wmpact.

n,,

Proof. By 11, 4.7 T has the strict order property or the independence property. Case I. T has the strict order property. So there is a formula ~ ( 28), which is a partial order, and not reflexive, and for every n there are q,.. .,?i& E Ml such that ktp[sir, Let P, = {ZP:1 < m,},( N , P) = nipr (N, P,); so by assumption IPI < 2A. Notice that each (M, P,), and hence (N, P),satisfies the sentence (VZ)[(3g)[P(g)A (VZ)(P(Z)3 F(2, 2 ) = q(8,2))l v (VY)(P(B) + 1q(% Suppose N is A+-compact, then we can define ti,, 5, E P 7 E A 2 2 by induction on Z(q) such that: (i) N t= ~[si,, 5,] and for every n (N, P)k (360,.. .,Zn)( 9 ( Z l , Zl+l)) A sin = 2, A b, = Zn A P(ZJ),

a))].

A

A

1 en

1 en

(ii) 7
Case 11. T has the independence property. So some y(x; 8) has the independence property. Hence we can find sequences ti: E IN1 (1 < K ) so that k(3z)[AlCk ~ ( z?$)l-"lEW)] ; for each w c k, so let bk E IMI, be such that F(bk,; Zr)i(lew).

'A

1Ck

Let p = min{n,,, nJD: n, <

w

for i E I and

n,,, nJD 2

No} < P,

P = l?rer nilD.

For every ~ E define I k ( i ) = [log,n,] if n, # 1, k ( i ) = 1 if n, = 1. Let P, = {a;(,): 1 < k ( i ) ) ,&, = {b","):w c k( i ) },so clearly n,/2 - 1 I 1Qf1 s 12,. Let ( N , P,&) = I L r ( M , Pi,QO, 80 clearly IS1 = Riel ISil/D s p, but clearly !PIs 1 01 and for every 1 < w

{ i d : lP,l

2 1 ) 2 { i d : n , 2 2')ED

n n , / D 2 No).Hence IPI is infinite; hence by p's definition = /PI= p. Let A1 = min(A, p) and choose I s P, 111 = Al. By the definition of the Pi's, clearly for every J E I, pr = {y(z; si)ii(dor):si E I)is (as

OH. VI,

5 41

373

KEISLER’S ORDER

a type in N. Now by the definition of the Qr, if pJ is realized in N, it is realized by some element of Q. Hence the number of types pJ,which are realized in N is IIQl = p (because J, # J, implies no element realizes both pJl and pJJ. On the other hand the number of such types is I{J: J c I>I = 21’1 = 2”. Clearly 2” > p, and by hypothesis and definition of p, 2A > p ; hence 2.,1 > p. So for some J G I N omits pJ,and as IpJI = A, 5 A, N is not A+-compact.

ntsI nfoI

EXERCISE] 4.1: Prove that if M = M,/D is a model of an unstable theory, D an ultrafilter over I , KO I m,/D < 2.,, then M is not A+ -compact. THEOREM 4.6: Let T be stable and urith the jnite cover property. Let M be a model of T and D an ultrafilter over I ; if KO I p = m,/D then M’/D is not p+-mpact.

nfsI

Proof. Let q(z,y; Z) be as in 11, 4.4, i.e., for every E E IMI q(z,y ; a) is an equivalence relation and for n < w , q(x,y ; E,J has ~n but < No equivalence classes. Let Neq(E, M) be the number of equivalence classes of q(z, y; 5 ) in M . Clearly for every family of models M, i E I in which q(z, y; 2) is always an equivalence relation

Choose E[i] as En[rl where n [ i ] = max{Z: Neq(E,, M ) Im,} if such exists, n [ i ] = 1 otherwise. Clearly No I Neq(C, MI/D) In f o , m , / D , so we finish.

EXERCIBE 4.2: Prove the parallel to Exercise 4.1 for Theorem 4.6. LEMMA 4.6: If T is @A-minimul,p 5 A, then T is @.,-minimal.

Proof. Assume T is not @.,-minimal. By 4.2(2) there is a regular ultrafilter on p, and a model ill of T such that M @ / Dis not p+-compact. Let D, be a regular ultrafilter on A, D, = D, x D, I = A x p, so D, is a regular ultrafilter on I , 111 = A, and M1/D, z (MA/D,)#/D.By 1.1 M = MA/D,so by 1.9 and 2.11 M p / Dis p+-compact iff (MA/D,)U/D is p+-compact so MI/D, is not p+-compact, hence not A+-compact. So T is not @.,-minimal. Contradiction.

[OH. VI, 0 4

SATURATION OF ULTRAPRODUCTS

374

THEOREM 4.7: If Krdt(T) = 00, D ie a regukcr dtraji2ter over I , M ie an IIl++-eaturatedmodel of T,t h n Mr/D is not IIl++-cornl)ccct (eee Definition 111, 7.2). Proof. If T has the strict order property the theorem follows immediately from 4.3, 4.2, 2.4 and 1.3. Next aasume K ~ ~ = ( T 00 ; )w.1.o.g. we can assume (see 111, Definition 7.3, 111, Exercise 7.4 and 1.11) 11M11 > p = aA+ where IIl = A and there is a formula cp(Z;g) and a{ E \MI,B < A, a < p , I @ { ) = Z(g) such that for every fl { c p ( f ; a:): a < p } is 2-contradictory, and for every 7 E " p {q@; atlE,): < A} is consistent, realized by En. Let P = {a:: /3 < A, a < p}, Q = {En;q E "}. We define equivalence relations E(E, g), Z(Z) = I(g) = I @ ) as follows E(iZ2, iZ2) iff p1 = pa; -,E(E, ?if I{ 17 $)P , E(E, a) if 8 , a $ P . IZ has X equivalence classes in P ( M ) and IA+ equivalence classes in P(Mr/D) each one of cardinality z p . Let Jf, i < A+ be distinct equivalence classes of E, then for every sequence (a,: a, E J,, i < A+ ) , {cp(E; at): i < A+} is consistent, and if M r / Dis A++-compactit is realized in &(Mr/D).p < p + s lJfl s 1Q(M'/D1 = IQ(M)Ir/Ds pA = p , contradiction. So we ' ) by 111, 7.11 and are left with the case K,,,(T) < 00, ~ ~ ~ ~= (00,2 hence 1.11 we can assume there are a formula p(Z;g) and a,, E IMI (7E 'p)Z(G,,) = Z(J) such that if 7,v E > p are <-incompatible then M C 7(3z)[cp(Z; a,) A ~ ( 2a,)] ; and for each 7 E E,, E 1M1 realizes {cp(Z; a,,,,,): n < w}. Let P = {an: r ) ~ @ > p }Q, = {En: r) EOUCL), E = {(a,,, a,): for some n < w, 7, v E "p}, and < ={(sin, a,): Z(7) < I(v); 7,v E >p},El = {(a,,-cf),an-<j)):i, j < p , 7 E >p}.Let Nl= ( N , P N , QN,EN,EF, < N ) = Mi/D, MI = (N, P , Q , E , El, <): so as Disregular for a < B < A+. Now for every q E there are P EP N ,N , l=iP< "*>,A we define E PN such that N , k E@W, zi((W)) and the @'a am distinct, and N , C (3Z)(&Z; a*) A cp(Z, 7i"1°), N , C E,(F^). We do it by induction on Z(7). For Z(7) = 0, Z(7) = a 1 there are no problems. For Z(7) = 6 limit let p,, = {y(Z; ZIa):a < 6); clearly it is consistent, 80 if it is not realieed, M1/Dis not even h+-oompaot,80 we 9 E ldfl realizes pnybut finish. So assume i

n,<,,,+

@

+

Mi l= (E)(W[Q(Z) A (V#)(P(#) A

(P@;

8) 4 d Z ; 811

v,

(bythe requirements on the p(Z; 4 ) ' s ) 80 we 0811 w m m e 8 E and the rest is easy. Similarly for every q E "+pthere is P E QN recllizing p,,, but s lQMIr/Ds pA = p; the pnysam pairwise explicitly contradiotory and their number is pA* > p ; contradiction.

OH.

VI,Q 4)

375

KEISLER~S ORDER

EXERCISE 4.3: (1) Show that in 4.7 it sufficesto assume D is uniform. (2) Show that by 4.7, there is a model Mo of T, 1 Mo1 = 1I 1 ,such that Mo < M implies Mr/D is not I II ++-compact. +

DEBTNITION 4.2: Let M be a model, D a filter over I ; for a E I Mx/DI, eq(a) = ((8, t) E I x I: a[8] = a[t]} and for Q a filter over I x I , MLJ Q is the submodel of Mx/Dwhose set of elemente is {a E lMx/D1: eq(a)E a). We always aasume {(t, t): t E I) E Q. DEBWITTON 4.3: T1 @* T, if for every set I , XI-incomplete ultrafilter D over I , filter Q over I x I , cardinal A, and (A+ + (I1+)-saturated models MI, Ma of T,, T, respectively: if MGDlGis A+ -compact then M',,lQ is A+-compact. DEFINITION 4.4: Let Q) = {vr(Z;2'): y < a}, Q), = {#,(g; 2,): y < p} be indexed sets of formulas (possibly with repetitions) from L(T,) and L(T,) respectively; Z(Z) = ml, l(g) = ma. (a1, m,) 5 (a,,ma) if there is q E such that for every model M , of T, and By E lMll, y < a, there are a model Ma of T, and 6~E 1Ma],y < /3 such that: for every w c a {g@;BY): y E w} is consistent with M, iff {#,,c,l(g;8nCrl): y E w} is consistent with Ma. We write (@, m,) 5 (Qi,, ma) by q.

Remarke. (1) Clearly by the compactness theorem (a,,m,) 5 (a,,ma) by 3 iff for every finite Q) E @,, (@, m,) < (a,,m,) by q w, where W, = {y < a: py(E;Ir)E @}. (2) In Dehition 4.4 we can take M,, Ma as k e d A-universal models (for large enough A). (3) We write Q) E L (in this section only) to mean @ is an indexed set of formulrts, possibly with repetitions, from L.

r

EXERCISE 4.4: (1) If (Q),, m,) 5 (Q),, m,), @l E Q),, 0, E Oa then (@lS m1) 5 ( P s ma). (2) If O1[P] is the olosure of @, [a,]under conjunction and disjunction, then (a,, m,) s (a,,ma) implies (@I, m,) 5 (P,ma) (why do we not eay "negation " 2) and (@a9 (@a9 %) then (@IS (3) If (@lS ml) 5 (@a9 ml) (J's, 7%). EXERCISE 4.5: If for every 0, E L(T,), 1@,1 = A, there is L(T,) and % < o such that (Q),, 1) s (Q),, ma) then T, @, TI.

Q),

E

376

SATURATION OF ULTRAPRODUCTS

[CH.

VI, 8 4

DEFINITION 4.6: Let @, §, m,, ma be as in Definition 4.4 (@, m,) S* (Oatma)if there is 7 E ap such that for every model M, of T , and Sir; E 1 M11 y < p, n < W , there are a model Ma of T , and 6; E 1 Mal, y < 8, n < w such that: for every w c a x o {tp,(E; a;) : (y, n) E W } is consistent with M, 8 { # , , E y , ( ~ ; 6:[~1):(y, n ) E W } is consistent with Ma. We write (Qs,, m,) r * (@, ma)by q.

EXERCISE 4.6: ( 1 ) I n Definition 4.6, we can replace o by any a 2 o. (2) ( @ I , m,) s* (@a, ma) by 7 implies (@I, m,) 5 ( 0 2 , ma) by 7. ( 3 ) (@I, mi) I* (@a, ma) implies (01, mi) 4 (@a, ma). ( 4 ) If @, @' contain the same formulas (with a different number of repetitions) then ( 6 ) If @l G @, Q2 E Oa then (a1, m,) S* (Oa,m,) implies ( @ I , m,) s * (Ga,ma). (8) (@,, m,) S* (@, m,) (by the identity map).

EXERCIflE 4.7: The following statements about T,, T , are equivalent. ( 1 ) For every c L(Tl) there are 0, G L(T,) and ma < w such that (@;, 1) s* (@, ma). (2) For every §, E L(T,), s (T,I + 1T,1 there are @, E L(T,) and ma such that (@,, 1 ) s (@, ma). ( 3 ) For every @, s L(Tl) there are 6,c L(T,) and ma such that (@I, 1 ) (@a, ma). ( 4 ) For every @, c L(Tl), 1@,1 s IT,I there are 0, c L(T,) and ma such that (@,, 1 ) s * (@, ma). (6) Let 0, be the set of formulas tp(x;jj) E L(T,) (clearly )§,) = I T1l).There are 0, E L(T,), ma such that (@, 1) s* (@, ma). +

EXERCIflE 4.8: ( 1 ) If 0, is the set of all formulas in L(Tl),and for some @, c L(T,), ma < w , (Go, 1) s* (@, ma)then T , @* T,. ( 2 ) In fact it suffices to demand that there are a,, i < i,, such that: if,M, is a non-A+-compactmodel of T,, then there is a type p over M,, p = ipa(z,7ia): a < a, < A+}, such that for some i < i, every tp,(x, jja) E 0,; and there are @,, E L(T,), ma,, < o such that (QZ,, 1 ) r * (@a,is ma,{)EXERCILSE 4.9: Suppose M, is a A-universal model of T , M, is a A + compact model of T , and D an ultrafilter over A. If N',/Dis A+ -compact

CH.

VI, 8 41

377

KEISLER'S ORDER

then M i / D is A+-compact.If D is uniform it suffices to demand MI is ( < A)-universal, M 2is A-compact.

EXERCISE 4.10: For M ,I , D, Q as in Definition 4.2, D an ultrafilter; M',la is elementarily equivalent to M, moreover a H (. . ., a, . ..)te,/D is an elementary embedding of M into M',]a. Generalize 1.1.

DEFINITION 4.6: A complete theory T is simple if (1) L(T) contains only one-place predicates, the equality sign = , and one other two-place predicate E ; (2) for every model M of T, EM is an equivalence relation over 1M1; (3) there is a model M of T such that for every ~ E W I[aIM , is infinite where [aIM= {b E M : M k bEa, and for every predicate P ( x ) of L(T)

M I. P(a) = P(b)}; (4) there is a model M of T such that for every a E 1M1, there are infinitely many b E ( M ( from Merent E-equivalence classes which realize the same type over 0.

EXERCISE 4.11: Let T be a simple theory. Show that: (1) If M is a model of T,a E IMI, then any permutation of is an automorphism of M. (2) Every formula of L(T) is equivalent to a Boolean combination of formulas of the following forms: (9 x = Y, (ii) xEy, (iii) P ( z ) , (iv) ( ~ Y ) [ x EAY A j < n pj(Y) A A j < m l p j ( ~ ) l . ( 3 ) T is stable in every A 2 2ITI, so T is superstable. EXERCISE 4.12: Suppose M is a non-A+-compactmodel of a simple theory T. Show that M omits a type p, which is of one of the following forms:

5 < to Imin(A, IT])} (i) p = {xEa} u {P,(x)n[C1: u {x # c,: 5 < l o s A}, (ii) p = { P , ( X ) " ~ < ~ :c0 Imin(A, ITI)} u1p0 u {TxEc~: 5 5 0 5 A},

378

[CH. VI, 8 4

SATURATION OF ULTBbPBODUCTS

where p o consists of formulas of the fourth form from the previous exercise and of negations of such formulas, and r] is a sequence of zeroes and ones.

EXERCISE 4.13: If M is a A-compact model of a simple theory T, and N = MblG (see Definition 4.2) is IT1 +-compact, show that N is h-compad. (In fact it is ALIU-compact.) EXERCISE 4.14: Show that: (1) A simple countable theory is @-minimal. (2) If N is a model of simple theory T,D a I TI -good ultrafilter on p, then N u / D is Kg/D-compact. Hence if D is p-regular, W / D is 2,compact. +

EXERCISE 4.15: Show that for every theory T,and cardinal h there is a simple theory T 2such that Tl@A T,@A T,.If I TII IA then also lTal IA. Moreover if D is a A-regular ultrafilter over p, AZ, a model of TI, M , a model of T 2then MU,/Dis A+-compactiff M $ / Dis A+-compact. (Hint: See Exercise 4.5.) EXERCISE 4.16: Show that for every set { T c : &! < to}of theories there is a least upper bound for each of the orderings @, @A. Its cardinality is 5 ITel.

zc

EXERCISE 4.17: (1)Show that for every A there is a simple theory Tj,, lThl = h such that TA is @A-maximal.Hence if A < p, TA @ T, but not T, @ TA. So thare is an (uncountable) theory which is not @minimal nor @-maximal. (2) Show that if there is a countable theory which is not @-minimal nor @-maximal (see 5.9) then there are @-incomparabletheories. EXERCISE 4.18: Prove case I of the proof of 4.4 by 4.3. EXERCISE 4.19: (1) Suppose xlnp(T)> l I l + , iK an IIl++-saturatd model of T,and D a regulm ultrafilter over I . Prove that M r / D is not III ++-compact(see Theorem 4.7). (2) Prove the parctllel of Exercise 4.3. QUESrTION 4.20: Can we in Exercise 4.19 replace K

~

T)~by( xcdt(T)?

8 61

CH.

SATURATION OF ULTRAPOWERS

EXERCILIE 4.21: Show that T 1 T i , Ta.

379

Ta for any complete theories

THEOREM 4.8: For any model M of an umtable theory T and 8,incomlplete ultrafilter D Over I,K = lcf(w, D), MIlD is not ~ + - ~ ~ m p a c t .

Proof.Choose an unstable formula ~ ( zg) ; and sequences q (1 c n < o) such that {rp(x, $ ) u ( l > k ) : 1 < n} is consistent for every k < n < w. Choose a distinct b, E: 1M1 and let: PM = {b,: n < w } , < M = {(bk, b,):

k, < n <

W}

and Ffd (1 c l ( @ ) two place functions such that 7i; = (Po@,, b,,), PI(&,k n ) , * * Let a).

N1 = ( N , PN,<*, F r y .. .) = ( M y PM < M y Ff,. . .)'ID, and c, E PN (i < K ) be such that N , C b, c c, c c, for n c w, i < j c and for no c E PN,N C b, c c c c, for every n c w, i c K. Let

u { ~ ( xPo(c,, ; GO), . . .): O < i < K}. Clearly p is finitely satisfiable, but N omit it (if a realizes it, then the

I, = {Tv(x, Fo(bny GO),

. . .): n <

K

O}

formula

P(Y) A

(VZ"(4

z

Y +?+, FO(YY CO)F,(Y,CO),

- - *)I

define a bounded subset of P with no last element).

VI.5. Saturation of ultrapowera and categoricity of pseudo-elementary classes THEOREM 6.1 : Let T be a countable theory, MI a rnocEel of T for every i E I, and D an K,-incomp&e ultraJilter mer I . Let N = nlEI MJD. (1) If T does not have tk f.c.p., A = Ki/D, then N is A-saturated. (2) If T is stable and hua the f.c.p., then N is A-saturated, but not

nlEI

nllD 2 Xo}. ( 3 ) If T does not have the f.c.p., each Aft is p-saturated, and A = pl/D,

A+-saturated, where A = min{n,,, n J D :

then N is A-saturated. (4) FOTeveryjinite A s L(T)let &(A) = min{lpl: p is a A-I-type over lM,l in MI which is omitted by Mt} and t L(T),[A1 < H,}.

380

SATURATION OF ULTRAPRODUCTS

[m.VI, $ 6

nisr

Let A be an (injnite) cardinal w h tlrat A = Ai/Dfor ~ o m eA:, i E I, w k r e for every jEnite A C _ L(T),{i:A: s &(A)} E D . Then if T aoe.9 nd liaVe t k f.c.p., N i8 A-saturated, but not (A*)'-Sat~rate&. Remark. (1) Clearly the results, except (a), m the best possible. For example in (l), if we choose the M i aa aountable models, then IlNll = K',/D = A, Eo N is not A+-saturated. On (4)see Exercise 6.10. (2) Instead of demanding T countable, we can require D to be ITI good. See Exercise 6.2. +

Procf. Notice that as T is countable, for every model H of T and cardinality K > KO,M is wcompact iff M is K-saturated. Now in part (2), N is not A+-saturated by Theorem 4.6. Similarly we can prove in part (4) that N is not (A*)+-saturated.So it remains to prove that in all the parts N is A-saturated. N is XI-saturated by 2.1, 2.3 and 2.11. By 111, 3.3, 3.9 and 3.10,aa T is countable and stable, it suffices to prove: if {c:: i < o} c IN1 is an indiscernible set, then it can be extended in N to an indiscernible set of cardinality A. For every i E I let us choose a family Sf of subsets of lM,l such that: (i) IS:l = 11~:11, (ii) every finite subset of IHtl belongs to St, (iii) for every finite A c L(T),n < o,if w E 8,is ad-n-indiscernible set, 0 s p s 11H,ll, and there is a A-n-indiscernible set w', w E w' c lMtl, Iw'I = p, then there is wNeSi, Iw"I = p, w E w" c lMtl, and wN is ad-n-indiscernibleset. Let /Mil = {af:j < IIH,II},St= {wi:j < llH,ll}. Let us define the relation E{ on lMil: E' = {(a;,ah): a: E wh}. We shall write z E y instead of ~(z, y). In the language L = L ( T )u {E}, clearly, for every finite A c L(T),n < o,there is a formula 'g,,Jz) meaning {y: y E z} is a A-n-indiscernible set. Now for every i E I we define Pi according to the part of the theorem we want to prove; in (1): F = {a:: lwhl z H,,}, in (2):F = {a;:a < 11M,11} = [Mil; in (3): F = {a:: 1w:I 2 p}; in (4): P f = {a;: lwtl 2 hi); where the A' are defined so that X/Dl = A, and for every finite A E L( T){i: A: s & ( A ) } € D. Now the following hold:

InlE,

OH.

8 61

SATURATION OF ULTRAPOWERS

381

For every finite d c L(T), n < w , there is m = m ( d ,n) < w , such that the set of i's for which the following holds belongs to D : For every A-n-indiscernible set w;,Iw;l 2 my there is a (**) A-n-indiscernible set wi,wk E wj and a: E P . Let us prove it. In part (2) it is trivial. In the other parts T does not have the f.c.p. so in part (1) it follows from 11, 4.6(3) and in parts (3) and (4) from 11, 4.6(2). Notice that except in part (4) (**) holds for every i. Now clearly (**) is equivalent to a first-order sentence in L' = L u {E} u { P } . Let N' = ( N , E ~ PN) , = (Mf,cf, P ) / D . N' is 8,saturated (by 2.1, 2.3 and 2.11). By (*), clearly the sentences corresponding to (**) (for all finite A , n, m) are satisfied by N'. Remember it suffices to prove that {cf: i < w } can be extended in N to an indiscernible set of cardinality A. As {cf: i < w } is an indiscernible set, it is a d-n-indiscernible set for every A, n. Hence every finite subset of p = {c, E x: i < w } u {rpd,,,(x):d E L ( T ) , lAl < No, n < w } u {P(x)}is satisfied in N', hence p is satisfied in N' say by b. As N' C qd.,,(b) for every A, n, clearly w = {u E INI : N' C u E b} is an indiscernible set, and of course {cf: i < w } G w.As N' =! P[b],and lwl 2 I{cf: i < w}I = 8,, clearly IwI 2 h (the check for each part is easy). So we have proved the theorem.

(*I

nfo,

It would be more satisfactory if in 5.1(4) h = A*. (Thisis possible if V i E I , M , = M ) . For this it suffices to prove CONJECTURE 5.1: Let Let ( J , < ) = (p, <)'ID. (<-the natural Suppose order on ordinals.) For a E IJI, let 1 . 1 = l{b E IJI: b < }.I a,, E IJI for n < w , la,,l = lao[.Then there is a E I JI, u 5 a,, for every n < w , and 1. = la,[ (when D is an 8,-incomplete ultrafilter). See Exercise 5.11 and 5.12. THEOREM 5.2: Let M be a h-compact model of T , D an ultrujlter over I , IT1 5 111,N = M'/D. If N is (21'I)+-saturatedthen N is hI/D-saturated.

Rernurh. (1) For countable T this theorem follows from 4.4, 4.6 and 5.1(3). (2) Here the proof works also for D an 8,-complete ultrafilter. (3) See Exercise 6.13.

[OH. VI, 8 6

SATURATION OF ULTRAPRODUOTS

382

Proof. As N in (21rl)+-saturated,by 4.4 and 4.6 T is stable and without the f.c.p., and so clearly every infinite indiscernible set can be extended to one of cardinality 2 (21rl)+. By 111,3.10 (remembering that by 111, 3.3, K ( T )5 I TI s 111 +), it suffices to prove that: If I is an indiscernible set in N, 111 2 (21rl)+,then there is an indiscernible set J, 11n JI 2 KO, lJl 2 JAr/Dl.Let {aB:fl < (2111)+} E I. The following statement will be proved later. +

(*)

There is an infinite w c (21'1) such that for every i E I, {aD[i]:fl E w} is an indiscernible set in M. +

We can assume A > I TI , as otherwise the conclusion of the theorem is trivial. For every i E I let P' be a maximal indiscernible set such that {a,[i]:flEw}E P c IMI.AsMisA-compact,A > ITI,clearlyIP'I 5 A. Let (N, P) = (MyP ) / D . Clearly ]PI = lPfl 2 Ax/D. Now for every finite A G L(T),n < w , the statement " P is a A-n-indiscernible set " is elementary, hence P is an indiscernible set. So {aB:fl E w} E P E INI, hence IPn 11 2 l { a B : f l ~ w }2I KO. So P satisfies the requirement for J. So we need only prove (*). As T is stable, by 11, 2.13, 1231 I211' implies IS(B)I I (21rl)lTl = 2Ir1. It is also clear that for B, c IN], lBtl 5 211 ' , for every t € 1 ; I!Xt,rfl(Bt)( = n t e r ISCBt>l I (2"')''' = 2'"Define for j 5 l I l + , sets wj c (21'1)+ by induction: (i) wo = 0, wd = u j < d wj for a limit ordinal 8, and w, G wj for i < j 5 Ill+. (ii) Let w, be defined. Then for every fl < (2Ir1)+ there is a unique y E w , + ~such that: for every i E I, a6[i],a,[i] realize the same type in M over {a,[i]:jE w,}. Clearly for every j , Iw,I I21'1. Choose a. < (2Iz1)+, a,,$~,,~+. For every a < lIl+, let j ( a ) be the ordinal such that for every i €1,a,,[i],aj(,,[i]realize the same type in M over {a,[i]:j E w,} and j ( a ) E w,+ Clearly for every i, a I 5 5 5' < 111 , aj(o[i],a,(o[i] realize the same type in M over {aj,,,[i]:y < a}. By 11, 2.18 for every i, there is y ( i ) < 1 11' such that {ajc,,[i]: y ( i ) I a < l I l + } is an indiscernible set. Let yo = sup,,,y(i), w = {j(a): yo I a < 111+}. Clearly this is the w required in (*).

nfeI

miel

+

Remark. We can in fact find such w of c d i d i t y (21'1)

+

. See Exercise I,

2.9.

DEFINJTION 6.1:A model Y is complete if for every n < relation on and every n-ary function from IN1 to

w , every n-axy

is the inter-

CH. VI,

5 61

SATURATION OF ULTRAPOWERS

383

pretation of some predicate symbol or function symbol, respectively, from L(M). Remark. Clearly for every set A there is a complete model M , ]MI = A , IL(M)I = K O + 21-41. DEFINITION 6.2: PC( T,, T) is the class of L(T)-reducts of models of T,, of cardinality 2 lTll, where T is complete, T s T, and T, has infinite models. The restriction “of cardinality 2 ITl]” is technical, and without it the class is denoted by PC*(T,, T).

THEOREM 6.3: If T is countable, superstable, and doap not h u e the f.c.p., then there i8 TI, T E TI, ITl] = 2 4 8Wh that PC(T1, T) i8 categorical in every cardinality 2 2No. Procf. Let M be a countable model of T.We expand M to a complete model M,, IL(M,)I = 2% Let L, be the language of M , and T, the theory of Ml ( i a , the set of sentences from L, that M , satisfies). Clearly T, contains its Skolem functions (see Definition VII, 1.1). Let N , be any uncountable model of T,, and let N be the reduct of N , to L(T). It suffioes to prove that N is saturated (as by I, 1.11 every two saturated models of the same complete theory which are of the same cardinality are isomorphic). So let p be any 1-type in N, 1p1 < IlNll, and it suffices to prove that p is realized in N. Let p1 be any extension of p to a complete type over INI, and let Q(z,~)E~ be, such that R [ q ( ~ ; i i ) , L ,= a ]R ( p l , L , a ) . Let 1M1 = {ai:i < w}, and let c,, i c w be individual constants in L, such that cfdi 3: a,. Clearly there is aoE IN,], ao # $1 for i < w . Define A = {FNl[ii,ao]:P is a function symbol in L,}. Clearly the submodel Nf of N,, INfI = A, is an elementary submodel of N , (by the definition of T, and the Tarski-Vaught test I, 1.2 or VII, 1.2). Let N* be the reduct of N f to L(T). Clearly N* is an elementq submodel of N. We now show (*)

Nf is N,-compact hence N* is K,-mturated.

So let q be a countable type in Nf and we must prove it is realized in Nf. Let q = {ql(z; a&,. . .,ak): i < w}. As every a: E A, for some F f , E L,, a: = Fi,jp,ao].So by substituting ii, ao):i < w}. Remembering that IMI = {a1:i -c w}, we get q = {I,~~(Z;

384

SATURATION OF ULTRAPRODUCTS

[CH.

VI, 5 6

C ~= I a,,M, is complete; it is clear that there is a function symbol Q in L, such that for every a,,,6, bo from lMl, QMi(a,,,5, bo) realizes {t,h,(x;6, bo); i < m} for the maximal possible m In. Clearly for every n

M , c (VWY)[

As

(A Y z ci A l
(32)

A t,h,(x, 2))

:
-+

A t,h:(Q(y, @, 211-

fen

a0 # $'* for i < w , clearly QN*(ao,a, ao) realizes q. So we have proved (*). As N* is N,-saturated, by 111, 1.2 and 111, 2.16, we can find B c "*I, IBI = KO,a E B, such thatp, 1 B is stationary, and we can define b, E IN*I for i < w , such that b, realizes p1 1 (B U {bj:j < i}).By 111, 1.10 Av({b,: i < a}, INI)= p l , so if I is an indiscernible set in N, b, E I for i < 0, then d(z, E) ~ p , implies , {b E I: N C 4 [ b , El} is finite. So clearly it suffices to prove that {b,: i < w } can be extended in N (not N*) to an indiscernible set of casdina;lity llNll (because thenall but s Ip I + KO elements of the set will realize p). Let 8 be a family of subsets of 1M1 such that: (1) 181 = KO. (2) Every finite subset of 1M1 belongs to 8. (3) If I is a finite A-n-indiscerniblesubset of 1M1,(A a finite subset of L), and I can be extended to an infinite d-n-indiscernible set in M ,then there is such an extension which belongs to 8. Let 8 = &: i < a},and noting JM)= (a,:i < w}, let EMI = {(a,,a,): a,E I,}, PMi = {a,: lIjl = No},where E, P belongs to L,, and let P E L, be such that for every a, E P i , PMi(x,aj)is a function from I, onto ]MI; we write x E y instead of E(Z, y). Clearly for every finite A c L(T),n < w , there is a formula vd,,,(x)in L, saying that {y: y E z} is ad-n-indiscernibleset. Let q = {pld,,,(z):A c L(T), n < w , lAl < No} u {b, E x: i < w } u {P(z)}.It suffices to prove that q is consistent with Nf. Since then, aa Nf is N,-compact, q is realized by some element b EN?. Hence I = {c E N , : N, t c ~ b is} an indiscernible set (as Nf C vd,,,[b], Nf is an elementary submodel of N , ) . Clearly b, E I for i < 0 . Also 111 = llNll aa N , I= P[b] (using P i ( % ,b)). Now in order to prove that q is consistent with N;f it suffices to prove that every fmita subset of it is consistent. By I, 2.3(3) instead of a finite number of q-~~,,,(x) we need take only one. So it suffices to prove u {b, E z: i < m < w}. By 11, the consistency of q' = { P ( x ) ,q.~~.,,(x)} 4.6(3) for every finite A, n there is T = r(A, n) < w , such that: i f m 2 r, { b o , . . .,b,} is a A-n-indiscernible set in M, then there is an infinite A-n-indiscernibleset in M which extends {bo, . . .,b,,,}. So for r 2 r ( d ,n )

OH. VI, § 51

M1

SATURATION OF ULTRAPOWERS

C ( V Z N V Y ~* * Yr)[ (A Yt *

1< f

+

~j

A qd.n(Z) A

A

385

~1

Z)

isr --+

(jy)(qd..(y)

A

P(Y)A

A Y; E Y ) ] . tsr

This clearly implies the consistency of q', as {bi: i < w } is an indiscernible set (in L ( T ) )and for every c,, . . ., cn E N , there is c E N , such that N , C: (VZ)(Z E c A?-, z = cf). Remarks. ( 1 ) TIhas no models in any uncountable cardinal < 2'0 and is categorical in KO. ( 2 ) By 5.1(2),VIII, 1.7 and VIII, 2.1 the theorem is the best possible. The following theorems have similar proofs so we omit them. THEOREM 5.4: (1) If T is countable, withut the f.c.p., and stable in KO (i.e., totally transcendental) then there is T,, T G T,, ITl! = No, m h that PC(T,, T ) is categorical in every h 2 KO, and every model in it is saturated. ( 2 ) If T has the f.c.p., is countable, and stable in KO,h 5 2N0 then there is T,, T G T,, lTll = h such that PC(T,, T ) is categorical in h and every model in it of cardinality h is saturated. THEOREM 5.5: If T is countable and superstable, then there is T,, T C T,, lTll = 2No such that PC(T,, T)is categorical in 2%, and every model in it of cardinality 2w0 is saturated.

Remark: We use the following fact: if M, is a complete model which expands ( w , c), N, is an uncountable model of the theory of M,, a s INl[, [{bE lN1l: b < a)I 2 KO,then I { ~ E lN1l:b c a}I 2 2% We leave it as an exercise. THEOREM 5.6: Let M be a model of a countable and superstable theory T , M, a complete expansion of M , N E PC(Th(M,), T).Then N is 2~0-saturated. THEOREM 5.7: If T is not @,-minimal, then it is rwt @,,-minimal for every p 2 min{21TI, A}.

Proof. If p 2 h the conclusion follows by 4.6 so we can assume h > p 2 2ITI; and by 4.6 again it suffices to prove the theorem for the case p = 2ITI. So let h > p = 2ITl, T is @,-minimal but not @*-minimal.

386

SATTJRJ~TIONOF ULTRAPRODUCTS

[CH. VI, f 6

As T is not @,-minimal, by 4.2(2)there is a regular ultrafilter D over A such that for every model N of T,N A / Dis not A+-compact. Let M , be a A+-saturated model of T, M <M,, llMll = (TI. Suppose first M A / Dis not IT)+-compact.Then there is A c ]MA/D1, IAI I; IT], such that MA/Domits a type over A. W.1.o.g. (by D s regularity) there is an equivalence relation, eq, over A, eq G A x A, such that for every a E A, eq c {(i,j) E h x A: a[i] = a [ j ] }and eq has IT1 equivalence classes. Let N be the submodel of N A / Ddefined by IN1 = {a E IM’/DI: eq c {(i,j): a[i] = arj]}}. Then N is also not ITI+-compact and clearly for some ultrafilter D, over IT1 N is isomorphic to NITI/D,, so T is not @lTl-minimalhence not @,,-minimal (as we can make D, regular). Assume nowMA/Dis ITI+-saturated.By 1.8,111, 3.10 there is an indiscernible set I = {a,:n < o} in M A / D ,dim(I,Mt/D) < A. Without loss of generality there is an equivalence relation eq over h with I;IT1 equivalence classes such that eq E {(i,j):a,,[i] = a , [ j ] } for every n < o. Clearly {aE wt/DI :eq G {(i,j): a[i]= ao]}} is the universe of an elementary submodel N of M t / D and I c It is also clear that for some ultrafilter D, which we can assume is regular, over ITI, N and MTI/D, are isomorphic. As Mo is A+-saturated, h > 2ITI it suffices to prove N is not (21TI)+-saturated.If it were, it would be A+-saturatedby 5.2, hence A 2 dim(I,Mt/D) 2 dim(1,N)2 A+. Contradiction.

m.

Now we shall try to deduce some results on @. THEOREM 6.8: Let T be countable. (1) T is @-minimal iff T does not have the f.c.p. (2) For h 2 2Mo T is @,-minimal iff T does not have the f.c.p. ( 3 ) I f No < h < 2N0,T is @,-minimal i,lff T ia stable. Proof. (1) By 4.2(3) and.Exercise 4.21, this follows from (2) and (3). (2) We use the criterion in 4.2(2); i.e., T is @,-minimal if for any model N of T and regular ultrafilter D over A, MA/Dis A+-compact. Suppose T does not have the f.c.p., N is a model of T and D is a regular ultrafilter over A. By 5.1(1) MA/D is K,”/D-compact and by 1.12 N$/D = N,”2 A + , hence MAIDis A+-compact. Suppose now T does have the f.c.p.; by 3.12 there is a regular ultrafilter D over A and n, < w (i< A) such that n l < , n , / D =‘2h (aa

OH. VI, 5 61

SATURATION OF ULTRAPOWERS

2M0 5 A) and by 4.6, for any model M of

387

T,YA/Dis not (2h)+-compact

hence not A -cornpaat. (3) Similarly. For M a model of T,T stable; if D is a regular ultrafilter over A, by 6.1(2) M A / Dis p-compact, where p = minm,,, n,/D: n,/D 2 KO}. But by 1.3(l), (2) D is 24,-incomplete, hexme by Exercise 2.10 pMo = p, hence p 2 2%, hence p > A, hence M A / Dis A -compact. Now let T be unstable, by 3.12 there is a regular ultrafilter D over A such that lof(No,D ) = K,, hence by 4.8 M A / Dis not K,-compeaf, but K, 5 A, so it is not A+-compact. +

nfeA +

THEOREM 6.9: Anwng countable t W e a (1) the t h i e a with& the f.0.p. fman equivalence class, (2) the stable them&?with % the f.c.p. fman equivahce c h 8 , (3) the t-ea with the etrict order property are all equivalent, (4) if T , , Ta and T , are a~ in (l), (2) and (3) rqectively, and T 8&8* n i of ~them, ~ t h T , @ Ta @ Tad @ T @ T,, but not T , @ T,, and not Thd @ Ta. If, e.g., M A + zn0 > K, holds then not T , @ Thd; m m e r K,~~= ( T00) impliea not T @ Thd.

Remarks. ( 1 ) The theorem holds for @A instead of @ for any h 2 2no; and for No < h < 2Ho, the only difference is that all the stable theories become equivdent . (2) For (4) see also Exercise 3.26.

Proof.Let T,,Ta, T , and T be as in (a), with respective models M,, Ma, bl, and M and i k f h d a model of Thd. (1) This is the previous theorem. (2) M?JD is A+-compact iff n,/D 2 KO * n,/D > A (by 6.1 (2)) hence they me all equivalent. (3) This is 4.3. (4) T , @ T , is clear from the proof of 6.8 and 6.9(2), T , @ Thd is n,/D 2 KO * n,/D 2 clear by the proof of 6.8(2) and 4.8 (as lcf(Ko,D)).Now Thd @ T can be proved by Exercise 4.6, and T @ T , follows by 4.3. NOWT , @ T , fails by 6.8, Tina @ T 2 fails, as by 3.12, for every A there is a regular ultrafilter D over A, n,/D 2 KO == n,/D = 2A, lof(Ko,D ) = N,, SO M $ / D is A+-compact by 6.1(2), but M&,/D is not N,-compact by 4.8. Now aaaume MA + 2Mo > K,. For not T, @ Thd use 3.10(2) (by 4.3 and 4.2(1) Mh/D is A+compact iff D is A+-good). For not T @ Tad when K&t) = 00, note

nieA

nf
nteA

n(
n,,A

388

SATUBATION OF ULTRAPXODUClTS

that in 3.10(2) D = D, x D, compact by 4.7.

SO

[m.

05

MA/D= (MA/D,)a/D,is not

EXERCIBE 5.2: Show that if we replace “ T is countable” by “ D is ITI +-good” in the hypothesis of Theorem 6.1,the conclusions still hold. EXERCIBE 5.3: Let M be a model of a countable and superstable theory T , D an 8,-incomplete ultralilter over I , U a filter over I x I , such that 11MLlUll > No, MLlU # M. Show that if T does not have the f.0.p. and M is h-compact, then MLlU is ALlB-compact. (See Definition 4.2.) EXERCIBE 5.4: Suppose E(Z, g; 2) is aa in II,4.4, M a countable model of a stable theory T which has the f.c.p., D an N,-incomplete ultrafilter over I and U a filter over I x I . n, = I{Z/E(E,#; a): E E 1M]}1, P = {n,: E E IMl}.Show that there is E E IMLlUl n, = A, iff there is 8 E I(w + 1, < , P)LlUl, such that P(8) and

EXERCIBE 5.5: Let M be a model of a stable theory T which haa the f.c.p. D an K,-incomplete ultrafilter over I , U a filter over I x I , and let A c L ( T )be finite. Let p be a A-1-type over N = MLlG which is omitted by N; but every q c p, 141 < lpl is realized in N; and lpl is regular. Show there is 8€(11itf11 + 1, <)&lasuch that 1231 = p: CllJfll + 1, 4blU l=t < 811. EXERCISE 5.6: Let M be a model, M, a complete expansion of M y T = Th(M), T , = Th(M,). Then {MLIU: D an ultrafilter over I , U a filter over I x I)is equal to PC,(T,, T ) . EXERCIBE 5.7: Let T be countable and complete. Then: for some countable T,, T E T , and PC(T,, T ) is categorical in No iff ID(T)I = No. (Hint: Like 6.3.) EXERCISE 5.8: Use Exercise 1.9 to get a better (2) presentation of the proofs of 6.1 and 6.3 (aee next exercise).

OH. VI,

8 51

SATURATION OF ULTRAPOWERS

389

EXERCISE 5.9: Let L E L,, P E L, a one place predicate, and L,model M , . Let M: be the submodel of the L-reduct of M I with universe P ( M , ) . Then for every theory T, c L,, { M f ; M , C T,,M , an L,model} is PC(T2, Ta)for some Ta (you can use VII, 1.1 and VII, 1.2). EXERCISE 5.10: In Theorem 5.1(4) let p = &I 11M,11,and A(d) E p r / Dbe (. . .,&(A), . .)t6j/Dand let A** = I{a E p r / D :for every finite d c L, p r / D C a < A(d)}].Prove that N is A**-wturated. [Hint: Use Exercise 1.9 and 6.8. There is (in N ; ) an order of type “ w ” of the formulas of L (including non-standard ones) and for every “set” I c N there is maximal n = nI such that I is tp,-indiscernible for every k < ni, It suffices to prove that each such I can be extended to an indescernible set of cardinality A**. We define by induction on a E N i , ii, E N such that nr, is maximal where I, = I u {?is I fi < a}. Then I u {aa:N ; C a < A(d) for each finite A } is ae required.]

.

EXERCISE 5.11: The regular ultrafilter D over I satisfies the conclusion of conjecture 5.1 when (J< ) C w 5 a, for each n iff lcf(K, D) > No,K s IIl where K = lcf(Ko, D). Remark: The conditions (J, < ) C w s a, are satisfied in the case we are interested in. [Hint: Let a, = (. .,%*, . . ) J D , and w.1.o.g. a: is a cardinal (see Exercise 1.1) and let X, = mink: {t: x < a:} 4 D}; clearly No 5 x,+, 5 x,, so w.1.o.g. X, = xo; and by 1.12 la,l = XI’ . If { t : qn = x} E D for infinitely many n’s, a = (. . .,x, . . . ) / D prove the assertion. So w.1.o.g. 4 c x for each n, t . Necessarily x is a limit cardinal, cfx s IIl. Let b, = (. . ., l$, . . .)ts,/D (j< K ) exemplify . . .)t,I/D; then (J,< ) Cc, s lcf(N,, D) = K ; and c, = (. . ., a, A c, < c,fori < j < Kandlet{dEJ:(J, < ) Cd 5 a,foreachn} = U,
.

.

EXERCIBE 5.12: Show that for good K,-incomplete ultrafilter D conjecture 6.1 holds. [Hint: If (J,< ) C a, < w for some n, by 2.3 there is c, (J,< ) C n < c < a, for each n, and use 2.13. Otherwise by 2.3. K I IIl * lcf(x, D) > 111 and uae Exercise 5.11.1 EXERCISE 5.13: Suppose that for some n,, p = nlEInt/D> Noand N = M1/D is (I TI + p ) +-saturated, and M ia A-compact. Then N ie

SATURATION OF ULTRAPRODUCTS

390

[OH. VI,

86

AI/D saturated. [Hint: As in 5.2, let A > [TI,T without the f.c.p., I E N indiscernible, 111 = ([TI+ p ) + .There are finite Pt E 1M1 such that IP"I I p, 11n PNI 2 KO, where (N, P") = n t , i ( M y P t ) / D . There is 7i E I such that stp(a, u I")E Av(1, N),and Qt = It E 1M1 an indiscernibleset over u Pt based on stp(a[t], U Pt),and (M, P,&) = ( M , P,,Qt)/D.Then Q is indiscernible, /&I = K / D , Av(Q, N ) =

nts, AV(1,

w.1

VI.6. Saturation of ultrslimits For every model M and ultrafilter D over I the natural elementary embedding of M into Mr/D,is defined by a H (a: i E I ) / D . Hence we can look at M'/D m an elementary extension of M ; and so we get an elementary chain of models by repeatedly taking ultrapowers and taking unions at limit stages. For simplicity, all the ultrapowers will be with the same ultrafilter, which will be assumed Kl-incomplete.

DEFINITION 6.1 : Let M be a model, D an N,-incomplete ultrafilter over I. We defhe the ultralimit, UL(M, D, a ) by induction on a,so that for < a, UL(M, D, /3) is an elementary submodel of UL(M, D, a). (1) UL(M, D, 0 ) = M . (2) For a a limit ordinal UL(M, D , a ) = UBSK UL(M, D , p). (3) For a = /3 + 1, UL(M, D, a )will be isomorphic to UL(M, D, @)'/D, and for each a E IUL(M, D, /3)1, the isomorphism ElB takes ( a : i E I ) / D E IUL(M, D, /3)'/Dl to a. So UL(M, D , /3) < UL(M, D , a).

/3

Notation.As M and D are fixed for most of this section we let M , = UL(M, D, a ) and FBbe the isomorphism mentioned in (3) of the above definition. Clearly we can msume that for every a,/3, UL(M, D, a + 8) = UL(M,, D,/3). We write T for Th(M). We shall try to find how compact the ultralimits am, for various properties of the ordinal, the is isomorphic to ultrafilter, and the theory of the model. As MK+, ML/D, we shall restrict ourselves to M6 for limit ordinals 8. THEOREM 6.1: If cf 8 2 p, and for every A < p, D is A-regular, then Md i8 p-COTn$XXCt.

Proof.Let p be a type in Mdof cardinality < p. Then clearly p is a type p is realized in M B + by 1.4, in M Bfor some /3 < 6. As D is ~p~-regular,

OH. VI, f 61

SATUIZATION OF ULTRALIMITS

hence p is realized in M,. So every type in realized in Md; hence Md is p-compact.

itid

391

of cardinality < p is

Proof. M , is K,-compact by 2.1, 2.3 and 2.11 (remember D is K,incomplete). As T is unstable, by 11, 2.13 and 11,2.2; there is a formula ~ ( zg) ; and sequences a,, a,,. . . a,, . . . from M , (an of the length of g) such that: for every m < w, {cp(z; 7in)M(nrm): n < w} is consistent with M , . As cf 6 = p, let 8 = U,<”u(j),wherej < i < p implies 1 < u ( j ) < u ( i ) < 8. We shall now define by induction on j sequences such that : (1) lMa(j)+Il,~’# l%j)l. a’)} is not rertlized by any (2) qj = {7cp(z; a,): n < w} U {~(z; element of Ma(j). (3) For every m < w, ppJ”= {cp(z; iE,)if(“Zm): n < w} u {cp(z;8):i s j) is consistent (with M w + l ) . If we succeed in defining the $’a then clearly by (3) p = {-,cp(z; a,): n < w} u {cp(z;Z): i < p} is consistent (with M,), because every finite subset of p is a subtype of ppJ”for some m,j.But i f p is realized in Hd, then it is realized in M B for some /3 < 6, and there is j < of 6, < u(j) < 8. Hence p is realized in Ma(,),in contradiction to (2). Hence p is a consistent type in M,, which M , omits, and IpI = KO + p < p+. So Md is not p -compact. It remains only to define $, assuming 8 has been defined for i c j. As D is K,-incomplete there are I,,E D, I,,, s I,, I , = I , I, = 0. Let us define 6j E lMk(,JDl: if i E I, - In+l,then @[i] = a,, 8’ = (6’[i]:i E I ) / D , and = Ba(j)(8j). Let us check conditions (l), (2) and (3) are satisfied. Clearly 8 E lMa(,)+,1.Now for any n < w, {i €1:6j[i]= a,} = I , - I , + , 4 D so $ 4 lMml. So (1)is satisfied. For proving (2), suppose, for somej < p, c, E lMa(,,] realizes pi. Then {i E I: Mw) b cp[P&(c,)[i], 8j[i]]}E D, that is, {i E I: M,, b ~ [ c , , P[i]]} E D. Hence for some n < w Ma(,) k cp[cj, a,], so c j does not realize qj, contradiction, hence (2) holds; and (3) has a similar proof. +

on<,

DEFINITION 6.2: p ( D ) is the least infinite cardinal p such that D is p-descendingly complete, that is, p ( D )is the least infinite cardinality p

392

SATURATION OF ULTRAPRODUCTS

such that I , E D, a < p, and a c (equivalently I , E D).

na<,

/I * I , c I , implies

[OH. VI, 5

6

no<,, I, # 0

Remark. If D is K-regular then K c p ( D ) ;also p ( D ) I ] I ] +Note . also that p ( D ) is a regular cardinal.

Proof. Let p be a type in Md, 1p1 < p, and we shall prove that p is realized in Md,thus proving the theorem.

As / p i < p

cf 6, p is a type in M a for some a < 6. Let lpl = We shall prove by induction on y I p that:

(*I

I

K,.

Every subtype of p of cardinality I K , is realized in

+ + 1K, = 1p1 < M a y

As /I I p 5 of 6, a + /I + 1 < 8, hence by proving (*)we shall prove that p is realized in Ma. Suppose we have proved (*) for every y1 < y. Hence every subtype of p of cardinality < K, is realized in M a + , (remember every model is 24,-compact, hence every finite subtype of p is realized in Ma). Clearly we can assume w.1.o.g. that p is a 1-type. So let q = {g~,(z;a,): j < H,} be any subtype of p of cardinality K,; we shall prove q is realized in M a + , + , . By the induction hypothesis, for every j < K,, there is c, E lMa+,1 which realizes {p,(z;ZJ: i < j } . As K, I IpI < p I p ( D )there I , = 0, with I, = I . is a decreasing sequence I,,j < K,, I , E D , Let us define C E l ( M a + y ) l / D [if: i ~ r \ , j ( n , < c I , - I?) = Ij+1 every j c H,, {i E I: M a + , C 9~[c[iI, E D. So q is realized in Ma + + ; so p is realized in i i f d .

n,
,,

DEFINITION 6.3: A model N strongly omits a type p (in it) if no subtype of p of cardinality I p I is realized in N. LEMMA 6.4: (1)If M strongly omitep, IpI = p ( D ) ,then M , also strongly omits p . (2) if Ma strongly omits p , 1p1 = p ( D ) , and a < 8, then M , also strongly omita p . ( 3 ) In (1) and (2) instead of Ip I = p(D), its suflcea to aesurne that there I , = 0; and lpl is are no I , E Dfor /3 < lpl, /I < y I , c I,, regular.

-

n,<,,,

OH.

VI, § 61

SATURATION OF ULTRALIMITS

393

Proof. We shall prove that

(1) as (2) and (3) have similar proofs. Suppose (1) fails, so c1 E jMll realizes q G p , 1pI = lpl. Let c1 = Po@), p = {(p,(z; a,): I < lpl}. So clearly for every 8 < 191 = 1131 {i E I : 211 k tp,[c[i]; a,]} E D. It is also clear that for every i E I, q(i) = {(p,(z; Be): M C (p,[c[i], a,]} is a subtype of q, hence of p, which is realized in M ; hence Iq(i)l < 1p1. As lpl = p ( D ) is regular, for every i E I there is a bound ((i) < 1p1 to (5: M b(p,[c[i];a,]}. Let for 5 < 1p1, I , = {i E I : [(i) 2 l}.Clearly I,, 5 < IpI is a decreasing sequence, and by the I, = 0. In addition each I, E D as I, = definition of [(i), {iE I : [(i) 2 t;} 2 {i E I : M =! (p,[c[i]; S]}E D. So we get a contradic-

n,
tion to the definition of p( D). THEOREM 6.5: If T is urntable, 6 2 p ( D ) , then M , is not p ( D ) + compact. (Moreover there is a type in of cardinality p ( D ) which M , StrO?& OW&%.)

Proof. As it is similar to the proofs of 6.2 and 6.7 we omit it. CONCLUSION 6.6: If T i8 unstable, p = min(cf 6,p ( D ) ) ,then M , is maximally p-compact. ( i e . , p-compact but not p+-compact).

Proof. Immediate by 6.2, 6.3 and 6.5. THEOREM 6.7: If K ( T )> p = min(cf 6, p ( D ) ) ,T is stable, then 116, is maximally p-compact.

Proof. By 6.3, M , is p-compact so we need to prove only that M , is , there are A,, not p+-compact. By hypothesis T satisfies p < K ( T )so 4 I p , E~W,,), ~ Q&, Sc), andai,,n, (I < p, n < w); such that: S < 5 * A, c A,; for every 5 < p, {i&: n < cu} is an indiscernible set over A,, ~ p Clearly . it suffices to E A , + l and -(pe(x; 7i,,o), (p(x; ii,,,, prove the theorem for the case L = L(27) is the minimal language containing all the formulas (p,(x; S,); so ILI I p. Choose a([) < 6 for 4 < cf 6 such that 6 = Uc
SATURATION OF ULTRAPRODUUTS

394

[OH. VI, 5 6

Let H , be the empty function, p(0) = a(0). For f a limit ordinal, He = U C < ~ H CrB(O , = max(a(€)iU<
(*I

u

hto a6,.

If p = No,this is true, aa for every N, NxlD is N,-compact, so /3 = /I(& 1 will suffice. So assume p > No.We now define, by induction on n, an increasing sequence of (a, Md)-elementarymappings Hn from Dom H , U {ae,,,,:m < n} into Md. If we have defined H", and cannot define Eln+', we can deduce Md is not p+-oompad [as it omits {'p(Z; H"(I7)):'p E L, I7 E Dom Hn, t &iC,,, El}] and so the conclusion of the theorem holds. So we can assume H" is defined for every n and let H* = Un
+

u

Define /3(( E D, n <

+ 1) = max(/3, a(&).

As D is K,-incomplete there me I , c I,,,, I, = 0. Define He+l(at,Onac.l) lMER+1)+1I &8 p8(t+1)@) where EM8(C)+l'/D is defined as follows: if i E I, - In+,,E[i] = H*(iLE,nn?iC,n+l). It is easy to verify that He+,, p(f + 1) satisfy the induction conditions (by 11,

I,

w,

I, = I, m < n

s

2.20).

Now P = {'p,("Y H,+l(a,,o)) l'p,(", H,+,(B,,,)): 5 < p} is a consistent type in ikf6(u), and it is strongly omitted by ME(,,).As Md = UL(ME,), D, 8 - /l(p)),by 6.4 Md strongly omits p, and this means ikfd is not p -compact. It is natural to conjecture that if K ( T )s min(pCC(D),cf a), and a, /3 < S * a + p < 6, then M is UL(X,, D, &)-saturated[uL(X,, D, 8) is the cardinality of UL(M, D, 8) for every countable dl] which would generalize 6.1(1). But this is not true. T may be superstable (~(5")= No) or even simple (see Definition 0.4) while M or M , strongly omits a type of cardinality p( D ). However: +

THEOREM 6.8: Supporre K ( T )s min(p(D),cf a), D is a ITI-regdar ultra$lter; a,$ < 6 * a+$ < 6. Then Ma is A-saturated, where A = UL(% D, 8).

CH. VI,

# 61

SATURATION OF ULTRALIMITS

395

Remarks. (1)For every 6 , there are S,, S such that 6, = S,+S; a,B < 6 =+ a + < 6 and UL(itf, D, 6,) = uL(Md,, D, 6). so the restriction on 6 is natural. (2) Clearly A > I2'1,so it s&ces to prove Mdis A-compact.

Proof.Let p be a 1-typein i t f d , 1p1 < A. It suffices to prove p is realized in M,. Let q be any extension of p in S(lMd)). Notice that if IBI < K ( T )s of 6, B E 1Md1, then for Borne a < 6, B G lMal. Hence by 111, 3.2.there is a < 6 and B E lMal such that q does not fork over B, and by 111,2.16 and I11 4.18, q 1 1Ma1 is stationary. So by 111, 4.2 there is a set B clHal, IBI s 12'1 such that for every 'p R(q 'p, 'p, 2) = R[(q 1 B ) 1 'p, 'p, 21. Now we can define a,, for n < w such that: (1) a,, realizes q 1 (B u {am:m < a}), (2) if 8 > ws a n E IJfo+n+ll* This is possible since D is 12'1-regular, by 1.4(2). Hence {a,,:a < w} is an indiscernible set and q = Av({a,,: n < w}, Ma). suppose for a moment 6 > o. Let P = {a,,:n < w } E lMa+al (ae a < 6, o < 6; a + w < 6). Let (Ma,Pd)= UL((M,+,, P),D, 6) (remember 6 = a + o + 6). Clearly Pd extends P and is an indiscernible set. So 'p(z, 6) E p implies 'p(z, 6)E q implies {a:a E Pd, I= -+z, 6)) is finite. So all except lpl- 8, < h members of Pdrealizep. As lPdl = UL(Ho, D, 6) = A, the theorem follows. So we remain only with the case 6 = O . But then we can define the an'ssimultaneously in M a + I and the proof goes in the same way. The following exercises will deal with the problem of categoricity in logics with generalized quantifiers.

Assumption: Let T be a complete theory in the logic L(Q1, ...,Q") (where ifg, is a formula so is (Q'xg,)).Let P,, ...,P , be unary predicates < ,um,Al, ...,A,, in L , VxP,(x)€T and let for any cardinals ,ul< K ( p , ,...,A,;..) = { M : M an L-model, that satisfies every $ E T when (Qzx)O is interpreted as "there are a t least A, x's such that B(x), and IRI = P ' Y ' Suppose further that ,uF = ,ul, each A, is regular and (Vx < A,)x'o <

4

*

Suppose (*)K(p,,...,A,,

...) has a unique model up to isomorphism.

[m. VI, § 6

SATURATION OF ULTRAPRODUOTS

396

EXERCISE 6.1 :We can assume w.1.o.g. that for any formula q ( Z ) in

I,(()', ...,Q"), forsomepredicateR, (VZ)[~(Z) = R ( 2 ) l ~ T . L eT'bethe t set of first order sentences of

T.

EXERCISE 6.2: Prove T' is superstable. [Hint : It is well known (see [CK 731) that K ( p l ,...,A,, ...) is closed under ultraproducts for ultrafilters over w . Let M belong to K ( p , , ...,A,, ...) (exists by a hypothesis), and let D be a non-principal ultrafilter over w . So M , gfM"/D and easily also M , gP UL(M,D,w ) belong to the class K ( p , , ...,A,, ...), hence by a hypothesis they are isomorphic. Now by 2.1, 2.3, 2.11(1), M , is K,-compact. By 6.5 if T is not superstableM, is not K,-compact. We conclude T is superstable.] We now define:

W = { ( q ( z ,g),$(g),A,) :p,$ first-order formulas, (Vg)($(g)

=

1

(QLx)rp(x, g ) ) belong to T'}U { ( P z ( z z) , = x,p;) : I = 1 , m } .

EXERCISE 6.3 :Suppose T is stable (but not necessarily (*)). Prove there i s M E K ( p , , ...,A,, ...) which is theL-reduct ofM,, IL(M,)(= JL(, M , the Skolem Hull of UIZ:nI,,I,indiscernible &,,I,. [Hint: See VII, §2,5 and use V, 6.10.1 Now we apply the results of V, 96 to analyze the (n+m)-tuples of cardinals for which we get categoricity.

+

EzampZe. Let N = ("2,+ N , P:),<" where : P r = (7E "2 :~ ( n=)0}, is a two-place operation, 7 N v = p iff for every n r](n) + v(n)-p(n) is , f , ...,P r , ...), < w even. Let M = (WI,P M Q, M , E M , N , G N ,H N ,F M P where :

+

+

P" = N , QM=NxNxN, EM = { ( a , , b , , c , ) , < a 2 , b , , ~ , ) ) : a , , ~ , , ~ l=,az,b, ~ 2 , ~=~ , b2}, ~~~~,~l = N, + is a ,two-place function from P Mto P', H N ,G N are one-place functions from QM to P', H N ( ( a ,b, c)) = b, @"((a,b , c ) ) = a, P M is a partial two-place function, F M ( ( a b, , c,), (a,b, c z ) ) = c1 + N ~ Z , P r = {(a,b,c)ENxNxN:cEPt).

+

CHAPTERVII

CONSTRUCTION OF MODELS

VII.0. Introduction The character of this chapter is different from that of II-V as extending the theory does not change much; hence it is natural to use theories with Skolem functions, that is, when we want to construct models of T, we choose T,,T z T,, ITl! = ITI, and T, hrte Skolem functions; we construct models of T, and go back t o their L(T)-reducts. The importance of T, having Skolem functions is that: (1) we can find such T, for every T ;(2) every model of T can be expanded to a model of T,; (3) if M 1 is a model of T,, every submodel of M 1 is an elementary submodel. (This is done in 1.1-1.4). Saturation is a very useful device, but unfortunately if A < A<", T does not necessarily have a K-saturated model of cardinality A. We have constructed in I, 1.7 a saturated model by an elementary chain Mf of models of T of cardinality A such that in Mf+ we realize rte many types over M ias we can (remembering the cardinality restriction). We generalize this process (in Definition 1.6) and prove (in 1.7) that for many x's we get some weakening of saturation, e.g., of the form: if p is an m-type over M , 1p1 = A , and every q E p, IqI < p is realized in M then some r z p , Irl = x is realized in M. Notice that when A'" = A we get a K-compact model (when the length of the chain has cofinality Z K , of course); if (Vp < K,)(p<" < K,), 8,'" > K,, and h= our results become weaker as p increases, and we get none when 9, 2 y = K, > K,. However for the theory of dense linear order we can get something (see Exercise 1.7) and a generalization of this is utilized in VIII, 3.2. (There are also other applications of Section 1 to VIII, Section 2, and 3, but we do not need them in the main presentation.) From one point of view a saturated model realizes many types, but from a deeper point of view it realizes few. Let TP(S, M) be the set of 397

,

398

CONSTRUCTION OF MODELS

[OH.

VII, f 0

types realized in M over ii, TP,(M) = {TP(ii,M ) : I = (a‘: i < K ) , a,E [MI}.Then clearly for a A-saturated model M of a countable theory ( K < A), TP,(M) has cardinality I ; ~ ” + I * I , but for a not necessarily A-saturated model, it is possible that ITP,(M)l = 2a“. We deal with these problems in 1.10-1.12 and prove, e.g., THEOREM 0.1: If p < 8, =- pK < X,, t9 .c K + , T countuble, 11M11 s Na+Bthen M hua an elementary extension N of cardinality Xcr+Bfor which ITP,(N)I = 2”. In Sections 2 and 3 we deal with Ehrenfeucht-Mostowski models, generalizations in two directions, and various applications; which are preparatory for VIII. For a theory T , with Skolem functions, every set A is a set of generators of a model, and it is natural to look for “nice”, “uniform”, such sets. The clmsical example is a free algebra in which every permutation of the set of (the free) generators can be extended to an automorphism of the model. But for any extension of the theory of dense linear order we cannot have such sets. However, by Ramsey’s theorem and compactness we can find models of T l generated by indiscernible sequences, so the order of the generators is important. Clearly the elementary type @ of the sequence and the order type I determine the model up to isomorphism, so we denote it by E N 1 ( I ,@), and its L(2’)-reduct by EM(I, 0). Its importance is that we can easily find orders I with specific properties, and often the model inherits them; e.g., stability (of the model) and some of the generalizations of saturativity from Section 1. An application is THEOREM 0.2: B”0r stable T with the f.c.p., lTll = No,I ( X a y T , , T ) 2 214.

Unfortunately this is true only for cardinalities > I T,I, so when we try to construct models of cardinality lT,I (e.g., in VIII) we have to generalize this construction. We first show we can assume w.1.o.g. that in L(Tl) there are only countably many non-logical symbols which are not individual constants, then we form a model N from individual aonstants, and define EM1(I,N 1 ) for any order I, and elementary extension Nf of N. Now we can determine properties of E M 1 ( I , P )by those of I and W .

OH.

VII, $01

INTRODUCTION

399

At first sight, we cannot expect anything better than an indiscernible sequence of generators. However, we sometimes know of a set of elements satisfying some conditions, and we want to find such a set as homogeneous as possible, which still satisfies those conditions. For proving this we always need the proper generalization of Ramsey's Theorem. I n particular if T is not superstable there are v1E L(T),and sin (for q E such that for q E Ow, v E O'w, vl&&,,a,] iff v Q q . For dealing with such situations we d e h e the indiscernibility of {G8: 8 E I > depends only on the atomic for a model I : the type of a8,co, -. ,., us(n-l) type of (s(O), . . .,s(n - 1)) in I . The general treatment appears in Section 2 and the specific cases in Section 3. We deal there with theories with the f.c.p., and indiscernible sets indexed by trees. An important case, which does not completely fit with the general case, is uncountable D ( T ) for countable T,.What we get is {an:q E "2) so that the an's realize distinct types from D(T), and for every cp(Zo,...,Z,,,-l) E L(T,) for every n 2 n(v),if q1 1 n(l < m) are distinct, the satisfaction . .) depends only on (ql 1n: 1 < m ) and a. For uncountable of v(an0,. T's see a generalization VIII, 1.10. The reader can relax while reading Section &almost no background is needed. Our main aim is THEOREM 0.3: If h 2 2", p 2 2N0, T,countable (for simplicity), then T,has a p-universal model stable in p, of cardinality A. If I is a well-ordered set of cardinality A, M = E M 1 ( I ,0 )is stable in every cardinality 2 ITl] (see 2.9). So if D is a good ultraflter over K , M"/ D is K -saturated, hence K + -universal, but it still is stable in every cardinality h = A". So assuming G.C.H., we can prove the theorem for a successor p. However, without it, we need h e r methods. Remember that (by 111,4.10) a type p does not fork over IN1 iff p is finitely satisfiable in !MI (though p is not necessarily over IMI). For not necessarily stable theories the second condition is still meaningful, and we can prove for it the extension property, and that it is satisfied conby any type over IMI. So if M , (i < A) is a list of models of TI, taining, up to isomorphism, every one of cardinality s p , we can find Mi z M , such that for each E ]Mil, tp@, U,<,M ; ) is finitely satisfiable in No. Let llMoII = ITJ, if T1 has Skolem functions generates a model of TI. For ,u 2 221T11the model is ,u-stable ; but we work more and get it for ,u 2 2IT1l. By such a construction we prove our theorem. Notice that by such a construction we can get a sequence of indiscernibles. +

utw

400

CONSTBUCTION OF MODELB

[CH.

WI, $ 1

In the last section we deal with Morley numbers, i.e., Hanf numbers for omitting types. PROBLEM 0.1: It will be interesting to find new kinds of indiscernibility. This may require proving new partition theorems or trying to find an application of a known partition theorem, e.g., Laver's generalization of Galvin's theorem 7 4[7]&(i.e., 9 + [7]R(,,,)).

VII.1. Skolem functions and generalizations of saturativity DEFINITION1.1: (1) Lo has Skolem functions in T , ( = T , has Skolem functions for Lo) if for every formula p(z; 9) E Lo there is in T , a sentence of the form (Vi7)[(34&;

i7)

+

'p(P(!i); @)I

(P a function symbol.) Such a sentence is called a Skolem sentence of = 'p(z; g) and P is called a Skolem function of 'p(z; 0). ( 2 ) T , has Skolem functions if it has Skolem functions for its language L, = L(T,). 'p

THEOREM 1.1: (1) For every lamgwe L there is a lamgwge Lc1) I> L and a theory T ( l )= T(l)(L)in L(l)such that: (A) T ( l )has flkokm functions for L. (B) Every L-model can be expanded to an L(l)-modelof T"). (C) IL(1'I = p 1 q = ILI. (2) For every language L there is a language Lag z L and t h y TsK = TsK(L)in LsK 8ucJL thut: (A) TsKhas rSkolem functions. (B) Every L-model can be expanded to an LSK-modelof Tag. (C) ITSKI = ILsKI = ILI.

If M is an LCondition (A) is satisfied by the dehition of 2'"). model we expand it to an L(l)-modelas follows: we define $(a) as any element b of M srttisfying M C 'p[b,a] if M C ( ~ s ) ~ Ja); ( s ,and we define $(a) as an arbitrary element of M otherwise. Clearly we succeed in

OH. VII,

8 13

401

SKOLNM FUNCTIONS

expanding M to an L(l)-model of T(l),so condition (B) holds. Condition (C) is immediate. (2) We define, by induction on n, Ln and Tn: Lo = L, T o = 0; Tn+l = T(l)(Ln),Ln+l = (Ln)(l). We let LsK = UneopLn,TsK = Un< Tn. Clearly Ln is an increasing sequence hence any formula 'p in Lax is in Ln for some n, hence in Tn we can find a Skolem sentence of 'p, so in TsKthere is a Skolem sentence of 'p; hence condition (A) holds. As for condition (B), if M is an L-model we define by induction on n, the Ln-model Mn: M o = M, Mn+lis an expansion of Mn to an Ln+lmodel of Tn+l (by (1) this is possible). In the limit we get an LsKmodel of TsE = Un Tn.condition (C)is immediate. +

THEOREM 1.2: (1) Sugqme M is a model of T,, N a subnzodel of M , ( M , N are L,mOdels) L, = L(T,) and Lo E L,, and T , hae Skolem function%for Lo. Then the Lo-reductof N ie an elementary subnzodel of the L,-reduct of M . (2) Suppose M is a model of T,, T , hae Skolem functim, and N is a mbmodel of M . Then N is an elementary submodel of M. Proof. (1) By the Tarski-Vaught test I, 1.2 it suffices to prove that if a E IN1 'p(s,g) E Lo and M C (3s)'p(s,a) then for some b E IN1 M C 'p[b,a]. But by assumption there is in L, a Skolem function P of 'p (in T,), hence M C (3s)'p(s;a) implies M C 'p[P(a),a]. As N is a submodel of M , P(a)E INI, so we finish. (2) A particular caw of (1) where Lo = L,.

CONCLUSION 1.3: Suppxe Th(M) hag Bkolem fundim, A c 1M1. Then the Skolem closure (=BkolemHull) of A , { ~ ( a )a:E A, T a term} = do1 A = acl A is (theuniverse of) an elementary submodel N of M . So for every 6 = (bo,. . ,b,) E IN1 there are a E A , and t e r m T ~ ( E ) ,. ,T,(@ 80 that bi = .;(a). we Wite it in 8hort 6 = ?(a),? = ( T ~ .,. ., 7%).If A is (totally)ordered, we can aemme is an increaeing sequence. I n any m e we can aasume there are no repetitions in a.

.

..

THEOREM 1.4: Suppoee [MI = {ai: i < A}, L = L(M), h > ILI, h ie regular. Then the set C of a < h which satisfy the following, is a closed unbounded 8ub8d Of h: A , = {ai: i < a} is (tlte universe of) an elementccry idmodel of M . Proof. By I, 1.3(2) C is closed. Let us prove it is unbounded. By 1.1(2) we can assume w.1.o.g. that T = Th(M) has Skolem functions. So

402

OONSTRUUTION OF MODEL8

[OH.

V'II, 9 1

a E C iE A, is olosed under the functions of M. Clearly Idd A,1 < A hence for some /3 = /3(a), do1 Aal G AB.'SOfor m y a. c A, if a,,,, = /3(an),a = Una,,, then do1 A, = A,, so we finish. DEFINITION 1.2: (1)&(A) = { B : B c A, IBI < A}. (2) %(A)= { B : B c A, IBI 1 A}. (3) S A ( A )= { B : B G A , IA - BI < A}. (4) #$(A)= { B : B G A, IA - BI 2 A}. DEFINITION 1.3: ( 1 ) H is a set function if it is defined on sequenoes {ti:i < A} (ti any entities), H({t,: i < A}) c ( 8 : s c {ti:i < A}} and H({ti:i < A}) = {{ti:i ~ q8 E:H(A)}. (2) We sometimes use aa a& of indioee sets other than A, mainly orders. H is a pure set function if any permubtion of A maps H(A) onto itself. H , 5 Ha if H,(I) E Ha(I)whenever H l ( I ) is defined.

Renucrk. Clearly SA,S f , SA,S$ are pure set functions. DEFINITION 1.4: Suppose H,,Ha a m set functions, M a model, A a cardinality. Then (1) M is (A, H I , Ha)9aturated when for every set A s 1M 1 of cardinality 5 A, ordered in some way in order type A, and p c S m ( A ) ; if for every B E H,(A), p 1 B is realizes in M , then for some B E H,(A) p tBisre&edinM. (2) M is (A, H,, Ha)-compact when for every sequence r = {qt(2;a,) : i c A} (ai E 1611, q, E L ( M ) )if every p E H,(r) is realized in M , then some p E H a ( r )is realized in M.

LEMMA 1.6: M i s A--act iff it i s (p,SN,, S1)-wm~pact for m y p < A. M is A--eneoua iff it k (p,rSn,, 8')-eaturated and No--, for every p < A. (2) If H1 5 H1, Ha 5 H ' , M k (A, H I , Ha)-saturated ]wmoc - [ then M is (A, H1, H " - s a t u r a [-compact]. (3) (A, H I , Ha)-Baturatim and mpa&ne%8 are p e r v e d by &fin;tional expansion. (See I11 6.14.) (4) (A, H,, H 2 ) - ~ p a & n e % is 8 preserved by adding individual constants and by taking reducts. Proof. Immediate.

CR.

VII, 5 11

SKOLEM FUNCTIONS

403

DEFINITION 1.5: Let M be a model of T,(L = L ( T ) = L(M)) 1 1M11 = A 2 K + ILI,K is regular. Then M is called (A, K)-wturatedif the following holds: (R(I?),E) is a model of ZFC, consisting of the seta of hereditary power < I?. Some N E R(I?)is a (2”) -saturated model of T and as a set N is of hereditary power 5 llNll, 94, = SA,,(a 5 K ) is an increaeing and continuous sequence of elementary submodelsof 93 = (&I?), E,N , A) (N,A serve as individual constants) such that: (1) 93, has cardinality A. (2) If a E l93,l, and a is a set (in R(R))of cardinality 5 A then b E a b E ISal. (3) (IS41: p 5 a) ~ 9 3 , + ~ (hence, e.g., there is an ordinal in 93a+l bigger than any ordinal €BE, hence the set of ordinals €!BKhas cofinality K). (4) Let N , be N as interpreted in 93., Then M is isomorphic to N,. (Notice N , E by (3), and that N, is an elementary chain as 93, is, and L E S,.) +

-

Remark. It almost always suffices to msume N is A- or at most A + saturated. We can replace “(R(I?), E) is a model of ZFC” by weaker conditions to ensure the existence of such a’s, but our theorems will not be changed. THEOREM 1.6: (1) If A 2 K + ITI, K regular, M a nzodel of T of cardinality 5 A, then M has an elementary extension N which is (A, K ) eaturated. (2) If Lo c L, the saturation of DeJinition 1.5 holde, and Lo E 23, then the reduct of M g N , to Lois ako (A, %)-saturated. ( 3 ) Let M be (A, ~)-saturated. If 2IL1 IA, M is K,-saturated. If Ax = A, K > x , M is X+-compact and X+-honzogeneowr. If 21LI 5 A = Ax, M is X+-BdUV‘d&. M is No-hO?TlQgent?W.

Proof. (1) and (2) are immediate. For (3) see the proof of Theorem 1.7(1) below.

-

THEOREM 1.7: Suppose M is a (A, K)-saturatedmadel, p a cardinal s A. If A@ > A > 2, let N, = mink: f 2 A} (hence 2, < N,, cf(H,) 5 p ; p < a N$ < 8,). Let A = K,. Then: (1) If A = A<#, K 2 p then M is p-compact and p-homogeneous. (2) If 2” < A < A”, y - a < cfp, cf K, # cfp, and K # cfp then M is (p, sn,, s : ) - c o m p ~ . (3) If 2’ < A < A,, y - a < cfp, cf N, # cfp, p < K , then M is (p, s, S1)-compact.

404

O O N S T R U ~ O NO F MODEL8

[CH. VII,

g1

(4) If 2 Y < A < AY; and for any 8, a < /3 5 y =+ /3 - a < Ks; p < x 5 A < K a + x y x # K then M ie (x,SHo,S;)-~rn~)a~t. (5) (i) If 2fl < A < AN; and for every 8, a < 8 5 y =-8 - a < NE and t h a M is (x, S:, S1)-compact. p <x
but for (1) we s b l d aesume 2ILI s A, in addition.

Remark. In parte (4) and (6)(i)we can replace

''8 - a"

by

"/3".

Proof. W.1.o.g. we shall use the notation of Definition 1.5 and assume M = N,. (1) Suppose p is a sequence of m-formulas over M of length p, < p (i.e., p = (~,('i(t; 8,): i < pl), yr E L , a, E M , 2 = (xO,. . .,x ~ - ~ which >) is finitely satisfiable in N. Aa K is regular and >pl, p is over Na for IL( 5 IINaII = A, hence the set @ of some a < K. Now L, Na E E Sa+ ,, but [ & ( @ ) I = m-formulae over N, belongs to b,+,, hence SN(@) A'" = A, so p E BY(@)==p 6 (by (2) in the definition). So p is realized in Na+, (as Sa+,is an elementary submodel of (R(K),E, N, A) and N is (2"+-saturated; we use such reasoning without saying). Hence p is realized in M = N,. The p-homogeneity is proved similarly.

CLAIM 1.8: Suppose u E IS^,^^, u c 1SASEl, S c u, IS1 < Kt 5 IuI. then for some V E S ~ , , , S G v, lvl 5 Kc and if (1) If IuI < cf Kc > JSleven IvI < Kc. (2) If IuI s Kc+,, cf 181 > j, then for s o w v E S AIS ,~ n v( = (81, IvI 5 Kc and if cf Kc # cf (Sleven lvl < Kc. (3) V c f K c > 181;and x < KI;=xlSl < A, then in ( 1 ) K p , < A+K5+, so S E %A,B; and if cf Kc# cf 181, x < Kc ==xlSl < A, then in ( 2 ) we can aesume v c 8. Proof of 1.8. (1) Let lul = NC+,, and we prove it by induction on n. Clearly Kc+, 5 A, and there is in a one-to-one function g from Kc+, onto u. For I < Kc +, let u, = {g(i) : i < so clearly u, E SAPE and if 181 < cf Kc+,, then for some I , S E u,; as )u,J < Kc+, the conclusion follows by the induction hypothesis. If (812 cf Kc+,, n = 0, u = v ie sufficient. (2) Let lul = Kc+ and we prove it by induction on I.

e},

,,

OH. VII,

5 11.

SKOLEM FUNCTIONS

405

If ( = 0 or ( is a successor the proof is as in (1). If ( is a limit ordinal < i, < cf IuI < cf l#l;hencedefiningu,,i < Nc+easin(l),there~((i),i cf 181so that u = U u,,,,, hence for some i, 18n ue,,,l = 181, and of course lue,,,l < IuI, so using the induction hypotheeis on ueco,8 n ueco we get the desired conclusion. (3) Immediate, by Definition 1.6, condition (2).

,,

Continuation of the proof of 1.7. (2) Let p be a set of m-formulas over N,, lpl = p , p finitely satisfiable. We should find some p' G p, Ip'l = p which is realized in N,; so we can replace p by any subset of cardinality p. As of p # K for some B < K,the set of formulas in p which am over N,, ply has cardinality p. So p1 is a subset of @ = the set of m formulas over Np, which E % ~ , @By + ~Claim . 1.8(2),(3) (taking p, = S , @ = u , 5 = a) and the assumptions, some p, E ply !pal = p, is a member of S,,, + ,, hence realized in N, + hence in N,. (3) Let p be a set of m-formulas over N, of cardinality p sucih that every p' E 8Jp) is realized in N,. We should prove p is realized in N. Let p = {p,(Z,ail: i < Cl) and choose 6j EN, which realizes pj = {p,(Z,at):i < j}. As K > p there is 8, a 5 8 < K such that a,, bj E N,. Using 1.8(2), (3) with {6f:j < tc) = 8, the set of finite sequenoes from N, = u and ' = p, belongs to 8,. As 93 satisfies a = 5; some AS" r {Ef:j < p}, 181 "there is a sequence 6 E N which satisfies any formula p(Z, 7i) (a E N6) which is satisfied by 6i for every big enough ~ E S also " SF+,satisfies it, so this 6 necessarily realizes p (in N,). So we have proved (3). Now we need

CLAIM 1.9 : Let Sc be as in Definition 1.5, N, = min{N, : Nz > N,}, 2 ~ N,< < N t ; a < / 3 ~ ~ * / 3K < B ; p < ~K 5 r< Na+x.If~~18tI, S E u E ISc(,lul < K,+,, IS1 = then there is S' E S , JS'I = p, S' E lScl.

x

Proof of 1.9. We prove i t by induction on lul and for fixed u by induction on X. Always there is in 93, a one-to-one function g from lul onto u, and we let ue = {g(i): i < 8 for I < lul. Case I : lul s 24,. Then, of Ha s p < x, for some < Ha lu, n81 2 p, hence (by Definition 1.6(2)) 8' = ue n 8 E SC,so we get the conclusion. Case 11: lul > Ha, x a limit cardinal. Then for some ~ ( 1 < ) X, ) IuI < Na+xcl,,so choose 8, E 8 , l8,l = ~ ( 1and ) use the p < ~ ( 1and induotion hypothesis on u, 8,.

OONSTRUCTION OF MODELS

406

[OH. VII,

41

x is a successor,1 . 1 > x, IuI singular. As X u < < Xu+x, x regular, necessarily IuI has confinality <x, hence for some 6, Iu, n Sl = x so use the induction hypothesis on u,, u, n S . Cme IV: 1 . 1 > Xu, x is a succesaor,1 . 1 > x,1 . 1 regular. Then for some 6, u, z 8,so use the induction hypothesis-n u,, S. Cme V: 1 . 1 > Ha, x is a successor, IuI = x. Let1 . 1 = A:, then for some 8, luel = IuenSl = Al, so uee the induction hypothesis on uey u, n 8. So we finish the proof of 1.9.

1.1

Cme 111: IuI > Xu,

Continuation of theprmf of 1.7.(4) Let p be an m-type over N, of cardinality x (which is finitely satisfiable). If cf x # K then there are /I< K , p* G p , Ip*l = x, p* is over N,; and then we proceed as in the proof of 1.7(2)using Claim 1.9 instead of 1.8. If cf x = K , x # K , x is singular (so a limit cardinal) hence there is x(1) < x such that p < x(1) s A < Na+xcl,,x(1) is regular and # K . So replace p by p* c p , 1p*1 = x( 1) and proceed as before. ( 5 ) (i) Let p be a set of m-formulas over N of cardinality x; such that every q E S ; ( p ) is realized in N,. Let p = pr,pi has cardinality x, the pi's are pairwise disjoint. Let E IN,] realize p - pf,and aa K > x

zf

uf

for some < K , 6f E lN,l and p is over N B .The rest is like the proof of 1.7(3) Using Claim 1.9 instead of 1.8. (ii) Now the proof is trivial. (6) The proofs for eaturativity are similar.

DEFINITION1.6: (1) Let i be an infinite sequence from N, ii = T P ( i i , N )= {tp,(in6,0,N):6E~bI((6afinitesequence)}. (2) T P A ( N = ) { T P ( i ,N): i is a sequence from N of length A}. DEFINITION1.7: For X = S , C

TPX"(a',N )= { E r is a set of formulas p(3; qCl,, . . .) such that for some 6 E IN[, N t p[6; afo,,. . .] for every p E and X = C * Irl s K , X = S ==only S K variables appear in r} TPX;C,(M)= {TPXK(iiy N): 5 a sequence of elements of N of length A}. Remark, Clearly from TPSi(M) we can compute TPA(N) and vice voi%a. .

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407

LEMMA 1.10: Let L(M) = L(N) = L, and add new variablm q,i < A, to L. (1) 2A s ITPA(M)I s min[2aA+'L'y llMllA]. ( 2 ) TPX$(M)is a monotonicfunction in K , A ; but if K 2 A , [X= C * K 2 1 1.13, it is a constant function in K ; and if K 2 ILI:ITPC;(M)I = ITP&(M)I (infact, one is computablefrom the other); and ITPSi(M)I = I TPA(M)I (infact, m e is computablefrom the other). A180 1 TPQ(M)I I ITPmwI* (3) 2A s ITPfl(M)l s min[2(A+ILI)x, I Jf 11"(4) 2A s ITP8;(M)l s min[2(Ax+a'L'11~11". ( 6 ) If M i s K+-compact, ITPCz(M)I s 2A+ILI,and if N is elementarily equiwalent to M and is K+-compct, TPCZ(M) = T P a ( N ) . (6) If M is K+--eMOUB (e.g., K+-sdUrded) t h ITP&(M)I S )Y

2A+ILI.

( 7 ) If M is stable in A (i.e., m < o, A ISm(A,M)1 s A) t h ITP,(M)l s 2A+lLI.

E

lMl, IA1 s A implim

Proof. Immediate. The next theorem will be used in Ch. V I I I in computing the number of models for an unsuperstable theory. THEOREM 1.11: Let 2" < A; K < A =- K" < A; M a n L-model, llMll s h, ILI s p, then there are x I p, a hngmge L, = L u {Q,:i < x s p} (&,--one place predidm) and an L , - d l M, w h tirat (1) M , b p+-lumLOgeneow,; weover if Lo c L,, &, E Lo, then the L,-reduct O f MI i8 p+-homogeneOU8. (2) The reduct of M, to L is an elementary extenAm of M . ( 3 ) M , ?macardinality A. (4) p l l = Uf<X&f(Ml)Y m d i < j =+ QdMd Q,(Ml).

=

Proof. W.1.o.g. llMll = A. If A' = A, take x = 1 and extend M to a p+-saturated model M , of cardinality A. So assume A p > A, and let x = cf A s p. Choose an increasing continuous sequence Mf, i s x so that L(Mf) = Lf = L u {&,: j < i}, Mf < (My&,),<, where &, = 1Mj1, and IM*l = 1611,/[Mi11 < h for i < x. Expand M to an L,-model M,by defining Q,(M,) = lMfI. Let D be a good N,-incomplete ultraflter over p, (exists by VI, 3.1) M, = MY,/D,and let M , be the submodel of M , with universe U,<,&,(M,). By VI, 2.3 M , is p+-saturated. As Q,(M,) is (the universe of) an Lf-elementarysubmodel of M,, &,(M3)is (the

408

[m. MI,0 1

OONSTRUCmON OF MODELS

universe of) an LWementary submodel of M,, hence M, is an L,-elementq submodel of Ma hence an L,-elementary extension of Ma. As IQr(Ma)l < h a h IQWdI = IQWdI’/D < &hence 11JfiII = A Now if A c ]Mil, IAI 5 p, p E P ( A ) thenp , is realized in M, iff it is realized in Ma in some Qi(M,)iff it is finitely satisfiable in some Q,(M,) (by the p+-saturativity of Ma);and this depends on the type of A and on p only. Hence Ml is p -homogeneous. +

Remark.If the other oonditions are satisfied but p < ILI 5 h we can get M, satisfying (2), such that for any Lo c L,, lLol 5 p, Q,E Lo the redud of M, to Lo is p -homogeneous. +

(By 1.10(2),we can wume p 5 x). Remark. If h 2 2 X + I L I , we oan find suihble K , or hX+ILI = X and the conclusion holds by 1.10(6)aa a (A, A)-saturated model is X+-s&turated by 1.6(3). But we cannot always satisfy “ y - a < x+”.

Prmf. We use the notation of Definition 1.6. Note that aa 2ILl < K 5 A, M is Ko-saturated by 1.6(3). By 1.10(2), (3) it suEces to prove ]TP.8;(M)l 4 2X+ILI. Suppose not; then there me sequences it from M of length x for i < (2X+lLI)+such that for i # j TPSfi(i,,M)# TP.8’(iijSM ) . As ( 2 X + l L I ) + < K , and TPS”(i,,M) ha9 cmdinality s 2x+IL1< K , for some C < K , i,is from N , and TPS’(iif,M )= TP.8’(iii,Nc). W.1.o.g. N, = min{NB: Nj 2 A}; so we can assume of Nu 5 p (otherwise N: = N, hence h = N,, hence M is p+-saturated, so our conclusion follows from 1.10(6).),and ( 2 X + l L I ) + < Nu. Let D be a regular ultrafilter over p(1) = p + ILI (exists by VI, 1.3), which belongs to So;and its power is 2’+ILI s 2x+lLI s K s h so it is ~ 8 , . As Sc+, C “ N c haa cardinality A”, in SC+,there is an elementary embedding of Nf(l)/D(asinterpreted in 9,+ l ) into Nc+ = PC + 1, which is an inverse to the natural embedding of Nc into N f ( l ) / D . Now define TPS’(S, B, M )as we defined TPSN(i,M),but restricting ourselves to 6 E B, end let TPS;(C, B, N )= {TPSM(’a,B, N ) : ii a

,

CH. VII, 0 11

SKOLEY BUNOTIONS

409

sequence of length x of elements from C). Let us prove by indudion on PI IfC, BE^,, C, B E N, then there is B ' E ~ , B , c B' E (*I N,, 123'1 < IBI+ + N, and TPB;(C, B , N , ) haa adinality s 2X+ILI.Moreover there is a formula B E L(93),with parameters from 93,+1 such that for suitable B, C, 93, ( x q ( 2 ,B, c),and 9, c e[E,B, q.

Proof of (*). For the definition of B use the construction below and a choice function (in 93,) for the family of sets of hereditary cardinality I llNll (we essumed N is in it). Case I: I BI < K,. B will be the image of Bfi(l)/D(ultrapower computed in 93), by an elementary embedding f of N$(l)/D(ultrapower in 93,) into N , where B u C c IN*] N , < N , N , E b,, N , is KO-saturated, '29, C " llN*II = A"; where f is an inverse to the natural embedding of N , into N$l)/D. As IBI < K,, all functions from p( 1) into B with range of cardinality s p are included, aa IBI" < N,, 2fi(l)s A. As D waa regular, clearly any m-type over A E C, IAI c p, is realized in B' iff it waa finitely satisfiable in B. It is eaay to check that B' satisfies our demands. CaseII: 1B1 2 K,, IBIisregular.SoIBI > (2x+ILI)+.In93,thereisa well ordering of B, so let B = {bi: i < IBI}, Bj = {bf: i < j},and let Bt)'; and B' = Ul (2X+ ILI) +, )B,u Uf
DEFINITION 1.8: H is a bi-set function if H({ul:i < A}) is a family of pairs of subsets of {al:i < A}, and H({ui:i < A}) = {(Al, A2):for some ( w l , wa) E H(A),A , = {af: i E q}}. We can use index sets other than A. A set function H is used also aa the bi-set function 8,1?({af: i < A}) = { ( A ,0): A E H({a,:i < A})}.

410

CONSTRUCTION OF MODELS

[CH. VII,

51

DEFINITION1.9: M is (A, H,, Ha)-comp&, provided that for m y set F = {vf(z,4): i < A} of m-formulas over M , if for every (F,,Fa) E H,(F) some sequence in M realizes (I', Fa) (i.e., realizes F,,and fails to satisfy any formula in Fa),then some sequence from M realizes some pair from H2(I'). DEFINITION 1.10: For an ordered set I: (A) DC(I) = {Ia:a E I), 1, = ({b E I : b < a}, {b E I : b > a}). (B) The pair (A, B) is a (A, p)-Dedekind cut of I if A < B (h., a E A , b E B =- a < b) and A v B = I, A is the cofinality of A, p is the lower cohality of B (i.o., the cohality of B with inverse order). (C) If W is a family of pairs of regular cardinals DC,(I) = {(A, B): ( A , B ) is a (A, p)-Dedekind cut of I , (A, p ) E W), DC*,(I) = ( ( A ,B ) : A , B E I , A < B, and for no c E I A < c < B and (cf(A, <), cf(B, >)) E W } . DIPINITION 1.11 : (1) A (A, +family is a family of sets of cardinality A such that the intersection of any two distinct members has cardinality < K. (2) AD(x, A, p, K) holds if there is a (p, K)-family of x subsets of A; where x > A 2 p 2 K.

Remark. Baumgartner proved the consistency of AD(2n0, N,, 2*0 > N,.

K,, No) -t

EXERCISE 1.1: (1) If AD(x, X,p, K) holds, x' 5 x, A' 2 A, p' 5 p, K' 2 K , and x' > A' 2 p' 2 K' then AD(x', A', p', K') holds. (2) If AD(x, A, p, K ) holds then A" 2 x; if x = 2Athen A" = 2A. (3) Prove that A D ( A + , A, p, p) holds when of A = p. (4) Use (3) to show 1.8 cannot be improved in this case. CONJECTURE 1.2: Claims 1.8 and 1.9 cannot be improved (at least in ZFC). EXERCISE 1.3: Find the order between the set functions S,,, S:, S', 8: (dso Werent p's). QUESTION 1.4: Investigate the logical connection between the various notions of saturativity (implications, and independence). QUESTION 1.5: Investigate the various kinds of compactnessfor dense linear order.

OH. VII, 8 21 GENERALIZED EHRBNFEUCFI“-MOSTOWSKI MODELS

411

z,,,

QUESTION 1.6: Let I = I,. Investigate the connection between the compactness (and saturativity) (with various notions) of I ; J, I,. EXERCISE 1.7: For every order J, IJI IIl = K such that DC((A,fl)l(I) = 0 if A, p

I; K

there is an order I 2 J,

+ KO,A # p.

QUESTION 1.8: If we use the construction of Definition 1.6 but without assuming that N is saturated, does the big model still bequeath its properties (such as (p,Sh,S,*)-compactness,-saturation, etc.) to its submodela. EXERCISE 1.9: Show that in 1.6(3), 1.7(1) and 1.7(6) we can replace (6 X-homogeneous” by “strongly X-homogeneous”, if K s x or N is strongly X-saturated. VII.2. Generalized Ehrenfeucht-Mostowski models Here we shall deal with the pseudo-elementary class PC(T,, T) where

DEFINITION 2.1: (1) PC(T,, T ) is the class of L-reducts of models of T,, of cardinality 2 IT,!, L = L(T),L, = L(T,), T is aa usual complete, T G T,, T , has infinite models. (2) T , is a conservative extension of T , if any L,-model of T 1 of cardinality 2 IT , I can be expanded to a model of Ta. Clearly if T , is a conservative extension of T , and I Tal = IT , I, then PC(T2, T ) = PC(T,, T ) ; hence we can replace T , by T , without loss of generality. Also if we want to prove the existence of some models in PC( T,, T )we can replace T , by any extension of the same cardinality. Hence if T , has an extension of power IT , I which hae some additional properties we can assume T , has them. We list below some such assumptions on T , . Note that if T i+ is a conservative extension of T,, T , = U,<, T,, ITiI = IT,I, then T , is a conservative extension of To. Assumption I: T , has Skolem functions. (SeeTheorem 1.1(2).) A s s u m p t h 11: T , is complete. Assumption 111: There is a countable LC, E L, such that for every formula ~ ( 3and ) term 7(2) in L,, there are a formula R ( 3 , y ) and

OONSTRUOTION OF MODEL8

412

[CH.

VII, 8 2

function symbol P(Z,z)&om LE and individual constants c,, c1 in L, such that c1)3 E T1, (1) (W[Cp(Z) = (2) (W)[.(Z) = P(Z, C')] E Ti. For this it suffices to prove:

m,

CLAIM 2.1: 80me coneervative ezteneiOn T , Of TI, Aem?n#on

m.

=

8&b$eS

Proof.It suffices to define by induction on n theories T",and countable languages L*s L,,= L(Tn)such that To = T,, T"+lis a conservative extension of T",IT"I = lLnl = lTll and for every p(Z), ~ ( 2in) L, there are R(Z,y), P(Z, y) in L,*+, and individual constants cl, c1 in (Then such that (1) and (2) in assumption I11 holds for Tn+l. T, = P , L; = U b*satisfies the claim.) If T" is defined let &*+ consist of (m + l)-place predicates RE(Z,y) and function symbols P:@, y) for m < w , and

,

Tn+' = T" U {(W)[q(Z) e ( Z , c:)]: (p(Z) EL,, Z(Z)

u {(vZ)[T(Z) =

= m}

4)]: T ( Z ) E L,, d(5) = m}.

Clea,rly d the demands a m satisfied, so we have proved the olaim.

I denotes an index-model, i.e., a model whose elements serve as indexes; usually I is just an order. DEFINITION 2.2: If J c I , 8 E I , then the atomic type of 3 over J (in I ) is atp(Z,J, I ) = {tp(Z; i): Z(Z) = Z@), Z = ( ~ 0 , . .), i E J , I C ~ [ 3 Z]; and cp(Z;jj) is an atomic, or negation of atomic, formula in L ( I ) } We omit I when its identity is clear. Clearly atp(3,J, I ) = tpd(3,J , I ) where A = the set of quantifier free formulas.

.

-

DEFINITION 2.3: If 8, t E I , J c I , 3 t(mod J) (in I) if atp(3,J) = atp(t, J ) . In both definitions, if J is empty we omit it. EXERCIBE 2.1: Check the meanings for I an order.

..

N o W h . If {&: 8 E I}is an indexed set, 3 E I, 3 = (8(O), ~ ( l ) ,.) then = ti#(o)-6r(l)n....

sj

CH.

5 21

QENERALIZED EHRENFEUCHT-MOSTOWSIU MODELS

413

DEFINITION 2.4: (1) The indexed set {&: 8 E I) (& E 6 ) is called A-nindiscernible over A : (A) If 8 t then I(&) = l(6J. (B) If 3 t, Z(8) = l(t) = n, ~ ( 3g);~ d a ,E A then Q C &; a] =

Q[k a].

--

(2) If 8 # t implies 6‘ # & we d our indexed set non-trivial (we use (2) rarely-mainly in estimations of cardinalities). (3) We adopt the same conventions for shortening aa in Definitions I, 2.3 and I, 2.4. (Note that this definition and Dehitions I, 2.3 and I, 2.4 am Consistent.)

-

Assumption IV: When I is an index-model we w u m e for every 3 E I there is a quantifier-free formula d(Z)in L(I) such that for 8 E I, B 8 iff I b 0[8]. DEFINITION 2.5: K ( I )is the c l w of L(I)-index models I, such that for any I E I, there is 8 E I for which atp(3,0, I , ) = atp(t, 0, I).So for every I there is I, E K ( I ) , 1111 s IL(I)I, such that &I,) = K ( I ) (withoutassumption IV, we would have only /Ills 21uOl). The c l w of orders is Kord= K ( ( w , c)) = K(w).

{a,

LEMMA 2.2 : If I , E K ( I ) , :s ~ l is) an indiscernible indexed set in a ntodel M of T , , then there is an indhcernible indexed 8et {a8:8 E I,} in a d M‘ of T,, ~h tirat if I E I , 8 E I,, atp(8,0, I) = atp(t, 0, I , ) t h tp(&, 0, bl) = tp(Zj, 0, MI). A8 T , h Bk&m f u n d h a we mn a88um lM’1 = c1{a8: B E I,} and then M’ is uniquely determined (wp to ieow p h h pmerving the 4). Proof. Immediate, by the compactness theorem and the dehition of

W). DEFINITION 2.6: If {68:8 E I)is non-trivial and indiscernible in M ,then the # of {g8: 8 E I)which we define as {&): M k &I} and the indexmodel I, E K ( I ) determine M’so we denote M’ by N1 = EM1(Il, #), and its reduct to L by N = EM(I,, a).Such # is called proper for (I,T,).We call {a8:8 ~ 1 , the ) skeleton of N1 (and N) (of course, really we cannot reconstruct the skeleton from the model, but we behave a8 if we can).

LEMMA 2.3: T h e is #proper for (w, T , ) .

[=.

OONSTRUOTION OF MODEL8

414

82

Pruof. It suffices to prove in some model of T,there is a non-trivial indiscernible sequence {a,:i < w}. Let M, be a model of T,,& distinct sequenoee &om M,.It suffioee to prove

I' = Ti u {v(%)

~ ( ev(E) ) :E L,, I

U {zs# Z t : 8 # t

N

8; I , f E W }

<w}

is consistent, for this it suffices to prove the consistency of a finite subset I" c T1 u {vl(Zal)) ql(Zi(1)): Z < n} u {iE, # 3:I # t < w } where I(Z) f(Z). Let A = {pl:Z < n}, m = maxk<,,Z(b(k)), a d let {& k < w } be a A- S m-indiscernible sequence (exists by I, 2.4 and I, 2.3(3)) and interpret Ek by 6Nk); 80 in M,,r ' is satisfied.

-

Asmmpth V: If T is unstable, for some formula 3 < g in L and some 6f,i < w , (I(!$ = I ( @ = Z(q)),6"< 6k iff n < k. Then (Z < ~ ) E L PIf.

T has the f.0.p. then T n L; has the f.0.p.

LEMMA 2.4: If T W unstable, thre W 9 proper for ordered by < ,i.e., (3,,< ?Em) E Q iff n < m.

(0,

T,) whkh W

Pruof. AS of 2.3, only to I'we tldd {(Z,,< Em)u(n<m):7~ < m < w}, and use the &,'a from assumption V. Similarly we can prove

LEMMA 2.6: If 9, i8 proper for ( w , T , ) and T , is complete, T, 2 T , , t h there W 91 2 whkh is proper for (w, T,). For Q, proper for (w, T,), we can investigate properties of EN(1, Q,) (IE R(w)),but for casdinals s IT,I we cannot get much. Hence we first introduce a generalization.

DE-ON 2.7: If 9 is proper for (I,T , ) let N1 = EM1(I,@), N' be the submodel of N1 with universe the set of interpretations of terms (with no free variables). (AsT , has Skolem functions, N' is an elementrtrg submodel of N1.) Let N* be N' expanded by adding the relations:

R ~ ,=~{a ,E ~N': ~1 c 'p[~(q, a)], when I E I, I c em} for every atomic formula 'p(Z) E Lf, term c(3, 9) E LE(l(Z) = l ( ~ )and ) quantifier free formula e(3)E L ( I )satisfying assumption I V (we cthoose = L(N,), one e(Z) for each atp(B, 0, I),B E I).Let T* = Th(N,), = Lz u {RB,o,s: 8, 'p, c as mentioned above}, so IQI s No + 1L(I)l and T* satisfies assumption 111when L(I)is countable.

OH. WI, f

21

GENERALIZED EHRENFEUCE’FMOSTOWSKI MODELS

415

We behave as if T, determines 9, and say, T,, or a model of T , has a property, if 9 has it.

LEMMA 2.6: 8wppoee N i s a maEel of T, (heme an elementary extenaim of N,) where 9 is proper for (Io,T,), and I E K(Io). Then there is a unique L,-modeZ W , = cl(m U { a # : t ? ~ I ) ) m h that (1)ForeveyR,,,,;~L, - L , , ~ ~ ~ E E I N I , I E I

w1

(*I

when I C ep],

N C R,,q,t[E]o M 1C p[z(h, a)].

We denote this d by EM1(I,I?), and it8 L-red& by M = E h f ( I ,N ) . We call {a8:s E I } the skeleton and N the ba& of M 1(and of M ) . ( 2 ) M’ is a &el of T,, and an elementary extension of the L,-reduct of N . The indaed set {Z8:8 E I } is indi8cernible over IN1 in AP. ( 3 ) For every E E IN1 I there are I E II I, 6 E IN I and z E Lf(we k d e r them aa determined by E ) m h that i5 = T(%, 6).Also (*) holds f o r any T , p E L, f o r suitable f m d m (wring mitable individual mtant8). The elementay type of Z is determined by atp(I, 0, I ) , tp(6,0, N). Also = IlNll + 11I11* (4) If in I only finitely many atomic types of sequences of length m are realized f o r each m: then f o r every fornula V(T(%, g)) ( p , z E L,, I varied over I , $over i N ) there is a Boolean cornbination #(S, g) of formdm O,(a) E L(I), e,(g) E L, m h that M 1 C p(z(q,E ) ) ifs #(& a). [Being formalistic # is not a well defined formula, but its meaning is clear.]

lP1ll

Proof. There is no problem in the proof. Remark. If M , satisfies (1) it automatically satisfies (2), (3)-and (4).

DEFINITION 2.8: I is A-atomically-stable if J E I , IJI 5 A implies 18T(J, I ) ! 5 A where A = &(L(I)) is the set of quantifier free formulas of L(I).(Notethat if IlIll 5 A, I is h-atomically stable.)

LEMMA 2.7: ( 1 ) If N is proper f o r ( I , T,), N stable in A, and I hatomically-stable then M 1 = EM1(I,N ) is stable in A. ( 2 ) If N is proper f o r ( 0 ,T,), IlNll 5 A and I is a well ordering then EM1(I,N ) is stable in A.

[CH. VII,

tYONSTRUCTION OF MODELS

416

02

Proof. (1) Let A E IWl, IAI 5 A, then there me J E I , IJI 5 A and B E N, IBI 5 A, such that for every a E A there are % ( a ) E J ~ , , E B and 7, E Lt such that a = 7,(7ii(,), 6,). Noticing that if at = ~(79,,,,h1) %(l)E I , 6I E IN1 for Z = 1, 2, and %(1) S(2) modJ, tp(6,, ByN ) = tp(EayByN),then tp(@, A, iK1) = tp(aa, A, Ml); the conclusion is clew. (2) Immediate from (1) and part ( 5 ) of the following claim.

-

-

CLAIM 2.8: Let I E K(w),J s I . (1)

81

and t >

81 mod J

81 0t

..

>

-

81.

if for any t E J , t < 81 o t < 81, t

...

-

=

81 o

t = 81

..

(2) ( 8 0 , .,8,) (to, ,tm) modJ ifst ti mod J for i = 0, .,m andat < 8 , 0 t t < t, for 0 5 i , j 5 m. (3) The number of atom&?typea of 8eqwnces of length m from I i8$nite (even <(2m)!). (4) I i 8 h-&?&dy 8tabk if 1&(J,I)I 5 IJI f or J E I , IJI 5 A, where d = &(L(I)). ( 6 ) If I k well ordered,I k A4omictally 8tabk for euch A 2 No. ( 6 ) If A 2 2b, I k A-Ut~WnMy-8tabk iffI k A-etabk. (7) If I = In; J, I , are A - a t ~ W l y - s t a b kOrders t h I i8 A-

zae,

atoncieauy ekrbk.

Proof. (l),( 2 )and (3) are immediate. (6) will not be used hence we leave it as an exeroiae fo the reader. (Hint: use the Feferman-Vaught theorem, see, e.g., [CK731.) (4) Suppose I , J are a counterexample. So IJI = A, J E I , p i = atp(%{, J , I ) l(SJ = myfor

i < A+, pi # p , for i # j.As we can replace {p,:i < A+} by any subfamily of same ccudinelity we can aasume w.1.o.g. ( A ) atp(Siy0 , I ) is constant. (B)Letting Si = (a!, ..,Y-l>, then for each k, 0 Ik < m, atp(6, J, I ) dependa only on k (for this use m times the aasumption I(atp(8, J, I ) : 8 E r)( 5 A). So we get a contradiction by (2). (6) Using (4) we prove lRj(J, I)I 5 A when I JI 5 A. Define a function h on I : h(8) = min{t: t E J , 8 < t } and if for no t E J t > 8, let h(8) = a0 (h is well defined aa I is well ordered). So the range of h has cardinality I JI + 1, and if 8, t 4 J, h(8) h(t) then by (1) atp(8, J, I ) = atp(t, J, I ) . Hence we finish. (7) by (4) it is immediate.

.

3 :

CIE.

m, 5 21

O E N E U E D E~ENFEUCET-MOSTOWSKI MOD-

417

CONCLUSION 2.9: (1) If A 2 ITl] then there ie M E P C ( T ~T,) of cardinality A, which W 8tubk in evey p 2 IT1];80 if T W unatabk in lT1l, Af W not IT,I+-univereal. (2) I f p 2 2”, h 2 lTll then there is Af E PC(T1,T ) of cardinality p, whhh ie A + - ~ n i ~ s ax-etabk l, for x 2 2”, h when T W tlnekrble Af is ?lot (2”) -universal. +

Remark. In ( 2 ) if 2” = A+ the mult on universality is sharp, but in general not. If pA = p we can assume M is A+-satmted x-stable for x = XA (using ultrapowers). Sharp results, using finer methods, appear in Section 4.

Pvoof. (1) Let N be proper for (w, T l ) ,and be of mrdinality ITII.Then EM@, N ) is of oardinality A by 2.6(3); it is stable in p 2 lTll = llNll by 2.7(2). If T is unstable in IT , I, there is a model M of T which is unstable in IT lI, of mrdinality ITl I , hence Af cannot be elementarily embedded in EM@, N ) . (2) Use a A+ -saturated model N of cardinality 2” whiah is proper for ( w , TI). Take EM(p,N) as the required model and use Theorem 11, +

2.13(2).

LEMMA 2.10: If M 1= E M 1 ( I ,N ) ,N p o p ~ f ~ T( Il ), ; t hITPX;(Afl)l s ITPX;(N)I + lATPXf(I)lwhere X = 8,C and where A T P metam3 that in &$nition 1.6 we replace t p by atp.

Proof. Similar to that of 2.7. LEMMA 2.11: S-e I is a h e and h+-saturated or& and N W h+-cumpacta n d pfor ~ (0, T l ) .Then Afl = EAfl(I,N)W (A,SN,, S1)compact.

Proof. Let p be an m-type over M 1 of cardinality A, and every q E SNl(p) is realized. W.1.o.g. h 2 lLll. (*)

There is F = ~ ( gZ), E Lt such that any q ~ & ( p ) is realized by some F(&, 6), 3 E I , 6 E N .

Otherwise for each F EL: there is a counterexample qo so q = Usqr is a countable subtype of p , and is not realized, contradiction. So for every q E SNl(p) there are S(q) E I , 6(q) E N such that T ( % ( ~ , ,b(q)) realizes q. Let D be an ultrafilter over Sh(p), such that { q ~ S ~ ( p ) : ~ E ~ } for E any D v ~ p Let . J EI, IJI s A, B E INI, IBI s h be

418

[OH. WI,

00NS"RUUTION OF MODELS

82

such that p is a type over dcl(B u {a8:8 E J}).We can find, by the A+-sa.turationof I and N , f E I and E E IN1 such that: (1) For every 'p(iZ, E), 5 E B, 'p E L ( N ) ,

N C v[E, 51 * {q ~ S b ( p )N: b &q),

51) E D-

(2) For every # E L ( I ) , 3 E J ,

I

#[t, 31 0{q E SNo(P):I

f= #P(q),311 E

D.

It is easy to check ~(s, E) realizes p. QUESTION 2.2: Investigate the connection between (A, H,, Ha)compactness (or saturativity) of N , I and of E M ( I , N ) ; in particular when the H a am S,,,S:, Su and S$. QUESTION 2.3: (A) Investigate the connection between non-(A, H,, Ha)-compactnees (or saturativity) of N , I and of E M ( I , N ) when N C T,, @ is ordered (see Lemma 2.4). (B) Show, for unstable T , the existence of models which satisfy simultaneously various compactness and non-compactness conditions (also for mturativity; we mean of course for the various (A, H,, Ha)compactness notions). EXERCISE 2.4: Suppose A > p > lTll am regular, bl = E M ( I , @) I a dense order, @ proper for ( w , T , ) .Show N is ( A + p*, DC, DCo,,,)compact, when I has no (i) (A, p) or (p, A) Dedekind cut, (ii) (1, A) or (A, 1) or (0, A) or (Ay 0 ) and (1, p ) or (py1) or (0, p) or (p, 0 ) Dedekind cut. EXERCISE 2.5: Show that if T1 i < 8 is an increasing sequence of 4 5 @, for i < j < 8, then complete theories, @: proper for (I,TI) U l < d @: is proper for ( I , U l < d T1). DEBTNITION2.9: I hw the extension property i f When T,is complete, Q1 proper for ( I , T , ) , Ta 2 T I , then there is @a z proper for ( I , TI ) .

EXERCISE 2.6: Show that if is proper for (I,T,), I has the extension property, then there are T , 2 T,, ITaI = IT,Iyd& z a, proper for I,T,,such thrtt for any J E K ( I ) EM(J, , 0,) is KO-homogeneous.

OH.

VII, 8 31

ON THE

f.c.p., AND UNIFORM

419

TF~EES

.

EXERCI8E 2.7: Suppose T 1 = Th(M,), M , = (w, < ,R,,. .), T,has Skolem functions, and that we get @, proper for ( w , T,)&B in 2.3 for b, = n. Then for any I E K(w),N = E M ( I , @), 8 E I I{t €1:t <

8}1

5

l{b E I N ] :N b b < aa}l

5

I{t E I : t <

8}1

+ min {2No,

1T11}.

Moreover for each K we can find such a T,,lTll = K , such that the second inequality becomes an equality. (Hint: See the proof of 3.1.)

EXERCI8E 2.8: Let w denote here a finite set, and say M is (A, H,, H,)*-compctCt if for any indexed set {y,,,:w c A} of formulas over jK if for every 8 E H,(A), ps = {yW:w E 8)is realized in M then for some 8 E Ha(A),ps is realized. (1) If H,(A) is closed under countable unions, N is A+-compact,I is (A, If1,El,)*-compact (or vice versa) and N is proper for (0, T,), then E M ( I , N ) is (A, a,, Ha)*-compact. (2) Generalize from I E K ( o )to I E K(Io). (3) Generalize to saturativity and Definition 1.8. (4) Check the implications among these notions.

VII.8. On the f.o.p., uniform trees, and ID(T)I > IT1 = KO We continue the conventions of Section 2, with various additional assumptions on T. THEOREM 3.1: 8wpp8e T 7uza the f.c.p. (finite cover property). If @, i 8 proper for (w, T,)then there are T , 2 T,,lTal = ITl] and @a and y(z; g) E L,O(g; Z) E La m h t7&: (1) @2 2 0, is proper for (w, Ta). (2) If N 1 = EM1(I,@a), 8 E I , then O(N1;4) 7uza cardinality

I{t E I : t <

8}1

+ min{lTll, 2&}.

(3) {rp(z, a): a E N 1 , N 1 C O[E, an]}is not realized in N 1 but any proper of it is realized in N 1 .

&set

Proof. Let M' infinite set W

= EM1(w, 0,).By assumption for some y(z, 8) E L and (i< n) such that E w for any n E W there are

M' C A

fen

(W[A d z , 5,,.,)] j-8

jrrt

A

-(W[

A t < nd z , 5,J] -

420

O O N S T B U ~ O NOB MODELS

[OH.

VII, 9 3

v,

We expand M1 by: P = {ti,,: n E sequence of functions F, P(7i,,~i0= 6n,1andQ = {6n,,:i < n , n W}andR ~ = {6n,1n~n:i < n, n E W),and by adding Skolem and other funations so that we get a model M" such that T a = T h ( M a )satisfies all the assumptions from Seation 2. Now for any finite A E La = L ( T 7 , by Ramsey'e theorem (eee I, 2.4) we can find an infinite J A c {Zi,: n E which is A-indiscernible. So we culn find a maximal consistent set of formulas 0, in La in the variables itn, n < o, such that for any formula $(Zo, .$4) E La if for some finite A, E La for all finite A, A, E A c La, and all Z,,(o,, .. ., E Jb(n(0) < n(1) < .), Ma C $[aM0), . ..]then $(%, . . .) E 0,. So clearly 0, 2 0,, and is proper for (o,T,). Let B(E; g ) = R(g;E), then (3) is easy to cheok so only ( 2 ) remains. suppose 8 ie a counterexample. Using P we see 1 e(lN11 ;a,)! r I{F(q,q):t < 811 = I{t: t < 811. By d e w g Ma properly we am ensure that 1 B(lN1I ;G,)1 r m i 41 T 1I,%). So we prove one inequality. Now suppose I B(N1;4)1 > A,, = I{t : t < 811 + rnin{lT I1, 2Ho). So there are distinct ri(b(,))E B(IN1l;4) for i < &+. As we can replace them by any subset of A,,+, we can assume all the J ( i ) are similar over {t: t < 8). &ality Also, if A, r ITII we can assume 7, = r for any i. If A, < ITII n e d y A,, r %. Define an equivalence relation among the 7,: 7, r j if for any f E W, rl(&) = ~ ~ ( i or 3 ~rl(i&), ) , r j ( B f ) Q in the model Ma. Clearly the number of equivalence classes is s 2n0, so we can assume all the riys am equivalent. As q(%(,,! E Q(N1). q(7i8(,,)= r0(8,,,); we can assume a ( i ) = tn(8)"€(i), f[l] c 8, t ( i ) [ j ]> 8. Now if R@, rO(Z,8, Zl)] + [rO(it,g, zl) = r0(itYg , ~ , ) ]E A and E, m, By i E {i: a, E JA), (I(@= I(€), z(E) = ~ ( i=)Z(t(i)),and ~ [ i<]m < E l i ] , iu]then necemady

v

..

.

N

(otherwise n 1 I{Z: R(amya))] r l{rO(a;,am,aE):suitable B}I r No). So all the rO(ZZHi)) are equal, contradiction.

THEOREM 3.2: 8 u ~ Te is stable, M1 = EM1(I,N ) , N proper for TI). ( 1 ) Let v(E,8) E L, E, T E El. If tl, f a , J1, ga are i n ~ r e ~ nsequences g from I , = l ( 6 )= m,,l(s,) = l ( 4 )=.m2 and g1[i]= g 2 [ j ] o t ; [ i= ] ,, f a [ j ] ccnd , 6, E N then M1 t T[T(%,,a), 6(qa,611 = v[.r(aZl,a), d ( ~ i ~6)]

...,&.)

(Remark: I7cstead of if,,aa, we could have taken if1,

CH. VII,

8 31

ON TILE f.C.p., AND UNIFORM TREES

42 1

(2) #uppose A > 1 N 1 is regular, and every interval of I hm cardinality # A (including the intervds with an endpoint &a). Then M , the L-reduct of M', is ( A , # ~ , # ' ) - c r n n ~ ; and moreover (A,#A,#l)-mpact. (The aame h0ld.a for saturation.) (3) In (2), instead of A regular, it aumes to msume I hQs ru) interval of cardinality cf A; or for some A, < A, any interval of cardinality 2 A, has

cardinality > A. We can replace " A > 11NII

"

by " N h+-compact".

Prmf. (1) Here we can w u m e I is dense. Suppose we have a counterexample. Then, by assumptions we can find go, . . ., Ik such that: (i) So = and glniik 81-82. (ii) a2[i]= 8,[j]-Qi] = <[j] (0 < 1 < k, 0 < i < m,, 0 < j < m2). (iii) For each I there am i(Z),j(I) such that for n # i(1)S"[n]= g'+'[n] and gl[j(Z)] < a'[i(l)]if€ not gl[j(l)]< i?+'[i(I)], but for j # j ( l ) ,S,[j] < d[i(I)]iff gl[j] < #+'[i(l)]. BY (i)

-

M' c 'p[t(%,, a),

6(ha, 6)]0M'

c 'p[t(%,, a), 6(Q, i)],

M' c 'p[t(a~,, a), 6@&, 6)]0M' c 'p[t(%,, a), 6(%&, 6)]. But as we are dealing with a counter example

M' c 'p(C(%,, a), qqa, 6))0M' c 7'p(e(afl,a), 6(afp,6)). So for some 1 s k M' c 'p(t(%,, a), 6(%, 6)) =

l'p(?(%,,

a), 6(7ij+l,6)).

Let U, E I, U, < U,+', and U, is between #[i(l)]and $+l[i(Z)].Let vt E I for n < o be defined such that: they are of length l(gl), I(&J if j # j ( l ) , V:[j] = S,[j], if j # i(Z),v 3 j ] = 8 [ j ] and "[j(4,] = U,, vi[i(Z)]= U,. It is eaay to check that for n # k M 1I='p[d;, a;] = +a:, where a: = ~ ( a), 9 at " =~6(az,6). Hence for # = 'p or # = 7'p M 1 C #[a:, A d: # d; iff n < k, ao T has the order property contradiction by 11, 2.2 and 11, 2.13. (2) Suppose every q E &(r), 'I = {'pf(3,&): i < A} is realized in M1. So let ~ E W M I ,' + r p t [ ~ , b , ]for i <j. So let ci = ~ ~ ( a ~($EM, (,$) TtE LE). Of course, each ct is increasing. As A > llNll by renaming we can assume that for i < A cf = c, 8 = 8, (as we can replace Ef by a, when j > i). In the same way, we can assume that for each I, either all Zif[Z], i < A are equal, or for every i, j S,[l] = q z ] =$ i = j.

v:,

a:]

a:]

422

CONSTRUCTION OF MODELS

[CH.

WI, 5 3

- -

Let Z(0) < - < Z(k) be those for which the firat w e occurs, and again we can wume that for n, m # Z(O), . . . ,Z(k) i,[n] = 8,[m] iff i = j , n = m. So Z(n) + 1 < Z(n + 1) implies {e E I:eo[Z(n)]< s < 8o[Z(n + 11) hae cardinality T A hence > A (for - 1 s n s k + 1, where we stipulate I( - 1) = - 1, Z(k + 1) = the length of a,,). Hence we %[Z(n)]= i#(n)], and if n # Z(O), . . .,Z(k), can define i E I, @) = Z(io), then a[n] does not appear in ti, where b, = ah, 6,).Then by part (1) ?(ai,a) realizes p. (3) Proof similar to (2). CONCLUSION 3.3: 8 p e T i8 S a k , with the f.0.p. A 2 lZ'11, and 8 a set of c a r d i d 8 A E 8.T k n there is M E PC( T , , T)such that: , for s m v(z,a)E L , bl (1) If min{lT1l,2no} p 4 A, ~ € 8then omits a T-type p of cardinality p, m h that any proper &type of p b realized in M . (2) If m i n { * o , lT1t}s p < h , p $ 8 , p i e r @ a r o r p = min{*o, lTll}, then M ie ( A , 81)-compt.

Proof. When lTll < 2Mo this is immediate by 3.1 and 3.2, aa there is an order I, IIl = A such that for every p, lTll 4 p 4 A: p E 8 * for some s E I I{t E I: t < 8 ) ) = p, and p $8 ==I haa no interval of cardinality p. When lTll 2 2% we can use Exercise VI, 6.4, 6, Exercise 2.9 and take an elementary submodel, or 3.2(3), laet phrase. CONCLUSION 3.4: 8uppo8e T i s s a l e with the f.c.p. If Nu 2 N, + lT1l,t48 = min{lT1l, P o } , then there are in PC(T,, T) at Zeaat 2Ia-B1 n o n - i e ~ p h i models c of cardinality Nu.

Proof. If a - 8 2 w , this is immediate by 3.3. When a - 8 < w we look at &: for some finite A c L, n < w , there is a finite A-n-indiscernible set I c N such that dim(1, A, n, p ) = p}. For suitable T,this is not difficult, and we leave it to the reader. (Compare with VI, 3.9.)

DEFINITION 3.1: For an order J, I is called a (J,/3)-tree if I is a model with universe G JsB = (7:7 a sequence of elements of J of length sp}, such that 7 E 111 =- q a E 111;and with the relations Pa = (7E 111: Z(7) = a} for a 4 8, the lexicographic order < ,and order 4 of being an initial eegment, and the function h, h(7, v) = the longest common initial segment of 7, v. I is the full (J,/3)-tree when 111 = JsB. We denote I by JsB. Note that up to isomorphism the claas of (J,/3)-trees for a11 J's is K(w'8). If for some J, I is a (J,/3)-tree,then I is a 8-tree.

OH.

VII, f 31

ON THE

f.c.p.,

AND UNIFORM TREES

423

Remark. If I , f E I ; I ,. f iff Z(I[i])= l(&i])and for any i , j ; ai,j =def max{a: B[i] 1 a = B[j] 1 a} = max{a: t[i] 1 a = f [ j ] a}. (In another notation, l ( h ( g [ i ]~ , [ j ]=) Z(h(E[[i], ) E[j])) a n d ~ [ i < l g[j]e-E[[i] < E[j]).

DEFINITION 3.2: (A) T has a uniform ,%tree of the form ((cpa,ma): a < 8, a successor) (cpa E L, ma s w ) if there are a,, for q E A S S such that : (1) a,, realizes p,, = {cpa(f, anla): a < 8, a successor} for q E B A . (2) For any a, r ] E Aa,v E AB,7iv realizes <ma 1 of the formulas ~ ~ + Sin-
+

(B) The tree (and the form) are called strong if for any set w s A,

LEMMA 3.6: (1) Definition 3.2 holdejor s m e A iff for every A. ( 2 ) If 8 < K ( T then ) T h a uniform ,%tree, ma = 1 for all a. ( 3 ) If T is stable, /3 < K ( T)then T h a strong uniform 8-tree, ma = 1, for all a. (4) If T hua the strict order property then T h a strong uniform 8-tree, with ma = 1 for all a. ( 5 ) If D ( s = 2, L, GO) = GO then T h a strong uniform Ho-tree, ma = 1,for all a. (6) If K < KrOdt(T), K is regular, then T h a strong uniform K-tree, ma = 1.

Proof. Immediate, see 111, 3.1 for (3), 11, 3.9 for (6) and 111, 7.10 for (6). Clearly, (1) and (4) are immediate. For (2), note that if on J = As* x {0,1} we define an ordering < : ( 7 , i ) < < v , j > i f q v, i = 0 or q = v, i = 0, j = I, 01 p Q q , v where p = i ( q , v ) # q , v and q < v (in the lexicographic order). Then assuming T is unstable (as (3) has been proved) by the order property, there are cp E L, a, (8 E J) such that tcp[Z8,at] iff 8 < t. Then let

2

a

= ~<,o>-~<,*l>

and cp(Jln~,,

a,)

= [cp(fl, B<,,,OJ

= -lcp(fl,

~<W,I>)l.

THEOREM 3.6: (1) If T h a uniform @-treeof the form (cp,, ma): a < 8, a a 8ucce88or), then it h such a tree which i s indisernible ( w h

CONSTRUCTION or MODELS

424

[OH.

VII, 8 3

we ukw the 8-tree acr a n index set). In fact, JsB h a the extendm property (see Definition 2.9). (2) B y the assumption of ( I ) , there is a @ proper for 8-trees and T I , euch that V M = EM(hs4, @), then {a,,: q E hsB} is a uniiform 8-tree of the form ((v,, ma): a < 8: a memasor). (3) If we dart in ( 1 ) and ( 2 ) with a strong tree, we get one. (4) BY rep hi^ c p , ( f ; g ) by & ( f ; 81, gal = v a ( f ; a11 A i , ( f ; Ya), we can wmme in (2) t7& if M = E M ( I , @), q E I n ha, no seqin M satis+ infinitely many formulae rpa+ a,,-<,>),qh(i ) E I . That is, we get a strong uniform tree of the form ((cp,, No): a < 8, a a memasor).

Proof. ( 1 ) Suppose {Si,: q E hsB} is a uniform 8-tree; and choose h 2

sa+,, a, r IT(.By adding dummy variables to the qaYswe can assume

the a,'s ctre distinct. X t sufXces to prove the consistency of

u {7va(gn,g v ) : a v

=y

+ 1,l(v) = a,l(q) = 8,

r Y = q r Y YV ( Y ) Z r](Y)l

u {!I, #,!I" :r ) st l(!I,,) = l(!J")). v7

r.

So it suffices to prove the consistency of all finite I" s In I" the set of a which me l(q)where appeam in I" is finite. So by renaming we can aasume all such a's me <no. (We rename in a corresponding way the anye.)By the combinatorial theorem 2.6 of the Appendix we can fhd an aeeignment showing the consistency of I". (2), ( 3 ) Follow easily. For (4) notice that for the @ from 2, for no 6 E &I 80 S I{i: cpa(6,a,,-) 7d6; a,,-<~+~>)}l.

THEOREM 3.7: 8 w p p ~ eIT1 = KO and let v,,(P)E L ( T )for q E 2<@be euch thud (a) q 6 v =- w ' a v v ( f ) + v,(f)l, (8) W 3 ~ ) [ ( ~ n - < 0 >A( 5~ 3, , - < i > ( f ) I , (7) b(3x)vn(f). ThenTha;sanzodelMand7i,~Mforq~2",andafunctionh:2<"+ 2<" such that: ( 1 ) q 4 p impliu h(q) 6 h(p), and h(qh(0)), h(qn
OH.

m, 5 31

ON THE f.C.p., AND UNIPORlK TREES

425

-

(2) a,, realizes p: = pi(,,),p t = {‘p,+,(i~): n < 0). (3) Call i j , i E 2’ 8imiZar over myi j 6(mod m ) if (i) ij[ll m z ij[kI t .zfor k # 4 (ii) $1 m = $3 1 m for any 1. Thenfor p(0)E L there i 8 m, euch t7td if i j , i are 8imilar over m 2 m,,

r

then M C cp[a;] = cp@~]. (4)If T E L, 7(aq)realizes p: then v = q[Z] for some 1. Moreover there ie nz, such tW if i j +j(mod m), m > mi and v E 2m, v# m, ., then MI= -tpncV,[~(aq)]. (6) If are j h i t e set8 of formulas in L, E UneP, i8 the set of formdm of L,then in (3), if ‘ p ( j j ) E we can chooee m, = n.

go]r

r,

-

..

r,, r,,,,, r,,

r,

Proof. Choose a model M , of T , and Z,, E lMol , M , C cp-,[Z,,] for r) E 2. We use 2.4 ofthe Appendix for the reasonablefm’s (for each ‘p(Zl, . . ,Z,) there is m m such that f ( i j ) = 0 if€ f ( i j ) # 1 iff M C Now letting M = Mb/D, a,, = (. . .,E,,i~,,,nly . . ) / D , D be any non-principal ultrafilter over w, clearly (l),(2) and (3) are satisfied. Now it is trivial to get (6) by renaming, and (4) is quite easy.

.

‘p[s]).

.

EXERCI8E 3.1: We can assume in 3.7, that there are distinct b, E M such that (i) {bn: n < w} is indiscernible over U {a,,; r) E ‘2}, (ii) ( M b,, y b,, . .) satisfies (1)-(6) of 3.7. [Hint: Let N < M yb, E IMI,{b,: n c w} indiscernible over N. Apply 3.7 to T’= Th(M, P,boy . . .) (for P = N) and cp,,(z) such that CP[a,,] (more exactly, we have to repeat the proof).]

.

EXERCISE 3.2: Show that in 3.2(2), we can replace “llNll < A” by “ N is (A, SAY S1)-oompact.” EXERCIBE 3.3: Suppose #,, E L ( T ) , T countable, p an m-type over 0, N a, model of T , omitting p , and {{I,$,(??): N C #,[6]}: 6 E INI} has cardinality 2Mo and 2Ho is real valued measurable (see [So 711). Show that T has a model M omittingp, and there m e ‘p,, E L(q E @’2) which are Boolean coqbinations of the #,’a, and a,, E 1M1 aa in 3.7. [Hint: Let D be a normal measure over 2n0, and 5, E INI, (i < 2Wo) such that p , = {#,(if): N C1,4,,[5~]} are distinct; and w.1.o.g. T has Skolem functions. Now we define inductively Ela E D (a < wl) such that S6 = are S, c 8,,8, = 2n0, and: if i(O), . , .,i ( n - 1) distinct, and ‘p E L, N C p[b,(,,, . . ., 6,(,- ,)I, then for some jE Su - Su+

nf
.. .,&,-l)]. NOWwe shall define indu~tivelytp,,(. such that (i) for every a for some i(9) E Is, N C t p ~ .. ., & ,), .I; (ii) If v ( 7 ) E {f(O), ~"(1)) for 7 E 2', then

N C cp[g,, z,,,

[m.VII,

CONS!l'RUCTION OF MODEL3

426

...

&l),

84

. .,

..

tpn+i(

- .., - - . zD,

)pea-+l

I- tpJ.

- - ~ v ( n ) ,- . .,z",.. .) is an approxima9

)"ep;

(iii) When 7 4 v(r))E 2" for r) E P, tp,,(. ~n7iv(,,)n--- (i.e., we allow only tion to the desired properties of finitely many terms, and note here we should decide why a term does not realize p). For further help SIX [Sh 761.1

M . 4 . Semi-definabilify

DEFINITION 4.1: (1) If D is an ultrafilter over I (I E 161") then Av(D, A) = {tp(Z; a): si E A , (6 E I: ktp[6; a]}E 0) (Clearly it E Ism(A),it is consistent as D is closed under intersection). (2) An m-type p is semi-definable over I if there is an ultrafilter D over I such that p c Av(D, Dom p). (3) An m-type p is semi-definable over A if it is semi-definable over Am. (4) tp*(C, A) is semi-definable over B c A, if for any E E C, tp(E, A ) is semi-definable over B E A. (Thisnaturally extends to not-necessarily complete t y p e s with infinitely many variables.) ( 6 ) p E Ism(A) (B c A ) is stationary over B, if p is semi-definable over B, and it has no two contradictory extensions which are semidefinable over B. (Thisis not necessarily consistent with the previous definition of stationary Definition 111, 1.7, but our meaning will always be clear.)

Remark. Notice that if T is stable I an indiscernible set, D a nonprincipal ultrafilter over I then Av(D, A) = Av(1, A).

LEMMA 4.1: (1) A type p ( p s i b l y with injnitely many variables) ia semi-dejnable over I(A) iffeveryjnite subtype of p is realized in I(A). ( 2 ) Every type over a model M , i8 semi-dejnable over ]MI. ( 3 ) If p is a type over A , and is semi-dejnuble over B, then there is a complete type over A extending p which is semidejnable over B (the 8ame hold8 for I instead of B).

CH.

VII, 5 41

SEMI-DEFINABILITY

427

Remurk. Part ( l ) , implies, by 111, 4.10(1) that if T is stable, B = lUl, then p is semi-definable over B i E p does not fork over B.

Proof. ( 1) Let p be an m-type semi-definableover I. If p c Av(D, Dom p) for some ultraflter D, and q c p is finite, then {C

I: c realizes q}

=

{C

I: b Q [ ~ ;a]}E D

,(Z;;a)sq

(as D is closed under intersection). So {C E I: 5 realizes q} # 0 80 q is realized in I. Suppose every finite q c p is realized in I. For each Q EP let J, = {EEI: C satisfies Q}. By clssumption the intersection of any finitely many J, is non-empty, hence there is an ultrafilter D over I such that Q ~p J@E D, by 1.1(2) and (3) of the Appendix. Clearly Q ~p * Q E Av(D, Dom p). For types with infinitely many variables the proof is similar. (2) Every type over a model is finitely satisfiable in iK,hence the result follows from (1). (3) It suffices to note the following; and use (1): (i) If p,, i < 8, is an increasing sequence of types, which are finitely satisfiable in B, then Uledp , is finitely satisfiable in B. (The proof is immediate.) (ii) If p is finitely satisfiable in B, ~ ( 33) ; a formula, then p1 = p U {v(Z;a)}or p , = p U {-q(z;a)}is finitely satisfiable in B (or both are). Because otherwise there are finite q1 G pl, qa G pa which me not satisfiable in B. So q = (ql u qa) np 5 p and is finite, hence satisfied by some 6 E B. If b ~ [ 6 ;a] then 6 realizes ql, and otherwise it realizes q2, contradiction.

LEMMA 4.2: (1) If C is a subsequence of 6, tp(6, A ) is semi-&$nable over B t h n tp(C, A) is eemi-de$nuble over B. (2) If p k q (e.g., q c p ) and p is semi-dejnable over B, t h q is semidejinuble over B. ( 3 ) If B' c B, p is semi-&$nuble over B , t h p is semi-&$nuble over B.

Proof. Immediate. Note this lemma shows the consistency of the definitions of semi-definability.

428

OONSTRUCTlON OF MODELS

[OH.

VII, 6 4

LEMMA 4.3: (1) If p is semi-de$nable over Bythen it doecr not &it over B. (2) If p is a complete type over A , eemi-&$dle over By B c A , and every qESm(B),m < w , is realized by some F E A ,then p is stationary over B.

Proof.(1) Supposep c Av(D, A), and tp(6, B) = tp(E, B), 6, E E A . If v(Z;6) E p , then 8 = {8 E B: hpp;51) E D. But for any 8 E B bp; 61 = g(G; Z], hence 8 = {8 E B: b[8, hence v(E;Z) E Av(D, A) hence +P; 5 ) $ p , so p does not split over B. (2) Suppose plypa are contradictory extensions of p which are semidefinable over B. By 4.1(3) we can m u m e they are complete types over some C 2 A, SO them is iz E C q ( E , 8 ) €ply -Q(Z, 8 ) Choose 8, E A, tp(8,, B) = tp(iz, B) and w.1.o.g. p(2;a,) E P . So p 2 splits over B, contradiction.

a},

DEFINITION 4.2: q = { ( p ,B ) :p E ~ ~ ( BA E ) ,A , IBI < A, p is semidefinable over dcl B}.

LEMMA 4.4: (1) satiejk9 the fobwhg a&ma (eee IVY8ectbn 1): Ax (I), (11, 11, (IIY2)Y (III,l), (111, 2), I V Y (VII), (VIII), (IX), (XI, 1) and (XI, 2).

A(Fi) = A; p(Fi) = a. (3) If T hm 8kolem functions, then also Ax(X.1) and (X.2) hold. Moreover if IDompI < A, p i s over B E A , IBI < A, t h there ie a complete type q over A &ending p mch tluct (q, B) E Fi.Also Ax(II.3) and (11.4)hold. (2)

Proof. (1) For Ax(II.1) notice that if 8 E B c A , l(8) = m; then D = (J c B"': ii E a) is an ultrafilter over B"' and Av(D, A) = tp(iz, A). Ax(1V)is Lemma 4.2(1). For Ax(VII),it suffices to prove the following: suppose dcl B = By dcl C = C, B c C; and B s A, tp(Z, A u C) is semi-definableover B u C,tp*(C, A) is semi-definableover Bya E A and v(Z; a) is realized by some Z E iz u C. We should prove q ( Z ; a) is realized by some i? E B. We can assume Z = 8-Zl, El E C,so Cq[Z, E l ; a], hence for some izl E C C @,,El; a]. As Z,,El EC, for some a;, Z; E B,

a] and we finish. The other axioms are trivial. (2) Immediate. (3) If T has Skolem functions, for any By cl B is the universe of a model. So by 4.1(2) every type p over cl B is semi-definable over cl B. So 4.1(3) implies our conclusion. The proof of Ax(II.3) and (11.4) is easy. C &, 5;;

CH.

VII, Q 41

429

SEMI-DIEFINABILITY

LEMMA 4.6: Let P = I N [ , and (N*, P ) be an elementary submodel of ( Q , P ) m h tlrat IN1 E IN*I. Thenfor any sets A , B there are elementary mappings f, g which are the klentity over IN!, IN*! r q . m h that if A' = f ( A ) , B' = g(B)then: (1) tp,(B', IN*I u A') is eemi&$nuble over IN*I, (2) tp*(A', IN*I u 8) i.9 smi-de$nuble over INI.

Prmf. By 4.1(2) and (3) we can extend tp*(A, INI) to a complete type over IN*I which is semi-definable over INI. Hence we can define an elementary mapping f, which is the identity over INI, such that where A' = f ( A ) .Now let tp*(A', IN*I) is semi-definable over IN],

I' = tp,(B, IN*[)u 0 u Y, where @ = {-,cp(Zs;

Y = {l$(Z6;

a, 5): 6 E B, E A', E E IN*I (Q E L) and Q@E; a, 2) is not realized in N*}, B, 5): 6 E B, a E A', E E IN*I (+ EL)and #(6,g, a) is not realized in N).

Clearly it suffices to prove I'is consistent (if x,, I+ b' is an assignment satisfying I', let g map c to c for c E IN*I and b to b' for b E B. Now g is elementary aa tp,(B, IN*I) E it satisfies (1) aa @ E m d (2) &B !PGr). Let r' be a finite subset of I'.As tp,(B, IN*[),0, Yare closed under conjunctions, we can mume, by adding dummy variables,

r

r' = {6(36,q, 7q(5,a,q, -@(s, a,q> where a E A', 6 E B, E E IN*I, CO[6, El, +(6,gy5 ) is not realized in N. So, as P = (Q,

"1,

a, a) is not realized in N*,

P)C (W[A P(#[iI)-+ +@,a, El]

e(6, el.

A

t

As (A'*, P)is an elementary submodel of (a, P)and 6' E IN*I

E

IN*I, there is

By the second conjunct, 6' satisfies O ( q , a); by the first one for every a', 131,but tp@, "*I) is semi-definable over IN1 hence a, So in order to by 4.1(1) finitely satisfiable in N,hence Cl+[6', realize r',6' has to satisfy l Q ( f 6 , a, a), but as 6' E IN*I this follows from W s definition.

a' E INI, C7+[6',

a.

430

[m. VII, 8 4

THEOREM 4.6: Por every p 2 2A1, A, 2 2IT1I, PC(T,, T) has a A,universal model M of cardinality p which is stable in A,. (Hence if T is A,-unstable, M is A,-universal but not A:-universal.)

Remark. The construction here w i l l serve additional theorems.

Proof.We shad describe here a construction for a cmdinal A > IT,], 2<"

s p such t h t for A = A t we get the desired model. Of course, we

can assume T,has Skolem functions. Let N be a model of T,of cardinality lTll, P = INI, and (N*, P) an elementary submodel of (El, P)of cardinality ITII. Let M1, M a be p-saturated models of T,extending N. By Lemma 4.5 (and 4.2) we can aasume tp*(lM1l, IM'I), tp*(1HaJ,lM1l) are semi-definable over IN*I, IN1 reap.; hence do not split over them (by 4.3(1)). By 4.3(2) tp,(Wl, PI)is stationary over N. Note that if a, 6~ t p ( q W*l) = tp(6, p*l)then t p ( a , w l ) = t p ( 6 , w l ) (as for any CEWIand tp(z,g),t=tp(a,c)EE ~ ( 6 , cas ) tp(q PI)does not split over p*l); hence the number of ~ E S " ( Wm ) ,< w , realized in 1M2 is < IUmSm (N*)l Q 2ITll. We can interchange the roles of JP and M2 (with N instead of N*). Let {M,: i < p} be a list of models of T,of cardinality < A, such that up to isomorphism, each model of T, of cardinality < A appears in it. As M a is A-saturated, we can assume all the M i are elementary submodels of Ma. We define by induction on i models Mf such that: (i) tp,(lM:l, lM1l)= tp,(lM,I, IMlI); more exactly there is an fi maps M , onto M?. elementary mapping f,,f,is the identity over M1, (ii) The type of iKf over M' u UjCi M; is =mi-definable over IN!. Let M be the Skolem closure of M:. Clearly M has cardinality s p (as 2 C As p) and M is x-universal for every x < A. Suppose x+ 2 A so llMfll s x for each i, and we shall show M is X-stable. Let A E 1x1,IAI s x, but @(A, M) has cardinality >x. W.1.o.g. A is the Skolem closure of {M:: i E w} where IwI s x. Suppose Zi E 1611,i < x+ realize Werent types over A, so by the definition of M a, = T,&, ..., 6,,J where n = n,, 6,,,E 1M&,,l. As we can replace {a,: i < x+}by any subset of the same cardinality we can assume n, = n, fi = T . Let 6,,,= f,c,,,,(E,,.l) E,,, E M,,,,, hence Z,,, E Ma. As the number of complete types sequencesfrom M a realize over M 1is s 2IT1I,we can assume, if x 2 21T11, that tp(E,,,, lM1l) = g,.As the order p is atomically stable in x (see 2.8(6)) we can aasume the atomic type of @(j, I), . . .,p(j, n ) ) over w is constant, and p(j, 1) E w * b,,, = b,. So it suffices to show that under

wl,

ufCg

CH.

VII, 8 41

43 1

SEMI-DEFINABXUTY

a11 those conditions all the B, realize the same type over A. For this it suffices to show: (*)

If i(1) < *

* *

< i(m) < p,j(l) < *

'*

< j ( m ) < p, 81 E 1M,(1)lY

Proof of (*). We prove it by induction on m. Form = 1, tp(si;, lM1l) = tp(7i1, lM1l) = tp(6,, lM1l)= t p ( g , lM'1). Suppose we have proven for m, and we shall prove for m + 1. By the symmetry we can w u m e i(m + 1) 5 j ( m + 1).Now tp(7ik+l, 161'1) = tp(pm+l,1M:l) is stationary and tP(Za+lY p ' l u {pq: j < i(m + 1))) tp(%+,, pfll u U {IM,l:j < j ( m + 1))) are semi-definable over INI,hence the former is a subtype of the latter, and the latter does not split over INI,so by the induction hypothesis we get (*) and finish the proof. (By checking that when h = A,+, A, 2 2ITil, we get the result easily.) Note that w.1.o.g. Mg $ N , hence llMll = p.

u

EXERCISE 4.1 :Let Tbe the theory ofthe rational order, IMI = (0, l ) , a = 1, b = 2. Show there is no a', such that: t p ( a ' , M u b ) extends tp(a,M) and is semi-definable over M and tp(b,M U a') is semidefinable over M. EXERCISE 4.2: Show that Fg cannot (in general) satisfy more axioms than stated in Lemma 4.4. (Hint: Ax(II.3, 4): Choose A = B = 0, aEac1A - dclA.) Ax(V): Let M = EM1(wly0 ) where we get the theory T1and @ as in Exercise 2.7. So M Ca < a,, implies: M Ca < a, for some i < 6, and a E acl{a,:j < 6) ({a,:j < w,} is the skeleton). Let B = acl{a,: i < w or i > w + a}, A = acl{a,:i s w or i > w + w } , 7i = (a,,,), 6= (a*+7)-

Ax(1V) : By the previous example, for A = B

6 = (a,), z

= C = acl{a,: i

<w}

= (a,,,). Ax(X.l), (X.2): Choose A = dcl A = 0 (Ao = dcl Ao),g, = 5 = 5. Ax(XI1):C1 = acl{a,: i < w } C, = acl{a,: w . 2 < i < w.3) B = C, U CayA = acl(B u {ad),p = tp(a@+,,A) (in the notation above).]

432

CONSTRUCTION OR’ MODELS

[aVII, . Q5

VII.5. Hanf numbers of omitting types K

need not be infinite

DEFINITION 6.1: (1) We define p(A, K ) as the first cardinal p such that: ’5 K ; if lLol 5 A, To a theory in Lo, r a set of ( < No)-typesin Lo, and for every x < p there is a model in EC(To,F)of cardinality 2 x, where EC(To,r)is the class of Lo-modelsof To omitting each p E r, then there are in EC(To,r)models of arbitrarily large cardinality. (2) We define &A, K ) as the &at ordinal a such that: if lLol 5 A, < €Lo, Toa theory in Lo, f a set of ( < N,)-types in Lo, If I < K and there is a model M in EC(To,r)such that (1611,< M ) has order type 2 a, then there is M E EC(To,r ) which is not well-ordered by

Remark. The difFerence between the two definitions (“for every x < p” but not “for every S < S(h, K ) ” ) is inessential, sea 6.2. LEMMA 6.1: (1) If in EC(To,r ) there i.3 a nzodel of cardinality A, IToI 5 p s A, t h n in EC(To,r)t h r e is a model of cardinality p. (2) If s Al, K s ~ 1thenp(h, , 4 s Ah, 4,a d W,4 5 W ,~ 1 ) ( 3 ) S(A, 1) 2 A+. (4) S(A, 1) = S(A, A). (6) &A, 0) = w.

Proof. (1) Because N < M , M e EC(To,r ) implies N E EC(To,r) using the downward Liiwenheim-Skolem theorem I, 1.4. (2) Immediate. (3) For a < A+, let M , be (a,<, . . .,i,. . .),<, T, = Th(M,), p = {z # i: i < a}. Then every model in EC(Ta; {pa})is well ordered and has order type a aa it is isomorphic to Ma. So S(A, 1) > a;and as this holds for every a < A+, 8(h, 1) 2 A+. (4) By (2) S(A, 1) 5 S(A, A). To prove the converse, assume ]Lo! s A, Toa theory in Lo, r a set of ( < #,)-types in Lo, I rl 5 A, M E EC(To,r ) has order type S(A, 1) and we shall prove that there is a non-wellordered model in EC(To,r). Let r = { p a :a < a, 5 A}, pa = {tpf(~,): i < i, 5 A}, pa an matype. Let $(a < ao,i < i,) be new individual constants, &, P, (a < ao)

.

CH.

VII, 5 61

HANF NUMBER8 OF OrdITTINQ TYPES

433

new one place predicates, and for a < %, Fa a new ma-place functionsymbol. Let L, = Lo U {Q} u {Pa,Fa: a < ao} u {q:i < i,, a < ao},

T I = To U {Q(cf),P,(cf): i < i,, a < ao} u {(VZ,)P,(F,(Z,)): a < ao} u {(VZa)[F,(Z,) = cf + -,Q~Z,)]: i < i,, a < ao}

"{(W(Pa(4

Q(4): a < a01 u {(wrp,(4+ 1p&41:a # B < ao}, p = {Q(z) A z # 4:i < i,, a < ao}. -+

Clearly every model in EC( To,r ) of cardinality 2 h can be expanded to a model of EC(T,,{p}),and the L,-reduct of every model of E C ( T , , b } ) belongs to E C ( T , , r ) . So we can expand bl to W E EC(T,, { p } ) ;and then aa T,, p we in L,, lLll s A, and bl' haa order type S(h, l), there is N' = W ,N'EEC(T,,{ p } ) which is not wellordered. The Lo-reduct of N',N,is the desired model. ( 5 ) If in (3) we take a < w, then every model in EC(T,, 0 ) is isomorphic to Ma,hence &(A, 0 ) > a; so S(h, 0 ) 2 w. By the compactness theorem if for every n < w To haa a model bl, of order tyoe 2 n, then To u {c,+, < c,: n < w} has a model, so To hae a non-well-ordered model.

LEMMA 6.2: 8 w p p s e P, < E Lo(P--a one plccce predicate, < a two place predicate) lLol 5 A, To a theory in Lo, r a set of s K ( < N,)-typee. If for every a < 6 = S(h, K ) there is in EC(To,r) a model bl, such t?@ (P*u, 4%) laas order-type l a t k n in EC(To,r)there i s a model bl in which cMdoecr not well-order PM.

Procf. For K = 0 this follows by the proof of 5.1(5), so let K > 0. Let P be the function such that F ( a ) = M a for a < &(A,K), w.1.o.g. the are pairwise disjoint and F , be the function such that for each a < & ( A , K ) F,(a,z) is an isomorphism from (a, < ) into (PMu, Let 93 be an elementary submodel of (R(E),E, F , P,,8, To,. . .,Q, . . . of cardinality 181, {i: i < S } = S E ]%I, and let # be a one-to-onefunction from 6 onto 193 I, and <* an order on 1931 such that # is an isomorphism from ( 8 , <) onto (1931, <*>. Let 93, = (99, <*, #) and T, = Th(99,). For Q = ~ ( E f Lo ) let Q, = 'pl(y,Z) be an L(S,)-formula saying that y is M a = F(a) for some a < S and f is from lM,l and Q(Z) is satisfied in Ma.For p E I', p = p ( f )let p l = {Q,: Q ~ p }and , r1= { p l : p E u {{z E Lo A z # Q : Q E Lo}}.SO in EC(T,, rl)there is a model (99,) such that <* orders it in order-type

r}

[CH. VII,

CONSTRUCTION OF MODELS

434

86

6 = &A, K ) , lLll s A, lF1l s K (or if K < A, IF1(s A, so by 6.1(4) 8(A, IFll) = E(A, K ) ) hence there is 9’ E EC(Tl, Fl)which is not wellordered by <*. Hence ord(23*) = {a E IS*]: $* C “ a is an ordinal < 6”) is not well ordered by < (use a).Choose a E ord B* so that {b E ord 23*: b < a} is not well ordered. Then P(a)is, in fact, the required model.

THEOREM 6.3: Suppoee lLol s A, To a theory in Lo, r a get of IK ( < Ko)-typeain Lo and P a one-place predicate in Lo. If for every a < 6(A, K ) there irr H a € EC(To, IIM,II 2 aa(IPM“I)then for every p 2 ILol, there is H E E C ( T ~F), , 1 1 H 1 1 = p , IP”I = ILol.

r),

Remark. Notice that we can replace a a ( l P M ~by l ) aa(lPMu1+ A) for K > 0, because ~ ~ (20 A,) SO for a 2 A + w they are equal.

Proof. We proceed as in the proof of 6.2. Let P be a function from 6 = 6(A, x), P(a) = Maandlet23, = (R(;), E, P,8, To,. . , tp,. . .)(PEL,,. So again we can find 23 elementarily equivalent to 23, such that in 23 there are “ordinals” a,, < 6 , a,,+,< a,, and for each a < 8, P ( a )is a model of To omitting each p E F. Now w.1.o.g. we can assume that To has Skolem functions and 93 k “a,,+, + n + 1 < a,,”. Let < be, in 23, a well ordering of P(ao).Now define by induction on n, elements X, of 23 such that 23 C “X, is an increasing sequence of elements from the universe ofF(a,),it has cardinality 3aJP(F(ao))l), and is n-indiscernible over P(P(ao))”and 8 C “Xn+l is a subset of X,,”. This is possible by the Erdos-Rado Theorem (2.6 of the Appendix) (more exactly-as 23 Hatisfies a first-order sentence saying it). E @, then Now we define @ proper for (w, To)so that if p(zl, . . ., z,,) : a, < at+,”then 23 C “tp(al,. . .,a,,)is satif 49 C “A:* a, E X, A A? isfied by P(ao)”.Clearly there is such @, and E M ( & @) is the required model. (Itdoes not realize types omitted by F ( a o )because of Lemma 2.2; hence, it omits the set of types r.)

.

,

THEOREM 6.4: p(A, K ) = =

ad(h,rc) for

K

> 0 ; and p(A, 0) = No, 6(h, 0 )

0.

Proof. For K = 0 this is by 6.1(6) and the compactness theorem, So let K > 0. By the last theorem it is easy to see that p(A, K ) 5 a6(A,rc). For the other direction assume that every model in EC(To,F)is well ordered by < ; To, F a r e in Lo,lLol = A, IF( 5 K . Now let Q be a new O O= one place predicate, for every # let $0 be # relativized to Q, T {p: E ~ ~ 1= {p: , 4 Ep), ro = {pa:p E r}.

+

CH. MI,5 61

EANF NUMBERS OF OMITTIWO TYPES

435

Let E be a new two place relation, P a new one place relation, and P a new one place funotion symbol, and ct, i < KOnew constant symbols. Let

(VZ, y ) [ - ~ p ( Z ) A

- I P ( Y )A

(vZ)(Z

(WQ(m)), (VZY Y ) ( Z w + m ) F, = YOU {{P(X)A

2

#

C,:

E2

Z Ey)

+X =

3/19

< P(Y)))UWet): i < So},

i < KO}}.

Clearly if iK E EC(Tl, F,) then its submodel with universe QM belongs to EC(To,r ) and if the order type of the latter is a, then lldlll 5 D ~ Conversely, . if some N E EC(To,F)has order type 2 a + 1, then some bl E EC(Tl, r,)has cardinality aa ITII.

+

THEOREM 6.6: Let K > 0. (1) 8(hyK ) is a limit ordinal of cojdity > A. (2) h+ 8 ( h , K ) 5 (2’)’. ( 3 ) If cf h > KO then 8(hy1) > A+. (4) If of h = KO,p = ZX<’ 21 then S(h, 1) s p+. h i8 a &Ong limit cardinal O f W f i d i t y KO,t h ( 6 ) If h = No 6(h, 1) = A + . (6) &(A, 2’) = (2’)+. (7) &(A,1) < (29+ (and &(A,K ) < (2’)+ when 2“ < 2’)). (8) For a strong limit cardinal of cojinality > KO,2’ < & ( A ) ;and generally cf h > KO,(vp < h)(p”‘A < A) implies &(A, 1) > A”‘”.

Proof. (1)Easy, left as an exercise fo the reader. (2) We have already proved that 8 ( h , ~2) A+ (in 6.1(3)). For the other inequality let lLol 5 A, To be in Lo, F a set of types in Lo, s K and suppose iK E EC(To,F) has order type 2 (2’)+ (ordered by < ). We can assume Th(M) has Skolem functions, and let a, E IiKI, i < (2’)+ , be such that for i < j,dl t at < a,. Now we d e h e by induction on n < o,sets & E (2’)+, = (2”+, and for each iE&, i < al(n,n - 1) < which ordinals ai(n,0 ) , . . .,a,(n,n - 1) for < al(n,0 ) such that: if i E 8,,j eSm, n 5 m then tp((aa,,,,,,, . ., = t~((aaj(m.o), * * aa,(m,n- s)1 0, Jf). aa,(n,n- I)), 0 , For n = 0 let So = (2”)+, (we have no at(O, 1) to define). If we have defined for n; for each j < (2’)+ we choose an i, j < i €8,and let a,(n + 1 , l ) = at(n,1) for 0 s 1 < n and a,(n + 1, n) = i. The number

.

9

--

OON8TRU(rmON OB MODELS

436

[a. VII, 5 6

ofpo=ibletypeep;+’ = tp( q(n, 1) > clearly N, C a,, > a,,,,, so N, is not well ordered. All this proves that 8(A, K ) s (2”)+. (3) Let 49, = ( R ( i ) ,E, c , A+ , A, i)‘< A ( c -the order between the ordinals s A + ) and let p = {z c A A z # i: i c A}, and F = {p}and T = Th(99,). Suppose 99 E EC(T, 0, and we shall prove that {z:99 C c A+} is well ordered, thus by 6.2 proving that 8(A, 1) > A+. For suppose 99 C z,, + < x,, c A+. Clearly in 93 there is an element f such that 8 C“f is a one-to-one mapping from {z:2 < z,,}onto {z:z < A}”. Let 9 C “f(z,,) = a,,”. As of A > No,49 omits p , and {a,,: n c W} is countable it is bounded by some a* c A. As b C “the set of x c A+ such that f(z)c a* has cardinality c A” there is g in 49 such that 53 C “g is an order-preservingfunction from {z c zo:f(s) c a*} into A” so 99 C “g(z,,+,) c g(z,,) c A” contradiction to the omitting of p by 9 (aa A is well-ordered). (4) Let M €EC(To,r ) have order type zp+, and F = {p},p = {v‘(Z): i c i, s A}, Z(Z) = m. Let ?;(z0, . . .,$,,-,), a c A be a list of all terms from Lowhere Z(f:) = m. Let for a E “1611, G(si) = min{i: M C 7cpi(ii)}. For A > Noaa of A = Nolet A = A,,, A,, c A,,+, c A, and let H(7i) = min{n: G(a) < A,,}. The rest of the proof is just like that of (2), only p+ replaces (2”)+, and instead of tp((aul(,,,o), . ,aul(n,n-l)), 0, M) we use when A = No +

+

.

...,

---

,

z,,

..

((1, and when A >

FLY Q(~Lfr(aul(n.O)Y*

KO,

-

* Y

~u,(n,I-l)))):

2 5n Y a 5 n}

+

OH. VII,

5 61

HANF NUMBEBB OB OMITTINO TYPES

437

A t the end we choose Nlas a model of

Th(H)u {&+l

< am:n <

w}

u { l p p ( ~ ( U o..., l an-’)): p = G ( q % * ( t 8 , 0 p...1) for every i E&, n big enough}. (6) It is immediate by parts (2) and (4). (0) By (2) it suffices to prove S(A, 2”) 2 (2”)+, or that for every a < (2”)+, 6(A, 2”) 2 a. So let a < (2*)+ and choose for each /3 < a a Of subset SO that /3 # y * 8 6 # 8,. b t for every /3 < y < a P6.7 = {+ c y)} u {P,(2)1f(fES0): i < A} u {P,(y)u(W:i < A} and for every c A p s = { P , ( P ) U ( f € ? i < A}. Let Lo = {Pf:i < A} u { < y =},

s

To just “says” < is a linear order,

r=

{&7:

/3 < y < a} u bS: S c A, 8 # 86 for each /3 < a}

Pi($/): i < A}. Clearly there is M, E EC(To, r ) of order type a (lHl= a, /3 E Pf 8 i E 86,< is the usual order) and every model in EC( To, r ) is a submodel U {P

# y A Pi(%)

of M , (up to isomorphiem) hence is well ordered. (7) Immediate by cardinality considerations (there are essentially only 2* pairs (Tol{p})and similarly for S ( ~ , K ) ) . (8) Let K = cf A, and D be any H,-complete filter over K . For any function f from K to A (in fact, to the ordinals) we define its D-rank R D ( f ) :R D ( f ) 2 0 always, R D ( f ) 2 6 if for every a < 8, R D ( f ) 2 a, and R D ( f )2 a + 1 if there is g : K 4A. g/D < f/D(i.e., {a < K : g ( a ) < f ( a ) }E D ) and R D ( g ) 2 a. NOW R D ( f )= a if R D ( f )2 a but not R,(f) 2 a + 1; as D is 24,-complete, there is no descending sequence f n / D(n < w) SO R D ( f )is always an ordinal. Let a(A) =

sup{RD(f):f:

K

+ A}.

Let us define now a model H’: its universe is the disjoint union PM u Q M u &fwhere ignoring trivialities: PM = a(A), &M = {f:f a function.from K to A},

=

&f = { A : A K}, c M = the order on ordinals, F = a partial function such that F ( f ,i) = f(i) for i < K , f~ ah, i (i 5 A) = an individual constant, A (A c K ) = an individual constant,

UONSTRUUTIOW OB MODEL8

438 €M =

{(i, A ) :i < K , A E

Qi = D, B

K,

[a. VII, $ 6

i E A},

= a partial function such that U ( f ) =

B D ( f )E P’.

Let T = Th(M) and p = {x < A A x # i: i < A} and let N be a model of T omittingp; so by renaming we can assume the interpretation of i (i 5 A) is i ifself and any member of QN is a function from K to A, and PV,i) = f(i), m d

QF = Qy,

EN

= { ( i , A ) i: c K,A E K , A E Q F , i , E A ) ,

so Qf = D. Now we prove PM is well ordered. If cn > c , + ~ (n < w) is a counterexample, then there is fo B ( f ) = co. (As (Vx E P) (3y cQ)(B(y) = 2)E T,as PM = a(A).) Similarly, by the rank’s defi, that U(f,,) = c,,, {i < K : f,,(i) > nition, we can define fn E Q ~ such f,,+l(i)}E Qf = D ; and we get a contradiction. So in every model N E EC(T, {p}), PN is well-ordered by No, 2’5 < A; and if D is easily defined (e.g., the filter of closed unbounded subsets of K , or DEb the filter of co-bounded subsets of K ) the last restriction is not necessary. Now we prove a(A) 2 Aofh, e.g., for D = DEb thus finishing. First note that there are Aof A functions f, : K --+ A such that (Vi # j ) ( 3 a < K ) (VB)(a < B < K -.fl(B) # f,(B)) (let A = 2 , < x & Y & < A, and let for i < K ha be a one-to-one function from A, into A (exists as (Vp < A)(Vx < cf A)(px < A)). Let df:;i < AofA} be a list of all functions from K to A, and define f l ( a )= ha((ft(y): y < a)). Thefl’s are as necessary. If a(A) < Ax, there are ( 2 9 + fl’s with the same D-rank, say cfl: i < (29+}. For any i < ( 2 9 + , c f i = K , and a < K + , we define by induction on n < w, &(i,a ) such that (i) d,,% a ) < dfb+l(i,a ) < i, (ii) for d = d,,(iY a), f&4> f i b ) , (iii) for any m < n, letting f = &(i, a), = &(i, a )

n,ca

f d a ) > f , ( a )+fc(a)

By (iii) and (ii),fcncl,6(a)(n <

w)

’fM.

is strictly decreasing hence for some

n = n(i, a) &(i, a ) is not defined. Let Sk = {&,(i,a ) : n < n(i, a)},Si = Uac+ St,80 lS1l 5 K , 9’c i. As cf i = K + some h(i) < i bounds it, so by 1.3 of the Appendix, for some y {i < ( 2 9 + : h(i) = y } is stationary

OH.

VII, f 61

HANF NUMBERS OF OMITTING TYPES

439

and also for some S : W = {i < (2")+: h(i) = y , S6 = S} is stationary. We can find j < i, j,i E W such that for every 6 ES,t9 < K ,

fc(B) 'fdB)+fc(B) > fr(B) If for some a,f,(a) > h(a),j will satisfy the conditions on fna,a,(i, a), contradiction. So for every B fj(j3) If,(/3); but l{P: f j ( B ) = ff(/3)}I < K hence {/3 c K : f j ( P ) c fi(/3)} E D( = D;Ib).But this implies f,/D < ff/D hence R,(fj) < R,(f,), contradiction.

Remark. Barwise and Kunen [BK 711 have shown that it is consistent with ZFC that: (1) A + < 2 A y 6(h, 1) < A + + , (2) A+ < 2 A < 6(h, 1). EXERCISE 5.1: Suppose lLol Ih, To a theory in Lo, r a set of ( < No)types in Lo, s K . Show that if for all a c 6(h, K ) there is a model Ma in EC(To,r)such that IP"aI 2 >a and llMall 2 > ( ~ P " Ua) [ ,then for all h 2 p 2 ITo[ there is a model M in EC(To,r) such that llMll = h, IP"I = p. If we omit " IPMuI 2 say' we can still get IP"1 I

PI. EXERCISE 5.2: Show that if in the definition of p(No,1) we require T o to be a complete theory and p ( F = {p})to be a complete type the value of p( No,1) is unaltered.

CHAPTERVIII

THE NUMBER OF NON-ISOMORPHIC MODELS IN PSEUDO-ELEMENTARY CLASSES VIII.0. Introduction The essence of this chapter is the construction of many non-isomorphic models in a pseudo-elementary class PC(T,, 2’) and we shall use methods developed in VII; the proofs have a combinatorial flavor. At the beginning of Section 1 we make two important observations (Lemmes 1.3 and 1.4). (1) If after adding K individual constants to T,there are p many ( > As) non-isomorphic models of cardinality A in the pseudo-elementary class, then the adding does not change this number. (2) If we have h 2 lTll types in D ( T ) which are independent (i.e., for any subset of them, there is a model reelizing them but not the others) then I(h,TI, T)= 2” (aee Definition 1.1). Now we can show that some properties of T imply the existence of large families of independent types, sometimes over a set A (1.6) and then by (1) and (2) we prove that I@, T,,T)is big (1.7). A more difficult theorem is 1.8 (the main crme is 2” = 2n0, ID(T)I = No). THEOREM 0.1: If T ia countable, not Ko-8table, lTll = No, KO < A 5 2n0, then I(h,T , , T ) = 2”. If instea&of ITII = No we demand ITII < 2% we atill get a r d t when lTll s h < P o : (1) If A satisjies the combinatorial condition (*) (“not AD(2No,A , A , No)”), then I ( A ,TI, T ) 2 2Ho (see 1.9). (2) If Martin Axiom (see [MS 701) holds, I(A,T , , T )2 2A (see Exercise 2.6). More interesting is the fact that we can generalize the theorem to higher cardinals, but the hypothesis is quite strong:

440

OH.

m,8 01

44 1

INTEODUOTION

THEOREM 0.2: 8ypptxe (i) T has A independent fornzulae, (ii) T E TI, A = IT11 = IT!, (iii) There is a A-Kurep tree with x branch (e.g., A strong limit, or cf A = No, x = ANO), Thenfor p 2 A, I ( p , TI, T)2 min{2xy2”).

It would be nicer to get such a theorem aseumingonly ID(T)I = x > A =lT1l. The main aim of Sections 2 and 3 is THEOREM 0.3: Any unsulperstable T , has 2Anorc-isommy.~hk Cardinality A, for any A 2 IT1 + N,. We prove it also for pseudo-elementary 01countable T,T,). The proof is by cams. The easiest case is:

in many

of (e.g.,

THEOREM 0.4: If T i s uwwperstabk, A is regzrlar and > lT1l, then in PC(T,, T)there is afamily of 2” nzodele of cardinality A, no one ekmedarily embedduble in anothw. In the proof note that if M = UleA Ml (1 = 0, l), Mt (i < A) inincreaaing and continuous, Aregular IliKll = A > IIMf 1 then{i: Mf = Mt} is a closed unbounded subset of A. Hence if P is a property of pairs of y f ) satisfies P} is determined, mod D(A)by models then {i < A: ( M M the isomorphism type of M ;and in fact we can make P to depend on more information. We urn the indiscernible tree of sequences we have constructed in VII, Section 3. Choose for each S < A , cfS = w , an inmawing sequence of ordinals of length w converging to it, 76, and for B G {S < A: of 8 = w} let I, = @’A u {q6: S €8).Now from the isomorphism type of EM(1,) we can reconstruct S / D ( h ) . The other cams have each one its specific trick, and the proofs are somewhat complicated. If p < A 5 pNo, 2” 2A, the method of Section 1, on independent types, applies. If A is singular, but no previous m e appliee we choose a proper p < A s 2’, and make a construction similar to the first one, but A-fold. If A is a singular strong limit cardinal, we use games. For an elementary c l w , T stable, A = IT], use Fk-primclry models over indiscernible trees, and repeat the previous oaaw (all this in Section 2).

THE NUMBER OF NON-ISOMORPHIC MODEL8

442

[a. WII, 8 0

Then for A = IT , I, T unstable we note that skeletons of EhrenfeuchG Mostowski models rn N,-skeleton like (Definition 3.1 or below), and if in an insomorphism of one such model onto another, the intersection of one skeleton with the image of another is big, the orders of the skeletons are not contradictory (Definition 3.2). Proving the existenoe of large families of pairwise contradictory orders, we finish exoept when A satisfies a strong set theoretic condition, which implies it is < 2h or is high in the sequence of 8,’s (this is 3.2, case I). Note that M 1 = E M l ( I ) eatisfies: (A) (a,: 8 E I) is X,-skeleton like, i.e., for every 6 E lM1l there is a h i t e J E I, such that if 8 , t ~ I J , (VUEJ) (8 < W E t < u) then 63,, gnat realizes the same type. (B) Hence if I = I, + I,, then for every 6 E M,,for some t, E I,, reah08 the mme type for all 8 E I Satisfeg tl < 8 < t,. The m e left is T unstable, 1 T,I = A, AD(2”, A, A, No);we can assume T is countable. For understanding the proof let us concentrate on showing I(A, T,,T) > 1 for Aregular > N,. Let N be a (A, N,)-eaturatd N,: and M = E M ( I ) , model (see Definition VII, 1.5), N = Ul
N,, for some 5 < o,, J : UnErl b, G NC, UIEj6, E Nc, J c I , , I JI = A. In Sc+lthere is a sequence (Aa:a < A), increasing, continuous, 1A.I < A, U,<” A , = INC[. We can h d 8 < A, such that UaEIl 6, E A,. AS (6,: 8 E I) is H,-skeleton like all except < lA,l + 8, members of {6,: 8 E J}realizes the same type over A,, let 6: be one of them and J, = (8 E J: tp(a,, A,) = tp&, A,)}. Now it suffices to prove some consistent type extending {Z < 6,: 8 E I,} U {6, < Z: 8 E J,} belong to St+,.But we can define such a type using N,, A,, and 6*: +

p

=

{Z < 6: 6 E A,, Nc C St < Q U (6 < 1:6 E INC] , and for every 6’ E A,, N c C 6: < 6’implies N c C 6 <

6’)

(every b i t e subset of p is realized by all a,: 8 E I2except hitely many). In the actual proof we have to take care for getting 2Anon-isomorphic models, and making them K-compact, for suitable K . In Section 4 the reader can relax; here we mainly use previous results to prove theorems e.g., on categoricity, we prove:

CH. WII,

8 01

INTRODUCTION

443

Tireorem. If PC(T,, T ) is categorical in A > 1T11, then T is superstable, without the f.0.p. and stable in lT1l.

We also give a partial solution to the problem: doea the categoricity of PC(T,, T)in h imply its categoricity in p ? Show its equivalence to a two-cardinal theorem with omitting a type. We get similar results when we replace “categorical in A” by “each model in PC(T,, T ) of cardinality h is homogeneous ”. Then we use the construction of VII, Section 4 to prove results on universality, i.e. : THEOREM 0.5: If T ie unawperetable, A is regzllar >2IT1l t h in PC(T,, T ) there are arbitrarily lwqe ( < A)-univereal not h-universal nzode2e.

On the other hand we shall show that if of h = Nothe conclusion does not hold. We also characterize the class of cardinalities in which a theory h w saturated models, by using various previous results. THEOREM 0.6: T hua a eaturated model of Cardinality A iff X = ID(T)I or T ie h-stahle.

+

PROBLEM 0.1: Though we get a complete reeult for elementary classes, this is not true for pseudo-elementary clmms, when we look for , when we look for 2* pairwise monK-compact models K < K ( T ) and elementary embeddable models; it will be interesting to complete them. PROBLEM 0.2: It will be desirable to simplify and unify the proofs. A possible way is in the proof of 2.1 to try to replace the filter of closed unbounded subsets of A, by a filter on S,(h) or on the family of increasing sequences of members of S,(h), of length S (see Kueker [Kk 771 and Shelah [Sh 7681). PROBLEM 0.3: In many cams we have parallel proofs; in 2.7 an F&,-constructibleset, replaces the Skolem Hull. Also clearly different proofa uae different sets of assumptions on the construction. Maybe an axiomatization is in order. PROBLEM 0.4: For uncountable T,, we know little on I(h, T I ,T ) when L,A = JD(T)I> IT,I. It is natural to conjecture that for h > ITJ, l ( h ,T,, T)2 min{2A,W } , but it is reasonable to try also an independence proof.

444

“HE NUMBER OF NON-ISOMOBPHIU MODELS

[OH.

WII,

81

VIII.1. InaepenaenCe of tspes DEFXNITION 1.1: (1)I ( A , T , , T )is the number of non-isomorphicmodels in PC(T,, T ) of cardinality A. (2) IE(A, T,, T ) is the maximal p such that there is a family of M f E PC(T,, T ) , i < p, llMtll = A, such that there is no elementary embedding of Mt into M,for i # j. (3) If we omit T , this meam T , = T . Remark. The “maximal” in (2) is inaccurate, as there may be none, and we may get, e.g., 8, < IE(A, T , , T ) < K, for any n. Remember D( T ) = Urn Sm(0). LEMMA 1.1: (1) 2” 2 I ( A , T , , T ) 2 IE(A, T,, T ) 2 1for A 2 IT,!. (2) If T s T’ E Tl c T,, then I ( A , T , , T ) I I ( A , T i , T ’ ) and IE(A, T,, T ) s IE(A, T i , T’). Proof. Remember TIis consistent, and any model of T , is isomorphic to a model with universe {i:i < A}, and there are only

n

2(Am(”)

(2”)“ = 2”

RELl

such L,-models where m(R)is the number of places in R (we consider here m-place functions as (m + 1)-placerelations). The rest is even more trivial.

Remark. We will be mainly interested in I and not in IE.

Proof. By VII, 2.9(1), for every p E D ( T ) there is a model M , E PC(T1,2’) of cardinality A which realizes p and is stable in lT1l. (Replace T,by T,u {v(E): v E p}, E a sequence of individual constants.) Now M, r Mq induces an equivalence relation over D ( T ) . Each equivalence clam has cardinality s I T,I as I{q E D ( T ) :M q M,}l I I{q E D ( T ) :q is realized in M,}1 s lTll (by the IT,I-stability of MP). Hence ID(T)I is at most ITl! plus the number of equivalence classes (by cardinal arithmetic). As ID(T)I > lTll,ID(T)I is equal to the number of equivalence classes. So there are 1D(T)1 non-isomorphic

OH. n I I , § 13

445

INDEPENDENCE OF TYPES

models M,,so I ( A , T,,T)2 ID(T)I. Let g(p) = {q: q E D(T), M , has an elementary embedding into Mp}.So g ( p ) c D(T)and again Ig(p)l s ITJ;hence, when proving (2),by theorem 2.8 of the Appendix there IS1 = JD(T)I,such that p # q E S implies p $ g ( q ) , hence is S c D(T), IE(kT,,T)2 1{2M,:pEs)l= ID(T)I.

PROBLEM 1.1: Does (2) hold when ID(T)I

=

ITl[ ? +

LEMMA 1.3: If A 2 1TlIy p = I ( A , T,u T ( A ) ,T(A))> AIAl then I(A, T,,T ) 2 p; where T ( A ) = T u { ~ ( a si) :E A , Cp[si]) (the a E A serve also i n d i v i d d constants). Proqf. Every model M, of T, of cardinality A may be expanded to a model of TIU T ( A )in < hlAl forms, and we can get each model of T,u T ( A )in this way, from just one M , (up to isomorphism) hence p = I(A,

Ti U T ( A ) ,T ( A ) )5 I ( & Ti,T)+

But p > N A l , so our result follows.

Remark. Notice that we cannot use 1.2 for 1.3 a~ the estimation in 1.2 is too weak. Hence the following concept is interesting:

DEFINITION 1.2 : (1) If S is a family of types over A (in L), T s T,, then S or (8,A) is called (T,, T)-free (or free in (T,, T))if for every S' c S there are models M , of arbitrarily large cardinality, A 5 M , (of c o m e M, r L 4 6)such that p E S is realized in M , iff p E S' (hence in every A 2 lTll + IS'l there is such a model). (2) S is free if it is (T,, T)-free for every T,. (3) The pair (T,, T)has (p, A)-freedom if there are A, IAI = A, 8, IS1 = p satisfying (1). If this holds for any T,with the same A, S we omit T,.

LEMMA 1.4: (1) If (T,, T ) irae (p, x)-free&Om,p 2 A 2 lT1l,2A > Ax, then I(A, T , , T ) = 2A. (2) If (Tl,T) has (p, x ) - f r e e h , A 2 p, 2' > (p + p,p, then I ( A , T,,T)2 2y. ( 3 ) If (S,0 )is (T,, T)-free,A 2 lTll, thenIE(A, T,,T)2 min{21SI,2A}. '

Proof. For ( l ) , we estimate I ( A , T,u T ( A ) ,T ( A ) ) from , below, (where ( S , , A ) is (T,, 2')-free, I S,l = p, IAI = x ) by looking at { ~ E S , :M realizes p}, and then use 1.3. Part (3) is immediate as by 1.6(2) of the

446

THE NUMBER OF NON-ISOMORPHIC MODELS

[CH.

VIII, 8 1

Appendix there are 2151 subsets of 8,no one is a subset of the other. We are left with (2); by assumption for every S E 8,there is a model M1(S)of T , of arbitrarily large cardinality, A E M1(S), each p E S is realized in M1(S)by as = a(p,S), and no p E So - 8 is realized in it. By VII, 6.3 proof there is a model Ma(&)of T , = T , u T ( AU {ap:p €8)) omitting each p E .So - IS, which is EM(A, 0 ) @ proper for ( w , Ta).Let M3(S)be its L-reduct. By VII, 2.10 and 2.8(6) TP,(M3(S)) has cardinality 5 ( p + I T,I)X which is < 2". As TP,(M3(IS))depends on M 3 ( S ) only up to isomorphism and the number of possible 8's is 2', we finish.

THEOREM 1.6: (1) T has (A, p)-freedomif T is unstable and A < Ded p (i.e., there is an order of power A, with a dense subset of power p ) . (2) T has (A, p)-freedom if K < K ( T )and there is a (A, +tree I , [InA S I C [ 5 p < [ I n AICl = A (e.g.,if X , < K ( T ) A, 5 pwo). ( 3 ) T has (A, p)-freedorn if T has the independence property and p 5 A 5 211. (4) ( T , , T ) h (2b, X,)-freedom if (2%, 0)-freedcm when ID(T)I > 8,.

lTll

=

No, T X,-umtable. It has

Proof. (1) Let J E I be orders, I JI = p , 111 = A, J dense in I. Let J c J, c I, and we can w u m e J, I are dense orders (by extending),and let I, be an order of type x. By VII, 2.4 there are (Z < 8) EL, and @ proper for (0, T , ) such that in M = E M ( J , + I,, @), M C 7is < at iff s < t. Clearly llMll 2 x which waa arbitrary. Let A = {a8:s E J } , S = { p 8 :s E I - J),where p s = {(Z < 7it)if(s
-

CLAIM 1.6: If for every T I z T and x there are a pair (8, A), S E UmSm(A), IS[= A, IAl = p and for every s' E S a model My 11M11 2 x, M realizes p E S i f p €8'then T h (A, p)-freedonz. Proof. We should prove that we can choose (8, A) independently of T , . Up to isomorphism (of Q ) there are s 2Asuch pairs, so if each such pair (8, A) fails for T,(S,A), s'(8, A ) and x(S,A ) then (as T is complete) we can find T , z T , such that up to change of names of predicates and function symbols not in T,it extends each T,(S, A) (such T,,is consistent aa Q can be expanded to models of T I @ ,A) for any S, A by

CH. WII,

8 11

447

INDEPENDENCE OF TYPES

I , 1.12).For T,we get some suitable (8,A) for x = (8,A)}; contradiction.

2 h(S,A): possible

Continuation of the proof of 1.5. (2)Let I be the (A, K)-tree mentioned there, and x L A. T has a strongly uniform K-tree of the form ( ( q ~ ~ ,ma): a successor, a < K ) and let @ be as in VII, 3.6( 1) and (2).Then let

A = {a,,: 7 E I n A<"},

8 = {p,,: 7) E I n A%}, p,,

= { q ~ ~ sinla): (z, a

<

K

successor}.

If s' c 8 let I(&') = x<' u (7:q E I n A", p,, €8')and M = EM(I(8'),@). Clearly lldlll L x, A G IMI, and if I),,E s',a,, E 1dl1 realizes p,,. If v E An ?(a,-)realizes p y ,p v $ s' there is a < K such that q[Z] 1 a # v 1 a (or q[Z] has length
{[d% aN1)) = l q ~ ( Z~, a ( l ) + d I2: < n) (when, e.g., - a(Z)# a(k) + 1 for 1 # k.) Let for i < 2", U G p, I Ul < KO 6,(U) = B',,(. . .,a,, a a a + i , . . .)asw(t)n-u where n = Iw(i) n V l . So for a E U ;a E w, iff qJ[b,(U),a,,] = TqJ[b,(U), Let D be an ultrafilter over W = Sn0(p),such that for any finite w, s p, {w E W :w o s w} E D ; and let M* = Mw/D, making the natural identification of a and (. . .,a,.. . ) / D ; and for i < 2u let 6, = (. . ., 6,(U),. . .)&D. For 8' c S let M(S') be the Skolem closure of Idl] u {6,:p , E 8')in M*. Clearly A c M-( s ' ) ,and each p , E 8' is realized in M ( 8 ) . Suppose p , $ s', but 6 = .?(as,b,,,,, . ., 6,,,,) realizes p,, I EX. By assumption on the w,'s, there is a $ UISn wj(l),2a $ 8, 2a + 1 $8, a E wl.

.

448

TEE NUMBER OF NOW-IBOMOBPEIO MODELS

[OH.

n I ,81

We ca,n assume that for every U E W ,6[U] = T(~z,, 6,,,,[Uj,.. .) 80 if a E U E W 6[[u1=T(af) where i ~ { 2 p 2/?+1: , /~Ew~(Z ~ )G, n } U 8 , hence 2a, 2a+ 1 $t; hence MCtp(.?(at),aZz,) = tp(?(af),aZt2.+J so {YEW :k&U], ZSu) = q(6[U],Baa+,)} 2 {U:a E v) E D hence M* C q(b, aSU)E q(6,aSU+,), 80 6 does not redhe pfycontradiction. (4) Suppose T,I> T hasSkolem functions, lTll = No.If ID(T)I '> No by 11, 3.16 there are formulas q,,(E) E L for r) E 2<" such that (3Z)q,(P) E T ; v Q 11 (W(V"(@ VY(W E T and (Pv-(Z) am contradictory. We use theorem MI, 3.7, Exercise 3.1 on T,,q,,(and its notation), we let S = {p:: r) E 2"). Now (Is, 0) is (T,, T)-free, aa for any x and S' c Is let (Ml, .. .,b,,, . .),,<" satisfies the conclusion of M I , 3.7 where (in M,){b,: i < w } is an indiscernible sequence over {an:r) E 2"}, and M a an elementmy extension of M 1in which {bt: i < x } is an indiscernible sequence over {a":r ) E 2"). Then let M(Is') be the Skolem closure of {an:p: E S'} u {bt: i < x}. Clearly each p: E Is' is realized by a,, E M(S'), and by VII, 3.7(4) no p: $8' is realized in M(S'). If T is unstable in Nowe choose A, IS(A)I > IAI = No,and proceed as before with T(A). Clearly 1.6 and 1.4 have many conclusions, e.g. --+

.

CONCLUSION 1.7: (1) If T b not eujperstuble, P o 2 p 2 h + IT,l, 2u > 2" t h n I ( p , T,, T) = 2 Y . (2) If T , z T , T , countable, T No-unstubk then h 2 P o implies I(h,T,, T)2 2a'0y and 2" > 2'0 2 h implies I(h,T , , T ) = 2". THEOREM 1.8: Suppoee T, 2 T are countable, T K,-unstable 2'0 2 h > KO,then I(h,T , , T)= 2".

Proof. We assume T,has Skolem functions. If 2" > 2'0 or ID(T)I > No the conclusion follows from 1.7(2), 1.6 and 1.4(3); if T is not superstable by the next sections (Theorem 2.1). So assume T superstable, No-UIIBt&ble,2" = 20' and ID(T)I = No. The first two assumptions imply, essentially by 111, 6.1 that there am equivalence relations En@;jj) E L with finitely many equivalence classes (in a) En+,refining g,,and E,, r) E 2<" such that r) Q v, r) E 2,, implies En@,,, 4) but r ) # v E 2" implies 4 & ( Z V , 4). Let q,,(Z) = En@;4) where 4 will be individual constmta of T,. Using VII, 3.7 (to T,), and renaming, there is a model M of T, and a,,, r) E 2" satisfjdng (1)-(6) from that theorem, and by renaming we

CH.

m I I , 8 11

INDEPENDENCE OF TYPES

449

get h is the identity. We can dso w u m e (for (*2) by taking a subtree of 2<" and renaming) (*)

..

T&,. ,Z,,,)E L, there is n, such that for every n 2 nf: (1) There are in L, individual consbnta c? i < ni representing the E,,-equivdence classes such that

For every

-

~ I d V l Y* *

Y

f), $1

E

r,,

(see VII, 3.7(5) for the F,,'s). (2) If +j is a sequence of length m of distinct members of 2", and the same holds for 1, and there am k, n s k < w , and +jl, 1, such that Z(+jl) = Z(1,) = my +j[Q 4 +jl[Q E 2OP, C[Q Q FJZ] E 2", and +j1[21] = Sl[Za] 0+j[ZJ = F[Za] and lEk(t(?ZGl)y T @ ~ J ) ,then this holds for k = n, and any such +jl, 1,.

Remark. We can first take care of (2) then of (1). Note that M C E,,(sin,a,) (7,v E 2") 8 7 1n = v 1n. Now for any 8 c 2" let M , ( 8 ) be the Skolem closure of {an:7 €8) and N ( 8 ) its L-reduct. Suppose f :M ( 8 )-+ M(8*) is an elementary embedding; 8,8* c 2" uncountable. For any v E 8 there am T = T~~ n = n, and +j = +jv of length n such that M,(8*) b f ( a v ) = ?,(af). We can msume +j[Z] # +j[k]for 2 # k. As 8 is uncountable there is an uncountable 8, c 19 and t,no, k,, +jo such that: (**)

+jv[a

For any v E 8,,T, = T , n, = no,and ko = ijOIZ]1 Eo and +jo[Zl] 1 ko # +j&] 1 ko for Zl # 2,. We can w u m e also that n, (from (*)) is 5 k, and that for 7 # v E 8,, h(7,v) > k,, where we let h(7,v ) = max{n:q r n = urn}, so q(n) # v(n) when n = h(7, v) .

Supposep # v E 8,h@,V ) = n, 80 B(8)C En@,,a,) A T E , , + ~ ( ~a,)., If for every Z h(+j,[Z],ij,[Z]) < n then by ( ~ 2 74,+lLf(a,),f(~,)] ) implies -En[f(q,),f(q,)l.Suppose for every Zh(qv[Z],qp[Z])# n, so ~ , , [ lr ]n = qp[l]r n implies q,,[Z]r ( n + 1) = qp[Z]r ( n + 1). Now define a sequence +j: if +j,[Q I n # +j,[Z] I n t h +j[Q = +j,[Q, and otherwise +j[Q = +jv[Q. Let 6 = .?(a,);notice that by the suppoaition above q[Z] (n + 1) = +j,[l] 1 (n + l), hence by (*1) M(Pi*) C E,,+,(6,f(aD)),hence M(8*) C lE,,+1(6,f(Bv)).Now 6 = ~ ( i Z 6 ) , f ( i Z , )= ?(a;")and for any Z+j[Z] = +j$] or h(+j[Z],+j,[Q) < n (by +j's definition)

450

THB: NUMBER OF WON-ISOMORJ?EIC MODELS

[CH.

WI, f 1

+

and ko < 12, +$41 t ko # +a1 t k, S V [ 4 1 t ko # .)?v[JaI t ko for 4 la (by (**)). SO by (*2), 4 , , + , ( b , f ( G i Y ) ) implies TEn(6,f(a;)). But kE,,[6,f (a,,)]A EnV(a,),f(7iv)]contradiction. So we proved that for any v # p ~ 8 ,for , some 1 h(fjv[Z],+j,,[Z]) = h(v,P I . Choose infinite subsets w,, i < 2N0, of o,such that for i # j Iw, n wjl c KO, and let 8,c {r] E 2@:$13 = 0 when I $ w,}, be a set of cardinality A; and M,= M(8,). Clearly IIM,)I = A. Suppose i # j and there is an elementary embedding f : M,--+ M,.By what we have proved till now there is an uncountable 8;E 8‘ such that for r] # v €8;there are r]’ # v’ E 8 j such that h(r],v ) = h(r]’,v’). NOWw = {h(r],v ) : r] # v E S ~ } is infinite (as 8;is) and w is a subset of w, (by the definition of a)and is a subset of {h(r],v ) : r] # v E 8j} (by the above) which is a subset of wj (by the definition of wj). Hence Iw,n wjI 2 IwI = KO, contradiction.

so

I(A, T,, T)2 IE(A, TI, T)= 2 n ~ . But we have restricted ourselves to the w e 2N0 = 2” in the beginning, so we finish. QUE8TION 1.2: Is thereaparallelfor 1.8 when 2No 2 A > ITl! > No? (Add i f n e c e s q ID(T)I > lT1l).

THEOREM 1.9:8uppoae 2wo > A > No, and there is no family of 2No aubaeta of A, (*I euch of power A, the intersection of any two of which iajnite. (This is just rtot AD(2N0,A, A, KO),) If T , 2 T,T superatable but not totally transcendental A 2 lTll, them I(A, T,,T)L 2% Remarks: (1)Baumgartner [Ba 761 proved the consistency of “ A = K, satisfies (*)” and of “ A = K, does not satisfy (*)” with ZFC. (2) The msumption on superstability is needed only for proving the existence of the En.

Proof. Let p 5 2no be regular. As in 1.8 we can find epivalence relations E,,(%,g)(n < o) and a,, r] ~2~ such that M , kE,,(ii,,,av)iff 7 n = v n; in some model M,of T,.Assume for simplicity, l(ii,,) = 1 so a,, = a,,. Let, for B c 2”, M ( 8 ) be the Skolem closure of {a,,:r] ~ r 9 ) As . in 1.8 choose w, c w for i c P o , w, infinite but Iw, n w,I < KOfor i # j , and

CH.

VIII, 8 13

451

INDEPENDENCE OF TYPES

let Sf E {q E 2O: $11 = 0 for I # w,}, of caxdinality A, Mt = M(Sf). So IIMiII = A, Y: a model of T, , and let M,be the L-reduct of M i . Suppose I(A, T, , T ) < 2No; as p s 2 h is regular, for some i W = {j:M j M,}has cardinality 2 p, and let fj:M,j . bl,be the isomorphism (we use only its being an elementary embedding). Let, for j E W ,A, = {f,(a,): q EL$}. Clearly IA,I = A, A, G lMfl where llMfll = A; the number of the A,'s is p, and for a # p A, n A8 is h i t e (otherwise this contradicts Iw, n w81 < So).If 2No is regular choose p = 2No and we get a contradiction. Otherwise combine our conclusion on all p and by 3.1 of the Appendix we get a contradiction to (*).

THEOREM 1.10: 8uppoee T s T,,ITl[ = A, and there is a family {tp,(Z): i < A} of A independent fomnulae of L(T) (i.e., for every $nite w E A and h :w (0, l}, T I- (33) AfEW tpf(3)h(f)). Suppose J is a subtree of 2 s A (closed under initial eqm.enta) such that, letting J , = J n 2", for EL < A, IJ,I c A, and x = IJAI (for x > A, the interesting m e , this is a K u r e p tree). T h (1) For eweryp 2 A,IE(p,T,, T ) 2 min{2",2X}.Moreower i f (3u)(u< A < 21"1 A 1011<"(O) = loll), there are, for each p = p<"(O)2 A, min{2",2X} models of T,, of cardinality p, L(T)-models,the reducts of ~(0)-compact m one elementarily embeddable in another. (2) There are M, C T , , g,, E M l ( q E JA),zf E M , (i < A ) such that ( A )For each formula P)(z~, ...,Z ~ ~ ~ ) , ~ , . . . , Xthere ~ ( ~is) ) E L ( T ~ a, < A such that if v,, . . ., vn(,),ql, . . ., qn(,)E JA, p 2 a, and vl 1fl = 71 rpbu tforl # k, vl rp # vk lg, then j .

M1

-

-

-

- -

C ~ @ v , , * * 9 #vn(l), 21, * * 9 zn(aJ tp&ql, * * 9 #vncl), 21, * 9 zn(,)l. ( U ) {zl :i < p} is an indiscernible sequence ower {Yq:7 E JA}. ( C ) For any v E J,, a < A, there ie tpv(Z) E {tpf(3):i < A} such that v Q 7 E JA implies Mi C V~&,J"(~).

( D ) For every term T ( Z ,...,z,,xl, , ...,xm)there is a 01, < A, such that if rll, . . ., qn+, E JA, p 2 % and q, rp ( I = 1, n + 1) arepairwiee distinct, then ~(g,,,. . .,&, zl, . . .,z,) doea not realize I ) , , ~ + where ~, p,, = {cpf(P)t:Ml =! Vf@,It, i < A, t E (0, 1)).

( E ) For each K < A such that 2" = A choose a good ultraJlter D,ower and assume (Vu < A ) ( ~ U ~ < ~<( OA )) (and ~ ( 0 is) regular). We can assume that ifS E JA,IS1 < K ( O ) , M; is the Skolem Hull of { q , , : v ~ SU} { q : i< K(O)}, V E J n - S then p v is not realized in MICs(O),where M i ( K < ~ ( 0 is ) )increasing and continuous, M", = M,, M"! is isomorphic to (M,)"/D, ower M,. K,

THE WUWBER OF NON-ISOMORPHI0 MODELS

452

[OH.

an,5 1

Remark. When does such a tree exist? If A is strong limit (i.e., a*) J = 2sA w i l l serve, 80 x = 2A. Clearly J c 2s* is not really ncessary, it suffices J c asB, cfg = cfh, y < g+ lJYl < A, x = IJ,I. So if of A = No, J = A,, where A,, < A = A,, is sufficient, 80 x = A h > A.

n,,

z,,
Instead assume J is a A-Kurepa tree (i.e., a < A => IJ,I < A), it suffices to assume : there are Ji c J,(a < A), I < JiI < A for 8, < A, and for every distinct ql, ..., qn E J,, {a < A: ql la, ..., vn ra are in is unbounded (for part (E) we should replace n < o by K < ~ ( 0 ) ) . For = 2Athis follows from 0,(A regular), and if A = ACA it holds with = A+.

u,

c}

x x

Proof. ( 1 ) Follows by taking Skolem hulls of {tj,:qcq U { z i : i < p } for J c J A ,using (2) (b) (the K(0)-compactcase by (2) (E)),using 1.4. ( 2 ) Case I : There is K, K < A < 2". Let 8 = {w(i):i < A} be an independent family of subsets of p (see 1.5 of the Appendix). For j < K let a, realize {q+(~)if~Ew(i)):i < A}, let I = {a,:j< K}, and let M , be a model of T , , a5E (2M,(, llM,II = p and {z, :i < p} be an indiscernible sequence over U,,it, (inM,). We define by induction on a < A, for each r] E J n 2" a filter D, over K,and a set a,, E 8 independent mod D,,, (see Definition VI, 3.1) such that 18 8,J < K + 11(7)1; and v 4 7 implies D, E D,, S, G S,. For Y E JA let D, be a completion of A D,,I to an ultrafilter. Let M , be a (x + A) saturated model of T,,N o 4 M I , and for each r) E JAY will realize Av(Di, B,,) where B,, = U {&: v < r ) (in lexicographic order) v E JA} U U5
-

(I

a,,

+

CLAIM 1 . 1 1 : Let A, K, S, I, M,, be as above, and let D,, ..., D , be $filters over K, S, independent modD,, S, c S, and FEW,^, cpcL(T1).Then we can find fltm8 D:, and families 8: m h that (A) D, c 0:; (B) Si C S,, IS,-Sil < K ; and for 1 = l , n , S t is independent mod 0: ; (C) There is tE{O, l } , such that { ( i ( l ,..., ) i ( n ) ) : i ( l )..., , i ( n )< K, * x 0 : . MI t v[Gf,,), . . .,at,,,);ZJt}E D: x 0:x *

CH. VIII, Q 13

453

INDEPENDENCE OF TYPES

Prmf. We prove by induction on n. For n = 1 this follows by VI, 3.3. For m = n + 1, we define by induction on a d K , Dt,a,St,afor 1 d I < n such that (1) Dl E DlSB E Dl,, for B < a. E Sl,p E S,, lSl-Sl,al < K for B < a. (2) (3) Sl,, is independent Dl,,. (4) There is t(a) E (0, 1) such that {
-

Case 11: A > No,not I; so h is strong limit. Here in the induction stepa,we have a set r, (a < A) of formulas of L, in the variables gfl ( r ) E J,) s 1 . 1 + No, a model M! of T,,and zf(i < w ) and set U, c A, sequenoes 6; (r) E J,, V s A - U,)and cf E My (i < A) such that: (i) for every p)(gfln,,. ., g, z,, . . .,)2, E Fa,and V(l), . . ., V(n) A - U,, M f k p)[6ri1),. . ., 6;>), o,, . . ., c,]; and M! C p)f(6fl)u(fev)for

=

.

iEA

- u,, v s A - u,,

(ii) {cf: i < w } is indiscernible over {E:: r ) E J,, V E A - U}. The connection between r, and r, ( a < 8) is that p)(g,,,,, . ..,&, zl,.. .,z,) E the ql distinct, q1 Q v1 EJ, implies p)(gVl,. ..,&, z,, . . .,z,) E Again the point is the assertion parallel to claim. More exactly we want to add p)(Zfll,. , ,,Zflnn,zl,. . .,z,) or its negation, extend U, by sx = 1.1 + Noindices, and preserve (i) and (ii). Let N be the model (E1((2"+), E) expanded by individual constant for M? (assuming E Jh, - u,}, w.l.0.g. 11JfyII I 2')s ((7, $,): 7 E J,}, {(v, ,'J 6;): {zf: i < A}, Our desire can be expressed by an (L(N)),+,,+-sentenoe, hence if it fails, it fails in some N , < N , llNlll = 2,, llNl n All = 2 5 and b s lNll A lbl s x =. b E N , . But then we can apply the claim, and find the truth value of p), and the subset of A - U, of cardinality 5 x we want, contrctdiction.

r, r,.

r.

v=

Case 111: A = No.Thie follows by 1.6.

THEOREM 1.11: Swppwe (*) from 1.9, A 2 IT,l, T stable not q e r stable. Then I ( & TI, T)2 2Ko.

454

THE NUMBER OF NON-ISOMORPHIC MODELS

[OH.

WII, 6 1

Remurk. We a n omit " T stable" by 3.4 and replace 2% by 2" by 1.7(1), and replace "not superstable" by "not totally transcendental" by 1.9.

Proof.Let M = EM (AscD,@) where @ is as in 1.6(2) and w.1.o.g. we can aasume that if A = {a,: q E A<@}, A , = UnccD a,,,,dim(p,,,A,, M') is A, where M' = EH(A
u

u = {j < 2%: N,2 M,(,)} has cardinality 2 p, and let f, : 111, + M,(,, be the isomorphism. Let M,,,, = U,,ABa, B" increasing, 11BU11< A . For each j~V there is a ( j )< A such thatf, maps A, ( q ~ 2 < "intoB"(-", ) provided that cfA > M,, hence for some /3, U' = {jE U :a(j) = /3)} has cardinality ~ pFor . each j e U' and r] ES, choose q ~ f , ( I , ,such ) that t p ( 8 , lBll) does not fork overf,(A,,). So for somej(1) # j(2)E U' { 6 ( 1 9 ) : E B,,)}

n {5{(%):9 E 8j,2)}

is infinite. But clearly for ~ E U 'tp,,,(?$B!) = tp(&Bl)iffq rn = v n, hence we get a contradiotion as in 1.8 and 1.9. So aasume cf A = X,, A = ZnemA,,, A,, < A. For each n, j there is a(j,n ) < A such that l{q €8,: Ifj(In) n dla('sn)I

2 Xo}l 2

A,,.

We oan also find P(n)such that Un = { j E U:a(j,n ) = /3(n)}has cardinality z p . So as before we get AD&, A, A,, No),and aa thie holds for each n, p it is easy to prove AD(Po, A, A, X,), i.e., not (*) of 1.9. QUE8TION 1.3: Can we in 1.4 deduce something about I E ? EXERCISE 1.4: Prove in 1.9 and 1.11 that if 2No is a singular oardinal T)2 2% and if 2 b is regular, x < 2w0, and in (*) there then IB(A, TI, is no such family of cardinality x then IE(A, T,,T)2 2Wo. QUESTION 1.5; Can we prove in 1.8 that IE(A,TI, T)= 2 A ?(The remaining case is: T superstable, KO-unstable,2N1> 2'0, and ID(T)I = KO.)

QUESTION 1.6: Try to improve 1.7(2) to results on IE(p, TI, T). EXERCISE 1.7: Suppose T has (A, p)-freedom by one of the cases 1.5 A 2 IT,\, A = 2 p = A". Then IE(A, T I ,T ) = 2A.

Remark. Instead 1.5 it may suffice to assume that in Definition 1 . 1 , there is a model M , 2 A, and a P € M realizes p, {b, :i < w ) is indiscernible over A U {a, : pE S ) , and for each p, p is not realized in the Skolem Hull of A U {ag:q # p) U {b, :i < w ) . EXERCISE 1.8: Rewrite the proof of 1.8 where you prove only that for any v p E S1 for some lh(v, p) 5 h(lj,[l]), .ii,[l]) I; h(v, p) + 100, assume (*1) holds for even n's, (*2) for odd n. Is it simpler ?

+

EXERCI8E 1.9: Show that in Definition 1.l, B and A can be chosen independently of the cardinality.

Our main theorem here is THEOREM 2.1: I(A, TI,T) = 2"owidd holds:

(*I

that t h following condition

T is urcswperstable, A 2 ITII + K1 and at leaat one of the following me8 occurs: (i) A > I TI/; (ii) Po = A; (iii) A = A,, A, < n, A> = A,.

x,,,

Proof. By the subsequent theorems. If A is regular by 2.2 for p = A. Also if there is p < A, 2" = 2% 2.2 implies our conclusion. If there is p < A 5 pNo, 2" < 2Qhe result follows from 1.7(1). If none of the previous cases occur end there is p < A, 2Y2 A, 2.3 gives our result. If p < A =- 2' < A, but A is singular, there is alwaya X, of A 5 x < A, x = KO or x a strong limit cardinal of cofinality KO. Noticing that Aof" Theorem 2.6(1) gives our result. We covered all possibilities, thus prove the theorem. When (*) (from 2.1) holds there is a model No of cardinality ITII, proper for (us", TI), such that the skeleton of EM1(ws", N o ) is a uniform KO-treeof the form <(cpn, 1): n < w), vn E L (by VII, 3.6(2) and 3.5(2)); let cp,, = cpn(x;gn) = vn(J;8). In Case (ii) we can aaaume

456

TEE NUMBER OF NON-ISOMORPHIC MODELS

[a. VID, 8 2

N o is K,-compact; (by I, 1.7, and as we can replace N o by any elementarily equivalent model of the same cardinality). In Case (iii), by VII, 1.11 we can assume there are P, E INol, (n < w ) P, E P,+,,lNol = Un
.

THEOREM 2.2: Suppocre T is not auperstuble. (1) If A 2 p > lT1l, p regular then IE(A, T,, T ) 2 2’. (2) If (*) (ii)or (iii) ho& No < p 5 A, pregular, t 7 m I(A, T,, T ) 2 2”. Proof. For every ordinal 8 < p cf 8 = Nochoose a (strictly) increasing sequence of ordinals 7s = $8) whose limit is 8. For every w E p let I , be the KO-tl’Wp<” U {qd: 8 E W } u {(a): p 4- IT11 5 a < A} ( p -I-lT1lcardinal addition, so in (ii) and (iii) the third psrt disappears). Let ilfk = EM1(Iw,N o ) and M, be the L-reduct of Mt,(we use the notation of 1.1). By 1.3 of the Appendix there are pairwise disjoint stationary subsets of p, u, E (6 < p: cf 8 = KO} (i < p), and by 1.5 of the Appendix there are B(a) E p, (a < 2’) such that #(a) G 8(p) * a = p. Let ~ ( a=) u,. Our family of models is {MWca,: a < 2’). To prove that it exemplifies our conclusion, it s d c e s to prove: supposing w , u c { & < p : c f 8 = No},w-uu-warestationary,andf:ilf,+ilfu is an isomorphism for (2), and for (1) an elementary embedding, we get a contradiction (“ u -w stationary ” is not needed for (1) and can be weakened to “A-w stationary” for (2)). Clearly for every q E I , there are T,, E LE, F(q) = F, E p”” n I , and I ( q ) = I, E {(a): p IT11 s a < A}, En E No such that

+

Now we can define by induction on i c p, A, c INol, a ( i ) < p and X,_C {(p): p + ITl] 5 p < A} such that: (A) X,, A, are increasing (by G )and continuous sequence in i. (B)1-q< p, 1 41< p, and if IT11 < tc, A, = INol. (C) a ( i ) is a (strictly) increasing and continuous sequence, a ( i ) 2 i. (D) I f y E l , n uB
CH.

MII, 8 21

UNSUPERSTABLE THEORIES

457

(E) If r ) ~ a (+i l ) < *and fl < p but fl 2 a(i + l), then there are infkitely many y < a(i + 1) such that: ( a ) CC > I T11 =- % = %-* (/3) i+ modX1 (in I,, so, in fact, in the ordered set {(a>+ + IT11 5 a < ( y ) fi,,n^ + mod a(i)*". (6) +,,n^ = + V < Y ) It is easy to define a ( i )inductively; for i 1 notice that the demand U is only that in the Skolem closure ofA,+l U { q , : v d w n X,+, there will be some < y elements, and as y > KO is regular, clearly such a(i+ 1) exists; for E) notice that I, is X-atomically stable for any 2. Now clearly S = {i < p :j < i =- a(j w ) < i} is a closed unbounded subset of p, hence (w - u) n S is stationary. N

$n

w.

N

+

ul
+

Prooffor (1). (p > lT1l).Choose 6 E (w - u) n S . As 6 4 u, r), 4 I,, hence if u1 = ij,,,[Z] E p", then (as it is increasing) there is at < 6 such that vt[n] < al e v1[n]< 6 e-n < nt where nl s w . If v1 = ind[l] ~ p < there * is a, < 6 such that q[n] < a, o v l [ n ] < 6. Let a* = g(6) be < 6, 2 at and such that x1 = min{z E x d : 2 anna[l]} E X,. or x1 = a0 (exists as Xd = < X1).By the dehition of Sthere me 8', n; a* < a ( i ) < 6, r ) d [ r n ] < a ( i ) for rn < n but ?la[?&] > a(i + 1). Let p = v8 n,/3 = v8[n],then by (E) there are, for m < .w, distinct y ( m ) ,u* < a ( i ) 5 y(m) < a(i + 1) < fl, such that ZDn gDn mod Xi, = ZDn ijD-<,,> mod a(i)$* and +p; SDn
u{,,

-

-

N

Proof of (2). As p 5 I T,I clearly Xi= 0. We define g as in the proof of (1).As (w-u)n SisastationarysubsetofyandforeachSinitg(8) < 6,by 1.3 of the Appendix for some a* < y , w o= { 6 : 6 ~ ( w - u n) S , g(6) = q}is stationary. As the number of r) E (a*)'* n I,, is < p ; and if we partition a stationary set to < p parts, at leaat one is stationary (see 1.2 of the Appendix), there is a stationary w1 E w,,such that for all 8~ w1: = T is constant (as Tv(8) E LE, LE is countable) ; ( F7(8)[Z]) n is constant or always 4 (a*)s*; the similarity type of $,,(a) is constant;

r

T E E NUMBER OB NCYN-ISOMORPHIC MODEL8

458

[OH. VIII,

82

and there is a constant n(6) = n(*) such that ZMa, E Pno,(N,). Clearly we can find a closed unbounded 8, G 8 such that (1) if a 7 E a<” and there is 6 E w1 such that 7 Q 7 6 then there is such 6 < a. (2) if a E S1, 7 E a < @ and (6 E w1: 7 Q 7 6 ) is s t a t ~ a n then q there are v E a<”, with arbitrarily large ( < a )last element such that (6 < p: 7 - y Q vd}is stationary (the existence of such v’s follows from 1.3 of the Appendix). (3) If SES, n wl,r]Ea<W,r]Q rs, then SEW,:^] Q q6}is stationary. So for each limit point S* of S,, cf S* = No,we can easily find a strictly increasing 7 of length w with limit 6*, and S(n) < a*, 6(n)E wl,and S(n)< P(n) < S(n 1 ) where P(n)€8,; such that 7 n = r]b(n) n (define S(n), /3(n)simultaneously by induction on n w , using the properties ofS,). Choose also S > S*,SE wl. Then we can check that ?(&(,d)), Zv(d(n))) satisfies f (%$tI)): < (as Gvlmhfiv(d), FnrmhGMd(n)) are similar, by a*’s definition). Now choose E E P,(*)(N,)which satisfies every formula of L, all but finitely many Zv(d(n)) satisfy (a exists by N1’s definition). So clearly ~(7i~(,,,,, E) realizes {rp,(E, f (8v,m)): m < w}. As in (2) f is an isomorphism, in M , {vm(E; Bqrm):m < w } is realized. Hence S*EW. As S* was arbitrary S, G w.So we finish the proof of (2) too.

r

r

+

THEOREM 213: Suppose (*) from 2.1, and that p 5 h is regular; h > IT11 * p > lTll; 2 Y 2 A, p > KO; and x < p * XNO < p. T k n I(h, T,, T ) 2 2A. Procf. For 8 a sequence of subsets of (6 < p : cf 6 = No}of length A, let I E be the Ko-tree{q E A<”: $01 < A, (Vn > 0 ) ( ~ [ n<]p)} u (7:7 E A”; n > O-r[n] < v[n+i]< p ; 7’s limit is 6 and S ~ t @ [ r ] [ O l ] } . Let Mi = EM1(&,, N o ) and MG its L-reduct. Choose for each 6 < p, cf 6 = Ho a (striotly) increasing sequence of ordinals of length w , 78, whose limit is 6, and 7d[o] = 0. For a sequence 7 define q* such that 7 = (7[0])-7*. Let u,, i < p, be pairwise disjoint stationary subsets of (6 < p : cf 6 = No},and {#(a):a < 2”) be a family of subsets of p, no one of which includes an intersection of finitely many others. (See 1.3 and 1.5 of the Appendix for existence). Let {ui: i c 2A} be a family of subsets of A, no one a subset of the other. Let a,be a sequence of length h whose range is u,:a < A, a E u,}. Our family is {ME,,,: i < 2”). So it suffices to prove that iff is an isomorphism from M E onto Mg then for every i < h there are n < w ; j,, . ,j, < A and closed un-

..

CH.

n I I , 8 21

UNSUPERSTABLE THEORIES

459

-.

bounded8 E p such that @[i]2 @[jl] n. n @[j,,] n8. For simplicity for YE&,, (so i$ = F(7) let i = 0, p > IT,\ and let f(a7)= T7(ti&(7),~7) E I,, Z,, E N o ) .Now define by induction on i < p, ordinals a ( i )and sets Xisuch that: (A) Xi is an increasing and continuous (by G ) sequence of subsets of h of cardinality < p . (B) a ( i )is a (strictly) increasing and continuous sequence of ordinals
(C) If 7[0] = 0, 7 ~ n U,,,pSu 1 ~ then (F7[Z])[O]~X,, and (F#])*E B“” (this is possible as x < p * X’O < p ) . We know (see 1.2 and 3 of the Appendix) that if w c p is stationary then if g is a function from w to a set of cardinality < p then for some t , {OIEw : g ( a )= t} is stationary ; and if g is a function from w,g(a)EX,;X,is an increasing continuous sequence of sets of cardinality < p then for some t, {OIEw : g ( a )= t} is stationary. By successive use of this we can find a stationary w1E 301 and a(*) < p such that : (a) For d 6 E w1 T,,(d) == ?, Z,,(d) = c. (p) For all SE w l ,v7(&)has the same similarity type. (7)For all ~ E W ,for each Z,Bj = (FvO[Z])[O] is constant and EX=(*) or always it # Xd, but min {fi E xd:fl > fij} is fixed (maybe as GO).Let pi: 0 n Xam be (jl, * ’ ,jn}. (8) Let 4 = (fiMd)[i])*. Then for some n,, vf I n , is constant and E a ( * ) s ” ,and vi[n,], if defined, is 2 8. Now, &B in the proof of 2.2(2) we can find a closed unbounded 8 E p, such that for every 8 E 8 n Qj,] n. . n @[jn] for some 7 E pa, 7[0] = 0, 7 strictly increasing with limit S, {vn(~,f(lZvr,J) : n < w } is realized in Mu,hence 8 EW[O], what we want to prove (if p s lTll the changes are like the proof of 2.2(2) and we can have llN,II = JTJ’o).

-

DEFINITION 2.1 : For models M, N cardinal x and ordinal a we define a game GE;(M, N) [GI;(M, N)] between the players I and I1 as follows: in the @h move player I chooses i, < x and af E M i < i,, and then player I1 chooses bf E N, for i < i,. The play ends after a moves, and then player I1 wins if tp*({bf:i < ifl,/3< a},f3,N)= tp,({uf:i < i,,

p < 4, B,M),

[TPP({bf:i < i,, fi <

a}, 0, N

) = TPSX({af:i < i,, /3 < a},0, M ) ]

and player I wins otherwise. A strategy (of a player) is a sequence of functionsf, (fi < a) which “tells” him what to do (f, for the fih move) depending only on the previous choices in the play. A winning strategy

460

"HE NUMBER OF NON-ISOMOBPHIO MODELS

[OH. WII,

82

is a strategy such that in any play in which the player "behaves" according to it, he wins. A player wins in the game if he has a winning strategy.

Remark. Sometimes we denote that player I chooses sequences Zf instead of elements.

LEMMA 2.4: (1) If there is an elementary embedding f of M into N ; then player I1 wins in QE,O(M,N ) . (2) If M , N are isomorphic, pluyer I1 wins in QI;(M, N ) . ( 3 ) In the gamea from DeJinitim 2.1, at most one player wins. (4)If player I1 wins in GX$(M,N) he wins in GXI;(M,N) where /3 2 a, X E { I , E}. ( 5 ) If X E {By I } , player I1 wins in QX;(N,,M I + , ) 1 = 1, 2 then player I1 wins in QX:(M,, Ma). Proof. (1) In the pth move player I1 chooses be = f(af).Also the other proofs are immediate.

To clarify the proof of 2.6, we prove first: LEMMA 2.5: Suppose A = Ano 2 I T l l , Mi are models of T of cardinality 5 A, for i < A. 8uppo8e T i s unswperetable. Then there is N E PC(T,,T) such thut player I Win8 i n C#I&(N,Mi)for m h i c A. Proof. Let g be a one-to-one function from {(i, a): i < A, A. Let g ( g l ( a ) , g,(a)) = a for a < A, and

I = A'"

U {r] E A": gl(q[n])=

g0(7[n1)E

si E

lMil} onto

i for every n, and

1 ~ and~ {vn(s; 1 ~ ~ ( q r n i ) )<: w>

is (well defined and) omitted by M,} and let N = E M ( I , N o ) (No-proper for (us", T,)and of cardinality lTll). Let us show how player I wins in GI$(N, Mi): for p = 0 player I chooses the sequence a,, r] = qo = ( ); for p = n + 1, if in the mth move player I1 chooses 6m for 11c 5 n, player I now chooses a,,, r] = v,+, = ( g ( i , 6O), . . .,g(i, 6")). If in such play player I1 wins, necessarily N realizesp, = {v,(z; a,,,): n < w } iff Mi realizes {v,,(s;gn):n < w}, (letting Y E A", r], 4 Y); then N realizes p v iff Y E I iff Mi omits {v,(z;gz(v[n])) : n < o}= {q,(z; 6"): n < o},contradiction.

CH.

VIII, Q 21

46 1

UNSUPERSTABLE THEORIES

THEOREM 2.6: S p e A 2 p = fl = 21, (VK < A ) ( K X s A) and A 2 lTll. Then: (1) I ( A y T i , T ) 2 A". (2) There are models Mh E PC(T1, T )fOr h : p --+ A M h that if fop 8Orne i hl(i) < h,(i) then in aI:(Mhly Mh1)player I wim. Proof. Clearly (2) implies (1).By the assumptions on A we can assume No,Nlare as mentioned in 2.l(ii)or (iii)except that maybe lliV0II > IT,I (but llNoII s A). We can assume x = xn, xto = xn or x = K O and let xn = n. We define by induction on a < A pairwise disjoint wellordered sets JL (i < p), lJ6.1 = IaIx + p s A, and for a > 0 the universe of JL is the set of quadruples 8 = (a,5, u,y ) where 5 < x and u is a subset of (J;)""of cardinality < and y < a,and let gl(s)= a,g,(s) = 5, g3(8)= u , g4(s) = y . For a = 0, we let J i = ((0, g, i, - 1) : 5 < x} and, to simplify notation, iet distinct Ji's have distinct empty subsets. ~ "His(q) be the The well ordering is arbitrary. For ~ E ( J ; ) let , V E S n, < Z ( v ) , p ~ g ~ ( v [=ns]-)p ~ S . smallest set S such that ~ E Sand Let His*(q) = { ~ [ n Y ] :E S n, < w } . Let DP(q) be the order type of T(q)= {p:His*(q) n J i # (4 for some i or gJs) = /3 for sEHis*(q)}. Clearly His*(q),His(q) have cardinality < x hence DP(q) < x+. WeshalldefinesetsSI c xmfori < p , y < X+.Nowlet,forh:p+A, J h = &
zncU

x

uf<#

v,,, = (J;,ct,)san {;,,[I]: 1 < I(;,,),

Let uo = { ( ( U o ,

E u,,,}.

t, 0, hdi))): 5 < XO}, and

un+1=

(7 -(
CS,vn,

h ( i ) ) )CS < Xn+1, V E U n } .

Clearly the strategy is well defined and the nth move of the player I depends only on what player I1 haa done in the previous moves. Let u,, I < I(;,,)}; the history R of this play be (J{His(Sn[Q):r) E Uncm (Clearly lEIl 5 x) R* = U {His*(;,,[l]): r) E Uncm u,,,I < I(;,,)). So it suffices to prove we can define the Sr's so that those strategies me winning strategies for player I. Suppose we have another such play, (where player I plays by the

strategy we mention) with hi, ...,Ek, . ..,H ' , instead of h,, ...,En, ...,H. We call the two plays isomorphic if there are functions f : H +H' fY(y < p) with domain {a:JL n H , # 0 or a = g&) for f*:H*+H*,gn:un+u6 such that: (1) f,fa, g,, are one-to-oneand onto H',Ha, u;,resp.,

8 E Ha},

If 8 E Js,f(8) E J ~ , ( and I ) f, is inCreeg. (3) If s = (B, 5, u, a>E JJ then fb)= (fy(B), S,f*(u),fy(a)> where (2)

f* (4= {f*(0:t E 4.

(4) (f(s))[nl = f*(r)bI)Y f @ o ) = Y 4 i = i'. ( 5 ) s0(((a0, 6,6,h,(i)>>)= ((a;,5, B, h ; m > , and

gn+l(rlh<(a0,

5, V,,, hdi))))

= gn(Vr<<4Y

5 , 4 , Y(i')>>.

If r) E u,, f*(%$I) = q l where p = g"(r)). (7) If q(Z) E U , ( ~ ) Z s n then; denoting p(Z) = g,,,(ll(r)(Z));EM0,^. nCM,,), E;(o)- * * h-' cp(,,) realize the same type in the L2-reduct of N,. (8) f* J; :J; -+JJYu) is order preserving ; moreover if sl, s2E J; (6)

-

r

-n

Domf,, then

&{&Y

I{t EJ?: 81 S t < 82}1} = &{&I,

[{t EJSI0:f*(81) 4 t

< f*(82)}l}.

Clearly, by the dewtion of the Ih's, the winner in two isamorphic games is the same (asthe La-reduct of iVl is 8,-homogeneous). It is also olear that the number of isomorphiam types of such playa is s 2x = p. Let {at:j < p} be a set of representatives. We define by induction on j,a set I',of non-oontradictory"requirements " E 8r, $ 8r,euoh that II', - I'ol s ljI x and I',+, c'ensure"' the victory of player I in the play @. Let To be (7E&: r ) E xm, is eventuaJly zero}. If I', is defined, let us use for at the notation in the definition of the strategy of player I. Chooae r ) E x,, suoh that r ) does not "appeaz" in I', (thereis one by cardinality wnsiderahions). Let r)o be defined by yo 1n E Ua, and ( . .,g2(r)O[m]), .) = r), and let p = {V,,(Z; 7in,r,,): n
+

n,,<,

.

..

OH. WII,

8 21

UNSUPERSTABLE THEORIES

463

r e h e q] hence by the strategy definitions, Hie*(r)) z Hb+(i@]), a. = max T(q)4 Hk*(c[m]),so /3 < y. So F,+, ensurea q is realiz;ed in H,,,, but p is not realized in M,,, hence player I wins. But if there are no such #$'a, let F,,, = F, u (7 E For limit 6, F, = Uje6 F,;so Fu shows us how to define the 8's properly:

s; = {?j:{T/€AS;}Eq}.

Pro$ The proof is similar to that of 2.2. By 111,Exercise 4.13 we can find a,,E for r] ~ that:

yand~pnE" L such

(1) F o r q E p m , v E p n ,(la > 0) C 9 7 , , ~ , , ; t Z v 3 i E v 4 r ) . i < p are pairwiae contradiofory. E pn, the Q,, + ,(Z; (3) For r) c p S m ,tp(tZ,,, A,') does not fork over A: where A: = U {G,,,,,: ta < Z(r))}, Al, = (8":v # q but not q Q v}. Let, for S < y, cfS = w , 7, be an increasing sequence of length w of ordinals with limit 6, and for w E {S < p : cfS = KO} let M , be a F&-constructible model over A, = Ck of cardinality A, where CL={cT7: ~ ] E U ( " or 7 = y d , ~ E W &
u

( n , + l ) $ S c u for Z < m ; then C = U { a v : v Q ~ z ~ n z , Z or < mv ,= r ] , n, = w } is finite, and tp(Eq,Cb,)does not fork over C, where r] = (q0, ...,T ~ - . ~ >hence, , by 111, 4.13, and as B," c Cb, U A;, also tp(cTq,Cb,U Ab,) does not fork over C. Now by IV, 3.2 if ~ E A U, Ag there is finite B c A, U Ab, such that tp(b,A, U Ab,) does not fork such that B n A, c aq,%nd there is finite over B, and there is C as above; so tp,(6 U B, Cb, U A ; ) , hence tp(b,C: U A:) does not fork over C U (B n (Cb, U A;)). So for every ~ E AU, AL tp(b,Cb, U Ab,) is F&-isolated. Ae in 2.2 it s&ces to prove that i f f : & -+Iw M uis an elementary embedding,then for some closed unbounded8 s p, w n8 E u n8. By renaming we can aasume f maps A, u A5 onto A , u A t (see IV, 3.3), hence 8, = (6: 6 < p, f maps Cd,u A: onto 0: u A:}, 8 = (6 ~ 8 ~ : ~ E , U ( O

464

THE NUMBER OF NON-ISOMORPHIC MODELS

[CH.

nII, 8 3

6 a limit point of S,} are closed and unbounded. If 6 E w n S but 6 $ u, then 6 = .f(iZnc6J will give us a contradiction, as tp(6, C$ u Ad,) splits strongly over any CE u A4, for any a c 6 (because {ii,,-(t>:6 Ii c p} is indiscernible over Ck u A&,also {f(iZ,,-):6 Ii c p} is indiscernible over Ci u A t for 6 E 8). EXERCISE 2.1: Suppose A = lTll = (VK < ~ , ) ( K ~ < O K,) A 2 = = 2%;/3 < x+ or 2”3l I 2%;T not superstable, then I(A, T,, T ) 2 2’. PROBLEM 2.2: Uniformize the proofs (maybe use the filters from Kueker [Kk 771 and Shelah [S761 Section 3). CONJECTURE 2.3: IE(A, T , , T ) = 2Afor T unsuperstable, A > IT,I. EXERCISE 2.4: Show that for A regular > IT], T not superstable the partial order (P(A),G ) can be embedded into {M:M C T,llMll = A} quasi-ordered by elementary embeddability. EXERCISE 2.5: For each cardinal A = P o , show there is no Boolean algebra M, of cardinality A, such that any other Boolean algebra M of cardinality I A has an embedding in Mo,preserving countable intersections. (Hint: See 2.5, Th. 12, Grossberg and Shelah [GSh 831.) EXERCISE 2.6: Generalize VII, 3.7 and 1.7 to the cme IT,] c 2b, assuming MA (Martin Axiom).

VIII.3. Saturated models and the case A = ITl[

DEFINITION3.1: The indexed-set (4: 8 E I) (I an index-model) is called K-skeleton-likein (a model) M when: (1) It is indiscernible. (2) For any E E M there is J c III, IJI c K such that if By 8 E I , B t mod J , then for any p E L(M) M C p[@,E] E p[?if,El. N

DEFINITION 3.2 : The orders I , , I , are called K-contradictory if there is no model M ynz c w , an antisymmetric formula in L(M) [E < j’j](Z(E) = 1(jj) = m), an order J with lower cohality 2 K and sequences iZ8 E & I for 8 E I , u I , u J (we w u m e for notational simplicity I , , I,, J are disjoint) such that:

OH.

=,8 31

SATITBATED MODELS

+

465

+

(1) (a*: 8 E I1 J ) , ( g 8 :8 E I , J ) are K-skeleton like in M . (2) For 1 = 1, 2, s, t E I , J , M t= [a, < iLt]u(*rt).

+

LEMMA 3.1: (1) 8wppose a,, M are d e b of T,, K is regular 1M1 = l d l o l U {a,: i < a},4 E lM,l for 8 E I and (a8:8 E I ) i8 ~-8kehtonlike in M,. If the type each a, realizes over lMol U {a,: j < i} ie F~-isolaterl,or even F~-isolated,then (a8:s E I ) is ~-8keketOnlike in M . (2) The skeleton of E M 1 ( I ,@), or of EM1(I,N ) are ~-8kehdt-mlike, for every K 2 8,. (3) If (a8:s E I ) is the skeleton of M = EM1(I,N ) , M , t k subntodel of MAID(D ultraJilter over A) whose universe is {h/D: for 8ome J E I , IJ( < K , { h ( i ) i: < A} E dcl{gS:sEJ)}(clearly it exists), then ( g S : s € I ) is K-skeleton like in M , and M , < M A / D ( d c l i nL,). Proof (1) Prove for CE POI U {a,: i < p} by induction on 8. (2) and (3) immediate.

THEOREM 3.2: If A 2 lTll + K + , A
DEFINITION 3.3: (*)! means that there is a family of 2” subsets of A, each of mdinality A, the intersection of any two is < K . Remark. This is just not AD(2”, A, A, K ) (see Definition V I I , 1.11). A related property appears in 1.9. On independence results concerning it see Baumgwtner [Ba 761. Proof. Let 0.be proper for ( w , T,), such that the antisymmetric formula, 83 E L orders the skeleton of EM(w, @). We assume the assumptions from VII, Section 2. [Z <

Case I : not (*)!. By 3.3 of the Appendix there axe 2Apairwise K-contradictory orders I,, a < 2“; each of cardinality A. Let, for a < 2A,i 5 A If, be an order isomorphic to the converse of I,, and sf,E I,. Let, for a < 2”, J , = z,sAIf,. Let N: = EM1(J,, @), SO llNiII = A (T,has

Skolem functions). Clearly (as:8 E J,) is the skeleton of N;. Let N i < M i , M i satisfy (2) and (3) from the theorem, but such that (as:8 E J,) is still K-skeleton like (this can be done by 3.1 and VII, 4.4), and h = IIMi(I (by cardinality considerations this is possible). Suppose the number of non-isomorphic M,'s is < 2Aand p 5 2Ais regular, where M a is the L-reduct of M i . Then for some M S = {a: M a z M) has cardinality 2 p and let fa : M a+ M be such an isomorphism. Let & = f,(a8) for s E J,, a E S. So ( g : s E J , ) is K-skeleton like in M. Let W, = {@: 8 = s f , i < A}. Suppose a # p; a , p E S , [ W a n W,l 2 K; then, as h is well ordered, there are for < K i ( 6 ) < A, j ( f ) < h such where a(& = @), t ( f ) = g e ) , and i ( f ) ,j ( f ) are strictly that 8&) = 6tCe) increasing. h t &I) = SUp{i(f): f < K} 5 h, 8(2) = SUp{j(&: 5' < K } Ih. NOW clearly

are r-skeleton like in M. Looking at Definition 3.2 this clearly proves I,, I, are not K-contradictory; contradiction. So a # / 3 ~ 8 *IW,n W,I < K ; also IW,l = A, and all those W, me subsets of ]MIrnfor suitable m, which has cardinality A. If 2A is regular, choose p = 2", so we have proved (*)f, hence finished, otherwise use 3.1 of the Appendix to get the same contraxiiction.

Owe 11: h = p+, hand p me regular. Let I, be the set {(i,j,y ) : i < A, j < A, y < p} ordered by: (il, jl, yl) < ( i s , j,, 72) iff il < ia OT il = ia, j, > j2, or i, = i,,j, = jz, y1 < y2.We identify (i,j,0) with ( i , j ) . For every I c I, let M,(I) = EM1(I,@), M,(I) = Mo(I)A/D where D is a regular, good ultrafilter over A (exists by VI, 3.1). Let d,((i,j, y ) ) = {i,j,y},andforJ c I,,d,(J) = (JseJd,(s).ForeveryI G I 0 , I t / D ~ M 1 ( I ) let dl(h) = {s E I: for some i < A, h(i) = T(@, 8 minimal, and 8 = t[Z] for some 11, (note w.1.o.g. Cis uniquely chosen). Now for I G I,, let M2(I)be the submodel ofM,(I) whose universe is { h / D: Jd,(h)J< y}. By 3.1(3) M,(I) < Ml(I) m d (a,: 8 € 1 )is p-skeleton-like in Ma(I). It is also Clem that M , ( I ) is K-saturated because if J c I, IJI < p then M 2 ( J )= M , ( J ) < M 2 ( I ) is even y-saturated (by VI, 2.3 and VI, 2.11). In the same way we can prove that for such J, if p is an m-type over M 2 ( I ) finitely satisfiable in M 2 ( J ) ,Ipl d h then p is realized in N,(J). Let 93 = (R(k),IS), and define 93, for a 5 p such that:

OH.

VIII, 8 31

(1)

pall

(3) (4)

@n:B

467

= A, %<9.

(2) If a E 193,l

b E I%l*

SATURATED MODELS

and a has cardinality 5 A (in R ( k ) ) , then b E a; e-

I%+il, I%l

L,I,, D, Mo(Io),J

H

=

Ui
19d.

M l ( J ) , (a8:8 E I,) all belong to 9,; and

IE91.

( 6 ) In fact, 93, depends on I so when confusion may arise we write 8, = S,(I), but sSo(I) = SO(I0). Let for a 5 p, M i ( I ) be Ma(I)as interpreted in 3,. Clearly Ni(1) M a V ) and JfiV) = U;
<

Let N;(I) be the L-reduct of Mg(I). Notice that when cf 6 2 K, Ng(I) satisfies conditions (2) and (3) from the theorem. For w E A let I(w) = {(i,j , y ) E I,: i $ 2 0 * j < p } . Suppose wl,w 2 c (6: 6 < A, of 6 = p}, w = w 1 - w, is stationary and there is an isomorphism f from N t onto N{ where Nf = N;(I(w,)) (for I = 1 , 2 ) . We shall get a contradiction, and this is sufficient t o prove the theorem in this caae (as in the proof of 2.2). Let 68 = f(a8). We can find a stationary w' E w,and 5 < p such that for a E w', 6(,) E% and U, = {i < A : t?(a,t) E@} is a stationary subset of A. Let, for a E Ml(I0),d2(a)be a set J c I of minimal cardinality such that d,(lr) c J where a = h/D (we can aasume the function d , belong to So). Also, for E Ml(IO), da(a) = Ul da(a[Q).SO for each E dla(Io), Id&)] < p. Let d&) = (a < A: for some 8, (a,8) E &(a) or (8, a) E da(a)}. SO clearly, for a E Ma(Io), d,(@ is a subset of A of cardinality )n da(6<~,o>)E V , V a n Vn E V , and V E VaY Van p = V, (ii) if i E dg(6<,,,>), i 4 V then i 2 a, (iii) if i ~ ' d 3 ( 6 < , , 0 >then ), i < 8. (3) L e t a E w " , i , j E U ~ , < i j. (i) &3(6)n d3@<,,,>) Va, (5)if f E d3(5),f 4 V,, then t 2 i, (iii) if f E d3(6<,,;>),then i, f < j. (4) If i E U;,then i > sup V , and i > a + a, and a > p. ( 5 ) There are sets B , E ~ $ +( ~ a < A ) , B , G N 2 , IB,l < A ; P E W ' 8 < a =- 6<, E B,, and B, = Used B, (this ody because 9, ~93,+1).

=

468

THE: NUMBER OF WON-ISOMORPHIC MODEL8

(6) a E w", c E B,

[OH.

WII, 5 3

=- d3(c) c a.

(7) l o > SUP(V n p ) , l o < p. Cliooee 6 E w" which is an accumulation point of w", (so bigger than lo,and than p). So 6 E wl,6 # wa, cf(6) = p and (6, i) E I(w1)o i < A but (6, i ) E I ( W a ) o i < p. Let J o = {(i,j,y ) : (i,j, y ) E 1 0 ; i,j,7 E V } . Define J = J, u ((6, lo+ n ) : 0 < n < w} u {(a, 6, lo+ n): n < w, a E V , a E wa}. Clearly I JI < p, J E I ( W 2 ) . Let (for a < 8 ) A = {a:U € N 5 , , d 2 ( U ) E { ( a , i , y ) : a= s*i <&,y < &}},

A,= {a E A : d&) c a v ( A p =

{'p($;

E): E E A , 'p E L, and for all sufficiently large a < 6,

% ' PO'p

a)},

'

db;

1' 1,

IAK'

As f is an isomorphism, (b8:8 E I ( w , ) ) is p-skeleton-like in ivg, so p is a complete type over A. We now show that p is realized in N2(I(w2)), andeveninM,(J).As IJI < p, IpI s h,its~cestoprovepiehitely satisfiable in H,(J); so as p = u a < d pa, p a inoreasing it suffices to prove any finite q s pa is realized in it. By the choice of 6, p there is B E W " , a < fl < 6, B > p such that 6<8,,)realizes q and let it be (ho/D, ,hn.,-JD)where dl(hl) s cEa(6<
.

...

-

(B) Z E J , ( y ) if 8[m] E J,, then t[m] = arm], (6) if B[m] = , i > 8, then i[m] E ((6, to k): 0 < k < a}, ( 8 ) if 9[m] = (i, j ,y>, i < B (so i E V),9[m] # J, (so/3 s j < 6) then t[m] ~ { ( i6,, lo k): k < w}. Clearly this can be done, and 3 imod {(i,j, y ) : i = 6 j < to,and i < 6 * i s a} hence if 6 = (hb/D, ,hk-JD) then 6 O realizes q, and

+

+

- .. .

by the above 6 c M 2 ( J ) . SO by previous remarks there is 6 ~ M , ( l ( w ~realizing )) p. As (6#:8 E I(wl)) is p-skeleton-like in N;, l B d l < h, there is &1) < h such that all 6
z: E E B d , N&k E <

{z < E:

6
A , tp(E, Bd, N!l) = t P ( ~ < ~ , ~B(d l, )%)}* >,

CH.

VIII, 31

469

SATURATED MODELS

,

It is easy to check that q E p (by ( 6 ) )so 6 realizes q, and q E 8,+ . So, as 99 C "in M2(I(wa)) there is an element realizing p" also CZJ,+, satisfies this, so some 6' E Nb+1 realizes q, so (&: 8 E I(w,)) is not p-skeleton-like (as U;is an unbounded subset of A, a d i E U;=6 E A). Contradiction. Cme 111:A = A, is a limit cardinal, or A = A,+ ,A, singular; and for some p < A,, 2' = 2A. We can choose regular p < A, p > K , p+ # of A,, 2'= 2A. The proof is similar to the previous case. W.1.o.g. h = lEM(P,@)I. Define I,, M,(I), M , ( I ) as in Case 11. Let, for J E 1 0 p r l ( J ) = {a: for some 8, (a,8) E J } and M,(1) will be the submodel of M l ( l ) with universe {h/D:pr,[d,(h)]has cardinality e p , d,(h) has cardinality sp}. 23,, M ; ( I ) , are defined aa in case 11. For any function g : p+ + A, such that each g ( i ) is a regular cardinal > p+ , < Al, let I(g)= {(i,j)E l o :j< g(i),i < p'}. Now we suppose g,, 9, are such functions, and for every x .c A, {i < p + : c f i = p, gl(i) > g , ( i ) > x} is stationary, and f is an isomorphism from iVt onto N t , and get a contradiction, and this is sufficient [where iVF = N;(I(gI))]. Easily we can find 5 < p, B E B,, p+ < IBI < A, and S = {a < p + : 6<,,,> E B, cf a = p} is stationary, and let i3 = pa:a < p + } where pa is increasing (so 8, 2 a). Let w' = (6 < p + : cf 6 = p, gl(S) > ga(6) > IBI} (so w' is stationary) and let w", V be such that (exists by 1.8 of the Appendix): (1) w" is stationary, w" E w'. (2) V c p + , IVl < p' (3) If a E w",i ~ p r ~ [ d ~ ( 6 , ~i, 4, , V~ ,)then ] , i 2 a. (4) If a 6 w", i E V , then i < a. (5) Let B, = B n {c: c EN;, p l [ d 2 ( c ) ]E a} so B, E 23,+,, B, is increasing and continuous. ( 6 ) If 6 E w", a < 6, then 6<8a,o> E B,. Choose 6 E w",6 an accumulation point of w" (so cf(6) = p, gl(S) > ga(6) > IBI). Let J, = {(i,j): (i,j) €IO,i E V; and for some a E w", a < 6, (i,j) ~d,(6<~,,,>)}.Now there are 6, c gJ6) and U E gl(S), Iul =: gl(6) such that for i E u,d2(6 H,). Let A = {a: u EN$,d,(u) n ((6, i): So 5 i < ga(6)} = 0}.

470

THE NUMBER OF NON-ISOMORPEIU YODEL8

[OH.

m,8 3

A,, p , p a are defined as in Case 11, and so is the existence of 6 M2(I(g2))which realizes p (only part (e) falls).The resi is like Caee IIwe prove q is realized in N$ and so get the contradiction. So it suffices to prove that Cases 1-111, exhaust all possibilities. Suppose not (*)R, and it suffices to prove that for some regular p, p+ I A, 2y = 2A. By 3.2(1) of the Appendix AK = 2A,so if A I 2K, 2'5= 2A so p = K will suffice. If A > 2=,by 3.2(2) of the Appendix for some X , K + < < h , ~ 2" h so = 2 A )and 2X = 2A.So we finish.

x+

COROLLARY 3.3:If A 2 I T,I

x"

+ N,,

T uwtable, then I(A, T , , T)= 2A.

+ N,,

T not Bulperatuble, then I(A, T ) =

Proof. Take K = Noin 3.1. COROLLARY 3.4: If A 2 lTll 2A.

Proof.By 3.3, 2.1, 2.7 and 1.7(1).

+

THEOREM 3.6: 8 m e A 2 p 2 lTll K + , Aca = A, K < K ( T )K, , p areregular;andx < p = + x < I C< p.8upposealsotlurtp > lTll&x < p =- f s A. Then there are rnodek Hi,i < 2fi of T,, with L - r e d M,, ~ of ccrr&indity A 8wh t h d : (1) The. H,'e are p d r w h e non-imnmp~. ( 2 ) If p > ITJ,then no M, is elementarily embeddable in M, for i # j . ( 3 ) A¶, ia K - c o T ~ ~ & , ~-liomogeneous,and, if lD(T)l < p, ~-edureted. 8imilarly for

w.

Proqf. For simplidty we aasume p > IT,!.Using VII,3.6(3) and 3.6 we can find @ proper for (us",T,)such that the skeleton of E H ( o s a ,0 ) has the form ((cp,, 1): a < K a successor). For 8 < p, cf 8 = K choose qs e p K increasing with limit 8, and I, = psaU {qd: S E w}, and let Nt,= EM1(I,, a), Hr a IFb,-primq model over NL with constructing sequence {ur:i < A}, eatiafying (3). (For proving (2) when A s 2ITd we ehouldaeaumethatiffhetypeofaroverBp { a v : r ) ~ I , , , } u { u<~i;}j < K, then I{p: Bfo = B}I < A.) is rAv(DfO,d d Bfo), Let Hw be the L - d u c t of i& f :MW+H,, an isomorphism (for 2) elementary embedding). For (2) let p = A and the proof is aa in 2.2. For (1) there is V c A, IVl = p such that f maps {itn:r) E I,] u {ap:i E p) onto {a,,: q E I,,} u {ur:i E V). Let V = {i,: a < p}, Then there is a closed unbounded 8 E p such 3 :

OH.

Vm,8 41

CATEQOBICITY, SATURATION AND HOMOQENEITY

that for a E Sf maps {a,,:r) E I,, onto {a,,:7 E I,, r ) E Ubcr p'"} like 2.7.

471

UBeK 8'") u {a?:i = i, E V ,8 < a} U {a;: i = i, E V , 8 < a}. The rest is

r) E

+

THEOREM 3.6: 8-e A 2 p + IT1 K + ; A<' = A, K < ~ ( 2 ' )K;, p are regdur, x < p =- xcu < p. Then T has 2' n o n - k m p h i c models of cardinality A, which are ~-lumtogeneowr,K - C O ~ Z ) ( C C ~and , if ID(T)I s A also K-saturated.

Proof. Similar to 3.6 and 2.7. CONJECTURE 3.1: (1) If A = A"K 2 lTll + K + , K < K ( T ) ,K is regular, then there are 2" non-isomorphic models M E PC(T,, T ) of cardinality h satisfying (2) and (3) from Theorem 3.2. (2) Moreover, if h > I T,I, no one of them is elementarily embedded in any other. EXERCISE 3.2: Prove conjecture 3.1(1) for A strong limit cardinal > D ~ .(Hint: look at 2.6.) EXERCISE 3.3: Prove conjecture 3.1(1) when there is p, p c K = p < A, 2' = 2".

EXERCISE 3.4: Prove conjecture 3.1(1) when regular, or p < A s p", 2fi < !P.

K

< K ~ ~ and ( T )A ie

PROBLEM 3.6: Axiomatize the proofs in Sections 2 and 3 (toinclude a180 1.7(1)). Remark. Clearly in 2.6, 6 and 1.4, 7 we have less than in, e.g., 2.2. course, it will be d d a b l e to use fewer axioms in each proof.

Of

VIII.4. Categoricity, saturation and homogeneity up to a cardinslity THEOREM 4.1: Suppose PC(T,, T ) i s categorical in A. (1) If A 2 lTll + K,, then T k8tabkandif A > 2No lrae nd the f.c.p. (2) If A > lTll then T k eulperstubZe without the f.c.p., and dabk in every p 2 lT1l.

Proof. (1) T is stable by 3.3; hence hae not the f.c.p. by VII, 3.4. (2) T is superstable by 2.1. By VII, 2.7 there is M E PC(T,, T)of cardinality A, stable in lT1l. If T is not stable in ITII, there is M E PC(T,, T ) unstable in ITl!, 11M11 = A, contradicting the categoricity. So T should be stable in IT ,I, hence by 111,O.1 stable in every p 2 I T,I. By VII, 3.4, T is without the f.c.p.

THEOREM 4.2: (1) If T is not mperstable, then for every A > lTll there is a nvn-I T,I +-model--model in PC(T,, T ) of c a r d ~ d d yA (80itknot IT,I+-lurmogeneoue,~-~~~~ SM ) . below. (2) If T i8nOteulpW8tabk, IT11 < p 5 A,p*@rY (VX < p)(xWO5 A), then there are rnodels Mt of T , of cardinality A, for i < 2y thd (i) for i # j,the L-redwt of 211: cannot be elernentady ernbedded into the L-reduct of Mi, (ii) the set of ieommphhni typerr of elementary submodels of Mi of cardinality < p doea not t-kpnd on i. Remark. M is A-model-homogeneousif M,< M ylldl, 1 < A, f : dl, Ma, Ml < N , < M y llNIII < A, then for some f ' extending f, and N,, M2 < Na 4 N , f ' : N , Na. Proof. (1) We use the terminology of 2.2. Dehe a ( i ) for i < I!l',l+ as follows: a(0) = 0, a(i + 1) = a ( i ) + i , a(&)= Uleaa(i). We SSY u = /I + 9 i f u [ i ] = /I + q [ i ] for any i < l(u) = l(q). Let

I,

= A<"

u {a(in-l)+ ...+a(i,)+7ja:s < a(i,) < ... < a(in-l)< ITII+},

I , = IT,J<"U(9a: 8 <

pll+}Y

Ml = EM1(PYNo). As a ( i )is an inrreaaing and continuous function (6 < IT , J :there is +

9 E Io, r ) E a', sup,, ~ [ n= ] 6) is not stationary (each a ( i )does not belong to it). As in 2.2 this implies that M 1 1 L cannot be elementarily embedded into Mo L. Let, for 7 < lT1l+,I : = I 1 n y s m ,131: = E M 1 ( I i , N o ) . Clearly every elementary submodel of M1 of cardinality 5 IT,I is an elementary submodel of some Mi, but Mi a n be elementarily embedded into Mo, by the embedding induced by a,, H t5crtr+l)+n. Easily, those two facts imply N o L is not IT 1I +-modelhomogeneous. (2) For w c p let I, = A'" U {vd:8 E w} u (I. + a ( i ) + 7:i < p, YEP for some p < a(i)}.The rest is like 2.2 and 4.2(1).

CH.

n I I , $41

CATEGORICITY, SATURATION AND HOMOGENEITY

473

THEOREM 4.3: The following assertions on the cardinals,p, A, x, where A > x, p > x, are equivalent: ( 1 ) If ILI Ix, L ( T )E L, p a 1-type in L, P a one place predicate in L, and T haa a nzodel M omitting p , 11M(1 = A > IP(M)I, then T hae a mo&el N omittingp, llNll = p > IP(N)I. (2) If T c TI, (TJ< x,PC(T,,T ) is categorical in p , then PC(T,, T ) is categorical in A:(3) If T c T,, ITl! 5 x, and every model in PC(T,, T ) of cardinality p is h g e n e o u s , then every model in PC(T,, T ) of cardinality A is hornogeneowr. (4) If T E T,, ITl] I; x, p a 1-type, IpI s x and every model of T , of cardinality p omitting p has homogeneous L-reduct then every model of T , omitting p of cardinality p haa hmnogeneozls L-reduct. CONCLUSION 4.4: If A (1)-(4) from 4.2 h 7 r o d ls .

=

ad, ( V C ~<

S)(a

+ S(x) s

S), A, p > x then

Proof of 4.4: By V I I , 6.3, ( 1 ) holds in this w e .

Proof of 4.3. ( 1 ) =+ (4) Suppose M 1 is a model of T,,omitting p, of cardinality A, with a non-homogeneousL-reduct M.So there are K < A, a, E M (i K) b, E N (i < K) such that tp*((b,: i <

M) = tp*((a,: i <

K),

K),

M)

but for no b E M tp*((b,: i <

K)^(b),

Y)= tp*((ai: i

I; K),

M).

Let M* = ( M , P,P,c) where P = {a,: i < K} (one place relation), P(a,) = b, for i < K and P(a) = c for a # P, c = a,. So by ( 1 )there is a model N* elementarily equivalent to M * , ((N*I(= p > (P(N*)(and omitting the types p and Y =

{(vY~,..- - Y J [ A P(Y:)

-+

i

(because omitting p and q is equivalent to omitting r = {~(z) v #(z): ~ ( z€21, ) #(x) E q}). Clearly (by q) N is not homogeneous. So we prove (4). (4)

* (3)

Immediate.

THlD NUMB=

474 (1)

3

OF AON-ISOMORPHI0 MODlDLS [UE.

(2) Suppose PC(T,, T) is categorical in p.

By

4.1

mI,5 4

T is stable

in every X 2 IT1[, hence in p , hence by 111, 3.12 T haa a saturated model in p, so by I, 1.13 PC(T,, T)has a saturated model in p, hence every model in PC(T,, T) of cardinality p is saturated. Suppose M 1is a model of T, of cardinality A and its L-reduct H is not saturated. So there me A E lMl,a n d p €&'(A,M); p is omitted by H,IAI c A. For every 'p(z; 5) E L let R, = {a: 'p(z; 7i) ~ p } and , P = A; let M* = (M, P,. . .,R,, . .),EL. By (1) there is N* elementarily equivalent to M*; IN*] = p > IP(N*)I and it omits

.

Q

=

W5W,(B)

-+

9J@,

0)l: 'p E L}.

Clearly N* L E PC( T,,T) and it is not saturated as it omits p' = {'p(z, a): N* t R,[7i], 'p EL, E IN*I} (p' is consistent as N*,H* a m elementarily equivalent). Contradiction. So every MEPC(T,, T)of power A is saturated. Hence (by I, 1.11) PC(T,, T)is categorical in

A. not (1) 3 not (2), not (3). Suppose L, T, p, P form a counterexample to (1) and M is a model of T omittingp, 11M)1 = A > IP(M)I, and let I, = {vi(z):i -c x}. Let P be a one-to-one function from 1M1 x onto M - P ( M ) ,with converses P,,Pa [i.e., for a,b E IMI,c E lM1 P ( M ) ,P,(P(a,6 ) ) = a,P,(P(a, 6 ) ) = b, P(Pl(c),Pp(c)) = c]. Let Qi = : a 4 P ( M )and M I='p,[P,(a)]for j c i, but M I=7v,[Pl(a)]}. {a E Let Mf = (M, J', FI,FS,. .,Qi,. . ) { e x and Lf = L(M*), Tf Th(M*), L* = {Q: i < x}, T* = Tfn L*.Clearly eat& Q,hee cardinality A or is empty, and 13.21 - Ui<,Qi= P(M). Be IP(Mf)l c A = IMfI, the L-duct of Mfis not saturated nor homogeneous. (If P ( M ) = {hi: i < K), let at = a, = b,, and then we m o t find 8 euitable b,.) Suppose now Nfis a model of Tfof cardinality p. Being a model of Tf,each Q,(Nf)is empty or has cmdimlity IINf 11 (use P,Pl,3'2). By aesumption Nf realizes p or IP(Nf)l = A. In eaoh 0&88 A = Nf Ui<,Qi(Nf)has oardinality A (in the first ( ~ ~ 8 8if, a realizee p , as {P(a,c ) : c E Nf} E A; and the second ( ~ ~ as 8 8P(Nf) E A). As Tfhas elimination of quanti6ers, it is easy to check that the L-reduct of Nf is eaturated, hence homogeneous. So T?,T*provides the countmexample for (2) and (3). Be we prove (1) * (4) * (3) * 1, (1) (2) * (1) we h i & .

.

THEOREM

.

4.6: T?MM i8 Q 4 1 M lof Tl of CMClinaljty p which is ( < A)-univemd, but its L-reduct i 8 not A-univereat, if: ( 1 ) T unsuperstable, 2IT1l < A, 2
Ca. WII, 8 41 OATEOORICJITY, SATUEATION AND HOMOUENEITY

475

(2) T = T , is stable but not superstable, IT1 Q A , 2 c AQ p ; and A is regular or hNo= A.

Proof. Our model M , will be the model constructed in VII, 4.6. Clearly it ie ( < A)-universal. We should show only that its L-redud, h f o is not A-universal. (1) If A is regular, the proof is similes to that of 2.2. If P o = A, o h m @, cpn aa in the proof of 2.2, and let No = EM(Asm,@), so N is a model of T of oesdinalify A, henoe it s d m s to show No is not elementarily embedded into M0. Suppose f is suoh an embedding, and 8, = f(ii,) for- q E As@. W e mn 6nd finite w, c p suoh that v Q r ) w, c w,, b, = q $ . o , * * Y Z,,k(,)), 4 . 1 E Jf&,I), w, {i(r),0: 1 < k(7)). We now define by induotion q,, = q(n)E An, suoh that r),, Q qn+l and

=

-

(7 E An+':

7,

=

TMn+l),

01, *

-> *

k(q) = k ( ~ n + d ;

( i h + l ,0)s * * * > mod WMn);

ZM,,+l),l realize the same type over M1,moreover if

i&,

u {z,,~,,: i c nyj 5 k(7 1 i),i(7 1 i,j)= ~ [ n l } , then 7i"Z,,1, anZM,,+l),l rea.lizee the same type over lM1l}

a

=I

* i(q,Z1) # i(q, la)).

haa oardidity 2 No.(Of o o m , 2, # 1,

Let 7 E A@, q,, 4 q for every n. Then {v,,(Z,8,,,,): n < w } is realizedby b,, but oannot be realized in M o by the dehition of q, oontradiction. (2) A similar proof.

LEMMA 4.6: (1) There are complete countable t h r i e a T I z T , T unetable, 8uch that: if M E PC(Tl, T ) i8 ( < A)-univered,A etrong limit, cf A = KO then hf ie A-univered. ( 2 ) There are complete countable theories T , 1 T , T etable but not ezclper8tableY8uch that: if bl E PC(Tl, T )iS ( c A)-universa2, A etrong Jim$, cfA = KOthen M is h-universal. Proof. (1) Let T be the theory of the rational order, and T 1 eays that any two intervals are ieomorphio. We leave the proof to the reader. (2) Define the relation E,, over w @ :71,,v o 9 n = v 1 n. Let M = (wcu,1 0 , 4,. .., 4,. . .). Let P,,(v, p ) = ( ~ [ o ].,.,. v[n - I), p[n], p[n + 11,. . .), and MI = (M, Po, P,, . .). Let T = Th(M), T , = Th(M,). We leave the proof to the reader.

r

.

476

THE NUMBER OF NON-ISOMORPHICMODELS

[CH. WII, $ 4

THEOREM 4.7: Thefolknning cunditiom on A, T are equivalent: (1) A = A < A + ID(T)I or T is stable in A. (2) T laas a saturated nzodel of cardinality A. (3) If T, z T,lTll 5 A t h n there is a saturated bl E PC(T,, T)of cardinality A.

Proof. (1) * (2) If A = A < A + ID(T)I,the proof of I, 1.7 shows that (2) holds. If T is stable in A 111, 3.12 shows that (2) holds. (3) + (2) This is immediate. (2) + (3) This is I, 1.13 (in its proof we use (2) * (1) only, so no vicious circle arises). (2) (1) Let M be the saturated model of T of cardinality A. So clearly A = llblll 2 ID(T)I, and if IAI < A, I,S(A)I 5 A (as A2 is universal w.1.o.g. A E M ynow bl is saturated). Now we prove by cams. Case I : T unstable, A < 2<”. By 11, 2.2(6) there are Q, and 7E A>2 such that for each r ) E A2 { ~ ( z7i,r,),[al: , a < A} is consistent. Now we define 6, E bl by induction on 1(7), - such that if q0 4 7, 4 - 4 7, then G,(0)hGMl)h- - - na,(,,), and 6,(0)hb,(l~n-. --6,(,,)realize the same type. (Thisis possible as the only requirement on 6, is that it realizes a certain type over U {6,: v Q r)}. The type is consistent by the induction hypothesis, and is realized aa M is A-saturated). So for every 7 E ”’2, some c, E M realizes &, = {I&, ii,ra),caJ: a < l(7)). If l(7) = l(v) then c, # c, [for if a = min{a: y [ a ] # v [ a ] } , p = 7 a = Y a,then ~ ( c , , G,,) = ~ c p ( c V , ~ J ]So. for every a < A, h = llMll I ( c , : 7 ~ 2 ”=} 127, hence A 2 2<”, contradiction.

--

Case 11: T unstable A 2 2
=

A<”,

-

Case 111:T unstable, A 2 2 < A ,A singular. It is easy to check that p < A 2” < h (otherwise 2~+cPA > A , ,u + cf h < A ) . So it suffices to prove (as a saturated model is universal by I, 1.9(3)). LEMMA 4.8: 8uwo8e A 2 IT,I, A 8trong limit cardinal and cf A c K ( T ) . Then thre i8 H E P C ( T , ,T),lldlll = A, mch that no elementary eztemion N Of bl O f Cardinality is (Of h)+-&UTded.

Proof of Lmmu 4.8. Let K = of A. We can define suitable M and ii, E M for q E x>Ay such that for every 7 E “A, q, = {Q@, Gv)u(n9v): v E x>A} is

CH. VIII,

8 41

CATEGORICITY, SATURATION AND HOMOGENEITY

477

consistent (by 111, 7.6, 7 and the definition of the independence “property”). Suppose M < N, llNll = A, so let IN1 = U f e e A f , !A,!< A, hence I.S(Af)I s 2141 c A. Now defme by induction r),, u, for a s K such that a c =- r)“ Q 78, l(v,) = a, Z(r),) = a, r), Q and tP(av(a+1),A,) = tp(liv(a+l),A,), hence no element of A, realizes F J h %)b+l))A T ( Z , %+l)). Clearly Q = M . 9 av(a+l))9 lQ)(Z, a < K } is cqvcrc1, hence consistent, but is not realized in N &B IN1 = (JaerC A,. So N is not K+-saturated.

Case IV: T stable, A A. It suffices to prove the following lemma. LEMMA4.9:SuppoaeA 1 lTll,Ax > AandK < K ( T )< o o o r T h m t h e atrict order property or K < KrOdt(T).Then there is M E PC(T,, T ) , 11M11 = A auch that ru) elementury extension N of M of cardinality A ia K -Saturated. +

Remrk. In fact if K < K ( T )< then K c mdt(T).

00

or T has the strict order property,

Proof of Lemma 4.9. We can aasume K = minb: A# > A}, hence A = A<’. So there is H E PC(T1,T ) , tp, E L, and a,, E bl, for 7 E ‘“A such that: (1)

{tpa(it;

av)il(v*V): a < K , r ) E “A,

a successor}ie consistent, for every

v E “A.

“A, i c j < A {~.I,+,(?z; i ~ , - ( ~ >5) :E (4, j}} is inconeistent. (Thisis by 111,7.7, 111, 7.6(1) and ( 5 ) . ) Suppose M i N , I(N(1= A, and for ~ E ‘ A let p , = { F J ~ ( z , @ ~ ~ ~ ) (2) If

r) E

a <

Z(r)), a a successor}. If N is K -saturated, each p nis realized in N; but the p i s me pairwise contradictory. So IlNll = Arc contradiction. So we finish the proof of Theorem 4.6. +

THEOREM 4.10: .Suppo8e A 2 lTll + K,, and every model M PC(Tl, T ) of wrdimlity A ia K,-homogeneoua. Then T is swperatable.

Proof.Our assumption implies that every M

E PC(T,,

E

T ) of cardinality

478

THE NUMBER OF NON-ISOMORPHIC MODELS

[OH. VIII,

84

2 A is K,-homogeneous [Because if (a,: i II w), (bi: i < w) is a counter-

example, i.e., tp,((a,: i < 0)) = tp,((b,: i < w > ) but for no b, E M tp*( A of cofinality No;so by 4.8 T is stable, and by 4.9 superstable (remember I, 1.9(4)).

CONJECTURE 4.1: If PC(T,, T) is categorical in A, A 2 lTll then T is superstable (but see [Sh 801).

+ N,,

PROBLEM 4.2: Can we improve 4.2(2) (i.e., weaken the conditions on p) ? PROBLEM 4.3: Close the gap between 4.5 and 4.6. That is characterize the cardinals A, p , x such that: if IT I ] 5 x, T 5 T,,T unstable [stable but not superstable] [has the independence property], then there is a model M , of T , of cardinality p which is (
CONJECTURE 4.9: Remove the exception in 3.4. CONJECTURE 4.10: I n 3.5 waive the condition (Vx < A)(x‘“ < A).

CATEGORICITY AND THE NUMBER OF MODELL3 IN ELEMENTARY CLASSES

IX.0. Introduction In this chapter we return to elementary classes so in some respects it is a natural continuation of Chapter V. Part of Section 1 is devoted to categoricity theorems, eta, which are, in a sense, a summation of previous reaults with some additions. We characteripe the countable theories categorical in No (1.6). We prove that for countable T,T is categorical in some h > Noiff T is categorical in every h > No iff T is totally transcendental and for zi E 1421 v ( M ,a) 2 No* I&lZ, a)! = 114211 ( ~ e e1.8 m d 1.4). For not neceeearily countable T: for some A, T ( A )is categorical in some X > T ( A ) iff for some A, IAI 5 2ITI, T ( A )is categorical in every h > IT(A)I, iff every 2~T~-universal model of T is saturated, iff T is superstable and undimensional. If those conditions fail, for every h 2 2’, p 2 2ITl T has a p-universal not p+-universal model of cardinality A (see 1.14). We prove also that T is categorical in some h > I TI iff T is categorical in every h > IT],and that if T is categorical in IT1 > KO it is a definitional extension of some T’E T,IT‘I < ITI. Note that the property “T is categorical in I TI is not absolute. The properties “T categorical in N,,” for countable T,and “ T ( A )is categorical in I T(A)I for some A ” are absolutes by the proofs of the previous theorems. We also prove that if every model of T of cardinality h is IT1 universal, T is I TI -categorical; (see 1.16) and if every model of T of cardinality ho > IT1 is homogeneous, then: every model of T of cardinality > A( T)is homogeneous, and T has a non-homogeneous model in A when IT1 + N, s h s h(T)(see 1.16). The method of those proofs is as follows: by Chapter VIII we get the +”

+

+-

+

479

480

CATEOORICITY AND WUMBEE OF MODEL8

[CE.

Ix,6 0

results on unstable and unsuperstable theories, and by V, Section 2 on stable non-unidimensional T. So we concentrate on unidimenaional theom, and by V, Section 7 we find weakly minimal formulas cp(s,a). By the unidimensionality for no M 4 N , M # N is 7i E 1611,cp(N, a) = cp(Y,a), so bi is “almost prime” over a u cp(M,a), w h e w it is eaay to handle cp(M,Zi) by the weak minimality. For using this we need theorems assuring, for a suitable A, the existence of M , A E 1M1, ~ ( b a) l , E A, see 1.1, Exercise l.lO(2). In Section 1 we deal with two more problems. We prove that if IT1 < A(T) (=sup(lS(d)I: Id1 5 ITI}) then for every p 2 ITI, I ( p , T) 2 min{2’, 2*(m}. In VIII, 1.7(2), 8 we have proved this even for PC(T,, T) but we demand T, is countable; the most diflicult caa is 2’’ = PO, ID(T)I 5 !TI. The second result is that for superstable T, and if, e.g., IliIfll > ITI, 6 < llblll+, then bl is the union of a strictly increaeing elementary chain of length 8 (see 1.3). (For stable T this may fail, sea Exercise 1.2.)

In Section 2 we try to compute I ( A , T) in some cams. We try to count the models by their dimension (or dimensions). We prove that for T totally transcendental, if IT1 = K, I K,, T not + 11 + No categoricalin ITI+,thenI(N,, T) 2 IS + lI.I(X,, T) 2 except, possibly, when T is countable, categorical in KO (the idea is that T is not unidimenaional by 1.8, so we have a freedom to define one dimension and for appropriate T, also the finitedimensionsare possible). We also prove that for superstable countable T, not categorid in KO, I(Ko, T) 2 KO (the idea is that if p E P ( 0 ) is omitted by the prime model of T, there is a model M,,, in which {aI:1 < n} E p(iIf,,)whioh are prime over I,, = {ar:1 < n}, when I,,is included in an infinite indiscernible set baaed on p, and dim(p, M,,) is n. In V, Section 6 we deal with multidimensionality. We prove multidimensional theories have many models (in 8, > IT1 at leaat 214). For non-multidimensional T we prove they have quite few F&(msaturated models. For superstable T, we get here a sharper result, of course for non-multidimensional T.We also get a better structure for the models. Every Fio-sctturatedmodels is (in VQ)F;f,-primeover J,, each J , baaed on 0, Av(J,, J,) semi-regular, and the Av(J,, J , ) am pairwise orthogonal (we see somewhat more by the inductive definition of the J,’s). In 2.4 we get a similar theorem for T totally transcendental, for all models, or for the N,-compaot models.

u

u, u

CH.

Ix,

5 11

48 1

SUPEXSTABLE THEORIES AND OATEQORICITY

IX.1. Superstable theories and categoricity

THEOREM 1.1: Suppose p = tp(zi,, A) does not fork over B c A , p is stationary, PI < K ( T )< 00. If A' is Fi-constructible over A then p is not realized in A', moreover p has a unique extension in Sm(A')provided tluct: (*)A

+

Foreweytyper E p ,1.1 < handC E A , ICl < h K J T ) and finite A , there is 7i E A which realizes r guclc tW

P ( p ,A, 2) = R"[stp(a, C*),A, 21 for some C*, B V C c C* c A . FACT 1.1A: If p = stp(ao,A)satisjies ( * ) A for some B , h or even is just $finitely satis-able in A , then p is statiosmry [because for every E E FE"(A), E = E(z,y, F), choose by induction on 1 < w , ~ E A realizing {-E(z, a",6) : m < 1, -E(ao,a"l, b)},so for some 1 E(a0,a', b), hence p -I E(Z,a', b)]. So we can omit " p stationary " from the assumptions of 1.1.

Proof, This is proved by induction on the length of the construction; for limit length, or length zero it is immediate. For the successor step, note that if$ has a unique extension pr E Sm(A')then n e c e d y pr does not fork over B. So clearly it suffices to prove the following two claims. For simplicity we assume p € S m ( A ) Note . that as p is stationary, R"(p, A , 2) = R"(stp(ao,A), A , 2) for every A. CLAIM 1.2: Swppoae A E A', p c p1 ES"(A'),p1 does not fork over A (and the conditions of 1.1 holds). Then ale0 B, A', p' satiefv (*)A.

CLAIM 1.3: Suppose B, A , p are as in 1.1 and tp(6, A ) is F;-isolated. Then p has a unique extension in Sm(AV 6). Proof of 1.2. Let T' E p l , lrll < A, r1 is over C', C1 E Al lC1l .c h + K,(T),and A be finite, and let T' = {pi(%; a,): i < a < A}. By IV, 4.2 (with B, A here for C, A there) for each i there is a formula y$(Z; 6f) E p such that for every a E A and 8 p,(Z; a) k B(Z;a) implies $,@; 6,) k 8@, a). SOfor any a E A h,@; 6'1 + pt[a; a,] (use 8 = (3# a)).Choose C, B E C c A , ICl < h + K,(T)such that tp*(C1, A) does not fork over C, and 6,E C, and Rm(p,A , 2) = Rm(p1C, A, 2). By the hypothesis, (*)A holds, and clearly T = {#'(Z;Ei): i < a} E p and C satisfy the

482

OATEaORICITY AND NUMBER OF MODEM

[OH.

IX, $ 1

assumptions of (*)! Hence there is 2 E A c A1 realizing r such that for some C*, C G C* c A. P ( p , A, 2) = Rm[stp(a, C*), A, 21. By the choice of C, tp(C1, C* u a) does not fork over C, hence over C*, hence tp(a, C* u C1) does not fork over C*. So by 111, 4.2, P[stp(a, c*), A, 21 = ~ m [ ~ t p (C* a , u cl), A, 23. We can conclude that (again by 111, 4.2), Rm(pl,A, 2) = P ( p , A, 2) = IP[stp(Ti, C* u C1), A, 21 and as E A mlizes r, by the choice of the t,hi(Z;6i)'s it realizes rl, so we finish. Proof of 1.3. Suppose not, then some realizes p but tp(a, A u 6) forks over A (rememberp is stationary). So (by III,4.2) for some hite A and formula cp(f, @, a), 6, a, 'a E A and

As tp(6, A) is FR-isolated there am q c tp(6, A) Iql < h and C s A,

ICI < h such that q is over C, and no type over A extending q forks over C. We can aasume q is closed under finite conjunctions, and that for any 6',a', Rm[9)(x;6', Z'), A, 21 < n* (by 11, 2.9). Let r = {(3g)[cp(f,a, E ) A #(g, #(@,a) E q}, so Irl < A, r is over C, and clearly r E p. Hence by (*)" there are a*, C*; C c C* c A, a* realizes r, a* E A and n* = P[stp(a*, C*), A, 21. Clearly q1 = q u {tp(a*,5, Z)} is consistent and let 6* realize it; aa 6*, Z], and P[tp(Z, 6*, a), A, 21 < n* = P[stp(a*, C*), A, 21 it follows that tp(a*, C* u 6*) forks over C*, hence tp(6*, C* u a*), fork over C*, so also over C, contradiction.

a)]:

THEOREM 1.4: B u w e M ia K-compad, cf 8 2:

K

2 ~ ( n and , at

least one of t h following conditha (i) t h e ia p = p<" m h tluct 181 + IT1 s p < 11M11, (ii) t h e ie an indiscernible eet I c M, 181 s 111, (iii) t h e i s C c 1611, and I r JMl independent over C, and IT1<" + p]<x < 181 S 111. Then we can jEnd a strictly increaeing e k d y chain of K-compad mode2eM,,i < 8,suchtluctM = (J1,,M1.

Remark. Theorem 1.4, Example 1.4, can be adopted to prove the non-existence of Jonsson models.

OH. Ix, 8 13 SUPEBSTABLE THEORIES AND OATEQOBIOITY

483

First we prove Claim 1.6. CLAIM 1.6: (1) 8wppo8e A, (i < 6) i8 increasdng, U,
u,
(*I

1.1

E A,, < r and any $nite

For every type r over any C

a E Uj
K, I Cl < K , for any A , there is a, E A,

rdizing r m h thd R[stp(iZ, C),d, 21 = R[stp(li,, C),d, 21.

Proof of 1.5. (1) Let us define B, inductively: B, is maximal subset of A such that U,<,Bj u A, E B,, and tp&, u j < d A,) does not fork over A,. SO we have to prove O ~ that Y u f < d B, = A. SUPP CEA u # < d B,. For every i, aa c $ B,, neceesclrily B, u {c} does not satisfs our requirement, so tp,(B, u {c}, u,
Proof of 1.4. We prove Case (i), and leave (ii) and (iii) to the reader. We can easily find a strictly increaaing sequence M, (i < 6) such that llM,ll 5 p, M , is K-compact, M,E M yand for every m-type r over any C E lM,l, Irl < K , ICl < K, and any finite d, for m y li E lMl realizing r there is iz, E 1M,1realizing r such that R[stp(7iyC),A, 21 = R[etp(Zi,, C),A, 21.

Now apply 1.6(2).

484

CJATEQORIIOITY AND NUMBER OR’ MODELS

[CH.

Ix,8 1

EXERCIBE 1.1: (1) Show that in l.l( * ) Awe can replam “and finited” by “and finite A, there is a finite A 2 A,, and” (and 1.1, 1.2 and 1.3 holds). (2) Show that also we can replace “Rm(p,A, 2 ) = P[stp(Ti, C*), A, 21” by R”(p, A, KO)= Rm[tp(a,C*),A , KO]” and remember R(tp(6,I?), A , KO)= R(stp(6,q,A , KO). (3) Show similar asktione on 1.4. EXERCIBE 1.2: Find a countable stable theory T such that T haa a model of power A, which is not the union of a strictly increaaing elementary chain of length p, p regulaz when at least one of the following holds: (i) there is such countable T’(not necessarily stable), (ii) An = A, or (iii) of A # p < A, and x < A =- f ‘ < A. [Hint: Let M = ( A u “‘A, P”,Q”, . ., Fn,. ..),<, where PM = A, QM = @A, F,,(q) = q[n];FJa) = a. T .= Th(M) does not depend on h and have elimination of quantifkm. So let, e.g., A = An and construct a model N,IP(N)I = = A, such that for every striotly i n& sequence (At: i < p), A, G P(N),lAtl s p there is some a E PN,suoh , = a, Pl(c) € A t - U,<(A,, then {Pn(c):n < o}n that if c E Q ~Fo(c) (U,+A,-- UlliA,) # 8. This suffices for the relativized version : if N = U i < M N LNt , increasing, then V , < . , P NG A , . To get the full version we have to use a more complicated M: it is the disjoint union of P?, P,M = A , Py+,= “(Py), with all the projection functions.]

.

PROBLEM 1.3: Charwterize, or at leaat investigate the possible {(A, p): A 2 p, p regular, A 2 I TI, and every model of T of oardinality A is the union of a strictly inoreasing elementmy chain of length p}. THEOREM 1.6: The fouoWing are eqvuivalentfor a countclbk T : (1) T w categorical in No. (2) For euch m, D,,,(T)k j n i t e . (3) Every p E D( T)is Fh,-ieolaterl. (4) Every model of T of m r d i d i t y KO is atoncic (i.e., F&,-atomic). ( 6 ) Every d of T of m r d i d i t y Nok 8aturated.

-

Proof. (1) (3) If p E D(T)is not Fk,-iaolated T has a countable model which realizes it, and T has a countable model which omits it (by IV, 6.3). (3) s (2) Let

.

F,,,= {p(Z): Z = (zo,. . ,z,,,-,> and ~ ( 2k)r for some r E D,,,(T)}.So by (3) {7v(Z): v(Z) E T‘,,,}is inconsistent, so there is a

CH.

Ix, 8 13

SVPER8TABLE THEORIES AND CATEOORICITY

485

finite subset I'k of ,'I which is inconsistent; clearly IDm(T)II II'kl < 8,. (2) * (3) For every my p # q E D, choose Q~,,(Z) ~ p such , that 7(pp,,(Z) E q. Clearly p E Dm(T) is isolated by A {tpp,,(z):q # p , q E Om(T ) } (which is a formula by the finiteness of Dm(T ) ) . (3) * (4) By definition. (4) * (1) A particular case of IV, 5.2(2). (3) =- (6),(6) * (1) Trivial. THEOREM 1.8: The following conditions on a totally transcendental T are equivalent: (1) T is categorical in at lea& one h > IT]. (2) T is catqin every h > ITI. (3) Evey model of T of cardinality > I TI is eaturated. (4) T ie undimensional. ( 6 ) I f M < N,M # N,B E M yQ(X, 8)is not algebraic, then Q(M,a) # Q",

a).

( 6 ) T h e is no Q ( X , 0) m h tirat for every h > p 2 ITI thre is a p-eaturated model M y1 1M11 = h and E M yIQ(M,7i)l = p. ( 7 ) Working in aeq,there ie a etrmgly eemi-minimalformula ~ ( x )such ,

that for every M yM is prime over Q(M),Q alnmt over 0.

Proof. (1) =- (4) Immediate aa V, 2.10(1), Definition V, 2.2

says not (4) implies not (1). (4) * (6) Immediate. (6) =+ (6) By V, 6.14. (5) * (7) Let fi(z, a)be a minimal formula, end N be a satureted and let p = tp(a, M )not fork over 0, and not algebraic. model 7i EX, In the prime model N over P I U {a}there is C E +(N, @)-/MI (by ( 5 ) ) hence p , tp(c, M )rn not orthogonal, hence by V,Exercise 4.10(3) there is E such that stp(a/E, 0) is semi-minimal, (working in P)and there is

in it a strongly semi-minimal p(x), and there is a F E M such that +(M, @) is included in the algebraic closure of F U ~ ( ( 5 ~ 9 There ). is no infinite indiscernible set I PI over fi(M,a) (otherwise the prime model over wl U b , b realizing Av(I,(NI) contradicts (5))hence in Meq there is no infinite indiscernible set over q$kleq). Let M be a model. Let Neq be FMo-prime over cp(Meq),Neq < Meqyhence by (6) Neq = Meq, hence for every 8 E M- tp(8, v(Meq))is F~o-isolatedand as in M there is no infinite indiscernibleset over ~(61"); by the characterization of prime models (IVY4.18) 211" is prime over Q ( M ~ ~ ) .

UATEQOItICITY AND NUMBER OF YODELS

486

(7)

[UH.

Ix, 5 1

==(3) For eaoh bl, 11M11 > IT1 olearly:

(*)

If p is a type of Wrdinalitg < p = 1 MI1 ,finitely &isfiable in 'p(Mea),then p is realized.

Let Napbe FM-prime over 'p(Mm);then clemly B = 'p(Nw)= 'p(Mm), and we know by (7) that Nw and Map am prime over B, hence isomorphic over B, hence Mw too is saturated, hence M is saturated. (3) =- (2) By I, 1.11. (2) 3 (1) mvial.

CONCLUSION 1.9: If IT1 < 2w0, T categorical in at leaet one h > IT1 then T i s categorical in e v e y h > ITI. Proof. By VIII, 4.1 T is stable in IT1 hence by Definition 11, 3.1 and II,3.2 T is totally transcendental, hence the previous theorem applies.

LEMMA 1.10: Swppae T ie atable and unidincensional. (1) ~~anyJi.niteA,modeZM,ancE(A,m)-ty~pover IM1,1p1 < 11M11, p ia realized in M. (2) The conclueion in (1) implies thd T doer, not h v e the f.0.p. and if EM $ N , 'p(z,a) i e nd algebraic then 'p(M,a) # cp(N, 3); and for every 'ps for 8Ome k, I'p(Jf, 2 k, * I'pW, all = pq. Proof. (1) Suppose p , a (A, m)-type over M ,is a counterexample. We can assume p E @ ( A ) , A c M,IA I < llMll , and for each 'p(Z; g) E A let a@)define P f 'pFor every set A we define a theory T,( A ,), consisting of the following (i) T. (ii) {aB: < a} is a non-trivial A,-indiscernible set over &. (iii) {'p(Z; b,, b,, . . .): 'p E A , b,, ... E &, k#,(b,,, . . .;a,)} is consistent; a, E &. (aB,a, serve as additional individual constants, & as a new one-place predicate.) (iv) The type described in (iii)is omitted. For every finite A, TJA,) is consistent: we expand M to a model of T,(A,) by interpreting & aa A, a, aa itsee and for uB-by 11, 2.19there is in ilf an infinite set indiscernible over A. As T,(L) = U {TJA,): A, s L finite} there is a model M 1of TITI+(L), and we can assume M 1 is ITI+-saturated IlMlll = 2ITl. So we can find a ITI+saturated model M of T, A c M , q E @ ( A ) is omitted by M , and I = {a,: a < ]TI+}is indiscernible over A, and 11M11 = 2ITI. Hence the g,,+ -prime model over A u I omits q. Let J be such that J u I is indiscernible over A, IJ1 = (2ITl)+; so clearly also the FfTI+-prime model over A u J omits q. We find that T has a IT1 +-saturated model

#,a

,

OH.

Ix, $11

487

SUPERSTABLE THEORIES AND OATEOORIOITY

which is not saturated, and has cardinality ( 2 9 + (as IAl < (IM(I< 2 1 9 ; hence T is not unidimensional. (2) The firat phrase follows by VI, 4.4, 6, the second by V, 6.14. For the third phrase note that if there are a, EM,, n < ~ ~ ( M , , , i z< , , KO, )~ w.1.o.g. llM, 11 > 2Ho and by an ultraproduct get a contradiction to (1). CONCLUSION 1.11 :Bqpoee T is iwper8tuble and unidimwbnal. Then: (1) RyE = 53, L, a)< w . ( 2 ) For every tp(3;jj) and n there is a formula $ ( j j ) euch that R[tp(%;a),L, 003 = n if k$[6]. ( 3 ) For every model M of T there are tp, and ii E A m h that tp(x, 6 ) is weakly minimal. (4) If T ia totally t r a m d , in ( 1 ) and ( 2 ) we can r e p h 00 by KO,and in ( 3 ) ccweaklyminimal" by "minimal".

Proof. By 1.10, V, 7.6, V, 7.10, and compaotness.

LEMMA 1.12: ( 1 )B e e T is iwperstuble and unidi-, is a form&, 6 € A and (*)rc

O(x, 6 )

If p i13a 1-type over A , p k O(z,6),IpI < K , t h 80m.e K = KO we can a 88um p = { ( ~ ( x6)}, ; 6E A).

c E A realim p (for

Thenfor 13orneK - w T ~ ~ & Fc,-mt&ible (2) If w8 a88um inete4bd (*)$

rnodel M over A , O(M, 6 ) E A

if p i8 an F:-type over A , p k O(x; ii) t h realizes p ,

80112e C E

A

then for every c-primnw&l M over A , O(M,a) E A . Proof. (1) It suffices to prove that if tp(b, A) is Fc,-isolated, for some

q E tp(b, A), 1q1 < A and for no extension q E B(A)of q is R(q,, L,00) < R(q, L, a), then A u b satisfies and b $O(C, a) (i.e., O(C, 6 ) n A = e(c, 6 ) n ( A u b)). This is because we can find a K-compact M , lMl = A U {a,: i < a}

such that tp(a,, A,) is as above, where A, = A u {a,:j < i}. Now we prove by induction on i that A, satidles (*)Icand O(C, zi) n A, E A . The induction step is by the above mentioned ammrtion. For limit stages notice that by the dehability Lemma 11, 2.12 A satisfies (*)n iff A eatisfies and B u (A n O(6,6 ) )satisfies (*)rc. Hence the property " O(Q, 6 ) n A E A,, E A and A satisfies is preserved by union of increasing chains of A's.

488

CATEGOFLICITY AND NUMBER OF MODEU

SO let B c A, IBI < K ,

[OH. Ix, 0 1

B, 1q1 < K where p = tp(b, A) be such that q has no extension q1 in &A) for which R(q,, L, 00) < R(q, L, 00). We can aasume b $ A , hence q cannot be realized in A, hence b $ O(@, a). Now suppose A u {b} does not satisfy (*)rs,SO there is a type r over A U {b}, lrI < K , r t- O(z, a), but r is not realized in A. Let c realize r , then tp(c, A u b) forks over A (otherwise by IV, 4.2 r is realized by some c' E A), hence tp(b, A u c) f o r b over A , hence by 1x1, 1.2(4) n =def R[tp(b, A u c), L,a]< R[tp(b, A ) , L, 003 I R(q, L, 00). So we can find iZl E A and 'p such that I=& c, a,] and R['p(z,c, B1),L, 001 = n. Let 'p*(y, Z) be such that R[cp(z,c', a:), L, 001 = n if€ C'p*[c',a;] Q E

p

(by 1.11). We can assume q is closed under conjunctions and let

r* = ((3Y)('p(yt 5, ad

A 'p*(Y,

a*)

A e(z, a)): 'p*(y, a*) E g}.

So r* 1 O(z, a), Ir*l < K , r* is consistent (c realizes it: choose b as y), hence some C * E A realizes it. So q ~ { ' p ( z , c * , ~ ,is) } consistent, R['p(s,c*, a1),L,003 = n, and this contradicts the choice q. (2) A similar proof, using IV, 4.2. CONCLUSION 1.13: If T is supemtable and unidimen&nud a E M , e(z, a) is not algebraic and IMI,O(z, a) sat6fy (*)= then M is K - c o ~ ~ ~ ) ( c c ~ [(i)! then

M is E-~aturated].

Proof. Otherwise let N be an FZ-constructible, over M ,K-compact, such that 8(N,a) E 8(M,a) (exists by the proof of 1.12). So N # M , and we get a contradiction to 1.11.

CONCLUSION 1.14: The following pqertk8 of T are q u i d : (1) T is superstable and unidime7tsional. (2) For some A,,, every A,,-univemalmodel of T is saturated. (3) As (2) for &, = 21'1. (4) For some set A , IAI I; h(T) I; 21'1 (h(T)6 the first cardinal in which T is stable) T ( A )6 categordcal in every h > h(T). ( 6 ) For some h , p , A 2 2#, p 2 21'1, T haa no p-universd, nunp+ -universal 4 of cardinality h (for e2llpe*8table T , h > p 2 A( T ) is enough).

Proof. Trivially (3) * (2) s (6), and by VIII, 4.7, VIII, 4.1 (4) * (3). Assuming (6), by VIII, 4.6 T is stable and even superstable, and by V, 2.10 (as &,-saturated implies &,-universal) T is unidimensional; 80 (6)

* (1).

CH.

Ix, $13

SUPERSTABLE THEORIES AND CATEQORICITY

489

So assume (1) and we shall prove (4). Let A = 1M1, where M is a saturated model of T of cardinality h(T) (exists by VIII, 4.7). It suffices to prove that any N 2 M is saturated. By 1.11(3) for some Ti E M, and 8, 8(z, Ti) is weakly minimal. Now if r is an F & & p over IN[, which is omitted, and r k 8(z, a) choose rl E ~ ( I N I r) ,5 r,; so r, is not algebraic, and O(z, Ti) E rl, hence rl does not fork over Ti hence of (using IV, 4.2) r is realized in M , contradiction. So N satisfies 1.12; hence by 1.13 N is F:(,,-saturated; hence (as T is unidimensional, and stable in every h 2 h ( T ) )N is saturated.

THEOREM 1.16: 8uppose T i s &egm*culin 80me h > IT1 or every model of T of cardinality A (for some h > I TI) is 1 TI +-universal. Then T i s categorical in every p > ITI, and every model of T of cardinality > ITI ia saturated.

Proof.If T is categorical in h > ITI every model of T of ca,rdinality A is necessarily I TI +-universal, and this implies T is stable in ITI (by VII, 2.9(1)) and superstable (by VIII, 4.5) and unidimensional (by V, 2.10). Now it suffices to prove that every model M of cardinality > I TI be saturated. So suppose M is a counterexample, and by 1.11 there are E bl and 8 such that 8(z, a) is weakly minimal. As M is not saturated, itisnot ITI+-saturated(byV,2.10)by 1 . 1 3 t h e r e i s q ~ S ( B ) , ar~M B, 8(z,a)Eq, 1B1 s IT], M omits q. As 18(M,Ti)I = IlMll > ITI,there is a set I c 8(N,a), indiscernible over B, 111 > IT!.Let J be any indiscernible set over 1M1 such that I u J is indiscernible, I JI 2 h and let A = acl(1M1 U J).If we prove A omits q, and we prove t b existence of N ;A c N,8(N, a) G A, then we have a non-ITI +-saturated model of cardinality zh, hence there is such a model of cardinality A. As T is superstable and stable in I TI, hence in A, and by VIII, 4.7, T also has a saturated model of cardinality M , of power ITI+.If M , < N we can see that every type T over N, Irl < ITI,O ( x , & ) ~ ris, realized in N (either it is algebraic or it is realized in M,), contradicting 1.13. By 1.12 in order to prove the existence of N, it suffices to prove only the following: (i) A = acl(1M1 u J) satisfies (*)& of 1.12. For suppose E E A, 'p(z, E) I- 8(z, Ti): if & q E ) forks over li, it is algebraic, hence 'p(&, E) c A (as acl(ac1B) = acl B by 111, 6.2(4)). If 'p(z, E) does not fork over Ti, it is realized in M , by 111,4.10. (ii) No c E A realizes q. Otherwise there are a,, . . .,a, E J , 6 E M , and 'p such that 'p@, a,, . . .,a,, 6) I- tp(c, U J ) (by IV, 2.4(1)).

UATEOOBICITY AND NUMBEB OF MOD=

490

[OH.

Ix,8 1

Clearly there is a finite I , c I such that J u (I - I,) is indiscernible over B U I , U 6. Choose distinct a;, .,a; E I - I,. Then ~ ( za;, , . ..,a:, 6) I- tp(c, B), hence some c' E M realizes q, contradiction.

..

QUEBTION 1.4: Suppose T is superstable, not IT1 +-categorical.Does T have a non-N,-univemal model in every wrdinality ? THEOREM 1.16: Buppme h > IT1 and every nzodel of T of cardinality h i8 laornOgeneo~r,OT at l w t ITI + - ~ ~ v M oTwh. ( w h A( T ) M th $r8t cardinality in which T i8 8table): (i) euey model of T of cardinality > A( T ) ia Iwmogeneowr, (ii) iflTl 24, 5 A Ih(T),then T h a anon K,-lwmogeneowr modez of caraid* A.

+

Proof. By VIII, 4.10 T is superstable. If T is not unidimensional, T has a model M ,1 1M11 = A, and a maximal indiscernible set I E M,111 = No; 80 M is not N,-homogeneow. Now B U P IlMll ~ > h(T), M is not homogeneous, so it is not p+-homogeneous, for some p < IlMIl, p 2 h(T). Choose a weakly minimal O(z, a), a E M , and a h N < M such that = p, and if c E O(M,a) - IN!,then tp(c, IN!)is realized by r p + elements of Af (possible BB each such tp(c, INI) is minimal), and there are b i ~ N ( I = 0, j s p or 1 = 1,j < p). tp*((b!:j < p), 0) = tp*((bf: j < p), 0) but for no bj E M tp*((b,O:j 5 p), 0) = tp*((bf: j 5 p), 0). Now choose M,, N < M, < M , llMIII = p + , and for every c E e(M, a) - INI, tptc, INI) is realized by r p + elements of M,.Now choose hl > p + , hl = ad, 6 is divisible by (21'1)+ x W; and let M, be a rulturated model of T,IIM,II = A,, M, < M,. Let A = 1M,1 u {c EM,:tp(c, INI) is realized by some c E B(M,; a) - "1) and it is easy to check that A satisfies (*)E(o from 1.12(1) hence there is a model M,,A c M,,B(M,, a) G A , 80 IIM,II = A, hence by VIII, 4.4 M, is homogeneous. So there is M,, M, < M, < M,, 11M411= p + , M4is homogeneous (possible &B T is stable in p + ) . Clearly each c E 8(M,, a) IN1 realize over IN1 a type realized by p+ elements of MI. Now it is e&sy to find an elementary mapping P,Dom P = IN lu QM,,a), Rng P = IN1 u O(M,, a), P }NI= the identity. Now it is not hard to extend P to an elementary embedding P, of M, into M,. By 1.10 Rng P, = IM,l, hence bl, is homogeneous too; contradiction. So we have proved (i). For (ii) note:

-

CH. E, $13 SUPER8TABLE THEORIES AND CATEOOELIOITY

49 1

LEMMA 1.17: If IT1 + N, s A s A(T) (=the$rst cardinal in which T is stable), No= K ( T )< K < A, then T has a model MAof cardinality A , in which there is an indiscernible set of cardinality K , but m more.

Proof. Let I be indiscernible, III = K, and let M be F&,-primary over I , clearly 11M11 = A(T) and let N < M be such that I c N , IlNll = A. N is aa required (see IVY4.9). 80

QUEBTION 1.5: Can we in 1.16 replace “ITI+-homogeneoue” by “ &homogeneous ” 1 EXERCIBE 1.6: Show that any counterexample to Question 1.6 ie supemtable and unidimenaioml. PROBLEM 1.7: Can we in 1.16 replace homogeneous by modelhomogeneous ? CONCLUSION 1.18: If T is countable,and every model of T of cardinality N, ia Iwmogeneous, then T M c d q h c a l in K,.

Remark. We can replace 8, by any A,

KO < A s

2Mo.

Proof. By 1.16 A(T) < K,, so T is K,-stable. By 1.16 T is also unidimensional, so by 1.7 T is categorical in 8,. THEOREM 1.19: If T is categorical in I TI > No,then T i s a & $ n i t i d exteneion of some T’ s T, IT1 < IT1 (see 111, 6.14).

Proof. By VIII, 3.4 T is superstable, so by 1.17 n e c e d y A(T) < IT1 but ID(T)I s A(T); so we can apply 111, 6.14. PROBLEM 1.8: Suppose IT1 > KO, IT1 non-univeml model of cardinality ITI.

5

ID(T)I; prove T haa a

EXERCIBE 1.9: Suppose cf A = KO,and (Vp < A)(2” < A) (e.g., D ~ ) , hence 2” = P o . (A) If IS”(A)I > IT1 + IAI 2 A then IB”(A)I 2 2” (instead A strong limit it suffices to wume that for A‘

c A,

L’ c L(T),

and get IS”(A’)I 2

Po).

IA‘I

+ IL’I < A,

[Hint: Note that if 8

1BC(A’)I s IT1

+ IAI

c Sm(A),IS1 > IT1 +

0ATEQOR"FY AND ITUMBER OB MODELS

492

Ix, 0 1

[W.

IAI, p < A, and we define 'p,(Z, a,) by induction on Z(q) for 7 E "'2, such that, when 7 ) JV ~ = {v:

I ~ P E Bb: a < z(V))(cp,la(~,a,la)Ep)l > + IAI), then qn(0), qn(l) E W; and then I W n '21 2 p and even q E W n "'2 implies IW n {V E "2: 7 Q v}I 2 p.] (B) Suppose A < IT1 < 2AyJD(T)I= 12'1 then for every L' E L(T), ID(TnL')I 5 IL'I A. (C)If T is as in (B), IT1 regular, then there is an m-type p over 0, IpI < A, and p has a unique extension q in D,(T) which is not Fiisolated. (Hint: Otherwise, act like in (A), where " = 121' " replaces " > I T1 IAl".) (D) Prove that if A < I TI < 2A, ITI regular, then T has a model of cardinality I TI which omits sometype (from D( T))hence is not universal (except when ID(T)I < 1 TI hence T is a definitional extension of some T' c T, IT'I < ITI).

+

+

DEBWITION 1.1: Let TVf(8,A) where 8 = 8(z, a) is a formula over A, means that any Fz-type p over A, B ~ pis ,realized by some c E A. EXERCIBE 1.10: (1) TVL(8,A) if€ TVho(8,A) and TVt,(6,8(A,a) U a) for stable T (use 11, 2.12). (2) If A, is increasing (i < a) B(A,,a)= 8(Aa,7i) and TV;(e,A,), then TVL(8,Ute, A,) provided T is stable. (3) Suppose TP,(8, A) K 2 K(T),and N is an Fc,-constructible model over A. Then 8(M, a) c A when one of the following holds: (a)for every finite A, C c A , ICl < K , a type p over C , 1p1 < K , 8 €13, and c E B(C, a) realizing p , there is ct E A realizing p such that &tp(c, C),A , 21 = mtp(c'Y C ) ,4 21,

(8)every equivalence relation ~ ( sy,,a), over 8(C, a),

< KOor > IT1 equivalence class i np ( A)whenever p is a l-type over A, IpI < K , 6 E P (use Exercise 111, 4.11). (4) If T is unidimensional and stable, TV~o(fly A), then every equivalence relation 'p(z, y, a) over O(C, a), E A, has < KO or1 8(A,a)] equivalence classes in B(A,7i). (6)If T is unidimensional (and stable), K 2 K(T), TVL(8,A ) , lB(A,a)l > IT1 then for any E"c,-constructible model N over A, e(iv, a) E A . E A, hae

CH.

I x , 8 13

SUPEBSTABLE THEORIES AND CATEOORICITY

493

EXERClsE 1.11:(1)Use Exercise 1.10(3)(p)to get another version of V, 6.14 and V, 6.16.

(6))is needed in Exercise l.lO(3).

(2) Show some condition (as (a),

[Hint: Let where :

M

= (WI,P",Py,Q",

P"

u

= "'2

Qy,F,M,c y ) ~ o ' 2 , E yT2 ,= Th(M)

("2 x w ) ,

QM=w2~w~w,

Py

= { v : v e r] or

W(= P M U Q", v or r]eveW>2}u { ( p , n ) : r ]4 p e W 2 , n< w } ,

Qy={(p,n,m):r]~pe~2,n<w,rn<w},

F,M is a partial one-place function: its domain is Q", prk

F,M(
{

( p d

Prk=vrk, Prk p = v,

+ w,

I$" = r] (which is in P"). Building an automorphism of saturated models we can prove that T is superstable. Also for some model N of T,IlNll > IPMJ(remember F , is not one-to-one even on (for any n)the set {v} x {n}x w which we can "blow up"). Let A = "'2,O = P ( x ) , now for every aeQe, for -A. TVNo(B,A)holds [note that it some V E " ~ , F ~ ( U ) E P Lastly suffices to prove this for the reduct to every finite sublanguage, where we have enough automorphisms].]

EXERCIBE 1.12: Through the following series of awrtiona we reprove 1.18.

(A) If every model of T of cardinality h is homogeneous I TI < 2h, h > ITI;then for every p , CR(p, 2) < a0 (see Exercise 11, 3.8) (use EM(h)). (B) If for every p , CR(p, 2) < 00 then there is an infinite set of indisaernibles, and for every A, there is an F",,-atomic model M over A. (C)Use (B) to show that under the assumptions of (A), T has a non8,-homogeneous model M of T, which is N,-saturatied, hence

IlJfll 2 IWQI. (D) Show that under the hypothesis of (A), there ITII = ID(T)1 Suoh that PC( T,,T)ie a h g o r i d in A.

is T, z

T,

494

OATEOORIOITY AND NUMBER OF YODBLS

[m. Ix,8 1

EXERCISE 1.13: ( 1 ) If T is countable and K,-categorical prove that every model of T is F%-saturated (see 111, 6.16). ( 2 ) If T is totally transcendental, every N,-saturated model is F&,saturated. EXERCISE 1.14: Find a totally transcendental T such that for some < A} modelMdT, llMll < IT1 = (D(T)I.[Hint:LetIbeatree,(B,:i be a set of branches of I, let L = {P1: i < A}, Pi a one place predicate, and T = Th(M), where = I , P,(M) = B,.] EXERCISE 1.15: Suppose h = IT1 = h e Ais regular, then the following conditions are equivalent. ( 1 ) AU Fi-compact models of T of power IT1 are isomorphic. ( 2 ) Every p E D( T)is Fi-isolated. (3) Every FA-compa&model of IT is A-saturated. THEOREM 1.20: Suppo8e T ia eu~perstableand A > IT1 b the $rat cardinality in which T b atable. Thenfor every p 2 I TI + K,, I ( p , T) 2 min{P, P}.

Proof. Let x be any regular cardinal I TI c x 5 A, and choose a formula ~ ( za),(existing by 111, 6.16) such that: (i) For some A , E A , IAI s ITI, and I{p ES(A):tp(z,a) E ~ } 2 I x. (ii) a* = R(&, a), L,a)is minimal (for 'p's s a t i s m g (i)). Choose a model Mo, llMoll = ITI, T i € lMol, such that if B , A , n , p are finite, B 5 lMol, p a finite type over B, and for some c realizing p , n = R[stp(c, B),A , 21 then there is such c E Mo. Choose also I indiscernible over IMol, I I I = KO.Note that by the regularity of x and the minimality of a* : (*)

If a E A, IAI < x, then { p ES(A): ~(z, a) €21, R(p, L, a)< a*} has cardinality ex.

Now there are two possibilities.

When Possibility A occurs, there is a set A*, lA*l

= A(q*) (see D e w -

CH.

Ix, $ 1 1

SUPERSTABLE THEORIES AND CATEOORICITY

495

tion 111,5.1, Lemma 111,5.1, 3) such that every extension of q* in #(A*) which does not fork over ais stationary ;and I ( ~ ) E S ( M :q*~ C)_ p , p does not fork over = 2h('J*). We can assume A* 5 M , and that for any E(z, y, iz) E E(A) and G realizing q*, there is an indiscernible set I c A*, b d on Borne q, z q* u {E(z,c, B)}. If Possibility B occurs let A* = a. Now we define by induction on a < x, pa and a, realizing pa such that (a)~(z, a) € p a ES(A*), R(pa,L,a)= a*, and if Possibility A holds q* c pa, (/3) no extension of pa in S( lMol u I u {al: i < a}) forks over iz. Now pa exists as the number of p's satisfying (a)is 2 x (check each possibility) whereas by (*) [ { p€#(A*):p satisfies (a),but not

@)}I < x

because lMol U I u {Bf: i < a}I < x so we can choose an appropriate p a and also a,. Now w.1.o.g. for every a,no extension of p , in S( U I U {a,: i < i # a})forks over A* [if Possibility A holds, as pa is U I U {ai :i < a});if Possistationary p , = tp(a,,A*) I- tp(@,, bility B holds, we choose by induction on a,y, < and a,,Jy < y,) such that (i) holds, and we replace: (ii) by (ii)' no extension of pa in S(wolU I U (ap,y:/l < a,y < yp) fork over A*, and (iii) realizes p,,{i~,,~:y < y,} is independent over U I U {@p,r:,8 < a,y < yp}, and every p~Sl(!.Ilf,I U I U {ap,y:p < a,y < yp}) extending p , is deP realized by some a,,y,this is possible and it, = a,,o are as required: p < /3 # a} is an extension of p , forking if qeS(wol U I U over A*, for some y < y,, q I- stp(a,,,, A*) and by IV, 3.13 applied to FZ we get a contradiction]. Clearly, I is indiscernible over P O I u {a,:i < XI. So let p 2 12'1 + N,, and I, be an indiscernible set over IM,l u {af: i < x}, I E I,, II,I = p. Clearly for every a, no extension of p a in S(lMol U I, u {al:i < x, i # a})forks over iz. For every 8 c x let M; be any F&-constructible model over lMol u I, u {al: i E S}, so by 1.1 M$ realizes pa iff a E S (for a 4 S if Possibility A holds, each pa is stationary, if Possibility B holds, show that for each c realhing pa, stp(c,A*) is omitted). Clearly 1M;l = p + 181, thus we have proved that (T,2') has (x, IA*l)-freedom. All this depends on the parameter x which we suppresss up to now. Let K = mi.&, A}; and we split the proof to cams.

woI

x,

wo(

x

woI

x,

Cme (i).There is a regular x 5

K,

2X = 2". Apply the above construction

' +. If Possibility B holds, for some k < No, (T, T)has for x + 121 (X,X)-freedomhence by VIII, 1.4 I ( p , T)> 2X = 2"= min{2c, 2").So,

CATEOORICITY AND NUMBER OF MODELS

496

[CH.

Ix, f 1

suppose Possibility A holds. If 2x > 2IA'I again by VIII, 1.4 101, T)z 2x = min{2', 2'5), 80 aasume 2x = 2IA'1. Now IA*I = h(q*) 80 if h(q*) > KO,by III,6.13'sproof ID(T)I 2 2A(a') = 2X = min{2A,2') > IT1 hence by VIII, 1.2 I ( p , T)2 lD(!l')l 2 2"q" = 2IA'1 = 2 X = min{2", P}. SO we can assume h(q*) 5 No(hence h(q*) = No),so 22 = 2u0, IA*l = No, but 2' = 2%hence IT1 K < 2Wo. We can also assume lD(T)l < 21, hence by 11, 3.2. JD(T)Is 12'1. By 11, 3.15 as h > ITI,in fact h = 2n0, hence K = p < P o . We shall deal with this in Case (iii).

Caee (ii). Not Case (i). So K is singular and x < K * 2x < 2'; let K = ~ ( i of ) ,K < x(i) r: K , x(i) regular, and increasing. For each i there me asf), Mx(f),pX(,)(x; FZX(,)) as in the construction above and lo llMoII , = of K .t ITI,Ufedk M,, E Mo, as before. Now choose &I, x(j) 5 we define inductively on a 4 K elements a,, euch that if I,<, a < x ( i ) then Rrn(PU,L,00) = as0 I= VX(f)[%, ZX(f)I = tp(a,, IMol) has a unique extension in 8(lMol u I u {a,: i < a)).Now let I,,be an indiscernible set over lMolu {a,:i c K } , I c I,,, II,,l = p, and for each 8 G K let M i be any F&,-constructible model over lMol u I,,U {as: E 19). As before iKi realizes pa 8 a E 8 (by 1.1) hence (Ip, 2') has ( K , cfK+ITI)-freedom as 2cf"+lTJ < 2" (provided that IT1 < K ) by VIII, 1.4 I ( p ,T)2 2" = min{2A,2 p ) . Suppose IT1 = K . We can assume ID(T)I 2 h (otherwise we return to Case (iii)), and as h > ITI,we can work with x = I2'1 , so if 2IA*1< 2ITl we can finish as in Case (i). Otherwise, D(T)2 2A*(*)2 21A'I = 2ITI, so by VIII, 1.2 we finish.

zf

+

Caee(iii). ID(T)I 5 IT1 s p < 2W0,2" = 2~o.Herewechoosex= IT]+, and &;a) w before; by VIII, 1.3 we can Mume FZ is empty, so 'p = p(z). By the choice of 9,if $(z,6) 1 p(z), #(z;6) forks over 0 (or even R[#(z;6), L, a01 < a*) then for every A, 1{p ~r9(A):#(z;6)~ p } l s IT1 + IAI, hence by 11, 3.2 R[#(z;6),L, No] < 00. Choose q, p(z) E Q E D,(T),suchthatforsomeA,IA~ = IT1 andj{pES(A):q s p}] = 2b; by 111, Section 6 MI h*(q) 5 No, hence h*(q) = KO,there are E, E E(0) such that En+,refines E,,, and for each c realizing q, q U {E,,(x,c ) : 7c < W } is stationary. Choose b, (Z < W ) realizing q, such that for every b realizing q, and n, for some I > n, I=E,[b,,b], and {b, :I < w } is independent over $3. Now we define by induction on I, a natural number n(Z)and elements a, (7 E l2) realizing q such that (i) if q # v E '2, then I=4JMl)(urr, av), (ii) if v Q 7 , 7 E '2, v €82, then kE,(k)(an, av),

OH.

IX,5 21

LOWER PARTS OF THE SPEO!CRUM

497

(iii) {bi: i < w } u {a,,: r) E It2} is independent over 0, (iv) if for each r) E ‘2, I.ln0,(a,,, a;) and {bf: i < I ) u {uv:Y ~ ~ ’ u2{a;: ) r) E ‘2) is independent over 0, then

tp(( ..., a,,,... ),,e~2,{bf:i < 1 ) u { a v : v E ’ > 2 } ) =

tp(( ..., a; ,...),,sta,{bi:i <

~ } U { U ~ : Y E ~ ~ ~ } )

(for (iv) use ID(T)I I IT1 < 2No). Now for each r) E O2 choose a,, such that {bI:1 < a}u {a,,:r ) E 012} is independent over $3 and a,, realizes q U {En(l)(x, a7,1): 1 < w}. For each S c “2, (81= p , let M , ( S ) be an Fgo-prime model over A , = {bI: 2 < w } u {a,,: 7 ~ qand , let B, = {c E HI(&; c realizes q, and tp(c, 8) is realized in MI(#) by 5 I TI elements for some 8 E A,} 80 A, E B,. Now let M ( 8 ) be a Fg,-constructible model over B,. The proof is now like that of VIII, 1.8, when we observe the following, whose proof we leave to the reader. Let c depend on C (here always C u {c} s q(6))if for some 8 E C, in the Fg,-prime model over 8, tp(c, a) is realized IITI t’imes. Fact 1: If tp(c, C) forks over 0, then c depends on C. Fact 2: This dependence relation is transitive and of finite character. Fact 3: Any FLI,-type over B, which forks over 0 is realized by some C E B , ,hence by 1.1 I{cEM(S): c realizes q}l = B,. Fact 4: For any Z, {stp(c,0): c realizes q, c depends on 8} is countable.

M.2. On the lower parts of the ~pe~trum Remark. We have used here, for p€S”(B),B c A , “a stationarization of p over A ” as “a complete type extending A not forking over By’.

THEOREM 2.1 : (1) If T ie euperstable, countable and not categorical ie KO,then I ( N o , T)2 No. ( 2 ) If IT1 = 8, (and of course IT1 I ID(T)(),then T)2 la + 11; and i f T ie totally tranecendental except w h a = 0, T categorical in No,then I ( N , , T)2 la + 11 + KO. (3) If T ie totally trawcendental, IT1 = N, (and of c~uree IT1 s

1s

ID(T)I),P 2 a a n d T i e n o t c a t ~ ~ i n I T I + , t h e n I ( N 6 2 , T ) + 11 and, except when a = 0, T categorical in No,I ( N 6 , T ) 2 + 11 + No. Remark. On IT1

IID(T)I

see 111, 6.18.

UATEGOE1ICITY AND NUMBER OF MODELS

498

[CJH.

Ix,# 2

Procf. (1) If JD(T)I> KO, the conclusion follows by VIII, 1.2. So aasume ID(T)I 5 KO,hence by IVY5.1 T haa a Fi,-prime model M,. As T is not N,-categorical, there is a p E Om(T)which is not isolated, hence M , omits p ; and let p , E Bm(M0)be a stationarization of p. Define inductively ii,, (n < w ) suoh that p,, = tp(si,,, 1M,1 u U,<,,8,) is a stationarization of po (hence {ii,,: n < w } is indiscernible over lMol); and let N, be an FL-prime model over Ul<,, ?Zl (exists by IVY5.1(4)). Now we shall prove that Nn omits r l : rl a stationarization o f p over

Uii, I
..

.

Otherwise there is cp(E, a,, .,a,,- k r, ~ ( 5 3 ,a,, . . ) does not fork over 0. So cp(Z, iio, . ..) does not fork over lMol hence some Z E M,,reaJize it (by 111, 4.10(1)). But N o omits p, hence for some #(if) ~ p C,#[C,] hence t(3Z)(cp(Zya,, ,a,,,) A 7y5[Z]), contradiction. So in N,, In= {a, .,a,,- 1} is a maximal set independent over 0 of sequences realizing p. Hence by V, 3.13(2),

..

...

&(P, 0, N n )

5

lInl

=n

s W(P)~ W 0, PN,n )

and by V, 3.9(1) w(p)is hih. We can conclude that {n: N,,r Nn0}is finite; hence I(No,T)2 No. (2) The assertion I&, 2') 2 Ice + 1I , follows by VIII, 3.4 if T is not superstable, by 1.20 if T is superstable, not stable in IT1 = Haexcept when a = 0, and then the aesertion is trivial. Now if T is superstable, T stable in IT1 (see 1.17, IVY4.9) for every /3 s a T has a maximally KB-saturatedmodel of cardinality K,, hence I(Kay T)2 la + 11. So now we assume T is totally transcendental, and IT1 > KO OT T not categorical in No, and have to prove, in fact, I&, T)2 No. As IT1 I ID(T)and by 1.6, for some m D,(T) is infinite, hence aa in 1.6 some p E Dm(T)is not isolated, hence M,,the prime model of T , omits p; and M, exists (by IVY3.1,lO and IV, 2.18). We can proceed aa in the proof of (1) but maybe llN,, 1 < I TI.If for some 6,the prime model over 6 has cardinality ITI, we define B,, such that tp(a,,, lMol u 6 u U l < ,4) is a stationarization of po, let N,,be a Fh,-prime model over 6 u Ul<,, then I,,= {al: 1 < n}, is, in N,,,a maximal set independent over (by0) of sequences realizing p , hence by V, 3.13(2) and V, 3.15(2), d W p , 0, N,,)5 lI,,l = n s w(p)dim(p, 0, N,,) + w(6,0) hence as in (1) I(Ha, T ) 2

KO.

OH.

IX, 8 21

LOWER PARTS OF THE SPECTRUM

499

So suppose for every 6, the Fko-primemodel over 6, has cardinality < \TI; we can also assume, w.l.o.g., 0 < a < w (by (1) and the first part of (2)).For each non-algebraic p € D ( T ) ,let I, be an indiscernible set based on p of cardinality N,, I, = {.", :i < Na}.Let M , be Fko-prime ) , realizes r}l = /{TED(!!') : for some n, r over I,, so clearly J ( r € D ( T: M is realized in the Fko-prime model over < N,. Also the , d llM,II < N,. As N, d (D(T)I,clearly number of algebraic q € D ( T )is there are 2 N, non-isomorphic Mp's, so we finish. (3) By 1.8, T is not unidimensional, hence by V, 2.10 for every y 5 8, T has a maximally Kt,-saturated model of cardinality K,, hence I(&, T)2 Ip + 11. So m in (2) it suffices to prove I&, T)2 K,. As in (2) let M , be a F&-prime model of T,p E D,,,(T) omitted by N o .We can assumep is stationasy. (By 111,2.9, for some&(%,y) E FE(0), p u { E ( q a)}is stationary for any a realizing p ; we add the names of the 1F-equivalence classes as new relations. As eaoh model can be so expanded in finitely many ways the proof is suffioient.) As in (2) we choose appropriate 6, choose as before a,, N,(n < w ) , and then find N t , 6 V U l < ,al c N$ llNtll = K,, such that no iZ E N: realize.^ a stationarization of p over 6 U U,<, a,, or at leaat dim(p, 0, N t ) < KO and the rest is aa in (2). So what remains are the choices of 6 and Nt;. Case A. For some 6, in the Fbo-primemodel over 6,Ms there is an infinite indiscernible set I. Clearly we can find an idnite indiscernible set I, c INnl,and by 111, 3.6 we can assume it is indiscernible over 6 v U l < ,4. Let J , be an indiscernible set over 6 U Ute,,a,, I,, c J,, lJnl = K,, and let N: be Ft,,-prime over J , u 6 U UlenZ l ; clearly IlN~Il= K,, and every type over 6 U U l < ,8l realized in N;,is realized in N,; so N i is aa required. Case B. There is a non-algebraictype orthogonal top. Let q be such a complete type, p ~ S ~ ( b such ) , that q is stationary; choose {q:i < NB} be independent over (6 u Ulea, a,, 6), c, realizing q, and let N; be Fk,-prime over 6 U Ul<,,Zlu {cl: i .c 24,). If rj is any stationarization of p over 6 u U1
ui
500

CATEGORICITY AND NUMBER OB MODELS

[OH.

Ix,5 2

As K ( T ) = KO,we can assume that r and q me complete types over some 6, and they are stationary; and the stationarization of p over 6 is not weakly orthogonal to r nor to q (by not Case B) and as T is totally , E r is a minimal formula. Let J be an transcendental, some ~ ( z6) indiscernible set over lNnl based on q, and let N f ;be Fi,-prime over lNnl u J. Clearly the stationarization of r over N , has a unique exten,,,#t 6). sion over INnl u J , hence is omitted by Nf;, hence rp(N:, 6) = If dim@, N:) = KO, then dim(p, 6, N i ) = KO, so there is {an:n < W) exemplifying it. As tp(P, 6), r are not weakly orthogonal, there are c, E acl(2 u b) - acl (6), c, E rp(Nf;,6); and clearly {c,,: n < w } is independent over 6, hence indiscernible, contradiction to not Case A. EXERCISE 2.1: In 2.1(1) replace “superstable” by “stable and K ~ ~ *= ( T8,’’ ) and show N,, g N,,, ifF n = m.

THEOREM

2.2: 8wppose T is countable and categorical in (1) If T is not categorical in KO,then I(K,, T ) = KO.

K,.

(2) Every 4 e . l M of T has m i n i m l and prime proper extension N ( m i n i m l means for no N’, M $ N’ $ N ; prime means that for each N’, M < N’ implies N can be elementary embedded into N’ over M ) . Note N is necesearily unique. (3) Suppose T is not categorical in KO. There are countable models M a (a IW) of T such. that: (i) Every countable model of T is isomorphic to one of them. (ii) They are pairwise nun-isomorphic, M , is saturated and each M , is homogeneous. (iii) M , + , is the prime minimal proper extension of M,, and M , = Un<,M,, and M , is the Fk,-prime model of T . (iv) Ma can be elementhry embedded into M , iff a I/I (for a,/I I w).

Proof. (1) By VIII, 4.1 T is totally transcendental, hence by 2.1 I@,, T ) 2 KO;the other direction follows by (3)(i). (2) Let M be any model of T , and let d(z, a) be a minimal formula, a E M (at least one exists by l . l l ( 4 ) ) . Let b E O(Q, a) - [MI, and N be F&,-prime over M u b. If M N‘ then by 1.8(5) O(M, a) # O(N’, a), hence there is CEO(N’,E)- d ( M , a ) . Let B’ be the following mapping: P t = the identity, P(b) = c. Clearly P is elementary, and we can extend it to an elementary mapping from N into N‘ as N is F&-prime over /MI U b. So we have proved the primeness.

OH.

Ix,5 21

LOWER PARTS OF THE SPECTRUM

50 1

Now suppose M $ N' $ N; so again choose c E 8(N', 8) - [ M I ; hence tp(c, IM I u b ) forks over [MI,hence tp(b, [MI u c) forks over IMI, hence b E acl(]MI u c) hence B(N, Ti) E acl(lM1 u b) E acl([M]u c) E IN'];hence by 1.8(5)N = N'; so we prove the minimality. The uniqueness follows as in IVY4.20. (3) Define M a by induction: let M , be the F:,-prime model of T,and B(z, Ti), 8 E M,, be a minimal formula. If M , is defined let b, E 8(Q, 8) M,, and M,+, be F&-prime over ]M,l u b,, and let M , = Un<,M,. By 1.8(6),n = dim(tp(b,, a), M,) c No( M , is not saturated, as T is not No-categorical)so w.1.o.g. n = 0. Clearly M , is the F&,-primemodel over 8 u {b,: Z < n}. If M is a countable model of T,we can assume M , E M , and let I c 8 ( M , 8) - acl 8 be a maximal set independent over 8; and clearly M g Ma where a = III , so (i)holds. Clearly (iii)follows by ( 2 ) . Now let us prove (ii); by (iii) if H, g M,, n c k then for every 1 M,+, g M k + , (proof-by induction on Z), hence every M,,,,, is isomorphic to some M l , 1 5 k , contradiction to what we proved from (1) by (i). Hence M , (n c w ) are pairwise non-iaomorphic. By 1.8(6) it is easy to see that if 8(z, 8) is minimal, I E &Cry Z)-ad Z is independent over 8, 111 = KO and 8 u I E [MI then M is KO-saturated.Heme M , is saturated, and any countable model into which No can be elementarily embedded is saturated. So from (ii) we have to prove only the homogeneity of M,. Assume Til EM,, tp(8,, 0, I M J ) = tp(Z2, 0, [M,1) and we shall prove that M , has an automorphism P,#'(til) = 8,. We define by induction on k Mf, < M,: Mb is an F&,-prime model over Ti,, w l <M,. If Mf, is defined it is not saturated (as M , is not) and, by ( 2 ) if Mf, # M , there is a prime minimal proper extension ML,, of it, Mf, $ Mf,+ < M,. If for some I , Mf, is defined for every k again U k < o Mf, is saturated, hence M, ie saturated, contradiction. So let Mf, be defined just for k Ik(Z); clearly there is an isomorphism F, from MA onto M i , Po(Til)= Ti2, and we can extend it inductively to the isomorphism Pmfrom M i onto M i . If k ( 1 ) = k ( 2 ) we finish. Now let MA g ill,,(,) hence Nb g M,,,,, hence (by (iii)) Mf, g L W , ( ~ ) +so ~, k( 1) # k(2) implies M,(,) +,(I) M,,(,)+ r(2), contradicting,the part of (ii) we have proved. So only part (iv) remains, for a = w we have already proved it. Now suppose M a g N < M B defines inductively N,:N o = N . N , + , < M B is a prime minimal proper extension of N l . Tf N, is defined for every 1,

CATEOORICITY AND NUMBER OF MODELS

502

[CH.

Ix,$ 2

EXERCME 2.2: Extend 2.2(2) to any ITI+-categoricaltheory when llMll > IT1 or M is FiO-saturatedor M extends such a model. EXERCISE 2.3: Suppose T is totally transcendental IT1 = K, L N,, and let M be the F&-prime model of T . Then ( 1 ) o (2) =s (3), (1) =- (4), cf X, > KOimplies (4) * (1)and when N, is regular they are equivalent. (1) M is not K,-saturated. (2) Some p E D( T ) is not F&-isolated. (3) For some m IDm(T)I L N,. (4)M is not Np-saturated.

DEFINITION2.1: Let I(A, F, T ) be the number of non-isomorphic Fsaturated models of T of cardinality A. When F = F;T, let I=(&K , T ) = I(A,F;, T ) .

MAIN THEOREM 2.3: Let T be superstable not multidimensional (see V , 6.8)then there is a cardinal p = N D ( T ) (=the number of dimensions of T ) and a group B = a(T ) of permutations of p , such that: (1) 1 Ip I 2ITI andevenp I; A(T),andwhen T iscountable,p I KO o r p = 2No. (2) p = 1 iff T is unidimensional. (3) If X, > A(T), or K, = A(T), a - p+ p 2 w , Ia(N,, N,, T ) = la - /3 + I l f i / l a (see below). (4) If Na = A(T), 0 5 a - 9, < w , p < KO, then Ia(Na,N,, T ) = Z:=B IY

-p +

lIP/B-

( 6 ) If 8, < A(T),then Ia(N,, K,, T ) = 0, where f/G = { f / - G : f : p -+ x, 0 E Range f } ; for 0 a group of permutations of p ; x, p possibly jinite and is the equivalence relation, f l fa iff for some 0 E B, f l = faa.

-

-

THEOREM 2.4: Suppose T is totally transcendental,p = ND(T).There are functions hl, ha : p + {HE:No I N, I ]TI+}8 ~ that ~ hl(i) h = h,(i) is KO or a successor except when cf[hl(i)] = No, hg(i) = hl(i)+ and hl, h2 preserved by the permutations of a, satisfying: (I) The following cardinals are equal or both ,finite, when, N, 2 IT1 + K , , p > 1:

OH.

I x , 8 21

LOWER PARTS OF THE SPECTRUM

503

(2) If in (1) they are$nite, every model is Fto-saturatedas h,(i) 6 K, for every i. If for no i, K , = h,(i) < h2(i),then every K,-compact model of T is FGB-saturated,hence It&, NP,T)= I“&, K,, T)and 2.3 applies. In the remaining case necessarily a < o,and if K, > ITI, easily

It(K,, K,, T ) =

n K,/G(T)- n

t
(K1-

1)/G(T)*

t
> IT1 then I(Hay N,, T)= 1; if /& = 1, 8, = IT1 = ID(T)I,then I t ( s a , K,, T)= KO. (3) I f a 2 /3 w , OT a > 8, p 2 No, 8, 2 ITI,then It&, N,, T)=

If

la

/&

-p

= 1, N,

+ 11”.

+

Proof. The proof is broken to several stages, which we may use later. In fact, they contain information not summed up in the theorem. For 2.3 let x = Q and for 2.4 x = t , but we concentrate on the first. 8trzge A. By V, 1.13(1)the relation of non-orthogonality among regular types is an equivalence relation. Let {pi: i < p} be a set of representatives and ND(T)= p, and let I,be based on p,. Every automorphism P of Q induced a permutation HFof p: HF(p,)= pf iff P(p,), pf are not orthogonal. The set of permutations HFof p is a group we call a = a(2’). As T is not multidimensional cleaxly p s 2ITI. Now in order to get the precise results we shall find “nice” p,’s. Btuge B. For any cp(s,a) there is k, such that for any model M , I{ltp(M,a)l:&~M)I < ttV (by V, 7.5 for the trivial K). By VII, 3.4, V, 5.8, 9. T does not have the f.c.p.; and for any suitable K (see V, Section 7) and weakly K-minimalp(x; a)the conclusion of V, 7.6 (2)holds. The proof is easy by the non-multidimensionality of T. 8Qe C. We shall define by induction on a formulas pa@;fja), h(fj,), &, and a natural number %(a)for a < a. < p+ . Let

alh WJ>: Y < 4 80 by V, 7.1(2) we can find a K, satisf+g C1-6 from V, Section 6, Wg = W,,80 R,-minimality, etc., axe defined. As T is superstable, w a = {(9Jt(%

either every formula is K,-algebraic, or there is a non-K,-algebraic hence p(x; a) is weakly formula ~ ( x7i); with minimal R(cp(s;a),L, a),

504

CIATEGORIGITY AND NUMBER OF MODELS

[CH.

Ix,8 2

K,-minimal. If T is totally transcendental we can assume p(x; a) is R,-minimal. By V, 7.7 (possibly after changing p) there are formulas #(g), 1 such that: (i) B is an equivalence relation on #(a) with n(a) < X, equivalence classes. (ii) k#p']implies p(x; a') is K,-weakly-minimal and, for totally transcendental T, p(z;B') is even K,-minimal. Also there is a", kE[a', a"], tp(a", 0) = tp(Si, 0). (iii) If k+[aJ, then p ( ~7il) ; 5 Ka p ( ~a,) ; iff kB[Sil,aa]. Let pa, #, E,, n(a) be p, # , E , n reap. Now a. exists aa clearly p + , bemuse if I, G p,(B, a,), C#,[iZ,] HKa(I,)= 00, then the 1,'s a. are pairwise orthogonal. Clearly each (p, #) appear at most once, so a, < ITI +. Stage D. We define by induction on a, a, = ?i,,on. . .n~,,n(,)-l p"(s;a,) = Vl<S(f#) pa(z, S . l ) such that is weakly K,-minimal when x = a, (i) q,(z, (ii) kA\l
v,(Q,au.1)-

Clearly the I f s are pairwise orthogonal, and regular. Let us work in P,and let A* = aol(A,,) 4nd p t = Av(I;, A*). Stage E. Let N be any FGo-saturatedmodel of T.By the inductive choice of the am'swe can assume A* E P I , and for every a,y,let J; c N be a maximal indiscernible set over A * based on p;. Let M E N be Fc0-constructibleover A* U {J; :y < y(a), a < ao}, and we shall prove N = M.We prove by induction on a that pa(N;a,) G M. If a is the first for which this fails choose c E @(N,a,) - 1M1. Clearly M < K a N (by V,6.4) hence IZKa(c,!MI) = 00, hence tp(C, !MI) is not orthogonal to some p;, hence if x = a by V, 1.12 Av(J;, 1611)is realized in N, contradicting the maximality of I;. If x = t, T is totally transcendental so for some 1 < y ( a ) = n(a),c E p,(cE, and c realizes as p,(x, a,,l) is K,-minimal we get again a contradiction. Av(If, ]MI), Stage F. By Stage E, the function h$ defined by h$(a, y ) = IJtI = dim(p?, M ) determines M up to isomorphism (remember the p ; are

u

OH.

I x , 8 21

LOWER PARTS OF THE SPECTRUM

505

stationary (by 111, 6.9) and pairwise orthogonal). Moreover no regular type is orthogonal to all p; (otherwise let M be IZ'~+-snfurated, tp(c, lM1l) such a type, Ml an F&-prime over 1M( u {c}; then necessarily not all J t are maximal in Ml). So necessarily by Stage A, p = C{ly(a)l:a c ao},and the p; are a set of representatives. We can conclude I"(N,, X,, 2') 5 la - 81" + HE. Stage G. The two first assertions of 2.3( 1) are immediate. As for the third as a. < IT1 = N, it suffices to prove for each a, Iy(a)l 5 KO or ly(a) = 2Ko. Choose N countable, G,,EN, and let S = { ~ E S ( :gY(x, N) G,) ~ pp is , not K,-algebraic} ; we know each p E S is regular, hence non-orthogonality is an equivalence relation on S, and y, is the number of equivalence classes. Now S is a closed set (in the natural topology on S(N)) and non-orthogonality is a Bore1 equivalence relation on 8, hence by a theorem of Silver (see e.g. [Sh €441) Iy,l < EC, or Iy,l = 2'0. Stage H. Suppose p < No,then we can find a* E A* such that each p; does not fork over a* and p; 1 a* is stationary, and let us rename the p; 1a* as {pi: i < p}, 'and we allow ourselves to write p , = p&, a*). Now for every sequence (A,: i < p), A, 2 No there is a unique, up to isomorphism, model M = M(si*, (A,: i < p)) such that dim(p,, M) = A,. But we have here an arbitraxy choice of a*. However it is eaay to check that M(a*, (A,: i < p)) M(a*, (A': i < p)) iff there is 6*, tp(a*, 0) = tp(6*, 0) and a permutation u of p such that A, = Xu") and p&, a*) is not orthogonal to p0(&, 6*) (because tp(6*, a*) ie always Q,,-isolated) and 111, 3.6. (The proof here does not work apparently for p 2 No aa it is not clear why B* is F$-constructible over A*). It is also eaay to check that +

11M(a*, (A,: i < p))II =

2 Ai + A(T).

i< u

This clearly proves 2.3(3), (4) and (6). To make the proof work for p 2 KO (and a

- /.I small) we use a better canonization. In stage I assume x = a. Stage I. We repeat stage C more carefully, working in P a . We define by induction on a, formulas e,(z) (with no parameters) ordinals ?(a), types GES(aclp), g S c g(aeQ), such that (letting W, = {(O,(z), ):/.I < a} and K, is defined by V, 7.1(2) to satisfy C, 1-6 from V, Section 6, Wzo= W,) using V, 7.14: N,M yN are P&(n-aaturated, then O,(N) = OJN) iff (i) If Af = #(N)for every Y < Y ( 4 Y

!?w)

506

CATEBORICITY AND NUMBER OB YODELS

Ix,8 2

[CH.

c aclKi(C(EeQ) U (ii) is K,-minimal, @ is P;-simple and @(aeq) where = cP(c} and c, are orthogonal if i # j.Hence (iii) For A E g(E), i #j,we have low,;(A.acl~) = 0. For a model M let h;(M) = lowp(qi(M),j3), which is equal to dim($, q,M) (as q;(M) c aclKo(G(M) @ using V, 4.18 or V7.14(3)). Now we shall prove that for Fi,-saturated M,,Ma, they are isomorphic over acl0 8 A;(Ml) = A;(M2)for every a < ao,y < ~ ( a The ). only if part is clear. For the if part we define by induction on a 5 a. an elementary mapping Pa, Dom Pa = u B < a 8,(M1) u acl 0 onto uB
6

u{e

Dom Pi = Dom P8

U

u {Z;:

y

< y(a)}.

Using A$(Ml) = A ; ( M g ) , $ K-minimal, G(Ml) c aclKu($(Ml)u Z?) and the parallel fact for Ma, and (iii) we can extend PL to Pi, an (M,, bla)-elementary mapping extending P i , Dom Pj = Dom PB U u,$y(MT), Range F i = Range F , U u , q 3 M 2 ) ; you may wonder what to do when q? has small dimension, then e.g. use V7.14(3) or Example X, 5.3A (or [Sh 88a,1.41). LetN,<M, (N2 <M2) be Fio-prime over Dom Pi (Range Pi),FZ an isomorphism from N , onto Ngextending Pi,and Pa = P; 1 (UfrB8,(Nl))is onto Ufrs8,(N2). So cleaxly Pa is as required (as B8(N,) = 08(M,)). It is also clear that for each sequence A = (A;: y < y(a), Q < %> of infinite ctudimls, there is an %,-saturated model M = M(X), A;(M) = A;, and 11M11 = A(T) + A; and M is F&-satumted iff K B 5 A; for any a < ao, y < y(cc). The only remaining question is when M(X) z M ( p ) . Notice that any automorphism of Qeq preserve the BU(z)’s,but may take @ to another type. If P:M(X) M @ ) and w.1.o.g. E; E M(X), then f(<) should be not orthogonal to some A; = p ~ u t yand ~, P f acl0 determinefJy). The rest is easy. Stage J: Now we return to the case z = t , T totally trmcendental, and we repeat stage I, using now V, 7.14(2), using EY,, instead Fgoy with no special problems. The only point that remains is the possible values of A;. Let I;*obe an indiscernible set on 7ia based on p;, 1; = {a;,t: f s w}. Let Wl(a,y ) be the h t cardinal K such that for every n tp(a;,,, 7i, u {a;,,:I < n}) is

za,y

CH. IX, f 21

LOWER PARTS OF THE SPECTRUM

507

-

Fk-isolated and hp(a,y ) is the first K such that tp(at,,, lia U {a;,l:I < w}) is Fk-isolated. For (1) we do not divide by G , as we can easily prove that the result is the same : as T is totally transcendental, each ~ ( ais) finite; so the equivalence relation on p : i -j iff for some are not orthogonal (see Stage A) has automorphism F of 6,B’(I,),I, only finite equivalence classes, and i j=-h(i) = h ( j ) ,and if we take the product on a maximal set of non- -equivalent ordinals we get the same result. Also, the fact that the model should have power K, and not less does not influence the computation.

-

(n,,,)

N

EXERCISE 2.4: In 2.4 replace “totally transcendental” by “Aa(T)I K,, T superstable”. Also replace it by A,(T) = No, and investigate Ia(Na,K O , T ) . EXERCISE 2.5: (1) For fi > 0 , Ia(Na, x,, T)= Ia(Na,K,, T). (2) For countable K0-categorim1T, Ia(Ka,K,,

x,,

= It(Na,

T).

EXERCISE 2.6: (1)In 2.3, prove that for countable T,la - fi + ll@/U = la - fi + 11” except, possibly, when p and a - fi axe finite; and Ir(.>l 2 No * Il44l = 2% (2) Show that for every x,finite p, and a group U of permutation of U, for some T,IT1 = X, ND(T) = p, U ( T )= a. (3) In (2) we can also have arbitrary h,, hp (when they satisfy the conditions mentioned in 2.4 and hl = ha). (4) Show that if there is a Boolean algebra B, with s x elements and p ultrafilters (this occurs, e.g., whenever p Ix or p = 2”, or p = f for some K Ix), then for some T,ND(T) = p, IT1 = x. (5) I n 2.4(2) complete the case K, = IT1 < K , , ~ , K < ( KO (look at the proof of 2.1(3)).

CHAPTERX

CLASSIFICATION FOR Fi;SATURATED MODELS

X.O. Introduction We introduce here two dividing lines: having the dop, and being deep. They are meaningful for stable theories, but a t present important for superstable theories. The dop (dimensional order property) says that the relation: “there is an indiscernible set I realizing the type p = p ( ( z o ,xl,. . .),ai, a,) with dim(I,M) > IT1 ” (where aiE M )can define an arbitrary two-place relation on {at:i < a}. This is very similar to the order property from Chapter 11, only the order is not defined by a first-order formula (hence, does not imply the complexity of pseudo-elementary classes) but it implies T has 2A pairwise non-isomorphic Fio-saturated models of power h > h(T). The positive property equivalent to “not dop” is that for F:r(,,-saturated M , < M , , M , such that {Ml,M2} is independent over M,, the F:(,,-prime model is F:r(,,-minimal. This is developed in Section 2. Its importance is revealed in Section 3 where we prove the decomposition lemma for superstable T without the dop :any Fio-saturatedmodel is Fio-prime,Fio-saturatedover UTGINv, where (N, :7€1)is a non-forking tree of Fgo-saturatedmodels, I E ‘“’a, for some a, is closed under initial segments (non-forking trees means tp*(N,,,,, U { & : ~ - ( a ) $ v } ) does not fork over N,). Why do we assume T is superstable ? Because we use regular types to show that the tree “exhausts” the model. Note that the class of Fio-saturated models is very nice from our point of view. The saturation demand is weak, local (e.g. preserved by increasing unions), it is still easy to prove for unsuperstable the existence of many non-isomorphic models, and superstability implies the existence of an Fio-primarymodel over any A . The existence of such a decomposition is a structure theorem, e.g. it bounds the number of pairwise non-elementarily embeddable Fie508

CH.

x, $11

PRELIMINARIES

509

saturated models. Thus, the question naturally arises as to whether I may contain an infinite branch. If this is the case, T is called deep, if not, T is called shallow. Shallow (superstable without the dop) T has quite a few Fio-saturated non-isomorphic models, whereas deep T has 2AFfto-saturatednon-isomorphic models of power A 2 A( T)+ N,. In Section 4 we investigate the depth (which measures how far are we from the deep case), which really is the depth of a type, and get the upper bound. I n Section 5 we prove that any deep T has many models. For simplifying the proof, in Section 7 we introduce “trivial regular types ” (when the dependency relation is trivial : tz depends on I iff it depends on some &E I).In 7.2 and 7.3 we investigate them : if T does not have dop, Dp@) > 0, p regular, then p is trivial, and we have better canonization theorems (e.g. if stp(a,A) is not orthogonal to a trivial p, then for some E almost over A, stp(tz/E,A ) is c13(p)-simplewith weight 1). I n Section 6 we compute exactly the number of non-isomorphic Ffto-saturatedmodels of T of power A, when T is superstable, without dop, shallow of depth 2 w . Note that for countable No-categoricalT every model is Fi0-saturated,so we can (essentially) compute I(A,T). However, whereas usually the depth can be any countable ordinal, here it is finite. This is as if T is superstable and K,-categorical, it is No-stable (see 111, 5 . 1 7 ) . Hence by Cherlin, Harrington and Lachlan [CHL 861 R ( x = z , L ,No) < w , so by 4.8 Dp(T) < w .

X.l. Preliminaries

I n what follows, I,Jare sets of sequences from 6, I, J are sets of indexes (ordered sets, or trees or sets of sequences of ordinals, usually). (Hyp) I n this section T will be stable. CLAIM 1 . 1 : If A E B, ( 1 = O , l ) , tp(B,,B,) does not fork over A , peSm(B0) is orthogonal to A , then it i s orthogonal to B,. Proof. W.1.o.g. we work in Eeqand w.1.o.g. B, = aclB,, A = aclA. Choose any qeS(B1).Construct an infinite indiscernible set I of elements realizing q such that tp,(I,B, U B,) does not fork over B , and Av(I, I) is a stationarization of q. Similarly, choose a set J of

510

CLASSIFICATION

FOR

F

g

o

-MODELS ~ ~ [CH.X, ~ Q 1~

indiscernibles over B, realizing p . Since tp,(I,B, U B,) does not fork over B,, and tp(B,,B,) does not fork over A, by 111, 0.1 tp,(B, U I,B,) does not fork over A, hence tp,(I,B,) does not fork over A. As ~ E S " ( B ,is) orthogonal to A by V, 1.5(1)tp,U,B,) is orthogonal to A. Hence tp,U, B,), tp,(I, B,) are orthogonal, hence by V, 1.2(1) (since we work in Eeq)weakly orthogonal, hence by V, 2.7 Av(I, U I),AvU, U.J) are orthogonal, so p, q are orthogonal. DEFINITION 1.2 : For A G B E C we say B < A C iff for every CE C , tp(c,B) is orthogonal to A. LEMMA 1.3 : (1)Let N C M C A, M < A, M and N are F; -saturated, K 2 K,(T)and M' is F:-prime over A, -thenM < NM'. (2)If B < A C , A G Ao,B C Bo, A , EB, E Co, Co = C UBo,tp*(C,Ao) does not fork over A , and tp,(C, B,) does not fork over B, then B, < A o C,. Proof. ( 1 ) By V, 3.2 and V, 1.2(3). (2) By 1.1 it is easy.

CLAIM 1.4: Let K 2 K,(T).If N is Fz-saturated, p € S m ( B )is regular, stationary, not orthogonal to N,then p is not orthogonal to some regular q E S" (N)* Proof. W.1.o.g. p € S m ( M )N , c M and M is Fz-saturated. Let C E M , (C(< K , p does not fork over C, p C stationary. Let A E N,IAI < K , tp,(C,N) does not fork over A, tp,(C,A) stationary. Choose by induction on i < w elementary mappings f,, Domff= A U C, fi /'A = id, stp,( ff(C),A ) = stp,(C, A), and f o = id, Rang(f,) E N,and for i < w , i 2 2, tp,( f , ( C ) , MU f,(C))does not fork over

uj
A.

=.mr

Let p1 ( A u c)). Clearly for i < j,p, is orthogonal to p, iff p o is orthogonal to p, (by indiscernibility ) . Extend the domain of fz to N U C by fixing N-C. By V, 3.4, p,,p, are not orthogonal (p,f,f(p), A, B there, stand for p,, fz, p,, N, NU C here). Hence p,, p , are not orthogonal. Now p is parallel to p,, not orthogonal to p,, parallel to the stationarization q of p , in Sm(N),q regular. So we finish.

~

~

CH.

x, $11

PRELIMINARIES

51 1

DEFINITION 1.5: Let A C B, p€S"(B),then we say that p is almost orthogonal to A if for every C, s.t. tp@,B) does not fork over A and every 6 realizing p, tp(6,B U C) does not fork over B.

CLAIM 1.6: (1) Let A C B, then tp(5,B) is almost orthogonal to A iff tp(6, aclB) is almost orthogonal to A if tp(6,B) is orthogonal to A. (2)If A G B, p = tp(6, B), then p is almost orthogonal to A iff for any C, tp(C,B) does not fork over A implies stp(6,B) I- stp(6,B U C) iff for any C, tp,(C, B) does not fork over A implies stp(a,B) I- stp(a,B U C). (3) Let A E B, p = tp(6,B), then p is almost orthogonal to A iff for every &, tp(a,B) does not fork over A implies tp(a,B U 6) does not fork over A. (4) For every a, tp(a,B) does not fork over A implies tp(6,B)Itp(6,B U n) iff tp(6,B) is almost orthogonal to A and in aeq,tp(6,B) ttp(6,B U aclA). (5) If A = E B, M is Ff-saturated, p€S"(B) is Ff-isolated, A 2 ~ , ( 7 )then , p is almost orthogonal to A. (6) If A = WI E B, M is Fi-compact (i.e. A-compact), pcS"(B) is Fi-isolated, then p is almost orthogonal to A. CLAIM 1.7 : If N is Ff-prime over $3, A 2 K,(T),IAl < A, cf A and M is Ff-prime over NU A , then M is F,"-prime over $3.

2K(T)

Proof. We concentrate on the case h = No,so A = a. W.1.o.g. M is Ff-constructible overNU A. By IV, 4.9(3),IV, 4.10(2),IV, 4.18 we can find I c N , I an infinite indiscernible set, Av(I, N) = tp(@,N),and Av(I, U I)I- Av(I,N). Now IIIII = A as N is F,"-prime over $3 (see IV, 4.9(2)).By IV, 4.18, N is Ff-prime over UI,and by IV, 4.10(2) Ffconstructible over I U {@}.Hence M is Ff-constructible over I U {a},but as II U {@}I = A, IU {a}is F:-constructible over 9. Hence M is Ficonstructible over 9, hence M is F:-prime over $3. DEFINITION 1.8: ( 1 ) I(A,F, T)is the number of F-saturated models of T of power A, up to isomorphism. If F = Ff we write I f ( A , T). (2)IE(A,F, T)is the maximal number of pairwise non-elementarily embeddable F-saturated models of T of power A. If a maximum is not obtained, and the supremum is a limit cardinal p, we write the value as p-. If F = Ff we write IEt(A,T).We omit A if we do not restrict the cardinality (so the value may be m or 00-).

512

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MODELS

[CH.

X, $2

X.2. The dimensional order property (Hyp) I n this section T will be stable and

K

be K , ( T ) .

Remember that T is unstable iff it has the order property, and unstable theories are complicated in some respects, e.g. they have many non-isomorphic models. However, a stable T may have an order hidden in it. For example, consider for A > KO,A E A,, the theory T of the model (B,F,, F,)where

B

= A U {(a,P,y > : a , P ,y

< A , and ( a , p > ~ * A y < 4,

Clearly T is not only stable, but even KO-stable,and N,-categorical ; however, by cardinality quantifiers we can define an order (if A is an order). We shall consider here a property, which clearly means there is a hidden order property. From later sections we can see that for T superstable and Fio-saturated models it is the only one. Note that here the order and independence properties coincide. DEFINITION2.1: T has the dimensional order property (dop for short) if there are m o d e l s 4 (1 = 0,1,2),each Ff-saturated, 4 E M , , 4,and {M,,@} is independent over 4,and the Ff-prime model over M, U@ is not F;-minimal over M, U 4 . In this section we first develop a number of equivalent forms of the dimensional order property. These are summarized in Lemma 2.4. Condition 2.4(d),o is the form of the dimensional order property used in Theorem 2.5 to show that if a superstable T has dop, then Iio(T,A)= 2A(when T is stable in A).

LEMMA 2.2: Let &&< M , , 4 , each 4 F;-saturated, A 2 ,u 2 K , {M,,&} independent over 4,M F;-atomic over M, U@ and M is F,”saturated (and M, U M , c M ) . Then the following conditions are equivalent : (a) M is not F,”-minimalover MI U 4. (b) There is an injinite indiscernible I E M over M, U&. (c) There is p € S m ( M )orthogonal to M, and to 4,p not algebraic. (d) There is an inJinite I S M indiscernible over M, U@ such that

CH.

x, $23

513

THE DIMENSIONAL ORDER PROPERTY

Av(I,M) i s orthogonal to Ml and 4.Notes by 1.6(5), tp,(I,M, U 4 ) i s almost orthogonal to M, and to 4. Proof. The equivalence of (a) and (b) is the content of IV, 4.21, and (d)*(b), (c) are trivial; we now prove (b)*(d), ( c ) = ( d ) . The "notes" in (d) can be proved by 1.6(5). (b)* (d). We can assume 14 = No, and suppose I E M is. indiscernible over Ml U 4 but not orthogonal to M, (by symmetry). So by definition, Av(I,I) is not orthogonal to some ~ E S " ( M , ) . W.1.o.g. I = J U {an:n < w } , < K , Av(I,I) does not fork over J, Av(I,J is stationary, and let {bn: n < w } be an independent set over (Ml U A& U I,Ml) of sequences realizing r (hence indiscernible over M, u A & U I ) . By V, 2.7 for some k, tp(bon ...^bk,M,U4UJ, tp(aon ..."ak,M1 U& U J are not weakly orthogonal. As M is F,-atomic over Ml U 4 , for some B, E M, U 4, 1 41< p, tp,(I',B,) I- tp,(I', M, UA&) where I' = J U {al:1 5 k} (for p singular use 2.3(1)). Hence for some C EM, ~ 4 ICI < , KO,tp(bo^ ..." b k , 4 U C U J) and tp(a0-. .. 9,B, U C U J are not weakly orthogonal. Now tp(bo^ ..." b,,M, lJ4 U J does not fork overM, (by the choice of the 6 s ) and M, is F;-saturated and IB, U C U < p KO K = p , so some co" . . . ^ c k ~ M lrealizes tp(bon ... "bk,B,u C uJ) (see III, 0.1). Sotp(aon ..." Gk,B,UCUJ),tp(co" ..."ck,B,UCUJarenotweakly orthogonal, hence A

+ +

tp(a0

*

.. 974 u c u Ju 4 ... - c k ) U C) y tp,(I',B, U C U (roe...^Q),

"974 u c u J)Ytp(a0

en

5

hence (byAzV2forF8,)tp,(I',B, hence by monotonicity tp,(I',B,) y tp(I',M, U 4 ) contradicting the choice of 4. (c)*(d). So let rESm(M)be (not algebraic) and orthogonal to M l , 4 , It suffices to prove that r (B U M,, U 4 )is realized in M for every B c M , IBI < K . [This is because then we can choose I& E M , 141< K , r does not fork over B, r rlj, stationary, and 5nE M realizing r r ( l j , U {bl:Z< n} UM, U 4 ) , and then I = {bn:n< w } is as required indiscernible by 111, 1.10(l).] We can, of course, increase B as long as IBI < K . So w.1.o.g. r does not fork over B, r rB is stationary, for 1 = 0,1,2, t p , ( B , 4 ) does not fork over B n 4 , and let F realize r , 4 = B n Ml and tp,(B,M, U A&) does not fork over B, U 4. As IBI < K , M is Ff-saturated, it suffices to prove

r

stp(c,B ) I- r r (B u M, u 4).

514

CLASSIFICATION FOR

For this let

F

g

o

- MODELS ~ ~[ C H . ~X, 92~

F2€4(1 = 1,2) and it suffices to prove

r

stp(c,B)I- r (B u 6,u 6,). We can find C E&,, ICI < K such that tp(b,,&, UB;) does not fork o v e r l $ U C f o r l = 1,2. Now tp,(B,&) does not fork over B nib$ = 4 and C E 4 so tp*(B,& U C) does not fork over &. By symmetry, tp*(C,& U B) = tp*(C,B) does not fork over 4. Extend tp*(C,B) to a type q over M which does not fork over 4.Then qrM, is orthogonal to r and parallel to stp,(C,B) so by V, 1.2(4),stp*(C,B) is orthogonal to r. Hence, (1) Stp*(c,B)I- stp*(c,B u C ) . Now tp(6, U B , , 4 ) does not fork over 4 [as it is Gtp,(M1,4)] hence stp,(6,,4 U B,) does not fork over &, U 4.Also tp(6,,&, u 4 ) does not fork over B,U C by the choice of C , hence by 111, 0.1(2), t p ( 6 , , 4 U B) does not fork over B, U C. Now B, U C EM,, B, U C E B U C E 4 U B, hence stp,(61, C U B) is parallel to some complete type over M,, hence is orthogonal to r, hence to stp*(c,C u B). So (2) stp*@,c u B ) I- stp*(C,c u B u 6,). Now tp,(& U B,,M,) does not fork over &, [as it is E tp*(M,,Ml)], hence tp(6,,M1 U B,)does not fork over 4 U B,. Also tp(&,,&, U B,)does not fork over B, U C (by C's choice), hence tp(b,,M, U 8 )does not fork over 4 U C. As B, U C E 4,and B, U C E B U C U 6,G Ml U B, clearly stp(6,, C U B U 6,) is parallel to some complete type over hence is orthogonal to r , hence to stp(c,C u B

u 6,).so

c

c

(3) stp(c, u B u 6,) t- stp(c, u B u 6,u 6,). By ( I ) , (21, (3) clearly (4) stp(c,B)t- stp(€,Bu 6, u 6,) which, as mentioned above, is sufficient.

CLAIM 2.3 : Suppose A 2 K , 4 (1 = 0, 1,2) are Ff-saturated, & <M,, 4,and {M1,4}is independent over 4. Then

(1) Every Ff-isolated PES"(M, U 4 ) is Ff-isolated. (2) A model M is Ff-atomic over M, U 4 iff it is Ff-atomic over M, U 4.Hence if K 4 p, x 5 A , and an F:-prime model M over M, U 4 is F,"-minimal,then M is F:-prime over M, U 4 (and F:-minimal). Proof. (1) Suppose B E M, U 4 , IBI < A, F realizes p and stp(c,B)tp = tp(F,M, U 4 ) . We can assume that tp,(B,&,) does not fork over

B n&,.

~

~

CH.

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THE DIMENSIONAL ORDER PROPERTY

515

Let C G B, ICI < K be such that p does not fork over C, tp,(C,&) does not fork over C n&. Now by 111,4.22 it suffices to prove:

(*I

for any EM, U 4 , there is &EM, U 4 such that stp(a',B) is a stationarization of stp(it, C).

We can find A E & such that tp,(a U C,&) does not fork over A , its restriction to A is stationary, IAl < K , and let = a,- a,, EM, (1 = 1,2). Let A = {at:i < i(O)}, and we can find a;€&, such that stp,((a;:i < i(O)),Bn&) extends stp,((a,:i < i(O)),Cn&) and does not fork over C n &, hence over C. Now find €4such that stp,((a,:i < i(O))-a;,BU a;-,) extends stp,((a,:i < i(0)) -at,C) is as and does not fork over C. It is easy to check that it' = a; required. (2) trivial from (1).

LEMMA 2.4 : Thefollowing properties of T are equivalent for A 2 x 2 K : (a) T has the dop ( = dimensional order property). (b)A,x There are F,"-saturated models & , M , , 4 , A&, <MI,&, {M1,4} independent over 4,such that the Fi-prime model 4 over M, U 4 is not Fi-minimal. ( c ) ~There , ~ are F,"-saturated models i & , M l , 4 , & <M,,&, {Ml,4} independent over 4,and there is a n Fi-atomic non-Fi-minimal model ik& over M, U 4 . (d), There are sets A,, A,, A , such that A, E A,,A,, [All< A , {A,,A,} is independent over A,; and there is a n infinite I indiscernible over A , U A,, orthogonal to A , and to A,, and tp,(I, A , U A,) is almost orthogonal to A , and to A,. Moreover, if it, (I = 0 , 1 , 2 ) are such that { A l ,A,, a,} is independent over A,, and tp(al,A, U A , U A, U a, U a3-,) doesnotforkoverA,Ua, f o r 1 = 1,2, then stp,(I, A , U A,) I- tp,(I, A , U a, U A , U a2).

We can replace a, by B,. Remark. Note that (a) does not depend on the cardinals A,x. Proof. As (a) is (b)K,K it suffices to prove: (i) (')A,* * ( d ) K ; (ii) (d)K * (d)x; (iii) (d)x * ( b ) A . x ; (iv) ( b ) A , x * (')A,,* (d)K.So let &,M,,M&& exemplify ( c ) ~ Now , ~ . by 2.2(d) (i) (c),,, there is an infinite I indiscernible over M, U 4 such that Av(I,&) is orthogonal to M, and to 4.First assume K > KO,and w.l.0.g. 14 = KO.

516

CLASSIFICATION

FOR

F

;

o

- MODELS ~ ~[CH. ~X, $2 ~

By2.3(1),tp,(I,M, U&)isFz-isolated, andsoforsomeA E M , U&, IAI < K , with stp,(I,A)t- stp,(I,M, U&). We can also assume that tp,(A,&) does not fork overA fl4and tp,(A,A fl4) is stationary. Let A, = A n 4 . It is easy to check that {A,,A,} is independent over A, (as {Ml,&} is independent over 4,and use 111, 0.1). Clearly tp,(I,A, UA,) is almost orthogonal to A, and to A,, and even the stronger assertion in 2.4(d) holds. If K = No,T is superstable and so there is a finite J G I such that Av(I,I)does not fork over J and Av(I,J) is stationary, and continue as before with J instead of I, noticing that tp*(I,A, U A, UJ) does not fork over J, is stationary, and is orthogonal to M, and to 4. (ii) (d), * (d),. As K 5 2, if A, (2 = 1,2,3), I exemplify (d),, then they exemplify (d),. (iii) (d), =s (b)A,X. Let A,,A,,A,, I exemplify (d),. Let 4 be an F;-saturated model of T , A , E 4. By using automorphism of 6 we can assume tp,(Al,& UA,J does not fork U A, EM,, over A,. Next choose an F;-saturated M,, such that tp,(M,,& U A 2 )does not fork over 4,and an F;-saturated 4,such t h a t 4 U A, E & and tp,(A&,M,) does not fork o v e r 4 (this is easy). So clearly {MI,&} is independent over 4,and by the latter part of

u

u

(4,

stp,(I,A,

u A , ) -I

stp,(I,M,

u 4).

So tp,(l,M,U&) is F,"-isolated, so there is & F,"-prime over M, U&, such that I E&. As M4 contains an infinite indiscernible set I over 4 U M,, it is not F,"-minimal (by IV, 4.21) so we have proved (b)A,,. singular, prove by induction (iv) (b)A,,* (c)~,,. Trivial (for on a that if (M, U 4 , ( a t :i < a ) ) is an F;-construction, then M, U& U {ai:i < a} is Fz-atomic over M, U 4 : for this use 2.3(1).

x

THEOREM 2.5 : Suppose T has the dop, T stable in A and K 5 p < A. Then T has 2Anon-isomorphic F;-saturated models of cardinality A. Proof. Let A , , A , , A , , I be as 2.2(d),, 14 = Noand let J be any set of indices. W.1.o.g. we shall work in Eeq(see 111, Section 6). Let A ; = aclA,; and we can define, for 1 = 1 , 2 , s € J , an elementary mapping f f, such that : (a)Domf: = A ; , (p)f f, PA; = the identity,

~

~

CH.

x, $21

517

THE DIMENSIONAL ORDER PROPERTY

( y )tp,(f#i)), U{fF(A;):(t,Ic) # (s,Z),~EI,~ doesnot E { ~fork ,~}} over Ah. Next, for every s , t ~ J(not necessarily distinct) we choose an elementary mapping f8,,, whose domain is A ; U A ; U I , and which extendsf,1,f:. (This is possible a s f f Uf," is an elementary mapping, which is true because tp(Al,Ah) is stationary (by 111, 6.9(1))and the independence of { A ; , A ; } and of {f,1(Al)lf$42)} over Ah.) Let A: =f:(A,) (so I&I < K ) , 1 8 , t =fs,t(l). Now:

(st) stp*(I,,,,A,1u A 3 -I StP,

(&,,?

u 4u u 4u u

VEI

veI

( V ,U )# ( 8 , t )

qu).

This holds by 2.4(d) for

B,

( v J )z (s,1), (42)) u { ~ u , v : {nuW , 4I = fVl 4 = & UA,' u u { 1 8 , v : v # t , V E J ) ,

= UV;:

ct

B,=&uUA,2UU(Iu,,:u#s,uEJ).

Let be any set indiscernible over A,' UA:, extending of power p+. Clearly (st) continues to hold for the IZtls. Let R be any two place relation over J , and let C, = U,,,A: U UR(8,t) I;,, and let MR be an F,"-prime model over C,. It is easy to check that IJI = A implies lM,II = A (remember T is stable in A , obviously IC,( < A , hence llM,II < A ; on the other hand, clearly A , $ A,, hence IC,I 2 A , hence llM,II 2 A ) . Now, using (st) we show: for s, t E J , (st 1) there is inM, an I of power p+ realizing tp,(C,,,A,1 U A : ) iff R(s,t ) . If R(s,t ) holds, clearly there is in MR an I (of power p+) realizing tp,(c,,Af u A t ) : itself. Suppose R(s,t ) fail, I E MR,14 = p+ realizes tp,(C,t,A,' UA,2).We shall work in Eeq. Let I' E I , (1'1= KO. If stp,(l', A,' U A:) -= stp*(l,,,,A: U A:) we get an easy contradiction : by IV, 4.9, dim(I', C , , M ) 5 p , and by (st), dim(I', C , , M ) = dim(I', A,' U A:,M) which should be 2 14 = p+ > p , a contradiction. However, if stp,(I',A,1 u A:) # stp,(c,,A,' u A:), as tp(l,A,1u A:) = t p ( c , , A t U A:),stillA,,A:,A:,Isatisfies 2.4(d),, so we can prove the assertion corresponding to (st). So in M, the relation R on (A: U A: : t E J ) is defined ; and [Ail < K , A = A'" (as T is stable in A , K = ~~(5")). So as in VIII, 3.2, we can prove that there are 2Apairwise non-isomorphic Fz-saturated models

c,

518

CLASSIFICATION FOR

F

$

o

-MODELS ~ ~ [CH. X, ~ $2~

of T of cardinality A. The case which most interests us, p = KO,T superstable (i.e. K ( T )= No),has the same proof as in VIII, Section 2 : we have just to choose the right @ (see there). For more, see (remembering A = A‘”!): [Sh 85e], 111, $ 3 and (for K = KO)[Sh 891, 111, $7 and [Sh 88aI.

Remarks. ( 1 ) For most h we can also get 2A such models no one ‘elementarily embeddable into the other, e.g. for h regular >(TI; more generally, see also [Sh 831, [Sh 891, new version, 111, $7. (2) In 2.5, if T is not stable in h we can still get similar results. For the reader unsatisfied with this, we give two cases in more detail.

FACT 2.5A. If T has the dop x 2 h > K,A > p 2 K , A regular, then there are sets A , (i< 2A),each of power x, such that, letting M, be F,”prime over A,, the following hold: (i) The 4 ’ s are pairwise not isomorphic. (ii) For i # j,4 is not elementarily embeddable into 4. (iii) For i # j,there is no elementary mapping from A , into 4. Proof. Let S* = (6 < A : cf6 = K}, and for each SES let qa be an increasing sequence of successor ordinals of length K converging to 6. For every S c S* let Js = x ,

R,

= {(qb(i),6) :~

A$ =

ES i< , K}, A , = CRS, M s = MRs,

u A: u u&,a>

: (4

a> E m ,

2<2

i
(note that A$ E A,). Now suppose S,, S, E S,S, -4 is stationary, f : A:, +Ms2 an elementary mapping, and we shall eventually get a contradiction. This clearly suffices. By renaming, we can assume that f is into

B = 4 U {ac:i < A}, where 4 = A,, U U A:, ,
2<2

tp(ai,4) is F,”-isolatedwhere 6:= 4 U {a, :j< i}. Let A* > 2X be regular, and choose 4 < @(A*), E), 114(< ( A, 4 n h = a,, (4:j5 i ) ~ & +and ~ ,f , A:,,B,&,(ac:i < h),R,, ( A : :1 < 2 , i < x),(C,:,:s,tEJ,,) (m = 1,2) belongs to No. Choose CE&-&, 6 = 6,. Next choose M < (H(h*),E),~ E M , n MK an ordinal 5 < K,

~

~

CH.

x, $21

519

THE DIMENSIONAL ORDER PROPERTY

(4:i < A) EM, and all the elements which we demand to be in No will be in M too. Now f(A: U A: U I ~ v c ~ ogives , c ~ us ) the desired contradiction.

u

c

FACT 2.5B. There are @, such that T c T,,IqI 6 A(T) and @ is proper for (0,Tl)(seeDef. VII, 2.6 and Lemma VIII, 2.3) such that for every linear ordering I : (1) EM(I, @) is Fio-saturated. (2) There is a formula

$@pq

= ( 3 ~ o , . . * , ~ ~ , - . * )A i < ~~+) c ( . . . , ~ t c ~ , z ) , . . . , g , ~ , a
P )jirst ~ order, g,z of length

< K , such that, for

s, t E I ,

+

EM(],0) $[6,,41, ins < t , where

6,

= (fc(at): i < IA, U A,I

and if

K

= No,6t = a,.

ct

Proof. Let J be {i:i < 3,+,}, where 8, = &,([TI), has power A, lM0, be F;-prime over u , , , A : U U { I z t :s < t} (so R is the natural ordering). Now expand Mo,-by A(T)functions and get M, such that: (a) For every b €6,EM, stp(6,a) is realized in the closure of by the functions of N,. (b) P = {it,:SEJ)where a8E A,O U A:, and if K = Nothen equality : i}. holds (i.e. the range of a8is A: U A:) and always A: U A: c { ft(a8) (c) f(- , at)(s < t ) is a one-to-one function from M, into (d) M, has Skolem functions. (e) If s < t , I' G Skolem hull of as U at is countable, infinite and realized over U at the same type as It,8 , then Av(I', Skolem hull of a8 U a,) is omitted in M,. Now apply the proof of VII, 5.4 to get EM'(w,@), a model of = Th(MR),which realizes only types Mk, realizes.

ct.

2.5C Corollary of 2.5B: Iio(A,T ) = 2Afor A 2 A(T)+N,.

c)

such Proof. From the @ of 2.5B we can derive @, proper for (w2w, that for some (not first-order) L-formulas P)~,n < w , V E % , q ~ " w

EM(I,@) t=qfl(cT7, tiv), if and only if v = q rn (just let I = " * o x (0, I} if 7 is a proper initial segment of v, I t

520

FOR

CLASSIFICATION

F

;

o

-

MODELS ~ ~[CH.~X,

$~3

, ( q , O ) < ( ~ , O ) ~ ( v , l ) < ( q , l ) , a n d i f r l ~ ~k, o ,then(qn ITII( A(T)) the theorems of VIII, Section 2 work. For A = A(T)by the proof in VIII, Section 3.1, Case I for almost every A we get the result, and we can complete also the remaining case. Anyhow our main interest is T). Remember that for T superstable, IT1 + K, 5 A 5 A(T),I(A,T ) = 2A was proved in IX, 1.20. Now we shall mention a topic, not necessary for the rest of the book, but naturally connected to the dop. Just as we have looked at “hidden order” we can look for “hidden unstability”, like the one caused by K ( T )> KO.

+

I

DEFINITION2.6 : T has the discontinuity dimensional property (didip for short) in the cardinal p (p regular) if there are Ff-saturated models Ma (a < p ) such that a < p*x <M, and the Ff-prime is not F,“-minimalover Ui<JQ model over

u,&

Now if T has the didip for p, then p < K ( T )and there are 2AFisaturated non-isomorphic models of power A if A > x 2 K , T stable in A, at least when (VA, < A)A:, < A, cf A = A. The proofs are parallel to the proofs in VIII on the number of Fi-saturated models of power A, when < K ( T )regular. Also, the parallel of 2.3 holds, and if > K is singular, T does not then the Fi-prime model over any set A is have the didip for cf minimal. Also, if each & is F,Q-saturated (i < a ) M <4,{ly :i < a} is independent over M, and N is F,Q-primeover UtJ& and in addition T does not have the dop nor any case of didip, then N is F,Q-minimal over Utda&.

x,

x x

X 3. The decomposition lemma (Hyp)I n this section T is superstable without the dimensional order property. The main result of this section, the decomposition lemma, states that (for superstable T without the dimensional order property) every Fg;saturated model is Fio-prime over a non-forking tree N,(qeI) (i.e., I E w’llMll is closed under initial segments, and

~

~

CH.

X, $31

THE DECOMPOSITION LEMMA

52 1

t p ( 8 , {q: 7 6 v}) does not fork over A$(L(7)-l),when l(7)> 0, and 7Q ~ q )This . is a kind of structure theorem, so this division line (superstable not dop) is significant. Though we shall eventually prove that some such theories (the deep ones, see Section 5 ) have many non-isomorphic models, all of them do not have a family of K Fio-saturatedmodels no one elementarily embedded into another, with K of arbitrary cardinality (this is with the help of [Sh 82bl). Recall that for A c B E C, B < A C means that for each F E C, tp(F,B) is orthogonal to A .

u

v-q

+

THE ATOMIC DECOMPOSITION LEMMA 3.1 : Suppose Nl < M are Fgo-saturated models. Then there are elements EM (i < a ) and models N2,c,M , such that: (a)

Y a,, <M,

<M,

(b) 4,N2.t are Fin-saturated, (c) tp(a,,Nl) i s regular, and for i #j, tp(a,,N,), tp(a,,N,) are orthogonal or equal, (d) a,EN,,,, and N2,tis Fio-prime over Nl U {at}, (e) N,,,
Proof. Let I = {a,: i < a} G M be a maximal set, independent over Nl of elements of M realizing over Nl regular types and by V, 1.12 w.1.o.g. satisfying (c). Let N,,, S M be FJto-prime over Nl U{ai}. So (c) and (d) hold trivially. By V, 3.2, we have:

u,

FACT A. tp(N,,,,

<=,, +&,)

does not fork over Y.

Now for each i < a, we define by induction on j < llMll+ an , {b,,,:y <j}such that tp(b,,,,N,,, U {b,,,:y <j}) element ~ , , , E M - N , , U is Fio-isolated or is orthogonal to Nl (we can assume that the second possibility occurs only if N2,$U {bt,,: y < j}is the universe of a n Ft0saturated model). There is a first P ( i ) < llMll+ such that bt,8cr,is not defined. Obviously (by l.6(5)): FACT B. N,,, U { b t , B : P< P ( i ) }i s the universe of an Fio-saturated model which we denote by 2y (clearly M , < M ) .

522

CLASSIFICATION

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F

g

-MODELS ~ ~ [CH.X, ~ $ 3~

o

Now : FACT C. tp,(4,

u,,,M;) does not fork over 4.

To show Fact C, we just prove by induction on 6 5 zi
We prove by induction on /3 6 p(i)that:

4.i< N , & , L

(*I

{ b , , y : y<

For /3 = 0 this is trivial, as well as for p limit. So let us prove it for 1. Let tp(c,&,,) be a type which does not fork over X Since 4 is F,"(,,-saturated, by V, 1.2(3) it suffices to prove that tp(c,&,J, tp,({b,,,: y 5 /3},N2,Jare weakly orthogonal. This is equivalent to

p+

tP(G&,,) I- t P ( m , , u { & , y : y

s pH.

By the induction hypothesis on /l tP(c94.t)I- tP(c>N,,,u @t,+ y < PI). Hence tp(C,Nz,,U { b t , y y: < /3}) does not fork over N2,,, hence (by transitivity, see 111, 0.1) does not fork over N,, and it suffices to prove it is weakly orthogonal to tp(b,,,,N2,,U { b , , y :y < p}). If the latter is Fio-isolated this holds by V, 3.2 and if the latter is orthogonal to N,, this holds by V, 1.2(3). FACT E. M is Fio-minimalover ui
u,,4

PROOF OF FACT E. Let M' <M be F,"-prime over (we know that there is such M'). Suppose M' # M which holds if M is not F,"prime over U f < = 4 and , M' can be chosen so that it holds if M is not we shall eventually get a Fio-prime and Ffto-minimalover ut
~

~

CH.

X, $31

523

THE DECOMPOSITION LEMMA

orthogonal to it, hence by V, 1.12 some b ' d 4 - M ' realizes the stationarization of p over M'. But this contradicts the maximality of I = (a, : i < a} as b' can serve as a,. So we can assume tp(b,M') is orthogonal toNl. Choose a setB EM', IBI < K ( T = ) Nosuch that tp(b,M') does not fork over B. SinceM' is Fgo-saturatedwe can also assume tp(b,B)is stationary. Also there is a is Fgo-isolated[as tp,(B, U i K a q ) finite S E 01 such that tp,(B, Utes&&) is Fio-isolatedby IV, 4.3a s B -C M', andM' is FgO-primeover So there isM* E M' Fgo-primeover and B E M so that tp(b,M*) is a regular type orthogonal to Nl. We prove that this is impossible by induction on 181. If IS1 = 0 then tp(b,M*) is not orthogonal to Nl as M* = N,, a contradiction. If IS1 = 1,s= {i}we get a contradiction to the definition of &&.Suppose IS( = n 1 and assume by =< n, if N* < M is Fk-prime induction that for any U S a with and rES(N*) is a regular type which is realized in over M , then r is not orthogonal to N,. Let ~ E and S choose M+ EM*, Fgo-primeover U,,,,q such that M* is Fio-primeover M+ U && (for ) . V, 3.2 (Ill+,&}is independent over Nl. Since T some fixed ~ E S By does not have dop, by 2.3,tp(b,M*) is not orthogonal to one of&&or M + . Let N denote the model tp(b,M*) is not orthogonal to. By such that q is not orthogonal Lemma 1.4there is a regular ~ES"(N) to tp(b,M*). But tp(b,M*) is orthogonal to every (regular) complete type over Nl. Since non-orthogonality is transitive on regular types (V, 1.13),it follows that q is orthogonal to every regular type in S(Nl),i.e. by 1.4, q is orthogonal to Nl. But q is not orthogonal to tp(b,M*) and b is in the Fio-saturated model M , so by V, 1.12,the stationarization of q on M* and hence q is realized in M . But then q and N contradict the hypothesis of induction. So we prove Fact E and we can check that the only part of 3.1 to be proved is the last phrase of (e) which we leave to the reader. For (g) use l.6(5), 1.6(1).

u$
uies&&

+

uiEu&&

THE DECOMPOSITION LEMMA 3.2 : For a n y Fg0-saturated model M we can f i n d a set I S u'llMll (Jinite sequences of ordinals < IlMll) closed under initial segments, and q , a , for ~ E and I p , ( q ~ I - { ( )}) such that : ( 1 ) N, <M is Fgo-saturated. (2) No is F&-prime (over 9). (3)pl,yr>= tp(a,-(i),N,) is regular, and for qn (j) €1, p,-(i), p,-(,> are orthogonal or equal.

524

CLASSIFICATION

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MODELS ~ ~[CH. ~X,

$~3

N,-(,) is Fgo-primeover N, U {a,yl)}. (5) tp*(N,-(i),U{N,:V E Ibut not 7 -(i) 4 v} does not fork over N, (for 7 -(i) €1). (6) M is Fgo-prime,Fgo-minimaZover u,,,N,. (7) tp*(U{N,:q 4 v ~ o , N ,is) orthogonal to qrn when Z(7) = n+ 1. (4)

Proof. We define by induction on n, a set L, of sequences of ordinals of length n, models N,, M;, and elements a, for each EL, such that : (1) r l € l , , m < n i r n p l i e s r ] r m E ~ , a n d & = { ( )}. (2) N, is F;to-saturated. (3) ) is Fgo-prime (over $3) M( ) = M . (4) is Fgo-primeover N, U {ayTt)}. ( 5 ) Let = t~(a,~,),N then , ) , it 1s regular, and paYr),py(5) are equal or are orthogonal. (6) M,, is Fgo-saturated,N, G q E M . (7) 7 4 v i m p l i e s N , z N , E M , G q . (8) M;, is Fg;prime over U,M;,y2). (9) N,-(t) < N , M , , I Y , ) ' (10) { ~ y :c7 ) (i) €0is independent over N,. (11) A , = {a,y2):7 n(i) € I ) is a maximal subset of&(, (or even set independent over of sequences from (12) A , is a maximal subset of M which is independent over N7 and if every element of it realizes over N, a type orthogonal to Ub
N<

-

3)

7#<

8.

>.

The definition is easy: for n = 0 trivial, for n+ 1, for each ~ E I , we apply 3.1 with N , , q standing for N,,M and get U ~ , N ~ , and ~ , &let a,-(r) = a,, N,,(,) = &,c,M,,?$) = & (so L,+, is the set of v's of length n 1 for which N, is defined). Now all the conditions of the lemma are obvious Let I = except " M is F,"-prime and F,"-minimal over UVEIN," and ( l l ) , (12). Let us first prove ( l l ) , (12). If (11) or (12) fails for 7, let a exemplify i t ; i.e. a€&&,tp(&,N,)is orthogonal to N,r(k-l) if Ic = Z(7) > 0, and &#N, and tp(a,N, U {a,yi):7 does not fork over N,;w.1.o.g. tp(G,N,) is regular. As in the proof of 3.1,

+

urnIn.

-(i)~o)

~

~

CH.

X, $31

THE DECOMPOSITION LEMMA

and as M,, is Fie-primeover U {hi!&,, :r] ^ (i) not fork over N,; by V, 3.2,

525

€4,tp(a,U,M,,?,)) does

This proves (11) (i.e. when GEM,,) ; for (12) note that by the above t p ( a , q ) does not fork over N,, hence is the stationarization of tp(&,N,)which is orthogonal to N,r(l(,)-l) when it is defined. Let k = l(r]),and we now prove by induction on 1 k: that t p ( n , ~ , , , _ , , ) does not fork over N,. We have proved for 1 = 0 and for I+ 1 notice that as t ~ ( a , q ~ ( ~ - ~ , ) does not fork over N,, i t is parallel to tp(a,N,), hence orthogonal to N,r(k-i), hence to qr(k-l-l). SO for j # q(k - 1- 1), t~(a,M,,~(~-,)) is orthogonal to tp* (%r(k-l-i)yj), qr(k-l-l)). As {qr(k-,-l)-(j):j < a)is independent o ~ e r N , ~ ( ~ -tp(a,%,,,_,,) ,-~), is t~*(Uj#~(k-l-l)M,,r(k-l-l)-(j),~r(k-l-l)), and

So as M,,r(k-,-l)

is Fio-primeover

Ujqr(k-r-l,-(j),by IV, 4.10(2),

so also the latter does not fork over N,. For 1 = k we get a contradiction to EM = M( ), as tp(n,M( )) does not fork over N,, and a#N,. Now we shall prove that M is Fie-prime, Fie-minimal over U,,,N,. So suppose M' G M, M' # M, M' is F:-prime over U,,,N,, and choose EM-M', R[tp(b,M'),L, a ] minimal, hence tp(b,M') is regular. Now tp(b,M') is orthogonal to each N,. If not, choose r] with minimal length, then by 1.4 for some regular q€Sm(N,),tp(b,M'), q are not orthogonal and q = p,?,) for some i, or q is orthogonal to every pVy,). Let qf€Sm(M') be the stationarization of q over M', so by V, 1.12 there is & E Mwhich realizes qf,so tp(af,M')does not fork over N,. By choice of T,I (being of minimal length) (if r] # ( )) tp(b,M') is orthogonal to Uk,l(,,N,rk,hence by V, 1.13 also tp(ar,M') is, so we get a contradiction to (12).We conclude tp(b,M') is really orthogonal to each N,. Choose a finite B EM' over which tp(b,M') does not fork, and a finite I*G I closed under initial segments such that tp,(B, U,,,N,) does not fork over U N,. As M' is Fio-primeover U,,,N,, tp,(B, U,,,N,) is Fie-isolated, so tp,(B, U,,,.N,) is Fio-isolated(see IV, 4.3).

,,,.

526

CLASSIFICATION FOR

F;,-SATURATED MODELS

[CH.

X, $3

u{q: uVEI*q,

Let N < M ' be F;,-prime over r ] ~ land * } B C N. We have so that there is a regular found an Fg0-primemodel N over orthogonal to each N, ( T , I E I *realized ), in M, and I*is type in Sm(N), finite, closed under initial segments. We get a contradiction to the statement in the last sentence by induction on II*I. If I*= { r ] r l : 1 < lg(r])}this is trivial, and if not choose distinct r ] , u in some I*n L, with n minimal. Let No = qr(a-l), N' < N be FJto-primeover r] Q CT E I*} a n d P c N be Fio-primeover {N,: not r] 4 a,but r E I*} such that N is Ffto-primeover iV' U P.Now by 2.3 (and as T does not have the dop) for some 1 E { 1,2}, there is a regular complete type over N z orthogonal to No, realized in M, and we get a contradiction to the induction hypothesis on II*I. I n fact our proofs also prove:

u

u{N,:

LEMMA 3.3: Suppose I E "'A is closed under initial segments, {NV:r]eOis a non-forking tree of F,"-saturated models [i.e. r] Q ~ - N , < N , , T , I E I - { ( )}*tp(N,,U{N,:uEI, not r] 4 v}) does not

fork over N,r(l(,)-l),and each N, is Fio-saturated]. If T does not have the dop, M F,"-primeover U I E I qthen , M is F,"minimal over U,,,N, and every q E S m ( M )is not orthogonal to some N,. Proof. First note

FACT 3.3A. If S;, S, are non-empty subsets of I which are downward is independent over UYESln closed, then

{uvEs,N,, uVEs2Ny}

Proof. By the local properties of forking, it suffices to restrict ourselves first to the case 4-4 finite, then to the case 8-4, 4-4 finite, and finally &,Afinite !$ (using 111, 0.1). Now we prove the statement by induction on IS; U &I; w.1.o.g. S; z S; n 4 , S, z S, n 4 , clearly )ES; n 8 . So choose 7 ~ 4 - 4of maximal length, and by the induction hypothesis and transitivity of forking (see 111,O.l) it is enough that V E S ;U S,, u # r ] } ) does not fork over N,r(z(,).-l),but this tp,(N,, follows from the hypothesis. Now return to 3.3. If one of the conclusions fails we can reduce it to the case I = I* is finite; then, as in 2.2, the two conclusions are equivalent. If both fail, there is M' i M, M Fi0-primeover UTEI.q, bEM-M', tp(b,M') orthogonal to every N,; and continue as in the last part of the proof of 3.2: from the choice of B.

<

u{N,:

CH.

X, $41

DEEPNESS

527

X.4. Deepness (Hyp) In this section T is superstable without the dop. As we know the number of non-isomorphic Fi@-saturatedmodels in every N, 1 IT1 N, for unsuperstable T and for T with the dop, we concentrate on the case in the hypothesis. By the decomposition Lemma 3.2 we know every Fio-saturated model is Fi0-primeover a non-forking tree of F:-prime models. Clearly, if there are few trees then there are few models, but the converse is less clear. Anyhow, clearly the most important distinction is whether the tree ( I in 3.2) is always well-founded. If it is always well-founded, naturally some rank is defined (called here the depth), and if the rank is small the number of models is small. I n this section we introduce the basic relevant notions and the simple facts about them. Notice that we could have chosen some other variants of the notions, but by later sections we can show that they would be equivalent. We define the depth so that the results on the number of models can be stated smoothly (this is why 4.l(iii), 4.2(1) are such that the depth is not a limit ordinal). Let us make some more specific remarks. Note that if the tree is Fio-prime over 4 U {a,}, not well-founded there are 4 < q+14+l tp(al,&) regular orthogonal to &-l. So the rank is defined as an attempt to build such a sequence (i.e. the rank of (N,N',a)is 00 iff there is such a sequence, N = No," = Nl,a = ao). In Definitions 4.1 and 4.3 we give some variants of this, in 4.2 we define the relevant property of a theory (deepness), in 4.4 we prove various facts, and in 4.5 the essential equivalence of some variants is given. Now 4.6 says that, looking for high depth, it is enough to look at types not orthogonal to $3. For "canonical" examples see 4.9, 4.10.

+

DEFINITION4.1 : Let K' = {(N,N',a ) :tp(&,N) is regular, and N' is Fio-primeover NU {a}and N is Fio-saturated}. For every member of K' we define its depth, an ordinal (zero or successor but not limit) or infinity 00, by: 2 o iff (N,N',~%)EK'; (i) DP(N,N',~)

528

CLASSIFICATION

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E

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(ii) Dp(N,N’,a) 2 a+ 1 (azero or successor) iff for some N”,c’: (N’,N”,a’)€K‘, N’ < N N ” and Dp(N’,N”,a’) 2 a ; (iii) Dp(N,N’,a) 2 S+ 1 (6limit) iff Dp(N,N’,a) 2 /3 for ,8 < 6; (iv) Dp(N,N’,a)= co iff for every ordinal /3 Dp(N,N’,a) 2 /3, Dp(N,N,E) = a iff D~(N,N’,E) 2 a but not Dp(N,N’,a) 2 a+ 1. DEFINITION 4.2:(1) The depth of the theory Dp(T) is U{Dp(N,N’, a ) : (N,N’,a )EX}when this is finite and U{Dp(N,N’,a ) : (N,N’, a ) EK‘}+1 when this is an infinite ordinal. (2) The theory T is deep if its depth is 0 0 ; otherwise it is shallow. DEFINITION4.3: (1) KA= {(N,N’,a):N,N’ are F,”-saturated, EN', a#N,N’ F,”-atomicover NU {a}}.Dp((N,N’,a),K)is defined as in 4.1 for any set K of triples. If (N,N’,a)#K we interpret Dp((N,N’,a),K) , a)}(not closing under isomorphism) and if as Dp(N,N’,a),K U { ( N N’, K = K’ we omit it. Let K” = {(N,N’,&)EK’:tp(a,N) is regular}. (2) For a tree I, we define DpA(7,I) (A a cardinal, ~ € 1 )DpA(7, : I)2 a+ 1 iff for A v’s, 7 Q v, Dp(v,I) 2 a, and for a = 0 or limit DpA(7,1)2aiffDp,(y,I) 2/3forevery/3
LEMMA 4.4: ( I ) If (N,N’,a)€K,(N’,N”,a’)€K,N’ <,N”, then Dp((N,N’,a),K) 2 Dp((N’,N”,a’),K) (the inequality is strict except when both are co) [this holds for any class K of triples]. (2)If a < D p ( ( N , N ’ , a ) , K ) < m ,( N , N ’ , u ) E Ka, not limit, then has depth a (inK). some (No,No,a0)€K ( 3 ) Dp(N,N’,n) = 00 iff there are&,al (1 < w ) , & + ~< N l & + 2 , & =N, 4= N’, a0= a, (&+1,&+2, at)EX.(Similarly for any “reasonable” K. 1 (4) If a = Dp(T) or a = Dp(N,N’,a),a < 0 0 , then a < (21’1)’ and in fact a < &(IT[) (look at VII, $ 5 on the definition of S ( l f l ) and information on it). (5) The depth is preserved by automorphisms of 6. ( 6 ) If (&,q,&,)~K;l~ (1 = 0,l)and tp(al,q) are parallel or not orthogonal, thenDp(No,N’o,ao)= Dp(N,,Nl, 4).Ifonly (Nl,Nl,al)cKNO, tp(al,Y) regular, still Dp(&,N’, ao)5 Dp(&,x, at).

~

~

CH.

x, $41

DEEPNESS

529

DEFINITION4.4A : We let, for regular p, Dp(p) be the depth of (N,N’, a)EK&,when tp(a,N), p are parallel. PROOF OF LEMMA 4.4. Easy. Note for ( 3 ) we need (6). (1) Trivial. ( 2 ) We prove it by induction on /? = Dp(N,N’,a ) ;for /3 = a.there is nothing to prove; for p > a, use 4.l(ii) applied to Dp(N,N’, a) 2 /?+1, to get N”, a’ with y = Dp(N’,N“,a’) 2 a,but by 4.4(1), y < /?,and use the induction hypothesis. (4) By (6),Dp(N,N’, @) depends only on tp(@,N)up to parallelism, and by (5) it is preserved by automorphisms of a, hence there are 52ITI possible depths. But by ( 2 ) the ordinals which are the depth of some triple form an initial segment of the set of the non-limit ordinals, hence

a = Dp(N,N’, a) < co - a < (21T9+. The first phrase has been proved above. We work as in VII, Section 5. For the second phrase, let 23 = ( H ( h )E, , T, S(lTI)),H ( h ) the family of sets of hereditary cardinality < A , and w.1.o.g. T c ITI. Now consider 23’ elementarily equivalent to 23, with “ T and IT1 ” standard : but non-well-ordered “ordinals” < S(T).(23’is known to exist.) I n this model we can consider various notions and check whether they are absolute, i.e. whether if %’ says something holds it really holds. Now this holds for (a) being a model of T, (b) R ( p , d , h )= n ( A finite, h 5 No), (c) being orthogonal types, (d) non-orthogonal types, (e) tp,(A,B) does not fork over C , and (f) A c B, p €Sm(B) ; p has a unique extension in Sm(NU B ) when tp,(N,B) does not fork over A . We can conclude by 4.4(1) that also if %’I= “Dp(p) 2 a*, a* an ordinal”, then p has depth 2 the order type of ( ~ ~ 2 3a”< : a*}. If a* 2 S(lT1)this is not well-founded so Dp(p) = co. ( 5 ) Trivial. (6) We prove by induction on y that [Dp(No,No,@o) -5 y or Dp(N,,T1,ao)5 y] implies the equality. We can choose F:-saturated N,, No UN, c N,, and by 4.4(5) w.1.o.g. tp(q,N,) does not fork over & (for 1 = 0 , l ) .By V, 3.2, No is F:-constructible over N, U ao,and let

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N2 be Fz-primary over N2 UW,= N2 UN, U a,. In N2 tp(al,N2) is realized, so w.1.o.g. alcY2,Nl G N2,Y2F,“-primeover N, U aland over N2 U X1(and over N2 U a. and over N2 U No). By symmetry it is enough to prove Dp(N,No,a,) = Dp(N2,r2,go), because checking the definition we can observe that Dp(N2,X2, ao)= Dp(N,,W2,al).Now the inequality 5 is trivial (with the induction hypothesis for y ) and for the inequality 2 we have to replace parameters with others of the same type. For proving the second phrase act as before, observing that by the first phrase Dp(Nl,Nl, a11 = DP(N,,W2,ao). (3) First suppose that there are &, @,, and let a, = Dp(&,&+,, al). By 4.4(1), a, 2 al+l,and if a, # 00, a, > a l f l .So if a, is # co, then a, (1 < o) is a strictly decreasing sequence of ordinals, a contradiction; so a, = 00, but a, = Dp(No,Nl,a,) = Dp(N,N’,a), so we have proved the “if” part of 4.4(3). Now suppose Dp(N,N’,a) = 00. We define by induction on 1, &+l, @, such that is Fz-saturated, Dp(&,&+,,a,) = 00, 4 0 ) . So let No = N , Nl = N’, a, = a. So the induction ~ ,00, ) there are &+2, hypothesis holds. For l + 1 as D P ( & , & + ~ ,= a,+,= (a,+2), such that (4+1,&+2, a t + 2 ) ~ KD’ P, ( ~ + , , & + ~ , 2 ( 2 9 + , hence by 4.4(4), D ~ ( & + ~ , q + ~ = , a00. , + As ~ ) we can carry the induction we have proved the “only if” part of 4.4(3). The following lemma shows that there is no real difference between the various Dp( -, K)’s, in particular, whether we use a or @.

LEMMA 4.5: (1) For any (N,N’,a)€KA Dp((N,N,a),&) = Dp(N,N’,a). (2) If K’, = { ( N , Na, )EK’&:N is Fio-atomic over NU {a}}, then on K’,, Dp(-,K‘), Dp(-,Kl) are equal. ( 3 ) For any complete type p , Dp(p) = sup{Dp(r):r a complete regular type not orthogonal to p } . (4) Suppose N<,N”, & E N , @“EN”, tp(a‘,N) tp(a”,N’) are regular, tp(N’,N U a’) is almost orthqonal to N and tp(N”,N’ u a”) is almost orthogonal to N’, Then Dp(N,N’,a“)> D(N,N”,a“) (or both are 00) .

Proof. ( I ) Remember that by 111, 4.22, if M is F:-saturated, M’ is F:-prime over MU a, K < A , then M is Fgo-atomic over MU a. 80

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easily by the definitions and 4.4(6) Dp((N,N',a),&) 2 Dp(N,N',@). So it suffices to prove by induction on a that: (*)

Dp((N,N',a),&)2 a

Dp(N,N',a) 2 a for (N,N',a)EK,.

For a = 0, a limit and successor of limit this is trivial; for a = /?+1, /? not limit, there is ( N ' , N " , ~ ' ) E KN', ,<,N", and Dp((N',N", @'),K,)2 /?, hence by the induction hypothesis Dp(N,N", a') 2 @. Apply Lemma 3.1 forN,N" standing forN,,M and get ai,N2,,,& (i < i(0)). By V, Def. 3.2, Th. 3.2, w.1.o.g. N2,,= N and i(0) is finite ( = w(@',N)), and let y = Max,,,(,,Dp(N',N,,,,a,) (so y is not limit); Otherwise, as y is not limit there is clearly it sufficesto prove y (N",N*,a*) E K I , N" <,N*, Dp(N",N*, a*) 2 y . As tp(&*,N") is regular, orthogonal to N', clearly as in the proof of 3.1 (see Fact E ) ,~), and q are not for some i < i(0) and regular q ~ h " ( N ~tp(@*,N") orthogonal. Let c realize q, N'2,i be Fio-prime over N2,,U {c}. So by

z/?.

4.4(6)

(a)

DP(N",N*,@*)= Dp(N,,i)&,C).

Now as q is not orthogonal to tp(a*,N"), it is orthogonal to any regular complete type over N' (as non-orthogonality is an equivalence relation among regular types, see V, 1.13) hence q is orthogonal to N'. Hence
DI>(N',N,,,,a,)> DP(N,,*>N'2,i,q.

As Dp(N",N*,@*) y's definition. (31, (2), (4) Easy.

2 y by (a),(b),Dp(N,N,,i,ai)> y , contradicting

LEMMA 4.6: If (No,T0,a0)€KIXo, tp(ao,No)is orthogonal to 9 and has depth
-= Dp(N,,N;,Q

Proof. By 4.4(6), w.1.o.g. No and No are Fio-primeover 9. Let B C No be finite, such that tp(ao,No)does not fork over B . There is B' realizing the stationarization of stp,(B,pl) over No, and let M' be FEo-primeover No U B'. By 1.7, M' is Fio-primeover 9. Hence by IV, 4.18, No,M' are Fio-primeover B , B , respectively, hence there is an isomorphism from M' onto No taking B' to B. So there is a model N < No such that tp,(B,N) does not fork over 9 and No is Fio-prime

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over NUB. Hence by V, 3.9 there is a finite set J = {b, :i < n} C _ No independent over N, of elements realizing regular types, such that No is FP, -prime over N U J. By V, 3.2 there are N z Dp(q) = Dp(&,&,ao) (see 4.4(1), 4.4(6), respectively) and tp(b,,N) is not orthogonal to $3 as it is not orthogonal to tp,(B,N) (because bgENo,NoF" - rime over NUB, '' 'deI tp(B,N) does not fork over 9). Now put ( f l N ; ,it,) = ( N , v ,b l ) . THEOREM 4.7 : T) (the number of non-isomorphic F;,saturated models o power N,) is at most ~,,(T,(la1~'~') for shallow T, so if it is <Sa(lTi,(lalZ' I) = Sa(lTl,(l~l) < 3(Z~~~)+(IaJ) and if T is countable

<~ J I 4 ) . Proof. Immediate by 3.2, and the bounds on S(lT1) (see e.g. VII, 5.5 and 5.5(2)).

LEMMA 4.8: If T is superstable without the dop, then R(x = x , L , 0 0 ) 2 Min{w, Dp(T)}.

< o,k < Dp(T). Then we can find N 7 ,as~ in 3.2's notation for qek"A (any A). Now we can prove by induction on k-Z(q) that for every bEq\N7

Proof. Suppose k

k-l(q)+R(tp(b,N,),L,a)< R ( x = x , L ,0 0 ) . Example 4.9. A very natural example of a superstable T without the !, = Th("'w,f) when f ( q )is q dop which is deep, is the following T:& if r] = ( ), and q r n if qen+lo. Notice that a model of Tdpconsist of trees, exactly one with a root (i.e.f ( x ) = x ) , in which every element has infinitely many immediate predecessors (i.e. y's such thatf(y) = 2 ) . A similar example is TZp = Th(O>w,... ,P,, f,,, ...) where P, = ,w, f,, is a partial function: f rP,. Both theories are N,-stable, and by expanding a little we can get elimination of quantifiers.

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Examples 4.9A. Examples of shallow theories can be obtained similarly to 4.9; we prefer to use the T"* from 11: the language consists just of the two-place relations E, (i < 01). The axioms of T state: each E, is an equivalence relation, for i <j,E, refines E,, moreover each E,-equivalence class is the union of infinitely many distinct E,-equivalence classes. Also E, has infinitely many equivalence classes and each E,-equivalence class is infinite. It is not hard to check that Dp(N,N',a ) = i iff i = y when y < u,i = y+ 1 otherwise, where y = min{j : there is b E N ,bE, a} (and y = u if there is no such j).

CONCLUSION 4.10 : For every ordinal /3, which is a natural number or a successor ordinal, for some T = T;, IT1 = 1/31+ K O , and 'io(Na,TI = ',(la1 + '0).

+

Proof. For p < w or /?= u 2 we use the previous example ; for /?= S+ 1, S limit, take the sum of models of TI' (i < 8 ) with disjoint languages. The computation is easy, but it is a worthwhile exercise for the reader as we are proving in the chapter that every shallow theory T is in some sense similar to

X.5. Deep theories have many non-isomorphic models (Hyp) In this section, T is superstable without the dop. Clearly, if T is deep, we can construct trees like the one we get in 3.2, and try to prove that we get many models. The freedom we have is to determine various dimensions. So when u = N, this is easy. Generally notice that we have much less freedom than, e.g., in 2.5 (when proving that the dop implies there are many models). This section is dedicated to the proof of: THEOREM 5.1 : If T is deep, A(T)5 K,, N, < K,, then Igp(Ka,T ) = 2n=,i.e. T has 2 N non-isomorphic ~ FiB-aaturatedmodels of cardinality Ka.

Remark. But the lemmas will be used also for shallow theories. We shall concentrate on the case N, > A(T), where A(T) is the first cardinal in which T is stable.

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DEFINITION 5.2 : We call (4,a,: r] EI , v E I+)a representation i f : (1) I is a tree with root ( ) of height s u ,and we let 7- be the def def unique predecessor of r] for r] EI+ = I -{ ( )} and let I- = (7- : r] E I + } . (2) N<) is Fgo-primeover 9. (3) If r] = v; = v;, then pUl= tp(a,,,N,,) is regular, and pul,pUp are equal or are orthogonal. If all p, with v- = r] are equal, q,, will denote their common value. (4) For ~ ] E I +N,,, is Fio-primeover U {a,,}. ( 5 ) For r] E I - , {a,: v- = r ] } is independent over Nq. (6) If T E I , ~ -is- defined, then tp(a,,,N,-) is orthogonal to N,,--.

Convention 5.2A : When we say ( ' pis orthogonal to N,,- but r] = ( ) ", we interpret this as being always true. Remark. We shall write in short
u,,,,N,.

LEMMA 5.4 : Let (Nq,a,,: r] E I ) be a representation, then : (1) tp,( U {q: r] Q Y}, {q: not 7 4 v}) does not fork over

qq-)(for

r]EI+).

(2) For Y E Ptp,(U,,,,N,,N,) , is orthogonal to q-. (3) For r ] , V E I p,,, , p , are orthogonal or equal (and then 7- = v-). (4) Each N,, is Fgo-prime over 9. (5) If M is Fgo-primeover UQEIN,, ~ E S ~ (isMregular, ) then p is not orthogonal to some p' E Sm'(N,)for some r ] . (6) (q, a,,:r] EI ) represent some model of power A(I) III. (7) Dp(N,,-,N,,a,,)is at least Dp(r],I) (see Def. 4.3(2))even at least ( - I ) + Dp(r],4+ 1. I

+

Proof. As in the proof of 3.2, or easy using 3.3, 1.7. (1.7 is needed for (4b) LEMMA 5.5: (1) Every F;to-saturatedmodel hus a representation. (2) If (@,, a:: ~ E I , )Fio-represents 4, 1 = 0 , 1, F :&+Il a n isomorphism (for partially ordered sets so it preserves the level),F : q + U , , s r l q is a n elementary mupping, it maps onto

u,,,,o

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(equivalently,for each r] €4,F‘ i s a n elementary mapping ontoPF(,,, DomF’ = U l e 1 , ~then ) , 4,MI are isomorphic. ( 3 ) If in (2)F , F are not necessarily onto, t h e n 4 can be elementarily embedded into MI. (We may demand F’maps onto or at least maps into PF(,) and tp,(F’(q), WF(,)-) does not fork over F ( q - ) .

q

q

Proof. (1) This is by 3.2 (and 5.4). (2), (3)Trivial.

(3,

LEMMA 5.6 : For p > 0, av: r] EI ) Fgo-represents an F&-saturated model iff for every r] EI , and regular p E Sm(N,)orthogonal to N,- for at least NBv’s, p , i s not orthogonal to p .

Prooj. Easy by 5.4(5),and usual arguments.

u,N,,

Let M be F&-prime over and suppose A C M , IAl < N,, P E S ( A is ) omitted. Let tp(a,M) be a stationarization of p over M ; clearly w.1.o.g. p is stationary. By V, 3.9 there are {a,: i < n}, independent over M , realizing over it regular types such that tp(a,M U {ac: i < n})is Fio-isolated: w.1.o.g. tp(a,A U {ar: i < n}is Fieisolated, and tp(a,,M ) does not fork over A and tp(a,, A ) is stationary. We now try to define by induction on i ,b,EM realizing stp(a,,A U { b , : j < i } ) .We have to fail for some i , so w.1.o.g. p is regular. By 5.4(4)p is not orthogonal to some regular qES(Nv),which is not , , not orthogonal. By orthogonal to some P,-(~,, so we know p , ~ , ~ , are V, 2.3, 2.4, dim(p,M) = dim(p,-(,,,M), which is ZN, by hypothesis, so we finish. LEMMA 5.7: If T i s deep, then for every tree I with root ( ) and height - w , there i s a representation I a, : 7 E I ) (infact q, i s well deJined for every ~ E I -i.e. , pvYI>= q,).

(3,

Proof. Easy. By 4.4(3)there are Fio-saturated y,y+l Fio-prime over & U {a,}, t p ( a l , y )regular, 4+1 < N,y+z. Complete the partial ordering of I to a well-ordering and let {yi: i < i*} be a list of the members of I in increasing order. Let n(i)= l(r]J and m(i) be maximal such that qr rm(i)~ { r ] , : < j i } , in fact q irm(i)= qlCi,(for i > 0). Now we define by induction on i an elementary mapping Fi. F, is the identity on No, Fris an elementary mapping with domain N&), extending F,,,, such that tp*(F,(N,,,,),q
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DEFINITION 5.8 : For a representation (N,, a,,: 7 E I) we let : (1) E (E of the representation) is the equivalence relation on I+ defined by :

yEv iff p,, = p , (iff p,, p , are not orthogonal). We write P , , , ~instead of p , and NTIEinstead of N,,-. (2) The representation is standard if q / E is uncountable for every 7EI. (3) We say J is a A-large subtree of I (relative to E) if for q E J + there are s;lE G q / E of cardinality < A such that J = { ~ E for I : no 1 < ~ ( 7does ) 7 ( I +1)E S ; r ( n + l ) I E } . (4) We define similarly “J is a (
r

CLAIM 5.9 : Suppose (3a, ,:7 E I ) i s a (standard)representation of M . Then for a n y stationary type q over a finite subset of M : (a) ifq is not orthogonal to p,, then dim(q,M) = 17/EI, (b) if q is orthogonal to every p,, then dim(q,M) = No. Proof. Easy.

CLAIM 5.10: Suppose ) 0, is an Fio-prime model over N , U N for some V E I - u ,v - E U , v # p (and = a,),and for every such v there is such 7, (iii) qo, is F;Io-primeover N u -6°F. (iv) if ~ E I V~ E,I are as in (ii) then for any sequence 7 , q A 7 ~ I l ~ v A 7 E and I , if ~ - F ( ~ ) E I thenIV&,) ~, is Fio-prime over Nv-T-(t) U N&r and {N,-.T-(t), is independent over N,,,,.

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MAIN LEMMA 5.11 : Suppose for 1 = 1 , 2 , (fl,,i$; 7 € 4 ) is a standard representation of i&, and F is an elementary embedding of Ml into 4. Let 4 be as in 5.8. Then there is a function h from 4 into 4 such that : (i) h is one-to-one h(( )) = ( ) and: rE,p * h(7)E2h(p); ~ f i , ~ ,is a countable subset of VIEl,s : / ~ a, countable subset of h(q)/E,, (ii) for ~ E I ;F(&),p&) , are not orthogonal, (iii) DP(Pi/El)= DP(P~(q)/E2)~ (iv) i f yO€Il,~ ] , E v , c q0/-&, vo uncountable disjoint to sioIEI,for ~ E weVhave ~ Z,,-C,E~, tp(d,“C,^ai,?-) is the same (for all ~ E V and for some a,[7 h(u,) €Il A pin(.,) is parallel tp(d,, 4)] then for all but finitely many ~ E V h(7) ~ , 4 h(7 -(a,)). (v) If F is onto 4,h is onto 4, and (iv) holds for h-’,4, Il too.

Proof. Note that by 5.9, V, 1.16 (and as the representations are standard) : (1) dim(pk, = 17/41 for 7 €4. First note that for any ~ E I : dim(pi,Ml) , 2 N, (as the representation is standard). Hence dim(F(p:),&&)2 dim(F(p:), F(Ml))= dim(p:,Ml) 2 dim(p:,Ml) 2 N,. Hence by 5.9 for some v = v,~Il,F(pi),p.:~ are not orthogonal. is unique and depends on Clearly, v, is not unique, but v,/& 7/E1 only, and (2) 17/41 = dim(pi,M,) G dim(F(P:),m = dim(p,p&) = Iv,/E,I. We shall now define hvlE,= h ( v / E l ): q/El +. v,/El for each Elequivalence class separately.

r

Class I. If Dp(pvIE,)= 0, hTIElis any one-to-one function from VIE, into v,/E,. Class 11. Dp(p:/,,) > 0. By 7.2, piE,is trivial, and also P:,,/,~is trivial by 7.3(4). Let b, = bilElG Ni- be such that prlIE,does not fork over 6ilEl, pilEl 6ilE,is stationary. Let 6tlE,E N :,lE2 be such that pf7/B,does not fork over 6:lE2, pf71E, 6;lE,is stationary. 4 is Fin-atomicover So there is a finite Now F(&~)E&, u = u,,~,E 4, V - E U , u, closed under initial segments and tp,(F(6i), -I tp(l”;6:), Hence tp(F(bi),U v E I 2 Nis~ Fgo-isolated ) N G 4,4Finovel*uvEu,q.So there is N, Fin-primaryover

r

dVEu x)

r

byE12x).

UvEI2x. u,,, x,

~ )

538

CLASSIFICATION FOR

F&-SATURATED MODELS

uvEIzx

[CH.

X, $5

q/El

primary over U N and F(6;)E N . Let q = be a maximal set of sequences from A$ realizing independent over 6 :. By V, 1.16 it is countable. Easily q U {a;:p E r]/El}is a maximal set of sequences from Ml realizing pilE, 6 : independent over 6.; As dim(F(p,’),F(6;),N) = KO,there is a countable subset s ; / ~ ,E q/&, s.t. {F(a:):p E q/E, -s;} is independent over (N,F(6$?,)). Also {q: v E v,/ E,, v 4 u, hence tp(a,,N) does not fork over (8-)} is a maximal subset of 4 of sequences realizing independknt over (N,N,-). Also s,blEz= { v : v E v,/E,, avEN) is countable. Now as p i , p,2 are ‘trivial, the relation “ {F(@;), @} depends on 5; U 6;” for r]/Elv E vt/E2-~,b/~,, defines a one-to-one function from { p :p E r]/El -s;} into {v: v E v , / E ~ -s:}. Choose s ~ / E,~,2/,~ s.t. s;/E, 5 s ; ! ~ ,5 7/E2, s ; / ~E , s % /E ~, Js;/~,I = (s;/~,I = do, and the function maps r]/El-s: onto v,/E,-s;. We define h, on r]/El-s: according to the function mentioned above and h, rs: as any one-to-one function from s; into st. If F is onto A& there is no problem making be onto v,/E,. Now let us check that the demands on h are satisfied. Demand (i).Holds by its definition (and h/El :Ii/El+I:/& is one-toE , )pairwise orthogonal). one as the types ptlE, ( ~ ] / E , E I ~ /are

r4,

r

GE

v,/&,

Demand (ii). Trivially, by the choice of h. Demand (iii). As p~IEI,p&,)IEp are regular, not orthogonal, by 4.4(6). Demand (iv). So suppose r ] , ~ I ri ,] , ~ v G , qo/E,,‘von s ; / ~=~9, vo uncountable, for r]~w~,a,, c d,,- E , E ~ tp(a,nd,AC,,q-) , = q (i.e. does not depend on r] E wo),tp(d,, ct) is stationary regular and parallel Let v1 = h”(vo). to some pi-(=,), by renaming w.1.o.g. to We have to show that for all but finitely many q € v o , h(r])4 h(r]^(O)), i.e. (see 5.10) F ( P & ( ~ is ) ) orthogonal to p i when h(r])R P. Now we proceed as in the definition of h. We choose 6 * ~ q such that for r] E vo, tp(G,nd,-c,,6*

u a,) I-

tp(@,-d,-E,,N,-

u a,)

(possible as q exists). , can be achieved by Now we would like to have F ( ~ * ) E Nthis increasing uBoIEI, to uiolElpreserving its finiteness. This, naturally, costs us the omission of a finite number of members of wo,vl:

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I U C ~ / E , - U ~ ~ / E , I by 5.10 we finish (note that tp(Z7,c7) ( 7 E v o ) are pairwise orthogonal).

Demand (v). The same reasons for the existence (and uniqueness) of +A/&, show that it is onto. h/E, : In order to get demand (iv) for h-’, note that if we allow KO exceptions, the proof of (iv) dualizes. For the finite version note that the proof of demand (iv) for h-’ needs only that s:,st are large enough in order to guarantee the existence of the “dual” of NVlE,.

COROLLARY 5.12 : Under the assumption of 5.11, the function h there satisfies : (1) For some large (seeDefinition 5.8(3))subset J1of 4, h J1satisJies: it is (a one-to-one function from J1 into 4) preserving for 7 , P E J l , 7 4 P e h ( 7 ) 4 h(P). (*) (2) If in addition F i s onto 4,for some large subsets J1,J, of 11,4, respectively, h r J1 is a one-to-one function from J1 onto J, satisfying (*) (and even h-’ satisJies the parallel demand). (3) For any given cardinal p, h maps If” = { ~ EIr]/E,I I ’ : p } into I:’= {vEA:Iv/&;I 2 p}. If in addition x is regular > K O , (EIf”}/ < x, then for some ( <X)-large subset If*X of I f P , h I’;.X satisjies (*). (4) If F is onto 4,h maps If”onto I:”, and for x as above w.1.o.g. maps I’;*Xonto a (<x)-large subset of I:”.

r

r

5.13 Proof of 5.1. So K, 2 A(T),8, > N,. By 5.7 (and 4.5) there is a representation a:: 7 EIO) where I’ = ,’Ka,q,, is well defined. By 4.6, w.1.o.g. qo does not fork over 9. As we want F;;saturated models, by our system of notation (which is not convenient in this case) we have to increase I0 to take ~ such care of this. So we can find J,Ic J, and N7,a, for r ] J--Io that: (1) (A$ a,, : 7 E J) is a representation, (2) for each 7 E J,q/E, -I has cardinality K, exactly, (3) each regular ~ E S ” ’ ( N , , ) J, , ~ is E not orthogonal to some pry,>. For each 6 < N, choose 4 such that (a) c 1, (b) if v 4 - { ( 6 ) , ( >>,then 7 1 4 c 4,

(q,

540

CLASSIFICATION FOR F&-SATURATED MODELS [CH.

X, $ 5

(c) DP,$5),4) = 5, (d) ~ € 1 :implies 7 1 = (5). (e) if ~ € 1 then ; I{a:q1(a)~15}1 = N,. For any set S E N, let

r

I,=

u4,

5ES

J, = { ~ E ~J E : Ior, for some m, q rmEI,, q r (m+ 1)e J - 4 , M, be a model represented by

(q,a, : a E J,).

By ( l ) , (2), (3) above, and 5.6, M, is F;;saturated and of cardinality N,. Suppose S( l ) ,S(2) E N,, P is an isomorphism from M,,,, onto M,,,,. Let h : J,(,) +- J,(,) be as in 5.1I . Clearly any regular type over M,,,, of dimension >N, is not orthogonal to some p,, q EI,(,). So necessarily h maps I& into I,(,). Similarly, h-l maps I,(,, into I,(,,. Hence w.1.o.g. it maps { ( f ) : k ~ S ( l ) } onto { ( 5 ) : 5 ~ S ( 2 ) }(as q<>does not fork over 9). Now use 5.12.

THEOREM 5.14 : If T is deep and Ha is 2 N, the Jirst beautiful cardinal, then

+A( T )but smaller than

IE"5( N,, T ) = 2'~. Proof. We use the theory of K-bqo as developed in [Sh 821. Let 9 - , ( w , = ) be the class of ( I ,f ) , I a well-founded tree (with root, no w-branch and a t most w-levels,f:I+-w) ordered by (Il,fl) < (I,,f,) i f and only i f the first is embeddable into the second, i.e. there is a function G : A-+h,I, I= s < t implies: I , != G(s) < G(t)and fl(s) = f,(G(s)). By [Sh 821 there are 2'. pairs ( I ,f ) each I of power N,, no pair embeddable into another and w.1.o.g. each r"'(N,) (use [Sh82] 4.10, and translate it into trees as in the proof of [Sh 82, 5.71 for example). W.1.o.g. for each such I there are sequences in I of arbitrarily large (finite) length. So let {(I:,f :) : i < 2'~} be this family, so I: E "'(N,). Let g : Ha+- N, be such that (Vi < N,)l{j < N,:g(j) = i}= N,, I: = { q ~ " > ( N , ) : g ( vgf ) (g(v(Z)) : 1 < Z(q)) € I : } , f :(q) = f:(g(v)). We leave the case a = 1 to the reader, so by obvious monotonicity w.1.o.g. B 2 1. Let for i < 2Na, 4 = { q : v ~ I :or , for some m, v r m E I : , N , < q(m) < N,+K,, and for every 1 > m if (31c)[Z = L2+(fi(qrm)+m+2)2+m < Z(q)], then q(Z) < N,, otherwise q(Z) < NB}.

CH.

x, $51 DEEP THEORIES HAVE NON-ISOMORPHIC MODELS

541

Let I = "'(N,+N,), and by 5.7 there is a representation (N,,,a,: ~ € 1such ) that q,, is well defined. We choose J , I E J , IJI = K , and N,,,a,,for V E J - I such that ( 1 ) (N,, a, : 7 E J ) is a representation, (2) for each V E J - I , q/EJ has cardinality K, exactly, (3) for each regular ~ E S ' " ( N , , ) , . ~ )isE not J , orthogonal to some PrYa>.

L e t f o r i < 2 x ~ J, b = { r l ~ Jrl~l,orforsomeZ,rrZ~4,r]r(Z+l)41). : Let 2y be represented by ( N , , a , : y ~ J b Clearly ). each 4 is FiBsaturated and has cardinality Nu. So suppose i #j < 2'~, F is an elementary embedding of 4 into 4.We shall use 5.11, so we have a function h as described there. Now applying (iv) on hr4 we can conclude that for some &-large subset If of 4, h I ; preserves Q . By the choice of 4 (and 4-I,), necessarily h maps If into I,. We shall show later that w.1.o.g. 8 for some increasing function k : w + w , for q ~ I ; , l ( h ( v= )) k(l(rl)). Now by the way we have defined 4,J (and as I:,Ii are well founded) (*I0 k is the identity. [Why? For any ~ , v E ~ ( N , + N , )s.t. A ~ < ~ r l r z € ~ f , h ( r l r z ) ~ { v r m : m < w } , then max{m:q r m E l : } = max{m: v r m ~ l ; (by } the choice of 1-1;). Hence for ~ € 1II;I:, Z(h(7))= Z(7). But with an assumption on the ( I ,f)'s we finish.] Now as for every 7 E I: -{ ( )}, I{v €1::v- = q-}l = N, and similarly for I:, clearly by the definition of 4, I, : h maps I;l n I: into I;, and (*)I (*I2 for r l E C mf:(rl) =fi(h(rl)). Now as 1;n I: is an No-large subtree of I:, clearly there is a function g* from I! into I;l n I: preserving level and 4 and = , such that for q E I y , g ( g * ( q ) ) = 7. Now the composition g+ Ef (ghg") is a function from into I; preserving Q (but is not necessarily one-toone) for v 4 : f , ( g + ( r l ) ) = (f,g)h(g*(rl)) =f:(h(s*(rl))) = f i l ( g * ( r l ) ) = fi(g(g*(a)))= fi(q), contradicting the choice of the We still have to prove that w.1.o.g. 8 holds. We shall prove later:

r

c's.

CLAIM 5.15: If T is deep, then we can find (4,a , :1 < w ) as in 4.4(3) such that: for every 1 > 0 for some q,and 6,€4, tcp,[a,, 6,]and for every

542

CLASSIFICATION FOR

F

;

o

-

MODELS ~ ~ [CH.~X,

$~5

~

m > I , tp(a,,q) is orthogonal to every type to which cpl(x,6,) belong, (*), tp(ao,No) is not orthogonal to $3, (if we waive regularity, it does not fork over 9). Completion of the proof of 5.14. I n the proof of 5.7, we can use the (4,a, :l < w ) from 5.15 (instead of 4.4(3)). Also in our present proof we could have used (N,, :r] E "'(N3, N,)) derived in this way. So there are formulas cp,(x,Cl)( I < w ) 6,EN( ) such that for each i < 2 u ~r, ] ~ I l ,

+

cpl(q,(",

h,,J€!I,,,

l(r])< m < w * q,, is orthogonal to every type to which cp,(,,,(x,6!(,,J belongs. W.1.o.g. u,b, C No. Now look at the proof of 5.11 (applies to B':N+q). So for some countable u1c 4,u2E 4, closed under initial segments, Fjo-primary over U,,EuSq the following hold : and N c 9, (4 F(U,<,b,) c N , (b) tp,((6,: 1 < u ) , F ( & ) ) does not fork over F ( A )c N c 4,and A G%,U,6Jc A, A countable. As q<) does not fork over $3, h maps {r] € 4:Z(q) = l}into { r ] E 4 : Z(r]) = l}. So w.1.o.g. [r] €1:* h(r]) 1#u2and r] 14u1].Let us prove that h r I : preserves equality of the level. Let ~ € 1 ;If. l(r])= 1, we have already noted this. Now :

r

r

Observation. If q1,q2€I:, Z(sl)= l ( q 2 )> 1, then for some automorphism H of 6 over N( ) , H ( ~ , , , ) , Pare , ~ not orthogonal. Hence,

Observation. If rl,r], € I : , Z(q1) = l ( q 2 )> 1, then (1) for some automorphism H of 6 over A , H ( p q l ) , p l l are l not orthogonal, (2) for some automorphism H of 6 over F ( A )U u,<w6t,H(F(p,,l)), H(F(pVp)) are not orthogonal. However :

observation. If V~,!~EJ, l ( v l ) # l ( v 2 )> 1, then for no automorphism H of 6 over Ul<"bt,are H(p,,),pUB not orthogonal.

As non-orthogonality is an equivalence relation, we finish.

~

CH.

X, $51 DEEP

THEORIES HAVE NON-ISOMORPHIC

MODELS

543

Proof of 5.15. Choose ~ ( x6),(&,a, : 1 < w ) such that ( 1 ) (&,u,:Z < w ) is as in 4.4(3). (2) !=v[a,,6]for 1 < w , and 6 ~ 4 . (3) Under (1) (2), u* = R[q(z,6),),L, 001 is minimal. We can prove that N, is By 4.4(3) we can find such ~)(z,&),&,a,. Fjto-primary over No U a, (see Exercises 5.1, 5.2), hence for CEN,, tp(c,2\’oU a,) forks over 4, hence t ( c , N , ) forks over 4. So for 1 > 0, R[tp(a,,N,),1, 001 < u*, so for some 6, €4, and q,: !=rp,[a,,6J and u* > R[v,(z,6,), L, a].Let A!$ = { m :1 < m < w , and tp(a,,Nm) is not

+

orthogonal to some (regular)type to whichT,(s, 6,)belongs}. Suppose = {n(Z,i) : i < w } , n(Z,i) < n(1,if 1).Now let P ,= there is a N n ( 2 . m), and for each rn, as Pmis Fio-saturated, regular r, ES(N:) to which cp,(x,6,) belongs and rm is not orthogonal So r , is realized by some U ~ E N , ( , , ~ Now )+~. to tp(a,,,,,,,P,). pi(”,El), (Pm, a: :m < w ) easily satisfies (1) (2) (use Exercise 5.1, Claim V, 3.3), a contradiction to the choice of u*. So each A!$ is finite. So we can find {Z(k):k< w } , O < Z(k) < l ( k + 1) < w , A ! $ ( ~ ) E {i:i < l ( k + 1)). Let and (x,a:: = &(k),a,* = k < w ) , ~ ) , ( ~ )6z(k)) ( z , are almost as required. For the requirement prove as in 4.6 (and then use “tp(ao,No)does not fork over Exercise 5.1).

A!$ is infinite. Let kj

+

x”,

EXERCISE 5.1: If No is Fio-prime over 9, &+, FiO-prime over 4 U 4,for 1 = 0, 1, and tp(Gl,N1) is orthogonal to No, then N, is F;fio-primeover No U a,. EXERCISE 5.2: If 4 is F;fi.-prime over A for i < i(*),&is increasing then is Fi.-prime over A . (Hint: Use the and i(*)< characterization theorem IV, 4.21.)

u,y

EXERCISE 5.3 :If No is F;fi.-prime over A, (N,: a < a(*)) increasing continuous, Fi7-pfime over Nu U a,, and for each a,(8:tp(aa,Na) not orthogonal to tp(a,,N,)} has cardinality GN,,, then Nu(,)is Fi7prime over A . (Hint : see [Sh 88aI.) EXERCISE 5.3A : Show that in Lemma 5.5(1) we can add: (8) The representation is standard. (Hint: Use Exercise 5.3.) THEOREM 5.16: (1) If T is shallow, then IE: (Nu,T ) < 3a(lT,, and i.e. sup{lKl:K a family of #:;saturated models, even IEiAT) < no one elementarily embeddable into another} < aaclTI).

544

CLASSIFICATION FOR

F~;SATURATED MODELS

[CH.

X, $5

(2) If T is deep, K~ the$rst beautiful cardinal > ITI, then IEibN,, T ) < K; and even IEio(T)< K;. (3)If T is deep, and IT1 smaller than the$& beautiful cardinal K ~ N, < K,,, then for /3 < a,IEijN,, T ) = Min{K;, 2w-}

,

IEiAT) = K;. Proof. By 5.5(3)we can translate the problem of proving the upper bound to problems on embeddability of labelled trees. That is, if (q,a,,:7 E I) represents M, we can define model q such that (a) 3 is a model of T with universe c { u :a < 21Tl} x (1+1(7)), (b) there are isomorphisms I,,from q onto q , fVrr E f,, for 1< So the function g, g ( 7 )+q (7 E I ) has a range of cardinality 22'T' (really, 2ITI suffices). [Now if for 1 = 1,2,Ml has representation (fl,,, a,,:7 € 4 ) and above we have chosen Mi, (7 EZ), gr as above, then

w.

if there is a function h from 4 into 4, one-to-one, preserving Q , b , (and the level) g2(h(7)) = gl(r]), then Ml can be elementarily embedded into 4.1 For 5.16(1),see below (i.e. 5.16C implies the result by 5.5(3), 4.4(4)).For 5.16(2),(3) use what we have proved and 5.14 (for N, 2 K ~ note , that the proof of 5.14 still works).

(*)

Now the following problem on A-well-ordering should have been dealt with in [Sh 821, but we forgot. We give below just what is needed here.

DEFINITION 5.16A:(1)A quasi-order Q (i.e. a pair (IQI, <),< a transitive, reflexive relation (but maybe x < y < 5,x # y ) is A-wellordered if for any qf E Q , ( i < A) for some i < j,qi < q,. (2) Q is A-narrow if for any qf E Q (i < A) for some i # j , q1 < q,. (3)9(Q) is the family of subsets of Q. (4)For a given Q , Go is the following quasi-order on 9 ( Q ) : A GOB if and only if for some function f from A into B, (VqEA)q
CH.

X, $51 DEEP

THEORIES HAVE NON-ISOMORPHIC

MODELS

545

(with root and ,
THEOREM 5.16C: If a,(A)+-well-ordered.

11Q11

= ZA,u> 0 an ordinal, then

Y
We shall prove 5.16C and 5.16D later.

LEMMA 5.16D :For any cardinal K , and ordinal 5, letting 8 = 2",there is a function F from P("()to '5 such that for A , B E P ( " ~ A )<, l B i f F ( A ) < F(B). Remark 5.16E.

'5, "5 are ordered coordinatewise.

DEFINITION 5.16F: We say A E "5 is of kind (w,p,p)if: (i) 7 is a sequence of ordinals ,<[ of length K , (ii) w is a subset of K, (iii) for every a= (ui:i< K ) E A ,[i$w A i < K * a t = yc][iEw=. 016

< Ycl,

(iv) if y: < yi for i ~ w then , { a ~ A : y <: a r < y i for cardinality IAI, (v) IAl = P*

~ E W )has

DEFINITION 5.16G : We call A E "5simple if it is of some kind, we call it ,
CLAIM 5.161: Any A E "E i s (,
u,

iiE$if

and only if

A t
(VEENg)(ai

< E = pi < E

A a(= E

= pa= 6).

546

CLASSIFICATION FOR

F

;

o

-

MODELS ~ ~[CH.~X,

$~5

Clearly the number of Ec-equivalence classes is < 11N11" = 2", so for some & * € A for no [ < K+ is h*/Ec simple (otherwise {a/Ec:a€A, &/Ecis simple but for no E < [ is &/E(simple} a partition of A to < 2" sets, each of them simple). Let y r = Min{EEN:a: < 5>

y = ( y t : i< K ) , w = {i < K:a:$2\3.

~ , ~ = < & r w < yrw). Let for @
r

EY(~,,

+c+

Proof of Lemma 5.16D.Straightforward. Proof of Theorem 5.16C. By induction on a for a non-zero, we prove that there is a function F : 9.J(2) +A(u)[ (for some ordinal 5) where h(a) = &+=(A) such that

(U) 6 (4,fz) ifW1,fl) < F(4,A). For a = 1, since in '2 there are 2Apairwise incomparable elements, this is easy. For a = /3+ 1,/3 non-zero use the induction hypothesis and Lemma 5.16D (the reduction to (9(YGp), < I ) is easy). For a = /?, /? limit, note that to deal with (9(u,,,YG,),< I ) , we < I ) for y < /3. can reduce it to problems on (P(YG,,), EXERCISE 5.4 :(1)Suppose T is countable, superstable without the dop. Prove w2 < Dp(T) < GO implies IEibT) = 3Dp(T). (2) The natural upper bounds when T is not necessarily countable and/or Dp(T) < w2. EXERCISE 5.5: Show that for a < w 2 , there is a countable superstable T without the dop of depth w a 1, and IEi$T) = 3u+1.

+ +

~

~

CH.

x, $51 DEEP THEORIES HAVE NON-ISOMORPHIC

MODELS

547

5.17. Discussion on invariants. So the decomposition (h$al : ! € I ) totally describes a model M which it represents ; however, it IS not read easily nor uniquely from (the isomorphism type of) M. We can try to correct this somewhat. A technical point. RE-DEFINITION : In the definition of a representation we replace

(8,(4) by: (2)' IIyII G W ) ,

(4)' tp(N,,,N,,-U a,,)almost orthogonal to Zt-.

(q,

DEFINITION5.17A : (1) A representation at :7E I) is derived from the representation ( q , a i : V E J), if J s I, at = a: for 7~ J, there are for 7 ~ J , s , E { v : q v~l>,[vn(a)~s, q A ~ " ( a )YES,,], Isl]
r

)EJ,T)

DEFINITION 5.17B : Two representations will be called equivalent if they have isomorphic-derived representations. We can prove

LEMMA 5.17C: Two representations represent isomrphic models if and only if they are equivalent. However, is the equivalence class of representation of M a reasonable invariant ? We may try to replace p , , ( ~ ~ I , l=( q1)) which w.1.o.g. does not fork over 9, by cP@,,)-simpletypes over 8. Now for types of depth zero we need just their dimension, types of depth>O are trivial, so we can represent a,, by an equivalence class for the natural dependency relation on the simple type (this is not so far from IX, 2.3's proof). However, the equivalence relation is not necessarily first order, so we do not know to continue this. A primi, we can use CeQwith a non-first-order equivalence relation, but we do not know enough corresponding theory to make this work, and get, as desired,

548

CLASSIFICATION

FOR

F$-SATURATED MODELS [CH.X, $6

(3,

a, : 7€ I ) , a non-forking tree of imaginary elements such instead that the model is prime over it and is quite unique. Note that in the representation (N,,u,,:TEI)not all types, e.g. p(8,r), are necessarily there in the tree. We could have replaced NO by N&, and we could have replaced by any Fio-saturated subU E(8,c). Moreover, if we try to continue to model including shrink N(,,,) the process is not necessarily well founded (when Dp(T) 2 w ) . We may be tempted to take an inverse limit of the possible representation of a model, but it is not clear we get anywhere. We can use standard representations (see Exercise 5.3A) and represent p,’s by a canonical base, but we still have non-orthogonal, non-parallel p,’s. For a solution which is satisfying for me, see XIII, Section 1. By q)-decompositions. the way, 5.17C is true also for ( G;,+,

q,,,)

X.6. Infinite depth THEOREM 6.1: If K, 2 A(T)+N,+,, Dp(T) 2 w , then Ii$Ka,T)2 &qNT)(l4

+No)-

Prmf. Compared with the proofs of 5.1 and 5.14 we have one becomes advantage : the clause in 5.11 saying DP(JI;,~,)= Dp(p:(,),E,) meaningful. For finite depth in say 5.12, it would be many times enough to show that h usually preserves the level (i.e. when the level in the tree determines the depth). For infinite depth this is not true; however, n+ Dp(T) = Dp(T), so we can “dedicate” some low depth to “mark” the level. Let = IDp(T)I+ No.First assume a 2 w . We can find a representation (q, U: : q c I * ) , s.t. (i) tp(a&,Nf)) is not orthogonal to 9 and has depth i, (ii) tp(a:-(,),N,) has depth i, (iii) if p is a regular type of depth i not orthogonal to 9, i > 1, then (i)EI*, (iv) if qcI* -{( )}, and there is a regular type over N, orthogonal to If,-of depth i, i 2 2, then y ^ ( i ) e I * , (v) if Dp(p:) = 2, then q - ( i ) ~ I * - i = 1. (vi) qn (0) $ I * and vn( 1) € I * implies Dp(p:) = 2.

x

CH.

X,$61

549

INFINITE DEPTH

Clearly there is such a representation, II*l = JDp(T)J< h(T) < N, (or both are finite), and (by 4.6) Dp(T) = u { i : ( i ) ~ I * } + l . Now we define by induction on i, for q E (I*)-,a set H ( q ) . Case I : q(Z(q)- 1 ) = 1, H ( q ) = K>. Case 11:Not Case I and q(Z(7)- 1) = 2, H ( q ) = { h ; h a function with domain H(qn(l)) = {N,}, and

h ( R )= {Nl(7)+l>. Case 111: Not Cases I, 11: ~ ( Z ( q ) - l )2 3, H ( 7 ) = {g : g is a function with domain { H ( q ^(i)) :q (i) €I*}, and range E { A : No< h d N,} U { O } , and I{ Y :g(Y ) # O>l d N,}.

u

A

An exercise in cardinal arithmetic is to prove IH(( ))I = 3Dp(T)(lal+N0) (using Dp(T) 2 o)(check by cases). Now for every Y E H ( ( )), we choose I,, (K,a,: U E I ~ ) , ~ Y ( U and E I ~ functions ) g : , g y , s.t. : (a) (q, a,: ~ € 1 is~ a)representation, (b) gY is a function from Iy onto I*, preserving level (equivalently length), Q , mapping Iy-I;,I;, into I* - (I*)-, (I*)-,respectively, (c) g: is a function with domain I F ,g:(r]) E H ( g Y ( q ) ) , (d) if 7 ^ ( u ) E I ; , then g$(q ^(a)) is in the domain of gt(r), (e) if ~ E I F -2, is in the domain of g y ( q ) , then ( { q m ( a ) ~ I y : ( a ) ) = Z>l = (g:(q)) (2); if 7 ~ 1 y - I ; - (so q(Z(q)- 1) = 1) then g:(q I{q^(a) : q - w EIyIl = N,, (f) js an isomorphism from q onto qycy,, extending f& for z < Z(v). There is no problem, and IIyl = N,+h(T). We now can find Jy,N~,a,(uEJy--Iy)s.t. a, : u E Jy)represent an F&-saturated model My of cardi(g) (q, nality N, (see 5.6), (h) Iy E Jy and for q E Jy of length m, if q #I, then

fY

I{v E J, : tp(a,, N,) = tp(a,, Nv-)>l d K,,

(i) if /3 = 0, Jy= Iy. For notational simplicity : (j) for Y # 2, Jy n J, = {( >>. So it is enough to prove:

(*I

if Y # 2 are from H ( ( )), thenMy,M, are not isomorphic.

550

CLASSIFICATION FOR

F

g

O

-MODELS ~ [CH. ~ X, ~ $ 7~

So suppose F is an isomorphism from My onto M,. We use the U E J,), ( e , a vU: E J,), and apply 5.11, representations (x,uv: obtaining a function h from J, onto J,. So there are #,-large J& G J,, Ji C Jz such that h J k is onto Ji and preserves 4 (by 5.12). A s q ~ J hA I { u : q Q VEJ:}~= N,*I{v:h(q) 4 VEJ,}~= N,clearlyh maps J& n (&-I;) onto Ji n (Iz-I;) (remember Dp(p,Y)= Dp(pi(,,)). Similarly (looking for types of depth 2 ) , J$ n &-I;-) is mapped onto Ji n (1;-I;-). However, looking at the definition of H ( q ) when q(Z(q)- 1) = 2, we see that on J& n (I;-I;-) h preserves the level. n I, preserves the level. As From this we can easily deduce that h J& it also preserves the depth, easily (by induction on Z(u)) :

r

= g,(h(v)).

for V E J : nI,,B,(V) The rest is straightforward too. If

01

< w use Exercise 6.2.

EXERCISE 6.1 .-If T is multidimensional then I ; (N,, T)2 )a+ wJfor N, 2 A ( T ) + N ~ + also ~ ; I ( N , , T)2 la+wl if N, 2 I ~ + H , . EXERCISE 6.2: Suppose A,(1< n+ 1) is increasing, al€Al(O< I < n), tp(a,+2,Al+l)is orthogonal to A,, tpI(A1+l,AlU al+l)is almost orthogonal to A,. Suppose also that there are p pairwise orthogonal regular types, each orthogonal to A,,, but not to A,+1, and lAll = N8. Prove that n < Dp(T) and for K, > A(T)+#,, a-p 2 p

G & K u , T)2 an(la-BIP)* And even (if n > 0) 2 2,,-1[Ia+wl""-B+'b)]. EXERCISE 6.3: Suppose that for y there are no A,,a,(I < n+ 1) as above for n = Dp(5"). Prove I{o(K,, 2') < =I,(lal
X.7. Trivial types (Hyp) T stable, K = K,(T).

DEFINITION 7.1 : We call p~h'"(B)trivial if it is regular stationary and the following holds:

~

~

CH.

X, $71

551

TRIVIAL TYPES

If I_ is an independent set of sequences realizing p , 6 realizes p but tp(b,B U I )forks over B , then for some F E I , tp(6,B U C') forks over B .

u

LEMMA 7.2: Suppose Dp(N,N',@)> O,P,ES"(B)stationary and parallel to tp(a,N) (which is regular). Then p is trivial provided T does not have the dimensional order property. Proof. Assume T is superstable. W.1.o.g. B = m,NFio-saturated,I is it,} is finite and let I = {az:1 < n}. Suppose there is no such C, i.e. {6, independent over N for each 1. Let M be Fio-primeover NU I U 6. As I U 6 is finite by Definition V, 3.2, Theorem V, 3.6(2),V, 3.9(1) there is a finite set J G M independent over N, of sequences realizing regular types, such that M is Fio-primeover NU J. FACT. There is b* E I U {6}such that for no J EJ, tp(6*,N U J ) forks over N . Otherwise there are EJ,ba E J,such that each of the pairs {a:,az} ( 1 < n ) and {ba,6}is not independent over N . So for each 1, tp(az,NU{a;:1 < n } ) forks over N and tp(b,NU {az: 1 < n}) forks over N , so by V, 1.14 tp(6,NU {a::1 < n}) forks over N . By the choice of ba, tp(bo,NU 6) forks over N, hence by V, 1.14 again tp(Lo,NU {a::1 < n } )forks over N . As /is independent ba E {&p :1 < n } , so let ba = a;. So, by our choice, tp(6,NU a:) forks over N, and tp(@;,NUal)forks overN, hence by V, 1.14 tp(6,NU aZ)forks overN; a contradiction to the assumption " I , 6 is a counterexample" to 5.10. So we have proved the fact. Let J = { Z l :1 < m,}. As M is prime over NU J, there are

u,&

4 Fie-

prime over NU d z , such that M is so-primeover (see V, 3.2). Let N* <M be Fzo-primeover N U b*. As 6*realizes p (as all members of I U {b} do), by 4.4(6) Dp(N,N*,b*)> 0, hence there is a regular q* E Sm(N*) orthogonal to N, and let q be its stationarization over M. For each 1 < m,, {b*,dz}is independent over N (by the fact), hence by V, 3.2 tp,(&,N*) does not fork over N. Hence by 1.1 q* is orthogonal to &. Hence q is orthogonal to each &. But as T lacks the dimensional order property, q is not orthogonal to some 4. This contradiction proves the lemma. N is (TI+-saturated, What if T is not superstable? W.1.o.g. B = let ]be a maximal subset of I independent over N , so tp(6,N U forks over M, hence is orthogonal over N , hence FP,,+-isolated,so there

m,

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552

CLASSIFICATION

FOR

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MODELS - ~ ~[CH.~

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is M Fh,+-primary over N U U J to which 6 belongs. Continue as above.

LEMMA 7.3:Let T be stable, K = K,(T). ( 1 ) Suppose r, E S ~ ( ~ ) (for A J1 = 0 , 1 , r, parallel to r,, r, is stationary, regular and trivial, then so is rl. ( 2 ) Suppose r izSm(A)is a (stationary)regular trivial type and {a,6} is independent over A. Thenfor any F realizing r , tp@,A U i z U 6)forks over A iff tp@,A U a)forks over A or tp@,A U 6)forks over A. ( 3 ) Suppose r€S"(A) is stationary regular and trivial. For any a there are do,...,dn-,E r ( 6 )such that :{do,...,d,,-,} is independent over A, tp(d,, A U a)forks over A for 1 < n , and n = low,(a, A ) . So for any J ~ r ( 6 tp(d, ) , A U a)forks over A iff tp(&A U dz)forks over A for some l < n. Also tp(a,A U u,,,~,) is orthogonal to r. (4) Suppose ro,rl are stationary regular and not orthogonal. Then r, is trivial if r, is trivial. ( 5 ) Suppose A c B, tp(a,B) does not fork over A, r g S r n ( B )is stationary regular and trivial, not orthognal to stp(a,A). Thenfor some eEacl(,4 U a),e$aclA, stp(e,A ) is c13(r)-simp1eof weight 1. I n fact there are e,, . . . ,en-, as above, tp(a,A U {el : 1 < n})orthogonal to r. If stp(ti,A ) is semi-regular, stp(e,A ) is regular. (6) Iffor 1 = 0,1, p, = stp(@,,A) is not orthogonal to r, r a stationary regular trivial type, then p,, p , are not weakly orthogonal; infact p , has an extension over A U a, which forks over A. Proof. ( i )By the definition of parallel, r, is stationary and by V, 1.8(l ) , r, is regular. Let A, U A, E M, M F:-saturated, h > (A, U A,) and r2 be the stationarization of ro (and r,) over M. Clearly r2 is regular and stationary. Suppose r, is not trivial, then there are b, a,, . . .,an-,realizing r l , tp(6,A, U a, U .. . U an-,)forks over A,, but tp(6,A, U am)does not forkoverA,form < n. W.1.o.g. tp(6fiaon...nan-,,M) doesnotfork over A,, and then clearly 6,ti,, . .. ,an-, exemplify r2 is not trivial (use 111, 0.1). So tp(6,M U a. U .. . U an-,)forks over M, hence over Ao,-and 6, go,.. .,an-l realizes r2 and r, G r2. So by V , 1.11, clearly tp(b,A, U I )forks over A, where I = r,(M) U {a,,...,an-,}.Obviously for every FEM, tp(6, A, U F) does not fork over A,. So 6,I exemplify ro is not trivial (except that I is not independent, but this can be discarded by re's regularity) ; a contradiction, hence r, is trivial as required.

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~

CH.

X, $71

553

TRIVIAL TYPES

(2) The implication -=is trivial. So suppose cis a counterexample to the other direction. Let M be an Ff-saturated model, A EM, tp,(M,A U a U 6 U does not fork over A. By some application of 111, 0.1 clearly {a,6} is independent over M , tp(c,MU a) and tp(c, M U 6) do not fork over M, tp(c,M U E U 6)forks over M, and tp(c,M) is a stationarization of r , hence by (1) is a stationary regular trivial type. So w.1.o.g. M = B. Note: nz may be infinite below. Let { d m : m< no}be a maximal set of sequences realizing tp(c,M) independent over M such that tp(dm,MUa) forks over M, and m < n’} be a maximal set of sequences realizsimilarly let {Ern: ing tp(c,M), independent over M, such that tp(cm,MU 6) forks over M . By V, 3.16A, no = low,(&,M), n’ = low,(6,M) and tp(d,^ ... -d3-’,M -U a) and tp(Eon... nEnl-l,MU 6)are Ff-isolated. So clearly {do,.. .,d,,o-l,e,, . ..,E , , ] - ~ } is independent over M and ntp(d, -... dno-’ e, ... en]-’, M U-E U-6)is Ff-isolated (see V, 3.2). So let N be Ff-prime over M U a U b, {do,. ..,d,,o-l,go,... ,a,-,} E N . By (10) low,(an6,M) = lowr(a,M)+low,(6,M), and - V, -3.16(8), { d o , ...,d,o-l, e,, ...,En-,} is a maximal subset of r(N) independent over M. Now what about c? F realizes r € S m ( M ) F, depends on &^6 (i.e., tp(C,M U a U 6) forks over M), hence by V,- 1.16(3), (1) tp(c,,M U r(N)) forks over M, hence tp(c,M U {Jo U . . . U d,o-l U coU . . . U forks over M. By the definition of triviality, tp(c,M U J l ) forks over M or tp(C,,MU Ez) forks over M for some 1. By symmetry, suppose the former. Clearly by V, 1.9(2) {&,c}is not independent over M , a contradiction. (3) Assume, for simplicity, T is superstable (otherwise use V, 3.16, 3.16A). Let A E M,M Ff-saturated, tp,(M, A U 8)does not fork over A. By V, 3.9 there are n,do,... ,dn-lsuch that :tp(Jon . .. Z,,-’,M u a) is Ff-isolated, realizes the stationarization of r over M, n = low,(&, M) = low,(a,A), and {do,... ,dn-l}is independent over ( M , A ) .By the first part, qz = tp(dz,MU a) forks over M, hence q1forks over A , hence for some 6,, tp(d,,A U 6,) forks over A , ~ , E M . The only property missing is tp(dz,AU a) forks over A. If this fails we get a contradiction to part (2) (with A, a,6,,dt here standing for 2,a,6,C there). The second sentence is by V, 3.12(a) and triviality. We will still have to prove “tp(@,AU ul,,dz) is orthogonal-to r ” . Suppose not. Let N be F:-saturated A > IAI + K , AU UZ<,,dlEN, tp(a,N) does not fork over A U Ul5,d,. As the conclusion fails, there is d, realizing the stationarization of

a

A-

- - - A

al

-

554

CLASSIFICATION FOR

F

E

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-MODELS ~ ~

[CH. ~

X, ~ $~7 ~

r over N , tp(d*,NU a) forks over N , hence over A. As above we get tp(@,A U d,,) forks over A. But this contradicts the second sentence in (3). (4) Left to the reader (see V, 1.14). (5) Let {do,. .., be as in part (3)of the lemma (replacing A by B ), I, = { d ~ r ( Ctp(d,B ): U d,) forks over B}, and Q),,6, be such that 6,E B , Q ) , [8,~ , , and Q),(E,a,6,) forks over B . By 11, 2.2(8) there are E A U I,,and such that for every d~ I,, ~)[d,@,6,] iff @,[d,cf,6,],and w.1.o.g. (by increasing 4) is a concatanation of sequences from I,.Let e,ECeQbe defined by q/&, where

u

By the second sentence in part (3), for every automorphism F of

6- which is the identity over B U a,F maps Il into some I,,,and if m = I, then F(e) = e. Hence el can have at most n possible images (varying P). Hence el is algebraic over B U a. Let, for i < [TI',f, be an elementary mapping with domain C = B U a U u,(d,U {e,} U Q,f, r ( A u a) = the identity, stp,(f,(C), (C)) does not fork over A U C, and extend stp,(C, A U a), and fo is

u,,,f,

the identity. SUBFACT : For i # j,{f,(dm),f,(d,,,)}is not independent over U,f,(B). For every i > 0, tp,(f,(B), u,,,f,(B) U U,,, d, U C ) does not fork over A U a, and tp,(f,(B), A U a)does not fork over A [as tp,(B, A U C ) does not fork over A by symmetry], hence by transitivity tp,(f,(B), uj,,f,(B) U Ul<,dl U a) does not over A. So by 111, 0.1 tp,(U,,, f,(B),BU U,,,d,) does not fork over A , hence over B, hence tp,(U,,, z,, U,f,(B))does not fork over B. So {d,:1 < n} is independent over (U,f,(B),B). Clearly ( f , ( C ): i < ITI') is indiscernible over A U a, hence for every i {ft(dl) :I < n} is independent over (u,&(B),f,(B)). We can also conclude that tp(a, U,f,(B))does not fork over A. Now we know (for i > 0) that tp(a,f,(B) uf,(d,,,))forks over f,(B),hence tp(C, u,f,(B) Uf,(d,,,))forks over f,(B), so by the previous sentence tp(C,Uj&(B)U&(dm))forks over u,f,(B). So by V, 3.12(a), {f,(dm): i < 12'1') cannot be independent over u,f,(B). Hence, for some i, tP(.fi(Jm),uj&(B)U Uj+,fj(dm)) forks over f,(B), and by the indiscernibility and the finite character of forking, for some n

~

~

CH.

X, $71

555

TRIVIAL TYPES

tp(f,(dm),u,&(B)u~-l&(dm)) forks over U,f,(B). We can choose a minimal n,so {f,(dm):j= 1,n} is independent over U,f,(B).Applying 7.3(2) several times we find that for some Z,1 < Z < n, tp(f,(dm), u,f,(B) Ufz(dm)) forks over u,f,(B). But quite clearly tp(U if,@)

:j# O,z},fo(B) Ufo(Jm) uft(B) Ufi(dm))

does not fork over A, hence {fo(dm),j,(dm)} is not independent over f,(B) Uf,(B). By the indiscernibility we can replace 0, Z by any i # j, thus proving the subfact. As r is regular, for I > 0, tp(d,,B U does not fork over B and easily tp(d,,B U um,,(dmAcmn(em))) does not fork over B , hence tp(dl,BU Um,,(dmnEmn (em)) U Ui,,f,(B U &-con(eo>)) does not fork over B. We can find an elementary mapping g :

um,,Im)

0

Dom(g) = auU,f, B U U d m n ~ m em) n , (

r U,f,(B U donC,

m

(e,)) is the identity and stp,(range(g), U,f,(B U aonC0 n(eo)) extends stp,(Dorng, u,f,(B U don ~ on(eo))) and does not fork over U,f@ U don qn(e,)). Now {dm:m < n} U {g(dm):O< m < n} is independent over u,fi(B) by the previous paragraph, hence it is independent over (U,f,(B),B). As we have noted above, e, is algebraic over B U a, hence e, = g(e,) is algebraic over g(B U a) = B U g(a). Checking closely, we see d a t for any automorphism F of CCeq which is the identity over B U aU g(a),F(e,) = e, (as F(e,)E{e,:Z < n} andF(e,) = F(g(e,)) E{g(e,) :I < n}).So e, is definable overB U a U g(a), say by +(x,a,g(a),b*) (b*E B ) .Let g

w o n

5,Yl

n-

‘‘1

z1) =

“

(W[+@A yo, ql,f,(K*N = +(x,y1, TJ,6*))1 holds for infinitely many i’s”.

By 111,2.5, 1.7, E is a formula which is almost over A, and let e* = a ”g(a)/E.Now exactly as in the proof of V, 4.11, e* #A, tp(e*,A) is c13(r)-semi-simplenot orthogonal to r and e* E acl(U,fi(BU {e,})) but here, by the subfact, any two of {fr(e,) :i} depend on U,f,(B),hence low,(e*,A) = 1. The only point left is “e* E acl(A U a)”, but for any i we know that (k large enough) e*Eacl(Uf:f&(B U {e})5 acl(A U aU u::f&(B)) (because e,Eacl(B U a ) ; see above). As this is true for every j and

556

CLASSIFICATION FOR

F

g

o

- MODELS ~ ~[CH. ~X, $ ~ 7

{ f f ( B:j} ) is independent over A U a, clearly e* E acl(A U a).The rest is obvious. (6) Let A E 4 for i < 17J+,Nl < ITI, {q: i < 111'1 independent over A, and r has a stationarization T,ES"(&). Let u,q c M , M F:saturated, and w.1.o.g. tp(aona1,M) does not fork over A , and w.1.o.g. r € S m ( M ) .By the assumption there is J realizing T , a; realizing tp(at,M)such that tp(ai,M U 2) forks over M. For each i there is Ji realizing the stationarization of ri over M, tp(& M u d ) forks over M . By V1.9(2), tp(& M U q) forks over M (for i < JTI, I < 2). As r, does not fork over 4 by 7.3(2) for each 1, i, tp(&;,YU J ) forks over Y. Again by 7.2(2), {a",, $} independent over iVf is impossible. As this holds for each i as A E Y,{q:i < ITI'} independent over A, easily tp(a;,A U &) forks over A.

~

~

CHAPTERXI

THE DECOMPOSITION THEOREM

XI.0. Introduction (Hyp) T is stable. This chapter is similar in spirit to Chapter IV, i.e. we give here an axiomatic treatment, not claiming for it any inherent meaning, just that it is a good way to organize the material. The notion we try to axiomatize is “ a decomposition”. The most elegant case was presented in Chapter X, Section 3, but in order to get the “main gap’’ for other cases we need other such theorems. I n Section 2 we present the axioms, notions and prove the existence of decomposition (and the possibility to extend a partial one). T is a set of “small” types, E * a notion of “strong submodel ”. So we try to get decomposition theorems for models M realizing every p E T over M, by G *-submodela. I n Section 3 we define various T’s and E *’s and prove that the axioms are satisfied for various pairs. In Section 1 we prove some more facts on “ p stationary inside A ”, (weak) orthogonality and conclusions of “ p eSrn(M)is not regular ” or “ P , Q E S ~ ( are M ) not orthogonal” (for M not necessarily Ffr(T)saturated).

XI.l. Stationarization DEFINITION 1.1 : (1) We say that p €Srn(B)is stationary inside A if

B E A and there is a unique extension of p in S m ( A )which does not

fork over B. (2) We say that p is stationary inside ( B , A )when p is an m-type which does not fork over B , B E A , and for every qESrn(B),if p U q does not fork over B, then q is stationary inside A . 557

558

THE DECOMPOSITION THEOREM

[CH. XI,

$1

FACT 1.2: If B G , A (see Dejnitions 3.1(1)and 3.1(4))and p ~ s " ( C ) does not fork over B , then p is stationary inside ( B ,A ) .

Proof.Suppose not. By Definition 1.1 w.1.o.g. p € S r n ( B )so , as p is a

counterexample, for some ~ E and A p p U { p ( ~ , aand ) } p U { - p ( ~a)} , does not fork over B . By 1.4 below, for some 6~B and 6, I= $[a,61 and every a' satisfying l=6[a',61 satisfies this. As B = , A there is such a' EB , easy contradiction. FACT 1.3: If p ' ~ S " ( A ) , p = q , forks q over A , then for some p(Z,6), q l-p(Z, 6), and p U {p(x,6')}forks over A for every 6'.

Q,

=

Proof. It is easy from the next subfact.

SUBFACT 1.4: For every ~ E S " ( A and ), A such that:

+ = + ( ~ , gthere ) is #@,a),

for any 6 , p U {+(Z, 6)} does not fork over A ifs !=6[6, a]. Proof. The set of qEST(l6l) such that p U q does not fork over A is finite, in fact the number is <Mlt(p, H,,). Let {ql,.. .,qn}be an enumeration of those q's. We know (II,2.2(l ) , (a),and 11, 2.13(1), (3))that for some Bl(g,al),

+,

+(z,6)E ql if and only if I= el[6,all. 6(g,a)= V:..,6,(S,al).So we can conclude (by 111, 0.1,

Let extension property) that :

the

p U {+@, 6)) does not fork over A iff for some 1, +(z, 6)Eql iff for some I I= 6,[b,at]iff I= 6[6, a]. The only point left is ~ E AHowever, . easily every automorphism to an equivalent of 6 which is the identity on A maps formula. By 111, 2.3(1) this implies that #@,a)is equivalent to a formula with parameters from A .

Remark. Alternatively, for 1.4 we compute R [ - , + , k ] for large enough k depending on p . FACT 1.5: Suppose {atn&:i < w } is indiscernible over C, tp(&,CU U,<.a, U U,,,b,) does not fork over F U a,, tp(&FU u,<wai) does not fork

CH.

XI, $13

559

STATIONARIZATION

overcand tp(6,,FU U5<,E), tp(d,aU u,,,a5) arenot - weaklyorthogonal. Then for some n, tp(bfl,dt,,a,) and tp(6,”. . . bfl-l, UiSfla-t ) are not weakly orthogonal. Proof. Clearly, tp(bt,aUu5,,a5 U uj,i6,) does not fork over F U ai (otherwise there is a formula witnessing it, and as {a,n 6,::j< w } is indiscernible over F, we may assume w.1.o.g. that the formula is over FU uj,,a5U u5<,6,,a contradiction to one of the hypotheses). :i < w } is indiscernible Also {a,: i < w } is indiscernible over and (as {ai over F, tp(d,F U u,,,a,) does not fork over aand by 111, 1.6(1)strong splitting implies forking). tp(d,FU are not weakly orthogonal As tp(6,,FU and l=p[ao, there are 6; and p such that % realizes tp(60,FU b,, al,at,.. .,ak-l,d, F ] , l=1p[aE,&,al, . . .,ak-l,d, El. By the indiscernb;(i < w ) such that G, ibility of {at:i < w } over aU d , there are b;( realizes tp(6*,cU and

u,<,@,),

u,<,a5)

6,

u5<wa,)

6,at+l,

d,

u,,,it,)

b7p[@i,b;(, ai+l,

d, F]. Now the sequence (EO”&,al” K,tizn&,,a3CG,...,a2tAb;;l, p[at9

* * * 1

@t+k-l>

* * * 9

al+k-l -

3

azi+l n6&+l,.. .) cannot be an indiscernible set (as the formula p(,. . .,d , C) contradicts 11, 2.20). So it cannot realize the same type ..). The conclusion is immediate. as (gon 6, alA61,.

Remark 1.5A. ( 1 ) This is essentially like IV, 4.12, p. 190. (2) Note that we can replace at by a: = ai-c; (3) We can let the sequences be infinite. FACT 1.6: Suppose in 1.5 that in addition there is 2t’ realizing stp(d,FU ui<,at)such that tp(d’,au U 6,)fork ower F U 4. Then there are n < w and & realizing stp(6,,FU Ut<,at)such that tp(&,, U ~ - , ~ , U F U U , , , Z ,ower ) ~ ~F~U~ ~ ~ .

u,,a5

Proof. Work in Ceqand replace at,F by an enumeration of acl(F -at), acl(C),respectively. Now apply the previous fact.

FACT 1 . 7 : Suppose tp(6,M U a) is not almost orthogonal to M , l= @(6,a). Then there are al,6,(l-= 0 , l ) such that: (i) tp(aon60,M)= tp(a^b,M), tp(al,M) = tp(a,M),l=@[61,al], (ii) the type tp(a1,MU ito U 60)does not fork ower M , (iii) l=p[ao,6,,iz,,61, F] and {p(a0,Z,a:, 6,: @} U tp(6,, a. U @ fork : . ower a, U M , for some p and FEM,for every a:, 6

560

THE DECOMPOSITION THEOREM

[CH.

XI, $ 1

Proof.As tp(6,M U a) is not almost orthogonal to M , we know that for some 8,tp(d,M U a)does not fork overM but tp(&M U U 6) forks over M. We can choose by induction on n < w , sequences %,,,6,, such that anh6, realizes stp(&^6,M),and tp(an”6,,MU dn6^CU Ul<,,a2^6J does not fork over M . By 1.6 for some n < o there is i!$, realizing stp(b0,MUlJl<,,,al) such that tp(&,CU U,,,al u U,“,,6J forks over M U 4 ; w.1.o.g. n is minimal. By 1.3 there is a formula p and CEM such that (i) bp[ao,i!$,, ai,&,. .,a,, b,, a n + l , an+2,-.. ,4, (ii) for every a:, .. . , 6 : , . ..,5:,

e,

{p(a”*,+1,...,a:,@}U a o ,-~-* , a i5* ,1 , a*2 , 6* 2 , . . . ,-* a n 6* , tp(b0,MU ao)forks over M U a0. -

...,ah}is independent over As {ao 6,a1,a2^b,, ...,a, 6,, M (by the minimality of n ) , tp(a2n62-. . .n a,, 6, an+,^ . . . ak, M U a. ^al) is finitely satisfiable in M, hence we can find in M a sequencep = a; “& a; & ^. . . a; b:, a;+, A . . . ^a;such that l=(39) [p(ao,i!$,, a,, 8,P,6) A +(g,a,)] (remember I==@[6,a]). Let & satisfy this formula, i.e. h

A

k p[Eo, i!$, ;Gl,

4,@, C] A +[&, all.

Clearly, ao,g,a,, 6 are as required.

FACT 1.8 : Suppose p €Sm(M)is not regular, @(2,&) ~ pThen . there are

c , c ~ ,such c ~ that (i) C,Co realizes p , and I= @[c,,a], (ii) {C, Co} is independent over M , (iii) tp(C,M U C, U c(,)and tp(C0,MU c,) fork over M .

Proof.A s p is not regular, by Theorem V, 1.9(l ) ,(3) there are cl(l< n ) , C and i P realizing p such that { E l : 1 < n} U {c} is independent over M, tp(@,M U UzGnC2) forks over M and tp(C,MU UZGn cl U @) forks over M. W.1.o.g. n is minimal. Still n 2 0. If for every I < n, tp(q,M U Em U F )does not fork over M , then tp(Con . . .n C,,,

u;,,+,

MU @) does not fork over M, a contradiction. By renaming and U P) forks over M. monotonicity of forking, tp(Co,MU By 1.3 there are formulas pl,qz and EM such that letting

uu:,,~~

d = Clh...ncn: (4 l=Vl[@, CO, d, S I ? (b) != V 2 P , e,4, d , %I,

CH.

XI,

5 21

THE AXIOMATIC TREATMENT

56 1

(c) for every i?*,d‘, {pl(i?*,Z,d,al)}U tp(co,M)forks over M, (d) for every P * ,d,G,{v2(Z, P*, ci,d ,al)}U tp(c,M)forks over M. So Wy)[vl@,cO,&al) A v 2 ( ~co,&al) , ~ , A @ ( ~ , a )(use ] @ as a witness) and {d,c0,c}is independent over M (as { q :1 < n} U {F} is independent over M). Hence for some d’ EM

b(3Y) [Q)i(Y, co,

$9

a1) A

v2(G

8,4,$, a1) A @(Y, a)].

So let F+ satisfy

l=vl[a+,co,d’,all A v2[c,c+, cO,d’,al]A @[c+,a]. Clearly, F,Fo, F+ are as required (for F, c0,el) in the fact. CLAIM 1.9 : Suppose p E Sn(N) is not orthogonal to some type q to which @(~,6) belongs, where b€N,l(Z)= m. Then (1) for some r , @(z,6)E T E S ~ ( N r is ) , not weakly orthogonal to p . (2) If N c N* and N* realizes p , then $(N*, 6) # @(N*,6).

Proof (1) Let N E M,M F&,-saturated, and tp(a,M), tp(c,M), are complete types over M which are non-orthogonal stationarization of p , q , respectively. By V, 1.2(3) tp(&,M),tp(c,M) are not weakly orthogonal, so w.1.o.g. tp(a,M U C) forks over M, hence over E] and for every c‘,d, { ~ ( Z , E ’ , N. So for some J E M ,EEN,btp[a,~,d, d’,i?)}Up forks over N. Now as t=(3Z)[v(@,Z,Z,?3 A @(~,6)],{a,d} independent over N for some d” E N, k ( 3 ~[)~ ( Z, a d‘, , i?) A @(z,b)]; hence for some F’, I=cp[a,F’,d’,F] A @[F’,b]. Clearly, r = tp(F’,N) is as required. (2) Follows by the proof of (1): if N EW,CZEWrealizes p , obviously k ( 3 Z )[v(a,Z,d’,i?)A @(Z,b)]and we can choose c ’ ~ @ ( W , b ) , such that kg~[a, c’, d‘, C] A @[?, 61. Now c‘#N as tp(a,N U c‘) forks over N.

XI.2. The axiomatic treatment We try here to axiomatize the various variants of the decomposition theorem. Notice that we have such theorems for Fi0-saturation, <( T superstable without the dop, in X, Section 3) for FE, -saturation, <(T totally transcendental without the dop, in [bh 82bl) for FL0-saturated, E, (in [Sh 86, Section 2]), T superstable satisfying

562

THE DECOMPOSITION THEOREM

[CH. XI, $2

the (
CONTEXT2.1: We shall consider a pair (T, E *) where T is a set of types (i.e. m-types for m < w ) and E * is a two-place relation on the family of sets A E 6. If T = Ti, we omit it (see Def. 3.2) and if E * is Ek0 (see Def. 3.1) we omit it. Let T(A) = ( ~ E Tp; a type over A). Ax(Ai) : C * is preserved by automorphisms of 6, A E * B implies AcB,ifA$BletA s*BmeansAc*BUA,andassumeAs*A. Ax(A2):A E * C a n d A E B E C i m p l i e s A E*B. Ax(A3): C * is transitive. Ax(A4): If A

c *$,

for i < a,$ increasing, then A E *

u,,,$.

Ax(A4;,) : If A C B and A C * C for every C E B of power is a non-forking tree, Arrkc * A l , then A , ) E * U,,IA,. DEFINITION 2.2: (1) A set is T-saturated if every type PET@) is realized by some ~ E A . (2) F(T)= { ( p , B ) : p ~ 8 ~ (for A )some m < w,B E A, and there is qcprB,qET,qw (3) A is T-atomic over B if A is F(T)-atomic over B. (4) A is T-primitive over B, if B E A and for every T-saturated A* extending B there is an elementary mapping F , F ( A )E A * , F I B = the identity.

CH.

XI, $21

THE AXIOMATIC TREATMENT

563

( 5 ) A is T-prime over B if A is T-saturated and T-primitive over

B. (6) A is T-minimal over B if A is T-saturated, B S A , and every T-saturated A*,B E A* c A, is equal to A.

Remark. Usually any T-saturated set is a model (followsby Ax(B2)). So usually we denote such sets by M,N even if this is not necessarily assumed. Note that T-primitive is not necessarily equivalent to F(T)primitive. Ax(B1) : T is preserved by automorphisms of

a.

Ax(B2): Every finite type is in T Ax(B3): I f M C A , p E tp(n,A),pET, tp(&,A)does not fork overM, and M is T-saturated, then p is realized in M. Ax(B3-) : Like Ax(B3) but M 5 *A. Ax(B3--) : Like Ax(B3) but M E *A U ii. Ax(B4): If B is F(T)-constructible over A, then B is T-atomic over A. Ax(B4+):If B is T-atomic over A , A G B , and tp(6,B) is T-atomic, then tp(6,A) is T-atomic. Ax(B5) : If p ( Z , Q) is a type, p ( Z , 6)E T and p ( Z ,6) U {cp(%, 6)) is consistent, then for some q(y)E T which brealizes, Domq s Domp(Z,y) and, for every 6' realizing q ( p ) , p (6~'),E T and p ( ~6'), U { c p ( ~6')) , is consistent. Ax(B6) : If p ( Z ,g) ET, then p(Z,6)ET. Ax(B7): If 6 is a limit ordinal, i <j < 6*M, C M , , each M i is M iis T-saturated. T-saturated, then

ui<6

Ax(C1): If N S Nl E C , N E *A, { A , N l }is independent over N , C is Tatomic over Nl U A, and Nl,N are T-saturated, then Nl c * C .

564

THE DECOMPOSITION THEOREM

[CH.

XI, $ 2

Ax(Cl-) : Like Ax(Cl), but A is T-saturated too and N C *Nl. Ax(Cl--) : Like Ax(Cl-), but C C acl(Nl U A ) . Ax(Cl+): Like Ax(Cl), but N not necessarily T-saturated. Ax(C2) : Suppose M C _ * PE N , all T-saturated and for some FEN, C # P tp(E,W) is not orthogonal to M . Then for some F E N ,a#IP and t p ( F , P ) does not fork over M .

Ax(C3) : Suppose M C _ * A ,A C N , A # N and M , N are T-saturated. If for every EN, &#A,tp(5,A) is not almost orthogonal to M , then for some EN, 5 $ A , tp(5,A) does not fork over M . Ax(C3+): Like Ax(C3), but we strengthen the conclusion to : tp(6,A ) does not fork over M and is regular. Ax(C3-) : Like Ax(C3), but we assume : M c * N . Ax(C3*) : Like Ax(C3+),but we assume : M S * N . Ax(C4) : If M G * N , M # N , then for some F E N F#M , and tp(i7,M) is regular. Ax(D1) : Every P E T @ ) can be extended to a T-isolated complete type over A . Ax(D2) : SupposeM E * A , A C N , ~ E T ( AandM,Nare ) T-saturated, then there is FENrealizing p such that tp(c,A) is almost orthogonal to M . Ax(D2--) : Like Ax(D2) but M C * N . Ax(D2-) : Like Ax(D2), but suppose also & E Ml c . . . c M, = M , each Ml is T-saturated and C _ * N and for each I < n(VB)[Ml+,c B G N A tp*(B,4+l)is almost orthogonal to 4 C * B ] and tp,(A,i&+,) is almost orthogonal to 4. Ax(E1) : Suppose & S *Ml(Z= 1, 2){M1,4}is independent over ik& and each &(Z < 3) is T-saturated. Then there is a T-prime T-atomic model M over Ml U 4.

CH.

XI, $23

565

THE AXIOMATIC TREATMENT

Ax(El+):Like Ax(El), but we omit the hypothesis require that they are T-saturated).

&& E*&

(but

Ax(E2): For Ml as in Ax(El), every ~ E S " ( MU, 4 ) which is Tisolated, is almost orthogonal to MI and to 4. Ax(E2+):Like Ax(E2),for&&c Ml T-saturated, {&Il,@.} independent over &&.

DEFINITION 2.3: ( 1 ) Let A( E *) be the first A such that for every A EB,t h e r e i s A ' , A E A ' E * B , J A ' J < A + l A l . (2) Let A(T) be the first A such that the following is impossible: p 2 A, A, increasing continuous for i < p+,lA,l < p, and for every i some p€T(A,) is realized in A,+1, but not in A,. (3) A(T, E *) is the first A 2 121' such that: if A,(i < A+) is increasing continuous, lAJ < A, then for (*I some i : A , s * U, l ) , (e) t p * ( 3 , q r kU E,rcr+l,) is almost orthogonal to qrk (where k < r]€I), (f) : r] (i) E I ) is independent over N,, (g) tp(@,,N,-)is regular, (2) we define a (T, E *, A)-decomposition similarly, replacing A(T, E*)by A.

y

w,

r

Remark 2.4A.(1) Remember T,- = r] (Z(r]) - 1). (2) All we shall prove for (T, E *)-decomposition,hold for (T, E *, A)-decomposition if A satisfies 2.3(3) (*).

566

THE DECOMPOSITION THEOREM

[CH.

XI, $2

DEFINITION2.5: (1) We say that ( q , a , : q ~ Iis) a ( T ,c*)decomposition inside M if (in addition to (a)-(g)): (h) N, c M for every t E I, (i) if N, E A E M , tp,(A, N,- U a,) is almost orthogonal to N,-, then Nn =*A, (j) M is T-saturated, (k)N<)E *M. (2) We say that (N!,a,,:7 E I) is a ( T ,c *)-decomposition of M if it is a maximal ( T ,E *)-decomposition inside M (maximal - under the natural ordering). CLAIM 2.6: (1)If (N,,@,:~ E Iis) a (T, c *)-decomposition,then (N,: ~ E I is) a non-forking tree. (2) A n y (T, c*)-decomposition inside M can be extended to a ( T , *)-decomposition of M (there is always a (T, E *)-decomposition inside M: the one with I empty).

=

Proof. (1) Use (e) and (f)of Definition 2.4. (2) Easy. CLAIM 2.7 : (1) [Ax(B2)] Every T-saturated set is a model. (2) Ax(B3) implies Ax(B3-) and (B3-3; and Ax(A2), (B33 imply Ax (B3 7 . (3) Ax(Cl+) implies Ax(C1) implies Ax(Cl-) implies Ax(Cl--). (4) Ax(C3+) implies Ax(C3) and Ax(C3*), each of which implies Ax(C3-). Also, Ax(C3') implies Ax(C4). ( 5 ) Ax(D2) implies Ax(D2-) implies Ax(D2--). (6) Ax(Dl), (B4) imply Ax(EI+) implies Ax(El), also Ax(E2+) implies Ax(E2). (7) Suppose that whenever &(I < 3) are T-saturated {M,,k&} is independent over 4,and 4 c *Ml, then acl(M, U k & ) is T-saturated. Then Ax(B2), (Cl--) imply Ax(Cl-). (8) Ax(Cl-), (D2-), (El) imply Ax(E2). (9) Ax(B3) and Ax(B5) imply Ax(E2); also Ax(B3-), Ax(B5) and Ax(Cl--) imply Ax(E2). (10) Ax(B3-), Ax(B5) and Ax(D1) imply Ax(D2). (11) For A < p, Ax(A4;,) implies Ax(A4;J. If Ax(A2), then Ax(A4) ifl AX(A~;,~). (12) Suppose T satisJies Ax(B3). If M is T-saturated, p ( z ,Q) E T ( M ) , b realizes p and tp(6,M U a)is not almost orthogonul to M , then there are al -b1 ( I = 0 , l ) satisfying (i), (ii), (iii) of 1.7 and aln 6, realizes p . A

CH.

XI, $21

THE AXIOMATIC TREATMENT

567

(13) if T is superstable, Ax(B3) implies Ax(B7). (14) Ax(B4+)implies Ax(B4). Proof. (8) Let M be T-prime and T-atomic over M, Ui& (exists by Ax(E1)). Let us prove that tp,(M,M, U 4 ) is almost orthogonal to M,. Let C be a maximal subset of M such that tp,(C,M, U 4 ) is

almost orthogonal to MI. Clearly, M, U 4 E C. By Ax(Cl-) M, E *M, and M, C*C hence by Ax(D2-) (even (D2--)) C is T-saturated. As M is T-prime over M, U 4 , M is isomorphic over M, Uik& to some M-,M, U 4 c M- c C. Now tp,(C,M, U 4 ) is almost orthogonal to M,, hence tp,(M-,M,U%) is almost orthogonal to M,, hence tp,(M, M, U 4)is almost orthogonal to M,. (9), (10) Use 1.3. (12) Repeat the proof of 1.7. THEOREM 2.8: [Ax(Al), (A2), (A3), (B2), (B3), (B4+),(B5), (B6), (B7), (Cl-), (C2), (C4), (D2-), (El), (E2) and: (A4) or (A6), (Cl)]. Suppose T does not have the dop. If (A$a,,:qeI) is a (T, c *)decomposition of M, then M is T-prime T-atomic and T-minimal over

U,,,N,. We break the proof to a series of claims. CLAIM 2.9: (1) [Ax(A3),Ax(Cl-)]. Let &,M(l< 3) be as in Ax(E1). Then 4 E *M for 1 < 3. (2) [Ax(B2), 0331, (B5), (B6)I. Suppose 4 E *Mi94 E *4, ik& E M,*,{M,,M,*} is independent over 4 and each model is T-saturated. Then every T-isolated p E Sm(MlU 4 )has a unique extension in Sm(MlU M ; ) . Proof. (1) Trivial. (2) Easy too. Let p be a counterexample. By Ax(B2), (B3) (or use XII, 2.3) M, U 4 EtMl UM; (see Definition 3.1) hence by 1.2 p is stationary inside MI U Mz. So (as p is a counterexample) some U M,*)forks over M, U 4. Let a realize such extension of p in Sm(Ml extension, so a realizes p and for some 6eM,*,FEM,, and cp t=cp[a, 6,F] and cp(a,6,Q fork over M, U 4 . By Fact 1.3 w.1.o.g. for every 6,F’, p U cp@, b’,c’)forks over M, U 4 . As p is T-isolated there is p, E p, p, I-p and pOeTand w.1.o.g. F E Domp,. Clearly, p, U {cp(Z, 6, q} is consistent, hence by Ax(B5)

568

THE DECOMPOSITION THEOREM

[CH.

XI, $2

there is q(q,Z)E T such that q(q,Z)E tp(6-E, Domp,), and for every b'- c realizing q(g,q, p , U{cp(Z,6',c')} is consistent. Now tp(6, Ml U 4 )does not fork over 4,E E M and ~ q(g,C) E T (by Ax(B6)),and q(g,C)E tp(6,M1U&). So by Ax(B3) some 6 ' ~ & realizes q(q,C), hence 6'^ F realizes q(q,Z),hence p , U {cp(~,6', C)} is consistent. As p , I-p, also p U {cp(x,6',C)} is consistent, by the choice of cp it forks ~ we get a contradiction. overM, U 4 . As~ES"(M,U & ) , E , ~ ' E MU&, A -I

CLAIM 2.10: [Ax(Al), (AZ), (A3),(BZ), (B3), (B4+), (B5), (B6), (B7), (Cl-), ( E l ) , (E2), and : (A4) or (A6), (Cl)]. Suppose I Gu' a is closed under initial segments, ( N , : q € I ) is a non-forking tree, each N, is T saturated and Nqrr E *%. Then there is a T-prime T-atomic model M over UqEIq,and 4 E * M for every ~ E and I tp,(M, U,,,N,) is almost orthogonal to each 4.

r

-

Proof. Let I = {vr: i < /3} be such that qr = 7, k i < j.We define by induction on i a model 4 such that : (a) u,<,+rq, c (b) 4 is T-prime and T-atomic over U,,,+,N, U q for each y < i, 4

9

For i = 6,limit. We let 4 = u j , d M j . Now (a),(d), (e) are trivial, (c) follows by Ax(A4) (or use Ax(A6), ((31):by Ax(A6) q,s * U { N , : a < l+i,vj, 8 q,} and then by downward induction on k < l(v,),using Ax(Cl), 3,E < 1 +i, q, k < va},so for k = 0, qjE * Ua<,+&,=;now use Ax(C1) with q,, standing for N,N,,A, C, respectively, using Mdis Tatomic over U,
r

*u{q=:a

uu..l+,qm,l&

u,<,+&,,

'~('7

q,,

CH.

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THE AXIOMATIC TREATMENT

569

I- tp@,U,,dA?. We forking tree, clearly (by XII, 2.3) tp(c, u,,l+,q,) hence tp(c, IS Tcan conclude that p I- tp(c, isolated. The same argument proved half of (b); M, is T-atomic over u,,,qj U My. For the other half (T-prime) let M+ be T-saturated, EM+, and we shall define an elementary embedding of M For this we define by induction on i < 6 an into M+ over elementary embedding 4 of 4 into M+,Fi rN = the identity for j < 1+a and 4 E I$ for j < 1+a. F, is the ident!iy ; for limit we take union, for successor we use (b) from the induction hypothesis. So Fb is as required. Really this proves (b) for y = 0 ; the general case is proved similarly. Mi is T-saturated by Ax (B7). For i a successor, let 1 + i = j+ 1, where j > 0 and let l(7,) = k,, 7,r (k,- 1) = 7,. By (g) for i- 1,{N,,,4-l} is independent over q,. Those three models are T-saturated (M+l by (b), the others by Def. 2.4), by (c) qaG * J I - , and by Def. 2.4. q m~ * l $ . So Ax(E1) applies and a T-prime T-atomic model 4 over NVjU M ) exists. By Ax(E2) tp*(4,NTjUM)) is almost orthogonal to N, and to 4, and by 2.9(1) q,G * ~ , M ~ ; * Now 4 . (a), (d), (e) hold trivially, (c) holds (by Ax(A3)). We can get (f) by 2.9(2), and it implies (g). Now (b) can be proved using (B4+).

u,,,N,)

u,,,q,),

u,
uj,I+ui'$.

CLAIM 2.11 : [Ax(Al),(A2),(All), (B2), (B3), (B4+),(B5),(B6),(B7), (Cl-), (El), (E2) and: (A4) or (A6), (Cl)].Suppose ( q : r ] ~ is I )as in the previous claim, and M is T-prime over Then tp*(M,u,,,N,) is almost orthogonal to N, and N, G *M for each V E I , and M as T-atomic over u,,,N,.

u,,,y.

u,,,N,

Proof. By 2.10 there is M+ T-prime over which satisfies the conclusion of 2.11. As M is T-prime over UTEII$ there is an elementary embedding ofM intoM+over U,,,q, so w.1.o.g. M c M+. Easily M inherits the required properties from M+.

CLAIM 2.12: [Ax(Al),(A2), (A3),(B2), (B3), (B5), (B6), (Cl-), (C2), (El), (E2) and: (A4) or (A6), (Cl)]. Suppose T does not have the dop. a,, :7 EI) is a (T, E *)-decompositionu,N, G M,M # M+, Suppose (3, EM+ c M,M+ is T-prime T-atomic over U V E I qfor , each V E I , N, G *M+and tp,(M+, U,,,,N,) is almost orthogonal to 4.Thenfor some FEM,F$M+ and ~ €tp(c,M+) 1 does not fork over N, and is orthogonal toN,- if 7 # ). [Actually we use only (a),(b),(c) ofDejnition 2.4(1).]

u,,,N,

<

570

THE DECOMPOSITION THEOREM

[CH.

XI, $2

Proof. Let h = ~ ~ M ~ ~ +and + ~ the T ~ following +, fact is true (under 2.12’s hypothesis).

q,

FACT 2.13 : (A) We can find Ff-saturated q ( g E I ) such that N,, E qrk 5 tp,(q,M U : g # v and not g 4 v}) does not fork over

q,

u {v