The Core Model (London Mathematical Society Lecture Note Series)

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor I.M.James, Mathematical Institute, 24-29 St Giles, Oxford 1. 4. 5. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 20. 21. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

General cohomology theory and K-theory, P.HILTON Algebraic topology: a student's guide, J.F.ADAMS Commutative algebra, J.T.KNIGHT Integration and harmonic analysis on compact groups, R.E.EDWARDS Elliptic functions and elliptic curves, P.DU VAL Numerical ranges II, F.F.BONSALL & J.DUNCAN New developments in topology, G.SEGAL (ed.) Symposium on complex analysis, Canterbury, 1973, J.CLUNIE & W.K.HAYMAN (eds.) Combinatorics: Proceedings of the British combinatorial conference 1973, T.P.MCDONOUGH & V.C.MAVRON (eds.) Analytic theory of abelian varieties, H.P.F.SWINNERTON-DYER An introduction to topological groups, P.J.HIGGINS Topics in finite groups, T.M.GAGEN Differentiable germs and catastrophes, Th.BROCKER & L.LANDER A geometric approach to homology theory, S.BUONCRISTIANO, C.P.ROURKE & B.J.SANDERSON Sheaf theory, B.R.TENNISON Automatic continuity of linear operators, A.M.SINCLAIR Parallelisms of complete designs, P.J.CAMERON The topology of Stiefel manifolds, I.M.JAMES Lie groups and compact groups, J.F.PRICE Transformation groups: Proceedings of the conference in the University of Newcastle upon Tyne, August 1976, C.KOSNIOWSKI Skew field constructions, P.M.COHN Brownian motion, Hardy spaces and bounded mean oscillation, K.E.PETERSEN Pontryagin duality and the structure of locally compact abelian groups, S.A.MORRIS Interaction models, N.L.BIGGS Continuous crossed products and type III von Neumann algebras, A.VAN DAELE Uniform algebras and Jensen measures, T.W.GAMELIN Permutation groups and combinatorial structures, N.L.BIGGS & A.T.WHITE Representation theory of Lie groups, M.F.ATIYAH et al. Trace ideals and their applications, B.SIMON Homological group theory, C.T.C.WALL (ed.) Partially ordered rings and semi-algebraic geometry, G.W.BRUMFIEL Surveys in combinatorics, B.BOLLOBAS (ed.) Affine sets and affine groups, D.G.NORTHCOTT Introduction to H p spaces, P.J.KOOSIS Theory and applications of Hopf bifurcation, B.D.HASSARD, N.D.KAZARINOFF & Y-H.WAN Topics in the theory of group presentations, D.L.JOHNSON Graphs, codes and designs, P.J.CAMERON & J.H.VAN LINT Z/2-homotopy theory, M.C.CRABB Recursion theory: its generalisations and applications, F.R.DRAKE & S.S.WAINER (eds.) p-adic analysis: a short course on recent work, N.KOBLITZ Coding the Universe, A. BELLER, R. JENSEN & P. WELCH Low-dimensional topology, R. BROWN & T.L. THICKSTUN (eds.) Finite geometries and designs, P. CAMERON, J.W.P. HIRSCHFELD & D.R. HUGHES (eds.) Commutator Calculus and groups of homotopy classes, H.J. BAUES Synthetic differential geometry, A. KOCK Combinatorics, H.N.V. TEMPERLEY (ed.) Singularity theory, V.I. ARNOLD Markov processes and related problems of analysis, E.B. DYNKIN Ordered permutation groups, A.M.W. GLASS Journees arithmetiques 1980, J.V. ARMITAGE (ed.) Techniques of geometric topology, R.A. FENN Singularities of differentiable functions, J. MARTINET Applicable differential geometry, F.A.E. PIRANI and M. CRAMPIN Integrable systems, S.P. NOVIKOV et al.

61. 62. 63. 64.

The core model, A. DODD Economics for mathematicians, J.W.S. CASSELS Continuous semigroups in Banach algebras, A.M. SINCLAIR Basic concepts of enriched category theory, G.M. KELLY

London Mathematical Society Lecture Note Series. 61

The Core Model

A. DODD Junior Research Fellow at Merton College, Oxford

CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON

NEW YORK

MELBOURNE

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NEW ROCHELLE

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521285308 © Cambridge University Press 1982 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1982 Re-issued in this digitally printed version 2007 A catalogue record for this publication is available from the British Library Library of Congress Catalogue Card Number: 81—17989 ISBN 978-0-521-28530-8 paperback

PREFACE

This monograph is intended to give a self-contained presentation of the core model, adding details to the brief account that Ronald Jensen and I presented in [10] and [11]. By "self-contained" I mean that any result not proved should be easy to find in a textbook: I have used Jech's book ([25]) for references wherever possible. I have included just about everything I know about K, sometimes without proofs, and a few details about larger core models. I am conscious, looking over the text, of many places where the explanations could be clearer, of important results that never quite surface as lemmas and of clumsiness in some of the proofs. I hope that proofs of everything that needs proving can be extracted from the text by a careful reader: I shall be most grateful for comments from anyone who finds the presentation incomprehensible (or erroneous). Two particular comments may help: firstly the exercises range from the very easy to the unsolved. Sometimes they are essential to later results in the text; I hope these ones are not too difficult! Secondly there is no good reason why appendix II is not inserted at appropriate points in the text, as, apart from one result in descriptive set theory, it is self-contained. Were I rewriting the text there would be more to be said about models of ZFC and less about R; but the reader will have to extract general proofs from those given in chapter 12. Numbers in square brackets thus [1] refer to the bibliography. I have added to this a set of notes on the history of the results in the text, and used this as an excuse not to give attributions of results elsewhere. But I should add the traditiional disclaimer about not being a historian of the subject; it will be apparent that I have relied heavily on the notes in Jech's book. There is one historical point on which I can offer no help: people occasionally ask why mice are so-called. I am afraid that neither Jensen nor I can remember why, but plausible explanations would be welcomed. Many thanks are due to those who have helped by their own explanations or by their criticism of mine. I should mention particularly Keith Devlin, Bill Mitchell and Philip Welch; and among the discoverers of errors in previous editions Peter Koepke

and Lee Stanley. Robin Gandy has been a constant inspirer of improvements from the time when as my supervisor he cast a critical eye over the almost incomprehensible first draft of my thesis up to a recent seminar talk on rudimentary functions which compelled me completely to rewrite part one. It is appropriate that I should dedicate this work to Ronald Jensen, since so much of it is his work. How much is not apparent from the references: one must also take into account his patient explanations in answer to my endless questionning. I hope the references do make it clear that, although I alone am responsible for the exposition, the major results are our joint work and many of the others are his alone. I was supported while writing by Junior Research Fellowships first at New College and then from Merton College. To have been able to work in two such delightful communities has meant a great deal to me and I owe more than I can express to the friendship and generosity of the fellows of each. Professor loan James first encouraged me to write this monograph, and to him, as well as to David Tranah and the Cambridge University Press, who gave much advice and help, and waited very patiently for the result, I express my gratitude. Tony Dodd. Merton College, Oxford.

CONTENTS

PRELIMINARIES

ix

INTRODUCTION

xiv

I: FINE STRUCTURE 1. 2. 3. 4.

1

A weak set theory Relative constructibility Acceptability and the projectum Models of R

2 10 22 31

II: NORMAL MEASURES 5. Normal measures and ultrapowers 6. Iterated ultrapowers 7. Indiscernibles 8. Iterability

40 41 49 54 60

III: MICE 9. Mice 10. Properties of mice 11. Acceptability revisited 12. 0 # 13. The model L[U]

69 70 76 86 95 104

IV: 14. 15. 16. 17.

114 115 120 128 137

THE CORE MODEL The core model Examples of K Embeddings of K Applications of K

V: THE COVERING LEMMA 18. The upward mapping theorem 19. The covering lemma 20. L[U] and L[U,C] 21. Applications of the covering lemma

144 145 160 169 177

VI: 22. 23. 24.

LARGER CORE MODELS Coherent sequences Strong cardinals The great blue yonder

186 187 195 201

APPENDIX I: SOME GENERIC MODELS

205

APPENDIX II: ABSOLUTENESS RESULTS

208

HISTORICAL NOTES

217

BIBLIOGRAPHY

222

INDEX OF DEFINITIONS

227

ix PRELIMINARIES

My intention in writing this book was that it should be accessible to anyone who had a reasonable background in axiomatic set theory up to Godel*s consistency results; and, if this failed, at least that readers should not be expected to hunt through piles of old journals in search of alleged folklore. The appearance of Jech's book ([25]), and more recently of that of Kunen, have made it easier to find these results and I have not hesitated to give references for results that lie away from the main development of the fine structural theory. Various odd facts have, no doubt, slipped in along the way unproved, but consultation of one of the textbooks mentioned should help. Chapter 0 of Devlin [6] also contains many of the things I feel I ought to have said here. A few notes on particular points are necessary. LANGUAGE The language L is the usual first order language for set theory. On official occasions its primitives are - , A , 3 , ( , ) with binary relation symbols = and € and variables v.. Formulae are also treated as objects (definition 1.16). A bounded quantifier is one of the form (3v.€v.), where (3v.€v.)cf> abbreviates (3v.)(v.€v. A <j>) ; a formula all of whose quantifiers are bounded is called restricted or £_. In induction on E

structure, therefore, (3v.€v.)$ is more

complex than v.€v. A <j>. Generally if variables are displayed after a formula then all, but not necessarily only, free variables are displayed. But this is not a hard and fast rule, and the absence of a variable list is certainly not an indication that we are dealing with a sentence. The language L

is L together with N additional unary

predicates, usually called A,...A . If they are to be given other names, or we are to add binary relations or constants or anything else, or if we want to rename the € predicate then the complete list of non-logical symbols is displayed thus: L_ R, i,c AXIOMS A sequence of weak theories is introduced in part one. For reference the axioms of ZF are the universal closures of the

following: (1) x=y *» Vz(z€x «* z€y) (extensionality) (2) x*0 ** (3yGx) (yflx*^) (foundation) (3) 3z(z={x,y}) (pairing) (4) 3z(z=Ux) (union) (5) . 3yVz(z€y ** Z€XACJ>) (separation) (6). Vx3y4> (x,y,p) •» Vu3v (Vx€u) (3y€v) cf> (x,y,p) (replacement) (7) 3y(y=P(x)) (power set) (8) 3y(y=w) (infinity). The axiom of choice is (9) Vx(Vz / z l (z, z 1 €x -» z=z'vznz'=0) =» 3yVz€x3 iww€yDz) . (l)-(9) is called ZFC. (l)-(6),(8) is called ZF~. The theory KP consists of axioms (l)-(4) together with (5), and (6), for restricted <J>. Its models are called admissible. Admissible sets are too narrow a collection for the fine-structural theory used in part one - the theory R which we use is weaker than KP - but similar considerations motivate the removal of, for example, full replacement from the theory. FUNCTIONS The ordered pair <x,y> is {{x,y},{x}}. A relation is a collection of ordered pairs. A relation R is a function provided RxyARxz -> y=z; then instead of Rxy we may write R(x)=y. This rather trivial point needs attention: usually the literature of fine structure has followed Go'del in identifying the function f with {:x€dom(f)} . I have used the different notation here because it seems to have become the more commonly used one. It is natural, given this usage, to define the ordered n+1-tuple so that <x....x ., >=<<x,1 ...x n >,Xn+1 ,, >; thus n+l-ary functions are functions. 1 n+1 Parts of the theory - lemma 1.5 for example - are sensitive to the choice of convention - although they would not be if we were to redefine <x,y>={{x,y};{y}}. I have tried to avoid reliance on the convention by adding unnecessary < > in places. Sometimes an inverse defined by x= < (x) Q ...(x) _. > is used, but care is needed as this is ambiguous. x represents a list and not a sequence: so x€X means x, ...x €X and not <x....x >€X. Exceptions occur if there is no danger of confusion. id is the function id(x)=x. ORDINALS AND CARDINALS X denotes the cardinality of X: this is always identified with the least ordinal in one-one correspondence with X (in all our weak systems other than R this can be shown to exist). The order-type of

xi <X,<> is written otp(<X,<>); < may be omitted when it is €. K

and

a) are used interchangeably. If K is regular and uncountable and C C K

then C is closed in

K if XcC => supX€CU{i<}. It is unbounded if sup C = K . S<=K

is stationary provided snc*0 whenever C is closed and

unbounded in K. If C , C

are closed unbounded in K then so is CDC'.

Indeed, if yC~ closed unbounded in K then

n C- is

closed unbounded in K. Fodor's theorem states that if S is stationary in K and f:S->K is regressive (i.e. for x€S f(x)<x) then there is $ € K and stationary S'cS such that f"S'={&}. STRUCTURES A structure for L is a pair <M,E>. We distinguish the structure <M,E> from the set M by underlining: M denotes <M,E>. There are some exceptions (other than carelessness) to this rule, though. One is discussed in chapter 4; another will be introduced in a moment. But we have tried to adhere strictly to the rule that underlining should not be used for any other purpose: so M and M will never be used simply as different variables. V denotes the universe, even if the theory is weaker than ZF; this looks bizarre until you get used to it. When VJ=ZFC, we call a structure M an inner model of a theory T provided M|=T and M is a transitive class containing On. It is usually safe to ignore the underlining convention with inner models, and in particular with V itself (as we did at the start of the previous sentence). Another abuse of notation is the writing of <M,A> to denote <M,AnM>. This never causes confusion.

If M=<M,E,A 1 ...A N ) then

2

M|X denotes < Mnx,EDX ,A..nx, . . . ,A^0X >. In listing structures E may be omitted if it is clear what it should be. If t is a term then t (y) denotes that x such that M(=x=t(y). If convenient a subscript rather than a superscript may be used. The Mostowski collapsing lemma (a theorem of ZF) says that if <X,E>|=axiom (1) and E is well-founded then there is a unique transitive set M and a unique isomorphism TT such that TT:< X,E >=< M, € >. If X is a proper class it must also be specified that for all x€X {y:yEx} is a set. THE LEVY HIERARCHY £

has already been defined. Suppose E

is defined: then n^ is

the collection of negations of formulae in E . E

+1

is the set of

formulae of the form 3y
n

T ={<j):T|-i|; for some i|/ in V} . T is always omitted; it is taken to be whatever set theory we are working in.

xii TT:N-*

M means that for all E n\

2J

X-<

v

<j> and x,...x,€N

M (where XcM) means id|X:M|x+

parameter p provided there is a E

JL

K

M. A relation R is £ (M) with rormula such that

R(y) ~ Z m (M) denotes the set of relations which are £ m (M) in some parameter p. Really we should use a bold-face E for this, but instead we specify explicitly if there is a restriction on parameters. Note that if M,N are models of ZF~+AC and ir:M->s N maps 0n M cofinally into On

N for all m. Suppose this proved for m n<m and let

( 3 )

then TT:M-*

N|-(3y€Vir(a))Tp(yfir(x1)I!l..ir(xk)). Since i r s M ^ N TT(V^) -V^ ( 0 ) ;

and (3y€z)\|;(y,x) is n m (see [25] lemma 14.2 (ii):

it is here that we

need the full force of ZF~) so M|= (3y€VQ) ip (y,xn . . ,x, ) , i.e. M h ^ x . .. ,x, ) . TT:M->

N for all m is abbreviated ir:M-> N (e for e m elementary). id|X:M|x-> M is written X-^M. The usual terminology, x

JC

~~ £j

N-^M may also be used. An=EnMIn.

A x formulae are absolute; that is, if <J> is A^^ and

XcM, x€X then M|= (x) «* M|xf=4>(x). A^ are absolute between models of T; for example, if M is an inner model of ZF and R€M is a partial order then R is well-founded if and only if M^=R is well-founded. More results of this kind are in appendix II. Note also that if M,N|=ZF and j:M+ N and j*id|M then there, is an ordinal K such that J ( K ) ^ K . For otherwise for all x j(rank(x))= =rank(x) so rank(j(x))=rank(x) and an easy induction shows that for all a€On

j|v M =id|v M . The least such K is called the critical point

of j. DESCRIPTIVE SET THEORY The reals are identified with P(OJ). A formula is arithmetic if all its quantifiers are restricted to range over w: ^ is E, if there is an arithmetic formula ip such that cf)(x) •* (3a€P(w)) ip (a,x) . if this holds with "II1" in place of n 1 1 1 1 1 Free variables "arithmetic"; and it is II + 1 if -<J> is £^+1* A n n may represent either natural numbers or reals: we may therefore also define E sets of reals and E reals. All notation is lightn n •, faced. Note especially that if cf> is, say, E, and we write A)=<j> this Generally, it is E * n

means that there is a real in A satisfying the appropriate condition; in other words, is not treated as a second-order formula. Our coverage of descriptive set theory is very skimpy and the reader should

consult Jech [25] for details.

xiii NOTATION Other than the above most notation can be found in the index of definitions. For the rest the following is a list of standard symbols that may not be immediately recognisable as such. (i) ^ is complement, but - is negation. (ii) U and PI are used both for the binary and for the unary union and intersection. (iii) | denotes functional restriction, f|X={<x,y>€f:x€X}; f'X denotes the range of f|x. (iv) •*-*• indicates a bijection, = an isomorphism. (v) » and On are used interchangeably. (vi) 0 is the null set; is phi. (vii) D marks the end of a proof. (viii) P is power set. (ix) # is a sharp, t is a dagger and U is a pistol. (x)

HK={X:TC(X)
INTRODUCTION

The core model arises from the mixture of two techniques that had once seemed incompatible: fine-structure and iterated ultrapowers. Fine structure was designed for use in Godel's L; but the major application of iterated ultrapowers was to measurable cardinals: and there are no measurable cardinals in L. First let us examine the two sources separately. 1: FINE STRUCTURE Godel's constructible universe L may briefly be defined as follows: L

a+fDef(La»;

L = U L (X a limit). A a where M is transitive. Such an M exists by the Mostowski collapsing lemma. Then M=L, for some A. We could get by with conditions on K and X much weaker than these. Now consider any acw, a€L. For some cardinal K a€L . Let X -< L ^ * — ' K K with X=N 0 , a€X. wcX, of course. Let TT:M^<X,€> with M transitive. Say M=L^.

Then say a=7r

(a) . acw and for all n€oj 7r(n)=n so n€a ** n€a.

L,=T. And L.=X=N_ , so X=N^: that is, A<0), . Hence a€L A

-

A

0

proved P (o))cL v^

*

2 °
0

1

. W e have

03i

. If we carry out the argument in L we deduce that

CO i

iv

=N. . By Cantor's theorem 2 °>N-. so the continuum hypothesis

holds in L. In fact an almost identical proof shows that for all infinite cardinals A P L (A)cL x +, so that GCH holds in L. Godel deduced that Consis(ZF) «* Consis(ZF+GCH). In fact the condensation property can be used to deduce more powerful properties such as 0 (see [6]).

A little close attention to the condensation property reveals many more details about the corollary that P (u))cL

. We have

already observed that we may replace co, by w,: and this must be the best possible, since L(=P(o)) is uncountable. But that does not mean that every a
.. This is called a gap. Indeed there can be gaps

much longer than 1 - see [39]. The existence of a gap is equivalent to some form of comprehension: for it says that all subsets of u) definable over L, are already in L^. We could (and shall) define a measure of the failure of comprehension over L, as the least y such that L,nP(y)=j= ^L, + ,nP(y) . Such measures are clearly related to admissible set theory (KP). In fact it pays to be even more precise. Maybe there are no new subsets of a) E,-definable over L^, but there are such Indefinable definable subsets? "We find such questions both interesting and important in their own right. Admittedly, however, the questions and the methods used to solve them - are somewhat remote from the normal concerns of the set theorist. One might refer to "micro set theory" in contradistinction to the usual "macro set theory". Happily, micro set theory turns out to have non-trivial applications in macro set theory"(Jensen [26]). We shall return to the question of applications at the end of this section. Jensen's paper just quoted contains a full fine-structural analysis of L. This involves putting "coarse" results into forms that involve "fine" definability distinctions. As an example consider the result of Gttdel that if a<=y, a€L^ + 1 ^L^ then X=y. It turns out that if a€E 1 (L.)^L^ then there is a function f Indefinable over L, of a subset of y onto X. (We have to say "a subset of" because if we tried to add trivial values for 3€y^dom(f) we might end up with a E 2 function.) It turns out to be convenient to replace the L hierarchy by a modified form, the J hierarchy. This leaves the total model the same but redistributes sets among the levels. One reason for doing this is that the pairing axiom fails in arbitrary L^, which makes for difficulties. As the text uses an odd definition it is worth giving the original here. A function is called rudimentary if and only if it is finitely generated by the following schemata: (a) f(x)=x ± ; (b) (c) f(x)={x.,x.}; (d) f(x)=h(g(x)); (())

(e) f(y,x)= U g(z,x). z€y Rudimentary functions were invented as part of a programme to generalise the notion of primitive recursive to arbitrary sets (see [17] and [32]). Now the J hierarchy is defined as follows: J o =«; J

a+l=R(JaU{ja})' J,= U J (X a limit). A a<X a ^ R(X) denotes {f(x): f rudimentary, x€X} . Then* (i) L= U J ; a£On a (ii) OnflJ =a3a (whereas OnflL =a) ; (iii) J a + JnP(J a )-Def(J o ). J a = L a if and only if coa=a. (iii) shows that R is very like Def; (ii) shows that in general it is longer. One of the most important technical devices of [26] is the "master code". One way of looking at this is to say that if there is some subset of y in E,(J^)\J. then there is a "universal" one. To put this more precisely, let us define the projectum. The E projectum of X, p?, is the least y such that P(ooy)nE (J..)ctj We A n A A shall restrict ourselves to E, for a bit. As we have already said there is a E, function of a subset of cop, onto J..Indeed there is J.

A

A

such a function uniformly, called the E, Skolem function. This is a partial E.. function h with the property that given a fixed enumeration of E, formulae with two free variables It may help if we sketch a proof of the assertion that h is a surjection when restricted to ojxwp^Un fact this is not quite exact because h may need a parameter argument as well, but we shall avoid this difficulty). Suppose a€P(u)p*) n (Ex (J,) ^J.) . Assume for simplicity that a has a parameter-free E, definition over J.:

3€a ~ J,f=4>(3) • Now let X=h" (ojxwp^) . it is easily seen that X -< v J, . L e t T T : M = < X , € > A

2J

i

A

—~

with M transitive. Now the condensation property holds in the J hierarchy for all levels - indeed there is a IT2 sentence saying, in any transitive set, "I am a J " - so M=J O for some B. TrrJn^v J i • n

a

p

p 2,1 A

Now for yCcop^ 7r(y)=y, and M H > ( Y ) ** J X N (* (y) ) «* J X H<J>(Y), S O a€E x (J B ) If 8<X we should have a€J Q j ,cJ., but a^J, , so 6=X. So TT:J.+ J. . p+l— A X X Ei X Furthermore given x€J, TT(X)€X S O fr(x)=h(i,y) say, with i€w and 1 y
xvii J 1 - we can then code it in wp, if necessary. In general we only p, A A jL know that for some parameter p h" (o)x (ojp x{p} ))=j • let p, be the least such p in the well-order of J\ used to prove Lf=AC. Then let A={< i,x >: i€w A X € J 1 A J.(=4> . (x,n) }. For every B€E, (J.)nP(J 1) P,

A

there is a r u d i m e n t a r y A is a E, m a s t e r

1

A

function

J.

A

p<

f such t h a t xGB «* f ( x ) € A .

code.

Having obtained

this result we

find

that the m e a n s are m o r e

interesting t h a t the end. For A d o e s n o t m e r e l y code s u b s e t s up,: it c o d e s the w h o l e of J,. For e x a m p l e , if TT:M-> <J 1,A> A A ^ - Li px there is a u n i q u e

y and a u n i q u e

TTZDTT such t h a t TT:J •>_ J,

of then

and,

letting M = < M , B > , B is a E, m a s t e r c o d e for J , M=J 1 and TTT(p ) =p, . A ~ ~~ y PU y Also TT:J •*•_ J, . T h i s is p r o v e d by r e c o n s t r u c t i n g J from B (see y L2 A y lemma

3.26). To generalise

define a r e l a t i v i s e d

this c o d i n g

projectum

of a structure <J

that p-v=p 1 A . S i n c e r e l a t i v i s e d A p •> , A consideration

in a m o r e g e n e r a l

to p, it is n e c e s s a r y ,A> and

p r o j e c t a w i l l need c o n t e x t we shall

to

to show

detailed

say n o t h i n g

more

about them h e r e . In

[26] the a p p l i c a t i o n s of these p r i n c i p l e s

c o m b i n a t o r i a l . It is d i f f i c u l t

that r e v e a l s the r o l e p l a y e d by fine try, for the a p p l i c a t i o n s Essentially

the p o i n t

all

s t r u c t u r e , and w e

that c o n c e r n u s are n o t

fine structural a n a l y s i s , E

,1 p r o p e r t i e s

properties of J n, w h i c h are then h a n d l e d P

of J,

not

combinatorial.

formulae b u t do n o t g e n e r a l i s e

a r e a source of such r e s u l s , as w e

these

shall

is t h i s : m a n y nice t h e o r e m s can be

about E, m a p s , sets and ultraproducts

are

to give a b r i e f a c c o u n t of

proved

to E :

shall s e e . B y

are r e d u c e d

to E,

n i c e l y and r e t u r n e d

to

X

E , form by a futher bit of fine structure. For example, for every E-, (J ) relation R(x,y) there is a E, function r(x) such that 3yR(x,y) =* R(x,r (x) ) . (E, relations are said to be "E, uniformisable11) . This is not so clear for E

+1

relations. We may prove it as follows, by induction

on n. n=0 is given. Suppose R is E + ,; recall that there is a E f mapping a subset of J n onto J, . Say R(x,y) ** R(f (x) , f (y) ) : then ^A

RcJ n is E .-.(Jy), hence E, (J n,A n ) where A n is the nth master code — p, n~t-x A 1 p, formed by the inductive process we have just described. Uniformise R by r, so that 3yR(x,y) =» R(x,r (x) ) . Finally f

is a E (J ) relation so by induction hypothesis has a

uniformising function g; in other words g is an inverse function for f. Let r(x)=f(r(g(x))). r is the required E + , function. For details of other combinatorial applications the reader should

xviii consult [26] or [6]. In section 5 of this introduction we shall see how the covering lemma uses fine structure. In this book we are dealing all the time with relative constructibility. That is, given a set A we define a function as "rudimentary in A" provided that it is generated by (a)-(e) above and (f) f(x)=x i nA. We define

J^= U j *

(X a limit)

where R

is like R with "rudimentary in A" in place of "rudimentary". A It is also possible to define an L hierarchy by using an operation

Def

replacing L by i, and <X,£> by <X,€,XflA>. Or, indeed, for any A

number of A 1 ...A^.

U Ja is always called L [ A ] . The most striking

j[ a€On

fact about the J hierarchy is that it does not have the condensation property: if X -< J A and TT:M=< X x G,Anx > with M transitive and M=<M,€,A> then certainly for some 3 M=J O . But A=AflJo does not necessarily hold. p

P

But in the main application cited, the proof that there is a ^i(J^) map of a subset of J 1 onto J, it was essential that the transitive p -v

A

1 collapse should leave us in the same hierarchy. Of course if wp,> >sup A this is possible so ordinary fine structure holds above sup A; this is not much consolation. In fact we speedily see that the E, map property fails. Suppose acco but a^L. Let A={a), +n;n€a}. Then A A for a1 AflJa=0 so J^=J a : hence afJ^. Also for all 3<w-L+n An£€L as it is finite? so J ^ ^ J ^

and a f j j i + 1 . But a e ^ t J *

+ 1

),

for

a={n:u), +n€A}. Yet it is impossible that there should be any map A A of a) onto J^

+1,

whether ^ ( J ^

+1)

or not.

Worse is to come. Suppose we had let A={< a),,n >:n€a}. Then we should have had a€J A ... : but since APIJA =0 , Z ^ a^E

(j

,A >) =Z (JA)cL so

) . Thus there may be undefinable new subsets of w in /

,,:

this means that the projectum is not a satisfactory index of formation of new subsets. And we no longer have R_(J A U{J A }) flP(JA) = A

2>

A

A

A

=DefA(JX>• We may still define a master code as before; but, for example, the E, master code may no longer code all the S, subsets of cop , . x

i.

A ,A

over J,. For example in the first example above we may assume L N-j=xn ; but no subset of u> could code N, reals in the sort of coding we have used. We said, though, that the major interest of the master code was that it coded the whole of J,. What structure do these "inadequate" master codes code?

xix A fact about L that we did not state is that if A is the E, master code of J, then <J 1,A> is amenable, i.e. for all x€J 1 X P P X X xflA€J 1. The coding would have made no sense otherwise, for P A important arguments in <J 1,A> would have failed. For example the P X E, master code of <J 1,A> would not be E, definable over <J 1,A>. A A A So we must know that for all x£J 1 xflAEJ 1 . For a start it is i

A f A,

A J J\.

clear from the definition of p, . that xflA€J, . But we cannot get A t Pi.

A

any further than that. For example, take a^L and A={o),+n:n€a} again. 1 A Take some a>0), with p =o), .Let B be the L, master code of J . Then a is rudimentary in B so if <J 1 ,B> is amenable a€J =J cL. P

A J 1 A

rv

A

0)1-

0)1

' is not a convenient structure to work with; it is too

iA

A

A

small. All the sets we were trying to capture in J 1 were in J, A Pi A 1 1 ' _A and bounded subsets of up, ; and up. is a cardinal in JV so we A f **

could get by with H=(H

0)P->

•»

A / **

A

) J X, the collection of sets in J^ whose

1 ,

A

A t r a n s i t i v e c l o s u r e i n JA .A i s of c a r d i n a l i t y l e s s than o)p1 A . in J , ; A, A A certainly will be amenable, where B is the Z n master code of On the other hand H is not a very tidy structure. It turns out, though, that H is J 1 , to which fine structural techniques can be P A,A applied. At least, they are equal subject to the acceptability constraint. For example, if H is to be included in J 1 i i cannot be more than o)p, _ bounded subsets of o)p. A,A

j.

A,A

there

Px

-& in J r A

We should

have no difficulties if GCH held in :r\ Here is another difficulty: to get J 1 cH we need not only a P A,A cardinality restriction (which is easy) but also we need to know that J B 1 c=j^. This may fail. (This is exercise 2 of chapter 3. Here P A,A~~ is a hint: suppose ^ 1 = ^ / 2 ° = X i b u t V*L. Let (x^za1) enumerate P(o)) and let A={< o) ,a,n >: n€x }.) Again GCH will prevent this. GCH can only be formulated if we have the power set axiom, which in general we certainly do not, so we formulate a weaker constraint, called acceptability:

P (Y) nj^ +1 £jA - vuepp (Y) n 4 + 1 J * + I h % " .

_

^

This implies a weak form of GCH: if P(y) is a set then P(y)=y ; otherwise if ucp(y) then uVy.Acceptability plainly fails in the case in the hint. (Actually as stated the acceptability property is not sufficiently uniform for the coding property claimed). "Generalised fine structure" is fine structure that applies to all acceptable J A , i.e. to all J, in which the acceptability axiom A A

XX holds. It is also generalised in that it applies to non well-founded structures. The reason for this latter extension should become apparent as we go along: in order to avoid excessive indexing we present the theory internally as an axiomatic development of sentences that say roughly "I am an acceptable J ". It is always easy to convert back to the terminology of this introduction when dealing with transitive models. Since our structures may have several added predicates, M=< M, €, A.. . . A N >, it is more convenient to write te p M than to list all the predicates by writing p, . ,A . JYl A , A , . . . A.Anyway , non transitive models of the axioms are not of the form J,l**" w so the other form would not work. Generalised fine-structure preserves the coding property of master codes - that if A^ is the £, master code of N then N can be coded in . But for N to be recovered exactly from the code P N another condition is necessary. Recall our counterexample in which P N =1 but N=N.. ; it would not be reasonable to expect A^ to code all of N. Since \

was defined to be {:i€w A X € H 1 A N^=4>. (x,p) } for Pvj

1

some p N will only be coded by A^ if every x in N is £^ definable from parameters in H lU{p}, that is, if N=hN"(o)x (a)pN*{p}) ) . Such N are called p-sound. All levels of the J

hierarchy are p-sound for

some p, but most of the structures that concern us are not. Given an acceptable N we may form its master code even if it is not p-sound and then decode the master code to obtain a p-sound structure N 1 .Apart from the fact that there is a l^ map of N 1 into N there is little to be said about the relation between N and N 1 in general. Provisionally N 1 will be called the core of N; later we shall give a rather different definition of this term. Although generalised fine structure has little to say about the relation of a structure to its core, the theory of iterated ultrapowers will reveal a very simple relation for the structures that we shall be examining. Unfortunately we encounter considerable difficulties when we try to iterate this construction to the projectum pJJ. The core in this case is defined differently. Another problem is that although we can form cores we cannot in general extend maps TT:-^ P ~ N ->M' to TT:N->M with M'=; unless N is p-sound for some p the usual construction breaks down. This does not matter for the structures considered in this book; it is a problem when more general structures - with lots of measurable cardinals, for example - are taken into account. Generalised fine structure does provide an extension of embedd ings lemma in this case but it involves a new hierarchy of formulae, called E* ; this would take us

xxi too far afield. The reader may ask whether it would not anyway be simpler to handle each case as it arises and not bother with a general theory. This raises the very general question of the necessity of fine structure. Recall Jensen's cited remark about the applicability of micro set theory: since those words were written most of the macro set theoretic results proved using fine structure have been proved without it. First there was the technique of "Silver machines", a slowed down construction of L that gave proofs of most of the combinatorial results proved using fine structure and of the covering lemma for L. Then Silver found an even simpler technique, needing nothing more than the notion of a £ substructure. This is the method used in Silver's proof of the covering lemma ( see [24]); it can certainly be generalised to the core model. There are three reasons why we have nevertheless used fine structure in this book. Firstly, like Jensen "we find such questions both interesting and important in their own right". The projectum is an excellent technical tool for describing various features of mice: and the simplification in not using fine structure is much less than in L. Solovay's crucial lemma - corollary 11.21, proved on the assumption u>p = K - is a fine-structural statement, but one with consequences that would have to be proved by some means or other. Secondly, and this is the most important point, we have indicated that the £

hierarchy is not susceptible to generalisation

to models that are not p-sound. As soon as we have to look at such models - for example in part six when we consider mice with many measures - we find that it is the properties of the projectum that generalise and the properties of the E n

hierarchy that are lost. »

In fact this hierarchy must be replaced by a new E*

hierarchy whose

definition requires the notion of projectum. It is strange that a technique that was invented to facilitate the analysis of E n formulae should turn out to be of more interest than the formulae themselves; but even a reading of the fine-structure of L soon reveals that the actual hierarchy of formulae is not very important provided we know that its union gives all formulae. Thirdly, and this point would carry no weight if the second one fell, fine structure has proved a most productive research tool whereas the various simplifications have not yielded any new results, although they have given more elegant proofs of old ones. Admittedly there could be all sorts of subjective explanations of this; and admittedly

it is difficult to make comparisons. We feel, though,

xxii that investigation of large core models can only really be tackled using the techniques of fine structure. If after the analysis is done it turns out that there is another way to produce the same results, so much the better. Had the aim of this book been simply to prove the covering lemma for K as briefly as possible we should have used Z

substructures. In summary, we feel that in a book

intended as an aid to research, scenic detours are permissible, and indeed desirable if the direct route conceals important structure. 2; ITERATED ULTRAPOWERS The question "what large cardinals are there?" is, although undecidable (unless there are none) surely a natural one. As soon as set theory is axiomatised by rules guaranteeing the existence of certain sets the question must arise whether all such rules have been included. In ZF set theory there are two ways of generating big sets: the power set axiom and replacement together with union. It is well known that from these two principles a regular strong limit cardinal that is uncountable cannot be obtained. We could consistently stop with these two operations - classical mathematics would be none the worse - but such a decision surely seems arbitrary. Not Lhat these strong inaccessibles obviously exist; but if caution vas to be exercised it should have been exercised a long way earlier. Anyone who is happy about unlimited application of the power set operation can feel few qualms about an inaccessible. And anyone who admits one strong inaccessible can scarcely complain about two. But there is presumably some limit to this process; for example, is there to be a strong inaccessible K that is the Kth inaccessible? A more natural, although stronger, axiom is that the set of regular cardinals below K is stationary: such a K is called Mahlo. This is a new principle for generating large cardinals; and it would be possible, for example, to ask whether the set of Mahlo cardinals less than some K is stationary in K. This process could go on indefinitely. It could be assumed consistently that it stopped, though. Not only is it impossible to prove the existence of 33 Mahlo cardinals from the existence of 32, but also it cannot be proved in ZFC+there are 32 Mahlo cardinals that (Consis ZFC+32 Mahlo cardinals) implies (Consis ZFC+33 Mahlo cardinals) (unless the former theory is inconsistent). Not only does the ad hominem argument have no technical force, it also does not take us very far in the direction we want to follow. Perhaps the feeling that it is unnatural to stop generating cardinals by some process before the process has been exhausted can

xxiii be made precise by looking at indescribability. This partly captures the sense that we must discontinue the large cardinal sequence at a point where more apparatus is needed for description of the property. 0

describes K provided that |=a but for all a
< V a ,€,unV a >^a, where UcV . K is A-indescribable provided that no a€A describes K. This classification has links with the old one; for example,K is strongly inaccessible if and only if K is £ -indescribable. 0)

Perhaps that is indeed a reason for restricting generating principles to those of ZF. But if we are determined to go further then we can still use indescribability provided we admit formulae with higher order quantifiers. An m+l-th order quantifier over X ranges over P m ( X ) . A formula with n alterations of m+lth order quantifiers beginning with a universal is said to be n m . On the present criterion if we are to admit any inaccessibles then we should not impose any restrictive criterion on cardinals weaker than II,-describability. Being II, indescribable turns out to be the same as being weakly compact, a very natural large cardinal property with many equivalent definitions. If we did not want to be restrictive at all we might investigate the existence of totally indescribable cardinals. For example, suppose j were an elementary embedding of V to itself, but not the identity. Then there is an ordinal K such that j|K=id|< but J ( K ) > K . Suppose a described K. Then ^=a and Va
terUf\V)^o.

Hence Va<j(K)
j(U)nVK=U. Contradiction.

>^a. But since j|«=id|K

So K is totally indescribable.

As far as the analysis using fine structure is concerned, embeddings are easier to deal with than properties of levels of the V hierarchy. It was noted in section 1 of this introduction that fine structure gives extension of embeddings properties for levels of the J

hierarchy. But the beautifully simple large cardinal

property mentioned - that there is a non-trivial elementary embedding of the universe into itself - is not consistent. There are weaker forms that are not known to be inconsistent, and one of them will concern us closely. A cardinal is said to be critical if it is the critical point of some elementary embedding of the universe into an inner model. This does not yield total indescribability, for if the inner model is called M the former contradiction turns into < V ^ G , j (UjDV^^a but |=a. Note that since K

K

—

j|V K =id|V K

then j (a) flV =a so V ., =Vi* , . Thus a cannot be

V =V*S indeed if acV K

K

K"T"J-

KTJ.

II for any n. Indeed if a were II, it would be downward absolute, so

xx iv 2 K is n,-indescribable. None of this discussion is intended to make the existence of these cardinals seem plausible; the cardinals that we are discussing will arise naturally in the study of inner models and covering lemmas. Unless they can be proved inconsistent they must be analysed however implausible they seem. But it does seem that critical cardinals are closely connected with other notions of large cardinality that arise naturally if not demonstrably. For " K critical" is equivalent to a property that had been invented long before indescribable cardinals came to be discussed. In measure theory a a-additive measure on a set S is a function y:P(S)+[O,l] such that (a) u(0)=O, y(S)=l; (b) XcY =» y(X)
(d) if X n c S

(n€o)) and n*m +> X^X^Q

03

then y ( U X R ) = E v U

n

).

The question was asked: could there be a set S with a a-additive measure y such that rng(y)={0,1}? Certainly not for S=R. In fact letting K be the least cardinal with such y it can be shown that K is critical, hence inaccessible; indeed a great deal more than inaccessible. It is trivially true that if X > K then A also carries a a-additive 2-valued measure. Suppose that (d) were replaced by (d) ' if X cS (a<3
if and only if y(X a )=l for some a<3;

call K measurable if K carries a measure y that satisfies and (d)' and

(a)-(c)

K is uncountable; then K is measurable if and only if

K is critical. The discovery that measurable cardinals are critical was made using the model-theoretic ultrapower technique. To take an ultrapower of the universe, first let V K ={f:f:K->V} where K is measurable. Rather than working directly with y it is customary to let U = { X £ P ( K ) : y ( X ) = l ) and call U a K-additive measure on P ( K ) . Let an equivalence relation ^ be defined on V K by f^g ~ U : f (£)=g(S)}€U and let M = V K A . Say fEg <* {£: f (£) €g(£) }€U and let E = E A . a -additivity implies that <M,E> is well-founded so we may take its Mostowski collapse IT:< M, € >=< M,E >. Let c

denote {< £,x >: £ € K } . Then if

j(x)=ir([c ]) j:V-* M with K critical. j:V+ M is a special case of which is Los 1 theorem; [f] denotes the ^-class of f. In fact we are not restricted to taking ultrapowers of the universe by U; we could take an ultrapower of any structure M. We usually insist at least that <M,U> be amenable. In this case we need only require that <M,U>^U is a a-additive measure. But there is one

d i f f i c u l t y : even i f M i s t r a n s i t i v e we cannot show t h a t i t s ultrapower i s well-founded. For there may be a sequence such that (a) {5:f jL+1 (C)'€f i (C)>€U all i€u>; (b) each f ± €M; but the sequence $M so that a-additivity cannot be applied. If U really were a a*additive measure this would follow, but in general we shall not be able to assume this. Hence our concern with non well-founded models in section 1. Also we can generally only prove j:M-* M 1 , if we are working with levels of some J hierarchy rather than models of ZFC. Our justification for fine structure was that E.. has nice properties that Z , does not; this is the nice property that concerns us particularly. There was a time when measurable cardinals seemed vast untameable beasts: all known results about them stated negative properties: they were bigger than such and such a cardinal. It was Silver's work on L[U] that began to make them respectable. If U is a normal measure on K (normality is a slight generalisation of K-additivity that it is helpful to assume) then urtL[U] is a normal measure on K in L [ U ] , But in L[U] GCH holds: the first really positive result about measurables that was known. (For a time it was hoped that large cardinals would settle GCH negatively; Silver's result showed that at least measurables do n o t ) . Silver transferred various comfortable results about L to L [ U ] , and Solovay showed that L[U] has a fine structure, of which more later. Our concern now is with the work of Kunen and Gaifman. Here is an obvious question: take the ultrapower of V by U, M say. Let U'=j(U) where j:V+ M is the ultrapower map. Could we repeat the process, taking the ultrapower of M by U'? It was with this in mind that we did not insist that U 1 be a normal measure in V, only in M 1 . We can repeat this any finite number of times. This gives models <<M n > n n < m < u ) > such that M Q =V, J n n + 1 is the ultrapower map of Mn to M n+1 ,. and jmnj. This is called a km=j. kn (k<m is amenable; (b) If some M is not well-founded then we must be able to take a an ultrapower of a non-well-founded structure. M is said to be iterable by U provided each M is well-founded.

xxv i V is always iterable by U. Kunen's paper ([35]) proved using these methods that if UflL[U] is a normal measure on K in L[U] then L[U] is an iterate of the unique L[V] with V a normal measure on the least ordinal that carries a normal measure in some inner model; also that UflL[U] is the only normal measure in L[U]. Not that there may not be two different normal measures U,V on K: but L[U]=L[V] and Uf1L[U]= VflL[U] . Iterated ultrapowers yield indiscernibles. Let <<

^a > a€On' a<6eon > b e t h e i t e r a t e d ultrapower of M. Let M ^ U U J where U i s a normal measure on K in L[U ] . Then for any formula c> f and a 1 <...
hierarchy (a

premouse) then the stated result holds only when (f> is £.. . However a combinatorial argument shows that if all of a... ..a ,3-,... 3 ,Q are limit ordinals then 4> may be E~. If they are limits of limits then (j) may be E». And so on. We shall see several applications of this idea. Suppose M is as before and 0 is regular. Then K =6. Furthermore given X<=0, X6M then X=7TaQ (X)

for some X and some a<9. By the

indiscernibility result K O € X <=> K D , ex (a<3<3'<9). If X€U Q then K €X so {K Q :a<3<e}cX . It follows that U ={Y€M:Yc0 A (3C)(C is ot p — — w — closed unbounded in 0 A CCZY)}. This is true irrespective of the choice of U. This is the heart of the proof that if U and V are normal measures on K in L[U],L[V] then Uf1L[V]=VnL[U] . If X is not in M but is l1 definable over M the proof still works. But if X is l^ definable then the limit point trick must be used. 3; MICE If Silver tamed measurable cardinals he also showed that from the point of view of L they are wild. Suppose there is a measurable cardinal. Then there is an embedding of V to M, j:V+ M, other than ^ M the identitity. So, since L =L, j:L+ L ( j is a class, but, since a non-trivial j:V-* V is impossible, cannot be a class of L; that is, there is no formula <J) such that for some p€L j(x)=y «* L|=(j> (x,y,p) ) . Hence V*L (so there are no measurable cardinals in L. This is the most indirect proof of the fact imaginable). Scott ([51]) had already proved this result directly. One might expect that the existence of a measurable cardinal would only affect the region of L above the critical point of j. So Rowbottom's result ([49]) - that if there is a measurable cardinal then P(w)flL is

xxv ii countable - is unexpected. How can j:L-»- L with so vast a critical point affect Although K may have been chosen least with j:V-> M with critical point K it may not be the least critical point of a j:L-> L. We shall show that if there is a j:L+ L which is non-trivial then there is such a j with countable

critical point. Since the critical point

of j must be an L-cardinal and J(O)..)=OD.. it is clear that u), L with K critical. Define U = { X C K : X € L A A K£j(X)}. Then U is a normal measure on K in L (it is not claimed that U6L). can be shown to be amenable; take a countable X-<. Let TT:<J ,U>=<X,UDX>. Let K Y

I<=IT""1(K). Then take the

iterated ultrapower of L by U: it can be shown that L is iterable by U so that the iterated ultrapower is of the form «L>

_

,< TT g >

Let K =71. (7). Then the class C = { K :a€On} will be indiscernibles for L. C will contain all uncountable cardinals and for every x in L there is a term t of i and there are Y T - - « Y L(=x=t(K

...K

such that

) . C are called the Silver indiscernibles for L.

Y Y l n Silver's construction was rather different: this proof was discovered by Kunen.

If we fix some enumeration {<(> :n€u)} of m(n)-place formulae we may let 0#={n:Lf=(|> (^..

.^ m ( n ) ) > • 0 # is a non-constructible real.

In fact every constructible real is recursive in 0 . u

The statement that 0

exists - which is equivalent to the

statement that there is a non-trivial j:L+ L -says that the universe is about as unlike L as could be imagined. For example, suppose K is a definable ordinal in L, say X = K <* L|=<j>(X) • If K>_N, then x=« x satisfies (VY)(<|>(Y) "• YiLx) '

so b

Y indiscernibility all

uncountable cardinals will satisfy this (in L ) . But that is impossible. Thus K must be countable; so X , (N^ ) , the smallest Mahlo cardinal in L and so on are all countable. u

This procedure can be generalised to any real a; a exists is equivalent to the statement that there is a non-trivial j:L[a]+ L[a] e # If a

exists then the universe is as unlike L[a] as can be imagined. In particular, define a sequence 0 n by 0°=0, 0 n

U)

= ( 0 n ) # . Let

m

0 ={< n,m>:n€O } : given a suitable coding of ordered pairs 0^ can be regarded as a real. This can go on for some while, depending on how suitable our suitable codings must be: certainly up to 0 a for any ot<(u),) . But plainly it cannot go right up to 0^1 which has instead to be coded as a subset of co, . This is not too complicated and the 0 sequence can be iterated right through the ordinals, giving a model L[A] where A={< $,a >: $€0 a } . If there is a non-trivial j :L[A]->eL[A] then (surprisingly?) there is one with a countable critical point,

xxviii so that the embedding can be coded by a real and we start all over again. Informally the process can be described as follows: take some model M. There cannot be a j:M-> M in M (other than the identity), but there may be one in the universe; if there is then add it to M and call the new model M 1 . Repeat indefinitely, or until a "rigid" model is obtained. This is all very well informally, but adding a proper class (and choosing which one to add) involves difficulties, and the coding we have discussed is needed to make the definition precise; even then the construction may break down long before we reach a rigid model. u

We saw that the existence of 0

was linked to the existence

of a certain normal measure on P(K)flL. Call this U: U is not in L for L must not contain a non-trivial j:L+ L, and anyway L has no normal measures. Nor will U be a normal measure in L [ U ] , for X€U => => X€L or K V X € L , whereas O # €L[U] so P (K) flL[U]*P (K) flL. U will be a normal measure in an intermediate structure, as we shall see in a moment. It turns out that there is a remarkable connection between the stages of this process of iteration by adding elementary embeddings and the fine structure of L [ U ] . This theory - invented by Solovay must now be summarised. Recall that in generalised fine structure the condensation lemma fails. Suppose K is measurable in L [ U ] , U a normal measure on K in L[U] and L[U] is acceptable. Take X-< J^ and let 7r:M=Jg|X; assume X is countable, M transitive. Suppose M=J-^. U will be a — -1 normal measure on K=TT (K) in M, but may not be a normal measure in L [ U ] . On the other hand M is an iterable premouse. Working in L[U] take the Kth iterate of M; this will be of the form jf, where — P F={X€L[U]nP(K):3C€L[U] C is closed unbounded in K A C C X } . But then F U FcU so Jg=Jg. If 3 were some simply definable ordinal - K + 1 say -

/s

P

P

JJ

then 3 would satisfy the same definition. Then J

. would be an

iterate of M. This adds a great deal to generalised structure. Recall our definition of the core of a structure; first code N.in the projectum and then uncode the projectum to get N 1 . We have shown that J

, is an iterate of its core. This result is of such

importance that it is worth establishing a more general condition for

it than 3 = K + 1 .

This condition is wp £ K . First of all, since the condensation U lemma holds for X -<JR with K C X we deduce as in ordinary fineU 1 structure that for some p N=J o =h_U" (cox (Kx{p}) ) v/hen IOD < K . If J P o N— 1 U 1 o)p = K the result is proved, for then Jg=h " (ux (ojp x{p} ) ) . Suppose

xxix then that wpjJj
Tr:N'=N|x. N' is an

As above M ' = J A for some 3. Suppose 3<8. Then firstly A is I ~~

1

-1

definable over N 1 in parameter TT

(p) , since a)p.TcX. But secondly 1

~~

1

-

1

let a:N'-* M 1 be the iteration map. Then since wp < K a>pM< IT (K) which is the critical point of a, so a|u)p =id|a)p . Hence A is E,u -1 u definable over j£ in oi\ (p) . So A€J Q , which is a contradiction. —P 1 ^ 1 U ^ A similar argument - using P (wpN) flN1 =P (wpj^) flJg - prevents 3>3. So 3=3. Thus N is the Kth iterate of N 1 : and N I =h N V (o)x (oop^xfir"1 (p) }) ) . This N 1 is called the core of N. And the indiscernibles C

given by

the iteration of N 1 to N turn out to be I^-definable over N. By the limit point trick they may be thinned to get E^-indiscernibles C for all n: and each of these sets will be in U

J o , . Indeed each will 3 1 U

be in Z^(Jg): yet any set X that is ^-definable over J^ can be "measured" by C n , because X€U «* O Y < K ) C n ^ Y E x -

Jt

follows that

UHJO ,, =FflJD ,, and that if we just close Jo under functions P+l P+l rj P rudimentary in UflJft the result is nevertheless closed under U functions rudimentary in U, so that it must be J R . . Hence P(K)f1J^1=P(K)nE ( J Q ) - a result that is also true in the fine p+l

(A)

p

structure of L, although it is not true in general fine structure. Obviously this is not going to work for all 3/ irrespective of whether or not cap_Uvn + 1 N given by generalised fine-structure with n ~ IT | o) p =id | a) p . Let M 1 be the Kth iterate of N 1 with iteration map a. Then M1 will be of the form Jg. And 3£3. But when we try to get a contradiction from 3<3 we encounter the fact that although a:N'+ M' L> i

we do not know that a:N'->

M 1 . And it is not difficult to obtain

n+1 an example where 3 is actually less than 3. In this case we replace the ordinary iteration map by a finestructural one. Suppose n>l is least with o)pJJ_K. Let N'=J

n-1 where A is the n-lth

master code of N. Then N 1 is a premouse, so we may take its ultrapower by U, r):N'"* M 1 and then use the extension of embeddings lemma to get f\^r\, fj:N->^

M. This process may of course be iterated:

M is called an n-iterate of N. Now the gap in the theory may be filled by replacing "iterate" by "n-iterate" throughout. J g is an niterate of its core, where n is least with top^u < K . 0-iterate means

iterate.

If

for

t h i s n o)prj51
(0

p

P+l

.

Now consider the other possible cases. Of course if a
and if ct>K 01

,

then P(K)flL[U]cj

"""

"""' ^

so nothing interesting TT

unj

happens.

T7 J

as

tner

If KK, all n, then e are no new a+iE a U U subsets of K and E ^ C r ) f)P(K) =J + l n P (K) . The remaining possibility is that 03pIJu=K for some n. This case must occur - indeed it occurs when a=K - anS gives us the "unpredictable" bits of the measure. In other words we do not expect Z (J )PIP(K)=J

+,nP(K).

This again puts in

doubt the suitability of the projectum as an index of formation of new subsets, but it can be shown in this case that for y0. The essential property of the projectum - "if there is a new subset of y then mpn
for some n" - is preserved. Even so there will be a lot of

undefinable new subsets of K. Solovay used this theory to show that various combinatorial results from L hold in L[U] (L[U] is always acceptable, by a result of Silver). Later we shall use them to get a covering lemma. Our u

present interest is in the sharp series. 0

was equivalent to a

certain normal measure U; we looked at the amenable structure < J K + , U > . Now consider J^ +1 # ={X:K€J(X)}

with U the measure given by 0 # . UflL=

as usual. Since J

, is the closure of J

functions rudimentary in U and <J +,U> is amenable J

under

+1

EL

+#

Clearly

equality does not hold. On the other hand some iterate N of J K+ q will contain L + and P ( K ) flJU . =P (K) PIN: so P (K) 0L=P (K) flJU , . »-.

K

K i x

„ K"T"-L

J

, is an easier structure to handle than <J +,U>. 0 is £-K U U definable over J ,,, the iteration indiscernibles of J ,, are #• % U exactly the Silver indiscernibles for L, and 0 fL so O $J + , . Hence p_U =1. K+1 # In other words the existence of O

gives an iterable premouse

N with P N =1. But also any iterable premouse generates an embedding of L to L that is not the identity. (The requirement of iterability is not redundant. Although it is a theorem of ZFC that if U is a normal measure then iterated ultrapowers by U are well-founded, it is not necessarily the case that J + , is a model of ZFC; indeed it is not.) J

+,

is a simple example. We may transplant the whole of

Solovay 1 s analysis of U that are normal measures in L[U] to the analysis of structures N=J

that are acceptable premice with

u)pj*
xxx i We had to say "acceptable" because although L[U] is acceptable we do not know that arbitrary premice are; indeed they are not. But we shall see that iterable premice are always acceptable; the proof puts together all of the fine-structural results that we have discussed. 4; THE CORE MODEL In a moment we shall return to the sharp series. First we need a lemma extracted from the proof that if J g + ^ is an iterable premouse, U a normal measure on K in Jo,n and prJu=K then for all P+l Jo JJ Y is p

P+l

—K

K

Jo p

a model of ZFC. This is not an inner model, but we can get such M K U of arbitrary length by iterating Jg + 1 » Then M=UM It is called K_U

is an inner model.

. If 3 = K then it is just L. Generally

P(K)HM=

3+1

The interesting relation with the sharp sequence is this: if there is an inner model with a measurable cardinal and we enumerate all 3 with K < $ < K

with u)pn=K, some n, and consturct K_U then we 6 J S+1

find that we have enumerated precisely the sharp sequence of models. The sharp sequence was a bit vague, but this is quite precise. An equivalent formulation of K U exists: K U is the union J J e+iu e+i 0 of mice which are constructible from J D l l but from which Jo,, is P+.L

[5+1

riot constructible, together with L in case there are no mice. (This is more general, K being definable for a large class of mice M ) . M So K U is the union of mice in K U (with L ) . We can turn this d J 3+l 3+l into a completely general definition: the core model is the union of all mice together with L. In the text a different definition, which makes K)=ZFC immediate, is used. Also K|=GCH. K._ was a definite model but K depends on the size of the universe. For example if 0

does not exist then K=L: if 0

exists

but 0 # ^ does not then K=L[O # ], The possible values of K are the sharp sequence of models, up to a point. Suppose that there is an inner model with a measurable cardinal. Then K has a maximal form: N that is, if N is an inner model with a measurable cardinal then K =K. So K is not K,, for any mouse M in this case. In fact K arises from M the inner model with the measurable just as K arose from M: T rn 1 K= U H L iJ where L[U.] is the ith iterate of L[U] and U. is a i
K

1

i

1

normal measure on K. in L[U.](recall that if U is taken at a minimal point then all inner models L[V] with V a normal measure in L[V] are of the form L[U.]). Alternatively K= PI L[U.]. 1

i
x

xxx ii Suppose that j:K-> K is non-trivial. What happens when we try to extend the sharp sequence by adding j to K? As usual j defines a normal measure U on P ( K ) D K , where K is the critical point of j, such that XGU «* KGj(X). Form L[U], If a £ ( K + ) K then J^cK, so U is a normal measure in J, +.K. But suppose a>K and a)pjU=K: then J would be a mouse (this is a gross over-simplification; n-iterability requires a lot of work) but not in K, and this is impossible. Therefore P(K)nL[U]c:K, so U is a normal measure in L[U]. In other words, if there is a non-trivial j:K+ K then there is an inner model with a measurable cardinal. It needs to be added that the unsimplified proof will give a normal measure not on K but on some larger point. u

Just as the core model contains 0 and its immediate generalisations, so other large cardinal axioms weaker than measurable hold in K provided that they hold in V. For example partition cardinals K-*-(CX) and Ramsey cardinals (these are defined in the text), as well as those (such as weakly compact) that relativise to L. 5: THE COVERING LEMMA Sections 3 and 4 broke off the historical development in order to discuss the core model in its own right: but the core model was developed in order to prove the covering lemma over K; which in turn was motivated by consideration of the singular cardinal hypothesis. At the risk of covering very familiar ground, we discuss briefly the history of GCH. Since finite sums and products of cardinals are trivially calculated it must have been infuriating for early set theorists that exponents of 2 could not be found - not even the simplest, 2 °. And since it was known that 2 a>N for all a 2 a=N was the simplest possible answer. Skolem seems to have been the first to advocate the view that there would be no answer on the basis of the ZFC axioms, and this was proved by Gc-del([18]) and Cohen([4]). Gfldel, as we have seen, proved GCH in L: Cohen's proof used - indeed, inaugurated - the method of forcing. Cohen gave a model in which 2 °=N 2 : Solovay([56]) showed (a) that 2 ° can be "anything it ought", that is, any K such that c f ( K ) * « 0 ; (b) that a similar result can be proved for any regular K : cf(2 K )>K is essential by Kflnig's inequality. Then Easton ([13]) showed that given a class function G satisfying (i) ct£3 => G(a)£G(B) ; (ii) c f ( « G ( a ) ) > K a which is absolute for ZFC models there is a model of ZFC in which,

xxxiii for regular N , 2 a=K , • the values of 2 a for singular N are ^ a G(a) ^ a determined by the values for regular K in Easton's model. Thus the matter stood for some time (Easton obtained his results in 1963-64) and the general view was that some forcing construction would be found that removed the regularity restriction from Easton's result. This was wrong, though, for Silver proved ([55]) that provided cf(K Q ) is uncountable and N Q is singular, then Va<$(2 a=N

) => 2 3=*-

. Actually his result is much broader than

this, but it is the fact that there is any restriction at all that is surprising. The proof of [55] rests heavily on the uncountable cofinality assumption: for example, the question whether 2 n=N ~

K

for all finite n and 2 UJ=N

, n-i j.

_ are possible in the same model

remained open. Jensen's "Marginalia to a Theorem of Silver" extended these u

results. On the assumption that 0

does not exist Jensen firstly

showed that for N o singular of uncountable cofinality a very general rule for the determination of 2 3, the singular cardinal hypothesis, holds, and also that V is L-like in various other ways: for example if N Q is singular of uncountable cofinality then N o is sinaular in P

P

L. Then (Marginalia II) he removed the restriction that cf(N Q ) be uncountable: so, provided that 0 does not exist, Vn€a)(2 n=N ,1 ) «• s>

n+J-

=*• 2 o)=N

. . Finally (Marginalia III) he adapted his earlier proofs

to get the covering lemma: if XcOn is uncountable then there is Y€L with XcY and Y=X, still on condition that 0 # does not exist (if u

0

exists the result fails; it also may fail for countable X ) . In

[8] these are all written up with many simplifications. It will be helpful to summarise the proof, although it is given in full in the text. Perhaps first of all one extra remark about the subsequent literature is needed. There have been many further simplifications to the proof of [8]. At the end of section 1 we said that most of the results proved using fine structure have later been proved without it; this is a case in point. Silver machines gave a short proof of the covering lemma, and subsequently Silver found an elementary proof that used only the idea of £

substructures. We

shall not repeat our reasons for nevertheless using fine structure; we have incorporated Silver's other simplifications. Now for the summary. We are to suppose that for all y
with Y=X and XcY. We cannot quite

prove Y€L; some modification is needed. Let TT:J—=J | Y , then. The main lemma needed states, roughly,

xxx iv :J

J

that provided T is a cardinal in J~ there will be a map ^ g"*"2i B ' with TTSTT for some 3*. Essentially this is done by coding Jg as a direct system: for every y<3 and 6<7 let X .=h_ " (u>x6) . Let a .rN ~=J |x . where N x is transitive. Then by the condensation Y6 _ _ y 6 —y 6 y ' y 6 lemma N ~=J

/Say: and because T is a cardinal in J g a

(T

<

T . Then

if we define < y,6 ><: < y1 ,6' > ~ Y a =a Y'6' a Y6 t h e n a Y 6 Y • 6 • : H Y 6 ^ o ^Y '6' a n d a y 6 y 6 ' € J 3 " Y 6Yl6' Let I=3 x x; then <J D ,.^x> is the direct limit of let

p

< < N i > i e i / < a ± . > ±<

1

lfc-L

. >. Let N*=Tr(N±) , a* =Tr(ai .) and let < M,< a* > ± € I > be

the direct limit^of < < N* >i€I/< a* . >i<

. >. Is M well-founded? This is

the most difficult technical point of the proof, and requires careful choice of Y; let us assume that Y has been chosen so that M is well-founded. Let M = J O I / then, and let TT:JO-* ~

^

~

P 2.1

J O I be defined so P

that <7fa.=a*7T for all i€I. So TT^71"* Now suppose that T is a cardinal in L. Then the above argument gives 7rz>7r, TT:L+

L: that is, 0

exists. So we may suppose that T is

i-i

not a cardinal in L. Let 3+1 be least with some fry^T, y<x in JQ ,, , n ^ f a surjection. So x is a cardinal in J o . For some n o)pQ<x, as _ P P there is a new well-order of some y<x in Jo,,. Take n least such. n—1 — ~ P+JLet p=p o (so oap>x) and let TT:J •>„ J , extend TT as usual. Let P — ^p h\ p ^:JB"*£ J 8 ' lDe t^ie e x t e n s i o n °f ^ given by the extension of embeddings lemma. Now we may prove the result. There is p such that J = =h" (u)x(a)pnx{p}) ) , where h is the E, Skolem function of <J ,A n >. v> 1 -, ^ P 3 Let h 1 be the Z, Skolem function of <J ,,A DI

>; let p=sup irHo)p .

Then p*<x since a)pg<x ; let Y'=h' " (cox ('px-Cif (p) }) ) ; so Y'€L. Moreover YcY 1 ,

for Y=rng(7r) and Y'3rng(Tf). Finally, in L, Y'<x. So there is

fGL such that fry-^-Y1 with y<x.Let X'=f"" "X: since X'cy there is Z3X1

with Zcy, I=X and Z€L. Let Z'=f"Z. Z 1 is the required cover for

X and the covering lemma for L is proved. Now let us see what this implies about singular cardinals. A sample application shows that if 3 is a singular cardinal then 3 is singular in L. For suppose X is cofinal in 3 and X<3. Then there is Y€L with XcYc3 and Y=X+« 1 : hence since 3>X and 3>« 1 , Lf=3>Y. So Lf=3 singular. It follows also that if 3 is a singular cardinal then ( 3 + ) L = 3 + : all provided 0 # does not exist. As for cardinal arithmetic, the following can be shown: if there is y<3 such that for all cardinals 6 with y<6<3 2 Y =2

then

2 = 2 Y (this does not use the covering lemma: it is a theorem of ZFC due to Bukovsky and Hechler). Otherwise 2°=(sup 2 Y ) + (Here 3 is a # Y<3 6

XXXV all exponential calculations are fixed by the values of 2a for regular a. Given any infinite cardinals K,A KX=2X

=K

=K

if

2X^K; X

if 2 < K and

X
otherwise.

Jech([25]) gives a brief formula that implies all of this: the singular cardinal hypothesis (SCH) is the claim that for all singular cardinals 3, if 2 c f ( B ) < 3 then 3 ° f ( 6 ) = 3 + . A simple lemma enables us to convert these results into impossibility results about forcing. If V is a generic extension of L then 0

does not exist. So we cannot by forcing over L get a model

in which SCH fails. We cannot even add a generic subset of a singular cardinal 3 without adding a subset of some smaller y. For suppose Xc=3, x^L but for all y<3 P(y)cL. Let f:3+ U P(y) be a y<3= constructible bijection. Let C be cofinal in 3 with C<3 and let 1 X'={f~ (xny):y€C}. Then X'<3 so there is Y €L with Y^X1 and Y=X'+N,. j

J-

Hence L|=Y<3. Let g€L map Y one-one onto y=(Y) . Then g"X'c:y, so g"X'eL. Hence X'€L, so X=

U f(y)€L. y€X' It was already known that by forcing over a model with a

measurable cardinal one could add a cofinal subset of the measurable of order-type OJ without adding any new bounded subsets of the measurable. The covering lemma can only be generalised by changing the u

model L: none of the conditions can be weakened. If a

does not

exist (acoo) then the covering lemma holds over L [ a ] . This helps us to get a little way along the sharp sequence. Since we have just seen that the lemma may fail over L[U] in a generic extension (U a normal measure in L[U]) there must be doubts as to how far we can go beyond this. Hence the core model: the furthest point we can reach in the sharp sequence without getting an inner model with a measurable cardinal. The original non-fine-structural motivation for the core model was that although the covering lemma fails over L[U] in a generic extension, yet we could iterate the measure out of the universe: K= n L[U.]. The term core model refers to this i
-1

characterisation of K as the core of the L[U.] that is left when all the U. are removed. This characterisation only works, though, when there is an inner model with a measurable cardinal. The general definition gets around the problem. From the proof of the covering lemma for L it can be seen that failure of covering gives an embedding of the relevant model to itself non-trivially and hence advances one stage up the sharp

xxxvi sequence; but the core model already contains the whole sharp sequence; so the only possibility is that the coding of the sequence by the core model has reached its end, i.e. that there is an inner model with a measurable cardinal. So we hope to prove the following: if there is no inner model with a measurable cardinal and X is an uncountable set of ordinals then there is Y€K with XcY and X=Y. There is one snag. At a crucial point the covering lemma for L used the fact that there is p such that J D =h_ " (cox (ojpDx{p}) ) . This P

Jo

P

"

P u fails in generalised fine structure. If we know that J o is a mouse P

- call it N and assume for simplicity that U is normal on K with ojp < K - then all we know is that there are indiscernibles C for N such that N=h " (OJX (cop x{p}xC) ) . We can always ensure, in the cases that concern us, that the order-type of C is less than or equal to oo. If it is less then C is finite, hence in N, and it may be added onto the parameter. But if C is infinite then a whole new construction is needed, giving directly an inner model with a measurable cardinal. We can deduce that if there is no inner model with a measurable cardinal then SCH holds. Surprisingly we can go further: although one may generically add C of order-type OJ cofinal in a measurable to L[U] (a "Prikry sequence") that is, in a sense, all the generic mischief that one can do. For it follows from the proof of the covering lemma over K that either the covering lemma holds with L[U] in place of L (where L[U] is the inner model with U a normal measure on the minimal point K) or there is a Prikry sequence for U, C, such that the covering lemma holds with L[C] in place of L, or there is a non-trivial j:L[U]-»- L[U] with critical point above K. This last possibility has an O -type equivalent, called 0 . O

does

not exist in any generic extension of L [ U ] . So despite initial appearances SCH holds provided that O

does not exist.

6: BEYOND # We identified 0 V in J

V with the mouse J #

,. Similarly we may identify a

or indeed a

,, where V i s normal on K aV with an a-mouse J

, with acoi:

for any acOn may be identified with Ja,-, provided a<=K.

So in particular O it than that: J

can be identified with U . But there is more to

, is a double premouse and could be iterated by U

as well as by V. The theory of these "double mice" is very similar to that of single mice and one gets a double core model K . For this we insist in all double mice that the projectum p n drops to the point K on which U is a normal measure, for some n. If there is no non-trivial j:K -»• K

then the covering lemma holds over K . If there

xxxvii ijs such j , with critical point K say, then as usual we get a normal measure U on P(X)PIK

for some \>K which is normal in L[U] (subject to

the usual technicalities). That is not very thrilling: K + is full of such models. To get anywhere we must add U to K

by taking K

as the

union of all double mice whose bottom measure is U. There is some difficulty in deciding which U to add; we ignore this. See if the covering lemma holds over K , or some Prikry generic extension of K. If not then there is an embedding of K U j:KU->eKU whose critical point is above K.This gives an inner model with two measures L[U,V]. Does the covering lemma hold over L[U,V] adding, if necessary, Prikry sequences to U or V or both? If not then there is j:L[U,V]+eL[U,V] with critical point high up, and this gives 0 f t , as we may call it. All of the models K + , K U , L[U,V] and the generic extensions satisfy GCH, so from the failure of SCH we may deduce that 0

exists. And so on for any finite number of t, the

provisions about Prikry sequences and the like getting more and more complicated. For a limit number of

f we have to be careful. When we are

looking at mice with u) measures it no longer suffices, if we are to generalise the proof from L, that each set of indiscernibles be finite; their union must be finite, that is, the mouse must be a finite iteration of its core. Although elaborate coding takes one a little further at the time of writing it does not go beyond the first measurable that is a limit of a sequence of measurables whose ordertype is inaccessible. At what stage is there such a riot of Prikry sequences that SCH fails? To get some idea about this, go to the other extreme of the universe, the very large cardinal axioms. A cardinal K is o

3-supercompact provided there is j:V-> M with K critical and M cM, i.e. M is closed under 3-sequences. K is supercompact provided it is 3-supercompact for all 3. Clearly supercompact cardinals are measurable: in fact the first supercompact (if there is one) is much greater than the first measurable. Silver showed (see Menas [41]) that if there is a supercompact cardinal K then there is a K + generic extension in which K is measurable and 2 >K . By adding a Prikry sequence to this model we may make cf(K)=co but preserve all bounded subsets of K and all cardinalities. Hence 2 K > K . On the other hand since K measurable =» K a strong limit 2 ^ < K for all y(sup 2 Y ) + . On the other hand GCH yK . : using an even larger cardinal he was able to replace 2 n
. ([38]). By Silver's theorem N

by

could not be replaced by

xxxviii N

. Some scrappy details are in appendix I. After Cohen's result on the independance of GCH appeared there

were many who hoped that large cardinal axioms would nevertheless settle the problem. The idea was that large cardinal axioms should be taken as true whenever they could (as far as was known) consistently be assumed: the reason for this being a heuristic assumption that set theorists should aim for maximum "richness" of sets. (Similar arguments may be used about forcing extensions, to justify, for example, the conclusion that "really" v is not L. This gave no clue about GCH, though, for just as it may be destroyed in a generic extension so it may be restored by o n e ) . In fact large cardinals have little effect on GCH: Scott showed that GCH cannot fail for the first time at a measurable cardinal, but no other very significant restriction on regular powers of 2 ever emerged. It is anyway hard to believe that anyone would have been persuaded to change their views about the power of the continuum by anything as recherche as a supercompact cardinal. The interesting point is that SCH, a weak form of GCH, is heavily influenced by large cardinal assumptions. For (i) Consis(ZFC+there is a supercompact) =*• Consis(ZFC+-SCH); (ii) Consis(ZFC+-SCH) => Consis(ZFC+there is a measurable). There is a wide gap between measurables and supercompacts; in part six of this book we try to set up some markers in it. What we should like to prove is that -SCH is equiconsistent with some large cardinal axiom: that is, to obtain axiom X such that Consis (ZFC+there is an X cardinal) <=» Consis(ZFC+-SCH). (We cannot hope to get there is an X cardinal ** -SCH). There is no concealing the fact that we are very far from getting such an X.

PART ONE: FINE STRUCTURE

The analysis of part one is a general theory of relative constructibility subject to the "acceptability" constraint in definition 3.1. Although it is called fine structure, really chapters 1 and 2 are coarse structure; fine structure begins when we define (definition 3.5) the projectum. In chapter one we consider a theory equivalent to "I am rud closed". The main technical result of the chapter is a finite axiomatisation of R. Our treatment is purely internal, all theorems being proved in R, with liberal use of classes. This would be a paradoxical way of developing the subject were it not that straightforward accounts already exist; our approach avoids excessive indexing and gives a few neat proofs later on.

^

Chapter 2 strengthens R to R , whose transitive models are J . R

enables us in addition to do various inductive arguments that

failed in R; division by co for example. The axiom of choice holds in R

as well. Throughout the chapter the reader should remember that

the intended models are the transitive ones. Chapter three contains the heart of fine structure theory. Again we strengthen the theory, this time to RA, by adding a form of GCH. In part three of the book we shall verify that the theory RA holds in the structures that make up the core model. In RA the central constructions are the master code A™ and the projectum p. Chapter four moves into ZFC so as to use the Mostowski collapsing lemma. It also tidies up terminology for the rest of the book and contains the "extension of embeddings" lemmas. The attraction of this general theory is that when we come in part six to generalised core models we do not have to change the fine structure, although the upward extension of embeddings lemma needs substantial generalisation. This is a modular fine structure theory that can be plugged in to any acceptable context. On the other hand this makes the going rather hard for a reader with no knowledge of fine structure. It would be better for a first approach to read [26] up to section 4 and then to return to this part. The reader who is familiar with fine structure may prefer to use part one for reference and to go straight to part two.

1:

A WEAK S E T THEORY

Let L be the language L augmented by N unary predicate symbols

Definition 1.1. The theory R is the deductive closure of the following axioms: (i)

VxVy(x=y

(ii)

«*

Vz (z€x«*z€y) )

Vx(x*0=*3y (y€xAxOy=0) )

(iii)

(extensionality) (foundation)

VxVy3zVt(t€z~t=xvt=y)

(iv) Vx3yVt(t€y*>3z (z€xAt€z) ) (v) where

VuVxVw3yVz(z€y~3t€w

t h eformula

z={s€u: ( s , t , x ) } )

<J> i s £ _ .

The theory is usually referred

(pairing) (union)

(I.- closure)

to simply as R.

The development of chapters 1-3 takes place in the theory R or in strenghthened forms of R. Lemma 1.2. iffy is ZQ then VuVx3yVt(t€y~t€uA<j>(tfx) ) Proof: By (v). there i s a s e t y1 such t h a t Vzfzey'^teu

(1.1)

z={s€u: (J>(s,x)}).

(1.2)

Let y= Uy';then y s a t i s f i e s Vt (t€y«*t€uA ( t , x ) )

(1.3)

so (1.1) holds.

D

The main r e s u l t of chapter 1 i s t h a t the theory R i s f i n i t e l y axiomatisable. Definition

1.3

The functions

F

F 1--- 17+

F1(x,y)={x,y} F2(x,y)=Ux F3(x,y)=x^y F4 (x,y)= xxy F5(x,y)=dom(x) F6(x,y)={x"{z}:z€y} F?(xfy)={< u,v,w>:

u€x A 6y}

F g ( x , y ) = { < u , w , v >:< u f v >€x A w€y} F 9 ( x , y ) = { < u , v >€xxy:u=v}

are

defined by

F^ (x,y)={€xxy: u€v} F 1 1 (x,y)=< x , y > F , 2 (x,y) =< x , v , w > i f y=; 0 o t h e r w i s e F, ~ (x,y) =< u , y , v > i f x=< u , v >; 0 o t h e r w i s e F, , (x,y) ={< x , v >,w} i f y=; 0 o t h e r w i s e F 1 5 (x,y) ={< u , y >,v} i f x=< u , v >; 0 o t h e r w i s e F 1 6 ( x , y ) = { < x f y >} F 1 7 ( x , y ) = { t : < y , t >€x}

F. are called basis

Lemma 1 . 4 .

functions.

F. is a total

function

whenever 0
j.

Proof: (1) By axiom

-

(iii).

(2) By axiom (iv). (3) x^y={s€x:s€y}; use lemma 1.2. (4) Let X={{u,v}: uexUyAv€xUy}. X exists by (iv) and ( v ) . Then xxy={{s€X:s€}: u€xAv€y} which is a set by ( v ) . (5) dom(x)={u€UUx: (3v£UUx)< u,v >€x}. The rest are left to the reader.

o

The definition of each F. can be written as a E

formula

of

the language L ; that is, there is a E_ formula ^ such that F.(x,y)=z <* (J>.(x,y,z). (1.4) N Let R 1 (or, to be precise, R 1 ) be the deductive closure of (i) and (ii) together with the axioms (vi).: ( v i ) ± VxVy3z(j)i(x,y,z)

(1.5)

for 0. To this end we prove: Lemma 1.5.

Suppose
formula of L . There is a function F, N (p

which is a composition

of F ...F such that 1 7f~N 1 R'f F^(a 1 ...a n )={<x 1 ...x n >ea 1 x...xa n :4,(x 1 ...x n )}

Proof: For convenience we call a formula which is Z

(1.6) orderly

provided that whenever the variable v. occurs free in the scope of a quantifier Vv. or 3v. then i<j. There is no loss of generality in proving (1.6) for orderly <j) since every formula is equivalent ( in predicate calculus) to an orderly one. Our first aim is to dispose of dummy variables: recall our convention that although all the free variables of <J)(x, ...x ) must lie among X.....X , not all of x,...x need be free in <|>(x^...x ) .

4 (1) If (p(x1...xn) is ij;(x1. . .x n _ x ) and F. satisfies (1.6) then there is F. satisfying (1.6). Proof: F ( J ) (a 1 ...a n )=F^(a 1 ...a n _ 1 )xa n =F

4 ( V a l'-- a n-l ) ' a n ) -

°(1)

(2) If
satisfying

(1.6).

Proof: F A (a n ...a )

Hence by induction: (3) If F

satisfying

(1.6).

Next the propositional connectives: (4) If F

satisfies (1.6) then there is F . satisfying

(1.6).

Proof:

F . ( a , . . . a )= ( a , x . . . x a ) \ F . ( a 1 . . . a ) a(4) - c p l n 1 n 9 l n (5) If <|>(x1...x ) and ij; (*-.• • - x n ) each have F.,F, satisfying

(1.6)

then, letting X ( X T » - - X )H>(x,...x ) AIJJ (x-.. • » x ) there is F satisfying Proof:

(1.6).

F x (a r ..a n )=F ( ) ) (a 1 ...a n )nF^(a 1 ...a n ) =F(j)(a1...an)MF(f(a1...an)^ya1...an))

D(5)

The next two claims use F^ and F g to manipulate ordered sequences. (6) If i|;(x, ...x ) has F. satisfying (1.6) and (p (x, . . « x n + 1 ) results from \p by replacing each free occurrence is F

satisfying

of x

by x + . then there

(1.6).

Proof: If n=l then (p (x 1 ,x 2 ) «*i|; (x 2 ) so F . (a 1 ,a 2 ) =a 1 xF . (a 2 ) . Otherwise (7) If ij;(x, ,x 2 ) has F, satisfying (1.6) and
(n>2) then there is F.

(1.6).

Proof: We may assume n>2. Then %(al---an)=F7(alx"-xan-2'%(an-l'an))We turn to the atomic formulae. (8) There is F v _

satisfying

°(7)

(1.6).

Xi —X2

Note: It no longer matters whether * 1 = x 2 is cp(x1,x2) or 2 f by (3). Proof:F is F q . D(8) (9) There is F satisfying (1.6). n n+1 Proof: By (8) and (7) o(9) Ivpw an induction starting from (9) and using (6) yields: (10) For all m,n there is F

_ satisfying (1.6) . m~Xn Proof: If mn use F x

X

If n=m then F

x —x

n

(a,...a )=a,x...xa . m

-L

n

J.

n

a(10)

_ n"xm

Using F._ we can establish by similar arguments (11) For mn. Say LO), (] By (10), (11) and (5), together with (3), there is F, satisfying (1.6). Let

(13) F o r i > 0 , i
satisfying

(1.6).

i

Proof: Suppose <j>(x,...x )<=>A.(x.). Then F

Ai(al---an)=alX---Xaj-lXF17+i(aj'aj)xaj+lx---xan-

D

<13>

Hence (14) If <(>(x1...x ) is atomic then there is F. satisfying (1.6) It remains only to consider bounded quantifiers. We may of course restrict ourselves to the existential quantifier. (15) If I|J(X-I . . .x, ) n

a sF

i satisfying (1.6) and <j>(xn...x )

3x, €x . ip (x, . . .x, ) , where k>n,then there is F^ satisfying

is (1.6).

Note: The assumption that k>n is justified by the fact that <j> is orderly. Proof: Let 0(x....x, ) be x,€x.. So there is F Q , satisfying 1

K.

K.

"2

(1.6);

v Alp

and F~ , ( a , . . . a , , , U a . ) = { ( x 1 . . . X i >:x, € x . , x . £ a . 0Aip 1 k-l j 1 K k j i i

(l
ty ( X , . . . X, ) } .

Then l e t F. (an . . . a )=dom " n ( F Q . (a, . . . a ,a,...a,,Ua (pi n oAip 1 n 1 1 Lemma 1.5 i s p r o v e d . Lemma 1 . 6 . if
then there

.)) j

is F, a composition

a (15) • of F . I

F ( a , x , . . , x . _ w X . + , . . . x ) ={x . €a:cp ( x , . . . x ) }

(1.7)

Proof: Let F, be as in lemma 1.5. Then n-i * dom11 X ( F , ( { X j } . . . { x . _ 1 } / a , { x i + 1 } . . .{x n >) =

Applying r n g ( r n g ( x ) = F 2 ( F g ( x , d o m ( x ) ) , x ) ) Lemma 1 . 7 .

gives the required s e t . n

If l
of basis functions

F* which is a composition

such that F*(u,v)=F ."u>
Proof: Suppose there were a function G. such that G. (u,v)={<<x,y)/t ):x€uAy€vAt€F . (x,y) } . Then F*(u,v)=F c (G.(u,v),uxv). So it suffices to show that G.is a

composition of basis functions. (1){< < x,y>,t>:x£uAyevAt€{x,y}}=((uxv)x u )U((ux v )x v ) (2){<<x,y>,t >:xGuAy€vAtGUx} = {< <x,y >,t >€ (uxv) xuUu: t€Ux} Since t€Ux is I Q the result follows from lemma 1.6.We shall not repeat this remark with the other clauses. (3)

{<< x , y > , t > : x € u A y € V A t € x \ y } = {< < x , y > , t >€ ( u x v ) x u u : t € x ^ y }

(4)

{< < x , y > , t > : x £ u A y € v A t € x x y } = {< < x , y > , t >€ ( u x v ) x ( UuxUv) : t £ x x y }

(5)

{< < x , y > , t > : x € u A y e v A t € d o m ( x ) } = {< < x , y > , t >€ ( u x v ) xiJUUu: t € d o m ( x ) }

(6)

{< < x , y > , t > : x € u A y € v A t e x " { z } } = {< < x , y > , t >€ ( u x v ) x u u U x : t € x u { z } }

(7)

{< < x , y > , t > : x € u A y € v A 3 < s , z > 6 y 3 w e x

t=<w,s,z>} =

= {< < x , y >, t >€ ( u x v ) x ( U u x U U U v x l l U U v ) : 3< s , z > € y 3 w £ x t = < w , s , z f t The

It

remainder

follows

Lemma

1.8.

are easily

by induction

then

Lemma 1 . 9 .

R'=R.

of basis

F * is a composition

We saw t h a t

it

D

that:

If F is a composition

=F"u x . . . x u

Proof:

deduced.

is

functions

of basis

sufficient

Given $ l e t F be a s i n lemma 1.6

and F* (u . . . u ) =

functions.

t o prove

(v),€R'

for a l l

£

$.

with

F ( u , t , x 1 . . . x m ) = { s € u : ( | ) ( s / t f x 1 . . .x m ) } Then F* ( {u} ,w, { x ^ . . . {x m }) ={F(u, t ^ .

(1.9)

. .x m ) : t€w}

:<|) (s,t,x]L. . ,xm) }:t€w}

D

It follows that R may be axiomatised by a single 11^ axiom. Definition

1.10. A function F is rudimentary provided F is a composition of

basis functions.

We have asserted that for each i greater than 0 with i<17+N there is a E_ formula (j>. such that z=F i (x,y)H> i (x,y,z)

(1.10)

If N=0 we can prove (1.10) for all rudimentary functions. So assume N=0 until further notice. We actually prove a stronger assertion. Definition x ...x

1.11.

,u) is t

A function F is substitutable

provided that whenever

there is a E formula ty such that (1.11)

Lemma 1 . 1 2 .

All rudimentary functions

Proof:It will suffice

are

substitutable.

to consider only EQ formulae of the form

Vt€yx(t,x 1 ...x n ) where x is ?.Q. For from this the result follows by induction on Z Q structure. Also substitutable functions are closed under composition. So we need only consider the basis functions. Note that x i €F(z 1 . . .z m ) <* -(Vt€F(z 1 . . •z m ))-( x i = t ) • We shall only consider a few cases. (1) Vt€{z 1 ,z 2 }x(t,x 1 . . .x n )**x( z 1 /X 1 . . .xn) A X ( Z 2 , X 1 . . .x n )

(2) Vt€Uz 1 x(t,x 1 . . .xn)**(Vs€z1) (Vt€s) x (t,x1. . .xn> (3) Vt€z 1 ^z 2 x(t,x 1 ...x n )«(Vt€z 1 )(t€z 2 vx(t,x x ...x n )) (4) V t € z 1 x z 2 x ( t , x 1 . . .x n )**Vt 1 €z 1 Vt 2 €z 2 x« t±,t2 ),x1. . . X R ) using (5) Vt€< z i ; z 2 >x ( t , x 1 . . . x ^ ^ x ^ z ^ } f X ^ . . x n ) A X ( { Z 1 , Z 2 > , x 1 . . . x n ) which i n t u r n u s e s ( 1 ) . The r e a d e r may check t h e o t h e r s . o C o r o l l a r y 1 . 1 3 . If F is rudimentary then there is a 1 formula $ such that z=F (x 1 .. .xn)«<J) (X 1 . . ,x n , z) If N*0 then corollary 1.13 may f a i l . For example, {x,y}€A i s not necessarily 2L in L1"n . For t h i s reason we need to consider a broader class of relations. D e f i n i t i o n 1.14 A relation R is rudimentary provided that for some rudimentary function F R(x 1 ...x n )**'(x 1 ...x n )* 0 Lemma 1.15 Every Z relation is rudimentary. Proof: Let <J) be a ZQ formula. Then by lemma 1.6 there i s a rudimentary function F' such t h a t F'(a,x1...xn)={xn+16a:(D(x1...xn)AXn+1=xn+1}. Let F ( x 1 . . . x n ) = F l ( { O } , x 1 . . . x n ) The function in (1.12) can, informally "effectively"

(1.13) D

speaking, be obtained

from cf>. To formalise this idea we introduce a coding

of the rudimentary

functions.

L e t Q={q: q is a finite function A q*tf> A dom(q)c2 A (s=t|dom(s),t€dom(q) A

•* s€dom(q))

A

A rng(q)cl7+N+l A

(q(s)*O => s/s0,s"l€dom(q) ) A (q(s)=O =» s"0, s"l^dom(q) )}

Let a ={f: f is a function A dom(f) ={s€dom(q) :q(s) - } } . Let F

(1.12)

be a function defined by

F (f)=z ** f€a

A (3g) (dom(g) =dom(q) A

A(Vs€dom(g) ) (q(s)=0 => g(s)=f (s) A A A g(O)=z)

q(s) *0 => 9( s ) = F ,Wc^ (^(s^O) /

The clause with F , . must, of course, be written out as a finite disjunction. For all q in Q F

is rudimentary; and F (f)=z is a A, formula.

Now let T be some function from (s6dom(q):q(s)=0} to n. Let F where f ( s ) = x a 1 1S in dom(T) q T ( X 1 " -Xn)=Fq(f) • T ( s ) ' Then every rudimentary function is of the form F n . And q n 'T F (x,...x ) =z is a A, property of x, . • .x ,z,q,x. For any cj) we can therefore find <3L/T, such that 4>(xn...x ) « F " (x ...x )*0 . (1.14) 1 n q <j>' < J > Fix such q^/T. for all E Q <J> (informally, do so "effectively").

D e f i n i t i o n 1.16.

The E

satisfaction relation, |=" $(x . ..x ) is the relation

This is a A, relation. Definition

1.17

The E^ satisfaction relation, (==£ ( j ) ^ . . . ^

is the relatio

We identify E Q formulae <J> with <0,>. If (J) is 3y 1 Vy 2 . . .Qyin\p(y1. . ,y m ,x 1 . . .xn) then we identify (f) with <m,>. The E satisfaction relation is itself E . We shall never have to m m refer to this representation explicitly: and later on we shall code the formulae by natural numbers. The superscript n may usually be omitted. Note that we have shown that every rudimentary relation is This is perhaps the moment to mention the role of classes in this theory. As we do not have a E, comprehension axiom there may be E^ definable bounded subsets of the ordinals - which, by the way, are defined in the usual way in R - that are not sets. They may even be like ordinals themselves: they may be initial segments of On. Such segments cannot have a largest element, of course. Definition {(X:
In

1.18. ) } , for

A E limit segment (v\) is a n some

E

n

d) and some

x,...x

In

E class of ordinals (i.e.

n ) such

that

(i) a<3€n =* a€n (ii) a€n => a+l€n. Every E

limit segment i s either On or an ordinal.

I t i s not even precluded that OJ might have a E segment properly included in i t . be disallowed, however.

limit

In our applications t h i s will

Exercises 1. X is rudimentarily closed provided x,y€X =• F ± (x,y)€X for all i with 0 f=R? 2. For which a does L

|=R?

3. Show that F Q is definable from F,..,FO and Fn . y

1

o

10

4. If X is a set, can it be proved in R that there is a smallest set Y=>X such that Y is rudimentarily closed? 5. <M,A> i s amenable provided t h a t f o r a l l x€M AflxGM. Show t h a t <M,A>}=R i f and only i f M f=R and <M,A> i s amenable. 6. Show t h a t i f u i s t r a n s i t i v e and 0
10 2: RELATIVE CONSTRUCTIBILITY

The theory R is not very strong. Although it is possible to define ordinal numbers in it there is practically no ordinal arithmetic. One way to strengthen R would be to add axioms making R closer to ZFC; for example, KP set theory. We want a theory that works in all levels of the constructible hierarchy, however, and not merely at admissible levels. We shall add to R axioms that say V=L[A1...AN] . 17+N

Definition 2.1. s(u)=uU U F ."u i=l By lemma 1.7 S is rudimentary, so R[- VU3ZZ=S(U) Lemma 2.2. If u is transitive then so is S(u). Proof: This is where the apparently redundant F ^

to F.. _ are used.

We check each F.. (1) x , y € u •» { x , y } c u c S (u) . (2)

x€u •» UxcucS(u)

(as u i s

transitive).

(3) x , y€u =• x ^ y c u c S (u) (4) x , y € u , s € x , t E y + < s , t >=F1]L ( s , t ) £S (u) . (5) x , y € u => d o m ( x ) c uc=S(u). (6)

x,y€u,

z € y , t = x " { z } =* t = F 1 7 ( x , z ) .

(7) x ^ e u ^ ' e x ^ v (8) x , y € u ,

1

, w ' >€y => < u 1 , v ' , w ' >=F 12 (u 1 ,< v ' w 1 » w1 €y =* < u 1 , w ' , v 1 > « F 1 3 (< u ' , v 1 >,w')

(u',v'>6x,

•i(9) F g ( x , y ) c F 4 ( x , y ) ;

(10) F 1 Q ( x , y ) c F 4 ( x , y ) .

(11)

x , y € u , s € < x , y > => s = { x } = F 1 ( x , x )

(12)

x,y€u,

y=
=» s = { ( x , v ' > } = F 1 6 ( x , v ' ) (13)

or

s={x,y}=F1(x,y).

s€< x , v f , w ' ) =* o r s={( x , v ' > , w ' } = F 1 4 ( x , y ) .

1

x , y € u , x = < u ' , v ' >, s€< u , y , v ' >=* =* s={< u 1 , y >}=F, 6 (u 1 , y ) o r s={< u 1 , y >, v 1 } = F , 5 ( x , y ) .

(14)

x,y€u,

y=,

s€{< x , v ' >,w' } => s=w' o r s = ( x , v ' ) = F 1 1 ( x , v l ) .

(15)

x,y€u,

x=,

s€{< u 1 , y > , v ' } => s = v ' o r s=< u 1 , y >=F1;L (u 1 , y ) .

(16)

x,y€u,

s € { < x , y > } =* s = F 1 1 ( x , y ) .

(17)

x,y€u,

s € { t : < y , t >€x}=*< y , s >€x=*seu;

Definition

2.3.

(17+i)

(i) f=s\x ** xEOn A f is a function

x , y € u => ^^ A dom(f)=x A

A (0€x "*• f(0)=) A (Vv€x; (v=\l+l «• f(V)=S(f(\l)V{f(\l)})

A

11 A (VX£x) (lim(X) => f(X)= U (ii)

y=Sv ** (3f)(f=S\dom(f) A

Definition

2 . 4 . R is R together with the axioms

(i) W3f f=Slv (ii)

Vx3v3y (y=Sv A

For later use we also define Definition

2 . 5 . rud(u)= USn (u), where Sn (u) is defined inductively

by

S0(u)=uV{u] Sn+1(u)=S(Sn(u)U{Sn(u)}) Clearly rud(u) is rudimentarily closed. But it need not be a set even if u is; and it may not be the smallest rudimentarily closed X3uU{u} (unless we restrict X to be a set). From now on we work in R . Lemma 2 . 6 . Each S is

transitive.

Proof: (This is a model for various subsequent inductions) Suppose not. Then letting E={v :S is not transitive}, E*0. Claim

E has a least member.

Proof: Take y€E. If Efly=0 y is the least member. Otherwise let f=S|y. Then Efly={v
(2.1) n(Claim)

Let y be least, then.y+O, of course, nor is y a successor by lemma 2.2. but if y is a limit then S = U S and each S is Y v
Similarly Lemma 2 . 7 .

y
Lemma 2 . 8 .

If X is a limit

ordinal then S, is rudimentarily

closed.

A

Proof: If x,y€S A then x,y€S

, some y
Let S_x =< S x ,€nS^,A i ns x , . . . /^flS^ >. Clearly SX)=R (A a limit)

Lemma 2.9. lim(X) =* X=Onf\s, .

12 Proof: If not then we could use the technique of lemma 2.6 to obtain a least X for which the lemma failed. Case 1

A is a limit of limit ordinals

Then 0nDS^=0nn U{SX,:X'<X and X 1 a limit} =U{Onns^,: X'<X and X1 a limit} =U{X':X'<X} =X. Case 2 There is a maximal X'<X which is a limit ordinal. Suppose Xc{:S, . Then there is a least v<X such that V ^ S , . Since X'=OnnS,,ES x ,v>X'. But X'=Onns x , and S^,€S X so X'€S X . Thus v>X'; so v is not a limit ordinal; say v=y+l. Then y£S,; but then yll{y}€Sx as §XJ=R. Contradiction! Suppose, then, that OnflS^i X. Then there is y£ (OnDS, )^X; since OnflS, is transitive, X€S,nOn. By the usual argument there is a least v<X with XcS nOn. And since S, ,DOn=X',v>X'. Say v=y+l. So XcfcS . Now — v A y by exercise 6 in chapter 1 UUUUS cS . But since X is a limit ordinal XcUUUUS v ; contradiction!

D

Lemma 2.10. lim(X) => S,\=R+ Proof: Again we may assume that X is the least where this fails and derive a contradiction. Case 1 X is a limit of limit ordinals. R

is axiomatised by a single IT2 axiom, so this is impossible.

Case 2 There is a maximal X'<X which is a limit ordinal. Suppose there is v<X such that s|v$S, . We may take v least such. u>X', since otherwise SIvES^.cS^. And S|X'=U{fesx, : Sx, |=f=S |dom(f) }so S|X'€S X . So v is a successor.

Say v=y+l. Then s|y€S,. But

S | v=S |yU{< y,S >} and S €S <=s, . So s|v€S,. Contradiction! y y v~~ A A But given x€S, there is v<Xwith x€S so S,f=3v3y(y=S Ax€y) . I

A

V

Thus §, f=R .

~~A

V

D

In ordinary set theory it would now be straightforward to call S S x, JJ\ howevei in R we must f i r s t x ;; however, a)X X ordinal a r i t h m e t i c . Definition 2 . 1 1 .

develop the necessary

T=v+n <=» x,v€0n A n€w A

A 3f (dom(f)chi A f(O)=V A Cii+l£dom(f))

(f (i+l)=f (i)+l)

A T=f(n)

This i s a E, formula. Generally we cannot prove that v+n e x i s t s . Suppose for some v i t did not; then {n:v+n exists} i s a E, limit segment of u). Definition stating

2.12.

The theory R+ is R* together with the axiom schema

that every definable

limit

segment of 0) is either

0 or 0).

13 From now on we work in R . Note t h a t the theory i s no longer n~ axiomatisable. Now we g e t : Lemma 2 . 1 3 . For all v and all nQjd \>+n

exists.

Proof: {n:v+n exists} i s co. R

strengthens R

•

by a strong form of induction on GO .

Lemma 2 . 1 4 . Suppose

(Vn&ti) (v+n
X= sup V+n ) n£(jo

Proof: L e t E = { £ < T :

f o r a l l n€u) v+n<£}.

Claim n=v+n ** £5 (=ri=v+n Proof: (=*) Suppose ri=v+n. Let f be as in lemma 2.11. Then f€S by an easy induction on n. (*=) Obvious.

•(Claim)

Thus E={£
If there is a largest

a limit

ordinal X then On={\+n:n€a)}UX.

Proof: Otherwise t h e r e i s x€On^{ A+n:n€oj} UX. So X+OJ e x i s t s , butA+a)>X, X+OJ i s a l i m i t o r d i n a l . Definition (ii)

2.16.

•

(i) f=D\v *=» f is a function

A v€On A dom(f)=v A

K K rvx f (X)= u T=[V/u]**Bf(f=D\dom(f) A

This is a T.^ formula. [ V/OJ] is the unique T such that v=o)T+k, some k€oj.

Lemma 2.17.

VV/"V/OJ; exists.

Proof: It suffices to show VvD|v exists. Claim If X=0 or lim(X) then (Vv<X)D|v€S x . Proof: Suppose not. As usual we may get a least X where the claim fails. X*0, then; and if X is a limit of limit ordinals then the claim holds. So there is a maximal X'<X,X' a limit ordinal. Thus by lemma 2.14 X = { X'+n:n€o)}UX' . Now D|v£S, , for all v<X', and D|X'={< v,x>:Sx,f=x=[ V/OJ] }

(2.2)

so D|X'€S X I . Then if n is least such that DlX'+n^S^,, n=m+l, sayc But then D|X'+m€S A , and D | X ' +n= (D | X ' +m) U{< m, [ X '/a)] >} so DlX'+nES^; contradiction 1

•(Claim)

Now if On is a limit of limit ordinals the lemma follows by the claim. Otherwise there is a largest limit ordinal X and 0n= = {X+n:n€a)}UX by lemma 2.15, and the result is proved just as in the claim.

•

14 Definition

(ii)

2.18.

(i)

X=OJV ** (T=OAV=O)V(T is

dh denotes {y.-ojy

(Hi)

J =S

a limit

and

V=[T/U])

exists]

(provided OJV

exists)

I t i s immediate from lemma 2.10 that J f=R . Clearly every limit ordinal x i s of the form OJV . In R

the structure of limit

segments i s simplified by

Lemma 2 . 1 9 . If r\cpn is a bounded E limit then T] has no largest Proof:

limit

segment which is not an ordinal

ordinal.

Suppose i t did and c a l l

i t X.

Claim X+n€n for a l l n€oj. Proof: Let k={n: X+n€n}. k i s a E

limit

segment of OJ SO k=oj D( claim)

Hence X+OJCT) . So X+OJ e x i s t s and i s a l i m i t o r d i n a l ; So n i s an o r d i n a l .

Lemma 2 . 2 0 .

If there is a largest

dh is a limit

segment.

limit

ordinal OJV then dh=\)+l. Otherwise

Proof: Clear

If

o

n is a limit

limit

segment we l e t OJH denote {y:3v€n y
segment. By lemma 2.19 every E

but not an o r d i n a l

Lemma 2.21.

thus X+oj=n.

a

limit

segment t h a t

i s bounded

i s of the form ojri.

S \=R* (ojr)c: On) -OJTV —

Proof: If can is an ordinal then this is lemma 2.10. If ojri is On V=S . Otherwise S is a union of models of R by lemma 2.19 and V=S . Otherwise S is a union of model lemma 2.10.

D

We can extend definition 2.18 to limit segments by allowing J =S p r o v i d e d nsCfri. So

v=

J0n-

A l s o J =S

THE AXIOM OF CHOICE Just

a s i n ZFC V=L [A] =• AC, s o R+|-AC. The e s s e n t i a l

technical

lemma i s

Lemma 2 . 2 2 . Suppose u is a set with a well-order ordered. Indeed there is a rudimentary function

r. Then S(u) can be well-

W such that W(u,r)

well-orders

S(u). P r o o f : W ( u , r ) = { < x , y >€(S(u))

:

(x,y€uAxry)v(x€uAy€u) v

v(x$UAy$UA(3s€u)(3t€u)(3i£l7+N)(x=Fi(s,t) A

A ( V S ' € U ) (Vt'€u) (s'rs «* (Vj£l7+N) y*F.(s',t')A A ( s ' = S A t ' r t =» (Vj£l7+N)

y*F. ( s 1 , t ' ) )") )")A

(2.3)

15

Since the defining clause is E Q W is rudimentary by 1.6. • Note that W(u,r) is an end-extension of r. We let f=w|v be the formula f is a function A dom(f)=v A v€On A

(2.4)

pU{Sp} f f (?) U{< t,S >:t€S }) ) A )=

U f(£')) A (v*0 =* f(O)=0)

And r=W v means 3f(v€dom(f) A f=w|dom(f) A r=f(v)) Lemma 2.23.

(i) For all v W exists

(ii) For all v W is a well-order of S . Proof: Very like lemma 2.17. First of all establish that if X is a limit ordinal then for v<X W v €S^. If this failed there would be a least limit X where it failed. X cannot be 0 or a limit of limits; hence there is a largest limit X'<X. W £S,, for v<X' and W., is V

S x ,-definable

A

so W,,€S.. If X'+n+l i s least with

w

A s w e Xi+n+1$ x

derive a contradiction by rudimentary closure of S,. Then the result i s proved unless On=XU{X+n:n€a)}, some limit X, in which case i t i s proved by rudimentary closure. (ii)

i s then a t r i v i a l induction.

•

C o r o l l a r y 2 . 2 4 . Every set can be well-ordered. ordering of the universe that is a £

Indeed there is a well-

class.

The well-ordering we have constructed is called the canonical wellordering. Lemma 2 . 2 5 . Suppose R is a Z relation.

Then there is a I, function r such

that 3yR(x 1 . . . x n , y )

=» R(x x . . . x n , r ( x 1 . . . X R ) )

(2.5)

for all x , . . . x . 1 n Proof:

Suppose

at t h e l e a s t ordering.

R(x.....x



Then,

A(Vy So r

1

/

z eSY)(((y',z

Let r'(x,...x

R ' ( x . . . x ,y,z) in t h e canonical

) =* well-

R1 i s £ Q ,

« R'(x1...x

l

1

n /

y/z)

>,>€WY

A ( 3y) « y ,z >eSy

=> - R ' ( X ] _ . . . x

n

A

(2.6)

, y , z) )

i s E, . S a y r(xx...xn)=y

(2.7)

that

provided

r1 (x1...xn)= 1

, y ) <=> 3 z R ' ( x . . . . x , y , z ) .

such

« 3zr'(xx...xn)=

i s a T.1 d e f i n i t i o n

and r

satisfies

C o r o l l a r y 2 . 2 6 . If f is a E- function fog=id\rng(f)

.

(2.7) (2.5)

•

then there is a E function g such that

16 Proof: R(y,x) «=> f (x)=y

is a £.. relation. So there is g 1 such that

3xR(y,x) •» R(y,g(y)) =* f(g(y))=y. Let g=g'|rng(f).

D

SKOLEM FUNCTIONS The Kuratowski pair that we have used up to now is usually satisfactory, but sometimes we require a pairing function under which the ordinals are closed. The Godel pairing function is such; but generally if we are working in R we cannot assert that every pair of ordinals has a Godel pair. Our official pairing function will nevertheless be an adaptation of the Gftdel pairing function. Definition 2.27.

(i) < is the well-ordering of On determined by

< a, 3 > « max (a, 3) <max (a ' , 3 ') v G v (max(a, 3) =max(a ' , 3 ') Aa
(2.8) v

v (max (a, 3) =max(a' ,3 ') Aa=a ' A 3 < 3 ') (ii) f=p\y «=> dom(f)=^f A f is a function A Va,3
A Va
p i s a £, function

~ (3f) (f=p\dom(f)

that

A f (y)=< a,3 >;

enumerates t h e GdJdel p a i r s .

Lemma 2 . 2 8 . Va3f f=p\a.

Proof: Observe that< a, 3 >=p(y) =* max (a, 3)£p (y) . From this the technique of lemma 2.17 enables us to deduce that if X is a limit ordinal then Vv£A3f€S, f=p|v, and hence to obtain the result by the A usual adaptation. D 2 p|a:a-*-a , but is not, generally, surjective. D e f i n i t i o n 2 . 2 9 . (i) Q={oi:p(a)=( 0,a)} . (ii) a is G-closed if and only if a is a limit point of Q. Lemma 2 . 3 0 .

There is a E map of On onto On .

Proof: Although the strategy i s quite standard we give the proof

in

detail.

Claim

If A is a limit ordinal then there is a Z_ (S^) map of X onto

Proof: Suppose not. Let E = { A : A is a limit ordinal and there is no Z 1 (S X ) map of A onto X 2 } . So E*tf>. Take some X€E; if XflE=0 then X was minimal. Otherwise since ^ ( S ^ J c S ^ when X1 is an ordinal less than X, we may use definition 1.17 to set

17 EnA={A'
f:A'-A |2 onto}

so that EnA is a set. So there is a least A where the claim fails. Clearly A$Q. Case 1

A is a limit of limit ordinals.

Suppose p(A)=. Then V , T < A . And p|A is E.. (£T ) and maps A one-one J-

A

onto {z:z< < V , T > } . Let y={z: z< < v, T >}; suppose A ' is a limit and V,T
—

A

«

By inductive hypothesis there is g€E,(S,,) mapping A1 onto A1 ; l A 2 so g€S,. So by corollary 2.26 there is g'€E,(S,,) mapping A1 A

J. —A

one-one into A 1 . So g'€S,. Let f « y , y ' » = g ' (
one-one into A'.Let v=rng(f); then v = g i n ( g I M y ) , so v€S,

Let h(y)=f" 1 (y) =<0,0> Then h:A-*A Case 2

if yev

(2.9)

otherwise

maps A onto A . Contradiction!

There is a maximal limit A'
Note that this is the only other case because a>€Q. There is a E, (S..,) map g:A'-*(A') i. —A

onto, and so by corollary 2.26

*y

a E1(S x ,) map g':(A') +A' one-one. So g'CS^. Since A={ A '+n:n€a)} UA' there is a bijection j: A«-*A ' I (S, ) . Let f (y ,y ') =g ' (< j (y) , j (y •) » . 2 1 —A So f:A -»-A' is one-one. Let v=rng(f); then v=rng(g'), so v€S, . h is then defined as in (2.9). Contradiction! a(Claim) As usual the lemma follows from the claim by applying the relevant case to On.

n

It is important to notice that the proof of lemma 2.30 was not uniform. In the definition (2.9), furthermore, we had to use a parameter v,and there is no way in general of making the map parameter free and keeping it E,. At times when we are being sensitive about parameters we shall use the Go'del pairing function. Definition 2.31. n is a Z -cardinal provided •

n

(i) T) is a limit segment of On; (ii) there is no E map F with v€r| such that X]C{F(i) :i
If r\ is a E cardinal

then r\ is closed

i€dom(F)}.

under p (so if T) is an

T\$.Q) .

Proof: Note t h a t n must be a l i m i t of l i m i t o r d i n a l s . But i f t h e lemma f a i l e d t h e r e would be A' €n with A1 a l i m i t and p"n<=(Al) . But then p " " 1 | ( A 1 ) 2 i s a E, f u n c t i o n with n E r n 9 ( P " 1 I ( A ') 2 ) • A n d t h e r e 2 i s a map of A'-^A 1 . C o n t r a d i c t i o n ! •

18 Let s be a map of u) one-one onto {:q€Q A A T:{3£dom(q):q(§)=O}+n,n<w}. Then the relation R m (i,<x 1 ...x <=> s (i) =< q , T . ,n > A kj} <j>(xn...x ) is E an

s

<> j q> ' Z 1 f i x e d f o r m n o w o n . $.

is

n is

provided s is picked £,; such

m

such

that

s(i)=
sometimes c a l l e d

Definition

2.33.

h(i,( x))=y

uniformises

R' (i,( x),y)

h is called

the Z Skolem

Lemma

2.34.

h" fup<x

s(i);

;^

n;

"*"

R denotes R . is the I

*• R (i,(x,y))

co

1 ,n),some

, T. •*"

1

cj). i s

>) ~

given

parameter

free

function

by the proof of lemma

that 2.25.

function.

h\

v.

Proof: (V denotes
> of course)

Let Y=h"(ooxx
Say

Y;L=h

( j±/< x±1. . . x ± k

»

where j.€o),0 (z,y, . . .y ) where 4> is E Q . Let

r(x 1 1 ...x n l ...x n k ,z) «< M z,h(J 1 ,<x 11 ...x lk »...

(2.10)

and suppose
(2.11)

3zR' (i,< x^., . . .x

>,z), so by definition k n R (i,< x i;L . . . x n k > r h(i,<x i;L ...x nk » ) . (2.12) n n Let z l = h ( i , ( x n . . . x n k >) ; so z ' £Y and by (2.12) R ' (i ,< x±1. . . x R k >, z ') n n 1

Definition 2.35.

ii(Xj denotes h" (wxx
Strictly speaking this is ambiguous, but in practice we always recognise functional applications of h by the natural number argument. Lemma 2.36 . v*h(On) . Proof: The usual strategy. Claim

If A is a limit ordinal and xGS^ then there are i€o> and

a, . . .a <X such that S, hx=h (i ,< a, . . .a >) . ~~A J. n -L n Proof: Suppose the claim failed. There will be a least X where it fails. Case 1 X is a limit of limit ordinals. Suppose xeS-,,A'<X, X1 a limit. Say i€(jo ,ot, < . . . ) . Then S>x=h(i,(a, ...a )) . A j. n ~~A i. n Case 2 There is a largest limit X'<X.

19 Then A=A'U{X'+n:nEw}. Suppose n is least such that there is xes

X « + +i

an

x

^

violates the claim. Say x=F.(y,z) with 0
y,z€S,,, U{S,,, }. It suffices to show that x is £, definable A

> n

A

•+• n

-L

from ordinals in S,. But x=F.(y,z) is E», so it suffices to show y,z ~~A

E, definable. If y,z€S,,, .L

A

"i~n

1

U

this follows by the minimality assumption.

But -A^ y = S A'+n ** 3f (f=S|dom(f) A f (XI+n)=y) (2.13) is a £, definition of S,,, from ordinals. This establishes the 1 A +n claim. D(Claim) The lemma follows from the claim as usual.

n

So there is a parameter free map of a subset of wxOn

w

onto V.

Lemma 2 . 3 7 . There is a I, (with parameters) map of On onto UJXOn 2

Proof: By lemma 2.30 there is a £, map F of On onto On . Define by induction on n
F 1 (a)=a F n + l ( a ) = < a i - " 3 n - l ' Y ' y l > w h e r e F n (a)=<3 1 ...3 n > and =F(3R) Plainly for all n€o) Fn:0n-K)nn onto. Say G(a)= F n (B) if F(a)= with n€a); 0 otherwise. Then G:0n-K)n W is onto. And G(a)=s ** 3n€a)33(F(a)=< n,3 >A 3f(dom(f)=n A f(0)=$ A

(2.15)

(Vm A f (m+1) =< 3 X . . . 3 m _ x , y, y ' >A A F(3 m )= )) Af (n-l)=s) Hence G is £,. Finally G1 :On->-(joxOn may be defined by G'(a)=F(a) if F(a)€caxOn; =<0,0> otherwise. Letting G" (a) =< n,G(3) > where G'(a) = G" is our map.

•

If F was the Gfldel pairing function then G is parameter free. In this case G is by abuse of notation called the G6"del pairing function. Often we write G(-ia,...a ^ ^ a ^ . - . a In

Corollary

2.38.

Lemma 2 . 3 9 . Proof:

>.

in

There is a I, map of a subset

of On onto V.

There is a I. map of On onto V.

L e t h * ( T ,< i , x >) = h ( i , x )

if

BzGS H ( i , < x > , y , z )

=0 o t h e r w i s e ,

where h(i,<x>)=y *> 3zH (i ,< x >,y, z) is some E, parameter free definition of h with H E Q . h* is l-^ and total. Furthermore h*:0nx(u)x0n w)->V onto. But using F and the map of lemma 2.37 there is a map of On onto Onx(cox0n w ) .

•

20 MISCELLANEOUS Lemma 2 . 4 0 .

RESULTS IN R*

TC(x) exists

for all x.

Proof: Let f=TC|y be the formula f is a function A

A dom(f)=y A y i s t r a n s i t i v e A

(Vz€y)f(z)=zU t y z f(t)

Then y=TC(x) « 3f(f=TC|dom(f) W3f

(2.16)

A f(x)=y). So it suffices to show

f=TC|S .

Claim W < X 3 f € S x

f=TC| S v when X is a limit.

Proof: If not there is a least X where the claim fails and X=X'U{X'+n:n€o)}

for some limit X'<X. Since

(2.16) is a 2^ definition

and TC|s€S,, Let n be least such that TC| s x»+ n +l , , ffor o r v<X', v < X , TC|s,,€S,. TC|s,,€S, V V

A A

X

A

L e t fQ = T C | S x , + n . L e t fi+1={<x,y>:x€Sx,+n+1

A xcdom(f ± )

A y=xUz^xf±(z)}

(2.17)

Then f o r x€S, , , , , we have x€dom(f, , , ) and f, , , (x)=TC(x) , where k A +n+J-

i s such t h a t

K + J.

K + J.

U x<=S, l + . But by e x e r c i s e 6 i n c h a p t e r 1 i f xGS, , +

t h e n UUUUxcS,,,

. So T C | s , , ,

-— A + n

, 1 =f c -, which i s i n S, .

A +n+j_

D

+ 1

• (Claim)

A

As usual the lemma follows from the claim Lemma 2 . 4 1 . Suppose X is the maximal limit ordinal

D and that A .<=S, for 0
^————^—^——

2

A

Then every y x<^S, is definable over S, . —X X Proof: Suppose x=F(y,...y ) for y,...y €S,U{S,} with f rudimentary: x n x n A A this is possible because each S , + is rudimentary in S,U{S,}. Now there is a function F 1 which is a composition of F i ' ' * F i 7

such

that x=F (y, . . . y ) *» x=F' (y, . . .y ,A, . . .A ) . By lemma 1.12 there is a E o formula $ such that t€F(y^...y ) <=> <J> (t, y^. . . Y n /A 1 . . . A N ) and we may assume t€F(y, ...y ) «* 4> (t,yn . . . y^ , ,S, ,A, . . .A ) (yn . . .y -,eSy) L n L n—x A J. IN x n"~x A By induction on E Q structure of (J) we may produce \p such that •
X

H(t^^ . .y ^ ) .

For *(t,y 1 ...y n _ 1 ,S x ,A 1 ...A N ) « <S X U{S X }U{A 1 ...A N },€,A 1 ...A N >H N(t,y1...yn_1,sx,A1...AN) since S^UlS^JuCA,...A

} is transitive: the construction of \p by

induction is left to the reader: the only non-trivial clause replaces (3y€S.) by 3y. A Lemma 2 . 4 2 . X onto

If X is a limit

o ordinal

then for all 7c€o3 there

is f mapping

Sx+k.

Proof: By induction on k. k=0 is corollary 2.38. Otherwise let

21 3, , be o n t o .

Define

g(a,Y,Y')=f(Y)

i f a=0

(2.18)

=F ( f ( Y ) f f ( Y 1 ) )

i f 0
=0 o t h e r w i s e s

T n e n

:X ^" x+v+i* onto.

u s i n g F_ of (2.14) a

d e f i n e d over S» ,

Lemma 2 . 4 3 . Suppose X is the maximal limit ordinal and xcP(SJ, and for 0
A .cs

—

l— A

There is p such that xc£ (S, ) . — p —A

P r o o f : T h i s i s a u n i f o r m f o r m o f lemma 2 . 4 1 . S u p p o s e x € S ^ + , , k € w . So x c S , ,, a s S A j l i s t r a n s i t i v e . — A+K A+K

S a y f:X->S,^, o n t o b y lemma 2 . 4 2 . A+K

We may assume x*0. So letting X={< Y,t>:t€f (Y) A f(Y)€x}

(2.19)

XcS and so by lemma 2.41 XGS (S,) say. Then for y£x there is Y < X — A p —A such t h a t y=f(y); then y={t:€X} which i s I . • Exercises 1. Show that rank(x) exists for all x. 2. Show that the condition A.cS in lemma 2.42 is necessary. 1— A 3. Is there nGco such that there is a parameter free I

map of On

onto V provably in R ? 4. A limit segment n is £

regular provided there is no E map

F such that for some v€n VY€TI3Y '
E, regular imply that n is a E-. cardinal? Vice-versa? If

22 3 : ACCEPTABILITY AND THE PROJECTUM

Not a l l models of R have nice fine structure. The possibility of fine structure depends on our being able to code a l l the bounded subsets of y which are £, by a single one, provided that y is the least ordinal for which the E, comprehension axiom fails.This is assured by the following Definition which

definition.

3 . 1 . RA is the theory R* together with the axiom of

acceptability

states: Suppose X is a limit

suppose u£S,

ordinal and 6<X such that for some ac6 ,
. Then there is f=K fy :6<E><\ >€S,

f.:^{Uu(P(^)nu)

such that

onto

(3.1)

In this chapter we work in RA. Lemma

3 . 2 . If X is a limit

then

Sy\=RA. —A

Proof: This i s immediate. • The axiom of acceptability can be thought of as the best form of GCH we can get without assuming the power set axiom. Lemma 3 . 3 . Suppose w i s a set. Let v denote the least

cardinal >v, or On if

V is the largest

class

cardinal.

Then there is v'
such

and a I

that

(a) {a^ .: i is a set for all T enumerate r . Then 1 v v

isa

^

property.

x = ^y «•» 3f( f is a function A dom(f)=i A rng(f)=rvflST A

(3.2)

A f is strictly increasing ) A T € F This is £.. as V PIS is uniformly I. in T. The technique of chapter 2 establishes <^Y:j€S r v ^ for all it,. + n order, where uKv,i
23 dealt with s i n c e v_>0). L e t g be t h e l e a s t such map. L e t a

gg ( n, i) ), j j= jf j, n ( i ) ' S o P W - U ^^: i i s a s e t , for i t i s E, definable over £ ^v + , so i s a s e t for

T
Set v'=oov*. ^ Suppose o)V*>v . Then v €On,so i s a c a r d i n a l . Let $=[ v /GO] ; then there i s a map of 3 onto v ; so 3=v . That i s , v =o)V . Hence v*>v+. For iv onto. Contradiction. 1 • The following c o r o l l a r y makes i t c l e a r why a c c e p t a b i l i t y i s a weak form of GCH. Corollary 3 . 4 . Suppose v>0). Then either ?(v) exists

and ?(V) -V or for all

Proof: I f P(v) i s n o t a s e t then given ucp(v) t h e r e i s a l i m i t X with u € S , + and P ( v ) n s , + ^ S , * 0 . By t h e axiom of a c c e p t a b i l i t y u£v. If P(v) i s a s e t t h e n P ( v ) £ v + by lemma 3 . 3 . But then P(v)=v + by C a n t o r ' s theorem. n Definition 3 . 5 . The projecturn p is defined by wp={v: f o r a l l £, AcOn ADv i s a s e t } 1

(3.3)

—

To justify this definition we need Lemma 3 . 6 .

{v: for all £ Acpn Af)V is a set] is a limit

Proof: If Aflv i s a set then Af1(v+1) i s a s e t .

segment

D

p is either an ordinal or a limit segment of Oh or Oh i t s e l f . As far as we are concerned the use of wp rather than p in (3.3) i s a nuisance, but i t s use keeps our r e s u l t s in line with the l i t e r a t u r e . The smallest possible value of p i s 1. Definition 3 . 7 . RA is the theory RA together with the axiom stating there is a I, AcVn such that for all x AnoJp+xflOJp. 1 If p is an ordinal then RA and RA

Lemma 3 . 8 . ujp is a Z

cardinal.

that

coincide. From now on we work in

24 Proof: Suppose F i s

E , y€cop , dom(F)cy and copcrng(f) . Suppose

_i_

i

$€y ^cop. So t h e r e

i s a function

from y whose range includes ojp, g. x. Say 6€A' « g(6)€A. Then A1

Take A 2^ such t h a t AHcop+xflcop, a l l is

1

E.^, A'cyEcop so A

is a set.

But Aflcop= (g"A') Do)p and g"A'

is a

set.

Contradiction! So y ^oap=0; t h a t lemma 3.3 t h e r e A is

a E,

is,

y -cop or y €cop.Hence y £rng(F)

subset of y, hence A i s a s e t .

Say A=G(£). Then

contradiction. 1 Corollary

3.9.

Corollary

3.10.

if

n

p*On then cop=p or p=2.

wp i s closed under the G&del pairing function.

Proof: By lemma 3.8 and lemma 2 . 3 2 .

Hence we could have defined Ac(0n)
and by

i s a E1 map G of y onto P ( y ) . Let A={£:C^G(^)};

2^, AfMvf" fine

a

wp as the s e t of is

structure,

v for which given

a set. in L for

of projectum by looking a t

example, we i t e r a t e

t h e s t r u c t u r e <J

the

,A> where

A codes up the E, s u b s e t s of cop; A i s c a l l e d the E, master code. In our p r e s e n t circumstances we could not be assured t h a t < J was amenable. We have to r e p l a c e J contains

all

by a s t r u c t u r e

that at

,A >

least

bounded subsets of cop.

Definition

3 . 1 1 . tf~ denotes {x:TC(x)€&} (This is well-defined

Definition

3.12.

P denotes {p€[ On]

:for some A
by lemma 2.40)

E in parameter p

there is no x such that AOb}p=AC\x] .

Lemma 3 . 1 3 . p*0 Proof: There is some p 1 with the property ascribed to members of P, since we are working in RA . But there is a parameter free map G say of

(0n) < a ) onto V. If p 1 =G (< o^. . . a n >) then let p={a 1 ...a n >.

a

On is not necessarily closed under the GOdel pairing function so p may not be reducible to a single ordinal. On the other hand o>p is closed by corollary corollc 3.10 so cop may be replaced by (wp) w in Definition 3.12.

Definition 3.14. r^fti,*): i€oo A X€H

So A P , T P are 2^.

A \=?, s(i)(xfp)};

25 Definition

3.15.

Let B ...B

be given.

Ik

0<j
_

(u) denotes 7 * * *

1 /r/V-f-J

J

^~

S_ 1" ' k =< S,£0S denotes

.(x,y)=B .Ox where

UUU{F."U :0
\r

k is defined from S_ B . . .B B 2 I k

J8!. '"Bk

Define F

as S was defined in definition 2.3. V B B

,A OS,. . . ,A OS,B OS,. . . ,B Os) where S=S 1" ' k S^l"'Bk

and J^l '"Bk

S^l '"Bk.

denotes

We make no existence claims in general. However: AP

Lemma 3 . 1 6 .

For all v€oop s

exists.

P r o o f : 6 = X+y m e a n s 3f ( f i s a f u n c t i o n A (Vv+l€dom(f)) (f (v+l)=f ( v ) + l )

A dom(f)<=On A f (0) =X A

A (VyGdom(f) ) ( l i m ( y ) =*

=> f ( y ) = s u p f ( v ) ) ) vX A (3y' <X ') (Vy<X ' ) If E*0 then it has a minimal element X 1 . ClearlyX'>X. It is easily seen that X 1 is not a limit of limit ordinals. Say X'=X"U{X"+n:n€w} where X" is a limit. If y'<X" then (3y<X") S ^ y 1 =X+y v y'<X. But then y={y:(3y'<X")(y'=X+y)} is a set in S,,, so there is a least A yiy;

and S, ,(=X"--X+Y~~A

Then there is a least X"+n+l such that (Vy<X') S^,^X"+n+l=X+n. Suppose S, ,|=X"+n+l=X+'y+l and Y + l < A I (we are using the associativity —A

,

of addition, which may be seen to hold in R ) . Contradiction.1 So E=0 . The claim follows by the usual adaptation.

D(Claim 1)

Now suppose a€S, + \S, and acy€a)p . Suppose On={X+y ' :y' €n }UX. Claim 2 y€n. Proof:Suppose n^y. We are going to deduce that there is a E, map of y onto P(y). This is done by establishing that if X' is a limit and X'>X then there is a E, (S,,) map of y onto P(y)ns,,. Otherwise J- — A

A

take a least X1 where this fails. If there were a largest limit X"<X' then P(y) PIS, , *P (y) DS, „; if X" = X this is given, and if X M >X A A there is a E,(SAII) map of y onto P(y)ns, n . But by acceptability we deduce

i —A t h ee x i s t e n c e

o f a E,(S,,) -L ~~ A

A map o f y onto

P(y)ns,,. A

So suppose X1 is a limit of limits. If X
denote the

in the canonical well-order (y a limit).

So €£,(§.,). But there is g€S,, such that g:y«-+y . Let F(v)=f A + v

(v2) if X+v 1 <X' and g(v)=

=0 otnerwise. Then F is a E^^ map, and since n£y, X '<={ X+y' :y'
F: y>P(y) ns^ , onto.

The usual argument now gives a E^ map F of y onto P(y). Let }. A is a set as y£o)p. But £€A ** ^$A. Contradiction!

26 _ Claim 3

D(Claim 2)

S a exists for all a€ S, , . y X+0)

Proof: Suppose A.1 is a limit ordinal, X'>X and X + Y ' < X ' . Then S^ , f=3f (f=S a | y 1 ) . To see this suppose it failed at some least X 1 . X 1 cannot be a limit of limits_; so X '=X" U{X"+n:n£u)} where X" is a Limit. If X + Y ' < X " then S a |y'€S,,.Since aes,, it follows that S a |Y'€S,, where X J - Y I = X " . So there is least: y' +n+l, where \+y'=\",

such that

S a | Y 1 +n+l^S.. , . But then S a ,€S,, and an e a s ^ induction gives S a , , € S , , , so S a |Y'+n+l = S a | Y ' +nU{< Y ' + n , S a , _ S

a

>}£S. ,; contradiction.1

Now it follows as usual that if X + Y exists then S a |y and hence exists. But Y ^ I by claim 2.

•(Claim 3)

suppose y is a limit and letS,^ a=A™riY. X be the least Now limit ordinal with a€S,^ordinal . If y=X then flP(y)Let *S, flP(y) . A +0J

1 Ajr OJ

A

^ ^ f:Y^(Y) Th Otherwise let f C S • Then let a=f "a, so a€P(y) O S .++ Hence by claim 3 S exists. So S exists for all y£uiQ, •

w

^

Claim 2 of the preceding lemma is often of independant interest: it implies, for example, that if X is the largest limit ordinal and the projectum of S, is 1 then the projectum p

must also be 1.

—A Lemma 3.17. Suppose that for all acry
Let e={p

(< £,T » :f (£)€f (T) }. v€Q so ecv.

We claim x€S e . For rank|TC(x):TC(x) +rank(x) onto so rank(x)€x. (Here we are using the result of exercise 1 in chapter 2.) Define g=coll|Y « y is an ordinal A g is a function A dom(g)=Y A A g(O)=0 A ( V X € Y ) ( A a limit =» g(X)= U g(C)) A B,<\

A ( W + 1 € Y ) (g(v+l)={< C,g(v) "{6:p x ( < 6 , O ) €e} >: C
A

A

P n (a>x (Y)< w )

is a

D

Setp

then for all Y ^ ^ P S a €J A .

then 6+y exists and 6+y€.bip because cop is a £, cardinal.

By claim 3 in lemma 3.16 S a €J

. So it suffices to prove that

27 _AP => a £ j . W e l l ,

+ y coop s o by lemma 3 . 3 t h e r e a r e i€u>, £€a)p

such t h a t a = h ( i , £ ) . S a y <J>(z,y,p) «• y € h ( i , z )

(3.4)

and s u p p o s e <> j i s <J> . ( a s a E, So

£ a e ^£ .<

j< > >

P

}

formula).

Then

(3.5)

D

p C o r o l l a r y 3 . 1 9 . H = . / ;H chfwpltfp};. *

cap p

Definition

oop—

+

3.20.

+

KA is R together with the axiom of strong

acceptability:

For all limit X and 6<X such that P(6)C\S^*P(&)n It i s easily seen,

u s i n g lemma 2 . 4 2 , t h a t RA 3RA.

p Lemma 3 . 2 1 . j_A \=RA + Proof: C l e a r l y J

Ap

~~P

+ Ap Ap |=R . Suppose aCS^ ^ S ^ , ac6,X a l i m i t . '

UJ

A "t"CO

A

~~

Say

a=h(i,^) where ^€6 . Let (J) be the formula 4>(x,y,p) •* y€h(i,x)

(3.6)

Suppose .. So a=(c<6:< j &, (5>€AP} . It follows that a€S^+. So X<6+ . Let f:6->X onto. It suffices to show that some such f€S^ +a) . Assume that f is minimal in the canonical well-order. Let (J) be the formula <|>(6,X,€,c,p) ** f is the minimal map of 6 onto X A f(£)=£

(3.7)

So if 4> is $ . =5 ~p< j,<6,X, But then f€S^ +a) . n

Definition 3.22. v*W* . _p VP(=RA, so within V p we can define a projectum. But we cannot assume that VP|=RA .

Definition

3 . 2 3 . p =on; ip

np

np

A0=T°=P&Q=;v°=v.

n

Suppose lf ,T ,A ,p

and PA^ are defined for

pn+1=(){projecta of V^P,

(which is contained in

(v"p')p")

28 CODING V I N V P

Lemma 3 . 2 4 .

(i) (^,1?)

1

is amenable;

I

x=y(\TP is Z^ (V1*) .

(Hi)

x = y f l T p *> ( 3 f ) ( d o m ( f ) € O n

Proof:

a function"€Ap

is

A ( 3 i ) ( 3 ^ . . . T k € O n ) ( " h ( i , < T1 . . . T R f p»

A "dom(f) =dom(h (i/
( f ( £ ) = { f ( ? ) € T C ( y ) : " h ( i , < T±...TR,p» (C) € ,< T 1 . . . T k , p » (C)"€A P ) A ( 3 y ) ( y = f ( Y ) AX={fU)€y: A i€o) A ^ i ( j ) i ( z / p ) " e A P } ) ) )

"f(C)= P

P

i n ,

Before

where

t h e next

"<|>. (YrP) " d e n o t e s

definition

<J> i s a E_ f o r m u l a x obtained

variable

>.

we n e e d a l i t t l e

D

temporary

notation.

with

r n g ( T . ) c m - l a n d s ( i ) i s t h e E, f o r m u l a (p — J. by r e p l a c i n g e a c h o c c u r r e n c e o f v, by

h ( ( ( v

t h e n d 0)k)0'< ( ^ k h ' V ^ "*(""-)" < i , < < i o , x Q >. . . < i m _ 1 , x m _ 1 > > >€T. S u p p o s e s ( i ) i s t h e f o r m u l a

(3y) s ( j ) ( < x , p > , y ) ; t h e n D ( j , x )

Definition predicate

denotes



3 . 2 5 . The theory MC is formulated

in L

e n o t e s

€T.

as follows

(calling

the

••$«i0,Xo)..Mm_1,Xm_1)>»-D(i0,Xo)...D(im_1,Xm_1)!

(ii)

if rng(T,)cm-l and n>m then "(j>f< i n , x . >.. .< i .,x (p — — 0 u m—1 0 o n-1 n-1 (Hi) D(i,x) •* "
(V) "<

> =
(vi)k

i 0 / x o »•

(vii) rnfffT

x

m-

nd"x(i

_>;"*» m—l

* "^i2,x2)«

A »=

-( j^X^" A "< V V * W " " A "( i

I A

(viii) i f 4> is xpy m—1

o f one

symbols B ...B ,A,T) :

(i)

(i

If

then "

1

^i>"

* "B Jt r; w

(0
If... ) . . .< i J I | . 1 / X J | | - 1 »" **

,*0>..

where ^ ^ . . . v ^ ^ ^ o m.y Let s(k ) denote the formula y=v. (xi) let Vy<|)Cy; Jbe a single II axiom for R together with the axiom; then D(i,x) =* "(j>C< i,x)) "; (xii)

D(kQ,xo);

(xiii)

"
(xiv)

" < k

o

"Bk({k0,x))"

,

o o l

« B^x/

acceptability

29 (xvi)

"€< *0/y>" =*• (3&y) ("
Let s(k ) be the formula (xvii) (xviii) (xix) (xx)

i,x>");

y=v ;

D(iklf0)); D(iirx))

«> "< i,x)=h(i,(

< kn,x),( 0 t i , x ) € T « * "
kn fO) )) " ; 1

).

The reader should check t h a t < VP,B . . .B N ,A P ,T P >|=MC. We s h a l l prove the following converse. Lemma 3 . 2 6 . Suppose MC holds. there is an isomorphism O'zCV9 )N^(vfB - 1 - 1 TTC7

Then there

O:( (V^)M, (Tp)M^V

is a class

t^RA and p€M such

and $=V=h(upUip}) / furthermore

. . . B ,A) and rif=V=h(up[){p'}) then there Ik

that if

is i!:M=N_with Tt(p) = (p') and

= 0 ' .

Proof: L e t M f = { < j , x > : D ( j , x ) } . Kilfx1>

Define a r e l a t i o n I on M1 by

« "=".

By ( i i i ) and ( i v ) I i s an e q u i v a l e n c e r e l a t i o n . L e t M=M'/I and l e t [ j , x ] d e n o t e some member of t h e e q u i v a l e n c e c l a s s of < j , x > . Say [i()fx0]E[iirx1]

~

"€";

B^([i/X]) « " B k « i , x » " . Then E, B^. are well-defined by (v) , (vi)k- Let M=< MfE,B^, . . .B^. >. where <$> is E,. Proof: By induction on structure. Atomic formulae are immediate from the definitions and (ii) . - follows by (vii) and A by (vi'ii) . If #3y<M[i o ,x o ]...[i m _ 1 ,x m __ 1 ],y) say ^

([ ±QrxQ] .. . [ i ^ x j ) . Then by

(ix) "3y<M...,y)". Conversely if "3y<|> « i Q ,x 0 >.. . • • • ^ m - l ^ m - l " " t h e n s u p P ° s e "•«io'xo>---'<^<xO--. . . x ^ ) ) ) " ; so#<J)([i 0 ,x 0 ]...[i m . 1 ,x m . 1 ],[j,<x 0 ...x n i . 1 >]) / so MH3y(j)([io,x ]...[i m _ l f x m - 1 ] / y) .

•(Claim 1)

By (ix) M|=R. Define a map a:V->M by a(x)=[k , x ] . By (xii) a is well-defined and by (xiii)-(xv) a:V=M|rng(a). By (xvi) rng(a) is an E-initial segment of M. Hence M is a model of R^ and by (ix) M(=RA.

Let e=(p) M , p = [ k l f 0 ] .

Claim 2 a"On=u)£. Proof: Suppose (x)^ca(y). Let ACE-^M) be such that Ana(y)^M. Suppose A « (J)(^f [i,x]) . Let a={ C (< k Q , C >/< i,x>) " } . For C
~ "(t)«ko,a
So Afla (y)-o (a) . Contradiction!

(claim 1)

30 Suppose t h e n t h a t Observe

a"OnczcoY€cop. By ( x v i i i )

M)=V=h ( r n g ( a ) U{ [k^O]]) .

that

< i,<$>>€A ~ < i,<£>>€T

(by (xx) )

"c()i(>,
~

~ M(=cj)i(a(?) ,£)

(by (xix)) (claim 1)

~ < ±,o(t) >€T^ A

AP

~ a(Ci,oȣAP. A P

So M(=rng(a)cJA

A

A P

(J* is a set in M as y€p) . But there is f E E ^ J )cM

mapping coy onto

j

so M}=V=h (coylKp}) . HencG there is a £-,(M)

map,

g say, of a subset of coy onto M. Let A={£ £€g(£) *> £,$&- Contradiction! (Claim 2) D Now we h a v e

seen t h a t

< i , x > € T => a (< i , x » £ (T p ) M

S i n c e M|=V=h (oopu{j5}) , Mf=RA left

(as in claim

to the reader.

Note t h a t

Definition

s d a : V=< M p , (T^) M >.

2 ) . The u n i q u e n e s s

claim i s

o

a l t h o u g h MC i s t e d i o u s

to state

i t has a single

II, a x i o m .

3 . 2 7 . MC is the theory MC;

Given MC , ri>0, define MC , as follows: Given a structure ( M,B_.. .B, fA) n n+1 — 1 k define a set T from A by the definition of lemma 3.24. Let Vyij;fyJ be the II sentence

asserting

that

MC+"MC "+ Vi,x(D(i,x) n

(M? > is amenable =• "\l)(
.

(where I|J is T..). Then MC _ is 1 n+1

Exercises 1.

Suppose t h a t in definition

3.15 F.. g . . .F, 7

Show t h a t corollary 3.19 would s t i l l

+ N

had been omitted.

hold.

2. Show t h a t lemma 3.16 i s not a theorem of R . CO

3. S u p p o s e TT:M-*£ N c o f i n a l l y a n d < M , T > is a m e n a b l e . L e t T' = = U Tr(xflT). S h o w t h a t TT:<M,T>-* < N , T ' > a n d < N , T * > i s a m e n a b l e . x€M Lo 4.

R e p e a t e x e r c i s e 3 f o r TT:M-*-V N . 2J2

5. Use MC to formulate an iterated version of lemma 3.26. n 6.

Suppose P ( 5 ) n J x * P ( 6 ) n J x + 1 ,

also that either

6€cop. Show t h a t coA+6 e x i s t s . Show

sup{coX+6: 6€cop} i s bounded in On or 06P.

31 4: MODELS OF R

I t i s more c o n v e n i e n t i f we use ZFC a s our b a s i c t h e o r y i n t h i s chapter. D e f i n i t i o n 4 . 1 A model M=(M,E,A . . .A > of R is standard provided (i) if M' is an initial segment of M (that is, x€M' and yEx => y€M') and EO(M')2 is well-founded then M' is transitive and Ef)(M')2=e0(M')2; (ii) U3+1 is an initial segment of M (or On =U) ; . Lemma 4 . 2 . if $=R is standard then M^=R . Proof: Suppose Mf= n i s a non-empty l i m i t segment of a>. Then n i s a non-empty l i m i t segment of u . So n=w . So M(=n=u>. D By c o n s i d e r i n g o n l y s t a n d a r d models we s h a l l n o t be embarrassed by the l o g i c a l l y complex axioms of d e f i n i t i o n 2 . 1 2 . The n e x t lemma i s a form of t h e Mostowski c o l l a p s i n g lemma: Lemma 4 . 3 .

Suppose t£=R and let

Jf is isomorphic

to a standard

z={n.-M^n6u)}. If Eftz

is well-founded

then

model of R.

Proof: Set T(x)={z:zEy , some n} where Mf=y =Unx. Set Z= ={x€M:T(x)2nE i s well-founded.} Z i s an i n i t i a l segment of M. For if x€Z and yEx then M^ycUx so for n€oa MJ=UnycUn+1x, hence T(y)cT(x), so T(y)2flE i s well-founded, i . e . y€Z. Z PIE i s well-founded. Suppose AcZ,A:t:0. Take some x€A. Let A'=T(x)nA. If A'=0 then yEx -» y€T(x) -» y$A so x i s E-minimal in A. If A'*0 l e t y be E-minimal in A 1 . If t€A,tEy then since y€T(x), t€T(x); so t€A'; c o n t r a d i c t i o n ! Hence y i s E-minimal in A. is extensional. For if x,y€Z and x*y then M^=3t(t€xAt^y) say. So tEx, hence t€Z. So by Mostowski's collapsing lemma there i s a t r a n s i t i v e Z' and IT such t h a t

ir:< z 1 , e n z | 2 >=< z,Enz 2 >. If (M^Z)nz'*0

(4.1)

replace M^Z by an isomorphic

copy disjoint from Z 1 .

Set M'=Z'U(M^Z). Let TT'(x)=7r(x) if x€Z', x otherwise. Say xE'y <=» (x,y€Z ' Ax€y) v (x€Z ' Ay^Z ') v (xryf Z ' AxEy) . We complete the proof by showing that <M',E' > is standard. N is an initial segment of Mf and N

is well-founded.

Suppose

Let N ^ T T " ^ .

32 N 1 is a well-founded initial segment of M': for if x€N, zEir1 (x) and z=7Tl(zl) then z'E'x so z'€N; hence z€N. Suppose x€N* . Then zET(x) => z£N'; so T(x)2flE is well-founded, so x€Z. Thus N'cZ, so NcZ', giving the desired result. Finally we are told that EDz

is well-founded, so ir:o)=z. If

Mf= " OJ exists" then M'|="u) exists". But E'n(a)U{oj})

is well-founded

1

so a) is a) in the sense of M . Thus OJ+1 is an initial segment of M 1 .• Next some terminology. Generally speaking t M(y,...y ) denotes that y such that M)=y=t (y. . . . y ) . The M may be a subscript or a superscript as convenient. For classes there is a small problem. Suppose n is a class of M defined by <$>. The natural choice for n

would be {x:M^=<j) (x) } .

Admittedly inside M we have written x€n - which should be read xE TJand now we are writing x€n in the real world. But x€n is only a convention with classes, whereas classes of M are sets in the real world, so that there is no ambiguity. Ambiguity does arise, though, when a class of M happens to be a set of M. In M we identified n with a set; in the universe the corresponding set is {x:xEr|}. Really this causes no trouble and we only mention it to account for some apparently bizarre uses later on. Three conventions: firstly the use of €., is restricted to cases where M is standard. This is handy when both a collapsed and an uncollapsed model are around together. If M is not standard we must explicitly give a name to its membership relation: it will usually be E. Secondly, we do not write vJJ^ but simply N n ^. If n=0 this is unambiguous as V =N. Finally, to avoid cumbersome expressions we do not underline structures that occur as subscripts and superscripts so that, for example, p N denotes p.,. Lemma 4.4. (The condensation lemma)

Suppose x-
—^——————

LT~

model of R. There is a standard

model M_ of R and a map TT with

r

n:M=NJX. If X is

a Em substructure of — N then in addition if:M+ — vL N.— P r o o f : R h a s a s i n g l e n 2 a x i o m s o N|x|=k. A l s o OJCX; s o t h e lemma may be p r o v e d by a p p l y i n g lemma 4 . 3 t o N | X . Lemma 4 . 5 . (resp.

If $=R

(resp.

RA) then f£=R

RA).

Proof: R 4.2,

Suppose TT:M+yN with M_,N_ standard.

•

i s IU a x i o m a t i s a b l e

s o Mj=R . S u p p o s e N(=RA. M^=R

and t h e a x i o m o f a c c e p t a b i l i t y

i s TI2 s o M^=RA.

There i s a l s o an upward form o f lemma 4 . 5 .

•

by lemma

33 Definition

4 . 6 . TT.-JV-^ cofinally

Lemma 4 . 7 . J f TT;JV-VV cofinally

provided

ir.-tf-^tf and Vx€w3y€tf tf[=xe7r(t/; .

then ir.-iv^N.

P r o o f : S u p p o s e N^=3x<J> ( x , TT ( Z . . ) . . . TT (Z ) ) w i t h 4> E Q . S u p p o s e N^4)(X,TT(Z1) . . .Tr(zn))

a n d NJ=x€ir(y) . T h e n N^3X€TT (y) <J) ( X , T T ( Z ; L ) . . . 7 r ( z n ) )

so N|=3x€y<J> (*r z ± . • . z n > • T h u s H=3x ( x f z1. . . z n > . Lemma

4.8.

If v.-M+^N cofinally

and t£=R then

a

$=R.

Proof:Use t h e f i n i t e axiomatisation of chapter 1. E x t e n s i o n a l i t y and foundation a r e IL, so they hold in N by lemma 4 . 7 . Suppose u,v are in N; we have t o show t h a t N^F^UjV) e x i s t s

for 0
R i s RK. Suppose NJ=U€TT (U 1 ) ,V€TT (v 1 ) . Then J^=F i "u l x v t e x i s t s lemma 1.7.

Say

M(=w=FiIIul

f

by

1

x v ' . Then M(=Vx€u VyEv 3t6w t = F ± ( x , y ) . So

Nf=VxeTr(uf) Vy£Tr(v') 3t€7r(w)t=F i (x,y) . In p a r t i c u l a r NJ=3t t = F ± ( u / v ) . a Lemma 4 . 9 . If TT;#-* tf cofinally

and irro) -HO T onto then if d=R there is a

standard Nff=i? with a.-N'=iV_ and a l - f J '

cofinally.

Proof: By lemmas 4.3 and 4.8. Lemma 4 . 1 0 . If M_,N_ are standard then NJf=R

(resp.

Proof: N(=S

a models of R,^:M^^N cofinally

RA) .

e x i s t s for c o f i n a l v in On.,, hence for a l l v€0n

x in N, if Nj=x€7r(y) and M|=y€Sv then N^TT (y) ES^ , * Nf=R

and $=R+ (resp. RA) Given

(this i s Z^

so

as N is standard. For N(=RA observe that the axiom of

acceptability restricted to X+a)€On is IL: so let A be the largest limit ordinal. Let 6n be least such that for some ac6,a€S,, , ais,. ~~ A"i~n A 6 =7r B u t if f Then 6n€rng(iT); say n (\)satisfies the acceptability axiom for U=S,JT-1 ,^* + n , 6=6"n in M then ir(f) satisfies it for u = S ^ + n , 6=6n in N. This suffices. Lemma $=R+

4.11.

a and ^=R+ (resp. RA) then

If M_,N_ are standard models of R^M^^N

(resp.RA) .

Proof: All axioms are IU.

n

EXTENSION OF EMBEDDINGS Lemma

4.12.

isomorphism) Proof: are

If r(:Mh^ylr and N^=RA is

with

< N P , T P >f=RA+MC.

unique

standard

then

there

M,p' with

L e t T = { x € M : T P ( ir ( x ) ) } ; Mp'=Mf

s o <M,T>|=RA+MC

Tp =T. T i s unique

Note: E, was only needed to ensure <M,T>(=R .

x

are M_,p' (unique

up to

Mr =M_ and Mf=V=h(u>pUp') .

by

a

and

3.24(iii).

there

34 T h e M in lemma 4.12 m a y b e taken a s standard. It will b e a m o d e l of RA . P Lemma

4.13.

Suppose M_,N_ are standard models of RA . Suppose Tt:l£**v, f£ and

f^=V-h(UiQ\Jp') . Then there is a unique T? such that TT.-tf+ E

and

^(P')=P' Also

.-^-J, W. P

Proof: By lemma 3.24 TT:< M P * ,TP'> -* , so for xn . . ,x €M — LQ — x n , < x 1 / P » » . . . h ( i n , < x n , p ' » ) «* N ( = ( | ) ( h ( i l f < 7 r ( x 1 ) , p »

..(4.2)

...h(in,))«hN(if
f

p ».

This is well-defined since M=vf=h(a)pUpf) ; and by (4.2) TT:M->£ N. We must show TOTT. Let s(j o ) be the formula y=x; then h (j <x,p* >)=x, h N (j Q ,)=y. So ^(x)=^(h M (J 0 <x,p' » ) =h N (J 0/ <7r(x),p») Similarly It remains to prove uniqueness. Suppose Tr'rM-*--, N^TT'^TT and 1

TT (p')=p. Take y€M and suppose ^=y=h(i,< x,pf >) with x€M p . Then N^TT 1 ( y ) = h ( i , < TT1 ( x ) , 7 r ' ( p 1 ) » = h ( i , < TT(X) , p » = i f ( y ) .

D

In fact we can improve this result if TT is a stronger map.

Lemma

4.14.

Suppose

TT.-VP •*•-. N^ where

P^v=h(up\Jp'),

and N^=V=h(up\Jp) .

Suppose

h

Ihm,

7 r : ^ v N and Hfp')=p. Zo

Then 7T;/f>v

N.

~ h+i

Proof: Suppose <J> is a E. + , formula. For def initeness take h odd: so say (\> •• 3x 1 Vx 2 . . . Bx^

where ^ is II- . Let

D={t j,y >:MhJ€o) A y€ M P '

A

h(j f
D={< j,y >:N|=j€a) A y€ N P

A

h(j,
Say R(x",z) •» ^=x±=< j ± ,x^ >€D A z ^ k ^ z M C D

exists}

(4.3)

>) exists} A

A ^ ( h ( J 1 , < x ^ p t »...h(j h ,<x i ; / p' » / ^

...h(k n ,
Define R s i m i l a r l y from N,D,p. Then M|=(j)(h(k1/
M

p l

so there is a A, formula x such that

|(z) .

(4.4)

And there is a A, formula 6 such that M(=xeD *» MP'(=<5(x) . Let x'( z ) ^ e

tne

formula

(4.5)

35 (4.6)

3xi(6(x1) A Vx2(<5(x2) => ...3xh(6(xh) A X ( X , Z ) ) (4.6) is is Z h and ^(h(kir<2^p' ». Similarly for N. Hence n

n

<* N P j= X <7T(z|) , p » . . . h ( k But

if(h(kir< z^,p» »)=h(k

Lemma 4 . 1 5 . Then there

Suppose

±

M, iv are standard

are JV, p, unique

n

,
, p » .

< TT(Z|) , p » .

D

models of R and TT;^ +V iv

up to isomorphism,

such that

cofinally.

N^^£ and N^V=h(wp\Jp) .

Proof: By lemmas 4.10 , 3 . 2 6 and c h a p t e r 3 e x e r c i s e 3 . n Lemma 4 . 1 6 . conclusion

Suppose M, If are standard

of Lemma 4.15

models of R and TT.-i^ ->-v N. Then the

holds.

Proof: By lemmas 4 . 1 1 , 3.26 and c h a p t e r 3 e x e r c i s e 4 . o Lemmas 4 . 1 2 - 4 . 1 6 a r e c a l l e d t h e e x t e n s i o n of embeddings lemmas. SOUNDNESS In order to iterate the extension of embeddings lemmas we must first look at the notion of soundness. Definition 4 . 1 7 . M is p-sound means $=v=h(u>p\Jp) . M M Suppose p-sound is defined for all p€.PA . Let (p....p _ )€PA ,._ Then n i n+i n+x . M is p-sound provided M is (p^-**p )-sound and M l " n is p .-sound. ~ — I n — n+1 Lemma 4 . 1 8

Suppose

lk=RA is p-sound.

Then t&=RA and

p€Pu-

Proof: L e t f€E,(M) with p a r a m e t e r p be such t h a t rng (f "up^) =P(On ) PIM Let A={^: £$f (£) }; A€£,(M) with parameter p . Suppose APloop =xntop w i t h x€M. Assume xcOn . Then i f x = f ( c ) , C€u)pM we have

Contradiction!

n

A similar proof shows t h a t if V=h(yU{p}), any p, then o)pcy. This was used in lemma 3.26. Lemma 4 . 1 9 . only if there

Suppose # is p-sound. is AS2h+1(M)

Suppose B£(wp)<(A). Then B isZ^Np)

with parameters

from pllfa)p;
if and

B^AOf^ (h>0) .

Proof: (in RA) F i r s t l y suppose A€E h+1 and B=Afl (cop)
36 x , p » ) «> V p (= X (
Z1(VP).

There is a E, map of some y€iop cofinally into On.

Call the map g. Suppose x€A P «=* 3y\|j(y,x,p) where i> is E Q . Say (f>(v,x) ~ (3y€Sv)i|;(y,x,p) . Obviously it suffices to prove the result if B is E Q ( V P ) . Furthermore, letting A'={< v,x ):\\) (v,x) } , since A p is rudimentary in
,A' >we may assume B is rudimentary in. But there wp

is a rudimentary function F such that |=x€B ~ F(x)*tf>.

(4.7)

Hence it suffices to prove that the function a(u)=A'nu is E 2 In fact it is IIlf for y=a(u) <* Vz(z€y «* A 1 (z) A Z € U ) Case 2

(4.8)

There is no E, map of y€iop cofinally into On.

Claim If H is E_ and u€V p then ——^— (_) Vx€u3yHxy => 3vVx€u3y6vHxy. (4.9) VP Proof: Suppose Vx€u3yHxy. Suppose u€S , y a limit, and suppose yP yP Y f:y+S

is E.(S

) and onto. Define g:y-K)n by

g(v)= the least y such that 3yGS Hf(v)y if f(v)€u = 0 otherwise. Then g is £,. So rng(g) is bounded in On;say g"ycy'€On. Then v=S , satisfies the claim.

•(Claim)

Now as in case 1 we must show y=a(u) ~ Vz(z€y « A P ( z ) A Z € U )

(4.10)

to be E 2 . (Vz€u) (Ap(z) => z€y) is certainly 1^. As for (Vz€y) (Ap(z) A Z € U ) , suppose A p (z) ** 3tip(t,z,p). By the claim (Vz€y) (3t) (i|;(t,z,p) A Z € U ) « (3v) (Vz^y) (3t€v) (^(t,z,p) A Z C U ) ; and this last is 2^. So y=a(u) is E 2 . Now an easy induction gives L e m m a 4.20. if

and only

if

D

Suppose M is p-sound, p£PAM. Suppose Bcru)p";
that

M

B=AOffp. (h>0) . Corollary 4.21. definable

Suppose M_ is

from parameters

in flop ) M

p-sound Up.

with p€.PA . Then every x in M_ is E

37 Now we a r e i n a p o s i t i o n t o i t e r a t e t h e e x t e n s i o n of embeddings lemmas. Lemma 4 . 2 2 , Suppose N_ is a standard MjP* unique up to isomorphism

model of R and TT.'AHy N_ . Then there are

with M==M_

and Af

Proof: I t e r a t e lemma 4 . 1 2 . Lemma 4 . 2 3 . Suppose M,N are standard Then there

is a uniqueitti

p'-sound.

D

such that

models of R and Tf:My ->•_ N_ , M

Lemma 4 . 2 4 . Suppose i\,M_,N_,"n,p,p' are as in lemra 4.23 and in addition p-sound.

Then TT-^Y ^ ' Provided n+h

that

f

Then there are N_,p unique up to isomorphism that M_ is p'-sound)

np

U:M

Lemma 4 . 2 5 . Suppose M_,N_ are standard (provided

N_ is

ip

•+•_ lf . h

models of R and if:tf*P •+•„ N_ co such that N=i^

and N_ is

finally.

p-sound

.

Proof: Use lemma 4.15 to get the f i r s t thereafter. Definition

p'-sound.

Tf(p')=p and T[\M m:M^Pm+y /r^Pm for all m
step and lemma 4.16

o 4 . 2 6 . Suppose TT.-M -> N cofinally

and M_ is p-sound.

TT:/f*v, N_ is the unique map of lemma 4.23. TT is called

the n-completion

Suppose of TT to M_.

In fact the n-completion depends only on M and TT. For suppose is the n-completion of Tr:Mnp->v N" and suppose M is p'-sound

TT:M-»V N p

and TT:M

^

N 1 where N^N 1 . Then it suffices to show that N is

7r(p')- sound and N ' = N n

p

. We may restrict ourselves to n=l. Then

since M is p'-sound we may assume p=h..(i,< £,p' >) (iCa),^€a)p ) so 7r(p)=h._ (i,< TT(^) ,TT(P' ) >) , TT(^) €a)p.T . But N is if(p)-sound so N is r c ^ r N N — — 7r(p') -sound. ^ 1 p To show that N ^ ^ ' ^ it suffices to show that TT:< M ' ,A P ' >->„ — — M Zo Let y=A p fix. Then V i , z ( < i , z > € y •• V^±(z,pM) So V i , z(< i , z >e7r(y)

A€x)

«• N ' p ( j ) . ( z , 7 r ( p ' ) ) A < i , z >€TT ( x ) ) , a s 7 r : M + E N ' . T h u s

Tr(y)=AjJ (pl) mr(x) . In fact 7r:M->N

is the unique TT ' :M-> N * such that n (a) every y in N* is E definable with parameters from (b) TT' "wp
THE CONSTRUCTIBLE UNIVERSE J i s t h e unique t r a n s i t i v e model M=(M,€> of R w i t h OnM=a. We may deduce a l l t h e u s u a l f i n e - s t r u c t u r a l p r o p e r t i e s of J from

38 our previous work; the following is the main result: Lemma 4.27. For all a jJ=RA+. Proof: By induction on a. Suppose it true for all $
have t o show

Since ac60. If u)p6 for all m then for some p a€E, (Sa)a p ) and ajp cn-lp>6 S 1 S toa1 ~ coa so a^S^ icS . Thus a)pg_<6 for some m. So it suffices to show Claim For all m there is such that —S coot is c-sound. Proof: Let m be the least for which this fails. m>0, of course. Now there is A€Em-./such thatvA *xna)pm, all x€Scoot , where pPdenotes #p pmo . Say A€E,(S " ^l*" m-l) in parameter pm, say, with pm minimal ± —coot in the canonical well-order (which really i_s a well-order in this m case). Let X=h_m-10(a)p Up m) . Then by lemma 4.4 let~~ M be transitive, o p

TT:M=S

with M=M by lemma 4.22. Then V J wa ^ |'x . Let TT:M-> —E * m —a A€I (M). But M=J O some 3<wa and a^S so 3=wa. M is-rr (p )-sound; TT (p )

. / P >-sound.

Corollary 4.28.

a(Claim)

i\=GCH

P r o o f : L^=the power s e t a x i o m . T a k e a s u c h t h a t Ja|=P(Y)=Y+ b y c o r o l l a r y theorem.

3 . 4 . S o i n L P(y)
P(y)€J

. Then

s o P(y)=Y+ by C a n t o r ' s

D

Definition transitive S

n

4.29.

J^ denotes the unique model <«,€,!'> of R* which is

and has On =taa , A!=Aj)M (0
These definitions

are unambiguous provided i t i s clear that we

are working in ZFC and not in some model of R. In a model with predicates, M=<M,€M,A> —

J

denotes a model with the same predicates.

—^

M

If there is any danger of confusion the l a t t e r must be indexed J . Definition

4.30.

Models of RA are called acceptable; of RA

strongly

acceptable.

Exercises

1. Suppose 7r:N-*£ M where N,M are standard models of RA. Do either of (a) p€P N *1TT(p)€PM; (b) 7r(p)€PM => p€P N

39 necessarily hold? 2.Show that there is a non-standard model of R. 3. Suppose M,M' are standard models of R. Suppose 7r:M+ and <M,A> is amenable. Show that there is a unique A TT:<M,A>+

1

M 1 cofinally

such that

<M' ,A* >.

4. Show that if N=J

then N n p = J n for all n€u>. PN

5. Say rud+(M)= the closure of MU{M} under

F

i'"Fi5+N«

Show that if <M,A>|=R+ and M is transitive then if M'=rud+(M) and M'=<M',e,A>, then M'f=R+ and M=s5?' .

40 PART TWO: NORMAL MEASURES

In this part we examine Kunen's theory of iterated ultrapowers. Most of the non-fine-structural apparatus is developed here; many important applications are deferred to part three, where finestructure is reintroduced. Much of our subsequent work could be done using the iterable premice developed in chapters 5 and 6. Indeed the core model will be seen to be the union of those iterable premice whose projectum lies at or below their measurable cardinal; and the existence of 0 # is equivalent to the existence of an iterable premouse. Nevertheless we shall prefer to use "mice", which are indeed iterable premice, but which satisfy a weaker condition on the projectum than that stated above. Reasons will be given at the appropriate point. Generally speaking the message of this part - especially of chapter 7 - is that if N is a premouse with critical point K then nothing that affects P ( K ) is altered by iterating N. The reason why we might nevertheless want to iterate N is supplied by lemma 8.18; iterating restores to premice the L-like simplicity of levels of construction where a<$ **> J =J 8 . In part three we shall see that we are still living in a reasonably L-like universe.

41 5: NORMAL MEASURES AND ULTRAPOWERS

The dominant theme of studies of large cardinals has been the existence of embeddings of the universe into itself. Large cardinal axioms originally formulated in other terms have been translated into statements about embeddings; and these have revealed their connection with other axioms. Measurability is one of the best examples. The original definition was motivated by classical measure theory. Early results proving measurables inaccesssible were purely set theoretic, with no "model theoretic" methods used. Scott used an embedding equivalent to obtain far stronger results - including the absence of measurables in L. From our point of view, as in recent applications of forcing in large cardinal theory, the embeddings are central. Definition

5.1

Suppose j:#*v. N where M, N are standard models of R. — Lo—

Suppose that € 0(j"K)

is an initial

If wj=£<jfK; , then K is called then K i s the smallest Definition

5.2.

— —

the critical

K such that

^

segment of 6^ and £=sup (j"K)

exists.

point of j . (If M_,N_ are

transitive

j(K)>K).

K is measurable provided K is

the critical

point of some

j:V+ M, where M is an inner model.

Note t h a t d e f i n i t i o n

5.2 i s made in ZFC; o t h e r w i s e "measurable"

not d e f i n e d . Also note t h a t d e f i n i t i o n

5.2 i s i l l e g a l a s

q u a n t i f i e s over proper c l a s s e s . A l e g a l e q u i v a l e n t i s provided Until further

is

it later.

n o t i c e K i s a fixed measurable c a r d i n a l and

j:V^ M with K c r i t i c a l , e Lemma 5 . 3 . K>W Proof: j(co)=o) and j ( n ) = n , a l l n€u). Lemma 5 . 4 .

o

K i s a strong limit

Proof: Suppose yK o n t o . So j ( f ) :Pw(y)+J(K) But i f

acy then j ( a ) e j ( y ) = y ;

3€a «

j(3)€j(a)

~ B€j(a)

and for 3
onto.

42 so

j(a)=a.

Suppose

of

course.

So j ( K ' ) = j ( f ) ( j ( a ) ) ,

Corollary

Lemma

5.5.

5.6.

Proof: Suppose

K is a

Kis

Suppose

K=j(f)(a)

with

a€PM(y).

i . e . K'=j(f)(a).

Kf=f(a); K'
Contradiction! D

cardinal.

regular.

y
cofinally.

j ( f ) ("3) > K , 3"
3'=j(3I)=j(f)

Suppose

Then

j (f) : y * j (K)

cofinally.

Then

(j("3))

=j(f)(3)

>K

>3'. Contradiction!

•

Corollary 5.7. K is strongly inaccessible. For a long time it was an open problem whether K might be the least such. The following lemma shows that it may not. Lemma 5.8. K is not the smallest strong inaccessible. Proof: Since K is strongly inaccessible it is strongly inaccessible in M. If K were the smallest strong inaccessible then M(="J(K) is the smallest strong inaccessible". But K<j(K) . Contradiction!

D

There is a direct proof that V*M (see corollary 24.4); but we will come upon a form of this result as we proceed with our alternative characterisation. NORMAL MEASURES Definition 5.9. Suppose f^=R. U is a normal measure on K in M_ provided (ii) xEu A XOY, ye? (K) => YEu (Hi) (i)'(iii)

XEU A YEU + characterise

a

filter.

(iv) XEU v KNK€[7, for all

xEPtK)

M (i)-(iv) characterise an ultrafilter. (v) $XJ Such an ultrafilter is called proper. (vi) for all y is (in M) a sequence with X Eu for all y < 6 « ; t h e n H X €£/ y<6 Y

43 Such an ultrafilter

is called

K-additive

- the influence

of classical

theory shows through in the terminology - and a K-additive ultrafilter

is called

(viii)

measure

non-principal

a measure, or a two-valued measure.

Suppose (X :y is

(in M) a sequence with X €.U for all Y
then {6:(VY<6)6ex }£u. {5; CVy<&)<S€x } is called

the diagonal intersection

of (x ;6.

There are redundancies in t h i s l i s t , of course. Some clauses are inserted merely for the intermediate d e f i n i t i o n s . Note t h a t there is a II, formula < > j such t h a t whenever <M,U>f=R we have: U i s a normal measure on K in M «* < M,U )\=<$> (K) . (5.1) We do not i n s i s t on <M,U>j=R as p a r t of the d e f i n i t i o n of normal measure, however. Note t h a t K€U (sincetf>$U); t h a t Y
6
Let

(i)

Suppose K is measurable. Then there is a normal measure on K (in V) U={X€P(K):K€J(X)}.

Immediate.

(ii) (iii) (iv)

K€j (X)

A

XcY => K€j (Y) .

j(XflY)=j(X)nj(Y) J(K)=J(X)UJ(K^X)

(v) K^j(0) . (vi)

(vii)

K { J ( { Y } ) = { Y > (Y
Suppose K^j (X ) , a l l Y<<$. Then j ( D X ) = n j (X ) a s <5
so K € J ( n x ) .

Y

^<5

Y

^

"

Y<6 Y (viii) Suppose KGJ (X ) , all y
But (VY
D

The converse to lemma 5.10 is one of our essential tools. That is, we shall obtain from U an embedding j:V-*eM. A word of warning, though: this procedure is not an inverse to that of lemma 5.10; if we start with j and let U={X€P(K):K€J(X)}, and then obtain an embedding from U we shall not necessarily return to j. Cases where this fails are encountered in part six. D e f i n i t i o n 5 . 1 1 . Suppose f^=R+AC is standard. Suppose j:M^y W with K Suppose U is a normal measure on K in M_. Then j ••#""*"# means (i) For all x€N, there is fOi, f:K+M such that $=x=j(f)(&) where

critical.

44 For all x£?M(K) M N_ is called the ultrapower (ii)

Suppose

x£u ** of M_ by U.

jrM-^N.

Lemma 5 . 1 2 .

j:H+v N

cofinally.

— Lo —

Proof: Suppose x€N. Say x=j (f) (£) with f€M, f:K+M. Suppose MJ=y=rng(f). Then so in p a r t i c u l a r N(=x€j (y) .

Corollary

5.13.

j--^v N_r f^ t->\

Proof: By lemmas 4.7 and 4 . 8 .

a

Suppose from now on t h a t N i s s t a n d a r d . This we may s a f e l y do a s M N i s standard/

and K>CO ( s i n c e £ cannot be w ) .

Lemma 5 . 1 4 . Suppose j':M+ N_'. Then there is an isomorphism O:N=N_' such

that

Oj=j' ,0(&)=&', where ^'\=&'=sup j " K . Proof: F i r s t of a l l o b s e r v e

that

Nt=JU) ( f c ) = j ( f ' ) (fc) ~ N ' h j 1 ( f ) ( f c ' ) = j ' (f ' ) (l^ 1 ) .

For

and

f

(5.2)

^

s i m i l a r l y f o r N 1 . B y t h e same

argument

N(=j(f) (f5)€j(f ') (g) <=» Nf|=j f (f) (($ f ) €j f (f f ) (^f) N|=A k (j(f) (£)) ** ^ ' ^ ( j

1

(5.3)

(f) (£')) .

Definea:N-vN' b y a (j (f) (£) ) = j ' (f) (£ ') . B y (5.2)-(5.4)

(5.4) this is a n

i s o m o r p h i s m . G i v e n x € M l e t c : K-»-M b e defined b y c (£)=x, a l l ^ < K . T h e n j ( x ) = j (c x ) (1^) . S o a(j(x))=a(j(cx)(«)) = j ' (c x ) (l^1)

Half of the above argument shows Lemma

5 . 1 5 . If j':n+ir with K critical and x£u «* (^'€j' (x) then there is

a;tf-ȣ N' such that Oj=j'.

T h i s g i v e s a " u n i v e r s a l " p r o p e r t y of j:M+ N . T h e a b o v e r e s u l t s w e r e u n i q u e n e s s a s s e r t i o n s : w e a l s o need a n existence

theorem.

45 Lemma 5 . 1 6 .

Suppose U is a normal measure on K in f^=R+AC, Then there

are

W,j such that jiMP-yfl.

Proof: Let oMf£M: f:K-*M}. Suppose f,g€T. We say f~g ~ {£:f (C)=g(^)}€U fEg « U :

(5.5)

Then ~ is an equivalence relation on T and a congruence with respect to E and A, . Let T=T/~ and define [fjE[g] « fEG (5.6) A R [f] ~ A k f where [f] is the equivalence class of f. Claim 1 For any £ Q formula <j> < T,E,A >f=cf> ( [f^] ... [f^]) «* {^:<|)(f1(5) . . .f n (?) ) }€U. Proof: By induction on E Q structure. Case 1 <j> is atomic. This is immediate. Case 2 $ is -ij;. Then < T / E / A H ( [ f 1 ] . . . [ f n ] ) «* < T , E , A ^ ( [ f 1 ] . . . [ f n l )

Case 3 (|) i s

I|^AX.

Then f=(j)([f 1 ]...[f n ])

** < T,E,A >H ([ f ±] . . . [ f R ] ) ,X (

Case 4 <j> is (3v.€v.)if> If < T f E , A ^ ( [ g ] , [ f 1 ] . . . [ f n l ) A [glCEfjlthen

j

Conversely suppose {^: 3v ± €f ^ (^)i|;(vif f x (?) . . .f n (5)) >6U. Let g(^)={x€f j (^):^(x / f 1 (^)...f n (C))>. Then g is a set in M. Since M^AC there is h in M such that Mt=h(S)€g(S) whenever g(^)*0. h:K^M. But U:g(£)*0}€U so {^:h(5)€g(5)}€U. Thus A i|;(h(5),f1(5)...fn(C))}€U so j A ^([h],[f1]...[fn]) o(Claim 1) Note 1: If M[=ZFC then g would e x i s t even without the bound on x. Hence claim 1 would hold for a l l cj>. Note 2: Claim 1 only u s e s the f a c t t h a t U i s an u l t r a f i l t e r . Recall the d e f i n i t i o n of c from lemma 5 . 1 4 . Let j ( x ) = [ c ] . Let N=< T,E,A>. So j:M+

H

N. For i f <J> i s EQ then

46 ^ ..xn) }€U

(by claim 1)

« M|=c|)(x1...xn) If M|=ZFC then j:M+ N. Claim 2 K is the critical point of j . Proof: First we must show that J"K is an initial segment of On.,. That is, if N|=x<j(3) and MJ=3€U by claim 1. Let Xftt = U : f (£)=3'}. So

fiIUftXft,€U;

p

hence for some 3'<3

Xft(€U. That

p

P

is, N)=[f] = [ c 6 , ] , i.e. x=j(B'). We shall now show that if \ is id|K then [i]=£=sup J " K . Clearly if 3 < K then {£:3. So £€ n Xft => f(^)>^. So 3 3 {E,z^e fl XR}£lJ, hence for some 3 X Q $U, thus {^: f (£) =3>€U. So [f] = [cft]. 3<^ P P P Hence [x]=f^. But [i]<[c K ], as { ^ : ^ < K } € U . n(Claim 2) Claim 3

X€U «* Nf=[ i ] €[c x ] .

Proof: X€U «* {^:^€X}€U. Claim 4

D (Claim 3)

N|=[f] = [c f ]([i]).

Proof: {^:f(C)=f(?)}€U.

a(Claim 4)

Hence j:M-> N.

D

Note that we can always replace N by an isomorphic copy in which 7r|K=id|K and [id|i<]=K. Unless we specify otherwise this is assumed from now on. Lemma 5 .17 . Suppose U is a normal measure on K. Then K is measurable. Proof: Note first that Ooo; OJ cannot carry a normal measure. Let j:V-*M. Then j:V-> M and K is critical. Take M standard. By the Mostowski collapsing lemma it is sufficient to show that M is well-founded (clearly for all y€M {x:x€ y} is a set). Suppose M were not well-founded and <x :n€a)>a sequence with x

€

n+l Mxn'

a11

n€a)

"

Then

su

PP°se

x

= n

J

Let X= n X ; then X6U, as K>O>. Take n€w n f n + 1 ( ^ ) € f n ( ^ ) ; contradiction!

^£X. Then for all n •

Now we give some computations in ZFC that are needed later. Suppose J V M (2K)M<j(K)<(2K)~h.

Lemma 5.18. K

M

Proof: ( 2 ) < J ( K ) as MJ=j (K) is a strong limit. But j (K) ={ j (f) (K) :

47 :

f:K-><} so J T K T £ K K = 2 K . Hence J ( K ) < ( 2 K ) + .

D

Corollary 5.19. V*M

Corollary 5 . 2 0 . v+L Proof: The only inner model o f L i s L. Lemma 5 . 2 1 .

D

if cf(\)*K then j(\)=sup j(a).

Proof: Suppose j(f)(K)<j(X). Then {£:f(£)<X}€U. Assume without loss of generality f:K-»-X and let y=sup f"K. If cf(X)>K then y<X and j(f)(K)<j(y) so j(X)=sup j(a). a<X Suppose cf(X)
If X>K and X is a strong limit cardinal with cf(X)*K then j(X)=X.

Proof: By lemma 5.21 it suffices to show j(y)<X whenever y<X. But JTyTlif: f:K+y}=yK and y K £ 2 Y # K < X , as X is a strong limit cardinal. Hence j(y)<X.

•

In particular if X > K is strongly inaccessible then j(X)=X. Lemma 5.23.

if cf(\)=K then j(\)>\.

Proof: Say X=supX ( X strictly increasing). Then j(X)=sup j(X).

^ But

X=sup

a
a

X<sup

a
a

"a
^a

j ( X) < j ( X ) ^ < j ( X ) . a

~

These r e s u l t s make e s s e n t i a l 5.3-5.7

a

K

use of

ZFC; on the other hand lemmas

can be proved in R, giving

Lemma 5 . 2 4 . inaccessible.

Suppose u is a normal measure on K in M_. Then ^ K is strongly

See exercise 7.

Lemma

5.25. •

Suppose

j:M+ N with — U—

U normal

on K in M. Then —

V ,(K)czP (K) . M — N

Proof: This is loosely formulated, and involves an assumption about the relation €.,; the reader should make the formulation N precise. From now on we shall be less careful about these statements because they can become very cumbersome. Suppose M[=acK. Then for y
•

48 If M^=ZFC then equality holds. In general Lemma 5 . 2 6 . Suppose j:tf+ N_ with U normal on K in M_ and <M_,U^=R. Then Proof:

Say a=j(f)(ic).

Then

3€a ~ B€j(f) (K) «

U:B€f(5)}€U.

A s <M,U>|=R { 3 : { £ : B € f ( £ ) } € U }

Definition

i s a s e t o f M.

If #=<#,€ ,uJi)\=R+ is standard and U is normal on K

5.27.

in M_ then M_ is called

a premouse. K is called

If M_=<Mfe^yu>then M_

is called

Lemma 5 . 2 8 .

D

the critical

point

of M_,

pure.

There is a IT sentence

0 such that for all

standard $=R+

M(=a *» M_ is a premouse. "troof:

Use t h e f o r m u l a

C o r o l l a r y* 5 . 2 9 .

If

of

(5.1).

-n:M+_ 2, 0 —N cofinally,

D

M is a premouse and — —N is

standard

then N is a premouse.

C o r o l l a r y 5.30.

If j:M+ N_ and M_ is a premouse then N_ is a premouse.

Therefore we can take an ultrapower of N. This is explained in the next chapter. Exercises 1. Suppose jrV-^M. Show that U^M. 2. Show that if K is uncountable and carries a measure then K carries a normal measure. 3. Suppose jrV+yM with K critical. Show K + = ( K + ) M . 4. Show that if K is measurable then K is the Kth strongly inaccessible cardinal. 5. Suppose K is measurable. What large cardinal properties will K have in L?

(In 1-5 assume ZFC).

6. Suppose U is normal on K . Suppose f:K-M< and {£:f(£)
49 6: ITERATED ULTRAPOWERS

We have just observed that if M is a premouse and j:M+ N then N is a premouse, so that if N= we may repeat the construction. And so on. We should thus obtain premice N

and maps j :N -> N n + 1 n

where N n = < N n 'u n * • T n e maps may be composed to give 3mn:Mm-**£o N n cofinally whenever m
6 . 1 .

A c o m m u t a t i v e s y s t e m i s a p a i r ( ( A ) . . > , ( IT

"

~ 0 t Ot
o

)

CXp

where A \=R, \>0 and

(iii)

iraa -id|Aa

Definition is a pair

6.2.


(i)

a€A there

Suppose

commutative system that

of a commutative

system

< < - ,< IT g>


for all

Lemma 6 . 3 .

limit that

(X CX^A

————————

such

A direct

> ., > s u c h

^a-Vjii

(iii)

(a<\)

is a<\ and a€A

such

^ , > and —

(X Ot
Ot

that

a=TT (a) .

. > are direct

limits

Ot^A

< , « . , ) . Then t h e r e i s an isomorphism —a ot
of

the ^

O:A=A'

a a _ _ Proof: L e t a(ir (a))=7r ' ( a ) . Claim a i s w e l l - d e f i n e d . Proof: Suppose TT ( a , ) =Ttg (a 2 ) , a£ 3 . Then ir B(iraB(al))"iro(al) =TT 6 (a 2 ) so

ir

a =a ag( 1) 2-

Hence ir^ (Trog (aj^J^iTg ' (a2) , so TTa'(a^^) =iTg ' (a2)

a (Claim)

The claim in reverse shows that a one-one; it is clearly surjective. If h={A,E,A1.. .A N> and A'=< A,E' f^' . . .AN" > then a )E a

V i V 2> - V ^ i ^ ' V ^

just as in the claim, and similarly for A, ,A, ' . So a i s an isomorphism, air =TT ' by d e f i n i t i o n . D

50 Suppose < < ^/^ 1T o^
Lemma 6 . 4 . it has a direct Proof:

If

a commutat

^-ve

system.

Then

limit.

A = Y + 1 t h e n l e t A=A , IT =TT . S O s u p p o s e X i s a

limit

ordinal. A function

f:X+ U A i s c a l l e d c o n v e r g e n t p r o v i d e d t h e r e a<X i s y<X s u c h t h a t f o r a l l 6 w i t h y£6<X f ( 6 ) = i r g ( f ( y ) ) . L e t A* = { f : f:X-> U A A f i s c o n v e r g e n t } . F o r f , g € A * s a y a<X a f~g i f a n d o n l y i f t h e r e i s y<\ s u c h t h a t y<_6<\ => f ( 6 ) = g ( 6 ) . Then

~ i s an e q u i v a l e n c e r e l a t i o n .

Let

A=A*/~.

Suppose A = . Let *^ —a a' a a a fE*g ~ (3y<X) (V6) (y±6<\ => f (6)Egg(6))

(6.1)

A*g « (3y<X) (V6) (Yl6<X => A^g(6)) Then f~f',g~g',fE*g =» f'E*g'; similarly for A*. Let [f] denote the ~

equivalence class of f. Then define [f]E[g] <* fE*g

(6.2)

A k [f] o A * f . Let A=< A ^ ^ j ^ . . -A N >. Define t a :X+ U A X

^<X t^(Y)=O if Y
by

Y

t (y)=7T x aY(x) Y-a where Y<^-/ X ^ A . Define IT :A ->A by TT (x)=[t a ]. We shall show that

<

^'<7rct> < x } If

=wa6(x)

is a direct 6>3 t h e n t ^ (6)=TTa6 (x) a n d t £ ( x ) ( 6 ) =7T36 (7Ta3 ( x a ft ft ^ t»(«)-t^(xj«)8O ^ ~ t ^ ( x ) , i.e. irp(iraB(x))-ira(x).

a£3<X so

Given

and

f€A*

suppose

6>;Y/<S<X =» f ( 6 ) = 7 T

g

(f(y)).

Let

x=f(y)-

) )

Then

Y f-t^ so [f]=-n (x) . x y It remains to show TT :A -*-V A for a<X. This is done by J a -a Eo-

induction on Z

structure.

Case 1 <j> is x=y. Aa|=x=y <» AHr a (x)=TT a (y). =7r

ay(y)/

some y>a.

Case 2 $ i s

But i f

TTa (x) =TTa (y)

then t^~t^

so TT^X) =

So x=y.

x€y.

Aaf=x€y -> AY(=TTay(x)€TTaY(y) a l l

y>a,y<\

If t a E*t a , conversely, then there is y>ct with A (=TTa (x) €TT so x€y. Case 3 is A,x. Similar Case 4 $ is -T|J . A a h ^ ** A a ^ «• ^ Case 5 <J> is

*=> A|=cf>.

=

(y)

51 £ a N ~ Z^H

a n d

Aa|=X ~ A ^ and |

f

Case 6 <J> i s If A a H ( y , x 1 . . . x n )

A y € z t h e n A^i|;(7r a (y) / TT a (x 1 ) . . . 7 r a ( x n ) )

A 7ra(y)€

€iro(z).

C o n v e r s e l y s u p p o s e Aj=i|i(iro (y) . i r ^ x ^ . . . i r a ( x n ) ) a n d iTg(y)e i r a ( z ) , <x
X )

^* W l "-

7r

(x

cxe n

a n d Trg (y) C i ^ g (z) s o

) ) SO

A |-*(x 1 ...x n )

a

The proof of case 6 also shows Lemma 6 . 5 . If TT O:A ->V A^ f o r ot<$<X then TT :A ->•„ A for all ot<X. a$ - a Ei-3 a - a Zx Lemma 6 . 6 . If ir O:A ->V A O cofinally 1

for a<$
Otp ~ 0 t Lo —P

• ••

for

01 "~0t 2JO —

—

a<X. Proof: Suppose y€A. y=TTg (y) f o r some $
J f TT O:A ->•„ Ao cofinally ap ~ct Lo ~~P

for a<8<X then A\=R. ~~

Lemma 6 . 8 . If TT O.-A •>- A f o r a l l a<3<X then TT :A ->« A. Otp ~0l 2J — — CX "~0l L — n n Proof: By induction on n. n=O is known. Suppose n=m+l. Then

"~~~~~~~~~~~"^

T T ^ A ^ J , A. Suppose A^=3xip (x,TTa(y1) . . .TTa(yn) ) with \p U^. Say m ^ ( T T (x) ,TTa(y1) .. ,TTa(yn) ) ,3>.a. I ^ A Q + J . A so A>i(; (x.^g (yx) . . . m •••^aP^n^ *

Hence

^ l = 3 x ^ ( X / 7 r a 3 ( Y l ) # •* 7T a3 (y n ) }

so

^ H 3 x ^ (x '^l * • • y n )

#D

ITERATED ULTRAPOWERS 6 . 9 . Suppose
Definition

A pair < < 2 V > - . ^ / < T T O > . O >-^ ) i s an i t e r a t e d ultrapower of N by U provided ~~QL Otton otp ot^ptc/n — are U such that (i) << V £7 a > a€Dn' a (ii) t^N_f uQ=u (Hi)


i s a

(v) << V V ' ^ a x W Lemma 6 . 1 0 . exists

coirmutative

for all a€0n

Vl* whenever X is a limit

there

i s a direct

for aU

limit

°f

ordinal.

Suppose (N_,u)^=R+AC where U is a normal measure on K in N_. There

an iterated

ultrapower

of N_ by U.

Proof: Induction based on lemma 5.16 at successors and lemma 6.4 at limits; each U

is a normal measure on ir. (K) by lemma 5.28. a

52 Lemma

6 . 1 1 . S u p p o s e < < A^ > a £ o n , < ^

^

^

>and<(^

>a£On,< i r ^ ^

^

>

are

iterated ultrapowers of N_ by U. Then there are isomorphisms Oa:Na=Na such that (i) OQ=id\N (ii) O n ; (iii)

CJgfTT^

Proof: Induction based on lemma 5.14 at successors and lemma 6.3 at limits. n D e f i n i t i o n 6 . 1 2 . Suppose < < N > c

,< TT O ) ^ o - > is an iterated

—Ot OLfcOn

of N_ by U. Each N is called an iterate of N_ by U. D e f i n i t i o n 6 . 1 3 . tf i s iterable founded .

ultrapower

Otp Ot^ptOil

of N_ by U; N is called an ath iterate

by U provided each iterate

of iV by U is well-

D e f i n i t i o n 6 . 1 4 . Suppose N=
< (N >

,< TT g > < g £

> is an iterated

ultrapower

of N_ by U.

(K) where U is a normal measure on K in N.Then

a oa (i) K is the critical

point of 7T o

Ot

Otp

(ii) tfJ=K (iii) Nj\=K,=sup K (\£0n a limit) a<X

Proof: ( i )

o On t h e

But K =7ra (K )>^K f o r a l l y with a
hand

other

W^a'^I >K

.

— a (ii) Immediate from (i). (iii) Suppose N ^ p 1 ^ . Then 3=^(3*) then $ > K , ; so —

A

f o r some

ct<X,F. If ^>

3"
OtA

(iv) From lemma 5.26.

Qi

o

In the above, and generally, we write, for example, ot<3 to mean N [=a<3 if there is no danger of ambiguity.

53 We should refer here to a s i t u a t i o n t h a t sometimes a r i s e s . I t may be the case t h a t M|=ZFC+RTPM (K) cj^ 1 and <J^,U>f=R but U$M; thus <M,U>^R. Let N=J^; then we can define an i t e r a t e d ultrapower of N but not, apparently, of M. But actually we can cobble something up: D e f i n i t i o n 6 . 1 6 . Suppose N=J^ , ?M(K)<=J*^ , f^=cf(\)>K A H^=Jy- and suppose that (

% > a C O n ' < 1 [ a 3 W o n > i s an iterated ultrapower of^by U. ((Mo, >OitOn _ ,< TT'otp>a
< Vir'a>a<X>

(iv) a limit (V)

i s a d i r e c t

1±mit

as

H^ =JX (K -IT (K)) a a in definition 6.9) °f

<<M >

a a<X'a<3<X>

when

X l s

MQ=M.

Lemma 6 . 1 7 . Suppose N=J\, P (K)GJ and U is a normal measure on K in N. Suppose f^=cf(X)>K A H,=J, . Then M has a weak iterated

ultrapower

by U.

~ M' Proof: Suppose IT ': M-»-TTM' and T M N + N ' . We claim N'=J. , for someX' . U - JJ- -A Let X'=TT'(X). Suppose TT1 (f) (K) 6J, , . Then without loss of generality f:K+J^, so f:K+J^, some 6<X, so f€N. Let a (IT1 (f) (K) ) =TT (f) (K) . a is the required isomorphism. Clearly cf (X') >TT(K) in M 1 and M'^J^^H^,. Thus we may iterate at successor ordinals. Now suppose y is a limit ordinal, and suppose M ,irR are defined for a£3
and x=7r-«- (x) with X € J ^ Y and yy A^

J^yN ) . The reader i s l e f t t o check t h a t A ^N-y l e t a(x)=ir-yy (a-(x) y

this

works. Lemma 6 . 1 8 . Lemma 6.11 applies

to weak iterated

ultra

powers.

Exercises 1. Suppose < < M a > a € O n , < i T a 3 > a < e e O n > is an iterated ultrapower of V by U, a normal measure on K , and each N transitive. Find a bound on the size of K^. 2. Let < < N a > a € O n , < ^ a 3 > a l 3 e O n > be the iterated ultrapower of an iterable premouse N. Suppose N =. Show that (A)

X€U w *» 3nVk>nKk€X where X€P N (K^) and K a is the critical point of 1^. CO

54 7:

INDISCERNIBLES

Our aim in t h i s chapter i s to show that {KiBxeN- In other words given a E, formula $ and x€N and any two increasing sequences

N f=(K. ...K. , TT^ (X) ) «* <J)(K. . . . K . , 7T^ ( X ) ) . -oT

ix

in

Oa

(7.1)

D n Oa

D l

An argument of Solovay is then used to extract E from these {K O :3
indiscernibles

p

For the time being fix a premouse N with iterated ultrapower <

,( TT o )

Lemma 7 . 1 .

Q

); let K

For all

x€No there p where a
be the critical point of N .

n is f€N , N \=f:K -+V,and a —a1

there

are

y....y 1

n

,

Proof: Assume a=0 without loss of generality. The result is proved by induction on 3. Suppose 3=Y+lf Say TT=TT

an

^t

n e

result is known for y.

o.

Take some x€N o . Then No(=x=iT(f) (K ) for some f€N , YP P ~P Y Y N [= f:< ->V. By the induction hypothesis N (=f=TT (g) (K . . . K ),say. Define a function g • €N with g ' : Km+1->V by g'(a1...am+1)=g(a1...am)(ain+1).

NOW

(7.2)

^03(gl)(KY...KY,Ky)=TT03(g)(KY...KYJ(KY)

=ir(f) ( K y )

=x

(in N^)

Suppose X is a limit ordinal and the result is known for y<\. Given x€N, , X=TT

A

(X) for some y<\; say N NX=TT_ (f) (K^ . . . K ^ ) . Then -Y OY y1 Ym

YA

Lemma 7 . 2 . Suppose (J> is a E formula.

Then there

is a £ formula

$* such

for all a

i

Proof:

in<...

(|>* i s c o n s t r u c t e d b y i n d u c t i o n o n n . I f n=0 l e t <{>* b e .

S u p p o s e n=m+l. S u p p o s e <|>(y-,...y

/ x ) i s ^0«

Sa

Y

that

55 iMyi...ym,x)

~ U:4>(y1...ym,S.x)}€u

and l e tty*be such

(7.3)

that x)

(7.4)

for all a, i,<...
. . . K

±

,*

< X > ) <* N

± + 1

I n

N ( ^ i

- . . K

±

,TToi

+ ]

_(X))

( 7 . 5 )

n •• N ± h U : * ( K ± n i

1 n n . . . K ± ,C/TToi ( x ) ) } € U i 1 m n n

...K

i

n

,7T oi

±

1

(x))

m n

<* N |=i/;* ( x )

for a l l a, i.<...* b e ij;*. S u p p o s e , t h e n , that 4>(y-i---y /X) ** 3y^(y y n . - - y « / X ) w h e r e ip ± n , ± n is E . Let x k e the formula 3y€x'ip (y,y, . . .y ,x) and let x* be such that Na(=X(K± - . . K ± ,7rOa(x),TrOa(x')) <^#=X*(x,x') 1 n Then

. . . K

i

±

(7.6)

** 3 x ' GN N ^ B y e i T ^ ( x 1 ) ip ( K ± . . .

,irOa(x))

I n

(7.7)

1 •••Kin'7TOa(x)) «

S o we m a y l e t

<J>* b e

Corollary 7 . 3 .

3x'€N NaHx(Ki

3x'x*(x,x').

{K i ;i
Proof: Suppose i . < . . . < i Na|=4)(Ki...Ki 1

.

•

indiscernibles


,7TOa(x)) n

for < ^ / ^ ^ ^ g ^ -

< a , w i t h c> j E, a n d x € N . Then

«*N|=(j)*(x)

(7.8)

-N^K.^.K.^U)).

a

Corollary 7.4. Z C ^ ; O P C K ; -Z CN

Proof: Suppose X€I, (N) , X C K . Suppose 4> is E, and ^€X «* N)=(J) (^ ,p) . Then let X = U < K : N J = <j) (^ ,TTQa(p) ) } . Clearly X=X; and X is Z 1 ( N a ) . Conversely suppose X€S, (N ) .Suppose £€X <* N [=(})(C,p) . By lemma 7.1 there is f€N such that N |=p=ir .--yn))a

N

a

i

(f) ( K ± . . . K . ) . Say 1 n Then (J)1 i s Z 1 . And

. . . K ± ,irOa(f))

(7.9)

** N ^<J)'*(C f f) . Hence X=(C
a

O c c a s i o n a l l y a s h a r p e r form of lemma 7.2 i s Definition defined

7.5.

Suppose U is a normal measure on K in Af. Then un<=P(Kn) is

by: U *U

X€un+1

needed.

<* « ^ 7 . . . £ :

In

>:{£:<£...£ ,

In

56 7 . 6 . x€un is a Z, property of X,n (over M, where U is a normal measure on

Lemma

_ K in M) . X€Un «* 3f(dom(f)=n

Proof:

A f(O)=X A (Vm+l€n) (f (m+l) =

= { < £ - _ . . . £ n _ m > : U : < £-_...S n _ m ,C>€f (m)}€U}) Lemma 7 . 7 .

Suppose (j> i s a I

NJ=4>(
formula.

Then there

. . . K . > ,TT (x)) mrirf*(n,x) 1 n

A f(n-l)€U)

is a Z

(i <...
formula

D $* such

that


Proof: I f <> f i s Z then * i s (n=0 A 4>(x)) v (n>0 A {< K± . . . ? n >: 4) « K± . . . S n >,x) }€Un) . The adaptation t o <> j Z1 i s a s b e f o r e . •

(7.10)

GENERALISED INDISCERNIBLES S o l o v a y ' s g e n e r a l i s e d i n d i s c e r n i b l e s employ a combinatorial nent t o o b t a i n £ iinnddi s c e r n i bbllee s . The c e n t r a l notion i s t h a t of argument a f u l l sequence of i n d i s c e r n i b l e s . Definition

7.8.

Every ordinal is 0-good.

a is n+l-good provided a is a limit itself)

of n-good points

(and hence n-good

.

Note: a i s n-good i f and only if a i s a multiple of w , of course. (We count 0 as n-good). Solovay's proof was formulated in ordinal arithmetic: but we feel that the present approach gives more idea what i s going on, and, more importantly, may be generalised to models with many measurables, where the ordinal arithmetic becomes hopelessly confused. Definition

7 . 9 . Every

increasing

sequence

————————————


a ...a

with

IK

)is n+l-full

a -0 is

0-full.

J.

provided

(i) is n-full (ii) if a,<3
7.10.

ch (a)=max(m
_____________________

JJ

__

Lemma 7 . 1 1 . Suppose —————— ft

a <$ is n+l-full. ti+i IK

Then ch (&)< n

Proof: Suppose ch (B)_Lcn ( a n + i ^ ~ m ' s a v # T ^en m
Suppose is an increasing

sequence

<

3 I » . . 3 h > with

Suppose n o t .

some < a 1 . . . a , >

sequence of ordinals.

l ^ . . . a ^ ^ B ^ . . .3^}

L e t k be minimal such t h a t

and let< $-_.. . 3 , , >be n - f u l l

There

is

and <*k=$h-

where

the r e s u l t

fails

{$1...Bh,}2

for

57 ,} and Gu'^k-l (P r o v ^^ e ( ^ k>l) . So there are no

2{a, ...ou

{3 h , +1 ...3 h } with 3 n =a k and <3 1 -..3 h > n-full. Assume that a^ is minimal for the previous sentence. Then a^ is not n-good, otherwise < 3-^.. .3J-I rOt, > is n-full. Let a=sup{3 are n-full as a
D

A few examples may help clarify this. A sequence is 1-full provided that whenever i, is a successor ordinal i, , =i,-l. Hence • 2 2 12 2 2 <0,u)2+l,a) +3 > is not 1-full, but < O,o)2,o)2+l,o) ,0) +1,0) +2,0) +3 > is. The latter is not 2-full, but would be if 0) were added. The essential fact about n+l-full sequences is that from their ch ,, values we may determine the ch values of their n-full n+1

•*

n

extensions. D e f i n i t i o n 7 . 1 3 . Suppose
Then

> means (m....m )+ (n....n. i p nfx i K (i) X={u^...u^} / 0
m =max(n.,n) uh h (Hi) uh<j
Definition

7 . 1 4 . A?X

•" • Lemma 7 . 1 5 . Suppose {aJ...ajt>c{3

(0
s={<

mn.. .m >.-<jn....m H

\ n . . .n /

1

< 3,...3

> is n-full

. . . B } and 3 ^ a ^ . Then,

< Chn (^),. .P.,chn (*J*BJ

C

p

1


1

k

and < a . . . a > is n+l-full.

letting

V j

p

a h = 3 u (0
(CLJ

Suppose

and X = { u J . . . u J t }

Ch^ («k) >

Proof: Immediate from lemma 7 . 1 1 .

n

Conversely Lemma

7.16.

Suppose i sn+l-full

———-————————

J.

a n d <m . . . m

K

1

>+ p


'TlfX

^ . ( a j . . . n+J. JL

,,.ch . 7 (OL,)>. Suppose X={u .. .u,} ,u <.. .
K

1

K

JL

K

< 3I, . . . 3 p > with 3u,=a,(0
h' =u, .

Then l e t 3 h ,=a h .

Case_^

ch n (3 h ,)=max(n,ch n + 1 (a h )) =m h"

Vhf
Let h " = h ' - l .

L e t m=m, , . T h e n m
+

i(av1+i)-

Hence

^+1

i s

58 So sup{$3nii. Then ch (3)^m. Suppose ch (3)>m; then sup{3'<3:3' is m-good}=3/ contradicting the minimal choice of 3. So ch n (3)=m. Let B h ,=3. It remains to check that <3-,...3 > is n-full. So suppose B^i is not n-good and 3n,,<3<3n, (h'=h"+l) Case 1

h' =u.

Then a, ,<3
Vhl
52=ii

Since 3_, i s not n-good ch (3, ,)
a

That concludes the disguised ordinal a r i t h m e t i c . We are a t l a s t in a position to prove a r e s u l t about i n d i s c e r n i b l e s . Lemma 7 .17 .

Suppose <J) is a E

formula. Then there is a I,

such that whenever is n-full

formula (j)*

then

NJ=<$>(< < a . . ,Ka >^0QL(x)) ** AH*C< chn(a2)..

-chn(ak) ),chn«x) ,x).

Note: As in lemmas 7.2 and 7.7 (J>* depends only on (j) and not on W. P r o o f : By i n d u c t i o n on n . n=0 i s lemma 7 . 7 . So suppose < > j is fX) and ty i s II . Then t h e formula 1

«Y 1 ...Y p >rX,x,f) «* 4>(f (Y1..-Yp)/< Y u ..-Y u >/X) A

(7.11)

is n n . By induction hypothesisthere is IIn i^1* such that NaH'«K3i...K3

>.XfirOa(x)firOa(f)) -

(7.12)

« N^i|;'*«chn_1(31) ...ch n _ 1 (3 p ) >,chn_1(a) ,X,x,f) provided 3,...3 ,a is n-l-full. Set: >

)

f

First of all, suppose N |=<j>((y,< K

<

. .mp+]j«in1. . .m p+ ^€

(7.13)

...K >/7U (x) ) . That is, there is y

...K >f^n (x)) . By lemma 7.1 there are f€N and

and < Kg, . . .K^, > such that M a Hy = 7 T O a ( f ) (K^, . . .K^,) . By lemma 7.12 there are (3-j^. . . 3 ,a>3{3|. . . 3 p u { a 1 . . .ak,a} such that
L e t a,=3 1 1 n

'^'i~

(0
p

and X={u, . . . u w p + l } . Then JL

K.

NaH'«K3...K3>,X^Oa(x),uOa(f))

(7.15)

so N(=V; l *«ch n _ 1 (B 1 ) • • • c h n _ 1 ( 3 p ) > / C h n - 1 ( a ) , X , x , f ' ) . L e t ni h =ch n _ 1 (Bh) (0
m

p+1=chn_1(a).

59

By lemma 7.15 <m 1 ...m

+1

>-*n-1

x

< c h n ( a 1 ) . . . c h n ( a k ) , c h n ( a ) >. So

N H > * ( < c h n ( a 1 ) . . . c h n ( a k ) >, ch n '(a) ,x) . Conversely suppose N^=<j>*(,ch (a) ,x) .

(7.16)

>be s u c h tnat Let n h s s c h n ^ a h^ (0€A ~ and N H > ' * « m n . . .m >,m - ,X,xf£) . By lemma ~ '' „

x

pt-j.

I• •

V+l

7.16 there is an n-l-full sequence < 3-j_ - - - B

> with a n = 3 u

where

X = { u 1 . . . u k + 1 } / u 1 < . . . < u k + 1 and chn__1 ( $ h , ) =m h ,, 0
1

(7.17)

so

k

Lemma 7 . 1 7 i s one o f t h e major t e c h n i c a l r e s u l t s we n e e d . We can immediately d e r i v e some c o r o l l a r i e s . Corollary 7 . 1 8 . PCKjOZ ,(N )C P(K)C\1 , (N) * n+l —a. — n+l — Proof: J u s t l i k e h a l f o f c o r o l l a r y 7 . 4 . Corollary

7 . 1 9 . Suppose

<j> is a E

, formula

n and
n+l

sequences such that: (i) < a , . . . a / a > , < 3 , . . . 6 ,a> are 1 P 1 p Xii) chn(o.h)=chn($h) (0
...K a ,*0(x(x)) p

a^

. . . K >,< K Q . . . K Q > are #^ p a p

p

n-full

«d)rK3 ...K 3 1 p

^^M).

Corollary 7 . 2 0 . Suppose a is n-good and C={Kg.-$
for <

V

W

W

Exercises 1. Suppose 0
ft

i s an

elementary embedding. 2. Show t h a t {K o :$
c u < . . . < a < 8 n < . . •<3V,
UCX

60 8: ITERABILITY

This chapter begins with an embedding theorem which yields relative criteria of iterability; continues to some absolute criteria; and concludes with some important consequences of iterability. Review the proof of lemma 5.17. We used the fact that M=V together with K-additivity to obtain well-foundedness of the ultrapower. The reason that we needed M=V was that otherwise <X.:i€(o> might not have been a member of M, even if each X. were. Observe that we only used the o^-additivity of U (i.e. X ^ U for all i€a) => .^Q X.€U). We may formulate a weaker condition: Definition

8.1.

U is countably complete provided that whenever X.£[/ for all

j^x^'

i€o) then

In the case that M=V this is actually equivalent to o^-additivity. Lemma

8.2.

Suppose U is an ultrafilter on K. Then if U is countably complete

then U is 0)-additive.

Proof: Suppose Xi€U,i€a), but ^ X ^ U . Let Let

Y

i+l

=X

i'

i€a);

SO

Y =K ^i^a)xi' 0

so

Y

€U 0

*

l^V*'

Looking at the proof of lemma 5.17 it is only the fact that the intersection of X. is non-empty that is used to prove well1 U foundedness; so it might be hoped that if N=J

is a premouse and

U is countably complete then, letting j:N-*M, M is well-founded. But we want to go further and prove that N is iterable. Unfortunately the obvious inductive proof fails. Let ^ M a \x€On'* u 8* <8G0 * ^ e a n i t e r a t e < ^ ultrapower of N; suppose K is the critical point of N. Let K=TT (K) , for a€0n. We know (lemma 6.15(iii)) that Ka)=sup{Ki:i€o)}; let Xi=Ka)^KjLr for i€u>. M ^ N ^ ^ ^ ^ ,U) 0)

Then X. €U 1

0)

for i€a) but . Q X.=0; so U ltO)

1

0

could not possibly be c

)

countably complete. This also shows that countable completeness cannot be necessary for iterability, for N

is iterable if N is.

The proof of iterability from countable completeness uses the following very versatile embedding lemma.

61 Lemma 8.3. 111

order-preserving.

Suppose artf-* M where N and M are premice. Suppose f:0n+0n is — Lo— — —

,< ir o > oc respectively. Let -a

Suppose

A M # , £ . [ / , A . . . . A ) and M=4 M,eM,U'

, A ' . . .A')

. Let

>,< c , < T T ' D > ^oc > he iterated ultrapowers of N,M K ,K' be the critical points of N ,M respectively. Then there

are unique maps O such that

(Hi) oa« then a d if If o,ff-&J[ then " cofinally. Proof: By induction on a . Let a o = 7 r of fO) " Suppose a a i s d e f i n e d . Define < V l ^ a + l ^ o ^ f ( a + 1 ) b y ClaJjnJ, a a + 1 i s well-defined, o Proof: Suppose i s SQ. Then

Mf ( a + 1 ) (8 2)

-

•••ltf(a)f(a)+l«Ja(fn))(Kf(a)))

where

Proof:

a(Claim 1)

°a+1*Oa+1=
f ( a ) f (a+l) a a 7 r 0a f (a)f (a+D^Of ( a ) 0 = *0f(a+l)a =K (6 a+1) Claim 3 g a + 1 ( K e > f ( e ) ^ Proof: Suppose B
o(Claim 2)

f {a+1)

Hence a a + 1 ( K B ) - O ? + 1 ( i r a a + 1 ( =ir

f (a)f (a+D'
7T

Claim 4 a

f(a+l)

a(K)

o(Claim 3)

=
Proof: Suppose (3£a) . Given x€N

«»o a + l = i r 6 f + 1

there are f'

. . . KK

<

1

a a n ((a+1) such t h a t

d 8 (ie

B B) )==<< f (6) B N a + 1 |=x=

62 =TT

+ . (f

') (K

K 1

o n =a. Then

T

) (a 1 <. . .
f(a)f(a+l)7r6f(a)a(fl)(Kf(a1)---Kf«xn))

=a . a+l(x)' where f ( y ^ . - Y ^ - ^ =f ' ( Y ^ . .Y n ) •

Claim 5 If

Wr*f

then

Q(Claim4)

V

Proof: Suppose <J> i s L , <> j <=» 3yip where I/J i s E Q . Then (x 55f ( a f l ) ^ (*'aa+l l> •' • •••^l^n" * Mf(a+1)Nyew'(a)f(a+1)(Z)*(y,aa+:L(x1)...aa+1(xn))/say/ZeMf(a)

^f(a+l)N * -

(a

a+l

(x

l> • • - a a + l

(x

n»}

w

^f(a+l)^y^f(a)f(a+l)(z)^y'Tf(a)f(a+l ••• • • •77rrr(a)f( r(a)f(a+l)(aa(fn))(l
where x k = i r a a + i (fk> (Ka) (0•_. M,, . cofinally1 then a ., :N .,->•_ M., .. . a —a Do— f(a) a+1 —a+1 Zo—f(a+1)

cofinally. Proof:

^fCaJfCa+D^^a+l^aa+l'

SO rng(a

a + l ^ r n ^ U f (a) f (a+l) a a ) ' D(Claim 6)

This concludes the successor case. In the limit case, where a is defined for a<X, we may define o x ( H a X ( x ) ) = ^ ( o ) f a ) a a ( x ) . For if \^x) =* BX (y) , a<6, then *a y). But V aVB < x ) - * £ { o ) f (f»,aa<x) so a e 7T a B (x)=a e (y). ; hence =7r

S

'f

f(a)f(X)acc(x)s i n c e a :N ->_ a

A 7 r OA = 7 T f ( 0 ) f (A) a 0 = 7 r f ( 0 ) f ( A) " o f (O) ° =7r

(iii)

Of ( A ) a #

a , ( K)=a,iTrv,1,(K ) A a A a+lA a =7T

f(

=7T

f ( a + l ) f (A) I* a)

=K'

a

.

+l)f(A)a

(a
°a 6 ( y )=

63

(iv) < V K A ) = a A ( l T O X ( K ) ) =ir

Of(X)(K<) f (X) •

(v) a, is the unique map with properties (i)-(iii). For suppose a':N.+M-,,. satisfies (i)-(iii); given x€N, suppose f€N and a,<...
7T

...K

DC.

UA

))

CX

6f(X)a(f)(Kf(a1)"-Kf(anr

=a x (x). (vi) If each oa:Ea^Mf{a) then a , : ^ Mf ( X ) . (vii) If V N a + I o M f ( a ) cofinally for a<X then o ^ N ^ cofinally. For rng(a.) 3rng(ir' f ,, , a) . a The lemma was expressed for premice but i t i s easy to see that i t works too for weak i t e r a t e d ultrapowers. Lemma 8.3 i s very useful: see exercise 1, for example. I t immediately proves: Lemma 8 . 4 . Then N_ is

Suppose M is an iterable

premouse and N is a premouse,

0:N+ M.

iterable.

Proof: Suppose < < V a € O n ' < *«($ W o i ^ ' ' ^ ^ O n ' ^ ' a B W e o n 1 i t e r a t e d u l t r a p o w e r s of N,M r e s p e c t i v e l y . Suppose N i s n o t well-founded;

l e t f:On-K)n b e t h e i d e n t i t y a n d l e t a :N •*•_ M b e a s

g i v e n b y lemma 8 . 3 . S u p p o s e < x . : i € u ) > i s s u c h t h a t x

then a ( - + i )

a r e

e

x

-+i

e

x N

- / a l l i€co;

x

M

° ( - ) / a l l i € w . So M i s n o t w e l l - f o u n d e d .

Contradiction!

•

Corollary 8 . 5 . If J^ is an iterable premouse with critical K<$
point K and if

~~p

Like lemma 8 . 3 , lemma 8.4 i s t r u e f o r weak i t e r a t e d u l t r a p o w e r s . L e m m a 8 . 6 . i f N i s a p r e m o u s e and <<W > c

,< IT Q > ^ o _ > i s a n i t e r a t e d

ultrapower of N, and N is well-founded for a<0). then N is ~~CL

1

iterable.

—

Proof: Suppose N i s n o t w e l l - f o u n d e d . Suppose <x.:i€a)> i s a

64 sequence

such

X.=TT

that

(f.)(K

x.+,€

N

i...K i

x . , a l l i€a). Suppose )

(8.3)

where K is the critical point of N

,a?"<...

S={cu:i€a> AO<j£p i >. Let 3 be the order-type of <S,€>; so 3 is countable. Let f':3-*-S be order preserving and surjective. Define f:On-K)n by f(Y)=f'(Y) (Y<3) (8.4) f(3+6)=a+6 (all 6) Then f is monotone. Apply lemma 8.3 with M=N and a=id|N. Then a PQ :N + N . Let ou=f " 1 (aj2:) (i
Then a^ (x\) =7rQa (f±) (Ka±. . .Ka± )=x ± . So for i€w x i + 1 € N x\; hence

1 p^ No i s not well-founded. But 3<w,. —p

3 a

l

We may modify this proof slightly to get Lemma 8 . 7 . N is iterable if and only if whenever 0:M+ N and M is countable then Wis iterable. Proof: Suppose n o t and l e t f . , x . be a s i n t h e p r e v i o u s lemma. L e t X-J N w i t h {f.:i€a)}cX. L e t a:M=N|x where X i s c o u n t a b l e and

M i s standard. Let f ^ a ' ^ f ^ .

Let 7< Ma >a€Qn,< TT^ ^

^ > be an

iterated ultrapower of M with K ' the critical point of M . We may now repeat the previous proof, getting countable 3 and x.€MD such _

t h a t aot (x.) =x I. , i€u).So — Mpo i s not well-founded. I

!

o

Lemma 8 . 8 . Suppose N=
-* premouse Proof: j

and j:M+ ,M' then suppose

< X . >.

Then X±eu.

enumerate Hence

i

there

U i s countably

are as in the statement

Let

K. U is

countable

~°

(M,E,U',A') Firstly

point

P

a1 ( j ( f ) (lc))=a(f) ( T ) . Then Ml^(|)(j(f1) (K) . . . j ( f n )

N such that Suppose

and l e t K be t h e c r i t i c a l

t h e X€PM(ic)

Qa)Xi*0.

is 0':M'+ complete.

Say

such

T € i

Qa)xi-

that

0'j=0. a , M,M* a n d

point

o f M.

X6U 1 . L e t X . = a ( X . )

Define

(i<w) .

a':M'+N by

f o r EQ $ ( K ) ) -» ( C : ( J ) ( f 1 ( ^ ) . . . f n ( £ ) ) } e u ' =* { ^ : ( ( ) ( a ( f 1 )

(5) . . . a ( f n )

(C))>€U

*+ T € { ^ : ( j ) ( a ( f 1 ) (C) . . . a ( f n ) =* N ( = ( j ) ( a ( f 1 ) ( T ) . . . a ( f n )

(8.6)

(?) ) }

( T ) ).

Conversely suppose the stated condition holds. Let <X.:i€o)> be a sequence with X.€U, all K w . Let X^Liv N with {X.:i€o)}cX, X countable. 1 — 1 —

65 Let a:M=N|x, M standard. Let X^=a" ( X i ) . Then M is a premouse; say M=<M,€ M ,U' ,A' >, K" the critical point of M. Let j:M+ ,M' and let a':M'->

N where a'j=a. Let T=a' (K") . Then i=a'(K)
And x=a' U ) €a' (j (X±) )

(as X ^ U )

=a(X ± ) =X.. So Lemma 8 . 9 .

If N is a premouse, N*
——— — countably complete then N_ is

— iterable.

ft

—

P r o o f : By lemmas 8 . 6 a n d 8 . 7 we n e e d o n l y show t h a t is countable with iterated

i f a:M-* N a n d M

u l t r a p o w e r <<M > cr. ,< ir Q) .ocr. >then M c —a atOn a (3 a£p£On —a

is well-founded for all a
•*_ N; in fact by induction we define a :M -*•„ N such that

—10 •, L o — 0

CX —CX L o —

^o ~°o (all 3
U

is defined a ,1 is given by

CX

CX"rX

lemma 8.8. If a is defined for all a<X and X is a limit ordinal a (y) (a<3) , then define a, (TT , (x) ) =a^ (x) . Then if TT (X)=TT A

CX A

(X

CXA

PA

—

IT g(x)=y so cTg(y)=agTr g (x)=a (x) ; so a, i s well-defined. Let

a%

-°U} • 1 I t i s necessary to check that M i s countable when aOIL we shall not even have N'cN. This difficulty could be got around if we lifted the requirement of standardness; the proof of lemma 5.16, for example, could then be done purely internally. But that only gives one stage; i t i s not likely that with a set of ordinals we can represent a proper class of iterates. But for well-foundedness we only need GO, iterates;so if 0nN>uK we might hope to code the f i r s t co, iterates and appeal to lemma 8.6. All the coding has already been done in chapter 7. Lemma 8 . 1 1 . whenever

Suppose

N is a premouse —

§ is a T. formula. with

iterated

There ultrapower

is a £

formula

(f) s u c h

((N ) c , < T T O > . O -._ > ~~cx cxtO/i cxp cx
that then

66 the relation R={(x,y):N \=$(x,y)} is well-founded if and only if R = ={< x,y >;/v(=(j) (x,yfQ.)} is, provided aU{a} is an initial segment of On . P r o o f : L e t T = {£N: f: (a)m-*N and u € [ a ] m some m}. Given v € [ a ] n , m < n we d e f i n e a f u n c t i o n f u v by fUV(Y1...Yn)=f(Yi

...Yi ) (8-7) 1 m where v={a 1 a n >, a 1 <...,f,g) ~ 4>,g(Y 1 ...Y n )). By lemma 7.7 there is a formula i|/* such that

(8.8)

- - - ^ >'\,a f,a) ~ ,€T a A ip*(f UW ,g VW ,n)

(8

a

'9)

(8.10)

where w=uUv. Suppose R is well-founded. Suppose €T , all i€a) and

and R"" ff« f«i +fl f u i +>1 >< , < f f ii , u i »

(i€w) . I f uu={aj; (i€w) If a* }} aaj ;j<;. .<. <
NaN«Kyi...KYi)^Oa(fUitlWi)^Oa(fUiWi))

so

a

O

a

i

+1

o

0

a

i

o

x p p±+l i so N a (=4)(x i + 1 ,x i ) (all i with R ( x . + , , x . ) , a l l i<0). Let X

1

i-0a(fi)(^--V<)

X

X ( a

l<"-
Let u i = { a ^ . . . a 1 } . Then R (< f i + 1 # u i + 1 >,< f ± , u i » a l l iCco.

( 8

-

U )

o

C o r o l l a r y 8 . 1 2 . There is a relation R uniformly T. (N_) , where N_ is a premouse and to U{w } is an initial segment of N_, such that N_ is iterable if and only if R is well-founded. Proof:

B y lemma

Corollary 8.13.

8.6, letting

<J>(x,y)

<=» x € y i n lemma 8 . 1 1 . •

If $=ZF~ and u^Of then

flf="iV is an iterable

premouse" <=» N_ is an iterable

premouse.

ITERABLE PREMICE

Two important results on iterable premice follow. The first i s

67 the c o m p a r a b i l i t y property.Although two i t e r a b l e premice J Ua , JU' a r e Q 1 "~" => J"U €U not n e c e s s a r i l y d i r e c t l y comparable, i n t h e sense t h a t a<$ J^, a yet they may be i t e r a t e d t o premice with t h i s p r o p e r t y . Lemma 8 . 1 4 . Suppose N_ is an iterable <<JV > c

,< IT

n

>

If X is a limit

premouse with iterated

ultrapower

>.Suppose N = and K is the critical

nc

ordinal

point

Proof: Suppose x€U,; say X=TT . (x) , so x€U . Hence K €TT K €7T

a

x

aX^~ *

A

Tha : i s

t

'

x€rn

Tr

OtA

K €x

^^ aX^ "* a "

If x$Ux then K ^ X € U .

n (x) CX CXCX"rx

Ot

Let Ybe least

SO 3y<XV6 (y<6<\ -> K^^X) .

Lemma 8 . 1 5 . Suppose

0 is a cardinal

premouse with critical

point

such

of N then 9 is the critical —

point

Q

>

Q

, so

that

•

and Q>P(K)C\N, where N_ is an

K. If < < N > c^ ,< IT —OL ottc/n

ultrapower

of N .

then for all

> i s t/ie

iterable

iterated

otp d^pton

of N~. —0

Proof: L e t K^ be t h e c r i t i c a l p o i n t of N^.An easy i n d u c t i o n on y ^ ' 6 regular. Then for all X€P(Q)0NQ x€.Ua •• x contains a closed unbounded set where i^=< N A , € / f / A / A A >. ~~0

0

0

0

Proof: By lemma 8.14 if

x€UQ then x contains a closed unbounded

set

{K :Y
a

Definition 8 . 1 7 . Iterable premice M_,N_ are comparable if and only if there is a such that either M=J^a or N-J^'. a Lemma with

8 . 1 8 . Suppose

iterated

respectively.

ultrapowers

M_,N_ are

iterable

<<M > _

,< TT

pure O

> ^oc

premice >, < _

,< I T '

Suppose 9 i s regular, Q>M,N. Then MafNa are comparable. — 0 ~~o

Proof:

9 is the critical point of both ML and N n by lemma 8.15. —o —o So the result follows immediately from lemma 8.16. a

Of course, lemma 8.18 does not hold if M,N have

additional

predicates A.. Does it follow from the fact that M =N Q that JL

either

—0 —0

M=N or M is an i t e r a t e of N or N is an i t e r a t e of M? Not in general: but we shall show in the next part that i t does in the cases that interest us.

68 The second p r o p e r t y of i t e r a b l e premice t h a t we need i s of obvious

significance.

Lemma 8 . 1 9 . Suppose N is an iterable Suppose a.-N-* M; suppose <
r/,

~~o> u t c / n

Suppose

So

5

tne

cxp o t ^ p t o n

n o t . Say a(^)
such t h a t

premouse and that M_ is an iterate

,< TT o ) w><-« ^s

say M=N . Then for all £€On , O(E,)>^ Proof:

less

iterated

ultrapower

of N_. of N;

—

(B,) . ( £ ) . By lemma

V,.B,O+Y<»|J«VBY'

8.3 t h e r e

a r e

a l l B
n£Na.n- L e t V ^ a . n ^ . o o ^ n * • We claim C n > C n + 1 - For C 0 >C 1 . because a(^)
n+2 =7r a(n+2) ,a.03(^n+2) =Tr

a(n+2) ,a.w (a a(n+l) ( ^n+l } }

=a

a.a)7Ta(n+l) ,a.a)(^n+l)

=a
a.w ( W (c )

(induction hypothesis)

=CiSo N

i s not well-founded; c o n t r a d i c t i o n !

Corollary 8.20.

Suppose M_ is an iterate

•

of N_, an iterable

premouse.

Suppose

7T.-W-* M_. Then Tf:N+y M_ cofinally.

Exercises 1. Derive corollary 7.3 from lemma 8.3.

2. Suppose U is a normal measure on K and <<M a ^ a e O n'^^aB^a<8€0n^ is the iterated ultrapower of V by U. Suppose 9 > K is a strong limit cardinal with cf(9)>K. Show that for all a<6

VOGL(Q)=Q.

3. Suppose M,M' are comparable iterable premice, N,N' iterates of M,M' respectively and N,N' comparable. Show that M€M' •• N6N 1 . 4. Suppose N ,M ,T\ QIT\X Q,O ,£ are as in lemma 8.3. Show that for all —(x —ot

otp

otp

ot

3
69 PART THREE: MICE

Part three brings together fine-structure and the theory of iterated ultrapowers. Chapter 9 will introduce mice, the main technical device of this study. The development here is rather different from that in [10 ] . We are concerned to show the parallel with the results of part two more closely because this is important for generalisations. To eliminate the operator -*jj makes the proof of the covering lemma harder. Chapter 10 reveals a simple structure theorem for mice: every mouse is an iterate of its core. So given two mice with a common mouse iterate one is a mouse iterate of the other. This fails to generalise even to mice with two measurable cardinals. Chapter 11 proves acceptability for iterable premice. This would seem to make the specification of acceptability in the definition of mouse redundant; but actually it is much simpler to put it in. Mice are used essentially in the proof of corollary 11.29. Chapter 11 uses many techniques that will be needed for the construction of the core model. Chapters 12 and 13 stand a little aside from the main development, relating to class models of ZFC rather than to mice. None of the results in these chapters require mice for their proofs, but the results are essential for the statement of later theorems, which is why they are included here. Chapter 12 gives a first example of a mouse and chapter 13 a very weak generalisation of mice. This is, moreover, only sketched; but the theory of double mice is not very different from the theory of single mice presented in this book. The descriptive set theory at the end of chapter 13 is included to show how mice may be used to prove results that predated their invention. It is not necessary for anything that comes later in the book.

70 9 : MICE

Consider an iterable premouse N. Let us assume for the moment that the critical point of N is K and that wp < K . Then it is clear that if < < N > e O n f< IT g > a for all a p

is

the

iterated

ultrapower of N then

= p N , since ?(K) flZ1 (N^) = P ( K ) nZ^ (N) by lemma 7.4, and

P ( K ) D N =P(K)P(N. In fact we shall show that A^=A^N and " ^ a ^ N ^ N ' a ot where p N is a standard member of P to be defined. Not so, however, if o)pN>K. Suppose oopJ*
have 7TQ :N-*-~ N

and we cannot conclude that pjj = pn. or that AJJ =A JJ«

For reasons we explained in the introduction we are not interested in premice with o)p^>K. But we shall need to consider the cases where alp. >K but o)p5JK we are going to iterate N n ^ rather than N to get TT:NnP->M, say. Then the mouse iteration map will be the

n

M 1 , and M ' n p ' will be M. In n+1 will be bigger than the usual iterate of N.

completion of IT, Tr':N+

general M 1

Fine structural preliminaries precede the main definition. Definition 9.1. Lemma 9 . 2 .

Suppose p,g€[On]
<^ is a well-order

.

of [On]
Proof: I f p<*q and q<*r, w i t h Y/Y1 r e p e c t i v e l y as i n t h e l e t Y = m a x ( Y / Y ' ) . Then p^Y=r\Y- rnY'*0 so rflY+0. Case 1 Y=Y-

definition,

Then either pD7=0 / in which case p<*r, or max(pn"y) <max(qn-y) . If max(qriY) =max(rnY) then p<*r. Otherwise qnY=qnY', rn-y^rriY1 so max(qflY) <max(rnY) . Hence p<*r. Case 2

"y=y ' .

If pny=0 P < * r - Otherwise qflY+0 so qfiY*0 . Hence rnax(qfl Y ) < m a x ( r n Y ) « The proof goes as in case 1. So <^ is transitive. Given p,q€[On]
71 q<*p otherwise. Finally, <* is a well-order. For suppose p. + ^<*p./ all i<w. Suppose y. is such that p. +1 ^Y•=P-^Y•i P-

ny.*0 and max(p^ + ^ny^)<

< max (p. fly. )or p.+-.ny=0.We may assume i£j «* y.
is infinite. Contradiction!

Lemma 9 . 3 . Suppose a;itf->v N, M,N transitive. ———————

— 2JO —

n Then o(p)>p,

all p€[On]

— —

w

.

"

Proof: Let p={a 1 ...a n >, a 1 < — < a n . T h e n a(p)={a(a 1 )...a(a n )}. Suppose for i>k a.=a(a.) If k>0 a,* a (a,)

(0
so a,
where a=a(a, )+1.

a

We can use <* to simplify definition 3.23. Definition

9.4.

Suppose N is a transitive

model of RA (hence a model of RA ) .

—

Then we define by

p

induction:

p

= the
9 . 5 . N is n-sound if and only if N_ is < p . .

Lemma 9 . 6 . Suppose N_ is p-sound, +1

Proof: a)fp

p=< p .. .p >. Then

.p")-sound.

pn = QNnP'

=n{o)pNnq:q€PA^}. A simple induction shows t h a t A m q l " # q m

i s rudimentary in N 1 1 1 1 ^ " ' ' ^ for m A€Zn (Nnp) , giving the r e s u l t . Lemma 9 . 7 . 0)p

D

Suppose N_ is a transitive

>K, where K is the critical

acceptable

premouse of the form J and

point of N_. Then N_ is n+1-sound.

P r o o f : By i n d u c t i o n on n . Suppose t h e r e s u l t h o l d s f o r a l l m M. By lemma 4 . 2 2 t h e r e a r e N , p ' w i t h N p 1 - s o u n d and N n p =M. By lemma 4 . 2 3 t h e r e i s o^a such t h a t a:N-> N and a ( p ' ) = In ° < p ^ . . . . p " > . By lemma 4 . 2 4 a:N-> N. NOWK+ICX; SO a | K+l=id | K+1 . ^ _ n+1 Y7 — I T Hence a(X)=X for XGNnP(K) . Say N=J^. Then XGUHN => X6U. Thus N=J^. By lemma 9.6 there is A€Z.. (Nn) such that Ana)p5J+1*x/ all x€N n , and A is definable using parameter p N

. Hence letting B have the

same Z. definition over M from a " 1 ( p ^ + 1 ) , Bna3p^+1=Ana)p^+1. But by lemma 4.20 Bflcop

B is Z1(Nn)

is definable over N. Thus

a=a , N=N. But then

so a"" 1 (p^ + 1 )€P^n=P N n. a " 1 (pJJ+1) 1 * P N + 1

b v l e m m a9

-3

72 so a " 1 ( p ^ + 1 ) = p J J + 1 .

But t h e n a i s t h e i d e n t i t y map so X=M=Nn.

Compare t h e proof of lemma 4.27 . A s i m i l a r argument Lemma

9.8.

top^fK,

Suppose

where

Definition

K is

N_ is a transitive the critical

point

acceptable

premouse

of N_. Then

l f

i

^ *

yields and cop >K but

+ 1

9.9. Suppose $=RA. N_ is strongly acceptable above y provided that

whenever X is a limit ordinal >y and aa5, a€S, NS, then S.

Lemma

a

9 .10.

\=\<max(y,S) .

Suppose N_ is strongly acceptable above y. Then whenever

£ *A, N N N N Proof: We have to show H n=J n. We know that J ncH n. Suppose

N a€H

"PN PN n N V "PN n n. We may assume that acy<(op , for H n=L){j n: a€N A acy<(opN} XT

ra X7

"NT

r"»

IN

IN

and a€J n =» J n<=J n since cop.T i s closed under addition p

N

P

N-

P

( unless n=0

N

N

when the r e s u l t i s t r i v i a l ) . N Suppose X i s l e a s t such t h a t X i s a l i m i t ordinal and a€S, . Then S,+ \=\<max (y, \i) . Since a)pN i s a E, cardinal in N and Y,y€topN, therefore X€a3p^. Thus a€J n. a N PN Lemma 9 . 1 1 . Suppose iV is a transitive acceptable premousef with K the critical point.

Then N_ is strongly

acceptable above K. (N_ must be of the form J ) .

P r o o f : S u p p o s e a<=6, a € S ^ , ^ S ? , X a l i m i t . C a s e 1 PCKJOSV 1 ^ W = P ( K ) n s , . N N n Then by lemma 2.41 a€E (S ) . Say a€S (S ) ; then cop^N<max(K /6) . But & a) — A n —x TL"~ N N we may apply lemmas 9.7 and 9.8 to S, to get an S, definable map of max(6,K) onto X. Case 2 Z (S^) ^ Suppose nP(K)cS\\ ^ L e t M be rud T . n e N(SH . Then by lemma 2 . 4 1 CO ~~A

—

A

Ullo^

A

P(K)nM=P(K)nEaj(S^)cS^. Hence <M,€,U>|=R+. But then S^+a)eM. Contradiction! So E (S,)nP(K)is,, so for some m wp m N
CO ~~A

o ^ ~~

case 1.

a

A

THE OPERATION n : N - ^ N ' u D e f i n i t i o n 9 . 1 2 . A premouse N_ is n-suitable provided: (i) N_ is acceptable. (ii) N_ is strongly acceptable above K , where K is the critical (Hi) (iv)

K€copJJ. W is p-sound

for some p€PA

point of N_.

73 (ii) is not necessary but may always be assumed in the cases we are considering. Of course, ( i v ) means that ltfpRA for m
(ii) and (iv) are redundant. 9 . 1 3 . Suppose N_ is an n-suitable

Definition

Then r\:N+nNj means that is the n-completion

of T) \ N

P

So n : N ^ 7 N '

n:N-> N

1

means

Lemma 9 . 1 4 .

N=
n \tinp:iJnp-*' N_'nT](p' and r\

to N. .

Suppose N_ is an n-suitable

there is an isomorphism

premouse,

for some p£PAN N is p-sound,

premouse.

If T):N+*\N_' ,T\' :J£*JV' then

a:tf'=W" such that (3T\±r\'.

Proof: By the remark after definition 4.26 we may assume that each o f n | N n p , n I | N n p : N n p - > u N ' n n ( p ) , N " n n ' (p). So by lemma 5.14 there c*n | N n p = n • | N n p .

is a such that d:N' n n ( p ) =N" n n ' ( p ) and

By lemma 4.25, then, there is o^6 such that a:N'=N". But then an satisfies the definition of the n-completion of n ' | N n P N, so by lemma 4.2 3 an=n'.

to

°

Lemma 9 . 1 5 . Suppose N is an n-suitable premouse. Then if N=(N,€ ,U,A) there ————— pj are r|/W' such that r):N+nN_'. Proof: By lemmas 5.16, 4.25 and 4.23. Note that n:N-> N'. D

" n

Lemma 9 . 1 6 . If r\:N+ N_' then N_' is

E

n+1~

n-suitable.

Proof: N 1 is acceptable by lemma 4.10 or 4.11. By the same argument N1

is strongly acceptable above n(K). N 1 is n(p)-sound, provided

N is p-sound, by definition, and clearly n (K) €a)pjj, .

D

Lemma 9.16 gives us a basis for iterating the operation n:N+yN'. But first we need a couple of general results about n-completions. Lemma 9 . 1 7 . n-completion

of f\* :N_'

nV[(p)

nV[ lT](p)

-*iV'

—

(N,(-n —

((N > . ,< IT O > . o , > ) is —$ ot^A d p Ot^p
> ., > a direct

(X CX^A

limit

of~the

of

N", n ' n ^ ' f i .

— Lo~

L e m m a 9 . 1 8 . Suppose —————————-

n ' is

then T)'T) is the n-completion

P r o o f : N" i s n ' n ( p ) - s o u n d a n d n ' n : N - >

ordinal,

of f\:f^P^N_'nT[(p)and

if r\:N+N_' is the n-completion

the

fi'fi.

°

~ a commutative

system,

Xa

limit

system and for each a<$<X —

TT r,:N ->NQ is the n-completion of TT' =TT o l ^ OOL (some fixed p such that A^0, N is p-sound). psound). Let TT '=11^ =11^||rf™0o.'p) ; then < ^0(p) ,< ^ > a < x >is a direct p€PA^0,

p; limit of < <£W j £ W p ; >a<x,< iriB > a l 3 < A > and TT is the n-completion of 7T. Proof: Suppose f i r s t of a l l t h a t x€N 0 ( p . Then x=7r a (x) f o r some a<X, x6N ( ^

74 It suffices to prove the result for n=l. Let Y={v€N:N£=v€Tr (6) , some a) . So a)pNcy. Hence NJ=TC(X)€Y, so N f=TC(x)€a)pN , without loss of generality. So X E N W ^ . For t h e second p a r t i t s u f f i c e s t o show t h a t N i s p-sound. If i t i s n o t then OJP N*Y: say TT^ (6) ey^o)pN, 6€u)pN . Let A be E-^N) in parameter q such t h a t AOTT (6)$N. We may assume q=7r (q) . L e t t i n g A have t h e same E 1 d e f i n i t i o n <> ( over N^ from q API 6 6 1 ^ . But then a

which i s ITO; and TT O : N ->> N o , a s IT o i s t h e 1-completion of IT ' o , so 2 a p —ot L 2 —P ap _ by lemma 6.8 TT :N •*• N. So APITT (6)=TT (AH6) ; c o n t r a d i c t i o n !

ap •

MICE

D e f i n i t i o n 9 . 1 9 . Suppose N_ is an n-suitable premouse. Then <<# > r / , ,( TT o > ^oe~ } is an n-iterated ultrapower of N provided ~0t

OtfcO/3

(*)

(ii)

Otp Ot^-ptC'72

<<

—

^a > a€O7i' <7r aB > a<3€On )

each N is n-suitable,

i s

a conanutati e

^

system,

N=t N ,€ / y a / A ^ • a

(Hi) .

jv «iv. m

^n

QLQL+ - a

U

a 1 a~ ' "

(v) < N, ,< TT , > ^ > Lemma 9 . 2 0 .

i s a direct

If N_ is an n-suitable

limit

of (( N >

premouse then N_ has an n-iterated

ultrapower.

Proof: Construct N by induction on a, proving t h a t N i s n-suitable. Let NQ=N. in the successor case we use lemma 9.15 to get N + , . N + ^ i s n - s u i t a b l e by lemma 9.16. In the l i m i t case N.. e x i s t s by lemma 6.4. To show t h a t N, i s ~~A

—~A

n - s u i t a b l e we observe that by an inductive argument a<3<X «• => TT g i s the n-completion of TT g|Nn7TOa ^ for some fixed p in PAn such t h a t N i s p-sound. The induction uses lemmas 9.17 and 9.18. Hence by lemma 9.18 TTQ^ i s the n-completion of TTQ^|Nnp so N^ i s TTQ, (p) -sound. The other clauses are e a s i l y checked. D Lemma

9 . 2 1 . S u p p o s e < c

n-iterated (i)

,< IT O > ^oc

> and < < # ' >

c

/< TT' > ^oc^ > a r e

ultrapowers of W. Then there are isomorphisms a :N =N' such that a 7T

3 aB=7Ta$aa

(ii) °^(Ka)ssK^

for

(K K a' a

a

-3* the

critical

points of ^ / ^ respectively) .

D e f i n i t i o n 9 . 2 2 . An n-suitable premouse N is called critical where K is the critical point of N. This n is written n(N) .

if K$a>pn+1 fT

75 Definition 9.23.

If N_ is a critical

of tf is called a mouse iteration

premouse then an n(N)-iterated

ultrapower

of N.

Definition 9.24.

Suppose < < N > c ,( T\ o) ^oc > i s an n-iterated ultrapower of yt ^ -a a€on a$ a<$€on N. Then each N is called an n-iterate of N, or a mouse iterate if n=n(N) . — —& — D e f i n i t i o n 9 . 2 5 . N_ is n-iterable provided each n^iterate of N_ is well-founded. If W is critical

and n(N)-iterable

then N_ is called a mouse (provided N_ is pure) .

The reason for introducing mice is quickly seen if we try to apply the ordinary iteration operation to critical premice with n(N)>0. Say n:N-> N 1 . Suppose n(N)=l; we do not know that a)pX7lK

and N is a

mouse.) But there are good reasons for excluding such premice; if they are admitted to the core model indiscriminately then we shall end up with measurable cardinals in the core model, and we saw in the introduction that this would be a bad thing. We saw that if N is a mouse r^N+j^N1 and n ' rN-^N" then N" is an initial segment of N 1 ; and

P (K) nN"=P(K) PIN1 where K is the critical

point of N. N 1 adds enough levels to N" to get a construction stage where ojp^,

)+

<: n d O . Of course it could be the case that N was an

iterable premouse but not a mouse, and then these levels might not exist. By the results of chapter 7 there will often be a,3 such that 7T Q :N -> N o when TT O is the ordinary iteration map. In this case ap — a ^ n ~ p

otp

the maps may or may not coincide. Exercises 1. Suppose V=L['U] . Assuming that L[U]is acceptable and U is a normal measure on K construct a premouse N with n(N)=l. 2. Is there a sentence a such that N(=a <* N is a critical premouse? Such that N^=a «=» N is a critical premouse and n(N)=0? 3. Suppose nsN^j^N1, n=n(N)>0. Do we necessarily have n(p^)=p{J,? 4. Prove the equivalent of Los theorem for n:N+yN'.

76 10: PROPERTIES OF MICE

In this chapter we shall prove the basic structure theorem about mice. Roughly speaking this says that if M and N have a common mouse iterate then either M is a mouse iterate of N or N is a mouse iterate of M. Indeed there is a smallest mouse N of which N is a mouse iterate, called the core of N; this mouse will be n(N)+l sound (indeed, as it turns out it will be m-sound for all m ) . All manner of corollaries are proved as we go along. PRESERVATION OF PARAMETERS Suppose until further notice that N is a mouse, n=n(N), and that < <> _^ ,< , > ooers __ >> is is its its mouse iteration. Let K be the —a a€0n' a8 a<3€0n a£3€0n a critical point of N . Let K = K • Lemma 10.1. E

,(N )0P(K)=1 n+1 (—ex

_(N)0?(K). n+1

n

—

n

Proof: E 1 (N )nP(K)=E 1 (N )nP(K) .

•

Lemma 10. 2 . Each N is n-sound. Proof: By lemma 9.7.

n

Lemma 10.3. tf*\=v=h(K Up12 ) Proof: By lemma 9.8.

•

Lemma 10.4. TT (pm)=pm UO. N

(m
N

—

Proof: By induction on m. Suppose ¥ +1
i€o), ye(a)p^ ) a

(p™. ) >*pJJ

• Then there are

such that Y,r))).

(10.1)

Hence

N"V(3r<,pg+1)(p^+1=h(i,r)). This is because r<^TTn (p

) => rUy<^Trn (p N

(10.2) ) . It follows that

N^ is r-sound, so r€P m. Contradiction! Suppose, then, that TT (p™. ) < *P^ • T n e n a s ^Q i s a n n-completion of TT- |Nn:Nn->N N p is TT (pJJ )-sound, where p= =< TT

(p )...TT 0 (pJJ) >. But by the induction hypothesis p=< p N ...pN> +1

so TTQ (pJJ )eP m. a a

Contradiction!

D

r

77 Note that the above is a general property of n-completions.

10.5. pj"^,

V

J

O<

-

a a Proof: The first claim is immediate from lemma 10.1. TJ T4 TJ Let q=p" -nK, r=p" -vK. Let r=p" -vaK . (K=K.) N N N O a a Claim 1 TrOa(r)=r. Proof: Suppose TT. (r) >*r". Then suppose i)) . So Nn i s fUq-sound. But fUq<*rUq; and rUq€P N n. C o n t r a d i c t i o n ! and A1 has t h e same d e f i n i t i o n

If A i s E,(N ) with parameter pJJ n

over N with parameter -nQ (p^ ) then A'nK=AflK; suppose AflK^Nn, .

+

So 7T

(Pvt

) ^P n ,

s o IT

(p

a _ Ta- (r)> A r. N a , that i s , To C l a i m 2 q=P^ + 1 HK.

a

Proof: As T r Q J P n + 1 ) €PNn and \

^

n+1

a

W O ^ r V

) >JLP

a

a D (Claim 1) + 1

)^

a

^

+

then

• T h u s TT

K

we know

K ( x

OaSaOaSOa

+1

t h e r e f o r e p[J nKaEK. Thus P N ^ V ^ O C X a a

(p

N+lnica) a

;

S O P

N+lnKa-*q* a a ( c l a

i m

2 )

a

Corollary 10.6 .

/1+1=Nn+1.

CORE MICE Let X=hNn(o)p^+1Up^+:L) . Let a:M'=N n |x where M 1 is transitive; so a:M'->

N n . Let M, p be unique such that M'=M n p , M is p-sound and

a:M+ N with oz>o and a(p)=. - ^n+1~ N N Lemma 10. 7 . M_ is a mouse. Proof: M is an acceptable premouse, and is transitive. By the construction a" (K)€oap^ so M is n-suitable. Suppose P J J + 1 > P N + 1 #

Su

PP°se

A i s

^x^N11) in parameter pjj+1 and

JJ +1 ^N n . Let A 1 have the same E., definition over M 1 from a" 1 Then for Y€U>PJ}+1 Y^A «* y€A' since o (y)-y.

Now A'nwpJJ^1 €M. Let

B=a(A'na)pJJ +1 ) ; so Bn(A)p^+1=Ana)pJJ+1€N; c o n t r a d i c t i o n ! Hence

PJJ+11PN+1

so o)p^+1
i t s

""iterated ultrapower.

M is n-sound

78 because i t i s t r a n s i t i v e . So each M^ i s TT' {< p M . . .pJJ >)-sound. And < < ^ ' a e o n ' * \ i e W o n >' where M;=M™6a (< % • • • ? £ > > , n

n

i s an i t e r a t e d

n

u l t r a p o w e r of M . By lemma 1 0 . 4 a:M -+y N ; so by lemma 8 . 3 t h e r e a r e maps

a : a M^->

n N £ such t h a t

a 7T g (Jte

lMl=7ra8aa'

when

a

l^

; a n d a

o=a#

Let a :M -*_, N be t h e n-completion of a . N i s well-founded, so n+1

Lemma 1 0 . 8 . Suppose Q_ and 0 ' are mice.

Suppose n=n(Q) and n'=n(Q_') . Let

^a€anaeaQ,Q'; let Q =J^a, Q'=J ,. Then Q^, QL. are -a - n - - n 8 6 comparable and Q Q' Qn*J~cfi

(some

U)$€£>IJ ** Qn^Za®

(some

0)3€£';.

Q

P r o o f : Q o =H 0n , s o P (0) 0 0 = P ( 0 ) flQQ ; s i m i l a r l y P (0) n Q ' = P ( 0 ) flQ' By lemma 8 . 1 8 Q 0 , Q ' a r e c o m p a r a b l e . So Q 0 , Q 0 a r e c o m p a r a b l e . — o ' — —— o S u p p o s e Q —J%Qf o)$€QA. QQ*QA s o Qft^OA» B u t i f Q ! = J Q 0 , t h e n P ( 0 ) n Q £ * P ( 0 ) n Q e a sa ) p ^ i + 1 < 0 . H e n c e P(0) OQ|P (0) nQ e . ~0 Contradiction! QI

a)p€O o

Q

Suppose Q =j;:0, o)3€Q'. Then by the last paragraph QA+JifQ* all —U —p

o)"p r €Q e .

Suppose

U

Q0=Q0.

Contradiction! Lemma

10. 9.

Then

U —p

P ( 8 ) flQ * P ( 6 ) flQJ

b u t P ( 0 ) n Q Q = P ( 0 ) fiQ^ .

D

For some

0^=2^.

M ^J — Proof: Let M'=J a, N'=J a . Let 0 be regular, 0>N.JBy lemma 10.8 —a — n' —a — n ^ either M O € N Q or M Q = N Q or N Q €M Q . If N Q =J^0, a)3€MQ, then N I = J ^ , say; ~~o —~u

~~u ~~u

~~o —*u

—"D ~~p

y

~~y ~ ~ P

but then TT0 a:Ml-^« M 1 is not cofinal, contradicting corollary 8.20. N Suppose M 0 =Jg0, then we saw in the proof of lemma 10.7 that there is A € E n + 1 (M)

such that Ana)pJ|+1 $N. But then AOa" 1 (K) £ E n + 1 (MQ)

so Ana"1(K)€N. But a"1 (K)2wpJJ +1. Fix 0 f o r a while. Corollary 10.10. pJJ Proof:

P^-PS^P

Lemma 1 0 . 1 1 .

a (p1?1) ^p"*

Contradiction!

79 ( p ^ x ) €PMn so

Proof: By t h e argument of lemma 1 0 . 7 a

' . But by lemma 8.19

But ^ogfPM^^PM* 1

b y leTma

10

-5

s o

Hence n+1

Corollary 10.12.

=An+1.

A M

N

C o r o l l a r y 1 0 . 1 3 . if1~f'1=tf1+1. Lemma 1 0 . 1 4 .

M_ is

n+1-sound.

+1

+1

Proof: We have t o show MnJ=V=h (u)p^ p^ +1UpJJ +1) . Suppose x€Mn; t h e n a(x)€X / N n h a ( x ) = h ( i , < Y / P ^ + 1 » , s a y . So Mnh=x=h

Lemma 1 0 . 1 5 . 1

JJ +1

TT^SW ' A.

•

0\J

0\J

Proof: M i s n+1-sound and
Proof:Let

respectively;

so K ' = K ^ . Let a be least such that K^>_K . Suppose

K'>K.

Suppose Mn(= =7T' ( f ) ( K ' . . . K 1 ) —ar K Oa a, a n In where fCM , f : K * V 1 and a 1 < . . . < a n < a M^K=^e(f)(K- . . . < ; ) 1 n so by lemma 1 0 . 1 5 $

o

e

K

;

K'

. . . K

1

. Then (10.7)

. . . K ; ). 1

But

(10.6)

(io.8)

n


j so K€rng(iT oe ). If K = T T O Q ( 6 ) and 6 < K then K = 6 < K . But if K = T T O 0 ( 6 )

and 6 > K then K=TT, . TT_, (6) >TT^, (6) >TT^, (K) >K . Contradiction! — xy ui — ui — ui Hence K ^ = K . Define T:M^->-Nn by +1 T(h Ma n(i, ) a

+

where i€a), y€<. Since

(10.10)

80 (10.11)

where <> j i s E ^ t h e r e f o r e T i s w e l l - d e f i n e d and irM^-^N 11 . But NJ=V=h(KUp?J+1) so T i s s u r j e c t i v e . Hence T:M^=Nn, so M*=Nn, so Ma=N.D Lemma 1 0 . 1 7 .

a=Tr' .

—————————

OQL

Proof: ir£oi n+1 D e f i n i t i o n 1 0 . 1 8 . M_ is called the core of N_. M_ is said to be a core mouse if and only if it is the core of some N_. Note t h a t if M i s a core mouse and N i s a mouse i t e r a t e of M then M i s the core of N. Lemma 1 0 . 1 9 .

Suppose W is a mouse with core M_. Let << M > _ ,< IT g > < g e O n ^

be the mouse iteration point

of M_ and suppose N^=M . Let n=n(M) . Let K be the

critical

of M . Then Y

Proof: By lemma 7 . 1 Nn=hNn(irOa"MnU{ic :y
D e f i n i t i o n 1 0 . 2 0 . Suppose iV is a mouse with core M_, and suppose the mouse iteration

of M_ is << ^ ^ ^ Z ^ g > a ™* «=«,- Then C^ denotes

where K is the critical

{Ky.-Y
point of M .

C are E. indiscernibles for < if1 fp1**1 ,y > - % n+1 where n=n(N) . N 1 — N ytu)pp U U P )E^PN P r o o f : I m m e d i a t e b y lemma 7 . 3 and t h e f a c t t h a t rng(irir )E^P P N N •D N

Lemma 1 0 . 2 1 .

c are Z ,. indiscernibles for < N,p* . .pf//Y> or!1*1' n=n(N).

Corollary 10.22.

N

Lemma 1 0 . 2 3 .

nTJ.

K€CN «
~~ N

>

K>(JL)

P^ J

N

ytCup


Proof: Suppose M i s t h e core of N with mouse i t e r a t i o n ^

K

y

t h e c r i t i c a l

i f N=M . Suppose K=Ky. I f K Y =h N n(i,
(hMn(i,< ?,pJJ +1 YUpJ5 )

+

>) ) so K €rng(ir

P°int

o f

V

(i€w,J€K Y )

So

CN

then

) , which i s impossible; so

. Obviously K Y >U) P JJ +1 .

81 Conversely suppose K^C... Let y be least such that K > K . Suppose K>a)pjJ+ . Then there are t«

such that

M^K=h(i,<£,pJj +1 >)

(10.13)

N n hc=h(i,
(10.14)

so

Thus K€h N n(KUp^

+1

).

D

Corollary 10.24. c €11, (Nn). •* •

•

fj

1

—

This fact will be of importance in chapter 11. The fact that the core of a mouse is itself a mouse may seem strange. We are accustomed to measurability being a large cardinal property; yet if M is a core mouse with K critical, n=n(M) and oopJJ

< K then Mf="K is inaccessible"; but there is a S +-, (M) map

of o)p^ onto K . This is not a contradiction, of course; although n+1 a)p = K implies that M is a core mouse, the converse is not true. Indeed, we shall later on give an example of a core mouse with p =1. This implies that M cannot be "continued", i.e. there is no premouse N with M=J

for some a)a€N. This does not only apply to core

mice: by lemma 10.19 there must be a lot of K

for w not to be

collapsed definably. In fact, continuability is a very strong N property: we shall see that if M=J , u3a€N, N a premouse with measure U, then C €U. This topic is taken up again in the next chapter. THE STRUCTURES N™, m>n To conclude the fine-structural analysis of mice we must look at the projectum below K . Let N be a mouse with mouse iteration

Lemma 10.25. For m>0

^

p

j

Proof: If m=0 this is trivial. p^ + m + 2 =n{p N n+m+lp: P ^ P A ^ + m + 1 ) . Now P N n+m+lp=p, n.m+lp' where p 1 are the last m+1 members of p. This follows from lemma 9.6. So pjj+m+2=n{p ^iijin+lp' : p ' C P A ^ } . But p?Jn+l=n{p ,.,n+l. mp": p"€PA :

p, n+lvmp"= p,Nn.m+lp

f

}. Hence it suffices to show

, where p" are the last m members of p 1 .

This would clearly be true if A^n N

n+1

n

for all p€N .

were rudimentary in

But A=A^n

is an iterate of M n , so A is l±(Mn)

f

is ^(N 1 1 ) , and N n

and therefore E 1 (M n ) with

82 parameters from p?. Ucop^

, and hence £, (N )with parameters from

-n+1Ua)p?i+1; but then i t i s rudimentary in N n + 1 . M

—

Corollary 10.26.

Let N_ be a mouse with n=n(N) and mouse

iteration

< _ ,< 7T Q) ^QCrs > . Then for m>n+l - a aEOn <x3 a<$€.0n

(iv) £*£. Proof: By corollary 10.13 N n+1 =N n+1 . Lemma 1 0 . 2 7 .

a

If N_ is a mouse and n=n(N) then W

is k-sound for all k,

and P^* + 2 =P K ^ • Proof: By induction on k. If k=0 i t i s t r i v i a l . Suppose m=n+k+l and suppose Nn i s k-sound. Then PNm=pNn+k+l=p^nll

(lemma 9.6)

= pJJ + k + 2

(10.15)

(lemma 1 0 . 2 5 ) . m

I t i s s u f f i c i e n t t o show N 1-sound. So l e t X=h N m(cop m+1 Up m+1 ). Let aiM'^N^Ix where M1 i s t r a n s i t i v e . So arM 1 -^ N™. L e t M be such t h a t M^M11^, M i s p-sound,

N, oz>o and a (p) =< p?.. . , p m >. 1 m

a:M-^v

Then j u s t a s i n lemma 1 0 . 7 , M i s a mouse. Furthermore p=

so M i s m-sound, and i n p a r t i c u l a r a c o r e mouse. Let t h e mouse i t e r a t i o n s of M,N be < < Ma > a£On ,< ^

e

> a l 6 e O n >,< < NQ > a€On ,<

^

respectively. Just as in lemma 10.9 there is 9 such that M Q = N Q . So M=core(M Q )=core(N n )=core(N). Say N=M . Then by corollary 10.26 — —y —y — — —ex

N^M 1 .

But M1 i s 1-sound by the argument of lemma 10.14 so N® i s

1-sound. Corollary 10.28.

D If N_ is a core mouse then N_ is n-sound for all n.

n-ITERABILITY We concluse t h i s chapter with some c r i t e r i a for n - i t e r a b i l i t y . Definition 10.29. E is defined inductively on Nnpl'"Pn

as follows

n

EQ(x,y) « x€y Em+1 (x,y) * x, where s(k) is the formula "x=^i,x) A y=(j,y) A E (h(i,(x,p , )) ,h(j,( y,p m m+i m+i

n

83 is rudimentary in N n p . Clearly

Then E

Lemma 1 0 . 3 0 .

Suppose N is p-sound,

only if (E ) np

p€.PA . Then N_ is well-founded

if and

is.

Most of the results of chapter 8 can now bw reproduced with E in place of 6. Lemma 8.3 presents a problem though; the natural approach is to build the maps a , as there, and take n-completions. However we do not know that a :N -»•_ M f . * cofinally. We do have the following, a special case of which was used in lemma 10.7. The proof is the same as the proof of n-iterability in lemma 10.7. Lemma 1 0 . 3 1 .

Suppose O:M+N is the n-completion

If N_ is n-iterable More useful,

though,

Lemma 1 0 . 3 2 . =JQ

to M_ and M_ is

p-sound.

is

Suppose N_ is an iterable

0:M+N'is the n-completion

provided

of O\M

then M_ is.

Af is n-suitable

(that

premouse and for some uxx€.N, letting

of O\M

to M_ (M_ p'-sound)

is, provided

A£ i s acceptable

. Then M_ is and the

N' =

n-iterable

critical

of AT is in cop21; .

point

Proof: L e t < a £ O n *<^o > a
>a€On'< * i 0 W o n >

critical point of M 1

U , U

1

b e t h e i t e r a t e d

b e a n

and let K

u l t r a p o w e r of N

n-iteration of M. Let K ^ be the

be the critical point of N_a. Let

be the measures of M ,N respectively. We shall define maps o^: {^Pa-»-j. N^ P a such that

^

(ii) oQ=o where P ^ i r ^ t p 1 ) , Pa=*0(x° (P1) • a

is to be ^Q a (3) so N^=TTOa (N1) . The maps are defined by

induction. oQ is a. Suppose a a is defined. Then define

Claim 1

a

a+i

:M

^+} + 1 "*" N a + i b Y

a . - :M^a+l->H a + 1 , where p=ojp", . a+1 a+T p Na+1

Proof: Since a (f)€NlPa

7T

aa+1

:N

'"* e N ( J t + 1 '

N

^ + 1i

s

by i n d u c t i o n h y p o t h e s i s ,

a c c e p t a b l e . Given so f o r a l l £
f€M^pct,

a ( ( f ) ( £ ) € N ' and

N^f=TC(aa(f) (£) )
84 and

(ir

^ + 1 ^ the claim. o(Claim 1) np Claim 2 aa+1 ^, :M a+l-> Eo—a+1 N' n p a+1. —a+I

(a

aa+l a

( f ) } (K

a

5 }< p

'

which

prOveS

Proof: Suppose $ is a I. formula; for simplicity, of one argument. ) (K

But

a ) } ~ M^Pa|=U:<J>(f ( S M K t T

^aa+l^^^Si+l 0 * 1 -

(10.16)

o (Claim 2)

' s o (i) h o l d s ' In the limit case l e t a, (TT' (x) ) =TT .a (x) . Details are left to A

(X A

Ot A Ot

i s rudimentary in N np

the reader. Since E

Mnpa f=En(x,y) «N^npa|=En(a a (x) , a a (y) ) .

(10.17)

But N' i s well-founded so M i s . C o r o l l a r y 1 0 . 3 3.

If N is an iterable

a premouse, O)B€N, W=J^, M acceptable,

_

K the critical oop M

point

p

of N_ and 0)p> >K then V_is n-iterable.


—

If in

addition

then M is a mouse.

~0t

Lemma 10. 34. If N_ is a transitive n-suitable premouse and its measure is countably complete then AT is n-iterable. Proof: As lemma 8 . 9 ; apply t h e argument t o E r a t h e r than € . • D e f i n i t i o n 10.3 5. E* denotes the uniformly Z relation of lemma 8.11 such that whenever < <— _ , ^nr. ) is an iterated ultrapower of— N then x CXCOJI <x CXCOJI

a p a
{<x,y):N^ \=En(x,y)} is well-founded if and only if {< x ,y ):N$=E^(x,y) } is (WjQi) Lemma 10. 3 6. Suppose N_ is an n-suitable N_ is n-iterable if and only if (E ) np is

p-sound premouse, p€.PA ,0) €.N . Then well-founded.

D e f i n i t i o n 10.3 7. Em is defined by induction on m>n by A < * ' ^ / ^ > € A m + I p l * * 'pm+l where s(k') formula

"x=
Lemma 1 0 . 3 3 . ———————————

A y^j/y)

Suppose

m

AE (h(i,<~x,p n

,)) m+l

N is (p,...p )-sound — 1 m

,h(j

,{~y ,p

is the

.)))". m+l

and

n-suitable,

m>n/Ud.€.N — 1

.

Then

85 # is n-iterable

if

and only if

(E ) nip) is well -founded.

Exercises 1. Show that if N is a mouse then the following are equivalent: (i) for some M,N=core(M). (ii) N=core(N) (iii) N is n(N)+l-sound (iv) C N =0. 2. Show that if M is a mouse with K critical and mouse iteration

then

VS ) n P ( l °£V

3. Suppose M+=J^+1 i s a premouse and M=Ja i s a mouse with n(J a )=O. Let

Show that in general N' + ru'M). Do N 1 , n (M) have a common iterate? 4. Suppose N is a mouse. Show that if A€£n(N)flP(p) arid A^N then

86 11: ACCEPTABILITY REVISITED

This chapter will prove that iterable premice are acceptable. The proof will be inductive; the reader can check immediately that the limit case is trivial: that is, if J, is an iterable premouse and X is a limit, and if JU is acceptable for all a<X a U then J, is acceptable. + U We shall be assuming that N = J a + 1 is an iterable premouse with K critical. We shall be able to assume that N=J is acceptable. This implies by corollary 10.33 that either w p S x for all n - and this case will present few difficulties - or N is a mouse. The two cases cop = K and cop < K are handled separately; there seems to be an overlap between them, but there is not. THE CASE iup^+1a€Qn,< ^ >a^eQn > be the mouse iteration of N. Suppose N=N,; let K be the critical point ~A

of N ; so C N = { K :y<*}a n d

K=K

x*

Let

K=K

o*

CL Reca11 tnat c en i(N N

) , so

C £ N

Lemma 11.1. c Eu. Proof: By lemma 10.23 C N ={K'cop^ + 1 A K'{h N n(K'Up* + 1 )}. Suppose K^C €U. Define f:K-»-K by f(y)=the least y'
(11.1)

=0 if there is none. Clearly {£:£€hNn(f(£)UpJJ+1)}€U; and f€N + . By chapter 5 exercise 6 there is 3 such that {£: f (£) =3>€U. Hence Y = U : C€hNn(3UpjjJ+1) }6U. But then Y€N + , Y ^ in N + and Y is cofinal in K . Contradiction! D This makes precise our remark in chapter 10 about continuability. Let C N = { K :y<X A y is m-good}. Then cJj€N . Lemma 11.2. For each m€co N

Proof: By induction on m. If m=0 use lemma 11.1. Suppose C N €U; CM IN

= { K ' € C M : K'=sup(K'flc!J)}. Let f:K->K be defined by IN

JN

87 so f€N + . £(£)<_£,, for all £.Suppose {£:f(£)<£}€U. Then by chapter 5 exercise 6 there is 3 such that {£:f (?) =3>€U. So U : 3=sup(£flCm.) }€U; hence C™ is bounded in K . But C™EVl So {£:f(£)=£}€U; that is, c"

€U.

D

N

Corollary 11.3.

A is m-good for all zn
So by c o r o l l a r y

7.20

Corollary 11.4. *

d"1 are E _ indiscernibles N m+1

for < i/^/ir., (x) > ^-n. — 0\ x€N

Corollary 11.5. *

C7" are I , indiscernibles N m+1

for
C o r o l l a r y 1 1 . 6 . d° are E Lemma 1 1 . 7 . ————————

Proof:

J f x€PCK;nw

indiscernibles and x£E _ CN; n+m+l —

n+

for
,-)

then

If C ^ y e X then X€U by lemma 11.2.

Conversely suppose cJJ^y^X for all y
s

n+m+1-

1

Then for some £ n + 1 term t p=t (K^^ ...K i ,P N »..

. . . P ^ 1 ^ ! - - - Y q ) where iK ± , or y>u)p^+1 if p=0. Pick j such that K .>y and K.€CJJ. By lemma 7.12 we may assume that is m-full. So if a € C N ^ Y ^ i - - - 1 ' a > i s m-full. Also if K a , K a , 6 C ^ y then ch m (o) =

K

=ch m (a')=m. Hence by corollary 7.19 K a €X **K a i e x follows that C^ycK^X, i.e. K ^ X C U . S O X^U.

Since C ^ y ^ X it

D

Lemma 11.7 says that we can predict whether a definable set will be in U just by looking at UDN. From this we deduce Lemma 1 1 . 8 . Let M=ruduf]N(N) . Then (M,u)is

amenable.

Proof: Suppose X€M, X C P ( K ) . By lemma 2.43 there is p such that X c E n + p + 1 ( N ) . Then xnU={Y€X:3y
But C €Z

P

(N) , s o C €M. Hence XflU€M.

(11.2) n

Lemma 11.9. N+=rudu(] (N). Proof: Every x in N

is definable from parameters in NU{N} by a

88 function rudimentary in N + . But <M,€,U>^=R so the same function has a value in M, and this value must be x by absoluteness. So x€M. Corollary 11.10. P(K)fli/=P(K)nz^(N). Proof: This follows from lemma 11.9 by lemma 2.41.

•

•

This result is important in many places. It does require the n+1 assumption cop < K ; after all, the measure must be unpredictable somewhere. Now put this case aside for a bit. THE KeepCASE the (notation introduced for the previous case. This time we shall show that H^=H^ , and therefore that ojp^=K for all m>n(N). Thus in this case the previous case cannot arise. A small change of convention is needed. Consider the case where N =J _, so N=J . The reader may easily check that J =J , so by U K U lemma 4.27 J is strongly acceptable. Also E (J )cj + _; hence since J +-,f=K ^ s a cardinal, cop =K . We want to apply the argument of the K present case in this situation; but strictly speaking J is not a premouse. So from now on we demand only that N be acceptable. L e t << > ^ a€0n' < 7 r a3 > a b e t h e i t e r a t e d ultrapower of N + ; each N + is of the form J Q a,, and we may let N =J Q a. Observe that N 3a+l -a 3 a -a is not asserted to be an iterate of N, or a mouse iterate of N; indeed, p M >p as TT_ (p )=p . Anyway, N is not even necessarily a premouse. Clearly TT g(N ) =Ng. Let K Since W p »

+1

- < , ^=
Lemma 11.11.

+1

Let H^ .

be the critical point of N . T ^

1

;

H;=<S a ,T a >.

ir O\H :H •+ HQ.

a3' a -a e ^ Proof: TT ft(H )=Hft. So otp — a

—p

^ ( x ^ . - . ^ t x ^ ) «N;HigH(^6(x1)...1raB(xm))»

(11.3)

Corollary 11.12. H -
—ot — p

Proof: IT O |H =id|H . For H =U{j a :acv
•

Lemma 11.13. H \=ZFC. Proof: H f=R , so we need only check separation, collection and power set. Note that H GH Q . For HQ(=VyJBBexists, where B Q = A £ + 1 . But ot

p

p

y

p

N«

89 \x6

(B

)=B

a

3

so B

a

=B

8

na)X(K

a

)
Hence

^ H ^

exists

-

(a) Separation. Suppose <{> is a formula and * € H a . Since H o h R there is y£H R such that (P €H a >

H3|="t€y ~ t€x A Hj=(t,p)" But H a H ( t , p ) ••HpNtt^P) ^

(11.4)

so

«* tex A <|>(t,p))

(11.5)

so the same statement holds in H . (It follows immediately that cop™ =K^ for all m>n.) s-

a,

A.

(b) Collection. Let x€H . Suppose u€x, a<3. If Hg^=3v(|> (u,v,p) A

/\

t h e n Haf=3v<j> ( u , v , p ) .

Say

A

A

/\

Haf=<j> ( u , v , p ) .

Then H ^ v € H a

A

<j)(u,v,p).

A

A.

Hence Hgf=3v(|) (u,v,p) «• OvGH^) <)> (u,v r p) . So for x€H a H3l=3yVuex(3v(u,v,p) >+ (3v€y) (j) (u,v,p) ) .

(11.6)

So the same statement holds in H . —a (c) Power set. Let xGH a . Let 0 be a regular cardinal greater than 2 X . Then K Q = 0 and P(x) nH Q <6. So P(x)f1HQcJB , some Y < 6 . S O u w w— y H0|=3y y3P(x) .

(11.7)

So the same statement holds in H . —a

o

In case N + = J ^ + 1 , Haf=V=L. This result will be generalised in part four. Lemma 11.14. H = H V Proof: H a cHg, so H a £H^3. Suppose x€H g and H ^ T C (x) <
Ha^Ha

is a limit cardinal, so x€H 3r some y
The strategy for the proof of the result we are after is as follows. Suppose we could construct all of N +

~

A

in some H . Then

~~ A

H cH a=H cN,and the result would be proved. H hZFC, so we can certainly construct N in H if ot>0. If were amenable we 2 — —a —a' A + should be done. But it isn't; for if it were U€H , so U€N , but see chapter 5 exercise 1. On the other hand, suppose <X :yGN

with X C K for all y
Since X 6U « KETT X (X ) therefore {Y:X Y €U} = {y:K€7r01(XY) } (11.8) But also < T O 1 ( X ) :y=7rol(< X :y
90 {y:X €U}€N + . As t h e r e i s a d e f i n a b l e map of K o n t o N i n N + working with K-sequences i s no hardship. Lemma 1 1 . 1 5 . ————————

N €#Q and is uniformly definable —<x p

over HQ from K provided a<$. —p a

Proof: H'€Hg and is uniformly definable from K . As H^f=MC

there

is M with McH a and p€PA^ + 1 such that M p = H a and M is p-sound, iterating lemma 3.26 n+1 times. And M is unique up to isomorphism; so M=N ; hence M is well-founded. But then N from M as its transitive collapse. Lemma 11.16.

is uniquely definable

o

{ K :a<3) are Z indiscernibles for Ho. a)

a

—p

,

Proof: They are £, indiscernibles for N g .

•

Let y^ b e the smallest elementary 1substructure of H , containing 3 a /s —a+m K U { K . . . K ^ , } . L e t IT :M m =H ^ 1IX^ where M is transitive; so a a a+m-1 a — a —a+m a —a ™rfn

L e t K m = H ^, where K ^ denotes the successor of < a in a

Lemma 11.17. fr^|/=id|/\

a

^

^

Proof: We have to ghow ^ O H ¥ + m transitive. Suppose x€X m , x€H H a a +m a K(X ~ H +^=x<j< there is f€H + m Then obviously xcH + . Since H +^=x<j< there is f€H + m mapping onto x. Let f be least such in the canonical well-order of H a

Then if y€x, y is definable from f and some y from x.


•

? but f is definable

D

M ^ R , for H a+m|=R. So K ^ R . X^=H a so M°=H a . By lemma 11.17 K m = X m n H H * + m ; and ^f H M are in X m + 1 . But H ^ ^ > Kr m = K so K m € K m + 1 . a a K a a+m a —a+m+l a a a a By lemma 11.15 N a €X^ ; and \=*a in H + 1 , since 0)pJJ+1=Ka. Thus N ^ K ^ . Let f™ be the least map of <^ onto

p

(Ka)nK^

in

t h e

canonical

well-order of j j ^ ^ . So f™ is definable over £ a + m + 1 in {<&. . and the definition is uniform for all a. Lemma 11.18. Proof: Let a:N , . n->r N O l ,, be the map of lemma 8.3 such that —a+m+1 Ei — p+m+1 (i)

aiT

aa+m+l=7Tap-+m+l

(ii) a ( K a + p ) = K 3 + p (p<m) So by the uniform definability of f™ f m (y)=a(f m (y)) for Y
91 (11.9)

Now Lemma 11.19.

^

^

Proof: U a nK™={f m (Y) ••y«a A K ^ " " + 1 ( Y ) } • But f m + 1 Sa+m+2

from

K

is definable over

a+l"-Ko+m+l •

Let K = U K m . K is transitive and rudimentarily closed. By m is amenable. Thus < K a , €,U a >|=R. But N a

Lemma 11.20. H <=H . + Ha Proof: N cK cH^+cH, where H=H a a K a w a Corollary 11.21.tf£«J£.

+ N

a~ . By lemma 11.14 H K cH . a a

THE CASE O)p^>K FOR ALL n<0) This is not very difficult. Lemma 1 1 . 2 2 .

Suppose upn>K, all n<0). Then N =rud

P r o o f : L e t M=rud

n

«W .

(N) . T h e n <M,U> i s a m e n a b l e . F o r g i v e n X6M, i f

X C K then X€N, since P (K) OM=P (K) fiE^ (N) by lemma 2.41. So given Y C P ( K ) Y6M we have Ynu=Y(1(unN) . But unN€M so YflU€M. Hence <M,€,U>(=R so N + cM. But clearly McN + . Corollary 11.23.

•

?(K)ONf=?(K)nZ (N_)=P(K)0N.

Proof: Like corollary 11.10.

o

So much for that. ACCEPTABILITY A lot of the difficulties that remain arise from the need to obtain a sequence of functions in S,,

rather than

individual functions. It is helpful to prove the simpler version first, though, as the strategy gets obscured by the details in the proof proper.

92 Lemma 1 1 . 2 4 . Suppose N_ is an iterable

premouse and N_ is acceptable.

Suppose

aoS, a € i / \ N . Suppose u€N* and u<=P(6) . Then tf*\=~u<&. (&€0nN) Proof: Case 1 For a l l n_K. Then by c o r o l l a r y 11.21 and c o r o l l a r y 11.2 3 6_>K. And t h e r e i s n such t h a t oopjj£6. By lemma 9.7 t h e r e i s a Z n o n t o N. I f u € S ^ + k , Case 2

(N) map of a s u b s e t of 6

OnN=0)X t h e n t h e r e s u l t f o l l o w s from lemma 2 . 4 2 .

For some n<w wpN
By c o r o l l a r y 1 1 . 2 1 l e t t i n g n=n(N), ojp^+ < K . Hence by c o r o l l a r y 11.10 a£E

(N). L e t M be t h e c o r e of N w i t h mouse i t e r a t i o n

<<M)

,( 7T

Q

)

Q

), l e t K be t h e c r i t i c a l p o i n t of M and

suppose N=M,. ~~ ""A

We may assume 6 < K . Let a be least such that 6
+ 1 (N)

by lemma 2.43. Then ucE + 1 (N n ) so by corollary 7.18

n

ucZ + . (M ) . For some m wp™ <_6 . We can get a subset of K that codes all the I + 1 (M n ) subsets of 6; call it A. Then A€N + by chapter 10 exercise 2. But then N~F=U
onto K_.

Now we must do it properly. Keep the terminology of the previous lemma. L e m m a 1 1 . 2 5 . AT is strongly acceptable above K (and hence the axiom of acceptability holds in i/ for all 6>K) .

Proof: Suppose ac6 , K_< 6 , and a€N ^N. 11.21 9.8.

By corollaries

and 11.23 wpJJ_<6 for some n so this follows from lemmas 9.7 and D

So the proof is complete if P(y)nN=P(y)ON assume that N is a mouse and that wpj^

for all y
Let M be the core of N and let <<M > cr, ,< IT O ) ^ocr. > be its — — -—ex otton otp ot^ptvjn mouse iteration. Suppose N=M, and let K be the critTcal point of ~~ — A

Ot

M . Suppose ac6
The axiom of acceptability

holds in N_ for 6
93 Proof:By lemma 11.25 we need only produce a sequence . In fact we produce sequences f,=
<JZ,,

(m>l) . Since for only finitely many m is p n + m + 1 < p n + m this



will suffice. Take uGN

. We may assume ucP(K)flN . So by lemma 2.4 3

u£Z n + p + 1
The sequence f . 1 The relation R ( K ,£>,i) is defined by R(Ka,p,i) «,^=TTaX(p) A M ^ . © where (f>± is the Z Up

N

+1)

+1

^±

(11.10)

formula s(i). Since ^ " M ^ h ^ K ^ U p ^ " 1 " 1 ) , letting (

P>

" Nn|3H±<e)

(^=^aX(P)); and the

sequence is definable over N, therefore R ( K ,^,i) is definable over N (we make no claim about its complexity). Let denote the least K

such that K >JE,. k is definable over N. Let Y,7,PN +1 >' i )>-

(11.11) +1

So g is definable over N. Since k(^)chNn(^UpJJ ) , H | 7 +1

^ . Let ( t p ) . < K be such that t : £+u)x (k (£) )


N


onto, or

Let f*(y)=gU,Y,i) if t^(y)= and =^

otherwise.

We must show f ^: ^(ur\?(E,) ) U ( U onto. Suppose a€unP(^), ^ < K . Then a € £ n + + 1 ( N ) , so a€Z lary 7.18 a€E

+1 (MjJ(r))-

\=$. (y,y ,p™+1

y€a ~ M ^

Sa

nn + 1 ( NN )).

By

Y Y€(k(^)) < a ) such that

).

(11.12)

y€a «* R(k(^) ^ Y , ^ , ? ^ 1 ) , ! ) .

(11.13)

So Hence ,i) Suppose < i , Y > = t £ ( ? ) . Then a = f ^ ( ^ ) . Case 2 The sequences f m + i

(m>_l)

Let u'=unP(o)p^ +m ) . So by corollary 7.18 u'cE u'cE

+1_m(M

the Z

+1_m

n+m

)

(11.14)

+1

(M n ) . Hence

(without loss of generality mi denote

formula s ( i ) . Let A={< i, y, r, >: i
A Mn+m|=(()i(C,Y,pJJ+m+1)>- Then A € Z p + 1 ^ m ( M n ) ; hence A € Z p + 1 (N n | X Q ) , so A € N + . Let t€N + , t:o) P £ +m+1 +u)x ( a 3 P ^ + m + 1 ) < a ) onto. Let

94 f m + 1 ( y ) = { C : < i,Y/C>€A} i f t h i s i s i n u =£

Proof:

(11.15)

otherwise. •

This suffices. C o r o l l a r y 11.27.

(t(y)=)

If N_ is an iterable premouse then N is acceptable.

I n d u c t i o n b a s e d o n lemma 4 . 2 7 , lemma 1 1 . 2 5 a n d lemma 1 1 . 2 6 . D

Corollary 11.28.

If N_ is an iterable

n top^
premouse with K critical

and for some

premouse.

If N is an iterable

premouse with K critical

and for some

uatN, Ka M=J with u>pn
a

ff—

—

Proof: By corollary 10.33 and corollary 11.27.

•

Exercises 1. Is every core mouse strongly acceptable? 2. Show that $N+ in lemma 11.2. 3. Show that if N is an acceptable premouse with cap < K then cf (ajpM) =cf (0nM) . Deduce that if K is regular and N is an acceptable U premouse then p =K,Show that if M=J + , and a)p .
such that if aey
minimally? 5. Find a,A,B such that J

is strongly acceptable, J

is not, but

95 12: O

At last we are in a position to construct a mouse. The existence of a mouse turns out to be equivalent to a well-known large cardinal axiom, that 0

exists. Each is equivalent to

the existence of a non-trivial map j:L-> L. We are going to prove u

(i) if there is an iterable premouse then 0 exists; (ii) if 0 # exists there is a non-trivial j:L+ L; e (iii)

if there is a non-trivial j:L+eL then there is a mouse.

D e f i n i t i o n 1 2 . 1 . A sharp is an iterable premouse N_ with K critical

and On--

Lemma 1 2 . 2 . If there is an iterable premouse then there is a sharp. Proof: Trivial.

o

Lemma 1 2 . 3 . If N_ is a sharp with K critical

then H <=L.

Proof: cop =K. SO by corollary 11.21 H c j cL.

•

Note t h a t in addition P(K)nLcN. Lemma 1 2 . 4 . If N_ is a sharp then JV is a mouse with

n(N)=O.

Proof: There i s a £, (N) map of K onto K+OO SO ojp
L e t6>N,N';

l e t <<^

then N_, tf' have a common

>a£On,< iraB >

a l B € O n

iterate.

>,< < N ^ > a £ O n , < i r ^ >

a l B e O n

be the iterated ultrapowers of N,N' respectively. So N Q ,N' are comparable. But 0n.T =On.7,=9+co. So N Q =N' .

N6

N

o

-e -e

0

Let M be the core of some, and hence of all, sharps. This is fixed from now on - obviously M is the unique core sharp - and K is its critical point. Lemma 12 . 6 . p =1; p =0 . 1 MM Proof: Let X=h M (K) . Then X ^ =a

__ E

M. Say a:N=M|x. S O a:N-*E M. Let K =

( K ) . Then N is an iterable premouse at K; so N is a sharp.

!

96 Hence N is an iterate of M. But

K £ K S O N=M. But then there is a

parameter free map of u) onto M; so P M = 1 / P M =0 -

D

Corollary 1 2 . 7 . If N_ is a sharp then p =2, p =. A crojMw)

W

could be coded as a real and would be a reasonable u u u definition of 0 . In fact 0 is a different real, but 0 €L[A M ] and M

A €L[(T ] , as we shall see. << ^a > a€On' < 7 T a3 > a<3€On > b e t h e i t e r a t e d ultrapower of M; let K be the critical point of M ; let C = { K :a€On}. We now apply the apparatus of the previous chapter. Let N =J a, so M =N . Note that H =N in this case. Hence by corollary 11.12 J -< J , a a a S when a£$, and J^ ^=ZFC by lemma 11.13. By lemma 11.16 {i
. The structures K m are defined as

indiscernibles for J

before; K m c K m + 1 , Km-<:J + and K = U K m . Also M cK and is a— a ' a — K a c a a— a a' a a mtu) amenable. It follows that M c j +; we shall soon see that it is not a member. Lemma 1 2 . 8 .

If x€j^ then x is definable in L from ordinals in

Proof: x€M , so M(=x=h(i,
COK^^.

. . . K >) for some a,<...
x=h(i,y) «» 3zH(z,x,i,y) be some definition of h with H £ . Then |=3zH (z,x, i,< K . . . K >) . Suppose z€K m , some z€M . We may 1 p assume z€S^a + n €K m , ,x€Km. But then x is the unique 1 p a M x€J such that 3z£K such that z€S a A H(z,x,i, ) . K

s"a

Ot

Kfrn

CX-i

(X

i s d e f i n a b l e i n and i s d e f i n a b l e from

v-Lemma 1 2 . 9 .

(a) C contains all uncountable

cardinals;

and for all uncountable 9 (b) cn9 is of order-type

9/

(c) CD9 is closed unbounded in 9 provided 9 is (d) CC\Q are indiscernibles

regular;

for J~;

(e) every a€Ja is Ja-definable

from Cf19.

Proof: (a) If 9>OJ is a cardinal then 9 = K Q by lemma 8.15. (b) Cn9={K a :a<9}. (c) C09 is closed by lemma 6.15 and unbounded by (b) (d) By lemma 11.16. (e) By lemma 12.8.

a

97 Lemma 1 2 . 1 0 .

C is

the unique class satisfying

Proof: Suppose C 0

a:J Q -KJ e

were a n o t h e r .

(a)-(e)

of lemma 12,9.

Let C ' = { K ' : a € 0 n } .

Define

for

regular

by

a(t L 6(K a ...K a ))=t L 6(K^ 1 ...K^ ) (12.1) 1 p 1 p where t is any term, a,<...
; i

6

By t h i s and property

(e)

p

of C, a i s well-defined. Also by

arJg-*- J Q ; and by property (e) of C , isomorphism, so a = i d | j Q . So C = C ' .

So cne=C'n6 for

all

(12.2)

Hence a i s an

r e g u l a r uncountable 0.

a

Definition

12.11.

Suppose there is a class

Let {({> ;n
# "0 exists"

a maps onto J Q .

^ means that

C satisfying

(a)-(e)

L, where < > j has m free

of lemma 12.9.

variables.

Let

n

a class

satisfying

(a)-(e)

of lemma 12.9

exists.

ft Lemma 1 2 . 1 2 .

If

there

is an iterable

premouse then 0

exists.

u

Note 1 : 0

is read "0-sharp" or "zero-sharp".

Note 2: Suppose 0 # exists: let M be an inner model with 0 # €M. a model of 0

Is M

exists? The terminology would be misleading if this

were not so; and later on we shall show how to construct an iterable premouse from 0*. Note 3: J^ -^L, of course, since L is a direct limit of ^ J*— ao *Qo ^ .« i^-^^ i( ^l JoQ^« reguxar J^

B u t

it would not be legitimate to replace

by L in definition 12.11. (0

Note 4: Suppose M were a constructible sharp. Since some iterate of M contains J.,

—

JSi

, P(w)nLcM. But p =1 so there is A€P(OJ) PIE, (M)^M. —

M

1 —

Since MEL, A€L; contradiction! Once we have proved the remark in note 2 we shall have shown 0 # ^L. In fact O # €E,(M)\M for all sharps M. Indeed, as we said, A^zo)* (00)
would have been a plausible

candidate for 0 # . u

Our next aim must be to extract from 0

a non-trivial j:L->- L.

This is rather easy. In fact any order-preserving map T:0n-K)n yields a map of L to L; simply define j (t L (K a . . . « a ) ) = t L ( K T ( a )-"
:a€0n}.

1

n

1

n

) ) r

98 u

Lemma 12.13.

If O

exists then there is a non-trivial j:L-> L.

Proof: Let x(a)=a+l and define j as above.

•

In the example in the proof the critical point of j is K Q , which is countable. So we could have added the requirement that j have a countable critical point. Third and most difficult is the construction of a sharp from j:L-*- L which is non-trivial .Since a sharp is a mouse this will complete our circle of inferences. Suppose then that j:L-»- L has critical point K. Let U= ={X€P(K)nL:K€j(X)}. The proof of lemma 5.10 shows that U is a normal measure on K in L.

Lemma

1 2 . 1 4 . <J +,u) is amenable (where K + denotes the successor of K in L) .

Proof: Suppose X€J +; assume X C P ( K ) . Then J +f=X_< K . Suppose f€J +, f:K-OC onto. Then j(f)€L. But f (y)€U ~ K6j (f (y))

(12.3)

~ K€j(f)(y) for y€L. But XnU€H^+ so xnU€J K +. D So J K + T E J +• Hence UPIJ

. is a normal measure on K in J K+ n • That is,

J K + , is a premouse. The difficulty arises in showing that it is iterable. We are going to prove that L is iterable by U, that is, that the weak iterated ultrapower of L by U preserves well-foundedness. Since we shall need to generalise this result later on we shall work with an arbitrary model M and a normal measure U on K in M such that (i) M|=ZFC + V=L[D](some class D) ; (ii) is amenable, where N=J^+; (iii)

N=H^+.

Definition transitive

12.15.

M_ is K-maximal provided that whenever -H:M+ M_', M_' is

and ir|K=id|K then M=M_' .

Note t h a t L i s 0-maximal.
Suppose from now on t h a t M i s >

b e

t h e

w e a k

Let < < M c t > a e o n ' a e > a < 6 e o n Let X be l e a s t such t h a t M. i s not w e l l - f o u n d e d . lemma 1 2 . 2 0 we are working for a c o n t r a d i c t i o n . Lemma 1 2 . 1 6 .

A i s a limit

K-maximal.

i t e r a t e d

ultrapower of M. U n t i l j u s t before

ordinal.

Proof: A*0, of c o u r s e . Suppose A=y+1. Since M i s t r a n s i t i v e M =M.So M={7T

0y(f)

(K

h *#"Kh

) : f € M A

h1<...

(12.4)

99 where K, is the critical point of TTUU + 1 • Now y*0. For we may define a:M,-+M by a(7T01(f) (K))=j(f) (K) (12.5) and check that a:M,+ M is well-defined, so M, is well-founded. —± Li — —1 Thus M={irol(g) (K) :g€M}.

(12.6)

Now by (12.4) and (12.6) M={

Voi ( f ) %--- K h p 'V : h i < --- < h P < y ' f€M}-

(12 7)

-

Define a:M+M^ by a(TT O p l r o l (f)(K h i ...K h p ,K y ))= T o x (f)(K h i ...K h p / % ))

Claim

#

!

'

(12.8)

H

where <|> is any formula (with one free variable for simplicity) Proof: Mxh(Tr0X(f) (Kh ...K h ,K p )) «{
(lemma 7.7)

^ MH=cj)(7TOyTrol(f)(Kh . . . K h ,K y )).

D (Claim)

So a:M-> M, i s w e l l - d e f i n e d . B u t a i s s u r j e c t i v e . S o a:M=M, . H e n c e *-~ e*~A —~ ~~A M, i s w e l l - f o u n d e d . C o n t r a d i c t i o n !

a

L e t X = { x € M : C K j < A =» T T O . ( X ) = X } . X i s a p r o p e r c l a s s b y c h a p t e r 8 exercise 2. Lemma 12.17.

K<=X and x-^M.

Proof: K C X is clear. Suppose x,...x €X and t is any term. Then for all j
12.18.

D

M_ is K-minimal provided that whenever 7T:M_-'> M, M_' is

and 7r|K=id|K then M= ' M_.

Note t h a t L i s O-minimal. Assume from now on t h a t M i s

K-minimal.

Let X. be t h e s m a l l e s t e l e m e n t a r y s u b s t r u c t u r e of M c o n t a i n i n g XU{K.:j
transitive.

Then i r j ^ M ^ M and ir||K=id|K so M|=M. L e t TT| .=TT\ ~ 1 7r|. Lemma 1 2 . 1 9 . Proof:

TT^ =TT . for all

Claim 1

i<j<\.

For acK, TT_ . (a) =TT' (a) DK . . — Ul U 1 Proof: a=7T^(a)riK as K C X . S O iroi (a) =iroi (TT^ (a) DK) =77^ (a) nK ±

100 (as TT^faKX).

• (Claim 1 ) .

K^SX±•

Claim 2

Proof: Suppose V < K . , V=TT O . (f) (K,

••••<, ) (h,<...
assume f:K P +K, so by claim 1 7rQi (f) =71^ (f) | K ? . Thus V=TT(!) (f) (« h Kh) 1* " P which is in X ± . a(Claim 2 ) . Claim 3

x O ^ =» Trk-(x)=xf all k,j with i
Proof: For 7r]c-(Kh^=Kh Claim 4

f o r h
«

• (Claim 3)

X.flK.=K. for all j^i* j<X.

Proof: Suppose y>K.,y€X., y_i, y
, , y+l<X. So

%u+i ( Y ) -%y+i ( K u ) = Vi > Y But IT U + 1 ( Y ) = Y by claim 3. Contradiction.1 Claim 5

a (Claim 4)

Tr!^.(Ki)=K..

Proof: Suppose Tr|.lKi) K . . So TT!~ TT!(K.)>K., i.e. TT!(K.)>>7TI. ( K . ) .

Note that TT!(K.) is the least member of X. greater than K . . Suppose IT! (K .) =t(K h

K h ,x)

(h1<...
M|=(3^1...^ < K .) (K .
But by claim Claim

6

(12.9)

to (12.9)

2 t (£,...£

If a<=K . t h e n — i

, x ) € X i . Contradiction. 1

• (Claim 5)

TT . . (a)=7r! . (a) . i] 13

Proof: a=ir! (a) DK . . So j

(a)=7ri. (Tr^(a)nK i )=7r^(a)nK.

=TT! . (a) . Hence X € U . «* K.€TT! . ( a ) ,

where

may define

a.:M+M b y

inductively

aQ=id|M

a

y(7riy(x)

)=TT

iy(x)

U. is the i-th normal

(12.10)

• (Claim 6) measure.

Now we

101

Note t h a t V a g

=7I

a

a B<X

ag a

-

-

We verify inductively that a. is well-defined, a.:M-> M and a. is surjective. For i=0 this is clear. For i=j+l a. is well-defined and a.:M+ M by claim 6. If x€M then TT!! (x)=t M (K a ...K a , K . , X ) , say (o^*...
. . . K a ,K_.,X)

=TT["1TT. (f) (K.) •* ^

where f(£)=t M (K a . . . K ,£,X) 1 p

= TT^i(f) (Kj) . Thus a. is surjective, as a. is. If i is a limit then clearly o± is well-defined and a^M-^M. Tr!(x)=tM(K

If x€M and

1

K , X ) (a,<-..
j
Hence each a. is the identity; so TT . .=TT! ., i£j<X.

•

But now define TT:M,->-M by TT (TT . . (x) ) =TT! (X) . Then

And

7T

i X ( x ) = = 7 T j X (y) =* T r i j ( x ) = y

(assume

=*> ir^(x)=7r! (y) .

So TT:M,->M is well-defined and 7r:NL->- M. Thus M, is well-founded. —A — —A e— —A Contradiction I In summary Lemma 1 2 . 2 0 .

Suppose

(i)

M^ZFC+V=L[D]

(ii) (N_,u) is amenable, where N=J + (Hi)

U is a normal measure on K in M_

(iv) N=H^+ (v) M_ is K-maximal and K-minimal (vi) j:M+ M, U={X:Kej(X)}.

Then letting

(<M )

each M is

well-founded.

c

,( 7T

Q

> ^Qc

> be the weak iterated

ultrapower

of M by U,

This is a long-winded way of proving a standard result, but the techniques will be useful later on. Return now to our premouse J + , . Lemma 1 2 . 2 1 .

jU

is

iterable.

102 Proof: J^+1EL- Let < < ^ > o 6 Q n , < ^ g > a £ B 6 O n > and

l e t

<

aeon'<1Tae>a<66On>

L by U. We c a n g e t e m b e d d i n g s o f M

b e

be an i t e r a t e d t h e

i t e

™ted

i n t o L, a , s u c h

ultrapower

u l t r a p o w e r of that

the proof is like many others and the details are left to the reader. (Actually it is not hard to show that each a

4) D Hence J

is id|M : see exercise

_ is a sharp.

Q # IS ABSOLUTE The proof that if 0 # €M then M|="0# is 0 # " can be done by an argument from descriptive set theory. Here we give a different proof that appeals to the absoluteness of well-foundedness. Lemma 12.22.

If M_ is an inner model and t£=x=O then x=0 .

Proof: M^=0

exists. Hence there is an iterable premouse in M. Hence # # there is an iterable premouse, so 0 exists. But then x,0 each satisfy definition 12.11 so x=0 # .

•

The other direction involves a bit of work. If x=0**, M_ is an inner model and x€M then W^x=0 # .

Lemma 12.23.

Proof: Let M be the unique core sharp; let <<M^> a € O n'*^aB > a<8€0n > be its iterated ultrapower. Let K

be the critical point of M" .

Let enumerate terms of L, and suppose t

U ={

' VV" K m n - l)!Koe V Kl- " K «>n )

interpreted in j ^ Claim 1

A

and < K :a€0n> enumerates C.

U'€M.

Proof: Let s

n

index the formula

"v^et ( v , . . . v O n l m

L e t A = { n : s n € 0 # } . A€M and t n ^ o " Say R ( p , q )

* t

(K . . . K

_±) < t p

1

^ -x)60'

1

R'=RnQ €M. R

**

) A t n€A

(v ...v i^^^n n OA m -m 1 — 0

-

(K . . . Km ^ X K . Then R€M. q

L e t Q={p: (Vq
is m -ary. Let

W'-'V1'"0*

_±) }.

Then QGM. So

i s a w e l l - o r d e r o f Q s o M|="R' i s a w e l l - o r d e r o f Q".

L e t f€M,

f : < Q , R ' >=. C l e a r l y

C=tn(Ka

..Ka

) . Then

5

^£K0*

G i v e n

^ < K n ' suppose

= t n ( K 0 . . . K m _ x ) . So K = K Q .

103 It follows that f(p)=t (K Q ...K m _ x ) . ...K

Let X R ={f(p):t

(KQ...

-,)€t ( K ^ . . . K . ) } . Then <X :n€M. B u t U I ={X : n € A } . m —l n O m —1 n n n P D ( C l a i m 1)

Claim 2

U'=U.

Proof: If t (K_...K , ) €U then K_£t (K.....K ) as in lemma 11.19. n 0 m n -l 0 n 1 mn So t (K^...K ,)€U'. Similarly for K ^t (K . . . K J. n u m —JL o n o m ~*J. n n suffices to show that if X€L, X C K

then X=t (K ...K _ , ) ,

But suppose X=t (K ^€t n (K a '"Ka n a a l m

...K ) . Then for £
So if O^exists then L[0 ]|=0

#

S O it

some n.

n(Claim 2)

•

exists. In chapter 15 we shall see how

to generalise these considerations. Exercises 1. Replace L by L[0 ] throughout the chapter. What equivalents for the existence of non-trivial j :L[0* ]-^eL[0# ] are there? 2. Show that if 0 # exists then P(o))flL is countable. 3. Show that if there is a measurable cardinal then there is a premouse in L.

u

4. Suppose 0

U

exists. Show that, letting U be such that JK+±

sharp, each weak iterate of L by U is well-founded

is a

(hence L) and

that the critical points of the weak iteration are the canonical indiscernibles for L.

104 13: THE MODEL L[U]

Suppose U is a normal measure on K; then UflLtU] is a normal measure on K in L [ U ] . So L[U] is a premouse. By lemma 8.10 L[U]|="each iterate of V by U is well-founded"; so each iterate really is well-founded; so L[U] is an iterable premouse. Fix U , K such that UflLtU] is a normal measure on K in L[U] . We need not assume that U is a normal measure in V. Lemma 13.1.

L [uj \=GCH

Proof: L[U] is acceptable by corollary 11.27. By corollary 3.4 for each v either P(v)=v

or for all ucp(v) u£v. But if the latter

held then since the power set axiom holds in L[U] , P(v)
in L [ U ] .

n

We know that L[U]*L. In fact Lemma 13.2.

L[U]*L[A] for all Aoc.

Proof: Let j:L[U]->0M. Then L[A]=L[ j (A) PIK ]CM S O M=L[U] . But see chapter 5 exercise 1. Lemma 13.3.

o

K is the largest measurable cardinal in L[U].

Proof: A slight generalisation of the previous lemma. If X > K and V€L[U], V is a normal measure on X in L[U] let j ^ t U ] - ^ . Then M=L[j(U)]=L[U]; but V$M by chapter 5 exercise 1. Lemma 13.4.

n

K is the only measurable cardinal in L[U] .

Proof: Suppose X < K and V is a normal measure on X in L [ U ] . Let ^ L t m + y L t U 1 ] ; then U 1 will be normal on J ( K ) in L[U']so L[U'] is an iterable premouse. Let Q,Q% be comparable iterates of L [ U ] , L[U'] respectively: then Q=0', since each is a proper class.

Now on the

X

one hand V€L[U] so V€Q cas K>2 (in L[U]). On the other hand V$L[U'] so V^Q 1 . Contradiction! Lemma 1 3 . 5 .

U(\L[U] is the only normal measure on K in L[U].

P r o o f : S u p p o s e V were a n o t h e r . L e t j : L [ U ] + V L [ U ' ] ; L [ U ' ] i s i t e r a b l e premouse s o L[U] , L [ U ' ] have a common i t e r a t e ,

Q,

S u p p o s e j:L[U]-»- Q, j ^ L t U 1 ] ^ Q a r e t h e i t e r a t i o n maps.

Let

an say.

105 J

5

J

T

(13.1)

By chapter 8 exercise 2 C is a proper class. Let X be the smallest elementary substructure of L[U] that contains K U C . Suppose a:L[U "]=L[U]|x. Then a|KU{K}=id|KU{K}, and if X C K then a(X)=X, so U M =UnL[U"]. It follows that L[U"]=L[U]. In L[U"] every subset of K is definable from parameters in KUCT" "C; S O in L[O] every subset of K is definable from parameters in K U C . Suppose X C K , X = l ^

1

n

...^

x€U

n

,c

l 1

m

...C

m

)

(q<...<Sn
j'j'(x)=t

Q

(^1...^

n /

c1...C

m

)

Then so

j(x) = j(x)=j'J(x).

<* K € J ( X )

** K€J"(X)

(13.2)

(K<J(K) SO J'(K)=K)

<* X€V. Thus U0L[U]=V.

a

L[U] is not a mouse but in many ways it behaves like one. In particular we can obtain a core L[U] of which all models L[U] are iterates. We need a preliminary lemma. Lemma 1 3 . 6 .

If L[U], L[V] are such that

L[V] respectively,

then

U,V are normal measures on K i n

L[U],

L[U]=L[V].

Proof: Let Q be a, common iterate of L [ U ] , L [ V ] . Let j,j' be the iteration maps. Let C={£:j(£)=j'(£)=£}. C is a proper class and just as in the previous lemma every x in L[U] which is a subset of K is definable from parameters in K U C ; similarly for L [ V ] . Consider x€L[U], X C K . Suppose x = t L [ U ] (^ . .. £ n , r>1. . . cm) (^<...<S n
so j(x)=

=j'(y). Suppose x€U. Then y£V. But for y
(13.3)

** y€j' (y) «=> y€y so x=y. Hence x€V. So UcV. Similarly VcU so U=V, L[U]=L[V],

a

Of course, lemma 13.5 could have been derived directly from the proof of this lemma. Now there may (so far as we know) be lots of different normal measures on K; but for any two, U,V we have shown that UDL[U]= VnL[V], Cutting down to L[U] loses the sets where U and V differ. In part six we shall return to the problem of inner models with

106 different measures at a point. Lemma 1 3 . 7 . Suppose U, V are normal measures on K,X in L[U], L[V]

If K<X then L[V] is an iterate

respectively.

of L[U].

Proof: Let < < L t U j > a€Qn ,< TT^ > a £ 3 € O n >be t h e i t e r a t e d ultrapower of L[U]; l e t K a = 7 r O a ( K ) • L e t a b e l e a s t s u c h t h a t K ^ X , and suppose (to get a c o n t r a d i c t i o n ) t h a t K >X. Let Q=L[UQ] be a common i t e r a t e of L[U], L[V]. a must be a w successor o r d i n a l ; say a=8+l. Suppose ^ =7T g a ( f ) (Kg) • I f f = =tL[UB](?1...5n,Cl...?m)

where

5l

<...<5n
C

l ' ' " C m e C = { ? : i r oe

(5)

=

1

=TT'(£)=£} and TT i s t h e i t e r a t i o n map from L[V] t o Q, then *Bo so t h e r e i s a term t

1

(13.4)

such t h a t

<51...5n,Vc1...cm> so X - f Q ( C 1 . . . 5 n , K B , c 1 . . . c n ) - i r - ( f

(13.5) L[vl

(51...Cn,KB,Cl...cm))

X€rng(ir'). C o n t r a d i c t i o n ! So K =X. The r e s u l t t h e r e f o r e by lemma 1 3 . 6 .

so

follows

D

Compare t h e proof of lemmas 13.7 and 13.6 with t h a t of lemma 10.16. The l a r g e c l a s s of fixed p o i n t s C enables us t o c a r r y through t h e argument. D e f i n i t i o n 1 3 . 8 . (i) If u is a normal measure on K in L[U] then L[U] is called a p-model. (ii) Let L[UJ be the p-model with minimal critical point. L[U] is called the core p-model. (Hi) The critical point of the core p-model is called the critical measure ordinal. 0+

~T

#

0 is intended to code embeddings of p-models just as 0 coded those of L. It may also give insight into why p-models behave like mice. The theory of double mice will not be developed here in any detail. One reason is that it is similar in most respects to the theory of single mice and it would be tedious to repeat all the details. Another is that we really do not need any of this material later on (other than the main definition 13.17) . D e f i n i t i o n 1 3 . 9 . Suppose M*<M,E ,U,v). M is a double premouse provided f^R " • " • " " " " " " — " • ^ — — — — — — — —

fq

107 U,V are normal measures on K,\ respectively Definition

13.10.

Let (h^-itOn)

in M and K<X.

be such that h£.2, all i€0n. Let M_be a

double premouse. Then <<# > -. , is called an iterated

ultrapower of

—a afcOn otp a
<< << ^a >>aeo

(ii)

^a a

n'

<7r

a3 > a<3€o n >

i s a commutative

system;

^M; ifh =0

a >'

if i s

a direct

V1' lim±t

of

<<

^a)a<X'<7Ta3>a<3<X>

when

X i s a

limit ordinal.

It is easy to see that for each with h.€2, all i€0n, there is an iterated ultrapower of M with index . Definition

13.11.

A double premouse M_ is iterable

hi0n+2 there is an iterated

—

h such that for all a M

Definition 13.12.

if

ultrapower of M, < < M > c^ ,< IT ~~GL CXtOn

and only if Q>

^Q_

for

> with

all index

Otp O t ^ p c O J l

is transitive.

A double premouse is a dagger provided that M_ is iterable

and, letting Af=
If there is a dagger then there is an inner model with a measurable cardinal; indeed/ every uncountable cardinal is the critical point of some p-model. Lemma 1 3 . 1 3 .

Suppose M=J '

is a dagger. Then U is a normal measure in L[U].

Proof: Suppose U,V are normal measures on K , X respectively in M. If U is not a normal measure in L[U] then there is a least 3 such that U is not a normal measure in J o . Let h:0n+2 be defined by P

h(a)=l, all a. Let < < M a > a € O n , < ^ 3 > a < 3 € O n > b e t h e i t e r a t e d ultrapower of M with index h. Say M =< M , € M ,u7V a >. Let * a = 7 T O a ( X ) * P i c k a with X >3. Then J O =J D so U is a normal measure on K in J Q . • ot p P p Now a literal repetition of the proof of corollary 11.21 shows that if M is a dagger and M|=X is the largest limit ordinal then H^ =J^ ; and so M is acceptable. Suppose M, M 1 are daggers. Let 8,6' be regular cardinals such that e'>e>M,M'. Let h:0n+2 with h(a)=0 (a<0), h(a)=l (a>0 ) . Let <<M > ^ ,<7r o) o^r, >,<<M'> ^ , or.^ > be the iterated —a a€0n a3 a<3€0n ' —a a€0n' a3 a<3€On

108 ultrapowers of M,M' respectively with index h. Then it is easily seen that M Q I =M',. (They are comparable just as in lemma 8.18; but On..

MQI

=6 '+a)=On.,, ) . So any two daggers have a common iterate. Me, Suppose M is a dagger. Let X=h M (0). Then if a:M'=M|x and M 1 is

transitive, M 1 will be a dagger. Clearly p ,=l,p M ,=0. It is easily seen that A , AM,

is £,(M) with no parameters and

^M. So for any dagger M, P M =1 and P M =0• In fact the

proof of lemma 10.16 adapts to show that M is an iterate of M 1 and,

indeed, an iterate with the special property that there is

an index h for the iteration that is monotone. Actually this is not a special property at all: every iterate has such an index. This is left to the reader. Now let M 1 be fixed as the core dagger. Let <<M

a ^ e o n ^ ^aB > a<6€0n > d e n o t e i t s iterated ultrapower with index h. Let 9, 6' be uncountable cardinals. Let h(a)=0 (a<0) and h(a)=l (a_>6) . Let M=M Q , . ( We should have to improve this terminology if we wanted to study double mice in any depth1.) Suppose M.%=(M% ,€,\J* ,V* > where U ' f V are normal measures on K,A respectively in M 1 . Let K =7r =7T 0a(X) aa r 00aa( (KK) ) XXaa=7T 0a as in lemma 12.9

Lemma

if

13.14 .

C has order

type

(c)

C is closed

unbounded

9 ' is

-

L e t

C = { K

: a < 6 } a

(a) CU{6}UD contains

(b)

all

9 ; D has order in 0 if

'

uncountable type

D = { X

a:

e

cardinals

±

a < e

'

less

}

J u s t

•

than

9'.

9'.

9 is regular;

D is closed

unbounded

in 9 '

regular.

(d) are

( a < e l )

If

<J> is any formula,

a < . . . < a , a ! < . . . < a ' are in C, $ < . . . < $ ,&'<...<&' 1 n 1 n 1 m 1 m

in D then

&H
a ,xB . . . x B ; « < K K 1 n 1 m (e) Every a J Q , is J^,-definable Furthermore C,D are the unique classes

Now l e t K ^ {K^:a<9}.

Lemma

be the c r i t i c a l

Then for all

point

of t h e a - t h p-model.

Let C'=

13.13 CcC1.

By lemma

13.15.

a ; - • • < „ . , x B . . . x ;. 1 n 1 m from CUD (where MM M,e,U,V>) . with this property.

Let $ be any formula

A , . . . A €p 1 m

of L and suppose U

a <...
Proof: Suppose XCD. L e t « L [ u ; ] > a € O n , ^ g > a l g e O n > be t h e i t e r a t e d u l t r a p o w e r of t h e c o r e p-model. I f A i s a l i m i t c a r d i n a l with cf (A)> >9

then

7r

^(0)=e

f

o r a l l a£3<9. But U i s a normal measure on 9 i n

109 L[U] so U'=U. Picking ^ . . . A ^ D c a r d i n a l s of c o f i n a l i t y >9, -9'

a1

an

1

m

1

n

V--V

}eu

-9

o

But then this is true for all X,...X €D. 1

In

•

in

Corollary 13.16. c'=c.

It follows that if we let h be the constant zero function and let K =TT

(K) then < K :a€0n> enumerates the critical points of all

p-models. In particular, K is the critical measure ordinal. Definition 13.17.

Let 9 , 9 ' be uncountable cardinals,

9<9'. Let C,D, and U

be as in lemma 13.14. Let ,< XD:$<9' > enumerate C,D respectively. Let a p {d>n :n
0

i s p l a i n l y independent of our c h o i c e of 9 ' . We may t a k e t h e

statement "0 Lemma 1 3 . 1 8 . embedded in

e x i s t s " t o a s s e r t t h e e x i s t e n c e of a d a g g e r . If 0 exists

then every p-model may be non-trivially

elementarily

itself.

Proof: Every f:0n-K)n which i s s t r i c t l y i n c r e a s i n g g e n e r a t e s a map TT:L[U]+L[U] such t h a t Tr|K=id|ic, TT(X )=X^, V where h:0n+2 i s t h e function h(a)=0

(a<9), h ( a ) = l

(a>9),M"=<M"

€ , U , V Q > , s a y , and

F i n a l l y we s h a l l o b t a i n a dagger from some j:L[U]-> L[U] t h a t i s n o n - t r i v i a l , where L[U] i s a p-model with K c r i t i c a l and j | K = i d | K . Let < a € O n , < . ^ 3 > a l 3 € O n > be t h e i t e r a t e d u l t r a p o w e r of L [ U ] . Let V={X€L[U]:X€j(X) A X C X } where X is the critical point of j . Then V is a normal measure on X in J^l-i- By lemma 12.20 if we let ^a>ae0n'a<6€0n> b each M i s well-founded.

<<

e a n i t e r a t e d

ultrapower of J ^ by V then I t follows just as for 0 that

J^|^nP(X)cL[U], so that J?+i

is a double premouse. The reader is

left to check (exercise 1) that iterability will follow once we know that for all monotone h and all iterated ultrapowers of J / , <<

^>ae0n'a «5

i s

well-founded

(all a).

110 Observe that there are maps J a : L t u a ^ e L t u a ^ such that (i) ;Ja|Ka+l=id|Ka+l (Ka the critical point of L[U a ])

a

^ T

(iii)

jQ=j.

L e t V a ={X€L[U a ]:XcX a A * a € J a ( x ) } ' where Xa i s t h e c r i t i c a l p o i n t of j . (In f a c t

X =TT0 (X) . I f Y<#n"0 (X) t h e n Y
X must be r e g u l a r i n L [ U ] . Say Y=
and f : K - 7 .

Then J a ( Y ) - j

"vOai£)iKa1"-Kaa)<<

B u t

MB > 6 e0n' < i r BY > 6

a

. ..K

) where a , < . . . < a
( ^ ( f ) ( K ^ . . . K ^ ) ) =7rQ"j (f) ( K ^ . . . K ^ ) =

3a(^oa(X))^Oa(3(X")>lrOa<X»)-Then

i s

t h e w e a k

i f

i t e r a t e d u l t r a p o w e r of L l U j by

Given h:0n->2 which is monotone let 0 be least such that h(0)=l (if there is o n e ) . If a£0 we shall embed M h into L[U ]; if a>0 we shall embed M h into M 0 Q . Suppose Mh=< M h , e ^ h , ^ , ^ >. For a<0 let —a —a—9 —a a M a ex — a n v a =id|j!r'^ if a=0; let a be defined by a

a(Tr3a(f)(S))=^a(a3(f))(KB}

where ^ a = ^ Q a ( K ) /

i f a=

^+1-

(13 7)

'

I f a i s a liinit a

(7T a

3a ( x ) }=7T6a ( a 6 ( x ) } *

Then ao:<M*?,€Ji,Uo >-*„ L[U Q ] and each M h is well-founded (a<6) . (Actually oQ is id|M^.)

Now define a Q + a ^ + 0 L ^ l by

(where X^ a

0+a ( 7 T 0+30+a ( x ) ) = T r 6a ( a 0+3 ( x ) )

( a

a

I t i s necessary to check that 0

9+a:«taVJ6'Ve> 8

(13

f

where 6=TT (XQ)" * and v " i s t h e a - t h i t e r a t i o n of V Q . (13.9)

"9

follows

from

in the usual way. So we must show a Q (xflVQ) =a Q (x) flVQ. This is proved by induction on 0. If 0=0 it is immediate. If 0=0'+1 then 7T

e ' 0 ( f ) ( ( V ) € V 0 ** {^:f ( C ) € V Q , } € U Q ** {C:a e , (f) (^)€V0I}€Ue *• {C:X e ,€j 0 f (ae,(f) (^))}€Ue -

ir ele (X el )€ir e , e (J el (a el (f)))(K el )

(13.11)

Ill - Xe€Je(ae(ir5,e(f)(Ke.))) ~VVe(f)(Ke,))eve. Suppose 0 is a limit. Then

7r£,e(x)€v£ ~ x€v£,

(13.12)

- a e l (x)€V 0 I •• ^ 0 i € D Q i **

x

€ T r e

0

t

e

(OQ I

( j

0

i

(x) )

(creI ( X ) ) )

(

- VWe V(x))) ^ Ve (c V (x))€ V The proof is complete. We have shown Lemma 1 3 . 1 9 . If there is a non-trivial into itself

whose critical

elementary embedding of a p-model

point is greater than that of the p-model then there

is a dagger. We could prove analogues of lemma 12.22 and lemma 12.23; hence there is a minimal model L[0 ] in which 0

exists. Hence every p-model

is contained in L[0 ] . SOME DESCRIPTIVE SET THEORY The rest of the chapter is not needed for any subsequent results in this book. Definition 13.20. A mouse M with p =1 is called a real mouse. "

~~

M

For example, a sharp is a real mouse. Note that an iterable premouse M with P M =1 must be a real mouse. Fix some recursive function T : o)<~*- (a) x (a)) w ) . Let A T denote {n:T(n)€A}, when Ac(o)x(o)) w ) . Lemma 1 3 . 2 1 .

There i s a Tl1 formula

<\>(a) ** there

<J> such

that

T T (some(some p€PA) is a real mouse M_ with A^.) a=(A^.) p€PAM)

Proof: By lemma 3.26 there is a II2 sentence o

such that

<JlfA>(=a «* there is an acceptable premouse M with P M =1

(13.13)

a)

and- T?,n (cox (co) )=A. a can be written as an arithmetic condition ty on A . The complexity arises from the i t e r a b i l i t y requirement. If Rco)

is a well-order of a subset of co then M

denotes the

112 (or rather an) a-th iterate of M when a is the order-type of R. <J)(a) is to be the formula \|; (a)

A VR(R well-orders a subset of GJ =*•

(13.14)

-> NU is well-founded) where M is some p-sound premouse with P M =1 and

(A^) =a.

Now "R well-orders a subset of u)" is (Vbcu)(3n€b)(Vm€b)(-Rmn)

(13.15)

together with the arithmetic requirement that R be a linear order. By lemma 8.6 we need only consider well-orders R of u>, and the lemma follows from the following claim. Claim There is a II, formula x such that X(a,R) •» there is a p-sound premouse M with P M = 1/ (T^n((ox(u))
...Y± ) 1 m where v={k n ...k },k,R...Rk and u={k. ...k. } (i < . . . < ! ) . 1 n 1 n i . i l m * 1 m There is a formula \p such that M h M f , g , n ) ^ MR|=iT0R(f) (Ka . . . K

>€TrOR(g)(K

In

...K a )

(13.16)

(13.17)

In

where IT.- is the iteration map of M to M_. and K

is the critical

point of the a-th iterate of M. Let Z ={:hM(i,< nn . . .n >) is a function of K-*K^ where CJ ± p M ± p K=K Q .} Let T'={<x,u>£J 1 :x€Z A U € [ R ] ^ (some q ) } . Define a r e l a t i o n on TR by uuUv vuUv S (13.18) S « x«, u>> , <
S is arithmetic in R,a. But x( a / R ) *» S is well-founded

R

so x( ' ) is n l .

a(Claim) •

Lemma 13.22. o # is A1. Proof: There is a recursive function t such that for any sharp M n€0 # «• t(n)€(A?J T

. And "a is (T^fl (o>x (co) <(A)) ) T for some sharp

M" is II2 , for it is (a) A i€a, where s(i) is the sentence asserting that the critical point is the largest ordinal. Call this 11- formula \p(a). Then

#

n€0 # *• 3a(\p(a) A t(n)€a) =» t(n)€a)

h

(lh (II^)

(13.19) °

Note 1: This is the best possible result, for if 0 # were n^ or then it would be in L by the SchOnfield absoluteness theorem.

113 # 1 Note 2: x=O i s Ji0, although t h i s i s not clear from our approach. # We should have done b e t t e r to define 0 as the unique a satisfying the formula \p of lemma 13.22. Lemma 1 3 . 2 3 .

Suppose a€lJ[U], where U is a p-model, and aQjO. Then there

is a

real mouse AT with a£M. Proof: a€J + where K is the critical point of L[U] and K is K U U calculated in L [ U ] . If a£J then a€J K n which is a real mouse. K h Otherwise let 3 be least such that a€J o =M, say. M is an iterable p+1

~~

premouse, hence acceptable. By chapter 3 exercise 6 a $+u)€M; contradiction! So P M =1.

if P M >1 then

This would be true even if we insisted P M =0• Let <

be the canonical well-order of the real mouse M. If N is

another real mouse, let <

be its canonical well-order; let M', N'

be comparable iterates with canonical well-orders < , , < , . Then for a,b€P(u)) riMON a<

Mb ~a<M'b ~a V

Lemma 1 3 . 2 4 .

b

**a
Tn L[V] there is a A^ well-order of the reals.

Proof: Say a<*b if there is a real mouse M in which a< M b. We have just seen that this defines a linear order. In fact it is a well-order, for given ucP(w),u€L[U], let y+1 be least>K such that Jo.+lnu*0.

Then P (GO) nj U #P (GO) n j U + 1 , so J ° + 1 is a real mouse. Take

a€u least in its canonical well-order. Then a will be <* minimal in u. (We are using the result of Chapter 8 exercise 3.) Now suppose M is a real mouse and let 3=(T^n (cox (OJ)

a)

) ) T . Then

a< M b ~ (3i,x,j,y)(a=h(i,x) Ab=h(j,y) A h(i,x)Wh(j,y))

(13.20)

which is arithmetic in £. So a<*b «* 3M(M is a real mouse A a< M b) «* VM(M is a real mouse =* a<..b) — —

(E^) (11^) . Mo

(13.21) •

This is known to be the best possible. Exercises 1. Suppose M is a double premouse. Show that if N is an iterate of M then there is h:0n->-2 monotone such that N=M for some a. — — —a 2. Show that Consis(ZFC+there is a measurable cardinal) => Consis(ZFC +" the critical measure ordinal is uncountable". 3. Show that 0

is A- Use the method of this chapter to show that

if V=L then there is a A 2 well-order of the reals.

114 PART FOUR: THE CORE MODEL

This short but essential part introduces the core model. Once mice are to hand the definition (14.3) is very simple. Chapter 14 gives the analysis of the Q

structure of K and proves that K is a

model for GCH and is O-maximal. Chapter 15 generalises the work of chapter 12 and shows on the one hand that all manner of # iterations can be coded by mice but on the other that before we get very far the situation degenerates into chaos. Mice are, we think, a better way to describe sharps than reals and we single out some "sharplike" mice M # . At this stage M # is only defined if M^=V=K; other cases require the covering lemma and are dealt with in part five. Chapter 16 introduces a new technique to show that elementary non-trivial embeddings of models M for V=K into themselves imply the existence of M

unless K has its "maximal" form when there is a u

p-model. The p-model is a sort of K

, and we have seen in chapter

13 that it behaves like a real mouse (it is assumed that the reader is becoming inured to the absurdity of the terminology). Chapter 17 covers as many applications as can be covered without the covering lemma. The covering lemma and the singular cardinal hypothesis are the most important applications of K and we reach these in part five. Chapter 17 is not needed for the proof of the covering lemma and could be omitted.

115 14: THE CORE MODEL

We may now define K, the core model. Perhaps the simplest definition would be to make K the union of all mice; we should then have to prove K|=ZFC, and this is dull work, so instead we adopt a definition that gives K(=ZFC immediately but leaves us to prove that K is the union of all mice. This has a couple of advantages: firstly we get a model even if there are no mice. Secondly, we get a i structure with levels, indeed, a model of R . Although it is not a model of RA we can still get GCH. Lemma 1 4 . 1 . (<

If M,N are core mice with mouse iterations < < M >c ,< TT' > .Qc- >, —— - a a€On a$ a<$€On V a e O n ' ^ a s W o i ^ respectively and if ^ , ^ have the same critical point K,

then Nf=M_. Proof: L e t

6>M,N

be r e g u l a r .

If

MQ€NO t h e n CM =CM flKeil., ( N R ( M ) ) D O M MQ 1 —U (A)

U

so C M EN^. But then Nj=cf (K) =to;but Nj= K inaccessible. Similarly 0)

N ^ M Q . SO M 0 =N e . So M=N. Definition

14.2.

n

D={<^ / K>; N_ is a mouse with critical

point

K, otp(CN)=m A

By lemma 14.1 N is unique if it exists. Definition 14.3.

K=L[D]

So Kf=ZFC. Lemma 14.4.

if N_ is a mouse then N£K.

Proof: We may assume otp(CN)=u) , since N£K if and only if some mouse iterate of core(N)€K. Suppose N=«J^« It suffices to show U€K. Suppose K is the critical point of N. Then for all x€P(K)nN x€U «* C N ^Y£X, some Y
-a -a

(14.1) •

116 Lemma

1 4 . 6 . K=L «» O

does not exist.

Proof: If 0 # exists then by lemma 14.4 O#eK. But 0#$L. So K=L implies that 0* does not exist. On the other hand if 0

does not

exist then there are no mice by lemma 12.12. So D=0; so K=L. a u

Suppose until further notice (corollary 14.14) that 0 exists. Definition

1 4 . 7 . Q =\j{N:N is a mouse with critical point K and cop" K

—


N

F ={XQC: X contains a closed unbounded set}. Q

i s only interesting if K i s regular

Lemma 1 4 . 8 .

If K is regular

>a>.

then Q =< Q ,€,FftQ) is a premouse.

Proof: Since K i s regular every mouse N with c r i t i c a l point K and ojpJJ K. So l e t t i n g e=OnflQ —~CL K F Q — S 0 K and F is a normal measure on K in Q . D Call this 8 0 . Since F PIQ is countably complete Q is iterable. Lemma 14.9. cop


£

_

Proof: Let 6 K =OJ6 and suppose 6 is a limit ordinal,otherwise the result is trivial. X=h

It suffices to show that QKf=On=h(K). So let

(K) . Suppose K
Then M is a mouse. Let ~K be the coth member of C... By lemma 14.1 y is the unique "y in Q

such that

(i) ojy>K

(ii) for some n copn < K

J K

f

(iii)

K i s the coth member of CJF J K Y

Hence yEX. Suppose K < 6 < 6 . There is ojy>6 with ojpJJ^M'+ < K , M = J F K , So K is definable over M from parameters in
from y. So

=K.

Proof: Suppose cop < K . Let C=C O . Since C=O{X€F : X€h

(cop Up )}

117 C must contain a closed unbounded set. Let Q =rud

(Q ) . Let

n

K

K

C°=C, C m + 1 = the limit points of C m . So each C m contains a closed unbounded set. Now given X€Q + , X C K there is m such that X€Zmm++11 (Q K >m

m

So either (3y
+

is a normal measure on K in Q . But <0 + ,F

K

> is

K

given X€Q , X C P ( K ) then X c E m + 1 ( Q K ) for some m. So y

K

So XflF €Q K

y

. Hence

.

(14.2)

Q =rud r

(Q ) . K

Thus

Q =< Q , € , F

> is

+ Since F Now P(u)p

is countably complete Q )0Q *P(cop

a

premouse.

K

+ is iterable. So Q

is acceptable.

+

)flQ ; if p Q

exercise 6 that 0+wp n €Q . Contradiction! So a)pn+
is a mouse and Q cQ . Contradiction!

•

It follows that "9 is a limit, so by the proof of lemma 14.9 Lemma 14.11. p =4. K

Lemma 14.12. K 3? . D

0

Proof: K =J Let . It^=sup suffices show K cH K; and so it suffices to show £ ,=OnnM'. £ ,. to Clearly £<6. K K K'
E, ( J ? K ) . So Dfly2€Q . ±

Lemma 14.13 .

K =H K K ^

c,

D

K

•

Proof: K ^ C H ^ K is proved. Since H Q K=U{j a : acy K and aGJV^, . Then J-K^. is a mouse with wp_F


with mouse iteration < < M a > a £ O n , < v ^ > a l 3 G O n > . Let K a be the critical point of M^. Since K K = K there is a
•

118 Corollary

14.14. K=\J{M: M_ is a mouse }

Corollary

14.14 is false if 0

u

assumption that 0 Lemma 1 4 . 1 5 .

u

does not exist: from now on the

does not exist is not needed.

If 3 i s an infinite

cardinal in K then P(&)c=K0+

(in K) .

—p

Proof:

Cln K) B y lemma 1 4 . 1 3

So i t suffices to show that i f ac$ then a6QQ+. Suppose a€Q , K + ~" 0 regular and greater than 3 . Let y be least such that a i s in CT;K. o Then as i n t h e proof of lemma 14.12 M=JTK i s a mouse and wpM£3. 1 + Let M be the 3 - t h i t e r a t i o n of the core of M. Then a€M' and ujpMl<3 so M'cQ + . i f o # does not e x i s t the r e s u l t i s known. D C o r o l l a r y 1 4 . 1 6 . F\=GCH

Corollary 1 4 . 1 7 . K=\J{N:N£KO P

—

p

u

If 0 exists and 3 i s an uncountable cardinal in K then

A N is a mouse}. —

K i s not a b s o l u t e . For example i f V=L[O#] then K=V but KL=L. In fact Lemma 1 4 . 1 8 .

If M is an inner

model

then K^=Kf\M.

Proof: N i s a mouse i f and only i f Mf=N i s a mouse.

•

In general K i s not K-minimal for any K. But K i s O-maximal. Lemma 1 4 . 1 9 .

Suppose j:K-* M where M is transitive.

Then M=K.

Proof: Since K(=Vx3N(N a mouse and x€N) t h e r e f o r e M|=Vx3N(N a mouse and x€N) . So McK.

(14.3)

Let K be regular and let Q'=j(Q ) . Q 1 is a mouse with point J ( K ) and a>pn,=j(K). Let N,N' be comparable

critical

iterates of Q ,Q*.

If N'€N then, letting TT1 be the iteration map from (V to N 1 , ir'j:Q •*•« N K

non-cofinally,

contradicting

corollary

8.20. So NCN' .

2JO

It follows that P ( K ) O Q C Q 1 . K 2 K 2 Now for any y y flD6Q cQ 1 , where K>y is regular. So y 0D6M; so KcM. D Exercises 1. Show that cf(0 )=K when K is regular.

119 u

2. Given that O

exists then by exercise 1

0 >_K+K. Could they be

equal? 3. Find core models K,Kf where Q K =Q K ' but K+K 1 . Oil

'd)i

•p

4. K , K ' are regular and K < K ' . J , K ' is the K'-th iterate of Q . K'
120 15:

EXAMPLES OF K

We may think of the Q of the previous chapter as follows: start constructing at regular K from F and continue for as long as ^ F possible while new bounded subsets of K are constucted and J K is a mouse. 8 is a measure of how long this process goes on. It can be used as a measure of the large cardinal properties of the universe: u

u u

for example, if 0 exists but 0 does not then 0 =K+K; if all the # sequence exist then 0 ^>K2 ; and so on. This can be made precise. Definition sharplike

1 5 . 1 . Suppose

provided

M_ is a mouse

with

critical

point

K. W i s

called

H EM.

The reason for the term sharplike is that every sharp has this property. In fact a l l sharplike mice relate to some model of V=K in the way that a sharp relates to L. Note that if M is a mouse with c r i t i c a l point K and On =K+U).2 then M is not sharplike. Lemma 1 5 . 2 . If J, is a mouse with critical if and only if 0)p U>K for all n. X

n

TU

point K then J\ ,, is sharplike

TU

TU

TT

Proof: If o)p"u>K for all n then H X=H A+l so H H16J, x1 . If J -\—

cop^lKK

f o r

some

K

n t h e n

K

P(cop^U) n j ? ^ { j ? . !

jC

s o

A+X

H ^ A ?

Suppose M is a sharplike mouse with K critical. Let <<

^ > ae0n' W a8 > a

b e i t smouse

iteration. Let Ha=H^a where <

is the critical point of M^. Then ^ Q (H

Lemma 15.3.

^a=^

Proof: Ha=U{S^: acy =U{sf: acy
=HM3 K a

—

ot

p

by lemma 3.17.

) =Hg and TT

|Ha=id|Ha so

121 (Sa exists in M

since aGJ^ , some u)6€Ma with y^PjN^; similarly for

Corollary 1 5 . 4 .

"^o-

=

===================

Proof: H^BCZjHgJcMg; and (H^S)MS=( iXS*: acY
a

It follows a s i n lemma 11.13 t h a t H^ZFC. Lemma 1 5 . 5

H \=V=K H

Proof: Let H=HQ. We must show t h a t if acyK. SO J g + 1 i s a mouse; c a l l i t N. Since H^N, 3+l€M. And o)pN<.Y so A^H. Thus core(N)€H, and H|=ZFC SO there i s a mouse i t e r a t e of N1 in H containing a. (N'=core(N)). o D e f i n i t i o n 1 5 . 6 . Let K = U H . Let D have the same definition ————————————

M

-

CL

in K as D in

M

M

aEon K. Let K^

KM,£,DM>-

Since is an elementary chain H -< K M for all a; so u a a M KM|=ZFC+V=K. I t follows t h a t KM|=V=L[DM] . Before carrying the a n a l y s i s further we introduce some more n o t a t i o n .

Definition

15.7.

Suppose M_, 2V are core mice. M_' ,N_' with M'EN'.

comparable mouse iterates Lemma 1 5 . 8 .

M
Suppose M_,N_ are core mice and M_'=^

t[',N_' have comparable

iterates

,N_'=1^ ^ . M
if

W",AT" with M"€.N".

Proof: Suppose M a € O n ,< ^ B > a < B6On >.<< »„ > a £ O n ,< ^ > a < B e O n > be t h e mouse i t e r a t i o n s of M,N r e s p e c t i v e l y . Suppose M ,N g a r e comparable and M €N O. L e t M ' = M n ( M ) , N ' = N n ( N ) . Suppose K i s t h e common Ot

critical

p —ex

also

P(K)0M

M €N' ex

p

—(X —Ot

—p —p

p o i n t of M ,NO. M',N' a r e comparable:

*P(K)nNo, CX p Thus M'€N' ex p

—p

and P(K)PIM cP(K)nND :

—ot — p

a s M i s a mouse. —CX

ex—

p

B u t P ( K ) PINO=P ( K ) O N ' s o p p

Conversely if M',N' are comparable then M , N o are comparable —ex

—p

—ex

—p

as P(K)flMex— cP(K)nMexl , P(K)DNpQ— c:P(K)nN'. Suppose M'EN' p ex p (otherwise

N'€M') p ex

a n d NO*M (otherwise —p — a

M'=N'). —ex — p

Then NOp^Mex

So M €NO. ex p

a

Lemma 15.9. < is a strict linear order on the class of core mice. Proof: < i s i r r e f l e x i v e . Suppose M,<M2 and M2<M3. Suppose Q,, Q2 are comparable i t e r a t e s of M-j,M2 with Q^EO^', s i m i l a r l y Q^.1^' L e t ® be r e g u l a r , 0>Q 1 ,Q 2 ,Q 3 . Let ^ z ^ ' — s b e t n e 0 t n m o u s e i t e r a t e s of M, ,M2,M3« N-, and N2 are comparable. Suppose N2cN, . Let TT:Q2->N2 be

122 N

the iteration map.Then ir | Q^-Q^^

(C^) C l ^ c N ^ So ^ l O i ^ " ^ x

non-cofinally. Hence N 1 € N 2 . Similarly N 2 € N 3 . So N 1 € N 3 . < is clearly connected.

n

In the next lemma it is important to note that a well-order of a proper class is not taken to require that the initial segments be sets. Lemma 15.10. < is a well-order Proof: Let A be a set of core mice. Let 0 be regular such that 8>M for all M€A. Let M'=J F 0 denote the 6th mouse iterate of M. Pick M a " M " with a., minimal and M will be minimal in A. a

M Definition 15.11.

— Suppose M_ is a core mouse. deg(M) denotes

{NjN_ is a core mouse, N_€L[M] and M€L[N] .} Lemma 1 5 . 1 2 .

Proof:

if N<M_ then N_€L[M] .

A £ ( N ) + 1 CM.

Hence deg(M) i s an i n t e r v a l of <. The relevance of t h i s t o KM w i l l be seen from t h e next lemma. Lemma 1 5 . 1 3 . Proof: A

M

€M

core(M) is

the <-minimal core mouse not in K .

S u p p o s e M€K M . K = U H . ; A^€K xir ( n = n ( M ) + l ) s o A ^ € H . s a y . B u t —

A

i ** M

M

M

.

^

l

M

M

—

M

l

€M; c o n t r a d i c t i o n :

Suppose N<M. Let N 1 , M1 be comparable mouse i t e r a t e s of N, M with N'€M'. So A^, ( N ) + 1 6M1 . But P(u)p£ (N) + 1 ) OM'cH^'n (N) +1 so

This i s a dual p r o c e s s : suppose M|=V=K; then provided M*K there i s a s h a r p l i k e mouse N such t h a t M=KN» Definition

15.14.

Supposefi£=V=K,M*K./ / denotes the <-least core mouse not in

/f (M_ an inner model) .

Note 1: 0 # and L # are interconstructible. L* is the core sharp. Note 2; If M is sharplike then K^=M. Fix M such that M is an inner model, M)=V=K but M*K. Lemma 1 5 . 1 5 .

M* is

sharplike.

123 # # Proof: Suppose a€M , where M has critical point K and acy
and a€E (N). But N€M so a€M.

U P(y)nM^cM. Also ? M (K)C:M # . For if A C K , A € M , suppose A€N

Hence

Y
where N is a mouse and N€M. Let M', N 1 be comparable mouse iterates of M # ,N. If M'CN 1 then by lemma 15.12 M # €M; so A€M f , hence A€M # . It follows that K is a cardinal in M. Mj=GCH; S O let f map K U P (y) , f€M. Let A={<£, C >: £€f ( r,) } ; so A€M # . Hence for Y < K # # M* SO H =H a • « PM#(y)€M . If OnM#=a)a+a) then for all Y < K P M # ( Y ) C J M M J If OnM#=a)A,A a limit, then there is a such that A€J , so H =H a .• onto

Y
jyi

ex

Lemma 1 5 . 1 6 .

K

K

K #±M. u

a

Proof: (z)) Let < < W > __ ,< i\ o) ^Qcr. > be t h e mouse i t e r a t i o n of M . — ot otcOn otp a_
U

by the argument of lemma 15.15. So a€H a, so a€K M . Thus for all y a P(Y)nMcK # . B u t t h e n f o r a l l m i c e N€M A..€K M #so M<=KM#. — M

„

—

NM

— JVL

M

(c) Suppose a€H a, acy,I. In fact K =U{M':M' is a mouse and core (M ) <M} UL. MM M — — — Lemma 1 5 . 1 7 .

If M_ is sharplike then M_ is minimal in deg(M) .

Proof: Otherwise suppose M'<M, M'€deg(M). Then M'€K M ; so L[M']5K M , so M€K M . Contradiction!

n

So each deg(M) contains at most one sharplike mouse. Some deg(M) contain no sharplike mice, however. The constructibility degree of «n,m>:n€0 # m } is one of them. Lemma 15.18. deg(M) contains no <-maximal element. Proof: Suppose NEdeg(M) were one. Let 0 be regular, 6>N, and let N 1 be the 0th mouse iterate of N. In L[M] N'€Q Q . Contradiction! —

—

—

n

y

In fact deg(M) is a proper class; hence the caveat about wellordered classes before lemma 15.10. EXAMPLES u

0

u

and L

u

are interchangeable: L

is the core sharp. Which # # sharplike mouse comes next; that is, what is L[0 ] ? By

124 if X > K and P (co) nj^ + 1 *P (<*>) n j K + 2

b u t a)

PJu=K A then a)X must be closed under K+y for y
chapter 3 exercise 6

assume for the sake of definiteness that K is some regular cardinal and U is F ) . So the question is: does cop U K

= K ? Since there is a

«+K

s i m p l e map o f K o n t o K+K cop U J

u

< K . I f cop U =y
K+K~

iterable premouse, argue as follows: suppose x€A U ~ J^ f=3t<|)(tfxfp) (some P € [ K + K ] < W , J

Let t

U

be least in the canonical well-order of J +

f(6)=the least y such that t , ~v€S U x (o; y If rng(f) were bounded in K + K then A_U J K+K

u so f:y->K+K cofinally. But f€J +..|=cf

Z ) .

K + K

K+K

cj>(t,x,p), if there is one. Let T : y-*~>-co x (y)

J

, i s an K + K + 1

K+K

w

such that

. Define f:y+K+K by

(if one exists). would be in J ; K TK

u

, and J

, is a premouse so

(

K)=y. Contradiction! So cop U = K . Since J K K+K iterable premouse it follows that it is sharplike.

+,

is an

u

What is K U ? Certainly it contains 0 . Suppose it also K + K+1 contained N (a mouse) with N$L[O ]; let N be <-minimal. Then N is U1 sharplike. But then for some K 1 , some mouse iterate N'=Jg of N with 1 K critical, $
Contradiction! So K U is L[0 ] . In fact in general if M is K+K+1 sharplike and a "successor" in the sense that for some mouse M' N<M =» NGLtM 1 ], we deduce that K =L[M']. —

—

J

—

U

jyi #

+

+1

#

is L [ 0 ] . Now apply the results of chapter 12 to see

that p U =1 and p U =0. We may, then, extract a unique class J K+K+1 K+K+1 C of generating in^iscernibles for L[0 ] and a set 0

. Furthermore

since L[0 ] is O-minimal and O-maximal we get the equivalence of the the following: (i) 0 # # exists (ii) there is an iterable premouse J

+,

#

with K critical #

(iii) there is a non-trivial j:L[0 ]+ e L[0 ]. L [ O # # ] = L [ O # ] # . There is an obvious generalisation to 0 # n for all n€co. Consider J Each J 0

n

+,

, then, and suppose it is an iterable premouse.

is an iterable premouse and is interconstructible with

. Furthermore J

is a premouse whenever J

+,

is a premouse for

125 all n. It is not sharplike (KOJ is a limit of points with projectum 1) but it (or rather its core) must be minimal in its degree. If we let O#(A)={:n(EO#m} we see that L[J U ] is L[0#a)] . U How far does all this go? Mice of the form J K ^ + 1

where X < K is a

successor ordinal are sharplike and correspond with the model of V=K with X-l iterates of #, giving the # of that model. Transfinite u

iteration of # will soon lose the property that 0 COJ, of course. ,-,,,-, for example: 0 # W l

Consider M=jJ?,

is {< a, 3 >:a€0#3 , B
We can no longer prove P M =1. For to do that we let X=hM(0),a:N=M|X with N transitive and iterated N,M to comparable Q, Q 1 . These will only be equal if the iteration maps preserve a).. ; so we must insist that w,cX. So we set X=h(a),) and prove P M =w n • This is not surprising. We get the equivalence for successor X of: (i) 0 # X exists (ii) there is an iterable premouse of the form J , + ^ with K critical (iii) there is a non-trivial j:L[O #X " 1 ]-^ e L[O #X " 1 ] with j|X=id|X provided X
((iii) is necessary for minimality and maximality).

For limit X
exists is equivalent to the existence of a

premouse of the form J . with K critical. The reader will see that this simple situation breaks down if X_>K. Certainly J 2 gives all the sharps up to K , and is not sharplike. Consider, then, J U 2^,. a)p_Uo=K, so J U 2^ n is sharplike. Where did all the 0

for a>K go? They can easily be retrieved by

iterating to K'>K; they simply will not occur in Q ,for J 2 + , is a different creature altogether. Call it M and consider K . Iterate ~ U1 U M to some K ' > K and it will be of the form J ,2 , (whereas J 2 only K U1 #a goes to J , ) and so exceeds all the 0 , a
then Hf="for all a

0 # a exists". On the other hand if N<M then N€L[O #a J for some a. K M is the closure of L under all sharps - sometimes called L

but

if CO

L

in our notation. It is not of the form L[M] for any mouse M.

It recovers properties of L lost along the way - it is O-minimal for example - and J 2 , is (L K

) . This corresponds to a subset U U

of a), and off we go again, up to J 2 2 + , - and presumably J 3 corresponds with some still grander model. In all the models defined so far the length has been nicely definable from the critical point. So in all our sharplike mice the projectum has fallen straight to the critical point: n(N)=0. This cannot continue all the way up to K . There is a murky middle ground where coding by # sequences fails. For suppose a sharplike mouse N does not have n(N)=0; or even worse, that

N

126 $€N but 0)3 is not the largest limit ordinal in N) . Certainly we shall get indiscernibles for K ; and in N they will be generating indiscernibles; but in K they will not. The argument that turned I-, in a sharp into elementary in L relied on £ heavily. It may be that the equivalence with non-trivial elementary embeddings breaks down as well. This, after all,relied heavily on minimality and maximality. Maximality is no problem: if M|=V=K u

then either M=K, which is 0-maximal, or M*K, so M exists. Minimality cannot be so simply disposed of. In chapter 16 we shall take a sledgehammer to the problem and show that if M|=V=K u

then M

exists if and only if there i s j:M-> M non-trivially,

unless

— e ————— there is an inner model with a measurable cardinal(which is sharplike all but in name). The case where there is a p-model is of considerable interest: it lies at the opposite end of the core u

model spectrum from O w p-MODELS AND K Let L[U] be a p-model with critical point K. Let < a ae0n' < 7 T a3 > a<3€0n > b e i t s i t e r a t e d ultrapower. Let K a be the critical point of LT~U ] . Lemma 15.19. K= U HL[Ua]. a T Cn1 Prodf: Suppose that x is in HT L aJ . There is BcK a ,B€L[U a ] such that a L[U 1 B H a =J so we may assume xcyM. Obviously M"€L[U ]. So the mouse iterate of M at K ,M is in L[U ]. So M€L[U ]; and so TTTTI

—

a—

a

—

a

MeHL[lV. D K a Lemma 15.20. K= n L[u ]. a€on

a

Proof: If X C V < K , x€L[U ] then x€L[Uo ] (3
(X

Ot

p

Ot —

p

3>a then TT Q ( X ) = X S O X € L [ U Q ] , a " " ===== LFU 1 Conversely if x€ PI L[U 3 then for some a TC(X)
~~

127 so this is impossible. Hence K|="there is no p-model" even if there is one. If there is a p-model then K is not K-minimal for any K . For take K < K

and let X=h (Klip) where N=J <JL and p is such that H

a

is Z, (N) in KUp. Let a:M=N|x with M transitive; so a :M-+y N. —l n + M|=V=h(KUa~ (p) ) and M=J D a for some 3 < K so M is a mouse; letting ~ i i rn i "~ ~^ — ot — H=a (H L ctJ) ^=Vx(xcy x€H) , so M is sharplike. If we let K a a ^aS > a<8€0n* b e i t s i t e r a t e d ultrapower and let maps

V M

be defined such that a a Z i a

o = a ' a 3 7r a3 =TT a3 a a

then

a :HMQt-> H L t U a ] where K 1 is the critical point of M 1 . Given x€HM<ji a Ka e Ka a a Ka let a(x)=o

(x). Then a:KM+eK but K M *K. On the other hand a|<=id|K.

The existence of a p-model is by no means necessary for this result. The existence of a mouse M with n(M)>0 suffices. It follows that there is no sentence that characterises the largest core model internally. Exercises 1. Given the existence of a p-model show that there is a sharplike mouse M with OnM=coX, A a limit. 2. Show that each deg(M) is a proper class. 3. Is deg(M)fl{N:N<M} always a set? 4. Suppose there is a p-model. Show that there is j:K+ K which is non-trivial. Suppose K M *K; show that there is j:K M + e K M which is non-trivial. (Hint: Let N = ( K M ) # and let W={f€N:3m€co f:Km+H^} where K is the critical point of N. For f€W let ?= U TT . (f) , where N has mouse i
O l

i t e r a t i o n < < ^ > a € O n ,< TT^ > a £ 6 € O n > and Ka i s t h e c r i t i c a l p o i n t of 1 ^ . Show t h a t : Vx€K M 3f€W3i_<...
(i)

KM(=(|)(¥1(K 1

. . . K

) . . . ?

(K

. . . K

))

a n d cj> i s m - a r y

«

P 1 ^-O m-1 0 m-1 ^ <|>(? ( K . . . . K . ). . . ? (K. . . . K . )) .) 1 D D p J D 0 m-1 0 m-1

128 16: EMBEDDINGS OF K

This chapter contains two important results: the indiscernibles lemma and the embeddings lemma. The indiscernibles lemma will be further used in chapter 17. The embeddings lemma is one of the main tools in the proof of the covering lemma. Definition

16.1.

Suppose A_ is a transitive model of R

and K=On . Let A

g

denote J\. I is a good set of indiscernibles for A_ if and only if for all y€J

(b) I^y is a set of indiscernibles for <^/^>r< •

The indiscernibles lemma states that if A=< K ,DflK*A^. . .A N >, I is a good set of indiscernibles for A (this requires that Af=R ) and cf(otp(I))>w then there is I'GK such that I' is a good set of indiscernibles for A and Iczl'. In a sense this continues the work of the previous chapter: all # are contained in K. It also follows that although K, the critical point of some p-model, cannot be measurable in K, it will be Ramsey. Model 2 in appendix 1 shows that cf (otp(I) ) >o) is necessary. Fix a structure A, then, where A=< K /DflK2,A, . . .A^, > and suppose I is a good set of indiscernibles for A. Suppose cf (otp(I) ) >a). Lemma

1 6 . 2 . If y€j then y is strongly inaccessible in A_.

Proof: Suppose y were singular in A. If cf(y)=6, 6y. Suppose f is the least cofinal map of 6 to y in the canonical order of A; f1 defined similarly for y 1 . Let 6'<6 be least such that f(6')>y. Say 6 n =f(6'), so 6"
A\=ZFC

D

129 From now on 3 + denotes 3 + calculated in A. If y€I let A n =A +|x where X is the set of x in A + definable in A from yU{y,y^...yn> ( t y ^ . • -Y n >£l n+1 ) . So A n ^ A +. But A n is transitive (if x€A n 6n
there is a definable map of y onto x) so A^=A.n, some Y y 6n<6n+1.

Lemma 1 6 . 4 . Proof: 6

n

i s A-definable

from yU{y , y , . . •Y n + -j} /

n

-definable A ={x€A + : x i s A Y ~Y Yn+1

since

from y U { y / y 1 . . . y } } . 1 n ^

(16.1)

Also AYn ^ YA n + 1 . L e t AY =A| U AYn , so A ~< A + . L e t 6 d e n o t e s u p 6 n = _ n€u) YY Y Y n€a) =OnflA . Y

Define maps T T ^ , :A n + e A n , for Y , Y ' € I , y£y • by **y,(tA(v/yfy1...yn))=tA(v/y'fy1...yn)

(16.2) n

where t is a term of L-. •> v€I . The choice U/A ~~x n l. n of Y 1 - . Y n is irrelevant. Letting i\ TT^, |y=id|y and TT

, (Y)=Y'-

,= U ^ , ,TT^^ , : A ^ e A ^ , ; also

Define U y by X€U y « Y€TT

X€A DP(y); the choice of Y ' > Y , Y ' £ I Lemma 16.5. (A ,U' > is amenable.

, (X) where

is irrelevant.

-Y y n

Proof: Let U =A n nu . It is sufficient to show U n €A iTnY,

is A-definable from Y/Y / Y x • • • Y n + 1 / as is A^ so U n € A n + 2 . D

I t also follows fact

that

IT 1, i s c o f i n a l YY Now l e t y * = s u p

be

for all n€o). But

1

t h edirect

limit

T T ^ , (U n ) =U^, , s o iryy , :< A y , U y )+u < A y , , U y , >. I n s o u , :1< A ,U )•*•„ . y y —y Y 2 i - Y Y I a n d I * = I U { y * } . L e t < < A * , U * >,< TT o f <£1,< *

because i t i s w e l l - f o u n d e d : c o n t a i n e d i n rng (IT n

A *=ir

n

* ( U n ) , any y€I

>* ( T h e d i r e c t

t h a t i s countable

limit

is

(choice

irrelevant).

u * is countably complete.

Proof: Suppose Xcy*, X€A *. Suppose X=TT ** X6U

C I

*) , some y € I , s i n c e cf (otp (I) ) >OJ) . Let n

*(A ) and U *=7T

Lemma 1 6 . 6 .

any acA *

i >< i

* >< * )

* y€Tr

cf(otp(I))>w.

* (X) , y€I. Then X€U * «

*(X) . So X6U * « I^ycX; this proves the result as D

130 For y€I* let M ^ l H j ^ : Lemma 16.7. M a

J^y€A y }. Then My=< M y , € , U y >=J^y say.

but M §A .

Proof: Suppose J^Y^A . Then s i n c e f=R J^ + 1 EA y • But i f M €A then J gY + iE M ' c o n t r a d i c t i o n !

•

My * i s i t e r a b l e a s Uy* i s c o u n t a b l y complete; b u t TTyy* | M _M y :My-*•LO y * so each M i s i t e r a b l e . Lemma 1 6 . 8 .

If y£T

and n€u then ?CY)VM ja*1.

Y Y nn Proof: We may assume y€I. Suppose P(y)nM cA . So M =JgY. AJ=ZFC+V=K Y Y Y Py n so A(=there are no p-models. Thus J y|=U is not a normal measure K Y n ( K = Y + ) . So for some T with 3 £T
Lemma 16.9. ?(y)f\M =?(y)0A for y€x . Proof: P(y)flM cp(y)flA

obviously. Assume y€I. So suppose aczy, a€A

but a^M . So a$L; so A|=ZFC+V=K+V*L. Hence A +|=there is a mouse N with ($>y critical and a€N. And A -< A + so the same holds in A . Suppose A [=N is a mouse with £ critical. Then A +^=N is a mouse so Af=N is a mouse. But a)..€A Cas cf(otp(I)) is uncountable) so N is a mouse. Iterate M , N to M 1 , N 1 respectively with 0 critical where 0 is regular and greater than M^,!?. If N'cM1 then a€P (y) DN=P (y) ON'c cP(y)nM'cP(y)nM ; but a^M . So M'€N f . Suppose NeK n . Then P(y)flM = =P(y)nM'cP(y)nN I =P(y)nNcA n , contradicting lemma 16.8. If y=y

the result is deduced easily.

•

Note that A €A . If y€I this is true from the definition: but Y Y * otherwise let A*=TT * (A ) : then A =J A . Hence A*= U A =A *. YY Y Y Y Y # y € I Let A ={(i,Y 1 ...Y n ): i€a) A A J=4>i (y1. . . y R ) Ay± . . . Y n . enumerates all

131 formulae - and d e f i n e a s e t X1 cry a s f o l l o w s : =Y -xm . So X° €U For x1<...<xm
(some y^...

...
so

Ay|=(t)i(x]L...xm_k_lfy1...yk,6')

=> Y ^ ' * ' X m-k-l={y: $± (xv . .

(16.4)

So if y{<...
l6

m

1

1<S

= X . Let X = {6 : V6 ' <6 (6€X

(16.5) 1

' ) } . Then X €U

and

fl X 1 are

see good indiscernibles for A . Also by the uniformity of the definition

Let I'=

fl X^*. Then Icl' , for X^eu Y Y Y

i€a}

so Y€TT * (X^) =X^*. YY Y Y

I 1 are good indiscernibles for A * and hence for A as A *^A. Finally I 1 were defined e n t i r e l y from J UY**+i which i s a mouse. So I'€K. We have proved the indiscernibles lemma: Lemma 1 6 . 1 0 . of indiscernibles

Suppose AM K ,DC\K ,A ...A for A_ and cf(otp(I)

and I' is a good set of indiscernibles THE EMBEDDINGS

) , A^R+. Suppose I is a good set

)>w. Then there

is I'€.K such that JOT'

for A_.

LEMMA

The indiscernibles lemma i s used in the proof of the embeddings lemma, which says that if j:K+ K i s non-trivial then there i s a p-model. The proof in chapter 12 will not give the r e s u l t : we have

132 seen that K may not be K-minimal for any K. What we now show is that a sufficiently drastic failure of minimality yields an inner model with a measurable cardinal. Suppose K had been minimal. Then we should get a normal measure U on some K in K such that is iterable. We can assume that U is countably complete, for its to,-th iterate will be. This cannot give a mouse for all mice are in K. This enables us to prove the following Lemma

16.11.

Suppose ( K + ,u) is amenable, U a normal measure on K in K and U

is countably complete

(K

calculated in K) . Then U is a normal measure in L[U].

Proof: Suppose not. So L[U]cj:K. Let a be greatest such that J cK. __

TT

_

*•*

Then let U=unj . If U were a normal measure in L[U] then U=U; for if a€L[U], acK but a £ j U and 3 is least with a€J^,n then E (J^)cL[U] so n ~~ U U "+ _ ^u ~" a)p_U>K, all n<0), so Jo,, =rudTTri _U( J Q ) ; thus a€L[U]so a € J . J

p +1

Ul IJ o p

p

Ot

Contradiction.1 (3 must be taken least for any such a in the above so that J^=Jo) _p p _ n So_U is not normal in L[U] . So there is $>a with acK, a€Jg + 1 , but a^Jg* Take 3 least such; so J g is a. premouse. J g is iterable because U is countably complete. And JD is a mouse because a€J o ,, U n ~~ U + a £ j Q , ac< implies wp T U is amenable so JOj^n<=K (for J O =J O ) . But 3+l>a; p

K

K

contradiction.1

P+j.—

p

p

a

Lemma 16.11 occupies only a small proportion of this chapter but it is one of the most important results in this book: its argument was one of the original motivations for the definition of K. It is irritating that in order to prove the embeddings lemma it has to be embedded between two slabs of technicalities. But then: in general K is not minimal. Suppose j:K-> K is non-trivial with critical point K. Let U={xeK:K£j(X) A X C K } . A S in lemma 12.14 is amenable (K = ( K )

here and for the rest of this chapter). K is K-maximal

by > __ ,< IT O > ocrs > be a weak iterated 2 lemma 14.19. Let <<M —a a€On' a3 a_<3€0n ultrapower. If M is well-founded then M =K. Let p be least such that M

is not well-founded. If p does not exist then letting U1

be the a),-th measure, satisfies the conditions of lemma 16.11 and so there is a p-model. By lemma 12.16 p cannot be a successor ordinal. Let X* = = {x€K:CKj IT . (x)=x) (this is the X of chapter 12). Let TT:M=K|X* where M is transitive. If M=K then the proof in chapter 12 still works, for TTQ.:K+ M1. implies that each M1. is K. So M*K; M=K N ,say.

133 We can generalise this a bit. Let X* = {xGK: i< j

TT . .(x)=x}. If there is i such that K = K | X * then the proof of chapter 12 works. So let Tr.:M.=K|x£ , M i = K N . This is what we meant by a sufficiently drastic breakdown of minimality. A technical lemma is of great help: the argument extends that of lemma 12.16. Let K =TT^ (K) . a Lemma

16 . 1 2 .

Proof:

Suppose

Suppose

i,j
Oa

Then

g€K. Then

i+j
and TT .ir .=ir .

g=7i_.(f)(K,

. . . K,

)

x

TT

(K ) = K .

( k n < —
U

Also x=7Toi(g) (Kh • ».Kh

.. Also

K K L J ^ p ) (h1<...
say.

P

^oi^oj(f)(Kh1--Khq'Kk1--Kkp)

where K^=IT O . (K,) . Define a:K-*M. a (

(

W o j

f

)

{ K

h - • •

K

h '

K

k -

•-

(h<j)

(16

'6)

. by K

k ')

p

For all formulae <J> (with one free variable for simplicity)

>:<)>
so

a

is well-defined,

So

a

is

the identity

Lemma 1 6 . 1 3 .

a:K-> M. + . .

Hence

and i r o l i r o j - i r o i + j

a:K=M.+.; and *

o i

so i+j
(Kh)=K

i + h

.

M.+.=K. a

Suppose that for all j with i<j

P r o o f : L e t X.=TT. . ( x ) ' . Then i f <' i s t h e c a n o n i c a l o r d e r o f K for

X=IT . . (x .) =TT . . , (x . , )

-• x . , ^ x . ,

i.e.

7Ti . , (x .) >" TT^ . (x .) =Tri . , (x . , ) . So f o r some j j ' >_j => x . = x . , . F o r

any k , i
ik+h

( x

j

and

7r..,(7r..(x.))>l7r..(x.),

j<j'

(16.9) (j=i+h say)

)

(

J

Now p i c k

some

T such

that

T is

regular,

x>^up

__

K.^ T>N., a l l

i
134 Lemma 16.14.

T€X*.

Proof: Suppose n
hx<.

. .
s o that m

m

A l s o i f f:K +T t h e n f:K ->nf

): f:

1 m ( i n K) . T h u s

============

some ri
^ " ^ 'f€K' IT . ( n ) < T .

TOT.O(. T( T) )==SS U UP

TTTT . (n),£T.

ThUS TTo

U (T+C\x*) is cofinal in T+ (calculated in K) . f \ + P r o o f : L e t 6=sup U ( O X * ) . 6$ U (T TIX*) , f o r X* -^ K so 6€X* => 6+16X*. x 1 x x 1 i

Lemma 1 6 . 1 5 .

i< >

TTQ^TT^ . (x) = i r . + h . + .IT 0 . when i , h , j < p , h < j . of

This i s a modification

lemma 1 6 . 1 2 : s u p p o s e X=TT_, ( f ) (K, . . . K, ) ( k n < . . . < k
oi7roj(f)(Ki+k1"-Ki+k

(lemma

>

^oi+j(f)(Ki+ki...Ki+kJ

(leMna 16.12)

•iri+hi+jiroiiroh(£)(lci+k1---Ki+k " ir i+hi+j ir oi (ir oh (f)(ie k 1 ---' e k

}

(lemma

J )

( l e m n a

Hence TT . "X^fczX* . (i,j
T h a t

i s

' ^oi^'^i+j*

: :

Suppose cfx.(6) l K. Then TT^ . (cf^ (6) ) =sup TT_.(V), all i
K

Oi

V<>Y

y=cfT.(6). If cf v (6)-v, so TT . (f) (K, . . . K, ) < T T O . ( V ) . 1 m Take f€K such that f:y->6 cof inally. So TToi(6)=SUp 7Toi(f)(v)

(p = TToi(Y))

(16.11)

=sup Tro±(f) (iroi(v) ) ( _ Given

v_
_

v
7T o i (f (v) ) <6. Thus

(some 6 • , j ) s o i\Q± (f (v) ) <7r oi (6 •) €X* + . . So

TT o i (6)=6, a l l i < p . So 6€X*

C o n t r a d i c t i o n . 1 So

Cf R (6)=K. Let f€K with f:K->6 cof inally. Then irlj(6)=sup 7r1:.(f)"K =^Up V
IT . (f (V) ) . X J

(16.12)

135 But

T T 0 . ( f ( v ) ) = T T 1 . ( 7 r o l ( f ( v ) ) > : TT 1 . ( f ( v ) ) ;

a n d T\Q. ( f ( v ) ) < 6

as before

v1 . (f(v))<6. Thus 7^.(6)=6, so 6€X*: contradiction! Now let <-6On'< 7T jk > j Let T

1

be the mouse

so

a

iteration of N ± .

be the critical point of N ± . . Let C ± ={T : 1": J < T } and let

C'= n C . Let C={n€Cl:7r"nc=n} (recall that TT:K__ =K|x*). N U i

TTQ . (Y)£TT 0 . (-IT (y) ) =TT (y)
Suppose f€K, f:y->n cofinally, yn . So TT.(n)>n, as n is K

j —w . —

singular in K.

j

j

—

Hence r\=i\. (r\') , some n '
j

On the other hand

7r(n')TT(TI ') . But X*cX*. Contradiction! So in particular in K cf(n)>K. It follows as usual that TT . (n)=sup TT . (y)=n. J y
a

Hence TTi|c=id|c. Lemma 16.17. C is closed unbounded in T. Proof: Each C. is closed unbounded in T as T > N . , and T>p so C

is

closed unbounded in T . C is closed, obviously. To get unboundedness it is only necessary to show that nK } is certainly cofinal; this suffices by the argument of lemma 16.14. Lemma 1 6 . 1 8 . indiscernibles

Suppose A£?(T)C\K.

a

There is y
for < K ,nnT ,A>.

P r o o f : A€K +. Because T i s r e g u l a r i t i s e a s i l y seen

that

+

|=R . By lemma 1 6 . 1 5 t h e r e a r e i

GKN

with

enumerate P ( T ) n j ^ N i . L e t

B={< v , v ' > : v ' € B v } , B=7Ti(B), B v , ={v:< v , v ' >€B> . I t s u f f i c e s

t o show

t h a t C^y i s good f o r f o r some y>v' where A=B , .

136 ' N!

from

IT+I

yUp_ T, w

iT+l

so

there is y with B Indefinable in

U{xT 1 }(v
Then we claim CNy is a good set of indiscernibles for . Let A=< K ,DriT2,B> and A=< J K N . ,DKN. Ox 2 ,B >. Then iri(A)=A. If y€C^y then we must show A =J^-< A. But given f
(y=Ti)

(16.13)

** N ± f="^=<j>(|) "*(by Z x -def inability of B) Secondly suppose ^ 1 <...<^
(16.14)

by a similar argument. But these results transfer to A, A

using ir.|C=id|C. n

Define Ucp(T)nK by A€U «* 3y<xCNycA. Then U is a normal measure on x by lemma 16.18. Lemma 16.19. < K +,U) is amenable. Proof: It is enough to show that x<_£<x+

=>

UflK^€K. Let €K

enumerate P(x)nK^. Let A={(v,v' >:v€A , } . Pick y so that CNy is good for . By lemma 16.10 there is I€K with CNycl, I good for . So UDK ={A v :v<x A (3y<x) (I^ycAv) }. So UnK-€K.

n

Corollary 16.20. There is a p-model. Proof: By lemma 16.11. U is countably complete because cf(x)>0).

•

At last we have proved: Lemma 1 6 . 2 1 . if there is j:K+ K that is non-trivial

then there is a p -model.

This complex construction i s perhaps necessary: see appendix I I . Exercises 1. Suppose L[U] is a p-model with iterated ultrapower V W o n ' < % 3 } a < 3 G 0 n >' S h o w t h a t 7 r Oa 7 r OB = i r O+Tr ( ) a ( 3 ) "

<< L [

2. Strengthen 16.21 to: suppose j:K+eK with c r i t i c a l point K; suppose x regular and greater than maxiK,^^). Then there is a p-model with critical point x.

137 17: APPLICATIONS OF K

There are two important applications of the indiscernibles lemma other than the embeddings lemma. These concern large cardinal properties and ultrafilters: the first is given in detail in this chapter, the second summarised very briefly. Also summarised briefly are the combinatorial properties of K. In part five, once we have proved the covering lemma, we shall obtain further applications; to cardinal arithmetic and concerning the Is of inner models that are not of the form K . LARGE CARDINALS There are no measurable cardinals in K. On the other hand many of the properties implied by measurability are preserved: to discuss these we must introduce the arrow notation. 1 7 . 1 . Suppose f:[A]n-+I.

Definition

and f"[H] ={i] for some i£l.

H is f-homogeneous

Suppose f: [A]

provided

-*-J. H is f-homogeneous

that H<=A and provided

that H

is f\ [A] -homogeneous for all n
17.2.

Suppose X is a limit

ordinal.

K-+(X) means that whenever

-*j there is an f-homogeneous set of order type X. Similarly

with n repla

by
K is Ramsey if and only if K+(K) W .

In the following, observe the similarity with the final argument in the proof of the indiscernibles lemma. Lemma 17.4.

If K is measurable then K is Ramsey.

Proof: Let U be a normal measure on K. We shall first prove by indue induction on n that whenever f:[K]n->2 there is an f-homogeneous set in U. If n=l then either {ct:f ({ct})=0} or {a:f({a})=l} is in U so this is clear. Suppose n=m+l, m>l and let f:[K]n->2. Define fa:[K]m->2 by fo(x)=f(xu{a>) =0

(a<min x) (otherwise).

(17.1)

138 By i n d u c t i o n

hypothesis

L e t X={3
n

a

homogeneous that

) } . Then X€U a s e a c h

and a]L<...
then

suppose

(0
f-homogeneous.

•

Definition

(y=0 o r 1 ) .

X €U. F u r t h e r m o r e i f a ]

m

s o fa

({a2...on>)

a n d ZGU.

f: [ K ] < a ) +2 a n d l e t f n = f | [ K ] n .

-homogeneous

. Let

Y={a:ya=y}€U

{a2...an>€[Xa

} ) = Y . So Z i s f - h o m o g e n e o u s

Finally

for f

. L e t Z=XflY. Then f o r { c ^ . . . a n > € [ Z ] n

, i . e . f ({a1.. .an))=ya

f({a,...a

f

is X

f o r some x € [ X a ] m . S u p p o s e

Ya=fa(x)

=Y a

there

a n d X €U. L e t X=

fl

L e t X be

X ; then

X€U a n d X i s


1 7 . 5 . Kfa; denotes the least

K such that K+(OL)<1X), for a a limit

ordinal.

K(a)

is clearly

a cardinal.

the following p a r t i t i o n

Definition

Actually

t h e K(a) a r e s p e c i a l

1 7 . 6 . K is OL-Erd&s provided K is a regular cardinal

OZK is closed unbounded in K and f:[CJ ^-^K satisfies (such f are called

c a s e s of

cardinals.

regressive)

f(a)<min(a)

then there is DOC with order-type

and whenever

for all a£[C] a that

W

is

f-homogeneous.

K(a)

is

a-ErdOs. To see t h i s we f i r s t

prove:

Lemma 17.7. if ie>i'a; • Proof: Let f: [ K ] 2}. Let f R denote f|[K] n

(n€a>) and let

fnk:[K]n->2 be defined by f

Let all

(17 nk({ar--an})=fn({ar-an})(k)-2) a:03X03-^0) be the Godel pairing function. Observe that a(n,k)>n for

n. Let g , . . : [ K ] a ( n ' k ) ^ 2 be defined by o \ nr K ) 9a(n,k)({V--aa(n,k)})=fnk({cV--an})-

<«!<•••
(17

"3)

Suppose g({a,...a })=g ({a,...a }) and let X C K be g-homogeneous of order-type a. Suppose {a1...aR},{$1...^n>€[X]n. Suppose f({a1...an>) *f({3 1 ..-3 n >);

X with

a

> a n + 1

n

Contradiction?

/

then for some k fnk( {ax...an>)*fnk({31-..3R>) so {3 r-- B n? f o rS O m e "n+l^ ' " < a m' ^n+l^ ' ' < 3 m c h o ^n+l^n ^ a i s a l i l n i t ordinal) where m=a(n,k).

So X i s

f-homogeneous.

Lemma 1 7 . 8 . Let A_ be a structure and constants order-type

s e n i n

D

with a countable number of

relations,functions

and with KCA, where K-+(OL) j\> Then A has a set of indiscernibles

a.

Proof: Let $ be the set of formulae in the appropriate language.

of

139 Let

f ({a1...an})={(l)€*:A|=<()(a1...an)}

(a1<...
Let

X be f-homogeneous

a. Then

of order

type

S o f: [ K]


+P (*) .

f o r €,a,<.. . < a

,

$,<...)

(17.4)

~ <j)€f ({3 1 ...3 n ))

«=> Af=(f)(3 1 ...3 n ) .

n

Lemma 17.9. K(OL) is a-Erd&s. Proof: Suppose f:[C] W->K is regressive and CC=K=K(CO K

is closed in

(i.e. y
suppose h :[v] a)-»-2 has no homogeneous subset of order-type a. Let f n :C n +K and gn:Kn-v2 be defined by f (V.....V )=f ({v, ...v }) n 1 n I n

(17.5)

W-V = h v n ( { V-- v n-l } ) Let A be the structure . Claim 1 There is a set XcC of order-type a such that X are indiscernibles for A. Proof: Adapt the proof of lemma 17.8 to C by letting f: [C]


->P (*) be

defined as there, and use the fact that C = K . D(Claim 1 ) . Pick a set X of indiscernibles for A of order-type a with min(X) minimal. Say X={b v :v
m

some k1<...
1 SO f ( {

m

\ ' " • b kj l } ) *

"^- Let V £ » ( d

is defined for all v
d *d , so d
If c

o
then let X"={c :v) so d o < b Q as f is regressive. But b =sup(Cflbo)

so b o > c Q , contradicting

minimality. So

c

=c 0

i'

a n d so c

=c v

fora 1 1

v,y
140 all v
and so by indiscernibility

{gn(d1...dn_1,co)/gn(dn...d2n_2,co),gn(d2n_1...d3n_3/co)}

has three

n

members. But g :K ->2! So h ({dn...d n >)=h ({d ...d, 9 } ) and it ^n c~ 1 n-1 c~ n 2n-z follows that {d :v
-homogeneous. Contradiction;

•(Claim 2)

Finally K(CX) is regular. For suppose it were singular, of cofinality Y, say. Let K(a)=sup{K.:i2 has no homogeneous subset of order-type a. Let f:[K] ^ K be defined by f ( { a r . . a 2 n } ) = 0 if f i ( { a 1 . . . a n » = f . ( { a n + 1 . . . a 2 n } )

(17.6)

where K . is least with K.>a, =1 otherwise f ({a 1 . . . a 2 n + 1 > ) = t h e l e a s t f

i s regressive on [K^y]

f-homogeneous.

w

i such t h a t

Now for a l l ct€X f ( { a } ) = i , s a y .

{ar..a2n}£tX]2n

Since K Ramsey => K=K(K)

Corollary 17.10.

this

2

/)

as f±:

[^1^2

•

implies

K i s Ramsey if and only if K i s K-ErdOs.

i s a large cardinal property.

Lemma 1 7 . 1 1 .

So XCK . . Also if

then f .( { a , . ..«„}) =f. ( { a n + r . . *

so X i s f .-homogeneous . Contradiction. 1

K a-Erdos

a]L
so we may take XCK^Y of order-type a

For example:

If K is a-Erd&s then K is strongly

inaccessible. n

P r o o f : We must show K t o be a s t r o n g l i m i t . So suppose 3<« b u t 2 _>K . L e t x:K+{f: f:3-^2} be o n e - o n e . L e t f ( { a , , a o } ) be t h e l e a s t a such 2 t h a t T ( a 1 ) ( a ) * T ( a 2 ) ( a ) . So f : [ K \ $ ] ->K i s r e g r e s s i v e . L e t XCK^B be f-homogeneous of o r d e r - t y p e a . Take c t , , a 2 , a 3 € X . L e t a=f ( { a , , a 2 } ) . Then T ( a 3 ) ( a ) = T ( o ^ ) ( a )

Lemma 1 7 . 1 2 . every transitive order-type Proof:

or T ( a 3 ) ( a ) = T ( a 2 ) ( a ) .

Suppose K is a regular cardinal.

Contradiction!

n

K is a-Erd&s if and only if

model J^=R with On =K has a good set of indiscernibles

of

a.

First

reflection

of a l l suppose K i s a - E r d o s .

principle

there

Let A

denote J

. By t h e

i s a c l o s e d unbounded CCK such

A ^ A f o r y€C. L e t g be t h e Godel p a i r i n g

function;

t h a t each y i n C i s c l o s e d under g . Now d e f i n e f ( { a 1 . . .a> 2n }) = tthhee l e a s t

X
that

we may assume

f:[C]

-*K by

f o r some 66 , < . . .

. . . < 6 , < a , , m
(17.7)

141 =0 if there is none. Then f is regressive. Let XcC be homogeneous of order-type a. Then if a r . . V a n + 1 . . . a 2 n , a 2 n + 1 . . . a 3 n are in X and f ({c^.. .a2n})*O then letting g(m,6,...6, )=f({ a,...a 2n >), S r ..6 k ,a n + 1 ...a 2 n ). (17.8) So either 5

or

A|=_ (6., . . . 6,., a_ , n . . . a n _ )

l--- 6 k' a 2n+l"- a 3n ) «=> 4>_ (6 n . . . <S n .,a o _^T . . . ^ 3 n ) •

Contradiction! So f({c^...a 2R })=0 and it follows that X are good indiscernibles for A. Conversely suppose f:[C] ^ K is regressive. Let f ( a ^ . • • a n ) = =f ({a][. . .an>) (a 1 <...: f R (o^. . . a R ) =y> and let A=J^. Let I be good indiscernibles for A of order-type a. Then given n€u) let a, ...a €1, a, <...
and let y=f (a,... a ) . Pick ' n l n

A^=GB =* A^=< n,c^ . . . a^,y >GB. Hence I is f-homogeneous.

D

We could have obtained a stronger result than this; for example K C A suffices, and R

was not really necessary. But lemma 17.12 suffices

for our main result, which follows.

Lemma 1 7 . 1 3 .

Suppose K i s a-Erd&s, whgre cf(a)>(ii. Then K is a-Erd&s in K.

Proof: Suppose A ^ . - A ^ K . Let A = j \ Then ACKK and |=R + . Since

K i s ct-Erdfls there i s a good set of indiscernibles I of order-type a. By lemma 16.10 there i s a good set of indiscernibles I 1 ^ 1 f ° r , I'€K. o t p ( I ' ) ^ a and I 1 are good indiscernibles for A. This

suffices by lemma 17.12.

D

C o r o l l a r y 1 7 . 1 4 . If K i s Ramsey then K is Ramsey in K. Lemma 1 7 . 1 5 . Suppose K is a-Erdb's, a>w . Then v*L. Proof: Let I be good indiscernibles of order-type a for J . Let X be the smallest elementary substructure of J containing I . Say a:JQ=J |x. Then 3>a)n , as a>u), . So P((jo)flLcJ So P(u))DLcX, as a|a)+l = | Suppose t is a term of L and suppose t K(a,...a ) cai,a^.. . ^ n e i • Take a'<...
•

(17.9)

142 Corollary 17.16.

u

Suppose K i s a-Erd&s, cf(a.)>U). Then 0

Proof: Otherwise K=L.

exists.

D

Much stronger r e s u l t s than t h i s can easily be obtained the same way. In particular

the # sequence can be iterated OJ. times.

Lemma 1 7 . 1 7 . Suppose K is a-Erdb's, cf(a)>u>. Then there is an inner model M such that K i s a-Erd6s in M and for any inner model N in which K is a-Erd&s M=N.

Proof: Suppose for some s h a r p l i k e N KJ= K i s a-Erdfls; then l e t M=KN where N i s < - l e a s t such. Otherwise s e t M=K. Suppose M1 i s an inner model, M'|=K i s a-Erdos . Then KM'|=K i s a-Erdos. I f KM'=K then KM'£M. Otherwise say KM =K , . Then by minimal choice of N N*>N or N'=N. So M=K cK ,cM'. • I t i s a l s o t r u e i f cf (a)>a> t h a t K+(a) Kf=K+(a) < a J . For K+(afw => =* K>K(OO. K(a) i s a-Erdc-s i n K so K|=K (a)-> (a)

Lemma 17.17 can be repeated with K-*(ct)

W


. Hence K|=K-> (a)


.

i n place of K a-Erdfis.

SLITHY TOVES The i n d i s c e r n i b l e s lemma i s a l s o used t o prove r e s u l t s i n t h e theory of u l t r a f i l t e r s . The b r i e f e s t summary i s given h e r e . Definition 17.18. (i) An ultrafilter

on a regular cardinal K is uniform if and

only if all elements of U have cardinality (ii) subsets

An ultrafilter

if there is a family {M :X€S] of

of K such that ~M
Lemma 1 7 . 1 9 .

Suppose there are no p-models. Let K be any uncountable

Then every uniform ultrafilter

Regularity i s a form of

Definition if

K.

U is (y,K)-regular

17.20.

on K i s (K,K ;

non-normality.

An ultrafilter

U on a regular cardinal

K is weakly normal

and only if U is uniform and whenever X€C7 and f:X->K is regressive

y
cardinal.

regular.

then for some

{Z:f(Z)*y}eu.

Lemma 17 . 2 1 . Suppose there are no o-models. Let K be a regular K=sup Y 2 . Then there is no weakly normal ultrafilter on K.

cardinal,

y
It is conjectured that the restriction on K is unnecessary.

143 COMBINATORIAL RESULTS Almost all of the combinatorial results commonly used in L can be proved in K. We mention only two. Definition 17.22.

0 is the statement: there exists a sequence of sets

S ,a
{a<w :xC\a=s } is stationary.

Lemma 17.23. K|= 0 The proof

i s not that hard and the reader who knows the proof

i s encouraged to work i t

in L

out. The same cannot be said of the next

principle.

Definitio _n limit

17.24.

•K is the statement:

there is a sequence C, defined for A

\
if cf(X)
point of C. then C =yfiC, .

A

*y A

Lemma 17.25. .K|=O for all K. They are only a couple of examples: stronger • principles and morasses are available as well. We shall not go into the applications of the principles, which are wide-ranging and not restricted to set theory.

144 PART FIVE: THE COVERING LEMMA

The applications of K in chapter 17 were discovered after the invention of mice, whereas K was designed with the covering lemma in mind. Much of the motivation for this part was discussed in the introduction. The historical notes to chapters 18 and 19 discuss the connection between the proof given here and the proof originally discovered. Chapter 18 opens with a summary of the proof. The reader is strongly urged to consult the proof for L at this point: for although a proof for L is embedded in our proof it can be substantially simplified. As we have said, we are concerned not to abridge proofs to such an extent that important techniques are lost, provided that it seems that those techniques show off the structure of K in an important way. It may well be that finestructure could be done without in this proof; we have explained why it is used here in the introduction. Chapter 18 is mainly occupied with a technical property called goodness; the reader who is unfamiliar with covering lemma proofs would be wise to take goodness on faith. Chapter 19 proves the covering lemma. Up to the case where C

is infinite the proof is very similar to the proof in L, but a

mucky argument is needed to settle this final case. The muck intensifies in chapter 20 when we extract a Prikry sequence from this final case and obtain a covering lemma on the assumption -0 . The pace relaxes in chapter 21 and there is no fine-structure in the discussion of SCH or patterns of indiscernibles that it contains. The covering claim for M is the statement: if XcOn is uncountable then there is Y€M with XcY and X=Y. In chapter 18 and 19 the term 'covering claim1 denotes the covering claim for K. From its failure we deduce the existence of a p-model.

145 18: THE UPWARD MAPPING THEOREM

This very technical chapter begins the proof of the covering lemma. What we should like to prove is Suppose TT:K—»•_ K T

ZJ

l

cofinally and M is a mouse such that

Vy
M1 such t h a t IT1 =>ir |MflK-

7r':M->

(18.1)

T

1

and

(n=n(M)+l).

T

^n

(18.1) is not true in general, but a watered down version is. Various closure conditions must be imposed upon K—; also we prefer to use n completions rather that general Z

, maps. But let us

see how (18.1) might have been used in a proof of the covering lemma. Recall that the covering claim says: suppose XcOn is uncountable. There is Y=>x with Y€K and X=Y. Suppose this failed, then. Take X with T=sup X minimal for such failure. By minimality T is a cardinal in K. Take Y=>X such that X=Y and Y - < K T . Let TT:K—=K |Y (here a closure property is being u s e d ) . There are two cases: (a) T is a cardinal in K. Then every mouse has the property

(18.1). Furthermore the various

extension maps fit nicely together so that there is TT':K+

K with

1

Tr'zrrr. But ir is non-trivial since TT is, so by lemma 16.21 there is a p-model. (b) T is not a cardinal in K. Then there is a mouse M in which "r is a cardinal but over which there is a definable map of y
definable then up

n ix


for the sake of argument suppose a)p!J>K, so n=n(M). The order-type of C

is no greater than w. Suppose it is less, i.e. that C^ is

finite. We know that

M

Let

TT'XM11-*

M

M' n

M

M

be

as

in

(18.1)

and

let

Z=hM,n(Tr' (pJJ+1)UiTI (CM)Usup TT1 "a)PJJ+1) (18.3) so Z€K. In K, ? < ! ; and YfixcZ as TcM n. So XcZ. Then t h e minimal c h o i c e of T e n a b l e s us t o deduce t h a t for some Z1 Z^=X, Z'€K and XeZ1 . There are lots of details to be filled in. Firstly all the

146 closure conditions: these are dealt with in this chapter. Secondly the case where K
18.1

K— is a y-base for M^R+ provided that y<0n and for all T — M

T

Suppose M=< M, £,A, .. .A_T > and suppose uzK—>

K

cofinally. Suppose

y is G-closed and K— is a Y~base f ° r MDefinition 18.2. TM/1*''i={f: f€M A dom(f)<=v ~ < x,y >€TT({< s,t >:f (s)=g(t) })

(18.4)

and let M'=T/~. [f,x] denotes the equivalence class of . Define [f,x]E[g,y] A £ « f , x »

~ < x , y >€TT({< s , t > : f ( s ) € g ( t ) }) ~ x€7T({s:Ak(f ( s ) ) } )

(18.5)

(0
It is easily checked that these definitions do not depend on the choices of f,g,x,y. Lemma

18.3.

Suppose <j> is lQ. Then, letting #'=< M' ,E,A^.. -A^K

M'\=<$>([f1,x1]...[fk,xk])

»<x1...xk>eTt(ls1...sk):lirt(f1(s1)...fk(sk))}).

Proof: Like Los theorem. The atomic formulae are given by definition. We use induction inductio on Z Q structure. Case 1
147

Now < x ^ . . . x k >€7T(dom(f 1 ) x . . . x d o m ( f k ) ) a n d d o m f f ^ x — . . . f k ( s k ) ) }U{< s

. . s R h M ^ (f

r

xdom(fk) 1

so 7r(dom(f 1 )x.. . UTT({< Sj^. ..s k >:Mf=-i|;(f 1 (s 1 ) . . . f k ( s k ) ) }) . I t

i s of c o u r s e

essential

that \\) be E , otherwise these sets may not exist. Hence (*) is equivalent Case

2

to

<x][.. . x

R

>€TT({< S±.

. . s R >:M|=cf> ( ^ ( s ^ . . . f

R

(s

k

))})

$ i s iJMX»

This is left to the reader. Case 3. $ is (3y€z)i|j. that M'H^y1 , [ f ^ x ^ .. . [fk/xk])

Suppose f i r s t [g,y].

A

y ' € [ h , z ] . L e t y1 be

Then

€TT({< t/urs1...sk>:Mf=i|;(g(t) ,f1(s]L) . . . f k ( s k ) )

(18.6)

A g(t)€h(u)>) so . 7) For the other direction a bit more work is needed. Suppose < z,x

x r ..

Let g(u,s,...s s,... k) be the least (in the canonical order) y1 such that y'€h(u) and ip (y' , f1 (s1) . . . f k (s k ) ) if there is one; 0 otherwise. Then g€M. (This uses the boundedness of the quantifier; if h€S ^

then

2

j^

u, s 1 . . . s k >:M|=^(g(u,s1. . . s k ) , f± (s^ . . . f R (sR) ) A

k

A g(u,s 1 . . .s k)€h(u) }) . Let g ' (4 u, s± . . . s ^ ) =g (u,s 1 . . . s k ) where -(V denotes the Godel pairing function. g'€T because y is G-closed. If y=Sz,x

.,xjr then it is easily seen that

€Tr({:MHMg' (t) / f 1 (s 1 ) ...f k (s k )) (18.9) A g 1 (t)Gh(u)}) so M'|=i|;([g',y]/[f1,x1]...[fk/xk]) A [g1 ,y] €[h, z ] . Let

TT'IM-^M 1

Lemma 18.4.

be d e f i n e d

D

b y IT1 (x) =[ {< 0 , x >} , 0 ] .

7T.-Af> M'. 2

Proof: Let <$> be E Q . Then >: (18.10) ^

. .sk =0})

. . .x k )

Lemma 1 8 . 5 . TT :M+^ M' ———————-

Proof:

Given

cofinally.

— 2,0 ~

[ f , x ] < x , 0 >£TT ( {< s , t >: f ( s ) £ r n g ( f )

A t=0}) , i . e .

148 [f,x]E[«O,rng(f) >} ,O]=TT' (rng(f) ) . Corollary 18.6. TT'.-M->

D

MJ and M_'\*R .

Now we construct a map a:y'-K)n , where y'=supTr"y. Suppose a: (id | 3) (s) £ €(id|3) (t) })=ir({< s,t >€32:s€t})={< s, t >€TT (3)2 : sEt} . So M'|=a (a) 6a (a') . Suppose M'|=[f ,x]€a(a) . Then < x,a >€TT ( {< x 1 ,a ' >: f (x1) 6a 1 }) . Assume fGK— (otherwise replace f by fn(dom(f)x3) , where ae{< s,t >€32:s=f (t) } so for all x < Tr(f) ,X>€TT({< s,t>€32:s=f (t) }) . Hence [id | 3 ,TT (f) (x) ] = [f ,x] i.e. o(&) =[f,x]. So rng(a) is an initial segment of On ,. Lety"=rng(a): so a:y'=y". So we may assume that M 1 is standard. We shall not change any notation, though, except fc for our convention about € ,. a must now be the identity and y'=y". Lemma 1 8 . 7 . TT'^rr|y. Proof:

[ { < 0 , 6 >},O] = [ i d | (6+1) ,TT(6) ] w h e r e

<5
F o r < 0 , TT ( 6 ) >€

€TT({< O , t > : 6 = t } ) . S o TT1 ( 6 ) =CTTT (6 ) . B u t a = i d | y ' .

Fix

t h e above

Definition

18.8.

(b) TT' will

Definition '

by a

(a) M_' will

be called

be called

'

denotes

of TT

'^ :Mn*)-*-(M )

M—

.

of course, but we may select any in whether or not M17'^'11 i s well-

In general i t i s not.

Definition M_ for which

(Hi)

and p-sound, p€.PA . Suppose O)pf>y. n

standard one. We are interested

(ii)

transitive

the n-completion

We should say an n-completion,

(i)

M^^;

—'

:A£*W

founded.

definition:

TTM'Y.

1 8 . 9 . Suppose ^F=RA is •'- •

• •

Then TT

terminology

n

18.10.

T" is it-good

provided

K— is a T-base:

if

if

M_ is a core

i\:K-^>-^ K^. cofinally

M_' '

n

is_ n-iterable

mouse and U}p™>T_then M_ ' 'm is

M*JO and O)pm>T then M~ p

and for

all

_

M_ is a mouse and n=n(M_) then

if

that

M^'^''m is —

provided

wp^>T;

n-iterable;

well-founded,

(iii) is only needed when K=L. In the above T was the least x such that for some uncountable Xct there is no Y=>X with Y€K and X=Y. Fix

149 T from now on. All roads lead either to a contradiction or to a p-model, but not until the end of the next chapter. Lemma 1 8 . 1 1 .

T is a cardinal in K.

Proof: Let 6=x K . Let f:6«-»-T. If 6<x then given X C T there is Y'3f" ln X with Y'=X and Y'€K (assuming f€K). So let Y=f"Y'. Then XcY and X=Y and Y€K. Contradiction.1 Lemma 1 8 . 1 2 .

n

Suppose XCT, ^ , YtK and 7<x in K. Then there is Y'£K with x=y'

and 7 f =X. Proof; Let 6=Y K . Let f:5-*-»-Y, f€K. Let X=f" l n X. There is Y€K with Yc6 X=Y, XcY, because 6<x. Let Y'=f"Y. The result intended to replace (18.1)

D is:

Suppose X C T . Then there is T">W and TT:K—• ~~

T

K 2JO

cofinally (18.11)

T

such that (i) Yen "7; (ii) the cardinality of T does not exceed max(Y,XJ; (iii) T" is ir-good. Even (i) and ( i i ) r e q u i r e a l i t t l e c a r e : j u s t as K may not be minimal, so Y -
A set Ycr is u-closed in i provided that whenever

__—__»_^__—__—_______ are such that a.€y for all i€w and sup a.
Lemma 1 8 . 1 4 . transitive.

i€03

2

Suppose Y-
Then there

i€a)

^

2

and cofinal

in T, and yflo)

is

are T,TT such that TT:JC— =K \Y.

— _"~X* "^

—

—

P r o o f : C l e a r l y 03ncY. Suppose TT: =K | Y , K t r a n s i t i v e . As OKCK ( K , ^ ) ^ " ^ i s a mouse" => M i s a mouse. Assume K * j result is trivial.

otherwise the

So K=U{M€K: M i s a mouse}. So KcK-.

Suppose N€K— i s a mouse w i t h K c r i t i c a l and otp(C )=o). Suppose K^0J2nY. Claim K i s s i n g u l a r i n K. P r o o f : L e t K'=sup TT"C . K ' €Y by 03-cloSure, so K'=TT(K). I t t o show K |="K' i s s i n g u l a r " . K'>U>2.

Now K ' < T SO t h e r e

is

suffices

I f K'<W 0 t h e n K'€w o nY, so KEw^nY. So B€K w i t h B3TT"C

and B = « 1 . And B < K ' SO

K1 i s s i n g u l a r i n K; hence i n K a s x i s a K - c a r d i n a l . L e t M€K be a mouse w i t h c r i t i c a l p o i n t above K such

•(Claim) that

MJ=K s i n g u l a r . L e t M',N' be comparable mouse i t e r a t e s of M,N respectively.

I f M'cN'then N'|= K s i n g u l a r so N|=K s i n g u l a r .

N'€M'. So A ^ ( N ) + 1 c ( a ) x ( K ) < a ) ) € K /

Hence

soN€K.

I f N€K— i s a mouse w i t h K c r i t i c a l and K€o)2nY t h e n ir (K)=K and

150 Tr~ l n C N =C N s o N6K. H e n c e

Lemma 1 8 . 1 5 . (i)

Suppose xcrz is cofinal.

There is Y^x such that

Y-
(ii) YC\T is U3-closed in T; (iii) Yf\ud is transitive; (iv) ~Y^max(X,K ) . Proof: Let Z* denote some Z'DZ such t h a t "Z'=Z and Z' •< K . Define XQ=X X

a+l=(XaUXiUU(Xana)2)}* X = U X (X a l i m i t ) A a<X a w h e r e X ' = { s u p a . :< a . : i€a) >€ (X ) W }fiT. L e t Y=X c a r d i n a l i t yJ , ' inductively If

X'<X a n d i f a— a

X
t h a t X^_<max(X, X^ ,

{ a . : i€w}cYnT t h e n . If

. T h e n Y-< K . As t o

U (X fluO
YjOC+fr^.

{a.:i€afcx

fix, some a
6.63€Yna)2 a n d BGX^, s a y , t h e n 6€U(X flw ) , s o 6€X

This gives (i) and (ii) of (18.11). The bulk of the rest of the chapter is devoted to getting "r to be ir-good. Suppose, then, that cofinally. Suppose M is a mouse but that M77'^ is not well-

TT:K—•„ K T

Zi o

~~

T

~~

founded - this is the simplest case, and we shall see that once we can handle this we can handle any relation E

and therefore

get the full result. Suppose n(N)=0. There are < f . ,x ± >€T M / 7 T / Y with [fi+1,xi+1]E[fi,xi]

(i€(jo) , where E is as in (18.5). It cannot be the

case that x.€rng(IT) for all i: for 7T(xi) ]

=> < T T ( X i + 1 ) , T T ( X ± ) >€7T ( {< U , V > : f

±

€f±(v)})

But this contradicts the axiom of foundation. Our strategy is as follows: start with X =X. Define X

inductively such that X^ is

co-closed in T , X 0u)2 is transitive, X -< K and M is a mouse for which K

and whenever TT :K

=K | X

is a y-base but M^a'^is not

well-founded add some sequence with the above property,{x.:i
, (add the elements, that i s ) . Unions are taken at limits. Let , TT:K—=K |Y. Suppose M is a mouse for which K— is a Y~base;

suppose M71'^ is not well-founded. Say f.,x. are as above. By a condensation argument we obtain for some a M for which K is a a _ Y-base, say, with each 7r(f.) in rng(7T ) (this cannot literally be the case, but we shall see how to arrange things) and, letting

151 f ± denote T ^ " 1 (IT (f± ) ) then for all i [fi + 1 ' x i + 1 l E M C ^ i ' x i ] • So the x. must all have been added at stage a of the construction and hence are in Y. But we have seen that this is impossible. There are several complications, though: firstly we need to deal with general rudimentary relations and not just €; secondly we cannot really add all the x. sequences because this would increase cardinality too much, so that we have to pick out nice ones that will be preserved under our maps; and thirdly, since f. is not necessarily in K— we have to add apparatus to code it. All this accounts for the complexity of the next definition. Note in particular that we want to ensure that M=u{rng(fi):i€a)} with notation as before; this is to make the condensation argument work.

Definition

18.16.

Suppose M is a transitive

model of R . Suppose T\:K—>V K

•

cofinally

T

and K— is a y-base

M such that <M,T) is amenable. Let R be a relation have the same rudimentary definition each i f .,x .)€T '

2J1

T

for M_ where y is G-closed. Let T be some relation

is R-vicios

rudimentary

on M_ and let

on

R'

over M_ . A sequence < < f . ,x . >:i€o)) with

for M,i[,y,T

provided:

f

(i) for all i [fi+1'Xi+1]R' t i'Xi]; (ii) if n\i,v€dom(fi) and 0
(Hi)

f.i(0)=(6.,f.,S

(a) & .<§ .

>

(d) f ^

where:


(b) dom (f^cScpig

under the G&del pairing

function;

(f ±) ; 2

;

) and

1+1

(e) sup 6 .*y. i<03

Lemma 1 8 . 1 7 .

Suppose, with the notation of definition

18.16, that R' is not

well-founded. Then there is a sequence which is R-vicious for W,7T,y' (some y':i€oa>that t o g e t << f | , x ^ > : i € 0 3 > s a t i s f y i n g 6Q>sup(dom(f

)) with 6

(i)-(iii).

c l o s e d u n d e r t h e Godel p a i r i n g

Let dom(f£)={U,v^:v€dom(f0)}uU2,v^ f^(A2fvlV)=vl.

satisfies

zv'<6Q}.

( i ) . We a i m

~ i s G-closed. L e t f £ (11,vV

Pick MQ with f ^ S ^ . Let x^=^l,x Q V.

Pick

function. ) =f Q (v) ,

(i)-(iii)

are

e a s i l y checked. Suppose, t h e n , t h a t 6 . , y . , f ! , x ! = -Hk,y,vV-: y,v€dom(f|) U{^17+N+2,v,O^: sup(dom(f1+.)) Let

v < ( S i + 1

are defined.

Let dom(f^)=

A 0
where 6 i + 1 i s picked g r e a t e r than 6^ and

and closed under t h e Godel p a i r i n g

function.

152

(

^k'y'vV)=Fk(fi(y)'fj_(v))

(0
H17 fN+1,v,0M =f ± + 1 (v) f[ + 1 (
y

,

such

that

,TPISS M €€SSM

y.,f'

1 ' -i-

1

l"i"JL

M•

. Let x!

M • _i i

= 17+N+l,x.

1 > J-

, , 0 . 1"T"JL

l=

Set y s u p 6 . . n i€ca Now suppose t h a t R i s d e f i n e d by a uniform r u d i m e n t a r y d e f i n i t i o n on M^ whenever M i s a mouse with n(N)=n. (m,n) i s c a l l e d t h e type of R. I f R i s d e f i n e d on M™ when M=Jg then t h e type of R i s m. Suppose R i s of type (m,n) and TT:K—• K c o f i n a l l y . Suppose t h a t f o r some M a mouse with n(M)=n, K— a y - b a s e f o r M , t h e c r i t i c a l p o i n t of M n o t l e s s t h a n y R1 i n (M111)11^ i s n o t w e l l - f o u n d e d . C a l l t h i s p r o p e r t y P(M,Tr,y) . L e t M be t h e < - l e a s t c o r e mouse some mouse i t e r a t e of which s a t i s f i e s P (M ,7r,y) f o r some y . rr A sequence Definition

is

vicious

18.18.

for

(i) M is

mm

M* , 7r,y i f f the least

it

is vicious

mouse iterate

for

M ,TT,Y,TM.

of M_ satisfying

P(M ,7r,y>) for some y. (ii)

Let y be least

such that P(M ,^,y).

The canonical R-vicious sequence

—7T

for it is defined inductively as follows: (a) Suppose :i€03 > for M_ ,TT,y with f'=f., x'.=xsuch .(j for Mm,T\,y with f'=f., x'=x . for

Lemma 1 8 . 1 9 . Suppose there are M_,y such that P(M_,T\,y) . Then there is a unique canonical R-vicious sequence for TT which is a vicious sequence for M^,v,y' (some Y'
Proof: Uniqueness

is clear. Existence

is also clear. The only point

in the definition of R-viciousness that is in doubt is (iii) and this is easily seen to hold for y'=sup6.. We mention now two slight modifications

•

that we sometimes need.

First

of all, if the type of R is (n,n) then we add to the definition of viciousness

the requirement

that fo(O)=pJJ+ . This is clearly

possible. Secondly if the type of R is n then instead of taking M as above we let M=J g where 8 is least such that for some y P 1 (Jg, 7T,y) , where P'(M,7T,Y) is the statement that K— is a y-base for M n and R1 in (M 11 ) 77 ' 7 is not well-founded.

Definition

18.18 is

153 redone with P1 in place of P. Lemma 1 8 . 2 0 .

Suppose «f ,,x. >:i€a)> is the canonical

7

Then M "^ U rng(f.)

R-vicious

sequence for TT.

, where R is of type (m,n) (or m) .

P r o o f : L e t X= U r n g ( f . ) . L e t Y = S U P < $ • (notation a s in d e f i n i t i o n 1 i€u) 1 8 . 1 6 ) , so that < < f. fx± >:i€o) > is v i c i o u s f o r M^,7r,y. S i n c e 6 . c r n g ( f . ) , Y £ x -L

e t

ooy=sup y.. S i n c e

C l a i m M™I X -
rng(f.)cS

, Xcj

(M=M ) . — —7T

Proof: First of all observe that X is rud closed. It follows that if (j) is E Q and x,t€X then {y€x: cf> (y, t) }€X. Hence it suffices to show that x€X, x*0 => xflX+0 to ^et the result for E Q . Suppose x£S ; the canonical order of S is in X so xDX*0 . i

jJd

_^

i

Mm

_^

For E , J

|=3y4>(y,t) -> (3i)(3yes )4>(y,t). n(Claim) m M Mmi m Let a:M'=M |X; so a:M'^y J . < J ,T > is amenable; letting x€T <^ — — — Li —y y M « a(x)€TMm , then, <M',T>|=MC M be m-sound with _ M m =M' and a:M-> M— — ' m . Let — _ _ Y,I ~ k k — such that o^o and a(Pj^)=Pjvi (k£m) . Then clearly M is a mouse with n(M)=n. Let K=a

(K) be the critical point of M. If a is a bounded

subset of Y then since K>_Y/ <>Y/

SO

or(a)=a. Hence K— is a y-base

for M. Each f. is in X; let f ^ a " 1 (f±) . 6i=a"~1(6i) and it follows easily that <:i€o)> is an R-vicious sequence for M ^ T I ^ Y . Iterate M,M to comparable N,N. If N€N then there is a £ noncofinal map of M to N. But P(M,TT,Y)/ S O N€N is impossible. Thus N=N. If M*M then M is a mouse iterate of M, since P(M,TT,Y) . But this is impossible since K X K . So M=M. But then ~E. <£.

in the

canonical order of M so f\ =f. , all i€a). M m = U rng(f\); hence M m = U rng(f.). 1 i€o)

•

The preceding lemma is a simple form of the next one, which is the essential property of vicious sequences that we need. Lemma 1 8 . 2 1 .

Suppose Y,Y'-
Y<=Y', TT:K- = K J Y, it':*:-, ~ K^\Y' and let

ft=TT' 7T; so il:K—• K~, . Suppose (( f , ,x . );i€o)) is the canonical

R-vicious

sequence

for 7T' and let b^dom(f^); i If

i i IT' (b .) ,7T' (h ,)€.Y for all i€a) then there

_ _ are f. such that < < f . ,x . >;i€co > is

154 the canonical R-vicious sequence for IT.

b^ft""1 (b±) , h ^ f t " " 1 ^ ) . L e t M ^ , . Let (m,n)

Proof: Let

be the type of R. Let Z= U f."ft"b.. 1 i€o) * Claim 1 Z is rud closed in Mr. Proof: Given s, t€Z,k<_17+N with k>0, suppose s=fi(ft(i)) and t= f .(ft(t)). Suppose i<j. We can show that there is s'€b. +1 such that ; in fact picking t such that 0=f±(ttt)) )

(18.12)

So let s 1 be -<3,s,th Using this result we can show by induction that there is s" in b. such that s=f . (ft(sM)) . So

f .+1(ft(-U,s",t>) )=f . +1 Hk,ft(s") ,ft(tfr )

(18.13)

=F,K (f .(ft(s")),f .(ft(t))) J

J

=F (s,t) Claim 2 S

n(Claim 1)

€Z for all i.

Proof: It suffices by claim 1 to show f i + 1 (O)€X. But 0 € b i + 1 >* O G b i + 1 so this is clear.

•(Claim 2)

Claim 3 a)p^=supy • . i€a) Proof: By lemma 18.20. Claim 4 M ^ Z - ^ M m .

•(Claim 3)

Proof: Just as in the claim in lemma 18.20, using claims 1-3. D(Claim 4) Let c^M'^M^X with M 1 transitive; so a:M'+

M™. By supposition M is

m-sound; let a:M^^, M such that aixr, M is m-sound, M ^ M 1 =p,

and 3^(Pk) =

for k£m.

Claim 5 M is a mouse; and a core mouse if m>n. Proof: M is n-iterable so it is only necessary to check that it is critical. Its critical point K must be a

(K) S O if m>n this is

clear. Suppose m=n. Given x€M n there are i€o),Y
Mn^3i£a)3Y
(18.14) (18.15)

155 ^<7.

a(Claim 5)

Claim 6 zny=rng(ft ) fly

( y - s u p ^ : i€u)})

Proof: Suppose v=ft(v) and v
D(Claim 6) Let y=sup 6\ . Claim 7 a|y=ft|y. Proof: Immediate from claim 6.

D(Claim 7)

Claim 8 If a€Z, acy'
and ac6. . Then (18.16)

f ± (C)€f i (v) •• {fi (^)

as

Let

in

C1)

. Then .

; so a

Claim 9 If a€Z, acy'
.

D(Claim 8)

a (Claim 9)

Claim 10 K- is a y-base for M™. Proof: Suppose acy'-oy. Then a(a)EM, so a(a)€K—,; and a(a)=ft(a). So a€K-. Claim 11

D(Claim 10) "Y is G-closed.

Proof: Each ~6 . is closed under the Gfldel pairing function. D(Claim 11)

Let f j L = a " 1 ( f i ) . Then dom(f ±) ^a" 1 (dom(f i ))=a" 1 (b ± ) =ft""1(bi) = b i . Also Ti±=f±riE±2. Claim 12 <:i€o)> is an R-vicious sequence for M^TTj-y. Proof: Since f i ,f i + 1 €Z and Z is rud closed {< s,t >:Mmf=f±+1 (s) Rf± (t) }€ €Z so by claim 9 (which clearly can be extended to subsets of y ) 1

={< s ^ ) : ^ f' i + 1 (s)Rf i (t) }. Hence - <xi+1,x±

,t>: m

(18.17)

:M Hf i+1 (s)Rf i (t)}) m

s,t>:M |=f i+1 (s)Rf i (t) }) "

<X

i+lfXi>e7r'

156 The verification of the other clauses is left to the reader. •(Claim 12) It follows that M^ is defined.

Let N be M^ and let Q,Qf be

comparable mouse iterates of M,N respectively. Q€Q' is impossible by the <-minimality of core(N). Suppose O'CQ. Let Q,Q',O" be comparable mouse iterates of M,N,M respectively. So Q'EQ. And there is a map "oiQ^y Q" such that cnf=7f"a, where TT,TT" are the respective iteration maps. But then a(Q1)contradicts

the

<-minimality of core(M). Hence Q=Q' so core(M)=core(N). By minimality again M is a mouse iterate of N. Claim 13 M=N Proof: Suppose not. Let k be the mouse iteration map from N to M and let K 1 be the critical point of N, so k(K')=K, k|K 1 =id|K'.Unless m=n the result is trivial since both N and M are core mice. So k"Nn=h-n(K'Up^"KL) . Let p=a (p^ +1 ) , £=supa"K • and let Y=h M n(£Up) . So ak"N n cY. And Y-*- M n , so we may let k:N'=M n |Y, with N 1 transitive; then k:N'+

M n and there is a mouse N with N n =N',n(N)=n. Then

letting d=k~ 1 ak / d:N n +, N n . The critical point of N is S(K') and K-, -

-

Li-

T

is a y-base for N . Iterate N, M to comparable Q,QM . Q'€Q is impossible because we should get a E Q non-cofinal map of N to Q; Q6Q 1

is impossible, because if< < g. ,y. >: i€co > were R-vicious for Nn,ir,

6 then < < d (g±) ,y ± >: i€a) > would be R-vicious for Nn,7r',6' (some 6 1 ) contradicting the minimality of core(M); so Q=Q% . By minimality again N must be a mouse iterate of M; but k:Nn->^ M n so N=M. So Mn=hMn((^Up) . It follows that M n =h~n(K' Up^ + 1 ) / so M n =k"N n . But K'^k"N n . Contradiction:

a(Claim 1 3 ) .

Hence < < l , x i >: i€u) > is R-vicious for M ^ T T J Y . Suppose it were not the canonical R-vicious sequence for TT. Then there would be an R-vicious sequence for M ^ T T J Y 1

such that either y' <:Y or for some i the

sequence < would satisfy j
an

<*

v

i<xi'

But

in

anv

of these cases < < a (g±) ,y± >: i€a) > contradicts the canonical status of

Lemmas 18.20 and 18.21 are proved for the case where R is of type (m,n). The reader is left to do the proofs for R of type n; all that needs to be done is to replace the iterate-and-compare arguments with direct comparison. Given X C T define a set Y as follows: Let Z* denote some Z'oZ with z;f=Z+K,, Z'flx w-closed in x, Zfriu)2 transitive and Z'-^K . (i) X Q =X*.

157 (ii) Suppose X

is defined and TT :K

=K |x . For each (m,n)

such that, for some m-sound mouse with critical point not less than T a for which K

T

is a y-base, (E™) • in ( M ™ ) 7 ^ ' 7 is not well-founded

let << fmn,xmn >:i€a)> be the canonical E^-vicious sequence for T^. For each n such that there is some M=J O for which K 3

T

is a y-base and

aa

(En) • in (M 111 ) 7 ^' 7 is not well-founded let < < f*,x£ >:i€a) > be the canonical E -vicious sequence for IT (E

is defined inductively by

< i , x >E n + 1 < j , y>«<s (4>) x x , y > > € A J J + 1 w h e r e <> f i s t h e f o r m u l a h ( i , < x , p £ + 1 » E h ( j ,< y , p £ + 1 » ) . Then M n M X a + 1 = ( X a U { x m n : x m n defined}U{x*: x n defined})*. (iii) X,=( U X ) * (X a limit). A a<X a Let Y= U X . Then YflT is a)-closed and cofinal in T (provided X is aPU>T" but M 77 ' 1 ' 111 is not n(M) -iterable. Therefore — M— — —

(M

111 71 1

) '

(Em) • in

n

is not well-founded. K- is a T-base for M and cop^T imply

that K~ is a T-base forrf11.Let < < f±,x± >: i€u) > be the canonical E^Jvicious sequence for TT. Let b. and h. be as in lemma 18.21 and pick a such that for all i:i€o)> is the canonical E m -vicious l

ii

n

sequence for TT . It follows that for all i€oj x.€Y. But let x.= =Tr" 1 (x i ). It follows that M™. f=f± + 1 (x ± + 1 ) E m f ± ( x i ) . But M^ a mouse. Contradiction!

is a

D

In summary: Lemma 1 8 . 2 3 .

Suppose ACT is cofinal

in x. Then there

such that (i)

XCTT"T;

(ii) cardinality of 'r <max(X,K^ ; (iii) 7 is -u-good. Finally we apply this to get

is

TXJO, TT.-JC—•• K

158 Lemma 1 8 . 2 4 .

Suppose -K:K—>-y K and 7 is n-good. Suppose that

in K. Then there

is

TT;JC-*-y K with TOir. ^1

T is a 0

(K)

Proof: Suppose K > T . Then K— is a T-base for 0 . Let IT 7r T

Q*=Q ' . Suppose fi**^* with K * = T T

(K)

cardinal _ T

=TT K ' ,

(K

( K ) . Then IT ' | K:K-*-K* cofinally.

V

For given TT ' (f) (x) < K * with X < T , dom(f)cY
defined), so TT

(f) (x)
(K)

(y) . (TT

(K)

(f) (x) =[f ,x],

is not

of course). But

l

(18.18) T

<(Y )T
e*>_0 and TTV : Q K + Q * cofinally. Given a<0 with core(J^K)=N and N€K K let N * = T T ( K ) (N) . Say N*=J U ^. N* is a mouse as N is; let N* be its Kth mouse iterate. Let C*=C N *. Then C* is closed unbounded in K . And so N*=J^£, some a'<6*. So O * = J ^ . If 6*>6 let a be such that a'> >6*. Then cap^F < K for some n, so a'<6; hence 0=9*. Let 7TK=7r^K) |K . K = H Q K by lemma 14.13 so / : K -»-r K Claim

and irK3ir.

If K
Proof: Let TT ' =TTK ' | K . Then TT1 :K ->v K . Any x€K can be written in the form TTK(f)(y) for f€Q K with dom(f )cy' <7, rng(f )cK K , y_ K ; for K Lo —K

given <J> lQ .-.IT1 (fn) (yn)) •• (Kl)

(18.19)

{Ki)

(fx) (y^ ..^

(fn) (Y n ))"

(fn) K

...7T (f n )

So a:K K =K K , hence a is id| K K Thus i\ CTT K K

.

Let TT=U{TT : K regular>x}.

So

irK (f) (Y) =TT'(f) (Y) and so TrK=7Tf . D (Claim) D

159 Exercises 1. K— is a Y ~ b a s e fo r M, n=n(M). Show that the ath mouse iterate of M^Y,11

is

M^' Y ' n where w^ is the ath mouse iterate of M.

2. Show that CM7r,Y,rr>«Y is countable. 3. Obtain lemma 18.24 for the case K=L. 4. Suppose Y -< K , each i
U Y.. For each i
=K |Y., and x. is IT.-good. Show

that there are TT,T such that TT:K—=K |Y and T is 7r-good.

160 19: THE COVERING LEMMA

Recall that we are assuming that the covering claim

fails and

that T is least containing some uncountable X that fails to be covered. Our aim is to get a p-model. Let x be such that (i) i r : K r Z i K T , (ii) cardinal of x <X; (iii) "T is TT-good; (iv) Xcrng(Tr) . This is possible by lemma 18.23. Note that X<x, otherwise T itself would cover X. u

Lemma 1 9 . 1

0

exists.

Proof: Otherwise K=L. I f "x were a c a r d i n a l in L then t h e r e would be TT3TT, TT:L->V L#Tr*id L so t h a t by the r e s u l t of chapter 12 0 """

exists.

2J1

Otherwise there is $>x with I (J o )nP(y)£j- (where y is the L__^

—"

U)

p

"*- T

rn-4-1

cardinality of x) . Let 3 be least such. So for some m cop- <x; take — — 8 1 m least. Let ^ , = (jp) *'T1' m ,1 W V T ' m . (J 3 ) m =hm+1 (J ^ ( c o p ^ U p ^ ) . So m ^f"(Jft) =h_m (ir-wp^UTftp ?* )). Let p=sup Tr"o)p , and let Y= =h,

3 3 ~3~ m+1 3 ,.m(pUTr(p" ) ) . So XcY and Y6L. Furthermore L|=Y<X, because

"p<x and x is a cardinal in L. By lemma 18.12 there is Y'€L with XcY 1 and X=Y'. Contradiction!

a

Hence K=U{M: M is a mouse}. The proof of lemma 19.1 foreshadows the proofs of the simpler cases of the covering lemma for K (and gives, of course, the covering lemma for L ) .

L e m m a 1 9 . 2 . Suppose x is a cardinal in K. Then there is a p-model whose critical point is less than x.

Proof: By lemma 18.24 and lemma 16.21 there is a p-model.JThere is x1 regular with T < T ' < T , O) 1 T (as Xa)1 as X>w 1 . So the restriction on the critical point follows from chapter 16 exercise 2.

a

— K From now on suppose that x is not a cardinal in K; if y=(x) then

161 P(Y)nK+K-. So there is N, a mouse with critical point >r such that P(y)nNcj:K-. Let 3 be least such that P(y) n j e+ii K 7 ; argument P(y) flE^ (Jg)<j:K-. Clearly J^ is a mouse.

then h

Y

t h e usual

Let P(N,T) be the

property: N is a mouse with critical point not less than "r and for some y
19.3.

P(M ,1). -a

satisfies __

Let M be the <-least core mouse some mouse iterate Then N— denotes the least mouse iterate —T

of M that —

of which satisfies

Lemma 1 9 . 4 . K- is a T-base for A£p Proof: By the <-minimality of core(N~) . Lemma 1 9 . 5 . c

a

cr.

Proof: Otherwise let C N _={K a :a<$} and suppose K ^ T . Let N be the ath mouse iterate of core(N—) . Then P(N,T).

a

Lemma 19.6. otp(C ;n. Let K be the critical point of N. Lemma 19.7. m=n. Proof: Suppose m>n. Then N is a core mouse; for by lemma 19.5 C N £ T ; but u)p£+1>"r and CMnu)P5J+1=0 so C =0. Let M^'11'™, n-iterable since 7 is ir-good. Also wpM>-ff(K) >_a)p ^

^

V

*

*

1

^=Tr N ' T ' m . M is so M is a mouse.

so ir^-h^dr-topJJ^UTrtp^ 1 )) Let

p=su P Tr"u) P" +1 f

) . Since M6K, Y€K. XcY and Y < T , S O by lemma 18.12 there is Y'€K such that XcY 1 and X=Y'. Contradiction! Lemma 19.8 otp(C ;=w.

__

o

_

Proof: Suppose C N were finite. Let M-N*' 1 ' 11 , 7r=TTN'T'n. Then M is n-iterable because T is ir-good. Every x in M n is of the form Tr(f) (y) for some y
>) (i
= (hMn(i,pjj+1
162 N n =h N n(wp£ + 1 UpJJ + 1 UC N ) so 7r"Nn=hMn(TT"ajpJJ+1U^(pJJ+1)Uir(CN)) . Let pf=sup Tr"a)p£+1, Y=hMn(
Contradiction.1

D

Corollary 19.9. T=K. Proof: Kifr^sup C = K .

•

Observe that lemma 19.8 would have given an outright contradiction but for the fact that if otp(C )=co then ^"C

may not be in K. Anyway

we could get a covering set in L [ T T " C N ] , although this is of limited use since N may have depended on the choice of X. In chapter 20 we obtain a single C that is TT"C N sufficiently often and prove a covering lemma for L[C]. In this chapter we are only trying to get a p-model. The idea of_the construction is that whenever 7 is jr-good we define a mouse N T with critical point T as N-' T ' n . Let C T =TT"C N __, so otp(CT)=u). If N T = J U

we find that X€U T *> 3y<xC T ^YEX. Subject to a

restriction to be stated all the N T are comparable. So define U=_U U T . For each x€K + we show that there is 7 with x€N T . So U is a normal measure on T in K^+ and < K T + , U > is amenable. Unfortunately U is not countably complete but anyway we deduce that U is a normal measure on T in L[U]. Two small modifications are called for. Firstly, although N— depends only on x N T depends on IT, which is not uniquely determined by I, so we must index our N by sets Y=rng(7r) rather than by ordinals. Secondly we want Y-^K + rather than Y-^K . Lemma

19.10.

Suppose YC3C +(T

calculated in K here and throughout the chapter)

iTlT cofinal in T and ~Y
K + such that

(i) V=rng(-n); (ii) (iii)

cardinality if

K £ max(Y,X ) ;

'r=-n~1(T) then f^K-

and T" is TT|IC— good.

Proof: There i s Y*3Y such t h a t (i) Y*-< K +; ( i i ) Y*riT i s u)-closed i n T; (iii) Y*nw 2 is transitive; (iv) Y*=max(Y,« 1 ). The proof of this is just like that of lemma 18.15. Suppose TT:K=Y* with K transitive. There is no hope of proving K to be of the form

163 K R since the proof of lemma 18.14 depended on the minimality of T . But letting Z=Y*flK and Z -
then Zflx is w-closed in x,_znco2 is transitive

so T T " 1 " Z = K - /

say. Since K T = H * X + , K - = H * ;

and T = T T " 1 ( T ) .

The

construction of K with IT T-good is just as in the previous chapter.n Let C = {Y:Y_DX and for some K,TT:K+ V K + Y=rng(iT), K satisfies _____ ^1 _£

lemma 19.10 and Y=X}.Let T y denote x, TT IT, N

(iii) of

N—. By lemma 19.8

otp(C N ) =co and by corollary 19.9 x y is the critical point of N y . Let N^Ny^Y1

T

Y'n(NY)

Lemma 1 9 . 1 1 .

a n d CY=7rY"CN

.

If

Y,Y'€C

a n d YcY1 l e t

TryYl=7rY}7rY.

Suppose Y,Y'£C,Y<=Y' and C ,^y'<=Y, some y'<x. Then C ^-ycf for

some y
(N

Y } , ft=TTyJ{ T Y ' n

(N

Y5 . M i s e a s i l y

seen t o be

a mouse (n(N )-iterable because if <:i€o)> were vicious then < < iTy (f.) f iTy, (x.) >:i€co> would be vicious for N'; and a mouse by the argument of lemma 19.8). Let 0,Q' be comparable mouse iterates of M,KL,, respectively. Q'^Q; for C JMy

=C . O T E E ( Q 1 ) , so if Q'€Q we should have C vj

CO

€Q, so

JNy

C N €M and M^cf (x yl ) =u). But T y , is the critical point of M. So QcQ 1 . Let TT MO be the mouse iteration map of M to Q. Let p*= =pjJ ( N Y ) + 1 . Pick Y>y', Y < T v l f such that either Q=Q' or {Q}U7TMQ'it(p*)cEhQ(n(Q') (YU(C Q | vT Y ,)Up^, ( Q l ) + 1 ) (remember that Q€Q' =* Q € Q

lll(Ql)

(19.1)

).

Suppose, for the sake of contradiction, that n>Y and n€C.T but N Y' TTY,(ri)^CY. But n€rng(7rYY,) so we may let n=frYY, (n) ; then n$C N . So Tieh N n(N Y ) (nUp*) . S a y n = h N n ( N y ) ( i , < ? , p * » . So n = h M n ( N y ) ( i , < 7r YY , ( ? ) ,ff(p*) » .

(19.2)

n = h Q n ( N y ) ( i , < T r y y , (t) ,7r MQ ft(p*) »

(19.3)

Hence

so that by (19.1) there are j , t
(19.4)

with n'>n and, usin<_j the fact that C.. W

yl

m

and C Q l ^n are T.1 indiscernibles for < Q ' n'=h Ql n(Q') ( J , < ? , P Q ,

(Q)+1

n(Q

'

}

,p£,

(Ql)+1

y'

-^

,6 > 6<

» .

By (19.4) n=n'. Contradiction! So C ,^TT , (y)__C .

=C n ,nT v , we get (19.5)

a

164 Lemma 1 9 . 1 2 . Proof:

Each

Suppose Y^C for i<^1 and i<j =» Y^Y.. Y.-^K 1

+,

so

U Y . -< K + b y T ii *

T

then for some i
lemma

6.8.

If

F.eC. {a.:i€u)}c U x "Kui

x

_

=X. Suppose

TT:K-=K |( T

T

X

1

U Y. is clearly transitive. Also since X is uncountable

_ i
Y.

U Y . R T is co-closed in x.

1

a)9n

Then ^

U Y.= i
U Y.flK ) . It remains to show that x is 1

i
-rr-good. But this follows from chapter IS exercise 4.

•

We shall say that Y is fine provided that (VY'eC) ( Y ' D Y ** (3Y"€C) (Y"3Y' A C ,,cY) ) .

(19.6)

Lemma 19.13. There is a fine Y£C. Proof: Suppose not. Define a sequence as follows: Y Q is arbitrary; Given Y 6C, Y is not fine. Hence there is Y ' D Y , Y'€C such that a a — a (VY"€C) (Y"2Y' => C «£Y ) . Let Y + 1 be some such Y 1 . If X is a limit then pick Y.ID U Y with Y.6C . a<X C is countable so Let Y= U Y ; then Y€C by lemma 19.12. a
C y cY a , s a y , a
and Y€C so

c

Yi

Y

a

-

Contradiction! D

F i x some f i n e Y€C and l e t C' ={Y'6C :Y'3Y A C ^ ^ y e Y , some y < x } . Then Lemma 1 9 . 1 4 .

For all

Y'ZCthere is Y"€C such that Y"z>Y'.

Proof: Say Y'UYcY, Y€C. Then t h e r e i s Y"€C, Y"=>Y , CY,,cY. So Y"€C' Y"3Y'.

D

Lemma 1 9 . 1 5 .

Suppose N^rjFf. Then for 'x€P(T)r\N^, xEuy ** 3y<x(C^y^i) .

P r o o f : Let N*=N^n *NY*. Fix X. There i s X€N^ such t h a t X€rng(7r Y ) / X=<X a :a<x>and X=Xa, some a<x; f o r suppose X€S^Y B) ; t h e n t h e r e i s X*= N

=<X*:a<x y > enumerating P(x Y )nSgY, by a c c e p t a b i l i t y : l e t X=ir y (X*). Let p * = p ^ ( N Y ) + 1 , and p i c k fi such t h a t X*€h N n(N y ) (6Up*) , 6<x y . Take 6 w i t h IT (6)>a w i t h o u t l o s s of g e n e r a l i t y . Then f o r n€C y , n>ir y (6) , n=TTy(n") f s a y , Ny|=\fa<6(n€X* ** X a €U N

]

'

(19.7)

** Xa €Uy ) .

(19.8)

Y

So ^

Thus if X€U y then C ^TT (6)cX. If X^U y but C y ^YEX then x^XCU C Mmax(y,y')) =0 . Contradiction!

so C^y'cx^X, some y'<x. So •

165 Lemma 1 9 . 1 6 .

Suppose Y'czy", Y',Y"€C.

Then for any mouse M for which K Ty

m ~ T -base and m>n(M) , top >T ,

is a '

P r o o f : S u p p o s e M^M111, ^ = ( 1 ^ ) ^Y1 Y " ' T Y I . S u p p o s e a c Y < T y l I , a€M2» Then a=TT y , yl , ( f ) (6) , s a y , w i t h 6 < T y l I . Assume y, 6 < T T Y , y I 1 (y) , and r n g ( f ) c P ( y ) . K

s o TT , ,,(f)€K 1

and s o a€K

. Thus

T

i s a T V I I - b a s e f o r ML. L e t M =(MO) ^Y " Y" . If

6'
Then f€K

domffjcy,

f€M 2 t h e n f = [ g , 6 ] w i t h < g , 6 > € T M 1 ' ^ Y 1 Y " . F o r s u c h g,6 and d e f i n e a ( [ [ g , 6 ] , 6 • ]) =7ry , (g) (TryB (6) ) (6 ' ) . Then f o r (19.9)

...7ryI,(6n)

...gn(Yn)(6n))»

So a is well-defined and a:Mo-> —J

M!rY'Y Y'Yll""TT Y l . It is clearly surjective

Lo —1

so i t i s an isomorphism. Lemma 19.17. if Y* ,Y"eC

D

then uyir\N^u

ON^, .

Proof: Take Y € C with YcY'UY". Then C y ^Y'E Y E Y ' (some Y * < T ) so by lemma 19.11 there is Y < T such that C y ^ Y ^ y i . Suppose xeu Y ,nN' Y . Then by lemma 19.15 there is Y n < ^ such that Y 1 >Y, C Y ,^Y'EX. Hence C ^ Y ^ X ,

S O XGU y .

Claim N^.cN'y,. Proof: By lemma 19.16 it is sufficient to show M=N Y ?Y'Y ' T Y l ' n ( N Y | ) cN y .

Vy

But suppose M=J^. Then for XGP(T y )OM, X€U *> ^ Y ' Y ' ^ N

-^

Y

some Y < ^ Y , by lemma 19.15. So M, N y are comparable. But if N y

f o r

= U Jg/

for some 3
a

C o r o l l a r y 1 9 . 1 8 . If Y1 ,Y"EC then there are V,a',0." such that N_'=J ,rH',,=J „•

166 L e t U=

U U ,. Y'€C Y

Lemma 1 9 . 1 9 .

Suppose x€K +, xcr. Then there

is 7'€C' such that x€.N^.f.

P r o o f : T h e r e i s Y ' e C 1 w i t h x€Y' . Say TT:K?Y', =HK and ir , =ir | K . L e t X ^ T T 1 ( X ) ; x€K. C l a i m x€N

K transitive.

So K T = Y1

,.

Proof: Kf=3M(M is a mouse, x€M and the critical point of M is not less than T Y , ) ; and a),€K. Let M be such a mouse. Let Q,Q' be comparable mouse iterates of M,N ,. Suppose Q'€Q. There is a€ €E (N__,)^K , acy
. So QcQ 1 . It follows that x€N v ,. l T

n(N l)

ft=Try^Y ' Y"

Y

a(Claim)

. Then for y
Y€ft(x) « y€^(xri6)

(some 6
(19.10)

Y

<=» y€Tr(x) <=» y€x. So x=ft(x)€Nyl.

a

Corollary 19.20. U is a normal measure on T in K and (K +,U> is amenable. Proof: If X € K T + , X C P ( T ) then, coding x as a subset of T , x€N y , for some Y'eC 1 . So xnu=xno_

€K +.

n

Were U countably complete this would suffice by lemma 16.11. But it plainly is not. The following result enables us to get a p-model anyway. Lemma 19.21. cf (T+) >CO . Proof: Otherwise take {Y.:i€a)} with sup OnflN' = T . Let x. , code all 1 1+± i€co l the subsets of T in N' ; we may assume x. ,€N* . There is Y •< K + Y 1+1 Y T i i+1 with Y € C and x.£Y, all i€o). As in lemma 19.19 x ± €N Y , all i€aj. So P(T)nKcN y and N y is a mouse. Contradiction!

n

The only facts used from now on are that U is a normal measure on T in K, that is amenable and that cf(T+)>u). If y
T

_j_

Suppose U is not normal in L [ U ] , Then there is 6>j

with

^ ,^J^. since P (T) flL[U]*P (T) OK.and Take >^T+ such P ( T ) flJ^ Then £ (J^)nP(T)^J^ J^ 6 is least a premouse. Letthat N=JT6\ +1 If U were

countably complete we should have a contradiction. In the

167 present case we also have a contradiction, but it takes more work. The argument to the end of lemma 19.2 5 will show that N is a mouse. (T+)N=T*.

Lemma 19.22.

Proof: Suppose T < Y < T + . Then KT+f=Y=T« If J +H"Y is a cardinal" then UnJ U +cJ U €K, so J U +€K. Hence Nf=y=x. X Y + X If x were not a cardinal in N there would be a well-order of T of order-type x + in N, hence in K. Lemma 1 9 . 2 3 .

N_ is

o

acceptable.

Proof: J + i s acceptable, as i t i s a limit of mice. An easy induction on co3€On a)pJJ

gives the result.

<.x, some n
Lemma 1 9 . 2 4 .

• A=A

N#

°*P—T ' ° ^ c o u r s e .

cf f(op;>co.

Proof: Suppose cf(u)p)=u); say o)p=sup p.. Let H be E o such that y=h N n(i,x) ~ 3zH(z,i,x,y).

(19.11)

Let t = - U t ) Q , ( t ) ^ be the Godel pairing function on x. Define f ± with dom(f i )cx

by

y=f i (t) ~ 3z€Sp H(z, (t)Q,< (t)1,pJJ+1>,y) A (t)0€o)n A 1

We may assume pfl

€S

N

P +

A y

'

t€S

(19.12)

p."

. Let a .=sup(rng (f.) fix ) . Since J + = U rng(f.) 0

X

X

T

+

i€03

1

sup a.=x . But cf(x ) >co so x =a.,say. So f. maps a subset of x x i€u) + + cofinally into x . But f. €N and N)=X is regular. Contradiction! D Let I={v
\Kp let

(this exists

because J =J ' ;. Letting ajv=sup v., v€I. i
is p-sound and standard.

Lemma 1 9 . 2 5 . V

wV€iC.

P r o o f : UflM =unNv, s o MV=J^ , s a y . a v £ 6 a s U i s normal i n M . B u t \i^= •» •» u y — + + so a <6. J +ira = T r b u t i n N x i s a c a r d i n a l so a <x . a v

168 It follows that J U €K. a

In K, U is countably complete, so M

is a mouse. Suppose N were not

a mouse. Then (E )._ is not well-founded. Let {x.:i€u)} be such that n N 1 N ^ = x ± + 1 E n x . , all i€o). Since N= U X v there is v€I with x ± €X , all i€w. Let x i = a V " 1 ( x i ) . Then ^ ^ x ^ - j J E ^ i ^ , all i€u>. But M V is a mouse and n(M V )=n. Contradiction! So N is a mouse and so N€K. Hence J +€K. Contradiction.1 So U must be normal in L [ U ] , Hence we have proved the covering lemma: Lemma 19.26.

Suppose there is no p-model. Then if xcpn is uncountable there is

Y£K such that Y=X and X=7.

Observe that although the proof made lavish use of the axiom of choice, we could obtain lemma 19.26 in ZF, since given X we may argue in L[X]. The assumption that X is uncountable is necessary: see model 1 in appendix I. Exercises 1. Prove the converse of lemma 19.26. K . Show that K-=H- T ( notation as in this chapter).

2. Let ir:K—> T

Zii

T

T

T

169 20: L[U] AND L[U,C]

We want to continue the discussion of chapter 19 for a bit. So we are still supposing that the covering property fails for some set XcOn with minimal sup x and that X is

uncountable. So there is an

inner model with a measurable cardinal. We are interested particularly in the case just considered where we get a p-model whose critical point is x. This involves assuming that N— exists whenever 7r:K—*

K , "x<x and T is ir-good. Let L[U] be the p-model we

T Li

X

have just constructed: note that V*L[U], as cf(x)=a) ; indeed the covering lemma for L[U] fails. Definition sequence

20.1

Suppose L[U] is a p-model with critical

for U provided

point

T. C is a

Prikry

that whenever x€P(T)C\L[U], X£u *• Bry
=0).

For example, if < e o ,< TT of L[U] and K

g

>
> is the iterated ultrapower

is the critical point of L[U ] then {Kn:n€o)} is a

Prikry sequence for L[U ] . Usually we shall be interested in the minimal p-model: model 3 in appendix I shows that this may carry a Prikry sequence. If M is an inner model and C€M then U€M: in particular L[C]= =L[U,C]. The aim of this chapter is to show that, letting C' be as in chapter 19, there is C'czC such that for all Y€C' there is Y'€C" with YcY 1 and for each YeC", C y is a Prikry sequence for U (C Y is always a'Prikry sequence' for UflN' , of course). To do this it is sufficient to stabilise C

so that for some y Y,Y'€C" ^ C y ^y=

=C Y ,\y. It looks as though a simple appeal to cofinality should do the trick, but it does not.

Suppose t h a t i s a sequence s a t i s f y i n g a£3 •> Y a E Y ft a n d Y a € C . Let TT :K =K | Y fiK , N =N , N ^ N ^ a ' T a ' n ( N a J . Let y denote + N~ (x ) a (or 0 n N +

a

if N f=x

a

a

is the largest cardinal) and y' denote

N

(i ) a(or 0n N , if N^|=x is the largest cardinal). Then y^= N =sup{u)Y+u):u)y€lsr AN P (TX ) nj a*P (x) nj N ^ 1 >. Assume that a<3 •» y^
170 some a<0)1

Let Y= U Y ; so Y€C. In f a c t Y€C' , for c l e a r l y CvcY a
a

a

so for some y
and y1

(x )

. Let y

(with the usual convention about largest

cardinals). Lemma 20.2. For a<0)

y'
_

Proof: Let tNir" 1 ^. Then N^= ((N^) *' V 1

n ( 1

V)

7 T / T

' n ( N a } . So there is a

1

E Q map from N to N ; but these are comparable, so for some a)3<0n—, N' N'=JO. Thus y ^ y 1 . B u tpI
—

p

ot—

a

OL+L—

ot

?(T)nrng(TtN'T'n(N))cy.

Lemma 2 0 . 3 .

Proof: Let TT:K=K + |Y, S O TT==TT | K—. let < < M o > a € 0 n . < ^ o 6 > a < 6 € 0 n > ^

Suppose a€N, act. Let M=core (N) and

its mouse iteration. So a=Trn(o(5), some

n€o),a. Let Ka denote the critical point of Ma . So acKn . a=afiKn €N so _ __ Z _ ~~ _ a€K-. Hence 7r(a)GY, say ir(a)€Y . Suppose IT (a) =iTa(b) . Let M'=core(N a ) and let <<M f > __ , .ocr. > be its mouse iteration. We may assume —a aEOn ap a
ir\'T(x'n^a) (b) =TT N/ T ' n ( N ) (a) . Let

: it suffices to show fl(b)=a.

Suppose b=h M ,n(N a ) (i,< Y,pJJ,(Na)+3» " ) .


So

(20.1)

Na

J

=h-n(N) (if<

a

€rng(TTna)). Sayft(b)=7rna)(a'). Then for 6<« n 66a 1

«* 6eft(b)

(20.2)

«* 6€a. So a=a'; hence ft(b)=TT (a)=a. nco Lemma 20.4. \i'=sup y'. Proof: Suppose not. Then there is a€P(x) nrng(7r ' ' a^N'

K

' ) such that

all a<w 1 . By lemma 20.3 a€Y so a€Y , some a. Let ft :K =K +|y

and suppose a=ft a (a). Let 3-TT a' a'

or (a) . For 6
171 (20.3) «* 6€a. Thus a^EN^. Contradiction! Corollary 20.5. cf(\i')>w. Now let p=p~,. wp^y^T and N ' n ( N ' ) =h-,n(N') (xUp^| N'} + 1 ) so just as in lemma 19.24 cf(a)p)>io. Hence p is a limit ordinal. Let p=

p ) ;

s o S = h _ i n ( M ) ( T u 5 K Let

i^wV V n ( V

u

nfNM+1 N' X1UN ; For v

N'

Lemma 20.6. Q Gj ,. _ _ Proof: Xv6N« and N ' h ^ l T . So QV€H^',=J^ . Lemma 20.7. j^'c u Q . y V

The purpose of this construction is shown by the next lemma. Lemma 2 0 . 8 .

There is a
Proof: First of all we may assume that there is y<x such that for all a<031 CY^ycC Y.Otherwise suppose that for all y X ={a<0)1 :Cy\ycC } a ^ a is bounded in a), . Let x =sup X : let be cofinal in x. Then Y y i 1 <x x :i<0)> is cofinal in o)n . Contradiction! Y i Secondly we may assume that for all v€I there is a such that v

v N a If the lemma fails then for each a there is £ with £ GC-y^ Y a but E, ^C . But supposing, as we may, that for all a C ^ Y ^ v ' t n e r e a 0 are only u) possible values for £ .So we may assume E, =£ all a +1 > ) (20.4) =^=h(i,< TT(?) ,p» .

(20.5)

172 (20.5) i s a E, property of £,7r(?),p; c a l l i t <J>(€,TT(?) ,i>) . There i s jsn(N') v€I such t h a t J f= (£, IT (6) ,p) ; so N.n(N') V J* |Xv|=(j)(^,TT(?)fp)

(20.6)

so that QvH(|)(^Tr(?)/a~1(5)) .

(20.7)

But Q v eJ^! so by lemma 20.4 Q ^ J ^ I ,

1

say. Assume a~ (p),Q v €

f

€h N ,n(N ) ((ynrng(Tra)) Up a ) . So

N;h"2vH> (£,*(?)/^(p))"

(20.8)

is a l1 property of £,TT(£) , c e y n r n g ^ ) ,P a : call it \\) (£, ,TT(?) ,-na U ) ,p^ Assume TT (?) =77^ (6") , ^ = 7 r a (T) • So +1

It follows,

since

>-

Tec

(20.9)

t h a t i f we take

then (20. 1 0 )

Reversing_the argument ( 2 0 . 5 ) - ( 2 0 . 1 0 ) (20. 11) So b y ( 2 0 . 5 )

S=TT ( I 1 ) . So 1 = 1 f . C o n t r a d i c t i o n . 1

•

Lemma 2 0 . 9 . There are y cofinal in T. Let Y 6 C be arbitrary. Given Y ^ C there is Y'€C' such that VYM3Y' C y n ^ + C ^ Y . Let Y i + 1 € EC1 be such that Y.UY'CY.,,. Let Y6C' contain

U Y.. Then whenever

Y*3Y, Y*3Y i + 1 so Cy^^yi* C y ^Y. Hence .

(20.12)

Now define a sequence with Y ' € C . Given any Y€C there is Y*6C with Y*3Y and P(T) n m g ( y Y * ' T Y* ' n ( N Y* ) ) cY* . This is shown by the argument of lemma 20.3, constructing an co..-chain with ZQ=Y, Z a+1 2P(T)nrng(7T^Z a ' T Z a ' n(N Z a ) ) , Z ^ U Z^ (X a limit) ot a<X with each Z in C . Y*= U Z has the desired property. a

ai

a

Let Y be arbitrary and let Y^=Y*. Given Y^ there is Y'^Y^ such that VyVY"DY'C ..sy*Cy,^Y• We maY assume Y 1 contains a well-order of T of order-type y . Let Y I + 1 = ( Y 1 ) * . If X is a limit ordinal let Y'=( U Y ) * . Let Y'= A

a<X a

U Y . By lemma 20.8 C V I ^ Y = C V ^ Y

ai

a

Y

Y

for some Y < T .

173 Contradiction, as Y'=>Y Lemma 2 0 . 1 0 .

^

n

If Y is as in lemna 20.9 then C is a Prikry sequence for U.

Proof: Take x€P(T)flKT+. Suppose Y ^ C f X E N y , and C y l ^Y=C y ^Y. Then x€U ~ x€U y ( ~ 3yf (Cy,vy'cx) ~ 3y* (Cy^y'cx) .

D

A NEW COVERING LEMMA Now we turn to covering properties again. Suppose L[U] is the minimal p-model and K is the critical measure ordinal. Suppose that there is some T with X C T uncountable but no Y€L[U] such that X=Y and XcY. Take T minimal. Lemma 20.11. if 0 does not exist then K=T. Proof: Suppose T < K . Then since KcL[U] the covering lemma for K fails for X C T ; S O there is a p-model whose critical point is not greater than T . But this contradicts the minimality of K . Suppose T > K . T is a cardinal in L[U], just as for K. Take Y -<JT with XcY and X=Y. Let TT:J~=J^|Y. Assume without loss of generality that if X>K then K C X . We may assume IT to be 7-good. Let a be the critical point of ir. If T were not a cardinal in L[U] then we could follow the strategy of chapter 19 and get a set covering X. So x is a cardinal in L[U], Hence there is a mapft:L[U]->L[U] with fto-rr. Case 1 a>K

(calculated in L[U] here and from now o n ) .

Then U=U. Soft:L[U]-*L[U] is non-trivial, and so by lemma 13.19 0 f exists. Contradiction 1 Case 2 K < O < K

.

a is a cardinal in L[U]; for if $ft(a) onto. Say a=ft(f)(y), Y < 3 . S O a€rng(ft) . Contradiction! Hence ft(a)>_K+. If ft(a)>K+ then a> (K + ) L [ U 3_But K+€rng(ft) ,a£K + so this is impossible. So ft(a)=K+. Hence a=(K + ) L ^ U ''. So U=UflJ^. But then there is 3>a with J^,_ O P ( K ) * J^flPdc) . Taking 6 least such ~"n ^n U ^ — u P(K)flZ (Jo)*P(K)njQ and J Q = J Q . Hence U is not a normal measure in _ w ~P P P P L[U]. Contradiction! Case 3 a_
say K=ft(l<) . 1<=K by m i n i m a l i t y of K. I f X>K then KCX SO

Tr|K=id|K, so a=K=K. But TT(1C)=K, SO ir(a)=a. Hence X
But 1C
C o r o l l a r y 2 0 . 1 2 . Jjf 0 does not exist

D then there is a Prikry sequence for U.

Proof: T i s the l e a s t point where the covering claim f a i l s over K. The conditions described at the start of t h i s chapter must hold, since otherwise there would be a p-model whose c r i t i c a l point was

174 less than T . Hence there is a Prikry sequence, by lemma 20.10. a Call such a sequence C. Suppose the covering property for L[C] fails so that there is T 1 with X C T 1 uncountable but no Y€L[C] with Y=X and Y3X.

Take T 1 least such. Suppose 0

Lemma 20.13.

does not exist.

T'=K.

Proof: If T ' < K then K is not the critical measure ordinal. If T ' > K then just as in lemma 20.11 0

exists.

n

Lemma 20.14. There is ACT' with otp(X)=u such that there is no Y€L[C] with Y=X and 7=N . Proof: Suppose not. With X some uncoverable set, suppose C={Y.si < ^} and let Xi=xnyjL. Take Y ^ K with Y i = X ± + « 1 and X±SY±*

Let 6=?"; then

6 < K ( X < K S O 6 £ K : but cf(K)=u>). There is f€K with f:K«--*H*, since H ^ = K K . Let v i = f ~ 1 ( Y i ) . By hypothesis there is Z€L[C] with Z^v^ieco}, Z C K and 1=*^. Let Y=U{f(v):v6Z A f ( V ) K < 6 } . Then Y€L[C] . If f (v)K<6 then f (v)<X so Y=X. Finally Xc u X.c u Y.cY. Contradiction.1

a

i€co X i€a) 1 Now we derive a contradiction by showing Lemma 2 0 . 1 5 .

Suppose xgc, X=co. Then there is Y€L[C] with 7 = ^ and xcr.

Proof: This i s t r i v i a l

u n l e s s K=sup X. There i s Y ' e C with XcY1 and

(some y) - Let TTY , :N n , (N Y' ) ^

N^(NY'] , n=n(NyI).

Since

UTT l(p*+1))chT_1n(CvlUsupTrVI"copJJ+1U7rv, (pJJ+1)) . Call t h i s l a s t set Y*. S i n c e N__, i s a m o u s e , N__ ,€K; and C_. ,^Y =CVv Y/

c

vinY L

€ L [ C ] , Thus Y * € L [ C ] . XcY a s Xcrng(fT , ) . And Y*

i s J

finite

s o C_.,€

< k . L e t f:Y*-*-+6,

6 < K . L e t X ' = f " X . T h e r e i s Z^X 1 , Z c 6 , Z€L[C] w i t h I = « 1 . L e t

f€L[C], Y=f"luZ.

D

Thus we h a v e a more g e n e r a l c o v e r i n g lemma: Lemma 2 0 . 1 6 .

Suppose 0

does not exist.

Then either

the covering

property

holds over K or it holds over the minimal Q-model L[U] or there is a Prikry sequence for U such that it holds over L[C]. A few f i n a l o b s e r v a t i o n s about Prikry s e q u e n c e s . Let L[U] be the minimal p-model and l e t <

cr,

,< TT

O

> ^oc^

> be

i t e r a t e d ultrapower. Let C={K :n
C0~~

C

O

"

~

C

O

its sequence (C^Y^X)}.

C

O

"

^

175 Also McL[U ] for n€o): for, given n€o), C ^ K is definable in L[U + , ] as the critical points of the iterated ultrapower of V; but C H K is finite. So for no n is L[U ]cM, otherwise U €L[U ,,]. Hence n ~~

r

n

n+i

j.

M^=L[U ] is the minimal p-model. Also M(=0 does not exist, since 0 € 6L[U] is impossible ( since 0

exists =* the critical measure ordinal

is countable). But as C€M the covering lemma fails over L[U ] in M. This shows that the L[C] part of lemma 20.16 cannot be omitted. Consideration of L[U ] where y=N p-model was needed. For let D = { K ^ Prikry sequence for U

shows that the minimal

:a
(cf(K/)>a)) dropping "minimal" in lemma 20.16

would give D'€L[U ] with D I = « 1 and DcD 1 . But K >;y so L[U ]|=D'
is regular in L[U ] .

Finally although lemma 19.2 6 could be proved in ZF, lemma 20.16 requires AC. Let M= n 0 L [ U . ] ; then M|=ZF. Let A={g€L[U] :g:o)+a)2 is cofinal and strictly increasing}. For g€A let C = { K , v:n€o)}; let G={C :g€A}. Let G'={C:3C'e 3 X < K 2 C ^ X = C ' ^ A } . Then G'€M. Lemma 2 0 . 1 7 .

G' cannot be well-ordered in M.

Proof: L[U 2]cM, of c o u r s e , b u t G'nL[U 2 ] = 0 . Suppose G1 c o u l d be w e l l - o r d e r e d i n M, by ^ Vy
rng(g) Oy+rngCg1 ) Hy and for a l l g€A t h e r e i s g'EA 1 such

that

rng(g) fly=rng(g') fly. L e t brG'Ho be d e f i n e d by: b(C)

i s t h e l e a s t b<w such t h a t i f y

i s t h e b t h member of C t h e n C^yc{K :a<0) }. b can be coded as a map of

(u )

t o (o, and hence i s i n M. L e t X=U{C^y:C€G" A C^yc{K a:a
Then X€M. There i s a:Af->u) i n L[U] such t h a t Vg€CVn>;a(g) (K

(n)€X).

Let K={g(m): g€A' A m>^a(g)}. So K£M. Now suppose K i s of o r d e r - t y p e 2 a). S i n c e K i s c o f i n a l i n u) , l e t t i n g g:u)-*K be i n c r e a s i n g and o n t o g€A. So l e t t i n g g 1 (n) =g (n) + 1 , g'€A,

so C ,cX, and hence

rng(g')c:K.

C o n t r a d i c t i o n . 1 But l e t t i n g y be t h e oath member of K KflyCM. I f y i s K^ t h e n U. can be d e f i n e d Contradiction! Lemma 2 0 . 1 8 .

from Kfly. So L[U6 ]cMcL[U 6 + 1 ].

o

Suppose the covering

lemma holds over L[C] and for all y
=>a€L(C], Then P(K)<=L[C]. Proof: (in M ) . Suppose acK. Let C={yi:i€o)} and let a i =a0y i . Then each a.€L[C3. Let L[C]|=e=sup 2 3 . Then if f:6-^ u P(B), f€L[C] let i 3-^. Let A={g (v.) : i€o)} so A6L[C]. Hence {v . : i€o)}€L[C] so {a i : i€o)}€L[C] , so a€L[C].

n

176 But then if M|=the covering lemma over L[C] then since acry => a€L[U a ] (some K a >y) => a € L [ u a ) l

]

"• a€L[C]. So by lemma 20.18 G'cL[C]

and hence can be well-ordered in M, contradicting lemma 20.17. Exercises 1. Show that the converse of lemma 20.16 holds. 2. Let L[C]x,K be as usual. Show that if y>_K is an L[C]-cardinal then L[C]f=2Y=y+ (actually L[C]|=GCH: show this if you can.1)

177 21: APPLICATIONS OF THE COVERING LEMMA

SINGULAR CARDINALS Various properties of singular cardinals follow. Lemma 21.1 for example will say that if there is no p-model and T ^ O ^ is regular in K then c f ( T ) = T . Because no generic extension of the core model contains a p-model, this result tells us that it is impossible by forcing to change the cofinality of a cardinal unless we also change its cardinality (unless T<03 2 ), unless there is a p-model. Since we certainly do have a forcing construction to preserve the cardinality of a measurable cardinal while changing its cofinality to w the converse holds too. The first results of the chapter are best read as saying that certain large cardinals are necessary for certain forcing constructions. Lemma 21.11, on the other hand, makes no mention of inner models and is of significance apart from forcing considerations. Lemma 2 1 . 1 . Proof:

If

Suppose

there is no p-model and T>U) is regular 6=cf(T),

Y3X, Y C T ,

Y€K, Y=X+« 1 . S i n c e K(=T r e g u l a r ,

= 6+«1.

6,«1
But

in K then cf (T)^T.

6 < T \ Suppose XCT i s c o f i n a l

i n T and X=6 . L e t

K(=Y=T. SO Y=ir. Hence 7=

a

Now if we had been able to prove the covering lemma without the uncountability restriction then we could prove lemma 21.1 without the restriction T_>a)2« Model 1 in appendix I shows the consistency of u

(i) 0

does not exist;

(ii) cflu^)=u); (iii) w^=K 1 . Hence the uncountability assumption is essential. Corollary

2 1 . 2 . If

there is no p-model and 3 i s a singular cardinal then 3 is

singular in K.

Proof: 8>o>2« If 3 were regular in K then cf(3)=3=3, that i s , 3 is regular. D I t i s , of course, essential that 3 be a cardinal.

178 Lemma 2 1 . 3 . Proof;

(8+)K<8+.

Otherwise

= ( 3 + ) K by 3 is

If t h e r e i s no p - m o d e l and & i s a s i n g u l a r c a r d i n a l t h e n

lemma 2 1 . 1 .

singular.

Corollary 2 1 . 4 .

But

Since

(B*)K

(3+)K<8+=>

is

regular

(B+)K=8.

(&)=*$.

in K cf((8+)K) =

Thus c f ( ( 8 + ) K ) = 8 :

but

a

If there is no Q-model and 3 i s a singular cardinal then a . |3

Proof: Kha o by lemma 17.25. But then as (3 ) =8

°o.

a

p

P

Of course lemma 21.3 is another restriction on forcing; you cannot generically collapse the successor of a singular cardinal without collapsing the cardinal (provided there is no p-model). The reader is left to formulate modifications of the above on the assumption that 0 does not exist. Now we turn to the singular cardinals hypothesis. Recall that f(e) this states: if 8 is a singular cardinal and 2 c f ( 3 ) <8 then =8 . This is abbreviated SCH. The background to this hypothesis has been explained in the introduction. Before showing that SCH follows from the non-existence of 0 we shall show that it completely determines cardinal exponentiation. Lemma 21.5. -Suppose 8 is singular. Then 2®=(2<®)cf(^>)

(2^ denotes sup 2 Y ; . Y<8 Proof: Let y=cf(8), 8=sup 8., 8.<8 for all i
2 =2^i
Lemma 2 1 . 6 .

(21.1)

If SCH holds

then

2^*2*® if for some y<8 2**2** = (2
Proof: If 2 Y =2 < 3 then 2 3 = 2 Y # c f ( 3 ) by lemma 21.5; but Y£Y'-cf(B)<& so 2P=2 . Otherwise there is a sequence {ya:a
=(2<8}cf(2

<3

{21m2)

)^

But 2 c f ( 3 ) < 2 < 3 so by SCH ( 2 < B ) c f ( 2 < 3 ) = ( 2 < 3 ) + . Hence the value of 2 for singular 8 is determined by 2 a for regular a. In particular, SCH implies that if GCH holds up to a singular cardinal then it holds at that cardinal. For example (Vn€uO 2 n = K n + 1 = >

179 =>2 OJ=K

. . Of course, Silver's result quoted in the introduction

gives this result outright for singular cardinals of uncountable cofinality. Lemma 2 1 . 7 . If K,X are infinite Proof:

X

X X

X

X

K £(2 ) =2 J
cardinals o

Lemma 2 1 . 8 . Suppose SCH holds. otherwise

(K,\

and 2 X ^K then K =2 .

Suppose 2
=K+

infinite).

Proof: By i n d u c t i o n on K. Suppose K=V . Then assume K>2 ; ifvv>2 then v £v + =K(by i n d u c t i o n h y p o t h e s i s )

or i f v=2

then v =2 =V
E i t h e r way v £ K . NOW (v+)X={f:f:X-*v'T. And f:X-*v

(21.3)

i m p l i e s f:X-»-Yf some y
(v ) = U+if:f:X-^Y) +

£V .V

(21.4)

X

SO K X =K.V X =K. Now suppose K is a limit cardinal. If XY>/ so K X =sup yX. If 2 X < K and Y > 2 X then yXcf(K). Let Y = cf(K), K=sup K. with each K i
n K±)

X

(21.5)

= n KX iy £(sup a X ) Y . But s u p a X =K, s o K * = K C f ( K ) . a
2 c f ( K ) < 2 X < K s o K C f ( K ) = K + by SCH.

a

Lemmas 21.7 and 21.8 tell us that SCH completely determines cardinal exponentiation, given the values of 2 a for regular a. We turn to the exponentia.tior proof of SCH. Lemma 2 1 . 9 . Suppose 3 i s a singular

cardinal

and M is an inner model with the

property

that for all xc$ there is Yc$ with YtM, raf and 7*X+N . Then every

function

f:y+& (Y<&) is of the form g'g where g.*Y"*T+ and gr'Gf.

Proof: Let Y 3 r n g ( f ) , Y=rng(f)+« 1 , Y€M. Then Mf=Y
if there is no p-model then SCH holds.

180 Proof: Take 3 singular with Y=cf(8) and 2 Y <$. By lemma 21.9

where 6=cf(B) + . K|=B'5l2B"'5=2B=6+. And (y+)y<(2y)y=2y<$.

(21.7)

so e c f ( e ) =e + . Lemma 21.11. if 0 does not exist then SCH holds. Proof:Let L[U] be the minimal p-model.Suppose 8 is singular, y=cf($) and 2 Y <6. Let K be the critical measure ordinal. If 8 < K then the hypothesis of lemma 21.9 holds for M=K and the proof is as in lemma 21.10. Similarly if there is no Prikry sequence for U and 8 is arbitrary, since L[U]f=GCH. Suppose 8_>K anc* C is a Prikry sequence on U with the hypothesis of lemma 21.9 holding over L[C]. So (21.6) can be proved with L[C] in place of K. But L[C]|=6<Sl23-<S=23=3+ by exercise 2 of chapter 20; so 3 C f ( ^ = $ + .

(21.8) o

MORE ABOUT # A brief result motivates the following. First note that for any real a the covering lemma holds over L[a] unless there is a nontrivial j:L[a]-* L [ a ] . Exactly the usual proof works. Now suppose 0 n

e

exists but 0 ^L[a]. Then L[a](=the covering lemma over L. Suppose there were no non-trivial j:L[a]-> L[a]. So the covering lemma holds over L [ a ] . Given Xcpn uncountable there is Y€L[a] with Y3X and Y=X. And there is Z€L with Zz>Y and ¥=Y. So the covering lemma holds over u

L. But 0

exists, so this is impossible. So there is a non-trivial

j:L[a]-* L [ a ] . This can be coded by a real, just as for L; we call this real a*. So: if 0 # exists and 0 # ^L[a] then a # exists. We can define a "mouse over a" and show generally that if there is M not in K. and a is a real in M then there is a mouse over a M K . It would be nice directly to extract indiscernibles for those for K , but this we cannot do. We can show that there

a mouse not in a from is a

neat relationship between these two sets of indiscernibles. Unexpectedly this gives a result in descriptive set theory. We defined M # only for M(=V=K; but we shall let M # denote (KM)# for arbitrary inner models M. We want also to define a # for arbitrary aoi; this will not always

€ L [ a ] # . The following are equivalent:

(i) there is a non-trivial j :L[a]-> L[aj; e (ii) there is a class C with the properties of lemma 12.9 for L[a];

181 (iii) there is an a-mouse where by an a-mouse is meant an n-iterable premouse J a ' with critical point K and <^P^ lK(N=J a ' U , some n) . Definition 21.12.

If any of these conditions is satisfied

then a

denotes

u

Lemma 2 1 . 1 3 . Suppose a does not exist/then Proof: Just as for L.

the covering

lemma holds over L[aJ.

D

Indeed, letting K[a]=U{M:M is an a-mouse}UL[a], we get an inner model with a covering property conditional on there being no models of the form L[a,U] where U is a normal measure in L[a,U]. Lemma 21.14.

Suppose if^^K. Then a** exists.

Proof: Let M=L[a]? Suppose a

does not exist. Then the covering

lemma holds over L[a]. Now L[a]|=there are no p-models, because if L[U]£L[a]were a p-model then KcL[U]cL[a], so KL^a^=KflL[a]=K. Hence L[a]|= the covering lemma over K

. Thus the covering lemma really

does hold over K L [ a ] . Let < < V a C O n ' * \x 6 W o n >b e

t h e mOUSe

iteration of M and let K be the critical point of M . Then K is a a LTa] ~a regular in K for all a; but K U =K is a sinaular cardinal, contradicting corollary 21.2. • Corollary 2 1 . 1 5 .

If 0** exists,

a is a real but O**$L[a} then a**

exists.

Fix a o o , t h e n , such t h a t KL^a"UK and l e t M be L [ a ] # with mouse i t e r a t i o n < < V a E O n ' * %$ W o n > and Ka the c r i t i c a l p o i n t of M^. Recall that K L * a ^=

U H M a. Let N=J aV be an a-mouse with critical a a€On K a point T, u)a=T+w. As in chapter 12, exercise 4, we may show that the weak iterated ultrapower of L[a] by V is of the form T rai ^Qcn >. L e t x =TT' ( T ) , C ' = { x : a € O n } , K ' = K L a j . ap a < p € O n ot O a a (T+)K'=(T^)L[a].

Lemma 2 1 . 1 6 . Proof: T

e
L e t<5=(T*)

'a+i (X)};

Lta]

,

6 I =( T + )

K I

s o V =V a n d < J . , V

;

suppose

6'<6. L e t V a = { x e L [ a ] n P ( T a ) :

>i s amenable,

a n d s o V riK'=V

nKi,e

CL[a]. C l a i m < K ' , V flK' > i s a m e n a b l e . Proof: Suppose $<6f and let Xcx^ be such that {{y:ex}:C= =P(x )nK« f XCK 1 . Now

182 since X€K' Tr I a+1 (X) €K f ; and letting X ={y:< y, C >€X} , K'flVa= = {X^:C<x a A €TT^ a+1 (X)}€K l .

D (Claim)

Now the covering lemma holds over K1 in L[a]

so in L[a] cf(S l )=6 T >

>o). It follows that V OK1 is countably complete in L[a]. Hence L[a] has an inner model with a measurable cardinal by lemma 16.11. So K=K'. Contradiction!

D

Let C={K a :a€On}. Let K = K

and set W=P(K f )OK 1 . For X€W let X denote

U 7T ( X ) . i€On U 1 Lemma 21.17. K'= U x. x€w Proof: Suppose a€K* . Then there is x€rng(Tro ) such that a€x, since K a a TrQ :M->j, M cofinally. But then if y=xnK' then a€7r0 (y) and y€W. D

Lemma 2 1 . 1 8 . c= n x. x£u Proof: K

Suppose

X6U.

Then

for

any

a

"nQ ( X ) € U

,

so

K €TT 0 + T (

X

a n d

)'

s o

ex. so cc n x. "xeu Conversely suppose 6^C. Let a be least with K >6; a=3+lr say.

Then there is f€W such that 6=TT_ (f) (K Q . . . K Q ) , say, with $,<... Oa 31 3n 1 . . ^ B ^ a . Let X={^:V?<^^*f (I) }. If X^U then {£ :st <^^=f (I) }€U so for all a 3$
^) which

( X ) . If $Kg.

But if 3>a then S f T T ^ U ) as 6=7r Oa (f) ( K ^ -..Kg ) •> < s = 7 r 0 3 ( f ) So 6^X. But X€U so 6^ n X. a X6U

( K

3 •••IC3^

Lemma 21.19. There is i such that ^he/iever *0
for

1J

a2I V and all x£w.

Proof: Let r={n:n=T A cf(n)>K>. Let r)eV. Claim For all x6L[a] there is £ T T ! . ( X ) = X . — —

ij

_

Proof: First of all, whenever a is a limit ordinal there is a xCrngCTr-1- ) . Let < be the canonical order of L[a]; •t

(XCX

then, letting x a = < f ^ (x) , if ax so 7r (7r (x )) 7r (x ) i e x x But since < is

ia' ia a ~ ia' a' ' " ' a- a» *

there is ^ such that a>£ => x =x-. Now suppose C£i£j
(21.9)

183 ir^(x) •

=x. So TT|. (x)=x.

n (Claim)

Given X6W let £ v be such that ^__<±< j Tr!.(3fnv ) = x n v . Then since W=K
Lemma 21.20. y€c «* TT'. .(y)€c 2j

Proof:

Suppose

b y lemma 2 1 . 1 9

y€C.

w

Then f o r

TT| . ( t ) e x n v ^ ,

all ,

) t

X£U y€3f. so

Pick

TM.(y)€X.

ij reverse argument is similar.

v>y: Hence

then

y€XPIV v

TT| . ( y ) € fi X .

so The

X€U o

Corollary 21.21. c'^x. <=C. 1 0 Proof: Given i>i Q pick n regular with n>i. Then T = K =n. TT' (T^) = T = =K €C, so by lemma 21.20 x.€C.

n

This is not very surprising. If y is an ordinal then the for some f€W y=f(3 1 -..B n ) with €C. So by lemma 21.19 given TT' (y)=7r« (?) (IT- (t))

^

(21.11)

So ir|. is determined by its action on C. The next lemma is much more surprising. Lemma 21.22. Suppose T. =K . There is B such that l0

"

184 Proof: Let C±={y€Cn: 1
to show that

*[A"c1+^=

since then, l e t t i n g $ be the order-type of C. , 1 W

TT! . C. =C. so the order-type of each C. i s $ (j^.in ) Furthermore

(ii)

follows as ir|. i s

If 7T- ."C^C.

(i^ilj)

giving

(i) .

order-preserving.

then n j " C i + h = n j T i i + h " C i = n j + h " C i

( a s i n

chapter 12) = C + h . So it suffices to show ir|."Ci=C. (i^ilj) . This follows by an easy induction from TT! i+i'^t^i +1 • Suppose this failed, then, and Y€C i + 1 ^rng(ir| i + 1 ). Then there is f:T.-*T.+,, f€L[a] such that y=n!. -(f)(T.). We may assume f monotone since it is not constant on a set in V . . L e t

f

n

{v
(all X€U)

(21.12)

Hence

-

W V ) '

L e t &=s

L e t

Y n - * n + 1 ( T i 4 l l ) . Let S ^ f ^ T . ^ ) (so

"P&n- E a c h Yn£C, so each 6n€C, so 6€C. ntu)

Since sup T =T , 6=sup f w " T i + ^ . T A is regular in L[a] so n€o) L[a]^=cf (6) =x. + . But the covering lemma over K 1 holds in L[a]; and 6€C, so 6 is inaccessible in K*. So L [ a ] ^ = c f (6) = T . + . But since y>T i , <5>Ti+aj. So <5>(T^ +a) ) KI . But by lemma 21.16 <5> ( ^ + a ) ) L [ a ] . Contradiction!

a

What we have shown is that after a certain point the indiscernibles for L[a] arise from those for K1 by simply taking every 8th. It would be nice if we could find some significance for the ordinal 3 as a measure of complexity of a. It would also be nice to give an example in which $>1, but this involves a deep forcing construction. For example, if a is generic over L and 0

exists then 8=1.

In appendix II this result is applied in descriptive set theory. Exercises 1. Suppose all uncountable cardinals are singular. Show (in ZF) that there is a p-model. 2. Show that if there is a p-model then there is y such that for all singular cardinals 8 with 8>Y (8 + ) K <8t

185 ssumption that O 3. Formulate a covering lemma on the assumption 0

(the

smallest triple mouse) does not exist. 4. Suppose a

exists, acoj, K=K

but L[a]|=there is no p-model. If

we let M be the minimal p-model, show that the results lemma 21.16lemma 21.22 still hold. 5. GCH *> SCH

186 PART SIX: LARGER CORE MODELS

In this last part we briefly review what is known of core models beyond K. Much, it turns out, is known about models and mice and very little about covering lemmas and SCH. In the £inal chapter we discuss the present state of knowledge about SCH. It was once thought that large cardinal axioms might settle the GCH. This has turned out not to be the case: powers of regular cardinals do as they please, irrespective of large cardinals. On the other hand it looks possible that large cardinals might partially settle SCH (partially because we only get relative consistency results from the cardinals). The division of large cardinals into two camps may mark a fundamental change in the nature of cardinality axioms; or it may say no more than that Prikry sequences and fine structure do not go together.

187 22: COHERENT SEQUENCES

Exercise 3 of chapter 21 challenged the reader to formulate the covering lemma on the assumption that 0

does not exist. This could

be generalised still further to any finite number of t. In particular failure of SCH gives an inner model with n measurable cardinals for any n. Let us forget about SCH until chapter 24 and concentrate on increasing the number of measurable cardinals. How far can we go? A proper class of measurables might seem to be the limit; but actually that is a relatively weak assumption (relative, that is, to the cardinals some people use. For all we know, of course, all these cardinals could be moonshine and the existence of a p-model equivalent to that most powerful of large cardinal axioms,0*0). Suppose that at some point K there were two different normal measures. Indeed, suppose that U,V are normal measures on K and that if j:V-*-M then U6M. This implies U*V by chapter 5 exercise 1. M|=K measurable, so {y
Suppose U,V are normal measures on K . Let j:V-+ M. Then

U
A still stronger assumption would be the existence of three measures U,V,W with U
Lemma 22.2. if u M*. Define a: j, (K) +j * (K) by a(jx(f)(K))=j*(f)(K)

(f:K+K).

(22.1)

Then j1(f) (K)=j 1 (f) (K) ~ U : f U)=f ' (£) >eu

(22.2)

~ j*(f) (K)=j*(f) (K) . So a is well-defined. Similarly a is one-one and order-preserving. Finally K K C M 2 S O a is surjective. So

j,(K)=j*(K).

By lemma 5.18 M2|= j * (K) < (2K) + , and M2|=j2 ( K ) ^ ( 2 K ) + . So

32M>

188

So < is irreflexive and well-founded. Lemma 2 2.3.

< is transitive.

Proof: Suppose U
(XCK)

(22.3)

j ( X ) f l K G j (U) ( K )

Let j':V-> M 1 . Then V€Mf . Without loss of generality U : K + H W

S O U6M1 , K

hence U6M 1 . That is,U<W.

•

In general < is not connected, but we shall restrict ourselves to models where it is. There is another difficulty: if U
sequence

of measures

provided:

2

(i)

dom(U)=A, where bf=pn and €A A 3 '< 3 => < a , $ ')€A (call

(ii)

A A) .

U(OL,$) is a normal measure on a(all< a , 3 >€A; .

(Hi) «

2 2 . 4 . Li is a coherent

Suppose

3 ' < 3 ; and for

j:V+n,

o

,M. Then

ufa,p;

for

all

3 ' , letting

_ U~j(li),



€ A <=»

3 ' < 3U>a,3 ')=U(OL,$ ' ) .

So < a , 3 ' >,< a , 3 >€A A 3'<3 => U ( a , 3 ' )
I t can be shown t h a t

normal m e a s u r e i n L[U] i s of t h e form (J(a,3) i n L [ U ] ) , so < i s i n d e e d a w e l l - o r d e r , Definition

22.5.

Lemma 2 2 . 6 . Proof:

every coherent

and L[U]|=GCH.

oU(a;={3:< a, 3 >€AU}.

Suppose LI is coherent

Suppose

(where U i s

3 < o

U

( a ) .

in L[U]. Then L[U]\=o (0L)<0. .

L e tj : L [ U ] + , . , Q . L [ U ] . U v Ot , p )

S o f o r 3 ' < 3

U (a, 3 ') €PP(a) PIL[U] . Now given XcP(a), X€L[U] there is f such that X=j(f)(cO

and f(y)cP(y), ally. So L[ U]|=PP (a) flL[f f<_2a. Hence

<

•

L[U]|=f£2 ? and L[U](=GCH so L[U]|=3
Now why is this? Recall that, although when given an embedding j:V-> M we may always define U={X:K€j(X)}, a normal measure on K ,

189 yet if we let j :V-*Mf we shall not always have j = jrj« So the restriction on the number of normal measures at K is not a limitation on the number of elementary embeddings with critical point K. M 1 is the minimal model with a map j

such that j :V-> M 1 :

other models and maps may be too "big" to be coded by a normal measure. What we need is a more powerful mechanism for representation of embeddings. It may be noted that if j r V ^ M with K critical then

J(K)<(2K)+.

Since U j (K)<J (K) this is another form of the restriction in lemma 22.6, but it is a more useful starting point for K + generalisation. How could we possibly get J(K)_>(2 ) ? For some 0 we shall allow ordinals less than 0 to be us£d to generate M; that is, instead of M={ j (f) (K) : f: K->V} we shall allow M={j(f)(K,a):f:Kx[K]n->V A a€[0] n }. Then the restriction becomes K = + J(K)<(2

.0)

which can be made as large as we please by increasing

0. The representation of the embedding follows naturally from this for if we are to get a unique "ultrapower" M then properties of j(f)(K,a) must be obtainable in V from the representation. Instead of X6U *> K£J (X)

(22.4)

we shall require o < K,a >ej(X)

X£E

(22.5)

where a € [ 0 ] n , say, and X c K x [ K ] n . Now for the definition: it should be compared with definition 5.11. In this case we simplify by ignoring the possibility of non well-foundedness. Definition

2 2 . 7 . Suppose E=( E :a&[B]

there are j,M such (i)

). E is a B-extender

on K provided

that

that:

j:V+eM;

(ii)

K i s the critical

(Hi)

M={j(f) (K,a):

(iv)

for /2€O3, a£[Q]n,

point

of j and j(K)>B;

for some n f:KX[K]n->V and x€P(KX[K]n)

X£E ** <

In chapter 5 we obtained an alternative characterisation that worked generally and had the additional advantage that when we were dealing with models of ZFC the model M would be well-founded. In this case things are less neat. There does not seem to be any reasonable equivalent for definition 22.7, even in ZFC. In exercise 1 a characterisation for all but the requirement that M be well-founded is given. Note that if j',M' also satisfy the definition then the map

190 a:M->M' defined by a ( j (f) (K,a) ) =j ' ( f) (K, a) is an isomorphism. So j,M are uniquely determined by E. Definition

22.8.

j:V-+ M means that j ,M are as in definition 22.7.

Next we consider the generalisation of coherent sequences. Here we encounter a difficulty. In applications of coherence as in definition 22.4 we know that if we take an ultrapower by U(a,3) we shall get a model L[U] where H(al,3')=(i (a1,3') whenever a'€A

and for some 3>a €A . Then the extenders at a will

upset those at a': are we to demand that extenders E(a',B) are fixed by j:V-*v,

,M? Actually the answer is simple (no!) but the

resulting technical difficulties are substantial. Let us, therefore, restrict ourselves to sequences where the difficulty does not arise until chapter 24. It is helpful to insist that E(a,3) be a 3-extender on a. It is easy to change a 3-extender into a 3'-extender for many other 3', so this is not very restrictive. E will no longer enumerate all the extenders in L[E] but that was a lost cause anyway. Definition

22.9.

E is a coherent sequence of extenders provided that

(i) dom (E;=A, where N=On2 and €A A 3 '< 3 •» < a , 3 *>€A (call A A ); (ii) Efa,3; is a ^-extender on a (all €A; (Hi)

Suppose j:V-+r/ OM. t (CLfp) r

;

_

Then for all 3' €A

« 3'<3; and for 3'<3

where E=j(E) (or

U j(EOv.) if E is a proper class); 1 i€0n (iv) a'GA => 3y$A. Definition 22.10. OE(OL)={$:( a,3 >GAE}. The connection between normal measures and extenders is not hard to see. K has an extender if and only if K is measurable; we must go on saying measurable though, because extendible means something quite different. Suppose U is a normal measure on K. Provided 0 < J ( K ) U can be turned into a 0-extender on K by setting X£E

«=* Gj(X) . Since a

letting j:V-»» M M={ j (f) (K) : f: K + V } , certainly M={ j (f) (K ,a) : for some n f:Kx[ K ] n +V and ae[9] n }. So E is a 0-extender. If U is a coherent sequence then if j:V->n/

D .M

U {K, p;

certainly j (K) >3 by lemma 22.2, so —

coherent sequences of measures can be turned into coherent sequences

191 of extenders. Conversely, given a 8-extender E, we define U by X€U_

« Xx{0}€E..

(22.6)

(p

£J

1

1

Suppose j:V+ M, j':V+ M . In this case M and M are not necessarily E U E the same. They will be equal provided M={j(f) (K) : f: K-»-V} which will be true provided 6c{ j ' (f) (K) : f : K + K } = J ' (K) . If 6_
obviously true, and for quite a way beyond K + 1 . By chapter 5 eexercise xercise 3 3 K K +==((K K +)) M so t h e y w i l l be e q u a l p r o v i d e d 0<_K + 1 : and f o r quite a bit beyond. On the other hand j 1 ( K ) < K

S O 0=K

cannot possibly give

equality (assuming GCH). In general the equivalence stops somewhere between K

and K

The remainder of this chapter summarises without proof the equivalents of the main results in parts two and three for coherent sequences. It should be stressed that this is done just to show what can be done, and is not a full discussion. Definition

2 2 . 1 1 . Let 0 be the sentence asserting that E is a coherent sequence

of extenders (see exercise 1). A standard model of R +0 is called a premouse.

Remember the complexities in the discussion of 0

in chapter 13

caused by the fact that we had to keep track of which of the two normal measures we were iterating by. Naturally the complexities are much greater in the present case. Definition 22.12. Supposefe=«K^v^.-cKP. Then << V a e ^ l ' *

V

an iterated ultrapower of M_ with index k provided

a) v*, each M

(ii)

(Hi) (iv)

—(X

is a premouse of the form (M , € ,E ,A >; OL M

CL

QL

<<^> a € ^ I / a < e € ^ 1 > i« a commutative system; €A a and if .M n:M ->_ a a a a + 1 - a E (K , v )-u+l

(v) if X is a limit

(&<£,);

then < M, ,< 7T ,> ., > is a direct —A

OtA Ot^A

Remember: j:M-> M'must be r e d e f i n e d

limit

of

f o r non w e l l - f o u n d e d

structures

for t h i s discussion to make sense. Each M i s calldd an i t e r a t e of M;the TT g are called

iteration

maps. Definition 22.13.

M_ is iterable provided each of its iterates

is well-founded.

192 Suppose M i s an i t e r a b l e premouse. For s i m p l i c i t y suppose N=0. Lemma 2 2 . 1 4 .

If N_ is an iterate

of M_ with iteration

map TT and O:M+ N_ then for

all £&>nM (Compare lemma 8.19). I t follows t h a t TT i s the unique i t e r a t i o n map, although i t s index need not be unique. Definition

2 2 . 1 5 . 7T

i s the unique iteration

map from M to N.

MPj

Definition

22.16.

—

Iterable

—

premice M_,N_ are comparable provided

M=N_ or for

some a M_=JN or N=JM. Lemma 2 2 . 1 7 . If M_, n_ are comparable premice provided

they are

(Compare lemma

iterates,

8.18).

Lemma 2 2 . 1 8 . If M_,N_ have a common iterate iterate

then M_,N_ have comparable

sets.

Q_ and mg(TT

)^rng('K

) then N_ is an

of M_.

This r e s u l t , which generalises an argument in lemma 10.16, i s not true i f clause Definition

(iv) in d e f i n i t i o n

2 2 . 1 9 . An iterated

ultrapower

fe=<.-a<£> is normal provided normal iterate

22.9 i s omitted. ((M > ,_». , , ( IT n) ^or.r , > with —a aec,-fi a p a
that a<8<£ =» K «

g

index

. Each M is said to be a

of M.

Lemma 2 2 . 2 0 .

Suppose that M_ is an iterable

of N_. Then N_ is a normal iterate

premouse and that

/V is an

iterate

of M_, and the normal index for the iteration

is

unique. Lemma 2 2 . 2 1 . all

If x<^ M_ and K f K j g , and if T\:N=M\X with N=J^ and M_=J^ then for

y
.

This is a sort of condensation lemma. Next we state a criterion for iterability. Definition

2 2 . 2 2 . An extender

whenever Kb.:i£u>) is a sequence X .££

, all i€o), then there


for all i€u).

E=( E :a£(Q] with b.£[Q]

> is countably

complete

if and only

(all i€b}) and <X.:i€w> is such

is T and an order-preserving

map 5: U b .-*On such l€uj

that that

193 Lemma 2 2 . 2 3 . countably

Suppose N=J is a premouse and for all €A

complete.

Then N_ is

E(K,V)

is

iterable.

As we have already hinted, the situation is not quite like it was for a single measure: it is not necessarily the case that L[E](=E(a,3) is countably complete. Finally a word about generalised mice. Iterable premice have already been introduced. There is a bit of a problem in getting further. Our fine-structure enabled us to extend a map of N p to H to some TT:N->

M with H=M P , 7r=>7r and p'=Tf(p) on condition that N be

p-sound. Since mice N with u)pN2lK ( K critical) always have this property for some p the definition of mice was fairly simple. But if N=J

is an acceptable iterable premouse we may have K , K ' with

(joa>K>a)p > K ' , O ( K ) , O ( K ' ) > 0 . But then how are we to define ^E(K',O)N"

S a

^?

Actually generalised fine-structure does give an extension of embeddings result without the soundness restriction. And applying this to mice with extenders it turns out that mouse iteration, like ordinary iteration, can ±>e "normalised", so that the problem did riot really matter; we can always iterate below the projectum before we iterate above it. Our worries, though, are by no means over. As with simple mice, the main result that we need is that iterable premice are acceptable. Consulting the proof of lemma 11.2 4 the reader will see that this is essentially a question of indiscernibles; so we must get a theory of indiscernibles for generalised mice. Just as when N is a premouse with a single N measure and M=J g , o)3GN, and M is a mouse then M has indiscernibles of order-type a multiple of a)W, so in the general case M has a smoothness property giving nice indiscernibles. The reason why we supressed the ordinal arithmetic in the proof of lemma 7.17 was to give something of the flavour of this more general case. Our first major obstacle is the fact that the crucial corollary 11.21 can no longer be proved. It is possible that top = K , O ( K ) > 0 but H * H . This may occur when O ( K ) is a limit N K K F F ordinal. Indeed, if N is such that N=J , , o (K)=O) but for all X
by a structure H with P(K)flN=E (H) n

0 P ( K ) , in which our usual results hold. H is called a hydra; we may

194 have to repeat the process any finite number of times. The cardinal principle is that the hierarchy may only grow at a speed that enables the projecturn to act as an index of formation of new subsets of given ordinals; if it cannot do this the hierarchy must be slowed down by dividing up levels. Also irritating, though of less interest, is the case N = J .n E N where for some K O (K)>U)CX. Then the usual rule that N=rud r i._ (J ) fc | N a breaks down because some extenders are not available in N. All these problems can be solved and we can prove Lemma 22.24.

Iterable premice are acceptable.

But all the applications are cluttered up with cases about hydras and so forth; life is never quite the same again. Exercises 1. Formulate definition 22.7 without the requirement that M tie wellfounded. Show that with this definition E=< E :a€[6] is a a 9-extender on K if and only if (i) E a cP( K x[ K ] n ) where a € [ 0 ] n . Let a € [ 0 ] n , b € [ 9 ] m , acb. Then given u € [ K ] m , u a b denotes {u.

...u, },

n

where

u = { u , . . . u }, j. m

XcKx[K]n

For

X€Ea

(ii) (iii) i.e.

some

Xab

Ea

~ is

(a)

for

(b)

X,Y6E

all

\{s j

AC.Hi

a **

u,<...
X a

a={b,

l n ...b, !}. n., n

{<£,u>€KX[K]m:<£,uab>£X}.

denotes a b

b = { b , . . . b }, b , < . . . < b , ± m _ L m

€Eb.

non-principal

y
u£[<]

<* XflY€E \ N ^ I K. J

a

n

ultrafilter

{}$E

P(Kx[K]n)

(aG[9]n)

; a

;

^*A) C JCJ

on

•

a a ( i v ) S u p p o s e f : K x [ K ] n - * K a n d {< £ , u > € K X [ K ] n C € K x [ K ] : f ( ^ , u ) = c ) € E . a ( v ) L e t f : K x [ K ] - > K . S u p p o s e {< £ , u > € K X [ K ]

: f (£ , u ) < ^ } € E a . : f (^,u)

Then

for

<max ( u ) + 1 } € E

.

a Then there is v<max(a)+l such that, letting m=n if v€a, n+1 otherwise, defining f a (^,u)=f(^,ua ) , and supposing that aL){v} = ={a'...a'} with a!=v, u={u,...u } (a'<...
m

l

_.,,f.,il

m

l

m

l

m

2. Show that there is a sentence a such that L[E]|=a » E is coherent. 3. If E is coherent in L[E]show that L[E]has a A^ well-order of P(w) 4. Show that if o (K)=n+1 then K is a limit of K 1 with o (K')=n. Can this be extended to infinite n? What if o (<)=0n?

195 23:

STRONG CARDINALS

Definition

2 3 . 1 . A cardinal K is ^-strong provided that there is an

embedding j:V+ M with K critical such that ?($)<=M.

Definition

2 3 . 2 . A cardinal K is strong provided that it is $-strong for all 3

For $<_K K is 3-strong if and only if K is measurable. For 3 > K it turns out that

3-strong is connected with the existence of

extenders at K. This chapter will show that models of the form L[E] are inner models for 3-strong cardinals; that L[E]|=GCH (this follows from lemma 22.24, of course, but we shall try to supply some details in this case); and then we shall solve the equation x/L[E]=O # /L. But first consider iterability of L[E]. If (J is a coherent sequence of measures in L[U] then L[U] is always iterable, but this is not true for extenders. If N=J E is an iterable premouse let < k > enumerate A in — —a a a s c e n d i n g o r d e r (< K, v ><< K ' , v ' > <=* K A/ o Q > be t h e i t e r a t i o n of N w i t h i n d e x < k' > where - a a < 6 ' a3 a<3<6 , *+1 T + x n fe'=7T (fe ) . C a l l NQ N + . L e t N°=N, N° =(N°) and l e t NA be t h e a Oa or ~®n~~ _ - _ ~ a d i r e c t l i m i t o f < < N > a < x , < T r ^ B > a < p < x >, (X a l i m i t ) . Each N i s an iterate

of N d e f i n a b l e

i n L [ E ] ; and each i t e r a t e of N i s embeddable

i n some N a . So i f L[E](="V i s i t e r a b l e " Lemma 2 3 . 3 .

(iii) Proof:

M then M is

Every E(K,V) is countably complete in

(i)=>(ii)

is

well-founded;

L[E].

trivial.

(ii) . L e t X. GE 1

, all i€a), w h e r e

1

a ^ = j (a.±) . L e t R b e

E=E(K,V),

x

f o r a l l i € o ) < K , a . >€ j ( X . ) .

£j

the

set

1

of

all



such

that

f: U a!-K)n i s o r d e r - p r e s e r v i n g , a n d < K , f " a ! > € j ( X . ) . i < n

<X.:i€aj>

a

i a n d €L[E] . L e tj : L [ E l ^ L f E] . T h e n Let

equivalent

iterable;

For all €A , if j:L[E]^

Suppose

iterable.

Suppose E is coherent in L[E] . The following are

(i) L[E] is (ii)

then L[E] i s

X

X

1

nGw,

Given ,

-1 < g , m > € R < f , n ><_< g , m > «• n > m a n d g = f I U a ! . L e t t i n g f = ( j | U a . ) R n i < mx i < _ < f , n); s o < R , < > i sn o t w e l l - f o u n d e d i n L [ E ] a n d h e n c e n+i K n K

196 not in L [ ¥ ] . Say <:n€a)> is a descending < sequence in L["E] . n K Let g= U g . Then g: U a!-K)n is order-preserving and for all i€o) n€w n i€a) 1 < K,g"a| >€j(X±) . So (3T<J (K) ) (3g: U j(a.)-K)n order-preserving) (ViEw) i€a)

Applying j

""

(23.1)

«T,g"j(a.) >€j(X.)) .

-1

( 3 T < K ) (3g: U a.-KDn order-preserving) (Vi€co) (< T/gIIaj. >€Xi) . (23.2) (iii)=>(i) by lemma 22.2 3 and the remark preceding this lemma.• Suppose that L[E] is an iterable premouse. Lemma 2 3.4. L[E]\=GCH Proof: Work in LL[[ E E ]].. Given an infinite cardinal a we must show 2 a =a . We use induction on infinite a. _ Claim 1 For all aca there is an iterable premouse N=J g such that (a) a,a£N, o E ( a ) = 0 , N|=ZF~; (b) every x in N is N-definable from aU{p} (some pea, p€N) (c)

for x
_

(i) o E ( x ) < a + -> O E ( T ) = O E ( T ) ;

(ii) (iii)

o E ( x ) > a + => o E ( T ) > ( a + ) N ; V
-» E ( T , V ) = E ( T , V ) .

Proof: We may assume that o (a)=0, since otherwise it is sufficient to prove the result in L[E'] where TT:L[ E]-»v ,

* L [ E ' ] . Pick a

regular y>a with a^J ; we may assume also that

Y

U P(6)cJ . Let pea

_ = _ 5
code a, U ?(&) (this is possible as U P(6)=a). Let X -< J be least Y 6
•(Claim 2)

Now given aca there is N as in claim 1 with aGN. And N max(a ,sup o (x))
in Q 1 ,

197 Suppose K<(a+)L[

L e m m a 2 3.5.

]

,,L^-


Then

P«X)()L[E]<=L[E] .

Proof: (in L[E]) Suppose aca. Then as in lemma 23.4 pick y regular with y>o (K) and a€J and E,

letting N=J O

U P(6)<=J . With p,N,a as before a€N, and

F1 Y + 6 < a - E | Y , o (K)
— —P "E E" E" —. J . Then N and M=J have comparable i t e r a t e s y t i , K / V ; y y

—

TT:J -*v , 7T

=n

I

NcJ'^ ' "MO l ^

= i d

v

I *

I f

Q'^Q t h e n

TT

,7T7Tcr:N+

Q,Q' with

Q non cofinally;

Hence a€M.

s o Q<=p_\

_

Now f o r y < a , s u p p o s i n g

a=Tr(f) ( K , b ) , b € [ v ] n , f € J

that

y € a » y € i ( f ) (K,±>) <=> {<

^

u

)

(23.2) :

u

.

{

}

witji notation as in chapter 2 2 exercise 1. Hence a€J . • C o r o l l a r y 2 3 . 6 . Suppose K<(u')L[

]


C o r o l l a r y 2 3 . 7 . Suppose o (K)=On. Then L[E]\=K is strong.

Our aim is to get converses to these corollaries. Define inductively a sequence E

as follows: Suppose T , V < K and

1

E ( T , V ) is defined for all ( T ^ V ) with T ' < T or T ' = T and v'€A K and E has been specified F ^ or it has been decided that < T ' , V ' >^A K ) . If there is a coherent sequence E* of countably complete extenders with E * ( T ' / V I ) = E ( T ' , V I ) E* (or both undefined) for T ' < T or T ' = T and v'€A , then let €A K and E ( T , V ) = E * ( T , V ) for some such E*. Otherwise < T,V > { A E K .

Note that this construction gives priority to existing measurables: we cannot have a T'-extender on x and an extender on x1 if x<x': we add the x'-extender on x rather than the extender on x. E

K

L e m m a 2 3.8. Suppose K is 8-strong, &>2 is a cardinal. Suppose o KfxJ + K (all X < K ; . Then there is a coherent E with E(T,V)=E (T,V) (all X < K ; and o (K)>$+.

Proof: Suppose not. Let j:V+ M with K critical and P(3)£M. Let E=j(E^); then

E(T,V)=E ( T , v ) , all X < K . Suppose 6=o ( K ) £ $ + . Define a

0-extender on K by X€E a « €j(X) where X c K x [ K ] n , a £ [ 0 ] n . Let X
(23.3)

+

AE = A E U { < K , 0 > } ,

E + ( X , V ) = E ( T , V ) if

or X = K and v<0; E ( K , 0 ) = E .

Claim E

is coherent and E is countably closed.

Proof: Let a:V-> M'. Then there is a map a':M'->a 1 (a(f)(K,a))=j(f)(K f a); so M 1 is well-founded

M defined by (as M i s ) .

198 E'=o(E+).

Let

E'

E' ( K ) = 0 S O 0 > O ( K ) . On the other hand F~ E(K,v)6rng(a'), all v<0, so G=o ( K ) . Also a'(E 1 (K,V))=E'(K,v) (v<0) so E is coherent. E is countably complete by lemma 2 3.3. n(Claim) Now o'(o

E

(K))=O

If 0=3 we are done. So suppose 0<3 . Then E£M, since it can be coded as a subset of 3(3_>2K). So M|=E can be properly extended by a 0-extender on K , where (23.4) 0=OE(K).

Applying j E

can be properly extended by a 0'-extender on K ' , where(23.5) E 0'=o K ( K ' ) (some K ' < K ) But this contradicts the definition of E . D K

K Corollary

2 3 . 9 . Suppose K is

$-strong,$>2

inner model L[E] which is an iterable Proof: o

By lemma 23 . 8 t h i s

K(T)=K.

E'=j(E

K

),

Now by

Let j:V+ Mwith then

M|=o

E

'(T)>3

l e t <<M >

E'(T,O). EI

o

+

+

is a cardinal.

Then there is an

premouse with o (K)>$ .

is true

provided

P(3)C=M,K

VT
critical.

K(T)
So J ( K ) > 3

,< 7T „ >

< e

^

0

> be the iterated

L e t M =MK , E"=7r0K ( E ' ) . As K > T , K r e g u l a r ,

( T ) >3 , M ' | = o

SO suppose

. Hence

if

.

1

EI1

+

( K ) >TT_

+

(3 ) = 3

+

ultrapower of M TTO(£T)=K.

. Thus L[E"] i s t h e d e s i r e d

Since model.a

OK If K is a strong cardinal then corollary 23.9 gives, for each 3/ an L[E] with o ( K ) > 3 . To get L[E] with o (K)=On requires a bit of work. We may assume V T < K O

( T ) < K : for if o ( T ) = K let U={X€P(K):K€j(X)} and

let < _^ ,< TT o) ocr^ > be the iterated ultrapower of L[E] by U. —a aGOn a3 a£3€0n N f Letting E =TT (E ) we have o a(x)=7r ( K ) . Hence letting N= U H a a U a K U a a|On K a N(=ZFC (just as in earlier arguments) and if E= U E then o (i)=On. a€On a Just as in corollary 23.9 we deduce that there is an inner model with o (K)=On. We shall return to this argument in a bit. Lemma 2 3.10.

Suppose (i) E,E' are coherent;

(ii) T
V ECK,V;-E' (K,V)

A O (K)=o ' (K) ;

(iv) T>K => O (T)=O ' (T)=0.

Then L[E']OE=L[E]r\E' (hence L[E]=L[E' ]) . F

F F'

Proof: Take 6 regular, <S>o (K) . Take X -
(where EflL[ E' ] c j E ' , similarly for E% ,E) and X = K + . Let

TT:N=X, N transitive. Let J = T T ( J - ) ,

J E ' = T T ( J - 1 ) . N O W since P ( K ) D ( J U

UJ E ')cX and TT | J--: J^->eJE, P(K)flJEcj|: similarly for E',E"'. Thus for

199 a
. Hence j f = j |

, and

so

(23.5)

EDL[E' ]=E =Tr(E'nj|)

= ElflL[E].

Lemma 2 3 . 1 1 . Suppose K is strong. iterable

Then there is an inner model L[€] that is an

premouse with o (K)=On.

Proof: We may, as we said, assume o K(T)
f:K++-*V and Vy
+

be as in lemma 2 3 . 8 . Let B=

f(y) is a yextender on K } . Then B is a set.

For fGB let b f ={3>K + + :(Vy
is a proper class, some b,. must be a proper class. But then by

-P CD ^

lemma

Q D

2 3 . 1 0 $ , 3 ' € b

f

a n d$ < 3 ' i m p l i e s

p

E flL[E

D p

i

p

| A

& P

o

]= ( E

i P

r

|A

r c

P

)DL[E

o P

].

n

Then letting E= U E 3€b f

gives the required E.

•

PISTOLS Definition

2 3 . 1 2 . A pistol

is a structure

N_=( N,€,E,U > such that

(a)E is coherent in N_; (b) N^R+; (c) U is a normal measure on K(say) in N_; (d) for some T o ("r;=K; (e) On =K+U); (f) N is iterable

(i.e.

combination of the two is

every iterate

by an extender in E or by U or by a

transitive).

Pistols are like sharps and daggers. It can be shown that if M,N are pistols then M, N have comparable iterates, Q,£>'say. Q/0 1 will be pistols. But Q6Q 1 and Q'GQ contradict (e) so Q=O'. Take some pistol N and let X=h (0) . Then let TT:M=N|X, with M transitive. Since X is cofinal, TT:M-> pistol then since N, N

1

N. But if N 1 were some other

have a common iterate we may deduce that the

same M is obtained. M is the unique pistol that may be embedded in any

pistol by a I,-preserving embedding (in fact every pistol is an

iterate of M, but this is not easy to prove. Pistols are acceptable and for any pistol N p =1, P ^ ^ just as for sharps). Definition

23.13.

Let T be as in chapter 13. Then 0 = AT. M

IF # + So 0 co3. (This is not really consistent with 0 and 0 , which were only interconstructible with the corresponding master code. But after

200 a l l , t h a t only complicated our d e s c r i p t i v e set theory, and we are free

to make such d e f i n i t i o n s As with 0 , 0

M^O

11

as we p l e a s e ) .

i s A_ and if

a=0 , a€M, M an inner model, then

.

The following standard lemma can be proved Lemma 2 3 . 1 4 . (i) 0 (ii) class C

The following

are equivalent

exists;

There is an inner model L[E], an iterable

premouse, with o (K)=0n and a

satisfying: (a) C contains all uncountable cardinals

above K;

(b) Cfie has order-type

Q>K) ;

6 (all

cardinals

(c) if 9 is regular and greater than K then Cf)0 is closed unbounded in 0 (d) C are generating £ indiscernibles (Hi) with critical Proof:

for (L[E],y)

;

There is an inner model L[E] with o (T)=0n and non-trivial

j:L[E]-+ L[E]

point above K.

(i)=>(ii). L e t N be t h e minimal p i s t o l ,

N=< N, € , E,U >; l e t t h e

iterated

u l t r a p o w e r of N by U b e <a€On,< ^

o

L e t N = a n d l e t 0 n M =K + W . Then i f we l e t

(T)=K.

—a

a

a

a

N

U E a€0n a

Suppose

a

a

F E=

>a<3eOn>•

L[E]|=o (K)=0n. Let C = { K

:a€0n}. Only (d) presents any

a

difficulty, and that is proved just as in chapter 12. (ii)=>(iii) Any order-preserving map of C to C generates an embedding. (iii)=>(i)

x-minimality and T-maximality both follow from

Claim Suppose Tr:L[¥]^eL[ E] , o E ( x) =o E (T) =0n, 7r|T=id|T. Then L[F]=L[E]. Proof: For T ' < T "E (T ' , V) =E (T ' , V) is clear. Since TT(T)=T, 7r|T+=id|T+ so P ( T ) nL[E]cL[iE] . Hence F ( T , v) =E (T , v) (all v) .

•(Claim)

Hence there is U a normal measure on some K > T with <J +,U> amenable and such that L[E] is iterable by U. From this we may as usual construct a pistol.

n

Actually in any model L[E] with o (K)=0n E is wholly determined by E(T,v) with T < K . Pistols are special cases of more general mice (just as daggers are special cases of double mice), and these mice will be discussed in the next chapter. With this more powerful apparatus we see that the analogy with daggers is very close, and that we could write "every L[E] with o (K)=0n" in (ii) and (iii) of lemma 23.14. Exercise 1. Show that if there is a strong cardinal and A is a set then V*L[A].

201 24: THE GREAT BLUE YONDER

This final chapter considers two questions. Firstly, what comes "after" strong cardinals; and secondly, what about SCH? BEYOND STRONG CARDINALS Suppose we were to lift the assumption on coherent sequences that K0 => O ( K ) < K ' . Then we should get premice of which the pistols in chapter 23 were primitive forms. We should have to change the definition of coherence, although this is less of a nuisance than it might be because we can (at least for our immediate purposes) safely assume that K < K ' A K ' < O ( K ) => O ( K ' ) < O ( K ) , The main new feature we have to deal with is the "virtual extender" that arises as follows: let N=J Ml

7r:N^r7N' and let K < V < T T ( K ) . Then if E=E —

u—

K

, be a pistol. Let +

1

(T,V) we may define Tr • : N'-»-_M • —

hi—

in the usual way; but it is more natural to deal with the virtual extension TT":N->

M". Now obviously there is no hope of arranging

all iterations "normally", that is, with increasing indices. In fact iterations are nested (the depth of TT:N-> M beinq one greater than that of a) and each nest index is normal. This complicates the argument appallingly. The other restriction we might try to lift was that if E is a 0-extender on K and TT:V+ M then 6
—

than the one just discussed but it is essential to the definition of extender: for if X=TT (f) (K , a) and f:Kx[K]n-*V then TT (f) : TT (K) X [ TT (K) ]n+V so .any a beyond TT(K) could not be in dom(f). Consider what happens if there is a 0-extender with TT(K)=0. Actually this is trivially attained: we need to add that E must be coherent with o (K)>0 and TT(K)=0 where TT:L[E]->

, Q.L[¥]. E(K,0) fc(K,o;

is called a TT (K) -extender.

In L["E] O E ( K ) = T T ( K ) : and o E (IT ( K) ) =ir (o E (K) ) >0 so c e r t a i n l y 011

e x i s t s . A c t u a l l y t h e assumption i s much s t r o n g e r than 0 . Definition that

2 4 . 1 . K is

P(Y)<=M for

all

superstrong

provided

there

is

j:V +M with

K critical

such

y<j(K).

I t can be shown t h a t sequences E with TT (K) - e x t e n d e r s do f o r s u p e r s t r o n g c a r d i n a l s what sequences E with o («)=0n do f o r s t r o n g

202 cardinals. That is, if E has a TT(K) -extender then L[E]|=K

superstrong

and if K is superstrong then there is an E with a IT (K)-extender. It follows that although a superstrong cardinal gives an inner model with a strong cardinal, yet by chapter 2 3 exercise 1 there could be a model with a superstrong but no strong cardinal. This is the limit of the use we may make if extenders as so far defined. However, other considerations motivate the following definition. Definition 2 4 . 2 .

K is $-supercompact provided there is j:V-> M with K critical

and M 'cM. CM. K is supercompact provided K is $-supercompact for all $.

3-supercompact implies 3-strong, of course. supercompact embedding. For cofinal in

(J(K)+)M.

given j ( f ) ( K , a ) < ( j ( K ) + ) M ,

For

But

since M

and M)=J(K) A

+

C M J"K EM.

cM. Then J " K

S O M ^ c f ( j ( K ) + ) £ K + . But

j(f)(K,a)<j(y).

( K + ) M < j ( K ) < ( j ( K )+)

M

is regular. Contradiction!

final note on generalisations: K is n-superstrong if

J°(K)=K,

is

f:KX[K]n+K+

letting y=sup f " K x [ K ] n

without l o s s + o f generality and K

No extender can code a

suppose j:V-*- M and M K

j n + 1 ( K ) = j ( j n ( K ) ) ) there is j:V+ e M with K critical

P(j n (K))cM. And

(letting and

K is co-superstrong provided there is j:V-> M with P(X)cM where X=sup J R ( K ) .

with K critical and

But

Lemma 2 4 . 3 . K is not 0)-superstrong.

Proof: Claim Suppose X^w, X a cardinal, 2 =X ° . There exists a function f:XW->X such that whenever AcA, A=X and y<X then there is s€A W with f(s)=y. Proof: Let {:a<2^} enumerate {xcX:x=X}xX. Define s a

a

—

=

a

for a<2 v>

y

inductively so that s EA , s * s g (all $
let f(s)=y

if s=s ,0 otherwise. Given AcX, A=X and

y<X suppose A=A ,y=y . Then f(s )=y.

n(Claim)

Now suppose j:V^ M with X=sup j ( K ) , K critical and P(X)cM. n€o) * « j(X)=X, of course. Let G=j"X. So GEM. 2 =X °: for X=sup{ j (K) :nEco} . Hence .n -n 2X=2(n^a3j ( K ) ) = n 2 j ( K ) (24.1) £

IT X n£a)

(j

( K ) is measurable in M, hence a strong limit)

203 =2X. Let frA^+A be a s i n t h e c l a i m . So t h e r e i s

sCG^ w i t h

j(f)

L e t t ( n ) = j " * 1 ( s ( n ) ) . So s = j ( t ) . K€rng(j). This i s Corollary 24.4.

must have t h e same p r o p e r t y .

(j(f))(s)=K. Thus K=( j (f) ) ( j ( t ) ) =j (f ( t ) )

impossible.

so

D

There is no non-trivial

j:V+ V.

So there is a point where the sequence of large cardinal properties becomes inconsistent. SCH Given the existence of a supercompact cardinal the SCH may fail. Model 4 in appendix I explains why. Indeed SCH may fail in very simple forms: it is possible (given a suitable large cardinal) that 2 n = « n + 1 for all n<w but 2 u)=N axiom: at least as strong as 0 as 0 +

n

times

+2.

ZFC+-SCH is a large cardinal

by lemma 21.11. At least as strong

for any finite n. Beyond this difficulties arise.

We may no longer take K as the union of all mice in our generalised sense of mice. Consider just the case of double mice. Let M=J

. Are we to insist (supposing U a normal measure on K, V on

A) that u)p^
ensues. What we do is to fix a single measure U and only allow mice J

: then allow all such mice with a)p|[i<X. U is picked minimally (in

some sense). In general, just as we added a maximal sequence to get L[E] in chapter 23, now we add one

to get K[E]. If TT:K[E] + K[E]

is

non-

e trivial then as in lemma 2 3.8 use TT to get a new extender. The resulting sequence E

will be coherent in K[E ] (otherwise it

yields a mouse) but cannot coherently extend E(nothing can). So what has gone wrong? We must have violated the coherence condition by getting K < K ' , O ( K ) = K ' and o(K')>0. But then 0

exists! So perhaps

-SCH implies 011 exists? Perhaps. But not by the proof just given, for we have reckoned without our Prikry sequences. The covering lemma was proved (on the assumption -0 ) either over K or over L[U] or over L [ C ] . The more measures, the more Prikry sequences we need. And once we need an infinite number, we shall no longer be able to disregard finite

204 initial segments of C: this was essential to the proof of lemma 20.15. By careful choice of sequences we can get a covering lemma on the assumption that there is no inner model with a measurable that is a limit of measurables. Compared with the cardinals of chapter 2 3, that is not very far (such a cardinal need not even be of order 2 ) . The proofs that SCH may fail rely on an abundance of Prikry sequences given by large cardinals: and the question is; at what point do the Prikry sequences get out of control? If 0 does not exist can they be chosen neatly to preserve SCH? Alternatively is a strong cardinal sufficient for forcing proofs of independance of SCH? A related question concerns •, . If 0 does not exist then CK A A holds for singular A. If there is a supercompact below X it fails. As all that we need here is K|=a and (3 ) =3 for singular cardinals 3 this may be easier than the full covering lemma. A similar situation exists with the question whether all uncountable cardinals can be singular; see chapter 21 exercise 1. Unless the universe is so obliging as to exclude large cardinals that proliferate Prikry sequences overmuch, the best we can hope for is a precise boundary cardinal- an X cardinal - such that (a) if SCH fails then there is an inner model with an Xcardinal; (b) Consis(ZFC+there is an X-cardinal) => Consis(ZFC+-SCH). Exercise 1. Show that if L[E] is as in chapter 23 then L[E]|="there is no K that is K -supercompact".

205 APPENDIX I: SOME GENERIC MODELS

All of the following descriptions of models are of the vaguest kind, and are only intended as guides to the literature. The reader should consult Jech ([25]) for any unfamiliar definitions; also for full proofs. Before any models are given, observe one general point: forcing does not generate new large cardinal properties. Otherwise we should have relative consistency proofs ruled out by Go'del's second incompleteness theorem. More precisely, suppose that M is a generic extension of N; then K =K . For otherwise there is a mouse in K

but

not in K , N . Suppose M=N[G] where G is a generic ultrafilter on B. If X=B (in N) then B satisfies the (A + ) N -chain condition, so for all N-cardinals y>X y is a cardinal in M. Let < < V o € 0 n ' < * a 0 W c O n } b e the mouse iteration of N . If K is the critical point of N and ^ —a a M 9>X is regular in M then K = 0 . N O W K Q ^ , =0 so K Q J 1 is not a cardinal in M. But the K are indiscernibles in K and K =0 is a cardinal in a 0 N so K Q , , must be. Contradiction! For example, if M is a generic extension of L then 0 ^M. The above proof is easily adapted to show: (i) if M is a generic extension of K then M contains no p-model (ii) if M is a generic extension of L[U](a p-model) then o'^M. Hence SCH holds in every generic extension of L[U]. A further restriction on large cardinals in generic extensions is proved in appendix II. Now for the models. Model 1

cf (o)^) =w but

(u^)^..

This model was discovered by Namba([46]) A tree is a set T such that: (i) t€T +> teo)^; (ii) t€T A s=t|n -> s€T. T is called perfect provided it is non-empty and every t€T has N 2 extensions sz>t in T. The set P of conditions is the set of perfect trees: the ordering is inclusion. Let M be some model of ZFC+V=L. Let M[G] be a generic extension in which we are to define a cofinal subset of (aO

of order-type ai.

It is quite easily shown that there is a unique infinite path that goes through each tree in G, that is, an co-sequence fiuH-fu^)

such

206 that VT€G3s6T3ns=f |n. Also by genericity

f:o)->(o)2)M is cofinal. So

cf (o>2)=o>. The crux of the proof is the argument to show that u), is preserved in the extension. Model 2 K+(a) < a ) but K^K+(a) (a) which a is countable then Mf=K->-(a)

w

and M is an inner model in

. Now suppose K->(UJ,)

: so 0

exists. Let P be the conditions in L for collapsing a), to w; so P is countable, and hence there is a generic extension L[G] of L with ((JO,)=U) in L [ G ] . Since K-»-(a),) w certainly K->(O),) U , and since a), is countable in M=L[G], J+=K->(W^)
required. M

Model 3 K is measurable in an inner model M, H =H

and cf(K)=o).

This model was discovered by Prikry([48]). Take a model M w i t h K m e a s u r a b l e . We a r e g o i n g t o add a s e q u e n c e t o M. The s e t where U i s

of c o n d i t i o n s P i s

{<S,A>:S€[K]

W

Prikry

A A€U}

t h e n o r m a l m e a s u r e on K. < s , A > < < t , B > m e a n s :

(a) t=sna, some a; (b) AcB; (c) s^tcB. Let G be a generic filter on P. Let S={s:<s,A >€G, some A } . If <s,A>, < t,B >GG then either set or tcs so S is of order-type u). S C K , obviously, and by genericity S is cofinal. So cf (K)=QJ. S is indeed a Prikry sequence. For suppose X€U. Then {<s,A>:AcX} is dense in P: for given any €P < t,BPIX ><< t,B >. So there is <s,A>€G with AcX. Suppose < t,B >€G, <<s,A>. Then t^sCAcX. Hence S^scX, so S is a Prikry sequence for U. The difficult part of the proof is H =H . This cannot be proved by K-closure since P adds a new co-sequence. It follows in fact from a technical lemma: if a is a sentence of the forcing language and <s Q ,A 0 >€P then there is AcA Q , A€U such that <sQ,A>||-a or < s Q / A >||—a. A converse also holds: Mathias

([40]) showed that any Prikry

sequence C for U yields a generic ultrafilter G={<s,A>:scC A C C A } on P. Model 4 K measurable and 2 K > K . This model was discovered by Silver (see Menas [41]). Since GCH cannot fail for the first time at a measurable it is necessary not only to disrupt P ( K ) but also P(a) for many a
207 iterated forcing add a new subsets to a for all inaccessible a£K. The resulting model has 2 K =K , of course, but K may not be measurable. It turns out that provided that K was supercompact in the ground model we can obtain an elementary embedding of M[G] into an inner model with K critical, so that K remains measurable. Now take a Prikry extension of M[G], M[G][C] and obtain a model where cf(K)=co and 2 K > K + and H ^ [ G ] = H ^ [ G ] [ C ] . But since K is measurable in M[G] it is a strong limit in M[G], so for all y
208 APPENDIX II: ABSOLUTENESS RESULTS

This appendix gathers together some absoluteness properties for ZFC models that are not true in general for models of R. These are used to prove the £~-absoluteness of K (under certain conditions) and to give a negative result prophesied in chapter 16. The proofs in this area seem to me dreadfully indirect, but no doubt descriptive set theorists think the same about fine-structure. The absoluteness theorem in its most general form - lemma 4 below - is well-known, but I have not found it as stated here in the literature, so I have had to include a proof. It is a trivial modification of the proof sketched in Jech ([25]), based in turn on the proof of Barwise and Fisher (see [1]). All absoluteness results in this appendix stem from the absoluteness of well-foundedness; this is proved by using a wellknown consequence of the replacement axioms to turn the apparently II, property of well-foundedness into a E, one. Lemma 1

Suppose M is an inner model of ZF and E&M is a partial order. Then E

is well-founded if and only if f^=E is well-founded.

Proof: If E is well-founded then M|=E is well-founded. On the other hand, suppose M|=E is well-founded. Recall that P E (x) is defined inductively on well-founded E by p_ (x) =sup{pT-, (y) +1 :yEx} . So in M there is a function f whose range is a subset of On such that xEy => f(x)
D

The foundation of the absoluteness theorem is the following lemma: Lemma 2

Suppose M is an inner model of ZFC and QOi is a set of V3-sentences,

countable in M, in a language L •+ •*• with no function symbols. Suppose further EJQA,C

that 0 has a model N for which E which E

is

is well-founded.

Then 0 has a model N'€M for

well-founded.

Note: By an V3 s e n t e n c e we mean one which i s of t h e form Vx... . ,x m 3y, . . , y ij;(x, . . . x , y , . . . y ) where \\> i s q u a n t i f i e r - f r e e . Proof: I f 0 h a s a f i n i t e model we a r e done, so suppose i t does n o t . L e t < a, :k€u) >€M enumerate 0 . We may assume L_ •+ -*• i s c o u n t a b l e : i f K

Ji , A , C

<£, :k€o)> enumerates its constant symbols we may also assume that all

209 the constants of a, lie among c^.-.c,

, . Let P be the set of triples

such that, letting B* be some model of 0 with B * < K , E well-founded and p B*:B*-»-K

(i) k€o); (ii) B C K , B is a finite model of the type of L E + (iii) f:B->K is such that bEb 1 =* f(b)
''

A partial order < of P is defined by <£ provided: (i) k>k'; (ii) B 1 is a substructure of the reduct of B to L > ; — — i A , c^ . . . c, , _, (iii) f'cf; (iv) for all h
l#'#ameB'

then

tnere

are

b 1 ...b n €B such that B|=\JJ (a.^ . .am,b-L .. .bn) .

Then since €M, is in M. We want to show that (P,<) is not well-founded. To do this we define inductively an infinite descending chain in < ,<:n€a)> B n n as follows: let B be some B* as abovewith E

well-founded. We may

assume B C K . Pick some b€B arbitrarily: let B ={b} f B = t h e unique substructure of the reduct of B to 1+

with domain B_, f =

{}. Now suppose B ,f have been picked with 6P and f «= =P E B|B n . Suppose h£n+l; let a h be V x i - ' » x m 3 y i — 3 y n ^ h ^ X f i ^

with

^h

quantifier free. aB|«=a is a finitea set X, such that for all r e hb so bthere r m n | l ' - - n 6 X h ^h<*V • - m ' b l - " " V ' L ' e t B n + 1= B UX^U...UX U{c }. Let B , , be the unique s u b s t r u c t u r e of the r e d u c t no n n —n+1 of B t o i_ ^ with domain B , 1 . Then since ik i s q u a n t i f i e r —

.Ei / A , C— . * » ^

n+x

n

free and involves only c o n s t a n t s from among c-....c_

(h_
Va

r - - V a m € B n 3 b l - - b n € B n + l B n + l K ( a r • - a m ' b l - • -bn} • L e t fn+l= = P E B|B n + 1 . So < B n + 1 , f n + 1 , n + l ><< B n , f n , n ) . So i s not w e l l founded; by lemma 1, then, i t i s not well-founded in M. So t h e r e i s , <
XX

in M, an i n f i n i t e descending <-chain: c a l l

>:n€o) >. Let O XX

it

U C . Define EC by xECy ~ 3n(x,y€C AxECny) ; ^»

XX

XX

similarly for A. Define c, to be c,h where k, >k. This defines C. K K _ n P — Let g= U g ; then g : O 0 n such that cE c' =» g(c)k. < C h + 1 , g h + 1 , k h + 1 > « C h , g h , k h > are ^ . . . b ^ C ^

so k < k h + 1 and there

such that ^ 1+1 H'il« 1 . • . a m , b r . . b j . Hence

Note that we could have stipulated that N 1 be countable in M. The apparently essential restriction to V3-sentences can be removed by the following lemma.

210 Lemma 3 sentence

Let 0 be a sentence O' of L'

(i) (ii) Note: a 1

if A[=0 then there

such

of a language

L ' . Then there

is &€u) and an V3-

that:

are R . . .R

such that (A_,R . . .R %=0' ;

if (A_,R J? >[=a' then ^=0. 1 k is called the Skolem normal form of a.

Proof: Suppose a is in prenex normal form, 0.. x,. . . OIX,I|J (X... . .x,) where Q. is V or 3. Define sentences a,: °1 i s Vx 1 ...x k _ 1 (Q k x k *(x 1 ...x k ) .Rjl^...^)), a h is V x 1 . . . x k _ h ( Q k _ h + 1 x k _ h + 1 R h _ 1 ( x 1 . . . x k _ h + 1 ) (l
«Rh(x1...xk_h))

i s Q.. x, R k n (x-,) «* R, . Let a 1 be

R, AO, A. . .AQ, . I t remains t o prove t h a t a1 i s V3. We need only c o n s i d e r a, : t h i s i s of t h e form Vx.. . . . x, , (OxRx «* Sx) . This can be h 1 h-1 " w r i t t e n as a conjunct of an V and an V3 s e n t e n c e . Now for t h e a b s o l u t e n e s s theorem: i n essence i t i s due t o Levy: Lemma

4

Suppose

hereditarily free.

M model of ZFC and suppose _ is an inner in M. Let (a,. . . a ) . In In

countable

Suppose

Proof: We may assume n=l. Let x=x,.

a ...a in are with at most

x....x In

Let y=TC({x}) and l e t f:u)->y onto

with f€M. Consider a theory 0 in the language L_ _

,

_

N rj,r_/V_,\ n_:nt(jo /

(in

standard set-theoretic formulae E is to stand for~€): "~ (i) extensionality; (ii) y is transitive; (iii) f is a function with domain co and range y; (iv) n

f(n)=f(m)

(if f(n)=f(m));

(v) n ' f(n)+f(m) (vi)n m f(n)€f(m)

(if f(n)+f(m)); (if f(n)€f(m));

m

n,m - l - ' ^ ^ ( i f (viii) n+l=n+l; 0=0; n — — (ix) <|>(f(k)) (where x=f(k)) . Then
,€,f,y,n)

is a well-founded model of 0. Also 0£M. Now

the only axiom of 0 that is not necessarily V3 is (ix). But there are R, ...R, and (ix) ' which is V3 as in lemma 3: so
,€ , R, . . . R, , f ,y,n)

is a well-founded model of 01=(i)-(viii) +(ix)'. By lemma 2, then, there is a well-founded model of 0' in M, call it B= < B,E,S.. . . .S, ,g,z,m > _ . Thus |=0. By (i) is i. K n ntco n ^ extensional, so there is a transitive C with TT:< C, € >=< B,E >. Let g'=7r

(g),zl=7T~ (z) , ml=iT

(m ) . By induction using (viii) m n =n.

Since C is transitive z 1 is transitive (by (ii)) and g 1 is a function of u) onto z 1 (by (iii)). Let a:y->z' be defined by a(f(n)) = g 1 (n) . Then (iv)-(vii) show that a is well-defined and a:< y, € >=< z' , €>

211 so z=y' and f=g'. In particular g'(k)=x, so < C,€ >H> (x) . B u t <J> is l1 so M|=(j) ( X ) .

n

By the remark after lemma 2 this could b e extended to H M|=(|>( x )• F o r W l L example L L-^v V. Hereditary countability is e s s e n t i a l : if o>, ^o), then "a), is c o u n t a b l e " b u t L|^"O), is c o u n t a b l e " . N o w w e quote a standard result of descriptive set theory, d u e in essence to K l e e n e : Lemma 5

If $ is a E formula then there is a E formula ip such that

For a proof s e e Jech [25] s e c t i o n 4 1 . Hence Lemma 6

If (p is a E_ formula and a€MC\P(Li)) , M an inner model of ZFC, then

————————

2

P r o o f : T h e r e is a E, formula <J>' such that <|>(a) « H

|=<|>' (a) . Suppose

(j)(a) ; then H ^ |=cf>' (a) . S o b y lemma 4 H 1 ^ ] ^ 1 (a) , so MJ=<|>(a) • If/ conversely, Mh<J>(a) then HMM|=4>I(a) and HMMc=H U)l

0)1 — 0)i

so H

h(J)'(a); thus

0)i

In particular any true E 9 sentence is true in M. If a is a E 9 \ singleton - i.e. there is a E 2 formula (j) such that x=a «* (j) (x) - then a is constructible, because 3x(J>(x) =* 3x£L(j>(x) . On the other hand # 1 # 1 0

is a II2 singleton that is not constructible. "0

exists" is a E_

sentence, so lemma 6 is the best possible. Indeed we do not need 0 : is our l\. inner model was the core model K. The O # example Suppose would cause no difficulty, of course. K cannot be E-. absolute, for it could be L. To say that K is not L is to say that 0

exists;

in fact we must not let K be of the form L[a] for any acoj ; this is effected if we insist that the reals be closed under #. O is a H, t 1 singleton, so "0 exists" is E-> (but fails in K) . So we must also stipulate that 0 t should not exist. Given these restrictions we can show that K is E^-absolute. Since x€KflP(o)) is E^((3M)(M is a real mouse and x€M); we know that being a real mouse can be coded as a TI2 property of reals) P(OJ) =|=KnP (o))

is E^, so this is the best

we can hope for. u

We shall make considerable use of the discussion of a chapter 21. The main result we need is:

in

212 Lemma 7 exists

If (j) is a H formula

of one free

and L[a]\=there are no p-models

variable

and acw such that $(a) , a

then 3a€.K<$>(a) .

Proof: Suppose first of all that K L ^ a ^ K and let M be L [ a ] # with mouse iteration <<M ^j

of M . Let N = J

a +1

>crs , —ex ottun

< TT O) .QC~ >. Let K otp o t ^ p t u n

ot

be the critical point

be the minimal~~core a-mouse and let

< ,< TT' > a < g

> be the weak iterated ultrapower of L[a] by V.

Let x =TT' (T) . ~ S O C ' = { T

:a€On} are the canonical indiscernibles for

L[a], By lemma 21.22 there are a, 3 such that, letting K = T ^ ,

(b)

T

io+i,io+j(Ka+6i+v)=Ka+Bj+v

^

Suppose yif v; otherwise if v=a+3j+Y (y<3) then t(y,v) is <2,y>. Given Jo< • • • < J m _ 1 e O n ^ ^ o " ' ' ^m-1^

denotes

m

< t(0, j Q ) . . . t ( j m _ 2 , jm__1) >. Suppose t=< t Q .. .t m - 1 >€(3xon) : t is called (a,3)-apt provided: (i) t i = => Vj; (ii) t . = ( O j > =* Y or t ± =<2,Y>

-• Y<3;

(iv) t±=< 2,Y > «• i>O,ti_1=< 1,Y' > or < 2,y' > and Y ' < Y . SO t

a3(j0-'Vl)

^0<-"-<W0n)

iS

- a B ^ O - - - ^ - ^ P r i d e d t; B ( j Q . . . j ^ ) =t* 3 (j(^. . . j ; ^ , . Suppose i s ( a , 3 ) - a p t . Define inductively ^h/Yh' by: (i) if t,= then k, ,y, are undefined and 3h=Y/* (ii) if t h = then k h = k h - l + 1 / o r ° i f k h-l i s u n d e f i n e d ' Y h =Y and 3 (iii) if t h =<2,Y> then k h = k h _ 1 (which must be defined), Y h =Y and $ h =a+3k h +Y h . Now fix some m-tuple < J Q . - . J ^ ^ with t* $ (j Q . . . j ^ ^ =< t Q . .. t ^ Suppose J h = a + ^ h + V h ^ v h < ^^ ^ i f ^ h < a t h e n y h' v h a r e u n d e f i n e d ) •L e t k=i Q +k m _ 1 +l, ]j=io+y _^+l • We are to define a map a:L[a]->- L[a]. Work by induction on h. oQ is id|L[a]. Suppose a h :L[a]-> e L[a]. If k h is undefined let a, , n =id |L[a]. If k,=k, , then let o,ll=o,. n+1 ' n n—1 h+1 h k h = k h _ 1 # If k^_^ is undefined then ^ = 0 ; let a, unique map such that a ^ i r ' ^ ^ - i ^ ^ ^

+1:L[a]->

L[a]be the

and a h + 1 ( x ^ -x . ^ ^ If

k^=k, __,+l let a,+,:L[a]-> L[a] be the unique map such that Let o=o . m „ ~ Claim 1 If X€W and v€0n then a,(xnv ) »=xnv , , . _ h Proof: Let V
a h (xnv v )=a h ((xnv v ,)nv v )

Suppose

213 lemma

21

-

19)

21.19) =xnv a Claim 2

(v).

a(Claim 1)

yGC « a h (y)€C.

Proof: If y€C then for all X€U y€X (M=Jg). Pick v>y: so y€XflVv, hence a, (y)€Xnv

, ,cX. So a, (y)€C. The converse is similar. h

Claim

a(Claim 2)

3 0

h

0

h

Proof: Suppose £€Cn[x.

, x. (£^) for some n ) ; then £=TT! . 0 yh 1 0 y h + i ^ O yh in cn[x. ,x. ,-i) . There is a term t and a k€co, y
£

1

L [ a ] h ^ = t ( y , T i o / T i o + 1 . . . T i o + k ) . Hence L[a]h^=t (y, T ^ ^ , x ^ ^ ^ . . .

T k) (for + ( fc) k( ••• w Vo ^ V "VoVW V = (Ti )=T k)> But for k>0 VoV o w W^Wv^V-11 i1h + 1

).

And

^ ^ ^ ^ V i

1

'

So

D(Claim 3)

It follows at once that a, ,, ( K O ) = K . . And so CT(KO ) = K h+1 3 3 3 D h h h h h<m. Hence

for all

L[a]f=<|»(?(Kft ...Kfi )) « (j)(?(K. ...K. )) p 3 D ^0 m-l 0 m-1 (using claim 1 again). But 3 1 - - - 3 m - 1 depended only on t*g(J 0

) m-l Consider the the statement ip.. o (a) that says: _ Map Map acw and, letting K be the critical point of M 3

+g

J-f=(J)(a); and

m

O m - l a 3 O m - l ...KQ

t

)) for all formulae <\>.

y

m

~

3

m-l Claim 4 TI;_, O (a) is true. Map Proof: We need only show J—|=(j) (a) . We know <j> (a), so L[a]f=4>(a) by lemma 6. But

K = T . , and {x :a€On} are the canonical indiscernibles

for L[a] so J ^ L [ a ] .

D(Claim 4)

tyj. o is not a E-. property of M,a,3/a because "M is a mouse"is not Z^ (otherwise there would be mice in L! It is TI^) . And M,a,3 are not necessarily countable in K, so lemma 4 cannot be applied directly. But lemma 2 can be adapted to prove: Claim 5

There are M,a,"3,a€K, countable in K, such that ipi - -^(a) M,a,p and there is a:M->- M, where ^' differs from ip in only asserting that

214 there is a transitive a+3u>-th mouse iterate of M. Proof: A simple modification of the definition of ; details are left to the reader.

•(Claim 5)

But then M is a mouse. Let <<M^ > a € O n /<^ a g > a <eeon* b e i t s m o u s e iteration. Let K"^ be the critical point of M g and let ^='
So

From now on X is to denote

U 7Tn. (X) . Consider the language i€On U 1 L', which is L augmented by a constant a and by constants v_ for each ordinal v. For any ordinal v there are y <...
...K

).

Define a set of sentences S of the language L1 as follows: L

a

v.=f.(K 1

...K

* Y0

)

with n free variables. Suppose v.€ 1

(i
V-l

~°

n X

"

only if there^are 6Q< . . . ^ ^ a + ^ s u c h and, letting V . = ? . ( K «

••• K ( 5

-.) is to be in S if and that < y Q . . . Y ^ - ^ 6 Q . . . 6 ^

) ' J^cJ) (v . . .v _,) . The choice of 6.

are irrelevant; and S is consistent. S has definable Skolem functions: that is, given a formula x of L1 there is a formula x' that defines a function (i.e. s

hx' (x,y) A X 1 (x,z) => y=z) , call it h , such that s|-3yx(x,y) -»

«• X ( x / n (x) ) . (h (x) may be the L[a]-least y such that x( x /Y) when X X there is one, for example). So we may build a term model of S with domain {["h (v)"]:v€On}, where [] denotes the equivalence class of a term t under t~t' «* s|-t=t'. Call this model O. Suppose "3x€0nx(x)"ES, where x X(x) « X ^ X F ^ T I K

. . .K

)--fn

is a

formula of L 1 . Let

i ( K v - • 'Ky

)) where x 1

is a

Y Y ~l 0 m - 1 ~ ~ n 1~"Y0 Y m - 1 ~ formula of L^. Take 3 0 <...<3 m _ x of t ^ ( y Q . . . y m _ ^ as in the indiscernibility argument above. So 3 n • • • $ ,
J

fHx"(g(K

K 0

),?(K p-1 0

K

)) With 7B£...f^_1}={(S0...BIn_1>. c m-1

It is easily seen that there are{y'...y' i^^Y o » • -Y
O"-yp-l>'^YC)'<

So s

-^ such that

l-X(g(KY,...KY, )_). That is, there is

is an ordinal v such that s|~x(v.) . In particular for any Skolem term t(jJ) of S there is y€On such that Shrank (t (\>) ) =2- So Q is wellfounded. An indiscernibility argument shows that for x€Q {y:O|=y€x} is a set, so there is a transitive class 0 1 isomorphic to Q. So a ^s the Q 1 -interpretation of a.. Also, since clearly for V<~K J

^=X(v) « S|-x(v), j|-^Q f . So Q'|=V=L[a], i.e. O'=L[a], and Q'f=
By lemma 6 <\>(a) . If K=K Lta - 1 then M is to be the minimal p-model: as in chapter 21 the modifications are left to the reader.

D

215 Lemma 8

If O does not exist

§(a) and a

exists

Proof:

and (j) is a IT formula of one free

variable,

acw

then 3a€K$'(a) .

Suppose not,

so t h a t by lemma 7 L[a]|= t h e r e

Let L[a]|=M i s a p-model with c r i t i c a l embedding j : L [ a ] + L[a] with c r i t i c a l gives 0 • .

i s a p-model.

point K. There i s an point above K : but then j|M

•

Lemma 9

Suppose that

the reals

Then any true E sentence

are closed

under # but that

0

does not

exist.

is true in K.

1 # P r o o f : Suppose (J) (a) f o r some aeco and < { > i s 1^, S i n c e a exists there i s a€K s u c h t h a t (a). But t h e n K|=({)(a) by lemma 6 . So K|=3a(j)(a). a

Lemmas 7 - 9 a r e d u e t o J e n s e n unpublished,

Lemma 10

but no p-model with K

Proof:

By g e n e r i c a l l y

assume

that

there

some p-model, countable. Claim 1

and

If

j(f)(K)
Proof:

Let

p o i n t of

(K )

is

K=(KI+)L[U'].

f: K + ( K + ) L [ U ] . Now L [ U ] | = ( K + ) K =K + ;

K i s countable

asserting

• (Claim 1) that

U i s a normal

K ^ c j ^ and j:J^-»- J ^ ' . Then

there

i s a generic

cj>(U,j|j^)

f such t h a t £ i s

in M=L[U'][f]. K i s not the c r i t i c a l

of L [ U ' ] ,

point

of any p-model

t h e n by u n i q u e n e s s

Y>i< t h e n y i s a c a r d i n a l

L[U]cM.

i n M. If

y is a

of M, hence of L [ U ] .

So there is 6 such that j(6)=6 and for all a ^ i ^ ^ + c T ^ ' =

is

n e c e s s a r y we may

So K: i s c o u n t a b l e .

be t h e formula

Suppose i t were:

cardinal

if

i s the c r i t i c a l

and l e t K ' = J ( K ) .

i n J ^ , where £ = ( K + ) K ,

countable

K that

=> Consis (ZFC+ there

critical).

cardinals

we may assume

i s countable.

Since

the following,

countable.

Now l e t (f>(U,j)

Claim 2

proved

t h e p-model be L [ U ] , t h a t

Let j ^ [ U j - ^ L f U 1 ]

(K )

holds.

collapsing

i s a countable

and, l e t t i n g

£ is

measure

who a l s o

Consis (ZFC+there is a measurable cardinal)

j:K+ K with K critical

Proof:

([28]),

result.

Y ( ^ ( p + +) +)

Let

. y is isa strong limit of cofinality greater than K in

L[U] so j(Y)=Y- But

=YContradictionl

n(Claim 2)

Now by_lemma4 there are U,7 in M such that <j)(U,J) • Claim 3

<J^,U> is iterable.

Proof: Its first iterate is J~

which is transitive. Therefore its

(l+a)-th iterate is the a-th iterate of J~', which is iterable.

216 D(Claim 3) Now let ( ( J f l a L ^ ^ i r ^ ) . ^ ^ ) be the iteration of J^. For f€j5f f:Kn^K

U TT . (f) . Then ¥:Onn-*K is a i€On U 1 function. Given x€K, suppose x€K (where K =TT (K) ) . Then a U n there are a <...
let ¥ denote

K

For any formula (J> , a Q < . . .
...K

, T T O R ( X ) ) <=» <J)(K , . . . K ,

) . So

K

^<$>(f(K . . . K )) *=» K \=<\>(£(K ...K . ) ) . In particular a a K a a 0 m-2 a' 0 m-2 m-1 m-1 a<3 => K -< K , so for all a K -< K. Hence for a <. . . K

a

a'<...
K|=(J> ( ¥ ( K

...K

)) •» <|)(?(K

,...K ,

) ) . So we

0 m-1 0 m-1 may define a map x:K->K by T ( ? ( K ...K ))=?(K ,.. . K ,) . a a a X a 0 m-l 0 m-l L T is well-defined, T : K + K, T|K=id|K and T ( K ) = K ' > K . Hence M is the model we are looking for.

D

One can even add the stipulation that K be uncountable, but then the proof becomes considerably more complex. On the other hand Jensen has proved that if j:K+ K with critical point K and K > W 0 then there e «^ is a p-model with K critical. It is, so far as I know, an open question whether it is consistent that there be j:K-> K with countable critical point but no p-model with UK critical (by chapter 16 exercise 2 there would be a p-model with oa2 critical). I suspect that it is.

217 HISTORICAL NOTES

In these notes we give brief attributions of results where we know them. References in square brackets are to the bibliography that follows. These notes refer only to the body of the text; the introduction and the appendices give references for results as they go along. Part 1; Fine Structure 1:

The theory R is the BST of Gandy [17]. Rudimentary functions

were developed by Jensen [26] and Gandy [17]. The presentation in, e.g., [26] is generally more elegant than that using the basis functions: but as we need these functions later on we introduce them. These basis functions are a modification of those used by Gfldel to define L in [19]; the present form and the proof of lemma 1.5 are derived from Barwise [1], although the proof there is complicated by a different convention about ordered n-tuples. 2:

This chapter is a restatement of §2 of [26] allowing firstly

for relative constructibility and secondly for the weak theory. This presentation of fine structure would be rather perverse were there not already elementary expositions in [26] and [6]. Apart from the use we must later make of non well-founded structures it is interesting to see what can be done internally. 3:

The notion of acceptability was foreshadowed in Silver's proof

[53] of GCH in L [ U ] . Otherwise this chapter, together with chapter 4, contains the substance of fine structure theory from [26]; the relativised form here first appeared in [10]. The presentation of lemma 3.3 in [10] contains an error. The definition of V^ and the proof of corollary 3.19 are due to Solovay [58]. 4:

The extension of embeddings theorems were first proved in the

downward case in [26] and upward in [8], Lemma 4.27 4.28 are, of course, from Gfldel [19].

and corollary

218 Part 2: Normal Measures. 5: Ulam [60] gave a definition of measurability based on classical measure theory. Keisler and Tarski [33] found the definition used here. Corollary 5.7 was already known to Ulam but lemma 5.8 remained open for some time; it was proved in [20]. Definition 5.9 is Ulam's original formulation. Claim 1 in lemma 5.16 is known as Los1 theorem [37], Corollary 5.20 was proved by Scott [51]; the more general result mentioned after lemma 5.8 is proved by Kunen [36], Premice are defined in [10], where in addition they must be transitive. In this book transitivity is not insisted on until we deal with iterable premice. 6: Gaifman invented iterated ultrapowers: they were developed by Kunen [35], who proved all the results in this chapter. 7: Kunen [35] is the source of lemmas 7.1-7.7. The remainder of the chapter is due to Solovay [58]. The notion of remarkability in exercise 2 is from Silver [52]. 8: Most of this chapter is from Kunen [35]. Lemma 8.3 was used in [35] to prove corollary 7.3 - see exercise 1. The criteria for iterability are also in [35], The proof of lemma 8.9 used here was discovered by Jensen [31]. Comparability was studied in [35] . Lemma 8.19 appeared first, to our knowledge, in [31]. It is not used in [10]. Part 3: Mice. 9: Real mice were used by Silver in [54] to obtain a A^ well-order of the reals in L[U]. A version of this proof will be given in chapter 13. Silver's mice all had n(N)=0. The material here, with the main definition 9.25, derives from [10]. Our approach there depended more on well-foundedness; hence in the more abstract framework of the present work the definition has been complicated. The ordering <^ was discovered by Solovay [58], 10: Solovay proved the preservation results [58]. Core mice were introduced in [10]. Lemma 10.27 was proved by Solovay; the rest of the chapter is from [10], 11: The strategy of the case u)pJj'N' +
219

the source of the argument for the case cop5J

=K.

12: The implication (i) was proved by Gaifman [16]. Silver proved the same result using combinatorial techniques [52]. Solovay u 1 extracted 0

from lemma 12.9 and proved that it is a IT2 singleton

[57], and hence lemma 12.22 and 12.23. (ii) was presumably noticed by everyone. Kunen obtained 0

from the existence of a non-trivial

j:L-»- L; his proof is easily adapted to give (iii) . 13: 0

All the results in the first part were proved by Kunen [35]. was invented by Solovay, who as already remarked also proved

that 0

is A3> Silver [54] proved lemma 13.24; his proof was

essentially the one given here. The full theory of double mice was worked out by Hodgetts in [23]. Part 4: The Core Model. 14:

The results in this chapter are all from [10].

15:

The first part of this chapter is taken from [28]. Sharplike

mice are there called critical but this seems confusing. The examples are folklore. Koepke [34] showed that L#°° is really very like L indeed. The results on p-models are in [10]. 16:

The indiscernibles lemma was proved by Jensen [28],

generalising a result of Mitchell (see chapter 1 7 ) . The final argument is due to Rowbottom [49], The embeddings lemma was proved in [11] on the assumption that the image of the critical point is regular. This suffices for the covering lemma. Jensen then used the covering lemma to prove lemma 16.21. The present direct proof was discovered later by Jensen ([29]). 17:

Definition 17.2 is due to Erdo's and Rado [15]. Lemma 17.4 was

proved by Erdo's and Hajnal [14]. The definition of ct-Erdfls is in Baumgartner [2], Lemma 17.9 was proved by Schmerl [50]- Corollary 17.14 was proved by Mitchell [43]; the generalisation to lemma 17.13 is in [28]. Silver [52] proved corollary 17.16 directly. Lemmas 17.19 and 17.21 are in [28], as is the conjecture. 0 and •

were proved in K by Welch [61] : many corollaries are

examined in [7]. The results of [28] have appeared in [12],

220 Part 5; The Covering Lemma 18: The upward mapping theorem for mice was proved in [11]. The proof here incorporates simplifications discovered by Silver and Magidor - see [24], These do not shorten the work on viciousness but they do enable us to make considerable abbreviations in chapter 19. 19: The result for L was proved in [8]. Lemma 19.2 6 was first proved in [11]; the Silver-Magidor simplification of vicious sequences simplifies the argument. 20: The proof of lemma 20.16 was given in [11]. Jensen discovered an error in the original proof and repaired it in [30]. Prikry sequences were introduced by Prikry in [48]. Lemma 20.17 is due to Dehornoy [5]. Our self-imposed "no forcing" rule is a particularly severe restriction in this chapter and the reader should consult appendix I for the real definition of Prikry sequence. The form used in this chapter is an equivalent proved by Mathias [40], 21: The corollaries about singular cardinals are straightforward generalisations of results in [8]. A detailed account of the development of SCH will be found in the introduction. Lemma 21.6 was proved by Bukovsky [3]. Lemma 21.8 embodies many results of which the oldest - the successor case - is due to Hausdorff [22], Lemma 21.10 is the main result of [11]; for singular cardinals of uncountable cofinality Silver had proved a weak form without any assumption about large cardinals [55], Lemma 21.11 is also from [11], where it is proved by the simpler expedient of appealing to the fact that L[C]|=GCH. a was invented by Solovay [57]. The main result of this section, lemma 21.22, is an adaptation by Jensen [28] of a result of Paris [47], Jensen also proved lemma 21.14 [27], The model referred to at the end in which 3>1 was discovered by Solovay. Part 6: Larger Core Models. 22: The material on coherent sequences of normal measures was discovered by Mitchell [42]. Mitchell worked out the theory of mice based on coherent sequences [45] and also produced a generalisation of measure - called a hypermeasure - that is equivalent to the notion of extender - see [44]. Extenders were defined, and all the results on them stated here proved, in [31].

221 23: Mitchell proved the results up to lemma 2 3.11 in [44] using hypermeasures. The proofs here are taken from [31], as is the theory of 0 . Lemma 2 3.8 adapts an argument of Solovay [59]. 24: Superstrong cardinals were considered in [9] together with virtual extenders. Supercompact cardinals were defined by Reinhardt and Solovay. Lemma 24.3 and its corollary are due to Kunen [36], The covering lemma on the assumption of no measurable limit of measurables was discovered by Mitchell.

222 BIBLIOGRAPHY 1.

BARWISE K.J.

Admissible Sets and Structures. Springer-Verlag,

Berlin Heidelberg and New York, 1975. 2.

BAUMGARTNER J.E.

Ineffability properties of cardinals. In

"Logic, Foundations of Mathematics and Computability Theory" (R.Butts and J.Hintikka eds.) D.Reidel, Dordrecht-Holland, 1977. 3.

BUKOVSKY L.

The Continuum Problem and the Powers of Alephs.

Comment. Math. Univ. Carolinae 6(1965) 181-197. 4.

COHEN P.J.

The Independence of the Continuum Hypothesis. Proc.

Nat. Acad. Sci. U.S.A. 50(1963), 1143-1148; 51(1964), 105-110. 5.

DEHORNOY P.

Iterated Ultrapowers and Prikry Forcing.

Ann.

Math. Log. 15(1978) 109-160. 6.

DEVLIN K.J.

Aspects of Constructibility.

Lecture Notes in

Mathematics 354, Springer-Verlag, Berlin and New York 1973. I.

DEVLIN K.J.

The axiom of constructibility: a guide for the

Mathematician, Lecture Notes in Mathematics 617, Springer-Verlag Berlin and New York, 1977. 8.

DEVLIN K.J. & JENSEN R.B. |=ISILC Logic Conf.

Marginalia to a Theorem of Silver.

Lecture Notes in Mathematics 499, Springer-

Verlag, Berlin and New York 1975. 9.

DODD A.J.

Superstrong Cardinals.

Handwritten Notes, Oxford

1980. 10. DODD A.J. & JENSEN R.B.

The Core Model.

Ann. Math. Log.

20(1981) 43-75. II. DODD A.J. & JENSEN. R.B. Log.

The Covering Lemma for K.

Ann. Math.

To appear.

12. DONDER D., JENSEN R.B. & KOPPELBURG B.J. The Core Model. 13. EASTON W.B.

Some Applications of

Freie UniversitSt Berlin Preprint 115, 1981.

Powers of Regular Cardinals.

Ann. Math. Log.

223 1(1970) 139-178. 14. ERDOS P. & HAJNAL A.

Some remarks concerning our paper "On the

Structure of Set Mappings".

Acta Math. Acad. Sci. Hungar.

13(1962) 223-226. 15. ERDOS P. & RADO.R.

A Partition Calculus in Set Theory.

Bull.

Amer. Math. Soc. 62(1956) 427-489. 16. GAIFMAN H.

Elementary Embeddings of Models of Set Theory and

certain subtheories.

In "Axiomatic Set Theory", Proc. Symp.

Pure. Math. 13 II (T.Jech ed.) 33-101

Amer. Math. Soc.

Providence, Rhode Island, 1974. 17. GANDY R.O.

Set-theoretic Functions for Elementary Syntax.

In

"Axiomatic Set Theory", Proc. Symp. Pure. Math. 13 II (T.Jech ed.) 103-126 18. GODEL K.

Amer. Math. Soc.

Providence, Rhode Island, 1974.

The Consistency of the Axiom of Choice and of the

Generalised Continuum Hypothesis.

Proc. Nat. Acad. U.S.A.

24(1938) 556-557. 19. GODEL K.

The Consistency of the Axiom of Choice and of the

Generalised Continuum Hypothesis. 20. HANF W.P.

Ann. Math. Studies 3(1940).

Incompactness in Languages with Infinitely Long

Expressions.

Fund. Math. 53(1964) 309-324.

21. HANF W.P. & SCOTT D.S.

Classifying Inaccessible Cardinals.

Notices Amer. Math. Soc. 8(1961) 445. 22. HAUSDORFF F.

Der Potenzbegriff in der Mengenlehre.

Jahresber.

Deutsch Math.-Verein. 13(1904) 569-571. 23. HODGETTS A.

D.Phil. Thesis.

Oxford 1980.

24. HOLZMANN R.

Jensen's Covering Lemma - an Elementary Proof.

Handwritten Notes, Jerusalem 1980. 25. JECH T.

Set Theory.

26. JENSEN R.B.

Academic Press, New York, 1978.

The Fine Structure of the Constructible Hierarchy.

Ann. Math. Log. 4(1972) 229-308.

27. JENSEN R.B. Bonn, 1977.

224 Pendant to a Theorem of Paris. Handwritten Notes,

28. JENSEN R.B.

Applications of K. Handwritten Notes, Bonn 1977.

29. JENSEN R.B.

Embeddings of K. Handwritten Notes, Bonn 1977.

30. JENSEN R.B. A Correction to "The Core Model". Handwritten Notes, Freiburg 1980. 31. JENSEN R.B. & DODD A.J. The Core Model for Extenders. Handwritten Notes, Oxford 1978-80. 32. JENSEN R.B. & KARP C. Primitive Recursive Set Functions. In "Axiomatic Set Theory", Proc. Symp. Pure Math. 13 I (D.Scott ed.) 143-167 Amer. Math. Soc. Providence, Rhode Island, 1971. 33. KEISLER H.J. & TARSKI A. From Accessible to Inaccessible Cardinals. Fund. Math. 53(1964) 225-308. 34. KOEPKE P.

L # . Dissertation, Bonn.

35. KUNEN K. Some Applications of Iterated Ultrapowers in Set Theory. Ann. Math. Log. 1(1970) 179-227. 36. KUNEN K. Elementary Embeddings and Infinitary Combinatorics. J. Symbolic Logic 36(1971) 407-413. 37. LOS J. Quelques Remarques, Theoremes et Problemes sur les Classes Definissables d'Algebres. In "Mathematical Interpretation of Formal Systems (T.Skolem et al. eds.) 98-113 North Holland Publ. Co. Amsterdam 1970. 38. MAGIDOR M. On the Singular Cardinals Problem. I Israel J. Math. 28(1977) 1-31; II Ann. Math. 106(1977) 517-547. 39. MAREK w. & SREBRNY M. Gaps in the Constructible Universe. Ann. Math. Log. 6(1974) 359-394. 40. MATHIAS A. On Sets Generic in the sense of Prikry. Math. Soc. 15(1973) 409-414.

J. Austral.

225 41. MENAS T.K.

Consistency results concerning Supercompactness.

Trans. Amer. Math. Soc. 223(1976) 61-91. 42. MITCHELL W.J.

Sets constructible from Sequences of Ultrafilters

J. Symbolic. Logic 39(1974) 57-66. 43. MITCHELL W.J.

Ramsey Cardinals and Constructibility.

J.

Symbolic Logic 44(1979) 260-266. 44. MITCHELL W.J.

Hypermeasurable Cardinals.

1978" (M.Boffa et al. eds) 303-316.

In "Logic Colloquium

North Holland Publ. Co.

Amsterdam 1979. 45. MITCHELL W.J.

The Core Model for Sequences of Measures.

Typewritten Notes. 46. NAMBA K.

Penn. State 1980.

Independence Proof of (a),a) )-distributive Law in

Complete Boolean Algebras.

Comment. Math. Univ. St. Pauli

19(1970) 1-12. 47. PARIS J.B.

Patterns of Indiscernibles.

Bull. London Math.

Soc. 6(1974) 183-188. 48. PRIKRY K.L. Changing Measurable into Accessible Cardinals. Diss. Math. 68(1970) 5-52. 49. R0WB0TT0M F.

Some Strong Axioms of Infinity incompatible with

the Axiom of Constructibility. 50. SCHMERL J.H.

On K-like structures which embed Stationary and

Closed Unbounded Subsets. 51. SCOTT D.S.

Ann. Math. Log. 3(1971) 1-44.

Ann. Math. Logic 11(1976) 289-314.

Measurable Cardinals and Constructible Sets.

Bull.

Acad. Polon. Sci. 9(1961) 521-524. 52. SILVER J.H.

Some Applications of Model Theory in Set Theory.

Ann. Math. Logic 3(1971) 45-110. 53. SILVER J.H.

The Consistency of the GCH with the existence of a

Measurable Cardinal.

In "Axiomatic Set Theory" Proc. Symp. Pure

Math. 13 I (D.Scott ed.) 391-396 Rhode Island, 1971.

Amer. Math. Soc. Providence,

226 54. SILVER J.H.

Measurable Cardinals and A- well orderings. Ann.

Math. 94(1971) 414-446. 55. SILVER J.H.

On the Singular Cardinals Problem.

Proc. Int.

Congr. Math. Vancouver 1974 265-2 68. 56. SOLOVAY R.M. (Abstract). 57. SOLOVAY R.M.

Independence Results in the Theory of Cardinals. Notices Amer. Math. Soc.

10(1963) 595.

A non-constructible A 3 set of Integers.

Trans.

Amer. Math. Soc. 127(1967) 50-75. 58. SOLOVAY R.M.

The Fine Structure of L[y].

Handwritten Notes

Berkeley 1972. 59. SOLOVAY R.M., REINHARDT W.N. & KANAMORI A. Infinity and Elementary Embeddings.

Strong Axioms of

Ann. Math. Logic 13(1978)

73-116. 60. ULAM S.

Zur Masstheorie in der algemeinen Mengenlehre.

Fund. Math. 16(1930) 140-150. 61. WELCH P.

D.Phil thesis. Oxford 1979.

227 INDEX OF DEFINITIONS Note: n.Xm denotes exercise m in chapter n. 1.1

RN

3.11

1.3

Basis functions

3.12

P

1.3

F±

3.14

A?

1.10

Rudimentary

3.14

TP

1.11

Substitutable

3.15

S^

(in RA)

1.14

Rudimentary

3.15

CT

(in RA)

RA+ VP

(relation)

1.16

E Q satisfaction

3.20

1.16

i n

3.22

1.17

E

satisfaction

1.17

3.23 3.23

m

P

nn

1.18

E

limit segment

3.23

Anp vnp

1.X1

Rudimentarily closed

3.23

Tnp

1.X5

Amenable

3.25

MC

2.1

S(u)

3.27

MCn

2.3

4.1

Standard (model of R)

2.3 2.4

f=S S R

2.5 2.11

3.23

X

4.6

7r:N->

4.17

p-sound

rud(u)

4.26

n-completion

T=v+n

4.29

J^,S A

4.30

Acceptable

+

N cofinally

(in ZFC)

2.16

R f=D

4.31

Strongly acceptable

2.16

T=[V/O)]

4.X5

rud A (M)

2.18

T=0)V

5.1

Critical point (of j)

2.18

On

5.2

Measurable

2.18

J

2.12

V

5.9

Normal measure on K

2.27

5.9

Filter

2.27

f=p|y

5.9

Ultrafilter Proper (filter) Non-principal

v

P

5.9

2.29

Q

5.9

2.29

G-closed

5.9

K-additive

2.31

En cardinal

5.9

Diagonal intersectior

2.27

2.33

h

5.11

j:M+0 N

2.33

E, Skolem function

5.27

Premouse

2.35

h(X)

5.27

Critical point (of M)

2.X4

E

5.27

Pure (premouse)

3.1

RA

6.1

Commutative system

3.5

P

6.2

Direct limit

3.5

Pro jecrtum

6.9

Iterated ultrapower

3.7

RA

6.12

Iterate (of N by U)

regular

228 6.12

ath iterate

6.13

Iterable by U

6.14

Iterate

6.16 7.5

Weak iterated ultrapower n

7.8

n-good

7.9

n-full

(of N by U)

13.8

Core p-model

13.8

Critical measure

13.9

Double premouse

13.10

Iterated ultrapower

13.11

Iterable (double

(of N)

ordinal

u

(double premouse)

7.10

ch n (a)

7.13

< itu . . .m >->

7.14

A

8.1

Countably complete

8.17

Comparable

14.3

K

9.1

<*

14.5

K

9.4

14.7

Q

K

9.4

14.7

F

K

15.1

Sharplike

~4r


premouse) P

n,X

v< n1

1

i

9.9

Strongly acceptable

9.4

k

n > I""" k

9.5

9.4

n, > 13.12

n-soxind above y

9.12

n-suitable

9.13

13.17

Dagger O1*

13.20

Real mouse

14.2

D

ex

15.6

K

15.6

D

15.7

M

M < (on core mice)

15.11

deg(M)

15.14

M#

16.1

Good set of

9.19

n-iterated ultrapower

9.22

Critical (premouse)

17.1

f-homogeneous

9.22

n(N)

17.2

K+(A) "

9.23

Mouse iteration

17.2

K+(A) j

9.24

n-iterate

17.3

Ramsey

9.24

Mouse iterate

17.5

K(a)

9.25

n-iterable

17.6

a-Erdo's

9.25

Mouse

17.18

Uniform (ultrafilter)

10.18

Core(N)

17.18

(Y,K)-regular

10.18

Core mouse

10.20

C

N

indiscernibles

(ultrafilter) 17.20

Weakly normal

10.29

(ultrafilter)

10.35 10.37

E

17.22

0

E

17.24

12.1

Sharp

18.1

a K y-base for M

12.11

0*

18.2

T M 'TTfY

12.15

K-maximal

18.2

•pM, 7T,y

12.18

K-minimal

18.8

13.8

p-model

18.8

n

7T M ' Y

229 18.9

7TM'Y'n

22.10

o E (a)

18.9

M¥'Y'n

22.11

Premouse

18.10

TT-good

22.12

Iterated ultrapower

18.13

a)-closed in T

18.16

R-vicious for

22.12

Iterate

(extenders)

22.13

Iterable

(extenders)

22.15

vm

22.16

Comparable (premice for extenders) Normal (iteration) Countably complete

(extenders)

(extenders) M,TT,Y

18.18 18.18

Canonical R-vicious

19.3

N

sequence 20.1

Prikry sequence

22.19

21.12

a*

22.22

22.1

< (for measures)

22.4

Coherent sequence (of

(extender)

measures)

23.1

3-strong

23.2

Strong

22.4

AU

23.12

Pistol

22.5

O U (K)

23.13

of

22.7

8-extender at K

24.1

Superstrong

22.8

24.2

3-supercompact

22.9

24.2

Supercompact

Coherent seauence (of extenders)

22.9 We have not listed terms that arise from defined terms by our subscripting convention: p definition 15.6, that K

for example. Note, in connection with

arises from the convention whereas Kx;

is defined in its own riant.

M