Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II
The proceedings of the Los Angeles Caltech-UCLA “Cabal Seminar” were originally published in the 1970s and 1980s. Wadge Degrees and Projective Ordinals is the second of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics, and discussion of research developments since the publication of the original volumes. Focusing on the subjects of “Wadge Degrees and Pointclasses” (Part III) and “Projective Ordinals” (Part IV), each of the two sections is preceded by an introductory survey putting the papers into present context. These four volumes will be a necessary part of the book collection of every set theorist.
Alexander S. K echris is Professor of Mathematics at the California Institute of Technology. He is the recipient of numerous honors, including the J. S. Guggenheim Memorial Foundation Fellowship and the Carol Karp Prize of the Association for Symbolic Logic. He is also a Member of the Scientific Research Board of the American Institute of Mathematics. ¨ e is Universitair Docent in Logic in the Institute for Logic, Benedi kt L ow Language and Computation at the Universiteit van Amsterdam, and Professor of Mathematics at the Universit¨at Hamburg. He is the Vice-president of the Deutsche Vereinigung f¨ur Mathematische Logik und f¨ur Grundlagenforschung der Exakten Wissenschaften (DVMLG), and a Managing Editor of the journal Mathematical Logic Quarterly. John R. Stee l is Professor of Mathematics at the University of California, Berkeley. Prior to that, he was a professor in the mathematics department at UCLA. He is a recipient of the Carol Karp Prize of the Association for Symbolic Logic and of a Humboldt Prize. Steel is also a former Fellow at the Wissenschaftskolleg zu Berlin and at the Sloan Foundation.
LECTURE NOTES IN LOGIC
A Publication for The Association for Symbolic Logic This series serves researchers, teachers, and students in the field of symbolic logic, broadly interpreted. The aim of the series is to bring publications to the logic community with the least possible delay and to provide rapid dissemination of the latest research. Scientific quality is the overriding criterion by which submissions are evaluated. Editorial Board H. Dugald Macpherson, Managing Editor School of Mathematics, University of Leeds Jeremy Avigad Department of Philosophy, Carnegie Mellon University Vladimir Kanovei Institute for Information Transmission Problems, Moscow Manuel Lerman Department of Mathematics, University of Connecticut Heinrich Wansing Department of Philosophy, Ruhr-Universit¨at Bochum Thomas Wilke Institut f¨ur Informatik, Christian-Albrechts-Universit¨at zu Kiel More information, including a list of the books in the series, can be found at http://www.aslonline.org/books-lnl.html.
LECTURE NOTES IN LOGIC 37
Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II Edited by
ALEXANDER S. KECHRIS California Institute of Technology
¨ BENEDIKT L OWE Universiteit van Amsterdam and Universit¨at Hamburg
JOHN R. STEEL University of California, Berkeley
association for symbolic logic
cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Tokyo, Mexico City 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/9780521762038 Association for Symbolic Logic Richard Shore, Publisher Department of Mathematics, Cornell University, Ithaca, NY 14853 http://www.aslonline.org C
Association for Symbolic Logic 2012
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CONTENTS Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Original Numbering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
PART III: WADGE DEGREES AND POINTCLASSES Alessandro Andretta and Alain Louveau Wadge degrees and pointclasses. Introduction to Part III . . . . . . . . . . .
3
Robert Van Wesep Wadge degrees and descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
Alexander S. Kechris A note on Wadge degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Alain Louveau Some results in the Wadge hierarchy of Borel sets . . . . . . . . . . . . . . . . . . .
47
Alain Louveau and Jean Saint-Raymond The strength of Borel Wadge determinacy . . . . . . . . . . . . . . . . . . . . . . . . . .
74
John R. Steel Closure properties of pointclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Alexander S. Kechris, Robert M. Solovay and John R. Steel The axiom of determinacy and the prewellordering property . . . . . . . . 118 Steve Jackson and Donald A. Martin Pointclasses and wellordered unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Howard S. Becker More closure properties of pointclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 John R. Steel More measures from AD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 William W. Wadge Early investigations of the degrees of Borel sets . . . . . . . . . . . . . . . . . . . . . 166 vii
viii
contents
PART IV: PROJECTIVE ORDINALS Steve Jackson Projective ordinals. Introduction to Part IV . . . . . . . . . . . . . . . . . . . . . . . . 199 Alexander S. Kechris Homogeneous trees and projective scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 Alexander S. Kechris AD and projective ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 Robert M. Solovay A Δ13 coding of the subsets of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 Steve Jackson AD and the projective ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 Donald A. Martin Projective sets and cardinal numbers: some questions related to the continuum problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 Steve Jackson Regular cardinals without the weak partition property . . . . . . . . . . . . . . 509 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
PREFACE
This book continues the series of volumes containing reprints of the papers in the original Cabal Seminar volumes of the Springer Lecture Notes in Mathematics series [Cabal i, Cabal ii, Cabal iii, Cabal iv], unpublished material, and new papers. The first volume, [Cabal I], contained papers on games, scales and Suslin cardinals. In this volume, we continue with Parts III and IV of the project: Wadge degrees and pointclasses and Projective ordinals. As in our first volume, each of the parts contains an introductory survey (written by Alessandro Andretta and Alain Louveau for Part III and by Steve Jackson for Part IV) putting the papers into a present-day context. In addition to the reprinted papers, this volume contains papers by Steel (More measures from AD) and Martin (Projective sets and cardinal numbers) that date back to the period of the original Cabal publications but were not included in the old volumes. Jackson contributed a new paper Regular cardinals without the weak partition property with recent results that fit well with the topic of Part IV. The paper Early investigations of the degrees of Borel sets by Wadge is a historical overview of the process of the development of the basic theory of the Wadge degrees. Table 1 gives an overview of the papers in this volume with their original references. As emphasized in our first volume, our project is not to be understood as a historical edition of old papers. In the retyping process, we uniformized and modernized notation and numbering of sections and theorems. As a consequence, references to papers in the old Cabal volumes will not always agree with references to their reprinted versions. In this volume, references to papers that already appeared in reprinted form will use the new numbering. In order to help the reader to easily cross-reference old and new numberings, we provide a list of changes after the preface. The typing and design were partially funded by the Marie Curie Research Training Site GLoRiClass (MEST-CT-2005-020841) of the European Commission. Infrastructure was provided by the Institute for Logic, Language and Computation (ILLC) of the Universiteit van Amsterdam. Many people were involved in typing, laying out, and proofreading the papers. We ix
x
PREFACE
Part III Andretta, Louveau
Wadge degrees and pointclasses Introduction to Part III
new
Van Wesep
Wadge degrees and descriptive set theory
[Cabal i, pp. 151–170]
Kechris
A note on Wadge degrees
[Cabal ii, pp. 165–168]
Louveau
Some results in the Wadge hierarchy of Borel sets
[Cabal iii, pp. 28–55]
Louveau, Saint-Raymond
The strength of Borel Wadge determinacy
[Cabal iv, pp. 1–30]
Steel
Closure properties of pointclasses
[Cabal ii, pp. 147–163]
Kechris, Solovay, Steel
The axiom of determinacy and the prewellordering property
[Cabal ii, pp. 101–125]
Jackson, Martin
Pointclasses and wellordered unions
[Cabal iii, pp. 56–66]
Becker
More closure properties of pointclasses
[Cabal iv, pp. 31–36]
Steel
More measures from AD
new
Wadge
Early investigations of the degrees of Borel sets
new
Part IV Jackson
Projective ordinals Introduction to Part IV
new
Kechris
Homogeneous trees and projective scales
[Cabal ii, pp. 33–73]
Kechris
AD and projective ordinals
[Cabal i, pp. 91–132]
Δ13
coding of the subsets of
Solovay
A
Jackson
AD and the projective ordinals
[Cabal iv, pp. 117–220]
[Cabal i, pp. 133–150]
Martin
Projective sets and cardinal numbers: some questions related to the continuum problem
new
Jackson
Regular cardinals without the weak partition property
new
Table 1.
should like to thank (in alphabetic order) Can Baskent, Hanne Berg, Pablo ¨ Cubides Kovacsics, Jined Elpitiya, Thomas Gobel, Leona Kershaw, Anston Klev, Alexandru Marcoci, Kian Mintz-Woo, Antonio Negro, Maurice Pico ˜ Stephan Schroevers, de los Cobos, Sudeep Regmi, Cesar Sainz de Vicuna, Sam van Gool, and Daniel Velkov for their important contribution as typists and diligent proofreaders. We should like to mention that the original
PREFACE
xi
LATEX stylefile for the retyping was designed by Dr. Samson de Jager. Very special thanks are due to Dr. Joel Uckelman, who took over the typesetting coordination from de Jager in 2007. REFERENCES
Alexander S. Kechris, Benedikt Lowe, and John R. Steel ¨ [Cabal I] Games, scales, and Suslin cardinals: the Cabal seminar, volume I, Lecture Notes in Logic, vol. 31, Cambridge University Press, 2008. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris, Donald A. Martin, and John R. Steel [Cabal iv] Cabal seminar 81–85, Lecture Notes in Mathematics, no. 1333, Berlin, Springer, 1988. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978.
The Editors Alexander S. Kechris, Pasadena, CA ¨ Benedikt Lowe, Amsterdam John R. Steel, Berkeley, CA
ORIGINAL NUMBERING
Numbering in the reprints may differ from the original numbering. Where numbering differs, the original designation is listed on the left, with the corresponding number in the reprint listed on the right. In rare cases where an item numbered in the reprint had neither a number nor a name in the original, we have indicated that with a ‘—’. Volume I Notes on the theory of scales, Kechris & Moschovakis, [Cabal i, pp. 1–53] §2A Theorem 2A-1 Proposition 2A-2 §2B Theorem 2B-1 Theorem 2B-2 Corollary Corollary Theorem 2B-3 Corollary Corollary §2C Theorem 2C-1 Corollary 1. 2. 3. 4. §2D §3A Theorem 3A-1 §3B Theorem 3B-1
§2.1 Theorem 2.1 Proposition 2.2 §2.2 Theorem 2.3 Theorem 2.4 Corollary 2.5 Corollary 2.6 Theorem 2.7 Corollary 2.8 Corollary 2.9 §2.3 Theorem 2.10 Corollary 2.11 Claim 2.12 Claim 2.13 Claim 2.14 Claim 2.15 §2.4 §3.1 Theorem 3.1 §3.2 Theorem 3.2
Corollary 1. 2. Theorem 3B-2 Corollary Corollary 1. 2. Theorem 3B-3 Corollary Corollary §3C Theorem 3C-1 Corollary Corollary 1. 2. §3D §3E Theorem 3E-1 §4A 4A-1 4A-2 xiii
Corollary 3.3 Claim 3.4 Claim 3.5 Theorem 3.6 Corollary 3.7 Corollary 3.8 Claim 3.9 Claim 3.10 Theorem 3.11 Corollary 3.12 Corollary 3.13 §3.3 Theorem 3.14 Corollary 3.15 Corollary 3.16 Claim 3.17 Claim 3.18 §3.4 §3.5 Theorem 3.19 §4.1 4.1.1 4.1.2
xiv
original numbering
4A-3 Theorem 4A-4 Theorem 4A-5 Theorem 4A-5(a) Theorem 4A-5(b) §4B Theorem 4B-1 Theorem 4B-2 Theorem 5-1 Theorem 5-2 §6A §6B Theorem 6B-1 §6C Theorem 6C-1 Theorem 6C-2 Theorem 6C-3 Corollary Corollary Theorem 6C-4 Corollary §7A Theorem 7A-1 Corollary Corollary Corollary §7B 7B-1 7B-2 7B-3 7B-4 Open Problem §8A Theorem 8A-1 Theorem 8A-2
4.1.3 Theorem 4.4 Theorem 4.5 Theorem 4.5 Theorem 4.6 §4.2 Theorem 4.7 Theorem 4.8 Theorem 5.1 Theorem 5.2 §6.1 §6.2 Theorem 6.1 §6.3 Theorem 6.2 Theorem 6.3 Theorem 6.4 Corollary 6.5 Corollary 6.6 Theorem 6.7 Corollary 6.8 §7.1 Theorem 7.1 Corollary 7.2 Corollary 7.3 Corollary 7.4 §7.2 7.2.1 7.2.2 7.2.3 7.2.1 Open Problem 7.5 §8.1 Theorem 8.1 Theorem 8.2
Corollary §8B Theorem 8B-1 Corollary §8C Theorem 8C-1 Corollary Open Problem §9A Theorem 9A-1 Theorem 9A-2 Theorem 9A-3 §9B Theorem 9B-1 — §9C Lemma 9C-1 Theorem 9C-2 Corollary — §10A Theorem 10A-1 Corollary §10B Theorem 10B-1 Corollary §11A Theorem 11A-1 Corollary (a) Corollary (b) §11B Theorem 11B-1 Theorem 11B-2 1. 2.
Corollary 8.3 §8.2 Theorem 8.4 Corollary 8.5 §8.3 Theorem 8.6 Corollary 8.7 Open Problem 8.8 §9.1 Theorem 9.1 Theorem 9.2 Theorem 9.3 §9.2 Theorem 9.4 Claim 9.5 §9.3 Lemma 9.6 Theorem 9.7 Corollary 9.8 Claim 9.9 §10.1 Theorem 10.1 Corollary 10.2 §10.2 Theorem 10.3 Corollary 10.4 §11.1 Theorem 11.1 Corollary 11.2 Corollary 11.3 §11.2 Theorem 11.4 Theorem 11.5 Claim 11.6 Claim 11.7
Inductive scales on inductive sets, Moschovakis, [Cabal i, pp. 185–192] Main Theorem Lemma 1 Lemma 2 Lemma 3 —
Theorem 0.1 Lemma 1.1 Lemma 1.2 Lemma 1.3 Theorem 1.4
Lemma 4 Corollary 1 Corollary 2 Corollary 3 Corollary 4
Lemma 1.5 Corollary 2.1 Corollary 2.2 Corollary 2.3 Corollary 2.4
xv
original numbering Scales on Σ11 sets, Steel, [Cabal iii, pp. 72–76] Lemma Theorem 1
Lemma 1.1 Theorem 1.2
Theorem 2 Corollary
Theorem 1.3 Corollary 1.4
Scales on coinductive sets, Moschovakis, [Cabal iii, pp. 77–85] Infimum Lemma Fake Supremum Lemma
Lemma 1.1 Lemma 1.2
Lemma Theorem
Lemma 2.1 Theorem 2.2
The extent of scales in L(R), Martin & Steel, [Cabal iii, pp. 86–96] Corollary 1 Lemma 2 Theorem 1 Corollary 2
Corollary 2 Lemma 3 Theorem 4 Corollary 5
Corollary 3 Corollary 4 Theorem 2
Corollary 6 Corollary 7 Theorem 8
The largest countable, this, that, and the other, Martin, [Cabal iii, pp. 97–106] Theorem Corollary
Theorem 1.1 Corollary 1.2
Theorem Sublemma 2.4
Theorem 1.3 Sublemma 2.3.1
Scales in L(R), Steel, [Cabal iii, pp. 107–156] Definition 2.2 Lemma 2.3 Corollary 2.4 Lemma 2.5 Corollary 2.6 Theorem 2.7 Corollary 2.8 Theorem 2.9
Definition 2.4 Lemma 2.5 Corollary 2.6 Lemma 2.7 Corollary 2.8 Theorem 2.9 Corollary 2.10 Theorem 2.11
Corollary 2.10 Proposition 2.11 Claim 1 Claim 2 Claim 3 Corollary 3.8 Corollary 3.9
Corollary 2.12 Corollary 2.13 Claim 3.8 Claim 3.9 Claim 3.10 Corollary 3.11 Corollary 3.12
The real game quantifier propagates scales, Martin, [Cabal iii, pp. 157–171] §0 §1 Lemma 1.1 Lemma 1.2
§1 §2 Lemma 2.1 Lemma 2.2
§2 Lemma 2.1 Lemma 2.2 §3
§3 Lemma 3.1 Lemma 3.2 §4
xvi
original numbering
Lemma 3.1 Lemma 3.2 Lemma 3.3 §4 Lemma 4.1 Lemma 4.2 Lemma 4.3
Lemma 4.1 Lemma 4.2 Lemma 4.3 §5 Lemma 5.1 Lemma 5.2 Lemma 5.3
Lemma 4.4 Theorem 4.5 §5 Theorem 5.1 §6 Theorem 6.1 Corollary 6.2
Lemma 5.4 Theorem 5.5 §6 Theorem 6.1 §7 Theorem 7.1 Corollary 7.2
Long games, Steel, [Cabal iv, pp. 56–97] Example Theorem 1 Lemma 1 Claim Corollary Lemma 2 Corollary 1 Theorem 2 Lemma 3 Claim 1 Claim 2 Claim 3
Example 1.1 Theorem 1.2 Lemma 1.3 Claim 1.4 Corollary 1.5 Lemma 1.6 Corollary 1.7 Theorem 2.1 Lemma 2.2 Claim 2.3 Claim 2.4 Claim 2.5
Corollary 2 Theorem 3 Lemma 4 Claim Claim Subclaim Theorem 4 Lemma 5 Conjecture Lemma 6 Theorem 5 Corollary 3
Corollary 2.6 Theorem 3.1 Lemma 3.2 Claim 3.3 Claim 3.4 Subclaim 3.5 Theorem 3.6 Lemma 4.1 Conjecture 4.2 Lemma 4.3 Theorem 4.4 Corollary 4.5
The axiom of determinacy, strong partition properties and nonsingular measures, Kechris, Kleinberg, Moschovakis, & Woodin, [Cabal ii, pp. 75–100] Lemma Lemma Open Problem Theorem 2.4
Lemma 2.4 Lemma 2.5 Open Problem 2.6 Theorem 2.7
Theorem 2.5 Lemma Lemma 1 Lemma 2
Theorem 2.8 Lemma 3.2 Lemma 4.2 Lemma 4.3
Suslin cardinals, κ-Suslin sets and the scale property in the hyperprojective hierarchy, Kechris, [Cabal ii, pp. 127–146] Definition Theorem 1.2 Lemma Conjecture Conjecture Definition Definition Lemma 2.3
Definition 1.2 Theorem 1.3 Lemma 1.4 Conjecture 1.5 Conjecture 1.6 Definition 2.3 Definition 2.4 Lemma 2.5
Lemma 2.4 Theorem Theorem 3.1 Corollary 3.2 Corollary 3.3 Corollary 3.4 Corollary 3.5 §5(A)
Lemma 2.6 Theorem 3.1 Theorem 3.2 Corollary 3.3 Corollary 3.4 Corollary 3.5 Corollary 3.6 §5.1
xvii
original numbering Conjecture Problem §5(B) Question §5(C) Question
Conjecture 5.2 Question 5.3 §5.2 Question 5.4 §5.3 Question 5.5
§5(D) §5(E) Conjecture Conjecture Proposition
§5.4 §5.5 Conjecture 5.6 Conjecture 5.7 Proposition 5.8
A coding theorem for measures, Kechris, [Cabal iv, pp. 103–109] Theorem Corollary Corollary Corollary
Theorem 1.1 Corollary 1.2 Corollary 1.3 Corollary 1.4
Corollary Corollary Lemma
Corollary 1.5 Corollary 1.6 Lemma 2.1
Volume II Wadge degrees and descriptive set theory, Van Wesep, [Cabal i, pp. 151–170] Definition Definition Definition Definition Definition Definition Definition Definition Theorem 2.2 Definition Lemma 2.3 Claim 1 Claim 2 Lemma Definition Definition Corollary Definition Remark Theorem 4.1
Definition 1.1 Definition 1.2 Definition 1.3 Definition 1.4 Definition 1.5 Definition 2.2 Definition 2.3 Definition 2.4 Theorem 2.5 Definition 2.6 Lemma 2.7 Claim 2.8 Claim 2.9 Lemma 3.2 Definition 3.3 Definition 3.4 Corollary 3.5 Definition 4.1 Remark 4.2 Theorem 4.3
Remark Definition Lemma 4.2 Lemma 4.3 Lemma 4.4 Theorem 4.2 Definition Claim Claim Definition Theorem 5.1 Theorem 5.2 Theorem 5.3 Theorem 5.4 Theorem 5.5 Theorem 5.6 Claim Lemma Lemma
Remark 4.4 Definition 4.5 Lemma 4.6 Lemma 4.7 Lemma 4.8 Theorem 4.9 Definition 4.10 Claim 4.11 Claim 4.12 Definition 5.1 Theorem 5.2 Theorem 5.3 Theorem 5.4 Theorem 5.5 Theorem 5.6 Theorem 5.7 Claim 5.8 Lemma 5.9 Lemma 5.10
A note on Wadge degrees, Kechris, [Cabal ii, pp. 165-168] Lemma 1 Lemma 2
Lemma 2.1 Lemma 2.2
Sublemma
Sublemma 2.3
xviii
original numbering Some results in the Wadge hierarchy of Borel sets, Louveau, [Cabal iii, pp. 28–55]
Figure a Figure b Figure c
Figure 1 Figure 2 Figure 3
Figure d Figure e Claim
Figure 4 Figure 5 Claim 2.8
The strength of Borel Wadge determinacy, Louveau & Saint-Raymond, [Cabal iv, pp. 1–30] Definition 1 Definition 2 Definition 3 Theorem 4 Definition 5 Definition 6 Proposition 7 Theorem 8 Remark Definition 9 Proposition 10 Definition 1 Definition 2 Definition 3 Theorem 4 Lemma 5 Definition 6 Theorem 7 Lemma 8 Lemma 9 Lemma 10 Lemma 11 Theorem 1 Theorem 2
Definition 1.1 Definition 1.2 Definition 1.3 Theorem 1.4 Definition 1.5 Definition 1.6 Proposition 1.7 Theorem 1.8 Remark 1.9 Definition 1.10 Proposition 1.11 Definition 2.1 Definition 2.2 Definition 2.3 Theorem 2.4 Lemma 2.5 Definition 2.6 Theorem 2.7 Lemma 2.8 Lemma 2.9 Lemma 2.10 Lemma 2.11 Theorem 3.1 Theorem 3.2
Theorem 3 Theorem 4 Theorem 5 Corollary 6 Theorem Theorem Theorem Theorem 0 Theorem 1 Theorem 2 Definition 3 Definition 4 Theorem 5 Proposition 6 Theorem 7 Theorem 8 Corollary 9 Example 10 Theorem 1 Theorem 2 Definition 3 Proposition 4 Theorem 5 Lemma 6
Theorem 3.3 Theorem 3.4 Theorem 3.5 Corollary 3.6 Theorem 3.7 Theorem 3.8 Theorem 3.9 Theorem 4.1 Theorem 4.2 Theorem 4.3 Definition 4.4 Definition 4.5 Theorem 4.6 Proposition 4.7 Theorem 4.8 Theorem 4.9 Corollary 4.10 Example 4.11 Theorem 5.1 Theorem 5.2 Definition 5.3 Proposition 5.4 Theorem 5.5 Lemma 5.6
Closure properties of pointclasses, Steel, [Cabal ii, pp. 147–163] Claim Claim Claim
Claim 3.2 Claim 3.3 Claim 3.4
Theorem 3.2 Claim Theorem 3.3
Theorem 3.5 Claim 3.6 Theorem 3.7
xix
original numbering
The axiom of determinacy and the prewellordering property, Kechris, Solovay, & Steel, [Cabal ii, pp. 101–125] Theorem Theorem §2.3 Lemma 2.3.1 §2.4 Lemma 2.4.1 §2.5 Lemma 2.5.1 Lemma 2.5.2 Lemma 2.5.3 Definition Definition Theorem Theorem 3.2
Theorem 1.1 Theorem 1.2 §2.1 Lemma 2.3 §2.2 Lemma 2.4 §2.3 Lemma 2.5 Lemma 2.6 Lemma 2.7 Definition 3.1 Definition 3.2 Theorem 3.3 Theorem 3.4
Lemma 1 Lemma 2 Corollary 3.3 Corollary 3.4 Definition Definition Theorem 4.1 Definition Definition Theorem 5.1 Corollary 5.2 Corollary 5.3 Lemma
Lemma 3.5 Lemma 3.6 Corollary 3.7 Corollary 3.8 Definition 4.1 Definition 4.2 Theorem 4.3 Definition 5.1 Definition 5.2 Theorem 5.3 Corollary 5.4 Corollary 5.5 Lemma 5.6
Pointclasses and wellordered unions, Jackson & Martin, [Cabal iii, pp. 56-66] §0 §1 Theorem 1.1 Corollary 1.1.1 Corollary 1.1.2 Theorem 1.2 §2 Theorem 2 Lemma 2.1 Lemma 2.2 Lemma 2.3 Lemma 2.4
§1 §2 Theorem 2.1 Corollary 2.2 Corollary 2.3 Theorem 2.4 §3 Theorem 3.1 Lemma 3.2 Lemma 3.3 Lemma 3.4 Lemma 3.5
Lemma 2.6 Lemma 2.7 Lemma 2.8 Lemma 2.9 Lemma 2.10 §3 Theorem 3 Lemma 3.1 Lemma 3.2 Lemma 3.3 Lemma 3.4 —
Lemma 3.6 Lemma 3.7 Lemma 3.8 Lemma 3.9 Lemma 3.10 §4 Theorem 4.1 Lemma 4.2 Lemma 4.3 Lemma 4.4 Lemma 4.5 Addendum (2010)
More closure properties of pointclasses, Becker, [Cabal iv, pp. 31–36] Definition Lemma 5
Definition 5 Lemma 6
Theorem 6 Theorem 7
Theorem 7 Theorem 8
Homogeneous trees and projective scales, Kechris, [Cabal ii, pp 33–73] Lemma Remark Remark
Lemma 2.2 Remark 2.3 Remark 3.1
Theorem Theorem Theorem
Theorem 3.2 Theorem 5.1 Theorem 6.1
xx
original numbering
Theorem Corollary
Theorem 7.1 Corollary 7.2
Theorem Lemma A Lemma B
Theorem 8.1 Lemma 8.2 Lemma 8.3
AD and projective ordinals, Kechris, [Cabal i, pp. 91–132] Definition Definition Definition Theorem 2.1 Corollary Theorem 2.2 Definition Theorem 3.1 Definition Lemma Theorem 3.2 Definition Theorem 3.3 Claim Definition Definition Theorem 3.4 Definition Theorem 3.5 Theorem 3.6 Theorem 3.7 Claim Definition Theorem 3.8 Theorem 3.9 Theorem 3.10 Theorem 3.11 Theorem 3.12 Theorem 3.13 Theorem 3.14 Lemma 1 Lemma 2 Lemma 3 Lemma 4 Lemma 5 Claim Claim Theorem 6.4 Corollary 6.5 Definition
Definition 1.1 Definition 2.1 Definition 2.2 Theorem 2.3 Corollary 2.4 Theorem 2.5 Definition 3.1 Theorem 3.2 Definition 3.3 Lemma 3.4 Theorem 3.5 Definition 3.6 Theorem 3.7 Claim 3.8 Definition 3.9 Definition 3.10 Theorem 3.11 Definition 3.12 Theorem 3.13 Theorem 3.14 Theorem 3.15 Claim 3.16 Definition 3.17 Theorem 3.18 Theorem 3.19 Theorem 3.20 Theorem 3.21 Theorem 3.22 Theorem 3.23 Theorem 3.24 Lemma 5.2 Lemma 5.3 Lemma 5.4 Lemma 5.5 Lemma 5.6 Claim 5.7 Claim 6.4 Theorem 6.5 Corollary 6.6 Definition 8.1
Theorem 8.1 Theorem 8.2 Lemma A Lemma 8.3 Definition Definition 8.4 Lemma B Lemma 8.5 Lemma C Lemma 8.6 Theorem 8.2 Theorem 8.7 Theorem 8.3 Theorem 8.8 Theorem 8.4 Theorem 8.9 Lemma Lemma 8.10 Proposition 8.5 Proposition 8.11 Basic Open Problem Basic Open Problem 8.12 Basic Open Problem Basic Open Problem 9.4 Definition Definition 10.1 Remark Remark 10.2 Definition Definition 10.3 Theorem 10.1 Theorem 10.4 Definition Definition 12.1 Theorem 12.1 Theorem 12.2 Definition Definition 13.1 Theorem 13.1 Theorem 13.2 Theorem 13.2 Theorem 13.3 Corollary 13.3 Corollary 13.4 Corollary 13.4 Corollary 13.5 Lemma Lemma 13.6 Claim Claim 13.7 Theorem 13.5 Theorem 13.8 Theorem 13.6 Theorem 13.9 Definition Definition 14.1 Theorem 14.1 Theorem 14.2 Corollary 14.2 Corollary 14.3 Theorem 14.3 Theorem 14.4 Lemma A Lemma 14.5 Claim Claim 14.6 Lemma B Lemma 14.7 Claim Claim 17.3 Theorem 17.3 Theorem 17.4 Corollary 17.4 Corollary 17.5
xxi
original numbering A Δ13 coding of the subsets of , Solovay, [Cabal i, pp. 131–170] Introduction §A §A.1 §A.2 Lemma Corollary §A.3 Theorem §A.4 Theorem §A.5 §A.6 §A.7 §A.8 Definition Lemma 1 Lemma 2 §A.9 Definition A.9 Lemma 3 Lemma 4 §A.10 §A.11 Theorem A.11 Lemma 1 Lemma 2 Lemma 3 §A.12 Lemma §A.13 Lemma
§1 §2 §2.1 §2.2 Lemma 2.1 Corollary 2.2 §2.3 Theorem 2.3 §2.4 Theorem 2.4 §2.5 §2.6 §2.7 §2.8 Definition 2.5 Lemma 2.6 Lemma 2.7 §2.9 Definition 2.8 Lemma 2.9 Lemma 2.10 §2.10 §2.11 Theorem 2.11 Lemma 2.12 Lemma 2.13 Lemma 2.14 §2.12 Lemma 2.15 §2.13 Lemma 2.16
§A.14 Claim 1 Claim 2 Claim 3 Claim 4 Definition Claim 5 Claim 6 Claim 7 Claim 8 Claim A.9 Fact §B §B.1 Theorem §B.2 Lemma Lemma — §B.3 Definition Theorem Lemma §B.4 Theorem B.4 §B.5 Corollary B.5 §B.6 Theorem B.6 Claim
§2.14 Claim 2.17 Claim 2.18 Claim 2.19 Claim 2.20 Definition 2.21 Claim 2.22 Claim 2.23 Claim 2.24 Claim 2.25 Claim 2.26 Fact 2.27 §3 §3.1 Theorem 3.1 §3.2 Lemma 3.2 Lemma 3.3 Fact 3.4 §3.3 Definition 3.5 Theorem 3.6 Lemma 3.7 §3.4 Theorem 3.8 §3.5 Corollary 3.9 §3.6 Theorem 3.10 Claim 3.11
AD and the projective ordinals, Jackson, [Cabal iv, pp. 117–220] §I §II Definition Definition Definition Definition Lemma 1 Lemma 2 Lemma 3
§1 §2 Definition 2.1 Definition 2.2 Definition 2.3 Definition 2.4 Lemma 2.5 Lemma 2.6 Lemma 2.7
Lemma 4 §III Theorem §IV Theorem Definition §V Definition Condition D
Lemma 2.8 §3 Theorem 3.1 §4 Theorem 4.1 Definition 4.2 §5 Definition 5.1 Definition 5.2
xxii Condition C — Condition A Lemma Lemma Lemma 1 Lemma 2 Lemma 3 Lemma 4 Lemma 5 Lemma 6 Definition Definition Lemma 7 Lemma 8 Lemma 9 Lemma 10 Lemma 11 Lemma 12 Definition
original numbering Definition 5.3 Definition 5.4 Definition 5.5 Lemma 5.6 Lemma 5.7 Lemma 5.9 Lemma 5.10 Lemma 5.11 Lemma 5.12 Lemma 5.13 Lemma 5.14 Definition 5.15 Definition 5.16 Lemma 5.17 Lemma 5.18 Lemma 5.19 Lemma 5.20 Lemma 5.21 Lemma 5.22 Definition 5.23
Lemma Cofinality Lemma Main Inductive Lemma §VI Definition — — Theorem §VII Definition Definition — — — Lemma §VIII Theorem §IX —
Lemma 5.24 Lemma 5.26 Lemma 5.27 §6 Definition 6.1 Remark 6.2 Definition 6.3 Theorem 6.4 §7 Definition 7.1 Definition 7.2 Definition 7.3 Definition 7.4 Lemma 7.5 Lemma 7.6 §8 Theorem 8.1 §9 Definition 9.1
PART III: WADGE DEGREES AND POINTCLASSES
WADGE DEGREES AND POINTCLASSES INTRODUCTION TO PART III
ALESSANDRO ANDRETTA AND ALAIN LOUVEAU
§1. Introduction. One of the main objects of study in Descriptive Set Theory is that of boldface pointclass, that is a collection of subsets of the Baire space (or more generally: of a family of Polish spaces) closed under continuous preimages. Since in this paper we will have little use for the concept of lightface pointclass used in the effective theory, we will drop the ‘boldface’ and simply speak of pointclasses. Also, in order to avoid trivialities, we will always assume that a pointclass is non-empty and different from ℘(R). Despite the fact that the the concept of pointclass is both very simple and ubiquitous in modern Descriptive Set Theory, it is actually quite recent, at least in its modern conception. The French analysts at the turn of the twentieth century—Baire, Borel, and Lebesgue—and later Luzin, Suslin, Hausdorff, ´ Sierpinski, Kuratowski, always worked with specific pointclasses (such as the collection of all Borel sets, or the collection of all projective sets) defined by closure under set theoretic operations, and stratified into a transfinite hierarchy, e.g., the Baire classes Σ0α , Π0α , and Δ0α for the Borel sets, and that all these collections Σ1n , Π1n , and Δ1n for the projectivesets. The fact were closed under continuous preimages was probably considered a simple consequence of their definition, rather than a feature worth crystallizing into a mathematical definition. Even the fact that the Borel hierarchy (and similarly for the projective one) exhibited the well-known diamond-shape pattern
⊆
⊆
⊆
⊆
⊆
Σ0α
Δ0α+1
Σ0α+1 ...
⊆
· · · Δ0α
⊆
⊆
Δ02
Σ02
⊆
⊆
Σ01
Π01 Π02 Π0α Π0α+1 ←−−−−−−−−−−−−−−−−−−−−−−−− 1 −−−−−−−−−−−−−−−−−−−−−−−−→ apparently was not considered to be an indication of an underlying structure. Hausdorff showed that any Δ02 set can be represented as a transfinite difference closed) sets, and Kuratowski, by the trick of of open (or for that matter, refining the topology, extended this to all Δ0α+1 sets. Thus Δ0α+1 = <1 D Σ0α Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
3
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ALESSANDRO ANDRETTA AND ALAIN LOUVEAU
where D Γ denotes the class of -differences of sets in Γ, i.e., sets of the form {x ∈ < A : the least that x ∈ / A has parity different from } for some sequence A : < of sets in Γ. Again we obtain a picture similar to the one for the Borel hierarchy:
⊆
⊆
⊆
⊆
ΔD2 Σ0α
D3 Σ0α ···
⊆
⊆
ΔD2 Σ0α
D2 Σ0α
⊆
⊆
Σ0α = D1 Σ0α
Π0α = D1 Π0α (D2 Σ0α )˘ (D3 Σ0α )˘ −−−−−−− −−−−−−−−−−−−−−−−−−−− 1 −−−−−−−−−−−−−−−−−−−−−− −−−−−−→ ←−
Wadge in his Ph.D. thesis [Wad84] was the first to investigate in a systematic manner the notion of continuous reducibility on the Baire space . The motivation for his study, and the reason as to why these matters had not been studied before is explained in [Wad84, pp. 2–3]: The notion of reducibility, including many-one reducibility, plays an extremely important role in recursive function theory. One would expect the same to be true in descriptive set theory; but that has not (at least till recently) been the case. Of course, there are in the literature many instances in which continuous preimage is used to derive a particular result. In Sikorski (1957), for example, this approach is used to construct for each countable ordinal a set in the th but no lower level of the Borel hierarchy. Luzin ´ and Sierpinski (1929) used preimage to show that the collection of (codes for) wellorderings of is not Borel; and there are a number of other examples. Yet nowhere (to our knowledge) is the relation A = f −1 (B) for some continuous f ever explicitly defined and studied as a partial order, not even in exhaustive work such as Ku´ ´ ratowski (1958) or Sierpinski (1952). In the latter, Sierpinski discusses preimage in general, continuous image and homeomorphic image, but not (explicitly) continuous preimage, which is perhaps the most natural. One possible explanation is that the investigation of ≤ naturally involves infinite games, and it is only recently that game methods have been fully understood and appreciated.1 Wadge’s main objective was a complete analysis of all the Borel pointclasses, i.e., boldface pointclasses contained in Δ11 . Working in ZF+DC, he defined Borel hierarchy, he proved that a hierarchy of Borel sets refining the usual it is well-founded and computed its length, and, assuming the determinacy of all Borel games, he could show that every Borel pointclass fits in this classification. As explained by Wadge in [Wad11] in the present volume and 1 The relation ≤ is nowadays called Wadge reducibility and it is denoted by ≤ , and the W references mentioned are, in order, [Sik58], [LS29], [Kur58], and [Sie52].
INTRODUCTION TO PART III
5
in [Wad84, pp. 10–11], all these results were obtained before Martin’s proof of Borel determinacy [Mar75]. The problem whether Borel determinacy is needed to prove that all Borel pointclasses fall into Wadge’s analysis remained open for over a decade, until Louveau and Saint-Raymond answered in the negative, by conducting Wadge’s analysis within second order arithmetic (see the paper [LSR88B] in this volume). As we mentioned, all pointclasses considered by early descriptive set theorists were defined in terms of operation on sets, like taking complements, countable intersections, countable unions, Suslin’s operation A , etc. All of these operations can be thought as operations O : ℘(R) → ℘(R) assigning a new set to a countable sequence of sets, and with the property that there is a T ⊆ ℘() such that for any An : n ∈ ∀x ∈ R (x ∈ OAn : n ∈ ⇐⇒ {n ∈ : x ∈ An } ∈ T ) . A function O as above is said to be an -ary Boolean operation, or simply a Boolean operation, and the set T = TO which completely determines O, is called the truth table of O. We will say that such an operation is Borel, or Σ11 , etc., if its truth table is Borel, or Σ11 , etc., as a subset of 2. For example: countable intersections, or countable operations of taking complements, the unions, as well as their compositions are all Borel, while Suslin’s operation A is Σ11 . Wadge showed in ZFC that each non-self-dual Borel pointclass in is of the form {OAn : n ∈ : ∀n (An is open)} with O a Borel Boolean operation, and Van Wesep in [Van77], assuming AD and building on earlier results of Miller, Radin, and Steel, extended this result to all non-self-dual pointclasses, using of course arbitrary Boolean operations. Thus we have come to a full circle—non-self-dual pointclasses considered by early descriptive set theorists were defined in terms of (explicit) operations, and assuming AD every non-self-dual pointclass is defined in terms of operations on open sets. Boolean operations are operations on the collection of open sets that allow us to construct all sets belonging to complicated pointclass Γ, and they figure prominently in the work of Louveau and Saint-Raymond[Lou83, LSR87, LSR88B]. But Wadge also introduced certain specific operations on sets which yield complete sets for various Γ. Thus a non-self-dual pointclass Γ can be described either as obtained via some some appropriate Boolean operation 0 O, Γ = {OAn : n ∈ : An ∈ Σ1 }, or else as the set of continuous : X ≤ A}. These operations on preimages of a Γ-complete set A, Γ = {X W sets are quite useful to compute the Wadge rank of the various pointclasses,
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ALESSANDRO ANDRETTA AND ALAIN LOUVEAU
and were extensively used in [Ste81B] and [Van77]. Recently this approach to the Wadge hierarchy has been extended in the work of Duparc and others in connection with automata theory, see [Dup01, Dup03, DFR01]. In the next section we give a few basic definitions an review some basic results on the Wadge hierarchy. §2. Some basic facts about the Wadge hierarchy. The relation of Wadge reducibility, A ≤W B, is defined as A = f −1 (B) for some continuous function f. It can be defined for any pair of ambient topological spaces: X containing A and Y containing B, so that f : X → Y, but the general theory becomes somewhat uninteresting if the spaces are not zero-dimensional, as there may be, in general, very few continuous maps. Following Wadge, from now on we will focus on the Baire space , which—as customary in set theory—will be denoted by R. A continuous f : R → R is determined by a monotone ϕ : < → < such that limn lh ϕ(xn) = +∞. If we require that lh ϕ(xn) = n, then the resulting f is a Lipschitz function with constant ≤ 1, where we use the usual distance on . In this case we will say that A is Lipschitz reducible to B. Wadge introduced the Lipschitz game GL (A, B): it is a game on player I a0 player II
a1 b0
b1
··· ···
where player II wins iff ai : i < ∈ A ⇐⇒ bi : i < ∈ B . Thus player II has a winning strategy for the game GL (A, B) if and only if A ≤L B. Conversely, if player I has a winning strategy, then there is a Lipschitz map witnessing B ≤L ¬A. Note that in this case player I’s strategy yields a ϕ : < → < such that lh(ϕ(s)) = lh(s) + 1 hence the induced f : R → R is a Lipschitz map with constant 1/2, and in fact the converse implication (if B ≤L ¬A then player I wins GL (A, B)) in general does not hold. Assuming determinacy we obtain the following simple—yet fundamental— result known as: Wadge’s Lemma. Assume AD. Then ∀A, B ⊆ R (A ≤L B ∨ B ≤L ¬A) . The gist of the result is that any two sets of reals are almost comparable, and that ≤L is almost a linear order. Wadge dubbed this as the Semi Linear Ordering principle for Lipschitz reductions. As every Lipschitz reduction is, in particular, a Wadge reduction, Wadge’s Lemma yields trivially the Semi Linear Ordering principle for continuous reductions: ∀A, B ⊆ R (A ≤W B ∨ B ≤W ¬A) .
7
INTRODUCTION TO PART III
Since it turns out that these two versions are equivalent (assuming DC(R) and that all sets have the property of Baire—see [And03, And06]) we shall denote either version with SLO. The quasi-order2 ≤W induces an equivalence relation on ℘(R) whose equivalence classes are called Wadge degrees. The collection of all Wadge degrees together with the induced order is called the Wadge hierarchy. A set of reals A or its Wadge degree [A]W is said to be self-dual if it A ≤W ¬A; otherwise it is said to be non-self-dual. Wadge’s Lemma implies that a self-dual degree is comparable to any other degree, and that if two degrees are incomparable, then they must be dual to each other. In other words: all antichains have size at most 2. Martin—building on previous work of Monk—showed in 1973 that AD implies that the ordering ≤ is well-founded. In fact these results hold verbatim for the Lipschitz hierarchy, i.e., the collection of degrees [A]L obtained using Lipschitz reductions. By a result due independently to Steel [Ste77] and Van Wesep [Van77], AD implies that A ≤W ¬A ⇐⇒ A ≤L ¬A
(1)
and using this it is possible to completely determine the structure of the Lipschitz hierarchy: at the bottom of the hierarchy we have the non-self-dual pair {∅} = [∅]L and {R} = [R]L , followed by an 1 chain of self-dual degrees formed by all clopen sets different from ∅ and R. Above these there is the non-self-dual pair Σ01 \ Δ01 and Π01 \ Δ01 followed by an 1 chain of self-dual ofuncountable cofinality we have a nondegrees. In general: at limit levels self-dual pair, while at all other levels we have a self-dual degree. The length of this hierarchy [Sol78B] is def
Θ = sup{α : ∃f (f : R α )}. Thus the Lipschitz hierarchy looks like this:
•
• • • ··· ←− 1 −→
•
• • • ··· ←− 1 −→
•
······
cf =
cf >
↓
↓ •
•
······
(2) ······
• • • • ←−−−−−−−−−−−−−−−−−−−−−−− Θ −−−−−−−−−−−−−−−−−−−−−−−→ Each block of 1 consecutive self-dual Lipschitz degree is contained inside a single (necessarily self-dual) Wadge degree, and by the result of Steel and Van Wesep (1) nothing else is, so in the Wadge hierarchy self-dual degrees and non-self-dual pairs alternate, with the former appearing at levels of countable 2 A quasi-order is a reflexive and transitive relation, and it is also known in the literature as a pre-order.
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ALESSANDRO ANDRETTA AND ALAIN LOUVEAU
cofinality, and the latter appearing at the remaining limit levels: cf >
cf =
•
•
•
•
•
↓ •
······
•
•
•
······
↓ •
(3) •
······
• • • • • ←−−−−−−−−−−−−−−−−−−−−−− Θ −−−−−−−−−−−−−−−−−−−−−−→ The Wadge hierarchy is the ultimate analysis of ℘(R) in terms of topological complexity, assigning to each set A ⊆ R an ordinal AW , the rank of [A]W in the hierarchy. This is somewhat surprising, since AD forbids the existence of long transfinite sequences of reals. It is not hard to check that non-self-dual pointclasses are of the form {B ⊆ R : B ≤W A} for some non-self-dual set A, while self-dual pointclasses are all of the form {B ⊆ R : B <W A} for some arbitrary A = R, ∅. (Here and below, B <W A has the obvious meaning: B ≤W A and A W B.) Thus Wadge’s Lemma yields a semi-linear ordering principle for pointclasses: for any Γ and Λ, ˘ ⊆ Γ. Γ⊆Λ∨Λ It is a classical fact that any pointclass Γ of the form Σ0α , Π0α , Σ1n , or Π1n has belongs to Γ (once itis coded as a a universal set, i.e., a set U ⊆ R × R that subset of R via some canonical homeomorphism) and such that Γ = {U(x) : x ∈ R} where U(x) = {y ∈ R : (x, y) ∈ U } is the vertical section of U through x. This fact generalizes to all non-self-dual boldface pointclasses. To see this, fix some canonical enumeration x : x ∈ R of all Lipschitz maps R → R with the further property that (x, y) → x (y) is continuous, and let U = {(x, y) : x (y) ∈ A}, where A is any set in Γ \ Γ˘ . Then U is in Γ, and since B ∈ Γ ⇐⇒ B ≤W A ⇐⇒ B ≤L A (by (1)) we obtain that U is universal for Γ. under AD to arbitrary pointclasses is the Another property that generalizes following: a non-self-dual pointclass Γ is said to have the separation property, in symbols Sep(Γ) if for any pair of disjoint sets A, B ∈ Γ there is a set C ∈ def ˘ ΔΓ = Γ ∩ Γ that separates A from B, that is A ⊆ C and C ∩ B = ∅. By work
INTRODUCTION TO PART III
9
´ of Sierpinski Π0α has the separation property, while Σ0α does not; assuming PD Moschovakis showed that Σ12n+1 and Π12n have theseparation, while neither Σ12n nor Π12n+1 has it. (For Σ11 this is theclassical result of Suslin, and does not given a pair of non-self-dual pointclasses Γ and require PD.) Assuming AD, ˘ Γ, at most one of them has the separation property [Van78A], and at least one of them has the separation property [Ste81B], hence exactly one of them has the separation property. The pointclasses Σ0α can be detected inside the Wadge hierarchy by means of the rank of their complete sets. Starting from the very bottom, R and ∅ have least possible rank, which for technical reasons is set to be equal to 1, then the clopen set have ranks 2, and thus sets in Σ01 \ Δ01 have rank 3, From this point For example complete Σ0 sets on the Σ0α are more and more spread apart. 2 have Wadge rank 1 , complete Σ03 sets have Wadge rank 11 , and, in general, complete Σ0n+1 sets have Wadge rank ϑn where ϑ1 = 1 and ϑk+1 = 1ϑk . The 0 rank ϑ of a complete Σ set is not the sup of the ϑn s, i.e., the first fixed point of the map E : Ord → Ord
(4)
→ 1 , since this ordinal has countable cofinality, and hence it is the rank of a self-dual set. It turns out that ϑ is the 1 -st fixed point of the map E. The computation of the ranks of Σ0α with α ≥ is quite technical—see [Wad11] for a summary [Wad84, Chapter V] for complete proofs. For example, the of the results and length Ξ of the Wadge hierarchy of the Borel sets, or, equivalently, the rank of a complete Σ11 or Π11 set, is computed as follows: for any cub class C ⊆ Ord let C = { : = FC ( )} be the set of fixed points of FC , where FC : Ord → C is the enumerating function, and consider the sequence of cub classes C = C (0) ⊃ C (1) ⊃ C (2) ⊃ . . . given by C (α+1) = C (α) and C ( ) = α< C (α) when is limit. Then Ξ is the least element of C (1 ) where C is taken to be the class of fixed points of the map E defined in (4). Thus the length of the Wadge degrees of Borel sets is an ordinal of cofinality 1 strictly smaller than 2 . This is not just an happenstance, since under AD the length of the hierarchy of Δ12n+1 degrees is < 12n+2 . On the other hand, by a theorem due independentlyto Martin and the length of the hierarchy of Δ1 degrees is equal to 1 . Steel, 2n 2n+1 §3. The papers in the volume. Early investigations of the degrees of Borel sets by W. W. Wadge. This paper is an overview of the results of the author’s Ph.D. dissertation [Wad84] and gives a glimpse on how this area of Descriptive Set Theory was
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ALESSANDRO ANDRETTA AND ALAIN LOUVEAU
uncovered. Although it contains no proofs, this article gives a quick introduction to the techniques ((α, )-homeomorphisms, Σ01+ -separated unions, etc.) used to give a complete analysis of the Wadge degrees of the Borel sets, and a computation of its length Ξ. Wadge degrees and descriptive set theory by R. Van Wesep, and A note on Wadge degrees by A. S. Kechris. Van Wesep’s paper provides a good introduction to the subject, with complete (albeit terse) proofs, surveying what was known at that time (1978). The reader will find the proof of several of the results stated in the preceding section, including Martin’s proof of the well-foundedness of ≤L , the result by Steel and Van Wesep on self-dual degrees stated in (1), and the proof under AD that the hierarchy of Δ12n degrees has length 12n+1 . As this last fact is a is natural to ask for a proof assuming only PD. result on projective sets, it Such a proof is given in Kechris’ paper, where Det(Δ12n ) is shown to suffice. The rest of Van Wesep’s paper is devoted to the reduction and prewellordering properties. Recall that Γ is said to have the reduction property, in symbols Red(Γ), if given any twosets A, B ∈ Γ there are disjoint sets A , B ∈ Γ such ⊆ A, B ⊆ B, and A ∪ B = A ∪ B; The prewellordering property that A PWO(Γ) means that every set in Γ admits a Γ norm [KM78B]. For any non ˘ selfdual pointclass, Red(Γ) =⇒ Sep(Γ ) and if moreover Γ is closed under finite ) =⇒ Red(Γ) [KM78B, unions and intersections, then PWO(Γ Theorem 2.1]. 0 Every Σα has the prewellordering and hence the reduction property, and Lou Saint-Raymond have shown that for Borel pointclasses, the reducveau and tion property and prewellordering properties are equivalent, and have given a complete description of which Borel pointclasses possess this property— see [LSR88A]. For the sake of brevity, we say that a non-self-dual pair of ˘ ) satisfies the prewellordering property if either PWO(Γ) or pointclasses (Γ, Γ ˘ ), and we follow a similar convention for the reduction property. else PWO(Γ In ˘ Van Wesep’s paper it is shown that there are non-self-dual pairs (Γ, Γ) that fail to have the reduction property, and since (under AD, which will betacitly assumed from now on) the separation property holds at every level of the Wadge hierarchy, this shows that the separation property is weaker that the reduction ˘ ) satisfy the reduction property. Determining which non-self-dual pairs (Γ, Γ property is a non-trivial matter. In the paper under review it is shown that If Γ is non-self-dual and closed under finite intersections then (5) ˘ Sep(Γ ) =⇒ Red(Γ). (Notice that by the result mentioned below in (13), the hypothesis could be weakened to ΔΓ .) A result of Steel is presented: If Γ is non-self-dual and closed
INTRODUCTION TO PART III
11
under countable unions and intersections, then reduction holds for (Γ, Γ˘ ). This result was strengthened shortly afterwards by Steel himself in [Ste81B]: If Γ is non-self-dual and ΔΓ is closed under finite unions and intersec- (6) ˘ ). tions, then the reduction property holds for (Γ, Γ Finally the proof a theorem of Kechris and Solovay is given: Suppose Γ ⊆ L(R) is non-self-dual and closed under countable unions (7) and countable intersection. Suppose also ∃R Γ ⊆ Γ and ∀R Γ ⊆ Γ. Then prewellordering holds for (Γ, Γ˘ ). The axiom of determinacy and the prewellordering property by A. S. Kechris, R. Solovay, and J. Steel. This paper, as the title suggests, is devoted to the study of the prewellordering property under AD and, in a sense, it starts from where Van Wesep’s paper ended. Firstly a criterion for PWO is established: Suppose Γ is non-self-dual, closed under countable unions and inter- (8) either ∃R Γ ⊆ Γ or else ∀R Γ ⊆ Γ. Then the prewellordersections, and ing property holds for the non-self-dual pair (Γ, Γ˘ ) if and only if ΔΓ is not closed under well-ordered unions. Recall that a pointclass Λ is closed under well-ordered unions if α< Aα ∈ Λ ˘ and for any sequence Aα : α < of sets in Λ. Note that if A ∈ Γ \ Γ ϕ : A κ is a regular Γ-norm, then each Aα = {x ∈ A : ϕ(x) < α} ∈ ΔΓ , / ΔΓ , so one of the two directions of the equivalence is but A = α<κ Aα ∈ of Kechris and Solovay stated in (7) is thus extended immediate. The Theorem to the case when Γ is closed under only one real quantifier: Suppose Γ ⊆ L(R) is non-self-dual and closed under countable unions (9) and countable intersection. Suppose also ∃R Γ ⊆ Γ or ∀R Γ ⊆ Γ. Then ˘ ). prewellordering holds for (Γ, Γ ˘ has the preIf Γ is Σ1n or Π1n then (9) says that exactly one among Γ and Γ wellordering property—in fact by Moschovakis’ First and Second Periodicity Theorems [KM78B] we can actually determine which of the two pointclasses has this property, namely PWO(Γ) iff Γ = Π12n or Γ = Σ12n+1 . The authors namely Γs which pointclasses, establish an analogous results for projective-like are contained in L(R), closed under countable unions and intersections, and R R closed under exactly one among ∃ or ∀ . Any such pointclass can be taken to be the base of a hierarchy, obtained by taking complements and closure under ∃R and ∀R , and if Γ itself is minimal, i.e., it is not of the form ∃R Λ or the resulting hierarchy is maximal. Call such an ∀R Λ for some Λ ⊂ Γ, then object a projective-like hierarchy. The projective-like hierarchies are classified
12
ALESSANDRO ANDRETTA AND ALAIN LOUVEAU
into four distinct types, and for each type the appropriate pattern for the prewellordering properties is established, first for the base level, and then for the higher levels by Moschovakis’ periodicity. Since each projective-like pointclass is contained in a unique projective-like hierarchy, this yields a complete analysis of the prewellordering property for projective-like pointclasses. Pointclasses and well-ordered unions by S. C. Jackson and D. A. Martin. In this paper the general question of when a pointclass is closed under wellordered unions is addressed. First a couple of easy facts are recalled: if ˘ is not closed under well-ordered unions of length κ, PWO(Γ) holds, then Γ where κ is the length of a Γ-norm; if moreover Γ is closed under countable under ∃R , then Γ is closed under well-ordered unions and intersections, and unions of length κ. Then Jackson and Martin prove under AD+DC that Suppose Γ is non-self-dual and closed under ∃R and ∀R . Then either (10) ˘ is Γ or Γ closed under well-ordered unions. Thus if Γ is as above and moreover PWO(Γ), then Γ is closed under well Lemma 2.4.1 in the preceding ordered unions. This last result complements paper by Kechris, Solovay, and Steel, which proves the same result3 assuming that Γ is closed under countable unions, countable intersections, under ∃R but not under ∀R . Therefore If Γ is non-self-dual and closed under countable intersections and ∃R , (11) PWO(Γ) holds, then Γ is closed under well-ordered unions. and Clearly, for a pointclass Γ to be closed under well-ordered unions is a meaningful property inasmuch there are well-ordered sequences of sets in Γ to be , then considered. Moreover if Aα : α < is a sequence of sets in such a Γ by replacing each Aα with <α A and thinning out the sequence if needed, we may assume that the sets are strictly increasing. In this paper it is shown, assuming AD+DC, that If S(κ) has the scale property and cf(κ) > , then there is no strictly (12) increasing sequence of sets in S(κ) of length κ + , where S(κ) is the class of all κ-Suslin sets. The proof breaks down into two cases, depending whether κ is a successor or limit of uncountable cofinality. The strength of Borel Wadge determinacy by A. Louveau and J. Saint-Raymond, and Some results in the Wadge hierarchy of Borel sets by A. Louveau. Harrington proved in [Har78] that the semi-linear ordering principle restricted to the class of Π11 sets, SLO(Π11 ) for short, implies the existence of x # , for any 3 Actually
in that paper the assumption PWO(Γ) is replaced by the weaker Red(Γ).
INTRODUCTION TO PART III
13
real x, and therefore it implies Det(Π11 ). By work of Harrington and Martin Det(Π11 ) is equivalent to the determinacy of Boolean combinations of Π11 sets, hence it follows that SLO(Π11 ), the determinacy of all GW (A, B) and GL(A, B), 1 ), are all equivalent. In fact the determinacy of with A, B ∈ Π11 , and Det(Π 1 all Wadge games GW , the determinacy of all Lipschitz games GL , and SLO are all equivalent [And03, And06] and the same holds true when restricted to any pointclass with sufficient closure properties, such as the Π1n ’s; for these reason we shall refer to any one of these hypotheses as Wadgedeterminacy. By [Har78] and [Ste80], Det(Π11 ) is also equivalent to the following: ∀A, B ∈ Π11 \ Δ11 ∃f f : R → R is a Borel isomorphism and f(A) = B ) . All these results lent some credibility to the conjecture that a similar pattern should occur in the Borel context, namely that Wadge determinacy for Borel sets should imply Borel determinacy, which by work of Martin [Mar75] holds in ZFC and by work of Friedman [Fri71B] is not provable in second order arithmetic. But it is not so, as proved in the first paper by Louveau and SaintRaymond: Wadge determinacy is provable in second order arithmetic. The proof relies heavily on Wadge’s analysis of the Borel classes, together with a “ramification” technique which appeared in [LSR87] for the Borel classes: one associates to each non-self-dual Borel Wadge degree, as described by Wadge, a specific game, which is somewhat of an unfolding of a Wadge game. Its determinacy implies that any set which is of this degree is strategically complete, i.e., player player II wins with it the Wadge game against any other set in the class. The second paper [Lou83] is a bit different from the other papers, as it deals with the “lightface” aspects of the Wadge hierarchy. In a previous paper [Lou80], Louveau had proved that for hyperarithmetic sets, the Borel class can be witnessed hyperarithmetically. A similar feature is proved in the paper for each Borel class in the Wadge hierarchy of Borel sets. But a great deal of work is done on introducing operations in order to build all Borel Wadge classes, and define appropriate codings of both the classes and the sets in them so that the corresponding lightface statement makes sense. It was also the first—and for quite some time the only—place where a printed account of some of Wadge’s work could be found. Closure properties of pointclasses by J. Steel, and More closure properties of pointclasses by H. Becker. In several of the results mentioned in the paragraphs above, in order to prove that a pointclass Γ has some structural property, like reduction or prewellor dering, we must require that Γ (or perhaps ΔΓ ) be closed under some simpler countable) unions or intersecstructural property, like closure under finite (or tions. Notice that closure under finite union or intersections is never a problem
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with the Baire or projective classes, but in the realm or arbitrary pointclasses, closure under finite unions or intersections is a non-trivial matter. One might ask, for example: Under which assumptions on Γ does closure under finite unions imply closure for countable unions? Do closure properties of ΔΓ imply analogous properties for Γ or Γ˘ ? The paper by Steel proves several theorems under AD that address these questions. Here is just a sample of such results: (13) If ΔΓ is closed under finite (or countable) unions and Sep(Γ) then Γ is closed under finite (or countable) unions too. holds, If Γ is closed under finite unions and Sep(Γ˘ ), then Γ is closed (14) countable unions. under Suppose Γ is closed under finite intersections and countable (15) not under countable intersections. Then PWO(Γ). unions, but Thus (14) and (15) generalize a well-known fact about the Borel hierarchy, that is: Σ0α does not have the separation property but has the prewellordering Steel’s paper contains also an interesting conjecture. Recall that property. Suslin’s operation A is a Boolean operation with Σ11 truth table, and that compositions of the the Boolean operations that generate the Σ0α s are just operations of countable unions and countable intersections. Conjecture 3.1. Assume AD and suppose Γ is non-self-dual and closed un der both countable intersections and countable unions. Then either Γ or Γ˘ is closed under A . Becker’s paper deals with closure under measure and category quantifiers. If A ⊆ 2 × 2 then let ∀∗ y A = {x : A(x) is comeager} and ∀ y A = {x : (2 \ A(x) ) = 0} where is the Lebesgue measure on 2 and A(x) = {y : (x, y) ∈ A} is the vertical section of A through x. In other words, ∀∗ y A is the set of all x such that (x, y) ∈ A for comeager many y, while ∀ y A is the set of all x such that (x, y) ∈ A for -almost every y; their dual quantifiers are defined by ∃∗ y A = {x : A(x) is non-meager} and ∃ y A = {x : (A(x) ) > 0}. The measure and category quantifiers are very useful in many parts of Descriptive Set Theory—see for example [BK96]. In the present paper it is shown
INTRODUCTION TO PART III
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that if Γ is nonselfdual and closed under countable unions and countable in tersections, then it is closed under the category and measure quantifiers. In particular, Δ11 is closed under measure and category quantifiers. More measures from AD by J. Steel. One of the early consequences of determinacy is Martin’s result that 1 has the strong partition property, 1 → (1 )1 . This in turns implies Solovay’s result that 1 is measurable. In the following years the study of the strong partition property for cardinals < Θ became one of the main research topics of the Cabal Seminar. The construction of the normal measure from the strong partition property is usually achieved via the Boundedness Lemma together with an appropriate coding of elements of κκ. In the present paper it is shown that, assuming AD, for every regular κ < Θ there is a measure on κκ. The main technical twist is the use of the Recursion Theorem instead of the Boundedness Lemma. §4. Recent developments. In this last section we will try to survey some of the development that occurred after the papers in this volume were originally written. 4.1. SLO and weaker reducibilities. As we already mentioned, Harrington proved in [Har78] that SLO(Π11 ) is equivalent to the determinacy of all Π11 games. This was extended by Hjorth [Hjo96] to the next level, i.e., SLO(Π12 ) implies Π12 -determinacy—generalizations of these results to all projective lev els, and beyond, have been an elusive goal, as they seem to depend on further technical advancement of core model theory. Yet the results we have now seem to lend some evidence to the following conjecture, probably due to Solovay: Conjecture 4.1. Assume V = L(R). Then SLO =⇒ AD. Note that there is no obvious natural way to reduce a general perfect information, zero-sum game on into a Wadge game, so the proof—if the conjecture is true—will probably be quite indirect. Although progress on this conjecture has been essentially nil after [Hjo96], the Semi-Linear Ordering principle and some generalizations of it have been investigated in recent years. In [And03] it is shown that SLO is strong enough to prove the basic structural results on the Wadge hierarchy as embodied in diagram (3), and in [AM03], the analogue of the Wadge hierarchy using Borel functions was introduced: for any A, B ⊆ R let A ≤Δ11 B ⇐⇒ ∃f f : R → R is Borel and f −1 (B) = A . The induced equivalence relation yields the notion of Δ11 degree, and it turns out that the their structure is similar to the one of Wadgedegrees, i.e., it is wellfounded, the self-dual degrees and non-self-dual pairs of degrees alternate,
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with self-dual degrees occupying the limit levels of countable cofinality, and since the length of this hierarchy is Θ, then its picture is just (3). Since all uncountable Polish spaces are Borel isomorphic, this hierarchy is independent of the underlying space, a feature sorely missing from the Wadge hierarchy. In this case there are no analogues of the games GW or GL , and the proofs use Δ11 the principle SLO , the analogue of SLO for Borel reductions, ∀A, B ⊆ R A ≤Δ11 B ∨ ¬B ≤Δ11 A .
1
Δ1 (SLO )
1
Δ1 Note that SLO follows from SLO, hence from AD, and in [AM03] it is Δ11 conjectured that SLO =⇒ SLO. In [And06] a similar analysis is carried out Δ02 for the Δ02 reducibility: again SLO is able to civilize this hierarchy and the familiarstructure (3) is obtained, and moreover in this case it is shown that Δ02 SLO ⇐⇒ SLO. (A function is said to be Δ0α if the preimage of a Σ0α is Σ0α .) true of The results above seem to indicate that similar results should hold ≤F reductions, i.e., A ≤F B ⇐⇒ ∃f ∈ F A = f −1 (B)
where F ⊆ R R. Obviously the class F must satisfy some assumptions in order for us to obtain non trivial results, e.g., F must be closed under composition, and must contain the identity, so that ≤F is a quasi-order, F = R R, etc. Motto Ros in [MR07] has isolated a very general class of F as above, with F a collection of Borel functions, and has shown, assuming AD+DC(R) that the structure of the F-hierarchy can be either of Wadge-type or of Lipschitz type, i.e., the ordering of the F-degrees is as in (3) or as in (2). For example: when F is the collection of all Δ0 functions, the resulting hierarchy is of Wadge type; when F is the collection of all Δ0α functions for some α < , the resulting hierarchy is of Lipschitz type. 4.2. Connections with bqo theory. By Martin’s result, Wadge reducibility ≤W is one of a few examples of “natural” quasi-orderings which are well-quasiorderings (wqo’s), i.e., which admit neither infinite antichains, nor infinite strictly decreasing sequences. Other famous examples are the countable linear orders with embeddability (Laver [Lav71]) and the finite graphs with the minor ordering (Robertson and Seymour [RS04]). As the class of wqo’s lacks nice closure properties, it is usual to consider the stronger notion of a better-quasi-ordering (bqo): A quasi-ordering (Z, ≤Z ) is a bqo if, for any continuous (or equivalently Borel) map h : [] → Z there is an X ∈ [] with h(X ) ≤ h(X \ {min X }), where [] is the collection of all infinite subsets of identified with the set of all increasing elements of the Baire space, and Z is taken with the discrete topology. (For a nice introduction to bqo theory, see Simpson’s contribution in [MW85, Chapter 9].)
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It is not hard to check that under AD the quasi-orders ≤L and ≤W are indeed bqo’s. But one can get by similar techniques other bqo results. For example, van Engelen, Miller, and Steel prove in [vEMS87] that if (Z, ≤Z ) is a bqo and one orders SZ , the set of all functions h : → Z, by h1 h2 ⇐⇒
∃ϕ : → Lipschitz such that ∀x ∈ h1 (x) ≤Z h2 (ϕ(z)) ,
then SZ is bqo too. (≤L corresponds to the case Z = {0, 1}, with 0 and 1 incomparable.) This result in turn is used to prove other bqo results, in particular in [LSR90], where Louveau and Saint-Raymond extend Laver’s result about countable linear orders to Borel (or projective) linear orders embeddable in (R, ≤lex ), using AD. 4.3. Reducibility in higher dimension. An alternative way of looking at the Wadge hierarchy is to view subsets A of R as structures (R, A) in a language with a unary predicate, with the Wadge ordering being continuous homomorphisms between such structures. This of course opens the possibility of extending it to more complicated structures with domain R (or arbitrary Polish spaces) and, say, a n-ary relation on it. Concretely, in order to allow arbitrary Polish spaces as domains and still avoid purely topological difficulties, one prefers to consider Borel reductions rather than continuous in this context. So for X , Y Polish spaces and A ⊆ X n , B ⊆ Y n , let (X , A) ≤B (Y, B) just in case
¯ ∈ B) ∃f : X → Y Borel, and ∀a¯ ∈ X n a¯ ∈ A ⇐⇒ f(a)
(where we follow the convention from model-theory and write a¯ for the n¯ for (f(a1 ), . . . , f(an )); also, when the ambient tuple (a1 , . . . , an ) and f(a) spaces X and Y are understood, we simply write A ≤B B). These considerations provide a natural descriptive complexity for relations. This notion was first introduced by Friedman and Stanley in [FS89], who used it to provide a classification for first order theories, by comparing their associated space of countable models with domain , endowed with isomorphism. It was extended soon after by Kechris and Louveau [Kec92, Lou92] to equivalence relations and even more complicated structures. It should be noted that many properties, like being an equivalence relation, a quasi-ordering, . . . are downward preserved under ≤B , so that the subject breaks naturally into many sub-areas. And in each sub-area there is no satisfying alternative approach to descriptive complexity by using operations instead of reducibility, as in the one-dimensional case. This is because equivalence relations, for example, are not built from simpler equivalence relations, in general. And
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the only ways that have been proposed, like Louveau’s notion of “potential Wadge class” (see [Lou94]), may be useful but are too coarse (i.e., too close to the one-dimensional situation) to provide the right notion of descriptive complexity. A lot of work has been done on linear orders, quasi-orders, even graphs, but the main part of the activity in Descriptive Set Theory over the last two decades has been to understand Polish spaces with Borel, or more generally analytic, equivalence relations. We won’t try to give here an account of this theory, but refer the reader to the nice overview [HK01]. Let us just mention here that the situation for the higher dimensional theory is very different, and much more complicated that in dimension one: although some features of ≤B are nice, it is a very complicated quasi-order, ill-founded and with large antichains. And games are of little use in the new situation, so that one cannot really work by analogy with ≤W . 4.3.1. Definable cardinality. In the context of AD and using arbitrary reductions rather than Borel ones, the classification results for equivalence relations become results on cardinality of quotients: if denotes this coarser reducibility relation, any f witnessing E F induces an injection fˆ : R/E → R/F . Conversely, assuming ADR , for any g : R/E → R/F we can uniformize the relation g˜ = {(x, y) ∈ R2 : f([x]E ) = [y]F } by some f : R → R: then f witnesses E F and moreover fˆ = g. In other words, under AD the quasi-order on equivalence relations yields an injection of the quotients, and under ADR any injection of the quotients lifts to a -reduction on R. Many of the results on Borel or analytic equivalence relations using ≤B , can be recast under AD using with essentially the same proof: for example the Silver [Sil80] and Harrington-Kechris-Louveau [HKL90] dichotomies become: If E ∈ Π11 is an equivalence relation on R, then either |R/E| ≤ , or else |R| ≤ |R/E|; if E ∈ Δ11 is an equivalence relation on R, then either ≤ |R/E|. But in fact these dichotomies of |R/E| ≤ |R| or else |℘()/ Fin| admit a more substantial generalization: If E an arbitrary equivalence relation on R, then either (a) |R/E| < Θ, or else (b) |R| ≤ |R/E|.
(16)
If E ∈ an arbitrary equivalence relation on R, then either (a) |R/E| ≤ |℘(κ)|, for some κ < Θ, or else (b) |℘()/ Fin| ≤ |R/E|.
(17)
Dichotomies (16) and (17) were first proved under ADR by Harrington and Sami [HS79], and Ditzen [Dit92], and, independently, by Foreman and Magidor (unpublished). The consistency strength was then reduced to AD + V=L(R) by Woodin (unpublished), and Hjorth [Hjo95], respectively.
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4.4. The Wadge hierarchy in set theory. Although most of the research on the notion of continuous pre-images is concerned with the general theory of pointclasses, the Wadge hierarchy has important applications in the study of models of AD+ , a generalization of AD defined by: • every A ⊆ R is ∞-Borel, i.e., A = {y ∈ R : Lα [S, y] |= ϕ[S, y]} for some S ⊆ Ord and some formula ϕ, and • for every A ⊆ R, every < Θ and every surjection f : R, the ordinal game on with payoff f −1 (A) is determined.
(AD+ )
Clearly AD+ =⇒ AD, and both AD + V=L(R) and ADR +DC imply AD+ ; in fact every known model of AD does satisfies AD+ , and the general consensus seems to be that AD+ is the correct axiom for the study of models of determinacy. Assuming AD+ let Θ(A) = sup{BW : B is ordinal definable from reals and A} and let Θ0 = Θ(∅) Θα+1 = Θ(A) for some/any A such that AW = Θα Θ = sup Θα . α<
The sequence of the Θα ’s was introduced in [Sol78B] and it is called the Solovay sequence; note that it may not be defined for all α’s. For example, if V = L(R) then every set is ordinal definable from a real, hence Θ = Θ0 and the Solovay sequence is not defined for larger indexes; assuming ADR will ensure that the sequence is defined up to some limit ordinal . In general, if Θα is defined then L(℘ Θα (R)) is a model for AD+ + Θ=Θα , where ℘ (R) = {X ⊆ R : X W < }. The smallest model of ADR is L(℘ Θ (R)), and in this model Θ has cofinality . Even stronger theories are obtained when the model satisfies ‘Θ = ΘΘ ’, or ‘Θ > Θ0 is regular’—see [Woo99]. Thus the Wadge hierarchy and, in particular, the Solovay sequence of the Θα ’s can be used to measure the strength of models of AD+ . Unfortunately, this method of comparing AD+ models is not always successful since Woodin, in unpublished work, has shown that it is consistent that there are two models M and N of AD+ having the same reals and with divergent Wadge hierarchies. Finally we mention a fairly recent application of the Wadge hierarchy to the study, under AD, of cardinalities of pointclasses. As any pointclass Γ is the surjective image of R, i.e., it is in bijection with R/E for some E, and as any R/E can be embedded into some Γ, it follows that the cardinalities |Γ| are cofinal in the set of cardinalities of quotients of R. The general problem is to determine which Γ are cardinality pointclasses, i.e., such that |Γ| > |Λ|, of self-dual cardinality pointclasses are Δ0 , or the for any Λ ⊂ Γ. Examples 1 pointclasses of the form α< Γα with Γα increasing cardinality pointclasses
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and limit—call such a pointclass a tower, and say it has countable cofinality if cf( ) = . In [AHN07] a complete description of the cardinality pointclasses is given, and an interesting feature of the proof is that it uses the detailed analysis of the Wadge hierarchy. Assuming AD+DC(R), a non-self-dual Γ is a cardinality pointclass iff Γ is closed under pre-images of Δ02 functions; a self dual pointclass Δ strictly larger than Δ01 is a cardinality pointclass iff either it is a tower, or else it is the (necessarily self-dual) pointclass immediately above a tower of countable cofinality. Therefore assuming AD+DC(R) the results of Hjorth [Hjo98, Hjo02] α < =⇒ |Σ0α | < |Σ0 | are obtained as corollaries.
and
|Δ1n | < |Σ1n | < |Δ1n+1 |
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Harvey Friedman and Lee Stanley [FS89] A Borel reducibility theory for classes of countable structures, The Journal of Symbolic Logic, vol. 54 (1989), no. 3, pp. 894–914. Leo A. Harrington [Har78] Analytic determinacy and 0# , The Journal of Symbolic Logic, vol. 43 (1978), pp. 685–693. Leo A. Harrington, Alexander S. Kechris, and Alain Louveau [HKL90] A Glimm–Effros dichotomy for Borel equivalence relations, Journal of the American Mathematical Society, vol. 3 (1990), pp. 902–928. Leo A. Harrington and Ramez-Labib Sami [HS79] Equivalence relations, projective and beyond, Logic Colloquium ’78. Proceedings of the Colloquium held in Mons, August 24–September 1, 1978 (Maurice Boffa, Dirk van Dalen, and Kenneth McAloon, editors), Studies in Logic and the Foundations of Mathematics, vol. 97, North-Holland, Amsterdam, 1979, pp. 247–264. Gregory Hjorth [Hjo95] A dichotomy for the definable universe, The Journal of Symbolic Logic, vol. 60 (1995), no. 4, pp. 1199–1207. [Hjo96] Π12 Wadge degrees, Annals of Pure and Applied Logic, vol. 77 (1996), no. 1, pp. 53–74. [Hjo98] An absoluteness principle for Borel sets, The Journal of Symbolic Logic, vol. 63 (1998), no. 2, pp. 663–693. [Hjo02] Cardinalities in the projective hierarchy, The Journal of Symbolic Logic, vol. 67 (2002), no. 4, pp. 1351–1372. Gregory Hjorth and Alexander S. Kechris [HK01] Recent developments in the theory of Borel reducibility, Fundamenta Mathematicae, vol. 170 (2001), no. 1–2, pp. 21–52. Alexander S. Kechris [Kec92] The structure of Borel equivalence relations in Polish spaces, Set theory of the continuum. Papers from the workshop held in Berkeley, California, October 16–20, 1989 (H. Judah, W. Just, and H. Woodin, editors), Mathematical Sciences Research Institute Publications, vol. 26, Springer, New York, 1992, pp. 89–102. Alexander S. Kechris, Benedikt Lowe, and John R. Steel ¨ [Cabal I] Games, scales, and Suslin cardinals: the Cabal seminar, volume I, Lecture Notes in Logic, vol. 31, Cambridge University Press, 2008. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris, Donald A. Martin, and John R. Steel [Cabal iv] Cabal seminar 81–85, Lecture Notes in Mathematics, no. 1333, Berlin, Springer, 1988. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. [KM78B] Notes on the theory of scales, in Cabal Seminar 76–77 [Cabal i], pp. 1–53, reprinted in [Cabal I], p. 28–74. Casimir Kuratowski ´ [Kur58] Topologie. Vol. I, 4`eme ed., Monografie Matematyczne, vol. 20, Panstwowe Wydawnictwo Naukowe, Warsaw, 1958.
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Richard Laver [Lav71] On Fra¨ıss´e’s order type conjecture, Annals of Mathematics, vol. 93 (1971), pp. 89–111. Alain Louveau [Lou80] A separation theorem for Σ11 sets, Transactions of the American Mathematical Society, vol. 260 (1980), no. 2, pp. 363–378. [Lou83] Some results in the Wadge hierarchy of Borel sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 28–55. [Lou92] Classifying Borel structures, Set Theory of the Continuum. Papers from the workshop held in Berkeley, California, October 16–20, 1989 (H. Judah, W. Just, and H. Woodin, editors), Mathematical Sciences Research Institute Publications, vol. 26, Springer, New York, 1992, pp. 103–112. [Lou94] On the reducibility order between Borel equivalence relations, Logic, Methodology and Philosophy of Science, IX. Proceedings of the Ninth International Congress held in Uppsala, August 7–14, 1991 (Dag Prawitz, Brian Skyrms, and Dag Westerst˚ahl, editors), Studies in Logic and the Foundations of Mathematics, vol. 134, North-Holland, Amsterdam, 1994. Alain Louveau and Jean Saint-Raymond [LSR87] Borel classes and closed games: Wadge-type and Hurewicz-type results, Transactions of the American Mathematical Society, vol. 304 (1987), no. 2, pp. 431– 467. [LSR88A] Les propri´et´es de r´eduction et de norme pour les classes de Bor´eliens, Fundamenta Mathematicae, vol. 131 (1988), no. 3, pp. 223–243. [LSR88B] The strength of Borel Wadge determinacy, this volume, originally published in Kechris et al. [Cabal iv], pp. 1–30. [LSR90] On the quasi-ordering of Borel linear orders under embeddability, The Journal of Symbolic Logic, vol. 55 (1990), no. 2, pp. 537–560. Nikolai Luzin and Waclaw Sierpinski ´ [LS29] Sur les classes des constituantes d’un compl´ementaire analytique, Comptes rendus hebdomadaires des s´eances de l’Acad´emie des Sciences, vol. 189 (1929), pp. 794–796. Richard Mansfield and Galen Weitkamp [MW85] Recursive aspects of descriptive set theory, Oxford Logic Guides, vol. 11, The Clarendon Press Oxford University Press, New York, 1985, With a chapter by Stephen Simpson. Donald A. Martin [Mar75] Borel determinacy, Annals of Mathematics, vol. 102 (1975), no. 2, pp. 363–371. Luca Motto Ros [MR07] General reducibilities for sets of reals, Ph.D. thesis, Politecnico di Torino, 2007. Neil Robertson and P. D. Seymour [RS04] Graph minors. XX. Wagner’s conjecture, Journal of Combinatorial Theory. Series B, vol. 92 (2004), no. 2, pp. 325–357. Waclaw Sierpinski ´ [Sie52] General topology, Mathematical Expositions, No. 7, University of Toronto Press, Toronto, 1952, Translated by C. Cecilia Krieger. Roman Sikorski [Sik58] Some examples of Borel sets, Colloquium Mathematicum, vol. 5 (1958), pp. 170–171. Jack Silver [Sil80] Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Annals of Mathematical Logic, vol. 18 (1980), no. 1, pp. 1–28.
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Robert M. Solovay [Sol78B] The independence of DC from AD, in Kechris and Moschovakis [Cabal i], pp. 171–184. John R. Steel [Ste77] Determinateness and subsystems of analysis, Ph.D. thesis, Berkeley, 1977. [Ste80] Analytic sets and Borel isomorphisms, Fundamenta Mathematicae, vol. 108 (1980), no. 2, pp. 83–88. [Ste81B] Determinateness and the separation property, The Journal of Symbolic Logic, vol. 46 (1981), no. 1, pp. 41– 44. Fons van Engelen, Arnold W. Miller, and John R. Steel [vEMS87] Rigid Borel sets and better quasi-order theory, Proceedings of the AMS-IMS-SIAM joint summer research conference on applications of mathematical logic to finite combinatorics held at Humboldt State University, Arcata, Calif., August 4–10, 1985 (Stephen G. Simpson, editor), Contemporary Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1987, pp. 199–222. Robert Van Wesep [Van77] Subsystems of second-order arithmetric, and descriptive set theory under the axiom of determinateness, Ph.D. thesis, University of California, Berkeley, 1977. [Van78A] Separation principles and the axiom of determinateness, The Journal of Symbolic Logic, vol. 43 (1978), no. 1, pp. 77–81. William W. Wadge [Wad84] Reducibility and determinateness on the Baire space, Ph.D. thesis, University of California, Berkeley, 1984. [Wad11] Early investigations of the degrees of Borel sets, 2011, this volume. W. Hugh Woodin [Woo99] The axiom of determinacy, forcing axioms, and the nonstationary ideal, De Gruyter Series in Logic and its Applications, Walter de Gruyter, Berlin, 1999. DIPARTIMENTO DI MATEMATICA ` DI TORINO UNIVERSITA VIA CARLO ALBERTO 10, 10123 TORINO ITALY
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WADGE DEGREES AND DESCRIPTIVE SET THEORY
ROBERT VAN WESEP
The work to be presented here is taken principally from the three sources Steel [Ste77], Van Wesep [Van77], and Wadge [Wad84], listed in the bibliography. There is so far nothing published on this subject except the Van Wesep paper in the Journal of Symbolic Logic [Van78A]. In Sections 1, 2, and 3 we provide a general picture of the Wadge degrees. In Section 4 we prove some results of Steel concerning functions from the Turing degrees to ℵ1 modulo the Martin measure and apply them to a computation of the length of the Wadge ordering of Δ1n sets. Then in Section 5 we prove some results about the separation, reduction, and prewellordering properties for suitable classes of sets of reals. We work throughout in ZF+DC+AD, but the reader will be able to determine when and how the determinateness assumption may be relaxed in proving corresponding results about restricted classes of sets of reals. §1. Definitions. Definition 1.1. Baire space := . An element of is a real. An interval of Baire is a set [s] := {α ∈ : s ⊆ α} for some s ∈ <. Definition 1.2. For A, B ⊆ , A ≤W B iff there is a continuous f : → such that A = f −1 [B]
Clearly, ≤W is reflexive and transitive. Definition 1.3. A ≡W B iff A ≤W B and B ≤W A. A W-degree (Wadge degree) is a an equivalence class ≡W . Consider the game GW (A, B): Player I Player II
: :
α(0) α(1) · · ·
α(n0 ) α(n0 + 1) · · · (0)
α(n1 ) · · · (1)
α
Player I plays α(0), player II passes or plays, player I plays α(1), player II passes or plays, etc. Player II’s plays in order are (0), (1), . . . . Player II wins iff he plays infinitely often and α ∈ A ⇔ ∈ B. It is easy to see that A ≤W B ⇔ Player II wins GW (A, B) (i.e., player II has a winning strategy in GW (A, B). Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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WADGE DEGREES AND DESCRIPTIVE SET THEORY
Definition 1.4. Let GL (A, B) be the following game: Player I : α(0) α(1) ··· Player II : (0) (1) · · ·
α
Player I plays α(0), player II plays (0), player I plays α(1), player II plays (1), etc. Player II wins iff α ∈ A ⇔ ∈ B. Definition 1.5. A ≤L B iff player II wins GL (A, B). A ≡L B iff A ≤L B and B ≤L A. An L-degree (Lipschitz-degree) is an equivalence class ≡L . Of course, A ≤L B ⇒ A ≤W B. §2. The Lipschitz ordering. Lemma 2.1 (Wadge’s Lemma). For A, B ⊆ , either A ≤L B or B ≤L ¬A, a fortiori, either A ≤W B or B ≤W ¬A. Proof. Immediate from the determinateness of GL (A, B). We shall generally use upper case Roman letters for sets of reals, and lower case Roman letters for their (L- or W-) degrees. ˘ := {A ⊆ : Definition 2.2. If Γ is any class of sets of reals, then Γ ¬A ∈ Γ} is the dual class to Γ. Wadge’s Lemma 2.1 gives us the following information about the Lipschitz ordering: (1) If a is a selfdual L-degree and b is any L-degree, then b
L a. (2) If a is a nonselfdual L-degree, and b is any L-degree, then ˘ or (i) b L a and b >L a. Definition 2.3. Let Ai ⊆ , i ∈ . Then i∈ Ai := {iα : i ∈ , α ∈ Ai }. For A, B ⊆ , A ⊕ B := {nα : n ∈ , n even, α ∈ A} ∪ {n : n ∈ , n odd, ∈ B}. Definition 2.4. For A ⊆ , s ∈ <, s A := {s α : α ∈ A}, As = {α : s α ∈ A}.
Theorem 2.5 (Martin [1973]). The relation ≤L is well-founded. Proof. Suppose not. By DC there are sets An , n ∈ , such that for any n, An+1
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ROBERT VAN WESEP
• { }
•
• • ···
clopen sets, order type = 1
•
←−−−−−
•
∅ ⊕
difference of closed sets open sets
↓ •
···
• ↑
•
•
closed sets
···
• •
countable limit
↓ •
···
−−−→
{∅}
←−−−−−−−
joins of collections of sets unbounded below
co-diff. of closed sets
• ↑ level of the Hausdorff difference hierarchy
Figure 1. A picture of the lower L-degrees
F
···
• ···
• • G
WADGE DEGREES AND DESCRIPTIVE SET THEORY
27
n To any α ∈ 2 assign the sequence fα(n) : n ∈ of strategies and the n sequence α : n ∈ of reals, as indicated in the following diagram.
···
n fα(n)
An+1 An ··· αn+1 (0) αn (0) αn+1 (1) αn (1) .. .. . .
1 fα(1)
0 fα(0) A2 A1 A0 α2 (0) α1 (0) α0 (0) α2 (1) α1 (1) α0 (1) .. .. .. . . .
n , where, for each n, αn (0) is just the first move according to the strategy fα(n) n n n+1 and α (k) is the response of fα(n) to α k. Now, if α and α are in 2 and n ∈ is such that α(m) = α (m) for all m ≥ n, then αn = αn . For n ∈ , define T n to be the set of ∈ 2 such that an ∈ An for any n2. Then, for any s ∈ n2, we have T n if s contains an even number of 1’s 0 Ts = ¬T n if s contains an odd number of 1’s.
Now no T n is either meager or comeager, because its two details of rank one are complementary. But if T 0 is not meager, then (by the consequence of AD that all sets of reals have the Baire property) for some s ∈ <2, Ts0 = T lh(s) is comeager, a contradiction. Definition 2.6. We let ordL (a) be the order type of the L-degrees below a (having first coalesced each degree and its dual). Some of the properties of L-degrees may be inferred from their ordinals. Lemma 2.7. Suppose a is an L-degree. ˘ then the L-degree of joins of a set in a and a set in a˘ is the (1) If a = a, ˘ minimum L-degree above a and a. ˘ then the L-degree of 0A for some A ∈ a is the minimum (2) If a = a, L-degree above a. (3) If ordL (a) is a limit ordinal of cofinality , then a is the degree of joins ˘ of -sequences of sets unbounded below a. So a = a. ˘ (4) If ordL (a) is a limit ordinal of uncountable cofinality, then a = a. Proof. (1) and (2) are easy. (3). Let Ai : i ∈ be unbounded below a. Let A = i∈ Ai . Clearly, A
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ROBERT VAN WESEP
one. (Strategies are codable as reals, so we may use AC for countable sets of sets of reals, which follows from AD.) ˘ Let A ∈ a. (4). Suppose a = a. Claim 2.8. For all n, An
•
• ···
−−→
···
successor or cofinality =
• • ↑
•
···
cofinality >
Figure 2. Picture of the L-degrees
§3. The Wadge Ordering. It is not hard to see that if a is a selfdual Ldegree, then the next 1 L-degrees are ≤W a. Thus, the following Theorem 3.1 provides a complete picture of the Wadge ordering. Theorem 3.1 (Steel, Van Wesep). For A ⊆ , A ≤W ¬A implies A ≤L ¬A. Proof. Suppose A ≤W ¬A and A L ¬A. Let g1 be a strategy for player II which witnesses the former fact, and f a strategy for player I which witnesses the latter. Let g0 be the strategy for player II which instructs him to copy player I’s moves as they are made. Consider a sequence Si : i ∈ , where, for each i, Si is f, g0 or g1 , and consider the following diagram. ···
S1
S2 A
A
S0 A
A.
WADGE DEGREES AND DESCRIPTIVE SET THEORY
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We imagine filling in a column of numbers below each occurrence of “A”, the columns being referred to as numbered starting with zero for the rightmost column. The idea is that the entries in the (i + 1)th and the ith columns Lipschitz player I should be a possible play of the game in which plays Wadge player II
f player I’s according to Si = when moves are represented by g0 or g1 player II’s the entries in the ith column. A finite filling in of the diagram consists of a finite set of numbers arranged in columns under the A’s in such a way that for each i ∈ : (1) the ith and the (i + 1)th columns are a partial play according to Si with the player on the left to move under the convention stated in the preceding paragraph, or (2) for all j ≥ i, the jth column is empty. Lemma 3.2. For a given sequence Si : i ∈ , no two finite fillings in can clash, i.e., have, for some i and n, different entries in the nth place of the ith column. Proof. Since all but finitely many columns are empty, we may take i0 to be the greatest i such that the ith columns of the two finite fillings in clash. This leads easily to a contradiction. (Lemma 3.2) Definition 3.3. The filling in of the diagram for Si : i ∈ is the union of all finite fillings in. The filling in is well defined by virtue of Lemma 3.2 Definition 3.4. The filling in for a finite sequence Si : i ≤ n is the union of all finite fillings in for the sequence with the nth column empty. Since the Wadge strategy g1 has the option of passing, occurrences of g1 in the sequence Si : i ∈ must be sufficiently rare if they are not to block the complete filling in of the diagram. Define an increasing sequence of numbers ik : k ∈ as follows. Set i0 = 0. Suppose ik is defined for k ≤ k0 . For each sequence
= Si : i ≤ ik0 with the property that and
(1) (∀k ≤ k0 ) Sik = g0 or g1 (2) (∀i < ik0 )[(∀k < k0 ) i = ik ⇒ Si = f],
define ik 0 +1 to be the least number n > ik0 such that the filling in for ff · · · f (n − ik0 f’s) contains at least k0 + 1 entries in the zeroth column. It is easy to see that ik 0 +1 exists for each . Let ik0 +1 = max{ik 0 +1 : as above}. Now, if Si : i ∈ is such that and
(1) (∀k) Sik = g0 or g1 (2) for all other i, Si = f,
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ROBERT VAN WESEP
then by construction there are finite fillings in for the sequence Si : i ∈ with arbitrarily many entries in the zeroth column. Thus, the filling in for Si : i ∈ as above has an infinite sequence in the zeroth, a fortiori, in every, column. For any z ∈ 2, let Sz : → {f, g0 , g1 } be given by gz(k) if i = ik Sz (i) = f otherwise. Define h(z) to be the real produced in the zeroth column by the filling in for Sz . Let T = h −1 [A]. Proceed as in the proof of Theorem 2.5. (Theorem 3.1) Corollary 3.5. (i) {∅} and {} are the minimal W-degrees; (ii) each non-selfdual pair of W-degrees has a selfdual successor; (iii) each selfdual W-degree has a non-selfdual pair of successors; (iv) each selfdual W-degree is the union of an 1 -sequence of selfdual Ldegrees; (v) a level of cofinality is occupied by a selfdual W-degree; (vi) a level of uncountable cofinality is occupied by a non-selfdual pair of W-degrees.
···
←−−−
cofinality
•
• •
•
• •
cofinality > ↓ • • • • ··· • • • • •
···
Figure 3. Picture of the W-degrees §4. The Order Type of the Δ1n Degrees. It is fairly easy to see that the order type of the Wadge degrees is Θ = sup{ : is the length of a prewellordering of the reals}. We seek to know the order types of the set of degrees of Δ1n sets of reals; in short, the order type of the Δ1n degrees. 1 It is easy to see that the set of degrees preceding a Δn degree has order type codes as continuous less than 1n+1 (just look at the prewellordering of their preimages of the given set). So the order type of the Δ1n degrees is less than or equal to 1n+1 . In case n is odd, the inequality is strict.(Prewellorder the codes as preimages of initial segments of a Π1 -prewellordered complete of Δ1n sets n
WADGE DEGREES AND DESCRIPTIVE SET THEORY
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Π1n set.) We shall show that for n even, the order type of the Δ1n degrees is of the Δ1 1 , hence equal to 1 . So, for n odd, 1 < the order type ≥ n n n+1 n+1 degrees < 1n+1 . that the order type of the Δ1 degrees is 1 for n even, is due Our proof n n Turing degrees to to Steel and proceeds via a discussion of functions from ordinals which is of interest in its own right. There is also an independent direct proof of this fact due to Martin. Consider functions from D (the set of Turing degrees) into 1 , relative to the Martin measure, . Definition 4.1. Define fn (d) := n1 (d), where n1 (d) is the least ordinal which is not the order type of a wellordering of which is n1 in d. Let : D1 / ∼ = , ∈ Ord, be the canonical isomorphism. Remark 4.2. We have that ADR implies = Θ, and V = L(R) implies > Θ. Theorem 4.3 (Steel). For all even n ≥ 2, we have ([fn ]) = 1n+1 . Remark 4.4. The full axiom of determinateness implies that 13 = ℵ+1 . ]) = ℵ . Steel has shown that if gn (d) = nth d-admissible beyond , then ([g n n His method also shows that the union of the first ℵn Wadge degrees for n ≥ 2 is just the nth level of the hierarchy based on the operation A. Proof of Theorem 4.3. For convenience we take n = 2. Let W α be a complete Σ12 (α) subset of , uniformly in α. Let ≤α be the canonical prewellordering of W α , and let |m|α be the rank of m in ≤α for m ∈ W α . Definition 4.5. We say m, s is a code of f : D → 1 if and only if (1) m ∈ and s is a real coding a continuous function from to , (2) for almost all α, m ∈ W s(α) , (3) for almost all d ∈ D, f(d) = sup{|m|s(α) : α ∈ d}, (4) for almost all α, s(α) ≡T α. Lemma 4.6. For any [f] < [f2 ], there exists a code m, s of f. Proof. Consider the game where player I plays m, α and player II plays . Player I wins iff α ≥T , m ∈ W α , and |m|α = f([α]). Suppose player II wins by strategy . Let α be any real ≥T and high enough that f([α]) < 21 (α). Let m ∈ W α be such that |m|α = f([α]). (The length of ≤α is 21 (α).) Then, player I can win by playing m, α; contradiction. So player I wins, say by the strategy . Let m be player I’s first play by . Let be any real ≥T , and let player II play . Then, if m, α = ∗ , we have α ≡T , and |m|α = f([α]) = f([]). If s codes the function → α, then m, s codes f. (Lemma 4.6)
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ROBERT VAN WESEP
Lemma 4.7. The set of codes is a complete Π13 set of reals. Proof. A pair m, s is a code iff (i) s codes a continuous function, (ii) ∀α∃ ≥T α∀ ≡T (m ∈ W s( ) ). (iii) ∃α∀ ≥T α(s() ≡T ). So the set of codes is Π13 . To show completeness, let P := { : ∀α(m ∈ W α, )} be an arbitrary Π13 set. Let k be such that k ∈ W α, ⇐⇒ ∀α ≤T α m ∈ W α , . Let s code the continuous function α → α, . Then, the map → s is continuous, (Lemma 4.7) and ∈ P iff k, s is a code. Lemma 4.8. There are a Σ12 relation R+ and a Π12 relation R− such that if n ∈ W , then R+ (m, α, n, ) ⇔ m ∈ W α & |m|α ≤ |n| R− (m, α, n, ) ⇔ (m ∈ W α & |m|α ≤ |n| ) or 21 (α) ≤ |n| . Proof. Define R+ (m, α, n, ) ⇔ m ∈ W α & ∃f : {m : m ≤α m} → {n : n ≤ n} (f is order preserving). Define R− (m, α, n, ) ⇔ ∀f[(dom(f) = {n : n ≤ n} & ran(f) ⊆ W α & f is order preserving ⇒ ∃m ∈ ran(f)(m ≤α m )]. (Lemma 4.8) We now define a Π13 -prewellordering of the codes which is very nearly the natural one. Define m, s ≤+ n, t iff (i) m, s and n, t are codes, (ii) ∀α∃ ≥T α∀ ≡T ∃ ≡T R+ (m, s(), n, t( )). Define m, s ≤− n, t iff (i) s, t code continuous functions, (ii) ∃α∀ ≥T α s() ≡T , (iii) ∃α∀ ≥T α∀ ≡T ∃ ≡T R− (m, s(), n, t( )). Note that if n, t is a code and m, s ≤− n, t, then for almost all , for all , ≡T , we have s() ≡T ≡T ≡T t( ), so 21 (s()) > |n|t( ) , and R− (m, s(), n, t( )) ⇔ [m ∈ W s() & |m|s() ≤ |n|t( ) ]. Thus, if n, t is a code, then (i) m, s ≤+ n, t ⇒ m, s is a code, (ii) m, s ≤− n, t ⇒ m, s is a code, (iii) m, s ≤+ n, t ⇔ m, s ≤− n, t.
WADGE DEGREES AND DESCRIPTIVE SET THEORY
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The relation ≤◦ on the codes defined by ≤+ (equivalently ≤− ) is a prelinear ordering, and clearly, if m, s ≤◦ n, t, then the function coded by m, s is ≤ the function coded by n, t. The converse is not quite true, as if f(d) is almost always a limit ordinal the codes of f(d) may occupy two consecutive levels of ≤◦ . For suppose m, s codes f, and f(d) = , a limit ordinal. Then may be the proper supremum of {|m|s(α) : α ∈ d} or for some α ∈ d, we may find = |m|s(α) . A code for which the first case almost always obtains will be <◦ a code for which the second case almost always obtains. Nevertheless, the order type of ≤◦ is just the order type of the functions < f2 . Since ≤+ is Π13 , ≤− is Σ13 , and the set of codes is Π13 , this order type is 13 by Moschovakis [Mos70]. Thus, the theorem is proved. (Theorem 4.3) Theorem 4.9 (Martin, Steel). The order type of the Δ12n Wadge degrees is 12n+1 . Proof. Again suppose for simplicity that n = 1. Let be the order type of the Δ12 degrees. It is easy to see that ≤ 13 . (The prewellordering of codes of sets as preimages of a Δ12 set is Δ13 ). To see that ≥ 13 , we exhibit a map of cofinal in {g : D → 1 : g < f2 }. Recall that as we are assuming full AD, 13 is a regular cardinal. Clearly, AD(L(R)) would also suffice for this result. Definition 4.10. If m, s is a code which works above , i.e., ∀ ≥T (s() ≡T & m ∈ W s() ), let Cm,s, := {p, α : α ≥T & p ≤s(α) m}. α α = {p, e : p, {e}α ∈ Cm,s, }. Note that α ≡T ⇒ Cm,s, ≡T Let Cm,s, Cm,s, . α , α ∈ d. Let hm,s, : D → D be given by hm,s, (d) = d ⊕ degree of Cm,s,
Claim 4.11. Cm,s, ≤W Cm ,s , ⇒ hm,s, ≤T hm ,s , a.e. Proof. Let be a (code of a) map which reduces Cm,s, to Cm ,s , . Let d ≥T , and α ∈ d. Then there are e1 and e2 such that for all p, e α p, e ∈ Cm,s, ⇔ p, {e}α ∈ Cm,s,
⇔ (p, {e}α ) ∈ Cm ,s , ⇔ {e1 }(α, p), {{e2 }(p, e)}α ∈ Cm ,s , ⇔ {e1 }(α, p), {e2 }(p, e) ∈ Cmα ,s , . α ≤T d ⊕ deg Cmα ,s , = hm ,s , (d). This proves So hm,s, (d) = d ⊕ deg Cm,s, the claim. (Claim 4.11) 1 Define the map H : → 3 as follows. Given < , look at the set of f : D → 1 coded by m, s such that for some , m, s is good above
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ROBERT VAN WESEP
and Cm,s, has ordinal < in the Wadge ordering of Δ12 . Let H ( ) be the supremum of the ordinals of these f’s. Claim 4.12. For any < , H ( ) < 13 . Proof. We prove this claim by showing that for any m, s a code good above , there is a g : D → 1 , g < f2 , such that for any n, t a code for g good above , Cn,t, W Cm,s, . To this end let such m, s, and be given, let d ≥T s be given, and let α α ∈ d. Uniformly in e such that {e}α ≡T α, find ne ∈ W {e} such that α , ∈ d. Paste together {p : p ≤{e} ne } has higher Turing degree than Cm,s, {e}α 1 all the ordinals |ne | getting some 2 (d) ordinal. Do this for each α ∈ d. Let g(d) be the least ordinal so obtained. Clearly, g < f2 . We now show that g is as desired. So let n, t code g above . Let d ≥T s, , . There is α ∈ d such that {p : p ≤t(α) n} has higher , ∈ d, i.e., higher Turing degree than hm,s, (d). But Turing degree than Cm,s, α {p : p ≤t(α) n} = {p : p, α ∈ Cn,t, } = {p : p, e ∈ Cn,t, } where {e} is the identity. So hn,t, (d) ≥T {p : p ≤t(α) n} >T hm,s, (d). So hn,t, > hm,s, , (Claim 4.12) a.e., whence Cn,t, W Cm,s, , and the claim is proved. It remains only to show that H : → 13 is cofinal. For this we need to there is a Δ1 set A so that any see that if g : D → 1 , is less than f2 then 2 , m, s is good f < g has a code m, s with the property that for some above and Cm,s, ≤W A. So let n, t be a code for g good above , with the additional property that for all d ≥T , f(d) = |n|t(α) , α ∈ d (such codes are actually given by Lemma 4.6), and let A = {m, s, , p, α : α ≥T & α ≥T & |m|s(α) ≤ |n|t(α) & p ≤s(α) m}. By virtue of Lemma 4.8, A is Δ12 . Clearly, if f < g, m, s is a code for f, and ≥T is such that m, s is good above and f(d) < g(d) for d above , then p, α ∈ Cm,s, ⇔ m, w, , p, α ∈ A, so Cm,s, ≤W A. This concludes the proof of Theorem 4.9 (Theorem 4.9) §5. Separation, Reduction, and Prewellordering Properties in the Wadge Hierarchy. In this section we establish a number of descriptive set theoretic properties of nonselfdual classes closed under continuous preimage, assuming throughout ZF+DC+AD. In the following, then, Γ will always be a nonselfdual class of sets of reals closed under continuous preimages. We use continuously closed to mean closed under continuous preimage. The descriptive set theoretic properties we shall be concerned with are as follows. Definition 5.1. Γ has the first separation property, SepI (Γ), iff ˘ ∀A, B ∈ Γ(A ∩ B = ∅ ⇒ ∃C ∈ Γ ∩ Γ(A ⊆ C ⊆ ¬B)).
WADGE DEGREES AND DESCRIPTIVE SET THEORY
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Γ has the second separation property, SepII (Γ), iff ˘ ∀A, B ∈ Γ∃A , B ∈ Γ(A ∼ B ⊆ A & B ∼ A ⊆ B & A ∩ B = ∅). Γ has the reduction property, Red(Γ), iff ∀A, B ∈ Γ∃A , B ∈ Γ(A ⊆ A, B ⊆ B, A ∩ B = ∅ & A ∪ B = A ∪ B). Γ has the prewellordering property, PWO(Γ), iff for every A ∈ Γ there are ˘ respectively, and a prewellordering ≤ of A relations ≤+ and ≤− in Γ and Γ such that for any real ∈ A and any real α, α ≤+ ⇔ α ≤− ⇔ α ≤ . Before proceeding to the statement of the results of this section we take note of some simple facts. The first is that Wadge’s Lemma (Lemma 2.1) has the immediate consequence that if Γ is continuously closed and nonselfdual, ˘ then A is complete for Γ, i.e., for all B in Γ, B ≤W A. and A ∈ Γ ∼ Γ, It is also true that Γ has a universal set, i.e., there is A ∈ Γ such that for all B ∈ Γ there is α ∈ such that ∈ B ⇔ (α, ) ∈ A. One may take ˘ and f ∗ is the result of applying A = {(f, ) : f ∗ ∈ C }, where C ∈ Γ ∼ Γ (the Lipschitz strategy coded by) f to . Finally, we take note of a standard result of descriptive set theory, viz., if Red(Γ) and and Γ has a universal set, ˘ and ¬SepI (Γ), and hence Red(Γ) ⇒ ¬Red(Γ). ˘ Since the proof then SepI (Γ) of this fact is included in our proof of Theorem 5.2, we do not give it here. We now state the results. Theorem 5.2 (Van Wesep). For any continuously closed nonselfdual class Γ, ˘ we have SepI (Γ) ⇔ ¬SepII (Γ). Theorem 5.3 (Van Wesep). For any continuously closed nonselfdual Γ, ei˘ Thus by Theorem 5.2, either ¬SepI (Γ) or ¬SepI (Γ). ˘ ther SepII (Γ) or SepII (Γ). Using Wadge’s characterizations of the continuously closed nonselfdual classes of Borel sets, Steel has shown that one of each nonselfdual pair of such classes has the first separation property. In each of the following theorems, Γ is a continuously closed nonselfdual class. Theorem 5.4 (Van Wesep). For some continuously closed nonselfdual Γ, ˘ we have ¬Red(Γ) and ¬Red(Γ). Theorem 5.5 (Van Wesep). If Γ is closed under (finite) intersections and ˘ then Red(Γ). SepI (Γ), Theorem 5.6 (Steel). If Γ is closed under countable unions and countable ˘ intersections, then Red(Γ) or Red(Γ). Theorem 5.7 (Kechris-Solovay). If Γ is closed under countable unions, countable intersections, projections, and coprojections, and Γ ⊆ L(R), then ˘ PWO(Γ) or PWO(Γ). We now proceed to the proofs.
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˘ Define Proof of Theorem 5.2. Let A ∈ Γ ∼ Γ. A0 = {((f0 , f1 ), α) : f0 ∗ α ∈ A} A1 = {((f0 , f1 ), α) : f1 ∗ α ∈ A}, where (·, ·) is a reasonable pairing function for reals. Clearly, A0 and A1 are in Γ. Moreover, the pair A0 , A1 is universal for pairs of sets in Γ, i.e., for any B0 , B1 ∈ Γ there is a real such that B0 = {α : (, α) ∈ A0 }, and B1 = {α : (, α) ∈ A1 }. (To see this, note that by Wadge’s Lemma 2.1, Γ = {B : B ≤W A}. Then, note that this is true even if ≤W is replaced by ˘ ≤L .) Likewise, the pair ∼ A0 , ∼ A1 is universal for pairs of sets in Γ. ˘ Now suppose Γ has the second separation property. We shall show that Γ does not have the first separation property. Let B0 , B1 ∈ Γ be such that A0 ∼ A1 ⊆ B0 , and A1 ∼ A0 ⊆ B1 , with B0 ∩ B1 = ∅. Define C0 = {α : (α, α) ∈ B0 }, C1 = {α : (α, α) ∈ B1 }. Clearly, C0 and C1 are disjoint ˘ To see sets in Γ. We claim they are not separable by a set in Δ = Γ ∩ Γ. this, suppose C0 ⊆ D, and D ∩ C1 = ∅, with D ∈ Δ. Let be such that D = {α : (, α) ∈ ¬A0 } and ¬D = {α : (, α) ∈ ¬A1 }. Then, we have ∈ D ⇒ (, ) ∈ A1 ∼ A0 ⇒ (, ) ∈ B1 ⇒ ∈ C1 ⇒ ∈ / D, and ∈ / D ⇒ (, ) ∈ A0 ∼ A1 ⇒ (, ) ∈ B0 ⇒ ∈ C0 ⇒ ∈ D, which is a contradiction. Now suppose Γ does not have the first separation property. Let C and D ˘ We shall in Γ be disjoint and not separable by a set in Δ. Let A and B be in Γ. find A and B in Γ such that A ∼ B ⊆ A , B ∼ A ⊆ B , and A ∩ B = ∅. Consider the following game. Players I and II produce reals α and respectively. Player I wins iff
and
∈ C ⇒ α ∈ A ∼ B, ∈ D ⇒ α ∈ B ∼ A, α ∈ (A ∼ B) ∪ (B ∼ A).
Equivalently, player II wins iff, α ∈ A ∼ B ⇒ ∈ D, and α ∈ B ∼ A ⇒ ∈ C. Now, player I can not win this game, for if he did, say by a strategy f, which we shall view as a continuous function from to , then we should have ˘ which C ⊆ E and E ∩ D = ∅, where E = f −1 (A) = ¬f −1 (B) ∈ Γ ∩ Γ, contradicts the inseparability of C and D. Thus, player II wins the game, say by the strategy f. Let A = f −1 (D), and B = f −1 (C ). Then A and B are as desired. (Theorem 5.2)
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Proof of Theorem 5.3. Let A0 , A1 be a complete pair of sets in Γ (see the proof of Theorem 5.2). Consider the following two games. G0 : Players I and II play reals α and respectively. Player II wins iff α ∈ A0 ∼ A1 ⇒ ∈ A1 ∼ A0 , α ∈ A1 ∼ A0 ⇒ ∈ A0 ∼ A1 , and ∈ / A0 ∩ A1 . G1 : Players I and II play reals α and respectively. Player II wins iff α ∈ A0 ∼ A1 ⇒ ∈ A0 ∼ A1 , α ∈ A1 ∼ A0 ⇒ ∈ A1 ∼ A0 , and ∈ A0 ∪ A1 . Suppose Γ does not have the second separation property. Then player II does not win G1 . For if he did, say by the strategy f, we could let A10 = f −1 (¬A1 ) and A11 = f −1 (¬A0 ). Then A0 ∼ A1 ⊆ A10 , A1 ∼ A0 ⊆ A11 ; A10 ∩A11 = ∅. Since A0 , A1 is complete for Γ, this contradicts our hypothesis. ˘ does not have the second separation property, then player II Similarly, if Γ does not win G0 . Thus, by determinateness, we shall have proved the theorem when we have shown that player I can not win both G0 and G1 . So suppose player I wins G0 and G1 by the strategies f0 and f1 respectively. From the definition of G0 it is apparent that for any played by player II, or or
f0 () ∈ A0 ∼ A1 & ∈ A1 ∼ A0 , f0 () ∈ A1 ∼ A0 & ∈ / A0 ∼ A1 , ∈ A0 ∩ A1 .
In other words, ∈ A0 ∼ A1 ⇒ f0 () ∈ A0 ∼ A1 , ∈ A1 ∼ A0 ⇒ f0 () ∈ A1 ∼ A0 , and ∈ ¬A0 ∩ ¬A1 ⇒ f0 () ∈ (A0 ∼ A1 ) ∪ (A1 ∼ A0 ). Similarly, ∈ A0 ∼ A1 ⇒ f1 () ∈ A1 ∼ A0 , ∈ A1 ∼ A0 ⇒ f1 () ∈ A0 ∼ A1 , and ∈ A0 ∩ A1 ⇒ f1 () ∈ (A0 ∼ A1 ) ∪ (A1 ∼ A0 ). As in the proof of Theorem 2.5, for any ∈ 2, consider the sequence f(n) : n ∈ of strategies for player I, and consider the following diagram. 3 (0) 3 (1) .. .
f(2)
2 (0) 2 (1) .. .
f(1)
1 (0) 1 (1) .. .
f(0)
0 (0) 0 (1) .. .
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ROBERT VAN WESEP
The rule of construction in this diagram is: n (i) is the response of the strategy f(n) to the play n+1 i, i.e., the first i in the sequence n+1 . We consider membership of the n in A0 and A1 for various . Call (A0 ∩ A1 ) ∪ (¬A0 ∩ ¬A1 ) the middle and (A1 ∼ A0 ) ∪ (A0 ∼ A1 ) the sides. Claim 5.8. { ∈ 2 : 0 ∈ middle} is meager. Proof. Suppose not. Since we are assuming AD, all sets of reals have the property of Baire. So for some s ∈ <2, { : (s )0 ∈ middle} is comeager. But (s )0 ∈ middle implies 0 ∈ middle, because ∈ sides ⇒ fi () ∈ sides for i = 0, 1. So { : 0 ∈ middle} is comeager. Without loss of generality, assume { : 0 ∈ A0 ∩A1 } is nonmeager. For any in this set, (1)0 is in the sides. Thus, 0 ∈ sides for a nonmeager set of . Contradiction. (Claim 5.8) Thus, { : 0 ∈ sides} is nonmeager. Suppose, without loss of generality, that { : 0 ∈ A1 ∼ A0 } is nonmeager. Then for some s ∈ <2, { : (s )0 ∈ A1 ∼ A0 } is comeager. There are two cases. If s contains an odd number of 1’s, then 0 ∈ A0 ∼ A1 ⇒ (s )0 ∈ A1 ∼ A0 , and 0 ∈ A1 ∼ A0 ⇒ (s )0 ∈ A0 ∼ A1 . So { : 0 ∈ A1 ∼ A0 } is comeager, which contradicts our assumption. If s contains an even number of 1’s, then 0 ∈ A0 ∼ A1 ⇒ (s )0 ∈ A0 ∼ A1 , and 0 ∈ A1 ∼ A0 ⇒ (s )0 ∈ A1 ∼ A0 . So { : 0 ∈ / A0 ∼ A1 } is comeager. By Claim 5.8, then, { : 0 ∈ A1 ∼ A0 } is comeager. But in this set implies (1)0 ∈ A0 ∼ A1 , so { : 0 ∈ / A0 ∼ A1 } is nonmeager. This contradiction establishes the theorem. (Theorem 5.3) Proof of Theorem 5.4. We shall show that if Γ is a minimal continuously closed nonselfdual class including F ∪ G , then the reduction property fails ˘ In Section II.4.1 of Van Wesep [Van77] a more general for both Γ and Γ. result is proved, viz., if the Wadge order type of Γ is not of the form 1α + 1 and Γ F , then reduction fails for Γ. Let A be G but not F . Let A0 = {(α, ) : α ∈ A}, A1 = {(α, ) : ∈ A}. It is clear that A0 , A1 ∈ G ∼ F and A0 , A1 is complete for pairs of sets in G . Now, the pair A0 , A1 is not reducible by sets in G , for by its completeness if it were reducible by G sets, then we would have Red(G ), which is false. Moreover, A0 , A1 can not be reduced by sets in F ∪ G , for if C ∈ F were such that C ⊆ A0 and A0 ∼ A1 ⊆ C , then letting be any real not in A, we have, for all reals α, α ∈ A ⇔ (α, ) ∈ A0 ⇔ (α, ) ∈ C , which is a contradiction. We may take for A the set {α ∈ : ∀m∃n > m(α(n) = 0)}, so that for any s ∈ <, As ≡W A. Now, by the above considerations, for any s ∈ <, (A0 )s , (A1 )s is not reducible by sets in F ∪ G . We shall have proved the theorem when we have shown that A0 , A1 is not reducible by sets in Γ, and ˘ therefore, by symmetry, in Γ.
WADGE DEGREES AND DESCRIPTIVE SET THEORY
39
Suppose B0 , B1 reduces A0 , A1 , where B0 and B1 are in Γ. It is easy to show that for some s ∈ <, (B0 )s and (B1 )s are in F ∪G . But (B0 )s , (B1 )s (Theorem 5.4) reduces (A0 )s , (A1 )s , a contradiction. Proof of Theorem 5.5. We use the following lemma. Lemma 5.9. If Γ is closed under finite intersections and ¬Red(Γ), then for ˘ there are C , D ∈ Γ, so that C, D ∈ Γ, C ∼ D ⊆ C , C ∩ (D ∼ C ) = ∅, D ∼ C ⊆ D , D ∩ (C ∼ D) = ∅, C ∪ D = . ˘ Consider Proof. Let A, B be a complete pair for Γ, and let C, D ∈ Γ. the game G(A, B, C, D) defined as follows: Player I plays α, player II plays . Player II wins iff ∈ A ∪ B, α ∈ C ∼ D ⇒ ∈ A ∼ B, and α ∈ D ∼ C ⇒ ∈ B ∼ A. Equivalently, player I wins iff ∈ / A ∪ B, or α ∈ C ∼ D and ∈ / A ∼ B, or α ∈ D ∼ C and ∈ / B ∼ A; in other words ∈ A ∪ B ⇒ ( ∈ B & α ∈ C ∼ D, or ∈ A & α ∈ D ∼ C ). Now, player I can not win G(A, B, C, D), for if he did, say by the strategy f, then letting A = f −1 (¬C ), and B = f −1 (¬D), we would have A ∪ B ⊆ A ∪ B , A ∼ B ⊆ A , B ∼ A ⊆ B , and A , B ∈ Γ. Thus, letting A = A ∩ A , B = B ∩ B , we would have A , B ∈ Γ, with A ∪ B = A ∪ B , A ∼ B ⊆ A , B ∼ A ⊆ B , but this contradicts ¬Red(Γ). So player II wins G(A, B, C, D), say by f. Let C = f −1 (A), D = f −1 (B). Then C and D are as desired, and the lemma is proved. (Lemma 5.9) To prove Theorem 5.5 suppose toward a contradiction that Γ is closed under ˘ and ¬Red(Γ). We shall derive the absurdity Red(Γ). ˘ So intersection, SepI (Γ), ˘ separate ˘ Let C , D be as given by Lemma 5.9. Let E ∈ Γ ∩ Γ let C, D ∈ Γ. ¬C and ¬D . Let C = C ∩ ¬E, D = D ∩ E. Then C , D reduces C, D. (Theorem 5.5) Proof of Theorem 5.6. Assume the hypothesis of the theorem, and sup˘ We shall show that in fact Γ has the reduction pose that ¬Red(Γ) and ¬Red(Γ). property. Let C, D ∈ Γ. Define C−1 = C , D−1 = D, and for all n ≥ 0, let Cn and Dn be Cn−1 , Dn−1 as given by Lemma 5.9. We have C2n , D2n ∈ Γ, C2n+1 , D2n+1 ∈ ˘ Cn+1 ⊆ Cn and Dn+1 ⊆ Dn , for each n ≥ 0. So C = ˘ Γ, n Cn ∈ Γ ∩ Γ and C separates C ∼ D and D ∼ C . Let C = C ∩ C , D = D ∩ (¬C ). (Theorem 5.6) Then C , D reduces C, D. Proof of Theorem 5.7. By Theorem 5.6 we may suppose that Γ has the reduction property. We shall show PWO(Γ). It will be useful to look at the ˘ following way of coding sets in Δ = Γ ∩ Γ:
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ROBERT VAN WESEP
Let A, B be a universal pair for Γ. Let A , B reduce A, B. For any real , if ∀α[(α, ) ∈ A or (α, ) ∈ B ], then let C = {α : (α, ) ∈ A }. Then { : C is defined} ∈ Γ, and if C is defined, C ∈ Δ. Lemma 5.10 (Independently also due to Steel). Let Γ satisfy the hypothesis of the theorem, and suppose Δ is not closed under wellordered unions. Then Red(Γ) ⇒ PWO(Γ). Proof. Let be the least ordinal so thatfor some sequence A : < / Δ. Let Γ∗ = { < B : ∀ < B ∈ Δ}. It is of sets A ∈ Δ, < A ∈ apparent from the minimality of , then, that PWO(Γ∗ ). Thus, we need only that Γ∗ ⊆ Γ, to see that Γ∗ = Γ, and hence PWO(Γ). Let be the supremum of the lengths of prewellorderings of Rthat lie in Δ. / Δ. Then ≥ by Moschovakis [Mos70]. But ≤ . For if not, let < A ∈ We may assume that < < ⇒ A ⊂ A , by taking cumulative unions and then a strictly increasing subsequence, noting the minimality of . Now, A : < provides a prewellordering in Δ of length , a contradiction. So = . Now we note that the order type of the Wadge degrees in Δ is exactly . Showing “≤” is trivial. To see the other direction note that if ≺ is a prewellordering in Δ, then one may define by effective transfinite induction a ≤W -increasing sequence of Δ sets of length |≺|. The proof of this is a routine exercise given that there are continuous functions which, acting on codes of sets in Δ, give codes for the Δ sets derived from them by the operations of union, etc., which do not lead out of Δ. Indeed, the strategies which witness the corresponding closure properties of Γ provide such functions. Now let C be the set of α for which Cα is defined. The relation {α, : α, ∈ C & (Cα ≤W C & Cα ≤W ¬C )} is in Γ, as is the complement of this relation relative to C × C . Thus, by the Main Lemma of Moschovakis [Mos70], Γ∗ ⊆ Γ. (Lemma 5.10) To finish the proof of Theorem 5.7 we must show that Δ is not closed under ¨ wellordered unions. Let J1 , . . . , J8 be Godel’s operations with the property that, in the terminology of Jech [Jec71], any almost universal transitive class closed under J1 , . . . , J8 is a model ZF. Set Gi (x, y, α) = Ji (x, y), for i = 1, . . . , 8, and G9 (x, y, α) = x ∩ α. Let ( , , ϑ) be the rank of , , ϑ in the ¨ Godel wellordering and let I , J , and K be such that = (I ( ), J ( ), K( )). Then put ⎧ ⎪ if I ( ) = 0 ⎨{F (, ) : < & ∈ R} F ( , α) = GI ( ) (F (J ( ), (α)0 ), F (K ( ), (α)1 ), (α)2 ) if 0 < I ( ) ≤ 9 ⎪ ⎩ {F (J ( ), (α)0 ), F (K ( ), (α)1 )} if I ( ) > 9 Then {F ( , α) : ∈ Ord & α ∈ R} = L(R).
WADGE DEGREES AND DESCRIPTIVE SET THEORY
41
Now suppose that Δ is closed under wellordered unions. By induction on ¨ the Godel wellordering of pairs , one can show that for each and the relations = P , (α, ) ⇔ F ( , α) = F (, ) ∈ (α, ) ⇔ F ( , α) ∈ F (, ) P ,
are in Δ. But by the preceding paragraph this means that every set of reals in L(R) is in Δ, whence Γ L(R). This contradiction establishes Theorem 5.7. (Theorem 5.7) We list some other results which partake of the flavor of those presented here: (Steel ): Any two non-Borel analytic sets are Borel isomorphic. (Steel ): Jump operators on the Turing degrees are prewellordered by the Martin measure. (Radin, Steel, Van Wesep): Any nonselfdual continuously closed class of sets of reals may be obtained by application of a fixed -ary Boolean operation to sequences of open sets. The first two of these results appear in Steel [Ste77] and will be published elsewhere. The last appears in the union of Steel [Ste77] and Van Wesep [Van77]. §6. Conjectures and Problems. One should like to prove one of the following two competing conjectures: ˘ (i) If Γ is a nonselfdual continuously closed class, then SepI (Γ) or SepI (Γ). Thus, the classical separation principles serve to distinguish each nonselfdual continuously closed class from its dual. (ii) If S is a set of Wadge degrees, then for some nonselfdual degree a, we have a ∈ S ⇔ a˘ ∈ S. The theory of the Wadge degrees seems to sorely need results of the following sort, of which there are now essentially no examples: Some “closure” property of the order type of Γ ∩ Γ˘ (e.g.. that it is a cardinal) implies some closure property for Γ (e.g., closure under ˘ intersection) or even for Γ ∩ Γ. REFERENCES
Thomas J. Jech [Jec71] Lectures in set theory, with particular emphasis on the method of forcing, Lecture Notes in Mathematics, Vol. 217, Springer-Verlag, Berlin, 1971. Yiannis N. Moschovakis [Mos70] Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory. Proceedings of an international colloquium held under the auspices of the Israel
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Academy of Sciences and Humanities, Jerusalem, 11–14 November 1968 (Y. Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam-London, 1970, pp. 24–62. John R. Steel [Ste77] Determinateness and subsystems of analysis, Ph.D. thesis, Berkeley, 1977. Robert Van Wesep [Van77] Subsystems of second-order arithmetric, and descriptive set theory under the axiom of determinateness, Ph.D. thesis, University of California, Berkeley, 1977. [Van78A] Separation principles and the axiom of determinateness, The Journal of Symbolic Logic, vol. 43 (1978), no. 1, pp. 77–81. William W. Wadge [Wad84] Reducibility and determinateness on the Baire space, Ph.D. thesis, University of California, Berkeley, 1984. SINAI HOSPITAL OF BALTIMORE 2401 W. BELVEDERE AVENUE BALTIMORE, MARYLAND, 21215 UNITED STATES OF AMERICA
E-mail: [email protected]
A NOTE ON WADGE DEGREES
ALEXANDER S. KECHRIS
§1. It has been shown by Martin and independently Steel (see [Van78B, Theorem 4.2]), that the wellordering of Wadge degrees of Δ12n sets of reals their proofs has length 12n+1 . Although this is a result about projective sets, require full AD as they proceed by showing that if 2n is the length of the wellordering of Wadge degress of Δ12n sets, then cofinality( 12n+1 ) ≤ 2n , which by the regularity of 12n+1 (a consequence of AD) implies that 12n+1 ≤ 2n . As it is easy to see that 2n ≤ 12n+1 , by a direct computation,we have the of full AD here can be replaced, by trivial desired equality. Of course, the use absoluteness considerations, by Det(L(R)), i.e., the hypothesis that all sets of reals in L(R) are determined. Motivated by the fact that one only needs Det(Δ12n ) to establish the fact that the Wadge degrees of Δ12n sets are wellordered (see [Van78B, Theorem 2.2]), Martin has asked if onecan compute that also 1 2n = 2n+1 , using again only Det(Δ12n ). We provide such a proof below. It on a method of “invertingthe game quantifier” which may be also is based useful elsewhere. §2. Let us take n = 1 for notational simplicity. From now on we assume Det(Δ12 ). 2.1. For each A ⊆ let Γ(A) be a pointclass with the following properties: (i) A, \ A ∈ Γ(A), (ii) B, \ B ∈ Γ(A) ⇒ Γ(B) ⊆ Γ(A), (iii) A ∈ Δ1m ⇒ Γ(A) ⊆ Δ1m , ∀m ≥ 2, (iv) Γ(A)is -parametrized and closed under continuous substitutions, (v) There is a map A → CA , sending each A to CA , an -universal set in Γ(A) and for each m ≥ 2 there is a total recursive function fm such that if ε ∈ is a Δ1m -code of A, then f(ε) is a Δ1m -code of CA . For example, we can take Γ(A) = 2 ENV(2E, A) = the pointclass of all pointsets semirecursive in 2E, A and a real. Research partially supported by NSF Grant MCS-17254 A01. The author is an A. P. Sloan Foundation Fellow. Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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ALEXANDER S. KECHRIS
G
44
G
2.2. Next let us reall that if is the game quantifier, then Δ12 = Δ13 . (Here is a quick proof due to Addison: 1 Let P, Q be disjoint Σ3 sets. Say P(x) ⇔ ∃αP (x, α), Q(x) ⇔ ∃Q (x, ), P (x, α, ) ⇔ P (x, α), Q (x, α, ) ⇔ Q (x, ) where P , Q ∈ Π12 . Let and let S(x, α, ) in Δ12 separate P , Q . Then it is easy to check that ∃α(0)∀(0)∃α(1)∀(1) . . . S(x, α, ) separates P, Q.) Now let W ⊆ be Π13 and universal for Π13 and a Π13 -norm on W . For x ∈ W , put Hx = {y, z : (y) ≤ (z) < (x)}. Then let f be a total recursive function such that if x ∈ W , then f(x) is a Δ12 -code of a set, say Δx , such that Δx = Hx . (The existence of such an f is the precedingparagraph.) Put finally for x, y ∈ W : clear from the proof in G
x y ⇔ Γ(Δx ) ⊆ Γ(Δy ). Lemma 2.1. is a prewellordering. Proof. is obviously reflexive and transitive. is connected: Let x, y ∈ W . Then by Wadge, Δx ≤W Δy or Δy ≤W function. \ Δx , where X ≤W Y iff X is reducible to Y via acontinuous Say the first case occurs. Then Δx ∈ Γ(Δy ). But also \ Δx ≤W \ Δy , therefore \ Δx ∈ Γ(Δy ), thusΓ(Δx ) ⊆ Γ(Δy ), i.e., x y. is wellfounded: Given ∅ A ⊆ W let x ∈ A be such that Δx has least Wadge ordinal. Then for any y ∈ A, Δx ≤W Δy or Δx ≤W \ Δ y , therefore, (Lemma 2.1) as above x y. Let ϕ : W be the norm associated with . Lemma 2.2. ϕ is a Π13 -norm. Proof. Fix y ∈ W . We want to express x ∈ W ∧ x y in a Δ13 way uniformly in y. When both x, y are in W the condition x y is equivalent to Δx ≤W CΔy ∧ \ Δx ≤W CΔy , which is clearly Δ13 uniformly in x, y. So it is enough to find a total recursive function g such that (i) y ∈ W ⇒ g(y) ∈ W , (ii) x, y ∈ W ∧ x y ⇒ (x) ≤ (g(y)). Because then for y ∈ W : x ∈ W ∧ x y ⇔ (x) ≤ (g(y)) ∧ x y, which by our preceding remarks is Δ13 uniforming in y. In order to construct g we use the following Sublemma 2.3. There is a total recursive function h such that
A NOTE ON WADGE DEGREES
45
(i) y ∈ W ⇒ h(y) ∈ W (ii) x, y ∈ W ∧ Hx ≤W Hy ⇒ (x) ≤ (h(y)). Proof. Let h be a total recursive function such that if y ∈ W , then h(y) ∈ W and z ∈ CHy ⇔ a, z ∈ W ⇔ (a, z) ≤ (h(y)), for some a ∈ . Then if x, y ∈ W ∧ Hx ≤W Hy , but (x) > (h(y)), towards a contradiction, we have z ∈ CHy ⇔ (a, z) ≤ (h(y)) < (x), for some a ∈ , therefore CHy ≤W Hx ≤W Hy , a contradiction. (Sublemma 2.3) To complete the proof of Lemma 2.2, we construct now g as follows: Let f 1 be total recursive such that if y ∈ W , then f 1 (y) is a Δ12 -code of CΔy and let f 2 , f 3 be total recursive such that if y ∈ W , then f 3 (y) ∈ W and G
α(z, t, α ∈ CΔy ) ⇔ (f 2 (y), z, t) < (f 3 (y)). Let g = h ◦ f3 . Assume now x, y ∈ W and x ≤ y. Then Γ(Δx ) ⊆ Γ(Δy ), so t, α ∈ Δx ⇔ z0 , t, α ∈ CΔy , for some z0 , thus G
G
t ∈ Hx ⇔ t ∈ Δx ⇔ α(t, α ∈ Δx ) ⇔ α(z0 , t, α ∈ CΔy ) ⇔ (f 2 (y), z0 , t) < (f 3 (y)). G
So Hx ≤W Hf 3 (y) , thus (x) ≤ h(f 3 (y)) = g(y).
(Lemma 2.2) Using Lemma 2.2 we complete the proof of the result as follows: By Lemma 2.2 we have = 13 . Since by direct computation we can easily see that 2 ≤ 13 , ≤ . For that define for < : it is enough to show 2
where ϕ(x) = . Since
f( ) = Wadge ordinal of CΔx ,
ϕ(x) = ϕ(y) ⇒ Γ(Δx ) = Γ(Δy ) ⇒ CΔx ≤W CΔy ∧ CΔy ≤W CΔx , this is well-defined. Also f : → 2 , so it is enough to show that f is order preserving. Indeed, let < < and ϕ(x) = , ϕ(y) = . Then x y and y x, so Γ(Δx ) Γ(Δy ). Consequently, CΔx ≤W CΔy but CΔy ≤W CΔx ,
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ALEXANDER S. KECHRIS
therefore, by Wadge, CΔx ≤W \ CΔy , i.e., the Wadge ordinal of CΔx , which is f( ), is smaller thanthe Wadge ordinal of CΔy , which is f(). This finishes the proof of the result. REFERENCES
Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. Robert Van Wesep [Van78B] Wadge degrees and descriptive set theory, this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 151–170. DEPARTMENT OF MATHEMATICS CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA 91125 UNITED STATES OF AMERICA
E-mail: [email protected]
SOME RESULTS IN THE WADGE HIERARCHY OF BOREL SETS
ALAIN LOUVEAU
This paper has two goals: First, to provide construction principles, by means of boolean operations, of the Wadge classes of Borel sets, and in a second step, to use these construction principles to define lightface versions of the Wadge classes, and prove that the notion of Wadge class, roughly speaking, is Δ11 : If a Δ11 set is in some (boldface) Wadge class Γ, it belongs to the corresponding Δ11 -recursive lightface class. The necessary background concerning Wadge’s hierarchy can be found in Van Wesep [Van78B]. Let us recall that a family Γ ⊆ is a class if it is closed under inverse images by continuous functions from into itself. If Γ is of the form [A] = {f −1 (A) : f : → , continuous} for some set A, it is a Wadge class (the Wadge class of A). For a set A, A˘ denotes its complement, ˘ = {A˘ : A ∈ Γ} is the dual class of the class Γ. Δ = Δ(Γ) A˘ = \ A, and Γ ˘ A class is the ambiguous class associated with Γ, and is defined by Δ = Γ ∩ Γ. ˘ = Γ. Γ is self dual if Γ The Wadge hierarchy is obtained by (partially) ordering the Wadge classes by strict inclusion Γ < Γ if Γ ⊆ Γ and Γ = Γ . We similarly define Γ ≤ Γ if Γ ⊆ Γ . This ordering admits a game theoretical analysis: If A, B are two subsets of , let GW (A, B) be the game where players I and II play alternatively integers, player I constructing α ∈ , and player II having the possibility of passing, as long as he constructs ∈ . player II wins this game if α ∈ A ⇐⇒ ∈ B. It is easy to check that player II has a winning strategy in GW (A, B) ⇐⇒ [A] ≤ [B]. Let us now consider only Wadge classes of Borel sets. Then we can use Borel Determinacy (Martin [Mar75]). This gives that < is almost a linear ordering ˘ in case Γ (Wadge [Wad84]): The only class not comparable to Γ is the class Γ ˘ of non self-dual classes, is non self-dual. So by identifying twin pairs (Γ, Γ) we obtain a linear ordering. Moreover, by Martin [Mar73], the ordering < is well founded, so that we can associate with each Wadge class of Borel sets Γ an ordinal o(Γ). The pattern of self-dual and non self-dual classes is as follows (see Van Wesep [Van78B]): Self-dual and non self-dual twin pairs alternate at successor stages. The hierarchy begins with the twin pair {∅}, {}; at limit Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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stages of cofinality stands a self-dual class, and at limit stages of cofinality 1 a non self-dual twin pair. Of course, the Wadge hierarchy is a refinement of the classical hierarchies, the Borel hierarchy (Σ0 , Π0 ) <1 , and the hierarchy of differences of Haus But it contains also a lot of more “exotic” classes. dorff and Kuratowski. The picture presented in part 1 closely follows unpublished results of Wadge [Wad84], except that Wadge’s description exhibits, for each non self-dual class Γ, a set A for which Γ = [A], whereas we define a boolean operation which enables to construct Γ in terms of preceding classes. I would like to thank John Steel for giving me access to Wadge’s papers, and also for enlightening discussions about Wadge classes. §1. A Description of Wadge Classes of Borel Sets. First, let us make some heuristic comments on what follows. The usual Borel hierarchy of sets, {Σ0 :
< 1 }, is our starting point in analysing the Wadge classes. This obviously does not give a complete description, and we certainly must refine it by adding, between Σ0 and Σ0 +1 , the hierarchy of differences of Σ0 sets. This gives a first again happens to be insufficient, at least for ≥ 2. The first which refinement, expectation is that by refining again (may be a few more times), one should reach the complete picture of Wadge degrees. It is indeed what happens, and we shall define a set of levels by using successive refinements; but in order to obtain the picture between Σ0 and Σ0 +1 , refinements are necessary. And because the Wadge ordering is a well-ordering, these refinements are not well-ordered by inclusion. In fact, it turns out that the reverse ordering is well-founded, and so the ordinal we shall associate with the refinements will measure the “degree of simplicity” of each level, rather than its degree of complexity: the hierarchy of differences is given level , whereas the most complicated classes are those of level 1. At this last level occur all successor classes and all limit classes of cofinality . The “simplicity” of each class may be measured, in mathematical terms, by its closure properties. We shall see (Lemma 1.4 below) that the closure properties do increase with the level of the class. For constructing the classes, we have selected a small set of operations, which is certainly not the least possible one, but is convenient for a nice description of the classes. Starting from the classes Σ0 , and applying successively these operations will give the desired description. The main point here is that each operation increases the Wadge ordinal, but decreases the level: Intuitively, it means that the resulting class is more complicated than the original one. A final word of comment: By our general knowledge of the Wadge hierarchy, it is not necessary to give a construction principle from below for the self-dual classes, which are exactly the Δ-parts of successor non self-dual classes, and among the non self-dual classes, we can choose to describe only
SOME RESULTS IN THE WADGE HIERARCHY OF BOREL SETS
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Γ Γ
Γ
Γ
...
Γ
Figure 1. D (Γ), for even one of the twin dual classes. One mathematically nice way for doing this choice is given by a result of Steel [Ste81B]: Among each twin pair, exactly one class has the separation property. For technical reasons, we have chosen to describe the other ones, i.e. the non self-dual classes which do not possess the separation property. In particular, we begin the construction with the Σ0
classes, not the Π0 ones. Let us now begin with the definition of the operations. Definition 1.1. (a) Differences (Hausdorff, Kuratowski). Let ≥ 1 be a countable ordinal. If C : < is an increasing sequence of sets, the set A = D (C : < ) is defined by A=
{C \ < C : odd , < } {C \ < C : even , < }
for even for odd
[So D1 (C0 ) = C0 , D2 (C0 , C1 ) = C1 \ C0 , . . . . This definition is not the usual one, as given in Kuratowski [Kur66], which deals with decreasing sequences—and is applied to Π0 .] If Γ is some class, D (Γ) is the class of all D (C : < ) for some increasing sequence of sets in Γ. D1 (Γ) is simply written Γ. (b) Separated Unions (Wadge). We define A = SU(C n : n ∈ , An : n ∈ ) in case the sets Cn are pairwise disjoint, by A = n (An ∩ Cn ). The set C = n Cn is the corresponding envelope of A. For classes Γ, Γ , we let SU(Γ, Γ ) be the class of all SU(Cn , An ) with the Cn ’s in Γ and the An in Γ . The set Γ, SU(Γ, Γ ) is theset of pairs C, A where A = SU(Cn , An ) is in SU(Γ, Γ ) and C = n Cn is the corresponding envelope. (c) One-sided Separated Unions (Myers, Wadge). We say that A = Sep(C, B1 , B2 ) if A = (C ∩ B1 ) ∪ (B2 \ C ) [This is of course a particular case of (b)]. If Γ, Γ are two classes, Sep(Γ, Γ ) is the class of all Sep(C, B1 , B2 ) where ˘ and B2 ∈ Γ [This is not symmetric in Γ and Γ ˘ ]. C ∈ Γ, B1 ∈ Γ
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Γ
Γ
˘ Γ
Γ
˘ Γ
˘ Γ
Γ
Γ˘
Figure 2. SU(Γ, Γ ) Γ
Γ Γ˘
Figure 3. Sep(Γ, Γ ) Γ
Γ
Γ˘
Γ Figure 4. Bisep(Γ, Γ , Γ ) (d) Two-sided Separated Unions. [This is again a particular case of (b)]. We let A = Bisep(C1 , C2 , A1 , A2 , B), in case C1 and C2 are disjoint, be the set A = (C1 ∩ A1 ) ∪ (C2 ∩ A2 ) ∪ (B \ (C1 ∪ C2 )). If Γ, Γ , Γ are classes, Bisep(C1 , C2 , A1 , A2 , B) with C1 , C2 in Γ, A1 in Γ˘ , A2 in Γ and B in Γ . Moreover, if Γ = {∅}, we just write Bisep(Γ, Γ ) [The definition is symmetric in Γ and Γ˘ ]. (e) Separated Differences. For ≥ 2 a countable ordinal, we define A = SD (C : < , A : < , B) in case the C ’s and the A ’s are increasing, with A ⊆ C ⊆ A+1 , by A=
<
A \
<
C ∪ B \ C . <
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SOME RESULTS IN THE WADGE HIERARCHY OF BOREL SETS
...
Γ
...
Λ Λ
Figure 5. SD (Λ, Γ ) If Λ ⊆ Γ × Γ , and Γ is a class, SD (Λ, Γ ) is the set of all SD (C : <
, A : < , B) for B ∈ Γ and, for each < , the pair (C , A ) is in Λ. Again, for Γ = {∅} we write SD(Λ) [This operation will be used only for Λ = Γ, SU(Γ, Γ ) as defined in (b)]. We now proceed to apply these operations. Particular ways of combining these operations will be encoded by elements of 1 . Of course, we put restrictions on the allowed combinations. This leads to the following inductive definition of a “description” and a corresponding “described class”. Let u be some element of 1 . We let u = u0 , u1 , where u0 (n) = u(2n) and u1 (n) = u(2n + 1). Similarly we let u = (u)n : n ∈ where (u)n (m) = u(n, m). 0 is the element of 1 defined by 0(n) = 0. We now define inductively the relations “u is a description” and “u describes the class Γ” (written Γu = Γ). Descriptions are elements of 1 . The first ordinal u(0) will give the level of Γu (as informally discussed in the introduction), u(1) and sometimes u(2) the operation used to obtain Γu , and the remaining of u the description of the classes from which Γu has been obtained. Definition 1.2. The relations “u is a description” (written u ∈ D), and “u describes Γ”, are the least relations satisfying the following conditions: (a) If u(0) = 0, u is a description, and Γu = {∅} (b) If u(0) = ≥ 1, u(1) = 1 and u(2) = ≥ 1, then u ∈ D and Γu = D (Σ0 ). (c) If u = 2u ∗ , where ≥ 1, ≥ 1, u ∗ ∈ D and u ∗ (0) > , then u ∈ D and Γu = Sep(D (Σ0 ), Γu ∗ ). (d) If u = 3u0 , u1 , where ≥ 1, ≥ 1, u0 and u1 are in D, u0 (0) >
, u1 (0) ≥ or u1 (0) = 0, and Γu1 < Γu0 , then u ∈ D and Γu = Bisep(D (Σ0 ), Γu0 , Γu1 ). (e) If u = 4un : n ∈ , where ≥ 1 and each un is in D, and either for all n, un (0) = 1 > , and the Γun are strictly increasing, or un (0) = n
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and the n are strictly increasing with < supn n , then u ∈ D and Γu = SU(Σ0 , n Γun ). (f) If u = 5u0 , u1 , where ≥ 1, ≥ 2, u0 and u1 are in D, u0 (0) =
, u0 (1) = 4 [so that Σ0 , Γu0 is defined], and u1 (0) ≥ or u1 (0) = 0, and Γu1 < Γu0 , then u ∈ D and Γu = SD (Σ0 , Γu0 , Γu1 ). Our aim is to prove that the preceding descriptions give the complete picture of the Wadge classes of Borel sets. We begin with a simple fact. Proposition 1.3. The described classes are non self-dual Borel Wadge classes. Proof. The “Borel” part is clear. To prove that these classes are non selfdual, it is enough to exhibit a universal set, and this is easy by induction. The only fact to note here is that by using the reduction property, one can find a sequence of D (Σ0 ) sets, in × , which is universal for sequences of pairwise disjoint D (Σ0 ) sets of . It is clear that for each u ∈ D, there corresponds exactly one described class Γu . We now show that the level u(0) gives the closure properties of Γu . Lemma 1.4. Let u be a description, with u(0) = ≥ 1. Then (a) Γu is closed under union with a Δ0 set. (b) SU(Σ0 , Γu ) = Γu (written Γu is closed under Σ0 − SU). Proof. By induction. Case 1. u(1) = 1, so Γu = D (Σ0 ). (b) Let Cn be separating Σ0 sets, and An = D (An : < ) be the D (Σ0 ) n sets. Consider A = n (A ∩ Cn ). The A are clearly Σ0 and increasing, n and moreover SU(Cn : n ∈ , A : n ∈ ) = D (A : < ). This proves (b). (a) Let A = D (A : < ), with A ∈ Σ0 , and let B ∈ Δ0 . If is odd, then A ∪ B = E (A ∪ B, < ), and the A ∪ B : < n are an increasing sequence of Σ0 sets. If is even, let A0 = A0 \ B, and let A = A ∪ B for ≥ 1. Then again A ∪ B = D (A : < ), and the A are Σ0 and increasing. Case 2. u(1) = 2, so Γu = Sep(D (Σ0 ), Γu ∗ ), with u ∗ (0) > . By the induction hypothesis, Γu ∗ is closed under union with a Δ0 +1 set, and under Σ0 +1 − SU. Now intersection with a Σ0 +1 is a particular case of Σ0 +1 − SU, so Γu ∗ and Γ˘ u ∗ are closed under intersection and union with Δ0 +1 sets. This clearly implies ˘ u ∗ is closed under Σ0 − SU. that Γ
n n ˘ u ∗ , An ∈ Γ˘ u ∗ , Cn ∈ D (Σ0 ), and (a) Let An = Sep(Cn , A1 , A2 ), with An2 ∈ Γ 1
let A = SU(Cn : n ∈ , An : n ∈ ), where the Cn are pairwise disjoint
53 Σ0 sets. Then clearly A = Sep( n (Cn ∩ Cn ), n (An1 ∩ Cn ), n (An2 ∩ Cn )), with 0 n n n (Cn ∩ Cn ) ∈ D (Σ ), n (A1 ∩ Cn ) ∈ SU(Cn : n ∈ , A1 : n ∈ ) is n n ˘ in Γu ∗ and n (A2 ∩ Cn ) = SU(Cn : n ∈ , A2 : n ∈ ) is in Γu ∗ . This shows (a). ˘ u ∗ , A2 ∈ Γu ∗ , and let (b) Let A = Sep(C, A1 , A2 ), C ∈ D (Σ0 ), A1 ∈ Γ 0 B ∈ Δ . Then A ∪ B = Sep(C, A1 ∪ B, A2 ∪ B), and the induction hypothesis gives (b). SOME RESULTS IN THE WADGE HIERARCHY OF BOREL SETS
Case 3. u(1) = 3. (a) If An = Bisep(C1n , C2n , An1 , An2 , B n ) where Cin ∈ D (Σ0 ), An1 ∈ Γ˘ u0 , n A2 ∈ Γu0 and B n ∈ Γu1 , with u0 (0) > and u1 (0) ≥ or u1 (0) = 0, and A = SU(Cn : n ∈ , An : n ∈ ) n are pairwisedisjoint where the C Σ0 sets, then A = Bisep( n (C1n ∩ Cn ), n (C2n ∩ Cn ), n (An1 ∩ Cn ), n (An2 ∩ Cn ), n (B n ∩ Cn )) which, together with the induction hypothesis, proves (a). (b) Let again A = Bisep(C1 , C2 , A1 , A2 , B) with the sets in the same classes as before, and let D ∈ Δ0 . If u1 (0) ≥ 1, take B = B ∪ D, A1 = A1 ∪ B, A2 = A2 ∪ B. Then A ∪ B = Bisep(C1 , C2 , A1 , A2 , B ), which proves (b) in this case. If Γu1 = {∅}, so B = ∅, consider the sets C1 ∪ D and C2 ∪ D. These are D (Σ0 ) sets, by case 1. Let C1∗ , C2∗ reduce them. Then A = Bisep(C1∗ , C2∗ , A1 ∪ D, A2 ∪ D), which proves (b) in that case. Case 4. u(1) = 4. In this case Γu = SU(Σ0 , n Γun ), and the Γun are of level > (at least for n ≥ n0 ). (a) is almost trivial. For (b), 0let A = SU(Cn 0 : n ∈ , An : n ∈ ), with An ∈ p Γup , and Cn ∈ Σ , and let B ∈ Δ . Let B ∗ , Cn∗ : n ∈ be Σ0 sets reducing the sets B, Cn : n ∈ . Then ∗ , A ∪ B = SU(Cn : n ∈ , An : n ∈ ) where C0 = B ∗ , Cn = Cn−1 n ≥ 1, and A0 = B, An = An−1 , n ≥ 1. Case 5. u(1) = 5. (a) Suppose An = SD (Cn : < n, An : < n, B n ), where the pairs (Cn , An ) are in Σ0 , Γu0 , with u0 (0) = and u0 (1) = 4 and B n ∈ Γu1 , n and A = SU(C n : n ∈ , An: n ∈ ). Then A = SD ( n (C ∩ n n n C ), < , n (Cn ∩A ), < , n (Cn ∩B )), and the induction hypothesis immediately yields (a). (b) Let An = SD (C : < n, A : < n), and let D ∈ Δ0 . Then clearly A ∪ D = SD (C∪D : < , A ∪ D : < , B), and by the proof of case 4, (b), the C ∪ D are envelopes of the A ∪ D. This shows (b) in this case. Lemma 1.5. If A, B are two disjoint D (Σ0 ) sets, there are sets A∗ , B ∗ in D (Σ0 ) such that ∗ (a) A and B ∗ are disjoint
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(b) A ⊂ A∗ , B ⊂ B ∗ (c) A∗ ∪ B ∗ ∈ Σ0
Proof. Let C (resp. D) be the union of the Σ0 sets in a D (Σ0 ) definition of A (resp. B), and let C ∗ , D ∗ reduce C, D. It is easily checked that A∗ = (A ∩ C ∗ ) ∪ (D ∗ \ B) and B ∗ = (C ∗ \ A) ∪ (D ∗ ∩ B) satisfy the desired properties. Corollary 1.6 (Normal form for the Bisep operation). Suppose A ∈ Bisep(D (Σ0 ), Γu0 , Γu1 ), where Γu0 is of level at least + 1, and Γu1 < Γu0 . Then for some set C in Σ0 , A = (A ∩ C ) ∪ (B \ C ) where A ∩ C ∈ Bisep(D (Σ0 ), Γu0 ), C \ A ∈ Bisep(D (Σ0 ), Γu0 ) and B ∈ Γu1 . 0 Proof. Extend the two D (Σ ) separating sets, using Lemma 1.5. The union C of the extended D (Σ0 ) sets clearly works. Let us now define a notion of type for each description u. This type intuitively corresponds to the character (successor, limit of cofinality or limit of cofinality 1 ) of the class Γu , among the Wadge classes of level at least u(0). Definition 1.7. The type t(u) is defined by the following conditions: (a) If u(0) = 0, t(u) = 0 ({∅} is the first class) (b) If u(0) = ≥ 1, and u(1) = 1, u is of type 1 if u(2) is successor, and u is of type 2 if u(2) is limit (c) If u(0) ≥ 1 and u(1) = 2, u is of type 3 (d) If u(0) ≥ 1 and u(1) = 3, then ⎧ 1 if u1 is of type 0 and u(2) is successor ⎪ ⎪ ⎪ ⎨2 if u1 is of type 0 and u(2) is limit u is of type ⎪ t(u1 ) if u1 (0) = u(0) ⎪ ⎪ ⎩ 3 if u1 (0) > u(0) (e) If u(0) ≥ 1 and u(1) = 4, u is of type 2 (f) If u(0) ≥ 1 and u(1) = 5, ⎧ ⎪ if u1 is of type 0 ⎨2 u is of type t(u1 ) if u1 (0) = u(0) ⎪ ⎩ 3 if u1 (0) > u(0) Definition 1.8. We define D 0 = {u : t(u) = 0} = {u : u(0) = 0}, D = {u : u(0) = 1 and t(u) = 1}, D = {u : u(0) = 1 and t(u) = 2} and D 1 = D \ (D 0 ∪ D + ∪ D ) = {u : u(0) = 1 and t(u) = 3} ∪ {u : u(0) > 1}. +
SOME RESULTS IN THE WADGE HIERARCHY OF BOREL SETS
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We are now able to restate Wadge’s result in terms of our notion of description: Theorem 1.9. Assume Borel Determinacy. Let W = {Γu : u ∈ D} ∪ {Γ˘ u : u ∈ D} ∪ {Δ(Γu ) : u ∈ D + ∪ D }. Then W is exactly the set of all Borel Wadge classes. The (long) proof of this theorem goes roughly as follows. We shall define, for each description u ∈ D 0 , a set Qu of descriptions, satisfying the following properties: (A) If u ∈ D + , then Qu = {u}, Γu < Γu and the only Wadge class Γ such that Γu < Γ < Γu is Δ(Γu ). (B) If u ∈ D , then Qu = {un : n ∈ }, for each n Γun < Γu , and the only Wadge class Γ such that ∀n Γun < Γ < Γu is Δ(Γu ). (C) If u ∈ D 1 , then Qu is a set of descriptions of cardinality 1 , and Δ(Γu ) = ∪{Γu : u ∈ Qu }. This will finish the proof of the theorem. For suppose there is a Borel Wadge class not in W , let Γ be the <-least counterexample (or one of the two <-least counterexamples, in a non self-dual case, for both must be outside W by definition of it). As the sequence (Σ0 ) is cofinal in the Borel Wadge classes, there is a <-least described class Γu such that Γ < Γu , and clearly u ∈ D 0 . Now each of the remaining cases, u ∈ D + , u ∈ D and u ∈ D 1 gives immediately, using (A), (B) and (C), that Γ is in W , a contradiction. In the following proof, we shall only indicate the main steps. Some arguments are just sketched, and others are missing (they can be found either in Kuratowski [Kur66] or in Wadge [Wad84]). We first consider the case of a description u of type 1. This will take care of (A), but as we shall see also of a part of (C). Definition 1.10. For each description u of type 1, we define a description u by the following conditions [and we indicate the corresponding classes]: (a) u(1) = 1, u(2) = + 1 if = 0, u = 0 [Γu = Σ0 , Γu = {0}] 0 0 if > 0, u = u(0)10 [Γu = D +1 (Σ ), Γu = D (Σ )] (b) u(1) = 3, t(u1 ) = 0 and u(2) = + 1 if = 0, u = u0 [Γu = Bisep(Σ0 , Γu0 ), Γu = Γu0 ] 0 if > 0, u = u(0) 2 0 [Γu = Bisep(D +1 (Σ ), Γu0 ), Γu = 0 Sep(D (Σ ), Γu0 )] (c) u(1) = 3 or 5, t(u1 ) = 1 and u1 (0) = u(0) u = u(0) u(1) u(2) u0 , u1 Lemma 1.11. For each u of type 1, (a) Γu = Bisep(Σ0u(0) , Γu )
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(b) Δ(Γu ) = Bisep(Δ(Σ0u(0) ), Γu ) Proof. (a) Case 1. u(1) = 1, u(2) = 0. The equality Σ0 = Bisep(Σ0 , {∅}) is trivial. ˘ u and Γu are contained Case 2. u(1) = 1, u(2) = + 1 for ≥ 1. First Γ in Γu = D+1 (Σ0 ), so by the closure properties of Γu , Bisep(Σ0 , Γu ) ⊂ Γu . On the other hand, if A ∈ D+1 (Σ0 ), then for some C in Σ0 and some B in D (Σ0 ) = Γu , A = C \ B. So A is in Bisep(Σ0 , Γu ). Case 3. u(1) = 3, t(u1 ) = 0 and u(2) = 1. Then Γu = Bisep(Σ0 , Γu0 ) and Γu = Γu0 , so the equality is trivial. Case 4. u(1) = 3, t(u1 ) = 0 and u(2) = + 1 with ≥ 1. Then Γu = Bisep(D+1 (Σ0 ), Γu 0 ), and Γu = Bisep(D (Σ0 +1 ), Γu0 ). The inclusion Bisep(Σ0 , Γu 0 ) ⊂ Γu is again easy. For the converse, assume A ∈ Γu . By the normal form for the Bisep operation, let C1 , C2 in Σ0 , D1 , D2 in D (Σ0 ) C2 , be such that C1 \ D1 ∩ C2 \ D2 = ∅, D2 ⊂ C1 , C2 ⊂ D1 , and A ⊂ C1 ∪ A ∩ C1 \ D1 ∈ Γu0 , A ∩ C2 \ D2 ∈ Γu0 . Let C1∗ , C2∗ be Σ0 sets reducing C1 , C2 ˘ u, and remark that A ∩ C1∗ = (A ∩ D1 ∩ C1∗ ) ∪ (A ∩ (C1 \ D1 ) ∩ C1∗ ) is in Γ ∗ ∗ and A ∩ C2 = (A ∩ D2 ∩ C2 ) ∪ (A ∩ (A ∩ (C2 \ D2 )) ∩ C2∗ ) is in Γu , so that, as A ⊂ C1∗ ∪ C2∗ , we obtain A ∈ Bisep(Σ0 , Γu ). Case 5. (Induction step) Suppose u(1) = 3, t(u1 ) = 1 and u1 (0) = u(0). So Γu = Bisep(D (Σ0 ), Γu0 , Γu1 ) and Γu = Bisep(D (Σ0 ), Γu0 , Γu 1 ), and by the induction hypothesis, we can assume Γu1 = Bisep(Σ0 , Γu 1 ) (for u1 (0) = ). The inclusion Bisep(Σ0 , Γu ) ⊂ Γu is easy. Suppose A ∈ Γu . Then by the normal form for Bisep, we have A = (A∩C )∪(B \C ), where B ∈ Γu1 and C ∈ 0 Σ , A∩C , and C \A are in Bisep(D (Σ0 +1 ), Γu0 ). Now B = (B ∩C1 )∪(B ∩C2 ), ˘ u 1 , B ∩ C2 ∈ Γu 1 . Let where C1 , C2 are two disjoint Σ0 sets, and B ∩ C1 ∈ Γ ∗ ∗ 0 C1 , C2 in Σ reduce the pair C ∪ C1 , C ∪ C2 . Then A = (A ∩ C1∗ ) ∪ (A ∩ C2∗ ), ˘ u 1 ) = Γ˘ u , and A ∩ C ∗ ∈ and it is clear that A ∩ C1∗ ∈ Bisep(D (Σ0 ), Γu0 , Γ 2 0 Bisep(D (Σ ), Γu0 , Γu 1 ) = Γu , so that A ∈ Bisep(Σ0 , Γu ). The second case of the induction case (u(1) = 5, t(u1 ) = 1 and u1 (0) = u(0)) is entirely analogous, and we omit it. (b) is an easy consequence of (a). The inclusion Bisep(Δ(Σ0u(0) ), Γu ) ⊂ Δ(Γu ) is obvious. Suppose now A ∈ Δ(Γu ). Then by (a), A = (A ∩ C1 ) ∪ (A ∩ ˘ u , A ∩ C2 ∈ Γu . Similarly C2 ) where C1 , C2 are disjoint Σ0 sets, and A ∩ C1 ∈ Γ ˘ u, A˘ = (A˘ ∩ D1 ) ∪ (A˘ ∩ D2 ), where D1 , D2 are disjoint Σ0 sets and A˘ ∩ D1 ∈ Γ ∗ ∗ 0 ˘ ˘ A ∩ D2 ∈ Γu . Let C , D be Σ sets reducing the pair C1 ∪ C2 , D1 ∪ D2 . As sets C ∗ = C ∩ C ∗ , C ∗ = C ∩ C ∗ , D ∗ = C ∗ ∪ D ∗ = , C ∗ , D ∗ and the 1 2 1 2 1
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˘ ∪ D1 ∩ D ∗ , D2∗ = D2 ∩ D ∗ , are Δ(Σ0 ). But clearly A = [(C1∗ ∩ A) ∪ (D1∗ \ A)] ˘ which shows A ∈ Bisep(Δ(Σ0 ), Γu ). [(D2∗ ∩ A) ∪ (C2∗ \ A)],
Corollary 1.12 (Statement (A)). Let u be some description in D + , and let Qu = {u}. Then the only Wadge class Γ such that Γu < Γ < Γu is Δ(Γ(u)). Proof. Using Lemma 1.11 for u(0) = 1, we see that Δ(Γu ) = Bisep(Δ01 , Γu ). imUsing the game-theoretical characterization of the Wadge ordering this + mediately implies Δ(Γu ) = Γu . Definition 1.13. Let u be some description in D 1 , with u(0) = + 1, for
≥ 1, and t(u) = 1. We define Qu = { 3u, 0 : 1 ≤ < 1 } Corollary 1.14 (Statement (C) for u(0) = + 1, t(u) = 1). Let u be a description with u(0) = + 1, ≥ 1 and t(u) = 1. Then Qu is a set of descriptions (of level ), and Δ(Γu ) = ∪{Γu : u ∈ Qu }. Proof. (a) The case u(0) = + 1, u(1) = 1, u(2) = 1, i.e. Γu = Σ0 +1 is solved by the Hausdorff-Kuratowski theorem: Δ0 +1 = ∪{D (Σ0 ) : 1 ≤ < 1 }. (b) In the general case, we have that Δ(Γu ) = Bisep(Δ(Σ0 +1 ), Γu ) by Lemma 1.11. Using the Hausdorff-Kuratowski theorem, we obtain Δ(Γu ) = Bisep D (Σ0 ), Γu <1 = {Bisep(D (Σ0 ), Γu ) : 1 ≤ < 1 } = {Γu : u ∈ Qu } by the definition of Qu . We now turn to the case of u(0) a limit ordinal. What we need here is the analysis, obtained by Wadge, of Δ0 for limit . This analysis is done by iterating the SU operation. Definition 1.15. Let be a countable limit ordinal, and n n∈ be an increasing sequence of ordinals, cofinal in . (a) Let Γ be some class. We define, for each (n, ) in × 1 a class SUn, (Γ), by the following induction: (i) SUn,0 (Γ) = SU(Σ0 n , Γ) (ii) SUn, (Γ) = SU(Σ0 n , ∪{SUp, (Γ) : p ∈ , < }), for > 0. (b) Similarly, if s = un : n ∈ codes a sequence of descriptions, with supn un (0) = , we define a family un, (s) by: (i) un,0 (s) = n 4s (ii) un, (s) = n 4un, (s) : n ∈ , < for > 0.
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(c) If u = 11u , we set Qu = {un, (s) : n ∈ , ∈ 1 } where s = { n 110 : n ∈ }. Lemma 1.16. If s = un : n ∈ codes a sequence of descriptions with either un (0) = and the Γun increasing, or un (0) = n , then each un, (s) is a description, and Γun, = SUn, (Γs ), where Γs = n Γun . Proof. The only thing to check is that the levels of the classes on which SU is performed are acceptable, and we omit it. The next result is a theorem of Wadge [Wad84], and gives the analysis of Δ0 for limit , in a way very similar to the Hausdorff-Kuratowski theorem. Theorem 1.17 (Wadge). Let be limit, and n n∈ be cofinal in . Let Γ = < Σ0 . Then Δ0 = {SUn, (Γ) : n ∈ , < 1 } Hence, with our notations, if u is a description with u(0) = , u(1) = u(2) = 1, then Δ(Γu ) = {Γu : u ∈ Qu }. Definition 1.18. Let u be a description of type 1, with u(0) = limit, and n a cofinal sequence in . Define a sequence su = un : n ∈ by un = n 31u, 0. We set Qu = {un, (su ) : n ∈ , < 1 }. Corollary 1.19 (Statement (C) for t(u) = 1 and u(0) limit). Let u be a description of type 1 with u(0) = limit. Then Δ(Γu ) = {Γu : u ∈ Qu } Proof. By Lemma 1.11, we know that Δ(Γu ) = Bisep(Δ(Σ0 ), Γu ) = Bisep {SUn, (Γ) : n ∈ , < 1 }, Γu with Γ = equality
n
Σ0 n , by using Wadge’s theorem. So the only thing to prove is the Bisep(Σ0 n , Γu ) . Bisep(SUn, (Γ), Γu ) = SUn, n
(a) Suppose first that = 0. The left side of the equality is Γ = Bisep(SU(Σ0 n ), Γu ) and the right side is Γr = SU(Σ0 n , Bisep(Γ, Γu )). The inclusion from right to left is obvious. Let A be in Γ . For some disjoint sets C1 , C2 in SU(Σ0 n , Γ), some A1 , A2 in Γ˘ u , Γu respectively, we have A = (A1 ∩ C1 ) ∪ (A2 ∩ C2 ). Now, C1 = p (Hp1 ∩ Cp1 ) with the Hp1 disjoint in Σ0 n , and the Cp1 in Γ; and similarly we can find corresponding sets Hp2 , Cp2 for C2 . Let Kp1 , Kp2 , p ∈ , be Σ0 n sets reducing the sets Hp1 , Hp2 . Then
SOME RESULTS IN THE WADGE HIERARCHY OF BOREL SETS 59 A ⊂ ( p Kp1 )∪( p Kp2 ), and moreover A∩Kp1 = (A∩Kp1 ∩Cp1 )∪(A∩Kp1 ∩ C˘p1 ) is in Bisep(Γ, Γu ), and similarly for A ∩ Kp2 . This shows that A is in Γr . (b) Suppose now > 0. The left side is now Γ = Bisep SU Σ0 n , {SUp, (Γ) : p ∈ , < } , Γu 0 = SU Σ n , {Bisep(SUp, (Γ)) : p ∈ , < } by the same proof as in (a), and the right hand side is Γr = SU Σ0 n , {SUp, (Bisep(Γ, Γu )) : p ∈ , < } . The induction hypothesis then immediately gives the result. We now turn to the case of descriptions of type 2.
Definition 1.20. For each description u of type 2 we define a sequence su by the following conditions: (a) If u(1) = 1 and u(2) = is limit, with cofinal sequence n , let su = 1 n 0 : n ∈ . (b) If u(1) = 3 and t(u1 ) = 0, u(2) = is limit, with cofinal sequence n , let su = 2 n u0 : n ∈ . (c) If u(1) = 4, so u = 4u , let su = u . (d) If u(1) = 5 and t(u1 ) = 0, then • if u(2) = + 1, with > 0, so u0 = 4un : n ∈ , let su = 5u0 , u1 : n ∈ . • if u(2) = is limit with cofinal sequence n , let su = 5 n u0 , u1 : n ∈ . (e) (induction step) If u(1) = 3 or 5, and t(u1 ) = 2 and u1 (0) = u(0), then writing su1 = un1 : n ∈ , set su = u(0)u(1)u(2)u0 , un1 : n ∈ . Definition 1.21. For each description u of type 2, we define a set Qu of descriptions by the following: (a) If u(0) = 1, Qu = {su n : n ∈ }. (b) If u(0) = + 1, with > 0, Qu = { 5 4su , 0 : 1 < < 1 }. (c) If u(0) = is limit, with cofinal sequence n , Qu = {up, (su ) : p ∈ , < 1 }. Definition 1.22 (Partitioned Unions (Wadge)). We say that A = PU(Cn : n ∈ , An : n ∈ ) if A = SU(Cn , An ), and moreover the envelope C = n Cn is (so that Cn : n ∈ is a partition of ). PU(Γ, Γ ) is the class of all PU(Cn , An ) with Cn ∈ Γ and An ∈ Γ , for each n. Lemma 1.23. Let u be a description of type 2 and level ≥ 1. Then (a) Γu = SU(Σ0 , {Γu : u ∈ Qu }) (b) Δ(Γu ) = PU(Σ0 , {Γu : u ∈ Qu }
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(c) In particular, if = 1, the only Wadge class Γ such that ∀u ∈ Qu , Γu < Γ < Γu is Δ(Γu ). (Assertion (B)) Proof. (a) is by induction. Case 1. u(1) = 1 and u(2) = . Then Γu = D (Σ0 ), and we want to prove Γu = SU(Σ0 , < D (Σ0 )). From right to left the inclusion is obvious. If A ∈ Γu , let A : < be an increasing sequence of Σ0 sets with A = D (A ), and let A : < reduce A : < . Then A = < (A ∩ A ), and A ∩ A = A ∩ A ∩ A is clearly in D (Σ0 ) Case 2. u(1) = 3, t(u1 ) = 0, u(2) = . Then Γu = Bisep(D (Σ0 ), Γu0 ), and we want to prove Γu = SU(Σ0 , < Sep(D (Σ0 ), Γu0 )). Again the inclusion from right to left is trivial. So let A ∈ Γu , and let C0 , C1 ∈ D (Σ0 ) be the biseparating sets. Using case 1, C0 and C1 are in SU(Σ0 , < D (Σ0 )), and then it is obvious, using the closure properties of Γu0 , that A ∈ SU(Σ0 , 0 < Sep(D (Σ ), Γu0 )). Case 3. u(1) = 4, so Γu = SU(Σ0 , n Γun ), which is the equality we want. Case 4. u(1) = 5, u(2) = + 1, ≥ 1, t(u1 ) = 0. Then, Γu = SD+1 (Σ0 , Γu0 ) where Γu0 = SU(Σ0 , n Γun ) and we want to prove Γu = SU Σ0 , SD (Σ0 , Γu0 , Γun ) It is clear that SD (Σ0 , Γu0 , Γun ) is in Δ(SD+1 ((Σ0 , Γu0 )), so the inclusion from right to left is obvious. Suppose A ∈ Γ u . Then for some increasing now n n pairs A , C : ≤ , with A = n (A ∩ C ), n Cn = C , where An ∈ Γn and Cn ∈ Σ0 , we have A = (A \ < C ). Let A< ⊂ C< . Now let Cn∗ : n ∈ reduce the sequence C< ∪ Cn0 , and consider A ∩ Cn∗ = (A< ∩ Cn∗ ) ∪ (A \ C< ∩ Cn∗ ). It is clearly in SD (Σ0 , Γu0 , Γun ). Moreover A ⊂ n Cn∗ = C , so that A ∈ SU(Σ0 , SD (Σ0 , Γu0 , Γun )). Case 5. u(1) = 5, u(2) = is limit and t(u1 ) = 0. The proof is entirely analogous to case 4, and we omit it. Case 6. (Induction step) Suppose that u(1) = 3 (the case u(1) = 5 is analogous) and t(u1 ) = 2, u1 (0) = . We have Γu = Bisep(D (Σ0 ), Γu0 , Γu ) with Γu1 = SU(Σ0 , {Γu : u ∈ Qu1 }) and we want to prove that Γu = SU Σ0 , {Bisep(D (Σ0 ), Γu0 , Γu ) : u ∈ Qu1 } Let Γ∗ = {Γu : u ∈ Qu1 }, so that the left side is Γu = Bisep(D (Σ0 ), Γu0 , SU(Σ0 , Γ∗ )), and the right side Γr = SU(Σ0 , Bisep(D (Σ0 ), Γu0 , Γ∗ )).
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The proof that Γr = Γu is entirely analogous to the one in Corollary 1.19 (a). We omit it. (b) The For if A, A˘ ∈ proof of part (b) follows easily from part (a). 0 0 SU(Σ , n Γn ), so that A = n (An ∩Cn ), An ∈ Γn , Cn ∈ Σ , (pairwise disjoint sets) and A˘ = n (An ∩ Cn ), An ∈ Γn , Cn ∈ Σ0 , then choosing a sequence Cn∗ , Cn ∗ of Σ0 sets reducing the sequence Cn : n ∈ , Cn : n ∈ , we obtain A = n (An ∩ Cn∗ ) ∪ n (A˘ n ∩ Cn ∗ ), which, because n Cn∗ ∪ n Cn ∗ = , shows that A ∈ PU(Σ0 , n Γn ). (c) Assertion (B) then follows easily from the game theoretical characterization of Wadge’s ordering. The following next two lemmas take care of assertion (C) for descriptions u of type 2 and of level, respectively, a successor and a limit ordinal. Lemma 1.24. Assume u is a description of type 2 and level u(0) = + 1, with ≥ 1. Then Δ(Γu ) = n {Γu : u ∈ Qu }. Proof. By Lemma 1.23, we know that Δ(Γu ) = PU Σ0 +1 , Γsu (n) n and we want to prove that 0 0 Δ(Γu ) = SD Σ , SU Σ , Γsu (n) 2 ≤ < 1 n Let Γ∗ = n Γsu (n) . By the definition of su , each Γsu (n) is of level ≥ +1. From right to left, the inclusion is easy: If A ∈ SD (Σ0 , SU(Σ0 , Γ∗ )) for some n0 < 1 , let A , C , < 0 , be pairs witnessing this fact, A ∈ SU(Σ0 , Γ∗ ), C ∈ Σ0
and A = <0 (An \ < C ). ∗ 0 ∗ 0 For each A = A \ < C is clearly in SU(D2 (Σ ), Γ ), so is in PU(Σ +1 , ∗ Γ ). But the sets A∗ are disjoint, and separated by the C∗ = C \ < C , which are disjoint and in D2 (Σ0 ). This clearly proves that A ∈ Δ(Γu ) = PU(Σ0 +1 , Γ∗ ). For the other inclusion, let us look at the case = 1. Suppose A ∈ Δ(Γu ) = PU(Σ02 , Γ∗ ). Because each Σ02 set is the disjoint union of Π01 sets, we also have some sequence F A ∈ PU(Π01 , Γ∗ ), say A = n An , where An ∈ Γ∗ and for n of closed sets, An ⊂ Fn , and the Fn are a partition of . Define a transfinite sequence O , < 1 , by O0 = {O ∈ Σ01 : A ∩ O ∈ Γ∗ }, and more generally O = {O ∈ Σ01 : O ∩ (A \ < O ) ∈ Γ∗ }. The sequence O is increasing, hence is stationary after 0 < 1 . We claim that the sequence (A ∩ O ) ∪ < O , O , ≤ 0 witnesses that A ∈ SD0 +1 (Σ01 , SU(Σ01 , Γ∗ )). We have to prove two things: First, that A ∩
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(O \ < O ) is in SU(Σ01 , Γ∗ ) which will imply that (A ∩ O ) ∪ < O is also in it, and secondly that A ⊂ O0 . The first fact comes from the definition of O : O is the union of a disjoint family of open sets On with On ∩ (A \ < O O ) ∈ Γ∗ . This is what we wanted. The second fact isan obvious use of the Baire category theorem applied to the partition of ( < O ) induced by the sets Fn . The case of arbitrary can be reduced to the preceding case using Kuratowski’s technique of generalized homeomorphisms. An alternative proof would use the effective topologies we introduce in Section 2. We shall also omit the proof of the next result, which again uses an argument of transfer, as in the proof of Wadge’s lemma, and in the preceding lemma. Lemma 1.25. Assume u is a description of type 2 and level u(0) = a limit ordinal. Then Δ(Γu ) = {Γu : u ∈ Qu }. We now turn to the last step of the proof of Theorem 1.9: the case of a description of type 3. Lemma 1.26. Let u be a description of type 3, with u(0) = . Then Γu is closed under the intersection with a Π0 set. Moreover Δ(Γu ) has the same closure property. Proof. The second assertion is an immediate corollary of the first, and the fact that Γu is closed under union with a Σ0 set, which can easily be seen from the proof of Lemma 1.4 (b). [The only classes of level not closed under union 0 0 with a Σ set are the D Σ for n ∈ , n even.] For if A ∈ Δ(Γu ) and B ∈ Π0 , ˘ B˘ ∈Γu then A∩B ∈ Γu by the first assertion of the lemma, and (A∩B)˘ = A∪ by the preceding remark. To prove that Γu is closed under intersection with a Π0 set, argue by in duction. If u(1) = 3 or 5, then either we have u1 (0) > u(0), and then Γu1 0 is clearly closed under intersection with a Π set (in fact with a Σ0 +1 set, by Lemma 1.4), or u1 is itself of type 3, and we can apply the induction hypoth0 ∗ esis. The only other case is if u(1) = 2, so Γu = Sep(D (Σ ), Γu ). But then u ∗ (0) > , so Γu ∗ is closed under intersection with a Π0 , and the conclusion follows immediately. Definition 1.27. For each u of type 3, we define a set Qu of descriptions by the following conditions: Case 1. u(1) = 2, so Γu = Sep(D (Σ0 ), Γu ∗ ). Define Qu = { 3u ∗ , u : u ∈ D, Γu < Γu ∗ and u (0) ≥ }. Case 2. u(1) = 3 or 5, and t(u1 ) = 3 (inductive step). Define Qu = {u(0)u(1)u(2)u0 , u : u ∈ Qu1 }.
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Case 3. u(1) = 3 or 5, and t(u1 ) = 1 or 2 (so u1 (0) > u(0)). Then Qu1 has been previously defined, and define Qu = {u(0)u(1)u(2)u0 , u : u ∈ Qu1 and u (0) ≥ }. Lemma 1.28 (Assertion (C) for u of type 3). Let u be a description of type 3. Then Δ(Γu ) = {Γu : u ∈ Qu }. Proof. Case 1. u(1) = 2, so Γu = Sep(D (Σ0 ), Γu ∗ ), with u ∗ (0) > . We want to prove that Δ(Γu ) = {Bisep(D (Σ0 ), Γu ∗ , Γu ) : Γu < Γu ∗ , u (0) ≥ } Suppose first A is in the right hand side class. Then, by the closure properties of Γu ∗ , there are disjoint D (Σ0 ) sets C1 and C2 such that A1 = A ∩ C1 ∈ Γ˘ u ∗ , A2 = A ∩ C2 ∈ Γu ∗ , and B = A \ (C1 ∪ C2 ) ∈ Δ(Γu ∗ ). Now A2 and B are two Γu ∗ sets separated by the Σ0 +1 sets C2 and C˘2 , so A2 ∪ B ∈ Γu ∗ , and ˘ u ∗ sets separated by the A ∈ Sep(D (Σ0 ), Γu ∗ ). Similarly A1 and B are two Γ 0 Σ +1 sets C1 and C˘1 , so A1 ∪ B ∈ Γ˘ u ∗ and A˘ ∈ Sep(D (Σ0 ), Γu ∗ ). This gives A ∈ Δ(Γu ). For the converse, we suppose A ∈ Δ(Γu ), and we want to find disjoint ˘ u ∗ , A∩C2 ∈ Γu ∗ and A\(C1 ∪C2 ) ∈ D (Σ0 ) sets C1 and C2 such that A∩C1 ∈ Γ ∗ Δ(Γu ), for then the inductive hypothesis will give the result. Let C, C be two D (Σ0 ) sets such that A ∩ C ∈ Γ˘ u ∗ , A \ C ∈ Γu ∗ , and A˘ ∩ C ∈ Γ˘ u ∗ , A˘ \ C ∈ Γu ∗ . Let C1 and C2 be D (Σ0 ) sets reducing the pair (C, C ). Then ˘ u∗ , A ∩ ˘ is in Γu ∗ , (A ∩ C1 ) = A ∩ C ∩ C1 ∈ Γ C2 = C2 \ A˘ = C2 \ (C ∩ A) and finally B = A \ (C1 ∪ C2 ) = A \ (C ∪ C ) = (A \ C ) \ C is in Γu ∗ and ˘ is in Γu ∗ . This proves case 1. B˘ = (C1 ∪C2 )∪A˘ = (C ∪C )∪A˘ = C ∪C ∪A\C Case 2. u(1) = 3, t(u1 ) arbitrary. We have Γu = Bisep(D (Σ0 ), Γu0 , Γu1 ), and we know that u0 (0) > = u(0), and eitheru1 (0) > u(0), or u1 (0) = u(0) and u1 is of type 3. We may assume that {Γu : u ∈ Qu1 and u (0) ≥
} = Δ(Γu1 ): This is the induction hypothesis if u1 is of type 3, and if u1 is of type 1 or 2, a look at the definition of Q shows that {Γu : u 1 u ∈ Qu1 and u (0) ≥ } = {Γu : u ∈ Qu1 }, if u1 (0) > . But we have already proved that this last class is Δ(Γu1 ). So, we want to prove that Δ(Γu ) = Bisep(D (Σ0 ), Γu0 , Δ(Γu1 )). The inclusion from right to left is obvious. Suppose now A ∈ Δ(Γu ). Using the normal form for the Bisep operation, plus the fact that Δ(Γu1 ) is closed under intersection with a Π0 set (Lemma 1.26), we have A = (C0 ∩ A) ∪ (C1 ∩ A) ∪ B, where C0 , C1 are disjoint 0 0 ˘ u0 , C1 ∩ A ∈ Γu0 and B ∈ Γu1 , D (Σ ) sets, with C0 ∪ C1 ∈ Σ and C0 ∩ A ∈ Γ ˘ ˘ ˘ B ∩(C0 ∪C1 ) = ∅. Similarly we have A = (C0 ∩ A)∪(C 1 ∩ A)∪B , with similar ∗ ∗ 0 properties. Let C0 , C1 be two Σ sets reducing the pairC0 ∪ C1 , C0 ∪ C1 . Then
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∗ ∗ 0 ˘ A∩C0∗ is in Bisep(D (Σ0 ), Γu0 ), and A∩C 1 , so A∩C1 is in Bisep(D (Σ ), Γu0 ). ∗ ∗ ∗ So, we just have to show that A \ (C0 ∪ C1 ) ∈ Δ(Γu1 ). But A \ C0 ∪ C1∗ = A\(C0 ∪C1 ∪C0 ∪C1 ) = B \(C0 ∪C1 ). We clearly have A\(C0∗ ∪C1∗ ) ∈ Δ(Γu1 ).
Case 3, for u(1) = 5, is entirely similar, and we omit it. Lemmas 1.12, 1.14, 1.16, 1.17, 1.19, 1.23, 1.24, 1.25 and 1.28 put together, give a proof of the assertions (A), (B) and (C) of page 55 and hence prove Theorem 1.9. §2. Effective Results in the Borel Wadge hierarchy. The Wadge classes considered in the first part are boldface classes. We now are interested in their lightface counterparts, and in order to define them, we need a coding system, both for classes and for sets in each class. For the classes, there is no problem: it is enough to code by reals sequences of countable ordinals, and this is obvious: we say that α is a D-code, written α ∈ D, if for every n αn ∈ WO, and the coded sequence uα = |αn | : n ∈ is a description. Going back to the definition of descriptions shows immediately that D is a Π11 set. We shall denote by Γα the class Γuα (although it is a bit ambiguous, as some descriptions may be reals). Encoding the sets in each Γα is also easy, but technical. Fix some recursive real 1 in WO, with |1| = 1. Start from a pair (W, C ) in Π11 which is universal for Σ0 sets in the following sense: 1. W ⊆ × , and ∃(α, ) ∈ W ⇐⇒ α ∈ WO 2. C ⊆ × × is Π11 , and ∀α ∈ WO, Cα = {(, ) : (α, , ) ∈ C } is universal for Σ0|α| subsets of 3. C is Δ11 on W , i.e., the relation (α, ) ∈ W ∧ (α, , ) ∈ / C is Π11 1 is is It is then easy to construct a Π1 pair (W , C ) such that 1. ∃(α, , ) ∈ W is ⇐⇒ α ∈ WO ∧ ∈ WO is 2. Cα, is universal for < -increasing sequences of Σ0|α| sets (C is ⊂ × × × × ) is 1 is 3. C is Δ1 on W . [Define (α, , ) ∈ W ⇐⇒ ∈ WO ∧ ∀n(α, ()n ) ∈ W ∧ ∀n∀m∀(((n, m) = 0) ∧ (α, ()n , ) ∈ C =⇒ (α, ()n , ) ∈ C ) and (α, , , n, ) ∈ C is ⇐⇒ (α, , ) ∈ W is ∧ (α, ()n , ) ∈ C .] Similarly, one can define a Π11 pair (W ds , C ds ) such that (a) ∃(α, ) ∈ W ds ⇐⇒ α ∈ WO (b) For α ∈ WO, Cαds is universal for pairwise disjoint sequences of Σ0|α| sets. (c) C ds is Δ11 on W ds [Let (α, ) ∈ W ds ⇐⇒ ∀n(α, ()n ) ∈ W , and if C1 = {(α, , n, ) : (α, ) ∈ W is ∧ (α, ()n , ) ∈ C }, let C ds reduce C1 (with respect to n), in such 0 a way that for each α (C )ds α,,n is Σ|α| .]
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With these preliminary constructions, we can define a relation is a Γα code, together with “ codes the the set Γα, ”. Definition 2.1. For α ∈ D, the relations “ is a Γα -code”, and “The set in Γα coded by is A” (written Γα, = A) are defined inductively by the following conditions: (a) If |(α)0 | = 0 and (0) = 0, is a Γα -code and Γα, = ∅ (b) Suppose |(α)0 | ≥ 1 and |(α)1 | = 1, and = 1 ∗ . Then if ((α)0 , (α)2 , ∗ ) ∈ W is , then is a Γα -code, and Γα, = D|(α)2 | (A : < |(α)2 |),
(c)
(d)
(e)
(f)
is for the unique n which has order type where A = C(α) ∗ 0 ,(α)2 , ,n in <(α)2 . Suppose |(α)0 | ≥ 1 and |(α)1 | = 2. Let α0 = (α)0 1(α)2 00 . . . and let α1 = (α)n+3 : n ∈ . Then is a Γα code if = 20 , 1 , 2 , where 0 is a Γα0 -code, 1 and 2 are Γα1 -codes. Moreover Γα, = ˘ α1 ,1 , Γα2 ,2 ). Sep(Γα0 ,0 , Γ Suppose now |(α)0 | ≥ 1 and |(α)1 | = 3. Let α0 = (α)0 1(α)2 00 . . . and let α1 , α2 be such that uα = 3uα1 , uα2 . (Such α1 , α2 could be defined precisely, and are supposed to be recursive in α.) Then is a Γα -code if = 30 , 1 , 2 , 3 , 4 where 0 , 1 are Γα0 -codes, 1 , 2 are Γα1 -codes, 4 is a Γα2 -code, and Γα0 ,0 ∩ Γα0 ,1 = ∅; and then Γα, = ˘ α1 ,2 , Γα1 ,3 , Γα2 ,4 ). Bisep(Γα0 ,0 , Γα0 ,1 , Γ If |(α)1 | = 4, then let αn be (recursively in α) a sequence such that uα = 4uαn : n ∈ ; then is a Γα -code if = 4 ∗ , ∗∗ : n ∈ , where ((α)0 , ∗ ) ∈ W ds , ((α)0 , ∗∗ ) ∈ W and codes the union of the disjoint sequence coded by ∗ , and for each n n is a Γαn -code. Then ds : n ∈ , Γαn ,n : n ∈ ). Γα, = SU(C(α) ∗ 0 , ,n Finally, if |(α)1 | = 5, let α0 , α1 (recursively in α) be such that uα =
5uα0 , uα1 Then is a Γα -code if = 51 : n ∈ n ∈ , where 1 is a Γα1 -code, for each n n is a Γα0 code, say n = 4n∗ , n∗∗ , np : p ∈ and the sequence of pairs (A , C ), < |(α)2 | defined by A = Γ(α)0 ,n , C = C(α)0 ,n∗∗ for the only n of order type in <(α)2 , is an increasing sequence with C ⊂ A+1 . And then
Γα, = SD|(α)2 | (C : < |(α)2 |, A : < |(α)2 |, Γα1 ,1 ). It is clear from the preceding definition, that the coding relations α ∈ D α ∈ D ∧ is a Γα -code α ∈ D ∧ is a Γα -code ∧ ∈ Γα,
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and α ∈ D ∧ is a Γα -code ∧ ∈ / Γα, Π11 .
are all From the proof of the main theorem of part 1, it is also clear that some variant of the preceding coding would enable to prove “recursive” analogs of the Hausdorff-Kuratowski-type results we have quoted. Such a variant would involve coding by partial recursive functions, in the spirit of what is done for the class Σ0 . Anyway, we are more interested here in “Δ11 -recursive” results, for which the coding we defined above is good enough. From now on, Wadge classes will be written boldface, to distinguish them from their lightface counterparts. Definition 2.2. A described Wadge class Γ is a Δ11 -class if it admits a Δ11 code (i.e., Γ = Γα for some Δ11 real α in D). So in particular the Δ11 classes, among the Σ0 ’s are the Σ0 for < 1ck . Definition 2.3. Let Γα , α ∈ Δ11 , be a Δ11 -class. We define the lightface classes Γα , Γα (), Γ1α by Γα = {Γα, : a recursive Γα -code} Γα () = {Γα, : a recursive in- Γα -code} and
{Γα () : ∈ Δ11 } = {Γα, : ∈ Δ11 } Because of the coding we chose, it is not clear that the lightface class Γα is really well defined, i.e., does not depend on the particular code α for Γα , even in case α is recursive. But it can be seen that Γ1α does not depend on the particular choice of α ∈ Δ11 , but only on the class Γα . [This can be seen result below.] directly, but is also an immediate corollary of the main 0 1 0 In Louveau [Lou80], we studied (Σ ) = α∈Δ1 Σ (α), for < 1ck , and we 1 proved that (Σ0 )1 = Σ0 ∩ Δ11 , i.e., that every Σ0 set in Δ11 admits a Δ11 Σ0 -code. The main theorem in this section is the extension of this result to all Δ11 (non self-dual) Borel Wadge classes. Γ1α =
Theorem 2.4. Let Γα be a described Wadge class, with α ∈ Δ11 . Then each set in Γα admits a Γα code which is Δ11 , i.e., Γ1α = Γα ∩ Δ11 . In order to prove this theorem, we need some tools from Louveau [Lou80]. For < 1ck , we define T to be the topology on generated by the Σ11 sets which are in < Π0 . T∞ , the Harrington topology on , is the topology generated by all Σ11 subsets of . For this topology, is a Baire space (i.e., no non-empty T∞ -open set is T∞ -meager). Say that a property of reals is true Δ11
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∞-a.e. if it is false for a set of reals which is T∞ -meager. In the induction used to prove the result for Σ0 sets, we proved the following two results: Proposition 2.5. Let A be a Σ0 set. Then there is a Σ0 set A which is T -open and satisfies A = A ∞-a.e. Proposition 2.6 (Separation result). Let A, B be two Σ11 sets and suppose there is a Σ0 set C such that A ⊆ C ⊆ B ∞-a.e. Then there is a (Σ0 )1 set D ⊆ D ⊆ B. such that A Proposition 2.6 clearly implies the result for Σ0 classes; working analogously, we shall prove Theorem 2.4 by proving first a separation result for Γα by induction on α. Theorem 2.7. Let Γα be a described Wadge class, with α ∈ Δ11 . If A, B are two Σ11 sets, and thereis a Γα set C such that A ⊆ C ⊆ B ∞-a.e., then there is a Γ1α set D with A ⊆ D ⊆ B. Proof. It is clear that if α is Δ11 , the class Γα has been constructed from can prove 2.7 by induction. previous classes which are also Δ11 , so that we There are five cases (we forget Γα = {∅}!). Case 1. (α)1 = 1, so Γ = D (Σ0 ), with = |(α)2 | and = |(α)0 | (so and are recursive ordinals). We assume is an even ordinal, the other case being similar. We first define a sequence A : < bythe following: If is even, let A be the largest T -open set disjoint from A \ < A . If is odd, let A be the largest T -open set disjoint from B \ < A . Clearly the sequence A : < is an increasing sequence of Σ0 sets. Moreover we have: (a) The relation ∈ An , where n is the order type of predecessors of n in <(α)2 , is Π11 (in and n). (b) If C : < is any sequence of Σ sets with A ⊆ D (A : < ) ⊆ B ∞-a.e., then for each < C ⊆ A ∞-a.e., and A ⊆ D (A : < ) ⊆ B. 1 To prove (a). Suppose H is a Σ1 set, and < 1ck . Then the largest T∞ -open set O disjoint from H is a Π11 set. In fact x ∈ O ⇐⇒ ∃G ∈ Σ0 ∩ Σ11 (x ∈ G ∧ G ∩ H = ∅) ⇐⇒ ∃G ∈ (Σ0 )1 (x ∈ G ∧ G ∩ H = ∅) the second equivalence being obtained by using Proposition 2.6. This clearly implies that the relation ∈ An is Π11 . To prove (b). Using Proposition 2.5, we may assume that C are T∞ open. Thenby induction on , we prove that C ⊆ A . Suppose is even. Then C \ < C is disjoint from D (C : < ), so is disjoint from A, ∞-a.e. This implies that C is ∞-a.e. disjoint from A \ < C , and
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using the induction hypothesis, also from A \ < A . But this implies, because this last set is Σ11 by part (a), that C ∩ (A \ < A = ∅, using the Baire category theorem for T∞ . So by the definition of A , C ⊆ A . The odd case is similar, and we omit it. To prove the last assertion, i.e., that A ⊆ D (A : < ) ⊆ B, it is enough to prove that A ⊂ < A . But A ⊆ < C ∞-a.e., so using the preceding result, A ⊆ < A ∞-a.e. Finally, < A is Π11 and A is Σ11 , so using again the Baire category theorem for T∞ gives A ⊆ < A . The last part of the proof consists in replacing the Π11 sequence A : < by a Δ11 sequence with the same properties. For each recursive ordinal ϑ, say that a Δ11 sequence D : < ϑ of Σ0 sets is a ϑ-system for (A, B) if D is from B \ disjoint from A \ < D if is even, < D if is odd, and say that this ϑ-system covers H if H ⊆ <ϑ D . What we want to construct is an -system covering the set A. So it is enough to prove the following claim. 1 Claim 2.8. Let ϑ be some even ordinal, ϑ ⊆ , and let H be a Σ1 set, H ⊆ <ϑ A . Then there exists a ϑ-system covering H .
The proof of the claim is by induction on ϑ. Suppose first ϑ is a successor, so that ϑ = ϑ + 2 with ϑ even. By the hypothesis H ⊆ Aϑ +1 , so we can find a set D 0 ∈ (Σ0 )1 with H ⊆ D 0 ⊆ Aϑ +1 . But as Aϑ +1 is disjoint from B \ Aϑ , D 0 ∩ B ⊆ Aϑ , and so we can find a (Σ0 )1 set D 1 ⊆ D 0 with D 0 ∩ B ⊆ D 1 ⊆ Aϑ . By the same reasoning, we must have A ∩ D 1 ⊆ <ϑ A , so by the induction hypothesis, we can find a ϑ -system D: < ϑ covering A ∩ D 1 . We then extend this ϑ system by setting Dϑ = <ϑ D ∪ D 1 , and Dϑ +1 = <ϑ D ∪ D 0 . This clearly gives the desired ϑ-system. Suppose now ϑ is limit. As H ⊆ <ϑ A = even A , we can first choose a Δ11 sequence H : < ϑ, even with H ⊆ H , and for even, < ϑ, H⊆A . By the induction hypothesis, and Δ11 selection, there is, for each even < ϑ a -system D : < , covering H , and such that the double sequence is Δ11 . Define then a sequence D : < ϑ by D = D ∪ D <
< <ϑ
We claim that the Δ11 and increasing sequence D : < ϑ is the ϑ-system we wanted. It certainly covers H , so the only thing we have to prove is that it is a ϑ-system. Suppose < ϑ is even. Then D ∩ B \ D ⊆ D ∩ B \ D <
< <ϑ even
<
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so fix even, with < < ϑ. As D ⊆ D for any < D ⊆ D ∩ B \ D = ∅ D ∩ B \ <
<
Similarly if is odd D ∩ A \ D ⊆ D ∩ B \ D < ϑ even
<
<
and for each even, < ϑ, as D ⊆ D for any < D ∩ A \ D ⊆ D ∩ A \ D = ∅ <
<
This proves the claim, and so finishes the proof of case 1. Case 2. We now suppose α codes a description u with u(1) = 2, so that Γα = Sep(D Σ0 , Γα1 ), where = |(α)0 | and = |(α)2 | are recursive ordinals, and α1 ∈ Δ11 , so that by the induction hypothesis we can assume the theorem is true for Γα1 . Let us suppose that is even, the other case being similar. B, we define a sequence A : < of Σ0 sets by the following: Given A and
sets O such that for the union of all T open If is even, we let A be some set C in Γα1 , (A \ < A ) ∩ O ⊆ C ⊆ B ∞-a.e., and similarly if is odd,A is the union of all T open sets O such that for some set C in Γα1 (A \ < A ) ∩ O ⊆ C ⊆ B. We claim that the sequence A : < , which is clearly an increasing sequence of Σ0 sets, has the following properties: (a) The relation ∈ An is Π11 (in and n) (b) If C : < is an increasing sequence of Σ0 sets such that C = D (C : < ) satisfies that A ∩ C ⊆ H ⊆ B ∞-a.e. for some H in Γα1 , and A \ C ⊆ H ⊆ B ∞-a.e. for some H in Γα1 , then for each < ⊆ A ∞-a.e. C To prove (a), let (C, D) be the union of all T -open sets O such that for some set H in Γα1 , C ∩ O ⊆ H ⊆ D ∞-a.e. We want to prove that if C, D are Σ11 , (C, D) is Π11 . But it is Π11 , for x ∈ (C, D) ⇐⇒ ∃G ∈ Σ11 ∩ Σ0 ∃H (x ∈ G ∧ H ∈ Γα1 ∧ ˘ ∞-a.e. G ∩ C ⊆ H ⊆ D) ˘ ⇐⇒ ∃G ∈ (Σ0 )1 ∃H ∈ Γ1α1 (x ∈ G ∧ G ∩ C ⊆ H ⊆ D) The second equivalence is justified by the following fact: If G is Σ11 and x ∈ G, and for some H ∈ Γα1 G ∩ C ⊆ H ⊆ D ∞-a.e., then using the ˘ But then G is induction hypothesis ∃H ∈ Γ1α with G ∩ C ⊆ H ⊆ D. 1
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disjoint from C \ H , which is Σ11 , so we can find B in (Σ0 )1 with G ⊆ G and ˘ and x ∈ G . G ∩ C \ H = ∅, so that G ∩ C ⊆ H ⊆ D, To prove (b), use Proposition 2.5 to replace the Σ0 sets C by T -open sets. Then it is immediate, by induction on , that C ⊆A In order to finish the proof of case 2, we argue as follows. Consider A \ 1 ˘ < A . It is a Σ1 set, by (a), and by (b) it is a subset, ∞-a.e., of ( < C ), where C : < is a sequence as in (b) (we know that such a sequence exists). Now it follows that there is an H ∈ Γα1 with A \ < A ⊆ H ⊆ B ∞-a.e., so by the induction hypothesis, we can find such an H in Γ1α1 . Now A ∩ H˘ ⊆ < A , and so by an argument very similar to the one we used for case 1, we can replace the sequence A : < by a Δ11 sequence C : < , having the same properties, namely • it is a Δ11 sequence of Σ0 sets, increasing, and of T -open sets O such that for some H in • if is even, C is a union Γα1 (A \ < C ) ∩ O ⊆ H ⊆ B˘ ∞-a.e. it is odd, C is a union of T -open sets O such that for some H in Γ , • if α1 (A \ < C ) ∩ O ⊆ H ⊆ B ∞-a.e. • and finally the sequence covers A ∩ H˘ . We now define a Δ11 sequence H : < of sets: Suppose is even. Then as C is a union of T sets O such that for some H in Γα1 A \ < C ∩ O ⊆ H ⊆ B˘ ∞-a.e., we can, by the induction hypothesis and using Δ11 selection, find two Δ11 sequences C n : n ∈ and H n : n ∈ such that the Cn are pairwise disjoint Σ0 sets, of union C , and Hn are in Γ1α1 with A \ < C ∩ Cn ⊆ Hn ⊂ B. Let H = SU(Cn : n ∈ , H n : n ∈ ). ˘ H ∈ Γ1α1 , so that C itself satisfies ∃H ∈ Γ1α1 A \ < C ∩ C ⊆ H ⊆ B. 1 ˘ α with A \ C ∩ C ⊆ H ⊆ B. ˘ Similarly for odd, there is an H ∈ Γ < 1 1 1 Using Δ1 -selection, we can find such a sequence H : < in a Δ1 way. We now put H0 = C H ∩ C \ odd <
and H1 =
even <
<
C C H ∩ C \ ∪ H \ <
<
˘ 1α , and the set D = Sep(D (C : < ), H0 , H1 ) is in It is clear that H0 ∈ Γ 1 ˘ 1 Γα and satisfies A ⊆ D ⊆ B.
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Case 3. Suppose α codes a description u with u(1) = 3, so that Γα = deBisep(D (Σ0 ), Γα1 , Γα2 ), with , as before, and α1 , α2 are Δ11 codes for scriptions. We assume again is even. Starting with A, B in Σ11 , we first construct the sequence A : < as in case 2, and a similar sequence B : < , defined by exchanging the role of Γα1 and Γ˘ α1 in the definition. Π1 and Σ0 . Now using the Let A0 = < A , A1 = < B . These sets are 1
normal forms for the Bisep operation (Lemma 1.5), we know that there are disjoint sets C 0 , C 1 in Σ0 sets, with < C0 = C 0 , < C1 = C 1 , and sets ˘ α1 , H 0 , H 1 in Γα1 and H2 in Γα2 such H00 , H10 , H01 , H11 , H2 with H00 , H01 in Γ 1 1 that A ⊆ (D (C0 : < ) ∩ H00 ) ∪ (C0 \ D (C0 : < ) ∩ H10 ) ∪ (D (C1 : < ) ∩ H11 ) ∪ (C 1 \ D (C1 : < ) ∩ H01 ) ∪ H2 \ (C 0 ∪ C 1 ) ⊆ B˘ ∞-a.e. It easily follows that for each < C0 ⊆ A , C1 ⊆ B and C 0 ⊆ A0 , C 1 ⊆ A1 ∞-a.e. From this, it follows that A \ (A0 ∪ A1 ) ⊆ H2 ⊆ B˘ ∞-a.e., so using the ˘ inductive hypothesis, we can find a set D ∈ Γ1α2 with A \ (A0 ∪ A1 ) ⊆ D ⊆ B. Now by the argument previously used, one can shrink the Π11 sequences A : < and B : < into Δ11 sequences D0 : < and D1 : < with the same properties, in such a way that setting D 0 = < D0 and ˘ D 1 = < D1 , D 0 , D 1 are disjoint and D 0 ∪ D 1 covers the Σ11 set A ∩ D. 0 1 1 ˘ α and Again imitating the proof of case 2, we can find sets H0 and H0 in Γ 1 H10 , H11 in Γ1α1 such that A ∩ D (D0 : < ) ⊆ H00 ⊆ B˘ A ∩ D 0 \ D (D0 : < ) ⊆ H10 ⊆ B˘ A ∩ D (D0 : < ) ⊆ H11 ⊆ B˘ A ∩ D 1 \ D (D0 : < ) ⊆ H01 ⊆ B˘ So that the Γ1α set H = [H00 ∩ (D (D0 : < )] ∪ [H10 ∩ (D 0 \ D (D0 : < ))] ∪ [H01 ∩ (D 1 \ D (D1 : < ))] ∪ [H11 ∩ (D (D1 : < )] ∪ [D \ (D 0 ∪ D 1 )] separates A from B.
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Case 4.Suppose α codes a description u with u(1) = 4, so that Γα = SU(Σ0 n Γαn ), where the sequence αn : n ∈ is a Δ11 sequence of codes of descriptions. Starting from A, B in Σ11 , we let An be the union of all Y -open sets O such that for some H in Γαn A ∩ O ⊆ H ⊆ B˘ ∞-a.e. As before, it 1 increasing sequence of Σ0 sets, and that is not hard to see that An is a Π 1
A ⊆ n An . Let Cn : n ∈ be a Δ11 sequence of pairwise disjoint Σ0 sets with A ⊆ n An and Cn ⊆ An , and let Hn : n ∈ be a Δ11 sequence with ˘ This is easy to find using the induction Hn ∈ Γ1αn , and A ∩ Cn ⊆ Hn ⊆ B. hypothesis. Then the set H = SU(Cn : n ∈ , Hn : n ∈ ) separates A from B and is in Γ1α . Case 5. Suppose uα (1) = 5, so that Γα = SD (Σ0 , Γα1 ,Γα2 ), with α1 , α2 ∈ Δ11 , and , < 1ck . We again define inductively a sequence A : < : A is the union of all T -open sets O such that for some set H in Γα1 with envelope O, (A \ < A ) ∩ O ⊆ H ⊆ B˘ ∞-a.e. Again it can be seen that the sequence A : < is Π11 and increasing, and moreover that A \ < A ⊆ H ⊆ B ∞-a.e. for some H in Γα2 . The rest of the proof is analogous to case 2, and reader. we leave the details to the Remarks. 1. All the preceding results are of effective type. But as usual, they can be translated into non-effective, uniform results concerning analytic and Borel sets in the plane, using Δ11 -selection. 2. The results in Section 2 do not use Borel determinacy, but without it we are unable to show that Theorem 2.7 covers all non self-dual Wadge classes of Borel sets. In a recent paper, Thomas John [Joh86] has proved Wadge Determinacy for Π04 (i.e., that all Wadge games GW (A, B), with A, B in Π04 are determined), the stronger statement that every Π0 \ Δ0 set is Π0 -complete, for n in , and n n n in second order arithmetics. His proof uses as main tool the characterization Π0n ∩ Δ11 = (Π0n )1 . This fact, together with Theorem 2.7, is a bit of evidence in support of the conjecture that unlike Borel determinacy, Wadge Determinacy for Δ11 could be proved in second order arithmetic. REFERENCES
Thomas John [Joh86] Recursion in Kolmogorov’s R-operator and the ordinal 3 , The Journal of Symbolic Logic, vol. 51 (1986), no. 1, pp. 1–11. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978.
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Kazimierz Kuratowski [Kur66] Topology, vol. 1, Academic Press, New York and London, 1966. Alain Louveau [Lou80] A separation theorem for Σ11 sets, Transactions of the American Mathematical Society, vol. 260 (1980), no. 2, pp. 363–378. Donald A. Martin [Mar73] The Wadge degrees are wellordered, unpublished, 1973. [Mar75] Borel determinacy, Annals of Mathematics, vol. 102 (1975), no. 2, pp. 363–371. John R. Steel [Ste81B] Determinateness and the separation property, The Journal of Symbolic Logic, vol. 46 (1981), no. 1, pp. 41– 44. Robert Van Wesep [Van78B] Wadge degrees and descriptive set theory, this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 151–170. William W. Wadge [Wad84] Reducibility and determinateness on the Baire space, Ph.D. thesis, University of California, Berkeley, 1984. EQUIPE D’ANALYSE FONCTIONNELLE ´ INSTITUT DE MATHEMATIQUES DE JUSSIEU UNIVERSITE´ PARIS VI 4, PLACE JUSSIEU 75230 PARIS, CEDEX 05 FRANCE
E-mail: [email protected]
THE STRENGTH OF BOREL WADGE DETERMINACY
ALAIN LOUVEAU AND JEAN SAINT-RAYMOND
One of the nice consequences of Martin’s theorem that Borel games are determined is the so-called Borel Wadge Determinacy, the determinacy of all games G(A, B) of the following kind: player I produces α ∈ , player II produces ∈ and player II wins G(A, B) if α ∈ A ⇐⇒ ∈ B, whenever A and B are Borel subsets of . Borel Wadge Determinacy allows to get a complete description of all classes of Borel sets, i.e., of all families Γ ⊆ Δ11 which are continuously closed. Our main result is proved in Section 3. It is a sequel to two earlier papers: First, Louveau’s paper [Lou83] which analyzes the Borel Wadge classes in a way which does not depend too heavily on Borel Wadge determinacy. We quickly review in the first section the material from [Lou83] that we need. The second source is our joint paper [LSR87], where we prove the particular instances of Borel Wadge Determinacy which correspond to the Baire classes (Σ0 and Π0 ) and introduce the main device for the general proof, a specific of associating way to a closed game another closed game, that we called the ramification method. We present the material from [LSR87] that we need in Section 2. Both papers are rather long and technical, so the information we provide in Sections 1 and 2 is a bit sketchy. In particular, the existence of ramifications will be used as a black box here, and we will also leave to the reader the verification that the results from Louveau [Lou83] we use do not depend of Borel Wadge Determinacy. The main consequence of Borel Wadge determinacy is Wadge’s lemma which asserts that any Borel set in which is not in a class Γ always generates ˘ (of complements of sets in Γ). In by continuous preimages the dual class Γ Section 4, we show how Wadge’s lemma extends to arbitrary Polish (even Suslin) spaces in place of , even if no game is available in this general context. This part is much more topological in nature, and uses some transfer methods and selection results for continuous functions which might be of interest in other contexts. Finally, in Section 5, we develop a notion of Hurewicz test for a class Γ, in order to extend to all Borel Wadge classes the well known theorem of Hurewicz which characterizes among Borel sets those which are Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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not Polish as those which contain a relatively closed set homeomorphic to Q. Most results in Sections 4 and 5 are again sequels of our paper [LSR87]. §1. Descriptions of Borel Wadge Classes. A family Γ of subsets of is a class if it is closed under continuous preimages, and a Wadge class if it is generated by one set A ⊆ . If moreover A is Borel, it is a Borel Wadge class. Using as main tool Borel Wadge Determinacy, Wadge analyzed in his thesis [Wad84], all Borel Wadge classes. Relying heavily on Wadge’s work, Louveau proposed in [Lou83] an inductive construction of all Borel Wadge classes in terms of certain Boolean operations. Fortunately, these works do not depend too much on Borel Wadge determinacy: Although it is unclear (at this point) that the analysis in [Lou83] exhausts all Borel Wadge classes, one can still show directly that it almost does, in a precise sense (given by Theorem 1.4 below). And part of our proof of Borel Wadge Determinacy will in fact consist in showing that the analysis is exhaustive. Let us first introduce some notations and definitions. Definition 1.1. Let Γ be a class of subsets of ˘ = {A˘ : A ∈ Γ} be the dual (i) If A ⊆ , we let A˘ = − A, and we let Γ ˘ the ambiguous class of Γ. We say class of Γ. We also set Δ(Γ) = Γ ∩ Γ, ˘ that Γ is self-dual if Γ = Γ(= Δ(Γ)). We also define the ordering < between classes by Γ < Γ ⇐⇒ Γ ⊆ Δ(Γ ).
(ii) PU(Γ) is the class of all A’s ⊆ of form A = n (An ∩ Cn ), where An ∈ Γ and (Cn )n∈ is a partition of in clopen sets (PU stands for “partitioned union”). We will use this operation mainly in two cases: If ˘ And if Γn : n ∈ Γ is a non self-dual class, we set Γ+ = PU(Γ ∪ Γ). + is a <-increasing sequence of classes, Γn = PU( n Γn ). Definition 1.2. Let Γ, Γ be classes, , ordinals ≥ 1. (a) A ∈ D (Σ0 ) ⇐⇒ {Aϑ − ϑ <ϑ Aϑ : ϑ < , ϑ of a different parity than } for some increasing sequence Aϑ : ϑ < of Σ0 sets in . (b) A ∈ Sep(D (Σ0 ), Γ) ⇐⇒ A = (A0 ∩ C ) ∪ (A1 \ C ) for some C ∈ D (Σ0 ), ˘ A1 ∈ Γ. A0 ∈ Γ, (c) A ∈ Bisep(D (Σ0 ), Γ, Γ ) ⇐⇒ A = (A0 ∩C0 )∪(A1 ∩C1 )∪(B \(C0 ∪C1 )) for some disjoint C0 , C1 in D (Σ0 ), A0 ∈ Γ, A1 ∈ Γ˘ and B ∈ Γ . (d) A ∈ SU(Σ0 , Γ) with envelope C ⇐⇒ A = n (An ∩Cn ) for some sequence of pairwise disjoint Σ0 sets Cn , with n Cn = C , and An ∈ Γ. (e) A ∈ SD (SU(Σ0 , Γ)Γ ) ⇐⇒ A = ϑ< (Aϑ \ ϑ <ϑ Cϑ ) ∪ (B \ ϑ< Cϑ ) for some increasing sequence Aϑ : ϑ < of sets in SU(Σ0 , Γ) with
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respective envelopes Cϑ , such that Cϑ ⊆ Aϑ for ϑ < ϑ, and some B ∈ Γ . In [Lou83], Louveau selects particular ways of combining the operations of Definition 1.2, encoded by what he calls “descriptions”. In order to simplify later work, we will also use another notion of description. So we refer to the descriptions of [Lou83] as first type descriptions. The encoding is made by elements u in 1 . Such sequences are sometimes viewed as pairs u0 , u1 or as sequences un : n ∈ , via some fixed bijections between and .2, 2, respectively. We let 0 be the constant function zero. Definition 1.3 ([Lou83, 1.2]). The relations “u is a first type description” and “u describes Γ” (written u ∈ D1 and Γu = Γ) are the least relations satisfying: (a) If u(0) = 0, u ∈ D1 and Γu = {∅}. (b) If u(0) = ≥ 1, u(1) = 1 and u(2) = ≥ 1, u ∈ D1 and Γu = D (Σ0 ). (c) If u = 2u ∗ , with ≥ 1, ≥ 1, u ∗ ∈ D1 and u ∗ (0) > , then u ∈ D1 0 and Γu = Sep(D (Σ ), Γu ∗ ). (d) If u = 3u0 , u1 , with ≥ 1, ≥ 1, u0 and u1 in D1 , u0 (0) >
, u1 (0) ≥ or u1 (0) = 0 and Γu1 < Γu0 , then u ∈ D1 and Γu = Bisep(D (Σ0 ), Γu0 , Γu1 ). (e) If 4un : n ∈ , where ≥ 1, un ∈ D1 for all n, Γun < Γun+1 , and either for all n, un (0) = > or un (0) n∈ is strictly increasing with sup n > , then u ∈ D1 and Γu = SU(Σ0 , n Γun ). (f) If u = 5u0 , u1 with ≥ 1, ≥ 1, u0 , u1 in D1 with u0 (0) =
, u0 (1) = 4, u1 (0) ≥ or u1 (0) = 0, and Γu1 < Γu0 , then u ∈ D1 and Γu = SDn (Γu0 , Γu1 ). [Note: As noted by Van Engelen, there is a slight mistake in the original definition [Lou83, 1.2, case e.]] One easily checks that each u ∈ D1 codes exactly one class Γu . And each Γu is a non-self dual Borel Wadge class, as can be seen by (inductively) constructing a universal Γu set in × . The main result of [Lou83] that we will use, which does not need Borel Wadge determinacy, can be summarized as follows (it corresponds to [Lou83], Lemmas 1.11, 1.14, 1.19, 1.23, 1.24, 1.25 and 1.28). Theorem 1.4. Let u ∈ D1 . The class Δ(Γu ) satisfies one of the following three possibilities: (i) There is a description u ∗ ∈ D1 with Δ(Γu ) = (Γu ∗ )+ . (ii) There is a sequence un in D1 with Δ(Γu ) = (Γun )+ . (iii) There is a family (u ) <1 in D1 with Δ(Γu ) = Γu . In [Lou83], Theorem 1.4 is just an intermediate step towards proving that + any Borel Wadge class is of form Γu , Γ˘ u , Γ+ u or Γun for some u, un in D1 ,
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as the case may be. However, this is derived from Theorem 1.4 by using Borel Wadge determinacy. As the proof is instructive, let us sketch it briefly: First, Borel Wadge Determinacy is used to show that if A ∈ Γu but A ∈ / ˘ u . Similarly, if A ∈ Γ+ ˘ u , A generates Γu —and A˘ generates Γ , but A ∈ / Γ u + ˘ u , A generates Γ+ Γu ∪ Γ / n Γun , A generates u , and if A ∈ Γun but A ∈ Γun + . Finally, Borel Wadge Determinacy is used again to prove that {Γ ∪ ˘ : Γ a Borel Wadge class} is well ordered by inclusion (Wadge [Wad84], Γ Martin [Mar73]). One can argue then that if A ⊆ is Borel, there is a least ˘ u . If A ∈ / Δ(Γu ), the Wadge class of (for inclusion) class Γu with A ∈ Γu ∪ Γ ˘ A is Γu or Γu . If A ∈ Δ(Γu ), Theorem 1.4 applies. Case (iii) is impossible by minimality of u, and Cases (i) and (ii) are solved by the facts above. So in all cases the class of A is described. In the next sections, we imitate the proof above, except that we will at the same time prove instances of the facts above and the corresponding instances of determinacy; so at the end we will get both that the analysis is exhaustive and that Borel Wadge Determinacy holds. We now introduce the second type descriptions. Definition 1.5. Let ≥ 1 be a countable ordinal, Γ, Γ two classes. Then A ∈ S (Γ, Γ ) ⇐⇒ A = (An ∩ Cn ) ∪ B \ Cn n
n
for some sequence An in Γ, B ∈ Γ , and a sequence Cn of pairwise disjoint Σ0 sets. Second type descriptions are also elements of 1 . Definition 1.6. The relations “u is a second type description” and “u describes Γ” (written u ∈ D2 and Γu = Γ—ambiguously) are the least relations satisfying (a) if u = 0, u ∈ D2 and Γu = ∅ (b) if u = 1u ∗ , with u ∗ ∈ D2 and u ∗ (0) = , then u ∈ D2 and Γu = Γ˘ u ∗ (c) if u = 2u n with ≥ 1, un ∈ D2 , un (0) ≥ or un (0) = 0, then u ∈ D2 and Γu = S ( n≥1 Γun , Γu0 ). The D2 -encoding is clearly much simpler than the first one. However, Theorem 1.4 would be hard to get using this encoding. Our next step is to show that any class admitting a D1 -description also admits a D2 -description. Proposition 1.7 ([Lou83, Lemma 1.4]). Let u ∈ D1 , with u(0) = ≥ 1. Then (a) SU(Σ0 , Γu ) = Γu (b) Γu is closed under union with a Δ0 set.
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So in particular both Γu and Γ˘ u are closed under unions or intersections 0 with Δ sets, and if A = n (An ∩ Cn ) where An ∈ Γu (resp. Γ˘ u ) and (Cn ) is a partition of in Δ0 sets, then A ∈ Γu (resp. Γ˘ u ). Theorem 1.8. Every class admitting a D1 -description, and every dual of such a class admit a D2 -description. Proof. If Γ admits u ∈ D2 as description, Γ˘ admits u(0)1u as D2 description. So we prove by induction on u ∈ D1 , that Γu admits a D2 description v(u), with v(u)(0) = u(0). (a) Clearly if u(0) = 0, we can take v(u) = 0. (b) Let u(1) = 1, i.e., Γu = D (Σ0 ). We use induction on . For = 1, one has Σ0 = S ({}, {∅}) so we can take v(u) = 2vn with v0 = 0 and for n ≥ 1, vn = 010. If = + 1, one uses similarly the equality D (Σ0 ) = S ((D (Σ0 ))˘, Σ0 ) and if = supn (n + 1) is limit Dn (Σ0 ), {∅} D (Σ0 ) = S
n All these equalities are easy to check. (c) Suppose now u = 2u ∗ , so that Γu = Sep(D (Σ0 ), Γu ∗ ). We again on Γ ∗ and the argue by induction on , and use the induction hypothesis u following equalities ˘ u ∗ , Γu ∗ ) Sep(Σ0 , Γu ∗ ) = S (Γ 0 Sep(D+1 (Σ ), Γu ∗ ) = S (Sep((D (Σ0 ))˘, Γu ∗ ), Sep(Σ0 , Γu ∗ )) and for limit Sep(D (Σ0 ), Γu ∗ ) = S
Sep(D (Σ0 ), Γu ∗ ), {∅} < the proof of which is left to the reader. (d) u = 3u0 , u1 , so Γu = Bisep(D (Σ0 ), Γu0 , Γu1 ). Again by induc tion on , one uses the equalities Bisep(Σ0 , Γu0 , Γu1 ) = S (Γu0 ∪ Γ˘ u0 , Γu1 ) ˘ Γu1 ) Bisep(D+1 (Σ0 ), Γu0 , Γu1 ) = S (Γ ∪ Γ, where Γ = Sep(D (Σ0 ), Γu0 ) and Bisep(D (Σ0 ), Γu0 , Γu1 ) = S
Γ , Γu1 < where Γ = Sep(D (Σ0 ), Γu0 ).
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The equalities do not follow immediately from Proposition 1.7, so let us sketch one of them, say the successor case (the others are similar, and a bit simpler). Denote by Γ and Γr the left and right hand classes. If A ∈ Γr , A = n (An ∩ Cn ) ∪ (B \ n Cn ) with pairwise disjoint Cn ’s in ˘ Write An = (A0n ∩ Σ0 , B ∈ Γu1 and An in Γ = Sep(D (Σ0 ), Γu0 ) or in Γ. Dn ) ∪ (A1n \ Dn ) with Dn ∈ D (Σ0 ), Anε(n) ∈ Γu0 and An1−ε(n) ∈ Γ˘ u0 , where ˘ Let Dn0 = Dn ∩ Cn and Dn1 = ε(n) = 0 or 1 depending if An is in Γ or Γ. 0 1 0 Cn \ Dn . Both Dn , Dn are in D+1 (Σ ), and so are D 0 = n (Cn ∩ Dnε(n) ) and D 1 = n (Cn ∩ Dn1−ε(n) ) by an immediate computation. By Proposition 1.7 (as u0 (0) > ) A0 = n (Cn ∩ Anε(n) ) ∈ Γu0 and A1 = n (Cn ∩ An1−ε(n) ) ∈ Γ˘ u0 . And A = (A0 ∩ D 0 ) ∪ (A1 ∩ D 1 ) ∪ (B \ (D 0 ∪ D 1 )), so that A ∈ Γ . The other inclusion is a bit harder: let A ∈ Γ , i.e., A = (A0 ∩ C0 ) ∪ (A1 ∩ ˘ u0 C1 ) ∪ (B \ (C0 ∪ C1 )) with C0 , C1 disjoint in D+1 (Σ0 ), A0 ∈ Γu0 , A1 ∈ Γ and B ∈ Γu1 . Let D0 , resp D1 , be the largest Σ0 sets in some constructions of C0 , resp C1 , as D+1 (Σ0 ) sets, and let D0∗ , D1∗ reduce D0 , D1 . Clearly A \ (D0∗ ∪ D1∗ ) = B \ (D0∗ ∪ D1∗ ), so in order to show that A ∈ Γr , it is enough ˘ to prove that A ∩ D0∗ ∈ Γ = Sep(D (Σ0 ), Γu0 ) and similarly A ∩ D1∗ ∈ Γ. Let us prove the first claim, the second one being similar. By the choice of D0 , D0 \ C0 ∈ D (Σ0 ), hence D0∗ \ C0 too, as D0∗ ⊆ D0 . Now, A ∩ D0∗ ∩ C0 = A0 ∩D0∗ ∩C0 is in Γu0 , and A∩(D0∗ \C0 ) = A1 ∩(D0∗ ∩C1 )∪(B \(C0 ∪C1 ))∩D0∗ . ∗ ˘ u0 , and separated by the Δ0 set Both A1 ∩ D0 and B \ (C0 ∪ C1 ) are in Γ
+1 ˘ u0 and the equality C0 ∪ C1 , hence by Proposition 1.7 A ∩ (D0∗ \ C0 ) is in Γ is proved. (e) One uses in case u(1) = 4 the equality Γun , {∅} SU Σ0 , Γun = S
n n (f) The final case is when Γu = SD (SU(Σ0 , n Γun ), Γu ∗ ) and is proved as in the D (Σ0 ) case, by using the following (easy) equalities, where Γ = SU(Σ0 , n Γun ) Γun , Γu ∗ SD+1 (Γ, Γu ∗ ) = S SD (Γ, {∅}), S
n
and for limit SD (Γ, Γu ∗ ) = S
<
SD (Γ, {∅}), Γu ∗ )
Remark 1.9. There is a slight defect in the proof above: if one really wants to build a v(u) for u ∈ D1 , one needs at limit steps specific fundamental sequences below limit ordinals. This requires a form of the axiom of choice. The best
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way to avoid this—which would in any case be necessary for a formalization of the preceding discussion in second order arithmetics—is to replace ordinals by reals coding them, and accordingly descriptions by codes of descriptions in . The function v(u) becomes then definable in the codes. However, since working with codes would only create more notational problems to the reader, we will continue this slight kind of abuse. Let us denote by W1 = {Γu : u ∈ D1 } ∪ {Γ˘ u : u ∈ D1 } W2 = {Γu : u ∈ D2 }, and by W the set of non self-dual Borel Wadge classes in . The preceding theorem says that W1 ⊆ W2 . It is clear that W2 ⊆ W , as can be proved by inductively constructing a universal set for each Γ ∈ W2 in × . We will prove later that these inclusions are equalities. We finish this section with the study of the effect of functions of Baire class on classes in W2 . For , countable ordinals, with ≤ , we denote by − the unique such that + = . A function f : → is a Baire class function if for each open A ⊆ , f −1 (A) ∈ Σ01+ (so continuous functions are Baire class 0). One easily checks by induction that if f is of Baire class and A is Σ0 , f −1 (A) is in Σ0+ , with + = 1 + + ( − 1). Let also
− , for ≥ 1 and < , be defined by − = 1 + ( − (1 + )), so that + ( − ) = . Definition 1.10. We define for each countable and each u ∈ D2 a description u ∈ D2 , and in case u(0) > or u(0) = 0, a description u ∈ D2 , by the following clauses: (a) If u(0) = 0, u = u = u (b) If u = 1u ∗ , with ≥ 1 u = ( + )1 ∩ (u ∗ )
and u = ( − )1 (u ∗ ) (for u(0)—hence u ∗ (0)—bigger than ) (c) If u = 2un , with ≥ 1, u = ( + )2un and u = ( − )2un (for > —note that un is defined from some n0 on, as sup un (0) > ). It is clear from the previous definition that (u) = u, when u(0) = 0 or u(0) > . And one easily gets by induction the following. Proposition 1.11. (i) If f : → is a Baire class function, and A ∈ Γu for some u ∈ D2 , then f −1 (A) ∈ Γu .
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(ii) If u ∈ D2 is such that u(0) = ≥ 1, there are unique u with u(0) = 1, and (= − 1) such that u = (u) . In particular, D2 is the least subset D ⊆ D2 such that 0 ∈ D, u(0)1u ∈ D if u ∈ D, 12un ∈ D if for each n, un ∈ D, and for any , u ∈ D when u ∈ D. Let us finally say a few words about relativization: If E is a subset of , one can define for any class Γ the relativization Γ(E) by using traces on E of sets in Γ. And clearly for u a description of any type, Γu (E) is the same as the class described by u, starting from Σ0 subsets of E. And if f : → E is continuous and A ∈ Γu (E), f −1 (A) ∈ Γu . We will use these remarks in the sequel mainly for E = 2, or E = 2 × , viewed as a subset of . §2. Ramifications of Closed Games. In our first paper [LSR87] on the topic of Borel Wadge determinacy, we proved particular instances of it, namely that ˘ the Wadge game if Γ is one of the classes D (Σ0 ) and A ⊆ 2 is a set in Γ \ Γ, G(A, B) is determined, for any Borel B in 2. The main technical tool we introduced to get this result is a specific way of transforming closed games that we called ramifications. We now discuss what will be needed in the sequel about this notion. Ramifications act on the following kind of games: player I plays ε ∈ 2, player II plays ∈ , and the game is closed for player II, i.e., specified by a tree J on 2 × —that we will confuse with the game itself. A position (εk, k) is legal in J if (εk, k) ∈ J , and a run (ε, ) is a win for player II if for all k, (εk, k) is legal in J , i.e., if (ε, ) is a branch through J . < We denote by J ⊆ (2×) the (closed) set of all trees on 2 × . A strategy for player I in games in J is a function : < → 2, and we denote by Σ the < set 2. Definition 2.1. A ramification of games is a triple (r, , F ) of functions, with the following properties: < < (a) r = (r0 , r1 ) : (2 × ) → (2 × ) satisfies (i) r0 (u, v) = r0 (u) depends only on u ∈ <2 (ii) If n = lh(u, v), t(u, v) = {r(uk, vk) : k ≤ n} is a subtree of ≤n (2 × ). We let r act on J , by defining a function R : J → J as follows: (u, v) ∈ R(J ) ⇐⇒ t(u, v) ⊆ J [Intuitively when playing a position (u, v) in R(J ), the players are imagining a tree t(u, v) of positions in J , and their position in R(J ) is legal if all the imagined positions are legal in J .] (b) = (0 , 1 ) : 2 × → 2 × satisfies (i) 0 (ε, ) = 0 (ε) depends only on ε ∈ 2.
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(ii) For all ε, , (ε, ) is a branch through the tree T (ε, ) = k t(εk, k). [Intuitively again, among the positions T (ε, ) associated to a run in some R(J ), exists a complete run (ε, ). So, in particular if (ε, ) is a win for player II in some R(J ), (ε, ) is a win for player II in J .] (c) F : J × Σ → Σ associates to each game J and each strategy (viewed as a strategy for player I in the game R(J )) another strategy ∗ = F (J, ), which we view as a strategy for player I in J . And F satisfies: If is winning in R(J ), ∗ is winning in J . Note: In [LSR87], we put some more restrictions on the notion of ramification, in order to be able to inductively construct a nice family of them. But we will have no needs for these refinements. It is easy to build ramifications—e.g., the identity. But what we need are ramifications for which the function o : 2 → 2 is as complicated a Baire class function as possible. In order to make this idea precise, let us introduce some more definitions: Definition 2.2. Let Γ be a class. A set H ⊆ 2 is Γ-strategically complete if (i) H ∈ Γ(2) (ii) If A ⊆ is a Γ set, player II wins the Wadge game G(A, H ) [where player I plays α ∈ , player II ∈ 2 and player II wins if α ∈ A ⇐⇒ ∈ H] Definition 2.3. Let f : 2 → 2, and a countable ordinal. We say that f is an independent -function if (i) There is a : → such that for all ε, k, the value of f(ε) at k depends only on the values of ε on −1 (k) (ii) If = + 1 is successor, {ε : f(ε)(k) = 1} is Π1+ -strategically complete; and if is limit, then for some increasing sequence n with supremum , {ε : f(ε)(k) = 1} is Π1+k -strategically complete. The main result of [LSR87] about ramifications [LSR87, 3.2] can be restated as: Theorem 2.4. For each countable , there exists a ramification (r , , F ) which satisfies (a) and F are Baire class functions. (b) If < 1 and f : 2 → 2 is an independent -function, then 0 ◦f : 2 → 2 is an independent + -function. Note: As usual, one cannot pick (r , , F ) for each without some choice, so we should work with a family or ramifications indexed by codes of ordinals. But we will not bother about this in the sequel.
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For an increasing map : → , let ˜ : 2 → 2 be defined by (ε) ˜ = ε ◦. Clearly ˜ is continuous, and in fact an independent 0-function. The next lemma is easy to check. Lemma 2.5. If ˜ : 2 → 2 is as above and f : 2 → 2 is an independent -function, then ˜ ◦ f is an independent -function. We let R be the least set of functions: 2 → 2 which contains the functions associated with the ramifications of Theorem 2.4, the function ˜ for increasing: → , and is closed under composition. By Theorem 2.4(b), and Lemma 2.5 each f ∈ R is an independent -function, for some we call the order o(f) of f. Part of our goal now is to define, for each u ∈ D2 a set Hu ⊂ 2 which is Γu -strategically complete. But the inductive construction needs sets with a slightly stronger property: 0
Definition 2.6. Let u ∈ D2 . A set H ⊆ 2 is u-strategically complete if (i) H ∈ Γu (2) (ii) for each f ∈ R of order o(f) = , the set f −1 (H ) is Γu -strategically complete. Theorem 2.7. Let u ∈ D2 . There exists a u-strategically complete set Hu ⊆ 2, and for each pair A0 , A1 of disjoint Σ11 sets in a closed (for ε ∈ 2, player II produces player II) game Ju (A0 , A1 ), where player I produces ˘ × ) such that: α ∈ and ∈ , and a set Cu (A0 , A1 ) in Γ(Σ (i) If (ε, α, ) is a win for player II in Ju (A0 , A1 ), then (ε ∈ Hu =⇒ α ∈ A0 ) and (ε ∈ / Hu =⇒ α ∈ A1 ) (ii) If for some fixed α ∈ , is a winning strategy for player I in the game Ju (A0 , A1 )α (where player II plays this α), then: / Cu (A0 , A1 )) (α ∈ A0 =⇒ ( , α) ∈ Cu (A0 , A1 )) and (α ∈ A1 =⇒ ( , α) ∈ This result is the main result of this section. The proof is by induction on u ∈ D2 . Let us say that u is nice if it satisfies the conclusions of Theorem 2.7. Using Proposition 1.11, it is enough to prove that 0 is nice, that if u is nice so is u(0)1u, and u for each < 1 , and that if (un )n∈ are nice, so is 12un . Lemma 2.8. The description 0 is nice. Proof. We must set H0 = ∅ and C0 (A0 , A1 ) = Σ × . For A0 , A1 disjoint sets with associated trees T0 , T1 respectively × , let J0 (A0 , A1 ) be the game where player I plays ε, player II plays α and and player II wins if for all n(αn, n) ∈ T1 . Clearly for any f ∈ R, f −1 H0 = ∅ is strategically complete in Γ0 . J0 (A0 , A1 ) is closed for player II. If (ε, α, ) is a win for player II, α ∈ A1 , so (i) is satisfied. And if for some α ∈ player I has a / A1 hence (ii) is satisfied too. winning strategy in J0 (A0 , A1 )α, α ∈ Σ11
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Lemma 2.9. Suppose u is nice. Then u˘ = u(0)1u is nice too. ˘ u , and one checks that Hu˘ = H˘ u , Ju˘ (A0 , A1 ) = Ju (A1 , A0 ) Proof. Γu˘ = Γ and Cu˘ (A0 , A1 ) = C˘u (A1 , A0 ) work. Lemma 2.10. Suppose u is nice. For each < 1 , u is nice. Proof. Let (r , , F ) be the ramification of order , with 0 ∈ R. Let Hu , Ju (A0 , A1 ) and Cu (A0 , A1 ) be associated to u. We define Hu = (0 )−1 (Hu ). If now f ∈ R is of order , 0 ◦ f ∈ R is of order + , hence (0 ◦ f)−1 (Hu ) = f −1 (Hu ) is strategically complete in Γu + = Γ(u ) , and Hu is u -strategically complete. We now define Ju (A0 , A1 ) by Ju (A0 , A1 )α = R (Ju (A0 , A1 )α) for all α ∈ . This is meaningful, for in order to check whether (εk, αk, k) is legal in Ju (A0 , A1 ), we need to know if t = {r (εk , k ) : k ≤ k} is contained in Ju (A0 , A1 )α. But by the properties of ramifications, t ⊆ ≤k (2 × ) so the knowledge of αk is enough for that. Finally we define Cu (A0 , A1 ) by ( , α) ∈ Cu (A0 , A1 ) ⇐⇒ (F (Ju (A0 , A1 )α, ), α) ∈ Cu (A0 , A1 ). As α → Ju (A0 , A1 ) is continuous and F is Baire class , the set Cu (A0 , A1 ) ˘ u by Proposition 1.11. It remains to check that they satisfy (i) and (ii) is in Γ of Theorem 2.7. For (i), let (ε, α, ) be a win for player II in Ju (A0 , A1 ). So (0 (ε), α, 1 (ε, )) is a win for player II in Ju (A0 , A1 ). This gives ε ∈ Hu =⇒ 0 (ε) ∈ Hu =⇒ α ∈ A0 and ε∈ / Hu =⇒ 0 (ε) ∈ / Hu =⇒ α ∈ A1 For (ii), let be winning for player I in Ju (A0 , A1 )α. Then ∗ = F (Ju (A0 , A1 )α, ) is winning for player I in Ju (A0 , A1 )α. So α ∈ A0 =⇒ ( ∗ , α) ∈ Cu (A0 , A1 ) =⇒ ( , α) ∈ Cu (A0 , A1 ) and α ∈ A1 =⇒ ( ∗ , α) ∈ / Cu (A0 , A1 ) =⇒ ( , α) ∈ / Cu (A0 , A1 ). The preceding proof was trivial—as everything has been embedded in the notion of ramification. The next one is on the other hand long and tedious— but more or less straightforward. Lemma 2.11. Suppose that for all n, un is nice. Then so is u = 12un . Proof. Let Hn , Jn = Jn (A0 , A1 ) and Cn = Cn (A0 , A1 ) be associated to un . First we choose a bijection , between ( ∪ {∗}) × and such that each i = i, · : → is strictly increasing, for i ∈ ∪ {∗}. We view each ε ∈ 2 asa sequence ε ∗ , εi i∈ , with ε ∗ = ε ◦ ∗ , εi = ε ◦ i . Recall that Γu = S1 ( n≥1 Γun , Γu0 ). Intuitively, we want εi to correspond to ui . As we
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need repetitions, we choose ϕ : \ {0} → \ {0} such that ϕ −1 (i) is infinite for all i. Let also : → \ {0} be such that −1 (i) is infinite for all i. We set: ε ∈ Hu ⇐⇒ either ε ∗ = 0 and ε0 ∈ H0 or for n the least i with ε ∗ (i) = 1, ε(n) ∈ Hϕ((n)) . We first check that Hu is u-strategically complete. Define H0 = {ε : ε0 ∈ H0 }, and for n ≥ 1, Hn = {ε : εn ∈ Hϕ(n) }, and Cn = {ε : ε ∗ = 0 and for m the least i with ε ∗ (i) = 1, (m) = n}. Clearly, the Cn are pairwise disjoint open sets, H0 ∈ Γu0 and for n ≥ 1, Hn ∈ Γuϕ(n) , so Hu = n≥1 (Hn Cn ) ∪ (H0 \ n Cn ) is in Γu = S1 ( n≥1 Γun , Γu0 ). Let now f ∈ R be of order . By Proposition 1.11, Hu = f −1 (Hu ) is in Γu . Let : → be associated to f, and let ϑn be the increasing enumeration of −1 (ran n ), and ϑn∗ the increasing enumeration of −1 ({i ∈ ran ∗ : ( ∗−1 (i)) = n}). Note that the fact that ε ∈ f −1 (Hn ) depends only on ε ◦ ϑn . Let then Hn = {ε ◦ ϑn : f(ε) ∈ Hn }, and also Cn = {ε◦ϑn∗ : f(ε)∗ takes value 1 on some i with (i) = n}. By the hypothesis, Hn is strategically complete in Γu for n ≥ 1, in Γu0 for n = 0. Moreover, ϕ(n)
one easily checks that if g is an independent -function, {ε : g(ε) = 0} is Σ01+ -strategically complete. So each Cn is strategically complete in Σ01+ . Let then H ∗ ⊆ be any set in Γu , say H ∗ = n≥1 (Hn∗ ∩ Cn∗ ) ∪ (H0∗ \ n Cn∗ ) with pairwise disjoint Cn∗ in Σ01+ , H0∗ in Γu0 and wlog Hn∗ in Γu (this is ϕ(n) where repetitions are used). So player II has for each n a winning strategy n in G(Hn∗ , Hn ) and n∗ in G(Cn∗ , Cn ). Let then player II play in G(H ∗ , f −1 (H )) against α by playing his strategies n , n∗ at the right places—the ranges of ϑn and ϑn∗ respectively—against this same α, independently. The result is some ε such that ε ◦ ϑn wins against α in G(Hn∗ , Hn ) and ε ◦ ϑn∗ against α in G(Cn∗ , Cn ). This wins, for ε ∈ f −1 (Hn ) just in case α ∈ Hn∗ , and f(ε)∗ takes value 1 on some i with (i) = n just in case α ∈ Cn∗ . But as the Cn∗ are disjoint, there is at most one n in {(i) : f(ε)∗ (i) = 1}, and ε ∈ f −1 (Cn ) just in case α ∈ Cn∗ . This proves that Hu is u-strategically complete. We now define Ju = Ju (A0 , A1 ). We view ε as ε ∗ , εi as before, and similarly we decompose as ∗ , i . In Ju , as long as player I plays 0’s on his ε ∗ -moves, player I and player II must play the game J0 with ε0 , α and 0 . And once player I has played 1 on his ε ∗ -moves, at say step k of the game, then letting k0 = ∗−1 (k), n0 = (k0 ), m0 = ϕ(n0 ), the players switch to the game Jm0 , played with εn0 for player I, α and the part of n0 which is played after step k for player II (i.e., when switching, player I does not revise his previous moves on εn0 , when player II does for n0 ). This defines a closed (for player II) game. We now check (i) of Theorem 2.7: So suppose (ε, α, ) is a win for player II in Ju (A0 , A1 ). There are two cases: (a) Suppose ε ∗ = 0. Then ε ∈ Hu ⇐⇒ ε0 ∈ H0 . But then ε ∈ Hu =⇒ ε0 ∈ / Hu =⇒ ε0 ∈ / H0 =⇒ α ∈ A1 as (ε0 , α, 0 ) is a H0 =⇒ α ∈ A0 and ε ∈ win for player II in J0 .
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(b) for some least k0 , ε ∗ (k0 ) = 1. Let k = ∗ (k0 ), n0 = (k0 ) and m0 = ϕ(n0 ), and n0 ≥k the sequence of n0 -moves after k. Then (εn0 , α, n0 ≥k ) is a win for player II in Jm0 , and again ε ∈ Hu ⇐⇒ εn0 ∈ Hm0 , so that we get the same conclusion. This proves (i). We now define Cu = Cu (A0 , A1 ). Let < be an ordering type of on <, with u ⊆ v =⇒ u < v. Given α ∈ and ∈ Σ, let us say that a finite sequence w ∈ < is ( , α)-legal if the position in Ju (A0 , A1 ) corresponding to the play α lh(w), w of player II and the -answer by player I is legal. For all ( , α), let w( , α) be the least for < sequence which satisfies (i) w is ( , α)-legal; (ii) the answers by on the ∗-moves are 0 up to lh(w); and (iii) lh(w) corresponds to a ∗-move, and
(w) = 1 [in other words, we are at a legal switching position in Ju (A0 , A1 ) where player II follows α and w, and player I answers by ]. The function ( , α) → w( , α) is defined on all pairs ( , α) for which there is a legal beginning w with (w) a ∗-move with value 1. We now define, for α ∈ and ∈ Σ, a sequence of strategies as follows: First, to each w0 ∈ <, viewed as a play of player II in J0 α, associate the play w in Ju α consisting in playing w0 on the 0-moves, and 0 on all other moves, with length such that the next play is the next 0-move. And define
0 (w0 ) = (w). Note that F0 : → 0 is continuous: Σ → Σ. Let now w0 ∈ <. We define Fw0 : Σ → Σ as follows: We let Fw0 ( ) = 0 unless the answers by to w0 are 0 on the ∗-moves up to lh(w0 ), and lh(w0 ) is a ∗-move and (w0 ) = 1. And in this case, we associate to each w ∈ < a position w as follows: w is w0 up to lh(w0 ) = k, with (k) = k0 say. After that, w is 0 everywhere except on the k0 -moves, where it is w, and its length is such that the next play will be the next k0 -move. And we define Fw0 ( )(w) = (w ). Again each Fw0 : Σ → Σ is continuous. We can now define Cu = Cu (A0 , A1 ) ⊆ Σ × by ( , α) ∈ Cu
⇐⇒
either for every ( , α)-legal sequence w ∈ < the ∗-answers by are 0, and (F0 ( ), α) ∈ C0 or there is a ( , α)-legal sequence with ∗-answer 1 by , and if w0 = w( , α) and k0 = (lh(w0 )), (Fw0 ( ), α) ∈ Cϕ(k0 ) .
We first check that Cu ∈ Γ˘ u : let Bw0 = {( , α) : w0 = w( , α)}. Clearly, each Bw0 is clopen in Σ × , and the Bw0 ’s are pairwise disjoint. Let D0 = {( , α) : (F0 ( ), α) ∈ C0 } and for w0 ∈ <, Dw0 = {( , α) : (Fw0 ( ), α) ∈ Cϕ0 (lh w0 ) }. By continuity of the F0 , Fw0 ’s, Dw0 ∈ Γu0 and Du0 ∈ Γuk , k = ϕ ◦ (lh w0 ). And Cu = w0 ∈< (Bw0 ∩ Dw0 ) ∪ (D0 \ ˘ w0 ∈< Bw0 ), hence Cu ∈ Γu . It remains to check (ii) of Theorem 2.7. So we let α ∈ , and winning for player I in Ju (A0 , A1 )α. There are two cases.
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(a) First, if w( , α) is undefined, i.e., for any ( , α) legal sequence w, the ∗-answers are 0. Then ( , α) ∈ Cu ⇐⇒ (F0 ( ), α) ∈ C0 . But we claim F0 ( ) is winning for player I in J0 α, for if 0 ∈ defeats F0 ( ) in J0 α, the play corresponding to 0 on the 0-moves and 0 everywhere else is easily seen to defeat in Ju α. So we get α ∈ A0 =⇒ (F0 ( ), α) ∈ / C0 =⇒ ( , α) ∈ / C0 =⇒ ( , α) ∈ Cu and α ∈ A1 =⇒ (F0 ( ), α) ∈ Cu . (b) Otherwise, w0 = w( , α) is defined, and by definition ( , α) ∈ Cu ⇐⇒ (Fw0 ( ), α) ∈ Ck where k = ϕ((lh w0 )). Again we claim that Fw0 is winning for player I in Jk α, which as before will finish the proof. Suppose is a play of player II which defeats Fw0 ( ) in Jk α, and let player II play in Ju α first w0 , then 0 on all moves except the moves corresponding to k0 = (lh w0 ), where he plays . One easily checks that all positions are then legal in Ju α against , a contradiction which finishes the proof. Altogether Lemmas 2.8, 2.9, 2.10 and 2.11 prove Theorem 2.7. §3. Proof of Borel Wadge Determinacy. As we said in the introduction, we will prove a slight generalization of Borel Wadge Determinacy. Consider, for A ⊆ and A0 , A1 two disjoint subsets of the following extended Wadge game G(A; A0 , A1 ): player I plays α ∈ , player II plays ∈ , and player II wins if (α ∈ A =⇒ ∈ A0 and α ∈ / A =⇒ ∈ A1 ). The usual Wadge game G(A, B) corresponds to B = A0 = A˘1 . [We will also consider the similar game where A ⊆ 2, and player I plays α ∈ 2, that we will denote ambiguously G(A; A0 , A1 ) too.] So Borel Wadge Determinacy is a particular case of Theorem 3.1. Let A ⊆ be Borel, and A0 , A1 two disjoint Σ11 subsets of . The extended Wadge game G(A; A0 , A1 ) is determined. In order to prove Theorem 3.1, we first prove particular instances of it. Theorem 3.2. Let u ∈ D2 , and A0 , A1 two disjoint Σ11 sets in . ˘ u separates A0 from A1 (i.e., (i) If A ⊆ is in Γu , and no set B ∈ Γ A0 ⊆ B ⊆ A˘1 ), then player II has a winning strategy in G(A; A0 , A1 ). ˘ u , and there is a set B ∈ Γ ˘ u separating A0 from (ii) If A ⊆ is not in Γ A1 , then player I has a winning strategy in G(A; A0 , A1 ). In particular, if A ⊆ is in Γu \ Γ˘ u , G(A; A0 , A1 ) is always determined. Proof. Let Hu , Ju (A0 , A1 ) and Cu (A0 , A1 ) be associated to u by Theorem 2.7. Being closed, the game Ju (A0 , A1 ) is determined. If player I has a winning strategy in it, let (α) be the corresponding winning strategy in Ju (A0 , A1 )α obtained by fixing the α-moves. The set B ⊆ defined by
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α ∈ B ⇐⇒ ( (α), α) ∈ Cu (A0 , A1 ) is then in Γ˘ u , and by (ii) of Theorem 2.7, separates A0 from A1 . (i) Assume A ∈ Γu and no Γ˘ u set separates A0 from A1 . Then by the previous discussion, player II wins Ju (A0 , A1 ). And by forgetting the -moves in this game, player II has a strategy in the game G(Hu ; A0 , A1 ) which satisfies by (i) of Theorem 2.7, ε ∈ Hu =⇒ α ∈ A0 and ε ∈ / Hu =⇒ α ∈ A1 , i.e., is winning in G(Hu ; A0 , A1 ). And as A ∈ Γu and Hu is Γu -strategically complete, player II also has a winning strategy in G(A, Hu ). Composing his strategies gives a winning strategy for player II in G(A; A0 , A1 ) and (i) is proved. (ii) Assume now A ∈ / Γ˘ u , and let for n ∈ , A(n) = {α ∈ : nα ∈ A}. By Proposition 1.7, one of the A(n)’s must satisfy A(n) ∈ / Γ˘ u . Let n0 be ˘ u separate A0 from A1 . Applying case the least such n. Let also B ∈ Γ ˘ 0 )), we get that player II has a winning (i) to B˘ and the pair (A(n0 ), A(n ˘ A(n0 )). Let then player I play first n0 , and then follow strategy in G(B, the strategy against player II’s play. At the end, one gets n0 α and , and n0 α ∈ A ⇐⇒ α ∈ A(n0 ) ⇐⇒ ∈ / B, so that ∈ A0 =⇒ n0 α ∈ / A and ∈ A1 =⇒ n0 α ∈ A and this strategy is winning for player I in G(A; A0 , A1 ), and (ii) is proved. And the final statement immediately follows from (i) and (ii). + + Recall that we associated to each class Γ a class Γ by A ∈ Γ ⇐⇒ A = ˘ (B ∩ D) ∪ (C \ D) for some D ∈ Δ01 , B ∈ Γ and C in Γ. Theorem 3.3. Let u ∈ D2 , A0 , A1 two disjoint Σ11 sets in and A a set in + ˘ u ). The game G(A; A0 , A1 ) is determined. Γu \ (Γu ∪ Γ Proof. For each s ∈ <, let A(s) = {α : s α ∈ A}. Let TA = ˘ {s ∈ < : A(s) ∈ Γ+ u − (Γu ∪ Γu )}. Clearly, TA is a tree, ∅ ∈ TA , + and as A ∈ Γu , TA is well founded. Let TA0 ,A1 = {t ∈ < : no Γu ∪ ˘ u set separates A0 (t) from A1 (t)}. Again TA0 ,A1 is a tree, which now may be Γ empty or not well-founded. Let G ∗ be the game where player I and player II play integers, and player I loses if he gets off TA before player II gets off TA0 ,A1 . (So if in particular TA0 ,A1 is empty, player I wins before the game starts). As TA is well-founded, G ∗ is clopen. We claim that whoever wins G ∗ also wins G(A; A0 , A1 ): Case (a): player I has a winning strategy in G ∗ . Let him play it in G(A; A0 , A1 ). Then a position (s, t) must be reached such that s ∈ TA but t ∈ / TA0 ,A1 (we use ∅ ∈ TA here). This means that A0 (t) is separable from ˘ u , say Γ˘ u to be specific. As A(s) ∈ ˘ u , player I has /Γ A1 (t) by some set in Γu ∪ Γ a winning strategy in G(A(s); A0 (t), A1 (t)) by Theorem 3.2, and switching to it is clearly winning in G(A; A0 , A1 ). Case (b) is similar: By playing his winning strategy in G ∗ , player II reaches a position (s, t) with t ∈ TA0 ,A1 , but any extension of s gets off TA . Let then player I play n0 . The set B = {α ∈ A(s) : α(0) = n0 } is in Γu ∪ Γ˘ u , say in
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Γu to be specific, and A0 (t) cannot be separated from A1 (t) by a set in Γ˘ u , hence player II has a winning strategy in G(B; A0 (t), A1 (t)) by Theorem 3.2, and switching to it is clearly winning in G(A; A0 , A1 ). For un asequence of type 2 descriptions with Γun < Γun+1 , we defined Γun + = ( n Γun )+ . The next result is entirely similar to the preceding one, and we omit the proof. Theorem 3.4. If un is a sequence in D2 with Γun < Γun+1for all n, A0 , A1 are two disjoint Σ11 sets in and A ⊆ is a set in Γun + \ n Γun , the game G(A; A0 , A1 ) is determined. The last step in the proof of Theorem 3.1 is to show that Theorems 3.2, 3.3, and 3.4 cover all possible cases. And to do this, the last ingredient is the following precise version of Martin’s result on the well-foundedness of Wadge’s ordering. Theorem 3.5 (Martin). Let (An )n∈ be a sequence of Borel sets in . Then player I cannot at the same time have a winning strategy in all games G(An , An+1 ) and G(An , A˘n+1 ). Proof. By contradiction. Let n0 resp n1 be winning for player I in G(An , resp G(An , A1n+1 ) where by definition A0n+1 = An+1 , A1n+1 = A˘n+1 . Associate to each ε ∈ 2 a sequence αnε by induction by αnε (0) = nε(n) (∅), ε and αnε (k) = nε(n) (αn+1 k). This sequence is clearly obtained continuously in ε ε, as αn k depends on εn + k. Moreover αnε depends only on the values of ε ε for m ≥ n, and for all n the pair (αnε , αn+1 ) is a run in G(An , Aε(n) n+1 ) where A0n+1 ),
ε ε player I follows nε(n) , so αnε ∈ An ⇐⇒ αn+1 ∈ / Aε(n) n+1 . Let B = {ε : α0 ∈ A0 }. < The set B is Borel. On the other hand, if s ∈ 2, neither B nor B˘ is comeager on Ns = {ε : s ⊆ ε}, for if ε ∈ Ns and ε is defined by ε(p) = ε(p) for p = p0 = lh(s), and ε(p0 ) = 1 − ε(p0 ), then ε ∈ Ns , and αpε = αpε for p > p0 , and for p ≤ p0 , αpε ∈ Ap ⇐⇒ αpε ∈ / Ap , so that in particular ε ε α0 ∈ B ⇐⇒ α0 ∈ / B. This shows B does not possess the Baire property, a contradiction which finishes the proof. Recall that we defined, for u, u ∈ D2 , Γu < Γu if Γu ⊆ Δ(Γu ).
Corollary 3.6. The relation < is well-founded on the set W2 = {Γu : u ∈ D2 }. Proof. If not, let un ∈ D2 be such that (Γun ) is a <-decreasing se˘ un . By Theorem 3.2, player I wins all games quence, and An any set in Γun \ Γ G(An , An+1 ) and G(An , A˘n+1 ), contradicting Martin’s Theorem 3.5 Proof of Theorem 3.1. We argue by contradiction. The set W1 = {Γu , Γ˘ u : u ∈ D1 } is a subset of W2 (Theorem 1.8), hence is well-founded for < by Corollary 3.6 above, and is cofinal in Δ11 . So if Theorem 3.1 fails, we can
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find a <-minimal class Γ in W1 and sets A ∈ Γ, A0 and A1 disjoint Σ11 sets in such that G(A; A0 , A1 ) is not determined. Now Γ = Γu or Γ˘ u for some u ∈ D1 , hence Γ = Γv for some v ∈ D2 . ˘ Then A ∈ Γv \ Γ˘ v , and by Theorem 1.4 G(A; A0 , A1 ) Case (a): A ∈ Γ \ Γ. is determined, a contradiction. So A ∈ Δ(Γ), and we can apply Theorem 1.4. Note that Case 1.4 (iii), (i.e., Δ(Γ) = <1 , Γu for some u ∈ D1 ) is impossible by <-minimality of Γ. Case (b): For some u ∗ ∈ D1 , Δ(Γ) = (Γu ∗ )+ . Then by <-minimality of + ˘ ˘ Γ, A ∈ Γ+ u ∗ \ (Γu ∗ ∪ Γu ∗ ), hence for some v ∈ D2 , A ∈ Γv \ (Γv ∪ Γv ) and G(A; A0 , A1 ) is determined by Theorem 3.3, a contradiction. The last case where Δ(Γ) = Γun + for some (un ) in D1 is handled similarly, using Theorem 3.4. (Theorem 3.1) Let us finish this section with a brief discussion of some other results which are by-products of the preceding proof. 1. We get from the preceding proof that W = W1 = W2 . 2. The sets Hu , u ∈ D2 we constructed are in Γu (2). If follows that (a) If Γ is a non-self dual Wadge class, so is Γ(2); and (b) Γ = {f −1 (A) : A ⊆ 2, A ∈ Γ(2)}. 3. For Γ a class and a countable ordinal, we can always define its expansion Γ as the class of f −1 (A), A ∈ Γ and f : → of Baire class . Then one gets (Γu ) = Γu for all u ∈ D2 : inclusion ⊆ is easy (Proposition 1.11), and in the other direction, the set Hu which generates Γu is in (Γu ) by its very definition. Using this and the equality W = W2 , one gets that the class W of non-self dual Borel Wadge classes is the least family of classes containing {∅} and closed under complementation, S1 , and -expansion for all < 1 . 4. The argument we gave for Borel Wadge classes and Borel Wadge games on easily translate to the case of 2. In particular one gets that the non-self dual Borel Wadge classes on 2 are exactly the Γu (2) for u ∈ D2 . One also gets the determinacy of the games G(A; A0 , A1 ) played on 2, by noticing that the Hu ’s are strategically complete in Γu (2), and that Theorem 1.4 holds for 2 too. There is however a slight difference the two cases: For 2, between + case (ii) in 1.4 trivializes as Γn ( 2) = n Γn ( 2) by compactness. This also shows that fact 2 above fails for self-dual classes. Let us make now some comments on how further information can be obtained from the existence of the “unfolded” games Ju : 5. It is clear, by looking at their definition, that the games Ju (A0 , A1 ) are defined uniformly in A0 and A1 . In fact if one codes the pairs of Σ11 sets reasonably—e.g., by coding pairs of associated trees on × , one easily checks that for each u ∈ D2 the tree of the game Ju (A0 , A1 ) is continuous in the codes. Using this uniformity, one easily gets the following result, where for A ⊆ × and x ∈ , Ax denotes the section of A at x:
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Theorem 3.7. Let Γ be some Borel Wadge class. (a) If A0 , A1 are Σ11 sets in × BΓ = {x : (A0 )x is separable from (A1 )x by a Γ-set} is Π11 , hence in particular for Borel B = {x : Bx ∈ Γ} is Π11 . , A are Σ1 in × and for all x, (A ) can be separated from (b) If A 0 1 0 x 1 Γ set, there is a Borel set B with sections in Γ which (A1 )x by some separates A0 from A1 . To prove this, one can easily reduce the case where Γ is self dual to the non-self dual case, say Γ = Γu for some u ∈ D2 . One gets then a continuous map: x → Ju˘ ((A0 )x , (A1 )x ) = Jx , and by Theorem 3.2, x ∈ BΓ ⇐⇒ player I wins Jx ⇐⇒ player II does not win Jx and this last statement is Π11 . This gives (a). By general standard facts, (b) is a consequence of (a). Onecan also use that if BΓ = , one can find by the selection theorem for strategies in open games a Borel function x → x , with x winning for player I in Jx . And by Theorem 3.2 again B = {(x, a) : ( x (α), α) ∈ Cu˘ ((A0 )x , (A1 )x )} is a set which separates A0 from A1 , and is Borel with Γu sections. 6. We now discuss the uniformity of our constructions in u. This leads to lightface versions of our results. First by encoding countable ordinals by reals, one gets a coding of descriptions. Let us say that a description u (in D1 or D2 ) is HYP if some code of it is Δ11 , and that Γ is a HYP1 (a HYP2 ) class if Γ = Γu or Γ˘ u for some HYP u in D1 (Γ = Γu for some HYP u in D2 ). The proof of Theorem 1.8 easily gives HYP1 ⊆ HYP2 . Let also WHYP be the family of non-self dual Wadge classes which are generated by a Δ11 set. One also checks easily that HYP2 ⊆ WHYP . Fix now some HYP description u (in D1 or D2 ). One can then define the notion of a HYP-in-Γu subset of : Intuitively, these are the sets in Γu which admit a HYP construction as a Γu set. Formally, one has to go through codes. This is done precisely for first type descriptions in the second part of Louveau’s paper [Lou83], and it is proved there that for u HYP in D1 , Γu ∩ Δ11 = HYP-in-Γu . Our games allow to prove a similar result for u ∈ D2 : The point is that for recursive ordinal , one can choose the ramification (r , , F ) so that all functions are Δ11 -recursive, and then it is not hard to check that for HYP u ∈ D2 , (a) Hu ∈ Δ11 ; (b) for A0 , A1 Σ11 sets, Ju (A0 , A1 ) is Δ11 , and Cu (A0 , A1 ) is HYP-in-Γ˘ u . Using this, one gets Γu ∩ Δ11 = HYP-in-Γu for u HYP in D2 : For if A is Δ11 in ˘ by Theorem 3.2, and this game is open Δ1 for Γu , player I wins Ju (A, A) 1 player I, hence player I has a Δ11 winning strategy . But then one gets ˘ which proves that A is HYP-in-Γu . A = {α : ( (α), α) ∈ Cu˘ (A, A)},
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The second step is to get a “lightface” analog of Theorem 1.4. First one + easily defines the notions of HYP-in-Γ+ u and HYP-in-Γun for u HYP in 1 D1 , and un a Δ1 sequence of HYP descriptions in D1 . Then one can prove the analog of Theorem 1.4, analyzing the family Δ11 ∩ Δ(Γu ) = HYP-in˘ u ) in terms of HYP-in-Γu classes, for u ’s HYP in D1 with Γu ) ∩ (HYP-in-Γ Γu < Γu . The proof essentially follows the proof in [Lou83, 1.4], and although a bit tedious, necessitates no new ideas. By combining the two previous steps, one finally gets Theorem 3.8. (i) Every WHYP class Γ admits a D1 (and hence a D2 ) HYP description—as ˘ u , Γu + or Γun + , as the case may be Γu , Γ (ii) if A is Δ11 and in a HYP class Γ, it is HYP-in-Γ. Hence in particular the Wadge class Γ(A) of A is a HYP class, and A is HYP-in-Γ(A). Finally notice that when considering a Borel Wadge game G(A, B) with Δ11 sets A, B, the winning strategies in this game are recursively obtained, using Theorems 3.2, 3.3, and 3.4, from winning strategies for player II in associated closed games depending on the Wadge class Γ(A), so by the previous result which can be chosen Δ11 . One then finally gets Theorem 3.9. If A, B are Δ11 in , one of the two players in G(A, B) has a winning strategy which is recursive in Kleene’s O. Note that this result is the best possible along these lines, as by considering A = and B some non empty Π01 set with no Δ11 members, we see that there may be no HYP winning strategy. 7. A final word on the games Ju : The fact that the -part of player II’s play in Ju are in is not an essential feature, and one can define similar games with ∈ κ for κ some infinite cardinal. Such extensions were used in [LSR87] to study separation of κ-Suslin sets by Σ0 -sets, and separation of lightface projective sets by Σ0 sets, using strong set theoretic hypotheses. Similar results could be obtained, along the same lines, for all Borel Wadge classes. §4. Wadge Classes in Metric Separable Spaces. Unless the underlying space is 2 or , there is no clear notion of Wadge game available, and to extend Theorem 3.1 to more general situations, we must first rephrase (and weaken) it a bit. Let Γ be a Borel Wadge class in and A0 , A1 a pair of disjoint sets in some metric separable space E. We say that (A0 , A1 ) reduces Γ if for any B ⊆ in Γ, there is a continuous f : → E with f() ⊆ A0 ∪ A1 and f −1 (A0 ) = B. Note that this property is intrinsic, i.e., depends only on ˘ does. A0 ∪ A1 . We say A reduces Γ if (A, A) Let us say, for A0 , A1 in , that Γ separates (A0 , A1 ) if A0 is separable from A1 by some B in Γ. With this terminology, Theorem 3.1 gives:
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Theorem 4.1. (a) If Γ is a Borel Wadge class in and A ⊆ is Borel and does not reduce ˘ Γ, then A ∈ Γ. (b) If Γ is a Borel Wadge class in , A0 , A1 are pairwise disjoint Σ11 sets in and (A0 , A1 ) does not reduce Γ, then Γ˘ separates (A0 , A1 ). Part (a) is usually called “Wadge’s lemma”. It is of course a consequence of part (b), but we stated the two versions, for their fate will be slightly different in the sequel. The main point in order to extend Theorem 4.1 to other spaces is to define, for each Wadge class Γ on , a corresponding class Γ(E) of sets in E. This can be done in various ways, and most of the work will consist in showing that the various possible definitions lead to the same classes. From now on, we will consider only classes Γ in W , i.e., non-self-dual Borel Wadge Classes. In the first section, we briefly looked at the case of subsets E of , where one can define Γ(E) using traces. As any zero-dimensional metric separable space is homeomorphic to such a subset of , this gives one way of defining Γ(E) for a zero-dimensional space E. Another way is to consider continuous preimages, still another way to consider Hausdorff operations performed on open sets in E. And there is a more subtle possibility, coming from Wadge’s lemma: to define the class Γ(E) by the properties of its continuous preimages. The next result shows the equivalence of all these possible definitions, at least when E is a zero-dimensional Suslin space, i.e., is absolute Σ11 metrizable separable. Theorem 4.2. Let E be a zero-dimensional Suslin space, Γ ∈ W , and A a subset of E. The following are equivalent. There is a 1-1 embedding E → and B ∈ Γ, A = j −1 (B). For all 1-1 embeddings E → there is a B ∈ Γ, A = j −1 (B). There is a continuous f : E → and a B ∈ Γ, A = f −1 (B). There is a Hausdorff operation D and (Un ) open in E such that Γ = D(Σ01 ) and A = D((Un )). 5. Forall Hausdorff operations D with Γ = D(Σ01 ) there is (Un ) open in E with A = D((Un )). −1 6. For all continuous G : → E, g (A) ∈ Γ.
1. 2. 3. 4.
We then define Γ(E) as the class of sets in E which satisfy one of these equivalent properties. Proof. 1 =⇒ 3 is trivial, and 3 =⇒ 5 and 4 =⇒ 6 come from preservation of Hausdorff operations on Σ01 sets by continuous preimages. 2 =⇒ 1 comes from the existence of an embedding from E into , i.e., the fact that E is zero-dimensional. 5 =⇒ 4 comes from the existence of a Hausdorff operation generating Γ, i.e., the fact that Γ is non-self dual. It remains to prove 6 =⇒ 2.
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So let j : E → be an embedding, and let A be such that for all continuous g : → E, g −1 (A) ∈ Γ. As E is Suslin, let : E be a continuous surjection. Then −1 (A) is in Γ, so is Borel, and A0 = j(A) and A1 = j(E \A) ˘ and (A0 , A1 ), one gets that Γ are both Σ11 in . Applying Theorem 4.1 to Γ ˘ separates (A0 , A1 ), else A0 , A1 would reduce Γ for some f : → E, f −1 (A) ˘ would be the complete Γ-set in . But if B ∈ Γ separates A0 from A1 , −1 A = j (B) as desired. With the previous definition, it is clear that if f : E → F is continuous, for E, F zero-dimensional Suslin spaces, and A ∈ Γ(E), then f −1 (A) ∈ Γ(F ). And one immediately gets, by applying Theorem 4.2 to A0 ∪ A1 , the following extension of Theorem 4.1. Theorem 4.3. Let E be a zero-dimensional Suslin space, Γ ∈ W , and A0 , A1 two disjoint Σ11 subsets of E. If the pair (A0 , A1 ) does not reduce Γ, the class ˘ Γ(E) separates (A0 , A1 ). We now study the general case, where E is still Suslin but not necessarily zero-dimensional. There are clearly difficulties: There are no more embeddings in , and the only continuous functions into may be the constants, so that definitions by (1), (2), (3) of Theorem 4.2 cannot be used. The first remaining possibility is to use descriptions and the corresponding operations, but even this approach has to be changed a bit for descriptions u with u(0) = 1, as there might not be enough families of pairwise disjoint open sets in E. So we adopt the following definition. Definition 4.4. Let E be metric separable. For each u ∈ D1 the class ΓE u is ˘E defined (together with Γ , where˘refers to the complementation inside E) by u the following: If u(0) = 0, then ΓE u = {∅} 0 If u = 1u ∗ with u ∗ (0) = 0, then ΓE u = D (Σ (E)) 0 E If u = 2u ∗ , then ΓE u = Sep(D (Σ ), Γu ∗ ) E E E If u = 3 u0 , u1 and ≥ 2, Γu = Bisep(D (ΣE
), Γu0 , Γu1 ) 0 If u = 13v0 , v1 A ⊆ E is in ΓE u if there are D (Σ ) sets C0 and E E ˘E C1 in E, and B ∈ Γu1 , such that A ∩ C0 ∈ Γu0 , A ∩ C1 ∈ Γ u0 , and A \ (C0 ∪ C1 ) = B \ (C0 ∪ C1 ) 0 E (e) If u = 4un : n ∈ and ≥ 2, ΓE u = SU(Σ , (Γun )) (e’) If u = 14un : n ∈ , A ⊆ E ΓE u if every point x ∈ A admits a is in E neighborhood V with A ∩ V ∈ n Γun 0 E E (f) If u = 5u0 , u1 , then ΓE u = SD (Σ , Γu0 , Γu1 ) Note that in case E has dimension zero, one has the reduction property for open sets (it is actually equivalent), so that (d’) and (e’) above correspond
(a) (b) (c) (d) (d’)
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then to the usual definition, i.e., ΓE u = Γu (E) when E is a zero-dimensional Suslin space. Note also that although our definitions d’ and e’ above look natural, it is now unclear that they correspond to some Hausdorff operation on open sets—and in fact they do not—and it is also unclear that if u and v describe the same class Γ in , they also describe the same class in any E. This will be true for Suslin E by Theorem 4.6 below. Another possible way for defining Γ in a space E comes from (6) of Theorem 4.2. Definition 4.5. Let E be metric separable, and Γ ∈ W . We define Γ(E) as the family of those sets A ⊆ E such that every continuous f : → E, f −1 (A) ∈ Γ. Note that in Definition 4.5, one could replace “” by “all zero-dimensional Suslin spaces” by using Theorem 4.2. Theorem 4.6. Let E be a Suslin space, and u a first type description. Then ΓE u = Γu (E). Hence part (a) of Theorem 4.1 extends to arbitrary Suslin spaces, i.e., if A ⊆ E is Borel and A does not reduce Γu , A ∈ Γ˘ E u. The second statement follows from the first, as before. But the proof of the first statement is more involved, and uses a transfer method via open maps. Recall that f : E → F is open if f(U ) is open in F for all U in E. It follows from a result of Hausdorff that if f : E F is a continuous open surjection and E is Polish, then F is Polish too. Conversely, one has the following classical fact: Proposition 4.7. Let E be a (non-empty) Polish space. Then there exists a continuous open surjection : E. Proof. Let D be a metric for which E is complete, and construct by induc− lh(s) tion on lh(s), s ∈ <, non-empty open , sets Us in E with diam Us < 2 Us∩n ⊆ Us and U∅ = E, Us = n Us∩n . This is easily done. For each α ∈ , s⊆α Us reduces to a singleton {(α)}, and : → E defined this way clearly works. The next result is a variant of a selection theorem of Saint-Raymond [SR76, Section 4]. Theorem 4.8. Let E, F be Polish spaces, a continuous open surjection E F , and f a Baire class 1 function E → X for some Polish X . Then there exists a selection s : F → E of (i.e., ◦ s = idF ) such that f ◦ s : F → X is a Baire class 1 function. Proof. Let dE , dX be complete metrics on E, X respectively. Say that a function S from F into the non-empty closed subsets of E is lsc if for each open U in E {x ∈ F : S(x)∩U = ∅} is open in F . For example x → −1 (x)
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is lsc, as is open. We construct by induction a sequence Sn of applications from F into the non-empty closed subsets of E such that (i) S0 (x) = −1 (x), x ∈ F (ii) Sn+1 (x) ⊆ Sn (x) (iii) For each n, there is a countable partition (Fkn )k∈ of F in Δ02 sets such that Sn Fkn is lsc for all k, and if x, y are in Fkn , α ∈ Sn (x), ∈ Sn (y), then dE (α, ) < n1 and dX (f(α), f()) < n1 . Case n = 0 is immediate, by taking Fk0 = F . Suppose (Fkn )k∈ and Sn have been defined. Fix k, and construct by transfinite induction a sequence T of closed subsets of Fkn by letting T0 = Fkn , T = < T for limit . And if T = ∅, let H = x∈T Sn (x), and as f is Baire class 1, let U be open, of 1 1 diameter < n+1 , with U ∩ H = ∅, and such that diamX (f(U ∩ H )) < n+1 . Let then T +1 = T \ {x : Sn (x) ∩ U = ∅}. As F is separable, there is a countable such that T = ∅, and the (T \ T +1 ) < form a partition of Fkn into Δ02 sets. And if one defines Sn+1 (x), for x ∈ T \ T +1 , by Sn (x) ∩ U , Sn+1 is lsc on T \ T +1 ; so by rearranging the T \ T +1 for all Fkn ’s, we get Sn+1 and (Fkn+1 )k∈ as desired. Now n Sn (x) is a singleton {s(x)} for each x ∈ F , and this clearly defines a selection s : F → E of . It remains to show that f ◦ s is a first class function. Choose for all n, k a point αn,k in the set {Sn (x) : x ∈ Fkn }, and define gn : F → X by gn (x) = f(αn,k ) if x ∈ Fkn . As the partition of F is in Δ02 sets, gn is a Baire class 1 function. And limit of the g ’s hence is Baire class 1 by condition (iii), f ◦ s is the uniform n too. We now state the transfer result from which Theorem 4.6 will follow. Theorem 4.9. Let E be a zero-dimensional Polish space, F a Polish space, f a continuous open surjection E F, u ∈ D1 and A0 , A1 two Σ11 sets in F 0 ∪A1 (a) If Γu (E) separates (f −1 (A0 ), f −1 (A1 )), then the set A0 is in ΓA u (b) If either (i) u(0) ≥ 2, or (ii) F is zero-dimensional, or (iii) A1 = A˘0 , then if Γu (E) separates (f −1 (A0 ), f −1 (A1 )), ΓFu separates (A0 , A1 ). Proof. Part b(iii) is of course a particular case of part (a). Assume first that u(0) ≥ 2. One easily defines, by induction on u, a Hausdorff operation D such that Γu , Γu (E), ΓFu are respectively D(Σ02 ), D(Σ02 (E)), D(Σ02 (F )): the only point is that for ≥ 2 the D (Σ0 ) sets do have reduction inany Polish space F . Let then Cn be a sequence of Σ02 (E) sets such that C = D((Cn )) g : E → [0, 1] be a Baire class separates f −1 (A0 ) from f −1 (A1 ), and let 1 function with Cn = {x ∈ E : gn (x) > 0}. Applying Theorem 4.8 to f : E → F and g : E → [0, 1] gives a selection s : F → E of f with g ◦ s of Baire class 1. Let Bn = {x ∈ F : gn ◦ s(x) > 0} and B = D((Bn )). The set B ∈ ΓFu and as B = s −1 (C ), it separates A0 from A1 . This proves both (a)
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and b(i). So it remains to study the case u(0) = 1 (We forget about {∅}!). It is done by inspection of the various cases. 1. Γ = D (Σ01 ). If C ∈ Γu separates f −1 (A0 ) from f −1 (A1 ), let Cϑ ϑ< be an increasing sequence of open sets in E building C . The sequence f(Cϑ ) is an increasing sequence of open sets in F which build some set B ∈ D (Σ01 (F )), and clearly separates A0 from A1 . 0 ∗ 0 2. Γu = Sep(D (Σ1 ), Γu ∗ ) with u (0) > 1. Let C ∈ D (Σ1 ), built say from ˘ −1 (A0 ) ∩ C, f −1 (A1 ) ∩ C ), Cϑ ϑ< , be such that Γ u ∗ (E) separates the pair (f and Γu ∗ (E) separates the pair (f −1 (A0 ) \ C, f −1 (A1 ) \ C ). Let Dϑ = f(Cϑ ), and fϑ = fCϑ : Cϑ Dϑ . Clearly as Cϑ is open, fϑ is a continuous open surjection. And depending on the parity of ϑ, Γu ∗ (Cϑ ) or Γ˘ u ∗ (Cϑ ) separates −1 −1 the pair (fϑ (A0 \ ϑ <ϑ Dϑ ), fϑ (A1 \ ϑ <ϑ Dϑ )). And asu ∗ ≥ 2, we can apply the first case, and the same class separates in Dϑ (A0 \ ϑ <ϑ Dϑ , A1 \ ϑ <ϑ Dϑ ). And sticking the pieces together gives, for D built from the Dϑ ’s, ˘ Fu∗ separates A0 ∩ D from A1 ∩ D and ΓFu∗ separates (A0 \ D, A1 \ D), as that Γ desired. 3. Γu = Bisep(D (Σ01 ), Γu0 , Γu1 ), with u0 (0) ≥ 2. At first sight, this case looks very similarto the preceding case. But in fact it is where the problem arises: The same argument as before does give two sets D0 and D1 in D (Σ01 (F )) such that ΓFu0 separates (A0 ∩ D0 , A1 ∩ D0 ), Γ˘ Fu0 separates ∩ D ) and ΓF separates (A \ (D ∪ D ), A \ (D ∪ D )). But (A0 ∩ D1 , A 1 1 0 0 1 1 0 1 u1 we cannot conclude that ΓFu separates A0 from A1 without sticking the pieces together, and this time D0 and D1 are not necessarily disjoint. So we can conclude only if there is no sticking to be done, for we work in A0 ∪ A1 and the only possibility is A0 —this gives part (a) in this case, or if we can reduce D0 and D1 , which is possible if F is zero-dimensional, and this gives b(ii). [We will see later that this obstruction cannot be avoided]. 4. Γu = SU(Σ01 , Γun ) with un (0) ≥ 2. The proof is similar to case 3: Again D in F such that ΓF separates A ∩ D from A ∩ D , one gets open sets n 0 n 1 n un and A0 ⊆ n Dn . This is enough to conclude for part (a) and for b(ii). Note that in this case one can always stick the pieces together, by using locally finite refinements of the Dn ’s, and the closure properties of the Γun ’s, but we won’t need this. 5. Γu = SD (Σ01 , Γun , Γu ∗ ) creates no difficulty, the argument being as in Case 2, and is left to the reader. Proof of Theorem 4.6. We want to show that ΓE u = Γu (E) for any Suslin space E. One easily checks, using Theorem 4.2 that ΓE u ⊆ Γu (E). If now A ∈ Γu (E), let F be Polish with E ⊆ F , and by Proposition 4.7 let : F be a continuous open surjection. By the definition of Γu (E), −1 (A) ∈ Γu (−1 (E)), and we can apply Theorem 4.9(a) to , F, and A0 = A, A1 = (Theorem 4.6) E \ A, which gives A ∈ ΓE u.
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Concerning separation of analytic sets, Theorem 4.9 does not give the full strength of Theorem 4.1, part (b), even for Polish spaces. One still gets Corollary 4.10. Let E be Polish, A0 , A1 two disjoint Σ11 sets in E and Γ a class in W . The following are equivalent: (i) A0 ∈ Γ(A0 ∪ A1 ). (ii) For any Polish zero-dimensional space F and continuous f : F → E, Γ(F ) separates (f −1 (A0 ), f −1 (A1 )). (iii) There is a Polish zero-dimensional space F and an open continuous surjection : F E such that Γ(F ) separates (−1 (A0 ), −1 (A1 )). In most cases, (i) above can be replaced by the stronger “Γ(E) separates (A0 , A1 )”, e.g., if dim(E) = 0, or if Γ is Γu for some u ∈ D1 with u(0) ≥ 2, or some specific Γu ’s with u(0) = 1, like the D (Σ01 )’s. But the next example in general Γ (E) is not shows it does not work in general (and hence that u always obtained by some Hausdorff operation performed on open sets). Example 4.11. Let E = [0, 1] × 2. E is a one-dimensional compact metrizable space. Let D0 , D1 be two disjoint countable dense sets in 2, and let Γ = Bisep(Σ01 , Σ02 ) and A0 = {(x, α) ∈ E : (x < 1 and α ∈ D0 ) ∨ (x = A = {(x, α) ∈ E : (x = 0 and α ∈ / D0 ) ∨ (x > 0 and α ∈ 1 and α ∈ / D1 )}, 1 D1 )}. Then A0 ∈ Γ(A0 ∪ A1 ), but Γ(E) does not separate A0 from A1 . Proof. On [0, 1[×2, Σ02 separates A0 , A1 , and on ]0, 1] × 2, Π02 separates A ). Suppose, towards a contradiction, that C ∈ (A0 , A1 ), so A0 ∈ Γ(A0 ∪ 1 Bisep(Σ01 , Σ02 )(E) separates (A0 , A1 ). Let U = {(x, α) : C is locally Σ02 at V = {(x, α) : C is locally Π0 at (x, α)}. By assumption, (x, α)} and C ⊆ 2 U ∪ V . Now, for x = 0, A0 x = A˘1 x = D0 is not Π02 , hence {0} × 2 ∩ V = U = {y : (y, α) ∈ U } ∅, and similarly {1}×2∩U = ∅. Fix α ∈ D0 , and let α and Vα = {y : (y, α) ∈ V }. Uα and Vα are open non-empty by the preceding facts, and cover [0, 1] as [0, 1] × {α} ⊆ A0 . By connectedness, Uα ∩ Vα = ∅, so ∃x ∈]0, 1[, (x, α) ∈ U ∩ V , hence a neighborhood W of (x, α) such that W ∩ C and W ∩ C˘ are Π02 . But both W ∩ C and W ∩ C˘ must be dense in W by choice of D0 and D1 ,contradicting the Baire Category theorem. §5. Hurewicz Tests and Hurewicz-Type Results. Let Γ be a Wadge class in W . A Hurewicz-test for Γ is a pair (K, H ) consisting of a zero-dimensional metric compact space K , and a Γ(K) subset H of K which satisfies: For every ˘ Borel set B in some (non-empty) Suslin space E, B ∈ / Γ(E) ⇐⇒ there is a 1-1 −1 continuous map ϕ : K → E with ϕ (B) = H (in other words, E contains a homeomorphic copy of the space K on which B is the corresponding copy of H ). Our terminology comes from the well known theorem of Hurewicz on the characterization of Polish spaces, which states, with our terminology, that if D is dense countable in 2, (2, D) is a Hurewicz test for the class Σ02 .
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It follows from work of Steel [Ste80] that every class Γu with u(0) ≥ 2 and D (Σ02 ) ⊆ Γu admits a Hurewicz test—at least for Borel subsets of . In [LSR87], we independently reproved Steel’s result by a method which yields Theorem 5.1 ([LSR87]). Let u be a description with u(0) ≥ 2. The class Γu admits a Hurewicz test (K, H ), with K = 2. We extend here this result to all Borel Wadge classes Γ ∈ W . Theorem 5.2. Every class Γ in W admits a Hurewicz test. Hence in particular, the membership of a Borel set B ⊆ E Suslin in Γ(E) depends only on the zero-dimensional compact subsets of E. We will prove the result by induction on descriptions. For doing this, it is easier to work with type 2 descriptions, except that one has to reduce a bit the allowed combinations: Definition 5.3. (i) Let Γn be a sequence of classes in W . The sequence Γn is admissible if it is increasing and whenever A = (A0 ∩ C ) ∪ (A1 \ C ) with C ∈ ˘ 0, A ∈ Σ01 , A0 ∈ n≥1 Γn and A1 ∈ Γ n≥1 Γn . (ii) We let D2 be the least set of type 2 descriptions satisfying 0 ∈ D2 , u(0)1u ∈ D2 if u ∈ D2 , and 2un ∈ D2 if un ∈ D2 for all n and the sequence Γun is admissible. Proposition 5.4. Every class Γ in W admits a D2 -description. Proof. It is enough to check that any Γu , u ∈ D1 admits such a description. In fact, one can check that the function u → v(u) defined in 1.8 takes values in D2 , by induction on u ∈ D1 . Note that the only change between D2 and D2 descriptions occurs for u(0) = 1, because by the closure properties of the classes Γu , u(0) ≥ 2, any sequence Γun with un (0) ≥ 2 is automatically admissible. It is immediate to see that if K is a singleton and H ⊆ K is ∅, K, H is a Hurewicz-test for Γ = {∅}, and that if K, H is a Hurewicz-test for Γ, ˘ So the proof of Theorem 5.2 follows from K, H˘ is a Hurewicz test for Γ. Theorem 5.1 and the following: Theorem 5.5. Let Γn be an admissible sequence of classes in W which admit Hurewicz-tests. The class Γ = S1 (Γn ) admits a Hurewicz-test too. Lemma 5.6. Let E be a Suslin space, and Γ = S1 (Γn ). A Borel set A ⊆ E is in Γ(E) iff There is a sequence Cn n≥1 of Σ01 sets in E such that for n ≥ 1, Cn ∈ Γ0 (E \ Cn ). A ∩ Cn ∈ Γn (Cn ), and A \ n≥1
n
(∗)
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ALAIN LOUVEAU AND JEAN SAINT-RAYMOND
Proof. If A satisfies (∗) and f : → E is continuous, f −1 (A) satisfies (∗) in , and by reducing the f −1 (Cn ), one gets that f −1 (A) ∈ Γ in . As this is true for any such f, A ∈ Γ(E). If now A ∈ Γ(E), let f : → F be a continuous open map onto some Polish space F containing E. Applying Theorem 4.9, we know that some set D in Γ separates (f −1 (A), f −1 (E \A)) in , and if (Bn )n≥1 are Σ01 sets with D ∩ Bn ∈ Γn for n ≥ 1, and D \ n≥1 Bn = D0 \ n≥1 Bn for some D0 ∈ Γu0 , then by Theorem 4.9 again, the sequence Cn = f(Bn ) ∩ E witnesses (∗) for A. Proof of Theorem 5.5. Let (Kn , Hn ) be Hurewicz-tests for the classes Γn , and Γ = S1 (Γn ). We define a Hurewicz-test (K, H ) for Γ as follows: First we fix a bijection n → (n)0 , (n)1 between \ {0} and ( \ {0}) × , and a dense sequence (xp0 )p∈ in K0 . Let K be the set defined by (n, x) ∈ K ⇐⇒ n ∈ ∧ [(n = 0 and x ∈ K0 ) or (n = 0 and x ∈ K(n)0 )] that we topologize as follows: the topology is generated by the sets {n}×V for n = 0 and V open in K(n)0 , and by the sets Uq,v = ({0}×V )∪( {{n}×K(n)0 : 0 n ≥ q, x(n) ∈ V }), where q ∈ and V is open in K0 . One easily checks that 1 K is compact metrizable of dimension zero. Note that K (n) = {n} × K(n)0 for n ≥ 1, is clopen in K , and the sequence K (n,p) converges, for fixed p, to the point (0, xp0 ) of K 0 = {0} × K0 . We now define H ⊆ K by (n, x) ∈ H ⇐⇒ (n = 0 and x ∈ H0 ) ∨ (n = 0 and x ∈ H(n)0 ). Clearly H ∈ S1 ((Γn ))(K) as witnessed by the sequence of open sets Un = (p)0 =n K (p) , n ≥ 1. ˘ n (Kn ). We claim that H ∈ As (Kn , Hn ) is a Hurewicz-test for Γn , Hn ∈ / Γ / ˘ ): For let V ⊆ K be the open set of all points z ∈ K admitting a Γ(K neighborhood Vz with H˘ ∩ Vz ∈ n≥1 Γn . By the density of the sequence (xp0 ) in K0 and the convergence property of the K (n,p) , any neighborhood of a z ∈ K (0) contains sets K (n) for arbitrary large n, hence V is disjoint from K (0) . So if H˘ were in Γ(K), one would get H˘ ∩ K (0) in Γ0 (K (0) ), hence H0 in ˘ 0 (K0 ) a contradiction. Γ This proves that if E is Suslin and B ⊆ E satisfies ∃f 1-1 continuous: ˘ ˘ ). K → E with f −1 (B) = H , the set B is not in Γ(E)—else H would be in Γ(K ˘ It remains to prove the converse, for B Borel in E. So, we assume B ∈ / Γ(E). ˘ Let for n ≥ 1, Vn = {x ∈ E : ∃Vx open x ∈ Vx and Vx ∩ B ∈ Γn (Vx )} and let F = E \ n≥1 Vn . By Lemma 5.6, if B˘ ∩ F ∈ Γ0 (F ), one gets B˘ ∈ Γ(E), a contradiction. So we can find a 1-1 continuous f0 : K0 → F with f0−1 (B) = H0 . We now inductively construct a sequence fn , n ≥ 1 of 1-1 continuous functions, with fn : K(n)0 → E, such that (i) the images f0 (K0 ), fn (K(n)0 ) are all pairwise disjoint (ii) fn−1 (B) = H(n)0 , for n ≥ 1
THE STRENGTH OF BOREL WADGE DETERMINACY 0 (iii) If d is a distance on E, d (f0 (x(n) ), fn (K(n)0 )) ≤ 0
101
1 . (n)1 +1
If we can do this, we are clearly done, for f : K → E defined by f(n, x) = fn (x) is 1-1 by (i), continuous by (iii) and satisfies f −1 (B) = H by (ii). Suppose the fn have been constructed for n < p. Let y = f0 (xp0 ) ∈ f0 (K0 ). As for n < p, f0 (K0 ) is disjoint from fn (K(n)0 ), we can find an open U in E with y ∈ U , U ∩ fn (K(n)0 ) = ∅ for n < p and U ⊆ {z : 1 d (y, z) ≤ p + 1}. We claim that B˘ ∩ (U \ f0 (K0 )) ∈ / Γp0 (E): if not, 1 ˘ by admissibility of Γn , the set B ∩ U would be in n≥1 Γn , contradicting the fact that U ⊂ n≥1 Vn (as y ∈ U \ n≥1 Vn ). But then we can find a 1-1 continuous fp : Kp0 → U \ f0 (K0 ) with fp−1 (B) = Hp0 , and this fp clearly works. (Theorem 5.5) REFERENCES
Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alain Louveau [Lou83] Some results in the Wadge hierarchy of Borel sets, this volume, originally published in Kechris et al. [Cabal iii], pp. 28–55. Alain Louveau and Jean Saint-Raymond [LSR87] Borel classes and closed games: Wadge-type and Hurewicz-type results, Transactions of the American Mathematical Society, vol. 304 (1987), no. 2, pp. 431– 467. Donald A. Martin [Mar73] The Wadge degrees are wellordered, unpublished, 1973. Jean Saint-Raymond [SR76] Fonctions bor´eliennes sur un quotient, Bulletin des Sciences Math´ematiques, vol. 100 (1976), pp. 141–147. John R. Steel [Ste80] Analytic sets and Borel isomorphisms, Fundamenta Mathematicae, vol. 108 (1980), no. 2, pp. 83–88. William W. Wadge [Wad84] Reducibility and determinateness on the Baire space, Ph.D. thesis, University of California, Berkeley, 1984. EQUIPE D’ANALYSE FONCTIONNELLE ´ INSTITUT DE MATHEMATIQUES DE JUSSIEU UNIVERSITE´ PARIS VI 4, PLACE JUSSIEU 75230 PARIS, CEDEX 05 FRANCE
E-mail: [email protected] E-mail: [email protected]
CLOSURE PROPERTIES OF POINTCLASSES
JOHN R. STEEL
We work in ZF+AD+DC throughout this paper. Our aim is to show that certain closure and structural properties of a nonselfdual pointclass Γ follow from closure properties of the corresponding Δ together with the regularity of the Wadge ordinal of Δ. Let 3E be the type 3 object embodying quantification over the reals, and o(3E) the least ordinal not the order type of a prewellorder of the reals recursive in 3E. Our results imply that o(3E) is the least regular limit point in the sequence of Suslin cardinals defined in [Kec81B]. The methods and most of the results of the paper fall squarely within the province of “Wadge degrees”, which might more informatively be titled “the general theory of arbitrary pointclasses.” (Sections 1 to 3 of [Van78B] contain the necessary background material.) Surprisingly, AD is powerful enough to yield nontrivial theorems in this generality. Now, when working in such generality, it is natural to ask: which pointclasses (identified perhaps by means of Wadge ordinals) have the closure and structural properties which make them amenable to the standard techniques of descriptive set theory. Our results bear on this question. The author wishes to acknowledge the contribution of A. S. Kechris to this work. In a sense, the work was commissioned by him. Some notation and terminology: We let R be , the Baire space, and call its elements reals. If ≥ 1, then the product space k ×() is homeomorphic to , and we shall identify the two. For A, B ⊆ we say A is Wadge reducible to B, and write A ≤W B, iff ∃f : → (f is continuous ∧ A = f −1 (B)). The partial order ≤W is wellfounded; |A|W is the ordinal rank of A in ≤W . A pointclass is a class of subsets of closed downward under ≤W . The dual of ˘ is {−A : A ∈ Γ}. Here, as later, complements are a pointclass Γ, denoted Γ, ˘ then we set Δ = Γ ∩ Γ. ˘ taken relative to . If Γ is nonselfdual, i.e., Γ = Γ, If Γ is any pointclass, we set o(Γ) = sup{|A|W : A ∈ Γ}. If Γ is nonselfdual and o(Γ) a limit ordinal, then of course o(Γ) = o(Δ). We write Sep(Γ), Red(Γ), or PWO(Γ) to mean that Γ has, respectively, the separation, reduction, or prewellordering property. The closure of Γ under existential real quantification is given by ∃R Γ = {A : ∃B ∈ Γ∀x(x ∈ A ⇔ ∃y(x, y) ∈ B)}. Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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103
˘ Similarly, the class of wellordered unions of We let ∀R Γ be the dual of ∃R Γ. length α of Γ sets is Γ= A : ∀ < α(A ∈ Γ)
α
<α
˘ and α Γ is the dual of α Γ. The selfdual pointclasses Δ which we consider will often satisfy ∃R Δ ⊆ Δ. In this case, Lemma 2.3.1 of [KSS81] states o(Δ) = sup{rank(≺) : ≺ is a wellfounded relation in Δ} = sup{rank(≺) : ≺ is a prewellorder in Δ}. §1. Consequences of the separation property. The key to the transfer of closure properties from Δ to Γ is the separation property. We begin with some simple results in this vein. Notice that the hypothesis Sep(Γ) of Theorem 1.1 ˘ since by [Ste81B] and [Van78B], exactly serves only to distinguish Γ from Γ, ˘ holds. one of Sep(Γ) and Sep(Γ) Theorem 1.1. Let Γ be nonselfdual, and suppose Sep(Γ). Then (a) 2 Δ ⊆ Δ ⇒ 2 Γ ⊆ Γ and Δ ⊆ Δ ⇒ Γ ⊆ Γ; (b) ∃R Δ ⊆ Δ ⇒ ∃R Γ ⊆ Γ;
(c) (∃R Δ ⊆ Δ ∧ α < cf(o(Δ))) ⇒ ( α Δ ⊆ Δ ∧ α Γ ⊆ Γ). Proof. (a) Suppose 2 Δ ⊆ Δ and 2 Γ Γ. Since 2 Γ is a pointclass, i.e., closed downward under ≤W , Wadge’s lemma implies Γ˘ ⊆ 2 Γ. Let A ∈ ˘ − Γ, and A = B ∪ C where B, C ∈ Γ. By Sep(Γ) we have D, E ∈ Δ so that Γ B ⊆D⊆A and C ⊆ E ⊆ A.
But then A = D ∪ E, so A ∈ Δ, a contradiction. The proof that Δ ⊆ Δ ⇒ Γ ⊆ Γ is the same. (b) It is enough to show that if A and B are disjoint sets in ∃R Γ, then A is separable from B by a set in Δ. We use the idea of Addison’s proof of Sep(Σ13 ) for this. Let A(x) ⇐⇒ ∃yP(x, y), B(x) ⇐⇒ ∃yQ(x, y), where P, Q ∈ Γ. Define P (x, y, z) ⇐⇒ P(x, y), Q (x, y, z) ⇐⇒ Q(x, z),
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and by Sep(Γ) let D ∈ Δ and P ⊆ D ⊆ −Q . Define C (x) ⇐⇒ ∃y∀zD(x, y, z). R
Then C ∈ Δ since ∃ Δ ⊆ Δ. It is easy to check that A ⊆ C ⊆ −B. (c)We extend the proof of (a). Suppose α < cf(o(Δ)) and ∃R Δ ⊆ Δ, ˘ ⊆ ˘ but α Γ Γ. Then Γ α Γ by Wadge’s lemma. Let A ∈ Γ − Γ, and A = <α A where each A ∈ Γ. Since Sep(Γ) and α < cf(o(Δ)), we can find a set B ∈ Δ so that < α ⇒ ∃C ≤W B(A ⊆ C ⊆ A). Let ϕ : R α be a Δ-norm of length α. By the Coding Lemma of Moschovakis [Mos70] there is a relation R in Δ so that ∀x∃yR(x, y) and R(x, y)∧ϕ(x) = ⇒ A ⊆ By ⊆ A. (Here By is the set ≤W B via the strategy y.) But then z ∈ A ⇐⇒ ∃x∃y(R(x, y) ∧ z ∈ By ), so A ∈ Δ, a contradiction. (Theorem 1.1) The proof that α Δ ⊆ Δ is the same. Part (b) is due to Kechris, and part (a) to Kechris and the author independently. Notice that the proof of (c) gives slightly more: if ∃R Δ ⊆ Δ, α < cf(o(Δ)), and A : < α is any sequence of sets each of which is Δseparable from a fixed set A, then <α A is Δ-separable from A. This fact will be important in the proof of Theorem 2.1. R Theorem 1.1 leaves us the question: given that Sep(Γ) and ∃ Δ ⊆ Δ, must 2 Γ ⊆ Γ? The next theorem provides a class of examples showing the extent to which 2 Γ ⊆ Γ can in fact fail. The theorem results from analysis of the example of a Type II hierarchy given in [Kec77B], p. 260. If A ⊆ R and ϕ : A , we say ϕ is Γ-bounded just in case whenever B ⊆ A and B ∈ Γ there is a < so that ϕ B ⊆ . Theorem 1.2. Suppose Sep(Γ) and 2 Δ ⊆ Δ. Let A ∈ Δ and ϕ : A be / Γ. Σ11 bounded, where = cf(o(Δ)). Then for some B ∈ Γ, A ∩ B ∈ Proof. Let { α : α < } be cofinal in o(Δ). Let W ⊆ R2 be a universal set in Γ. Consider the Solovay game: Player I x Player II y, z Player II wins iff x ∈ A ⇒ (Wy = −Wz ∧ |Wy |W ≥ ϕ(x) ). Since ϕ is Σ11 bounded, player II must have a winning strategy . Let R(x, y) ⇐⇒ x ∈ A ∧ y ∈ / W (x)1 . ˘ is closed under intersection by Theorem 1.1(a), R ∈ Γ. ˘ But {|Rx |W : Since Γ x ∈ A} is unbounded in o(Δ), so R ∈ / Δ, and thus R ∈ / Γ. On the other hand, R(x, y) ⇐⇒ x ∈ A ∧ y ∈ W (x)0 ,
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so that R = A ∩ B for some B ∈ Γ. R 1 Theorem 1.2 implies, for example, that if ∃ Δ ⊆ Δ, Sep(Γ), and cf(o(Δ)) = n , then Γ is not closed under intersection with Π1n sets. [Let S ⊆ R3 be universal 1 1 Σn , and let A = {x : Sx is wellfounded}. Then A is Πn , and the map ϕ : A 1n , where ϕ(x) = rank of Sx , is Σ11 -bounded.] On the other hand, Theorem2.1 to follow implies that under these hypotheses on Γ and Δ, Γ is 1 closed under intersections with Σn sets. On a grosser scale, one can modify the proof of Theorem 1.2 slightly (replacing the Solovay game by the Coding Lemma) to show that if Sep(Γ), ∃R Δ ⊆ Δ, and o(Δ) is singular, then Γ is not closed under intersections with Δ sets. Theorem 1.2 implies that the hypothesis “α < cf(o(Δ))” in Theorem 1.1(c) is necessary. For consider any Γ such that Sep(Γ), ∃R Δ ⊆ Δ, and cf(o(Δ)) = a Π11 norm on a complete Π11 set. Define R as in 1 . Let ϕ : A 1 be Theorem 1.2. Then R = α<1 Rα where Rα (x, y) ⇐⇒ ϕ(x) ≤ α ∧ y ∈ W (x)1 . But each Rα is in Δ by Theorem 2.1. Kechris and Martin have located the pointclass Γ such that o(Δ) = 2 with the aid of Theorem 1.2. Namely, let Γ be the class of -Π11 sets, that is, sets of the form A2n − A2n+1 A= n<
where An : n < is a decreasing sequence of Π11 sets. Then Γ is nonselfdual, and both Γ and Γ˘ are closed under intersections with Π11 sets. By Theorem 1.2 associated to Wadge we have o(Δ) ≥ 2 . By analyzing the ordinal games games involving sets in Δ, Martin showed o(Δ) ≤ 2 . Thus o(Δ) = 2 . It is unpleasant to have a natural ordinal like 2 assigned to an unnatural class like -Π11 . Solovay has shown that we get a more natural assignment if we replace ≤W by the somewhat coarser ≤ , where B ≤ A ⇐⇒ ∃An : n < ∀n(An ≤W A) ∧ B ≤W An . n<
Π11 , 2
A(Π11 ), 3
to to A(A(Π11 )), etc. The ordinal 1 is assigned now to (Here “A” denotes Suslin’s operation A.) The ordering ≤ behaves much like the order ≤m of jump operators defined and studied in [Ste82]. For example, the wellfoundedness of ≤ can be proved by a direct diagonal argument like that of Lemma 3 of that paper. §2. Applications of the Martin-Monk method. We return to our closure questions. The limitations established by Theorem 1.2 clearly rely on the singularity of o(Δ). Suppose then that Sep(Γ), ∃R Δ ⊆ Δ, and o(Δ) is regular;
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does it follow that 2 Γ ⊆ Γ? We believe so, but at present have a proof only for the case that every set in Δ is κ-Suslin for some κ < o(Δ). This partial result will be enough for our characterization of o(3E), since by [Kec81B], every set A inductive over R is κ-Suslin for some κ ≤ |A|W . The key to our partial result is the following theorem. Theorem 2.1. Let Γ be nonselfdual and suppose ∃R Δ ⊆ Δ. Let A be κ-Suslin, where κ < cf(o(Δ)). Then for any B ∈ Γ, A ∩ B ∈ Γ. ˘ holds, so assume Proof. This follows from Theorem 1.1(a) if Sep(Γ) Sep(Γ). Let A, B, and κ be as in the hypotheses, and suppose for a contradiction that A ∩ B ∈ / Γ. Let be a winning strategy for player I in the Wadge game GW (A ∩ B, B) (cf. [Van78B]). Thus whenever (x) ∈ A, we have x ∈ B ⇐⇒ (x) ∈ / B. We shall use the fact that flips membership in B this way to get a contradiction like that in Martin’s proof that ≤W is wellfounded. Specifically, we define a sequence n : n < of winning strategies for player I in GW (A ∩ B, B). Let be the copying strategy for player II, i.e., let ∀x((x) = x). For any x ∈ 2, define
n , if x(n) = 0 n = , if x(n) = 1. Consider the diagram of games ···
···
2 1 0 x2 (0) x1 (0) x0 (0) . x1 (1) x0 (1) . . x0 (2) .. .. .. . . . x1 x0 x2
The rule here is: xn = n (xn+1 ). If x(n) = 0 for infinitely many n, then because each n is a strategy for player I, there is a unique such sequence xn : n < . We shall define the n ’s so that for any x ∈ 2, if x(n) = 0 for infinitely many n and xn : n < is derived from x in this way, then xn ∈ A for all n. Suppose we have done this; then the standard Martin argument leads to a contradiction. For let I = {x ∈ 2 : x(n) = 0 for infinitely many n}. Define M = {x ∈ I : x0 ∈ B}. Since M has the Baire property we have a basic interval [s] determined by some s ∈ 2< on which M is either meager or comeager. Pick i ∈ / dom(s),
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CLOSURE PROPERTIES OF POINTCLASSES
and let
T (x)(k) =
x(k) if i = k 1 − x(k) if i = k.
Then T is a homeomorphism and T [s] = [s]. If x ∈ I , then T (x)k = xk for k > i, and / B for k ≤ i. T (x)k ∈ B iff xk ∈
Thus T (M ∩ I ∩ [s]) = −M ∩ I ∩ [s]. Since I is comeager, this contradicts our choice of s. We now define the n ’s by induction on n. Let T be a tree on × κ such that A = p([T ]) = {x ∈ : ∃f ∈ κ x, f) ∈ [T ])}. (Here [T] is the set of infinite branches of T.) As we define n we shall associate to any n : i ≤ n such that n = n and ∀i < n(i = i or i = ), and any i ≤ n such that i = i , a sequence of ordinals 0 , . . . , n = ,i . We arrange that for any z ∈ , if the partial diagram n .. . z
n−1 .. . xn
.. . xn−1
··· ···
0 .. . x1
.. . x0
is filled in as before (i.e., xk = k (xk+1 )), but setting xn+1 = z, then (i) (xi n + 1, 0 , . . . , n ) ∈ T , and (ii) z ∈ / B ⇒ ∃f(f n + 1 = 0 , . . . , n ∧ (xi , f) ∈ [T ]). Moreover these sequences of ordinals cohere in the natural way, that is, (iii) k+1,i = ,i k + 1, for i ≤ k < n. satisfying It will be enough to define i : i < with associated ’s (i)–(iii). For then suppose x ∈ 2 and x(n) = 0 for infinitely many n. Let i = i if x(i) = 0, and i = otherwise. Define xn by: xn = n (xn+1 ). For x(n) = 0, let
0 ,...,k ,n . f= x(k)=0
Then f ∈ κ by (iii), and by (i), (xn k, f k) ∈ T for all k. Thus xn ∈ A. If x(n) = 0, then xn = xi for some i such that x(i) = 0. Thus xn ∈ A for all n, and we are done. We now define n . Suppose that i is defined for i < n, together with associates satisfying (i)–(iii) above. We define n in 2n steps, one for each i : i < n with i = i or i = for all i < n. After step we have a Δinseparable pair C ⊆ −B and D ⊆ B with D ∈ Γ. We will have C+1 ⊆ C and D+1 ⊆ D for < 2n .
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Step 0. For each ∈ κ n set
A = {x : ∃f(f (n + 1) = ∧ (x, f) ∈ [T ])}.
A . Since (−B) ⊆ A and κ < cf(o(Δ)), the remark −B ∩ −1 (A ) is Δimmediately after Theorem 1.1 implies that for some ,
and let inseparable from B. Fix such a , Thus A =
n
∈κ
C0 = −B ∩ −1 (A ) and ∈ T }. D0 = {x ∈ B : (x) n + 1, ) D0 ∈ Γ since Γ is closed under intersections with clopen sets, and one easily checks that D0 is Δ-inseparable from C0 . Finally, at this step we set ,n = for all = i : i ≤ n such that i = i or i = for i < n, and n = n . (Or more precisely, we commit ourselves to doing so once we have defined n .) Step k + 1. We have (Ck , Dk ) from the last step, and we are considering i : i < n. Let i < n be largest such that i = i ; if no such i exists set Ck+1 = Ck , Dk+1 = Dk , and go to step k+2 without defining any new associates. For each j < n such that j = j , let j = : ≤i,j . Besides defining Ck+1 and Dk+1 , we want to extend these associates. For each z ∈ consider the diagram
.. . z
n−1 .. . zn
.. .
··· ···
··· .. .
0
.. . zi+1
zi
···
z0
filled in as before by setting zn+1 = z. For j < n so that j = j , define A j = {x : ∃f(f (i + 1) = j ∧ (x, f) ∈ [T ]}. So (ii) of our inductive hypothesis on : ≤ i and its associates says: zi+1 ∈ / B ⇒ zj ∈ A j . Now notice that if z ∈ Dk (so z ∈ B) and (z) ∈ A, then zn = zi+1 ∈ / B, and thus zj ∈ A j for all j ≤ i. Define / A j )}. X = {z : (z) ∈ A ∧ ∃j ≤ i(zj ∈ Then X ∈ Δ, and Dk ∩ X = ∅. Since (Ck , Dk ) is Δ-inseparable, (Ck − X, Dk ) must be Δ-inseparable. Notice that for z ∈ Ck − X we have z ∈ / B, so (z) ∈ A, and thus zj ∈ A j for all j ≤ i with j = j . This enables us to use the argument of step 0 to successively thin down Ck − X and Dk , once for each j ≤ i so that j = j ,
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retaining at each step an inseparable pair (Cj , Dj ) with Dj ∈ Γ. At the step for j we also define the associate i : i≤n,j extending : ≤i,j so that (i) z ∈ Dj ⇒ (zj (n + 1), i : i≤n,j ) ∈ T , (ii) z ∈ Cj ⇒ ∃f(f (n + 1) = i : i≤n,j ∧ (zj , f) ∈ [T ]. We define (Ck+1 , Dk+1 ) to be the last pair in this process, and go to step k + 2. Now let (C, D) = (C2n , D2n ) be the pair we have on completion of the last step. Consider the game in which player I plays y, player II plays z, and player II wins iff y∈ /B ⇒z∈C and y ∈ B ⇒ z ∈ D. Then player I has no winning strategy in this game. For if s is such a strategy, then C ⊆ s −1 (B) and s −1 (B) ∩ D = ∅. Since s −1 (B) and D are disjoint Γ sets, they can be separated by a Δ set. But such a Δ set separates C and D, a contradiction. Fix a winning strategy s for player II, and let n = ◦ s. Given any j : j ≤ n with n = n , and given any j < n with j = j , the induction hypotheses (i) and (ii) for and ,j follow at once from (i) and (ii) for Cj and Dj at the step at which i : i < n was considered, and the fact that s (−B) ⊆ C ⊆ Cj and s (B) ⊆ D ⊆ Dj . The construction of n : n < , and hence the proof of the theorem, is complete. (Theorem 2.1) For Γ such that Sep(Γ) and ∃R Δ ⊆ Δ, it should be possible to specify exactly, as a function of cf(o(Δ)), those pointclasses Γ so that Γ is closed under intersections with Γ sets. Theorems 1.2 and 2.1 do this when cf(o(Δ)) = 1n ; such that Γ is closed under intersections with Σ1n sets, but not under intersections with Π1n sets. By combining Theorems 1.2 and 2.1 we obtain the following curious fact: let A be κ-Suslin and let ϕ : A be Σ11 -bounded. Then cf( ) ≤ κ. [Proof. ˘ where Let Δ be such that ∃R Δ ⊆ Δ and cf(o(Δ)) = cf( ). Let Δ = Γ ∩ Γ, Sep(Γ) holds. By Theorem 1.2, Γ is not closed under intersections with A. By Theorem 2.1 then, cf(o(Δ)) ≤ κ.] This fact is easy to prove for natural , e.g., = 1n , but we see no proof for arbitrary which does not use Theorems 1.2 and 2.1.
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The basic method in the proof of Theorems 2.1 is due to D. Martin and L. Monk; as we mentioned, Martin used it to show ≤W is wellfounded. At present this method and variants on Wadge’s lemma seem to be the only tools in pure Wadge theory. We need one further preliminary closure result. Again, the Martin-Monk method is the key. Theorem 2.2. Suppose ¬Sep(Γ) and 2 Γ ⊆ Γ. Then Γ ⊆ Γ. Proof. Let (A0 , A1 ) be a Δ-inseparable pair of Γ sets. Suppose that Γ ˘ ⊆ Γ, so that Γ lemma. Since 2 Γ ⊆ Γ, we have −(A0 ∩A1 ) ∈ Γ by Wadge’s ˘ and hence −(A0 ∪ A1 ) = Γ, n Cn for some sequence Cn : n < of Γ sets. Now (A0 ∪ Cn , A1 ) is a disjoint pair of Γ sets, and so the lemma of [Ste81B] gives Lipschitz continuous maps fn , n < , such that f2n (A0 ∪ Cn ) ⊆ A0 ∧ f2n (A1 ) ⊆ A1 , and f2n+1 (A0 ∪ Cn ) ⊆ A1 ∧ f2n+1 (A1 ) ⊆ A0 . We proceed to the usual contradiction. For any x ∈ let xn : n < be the unique sequence such that xn = fn (xn+1 ). Suppose that {x : x0 ∈ Cn } is nonmeager; say comeager on the interval determined by s ∈ < . Notice that x0 ∈ Cn ⇒ (s 2nx)0 ∈ A0 ∪ A1 , / Cn , a contradiction. On the other so that for nonmeager many y ⊇ s, y0 ∈ hand, suppose {x : x0 ∈ Ai } is nonmeager; say comeager on the interval determined by s. Notice that x0 ∈ Ai ⇒ (s 1x)0 ∈ A1−i , so that for nonmeager many y ⊇ s, y0 ∈ A1−i , a contradiction.
The hypothesis ¬Sep(Γ) in Theorem 2.2 cannot be omitted, as witnessed by the case Γ = Π0α . Theorem 1.2 shows that hypothesis 2 Γ ⊆ Γ cannot be omitted, even if we assume strong closure properties of Δ. However, the proof of Theorem 2.2 can be modified to show that if Δ ⊆ Δ and An : n < is any increasing sequence of Γ sets, then 2 An ∈ Γ. ˘ [Van78B] shows that for any nonselfdual Γ, either ¬Sep(Γ)or ¬Sep(Γ). ˘ then either ˘ ⊆ Γ. ˘ It Thus if both 2 Γ ⊆ Γ and 2 Γ˘ ⊆ Γ, Γ Γ ⊆ Γ or ˘ then either A(Γ) ⊆ Γ seems quite likely that if both Γ ⊆ Γ and Γ˘ ⊆ Γ ˘ ⊆ Γ, ˘ where “A” denotes Suslin’s operation A. More vaguely, one or A(Γ) might guess that there is always an asymmetry between the closure properties ˘ of Γ and those of Γ.
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From Theorem 2.2 we obtain theorem for classes closed a prewellordering under one but not both of and . The theorem is analogous to those of [KSS81]. Corollary 2.3. Let Γ be nonselfdual, Γ ⊆ Γ and 2 Γ ⊆ Γ, but Γ Γ. Then PWO(Γ). ˘ holds, as otherwise ˘ Proof. Corollary 2.3 By Theorem 2.2, Sep(Γ) Γ ⊆ ˘Γ. But then, by Theorem 1.1(a), ˘ ˘ Δ Δ, as otherwise Γ ⊆ Γ. Since Γ ⊆ Γ, we have Δ = Γ. For A ∈ Γ, let A = B where each Bn ∈ Δ, n n and set for x ∈ A Since
ϕ(x) = least n such that x ∈ Bn .
2
Δ ⊆ Δ, ϕ is a Γ norm.
The corollary generalizes the fact that Σ0α has the prewellordering property for α < 1 . §3. Bounded unions and prewellorderings. Let Γ be a pointclass. We say that a union α< Aα is Γ-bounded iff the associated norm ϕ(x) = α[x ∈ Aα ] is Γ-bounded. Theorem 3.1. Let Δ be selfdual, ∃R Δ ⊆ Δ, and cf(o(Δ)) > . Then the following are equivalent: ˘ (a) Δ = Γ ∩ Γ for some nonselfdual Γ such that PWO(Γ); (b) o(Δ) Δ Δ. Proof. (a) ⇒ (b) is clear. Suppose then that o(Δ) Δ Δ, and let ϑ ≤ o(Δ) be least such that ϑ Δ Δ. Since cf(o(Δ)) > , Theorem 3.1 of [Van78B] ˘ = Δ. Assume w.l.o.g. Sep(Γ). ˘ It follows gives a nonselfdual Γ such that Γ ∩ Γ ˘ ˘ that Γ = ϑ Δ, as otherwise PWO(Γ). Thus Γ ⊆ ϑ Δ. Define now Aα : ∀α(Aα ∈ Δ) ∧ Aα is Σ11 -bounded . Γ∗ = α<ϑ α<ϑ Claim 3.2. Γ ⊆ Γ∗ . ˘ and let S ⊆ R2 be a universal Σ1 set. Let Proof. Let A ∈ Γ − Γ, 1 C = {x : Sx ⊆ A} = {x : ∀y(y ∈ / Sx or y ∈ A)}. Since cf(o(Δ)) > and every Σ11 set is -Suslin, Theorems 2.1 and 1.1(b) that C ∈ Γ without using Theorem 2.1. imply that C ∈ Γ. (One can show For C can be defined in the form ”∀(open ∨ Γ), ” and it is easy to see that if Γ is nonselfdual and contains the Boolean algebra generated by the open sets,
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then Γ is closed under unions with open sets.) Thus C = Cα ∈ Δ. Let
α<ϑ
Cα where each
Aα = {y : ∃x(x ∈ Cα ∧ y ∈ Sx )}. Then each Aα ∈ Δ, A = α<ϑ Aα , and α<ϑ Aα is Σ11 -bounded by construc tion. (Claim 3.2) Claim 3.3. Γ ⊆ Γ∗ . Γ∗ = Γ. Proof. It is easy to check, using boundedness, that ∀R Γ∗ ⊆ Γ∗ . By Wadge then, if Γ∗ Γ, then ∀R Γ ⊆ Γ∗ . It is enough for a contradiction to show that ∗ R Γ ⊆ ∃ Γ. In fact, we show ϑ Δ ⊆ ∃R Γ. The proof is standard, granted our first claim. (Claim 3.3) Let Aα : α < ϑ be a sequence of Δ sets, and let ϕ : C ϑ be a Σ11 -bounded ⊆ R2 be norm, where C ∈ Γ. Such a ϕ exists by the first claim. Let W a universal set in Γ. Consider the game: Player I plays x, player II plays y. Player II wins iff x ∈ C ⇒ ∃ ϕ(x) ≤ ∧ Aα = Wy . α≤
Σ11 -bounded,
Since ϕ is player I has no winning strategy. Let be a winning strategy forplayer II. Then Aα ⇐⇒ ∃y(y ∈ C ∧ x ∈ W (y) ). x∈ Since
α<ϑ
2
Γ ⊆ Γ, we have
α<ϑ
Aα ∈ ∃R Γ.
Claim 3.4. PWO(Γ∗ ). Proof. Let A = α<ϑ Aα , where each Aα is in Δ and the union is Σ11 bounded and increasing. Define for x ∈ A ϕ(x) = α[x ∈ Aα ]. ∗
To see that ϕ is a Γ norm, notice, e.g., that if S is Σ11 and S ⊆ {(x, y) : x <ϕ y}, then T = {x : ∃y x, y) ∈ S )} is Σ11 , and T ⊆A. Thus T ⊆ Aα for some α < ϑ, and so S ⊆ {(x, y) : x ∈ Aα ∧ x <ϕ y} = Iα . Similarly, ≤ϕ is in Γ∗ . (Claim 3.4) The three claims yield the theorem. (Theorem 3.1) We use “IND” to denote the class of sets definable over R by positive elemen from parameters in R. tary induction Thus <ϕ is the Σ11 -bounded union of the Iα ’s.
Theorem 3.5. Let Γ be nonselfdual and Γ ⊆ IND. Suppose that ∃R Δ ⊆ Δ and o(Δ) is regular. Then Γ ⊆ Γ.
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˘ Proof. This follows from Theorem 1.1 if Sep(Γ) holds, so assume Sep(Γ). Let Γ∗ = Aα : ∀α(Aα ∈ Δ) ∧ Aα is Δ-bounded . α
α
Clearly every set in Γ∗ is a Σ11 bounded union of Δ sets, and so the proof of Theorem 3.1 implies that Γ∗⊆ Γ. Claim 3.6. Γ∗ = Γ.
˘ and let A = Proof. Let A ∈Γ − Γ, α<ϑ Aα where each Aα ∈ Δ and ϑ is least such that ϑ Δ Δ. We may assume the Aα ’s are increasing. The Coding Lemma implies that |Aα |W : α < ϑ is cofinal in o(Δ). For α < ϑ, let Cα = {(x, y) : y ∈ Aα+1 − Aα ∧ x codes a continuous function fx such that fx−1 (Aα ) ⊆ A}. Now Cα is defined in the form ”Δ ∧ ∀z(Δ ⇒ Γ).” Every set in Δ is κ-Suslin for some κ < o(Δ) by Corollary 3.5 of Kechris [Kec81B]. Thus our Theorems 1.1 and 2.1 imply that Cα ∈ Γ. The proof of Theorem 3.1 now shows that if C = α<ϑ Cα , then C ∈ ∃R Γ. Notice that ∃R ( ϑ Δ) = ϑ ∃R Δ = ϑ Δ. So since Γ ⊆ ϑ Δ, ∃R Γ ⊆ ϑ Δ, and we may write C = Dα α<ϑ
where each Dα ∈ Δ, and the union is increasing. Let z ∈ Bα ⇐⇒ ∃(x, y) ∈ Dα ∃ ≤ α(y ∈ A+1 − A ∧ fx (z) ∈ A ). is in Δ by Theorem 1.1(c) and thefact that ∃R Δ ⊆ Δ. It is easy Then each Bα to check that α<ϑ Bα = A. Finally, the union α<ϑ Bα is Δ-bounded, since any Δ set is of the form fx−1 (A ) for some < ϑ and some x. This proves the claim. (Claim 3.6) It is enough now to show 2 Γ ⊆ Γ; by Theorem 2.2 we then have Γ ⊆ Γ. ˘ Let A, B ∈ Γ towards showing A ∪ B ∈ Γ. Since Sep(Γ), we haveRed(Γ), and so we may assume A ∩ B = ∅. Let A = α<ϑ Aα and B = α<ϑ Bα , where the unions are Δ bounded and increasing, and each Aα and Bα is in Δ. It is enough toshow that the union α<ϑ (Aα ∪ Bα ) is Δ-bounded. So let C ∈ Δ and C ⊆ α<ϑ (Aα ∪ Bα ). Then by Theorem 1.1(a), C ∩ A ∈ Γ. On ˘ by Theorem 2.1 and the other hand, C ∩ A = C ∩ (−B), and C ∩ (−B) ∈ Γ the fact that C is κ-Suslin for some κ < o(Δ). Thus C ∩ A ∈ Δ, and hence C ∩ A ⊆ Aα for some α < ϑ. Similarly, C ∩ B ⊆ B for some < ϑ. But then C ⊆ A ∪ B , where = max(α, ), and we are done. (Theorem 3.5)
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The hypothesis that Γ ⊆ IND in Theorem 3.5 was only used to conclude, via Theorem 2.1 and Corollary 3.5 of [Kec81B], that Γ is closed under intersection with Δ sets.Thus the conclusion of Theorem 3.5 holds for arbitrary Γ such that ∃R Δ ⊆ Δ, o(Δ) Δ Δ, and o(Δ) is regular, and Γ is closed under intersections with Δ sets. One can also show that Theorem 2.1 and Corollary 3.5 of [Kec81B] apply for Γ a bit beyond IND. For example, one can weaken the hypothesis “Γ ⊆ IND” of Theorem 3.5 to “every Γ set is inductive in the complete coinductive set of reals.” We now define C = {o(Δ) : Δ is selfdual ∧ ∃R Δ ⊆ Δ}. Clearly C is cub in ϑ. Theorem 3.1 of [Kec81B] implies that for ≤ o(IND) such that = , the th element of C is the th Suslin cardinal. Thus our next theorem is actually the characterization of o(3E) promised in the introduction. Theorem 3.7. Let o(Δ) be the least regular limit cardinal in C . Then ˘ where Γ is the boldface 2-envelope of 3E (i.e., the class of sets of Δ = Γ ∩ Γ, reals semirecursive in 3E and a real). Thus o(Δ) = o(3E). Proof. Clearly C is cub in o(IND), and since o(IND) is regular, we have o(Δ) ≤ o(IND) and Δ ⊆ IND. (Actually, Kechris has shown that o(IND) is Mahlo, so o(Δ) < o(IND).) By Theorem 3.5 and its proof, we have ˘ where Γ is the class of Δ-bounded unions of Δ sets of length o(Δ). Δ = Γ ∩ Γ, Thus ∀R Γ ⊆ Γ, Γ ⊆ Γ, and by Theorem 3.1, PWO(Γ). In order to show that Γ contains the 2-envelope of 3E it suffices to show that Δ is “uniformly closed under ∃R ,” in the following sense. Let Q(x, y) and R(x, y) be disjoint relations in Γ. Define S(x) ⇐⇒ ∀y(R(x, y) ∨ Q(x, y)) ∧ ∃yQ(x, y) and T (x) ⇐⇒ ∀y(R(x, y) ∨ Q(x, y)) ∧ ∀yR(x, y). Then we must show that S and T are in Γ. To see this, let Qα and R = Rα Q= α
α
be representations of Q and R as increasing Δ-bounded unions of Δ sets. Define Sα (x) ⇐⇒ ∀y(Rα (x, y) ∨ Qα (x, y)) ∧ ∃yQα (x, y)
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and Tα (x) ⇐⇒ ∀y(Rα (x, y) ∨ Qα (x, y)) ∧ ∀yRα (x, y).
Clearly Sα ⊆ S and Tα ⊆ T . We show simultaneously that S ⊆ α Sα and that the union is Δ-bounded. For let D ⊆ S and D ∈ Δ. Let A = (D × ) ∩ Q, and let B = (D ∩ ) ∩ R. Then A and B are disjoint Γ sets, and complementary on D × . Thus A and B are in Δ. Let α be such that A ⊆ Qα and B ⊆Rα . Then D ⊆ Sα , and we are done. An identical argument shows that T ⊆ α Tα and that this union is Δ-bounded. Thus S and T are in Γ, as desired. It is well known that the class of sets of reals is recursive in 3E and a real has the closure properties we assumed of Δ, that is, it is closed under ∃R and its ordinal is regular. Since Δ was minimal with these properties, Δ is contained (Theorem 3.7) in this class. Thus Γ is contained in the 2-envelope of 3E. Theorem 3.7 implies that o(3E) is not Mahlo. It gives some evidence for the natural conjecture, due perhaps to Moschovakis, that o(3E) is the least regular limit cardinal. Proof of this conjecture awaits further progress in computing upper bounds for the 1n ’s. We shall close with some remarks on projective-like hierarchies which extend and simplify some of the proofs of [KSS81]. For us, a projective-like hierarchy is a sequence Γi : i < of nonselfdual pointclasses such that (i) ∀R Γi ⊆ Γi or ∃R Γi ⊆ Γi , but not both, for all i < , and (ii) ∀R Γ0 ⊆ Γ0 , and ˘ 0 , Γ0 = ∀R Γ . (iii) For all nonselfdual Γ ⊆ Γ0 ∩ Γ Our definition is slightly more liberal then that of Kechris-Solovay-Steel, mainly because we do not require Γi ⊆ Γi and Γi ⊆ Γi for all i. (Condition (iii) is slightly more liberal than theirs, too.) It is easyto see that if Γi : i < is a projective-like hierarchy, then Γi ⊆ Γi and Γi ⊆ Γi for all i ≥ 1. Thus our hierarchies differ from those of Kechris-Solovay-Steel only in that they sometimes have an extra class Γ0 tacked on at the beginning. Consideration of this class seems to simplify some proofs. Let Γ be a nonselfdual pointclass closed under one but not both of ∀R and ∃R . Let α = sup{ ∈ C : < o(Γ)}. Then α ∈ C , so α = o(Δ) for some Δ. It is easy to see that either Γ or Γ˘ is in the least projective-like hierarchy Γi : i < such that Δ ⊆ Γ0 . We now show that for each i < , either PWO(Γi ) or PWO(Γ˘i ), thereby reproving one of the main results of [KSS81]. We prove this by considering cases corresponding to the types I–IV of projective-like hierarchies defined in that paper. We assume throughout that o(Δ) Δ Δ. Case 1. cf(α) = . (Type I).
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In this case, Γ0 = Δ. It is easy to see, and in fact implied by Corollary 2.3, ˘ 0 ). The first periodicity theorem propagates prewellordering up that PWO(Γ to hierarchy Γi : i < . Case 2. cf(α) > . ˘ where we may assume by Theorem 1.1 that In this case, let Δ = Γ ∩ Γ, ∀R Γ ⊆ Γ, and by Theorem 3.1 that PWO(Γ). Subcase A. ∃R Γ Γ. In this case Γ0 = Γ, PWO(Γ0 ), and the first periodicity propagates prewellordering again. The case occurs with hierarchies of types II and III; type II in the case 2 Γ Γ, and type III in the case 2 Γ ⊆ Γ (and hence Γ ⊆ Γ). Subcase B. ∃R Γ ⊆ Γ. (Type IV). In this case, Γ0 = {A ∩ B : A ∈ Γ ∧ B ∈ Γ}. The usual differencehierarchy proof shows that PWO(Γ0 ). Again, first periodicity propagates prewellordering. REFERENCES
Alexander S. Kechris [Kec77B] Classifying projective-like hierarchies, Bulletin of the Greek Mathematical Society, vol. 18 (1977), pp. 254–275. [Kec81B] Suslin cardinals, κ-Suslin sets, and the scale property in the hyperprojective hierarchy, in Kechris et al. [Cabal ii], pp. 127–146, reprinted in [Cabal I], p. 314–332. Alexander S. Kechris, Benedikt Lowe, and John R. Steel ¨ [Cabal I] Games, scales, and Suslin cardinals: the Cabal seminar, volume I, Lecture Notes in Logic, vol. 31, Cambridge University Press, 2008. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. Alexander S. Kechris, Robert M. Solovay, and John R. Steel [KSS81] The axiom of determinacy and the prewellordering property, this volume, originally published in Kechris et al. [Cabal ii], pp. 101–125. Yiannis N. Moschovakis [Mos70] Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory. Proceedings of an international colloquium held under the auspices of the Israel Academy of Sciences and Humanities, Jerusalem, 11–14 November 1968 (Y. Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam-London, 1970, pp. 24–62. John R. Steel [Ste81B] Determinateness and the separation property, The Journal of Symbolic Logic, vol. 46 (1981), no. 1, pp. 41– 44. [Ste82] A classification of jump operators, The Journal of Symbolic Logic, vol. 47 (1982), no. 2, pp. 347–358.
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Robert Van Wesep [Van78B] Wadge degrees and descriptive set theory, this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 151–170. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CALIFORNIA 94720-3840 UNITED STATES OF AMERICA
E-mail: [email protected]
THE AXIOM OF DETERMINACY AND THE PREWELLORDERING PROPERTY
ALEXANDER S. KECHRIS1 , ROBERT M. SOLOVAY2 , AND JOHN R. STEEL3
§1. Introduction. Let = {0, 1, 2, . . . } be the set of natural numbers and R = the set of all functions from into , or for simplicity reals. A product space is of the form X = X1 × X2 × · · · × Xk , where Xi = or R. Subsets of these product spaces are called pointsets. A boldface pointclass is a class of pointsets closed under continuous preimages and containing all clopen pointsets (in all product spaces). The following results have been proved in Steel [Ste81B]: If Γ is a boldface pointclass, Γ˘ its dual, i.e., Γ˘ = {¬A : A ∈ Γ}, and Δ its ˘ then assuming ZF+DC+AD, ambiguous part, i.e., Δ = Γ ∩ Γ, ˘ (1) Either Γ or Γ has the separation property. ˘ (2) If Δ is closed under (finite) intersections and unions, then either Γ or Γ has the reduction property. Note that by a result of van Wesep [Van78A], if Γ is not closed under ˘ to have the separation complements, then it is impossible for both Γ and Γ property (assuming again ZF+DC+AD). Our purpose here is to investigate the situation concerning a stronger structural property of pointclasses, namely the prewellordering property. We establish in §2 the following criterion: Theorem 1.1 (ZF+DC+AD). Let Γ be a boldface pointclass, closed under countable intersections and unions and either existential or universal quantification over R, but not complements. Then the following are equivalent, ˘ has the prewellordering property. (1) Γ or Γ (2) Δ is not closed under wellordered unions (of arbitrary length). 1 Research partially supported by NSF Grant MCS79-20465. The author is an A. P. Sloan Foundation Fellow. 2 Partially supported by NSF Grant MCS77-01640. 3 Partially supported by NSF Grant MCS78-02989.
Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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By an application of this criterion and some further analysis done in §§3, 4, 5 we obtain for example the following, Theorem 1.2 (ZF+DC+AD). Let Γ be a boldface pointclass, closed under countable intersections and unions and either existential or universal quantification over R, but not complements. Then ˘ has the prewellordering property. Γ ⊆ L(R) ⇒ Γ or Γ Here L(R) is the smallest inner model of ZF containing all the reals. Finally we address ourselves to the following problem: If a pointclass Γ as above is closed under only one kind of quantification over R, i.e., either only existential or only universal (in which case it is reasonable to call such a pointclass projective-like), then in view of the inherent asymmetry in the ˘ is it possible to determine in which side of closure properties between Γ and Γ, ˘ the pair (Γ, Γ) we have the prewellordering property, as it is done by the work of Martin [Mar68] and Moschovakis (see Addison-Moschovakis [AM68]) for the projective pointclasses Σ1n , Π1n ? We provide an affirmative answer in §§4, 5: We first embed each projective-like pointclass Γ in a uniquely determined projective-like hierarchy. We then classify all possible projectivelike hierarchies into four types: I–IV, and demonstrate that (at least within L(R)) each of these types exhibits a unique prewellordering pattern, identical to that of the classical projective hierarchy for the types I and III, but dual to that for the other two types II and IV. Of course ZF+DC+AD is assumed throughout. The concept of and some results about Wadge degrees will be of the essence in this paper. The papers [Van78B, Van78A] and [Ste81B] provide all the necessary information. For general results in descriptive set theory needed below we refer to [Mos80]. Finally, it is convenient to assume ZF+DC throughout this paper and explicitly indicate only any further hypotheses as they are needed. §2. A criterion for PWO(Γ). Let Λ be a pointclass (i.e., an arbitrary collection of pointsets). We say that Λ is closed under wellordered unions if for any sequence {A }< of members of Λ, < A ∈ Λ. (Here is an arbitrary ordinal, and it is understood that all A are subsets of some arbitrary space X .) Recall also that a pointclass Γ has the prewellordering property if for any A ∈ Γ, there is a norm ϕ : A → κ such that the associated relations x ≤∗ϕ y ⇔ x ∈ A ∧ [y ∈ A ∨ ϕ(x) ≤ ϕ(y)], x <∗ϕ y ⇔ x ∈ A ∧ [y ∈ A ∨ ϕ(x) < ϕ(y)] are in Γ. Such a norm Γ is called a Γ-norm. We now have
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Theorem 2.1 (AD). Let Γ be a boldface pointclass, closed under countable unions and intersections and either ∃R or ∀R , but not complements. Then the following are equivalent: (i) PWO(Γ), (ii) Red(Γ) and Δ is not closed under wellordered unions. In view of the result of Steel mentioned in the introduction, we have Corollary 2.2 (AD). Let Γ be a boldface pointclass closed under countable unions and intersections, and either ∃R or ∀R , but not complements. Then the following are equivalent: ˘ (i) PWO(Γ) ∨ PWO(Γ), (ii) Δ is not closed under wellordered unions. Proof. If Γ has the prewellordering property, let A ∈ Γ \ Δ and let ϕ be a regular Γ-norm on A, of length κ. Regularmeans that ϕ : A κ.) For < κ, let A = ϕ −1 [{ }]. Then each A ∈ Δ but <κ A = A ∈ Δ, so Δ is not closed under wellordered unions. Finally we clearly have Red(Γ). (Theorem 2.1) So assume now that Δ is not closed under wellordered unions, and Red(Γ) holds. We shall distinguish three cases, depending on the closure properties of Γ. 2.1. Γ is closed under both ∃R and ∀R . We shall start in this case with a lemma which will be also useful later on. Call a boldface pointclass Λ strongly closed if it is closed under (finite) unions and intersections, complements and quantification (of both types) over R. For example Λ = n Δ1n is strongly closed. Note that Λ need not be closed under countable unions. Lemma 2.3. Let Λ be strongly closed. Then the following three ordinals associated with Λ are equal: (i) sup{ : is the length of a Λ prewellordering of R}, (ii) sup{ : is the rank of a Λ wellfounded relation on R}, (iii) sup{|A|W : A ∈ Λ}. Here |A|W is the Wadge ordinal of A ⊆ R. (Note that |A|W = |¬A|W .) Proof. Clearly (i) ≤ (ii) and (iii) ≤ (i). So it is enough to show that (ii) ≤ (iii). For each list of pointsets A1 , A2 , . . . , An , let Σ11 (A1 , A2 , . . . , An ) be the small intersections, ∃ , ∀ and ∃R , est boldface pointclass closed under unions and which contains A1 , . . . , An . By Moschovakis [Mos70] §3, Σ11 (A1 , A2 , . . . , An ) An ) be the dual of is R-parametrized i.e., has universal sets. Let Π11 (A1 , . . . , closed under ∃R , so that we Σ11 (A1 , . . . , An ). Note that Π11 (A, ¬A) cannot be build up Σ1 (A, ¬A), Π1(A, ¬A), . . . in the usual way. can 2 2
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Recall now the 0th Periodicity Theorem (see Kechris [Kec77B]), which asserts that if Γ is a boldface pointclass closed under countable intersections and unions, then if ∀R Γ = {∀α(x, α) ∈ A : A ∈ Γ} and similarly for ∃R Γ, we have: (i) ∃R Γ ⊆ Γ ∧ Red(Γ) ⇒ Red(∀R Γ), (ii) ∀R Γ ⊆ Γ ∧ Red(Γ) ⇒ Red(∃R Γ). (For the convenience of the reader we repeat this proof in an appendix.) From this and Steel’s Theorem (mentioned in the introduction), we can find for each pointset A an integer N > 1, so that Π1N (A, ¬A) has the reduction property. (N = 2 or N = 3 will of course suffice.) From these remarks it follows that, if ≺ is a wellfounded relation in Λ, there is a boldface pointclass Γ Λ such that Γ is R-parametrized, closed under countable intersections and unions and ∀R , such that Γ has the reduction property and moreover Δ ⊇ Σ11 (≺). (Note that if Λ is strongly closed and A1 , . . . , An ∈ Λ, Π1n (A1 , . . . , An ) Λ.) For a Γ with these structural properties we can introduce the following Γ coding of the Δ sets of reals: Let W 0 , W 1 be a universal pair for Γ (i.e., for each A, B ⊆ R in Γ there ¯ is ε ∈ R with A = Wε0 = {α : (ε, α) ∈ W 0 }, B = Wε1 ). Let W¯ 0 , W¯ 1 in Γ 0 1 reduce W , W . Put ¯
ε ∈ C ⇔ W¯ ε0 ∪ W¯ ε1 = R and for ε ∈ C Hε = W¯ ε0 (= ¬W¯ ε1 ). Then C ∈ Γ and {Hε : ε ∈ C } = {A ⊆ R : A ∈ Δ}. We can now finish the proof as follows: By the Recursion Theorem, we can find a partial continuous function f such that (1) α ∈ Field(≺) ⇒ f(α) ∈ C , (2) α ≺ ⇒ |Hf(α) |W < |Hf() |W . We use here the fact that Σ11 (≺) ⊆ Δ. Thus rank(≺) ≤ sup{|A|W : A ∈ Δ} ≤ (iii).
(Lemma 2.3) We proceed now to complete the proof of case 2.1. Let Γ satisfy its hypotheses. Let be the ordinal associated to Λ ≡ Δ by the preceding lemma. Let also be the least ordinal such that there is a -sequence {A } < of members of Δ with < A ∈ Δ. We claim that = . To see that ≤ , notice first that we can code Δ sets using Red(Γ) as in the proof of Lemma 2.3. (That Γ is R-parametrized follows from the nonclosure of Γ under complements—see Van Wesep [Van78A].) Let C , ε → Hε denote
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ALEXANDER S. KECHRIS, ROBERT M. SOLOVAY, AND JOHN R. STEEL
the set of codes and the coding map respectively. If < , let ϕ : R → be a Δ-norm and then for each -sequence {B }< of Δ sets let, by Moschovakis [Mos70], R(w, ε) be a Δ relation such that (i) ϕ(w) = ϕ(v) ⇒ [R(w, ε) ⇔ R(v, ε)] (ii) R(w, ε) ⇒ ε ∈ C (iii) ∀w∃ε[R(w, ε) ∧ Hε = Bϕ(w) ]. Then B ⇔ ∃w∃ε[R(w, ε) ∧ α ∈ Hε ]. α∈
<
Thus < B ∈ Δ. To prove that conversely ≤ , we use the minimality of . It easily implies that there is a -sequence of Δ sets {A } < such that A A , if < < , A = < A , if < is limit, and A = < A ∈ Δ. For each x ∈ A, let ϕ(x) = least , x ∈ A +1 \ A . Then ϕ is a regular norm on A of length . But if x ∈ A and ϕ(x) = < , then the prewellordering {(y, z) : ϕ(y) ≤ ϕ(z) < ϕ(x)}, of length , is in Δ, since it is equal to ¬A , A × <
<
while < . So < , thus ≤ . Let now A : ∀ < (A ∈ Δ) . Γ∗ =
<
First notice that, by an argument Γ∗ has the prewellorof Martin [Mar71B], ∗ dering property. Indeed, if A = < A is in Γ and we let (x) = least , x ∈ A , be the norm associated to this wellordered union, then is a Γ∗ -norm. This is because if ≤∗ , <∗ are its two associated relations, then x ≤∗ y ⇔ ∃ < [x ∈ A ∧ ∀ < (y ∈ A )], x <∗ y ⇔ ∃ < [x ∈ A ∧ ∀ ≤ (y ∈ A )], so these are in Γ∗ , by the minimality of again. So it is enough to show that Γ∗ = Γ. Since Γ∗ Δ, by Wadge’s Lemma it is enough to show that Γ∗ ⊆ Γ. For that let ε ∈ W ⇔ ε ∈ C ∧ Hε is wellfounded
AD AND THE PREWELLORDERING PROPERTY
123
(view Hε as a subset of R2 here), and for ε ∈ W , let |ε| = rank(Hε ). Then W ∈ Γ and {|ε| : ε ∈ C } = = . Given now {B } < , a -sequence of Δ sets, consider the following Solovay-type game: I ε
II α
Player II wins iff ε ∈ W ⇒ α ∈ C ∧ ∃[|ε| < < ∧ Hα = < B ].
If player I has a winning strategy f, then the relation (ε, x) ≺ (ε , y) ⇔ ε = ε ∈ f[R] ∧ (x, y) ∈ Hε
is wellfounded and in Δ, so if player II plays any α ∈ C with Hα = < B
for any > > rank(≺), he beats f. So player II must have a winning strategy g. Then x∈ B ⇔ ∃ε[ε ∈ W ∧ x ∈ Hg(ε) ], so
<
<
∗
B ∈ Γ, thus Γ ⊆ Γ, and we are done.
2.2. Γ is closed under ∃R but not ∀R . In this case the result follows from the following lemma which will be also useful later on. Lemma 2.4. Assume Γ is a boldface pointclass, closed under countable unions and intersections and ∃R but not ∀R (thus Γ is not closed under complements). Then if Red(Γ) holds, Γ is closed under wellordered unions. Proof. Let be least such that for some {A } < with all A in Γ, < A ∈ Γ, towards a contradiction. It is easy to check that is a regular uncountable cardinal. Put Γ =
A : ∀ < (A ∈ Γ) .
<
˘ and since clearly Γ is closed under Since Γ Γ, by Wadge’s Lemma Γ ⊇ Γ, ˘ Put ∃R , Γ ⊇ ∃R Γ. ˘ Γ+ = ∀R Γ, so that Γ˘ + = ∃R Γ, ˘ + . Let also therefore Γ ⊇ Γ ˘ + wellfounded relation}. + = sup{ : is the rank of a Γ ˘ + ) and Γ ⊆ Δ+ (because ∀R Γ = Γ+ ⊆ Γ, thus Since Γ Δ+ (= Γ+ ∩ Γ Γ ⊆ Γ˘ + by Wadge), Δ+ is not closed under wellordered unions, so let + be the least ordinal such that some union of a + -sequence of Δ+ sets is not in Δ+ . Clearly + ≤ .
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ALEXANDER S. KECHRIS, ROBERT M. SOLOVAY, AND JOHN R. STEEL
We have now that + ≥ + , thus ≥ + . (This is essentially an argument of Martin [Mar71B]: If + < + , then as in the proof in case 2.1 an application of the Moschovakis Coding Lemma shows that A : ∀ < + (A ∈ Δ+ ) = Γ˘ +
<+
˘ + has the prewellordering property, contraand thus again as in case 2.1, Γ dicting the fact that Γ+ has the reduction property by the 0th Periodicity Theorem.) Let = sup{ : is the rank of a Γ wellfounded relation}. By the standard argument (see for example Kechris [Kec74]) + > , thus > . ˘ Let now ≺ be a Γ+ wellfounded relation. Then, since Γ˘ + ⊆ Γ , ≺= ≺ ,
<
where each ≺ is a Γ wellfounded relation, and by the minimality of we can assume that ≤ < ⇒ ≺ ⊆ ≺ . For each x ∈ Field(≺), put 0 if x ∈ Field(≺ ) fx ( ) = |x|≺ ≡ rank of x in ≺ otherwise Then fx : → is nondecreasing, so since < and is regular, we have fx ( ) = constant ≡ (x) < , for all large enough < . then x ≺ y ⇒ (x) < (y), (Lemma 2.4) thus rank(≺) ≤ , so + ≤ , a contradiction. 2.3. Γ is closed under ∀R but not ∃R . Let again be least such that some union of a -sequence of Δ sets is not in Δ. Put A : ∀ < (A ∈ Δ) . Γ =
<
˘ so, since Γ Δ, Γ ⊇ Γ. Since Γ has the prewellordering property, Γ = Γ Call now a sequence {A } < Λ-bounded (for any pointclass Λ), if ∀X ∈ Λ X ⊆ A ⇒ ∃ < (X ⊆ A ) .
<
AD AND THE PREWELLORDERING PROPERTY
Put Γ∗ =
125
˘ A : ∀ < (A ∈ Δ) ∧ {A } < is Γ-bounded .
<
The result will follow from the following three lemmas. Lemma 2.5. Γ∗ is a boldface pointclass with the prewellordering property. Lemma 2.6. If Γ∗ Δ, then Γ∗ = Γ. Lemma 2.7. Γ∗ Δ. ˘ is closed under ∃R it is trivial to check that Proof of Lemma 2.5. Since Γ ˘ Γ is closed under continuous preimages. Let now {A } < be a Γ-bounded sequence of Δ sets. Let A = < A and let ∗
ϕ(x) = least , x ∈ A
the associated norm on A. Then B (x,y)
! x ≤∗ϕ y ⇔ ∃ < ∃ ≤ x ∈ A ∧ ∀ < (y ∈ A ) , C (x,y)
x
<∗ϕ
! y ⇔ ∃ < ∃ ≤ x ∈ A ∧ ∀ ≤ (y ∈ A ) .
Let B (x, y), C (x, y) be the two pointsets indicated above. It is enough ˘ to show that {B } < , {C } < are Γ-bounded. Take for example {B } < . ˘ Let Y ∈ Γ, Y ⊆ < B . If X = {x : ∃y(x, y) ∈ Y }, then X ∈ Γ˘ and X ⊆ < A so for some < , X ⊆ A . Then Y ⊆ B . (Lemma 2.5) Proof of Lemma 2.6. Assume Γ∗ Δ. If Γ∗ = Γ, then by Wadge’s ˘ Since, by the 0th Periodicity Theorem, Red(Γ) ⇒ Red(∃R Γ), Lemma, Γ∗ ⊇ Γ. we have by Lemma 2.4 that Γ∗ ⊆ ∃R Γ, so ˘ ⊆ Γ∗ ⊆ ∃R Γ. Γ ˘ be such that Let now A ∈ Γ x ∈ B ⇔ ∀α(x, α) ∈ A is not in ∃R Γ. Write A=
A ,
<
˘ Then where each A ∈ Δ and {A } < is Γ-bounded. x ∈ B ⇔ ∀α∃ < (x, α) ∈ A . ˘ So if x ∈ B, then {x} × R ⊆ < A , thus by Γ-boundedness, {x} × R ⊆ A
for some < . Thus x ∈ B ⇔ ∃ < ∀α(x, α) ∈ A .
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ALEXANDER S. KECHRIS, ROBERT M. SOLOVAY, AND JOHN R. STEEL
If x ∈ B ⇔ ∀α(x, α) ∈ A , then B ∈ Γ ⊆ ∃R Γ, so by Lemma 2.4 again B = < B ∈ ∃R Γ, a contradiction. (Lemma 2.6) ˘ Proof of Lemma 2.7. Let A ∈ Γ \ Δ. Let S be universal for Γ, and let B = {ε : Sε ⊆ A}. Thus B ∈ Γ. As Γ ⊇ Γ, we can write B=
B ,
<
where each B is in Δ. Put A =
{Sε : ε ∈ B }.
˘ ˘ Then each A ∈ Γ,
< A = A and {A } < is Γ-bounded. Since, by the minimality of , we can choose the sequence {B } < to be increasing and continuous, the same can be assumed to be true for the sequence {A } < . We claim now that E = { < : A ∈ Δ} is -closed unbounded in , which of course completes the proof since if { } < is its increasing enumeration and ˘ A = A , then {A } < is a Γ-bounded sequence of Δ sets with union A ∈ Δ. Clearly E is -closed. To see that it is unbounded, let < be given. ˘ A ∈ Γ and A ⊆ A, we can find, using Sep(Γ), ˘ a set X0 ∈ Δ, As A ∈ Γ, ˘ with A ⊆ X0 ⊆ A. By Γ-boundedness, let 0 > be such that X0 ⊆ A 0 . Repeat now this process with 0 to define X1 , 1 , etc. If = limi< i < (recall that must be a regular uncountable cardinal), then A = < A = (Lemma 2.7) i< A i = i< Xi ∈ Δ. §3. Inductive-like pointclasses and projective algebras. We shall examine now the extent to which the condition (ii) of the criterion 2.2 is satisfied. We consider first the case when Γ is closed under both types of real quantification. Let us give the following definitions. Definition 3.1. A boldface pointclass Γ is called inductive-like if it is closed under countable intersections and unions, ∃R and ∀R , but not complements. The typical inductive-like pointclasses are IND(R), the class of all inductive over the structure of analysis R pointsets, and its dual IND(R)˘. Definition 3.2. A boldface pointclass Λ is called a projective algebra iff it is closed under complements, wellordered unions, ∃R and ∀R . Then Corollary 2.2 implies immediately
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Theorem 3.3 (AD). Let Γ be an inductive-like pointclass. Then the following are equivalent ˘ (i) PWO(Γ) ∨ PWO(Γ), (ii) Δ is not a projective algebra. The following result now provides a good first impression about the concept of projective algebra. Theorem 3.4 (AD). Let Λ be a projective algebra and let L[Λ] be the smallest inner model of ZF containing R ∪ Λ. Then ℘ (R) ∩ L[Λ] = Λ. Proof. Let Λ be a projective algebra. let , be the bijection between ¨ Ord × Ord and Ord corresponding to the Godel wellordering of pairs of ordinals. Let , , = , , and = I , J , K . Let F1 , F2 , . . . , F9 be ¨ the (binary) Godel operations as given in Shoenfield [Sho67]. Put for α ∈ R, A ∈ Λ ∩ ℘ (R): Gi (x, y; α, A) = Fi (x, y),
1 ≤ i ≤ 9,
G10 (x, y; α, A) = x ∩ α, G11 (x, y; α, A) = x ∩ A. Then define F : Ord × R × Λ → V by ⎧ {F (; , B) : < , ∈ R, B ∈ Λ} if I = 0 ⎪ ⎪ ⎪ ⎨G (F (J ; (α) , (A) ), F (K ; (α) , (A) ); (α) , (A) ) I
0 0 1 1 2 2
F ( ; α, A) = ⎪ if 0 < I ≤ 11, ⎪ ⎪ ⎩ {F (J ; (α)0 , (A)0 ), F (K ; (α)1 , (A)1 )} if I ≥ 12, where (α)i (n) = α(2i · 3n ), (A)i = {α : [i]α ∈ A}. We have then L(Λ) = {F ( ; α, A) : ∈ Ord, α ∈ R, A ∈ Λ}. (Notice here that if M is an inner model of ZF containing R and Λ ⊆ M then actually Λ ∈ M , since either Λ = M ∩ ℘ (R) or else if A ∈ Λ has least Wadge ordinal, A ∈ M , so Λ ∈ M .) Let us say now that a relation Φ(x, A1 , . . . , An ), where x ∈ X , Ai ⊆ R ¯ if for each fixed P1 , . . . , Pn ∈ Λ the pointset and Ai ∈ Λ, is in the class Λ, ∗ ∗ Φ ≡ ΦP1 ,...,Pn given by Φ∗ (x, ε1 , . . . , εn ) ⇔ ε1 , . . . , εn code continuous functions fε1 , . . . , fεn and Φ(x, fε−1 [P1 ], . . . , fε−1 [Pn ]), n 1 is in Λ. ¯ contains Φ(α, A) ⇔ α ∈ A, and is closed under continuous Lemma 3.5. Λ substitutions, complements, ∃R , ∀R , ∃Λ , ∀Λ and wellordered unions.
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ALEXANDER S. KECHRIS, ROBERT M. SOLOVAY, AND JOHN R. STEEL
¯ Proof. Everything is obvious except perhaps ∃Λ . Let Φ(x, A, B) be in Λ and consider Ψ(x, A) ⇔ ∃B ∈ ΛΦ(x, A, B). Fix P ∈ Λ. Then Ψ∗P (x, ε) ⇔ ε codes a continuous function fε ∧ Ψ(x, fε−1 [P]) ⇔ ε codes a continuous function fε ∧ ∃B ∈ ΛΦ(x, fε−1 [P], B) ⇔ ε codes a continuous function fε ∧ ∃ < ∃B[|B|W ≤ ∧ Φ(x, fε−1 [P], B)], where = sup{|B|W : B ∈ Λ}. So it is enough to show that if S (x, ε) ⇔ ε codes a continuous function fε ∧
∃B[|B|W ≤ ∧ Φ(x, fε−1 [P], B)],
then S ∈ Λ, as Ψ∗P = < S . For that assume, without loss of generality, that is such that |Q|W = ⇒ Q is self-dual, and pick such a Q. Then Q ∈ Λ and S (x, ε) ⇔ ε codes a continuous function fε ∧ ∃B ≤w QΦ(x, fε−1 [P], B) ⇔ ε codes a continuous function fε ∧ ∃ε [ε codes a continuous function fε ∧ Φ(x, fε−1 [P], fε−1 [Q])] ⇔ ∃ε Φ∗P,Q (x, ε, ε ), so S ∈ Λ.
(Lemma 3.5)
¯ Lemma 3.6. For each fixed , the following relations are in Λ: ε (i) Φ , (α, A; , B) ⇔ F ( ; α, A) ∈ F (, , B), (ii) Φ=
, (α, A; , B) ⇔ F ( ; α, A) = F (; , B), (iii) Ψ1 , (α; , B) ⇔ α ∈ F (max{ , }; , B), Ψ2 , (n; , B) ⇔ n ∈ ∧ n ∈ F (max{ , }; , B). Proof. Routine induction on , . (Lemma 3.6) To complete the proof, let now S ⊆ R be in L[Λ] and find 0 ∈ Ord, α0 ∈ R, A0 ∈ Λ with S = F ( 0 ; α0 , A0 ). Then α ∈ S ⇔ α ∈ F ( 0 ; α0 , A0 ) ⇔ Ψ1 0 , 0 (α; α0 , A0 ). So if ε0 codes the identity function and (Ψ1 0 , 0 )∗A0 ≡ R, we have α ∈ S ⇔ (α, α0 , ε0 ) ∈ R thus S ∈ Λ. In particular for any projective algebra Λ, ℘ (R) ∩ L(R) ⊆ Λ,
(Theorem 3.4)
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129
and if A ∈ Λ and ∃B ⊆ R(B ∈ L[A]), or equivalently, by Steel-Van Wesep [SVW82], A# exists, then A# ∈ Λ. Thus any projective algebra is quite big. As a simple consequence we have Corollary 3.7 (AD). Let Γ be any inductive-like pointclass contained in ˘ L(R). Then PWO(Γ) ∨ PWO(Γ). Also we immediately obtain, Corollary 3.8 (AD). If Γ is any inductive-like pointclass, then Γ ⊇ IND(R) or Γ ⊇ IND(R)˘. In particular, IND(R) is the smallest inductive-like reduction (or even not pointclass satisfying satisfying separation). Proof. By Moschovakis [Mos74], IND(R) is the smallest inductive-like pointclass satisfying prewellordering. (Corollary 3.8) We conclude this section by offering some speculations on the extent of projective algebras. Let P∞ denote the smallest projective algebra, B∞ the smallest boldface pointclass closed under complements and wellordered unions (the class of ∞-Borel sets) and S(∞) = κ S(κ), where S(κ) is the class of κ-Suslin sets. Clearly S(∞) ⊆ B∞ ⊆ P∞ . By results of Kechris, Solovay, Martin, and Steel (see [MS83]), AD + V=L(R) ⇒ S(∞) = Σ21 . Using this it can be shown that AD + V=L(R) ⇒ B∞ = ℘ (R). A proof of this is given in an appendix. On the other hand, if ADR denotes the Axiom of Determinacy for games on reals and Θ is the sup of the lengths of prewellorderings on R, then the following question has been raised in Solovay [Sol78B]: ADR + Θ is regular ⇒ S(∞) = ℘ (R)? An affirmative answer would also imply that P∞ = ℘ (R) and thus extend Corollary 3.7 and further results of the present paper to all appropriate Γ’s. Even if the above question admits a negative answer, it is still conceivable that some reasonable hypothesis extending AD (not necessarily properly) might still imply that P∞ = ℘ (R).1 1 In
the 1980’s, W. H. Woodin showed that AD+ “Every binary relation on the reals can be uniformized” implies every set of reals is Suslin. His unpublished proof made use of results in this direction of H. Becker [Bec85].
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ALEXANDER S. KECHRIS, ROBERT M. SOLOVAY, AND JOHN R. STEEL
§4. Projective-like pointclasses and hierarchies. We proceed now to discuss the case of pointclasses closed under only one kind of real quantification. Definition 4.1. A boldface pointclass is projective-like if it is closed under countable intersections and unions, and either ∃R or ∀R but not both, and is not closed under complements. The prototypes of such pointclasses are of course Σ1n , Π1n . It is convenient for the work below to embed every projective-like pointclass in a (unique) projective-like hierarchy, where this concept is defined as follows: Definition 4.2. A projective-like hierarchy is a sequence Γ1 , Γ2 , Γ3 , . . . of projective-like pointclasses such that: (i) Γ1 is closed under ∀R , (ii) Γi+1 = ∃R Γi , if Γi is closed under ∀R , and Γi+1 = ∀R Γi , if Γi is closed under ∃R , (iii) {Γi } is maximal, i.e., there is no projective-like pointclass Γ0 closed under ∃R such that Γ1 = ∀R Γ0 . We say that a projective-like pointclass Γ belongs to the hierarchy {Γi } if ˘ i , for some i. Γ = Γi or Γ = Γ 1 1 Again Π1 , Σ2 , Π13 , Σ14 , . . . is the prototype of this notion. that Note first eachprojective-like Γ belongs to exactly one projective-like hierarchy {Γi }. This is easy to see from the following two facts, where we note that Γ = ∃R Γ, if ∀R Γ ⊆ Γ, and Γ = ∀R Γ, if ∃R Γ ⊆ Γ ( is the game quantifier) and we define |Γ|W ≡ |A|W , for each A ∈ Γ \ Δ: (i) |Γ|W < | Γ|W , (ii) If Γ , Γ are projective-like and Γ = Γ , then Γ = Γ (otherwise, by Wadge, Γ ⊆ Δ or Γ ⊆ Δ , so, let us say in the first case, Γ = Γ ⊆ Γ˘ = (by AD) ( Γ )˘, a contradiction). We shall classify now all the projective-like hierarchies into four types I–IV. Let us recall a basic fact about Wadge degrees first. Call A ⊆ R self-dual iff A ≤W ¬A. (The Wadge reducibility ≤W is defined by A ≤W B ⇔ ∃f : R → R (f continuous ∧ f −1 [B] = A).) Then Steel-Van Wesep (see Van Wesep [Van78A]) prove: If |A|W is limit, then G
G
G
G
G
G
G
G
G
G
A is self-dual iff |A|W has cofinality . We describe now the classification. (AD is assumed throughout.) Type I. {Γi } is such that Γ1 = Π11 (A), for some A with cf(|A|W ) = and with the property that Λ = {B :|B|W < |A|W } is strongly closed (recall the beginning of 2.1 here). We visualize this by Figure 1 in the Wadge hierarchy. By convention we accept the case |A|W = 1 as being within this type, so that the classical projective hierarchy is of type I. The next example is generated by
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AD AND THE PREWELLORDERING PROPERTY
.. . ˘3 Γ
Γ3
Σ12 (A) = Γ2 ˘1 Σ11 (A) = Γ
Π12 (A) = Γ˘ 2 Π11 (A) = Γ1 cf(|A|W ) =
A
Λ Figure 1. taking A to be a set of least Wadge ordinal above the projective sets, so that Λ = {B : |B|W < |A|W } = n Δ1n . Type II. {Γi } is such that Γ1 = Π11 (A) for some A with cf(|A|W ) > , with ¬A ∈ Π11 (A), and Λ = {B : |B|W < |A|W } strongly closed. The relevant picture is in Figure 2. .. . Γ˘ 3
Γ3 Π12 (A) = Γ˘ 2 Π11 (A) = Π11 (¬A) = Γ1
Σ12 (A) = Γ2 ˘1 Σ11 = Γ ¬A
A
cf(|A|W ) >
Λ Figure 2. The smallest example of such a hierarchy is constructed as follows: Let Γ , 1 ≤ < 1 be defined by Γ1 = Π11 , Γ+1 = Γ , G
Γ = all countable unions of sets in
<
Γ , if =
> 0.
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ALEXANDER S. KECHRIS, ROBERT M. SOLOVAY, AND JOHN R. STEEL
Let then A be such that
Λ = {B : |B|W < |A|W } =
Γ .
<1
Since cf(|A|W ) = 1 it follows that {B : B ≤W A} is not closed under both intersections and unions (see for example Steel [Ste81B]), so Π11 (A) = Π11 (¬A), thus ¬A ∈ Π11 (A). Type III. {Γi } is such that Δ1 = Γ1 ∩ Γ˘ 1 is strongly closed. (See Figure 3.) .. . Γ˘ 3
Γ3
Γ2
Γ˘ 2
˘1 Γ
Γ1
Δ1 Figure 3. A typical example is Γ1 =2 ENV(3E) ≡ the pointclass of sets Kleene semirecursive in 3E and a real. It follows from Corollary 5.4(i) (see also Theorem 3.3 of [Ste81A]) that this is the Wadge-least example of a type III hierarchy. ˘ where Γ is inductive-like, and Type IV. {Γi } is such that Γ1 = ∀R (Γ ∨ Γ), ˘ ˘ Γ ∨ Γ = {A ∪ B : A ∈ Γ ∧ B ∈ Γ}. (See Figure 4.) Again the smallest example is constructed by taking Γ = IND(R). We have now the following Theorem 4.3 (AD). Every projective-like hierarchy is of exactly one of the types I–IV. Proof. As no projective-like hierarchy can be of two different types, it is enough to prove that every projective-like hierarchy {Γi } is of one of the types I–IV. For the classical projective hierarchy, no strongly closed pointclass is included in Δ1 . We include this hierarchy under alternative I. Let Λ be the largest strongly closed pointclass contained in Δ1 . Let = sup{|A|W : A ∈ Λ}. Then exactly one of the four possibilities below must hold: I. cf( ) =
AD AND THE PREWELLORDERING PROPERTY
133
.. . ˘3 Γ
Γ3
Γ2
˘2 Γ
˘ ˘ 1 = ∃R (Γ ∧ Γ) Γ
˘ Γ1 = ∀R (Γ ∨ Γ)
Γ˘
Γ
Δ Figure 4. II. cf( ) > and if A is such that |A|W = |¬A|W = , then {B : B ≤W A} (and thus {B : B ≤W ¬A}) is neither projective-like nor inductive-like. III. cf( ) > and if A, ¬A are as above, then {B : B ≤W A} (and thus {B : B ≤W ¬A}) is projective-like. IV. cf( ) > and if A, ¬A are as above, then {B : B ≤W A} (and thus {B : B ≤W ¬A}) is inductive-like. If I holds, let A be such that |A|W = . By the maximality of Λ we must have Π11 (A) = Γ1 , so that {Γi } is of type I. If II holds, then (by Wadge) ¬A ∈Π11 (A), and Γ1 = Π11 (A), so that {Γi } is of type II. If III holds and, withoutloss of generality,{B : B ≤W A} is closed under ∀R but not ∃R , then Γ1 = {B : B ≤W A} and Δ1 = Λ, so that {Γi } is of type III. Finally, if IV ˘ so that {Γi } is of holds and Γ = {B : B ≤W A}, we have Γ1 = ∀R (Γ ∨ Γ), type IV. (Theorem 4.3) §5. The prewellordering pattern in projective-like hierarchies. We conclude now our analysis by establishing (under certain assumptions) that each type of projective-like hierarchy has only one prewellordering pattern. For convenience let us introduce the following terminology. Definition 5.1. A projective-like hierarchy {Γi } is of character Π iff PWO(Γ1 ) holds (iff PWO(Γi ) holds for each i ≥ 1, by the First Periodicity Theorem of Martin and Moschovakis). A projective-like hierarchy is of ˘ 1 ) holds (iff PWO(Γ˘ i ) holds for all i ≥ 1). character Σ iff PWO(Γ Definition 5.2. The ground of a projective-like hierarchy {Γi } is defined to be the largest strongly closed Λ such that Λ ⊆ Δ1 . Thus the ground Λ coincides with the strongly closed Λ occurring in the definition of {Γi } being of type I or II, with Δ1 if {Γi } is of type III, and with Δ = Γ ∩ Γ˘ if {Γi } is of type IV and Γ is given in the definition of this type.
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We now have Theorem 5.3 (AD). Let {Γi } be a projective-like hierarchy. Then (i) If {Γi } is of type I, then {Γi } has character Π. (ii) If {Γi } is of type II, then {Γi } has character Σ, provided its ground is not a projective algebra. (iii) If {Γi } is of type III, then {Γi } has character Π iff its ground is not a projective algebra. (iv) If {Γi } is of type IV, then {Γi } has character Σ, provided its ground is not a projective algebra. Corollary 5.4 (AD). Let {Γi } be a projective-like hierarchy contained in P∞ . Then (i) If {Γi } is of type I, III, then {Γi } has character Π. (ii) If {Γi } is of type II, IV, then {Γi } has character Σ. In particular, this holds if {Γi } is contained in L(R). Corollary 5.5 (AD). Let Γ be a boldface pointclass closed under countable intersections and unions and ∃R or ∀R , but not complements. If Γ ⊆ P∞ , in ˘ particular if Γ ⊆ L(R), then PWO(Γ) or PWO(Γ). Proof of Theorem 5.3. (i) Let A be as in the definition of a type I hierarchy, so that in particular Γ1 = Π11 (A). Let Λ = {B : |B|W < |A|W }. Since cf(|A|W ) = , Λ is not closed under countable unions, so if Γ0 = An : ∀n(An ∈ Λ) , n
then Γ0 is a boldface pointclass, closed under intersections and unions, countable unions and ∃R . Moreover we have PWO(Γ0 ). Since Γ1 = ∀R Γ0 , we have PWO(Γ1 ) by the First Periodicity Theorem. (iii) By Theorem 2.1 and the fact that Δ1 is not a projective algebra, it is enough to show Red(Γ1 ) and by Steel’s Theorem it is enough to show ¬Sep(Γ1 ). This is immediate from the following: Lemma 5.6. If Γ is a boldface pointclass not closed under complements, and Δ is closed under ∃R , ∀R then Sep(Γ) ⇒ ∃R Γ ⊆ Γ. Proof. Let P, Q be disjoint sets in ∃R Γ. Let P(x) ⇔ ∃α(x, α) ∈ A, Q(x) ⇔ ∃(x, ) ∈ B, where A, B ∈ Γ are also disjoint. Put A (x, α, ) ⇔ A(x, α) B (x, α, ) ⇔ B(x, ).
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Then A , B ∈ Γ are disjoint, so find C ∈ Δ separating them. Then S(x) ⇔ ∃α∀(x, α, ) ∈ C is in Δ and separates P, Q (this is a variant of an argument of Addison). So if ∃R Γ ⊆ Γ, then by Wadge Γ˘ ⊆ ∃R Γ, thus if A ∈ Γ then P = A, Q = ¬A are disjoint in ∃R Γ and thus can be separated by S ∈ Δ. So A = S ∈ Δ, a contradiction. (Lemma 5.6) (iv) Let Γ be the inductive-like pointclass appearing in the definition of a ˘ 1 = ∃R (Γ ∧ Γ). ˘ Assume without projective-like hierarchy of type IV, so that Γ loss of generality that Red(Γ) holds. Then by Theorem 2.1 PWO(Γ) holds. Then Π = Γ ∧ Γ˘ is a boldface pointclass closed under intersections and ∀R ˘ and ϕ is a Γ-norm and has the prewellordering property. (If A ∈ Γ, B ∈ Γ ˘ on A, then ϕA ∩ B is a Γ ∧ Γ-norm on A ∩ B.) So, by the usual proof, ˘ has the prewellordering property. ˘ 1 = ∃R (Γ ∧ Γ) Γ (ii) Let A be as in the definition of type II hierarchy so that Γ1 = Π11 (A) and A is chosen (from A, ¬A) so that Π = {B : B ≤W A} ˘ Then, by Lemma 5.6, Π is closed has the reduction property. Put Σ = Π. under ∀R and hence, since Π is R-parametrized, under countable intersections. Thus Σ11 (A) = ∃R Π. Λ = {B : |B| < |A| } is not a projective algebra, it is not closed Since W W under wellordered unions. Let be least such that some -sequence of elements of Λ has union outside Λ. (Again is a regular cardinal > , since Λ is closed under countable unions as |A|W has cofinality > .) Let = |A|W = sup{|B|W : B ∈ Λ}. As in 2.1, ≤ . Put Γ = A : ∀ < (A ∈ Λ) .
<
Then Γ is a boldface pointclass closed under countable intersections and ˘ 1 we are done, since clearly unions and ∃R . So Γ ⊇ Σ11 (A). If Γ = Σ11 (A) = Γ PWO(Γ ). Otherwise, Γ Σ11 (A), so by Wadge, Γ ⊇ Π11 (A) = Γ1 , thus Γ ⊇ Γ2 = there is a wellfounded relation R ∃ Γ1 . Now notice that ≺ of rank ≥ (≥ ) in Γ2 . Indeed, let ( , α) ≺ ( , ) ⇔ = codes a continuous function f & f −1 [A] is wellfounded & (α, ) ∈ f −1 [A]. So by Moschovakis [Mos80], Ch. 7, Γ2 is closed under wellordered unions of length rank(≺), thus Γ ⊆ Γ2 . So Γ = Γ2 .
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Put now Γ∗ =
A : ∀ < (A ∈ Λ) ∧ {A } < is Σ11 -bounded .
<
Note first that Γ∗ is closed under ∀R . The argument is similar to that in Lemma 2.6. Indeed let (x, α) ∈ B ⇔ ∃ < (x, α) ∈ B , where {B } < is Σ11 -bounded. Let x ∈ C ⇔ ∀α(x, α) ∈ B. Then as in Lemma 2.6, C ⇔ ∃ < ∀α(x, α) ∈ B . Let C = {x : ∀α(x, α) ∈ B }. It x ∈
is enough to show that {C } < is Σ11 -bounded. Let X ∈ Σ11 , X ⊆ < C . , so X × R ⊆ B, Then ∀x ∈ X ∃ ∀α(x, α) ∈ B , thus∀x ∈ X ∀α∃ (x, α) ∈ B
thus for some < , X × R ⊆ B , therefore ∀x ∈ X ∀α(x, α) ∈ B i.e., X ⊆ C , and we are done. So if Γ∗ ⊆ Λ, then either Γ∗ = Π in which case we have PWO(Π) (as ˘ 1 ) since Γ˘ 1 = ∃R Π, and we are done, or else in Lemma 2.5), thus PWO(Γ ∗ ∗ R Γ ⊇ Σ, so Γ ⊇ ∀ Σ = Γ1 . Then we must have Γ∗ = Γ1 (since otherwise ˘ 1 , so Γ∗ ⊇ ∀R Γ ˘ 1 = Γ˘ 2 , contradicting the fact that Γ∗ ⊆ Γ = Γ2 ). So Γ∗ ⊇ Γ if W ∈ Γ1 \ Δ1 , we can write W = < W , with W ∈ Λ and {W } < a Σ11 -bounded sequence. Let ϕ(x) = least , x ∈ W
be the associated norm, which by the argument in Lemma 2.5 is a (Γ∗ =)Γ1 norm. But A ∈ Δ1 , so let f continuous be such that x ∈ A ⇔ f(x) ∈ W . By the usual boundedness argument (recall that Γ1 is closed under ∀R ), there is
< , with x ∈ A ⇔ f(x) ∈ W , so A ∈ Λ, a contradiction. So we can complete the proof by showing that Γ∗ ⊆ Λ. For that we use the same argument as in Lemma 2.7. Let S be universal Σ11 and put C = {ε : Sε ⊆ A}. Then C ∈ Π11 (A) = Γ1 ⊆ Γ , so C = < C , where each C ∈ Λ. Put A = {Sε : ε ∈ C }. Then A ∈ Λ, {A } < is Σ11 -bounded, and A = < A ∈ Λ, so we are done. (Theorem 5.3) An alternative approach to the proof of Theorem 5.3 is given in §3 of Steel [Ste81A]. §6. Problems and conjectures. Assume AD in this section. Let Λ P∞ be a strongly closed boldface pointclass and {Γi } the first projective-like hierarchy not contained in Λ or in other words the projective-like hierarchy with ground Λ. Let be the ordinal associated with Λ as in Lemma 2.3, i.e., the supremum of the ordinals |B|W for B ∈ Λ. The question is whether determines the type of {Γi } ( is always a limit cardinal).
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Clearly if cf( ) = , {Γ1 } is of type I. It is easy also to see that if cf( ) > and is singular, then {Γi } is of type II (otherwise Λ = Δ1 and PWO(Γ1 ) for a type III {Γi }, so, since ∀R Γ1 = Γ1 , we have by Moschovakis [Mos70] that = 1 ≡ sup{ : is the rank of a Δ1 prewellordering}, thus is regular). We offer now the following conjecture: If is regular, then {Γi } is of type III or IV. (As pointed out above the converse is true.) As positive evidence we consider the fact that this conjecture is true for at least Λ ⊆ IND(R), as shown by Steel [Ste81A]. It is not clear what property of a regular would distinguish between types III and IV. However the results in §5(A) of Kechris [Kec81B] suggest that some combination of indescribability and Mahlo properties of could guarantee that {Γi } is of type IV. And we conclude with one more conjecture and two problems concerning closure of classes under wellordered unions: (a) Conjecture. If Γ is inductive-like and has the prewellordering property, then Γ is closed under wellordered unions (compare with Lemma 2.4). (b) If Γ is projective-like, can Δ be closed under wellordered unions? (c) Can the ground of type II or III hierarchies be a projective algebra? Appendix A. We give here a proof of the 0th Periodicity Theorem. Assume below AD. Let Γ be a boldface pointclass closed under countable unions and intersections, which has the reduction property. We show that (i) ∃R Γ ⊆ Γ = Red(∀R Γ), (ii) ∀R Γ ⊆ Γ = Red(∃R Γ). G
Take first (i). Assume ∃R Γ ⊆ Γ. Then note that ∀R Γ = { αA(x, α) : A ∈ Γ and A is Turing invariant on α}, where A(x, α) is Turing invariant on α iff α ≡T ∧ A(x, α) ⇒ A(x, ). This is because ∀αB(x, α) ⇔ α∀ ≤T αB(x, ). So let P, Q be in ∀R Γ and say x ∈ P ⇔ α(x, α) ∈ A, x ∈ Q ⇔ α(x, α) ∈ B, with A, B ∈ Γ and Turing invariant on α. By Burgess and Miller [BM75], we can find A1 , B1 ∈ Γ Turing invariant reducing A, B. [Indeed, let A , B ∈ Γ reduce A, B and then put G
G
G
A1 (x, α) ⇔ ∃α ≡T αA (x, α ) B1 (x, α) ⇔ ∀α ≡T αB (x, α).] G
G
Now let x ∈ P1 ⇔ α(x, α) ∈ A1 and x ∈ Q1 ⇔ α(x, α) ∈ B1 . Then P1 , Q1 ∈ ∀R Γ and they reduce P, Q. The proof of (ii) is similar, utilizing the equivalence ∃αB(x, α) ⇔ α∃ ≤T αB(x, ). G
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Appendix B. We prove here the fact that AD + V=L(R) ⇒ B∞ = ℘ (R). Assume AD + V=L(R). Assume also that B∞ = ℘ (R), towards a contradiction. Pick then A ⊆ R such that A ∈ B∞ . As every set of reals is ordinal definable from a real, let ϕ(x, , α) be a formula and α ∈ R, 0 ∈ Ord be such that x ∈ A ⇔ ϕ(x, 0 , α0 ). Fix now α0 and pick the least ordinal such that {x ∈ R : ϕ(x, , α0 )} ∈ B∞ . Clearly this is definable from α0 , so we conclude that there is a set of reals B definable from α0 , say x ∈ B ⇔ (x, α0 ), such that B ∈ B∞ . By Skolem-Lowenheim, let < Θ be least such that L (R) |= ZFN +DC+AD + {x : (x, α0 )} ∈ B∞ . Here ZFN is a large enough finite fragment of ZF. Let C = {x ∈ R : L (R) |= (x, α0 )}. We have that L (R) |= C ∈ B∞ . But then we claim that actually C ∈ B∞ . Indeed, if Δ = {D ⊆ R : L (R) |= D ∈ B∞ } and = sup{ : is the length of a Δ prewellordering of R}, then Δ is closed under wellordered unions of length . To see this, let {A } < be a sequence of Δ sets. Let S ∈ L (R) be such that all D ∈ Δ are Wadge reducible to S (S exists since L (R) |= B∞ = ℘ (R)). Let C = {ε : ε codes a continuous function fε and fε−1 [S] = A }. Notice now that there is a norm : R in L (R), thus by the Moschovakis Coding Lemma there is a function h in L (R) such that ∀ < (h( ) = ∅ ∧ h( ) ⊆ C ). Consequently, {A } < = { ε∈h( ) fε−1 [S]} < ∈ L (R), so < A ∈ Δ. Since Δ is closed under unions, clearly Δ is closed under arbitrary wellordered unions, thus Δ ⊇ B∞ , so C ∈ B∞ . We shall complete the proof by showing that C ∈ Δ21 . This leads immedi . To see that C ∈ Δ2 ately to a contradiction, since Σ21 = S(∞) and so C ∈ B ∞ 1 notice that x ∈ C ⇔ ∃ < Θ[L (R) |= (ZFN +DC+AD + {x : (x, α0 )} ∈ B∞ ) ∧ is least with that property ∧ L (R) |= (x, α0 )].
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As structures L (R), for < Θ, can be coded in a straightforward fashion by sets of reals, this shows that C ∈ Σ21 and a similar computation shows that C ∈ Π21 , so we are done. REFERENCES
John W. Addison and Yiannis N. Moschovakis [AM68] Some consequences of the axiom of definable determinateness, Proceedings of the National Academy of Sciences of the United States of America, no. 59, 1968, pp. 708–712. Howard S. Becker [Bec85] A property equivalent to the existence of scales, Transactions of the American Mathematical Society, vol. 287 (1985), pp. 591–612. J. Burgess and D. Miller [BM75] Remarks on invariant descriptive set theory, Fundamenta Mathematicae, vol. 90 (1975), pp. 53–75. Alexander S. Kechris [Kec74] On projective ordinals, The Journal of Symbolic Logic, vol. 39 (1974), pp. 269–282. [Kec77B] Classifying projective-like hierarchies, Bulletin of the Greek Mathematical Society, vol. 18 (1977), pp. 254–275. [Kec81B] Suslin cardinals, κ-Suslin sets, and the scale property in the hyperprojective hierarchy, in Kechris et al. [Cabal ii], pp. 127–146, reprinted in [Cabal I], p. 314–332. Alexander S. Kechris, Benedikt Lowe, and John R. Steel ¨ [Cabal I] Games, scales, and Suslin cardinals: the Cabal seminar, volume I, Lecture Notes in Logic, vol. 31, Cambridge University Press, 2008. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. Donald A. Martin [Mar68] The axiom of determinateness and reduction principles in the analytical hierarchy, Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687–689. [Mar71B] Projective sets and cardinal numbers: some questions related to the continuum problem, this volume, originally a preprint, 1971. Donald A. Martin and John R. Steel [MS83] The extent of scales in L(R), in Kechris et al. [Cabal iii], pp. 86–96, reprinted in [Cabal I], p. 110–120. Yiannis N. Moschovakis [Mos70] Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory. Proceedings of an international colloquium held under the auspices of the Israel Academy of Sciences and Humanities, Jerusalem, 11–14 November 1968 (Y. Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam-London, 1970, pp. 24–62.
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[Mos74] Elementary induction on abstract structures, North-Holland, 1974. [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. Joseph R. Shoenfield [Sho67] Mathematical logic, Addison-Wesley, 1967. Robert M. Solovay [Sol78B] The independence of DC from AD, in Kechris and Moschovakis [Cabal i], pp. 171–184. John R. Steel [Ste81A] Closure properties of pointclasses, this volume, originally published in Kechris et al. [Cabal ii], pp. 147–163. [Ste81B] Determinateness and the separation property, The Journal of Symbolic Logic, vol. 46 (1981), no. 1, pp. 41– 44. John R. Steel and Robert Van Wesep [SVW82] Two consequences of determinacy consistent with choice, Transactions of the American Mathematical Society, (1982), no. 272, pp. 67–85. Robert Van Wesep [Van78A] Separation principles and the axiom of determinateness, The Journal of Symbolic Logic, vol. 43 (1978), no. 1, pp. 77–81. [Van78B] Wadge degrees and descriptive set theory, this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 151–170. DEPARTMENT OF MATHEMATICS CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA 91125 UNITED STATES OF AMERICA
E-mail: [email protected] DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CALIFORNIA 94720-3840 UNITED STATES OF AMERICA
E-mail: [email protected] E-mail: [email protected]
POINTCLASSES AND WELLORDERED UNIONS
STEVE JACKSON AND DONALD A. MARTIN
§1. Introduction. Our basic notation and terminology is that of [KSS81]. In particular, recall that a boldface pointclass is a pointclass containing all clopen sets and closed under continuous preimages. A. S. Kechris [KSS81] proved the following theorem, assuming the Axiom of Determinacy (AD) and the Axiom of Dependent Choice (DC): Let Γ be a boldface pointclass closed under countable unions, countable intersections, and ∃R but not under ∀R . If Red(Γ), then Γ is closed under wellordered unions. Kechris’ theorem raises two sorts of questions: Q1. Does Kechris’ theorem remain true when we replace “but not under ∀R ” by “and under ∀R but not under complements”? It seems strange that a failure of closure is needed to prove another kind of closure. The second question is, roughly: Are there indeed any interesting wellordered unions for Γ to be closed under? If Γ is closed under countable unions, complements, and Γ has the countable intersections, and ∃R but not under prewellordering property, then it follows easily from AD+DC and the Moschovakis Coding Lemma [Mos80, 7D.5] that Γ is closed under wellordered Γ prewellordering. unions of length ≤ κ, where κ is the length of any Q2. Can one show (assuming AD+DC) that, whenever A : < is a strictly increasing sequence of Γ sets and Γ is as above, then there is a Γ prewellordering of length ? The second question is especially important when Γ is the class of κ-Suslin sets, for then it bears on the problem of reliable cardinals. A set A is κ-Suslin if there is a scale on A of length ≤ κ. κ is a Suslin cardinal if some set is κ-Suslin but not -Suslin for any < κ. A cardinal κ is reliable if there is a scale on some set A of length exactly κ. The second author was supported in part by NSF Grant #MCS 78-02989. Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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To see how Q2 relates to the problem of whether all reliable cardinals are Suslin, assume DC and suppose that there are no non-trivial unions of κ-Suslin sets of length κ + , i.e., suppose that there is no strictly increasing sequence A : < κ + of κ-Suslin sets. Let be the least reliable cardinal greater than κ. We show that is Suslin. If not, let ϕi : i ∈ be a scale of length on a set A. By DC, κ + does not have cofinality . Hence some ϕi has length ≥ κ + . We may then assume that κ + ⊆ ran ϕi . Let A = {x : x ∈ A & ϕi (x) ≤ } for < κ + . Since each A is -Suslin, we may suppose that each A is κ-Suslin. But the sequence A : < κ + is strictly increasing, contradicting our hypothesis. Following Kechris [Kec81B], we are led to the following special case of Q2: Q2 . If the class of κ-Suslin sets has the scale property, does it follow from AD+DC that there is no strictly increasing sequence of κ-Suslin sets of length κ + ? The genesis of this paper was as follows: Jackson answered Q1 positively (with the minor change of strengthening the assumption Red(Γ) to PWO(Γ)). Jackson’s method was completely different from Kechris’. By a minor variant of his proof, Jackson also answered Q2 positively when the κ-Suslin sets are closed under ∀R as well as ∃R . Martin next mixed Jackson’s methods with other ideas to settle Q2 positively for a large class of successor κ. J. Steel remarked that Martin’s proof could be modified to deal with many limit cardinals, assuming a certain technical lemma. Kechris then proved this technical lemma. These results, together with Steel’s analysis [Ste83] of the scale property in L(R), were sufficient to answer Q2 in L(R) and also to show that in L(R) all reliable cardinals are Suslin. (See [Ste83].) Chuang [Chu82] used Jackson’s methods to produce a full positive answer to Q1. He proved several other results about wellordered unions and Suslin cardinals. We used these further results to give a full positive answer to Q2 for κ of cofinality greater than . (The case κ = probably does not arise, but this is not yet proved.) In §2 we present the results on classes closed under both ∃R and ∀R . In §3 we deal with the case κ a successor cardinal. In §4, we indicate how to modify the proof of §3 when κ is a limit cardinal. (We state, but do not prove, Kechris’ lemma. See [Ste83] for a proof.) In §3 and §4, we make use of the results of [Chu82], [Kec81B], and [KSS81]. This is purely to make our results general. The reader not familiar with these papers may think of a concrete case (e.g., Γ = Σ12 in §3), whereupon obvious from older the lemmas using results from these papers will become standard results.
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§2. Inductive-like pointclasses. Theorem 2.1 (AD+DC). Let Γ be a boldface pointclass closed under ∃R and ∀ but not under complements. Then either Γ or Γ˘ is closed under wellordered unions. R
Proof. We let Γ be as in the hypothesis of the theorem and assume that ˘ are not closed Γ and Γ under wellordered unions. We let ϑ1 denote the least such that, for some ϑ -sequence of sets in Γ, A , < ϑ , we have ordinal 1 1
A = ( <ϑ1 A ) ∈ Γ. We similarly let ϑ2 be the least ordinal such that Γ˘ is not closed under wellordered unions of length ϑ2 . We let Γdenote the pointclass obtained by taking ϑ1 -unions of sets in Γ, that is B ∈ Γ if there exists a sequence B , < ϑ1 , such that B ∈ Γ for < ϑ1 and B = <ϑ1 B . We are then assuming that Γ ⊂ Γ. Hence itfollows (from Wadge’s lemma) ˘ ˘ ⊆ Γ. Hence we may that Γ write a Γ complete set, which we also denote by A as A = <ϑ1 A where A ∈ Γ for < ϑ1 . Since Γ is not closed under complements, we get a coding of Γ sets by reals using a universal Γ set. We let B = {x : x codes a Γ subset of A}. It is easy to see that B is Γ˘ using the closure hypotheses (which imply that Γ is closed under countableunions and intersections). Hence we may write B = <ϑ1 B
where B ∈ Γ. We now let A(2)
= {x : ∃y(y ∈ B and x belongs to the Γ set coded by y)}. (2) (2) We then have that A(2)
<ϑ1 A since each A ⊆ A and
∈ Γ and A = (2)
<ϑ1 A ⊇ A for each < ϑ1 since each A is a Γ subset of A. We may further assume that the sequence A(2)
is strictly increasing in the sense that (2) A(2)
< A for all < ϑ1 .
⊃ Now the union <ϑ1 A(2)
has the property that it is Γ-bounded, that is, if (2) B ⊆ <ϑ1 A and B ∈ Γ, then for some < ϑ1 we have B ⊆ A(2) . This (2) follows from the definition of the A . We now play the game where player I plays a real x and player II plays reals y, z. We say that player II wins provided that if x codes a Γ subset of (2) A = <ϑ1 A(2)
then y is a Γ-code for some A , < ϑ1 , where is larger than (2) (2) the least ordinal such that A(2)
⊇ the set coded by x, and z ∈ A − < A . We then have that player II has a winning strategy for the game above, for, if not, then there would be a Σ11 set C of codes of Γ subsets of A with the some member of the property that for any < ϑ1 there is an x ∈ C such that (2) 1 set coded by x does not belong to A . Since Σ1 ⊆ Γ we would then have a
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Γ subset of A unbounded in A = <ϑ1 A(2)
. This contradicts the fact that (2) A = <ϑ1 A is Γ-bounded. Hence we may assume that player II has a winning strategy s. We let B = ˘ and is Γ˘ -complete (otherwise A {x : x codes a Γ subset of A}, and so B ∈ Γ would be in Γ). We define an ordering on Bby x1 < x2 ⇔ x1 ∈ B & x2 ∈ B & s(x2 )2 ∈ the set coded by s(x1 )1 , where s(x)1 and s(x)2 are the y and z respectively played against x according ˘ and is easily seen to be a prewellordering. (x1 < to s. The ordering is in Γ (2) x2 ⇔ 1 < 2 , where s(x1 )1 codes A(2)
1 and s(x2 )1 codes A 2 .) It follows from the regularity of ϑ1 that the rank of the prewellordering < is ϑ1 . ˘ is closed under It now follows from the Coding Lemma and the fact that Γ R ˘ is closed under wellordered unions of length ϑ1 . Hence ϑ1 < ϑ2 . ∃ that Γ The argument above, however, is symmetric in Γ and Γ˘ , and hence we also get ϑ2 < ϑ1 . This contradiction establishes that either Γ or Γ˘ is closed under (Theorem 2.1) wellordered unions. Corollary 2.2 (AD+DC). If Γ is a boldface pointclass closed under ∃R and has the Prewellordering property, then Γ ∀ but not under complements and is closed under wellordered unions. R
Proof. This follows from the fact that, since Γ has the Prewellordering ˘ cannot be closed under wellordered unions. property, Γ (Corollary 2.2) Corollary 2.3 (AD+DC). The class of inductive sets is closed under wellordered unions. The class of inductive sets also has the scale property. We now turn to the question Q2 for such pointclasses. Theorem 2.4 (AD+DC). Let Γ be a boldface pointclass closed under ∃R and ∀R but not under complements. Let κ be a cardinal and suppose that some complete Γ set admits a Γ norm of length κ. Suppose also that every set in Γ is κ-Suslin.There is no strictly increasing sequence of Γ sets of length κ + . Our last two hypotheses hold if Γ has the scale property via a scale of length other hypotheses hold and κ is a Suslin κ. They also hold by [Chu82] if the cardinal and Γ = S(κ). Proof. We let Γ satisfy the hypotheses of the theorem and assume that A for < κ + is a strictly increasing sequence sets. We may assume of Γ . We have that A ∈ Γ A ⊃ < A for all < κ + . We let A = <κ+ A
from Corollary 2.2. Since there is a Γ norm of length κ on a complete Γ set, we have a Γ coding for ordinals < κ, which we denote by |x| for x ∈ P, the complete Γset. By the Coding Lemma, using the closure properties of Γ, we
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have that every subset X of κ is Γ in the codes, that is {x : |x| ∈ X } ∈ Γ. Thus there is a Γ prewellordering of length for each < κ + . We choose a Γ coding of Γ relations and let C = {x : x codes a wellfounded relation}. ˘ . We then play the game where player I plays x and player We have that C ∈ Γ II plays y, z and player II wins provided x ∈ C ⇒ y is a Γ code for some A , < κ + where > the rank of the wellfounded relation coded by x, and z ∈ A − < A . We have that player II has a winning strategy, for, if player I had a winning strategy, then there would be a Σ11 set of codes for wellfounded relations unbounded in κ + , and, since Σ11 ⊆ Γ, there would be a Γ wellfounded relation of height κ + . This, however, contradicts the fact that Γ is κ-Suslin, 2G.2]. since κ-Suslin wellfounded relations have height < κ + by [Mos80, Hence player II has a winning strategy s. We then consider the relation defined by x1 < x2 ⇔ x1 ∈ C & x2 ∈ C & s(x2 )2 ∈ the set coded by s(x1 )1 . We have, as before, that < is a Γ˘ prewellordering. It follows from the regularity of κ + (which follows from theCoding Lemma and the fact that κ + = sup{ : is the height of a Γ wellfounded relation}) that < has length κ + . It then ˘ is closed under wellordered unions of follows from the Coding Lemma that Γ + length κ , and hence κ, which contradicts the existence of a Γ norm of length (Theorem 2.4) κ on a complete Γ set. §3. κ-Suslin sets for κ a successor cardinal. For each cardinal κ, let S(κ) be the class of all κ-Suslin sets. Theorem 3.1 (AD+DC). Let κ be a Suslin cardinal such that κ = + for some cardinal . There is no strictly increasing sequence A : < κ + such that each A ∈ S(κ). Proof. We begin by cataloguing some useful facts about κ and S(κ). Lemma 3.2. Let be the supremum of the Suslin cardinals < κ. is a Suslin cardinal. Proof of Lemma 3.2. Since κ is a successor cardinal, κ has cofinality greater than . By [Kec81B], S(κ) is not closed under complements. ˘ Let ϕi : i ∈ be a scale of length κ on a set A ∈ S(κ)− S(κ). Since κ has cofinality > , one of the ϕi , say ϕn , has length κ. For < α < κ let Aα, = {x : supi ϕi (x) + 1 < α or (supi ϕi (x) + 1 = α & ϕn (x) ≤ )}. It is easily seen that Aα, is a |α|-Suslin. Since κ has cofinality > , A = <α<κ Aα, . If ϕn (x) = , there is an α, < α < κ, such that x ∈ Aα, and x ∈ Aα,
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for any < and x ∈ Aα , for any α < α and any . Thus there are κ distinct sets Aα, . If we order {(α, ) : κ > α > } lexicographically, we arrange the Aα, in an increasing sequence. Thus we have shown that there is a strictly increasing sequence Bα : α < κ of sets such that each Bα is -Suslin for < κ. Fix, for the moment, a Suslin cardinal < κ, by [KSS81], either ∀R S() ˘ or ∀R ∃R S() contains S(), is closed under ∀R and countable unions and intersections, and has the prewellordering property. Let Γ be whichever of these classes satisfies these conditions. If 1 and 2 are the next two Suslin ˘ ˘ cardinals after , ∃R S() ⊆ S(1 ) and ∃R ∀R S() ⊆ S(2 ). Thus every Γ + prewellordering has length < 2 , by [Mos80, 2G.1]. By [Chu82], there is no strictly increasing sequence of Γ sets of length 2 ++ . Hence either there is a greatest Suslin cardinal < , and we are done, or there is no strictly increasing sequence of length of sets in S(). We may then assume that, for each Suslin cardinal < κ, there are fewer than of the Bα which belong to S(). Since there are at most Suslin cardinals < κ, we get the contradiction that there are no more than of (Lemma 3.2) the Bα . ˘ Lemma 3.3. The cardinal is a Suslin cardinal. S( ) has the prewellordering ˘ ˘ property via an S( ) norm of length κ. S(κ) = ∃R S( ). S(κ) has the scale property. Proof of Lemma 3.3. Since κ has uncountable cofinality and is the greatest Suslin cardinal < κ, the lemma follows from [Chu82, Theorem 5] and its proof. (Lemma 3.3) ˘ Lemma 3.4. S( )∩ S( ) ⊇ Bκ , the closure of the open sets under wellordered unions of length < κ, wellordered intersections of length < κ, and complements. Proof of Lemma 3.4. The proof of [Mos80, 7D.9] essentially proves the lemma. (Lemma 3.4) Lemma 3.5. Let be the -closed unbounded filter on κ. is a normal, κ-complete ultrafilter. If j is the embedding associated with the ultrapower by , j(κ) = κ + . Proof of Lemma 3.5. The proofs are like those of [KSS81, Theorems 11.2, 14.3]. (Lemma 3.5) Our next aim is to introduce the Kunen trees associated with S(κ) wellfounded relations. These trees provide us with our method for representing ordinals < κ + . Let R ⊆ ()3 belong to S(κ) and be universal for S(κ) subsets of ()2 . Let T be a tree on × × × κ witnessing that R is κ-Suslin.
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Let (i, j) → nij be a one-one surjection of × onto . If ∈ let i (j) = (nij ) if nij < lh . Similarly define i (j) for ∈ <κ. We define our Kunen tree T by T = {( , , ) : lh = lh = lh & ∀i, k ≤ lh (if i k, i+1 k, and i k are all defined, then ( k, i+1 k, i k, i k) ∈ T )}. For x ∈ let Tx = {(, ) : (x lh , , ) ∈ T }. For α < κ, let T α = {( , , ) : ( , , ) ∈ T & ∈ <α}. Similarly define Tx α (= (T α)x ).
Lemma 3.6 (Kunen). The relation Rx is wellfounded if and only if Tx is wellfounded. If Rx is wellfounded, |Rx | ≤ |Tx |, where | | is the height function for wellfounded relations. Proof of Lemma 3.6. A branch through Tx is essentially an infinite descending chain in Rx with witnesses. For the second assertion, let Rx∗ = {((y, f), (z, g)) : ∀i(xi, yi, zi, fi) ∈ T }. Clearly the tree of all finite descending chains in Rx∗ can be embedded in Tx . Hence |Rx∗ | ≤ |Tx |. It is not hard to see that |Rx | = |Rx∗ |. (Lemma 3.6) Choose a universal S(κ) set and so a coding of S(κ) sets by elements of . + Assume, contrary to the theorem, that +A : < κ is a sequence of S(κ) sets with < A ⊂ A for each < κ . We play a game as follows: Player I chooses x ∈ . Player II chooses y ∈ and z ∈ . If Rx and Ry are both wellfounded, player II wins just in case |Tx | < |Ty | and z is a code for A|Ty | . If Rx and Ry are not both wellfounded, let α < κ be the least ordinal such that Tx α or Ty α is not wellfounded. Player I wins ⇔ Tx α is wellfounded. Lemma 3.7. Player I has no winning strategy. Proof of Lemma 3.7. Suppose s is a winning strategy for player I. Let α < κ. Let Bα = {y : ∀ < α(Ty is wellfounded and |Ty | < α)}. ˘ Bα ∈ Bκ and so Bα ∈ S( ) ∩ S( ). Let S be a tree on × witnessing that ∗ Bα is -Suslin. Let T be the tree defined as follows: T ∗ = {( 0 , 0 , 0 , , , ) : ( 0 , 0 ) ∈ S & ( , , ) ∈ T α &
agrees with the reply to ( 0 , 0 ) according to s}. Since s is a winning strategy Tx α is wellfounded whenever x = s(y, z) and y ∈ Bα . Thus T ∗ is wellfounded. Since T ∗ is a tree on max(α, ), |T ∗ | < κ. It is easy to see that if y ∈ Bα and x = s(y, z), then |Tx α| ≤ |T ∗ |. We have thus shown that there is an f(α) < κ (namely, |T ∗ |) such that if player II plays a y ∈ Bα and player I plays according to s, then |Tx α| < f(α). Let C ⊆ κ be closed, unbounded and satisfy ( ∈ C & α < ) ⇒ f(α) < .
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Now let ∈ j(C ), > κ. Let h : κ ↔ be a bijection. Let ϕ be a S(κ) norm of length κ. Let (z, w) ∈ E ⇔ h(ϕ(z)) < h(ϕ(w)). E ∈ S(κ). Let E = Ry . Ry is wellfounded and |Ry | = . Thus |Ty | ≥ . Let player II play y and z coding A|Ty | . Let player I play x according to s. There is a closed, unbounded set of α < κ such that y ∈ Bα . For each α in this set, |Tx α| is less than the next member of C after α. By the normality of , |Tx | is less than the next element of j(C ) after κ. Thus |Tx | < |Ty |. It follows that the play is a win for player II, a contradiction. (Lemma 3.7) Let t be a winning strategy for player II. Lemma 3.8. There is a closed, unbounded C ∗ ⊆ κ + such that, if ∈ C ∗ , player I plays x with |Tx | < , and player II plays according to t, then |Ty | < . Proof of Lemma 3.8. If α, < κ, let Bα = {x : Tx α is wellfounded and |Tx α| < }. Bα ∈ Bκ . As before we get an f(α, ) < κ such that if x ∈ Bα and player II plays y according to t, then |Ty α| < f(α, ). Let C ⊆ κ be closed and unbounded such that ( ∈ C & α < & < ) ⇒ f(α, ) < . Now let C ∗ = j(C ) − (κ + 1).
(Lemma 3.8)
Lemma 3.9. For each ∈ C ∗ , {x : Tx is wellfounded and |Tx | < } ∈ ˘S(κ). Proof of Lemma 3.9. For each x, let (y(x), z(x)) be player II’s play against x according to t. Let ∈ C ∗ and w ∈ A − < A . Suppose Tx is wellfounded. Then |Tx | < ⇔ |Ty(x) | < ⇔ w does not belong to the set coded by z(x). ˘ ˘ This last condition is S(κ). Since S(κ) is closed under ∀R , ˘ {x : Tx is wellfounded} = {x : Rx is wellfounded} ∈ S(κ). ˘ Since S(κ) is closed under intersections, the lemma is proved.
(Lemma 3.9)
Lemma 3.10. For every sufficiently large < κ + , every S(κ) set is Wadge reducible to {x : Tx is wellfounded and |Tx | < }. ˘ Not a Proof.1 Let ϕ be an S(κ) norm of length κ on a set H ∈ S(κ)−S(κ). Let R∗ = {(x, y, z) : y ∈ H & z ∈ H & ∃y ∃z (ϕ(y ) = ϕ(y) & ϕ(z ) = ϕ(z) & (x , y , z ) ∈ R)}. 1 This proof, as published in the original paper in 1983, is faulty. A corrected version may be found in the newly added §5.
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Define Rx∗ in the obvious way. Rx∗ is thus just Rx fixed up to be well defined on ordinals as coded by ϕ. Let f be continuous such that, for each (x, y), Rf(x,y) = {(z, w) : z ∈ H & w ∈ H & (x ∈ H ∨ (ϕ(x) > ϕ(z) & ϕ(x) > ϕ(w))) & (z, w) ∈ Ry∗ }. Let V = {(x, y) : x ∈ H & Ry∗ ∩ {(z, w) : ϕ(z) < ϕ(x) & ϕ(w) < ϕ(x)} is wellfounded}. To see that V ∈ S(κ), note that (x, y) ∈ V ⇔ ∃α < κ(ϕ(x) = α & Ry∗ ∩ {(z, w) : ϕ(z) < α & ϕ(w) < α} is wellfounded). Thus V is a union of κ sets, so it suffices by Lemma 3.3 and the Coding Lemma to show that each of these sets belongs to S(κ). {x : ϕ(x) = α} ∈ ˘ S(κ) ∩ S(κ), so it is enough to show that the second conjunct defines a set in S(κ). {(x, y) : Ry∗ ∩ {(z, w) : ϕ(z) < α & ϕ(w) < α}} is induced by a relation on α. If it is wellfounded, it has height < κ. Thus the second conjunct defines a union of κ sets of the form {(x, y) : Ry∗ ∩ {(z, w) : ϕ(z) < α & ϕ(w) < α} has height < }. ˘ These sets all belong to Bκ and so to S( ) ∩ S( ). For each (x, y) ∈ V , Rf(x,y) is wellfounded. If {|Tf(x,y) | : (x, y) ∈ V } were unbounded in κ + , we could, using the fact that V is κ-Suslin, put together all the Tf(x,y) for (x, y) ∈ V to get a wellfounded tree of height κ + . Thus let κ + > > |Tf(x,y) | for all (x, y) ∈ V . As in the proof of Lemma 3.7, let y be such that Ry = Ry∗ and is a wellordering of order type ≥ . x ∈ H ⇒ (x, y) ∈ V ⇒ Tf(x,y) wellfounded & |Tf(x,y) | < . Also x ∈ B ⇒ Rf(x,y) = Ry∗ ⇒ |Rf(x,y) | ≥ ⇒ |Tf(x,y) | ≥ . ˘ We have thus shown that H ∈ S(κ) − S(κ) is Wadge reducible to the required set. (Lemma 3.10) Lemmas 3.9 and 3.10 give the contradiction which proves the theorem. (Theorem 3.1)
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§4. κ-Suslin sets for κ a limit cardinal of uncountable cofinality. Theorem 4.1. Let κ be a Suslin cardinal, and a limit cardinal of uncountable cofinality. Assume that S(κ) has the reduction property. There is no strictly increasing sequence A : < κ + such that A ∈ S(κ). Note. The hypothesis that S(κ) has the reduction property can be eliminated. Proof. We prove the theorem from a lemma of Kechris proved in [Ste83]. Lemma 4.2. The cardinal κ is a limit of Suslin cardinals. Proof of Lemma 4.2. The proof Lemma 3.2 shows that either there is a greatest Suslin cardinal < κ or else κ is a limit of Suslin cardinals. In the (Lemma 4.2) former case, [Chu82, Theorem 5] implies that κ = + . Lemma 4.3. S(κ) has the scale property. Proof of Lemma 4.3. This follows from [Chu82] and [KSS81]. (Lemma 4.3) Lemma 4.4. There is an -closed, unbounded set of α < κ such that ˘ Bα + ⊆ S(α) ∩ S(α). Proof of Lemma 4.4. If is a Suslin cardinal < κ, there is a < κ such that every set in S() admits a scale of length which belongs to S( ). To see this, let A ∈ S() and let ϕi : i ∈ be a scale on A of length . Since {x : x ∈ A & ϕi (x) < α} and {x : x ∈ A & ϕi (x) ≤ α} are both -Suslin for each i ∈ and α < , {x : x ∈ A & ϕi (x) = α} is a difference of -Suslin sets. If 1 is the next Suslin cardinal after , {x : x ∈ A & ϕi (x) = α} ∈ S(1 ) ˘ 1 ) is closed under ∃R , under for each i ∈ and α < . Either S(1 ) or ∃R S( countable unions and intersections, and has the prewellordering property, by [KSS81]. Thus one of these classes is closed under wellordered unions, by Theorem 2.1 and the Kechris theorem quoted in §1. Hence we may take as the least Suslin cardinal > 1 . There is an -closed unbounded set of < κ such that is a limit of Suslin cardinals and, if 1 < , there is a < such thatevery set in S(1 ) admits a scale in S( ). For in this -cub set, let Λ = < S( ). Let Γ be the collection of countable unions of members of Λ . It is easily seen that every set in Γ admits a Γ scale of length ≤ . By first periodicity, ∀R Γ has the prewellordering property. Thus, as in Lemma 3.4, B + ⊆ ∀R Γ ∩ ∃R Γ˘ . But ˘ = S(), ˘ ∃R Γ˘ ⊆ S(). By [Mos80, Theorem 2E.2], S()∩ S() ⊆ B + , so ∃R Γ and the lemma is proved. (Lemma 4.4) Lemma 4.5 (Kechris). There is a measure on κ such that extends the closed, unbounded filter and, in the ultrapower by , the function f(α) = α + represents κ + .
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For a proof, see [Ste83]. We define the Kunen tree T exactly as in §3. The rest of the proof is like that in §3. Lemma 3.6 goes through as before. Our game is as before. The analogue of Lemma 3.7 is proved as before, except that f(α) < α + and we choose such that < κ + and > the ordinal represented by f. The analogue of Lemma 3.8 is proved as before except that f(α, ) is defined for α in the set given by Lemma 4.4 and < α + , and f(α, ) < α + . We let Cα ⊆ α + be closed and unbounded such that ( ∈ Cα & < ) ⇒ f(α, ) < . We let C ∗ be the subset of κ represented by g(α) = Cα . The analogue of Lemma 3.9 is proved as before, as is the analogue of Lemma 3.10. (Theorem 4.1) §5. Addendum (2010). The proof of Lemma 3.10 contains an error which we fix here. A correction for this was pointed out in [Jac90B], and Theorem 3.1 was there also extended a ways past S(κ) (in particular, the statement of Theorem 3.1 holds when the A are Boolean combinations of S(κ) sets). Hjorth [Hjo01], using techniques of inner-model theory, later obtained an optimal result showing that there are no 2 increasing sequences of sets A with each A in the pointclass –Π11 . The error in Lemma 3.10 occurs in the statement that the κ sequence of sets defined towards the end of the proof of the lemma are in Bκ . In fact, there doesn’t seem to be any obvious reason why this should be the case. The correction given in [Jac90B] uses ˘ the scale property for the pointclass S( ) of Lemma 3.3. While this can be shown from just AD, the arguments are more involved than those for just ˘ getting the prewellordering property for S( ). Because of this, we present here a correction which only needs the prewellordering property as stated in Lemma 3.3. Let T be the Kunen tree of Lemma 3.6, which we now view as a tree on × κ. So, {|Ty | : Ty is wellfounded } is unbounded in κ + . We define an auxiliary tree T as follows. As in the incorrect proof of Lemma 3.10, let ϕ be an S(κ) norm of length κ on an S(κ)-complete set H . For α < κ, let ˘ Hα = {x ∈ H : ϕ(x) = α}, so Hα ∈ S(κ) ∩ S(κ). From the coding lemma, the tree T is S(κ) in the codes with respect to the norm ϕ. That is, the relation G
B(a, z) ⇔ a = a0 , . . . , ak & z = z0 , . . . , zk & z0 , . . . , zk ∈ H & ((a0 , . . . , ak ), (ϕ(z0 ), . . . , ϕ(zk ))) ∈ T is in S(κ). Let U be a tree on × × κ with B = p[U ] (we make the slight abuse of allowing the integer a to be regarded as a real). Let V be a tree on ××κ with p[V ] = {(z, x) : z ∈ H & (x ∈ / H ∨(x ∈ H & ϕ(z) < ϕ(x)))}.
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T will be a tree on × × × κ × κ such that a branch (x, y, z, f, g) through T will be such that f witnesses that for all i that (yk, z0 , . . . , zi ) ∈ p[U ] (here z codes the reals z0 , z1 , . . . ) and g witnesses that (zi , x) ∈ p[V ] for each i. More precisely, (s, t, u, , ) ∈ T ⇔ lh(s) = lh(t) = · · · = lh( ) & ∃x, y, z, f, g extending s, t, u, , [∀i, k (yk, z0 , . . . , zi k, fi k) ∈ U ) & ∀i, k (zi k, xk, gi k) ∈ V ] is wellfounded iff where fi (j) = f(i, j) and likewise for g. Clearly Tx,y Ty ϕ(x) is wellfounded (where ϕ(x) = ∞ if x ∈ / H ). Also, an easy argument as in the proof of the Kunen-Martin theorem shows that |Tx,y | ≥ |Ty ϕ(x)|. For x ∈ / H , this becomes: |Tx,y | ≥ |Ty |. Define
W (x, y) ⇔ x ∈ H & Ty ϕ(x) is wellfounded ⇔ ∃α, < κ (x ∈ Hα & |Ty α| < ) ˘ by the closure of Δ under Now, {y : |Ty α| < } ∈ Bκ ⊆ Δ = S(κ) ∩ S(κ) S(κ) is closed under wellorderedunions < κ unions and intersections. Since (we actually only need κ unions here, which follows from the coding lemma), we have W ∈ S(κ). We can therefore put together all the trees Tx,y for (x, y) ∈ W to obtain a single wellfounded tree on κ which therefore has rank less than κ + . So, let 0 < κ + be such that 0 > |Tx,y | for all (x, y) ∈ W . Let y be such that Ty is wellfounded and |Ty | > 0 . If x ∈ H , then (x, y) ∈ W | < 0 . If x ∈ / H then Tx,y is still wellfounded (since Ty is) and and |Tx,y |Tx,y | ≥ |Ty | > 0 . Thus we have shown that the tree T has the property that every S(κ) set is Wadge reducible to {(x, y) : Tx,y is wellfounded & |Tx,y | < } for + all sufficiently large below κ . This corrects Lemma 3.10 of the paper. is wellfounded } Also, T has the “Kunen tree property” that {|Tx,y | : Tx,y + is unbounded in κ . Thus, the previous lemmas of the paper go through without change using T in place of the tree T of the paper. REFERENCES
Chen-Lian Chuang [Chu82] The propagation of scales by game quantifiers, Ph.D. thesis, UCLA, 1982. Gregory Hjorth [Hjo01] A boundedness lemma for iterations, The Journal of Symbolic Logic, vol. 66 (2001), no. 3, pp. 1058–1072.
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Stephen Jackson [Jac90B] Partition properties and well-ordered sequences, Annals of Pure and Applied Logic, vol. 48 (1990), no. 1, pp. 81–101. Alexander S. Kechris [Kec81B] Suslin cardinals, κ-Suslin sets, and the scale property in the hyperprojective hierarchy, in Kechris et al. [Cabal ii], pp. 127–146, reprinted in [Cabal I], p. 314–332. Alexander S. Kechris, Benedikt Lowe, and John R. Steel ¨ [Cabal I] Games, scales, and Suslin cardinals: the Cabal seminar, volume I, Lecture Notes in Logic, vol. 31, Cambridge University Press, 2008. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris, Robert M. Solovay, and John R. Steel [KSS81] The axiom of determinacy and the prewellordering property, this volume, originally published in Kechris et al. [Cabal ii], pp. 101–125. Yiannis N. Moschovakis [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. John R. Steel [Ste83] Scales in L(R), in Kechris et al. [Cabal iii], pp. 107–156, reprinted in [Cabal I], p. 130– 175. DEPARTMENT OF MATHEMATICS UNIVERSITY OF NORTH TEXAS P.O. BOX 311430 DENTON, TEXAS 76203-1430 UNITED STATES OF AMERICA
E-mail: [email protected] DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES, CALIFORNIA 90024 UNITED STATES OF AMERICA
E-mail: [email protected]
MORE CLOSURE PROPERTIES OF POINTCLASSES
HOWARD S. BECKER
We work in ZF+DC+AD. The theory of arbitrary boldface pointclasses, also known as the theory of Wadge degrees, is one of the topics in descriptive set theory (under AD) which has been a major area of study in recent years. This paper is a contribution to that topic. Specifically, it is about closure properties of arbitrary pointclasses. It will be shown that pointclasses closed under countable union and intersection are also closed under quantification by various types of measure and category quantifiers. The canonical reference for descriptive set theory and AD is Moschovakis [Mos80], whose notation and terminology we will generally follow. Oxtoby [Oxt71] is a good reference for the subject of measure and category. Van Wesep [Van78B] is an introduction to Wadge degrees, Kechris [Kec73] is about measure and category in descriptive set theory, and Steel [Ste81A] is a paper which is also about closure properties of arbitrary pointclasses. In this paper we work with the Cantor space, 2, which we also denote by C; all of our results are also valid for . Let denote the product measure on 2, where the measure on each factor space is the usual probability measure m on 2, defined by setting m({0}) := m({1}) := 12 . To begin with, we work solely with the measure ; at the end of the paper we will discuss more general measures. For any B ⊆ C 2 and any x ∈ C, let Bx := {y ∈ C : B(x, y)}. To say that a pointclass Γ is closed under quantification of the form “for a comeager set of y’s” means thatif B ⊆ C 2 is in Γ, then the set {x ∈ 2 : Bx is comeager} is also in Γ. A similar interpretationholds for other quantifiers. A pointclass be nice if it is C-parametrized, contains all open sets, and is closed Γ is said to under continuous preimages, countable unions, and countable intersections. Theorem 1. Let Γ be a nice pointclass. Then Γ is closed under quantification of the forms: (a) For a comeager set of y’s. (b) For -a.e. y. This is the main theorem of this paper. Before proving it we will point out some of its corollaries, which follow very easily. Research partially supported by NSF Grant MCS 82-11328. Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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Corollary 2. Let Γ be a nice pointclass and let Ni be the ith basic open under quantification of the following forms: set in C. Then Γ is closed (a) For a non-meager set of y’s. (b) For a comeager-in-Ni set of y’s. (c) For a non-meager-in-Ni set of y’s. (d) For a set of y’s of positive -measure. Corollary 3. Let Γ be a nice pointclass and let r be a real number. If Γ is closed under either ∃R or ∀R , then Γ is closed under quantification of the following forms (uniformly in r): (a) For a set of y’s of -measure > r. (b) For a set of y’s of -measure ≥ r. Proof. We prove (a), where Γ is closed under ∀R . (A) > r ⇔ ∀z[if (z encodes a G -set Sz and (Sz ) ≤ r) then (A\Sz ) > 0]. By Kechris [Kec73], the set {z : (Sz ) ≤ r} is Borel. (Corollary 3) Theorem 1 also has a different sort of closure property as a corollary. Corollary 4. Let Γ be a nice pointclass. If Γ is closed under ∃R , then Γ is closed under quantification of the form: For uncountably many y. Proof. (For uncountably many y)P(y) ⇔ ∃z [z encodes a perfect tree Tz ⊆ <2 and (for a comeager set of branches y ∈ [Tz ])P(y)]. (Corollary 4) In Corollary 4, closure under ∃ is necessary. If Γ is closed under “for uncountably many”, then it is also closed under ∃R , because if Q(x, y) ⇔ P(x), then ∃xP(x) is equivalent to R
(For uncountably many x, y)Q(x, y). I do not know whether the hypothesis of closure under ∃R or ∀R in Corollary 3 is necessary. Theorem 1 (and its corollaries) were already known for many specific examples of nice pointclasses. Call Γ a Kleene-type pointclass if Γ is nice, Γ smallest is closed under ∀R but not under ∃R and Δ is closed under ∀R . The Kleene-type pointclass is Env(3E), the class of sets Kleene semirecursive in 3E " [Mos67]). To use the terminology of Kechrisand a real (see Moschovakis Solovay-Steel [KSS81], Theorem 1 was known to hold for all inductive-like
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pointclasses, and for all projective-like pointclasses except Kleene-type classes and their duals (that is, except the first class in a projective-like hierarchy of type III). For these classes, Theorem 1 was essentially proved in Kechris [Kec73] (and Corollary 4 was essentially proved in Kechris [Kec75]). It was the open question for Kleene-type classes which motivated this work, but it turns out that Theorem 1 holds in much greater generality than that, as there are many nice pointclasses which are not projective-like, e.g., Env(A, 2E), the " A ⊆ C is smallest boldface Spector pointclass containing A and ¬A, where not Borel (see Moschovakis [Mos67]). Many of the properties of measure and category that were proved in Kechris [Kec73] for the projective pointclasses can be generalized to other classes using Theorem 1 (together with the techniques of Kechris [Kec73]). For example, if Γ = Env(3E), then every Γ relation with non-meager sections has a Δ " uniformization. As usual in the theory of Wadge degrees, the results here are all boldface. I do not know whether Theorem 1 holds for lightface pointclasses, or whether it holds for the particular pointclass Env(3E). But it does follow from the S-m-n Theorem that for nice Γ, Theorem 1 and its corollaries hold for the lightface classes Γ(x), for a coneof x’s. We now prove Theorem 1. Definition 5. For any A ⊆ C, let A˜ := {y : ∃y (y and y agree on all but finitely many coordinates and y ∈ A)}. For any B ⊆ C 2 , let #x )}. Bˆ := {(x, y) : y ∈ (B Lemma 6. (a) For any A ⊆ C, A˜ is either meager or comeager. Moreover, A˜ is meager iff A is meager. ˜ is either 0 or 1. Moreover, (A) ˜ = 0 iff (A) = 0. (b) For any A ⊆ C, (A) Proof. For the measure case, use Kolmogorov’s zero-one law (Oxtoby [Oxt71]). (Lemma 6) Proof of Theorem 1. (a) Let Q be the quantifier “for a comeager set of y’s”. (Thus, QB denotes the pointset {x : Bx is comeager}.) Consider the pointclass QΓ = {QB : B ∈ Γ}. To prove (a) we must show that QΓ ⊆ Γ. Let Since Γ is parametrized, QΓ is also parametrized, hence not self-dual. C ⊆ C be a set in QΓ such that ¬C is not in QΓ. By definition of QΓ, there is a that C (x) ⇔ D is comeager. Let B be the complement D ⊆ C 2 , D in Γ, such x
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of D. Then B ∈ Γ˘ . ¬C (x) ⇐⇒ Dx is not comeager ⇐⇒ Bx is not meager #x ) is comeager (by Lemma 6) ⇐⇒ (B ˆ x is comeager ⇐⇒ (B) ⇐⇒ x ∈ Q Bˆ The pointclass QΓ is closed under Q – this is the Kuratowski-Ulam Theorem (Oxtoby [Oxt71]): Qx1 Qx2 P(x1 , x2 ) ⇐⇒ Qx1 , x2 P(x1 , x2 ). ˆ clearly Bˆ is not in QΓ. Since B is in Γ˘ , Since ¬C is not in QΓ and ¬C is Q B, by definition of Bˆ and the closure properties of Γ, Bˆ is also in Γ˘ . Therefore Γ˘ is not a subclass of QΓ. So by Wadge’s Lemma, QΓ is a subclass of Γ, which completes the proof. (b) The proof of (b) is just like that of (a), using Fubini’s Theorem rather than the Kuratowski-Ulam Theorem. (Theorem 1) We now consider more general measures. For any -finite Borel measure on C, Corollaries 2 and 3 can be deduced from Theorem 1 by the same proof as for . So the only question remaining is whether arbitrary measures satisfy Theorem 1. (Note that Lemma 6 is false for arbitrary measures.) Theorem 7 below gives an affirmative answer to this question for all nice pointclasses which are closed under preimages by Borel-measurable functions. In the original version of this paper, I asked the following question: Is every nice pointclass closed under preimages by Borel-measurable functions? About a year and a half later, John Steel proved that this is indeed the case. The proof of Steel’s theorem (which will not be given here) is similar to the proof of Theorem 2.1 of Steel [Ste81A]. Hence the extra hypothesis on Γ can be removed from Theorem 7. Theorem 7. Let be an arbitrary -finite Borel measure on C, and let Γ be a nice pointclass which is closed under preimages by Borel-measurable functions. Then Γ is closed under quantification of the form: For -a.e. y. By a Borel measure we mean that every Borel set is measurable, hence by AD, every set is measurable (in the completed measure). Call a measure regular if (C) = 1 and for any x ∈ C, ({x}) = 0. Call principal if there is an x ∈ C such that for any A ⊆ C, 1 if x ∈ A (A) = 0 if x ∈ / A.
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Call a measure good if it satisfies Theorem 7; we must show that all measures are good. To prove Theorem 7, we need a result from measure theory (which is a theorem of ZF). Theorem 8. Let 1 and 2 be arbitrary regular Borel measures on C. There is a bijection I : C ↔ C such that: (a) I is a Borel isomorphism, that is, I and I −1 are both Borel-measurable functions. (b) For any (measurable) set A ⊆ C, 1 (A) = 2 (I [A]). Theorem 8 is a combination of Lemma 6.2 of Aumann-Shapley [AS74] and Theorem 1G.4 of Moschovakis [Mos80]. Proof of Theorem 7. Since Γ is closed under preimages by Borel-mea Theorem 8 that if one regular measure is surable functions, it follows from good, then all regular measures are. So by Theorem 1 (b), all regular measures are good. It is trivial that all principal measures are good. Now let be an arbitrary -finite Borel measure on C such that (C) > 0. There is a sequence 0 , 1 , 2 , . . . of measures on C and a sequence r0 , r1 , r2 , . . . of positive real numbers such that for any A ⊆ C, ∞ $ (A) = (ri · i (A)), i=0
and such that each measure i is either regular or principal. Then (For -a.e. y)P(y) ⇐⇒ (∀i ∈ )(For i -a.e. y)P(y), and since each measure i is good, is also good.
(Theorem 7)
REFERENCES
Robert J. Aumann and Lloyd S. Shapley [AS74] Values of non-atomic games, Princeton University Press, 1974. Alexander S. Kechris [Kec73] Measure and category in effective descriptive set theory, Annals of Mathematical Logic, vol. 5 (1973), no. 4, pp. 337–384. [Kec75] The theory of countable analytical sets, Transactions of the American Mathematical Society, vol. 202 (1975), pp. 259–297. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. Alexander S. Kechris, Robert M. Solovay, and John R. Steel [KSS81] The axiom of determinacy and the prewellordering property, this volume, originally published in Kechris et al. [Cabal ii], pp. 101–125.
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Yiannis N. Moschovakis [Mos67] Hyperanalytic predicates, Transactions of the American Mathematical Society, vol. 129 (1967), pp. 249–282. [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. John C. Oxtoby [Oxt71] Measure and category, Springer, 1971. John R. Steel [Ste81A] Closure properties of pointclasses, this volume, originally published in Kechris et al. [Cabal ii], pp. 147–163. Robert Van Wesep [Van78B] Wadge degrees and descriptive set theory, this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 151–170. 4840 FOREST DR., STE. 6-B COLUMBIA, SOUTH CAROLINA 29206
E-mail: [email protected]
MORE MEASURES FROM AD
JOHN R. STEEL
We assume ZF+AD+DC throughout this paper. Let κ < ϑ be regular. We shall define a measure on κκ using some ideas of Kunen. If κ has the strong partition property κ → (κ)κ , then our measure is just the standard strong partition measure. But while the usual proof that κ → (κ)κ requires a coding of elements of κκ with certain boundedness properties, our measure can be obtained without appeal to such properties. We show our measures have the properties of the strong partition measures that allow the Martin-Solovay construction of homogeneous trees to go through. As a corollary, we have that PD is equivalent to the existence of measures of the sort we construct defined on all the projectively coded subsets of any projective ordinal. This answers a question of A. S. Kechris. (Kechris’ question inspired our work here.) After we had done this work (in 1980), Steve 1 Jackson showed that PD implies that if n is odd, then 1n → ( 1n ) n holds with The converse is true respect to projectively coded partitions. (See [Jac11].) by Martin-Solovay, so Jackson’s work also gives a combinatorial equivalent of PD. Fix a norm ϕ : R κ, and let Γ = IND(≤ ϕ, ¬ ≤ ϕ) be the pointclass of relations on R which are inductive in the associated prewellorder and its complement. So Γ is closed under real quantification, and -parametrized. For e ∈ and x ∈ R, let [e]x be the eth relation on reals Γ-recursive in x. Definition 1. For α ≤ κ, we say (e, x) is good up to α iff there is a function α α : α → κ such that (y, z) ∈ [e]x iff fe,x (ϕ(y)) = ϕ(z). fe,x Definition 2. For any e, x, α : (e, x) is good up to α}. fe,x = {fe,x Thus fe,x is a partial function defined on a perhaps improper initial segment of κ. We write fe,x (α)↓ if α ∈ dom(fe,x ), and fe,x (α)↑ otherwise. For any f : κ → κ, we define fˆ : κ → κ by ˆ f( ) = sup fe,x ( + n), n< Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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for all < κ. Now, for A ⊆ κκ, consider the following game GA : Player I Player II
e, x i, y
Here e, i ∈ , and x, y ∈ R. The order of play is digit-by-digit: first e, then i, then alternating the digits of x and y. Player II wins GA iff (a) ∃α < κ (i) fe,x (α)↑, or (ii) fi,y (α)↑, or (iii) fi,x (α) < fi,y () for some < α, or (iv) fi,y (α) < fe,x (α), and if α0 is the least such α, then either (i) or (ii) holds for α0 , or (b) Case (a) fails, and fˆe,x ∈ A. Note that in case (b), we have fˆe,x = fˆi,y . Of course, GA is just the usual strong partition game, but relative to a crude coding for κκ. The usual proof of the strong partition property uses boundedness properties of more subtle codings to simulate a game of length κ in which the ordinals themselves, rather than codes for them, are played. In our arguments, the Recursion Theorem will do the work that was done by boundedness. Definition 3. For A ⊆ κκ, (A) = 1 iff player II has a winning strategy in GA . Lemma 1. (κκ) = 1. Proof. Player II copies player I’s play.
Lemma 2. (A) = 1 ∧ A ⊆ B ⇒ (B) = 1. For C ⊆ κ, we put Cf = {fˆ : f : κ → C }. Lemma 3. (∅) = 0. In fact, if C ⊆ κ is club, then (κκ \ C f) = 0. Proof. Let A = κκ \ C f, and suppose were a winning strategy for player II in GA . Pick t ∈ R so that {W : ϕ(W ) ∈ C } is Δ in t. Consider now plays of GA of the form Player I e, , t Player II ie , ye where player II plays by . We can recursively in , t find an e such that fe , ,t (0)↓ and is the least element of C , and fie ,ye (α)↓ ⇒ fe , ,t (α + 1)↓
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and is the least element of C > fie ,ye (α), and if is a limit ordinal, ∀ < (fie ,ye ()↓) ⇒ fe , ,t ( )↓ ∧ fe , ,t ( ) = sup fie ,ye () . <
(e is found using the S − m − n theorem and the closure properties of Γ.) By the Recursion Theorem, fix e such that fe, ,t = fe , ,t . It is easy to see that if player I plays e, , t, then he defeats . This is a contradiction. Lemma 4. Either (A) = 1 or (−A) = 1. Proof. Let be a winning strategy for player I in GA and a winning strategy for I in G−A . Fix y = , . For i ∈ consider the plays Player I ei , xi Player II i, y of GA by , and Player I ki , zi Player II i, y of G−A by . Given i, we can find effectively an i so that whenever ∀α < κ (fei ,xi (α)↓ and fki ,zi (α)↓), then fi ,y (α)↓ and fi ,y (α) = max(fei ,xi (α), fki ,zi (α)). By the Recursion Theorem, we can fix i so that [i]y = [i ]y . Consider the plays of GA , G−A by , when player II plays i, y for this i. By induction on α fi,y (α) = max(fei ,xi (α), fki ,zi (α)), and all are defined. Thus condition (a) operates in neither game, and fˆi,y is the function determined in both games. So fˆi,y ∈ A as won for player I in G−A , and fˆi,y ∈ / A as won for player I in GA . This is a contradiction. Lemma 5. (A) = 1 ∧ (B) = 1 ⇒ (A ∩ B) = 1. Proof. Let A and B be winning strategies for player II in GA and GB , but a winning strategy for player I in GA∩B . Set y = , A , B . For any (i0 , i1 , i2 ) consider the plays Player I Player II
ei0 , xi0 i0 , y
Player I Player II
i1 , y ei1 , xi1
Player I Player II
i2 , y ei2 , xi2
of GA∩B by ,
of GA by A , and
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of GB by B . Given (i0 , i1 , i2 ), we can find effectively on (i0 , i1 , i2 ) so that fi0 ,y = fei2 ,xi2 and fi1 ,y = fei0 ,xi0 and fi2 ,y = fei1 ,xi1 . By the simultaneous recursion theorem, we can fix (i0 , i1 , i2 ) so that fi0 ,y = fi0 ,y and fi1 ,y = fi1 ,y and fi2 ,y = fi2 ,y . Consider the associated plays of GA∩B , GA , and GB for this (i0 , i1 , i2 ). Since the strategies in question were winning, we see by induction on α that fei0 ,xi0 (α) = fi1 ,y (α) ≤ fei1 ,xi1 (α) = fi2 ,y (α) ≤ fei2 ,xi0 (α) = fi0 ,y (α), and all are defined and ≤ fei0 ,xi0 (α + 1). ˆ which must then be in A and B Thus all f’s above determine the same f, but not A ∩ B, a contradiction. Notice that Lemma 5 implies, by AD, that is countably additive. Now let W be a well order of order type ≤ κ; we consider only order type = κ for simplicity. Using the natural map from W κ ↔ κκ, we can transfer to a measure W on W κ. These measures are compatible with one another, in the following sense: Lemma 6. Let j : W → V be order-preserving and j ∗ : W κ ↔ j”W κ the associated map. Suppose V (B) = 1. Then W ({j ∗ ”(fj”W ) : f ∈ B}) = 1. Proof. Clearly we may assume V = κ, W ⊆ κ, and j is inclusion. So we want to see W ({fW : f ∈ B}) = 1. Let h : W ↔ κ be order preserving, and let h ∗ : W κ ↔ κκ be associated. Let A = h ∗ ”{fW : f ∈ B} and suppose for a contradiction that is a winning strategy for player I in GA . Fix a winning strategy for player I in GB . Fix also a real t so that {(x, y) : h(ϕ(x)) = ϕ(y)}
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and {x : ϕ(x) ∈ W } are Δ in t. Fix y = , , t. For any (i0 , i1 ) consider the plays Player I i0 , y Player II ei0 , xi0 of GB by , and Player I ei1 , xi1 Player II i1 , y of GA by . Given (i0 , i1 ), we can find effectively an (i0 , i1 ) so that ∀ < κ: (1) if ∈ / W , then fi0 ,y ( ) = sup (fei0 ,xi1 ()) <
and fi0 ,y ( + n + 1) = fei0 ,xi0 ( + n), (2) if ∈ W , then fi0 ,y ( ) = max(fei1 ,xi1 ( · h( )), sup (fei0 ,xi0 ())), <
and fi0 ,y ( + n + 1) = fei1 ,xi1 ( · h( ) + n + 1), and (3) fi1 ,y ( + n) = fei0 ,xi0 ( · h −1 ( ) + n). By the simultaneous recursion theorem, we get (i0 , i1 ) such that fi0 ,y = fi0 ,y and fi1 ,y = fi1 ,y . Consider the associated plays of GA and GB . We now see by induction on α < κ that fei1 ,xi1 (α + n) ≤ fi0 ,y ( · h −1 (α) + n) ≤ fei0 ,xi0 ( · h −1 (α) + n) = fi1 ,y (α + n) and that all are defined and ≤ fei1 ,xi1 (α + n + 1). Clearly, fˆi0 ,y (h −1 (α)) = fˆi1 ,y (α), / A. But fˆi0 ,y ∈ B for all α < κ. Since was winning for player I in GA , fiˆ1 ,y ∈ ∗ ˆ ˆ since was winning, and h (fi0 ,y W ) = fi1 ,y . This is a contradiction.
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It is not hard now to show that PD is equivalent to the existence of measures on κκ, defined on all the projectively coded subsets of any projective ordinal κ, which satisfy Lemmas 1–6. To obtain PD from the measures, one proves by induction on n that every Σ12n set is the projection of a projectively-coded, projectively-weakly-homogeneous tree. This is because the properties of the measures W given by Lemmas 1–6 allow one to carry out the proof of Theorem 4.14 of [Jac08]. Martin’s proof of the strong partition property for 1 builds on Solovay’s original use of boundedness arguments to simulate games in which ordinals are played. Solovay showed this way that Δ12 determinacy implies there is an inner model of ZFC + “there is a measurable cardinal”. Martin, Simms, and others extended Solovay’s arguments so as to obtain inner models with many measurable cardinals. (Cf. [Sim79].) It is also possible to replace the use of boundedness in these arguments by the Recursion Theorem, in the style of the arguments above. In this way, one can re-prove the main results of Martin and Simms in this area. REFERENCES
Stephen Jackson [Jac08] Suslin cardinals, partition properties, homogeneity. Introduction to Part II, in Kechris et al. [Cabal I], pp. 273–313. [Jac11] Projective ordinals. Introduction to Part IV, 2011, this volume. Alexander S. Kechris, Benedikt Lowe, and John R. Steel ¨ [Cabal I] Games, scales, and Suslin cardinals: the Cabal seminar, volume I, Lecture Notes in Logic, vol. 31, Cambridge University Press, 2008. John Simms [Sim79] Semihypermeasurables and Π01 (Π11 ) games, Ph.D. thesis, Rockefeller University, 1979. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CALIFORNIA 94720-3840 UNITED STATES OF AMERICA
E-mail: [email protected]
EARLY INVESTIGATIONS OF THE DEGREES OF BOREL SETS
WILLIAM W. WADGE
In this paper, I give an overview/summary of the techniques used, and the results derived, in my 1984 PhD dissertation, Reducibility and Determinateness on the Baire Space. In particular, I focus on the calculation of the order type (and structure) of the collection of degrees of Borel sets. §1. Introduction. I would like in this article to present a overview of the main results of my PhD dissertation, and of the game and other techniques used to derive them. My first thought was to print the entire dissertation but I quickly realized that it was too long—about ten times too long! Hopefully, this condensed version will still be useful. In producing such a drastically shortened account, I have omitted detailed proofs, and many less important or intermediate results. Also, the remaining definitions and results are for the most part given informally. In writing this I have in mind, first, colleagues (whether in Mathematics or Computing) who are not familiar with descriptive set theory but nevertheless would like to learn about “Wadge Degrees”. To make the material accessible to these readers I have included some basic information about, say, Borel sets that will be very familiar to Cabal insiders. However, my hope is that even experts in descriptive set theory may learn something, if not about my results, at least about the manner in which they were discovered. In particular, I would like to give some ‘classic’ notions, such as Boolean set operations, the attention they deserve. As already indicated, the approach will be technical but fairly informal. I will skip many precise definitions and statements of results; firstly, because the details can take up precious space and obscure the important issues; and secondly, because these detailed formulations can be found elsewhere. I would like to acknowledge above all the expert guidance of my PhD advisor, John W. Addison, Jr. He not only suggested the right questions to ask, but also time and again he introduced me to the techniques that in the end allowed me to answer these questions. I am also very grateful for the financial support I received, as a graduate student, from the Woodrow Wilson National Fellowship Foundation, from the UC Berkeley Science Division, and from the Canada Council. Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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Readers who need more precise formulations can find them in the dissertation itself [Wad84], which, if all goes to plan, will soon be published as a book. §2. Definability. My research grew out of a seminar Prof. J. W. Addison, Jr. gave in the theory of definability in the Fall of 1967, at UC Berkeley. The theory of definability (founded, according to Addison [Add04], by Tarski) studies the relationship between the grammatical complexity of definitions and the semantic complexity of the objects (typically sets) that they define. A perfect example is the theorem that formulas whose sets of models are closed under ordinary extension are exactly those equivalent to existential formulas. In definability, it is usually easy to show that an object has a definition of a certain degree of complexity—just come up with it. However, proving the contrary—that no such definition exists—can be extremely difficult. Many of the most important results of definability theory help with this problem by reducing proving nonexistence of a definition to proving existence (of something else). For example, we can prove that there does not exist an existential equivalent of a formula by proving that there exists a model of the formula that can be extended to a model of its negation. §3. Descriptive set theory. In the 1967 seminar we learned of intriguing analogies (due largely to Addison himself) between apparently distinct results in predicate logic, recursive function theory and descriptive set theory. Descriptive set theory is the oldest of these topics, and it grew out of classical analysis. The study of continuity, differentiation and integration, and the limit process revealed the existence of sets (of real numbers) and functions (over the reals) that failed to possess some highly desirable properties. Measure theory, for example, extended the Riemann integral to a much wider collection of functions. Analysts discovered, however, that the axiom of choice implied the existence of functions and sets that are not measurable. On the other hand, every set/function that actually arose in practice was measurable. Similary, it seems that every set actually encountered has the perfect set property—it is either countable or has a perfect subset (and hence has the power of the continuum). Nevertheless, the Axiom of Choice implies the existence of sets without the perfect set property. The conviction grew that any set that was somehow constructible or definable was much better behaved than the mysterious sets that the axiom of choice allows us to produce like rabbits out of a magician’s hat. This led to a systematic study of the ways in which sets of reals can be constructed, and to a comparative study of the power of different ways of defining these sets. For example, the open sets are almost the simplest, and are easily seen to have the two properties just mentioned. So do closed sets (complements of open sets) and in general any finite Boolean combination of open sets.
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§4. The Borel sets. In the 1967 seminar Addison suggested I work on the problem of providing constructive examples to verify the nontriviality of the famous Borel hierarchy. The class of Borel sets is the least class containing all the open sets and closed under complement and countable union. Borel sets can be very complex but they are all measurable and have the perfect set property (as well as many others). The Borel hierarchy is determined directly by the inductive definiton just given. The Borel hierarchy classifies Borel sets according to how many alternations of negation and countable union are required to construct them. The simplest are the open sets and their complements, the closed sets. Traditionally G denotes the class of open sets and F the class of closed sets. At the next level in the Borel hierarchy we find the collection F of countable unions of closed sets and its dual, the collection G of countable intersections of open sets (again, traditional notation). (The dual of a collection of sets is the collection of complements.) Continuing in this way we form the collection G of countable unions of countable intersections of open sets, and its dual F ; the class F and its dual G ; and so on. The finite levels by no means exhaust the Borel sets. The simplest sets not necessarily in any finite level are those that are the union of a sequence of sets each at some finite level. If the individual sets are from higher and higher finite levels, their union will not in general be at any finite level. It is not hard to see that we cannot close out under countable union until we have a level for every countable ordinal. Nor does the class of Borel sets include all sets that can be (somehow) defined. If B is a Borel set and f a continuous function, the image under f of B ({f() | ∈ B}) will not in general be a Borel set. These sets (called analytic sets) are nevertheless all measurable, and all have the perfect set property. Closing the Borel sets out under complementation and continuous image gives us the projective sets, and once again they form a hierarchy. The bottom level of the projective hierarchy consists of the class of analytic sets and its dual. On the next level we find that class of continuous images of complements of analytic sets, together with its dual, and so on as before (the hierarchy has only levels). Almost all examples of sets explicitly defined in analysis are in the lower levels of the projective hierarchy. §5. The analogy with recursive function theory. Recursive function theory is much younger than descriptive set theory. It began with the work of Church and Turing, who formalized the notion of “effective” and showed that (in modern terminology) there are recursively enumerable (r.e.) sets that are not recursive. Recursive enumerability is clearly a form of definability—an r.e. set is ‘defined’ by the Turing machine that enumerates it.
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In 1946 Tarski [Tar00] pointed out an analogy between the analytic sets and the r.e. sets. In this analogy recursive sets correspond to the Borel sets: every set that is r.e. and co-r.e. is recursive, just as (a classical result) every set that is analytic and coanalytic is Borel. However in 1950 Kleene [Kle50] found a pair of r.e. sets that were not separable by a recursive set. This upset the analogy, because (by a stronger form of the classical result) every disjoint pair of analytic sets is separable by a Borel set. Kleene asked his new graduate student Addison to look into the anomaly. Addison discovered that Tarski had, arguably, got the analogy wrong, and that it made much more sense to pair the r.e. sets with the open sets. What Addison [Add54] called the “fundamental principle” of the analogies is the fact that the continuous functions are those that are, in a very natural sense, computable. Addison’s principle is indeed of fundamental importance both for the study of infinite games and of infinite (nonterminating) computations. To explain it, however, I must first introduce the Baire Space. §6. The Baire Space. Originally descriptive set theory, which grew out of analysis, studied sets of real numbers (subsets of the continuum). However the set of real numbers, considered as a topological or metric space, was far from convenient. All the spaces formed by taking the product of the reals with itself a number of times were distinct (not isomorphic), although in terms of foundational issues, these differences were of no importance. For example, it is just as easy to prove that some subset of the real line is nonmeasurable as to prove that some subset of the plane is. Furthermore, the decimal expansion of real numbers is annoyingly irregular; for example, 1.0 and 0.99999999. . . represent the same number. Also, there are no nontrivial sets of reals that are both open and closed. The first descriptive set theorists soon found that by considering only sets of irrational numbers, little was lost and much was gained. The space of irrationals, with the induced topology, (soon called the Baire space) has many clopen sets and is isomorphic to all its finite or countable powers. Furthermore, continued fraction expansion (necessarily infinite) is much better behaved than the decimal expansion. In fact the continued fraction expansion was used so much they eventually worked directly in the space of all -sequences of natural numbers. This is what contemporary mathematicians mean by the Baire space. It is easy to describe the topology of the Baire space directly in terms of -sequences. Given any finite sequence s, the interval of Baire [s] is the set of all infinite sequences that extend s. For example, [7, 3, 2] is the set of all -sequences whose 0-index element is 7, whose 1-index element is 3, and whose 2-index element is 2. The intervals of Baire form a basis for the Baire topology: a set A is open iff every element of A is in an interval included in A.
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§7. Clopen sets as recursive sets. To see the connection between the Baire topology and computability, let us start at the other end and ask, which subsets of the Baire space are recursive? In conventional recursive function theory, a subset of is recursive iff there is a computer M that implements the membership test. This means that given any n, if we give n as the input to M , M will compute for a while, then eventually output either 1 (meaning n is a member) or 0 (meaning it is not). By a computer we mean a deterministic and purely mechanical device. A computer must be finite but may have an unbounded memory. There are several ways of formalizing these ideas, all equivalent; for example, we can take our computers to be Turing machines with infinite (or extendable) tapes, initially blank. We also need an inputoutput convention/protocol; we can agree that the machine will be started with nothing but its input (in binary) on the tape, and that when it halts nothing but the output will be left. There is no real difficulty in carrying this definition over to the Baire space, especially if we use the Turing machine model. An element of the Baire space is infinite but so are the Turing machine tapes. So we can say that a subset A of the Baire space is recursive iff there is a Turing machine that, when started with an arbitrary α (in ) on its tape, eventually halts with either a 0 or 1 on its tape, indicating that α either is (1) or is not (0) an element of A. We should be a bit more precise about the input-output convention. For example, we can specify that the components of the sequence α are written on the right half of the tape, and that when the machine halts the 0 or 1 is under the read head. (We obviously cannot require the machine to erase all its input). The really important point is that the machine must give an answer after only a finite amount of computation, during which it can have examined only a finite number of α’s values (let k be the largest number for which αk is so examined). That means it must give the same answer for any α that agrees with α on at least the first k values. Suppose the machine concludes that α is in A, and let α|k be the sequence α0 , α1 , α2 , . . . , αk−1 . Then not only α, but every element of the interval [α|k] is in A. Similarly, if the machine had concluded that α was not in A, then [α|k] would have to be a subset of −A. Putting it all together, we see that if A is Turing decidable then (1) every element of A is in an interval included in A; and (2) every element of −A is in an interval included in −A. Bearing in mind the fact that the intervals are a basis for the Baire topology, (1) says that A is open and (2) says that −A is open, i.e., that A is closed. This means that decidable subsets of the Baire space are both closed and open—they are clopen sets. Are all clopen sets decidable? No, and it is simple to find a counterexample. Let K be any nonrecursive set of natural numbers and let A be the set of -sequences whose first element is in K . Set A is clearly clopen but not machine-decidable. However there is still a sense in which it is ‘easy’ to decide whether or not α is in A: we ‘just’ look at α0 and ask whether it is
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in K . In other words, A is machine-decidable modulo a countable amount of information (the membership list of K ). We can make this precise by allowing our Turing machines to have an extra, read-only tape with a single sequence written on it. If membership in A can be decided by such a machine, we say that A is recursive in . It is not hard to see that a set is clopen iff it is recursive in some . (Alternatively, we could allow our machines to have a countably infinite number of states, and impose no constraint on their transition functions.) §8. Open sets as r.e. sets. If the recursive sets correspond to the clopen sets, which sets correspond to the r.e. sets? Since a set is recursive iff both it and its complement are r.e., it is likely that the r.e. sets correspond to either the closed or open sets. We know that a set is r.e. if there is a machine that enumerates its elements; but it is not clear how a machine could enumerate an uncountably infinite set of -sequences. There is another definition, however, that does carry over. A set K is r.e. iff it is half decidable, in the following sense: there is a machine M that, given n as input, eventually halts with 1 on its tape iff n is in K . (The machine is not required to halt if n is not in K .) Moving to the Baire space, a set A is analog-r.e. iff there is a machine M that, given α as input, eventually halts with a 1 on its tape iff α in A. Given this definition, it is not hard to see that analog-r.e. is open. Suppose first that A is analog-r.e. and α in A. If we start M with α0 , α1 , α2 , . . . on its tape, M will eventually halt with 1 on its tape. Before halting, M will have had a chance to examine only finitely many components of α, none (for some k) of index k or greater. This means M will do likewise for any α that agrees with α on at least the first k components; in other words, any α in [α|k]. Thus α is in an interval (namely [α|k]) included in A, so A must be open. Conversely, if A is open, we program a machine M (with an extra read-only tape, as above) to examine one by one the components of α until, for some k, [α|k] is a subset of A. §9. Continuous functions as computable functions. Finally, given that r.e. sets correspond to open sets, it should come as no surprise that recursive functions correspond to continuous functions. To see this, consider what it might mean to say that a function over the Baire space is Turing computable. In the case of the natural numbers, we say that a function f from to is computable by M if, for any n, if we give n as input to M , M eventually halts with f(n) as output. To carry this over to the Baire space, we need a protocol that tells us how to give an α to M as input, and how M presents (= f(α)) as output. For input, we can simply write the components of α on half of the tape, as above. Output is not so simple, because it is infinite.
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There are actually (at least) two ways to do this. One is to present using what computer scientists now call a “demand-driven” protocol. This allows us to compute any particular component of f(α) for any particular α. More precisely, M accepts an arbitrary α in and an arbitrary n in as input, computes, and eventually halts with f(α)n as output. The other approach uses what is now called a “data-driven” approach. We present M with α as input, and M computes 0 , 1 , 2 , . . . , in turn, writing them in order. In this protocol M never stops; it is a continuously operating device. Fortunately, it does not matter which protocol we use. In either case it is easy to verify that a function is machine computable (in some ) iff it is continuous. To see this, suppose first that f is continuous and that = f(α). For any k, [|k] is open, so f −1 ([|k]) is also open; and since α is in f −1 ([|k]), α must be in some interval (necessarily of the form [α|j]) in f −1 ([|k]). In other words, the fact that f(α) begins with |k follows from the fact that α begins with α|j. Thus an arbitrary finite amount of information about f(α) follows from a finite amount of information about α. Conversely, if f has this finitary property, it follows immediately that the inverse image of an interval containing includes an interval containing α. Since the intervals are a basis for the Baire topology, f must be continuous. §10. Luzin’s examples. As my contribution to the seminar, Addison set me the problem of proving that each set in a particularly simple series of sets is exactly as complex as its obvious definition. The first set S1 is that of all sequences in which 0 occurs at least once. This is easily seen to be open; could I prove that it is not closed? The next is the set P2 of sequences in which 0 occurs infinitely often. It is a G ; prove it’s not a F . The third, S3 , is the set of sequences in which some number (not necessarily 0) occurs infinitely often (a G ), and the fourth, P4 , is that of all sequences in which infinitely many numbers occur infinitely often. The examples are from the famous 1930 book [Luz30] by the Russian mathematician Nikolai Luzin, one of the founders of descriptive set theory (I have used modern terminology). Luzin and his colleagues had found topological proofs of the ‘properness’ of these sets. Addison wanted to know if I could do better (than a founder of descriptive set theory!). The first set was relatively easy to deal with. Since any initial segment of any arbitrary sequence has an extension in the set (with a 0 in it), it follows that any such arbitrary sequence is a limit point of the set—the set is dense. This means it cannot be closed—the only closed dense set is the entire space. Thus the set of sequences with at least one 0 is a ‘proper’ or ‘true’ open set.
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§11. The difference hierarchy. With the second set—that of all sequences with infinitely many 0’s—things already get much more complicated. Here we need an important result of Hausdorff [Hau57], namely if a set is both G and F , then it must be a difference of a number of open sets. Hausdorff’s difference hierarchy orders these sets (in modern notation, the Δ02 sets) into a sets that are hierarchy of 1 levels. At the third level, for example, are those of the form (G0 − (G1 − G2 )) for open sets G0 , G1 , and G2 . Hausdorff showed that one can identify the level a set appears in the difference hierarchy by taking what he calls adjoins and residues. This involves seeing how far back and forth one can take limits between the set and its complement. For example, the first adjoin of a set A is the set of limit points of A in −A; the first residue is the set of limit points of the first adjoin that are members of A; and the second adjoin is the set of limit points of the first residue that lie in −A. The point at which the residues and adjoins become empty determines the set’s position in the difference hierarchy. The set P2 , however, is dense and codense, so the adjoin/residue process can never terminate. It therefore cannot be a Δ02 set, and so must be a proper or true G . §12. Many-one reducibility. The proofs for S3 and P4 were much more complex, so I started looking for another approach. Addison always impressed on me the importance of analogies, and he pointed me to a paper by Hartley Rogers [Rog59] in which Rogers considers sets of natural numbers that were strikingly similar to those in Luzin’s book. Rogers assumed a fixed system for indexing r.e. sets, and considered the following sets of natural numbers: (1) the set of all indices of nonempty sets; (2) the set of indices of infinite sets; (3) the set R3 of indices of sets that contain the index of an infinite set; and the set of indices of sets that contain infinitely many indices of infinite sets. Rogers proved analogous results about these sets, but not using topology; instead, he used reducibility by recursive function (called “many-one reducibility”). A set K of natural numbers is (many-one) reducible to a set L of natural numbers iff there is a recursive function f such that for any n, n is in K iff f(n) is in L. In other words, the function f allows us to ‘reduce’ the question of n’s membership in K to the membership of another number (f(n)) in L. Since recursive functions are computable, this means that the computability complexity of K is no greater than that of L. Another way to put it is that K is the inverse image of L under a recursive function (namely f). For example, it is not hard to see that set R3 can be defined by an ∃∀∃ formula—in modern notation, it is Σ03 . Rogers showed that the set in question is Σ03 -complete: any other Σ03 set is the inverse image of Roger’s third set under a recursive function. It follows that if this set were also Π03 , then every Σ03 would also be Π03 , and the arithmetic hierarchy would collapse.
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§13. Continuous reducibility. It took no great insight to see that an analogy to many-one reducibility by recursive function might do the trick. Furthermore, given the algorithmic description of the Baire topology discussed above, it was obvious to me that the analogous notion must be reducibility by continuous function: A ≤ B iff there is a continuous function f such that for any α, α ∈ A iff f(α) ∈ B (or, more concisely, A = f −1 (B)). However, if I had not learned to think in terms of infinite games, I probably would not have taken it any farther. To see where games come in, suppose I claim that a set A is reducible to a set B, and you doubt me. I produce a machine that, I claim, computes the reducing function incrementally. You are still skeptical and decide to call my bluff. You start to enumerate the values α0 , α1 , α2 , . . . , of some sequence and demand that I turn on my machine and start enumerating the values 0 , 1 , 2 , . . . , of the corresponding sequence . What follows, to put it melodramatically, is an epic contest between a human and a machine. You will try to discredit the machine, by choosing an α that the machine fails to reduce correctly. You can do this by enumerating an α in A that causes the machine to produce a that is not in B; or by enumerating an α in −A that causes the machine to enumerate a in B. Obviously, in the course of the enumerations my machine has access to your values of α, because they are its input. Conversely, I lose nothing if I allow you, my machine’s opponent, to see the values of as they are produced—because I believe my machine will cope with any α no matter what its origin. §14. The game G(A, B). In that case the real nature of the struggle between you and my machine becomes apparent: it is an infinite two-player game of perfect information. More precisely, given any subsets A and B of the Baire space, the rules of the game G(A, B) are as follows: Players I (you) and II (my machine) play alternately, player I moving first. On each move player I plays a single natural number. On each move player II plays a natural number, or passes (plays nothing). Let α be the sequence of all player I’s moves (necessarily infinite) and be the sequence of all player II’s moves. Player II wins if is infinite and either in B and α in A, or else in −B and α in −A. Otherwise player I wins. And now the moral of the story is apparent: if A is reducible to B by a continuous function, then there is a machine as described above; and this machine can be used to win the game G(A, B). Arguably, the converse is true; if player II has a winning strategy, we can build a machine that implements the strategy, and that machine must compute a continuous function that reduces A to B. To formalize this result, we have to formalize the game. The melodramatic account just given is of course not formal mathematics, but neither is the set
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of rules just given; for example, it refers to the sequence of all of player I’s moves, presumably collected at the ‘end’ of an endless game. In fact, formalizing a game amounts to formalizing what constitutes a strategy for player I, what constitutes a strategy for player II, and what is the result of playing a given strategy for player I against a given strategy for player II. (Addison calls this result the clash of the two strategies.) A winning strategy for one player is one that defeats all strategies for the opponent. Fortunately, this is fairly easy for G(A, B)—one way is to transform it to an equivalent Gale–Stewart game by having the players take turns choosing the components of a single sequence. When we do this (I will omit the details) we find that a strategy for player II is specified by (and we can take it to be) a monotonic function on finite sequences whose result is never longer than its argument. To play using such a function, player II ensures that after each of his moves, the sequence of moves that he made up to that point is the result of applying the function to the ‘history’ of player I’s moves. Since the function is monotonic, player II never has to retract moves (which of course he is not allowed to do). Now suppose that player II has a winning strategy for G(A, B). Given any element α of the Baire space, let ∗ (α) be the union (limit) of the sequence
(), (α0 ), (α0 , α1 ), (α0 , α1 , α2 ), . . . . The limit exists because is monotonic, and is infinite because is a winning strategy. It is then easy to show that ∗ is continuous and reduces A to B. Conversely, suppose that there is a continuous function f that reduces A to B. For any finite sequence s of length k, let (s) be the longest sequence of length at most k that is an initial segment of f(α) whenever s is an initial segment of α. (One can think of t as the output obtained when the machine that computes is given s and allowed to run until (i) more input is needed; or (ii) k output values are produced.) What we just proved (informally) is the following game characterization of ≤; for any subsets A and B of the Baire space: A ≤ B iff player II has a winning strategy for G(A, B). (It is possible to give a concise formal definition of G(A, B) and a short proof of this theorem; the informal approach above was used for expository purposes. In particular, the formalism does not need machines, although I find them a useful heuristic guide.) Then is a winning strategy for player II for G(A, B). §15. Completeness of Luzin’s sets. Once the game was perfected, I was able to make short work of Luzin’s examples. In each case it was relatively straightforward to show that the set in question was complete for its ‘natural’ complexity class, and thus a ‘proper’ member of that class.
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Consider first the set S1 of all sequences in which 0 occurs at least once. To show that it is complete for the open sets, let A be an arbitrary open set; here is the winning strategy for G(A, S1 ). Player II simply plays 1’s until the interval corresponding to player I’s moves is included in A. In other words, until player I has committed himself to A because all infinite extensions of player I’s current (finite) sequence of moves are in A. When (if) player I decisively ‘enters’ A in this sense, player II switches to playing 0’s. If at some point player I enters A in this sense, player II’s final sequence will have 0’s past a certain point and be in S1 . And since player I entered A, his sequence must be in A, and so player II wins. Conversely, if player I’s final sequence is in A, player I must enter A at some point, because A is open. So if player I never enters A, his final sequence will be in −A; and if player I never enters A, player II will never play a 0, and thus player II’s final sequence will be in −S1 . Player II wins in this case as well. Next, we show that P2 (the set of sequences with infinitely many 0’s) is complete for G sets (Π02 sets). Let A be an arbitrary G set, the intersection of open sets G0 , G1 , G2 , . . . . Without loss of generality we can assume that each Gi+1 is a subset of Gi . Here is player II’s strategy for G(A, P2 ). Player II plays 1’s by default, as above, but takes into account every time player I ‘enters’ one of the Gi , in the sense described above. (To avoid pathological cases, assume player II pays no attention to Gi before the ith move.) Every time player I enters at least one new Gi , player II plays a 0. If player I’s sequence ends up in A, it must be in all the Gi ’s, and player II will act on infinitely many entries so that his sequence has infinitely many 0’s and ends up in P2 . On the other hand, if player I’s sequence is not in A player I can enter only finitely many Gi ’s, and player II will play only finitely many 0’s. Either way player II wins. It should be clear now how to proceed with the others; in particular G(A, S3 ) is just countably many copies of G(A, P2 ) played in parallel. Furthermore, it is not hard to continue the Luzin examples through at least the finite levels of the Borel hierarchy. Naturally I was happy at having bested the father of descriptive set theory; but it was all thanks to infinite games, which were unknown in Luzin’s day. §16. The Δ02 degrees. I was so pleased at having done an end run around topological arguments that I had not really thought ahead. highly technical Addison urged me to investigate the structure of the degrees. In particular, I should find out if it was just an accident that Luzin’s examples of proper sets were in fact complete for their classes. In the case of S1 , the complete open set, the question reduced to the analog of Post’s problem. Turing’s original proof yielded a set that was r.e. but not recursive, and complete for the r.e. sets: membership in any r.e. set can be
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decided algorithmically by a machine that has access to an oracle for the halting problem. Post’s problem was to decide whether or not all nonrecursive r.e. sets have the same property (and are therefore Turing equivalent to the halting problem). The reducibility involved here, Turing reducibility, is much coarser than many-one reducibility. If a set K is Turing reducible to a set L, that means in deciding whether or not n is in K we can ask any membership questions we want about L, and do whatever we want with the answers. In many-one reducibility we can ask about only f(n), and must simply repeat this answer. Nevertheless, Muchnik and Friedberg (using the priority method) proved that there are many different Turing degrees of r.e. sets; depressingly many; for example, every finite distributive lattice can be embedded in the r.e. degrees. It was only logical to expect that game arguments (very similar to priority arguments) would show that the class of open sets is similarly distressingly complex. In fact, nothing of the kind emerged. Recall the strategy for G(A, S1 ) with A open; it relied on the fact that the sequence 1, 1, 1, . . . is not in A, but that every finite initial segment has an extension that is in A. If we refer to discussion of the Baire topology given above, we see that 1, 1, 1, . . . is a limit point of S1 . Our strategy for G(A, S1 ) is based on the fact that S1 has a limit point that is not in S1 ; that is, on the fact that S1 is not closed. In other words, any set that is not closed is complete for the open sets; so that in particular, any ‘proper’ open set is complete and of the same degree as S1 . Encouraged by this unexpectedly pleasant result, I began looking at the degrees just beyond those of the degree of the true open sets and the degree of the true closed sets. In terms of the game, what counts is the number of times a player can ‘switch’ between entering a set and entering its complement. For example, suppose that, like S1 , there is a sequence in −B but that every |k can be extended to a sequence k in B. But suppose each k has the property that any initial segment can in turn be extended to something back in −B. Then in the game G(A, B) player II can ‘feint’ towards −B, then if necessary ‘feint’ towards B, and finally if necessary enter −B (the last time is not a feint). Player II can win the game if A is either open or closed, because open or closed sets allow only one feint. A sequence with the property described above is easily seen to be an element of −B that is a limit point of elements of B each of which is a limit point of elements of −B. In other words, an element of the second adjoin of B (in Hausdorff’s terminology). A set has such a point iff its complement is not a difference of two open sets. Furthermore, an easy game strategy shows that such a set is complete for those that are differences of open sets. Continuing in this way, I was able to show that the degree of a Δ02 set is determined by how many back and forth feints are possible, and this in turn
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is entirely determined once one knows which of the residues and adjoins are nonempty. The structure of the Δ02 degrees coincides exactly with that of the Hausdorff hierarchy. The contrast to the situation in recursive function theory is striking; the degrees are almost linearly ordered (the exception being incomparable dual degrees). In fact they are almost (in the same sense) well-ordered, the order type being Ω (= 1 ), the first uncountable ordinal. Furthermore, the nonselfdual pairs of degrees alternate with selfdual degrees. At the very bottom we have the degree of the empty set and its dual. Right above we have the degree of a true clopen set, which is selfdual. This degree has two incomparable successors, the degrees of a true closed set and the degree of a true open set. There is a selfdual degree above them (not discussed), then another dual pair, namely the degree of a proper difference of open sets and its dual. Above them, another selfdual degree, then the degree of a proper ternary difference of open sets and its dual. This is the pattern through the Δ02 sets, with selfdual degrees at (countable) limit ordinals. §17. The determinacy of G(A, B). Once the Δ02 sets were mapped out, I for the G sets. Is there faced the problem of solving the ‘Post’ problem a topological criterion for G -completeness? As it turns out, yes—a set is G -complete if it is comeager on a perfect set (these are classical topological notions). Using this, it is possible to show that any proper G is complete. But then, like Luzin before me, I faced extending this result to the F sets, to the G sets, and so on. Fortunately, games allowed me to make another end run. It just involved examining a little more closely the relationship between the players in G(A, B). We saw that if player II has a winning strategy, then the complexity of A is comparable to that of B: it is either of the same complexity, or strictly less. But what if player I is the one with a winning strategy? The game is not symmetric; the rules are looser for player II than for player I, because player II is allowed to pass. As a result, a strategy for player I cannot be specified by a simple monotonic function on finite sequences. We can avoid this minor complication by considering a ‘fairer’ game GL (A, B) in which player II cannot pass. Then winning strategies for both players are determined by monotonic finite sequence functions (strategies for player I are length preserving, strategies for player II must increase the length by exactly 1). Now suppose again that player I has a winning strategy for G(A, B). Since GL (A, B) is harder for player II, it follows that player I must also have a winning strategy for GL (A, B). This strategy corresponds to a monotonic function for which it is not the case that ∈ B ⇔ ∗ () ∈ A. Simple logic allows us to conclude that ∈ B ⇔ ∗ () ∈ −A. In other words, is a winning strategy for G(B, −A).
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Putting this all together, we see that if G(A, B) is determinate (one of the players has a winning strategy), then either A ≤ B or B ≤ −A. And if all such games are determinate, then ≤ is almost a linear order—given any A and B, A is comparable with either B or its complement. It is not hard to see that this result—called the Semi Linear Ordering principle (SLO)—solves the ‘Post’ problem in its most general form. Suppose that B is an initial class—a collection of sets closed downwards under continuous preimage (so that B in B and A ≤ B implies A in B). It follows directly that any proper element of B—any set in B whose complement is not also in B— is B-complete. In particular, proper F sets are F -complete and proper G sets are G -complete. Once again I was spared intricate topological proofs. Actually, I have simplified the narrative; soon after discovering the game (in late 1967, while Addison’s seminar was still in progress) I realized that its determinateness had the dramatic consequences just described. In fact it took several days for me to convince myself that what is now known as “Wadge’s Lemma” was correct. Initially I expected the degrees of sets of reals to be just as complicated as those of sets of natural numbers. The picture that was emerging seemed too good to be true. But it was true— assuming, of course, that the appropriate G(A, B) games are determinate. §18. The Axiom of Determinacy. How could a game not be determinate? There are no ties, so must not someone win? In fact, the determinacy of infinite games does not follow from a priori reasoning. It is conceivable that given any strategy for player II, no matter how good, there is always a strategy for player I that beats it; and that in turn, given any strategy for player I, no matter how sophisticated, there is always a strategy for player II that outplays it. A priori reasoning guarantees only that at least one player has a collection S of strategies with the following property: given any strategy for the opponent, there is a strategy in S that defeats . There is no reason to think that this player can combine all the strategies in S into a single master strategy that uniformly defeats all comers. This problem was recognized long ago, and Zermelo [Zer13] first proved that all finite games are determined. On the other hand, it is fairly easy, using the axiom of choice, to show that there must be a (Gale–Stewart) game that is not determined. This is hardly a satisfactory state of affairs. One approach is to take the failure of determinacy to be yet one more implausible consequence of the Axiom of Choice (along with, for example, the Banach–Tarski paradox or the existence of nonmeasurable sets). In 1962 Mycielski and Steinhaus suggested [MS62] we drop (or weaken) AC and replace it with the Axiom of Determinacy (AD): every infinite (Gale–Stewart) game is determined. AD certainly has the intuitive plausibility required of
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an axiom; perhaps its consequences are more palatable. We have one very pleasant consequence at hand: AD implies that ≤ is (almost) a linear order. The study of determinacy began in earnest in 1953 when Gale and Stewart formalized infinite games and showed that those with an open winning condition are determinate. By 1964, Morton Davis had shown [Dav64] that all G games are determinate. However, at the time I was working on my dissertation, no-one knew how far determinacy held (without assuming extra axioms). I would not have been surprised had it fizzled out at the third or fourth level of the Borel hierarchy. In 1968, just as I was beginning my research, encouraging news arrived: Martin showed [Mar70] that the existence of a measurable cardinal implied that all analytic (and hence all Borel) sets are determined. I therefore decided to assume Borel determinateness and embark on a detailed analysis of the degrees of Borel sets. My choice was vindicated in 1975, three years after finishing the research, when Martin [Mar75] proved Borel determinacy. Once again, too good to be true, but true nevertheless. §19. Degree arithmetic. The structure of the Δ02 sets suggested (correctly, as it turned out) that the degrees are semi-wellordered. That means that in principle, at least, we can define operations on the degrees that correspond to ordinal operations like successor, limit, sum, and product. Do these operations have game characterizations? I was able to show that many of them do—in fact enough of them to allow me to find the exact order type of the degrees of the Borel sets. The simplest operator acts as a (countable) least upper bound. Given B0 , B1 , B2 , . . . , we define B0 & B1 & B2 & · · · to be {i}i∈,∈Bi We can think of the game G(A, B0 & B1 & B2 & · · · ) as follows: Player I plays as usual, but player II’s first move i is not part of his sequence. Instead, it is used to single out Bi , so that after this first move the game is like G(A, Bi ), with player II’s first move a pass. It was not hard to verify that & induces a least upper bound operation on degrees. This operation had in fact been studied by one of my predecessors, John Barnes [Bar65]. He and Addison called it the Kalmar union. Barnes showed that the clopen sets are generated by closing {∅, } out under the Kalmar union, forming in the process a hierarchy with Ω levels. Probably the most surprising result is that there is a simple binary addition operation on sets that induces an addition operation on degrees, one that corresponds to ordinal addition. Given two subsets B and C of the Baire space, let B + C be the set { + 1}∈C ∪ {(s + 1)0}s∈Sq,∈B
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(where +1 and s +1 are the sequences formed by adding 1 to each component of and s respectively) . In other words, an element of B + C is either an element of C with 1 added to each component, or a finite sequence of nonzero numbers followed by a 0 followed by an element of B. Addition of sets induces a corresponding operation on degrees, which does indeed act like addition, provided the first argument is a lub degree: a least upper bound of strictly simpler degrees. If b is such a degree, we can show that for any degree d strictly greater than b, d = b + c for some degree c (assuming determinateness, unless we restrict the result to Borel degrees). In particular (as we shall soon see), the ordinal associated with d is the sum of the ordinals associated with b and c. In the game definition of addition 0 is used as a coding device to indicate a switch from C to B. Given any A, we can describe the game G(A, B + C ) as follows: it is like G(A, C ) except that player II has the option, at any point, of taking all his moves back, at which point the rules switch to those of G(A, B). Player II does not have to exercise this option, and can do it at most once. Player I is not allowed to take his own moves back. In particular, G(A, B + B) is like G(A, B) except that player II has the option of taking all his moves back once. Similarly, B + B + B is B with two take-back coupons, B + B + B + B offers three, and so on. The most powerful operator I found is —set B can be thought of as B + B + B + · · · . With this set, player II can take all his moves back as often as he wants. He can even do so infinitely often, although in that case his sequence is considered as lying outside B + B + B + · · · . I was able to show that this operation corresponded to multiplying (on the left) by Ω (not by , basically because the Baire space is not compact). In other words, for suitable degrees b, if degree a is less than both b and its dual, then a is less than b · for some countable ordinal . If we begin with the two minimal degrees, namely ∅ and its dual, we can, using the degree operations just described, generate an initial segment of the degrees of length ΩΩ . In this initial segment, every dual pair is followed by a single selfdual degree and vice versa. At limit ordinals of cofinality we find a single selfdual degee, while at limit ordinals of cofinality Ω we have a dual pair. §20. (α, )-homeomorphisms. I have characterized the operations as “powerful” but in fact they are very weak. The inital segment of order type ΩΩ just described takes us through the Δ03 sets. Conceivably we could go further with more powerful operators, but it was never clear to me what these operators could be. A few years earlier (in the 1950s, most likely) Kuratowski faced a similar problem extending Hausdorff’s result that the difference hierarchy over the open sets exhausts the Δ02 sets. He wanted to extend the result to all levels of the
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Borel hierarchy—to show that for any positive , the difference hierarchy over the Σ0 sets exhausts the Δ0+1 sets. Hausdorff’s original result used adjoins and seen, are closely related to game characterizations. residues which, as we have However it is not at all clear that there are notions analogous to those of residue and adjoin at higher levels of the Borel hierarchy. Instead, Kuratowski developed [Kur58] a very general technique for ‘lifting’ results from lower to higher levels of the Baire hierarchy: (α, )-homeomorphisms. He used in particular (, 0)-homeomorphisms, which are continuous in one direction but limits of limits of limits of . . . of continuous functions in the other. More precisely, a function is class 0 iff it is continuous; class 1 if it is the (pointwise) limit of an -sequence of continuous functions; class 2 iff it is the limit of an -sequence of class-1 functions; and in general, of class iff it is the limit of an -sequence of functions each of which is of class for some < . These homeomorphisms are a kind of point set microscope that blows up the complexity of a set. My plan was to measure the effect of this kind of magnification operation by discovering the corresponding effect on degrees. In other words, suppose R consists of all sets whose degrees are among the first κ degrees. When we enlarge R under the microscope, the result is (hopefully) all sets of degree less than for some ordinal possibly much larger than κ. If we can determine how depends on κ, we can lift our picture of the degrees of the Δ03 sets to give us a corresponding picture of the degrees of the Borel sets. §21. The expansion operations. Kuratowski’s basic result was that every Σ01+ open set is (, 0)-homeomorphic to a Σ01 set on a closed set. More precisely, given any Σ01+ set H , there is a Σ01 set G, a closed set E, and a one-one −1 class- map f from onto E such that f is continuous and H = f −1 (G). (Unfortunately we cannot always take E to be .) Let us adopt the following notation from my dissertation: given any class H of subsets of , the th expansion H of H is the collection of all sets that are, as above, (, 0)homeomorphic to an element of H modulo a closed set. Kuratowski’s basic result, then, is that G is Σ01+ . Kuratowski was able, with the help of (, 0)-homeomorphisms, to extend the Hausdorff difference hierarchy result to all levels of the Borel hierarchy. In fact he proved a stronger result, namely that the th expansion of a particular level of the difference hierarchy over Σ01 is the corresponding level of the difference hierarchy over Σ01+ . We can express this more concisely by letting ∂ denote the th set difference operation, and by extending the classical notation so that G∂ is the set of all -ary differences of open sets. Then Kuratowski’s result is that (G∂ ) = (G )∂ .
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This result is plausible but by no means obviously true. To see the problem, consider the simplest case: showing that (G∂2 )1 is (G 1 )∂2 . This amounts to showing that any difference of Σ02 sets is (, 0)-homeomorphic to the difference set). Let H and H be the Σ0 sets. We of two Σ01 sets (modulo a closed 0 1 2 know that H0 is (, 0)-homeomorphic to an some open set G0, and H1 is (, 0)-homeomorphic to some open set G1 . It does not, however, follow that H1 − H0 is (, 0)-homeomorphic to G1 − G0 . The problem is that the homeomorphism reducing H0 to G0 is not necessarily the same as the one reducing H1 to G1 . To make the argument work, we need to be able to reduce H0 and H1 to G0 and G1 simultaneously (uniformly) using a single (, 0)-homeomorphism. This is in fact possible, and follows from a result of Kuratowski much stronger than the one cited above; namely, that any -sequence of Σ01+ sets can be uniformly reduced to some -sequence of Σ01 sets by a (single) (, 0) homeomorphism (as usual, modulo some closed set). §22. The ordinal jump functions. Our basic strategy is to measure the power of the expansion operators in terms of ordinal functions. The selfdual Borel degrees are well ordered, and so can be enumerated by a sequence. Let Ξ be the domain of this sequence. Of course this sequence omits the nonselfdual pairs, but since they alternate with selfdual degrees, their omission does not (at least at limit ordinals) affect the order type. Thus Ξ cleary deserves to be called the “order type” of the Borel degrees, and by 1971 or so my main objective was to define or at least characterize Ξ. Let r ( ∈ Ξ) be the th selfdual Borel degree. For technical reasons (to simplify the statements of the results) we begin the enumeration at = 1, so that r1 is the collection of all clopen sets that are neither empty nor coempty. It follows easily from the results on degree arithmetic that rκ+ = rκ + r for any positive κ and in Ξ. Next, we define a corresponding Ξ-sequence of initial classes and study the result of expanding these classes. For any in Ξ, R is the collection of Borel sets whose degrees are strictly less than r1+ . Thus R0 is {0, }, R1 is the class of sets that are open or closed; and R2 consists of sets that are the difference of two open sets, or whose complement is. It follows from what was said earlier that RΩ is F ∪ G , RΩ2 consists of sets that are the difference of two G , or whose complements are; and that RΩΩ is F ∪ G . The plan is to show that the R sequence is closed under expansion; to show that for each in Ω there is an ordinal ‘jump’ function ϑ with domain Ξ such that for any in Ξ, (R ) = Rϑ ( )
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This is rather difficult, mainly because the vital properties of the expansion operation (as just described) do not (obviously) extend from the Σ01+ classes to arbitrary collections of subsets of the Baire space. For example, if H is Σ01+ for some , it follows from what we have said that H + = (H ) This equation would allow us to prove that in general ϑ + ( ) = ϑ (ϑ ( )) Unfortunately, there was (and as far as I know, still is) no reason to think that the second last equation holds for arbitrary subsets H of ℘(), even if we add the assumption that H is an initial class (closed under continuous preimage). It is not hard to see that the inclusion H + ⊆ (H ) is true—it follows directly from the fact that the composition of a class function with a class function is class +. However, the opposite inclusion, (H ) ⊆ H + is far from obvious. It suggests that a function of class + can always be factored as the composition of two functions of class and respectively. §23. Boolean set operations. Fortunately, there is a large collection of classes H for which the cited property (and others) do hold—those that can be defined as the range of values of a Boolean set operation applied to an arbitrary -sequence of open sets. A Boolean set operation is an operation on subsets of the Baire space that is, roughly speaking, purely set-theoretical; it makes no use of the underlying topology or other structure. Countable union and intersection are Boolean, as are the difference operations ∂ . On the other hand, the closure and interior operations are not. Our result about expanding differences of open sets can be understood as follows: the 1-expansion operation commutes with the binary difference operation ∂2 on open sets. Our result immediately extends to -expansion for any . It should be clear that, in the place of binary difference, we could put any other Boolean (purely set-theoretic) operation of finite or countable arity. For example, we could prove that a set is a countable union of 8-ary differences of Σ02 sets iff it can be 1-reduced to a countable union of 8-ary differences of opensets. This notion, of a generalized “set-theoretic operation”, was first formalized by Kantorovich and Liveson in 1932 [KL32]. (They called these operations “analytical”, but that term is overused and we prefer “Boolean” as more appropriate.)
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We say that an I -ary set operation Γ over the Baire space (a map from ℘() to ℘()) is Boolean iff, roughly speaking, membership of any α in Γ(H ) is determined once we know, for each i, whether or not α is in Hi . More precisely, Γ is Boolean iff there is a subset K of ℘(I ) such that for any α,
I
α ∈ Γ(H ) ⇔ {i : α ∈ Hi }i∈I ∈ K We can think of K as being the ‘truth table’ of Γ. For example, complementation is a 1-ary Boolean operation, difference is a 2-ary operation, and countable union is an -ary Boolean operation. If I is finite, the I -ary Boolean operations are exactly those that can be constructed using union, intersection and complementation. The first Boolean operation to be studied (other than finite ones and countable union and intersection) was Suslin’s operation A [Sus17]; in our notation, Fα|k A(F ) = α∈ k∈ Sq
for any F ∈ (). (In our terminology, operation A is Sq-ary.) Kuratowski’s uniform reduction theorem allows us to conclude, then, that expansion commutes with any countable Boolean set operation over the open sets. To express this principle as an equation, suppose that Γ is an I -ary Boolean set operation and H a collection of subsets of the Baire space; we define HΓ to be the collection of all sets produced by applying Γ to some I -ary family of H sets. More precisely, HΓ = {Γ(H )}H ∈I H Then Kuratowski’s uniform reduction result allows us to prove that (GΓ ) = (G )Γ (Notice that even Kuratowski’s uniform reduction theorem is a commutativity result; it says that expansions commute with countable product.) §24. G-Boolean classes. The G-Boolean classes are those that are defined or generated by a Boolean set operation in the way that the class of G sets is defined/generated by the countable-union operation. Their importance lies in the fact that they possess the properties we need. A class is G-Boolean iff it is the range of an -ary Boolean set operation applied to sequences of open sets; in other words, iff it is of the form GΓ for some -ary Γ. For example, the class of Π02 sets is the range of the countable-intersection operation applied to -sequences of open sets. In 0 our new notation, the class Π2 is G , which (happily) coincides with the classical notation.
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Suslin proved that any analytic set can be produced by applying the operation A to an Sq-family of closed sets; in our notation, that the class of Σ11 sets is FA . If H is G-Boolean, our additive expansion result then follows directly from the fact that it holds for the open sets, and that expansions commute with Boolean operations and countable product: if H = GΓ , then H + = (GΓ ) + = (G + )Γ = (G ) Γ = (G )Γ = (GΓ ) = (H ) Incidentally, our notational shortcuts obscure the fact we are using Kuratowski’s uniform reduction result. For example, spelling out the omitted steps, we have (GΓ ) = {Γ(G)}G∈G = {Γ(H )}H ∈(G) = {Γ(H )}H ∈(G ) = (G )Γ and the second last step uses uniform reduction. The G-Boolean classes have many other pleasant (and easily verified) properties. They are all nonselfdual initial classes with complete elements. They are closed under countable product and expansion, and furthermore, if H is G-Boolean, then so is HΓ . (Since the class F of closed sets is G-Boolean, it follows that the class of analytic sets, which is FA , is also G-Boolean.) The collection of G-Boolean classes is also very large; both Σ01+ and Π01+ are G-Boolean for any countable , as are Σ11 and Π11 and all the levels of the corresponding difference hierarchy. In fact, it is hard to think of any nonselfdual initial class of Borel sets that is not G-Boolean; for good reason, because, as we shall soon see, there are none. §25. Separated and partitioned unions. Recall that our strategy is to show that in general (Rκ ) is of the form R , and to calculate as a function of κ and . To carry out these calculations, we need to know that Rκ is well behaved in a strong sense, and to prove that R is therefore well behaved as well. The notion of “well behaved” (called “regular” in the dissertation) includes the requirement that Rκ be a union of G-Boolean classes, that the degrees of sets in Rκ are semi-wellordered, and that the initial class of any nonselfdual set in Rκ be G-Boolean. This last requirement is very strong, because it implies that, in particular, for any nonselfdual degree a of a set in Rκ , the initial class In(a) is of the form GΓ for some -ary Boolean operation Γ. In other words, it implies that every such nonselfdual degree is in a sense defined by some Boolean set operation. We prove this by induction, and to do this we show that degrees that are definable in this way are closed under degree operations and expansion. Closure under expansion follows from our commutativity results, but proving closure under degree operations involves finding classical, non-game characterizations of the initial classes of arithmetic combinations of degrees.
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Fortunately, the classical descriptive set theorists had already identified the required notions—namely, separated and partitioned unions. In general, the union of a sequence of sets, or even just two sets, can be much more complex than any of the sets involved—there is no way to predict this. The problem, it seems, is that the two sets may be very close together, which complicates determining membership in their union. However, if they are far enough apart, the problem is manageable. And we can limit the ‘distance’ between them by requiring that there exist a simple set that includes one and whose complement includes the other. We can extend this notion to countable unions by requiring a partition of the space into simple sets, with each union set included in its own partition. The simplest case is that in which the separating sets are clopen; a Δ01 partitioned union of sets from a class C is a set of the form C0 ∪ C1 ∪ C2 ∪ · · · where each Ci is in C, and there is a sequence D0 , D1 , D2 , . . . of disjoint clopen sets such that each Ci is a subset of the corresponding Di , and the union of the D sequence is . In our notation, the collection of all sets of this form is Pt0 (C) (in general Pt (C) is defined similarly, with the sets forming the partition required only to be Δ01+ ). In a sense, we have already met the operation Pt0 ; it is the Kalmar closure of C. In other words, Pt0 (C) is (given simple constraints on C) the least class containing C that is closed under Kalmar union. The Kalmar union, as we have already seen, is closely connected to the degree least upper bound operation, so we already have a link between the classical and game-based notions. If we want to go beyond the lub operations, we need two operators that take us just beyond Pt0 . In general, a Σ01+ -separated union of sets in C is a set of the form C0 ∪ C1 ∪ C2 ∪ · · · where each Ci is in C, and there is a sequence G0 , G1 , G2 , . . . of disjoint Σ01+ sets with each Ci a subset of the corresponding Gi . In our notation, this class − + is Sp+ (C). We also define a dual operator, Sp , defined like Sp except that the complement of the union of the separating sets is added to the union of the Ai ’s. It is easy to see that + − − Sp− (C) = Sp (C )
There is a simple and direct connection between the operators Pt0 , Pt1 , Sp+ 0 and Sp+ 1 on the one hand, and the lub, addition, , and ordinal multiplication operators on the other hand. Suppose first that b is a selfdual degree, the lub of the nonselfdual degrees a0 , a1 , a2 , . . . , and let C be i∈ In(ai ). As we already indicated, In(b) is
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the Kalmar closure of C; which, in turn, is Pt0 (C). Using similar (but more complex) techniques, we canshow that the In(b + 1) is Sp+ 0 (C), that In(b ) + is Sp1 (C), and that Pt1 (C) is ∈Ω In(b · ). These proofs make use of game − arguments and a vital lemma that to the effect that Sp+ 1 (C) ∩ Sp1 (C) is Pt1 (C), − which is in turn the result of closing C out under Sp+ 0 and Sp0 . − We can express this concisely by defining Sp0 (C) to be Sp+ 0 (C) ∪ Sp0 (C), and letting Sp0 (C) denote the th stage in closing C out under Sp0 . Then − In(b · ) = Sp+ (C) ∩ Sp (C) = Pt (C) = Sp0 (C) 1 1 1 ∈Ω
∈Ω
§26. Determining ϑ1 . We are finally in a position to calculate the ordinal jump functions, starting with ϑ1 . As with everything else, we do it by induction. Since (R0 )1 = {∅, }1 = {∅, } = R0 , we see that ϑ1 (0) = 0. Now suppose that ϑ1 (κ) = . What is ϑ1 (κ + 1)? By the definition of ϑ1 , we know (Rκ )1 = R . We need to calculate (Rκ+1 )1 . The class Rκ+1 is the collection of all sets of degree less than rκ+1 (to simplify things, we assume κ is infinite). Since rκ+1 is clearly the lub of rκ + 1 and rκ + 1− , Rκ+1 must be the union of the initial classes of these two degrees, and this in turn (as we have already seen) is Sp0 (In(rκ )), and since In(rκ ) is Pt0 (Rκ ), we see that Rκ+1 = Sp0 (Rκ ) (This is true even when κ is finite.) Taking the 1-expansion of both sides, 1 (Rκ+1 )1 = Sp0 (Rκ ) = Sp1 (Rκ )1 = Sp1 (R )
− − = Sp+ 1 (R ) ∪ Sp1 (R ) = In(r ) ∪ In (r )
However, r and its dual lie (by previous results) just below r ·Ω . Putting it all together, (Rκ+1 )1 = Sp1 (Rκ )1 = Sp1 (R ) = R ·Ω and this in turn implies that ϑ1 (κ + 1) = ϑ(κ) · Ω. Once we verify that ϑ1 is continuous (which we do) the conclusion is that, for positive κ, ϑ1 (κ) = Ωκ It should be clear that we are omitting many important details. In particular, as part of our induction we have to show that the nonselfdual degrees of sets in (Rκ+1 )1 are G-Boolean. This follows from our ‘classical’ characterizations of the initial classes of the results of the degree operations, but not directly.
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§27. The arithmetic degrees. Once I determined ϑ1 , I could calculate ϑn for finite n using the additive property ϑ + (κ) = ϑ (ϑ (κ)). For example, κ Ωκ ϑ2 (κ) = ϑ1 (ϑ1 (κ)) = Ω(ϑ1 (κ)) = ΩΩ , ϑ3 (κ) = ϑ2 (ϑ1 (κ)) = ΩΩ , and in general ··
·κ
ΩΩ
ϑn (κ) = Ω
there being n Ω’s in the ladder of exponents. Now that we know ϑn for every finite n, we have a nearly complete picture of the arithmetic degrees—the degrees of sets appearing at some finite level of the Baire hierarchy. As we have already seen, R0 is {∅, }, R1 is F ∪ G , and R2 is the class of sets that are differences of open sets, or duals of such differences. Let us define A± to be A ∪ A− . Then the first components of the R sequence are as follows R0 = 0± R1 = G ± R2 = (G∂2 )± R3 = (G∂3 )± and in general for any countable , R is the union of both sides of level of the difference hierarchy over the open sets. If we take the 1-expansion of both sides of these equations, we see that RΩ is (F )± and in general RΩ is the union of both sides of the th level of the difference hierarchy over the F sets. Continuing up the Borel hierarchy, for any finite n, (Σ01+n )± is ···Ω R ΩΩ where there are n Ω’s in the ladder of exponents; and the the union of both sides of the κth level of the difference hierarchy over Σ01+n is · · · Ωκ R ΩΩ where again there are n Ω’s in the ladder of exponents. (Note that for infinite , the order type of the collection of nonselfdual degrees of sets in R is itself.) Throwing together all sets at finite levels of the Borel hierarchy gives us an ordinal that could be written ΩΩ
··
·
To define this and subsequent ordinals more precisely, we must use the sequence of so-called “epsilon numbers”. The epsilon numbers are the fixed
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points of exponentiation by . The first, ε0 , is the limit of the sequence , , , . . . . There are uncountably many countable fixed points of exponentiation-by- so that Ω is the Ωth such fixed point; in other words, Ω = εΩ . The collection of degrees of nonselfdual sets at finite levels of the Borel hierarchy therefore has order type εΩ+1 , and Δ0() = Δ01+n = RεΩ+1 n∈ §28. Luzin’s problem. There are, however, many sets in Δ0 that are not because the in Δ0() (not in Δ01+n for any n). At this point I was stumped, classical descriptive set theorists never found a hierarchy for the Δ0 sets—the difference hierarchy over finite level sets collapses. Luzin himselfnoticed this gap and declared the problem worth studying; although as far as I could tell, no one had done so (successfully) before me. Fortunately my study of the degree operations had allowed me to make a good guess, using partitioned and separated unions. It is not hard to check that the Δ0 sets are closed under Δ01+n partitioned these unions exhaust the Δ0 sets? I unions for all n. Does closing under thought so. It would make a great story if I could say I discoveredan elegant game proof of this, but that is not what happened. Instead, I looked very carefully at the classical results concerning (α, )-homeomorphisms and discovered that the result for the Δ0 sets could be obtained by ‘lifting’ the result by closing under Kalmar union. that the clopen sets are generated For any class K, define Sp() (K) to be Sp0 (K) ∪ Sp1 (K) ∪ Sp2 (K) ∪ · · · and let A be the class Δ0() (also called the class of arithmetic sets). My conjecture, which I verified, was that the class of Δ0 sets is the union of the Ω-chain Sp() (A) ⊂ Sp() Sp() (A) ⊂ Sp3() (A) ⊂ · · · ⊂ Sp() (A) ⊂ · · · To calculate the ordinal to which this class corresponds, we need to calculate the power of Sp() in terms of ordinals. To simplify the notation, let be εΩ+1 , so that, as we just saw, A = R . Our original result, proved using degrees, is that Sp0 increases the R index by one; thus Sp0 (A) = R +1 Now take the n expansion of both sides (n finite); on the left we have Sp0 (A)n = Spn (An ) = Spn (A)
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because the arithmetic sets are clearly closed under n-expansion. On the other side, we have
(R +1 ) = Rϑn ( +1) = R Ω n
Ω·
··
Ω +1
with n Ω’s in the ladder of exponents. Thus Sp() (A) is R where is the limit of the sequence +1
+ 1, Ω +1 , ΩΩ , . . . and this is easily seen to be the next epsilon number (fixed point of exponentiation-by-Ω) after . Since is εΩ+1 , must be εΩ+2 . What we have just described is only the first step in closing the arithmetic sets out under Sp() ; the second step corresponds to the ordinal εΩ+3 , the third to the ordinal εΩ+4 , and in general the Δ0 sets are exhausted by the hierarchy A = RεΩ+1 ⊂ RεΩ+2 ⊂ RεΩ+3 ⊂ · · · ⊂ RεΩ+ ⊂ · · · The limit of the ordinal sequence εΩ+1 , εΩ+2 , εΩ+3, . . . , εΩ+, , . . . is εΩ+Ω . This last ordinal is therefore the order type of the (nonselfdual) degrees of Δ0 sets, and Σ0 ∪ Π0 = RεΩ+Ω §29. Determining Ξ. Once Luzin’s problem was solved, I was quickly able to generalize it. First, I found a general rule for ϑ : ϑ ( ) = ε Ω · (1 + ) for positive (ϑ (0) is of course 0). To generalize this result to, say, ϑ 2 , we need (in Veblen’s terminology [Veb08]) the higher derivatives of the epsilon series. The first derivative ε (= ε (1) ) enumerates the fixed points of ε in the same way that ε itself enumerates the fixed points of exponentiation-by-. The ordinal ε0(1) is the limit of the sequence ε0 , εε0 , εεε0 , . . . and can be thought of as εεε·· The function ϑ 2 is given by
·
ϑ 2 ( ) = ε (1) Ω · (1 + )
(for any positive ).
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I generalized this to arbitrary positive (countable) powers of : ϑ 1+ ( ) = ε () Ω · (1 + ) (for any positive ). Since every countable ordinal is a finite sum of powers of , this last result, together with the formula for ϑ1 and the additive rule, allow us to calculate ϑ for any particular . After that, determining the structure of the Borel sets required only tedious calculation. For example, the order type of the collection of degrees of sets in level six of the difference hierarchy over the class of Σ0 + 12 + 8 +4 sets is (after some simplification) Ω6 ε () ε (11) ε (7) ΩΩ because the class in question is + 12 + 8 +4 ± (G ) ∂ 5
Once we pass the arithmetic sets the ordinals get bigger even faster (if “big” and “fast” have any meaning in this context . . . ). In general, to describe the order type of the Δ0 1+ sets, we need the th derivative of the epsilon series. This suggests that Ξ is a fixed point of every countable ordinal derivative of the epsilon series; in otherwords, Ξ is in ε (Ω) , the Ωth derivative of ε, the sequence of all ordinals that are fixed points of ε () for all countable . At this stage the countable ordinals are all out of the running, so that Ω is ε0(Ω) . The next ordinal in the series is ε1(Ω) ; it is the least ordinal greater, for () every countable , than the order type ε(Ω+1) of the Δ0( 1+ ) sets. This, finally, is Ξ. If we assume, just for the moment, analytic determinacy, we can summarize this result with the equation Σ11 ∪ Π11 = Rε (Ω) 1 §30. The Borel degrees. Once the induction takes us through the Borel hierarchy and exhausts the degrees of Borel sets, a very orderly picture emerges (the final pieces of which fell into place in the summer of 1972, when I solved Luzin’s problem). (Of course, assuming Borel determinateness.) To begin with, the degrees are semiwellordered with selfdual degrees at successor and cofinality limit ordinals, dual pairs elsewhere. Furthermore, every nonselfdual degree G-Boolean. In other words (returning to the basic definitions), if a Borel set A is incomparable with −A, then {B : B ≤ A}B∈℘() is of the form {Γ(G)}G∈G . The surprising aspect to this result is that the single set A is somehow associated with a set operation Γ that in turn generates the whole class of sets that are no more complex than A.
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As for the selfdual degrees, they are clearly all least upper bounds of countable collections of nonselfdual degrees, so that even selfdual sets are associated with set operations. This in turn implies that every initial class of Borel sets is a union of GBoolean classes. One surprising consequence is that the expansion B of a class B of Borel sets can be defined directly and simply as B = f −1 (B) : f of class B∈B,f : → . We can also conclude that there are only two kinds of R classes of Borel sets. If is a successor ordinal, or a limit ordinal of cofinality greater than , R is of the form In(a) ∪ In(a)− for some nonselfdual degree a. And if is a limit ordinal of cofinality , R is a union of Rκ sets of the first kind, for countably many κ less than . Even more remarkable is the fact that, on the Borel sets at least, there is a definition of ≤ that makes no reference to games or even continuous functions. The idea, natural enough, is that we can measure the complexity of a Borel set by looking at how powerful a Boolean set operation must be in order to generate the set in question, from an omega sequence of open sets. More precisely, we say that A B iff A is in every G-Boolean class containing B; that is, iff B ∈ GΓ ⇒ A ∈ GΓ for every -ary Boolean set operation Γ. Then it follows from the big induction argument that A ≤ B iff A B, for all Borel sets A and B. This definition would have made perfect sense to the descriptive set theorists of Luzin’s generation. Imagine their surprise had they learned that (on the Borel sets) is well founded, and that for any Borel sets A and B, either A B or B −A! Finally, the obvious question to ask is, does this simple picture generalize beyond the Borel degrees (assuming enough determinateness)? We now know that it does, thanks to the work of Martin, van Wesep, Steel and others, beginning in 1973 with Martin’s proof that AD implies ≤ is well founded. Van Wesep and Steel even spotted an important phenomenon that I had missed (for the Borel degrees): that in every dual pair exactly one of the initial classes has the first separation property [Ste81B]. Of course, by now a great deal more has been discovered about ≤, but I will leave that story for others to tell. REFERENCES
John W. Addison [Add54] On Certain Points of the Theory of Recursive Functions, Ph.D. thesis, University of Wisconsin–Madison, 1954.
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[Add04] Tarski’s theory of definability: common themes in descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic, Annals of Pure and Applied Logic, vol. 126 (2004), no. 1-3, pp. 77–92. John F. Barnes [Bar65] The classification of the closed-open and the recursive sets of number theoretic functions, Ph.D. thesis, UC Berkeley, 1965. Morton Davis [Dav64] Infinite games of perfect information, Advances in game theory (Melvin Dresher, Lloyd S. Shapley, and Alan W. Tucker, editors), Annals of Mathematical Studies, vol. 52, 1964, pp. 85– 101. Felix Hausdorff [Hau57] Set theory, Chelsea, New York, 1957, translated by J. R. Aumann. L. Kantorovich and E. Livenson [KL32] Memoir on the analytical operations and projective sets I, Fundamenta Mathematicae, vol. 18 (1932), pp. 214–279. Stephen C. Kleene [Kle50] A symmetric form of G¨odel’s theorem, Indagationes Mathematicae, vol. 12 (1950), pp. 244– 246. Casimir Kuratowski ´ [Kur58] Topologie. Vol. I, 4`eme ed., Monografie Matematyczne, vol. 20, Panstwowe Wydawnictwo Naukowe, Warsaw, 1958. Nikolai Luzin [Luz30] Lec¸ons sur les ensembles analytiques et leurs applications, Collection de monographies sur la th´eorie des fonctions, Gauthier-Villars, Paris, 1930. Donald A. Martin [Mar70] Measurable cardinals and analytic games, Fundamenta Mathematicae, vol. 66 (1970), pp. 287–291. [Mar75] Borel determinacy, Annals of Mathematics, vol. 102 (1975), no. 2, pp. 363–371. Jan Mycielski and Hugo Steinhaus [MS62] A mathematical axiom contradicting the axiom of choice, Bulletin de l’Acad´emie Polonaise des Sciences, vol. 10 (1962), pp. 1–3. Hartley Rogers [Rog59] Computing degrees of unsolvability, Mathematische Annalen, vol. 138 (1959), pp. 125– 140. John R. Steel [Ste81B] Determinateness and the separation property, The Journal of Symbolic Logic, vol. 46 (1981), no. 1, pp. 41– 44. Mikhail Ya. Suslin [Sus17] Sur une d´efinition des ensembles mesurables B sans nombres transfinis, Comptes Rendus Hebdomadaires des S´eances de l’Acad´emie des Sciences, vol. 164 (1917), pp. 88–91. Alfred Tarski [Tar00] Address at the Princeton University Bicentennial Conference on Problems of Mathematics (December 17–19, 1946), The Bulletin of Symbolic Logic, vol. 6 (2000), no. 1, pp. 1– 44.
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Oswald Veblen [Veb08] Continuous increasing functions of finite and transfinite ordinals, Transactions of the American Mathematical Society, vol. 9 (1908), no. 3, pp. 280–292. William W. Wadge [Wad84] Reducibility and determinateness on the Baire space, Ph.D. thesis, University of California, Berkeley, 1984. Ernst Zermelo ¨ [Zer13] Uber eine Anwendung der Mengenlehre auf die Theorie des Schachspiels, Proceedings of the Fifth International Congress of Mathematicians (E. W. Hobson and A. E. H. Love, editors), vol. 2, 1913, pp. 501–504. COMPUTER SCIENCE DEPARTMENT UNIVERSITY OF VICTORIA VICTORIA, CANADA
E-mail: [email protected]
PART IV: PROJECTIVE ORDINALS
PROJECTIVE ORDINALS INTRODUCTION TO PART IV
STEVE JACKSON
§1. Introduction. In this paper we introduce and survey the theory of the projective ordinals, the 1n , and the theory of descriptions. We work through out in the theory ZF+AD+DC. Recall AD, the axiom of determinacy, is the axiom that every two player integer game is determined. The projective ordinals, introduced by Moschovakis, are defined by (see the next section for more background): 1n = sup{|| : is a Δ1n prewellordering of }. The projective ordinals are important since the theory of the projective sets (assuming projective determinacy), the Σ1n , Π1n sets, is developed in terms of -Suslin if there is a tree T on them. Recall a set of reals A ⊆ is said to be × such that A = p[T ] = {x ∈ : ∃f ∈ ∀n (xn, fn) ∈ T }. Suslin representations are essentially the same thing as scales (see Fact 2.1 of §2) and this forms a central notion in descriptive set theory. The scale property, isolated by Moschovakis, combines a Suslin representation for the set together with a notion of definability for such a representation (we give the definition in §2). A classical result (phrased in modern terminology) is that the pointclasses Π11 and Σ12 have the scale property. In some sense, an present in the Novikov–Kondo proof of the early prototype of thisnotion was 1 uniformization property for Π1 sets. The periodicity theorems of Moschovakis determinacy, the pointclasses Σ1 , Π1 give that, assuming projective 2n 2n+1 also have the scale property (these results cannot be proved in ZF). A consequence is that the Π12n+1 sets are 12n+1 -Suslin, and in fact the 12n+1 -Suslin sets are exactly the Σ12n+2 sets (we need full AD for this direction). So, the 12n+1 are Suslin cardinals, that is, places where new Suslin representations appear. In fact, general scale type arguments along these lines show, assuming AD, that below the supremum of the projective ordinals the Suslin cardinals are Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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precisely the 12n+1 and their cardinal predecessors 2n+1 . The 2n+1 -Suslin sets (cf. Theorem 2.9 below). Thus, the scale property gives sets are the Σ12n+1 a structural representation for the projective sets, and this representation involves the projective ordinals. We refer the reader to [Mos80], [Kec78] for more details and history along these lines. We also give a quick overview in §2. We emphasize that in the modern theory the projective ordinals (particularly the “odd” projective ordinals 12n+1 ) play a much more extensive role than just representations. In fact, the entire modern providing the sizes of the Suslin theory hinges on the properties of the 12n+1 , and in particular on their partition not only to inductively propagate the properties. These properties are used theory, but to analyze the cardinal structure between the projective ordinals. Descriptions are the combinatorial objects which are central to the modern theory of the projective ordinals 1n (and beyond) assuming AD. For example, of the 1 , establish their partition properthey are used to compute the values n the cardinal structure between ties, and provide the framework for analyzing them (in particular to compute the cofinalities of the cardinals). Through the projective ordinals at least, descriptions will be hereditarily finite objects which code how to build ordinals with respect to iterated ultrapowers by certain canonical measures. These measures are defined using the partition properties of the 12n+1 . The proofs of the partition properties for the 12n+1 in turn need the theory of descriptions, but at a lower level. Thus, the entire theory is developed inductively. This has the disadvantage that the theory (at least in its current form) breaks down completely past the least point where we are able to complete the full cycle of inductive arguments. The theory as outlined here and presented in detail in [Jac99] and [Jac88] extends with only trivial modifications to the 1α where α < 1 , and this analyzes the cardinal structure up to ℵ1 . The description analysis works past this point, but description cease to be purely finitary objects. These arguments, which have not yet appeared, are believed to work through the level of the first inaccessible cardinal. On the other hand, results of [Jac91] show that serious problems arise by the time one reaches κ R , the first R-admissible ordinal (which is the ordinal of the inductive sets and corresponds to the least non-selfdual pointclass closed under real quantification). Finding ways to propagate the theory further remains an important program. This paper serves in part as an introduction to the papers [Mar71B], [Kec78], [Sol78A], [Kec81A], and [Jac88] of this volume. It also serves as a selfcontained survey of and introduction to the theory of the projective ordinals. The papers [Mar71B], [Kec78] and [Kec81A] are directly concerned with the theory of the projective ordinals. These papers, written before the theory of descriptions, give the earlier development of the subject. Martin’s paper [Mar71B] represented a fundamental advance in the development of this theory. In fact, many of the ideas of the current program can be found in some
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form in that paper. In this earlier work the values of 11 = 1 (classical), 12 = 2 (Kunen, Martin independently), and 13 = +1(Martin) were com puted. Also, the strong partition relation on 11 was established (Martin), as was the weak partition relation on 13 (Kunen). [Mar71B] and [Kec78] present all of these results except the last, which is the subject of [Sol78A]. It was also shown that the even projective ordinals are obtained simply from the odd by 12n+2 = ( 12n+1 )+ (Kunen, Martin; see [Kec78] and the discussion in the next At this time, the value of the next odd projective ordinal, 1 , was section). 5 unknown as was the strong partition property on 13 . To develop thetheory further requires the notion of a description. In the early 80’s Martin proved a key result which in this area we refer to simply as Martin’s Theorem (Theorem 6.10 of this paper). This theorem gives the existence of the Martin tree which provides an analysis of functions f : 13 → 13 with respect to one of the normal measures on 13 measures on 1 corresponding to the three regu normal (there are three 3 lar cardinals , 1 , 2 below 13 ). Versions of this tree play a similar role for the general 12n+1 . This extended Kunen’s earlier theorem giving the Kunen tree (see Theorem 2.11) which had a similar role for the cofinality normal measure on 12n+1 . Building on this and some joint work with Martin, the author developed descriptions and used them to compute 1 5 . This can be viewed as bringing to fruition a plan developed by Mar and Kunen for computing the projective ordinals via certain ultrapowtin ers by homogeneity measures (again, some of these important ideas can be found Martin’s paper [Mar71B]). We will outline this program in more detail in §3. The notion of a homogeneous tree (see Definition 3.2) is thus important to the overall analysis. This notion arose independently in the work of Kunen and Martin, and the precise formulation was given independently by Kechris and Martin. The concept is explained in detail in [Kec81A]. In hindsight, the earlier theory of the projective ordinals can be viewed as an instance of the more general theory. When these earlier results are recast into the theory of descriptions, the underlying descriptions are rather trivial objects, basically just integers. Nevertheless, one can use these “trivial descriptions” to redo these results. The reader can consult [Jac10] for details on how the trivial descriptions can be used, starting from scratch, to compute 13 , establish the strong partition relation on 11 , and prove the weak partition relation on 13 (technically one starts from the fact that 11 = 1 and the weak partition relation on 11 , both of which are easily shown). Although we discuss here as well, our focus here is showing how descriptions the trivial descriptions allow these earlier results to be unified and generalized into a theory which gives an analysis of all the projective ordinals and beyond.
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Whether one presents the arguments of [Kec78] or [Sol78A] in their original form or as arguments with (trivial) descriptions is mainly a matter of taste, the underlying mathematics is essentially the same. However, there are some points which are important when extending these arguments. The earlier arguments are presented using the theory of indiscernibles for L[x] and the fact that every subset of 1 is (assuming AD) in L[x] for some real x. For example, both Martin’s proof of the strong partition relation on 1 (see [Kec78]) and Kunen’s proof of the weak partition relation on 13 (see [Sol78A]) rely on this method. These arguments are not known to generalize to higher levels. Instead, the description theory relies heavily on partition properties. Partition arguments replace indiscernibility arguments throughout. Descriptions and proofs involving them are thus closely connected to partition properties, and this gives the description theory a more combinatorial nature. In particular, establishing the strong partition relation on the 12n+1 becomes a central part of the inductive analysis. This paper also aims to help bridge the gap between the earlier pre-description theory and the full analysis of [Jac88]. In that rather long and technical paper the general theory of descriptions needed to analyze the general projective ordinal is laid out. Since then, other papers have appeared which present various aspects of this theory. In [Jac99] the complete theory at the first level past the previous results is given. Namely, 15 is computed, the strong partition relation on 13 is proved, and then the weak partition relation on 15 is proved. This constitutes the complete first step of the general inductive analysis. In [Jac88] the framework for the general projective case is presented. The corresponding proofs of the partition properties are not given in [Jac88] as they use the same machinery presented, and proceed as in the corresponding proofs of [Jac99] (using the more general descriptions of [Jac88]). In §2 we present some background material and fix some notation. The material in this section in only a brief overview, and the reader wishing to see further details could consult [Mos80], [Mar71B], and [Kec78]. In §3 we give a outline of how the inductive analysis of the projective ordinals goes. The precise definitions and proofs are not given in this section, rather it is an attempt to give the reader the overall picture. In the remaining sections we attempt to systematically fill in some of the details. In §4 we present a sketch of the “first level theory” using the “trivial descriptions”. This involves calculating 13 , proving the strong partition relation on 11 = 1 , and proving the weak partition relation on 13 . Again, all of these were known results from the earlier theory, but it is perhaps instructive to see them redone using methods that will generalize to the higher levels. Some of the proofs are given completely and others just sketched or illustrated. In [Jac10] the complete proofs can be found. In §5 and §6 we discuss the second level of the theory, which computes 15 , proves the strong partition relation on 13 , and prove the weak partition
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relation on 15 . Now the descriptions will be less trivial objects, although they will still be hereditarily finite objects. To ease the transition from the trivial to the non-trivial descriptions, we present this analysis in two sections. In §5 we introduce the “level-2” descriptions (the trivial descriptions of §4 are the level-1 descriptions), and in §6 these are extended in a rather trivial way to get the set of level-3 descriptions. The distinction is small enough that in [Jac99] it is not even explicitly made. Although the formal difference between the level-2 and level-3 descriptions is minor, they are used in somewhat different contexts. Here the distinction is useful as it allows to introduce all of the combinatorial machinery of descriptions quickly in §5 without the burden of the extra framework needed for full next step of the induction (e.g., defining the Martin tree, analyzing the measures on 13 , proving the measure domination and some of the later ideas we theorems). To illustrate the combinatorics use the level-2 descriptions to solve two ad hoc problems: (1) analyze the cardinal structure of the iterated ultrapower of n by measures from the canonical families of measures W1m , S1m (these measures, defined in §5, are measures on (1 )m ≈ 1 and m+1 respectively) and (2) compute the number of descriptions defined with respect to a fixed sequence of measures. In §6 we say how these descriptions (technically the level-3 descriptions) are used to carry out the next level of the actual inductive analysis. We don’t give complete proofs here, but we present all the necessary ingredients. We give an example of what a general measure on 13 looks like. In §7 we make some comments concerning how the extension to the higher levels goes which should ease the transition to [Jac88]. In §8 we make some concluding remarks including an alternate way to describe the cardinal structure below the projective ordinals which does not use descriptions for the presentation. However, the theory of descriptions is needed to show this alternate formulation works. Nevertheless, this alternate formulation may have applications of its own, as well as serve to make the theory more accessible and applicable. §2. Background and Preliminaries. For general background in descriptive set theory we refer the reader to [Mos80] and [Kec94]. [Kec78] also gives more background related to the projective ordinals. We recall here some of the more important definitions and results, and fix some notation. We let (n, m) → n, m be a recursive bijection from × to . We let n → ((n)0 , (n)1 ) denote the inverse (decoding) map. When there is no danger of confusion we frequently drop the parentheses and just write n → (n0 , n1 ). We also use this notation for variations of these coding maps. Let WO ⊆ be the standard set of codes of countable ordinals, that is, . x ∈ WO iff <x = {(n, m) : x(n, m) = 1} is a wellordering. For x ∈ WO, let |x| be the rank of <x . Let WOα be the set of those x ∈ WO such that |x| < α.
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By a tree on a set X we mean a set T ⊆ <X closed under initial segment. We identify trees on X ×Y with subsets of <X × <Y , that is, we may view an element of such a tree as a pair (s, t) ∈ <X × <Y with lh(s) = lh(t) (lh(s) denotes the length of the sequence s). We extend this convention to longer products as well. If T is a tree on X , we write [T ] = {x ∈ X : ∀n xn ∈ T } for the body of T . If T is a tree on X × Y , we write p[T ] for the projection of T , so p[T ] = {x ∈ X : ∃y ∈ Y ∀n (xn, yn) ∈ T }. If T is a tree on a higher product, say on X1 × · · · × Xm , then in writing p[T ] we should specify which coordinate we are projecting on. If T is a tree on × Y , then for x ∈ we let Tx = {s ∈ <Y : (x lh(s), s) ∈ T } be the section of T at x. If T is wellfounded (i.e., [T ] = ∅) then |T | denotes the rank of T . A frequently occurring case is when T is a tree on × for some ∈ Ord (or T a tree on / p[T ], |Tx | makes sense and will be an × 1 × 2 , etc.). In this case, for x ∈ ordinal < + . We let Tx α = Tx ∩ <α be the restriction of Tx to ordinals less than α. If Tx α is wellfounded and s ∈ Tx α, we let |Tx α(s)| be the rank of s in the tree Tx α. For notational ease we usually identify finite sequences of ordinals s with ordinals , and so write |Tx α()|. This is only for convenience and this assumption cause no harm (see the remark after Theorem 2.11). In fact, using the Brouwer-Kleene ordering on Tx we can assume that each Tx is actually a linear ordering. Recall this ordering is defined by: s
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Fact 2.1. For every cardinal κ, A ⊆ is κ-Suslin iff A admits a κ semiscale iff A admits a κ very-good scale. A pointclass Γ is a collection of subsets of (or ()n ) closed under continuous preimages. A norm ϕ (or prewellordering) is a Γ-norm if the norm relations <∗ϕ , ≤∗ϕ are in Γ where: x <∗ϕ y ↔ x ∈ A ∧ (y ∈ / A ∨ (y ∈ A ∧ ϕ(x) < ϕ(y))) x ≤∗ϕ y ↔ x ∈ A ∧ (y ∈ / A ∨ (y ∈ A ∧ ϕ(x) ≤ ϕ(y))) ˘. A Note that the initial segments of a Γ-prewellordering are in Δ = Γ ∩ Γ ∗ semiscale {ϕn } (or scale) is a Γ-semiscale if all the norm relations
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If {ϕn } is a Π12n+1 -scale on a Π12n+1 set P, then all the initial segments of Δ1 1 1 each ϕn norm are 2n+1 and so have length < 2n+1 (all of the n are easily limit ordinals). Thus, each ϕn has length ≤ 12n+1 and so P is 12n+1 -Suslin. P cannot be -Suslin On the other hand, if P is Π12n+1 -complete, then for any 1 1 < 2n+1 . For if so, then since there is a Δ2n+1 prewellordering of length , the tree on × within the pointclass lemma gives that we may code any coding 1 Σ2n+1 . It would follow then that P ∈ Σ12n+1 , a contradiction. So, 12n+1 is a Suslin cardinal. Moreover, Γ is a non-selfdual pointclass closed under ∀ , ∨, and ϕ is a Γ-norm on a Γ-complete set, then a general argument using the recursion theorem (see [Mos80, 4C.14]) shows that |ϕ| is at least the supremum of the ˘ wellfounded relations. It follows that any Π1 -norm on a lengths of the Γ 2n+1 set has length exactly 1 . This also shows the following Π12n+1 -complete 2n+1 for odd n. fact Fact 2.4. The ordinal 1n is the supremum of the lengths of the Σ1n well founded relations. Alternatively, one can prove the fact using the Kunen-Martin Theorem, which will also work for even n (this proof uses scales, though, while the proof above used only the prewellordering property and so is more general). We recall the Kunen-Martin Theorem. This result was proved independently by Kunen and Martin. Martin’s original proof appears in [Mar71B] (see also [Mos80, 2G.2] for another proof). Theorem 2.5 (Kunen-Martin). If ≺ is a κ-Suslin, wellfounded relation on , then |≺| < κ + .
For the even case of Fact 2.4, recall Σ12n has the scale property. If A ∈ Δ12n , A are actually Δ1 , and thus eachhas then the norm relations of a Σ12n scale on 2n 1 length less than 2n . Easily all of the 1n have uncountable cofinality, and it follows that A is -Suslin for some < 12n . So, every Σ12n set is also -Suslin. a Σ1 wellfounded From the Kunen-Martin Theorem, it now follows that 2n relation ≺ has rank < + for some < 12n . Since 12n is a cardinal, it follows that ≺ has rank less than 12n . note that a Π1 1 For the odd case of the fact, 2n+1 scale on a Δ2n+1 set is actually 1 1 a Δ2n+1 scale, so all the norms have length less than 2n+1 . As before, this every Δ1 , and hence every Σ1 1 shows 2n+1 2n+1 set is -Suslin for some < 2n+1 . Using the Kunen-Martin Theorem we now finish as in the even case. As a corollary of Fact 2.4 we have the following which was shown by Martin for the odd n and Kunen for the even n. Lemma 2.6 (Martin, Kunen). All of the 1n are regular cardinals.
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Lemma 2.6 follows immediately from Fact 2.4 and the following Lemma 2.8. We first make a definition which generalizes the definition of projective ordinal to a general pointclass. Definition 2.7. Let Γ be a pointclass. Then (Γ) is the supremum of the ˘ prewellorderings of the reals. lengths of the Δ = Γ ∩ Γ Lemma 2.8. Let Γ be a non-selfdual pointclass closed under ∃R and ∧. Then the supremum (Γ) of the lengths of the Γ wellfounded relations on is a regular cardinal. Proof. Suppose f : → (Γ) is cofinal, where < (Γ). Fix a wellfounded relation ≺ of rank inthe pointclass Γ. From thecoding lemma, that: there is a A ⊆ fld(≺) × in the pointclass Γ such 1. If A(x, y) then y is a code (via a Γ universal set U , which exists as Γ in non-selfdual) of a Γ wellfounded relation Uy of length |Uy | = f(|x|≺ ). 2. For all α < thereis an x ∈ dom(≺) with |x|≺ = α with x ∈ dom(A). Define (x1 , y1 , z1 ) ≺ (x2 , y2 , z2 ) iff x1 = x2 , y1 = y2 , A(x1 , y1 ), and Uy1 (z1 , z2 ). Then easily ≺ is a wellfounded relation of length (Γ), a contra diction. R Clearly (Γ) ≤ (Γ) for any Γ closed under ∃ and ∧ (actually this holds the definition of (Γ) given in Lemma 2.8 for all Γ with Δ ∧ Δ ⊆ Γ if we use for arbitrary Γ; we need the closure hypothesis to getthat the strict part of a Δ prewellordering lies in Γ). If Γ is closed under ∃R , ∧, and PWO(Γ˘ ), mentioned then the result of Moschovakis above [Mos80, 4C.14] shows that (Γ) = (Γ). general scale and pointclass arguments as the one just given, we Using just can in fact say more. The next result is proved in [Kec78] and exactly places the Suslin cardinals below the projective ordinals. In the statement of the theorem, the cardinals 2n+1 are defined. We will use this notation elsewhere as well. Theorem 2.9. For all n, 12n+2 = ( 12n+1 )+ (Kunen-Martin). Also, 12n+1 = where 2n+1 is a cardinal of cofinality (Kechris). All ofthe 1n are regular cardinals (Martin for odd n, Kunen for even n). The Suslin 1 cardinals below the projective ordinals are exactly 1 = , 1 = 1 , 3 = , 13 = +1 , 5 , 15 , . . . , 2n+1 , 12n+1 , . . . . The correspondingSuslin classes are 1 1 1 S( 2n+1 ) = Σ2n+1 , and S( 2n+1 ) = Σ2n+2 (Martin, Moschovakis). The scale property and Suslin cardinal analysis of Theorem 2.9 can be extended much further. Martin and Steel [MS83] and Steel [Ste83], assuming AD + V=L(R), determine the scaled Levy pointclasses and classify the Suslin cardinals throughout the Wadge hierarchy (a Levy pointclass is a non-selfdual pointclass closed under ∃ or ∀ ). In fact, assuming just AD one can get a + 2n+1 ,
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classification of the Suslin cardinals (see [Jac10] for the arguments; the main new ingredient is due to Martin). Another result which is used frequently in the theory of the projective ordinals, and which uses only general pointclass arguments is the following result of Martin which appears in [Mar71B].
Theorem 2.10 (Martin). Let Γ be non-selfdual, closed under ∀ , ∨ and under wellordered unions of length < (Γ). assume PWO(Γ). Then Δ is closed An immediate consequence of this result is that B< 12n+1 ⊆ Δ12n+1 , where B<κ closed is the smallest collection of sets containing the open and sets and closed under complements and wellordered unions of length < κ. If T is a tree on × 12n+1 and α, < 12n+1 , then Aα, = {x : Tx α is wellfounded of rank < } is in B< 12n+1 by a standard tree computation (see [Kec78]), and so is in Δ12n+1 . This computation is important in many of the arguments involving projective ordinals. We note that the other direction is true also, namely the Δ12n+1 ⊆ B< 12n+1 . This follows from the general Suslin-Kleene Theorem which and \ A are both κ-Suslin then A ∈ B + , using here κ = says that if A κ 2n+1 (see [Mos80, 3E.2]). As we mentioned before, the Kunen tree is an important tool at the bottom level of the projective analysis. We will give some examples of how this is used in the next section. For the sake of completeness, and because of the importance of the Kunen tree and its generalization the Martin tree in the theory of the projective ordinals, we give (following [Jac10]; see Lemma 4.1 and Theorem 4.2) the Kunen tree construction. The result may be stated as follows. Theorem 2.11 (Kunen). There is a tree T on × 1 such that for any f : 1 → 1 there is an x ∈ such that Tx is wellfounded and such that for all α ≥ we have |Tx α| > f(α). Proof. Let S be a tree on × with A = p[S] a Σ11 -complete set. Note that sup{|Sy | : Sy is wellfounded} = 1 as otherwise A would be Borel. Let U be the tree on × 1 defined by: ((a0 , . . . , an−1 ), (α0 , . . . , αn−1 )) ∈ U ↔ ∀i, j < n (i, j < n ∧ ai,j = 1 → αi < αj ) Note that p[U ] = WF, where WF is the set of x ∈ such that the binary relation {(i, j) : x(i, j) = 1} coded by x is wellfounded. View every real ∈ as coding a strategy for player II in a game on in some standard manner (e.g., the response of to s ∈ < is given by (s) where here s → s denote a bijection between < and ). Abusing notation slightly, for s = (a0 , . . . , an−1 ) we write (s) for the play ((a0 ), (a0 , a1 ), . . . , (a0 , . . . , an−1 ))
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by player II following the strategy when player I plays s. If a ∈ , we let (a) ∈ be the real extending all of the (an). Let V be the tree on × × 1 × × given by: (s, a , α, b, c ) ∈ V ↔ ( a , α) ∈ U ∧ (b, c ) ∈ S ∧ ∃ extending s (( a ) = b). Suppose f : 1 → 1 . Consider the game where player I, player II play out x, y ∈ respectively, and player II wins iff x ∈ WF → Sy is wellfounded ∧ |Sy | > sup{f() : ≤ |x|}. By boundedness, player I cannot have a winning strategy for this game [If player I had a winning strategy , then [] would be a Σ11 subset of WF and then easily we have α = sup{|x| : x ∈ } < 1 . Player II could then defeat by playing a y with Sy wellfounded of rank greater than sup{f() : ≤ α}]. So, let be a winning strategy for player II (more precisely, a real coding such a strategy as above). Then V is wellfounded and for all α ≥ we have |V α| > f(α). To see this, let x ∈ WF of rank α. Fix α = (α0 , α1 , . . . ) ∈ α such that (x, α) ∈ [U ]. Let y = (x), so Sy is wellfounded of rank |Sy | > sup{f() : ≤ |x|} ≥ f(|x|) = f(α). Thus, V,x,α,y = Sy has rank greater that f(α). So certainly V α has rank greater than f(α). So, V is our Kunen tree except for the minor fact that it is not quite a tree on ×1 . To get the actual Kunen tree T , simply weave the last four coordinates of V into the second coordinate of T . This does not decrease rank, and so T has the desired property. Finally, we recall some terminology and facts concerning partition relations. As we said before, partition properties play a central role in the theory of the projective ordinals, and in description theory in particular. The classical ˝ Erdos-Rado partition property is stated in the next definition. Recall (κ) denotes the set of increasing functions from to κ. Definition 2.12. κ −→ (κ) if for every partition P : (κ) → {0, 1}, there is an H ⊆ κ of size κ and an i ∈ {0, 1} such that for all f ∈ H we have P(f) = i. We say H is homogeneous for the partition P. We say κ has the weak partition property if κ −→ (κ) for all < κ. We say κ has the strong partition property if κ −→ (κ)κ . Any non-trivial (i.e., > 1) exponent partition relation on κ implies κ is regular, so we henceforth assume that. Also, any infinite exponent partition property is inconsistent with AC, so it is important that we are in the full AD context when discussing such relations. A simple but important reformulation of the partition properties is used heavily. In this reformulation, the homogeneous sets are required to be c.u.b. subsets of κ. In order to be able to do this, we must specify the type of the functions f : → κ being considered. The simplest type is what in [Jac88]
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is called of the correct type. This notion arises naturally in Martin’s proof of the strong partition property on 1 . In [Kec78] the notation C f is used to describe the set of functions from to C of the correct type. Definition 2.13. We say f : → Ord is of uniform cofinality if there is an f : × → Ord which is increasing in the second argument such that for all α < , f(α) = supn f (α, n). We say f is of the correct type if f is increasing, everywhere discontinuous (i.e., f(α) > sup<α f()), and of uniform cofinality . We say f has uniform cofinality almost everywhere with respect to a measure on if there is an f : × → Ord which is increasing in the second argument such that for all α is a measure one set we have f(α) = supn f (α, n). We say f is of the correct type almost everywhere if f is of uniform cofinality almost everywhere, and there is a measure one set A such that fA is strictly increasing and discontinuous. Saying f : → Ord is of the correct type is equivalent to saying that there is a g : × → Ord which is increasing with respect to lexicographic ordering on × which induces f in the sense that f(α) = supn g(α, n). This is also equivalent to saying that there is an increasing h : · → Ord such that f(α) = sup<·(α+1) h(). Both of these terminologies are used frequently. However, as one moves further into the theory of the projective ordinals it becomes important to have the more general notions of type and uniform cofinality. We give the general definition next. Definition 2.14. Let g : → Ord. We say f : → Ord is of uniform cofinality g if there is a function f with domain {(α, ) : α < , < g(α)} which is increasing in the second argument and such that f(α) = sup
Definition 2.15. We say κ −→ (κ) if for every partition P of the functions f : → κ of the correct type, there is a c.u.b. C ⊆ κ which is homogeneous for P. In [Sol78A] an easy argument is given which shows the essential equivalence of these two forms of the partition property. Specifically we have the following. c.u.b.
Fact 2.16. For all , κ, we have κ −→ (κ) implies κ −→ (κ) . Also, c.u.b. κ −→ (κ)· implies κ −→ (κ) . Henceforth, in writing κ −→ (κ) we will be referring to the c.u.b. version of the partition property.
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In particular, the definitions coincide for κ having the weak or strong partition property. More generally, we get a partition property for function of general types. For example, if κ has the strong partition property then for any g : κ → κ we have the c.u.b. version of the partition property for functions of type g. In proving the strong partition property on 1 , Martin established a general result which all subsequent proofs of partition properties from AD have used. It says, roughly, that to show κ −→ (κ) it suffices to have a coding of the functions f : → κ by reals with a certain boundedness property. A general version of this principle (the following general version is proved in [Jac10]; see also [Kec78]) can be stated as follows.
Theorem 2.17 (Martin). Let Γ be non-selfdual, closed under ∃ , ∧, and ˘ ). Assume there is a mapϕ : → ℘( × κ) satisfying the following: PWO(Γ 1. ∀f : → κ ∃x ∈ (ϕ(x) = f). 2. ∀α < ∀ < κ Rα, ∈ Δ = Γ ∩ Γ˘ , where x ∈ Rα, ↔ ϕ(x)(α, ) ∧ ∀ (ϕ(x)(α, ) → = ) . 3. Suppose α < , A ∈ ∃ Δ, and A ⊆ Rα = {x : ∃ < κ x ∈ Rα, }. that ∀x ∈ A ∃ < ϕ(x)(α, ). Then there is a < κ such 0
0
Then κ −→ (κ) .
Remark 2.18. With some additional pointclass arguments one can see that the hypotheses (1)–(3) of Theorem 2.17 and the closure of Γ under ∃ actually ˘ ) if we assume κ is regular (see [Jac10]; imply Γ is closed under ∧ and PWO(Γ ˘ side). if Γ is closed under quantifiers, the statement PWO(Γ˘ ) just defines the Γ We say a coding map ϕ : → ℘( × κ) is -reasonable if it satisfies the hypotheses of Theorem 2.17 (for some Γ). The countable exponent relations 12n+1 −→ ( 12n+1 ) , < 1 , follow from coding map ϕ. Namely, let : → Theorem 2.17 using a straightforward 1 be a bijection. Fix a (regular) Π2n+1 norm on a Π12n+1 -complete set P ⊆ . ∈ P and ((x) ) = . So, is onto 12n+1 . Define ϕ by ϕ(x)(α, ) iff (x)(α) (α) 1 Using Γ = Σ2n+1 , it is straightforward to verify (1)–(3) of Theorem 2.17. In 1 = has the weak partition property, and we may take this as particular, 1 1 the start of our inductive analysis of the projective ordinals. In [Kec78] the countable exponent relation for the even 12n+2 is also shown, a result due to Kunen (this does not use Theorem 2.17 directly, but uses a partition argument on the odd 12n+1 ). Kechris [Kec77A] has also shown the an argument using generic codes for countstrong partition relation on 1 by able ordinals. Kechris and Woodin [KW80] have developed a theory of generic codes for uncountable ordinals which allowed them to show directly the rela1 tion 12n+1 −→ ( 12n+1 ) 2n−1 . Both of these results also use Theorem 2.17. The
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Kechris-Woodin theory of generic codes also has numerous other applications in the theory of the projective ordinals and beyond. A notational convention. We describe a certain notational convention which is used throughout in the theory of descriptions and here as well. It is a convention for making statements about iterated ultrapowers. Suppose 1 , . . . , n is a sequence of measures. Suppose ϑ ∈ Ord and P ⊆ Ord (we view P as a property or statement about an ordinal). We write ∀1 α1 · · · ∀n αn P(ϑ(α1 , . . . , αn )) to abbreviate the following statement: if we fix a function α1 → ϑ(α1 ) representing ϑ with respect to 1 (that is, ϑ is represented by the equivalence class of α1 → ϑ(α1 ) in the ultrapower by 1 ), then for 1 almost all α1 we have that if α2 → ϑ(α1 , α2 ) represents ϑ(α1 ) with respect to 2 , then for 2 almost all α2 we have that . . . , for n−1 almost all αn−1 if αn → ϑ(α1 , . . . , αn ) represents ϑ(α1 , . . . , αn−1 ) with respect to n , then for n almost all αn we have P(ϑ(α1 , . . . , αn )). Thus, the statement ∀1 α1 · · · ∀n αn P(ϑ(α1 , . . . , αn )) is equivalent to saying ϑ ∈ j1 ◦ · · · ◦ jn (P). However, in practice the convention is always used in the form as stated. §3. Outline of the Arguments. We give in this section an overview of the arguments used in the projective ordinal analysis. We present the overall plan and leave it to the remaining sections to fill in more details. As we said before, the arguments are inductive in nature. At stage n in the induction our main inductive assumptions will be the following. We let (0) = 1 and (n + 1) = (n) (ordinal exponentiation). Inductive Hypotheses (stage n): For m ≤ n, 12m+1 = ℵ(2m−1)+1 (for m = 0, 11 = 1 ). 12m+1 for m < n has the strong partition property. 1 2n+1 has the weak partition property. In addition to these main inductive hypotheses at stage n we also carry along several more technical induction hypotheses. Specifically, we also assume: 1. 2. 3.
Auxiliary Inductive Hypotheses: (a) For m ≤ n, 12m+1 is closed under ultrapowers. That is, if is a measure and α < 1 1 on ϑ < 12m+1 2m+1 , then j (α) < 2m+1 . 1 (b) For m ≤ n there is a Δ2m+1 coding of the subsets of 2m+1 . That is, there is a map from onto ℘( 2m+1 ) such that for all α < 2m+1 we have that {x ∈ : α ∈ (x)} ∈ Δ12m+1 . (c) For all regular cardinals κ < 2n+1 there is a set P ⊆ , and a homogeneous tree T on × , ≤ 2n+1 , with P = p[T ] and such that if {i }i∈ is the semiscale on P from T , then {0 (x) : x ∈ P} is an unbounded
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subset of κ (recall i (x) is the ith coordinate of the leftmost branch of Tx , for x ∈ P). (d) Every measure on 2n+1 is Δ12n+1 in the codes given by (b). That is, {x : ((x)) = 1} ∈ Δ12n+1 . Remark 3.1. The hypotheses (b) and (d) above actually both follow from a single more technical statement which gives the analysis of measures on 2n+1 . We will illustrate this analysis in the following sections. Also, (a) follows easily from (b) and (d). Thus, we could replace (a), (b), and (d) with the statement that we have a reasonable analysis of the measures on 2n+1 . Statement (c) is necessary to propagate the existence of the Martin tree, which we discuss later. To complete the induction we must compute 12n+3 = ℵ(2n+1)+1 , prove the weak partition relation at strong partition relation of 12n+1 , and prove the 1 2n+3 an in addition establish (a), (b), (c), and (d) at n + 1. Consider first the problem of computing 1 . The upper and lower 2n+3 bounds are obtained by different methods. We consider first the upper-bound. Historically, obtaining the upper-bound for 15 was the first problem for which inadequate. The upper-bound the previous pre-description arguments proved for 12n+3 involves centrally the notion of a homogeneous tree, and this was the original idea of the Kunen-Martin plan. It turns out that the notion of homogeneous tree plays, somewhat indirectly, another important role in the theory. Namely, when we analyze an arbitrary measure on 12n+1 or 2n+3 (which we will do at this stage of the induction) that they are closely related to the measures appearing in the homogeneous tree construction on Π12n+1 , Π12n+2 sets respectively. With a sufficiently general an arbitrary measure is equivalent to a proddefinition of homogeneous tree, uct of such homogeneity measures. More precisely, the general measure will be a “lift-up” of a product of homogeneity measures. The lift-up process will involve descriptions and the Martin tree. In §4 we show arbitrary measures are generated on 11 = 1 and 3 = using homogeneity measures on trees for The lifting-up here will only use the trivial descriptions and Π11 and Π12 sets. tree. In §5, 6 we show how the next level (non-trivial) descriptions Kunen the generate arbitrary measures on 13 , using these description to lift-up what are on Π1 sets. essentially homogeneity measures 3 Because of the importance of homogeneous trees in the inductive analysis, we recall the definition and the important Martin-Solovay construction for propagating Suslin representations using homogeneous trees. If is a measure on a set X and f : X → Y , let f() denote the measure on Y given by f()(B) = (f −1 (B)). Definition 3.2. A tree T on × is homogeneous if there are measures s , for s ∈ < such that Ts = ∅, satisfying:
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1. s (Ts ) = 1 (recall Ts = {α ∈ lh(s) : (s, α) ∈ T }. 2. If t extends s, then s,t (t ) = s . Here s,t : lh(t) → lh(s) is the natural projection map: s,t (α0 , . . . , αlh(t)−1 ) = (α0 , . . . , αlh(s)−1 ). 3. If x ∈ and [Tx ] = ∅ (i.e., Tx is illfounded), and if {An }n∈ are given with xn (An ) = 1, then ∃f ∈ ∀n (xn, fn) ∈ An . Clause (3) is the key homogeneity condition. It says that T is homogeneous in the sense that whenever Tx has a branch, then a branch can be found in any sequence of measure one sets for the xn . We write (T, ) for the homogeneous tree T along with the measures witnessing homogeneity. We extend Definition 3.2 in an obvious way to trees on × × , etc., (in this case, the measures are indexed as s,t , where lh(s) = lh(t)). We say A ⊆ is homogeneously Suslin if there is a homogeneous tree T with A = p[T ]. We say A is weakly homogeneously Suslin if A is the existential quantification of a set B ⊆ × (i.e., A(x) ↔ ∃y B(x, y)) and B is homogeneously Suslin. So, A is weakly homogeneously Suslin if there is a homogeneous tree T on × × such that A = p[T ], that is, A(x) ↔ ∃y ∈ ∃f ∈ (x, y, f) ∈ [T ]. We can code the last two coordinates of T into a single ordinal coordinate, and this leads to the notion of a weakly homogeneous tree: a tree which is isomorphic to such a coded tree. In practice, we work directly with the homogeneous tree on × × . One can also give an “abstract” definition of a tree T on × being weakly homogeneous. We will not need this here, and refer the reader to Definition 4.4 of [Jac08] for the precise statement. We recall next the Martin-Solovay construction. Let {ti } be an enumeration of all t ∈ < × < with all sequences preceding any proper extension in the enumeration. Definition 3.3 (Martin-Solovay Tree). Let (T, ) be a homogeneous tree on × × (so A = p[T ] is weakly homogeneous). Let ≥ + . Then the Martin-Solovay tree ms(T, , ) is the tree defined as follows. We define (s, α) ∈ ms(T, , ) if there is an f : Ts → which is order-preserving with respect to the Brouwer-Kleene order on Ts such that ∀i < lh(s) (αi = [f i ]s lh(ti ),ti ). Here f i () = f(ti , ) (if Ts lh(ti ),ti = ∅, we set αi = 0). Remark 3.4. In specifying the Brouwer-Kleene ordering on Ts , we must say how × is identified with an ordinal. Usually we do this by ordering by reverse lexicographic ordering (i.e., order by the ordinal coordinate first, then the integer coordinate). Also, we may restrict the type of the function f, for example we may require that f be of the correct type. We officially adopt the correct type restriction for our Martin-Solovay trees. The basic property of the Martin-Solovay tree is given in the next theorem. We refer the reader to [Jac08, Theorem 4.10] for a proof (we give the proof in a special case after Corollary 4.4, the proof in the general case is similar).
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Theorem 3.5. Let (T, ) be a homogeneous tree on × × , ≥ + , and S = ms(T, , ). Then for all x ∈ , Tx is illfounded iff Sx is wellfounded (i.e., p[S] = \ p[T ]). To propagate the Martin-Solovay construction through the projective hierarchy requires showing that the Martin-Solovay tree is itself homogeneous. This requires the use of partition properties. The exact manner in which this is done varies depending on whether we are at an odd or an even level of the projective hierarchy. Let us consider the case of propagating from an even to an odd level. Suppose we assume inductively that every Π12n set is the projection of a homogeneous tree on × 2n+1 . Thus, everyΣ12n+1 set A is the Definition 3.3 projection of a homogeneous tree T on × × 2n+1 . We apply 1 = . Assume (according to our hypotheses at stage n of with = + 2n+1 2n+1 1 the analysis) that 2n+1 has the weak partition property. Then S = ms(T, ) a Π1 projects to \ A, 2n+1 set. It will be homogeneous according to the following fact. Lemma 3.6. Let T , , , be as in Definition 3.3, and assume → ()< . Then S = ms(T, , ) is a homogeneous tree om × 12n+1 with 12n+1 -complete measures. Proof. For s ∈ < , the measure s on Ss is the one induced by the weak partition relation on 12n+1 and functions f : Ts → 12n+1 of the correct type. That is, E ⊆ ( 12n+1 )lh(s) has s measure one if there is a c.u.b. C ⊆ 12n+1 such that for all f : Ts → C of the correct type we have (here j = lh(s)): α = ([f 0 ]s lh(t0 ),t0 , . . . , [f j−1 ]s lh(tj−1 ),tj−1 ) ∈ E. Here the subfunctions f i of f are as in Definition 3.3. The weak partition relation on 12n+1 easily gives that s is a 12n+1 -complete measure, and the 1 closure of 2n+1 under ultrapowers (one of our inductive hypotheses) gives that s is a measure on 12n+1 (note that the 12n+1 -completeness of the -c.u.b. from the c.u.b. version of the 1 filter on 2n+1 follows by an easy argument partition relation 12n+1 → ( 12n+1 )2 ). suppose x ∈ and S is illfounded, which by To show S is homogeneous, x Theorem 3.5 is equivalent to saying Tx is wellfounded. Let Ai ⊆ ( 12n+1 )i have xi measure one. Let Ci ⊆ 12n+1 be c.u.b. and witness that Ai has xi measure one. Let C = i Ci , so C is c.u.b. in 12n+1 . Since Tx is a wellfounded tree on , it has rank less than + ≤ . Moreover, we an get an order-preserving map f : Tx → C of the correct type. Let F i : Txi → C be the subfunctions induced by f as in Definition 3.3. Let αi = [f i ]x lh(ti ),ti < 12n+1 . Then by definition of S we have (x, α) ∈ [S]. The propagation from the odd to the even levels is similar, but with a slight difference. Now we start with a Σ12n+2 set, which is the projection of
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a homogeneous tree on × × 12n+1 . We employ now a slight variation in which we use = 1 of the Martin-Solovay construction 2n+1 (so = now instead of ≥ + ). That we can do this follows from a property of the Martin-Solovay tree T constructed above on a Π12n+1 set. Namely, the tree T ,...,α has the property that if (s, α) ∈ T , where α = (α 0 lh(s)−1 ), then for all i we have αi ≤ sup j (α0 ), where the supremum ranges over the measures in the homogeneous tree used to construct T (so the are measures on 2n+1 ). Because of this boundedness property it follows that for any α < 12n+1 that Txα = {α ∈ Tx : α0 ≤ α} is a tree on an ordinal less than 12n+1 . Hence, if Tx α 1 is wellfounded then for all α, Tx has rank less than 2n+1 . In particular, if Tx is to the Brouwer-Kleene wellfounded than the rank of any α ∈ Tx with respect 1 ordering is less than 2n+1 . Thus, we may use = 12n+1 in Definition 3.3. The proof of the homogeneity of S now follows as in even-to-odd case, except that to get the measures s on the equivalence classes of functions f : Tx → 12n+1 of the correct type we must now use the strong partition relation on 12n+1 (since Tx is a tree on 12n+1 now). Part of our job at stage n is to prove the strong partition relation on 12n+1 , so assuming we do this, this will propagate the existence of homogeneoustrees from Π12n to Π12n+1 and Π12n+2 . Note that the propagation of homogeneous the partition only trees uses properties of the 12n+1 , and makes no direct reference to descriptions. As an immediate corollary we get a certain computation for the upperbound for 12n+3 . This expresses the idea of the Kunen-Martin program. Corollary 3.7. Assuming the inductive hypotheses at stage n, + 12n+3 ≤ sup j ( 12n+1 ) , where the supremum ranges over the measures in the homogeneous tree on a Π12n+1 -complete set. Proof. Assuming the stage n hypotheses we have shown that every Π12n+2 the is -Suslin where = sup j ( 12n+1 ) where the measures are from homogeneous tree of a Π12n+1 set. Thus, Π12n+2 and hence Σ12n+3 is -Suslin. If Σ1 the Π12n+2 set is Π12n+2 -complete, then every 2n+3 will be -Suslin. This shows 1 1 + 2n+3 ≤ , and so 2n+3 ≤ . Note that the construction of the MartinSolovay tree for theΠ12n+2 set uses only the weak partition relation at 12n+1 (the strong partition relation at 12n+1 was only needed to get the homogeneity of this tree). 1 The computation of the lower-bound 2n+3 ≥ ℵ(2n+1)+1 is done using the must be proved at stage n of the strong partition relation on 12n+1 (which analysis). The lower-bound argument does not use descriptions, but proceeds
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independently from that analysis just using the partition properties of the 12n+1 . In this sense it is similar to the propagation of homogeneous trees. main tool used for the lower-bound is the following result of Martin (see The [Jac10, Theorem 4.17] for a proof). Theorem 3.8 (Martin). Suppose κ → (κ)κ . Then for any measure on κ, j (κ) is a cardinal. Assuming the weak partition relation on 12n+1 , one defines a collection of normal forms for the ordinals measures on 12n+1 corresponding to the Cantor below (2n + 1). One then proves embedding results which show that if 1 corresponds to a smaller ordinal than 2 then j1 ( 12n+1 ) < j2 ( 12n+1 ). From (one computes directly Theorem 3.8 the lower-bound 2n+3 ≥ ℵ(2n+1) follows 1 1 that the ultrapowers j ( 2n+1 ) are Δ2n+3 in the codes and thus are below 12n+3 , cofinality are therefore below and being cardinals of uncountable 2n+3 ). The 1 reader can see [Jac99] for the full details of these arguments below 5 . Also, see [JL] for the definitions of these measures in the general projective case. To compute the right-hand side of the inequality in Corollary 3.7 and to carry out the remaining steps of the analysis at stage n of the induction requires the theory of descriptions. We show how the n = 0 arguments are done in §4. As we said before, this will use only “trivial descriptions,” but will show how the previous theory fits into the modern point of view. In §§5, 6 we show how descriptions are used to do the n = 1 stage arguments. In §7 we make some comments about the general stage n. In the remainder of this section we make some general comments about what descriptions are and how they are used to do the inductive analysis. Using the partition properties of the 12n+1 we will define certain families of canonical measures. The first canonicalfamily, used in §4, will consist of the measures W1m , where W1m is just the m-fold product of the normal measure on 1 . The “W ” in the notation stands for “weak,” denoting that the measures are defined using the weak partition relation on 11 . The second canonical family will be the measures S1m , used in §5. Each S1m will be a measure on m+1 . The notation S1m represents the fact that the measures are defined using the strong partition relation on 11 . The third family will be the measures W3m , using the weak partition relation on 1 . In which are measures on 13 defined 3 on m of measures the general case, defined in §7, there will be a family W2n+1 ,m 12n+1 for each odd projective ordinal 12n+1 , and finitely many families S2n+1 , n+1 m 1 ≤ ≤ 2 − 1, of measures on 2n+3 . One can think of the measures W2n+1 as simplified versions of the measures occurring in the homogeneous tree on a ,m Π12n+1 set, and the S2n+1 as simplified versions of those occurring for a Π12n+2 set. It will be important along the way to show that these canonical measures dominate the more general homogeneity measures. This is made precise in two theorems called the local and global embedding theorems. We state these
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for the families W1m , S1m , and W3m in Theorems 6.11 and 6.15. Their complete proofs can be found in [Jac99], although we give an example of each in §7. The statements of the embedding theorems in the general case as well as their proofs can be found in [Jac88]. Descriptions are hereditarily finite objects and are defined with respect to a finite sequence of canonical measures K1 , . . . , Kt . They “describe” how to build an ordinal in the iterated ultrapower by these measures (we note that the iterated ultrapower is not the same as the ultrapower by the product measure K1 × · · · × Kt under AD). The level-1 descriptions will be the trivial descriptions, which are just the positive integers. They describe how to build an ordinal in the ultrapower by a single measure of the form K1 = W1m . The description d = i describes the function f(α1 , . . . , , αm ) = αi . This function represents the cardinal i in the ultrapower by W1m (see §4 for details). Thus, the cardinals below 13 are exactly the ordinals represented by the level-1 to the measures W m . The trivial descriptions are descriptions with respect 1 also used to prove the strong partition on 11 and the weak partition relation on 13 and thereby complete the n = 0 stage of the analysis. level-2 descriptions describe how to generate ordinals in the iterated The ultrapower by a sequence of measures K1 , . . . , Kt where each Ki is one of the canonical measures Ki = W1mi or K1 = S1mi . The level-3 descriptions will be just minor variations of the level-2 descriptions. They will be defined relative to a sequence K1 , . . . , Kt where K1 = W3m and the Ki for i ≥ 2 are of the form Ki = W1mi or K1 = S1mi . This pattern continues throughout the projective hierarchy. We give the exact definition of the level-2 and level-3 descriptions in §§5, 6. Again it will be the case that the cardinals below 5 exactly correspond to the set of level-3 descriptions. We note that in [Jac99] and [Jac88] the level2 and level-3 descriptions were grouped together and likewise at the higher levels. For expository purposes we find it convenient to separate them here. The general level descriptions are defined in [Jac88]. Aside from describing the cardinal structure, the level-2 and level-3 descriptions (and similarly at the higher levels) are used to analyze arbitrary ordinals below 2n+3 as well as analyze arbitrary measures on 2n+3 . Arbitrary ordinals below 2n+3 will be generated as the “lift” of a finite set of descriptions via the Martin tree. Roughly speaking, the main theorem on descriptions says that if an ordinal ϑ is less than the ordinal (actually cardinal) represented by a description d defined relative to the measure sequence K1 , . . . , Kt , then ϑ is less than the lift of a smaller description L(d ) by some function g : 12n+1 → 12n+1 . The function g is in turn dominated by the ranking function ona wellfounded section of the Martin tree. However, unlike the Kunen tree where the relevant ranking function is given by α → |Tx α| (see Theorem 2.11), for the Martin tree M the relevant function is α → Mx sup j (α),
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where the supremum ranges over the measures in a homogeneous tree for a Π12n -complete set. From the embedding theorems, we can replace the mea sures in this supremum by those in the canonical families. This forces us, however, to lengthen the sequence of measures. That is, the lift of L(d ) by g will be bounded by the cardinal successor of the supremum of the cardinals represented by the smaller description L(d ) with respect to sequences K1 , . . . , Kt , Kt+1 as Kt+1 ranges over the canonical measures. Putting this together, this gives gives an upper bound for 2n+3 in terms of the rank of a certain lowering operator L on the descriptions and measure sequences. Computing this rank is then a purely combinatorial problem, with the result being (2n + 3). So, for example, this gives 5 ≤ ℵ . A variation of the analysis of ordinals of the previous paragraph allows us to analyze arbitrary measures on 12n+1 and then on 2n+3 . In §4 we show how the trivial descriptions allow to analyze the measure on 1 and 3 = . This was done originally using the theory of indiscernibles for L[x] by Kunen (see [Sol78A]). In §6 we show how a typical measure on 13 is generated. The full details for this analysis, as well as the analysis of measure of 5 are given in [Jac99]. Finally, the analysis of measures on 12n+1 and 2n+3 can be converted, via a clever argument of Kunen, into an analysis of arbitrary subsets of 12n+1 and 2n+3 . This produces a coding of these subsets good enough to satisfy the hypotheses of Theorem 2.17 and get the strong partition relation on 12n+1 and the weak relation on 12n+3 . The argument of Kunen referred to above appears in [Sol78A]. We present this argument explicitly in Lemma 4.8. We note one technical point here. In proving the strong partition relation on 12n+1 , Martin’s method (Theorem 2.17) requires us have a good coding of the functions f : 12n+1 → 12n+1 . If we just view such a function as a subset of coding of subsets of 1 1 1 1 1 2n+1 × 2n+1 and use the 2n+1 (actually 2n+1 × 2n+1 ) 1 directly given from the analysis of subsets of 2n+1 , then the coding is not good 2.17. Instead we must modify enough to satisfy the requirement of Theorem the analysis of subsets to directly work with functions. The technical changes, however, end up being minor. In showing the weak partition relation on 12n+3 this technical problem does not arise. Aside from the descriptions, the other main ingredients that go into the analysis of measures are the notions of a tree of uniform cofinalities and a complex. A tree of uniform cofinalities is a code for building a basic type of measure, roughly speaking a measure which occurs in the homogeneous tree construction. A level-n tree of uniform cofinalities will correspond to a measure in the homogeneous tree on a Π1n set. A level-n complex will be a the basic measure to generate a level-n tree together with a way of “lifting” more general measure. This lifting process will involve descriptions, though at the bottom level the descriptions are trivial. These more general measures will capture all measures.
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In the next sections we fill in more details starting with the n = 0 stage of the induction in the next section. Again, there we will only use trivial descriptions. The reader wishing to see non-trivial descriptions as quickly as possible (and skip the measure analysis at the n = 0 stage) can skip directly to §5. In the following sections we will not always give complete proofs or even complete definitions. Rather, we attempt to illustrate the main concepts and arguments that are used in this theory. We will typically consider some illustrative examples rather than the general definitions or proofs. Hopefully this will give the reader a general understanding of the modern theory of the projective ordinals and how it generalizes and unifies the previous results. It should also give the reader a good background for [Jac99] or [Jac88]. We will reference the papers of this volume, as well as others, for many details. §4. The First Level Theory. As we previously mentioned, the first level (stage n = 0) theory of the projective ordinals involves establishing the strong partition relation on 11 = 1 , computing 13 , and showing the weak partition relation on 13 . Our starting hypothesisis the weak partition relation on we observed earlier). Martin’s original proof of the strong 11 = 1 (which partition relation on 11 can be found in [Kec78], and Kunen’s original proof 1 can be found in [Sol78A]. Again, those proofs both of the weak relation on 3 used the theory of indiscernibles for L[x]. We will show how these arguments can be viewed as “trivial description” arguments. If κ has the weak partition property and < κ is regular then there is a unique normal measure on κ concentrating on points of cofinality which we call the -cofinal normal measure. It is generated by sets of the form C ∩ S , where C ⊆ κ is c.u.b. and S is the points of cofinality (to see this is a measure, consider the partition of f : → κ of the correct type according to whether sup(f) lies in a given set). In particular, the c.u.b. filter on 1 is a normal measure. Similarly, the m-fold product of the normal measure is generated by sets of the form C m for C ⊆ 1 a c.u.b. set. We may describe this measure as being induced by the weak partition relation on 1 and function f : m → 1 . We make this into the following definition. Definition 4.1. W11 is the normal measure on 1 . W1m is the m-fold product of the normal measure on 1 . We let W1 = {W1m }m∈ be the family of these measures. Consider functions F : 1m → 1 . There are m canonical functions given by Fi (α1 , . . . , αm ) = αi . We can view this as an instance of a description evaluation by viewing the descriptions as the set of positive integers, and the description d = i corresponds to the function Fi . Although this is just a
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trivial notational change, it nevertheless allows us to introduce a notational framework which will generalize to the higher levels. We make this into the following definition. Definition 4.2. A level-1 description is a positive integer. We let D1 be the set of level-1 descriptions. We say d ∈ D1 is defined with respect to W1m if d ≤ m. Given α = (α1 , . . . , αm ) ∈ 1m and d defined with respect to m W1 , let (α; d ) = αd , which we call the interpretation of the description. We define (W1m ; d ) to be the ordinal represented with with respect to the measure → (α; d ). Given g : 1 → 1 we define the lift by W1m by the function α g by: (g; α; d ) = g(α; d ) = g(αd ), and define (g; W1m ; d ) to be the ordinal represented by α → (g; α; d ). Of course (W1m ; d ) is just the ordinal represented by the canonical function Fd mentioned above. If we identify functions and the ordinals they represent, we may identify the canonical functions with the (W1m ; d ). We let id stand for the identity function, so (id; W1m ; d ) = (W1m ; d ). < αi for W1m almost all Suppose now F : 1m → 1 is given and F (α) α. In the language of descriptions this reads [F ] < (W1m ; d ), where d = i. An easy partition argument shows that there is a function g : 1 → 1 such that F (α) < g(αi−1 ) almost everywhere (unless i = 1 in which case F is constant almost everywhere). To see this, consider the partition of tuples (α1 , . . . , αi−1 , , αi , . . . , αm ) according to whether > F (α1 , . . . , αm ). Easily on the homogeneous side this property must hold. Let C ⊆ 1 be c.u.b. and . homogeneous for the partition. Let g(α) = NC (α) = the least element of C greater than α. Then for almost all α we have by homogeneity of C that F (α) < g(αi−1 ). Putting this back into the language of descriptions this becomes [F ] < (g; W1m ; d − 1). This suggests introducing a lowering operator L on D1 defined by L(d ) = d −1, unless d = 1 in which case d will be declared minimal. We can then state our observations in the following “main theorem” for level-1 descriptions. Theorem 4.3. If [F ] < (id; W1m ; d ) and d is not L-minimal, then there is a g : 1 → 1 such that [F ] < (g; W1m ; L(d )). If d is L-minimal, then [F ] < 1 . Continuing the analysis, fix g : 1 → 1 such that [F ] < (g; W1m ; L(d )). From Theorem 2.11, fix x ∈ such that the section of the Kunen tree Tx is wellfounded and ∀∗W 1 α g(α) < |Tx α| (actually this holds for all in1
finite α, but we don’t need this). This now gives a map ϑ → ϑ from (id; W1m ; L(d )) onto (; W1m ; L(d )) > [F ], where is the function (α) = |Tx α|. Namely, If ϑ < (id; W1m ; L(d )), let ϑ be represented with respect = |Tx (ϑ(α))|, where = (α; L(d )). This shows that to W1m by ϑ (α) (W1m ; d ) ≤ (W1m ; L(d ))+ . As an immediate corollary we get the computation for the ultrapowers jW1m (1 ) of 1 by the measures W1m (for the lower-bound
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of the corollary we are using the strong partition relation on 1 which is discussed later in this section). Corollary 4.4. jW1m (1 ) = m+1 . Proof. The upper bound was shown above. From Theorem 3.8, all of the jW1m (1 ) are cardinals and the lower-bound follows. That this gives an upper bound for 13 follows from the homogeneous tree construction of Corollary 3.7 and the fact that the homogeneity measures for the bottom level homogeneous tree—the Shoenfield tree—on a Π11 set are isomorphic to measures of the form W1m . For the sake of completeness, and to illustrate the proof of Theorem 3.5, we construct this tree and the homogeneous tree on a Π12 set. This will also help motivate the definition of a level-2 tree of uniform cofinalities (Definition 4.23). Again, the reader can consult [Kec81A] for more details on the homogeneous tree construction. If A ⊆ × is Σ11 , then A = p[T ] where T is a tree on × × . The . on × × such that p [S] = B = Shoenfield tree S is a tree \ A (here 1 1,2 p1,2 [S] means the projection to the first two coordinates). Thus, the projection p[S] ⊆ of S to the first coordinate is the Σ12 set A2 = ∃R B, and S witnesses A2 is weakly homogeneous. Let {ui } enumerate < such that any sequence precedes any of its proper extensions in the enumeration. We may define S by: [(αi < αj ) (s, t, α) ∈ S ↔ α0 > max{αi } ∧ ∀i, j < lh(α) ↔ (ui , uj ∈ Ts,t ∧ ui max{αi }. For any s, t, there is a unique permutation s,t of length lh(s) such that α ∈ Ss,t iff α is order-isomorphic to s,t (that is, αi < αj iff s,t (i) < s,t (j), viewing s,t as a bijection from lh(s) to lh(s)). This last property is the homogeneity property of S. For s, t of length n, let vs,t be the measure on n-tuples of countable ordinals order isomorphic to s,t which is induced by W1n and the permutation s,t . Clearly if s , t extends s, t, then s ,t projects to s,t under the restriction map. The measures vs,t , which are isomorphic to W1lh(s) , witness that the tree S is homogeneous. We now carry out explicitly, starting from S, the Martin-Solovay construction of the homogeneous tree U for the Π12 set B2 = \ A2 . Order × 1 by reverse lexicographic order (i.e., orderby the second coordinate first). Let <s be the Brouwer-Kleene ordering on Ss , using this ordering on pairs. For notational clarity, let also t0 , ti , . . . also enumerate < with each sequence preceding its proper extensions in the enumeration (so lh(ti ) ≤ i).
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∈ U iff there is an f : Ss → 1 which is orderDefine the tree U by (s, ) preserving with respect to <s and of the correct type and such that for all i < lh(s) (i = [f si ,ti ] si ,ti ), where si = s lh(ti ), and f si ,ti is the subfunction of f obtained by restricting f to Ssi ,ti . By Corollary 4.4, U is a tree on × . We show that p[U ] = B2 = \ A2 = \ p[S], as in Theorem 3.5. So, we must show that for all x ∈ that Ux is illfounded iff Sx is wellfounded. If Sx is wellfounded, let fx : Sx → 1 be order-preserving of the correct type. This is possible since the rank of any (t, α) ∈ Sx is countable. Here we use the fact that α0 > maxi
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makes use of a result of Kunen (see [Sol78A]) who gave a general argument which shows how convert an analysis of measures on κ into an analysis of the subsets of κ. Kunen’s argument plays an important role in the proofs of the partition relations. We give Kunen’s result in Theorem 4.8. We use the following general lemma. Recall that Θ is the supremum of the lengths of the prewellorderings of (Θ is much larger than all the projective ordinals). Lemma 4.6. Let κ < Θ and let F be a countably additive filter on κ. Then there is a measure on κ extending F (i.e., F ⊆ ). Remark 4.7. From AD (which we are assuming) it follows that every ultrafilter on a set is necessarily countably additive. That is, every ultrafilter is a measure. So, Lemma 4.6 could be phrased as saying that every countably additive filter on κ < Θ can be extended to an ultrafilter. Proof. Since κ < Θ, there is a prewellordering of length κ. The coding lemma then gives a map : → ℘(κ) which is onto. To see this, let Γ be a non-selfdual pointclass containing which is closed under ∃R and ∧. Let U ⊆ × be a universal Γ set (From Wadge’s Lemma it follows a universal set). The coding lemma that every non-selfdual pointclass has implies that every A ⊆ κ is Γ in the codes given by . That is, the code set {x ∈ fld() : |x| ∈ A} is aΓ set. Thus, we can take (x) = {α < κ : ∃y ∈ fld() : |y| = α ∧ U (x, y)}. Recall from AD that there is a measure M, the Martin measure, on the set D of Turing degrees (the countable set of reals Turing equivalent to fixed real). A set A ⊆ D has M measure one iff it contains a cone of degrees. That is, it contains a set of the form {d ∈ D : d ≥T x} for some x ∈ , where ≥T denotes Turing reduction (there is a slight abuse of notation here as d is a set of reals; what we mean is y ≥T x for any y ∈ d ). For d a degree, let α(d ) < κ be the least ordinal in ∩{(x) : x ∈ d ∧(x) ∈ F}. This is well-defined by the countable additivity of F. Let = α(M) be the push-forward by α of the Martin measure (i.e., E ⊆ κ has measure one iff {d ∈ D : α(d ) ∈ E} has M measure one). Then is a measure on κ giving all elements of F measure one. To see this, suppose E ∈ F. Let (y) = E. Then if d ≥T y, that is y is in the degree d , we have that ∩{(x) : x ∈ d ∧ (x) ∈ F} ⊆ E and so α(d ) ∈ E. Theorem 4.8 (Kunen). Let κ < Θ, and suppose S ⊆ ℘(κ) is a base for the measures on κ. That is, suppose that for every measure on κ and every measure one set B, there is an S ∈ S with S ⊆ B and (S) = 1. Then every A ⊆ κ is a countable union A = i Si of sets Si ∈ S. Proof. Suppose A cannot be written as a countable union of sets in S. Let I be the -ideal on κ generated by κ \ A and {S ∈ S : S ⊆ A}. By assumption, A∈ / I. Let F be the corresponding filter, that is, F = {F ⊆ κ : κ \ F ∈ I}. So, F is a countably additive filter concentrating on A.
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From Lemma 4.6, let be a measure on κ extending F. So, (A) = 1. By assumption, there is an S ∈ S with S ⊆ A and (S) = 1. However, S ∈ I by definition of I. This is a contradiction since κ \ S ∈ F yet (κ \ S) = 0. The strong partition relation on 1 . To prove partition relations we must analyze sets (actually functions), and to analyze sets, in view of Theorem 4.8 we must analyze measures. So, getting the strong partition relation on 1 reduces to analyzing the measures on 1 . The weak partition relation on 13 will reduce to analyzing measures on 3 = , or equivalently the measures on the n (at the next level of the theory where we need to get the strong partition property on 13 , we will need to analyze the measures on 13 ). outline of how analyzing measures in general First we give a rough goes. Given a measure on an ordinal κ, we analyze through a sequence of “pressing down” arguments. At stage n we will have two functions gn , rn : κ → κ, that is, we have a map α → (gn (α), rn (α)). Roughly speaking, the function gn will constitute the known part of the measure, and the function rn the part yet to be analyzed (the “remainder”). More precisely, gn () will be a known canonical measure. The two values gn (α), rn (α) will together determine α (for almost all α). A pressing down argument will give at the next stage functions gn+1 , rn+1 . Some sort of monotonicity of the functions is important in the pressing down argument. The measure gn+1 () will be a more complicated extension of gn () (more precisely, both of these measures will be naturally measures on tuples of ordinals, and gn+1 () will project to gn ()). On the other hand we will have rn+1 (α) < rn (α) for almost all α. Thus, after finitely many steps the process must stop (it will stop when the r function becomes constant almost everywhere), and at this point the measure is analyzed. The exact manner in which gn (α) and rn (α) determine α will involve non-trivial descriptions at the higher levels, but for the strong partition relation on 1 and the weak on 13 will only involve trivial descriptions. To illustrate, let us consider in more detail the case of a measure on 1 . We prove the following result of [Jac90A]. Theorem 4.9. Every non-principal measure on 1 is equivalent a measure W1n . That is, there is a h : (1 )n → 1 which is one-to-one on a W1n measure one set such that = h(W1n ) (i.e., for all A ⊆ 1 , A ∈ iff h −1 (A) ∈ W1n ). Proof. To begin, let f1 : 1 → 1 be such that: 1. There is a measure one set A on which f1 is monotonically increasing. (i.e., if α ≤ are both in A then f1 (α) ≤ f1 ()). 2. f1 is not constant almost everywhere (i.e., there does not exist a measure one set B such that f1 B is constant).
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3. [f1 ] is minimal with respect to (1) and (2) (i.e., if [f ] < [f1 ] then f does not satisfy (1) and (2)). Note that the identity function satisfies (1) and (2), and so f1 is welldefined. Also, f1 (α) ≤ α for almost all α. Fix A of measure one such that f1 A is monotonically increasing. By (1) and (2), for all α < 1 . we have h1 (α) = sup{ ∈ A : f1 () ≤ α} < 1 . Let g1 = f1 . Note that g1 () = W11 as otherwise there would be a c.u.b. C ⊆ 1 such that / C ). However, in that g1 () gives C measure zero, that is, ∀∗ α (g1 (α) ∈ case we could let f (α) = C ◦ f1 (α) for α in A, where C () is the largest element of C less than or equal to . We would then have a measure one set A ⊆ A and an f which is strictly less than f1 on A and which also satisfies (1) and (2). This violates (3). So, g1 () is the canonical measure W11 . Fix a real x such that the section Tx of the Kunen tree is wellfounded and h1 () < |Tx | for W11 almost all , say for all ∈ C (actually for all infinite , but we don’t have this at the higher levels so we use only the weaker statement here). By thinning out A we may assume g1 (α) ∈ C for all α ∈ A. For α ∈ A let r1 (α) < g1 (α) ≤ α be least such that α = |(Tx g1 (α))(r1 (α))|. So, g1 (α) and r1 (α) determine α by this equation. This completes the first step of the analysis. If there is a measure one set B such that for all α, ∈ B if g1 (α) = g1 () then r1 (α) = r1 (), then is equivalent to W11 and we are done. To see this, suppose B of measure one is as stated. We may also assume that r1 < g1 on B as this holds almost everywhere. There is a c.u.b. C ⊆ 1 such that C ⊆ g1 (B) [Otherwise there would be a c.u.b. C with . C ∩ g1 (B) = ∅. But B = g1−1 (C ) must have measure one, and so B ∩ B has measure one and so is non-empty. But if α ∈ B ∩ B then g1 (α) ∈ C , a contradiction to α ∈ B and the definition of C .] For ∈ C , let () be the unique value of r1 (α) for any α ∈ B with g1 (α) = . Since r1 < g1 on B, we have that that () < for all ∈ C . Hence there is a C ⊆ C such that → () is constant on C , say with constant value 0 . Let B = {α ∈ B : g1 (α) ∈ B }, so (B ) = 1. For ∈ C , let h() = |Tx (0 )|. Then h is a bijection between C and B (the functions h : C → B and g1 : B → C are inverses). Also, if B has measure one, then by the argument above there is a c.u.b. C with C ⊆ g1 (B ), and this shows h(W11 ) = . For the second step, let f2 : 1 → 1 satisfy: 1. There is a measure one set A such that if α, ∈ A, g1 (α) = g1 (), and r1 (α) ≤ r1 (), then f2 (α) ≤ f2 (). 2. There does not exists a measure one set B such that if α, ∈ B and g1 (α) = g1 () then f2 (α) = f2 ().
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3. [f2 ] is minimal with respect to (1) and (2). Note that f2 exists since r1 satisfies (1) and (2). Also, f2 (α) ≤ r1 (α) < f1 (α) for almost all α. Next observe that f2 () = W11 . For suppose C . were c.u.b. and ∀∗ α (f2 (α) ∈ / C ). Then f2 = C ◦ f2 is strictly less than f2 almost everywhere with respect to . Also, f2 still satisfies (1) and (2), a contradiction (for (2) we use the fact that for any measure one set B, for W11 almost all we have that {r1 (α) : α ∈ B ∧ g1 (α) = } is cofinal in as otherwise we would have a measure one set on which r1 is constant, which contradicts the assumption that r1 satisfies (2)). Let now g2 (α) = (f2 (α), f1 (α)). So, g2 () = W12 . Fix a measure one set A for which (1) holds. Let h2 (1 , 2 ) = sup{r1 (α) : α ∈ A ∧ g2 (α) = (1 , 2 )}. It follows from (1) and (2) that h2 (1 , 2 ) < 2 for W12 almost all (1 , 2 ). [Suppose C were c.u.b. and for all 1 < 2 in C we have h2 (1 , 2 ) = 2 . Thinning A we may assume that g2 (α) ∈ C 2 for all α ∈ A. From (2), let α, ∈ A with g1 (α) = g1 () and f2 (α) < f2 (). Let (1 , 2 ) = g2 (α). By (1), if ∈ A with g1 () = 2 and r1 () ≥ r1 () then f2 () ≥ f2 () > f2 (α) = 1 . Thus, h2 (1 , 2 ) ≤ r1 () < 2 , a contradiction.] Let x2 be such that Tx2 is wellfounded and for W12 almost all (1 , 2 ) we have h2 (1 , 2 ) < |Tx2 1 |. For almost all α we may then define r2 (α) to be the unique ordinal such that r1 (α) = |(Tx2 f2 (α))(r2 (α))|. So, for almost all α, if g2 (α) = (1 , 2 ), we have α = |(Tx1 (2 ))(|(Tx2 (1 ))(r2 (α))|)|, and thus g2 (α), r2 (α) determine α. Note that r2 (α) < 1 = f2 (α) ≤ r1 (α) for almost all α. This completes the second step. The remaining steps are essentially identical to the second step. At the end of step n (if the argument has gone on this far), we have a gn with gn () = W1n and an rn with [rn ] < [rn−1 ] < · · · < [r2 ] < [r1 ] . Also, for almost all α, if gn (α) = (1 , . . . , n ) we have rn (α) < 1 < · · · < n ≤ α and: α = |(Tx1 n )(α1 )| where α1 = |(Tx2 n−1 )(α2 )|, α2 = |(Tx3 n−2 )(α3 )|, .. .
.. .
αn−1 = |(Txn (1 ))(αn )|, and αn = rn (α) By wellfoundedness, after finitely many steps, say n steps, it must be that the analog of (2) fails. As we argued above, is then equivalent to
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W1n . In fact, similarly to how we argued above, the rn function is constant almost everywhere, and is equivalent to W1n in the following manner. Let 0 be the constant value of rn almost everywhere. Recall x1 , . . . , xn are reals with the Kunen tree sections Tx1 , . . . ,Txn all wellfounded. For convenience, we change notation and call these same reals xn , xn−1 , . . . , x1 (i.e., we enumerate in the reverse order). Define h(1 , . . . , n ) = |(Txn (n ))(αn−1 )|, αn−1 = |(Txn−1 n−1 )(αn−2 )|, . . . , α1 = |(Tx1 (1 ))(0 )|. In writing this equation, we assume that 0 < 1 , each |Txi i (αi−1 )| is less than i+1 and that undefined. Then (A) = 1 iff |Txn n (αn−1 )| ≥ n , otherwise we leave h() n ∃ c.u.b. C ∀ ∈ C h() ∈ A. We call h(1 , . . . , n ) the lift of (1 , . . . , n ) via x1 , . . . , xn and 0 . So, h demonstrates the equivalence of W1n and . We make this into the following definition. Definition 4.10. We say a level-1 complex C is a sequence of reals x1 , . . . , xn such that for each i, the section of the Kunen tree Txi is wellfounded. Given a level-1 complex C and an ordinal 0 < 1 , let h = hC0 : (1 )n → 1 be the lifting function as defined above. Let (0 , C) be the measure on 1 equal to h(W1n ). That is, A ⊆ 1 has (o , C) measure one iff there is a c.u.b. ∈ A. C ⊆ 1 such that for all ∈ C n we have hC0 () We have thus shown the following. Theorem 4.11. Let be a measure on 1 . Then there is a level-1 complex C and an ordinal 0 < 1 such that = (0 , C). The above analysis of measures on 1 together with Theorem 4.8 gives a coding for the subsets of 1 . Following the terminology of [Sol78A] we make the following definition. Definition 4.12. A ⊆ 1 is simple if either A = {α} is a singleton or there is a c.u.b. C ⊆ 1 , a level-1 complex C, and an 0 < 1 such that : ∈ C n }. A = {hC0 () Theorems 4.8 and 4.11 then give the following. Theorem 4.13. Every subset of 1 is a countable union of simple sets. This in turn gives a coding : → ℘(1 ) of the subsets of 1 as follows. If ∈ , let C be the closed (not necessarily unbounded) subset of 1 defined by C = {α : ∀ < α |T | < α}. From Theorem 2.11 applied to the function NC (recall NC (α) is the least element of C greater than α), we get that for any c.u.b. C ⊆ 1 there is a with C a c.u.b. subset of C . View every real z as coding a real , reals x1 , . . . xn for some n, and a real w (we allow n to be 0, which we specify by some syntactic condition on z).
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Then z codes the set Az as follows. If w ∈ / WO, then Az = ∅. Otherwise, if C n = 0 then Az = {|w|}. If n > 0 then Az is the set of h|w| (1 , . . . , n ) such that n C ∈ (C ) and where C = x1 , . . . , xn . In writing h () here we mean the |w|
value as defined above assuming the necessary wellfoundedness. For example, in writing h(1 , . . . , n ) = |(Txn n )(αn−1 )|, we assume here that αn−1 is in the C wellfounded part of the tree Txn n , etc., otherwise we say h|w| () is undefined. Let then (z) = i Azi , where z codes the sequence z0 , z1 , . . . . The map is a Δ11 coding of the subsets of 1 in the following sense. Theorem 4.14. The map : → ℘(1 ) is onto. For every α < 1 , {z : α ∈ (z)} ∈ Δ11 . Proof. Theorem 4.13 gives that is onto. For the second part, note that if α ∈ Az , where z codes , x1 , . . . , xn , w as above, then w ∈ WO and |w| ≤ α. C This is a Δ11 condition on w as WOα+1 ∈ Δ11 . Similarly, if h|w| (1 , . . . , n ) = α then then |w| < 1 < · · · < n ≤ α and αi+1 = |Txi i (αi−1 )| < i+1 by our convention (that is, αi−1 is in the wellfounded part of Txi i and has rank less than i+1 ). For any fixed α, , < 1 , the set {x : |Tx α()| < } is Δ11 . It x follows that for fixed 0 < 1 < · · · < n ≤ α that {(x1 , . . . , xn ) : h0 () = α} is a Δ11 set. Also, { : ∈ C } ∈ Δ11 for any < 1 , and from these observations the result follows. To prove the strong partition relation on 1 we need a suitable coding of the functions from 1 to 1 . If we simply view functions as being subsets of 1 × 1 ≈ 1 , the resulting coding given by Theorem 4.14 is not quite good enough to satisfy Theorem 2.17. Instead, we must redo the analysis of measures above to work directly with functions. The changes are minor. Given a function f : 1 → 1 , instead of 1 we work on the space X = {(α, f(α) : α < 1 } which of course is isomorphic to 1 by identifying (α, f(α)) with α. In the fist step of the measure analysis, instead of the h1 used there we use h1 (α) = sup{max(, f()) : (, f()) ∈ A ∧ g1 () ≤ α}. The resulting function r1 returns pairs of ordinals as values. The argument proceeds as before, with the pressing down arguments (i.e., the fi ) done with respect to the first component of the pair and the functions hi taken to dominate both components. We define the lift for functions similarly to that for sets. Given a pair (0 , 1 ), reals C = x1 , . . . , xn with Tx1 , . . . , Txn sufficiently wellfounded = (α, ) where α = |(Tx (n ))(αn−1 )|, = and ∈ (1 )n , let hC0 .1 () n |(Txn (n ))(n−1 )|, etc., and α1 = |(Tx1 1 )(0 )|, 1 = |(Tx1 1 )(1 )|. We define the notion of a simple partial function f from 1 to 1 as before, that : ∈ C n } for some c.u.b. C (we allow also f to be a single is f = {hC0 ,1 () pair). Theorem 4.13 now becomes: every function f : 1 → 1 is a countable union of simple subfunctions. This gives a coding for the functions from 1 to
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1 exactly as in Theorem 4.14. It is straightforward to check that this coding satisfies the requirements of Theorem 2.17. The point of this modification is that when hC0 ,1 (1 , . . . , n ) = (α, ), then n ≤ α and not just n ≤ . This is necessary in verifying (3) of Theorem 2.17. The weak partition relation on 13 . We next briefly indicate how the analysis of be given from the current point of view. We measures on of [Sol78A] can first need to introduce some notation concerning functions from (1 )n to 1 . Let = (i1 , i2 , . . . , in ) be a permutation of {1, 2, . . . , n} beginning with n (i.e., i1 = n). Definition 4.15. Let = (n, i2 , . . . , in ) be a permutation beginning with n. We say f : (1 )n → Ord is ordered by if f(α1 , . . . , αn ) ≤ f(1 , . . . , n ) iff (αi1 , . . . , αin ) ≤lex (i1 , . . . , in ). for all α, ∈ (1 )n . We say f is ordered by almost everywhere if there is a c.u.b. C ⊆ 1 such that fC is ordered by . Remark 4.16. We adopt the convention that we may write the arguments to a function f : (1 )n → Ord in any order. This causes no ambiguity since (1 )n consists of increasing tuples. For example, if f is ordered by = (3, 1, 2), then instead of f(α1 , α2 , α3 ) we may write f(α3 , α1 , α2 ) (the arguments are now listed in their order of significance in determining the size of f(α)). We say f : (1 )n → Ord depends on all of its arguments (almost everywhere) if there does not exist a c.u.b. C ⊆ 1 such that fC n only depends on a proper subset its arguments. An easy partition argument shows that if f : (1 )n → 1 and f depends on all its arguments, then there is a unique starting with n such that f is ordered by almost everywhere. If we remove the assumption that f depends on all its arguments, then there is a partial permutation = (i1 , . . . , ij ), j ≤ n, with i1 ≥ max{i2 , . . . , ij } such that f is ordered by , that is, f(α1 , . . . , αn ) ≤ f(1 , . . . , n ) iff (αi1 , . . . , αij ) ≤lex (i1 , . . . , ij ). For f : (1 )n → Ord, the same results hold except we remove the restriction that i1 be maximal. The following definition is used frequently. Definition 4.17. Suppose f : (1 )n → 1 is ordered by = (n, i2 , . . . , in ). For 1 ≤ j ≤ n we define the jth invariant of f, f(j), to be function from (1 )j to 1 defined by: f(j)(αn , αi2 , . . . , αij ) = sup{f(αn , . . . , αij , αij+1 , . . . , αin ) : (α1 , . . . , αn ) ∈ (1 )n }. We also define f s (j)(αn , αi2 , . . . , αij ) = sup{f(αn , . . . , , αij+1 , . . . , αin ) : < αij , αij+1 , . . . , αin };
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where the supremum is over αij+1 , . . . , αin such that (αn , αi2 . . . , , αij+1 , . . . , αin ) is order-isomorphic to . For a general f, we may get a c.u.b. C so that f is ordered by on C n (perhaps using a smaller value of n if f doesn’t depend on all its arguments), and then apply the above definition. It is easy to see that the equivalence class of f(j) with respect to W1j is well-defined. Example 4.18. If f : (1 )4 → 1 is ordered by = (4, 1, 3, 2), then for all α1 < α2 < α3 we have f(3)(α1 , α2 , α3 ) = supα1 <<α2 f(α1 , , α2 , α3 ). For α1 < α2 we have f(2)(α1 , α2 ) = supα1 <<<α2 f(α1 , , , α2 ). Note that we also have f(2)(α1 , α2 ) = supα1 <<α2 f(3)(α1 , , α2 ). That is, f(2) is also the second invariant of f(3). Also. f s (2)(α1 , α2 ) = sup<α1 ,<<<α2 f(, , , α2 ) = sup<α1 f(2)(, α2 ). It remains to describe the possible uniform cofinality of an f : (1 )n → 1 . This is done in the next lemma, whose proof uses the Kunen tree and easy partition arguments (see [Jac10] for details). Lemma 4.19. Let f : (1 )n → 1 . Then almost everywhere f(α1 , . . . , αn ) has either uniform cofinality , α1 , . . . , or αn . These n + 1 possibilities are all distinct (i.e., f can only have one of these uniform cofinalities). Remark 4.20. A generic coding argument shows that for f : (1 )n → < Θ, the possible uniform cofinalities are those listed in the lemma and the constant functions g(α) = κ for κ a regular cardinal. We say a permutation = (i1 , . . . , in+1 ) of length n + 1 extends the permutation of length n if (i1 , . . . , in ) is order-isomorphic to . For example, (3, 1, 2) is extended by (4, 2, 3, 1), (4, 1, 2, 3), and (4, 1, 3, 2). We may assume f : (1 )n → 1 depends on all its arguments, as otherwise we may consider f as a function from (1 )m → 1 for some m < n. Say f is ordered by = (n, i2 , . . . , in ). One possibility is that f is continuous almost everywhere, that is, there is a c.u.b. C such that for all α ∈ C n we have f(αi1 , . . . , αin ) = sup<αin f(αi1 , . . . , ) (in this case f(α) must have uniform cofinality αin ). Suppose f is discontinuous almost everywhere. Then f must have one of the uniform cofinalities listed in Lemma 4.19. Suppose f(α) has uniform cofinality αj for some 1 ≤ j ≤ n. Then there is a unique permutation of n + 1 extending and a f : (1 )n+1 → 1 ordered by such that f = f (n). For example, if f is ordered by = (4, 1, 3, 2) and f(α) has uniform cofinality α3 , then = (5, 1, 4, 2, 3). Summarizing, we have the following cases. Definition 4.21. Let f : (1 )n → 1 . We say f is of type if f is ordered by , discontinuous, and of uniform cofinality . We say f is of type s is f is ordered by and is continuous (and of uniform cofinality at points of
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successor rank). We say f is of type (, ) if extends and there is an f ordered by with f (n) = f. We extend these definitions to their “almost everywhere” versions in the usual manner. We then have: Lemma 4.22. Let f : (1 )n → 1 depend on all its arguments almost everywhere. Then f is almost everywhere of type , of type s , or of type (, ) for some permutation(s) and . The analysis of measures on involves putting together finitely many functions of various types from (1 )n to 1 for some n. The object which describes how these are put together we call a (level-2) tree of uniform cofinalities. It is basically a finite tree (with root node ∅) which at each node assigns a possible uniform cofinality for the measure associated to the previous node. The node is terminal if the uniform cofinality is . The precise definition follows. Definition 4.23. A level-2 tree of uniform cofinalities is a function R with domain a finite tree satisfying the following. 1. R(∅) = (1), the unique permutation of length 1. 2. For each (i1 ) ∈ dom(R), R(i1 ) is either the symbol (to denote uniform cofinality ), or is the unique permutation (2, 1) of length 2 extending R(∅). If R(i1 ) = , then (i1 ) is a terminal node in dom(R). 3. In general, if (i1 , . . . , ik ) ∈ dom(R), then R(i1 , . . . , ik−1 ) is a permutation of length k (beginning with k) and R(i1 , . . . , ik ) is either a permutation of length k +1 (beginning with k +1) which extends R(i1 , . . . , ik−1 ), or else is the symbol . In the latter case, (i1 , . . . , ik ) is a terminal node in dom(R). Remark 4.24. We didn’t make the definition of a level-1 tree of uniform cofinalities earlier, but we could have. A level-1 tree Q could be taken to be a finite set {1, 2, . . . , n}. The ordering
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1 before the old element 1 (which is now identified with the new element 2 of Q2 ), and Q3 adds the new element 2 between the old elements 1 and 2 (the old 2 becomes the new 3 of Q3 ), then = (3, 1, 2). We frequently implicitly assume that {ik : (i1 , . . . , ik−1 )ik ∈ dom(R)} is an integer. Example 4.25. A simple level-2 tree of uniform cofinalities is the following (the nodes s of the tree are labeled with the values R(s)): ∅ (1)
• (2, 1)
•
•
•
•
(3, 1, 2)
(3, 2, 1)
We view the domain of R in this example as being {∅, (0), (1), (1, 0), (1, 1), (1, 2)}. Given a tree of uniform cofinalities R, we let max{α2 , . . . , αk }). For Example 4.25, the domain of 0). 4. f(α1 , i1 , . . . , αk , ik ) has uniform cofinality specified by R(i1 , . . . , ik ). That is: (a) If R(i1 , . . . , ik ) = , then f(α1 , i1 , . . . , αk , ik ) has uniform cofinality .
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(b) Otherwise, f(α1 , i1 , . . . , αk , ik ) has uniform cofinality { : (α1 , . . . , αk , ) is order-isomorphic to R(i1 , . . . , ik )}. (when this set has successor order-type, we interpret this as requiring uniform cofinality ). Note that the requirement in (4b) is equivalent to saying the map (α1 , . . . , αk ) → f(α1 , i1 , . . . , αk , ik ) is of type (, ) where = R(i1 , . . . , ik−1 ) and = R(i1 , . . . , ik ). If f is of type R, then for all (i1 , . . . , ik ) ∈ dom(R), f induces a subfunction f i1 ,...,ik : (1 )k → 1 by f i1 ,...,ik (α1 , . . . , αk ) = f(α1 , i1 , α2 , i2 , . . . , αk , ik ) where here (α1 , . . . , αk ) is order-isomorphic to R(i1 , . . . , ik−1 ). The strong partition relation on 1 induces a partition property for functions of type R, and this naturally defines a measure MR associated to R. Namely: Definition 4.27. MR is the measure on tuples (. . . , α i1 ,...,ik , . . . } (indexed by nodes in the tree dom(R)) given by: MR (A) = 1 iff there is a c.u.b. C ⊆ 1 such that for all f : dom(
Example 4.28. For the level-2 tree R of Example 4.25, MR is a measure on tuples (α 0 , α 1 , α 1,0 , α 1,1 , α 1,2 ) where α 0 < α 1 < 2 and α 1,0 < α 1,1 < α 1,2 < 3 . The measure concentrates on tuples satisfying α 1,0 (1) = α 1,1 (1) = α 1,2 (1) = α 1 , where α 1,0 (1) denotes the ordinal represented by the first invariant of a function of the appropriate type representing α 1,0 . It is not difficult to see that the top level splitting of the tree corresponds to product measures, so MR in this case is equal to S11 × where is a measure on tuples (α 1 , α 1,0 , α 1,1 , α 1,2 ). Here S11 is the -cofinal normal measure on 2 , which is the measure induced by the strong partition property on 1 and functions f : 1 → 1 of the correct type (and the measure W11 on 1 ). The general definition of the measure S1m is given in Definition 5.1. Theorem 4.29. Every measure on is equivalent to a measure of the form W1m × M(R) for some m and some (level-2) tree of uniform cofinalities R. To see how this equivalence takes place, we extend the trivial descriptions (which are just integers) slightly to extended trivial descriptions, defined relative to a fixed tree of uniform cofinalities R. Definition 4.30. An extended trivial description, defined relative to R and W1m , is a sequence of the form d = (d1 , i1 , . . . , dk , ik ) or d = (d1 , i1 , . . . , dk , ik )s where the s is a formal symbol (standing for “sup”). The di are trivial descriptions with di ≤ m, (i1 , . . . , ik ) ∈ dom(R), and (d1 , . . . , dk ) is orderisomorphic to R(i1 , . . . , ik−1 ).
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We write (d1 , i1 , . . . , dk , ik )(s) to denote that the symbol s may or may not appear. We order the extended descriptions lexicographically, and where (d1 , i1 , . . . , dk , ik )s < (d1 , i1 , . . . , dk , ik ), (d1 , i1 , . . . , dk , ik )(s) < (d1 , i1 , . . . , d , i )(s) if k < and s does appear on the left side, or k > and s does not appear on the right side. We interpret the extended descriptions as follows. Definition 4.31. Suppose d = (d1 , i1 , . . . , dk , ik )(s) is defined relative to R and W1m . If f : dom(
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The proof of Theorem 4.34 is very similar to that of Theorem 4.9. [Sol78A] proves essentially the same theorem using different terminology and a somewhat different argument. In [Jac10] the details of the current proof can be found. Theorem 4.34 and Theorem 4.8 give a coding for the subsets of , exactly as the proof of Theorem 4.14 for subsets of 1 . This results in a Δ13 coding of the subsets of . That is: Corollary 4.35. There is a map from onto ℘( ) such that for any α < , {x : α ∈ (x)} ∈ Δ13 . Corollary 4.35 is enough to get the weak partition relation on 13 , as shown B ⊆ × in [Sol78A]. Namely, we code functions from to 13 via relations × such that for all α < , Bα = {(, ) : R(α, , )} is wellfounded (we are using Corollary 4.35 for subsets of ( )3 , which follows easily from the version for by taking a bijection between ( )3 and ). For example, to show 13 → ( 13 ) for = we set, in the notation of Theorem 2.17, iff B is wellfounded of rank , where B = (x) ⊆ ( )3 . Using ϕ(x)(α, ) α Corollary 4.35 and Theorem 2.10 applied to Δ = Δ13 it is straightforward to verify (2) of Theorem 2.17. To see (3) of 2.17, suppose A ⊆ Rα is Σ13 , where Rα is as in Theorem 2.17. Then supx∈A ϕ(x)(α) is bounded by the length 0 of the following wellfounded relation ≺: (x, y) ≺ (x , y ) ↔ (x = x ∈ A) ∧ ∃, < ((y) = ∧ (y ) = ∧ (x)(α, , ) where is a Δ13 norm of length . From Theorem 2.10 it is easy to see that has length less that 1 . ≺ is Σ13 and so 3 for further details. We note that here The reader can consult [Sol78A] the minor annoyance of having to slightly modify the argument to work for functions instead of sets (as we had to do for 1 ) does not arise, although it does when when have to prove the strong partition relation on 13 in the next level of the induction. §5. The Second Level of the Induction. At the second level (stage n = 1 of the inductive analysis) we must compute 15 , prove the strong partition relation on 13 , and prove the weak partition relation on 15 . The arguments to use non-trivial follow in outline those of the previous section, but we begin descriptions. Before saying what these are, we introduce two new families of canonical measures, besides the family W1m we already have. Their definitions will use the strong partition property on 1 and the weak partition property on 13 respectively.
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Let n be the permutation (n, 1, 2, . . . , n − 1). We abbreviate the order <n on (1 )n by
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A level-2 description will have an index associated to it which will be written as a superscript. In the notation for the general level description this m would be written as d W1 , where the index W1m is a formal symbol (which, of course, suggests the measure W1m ). In practice, to avoid overly cumbersome notation, we will write d m as an abbreviation. The set of level-2 descriptions with index m will be denoted D2m . A description d m will be defined, as we said above, relative to a sequence of measures K1 , . . . , Kt . We will let D2m (K1 , . . . , Kt ) denote the level-2 descriptions defined relative to K1 , . . . , Kt will define the iterated with index m. As we said above, the measures K ultrapower in which the description is building an ordinal. The index m (or W1m ) tells us that the ordinal that the description produces, given fixed functions h1 , . . . , ht , will be defined by first generating an ordinal in the ultrapower by W1m . That is, it will produce an ordinal < m+1 . In the case of level-2 descriptions, the final ordinal the description produces can be viewed as a single iterated ultrapower by the measures K1 , . . . , Kt , W1m . For level-3 descriptions, however, the ordinal will used indirectly and we cannot view the final ordinal produced as a iterated ultrapower by K1 , . . . , Kt , W1m . This is why the measures in the index are separated from the K1 , . . . , Kt . For the rest of this section we will suppress writing the subscript 2 and just write, for example Dm (K1 , . . . , Kt ), which we frequently abbreviate as ). Dm (K We fix some notation to make the above discussion more precise. Given = K1 , . . . , Kt and a description d ∈ Dm (K ), the the sequence of measures K ordinal in the iterated ultrapower by the measures K that d defines will be ). The ordinal (d ; K ) will be represented denoted (d ; K1 , . . . , Kt ), or just (d ; K with respect to the measure K1 by the function which assigns to [h1 ] the ordinal (d ; h1 , K2 , . . . , Kt ). Here h1 : dom(
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So, S1m induces the ordinal measure S1m under the map h → [h]W1m . If Ki denotes a measure of the form S1ri or W1r1 , then Ki denotes the corresponding function space measure S1ri or W1r1 . Note that if Ki = W1ri , then hi : ri → 1 , and in this case if we write [hi ] = [hi ] we simply mean hi = hi . Definition 5.4. Let K1 , . . . , Kt be measures with Ki = S1ri or Ki = W1ri , and let K1 , . . . , Kt be the corresponding function space measures. Suppose F : K1 × · · · × Kt → Ord. We say F is well-defined in the iterated ultrapower sense if: there is a c.u.b. C1 ⊆ 1 such that for all h1 : dom(
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Example 5.6. Consider one more example which is close to (but not quite) the general form. Say K1 = K2 = S13 , K3 = W13 , K4 = S13 , and m = 4. We might then let (d ; h; α) = h1 (α1 , h2 (3 , h4 (2 , α2 , α3 ), h4 (α1 , α2 , α4 )), h4 (α1 , α3 , α4 )), where h3 = (1 , 2 , 3 ). The reader can check that this definition is welldefined in the previous iterated ultrapower sense. In defining the level-2 descriptions, there will be two kinds of “basic” descriptions and then non-basic description which, roughly speaking, will be compositions of the basic ones. One of the basis descriptions, which we denote ·i , describes the function (d ; h; α) = αi . The other is a pair of integers = hk (i) (k; i) where Kk = W1rk and i ≤ rk . This describes the value (d ; h; α) (recall hk : rk → 1 is an rk -tuple of countable ordinals). As we define Dm (K1 , . . . , Kt ), we also define a function d → k(d ) ∈ {1, . . . , t} ∪ {∞} ) which records the outermost function used in the composition on Dm (K described by d (with value ∞ if none are used). The actual definitions follow. Although it was not done in [Jac99] or [Jac88], we find it convenient (following a suggestion of a previous referee) to ) of pre-descriptions. The set D(K ) of first give the definition of the set D (K descriptions will be the pre-descriptions that satisfy an extra condition which guarantees that they produce well-defined ordinals in the iterated ultrapower. We will use pre-descriptions only as a step in defining descriptions. In the following definition the symbol “s” appears. This is regarded as a formal syntactical symbol. Its intended meaning is to stand for “sup,” as will become apparent when we interpret the descriptions. The formal definition is actually by reverse induction on the value k(d ) ∈ {1, . . . , t} ∪ {∞} (that is, we assume the set of pre-descriptions having k(d ) > k has been defined, and proceed to define those with k(d ) = k). Definition 5.7 (Pre-descriptions). Fix m > 0 and K1 , . . . , Kt where Ki = m S1ri or Ki = W1ri . The set of pre-descriptions D (K) and the function k are defined through the following cases (the first two cases define the basic descriptions). 1. d = ·i where 1 ≤ i ≤ m. In this case k(d ) = ∞. 2. d = (k; i) where Kk = W1rk and 1 ≤ i ≤ rk . We set k(d ) = k. 3. d = (k; d0 , d1 , . . . , d ) where 1 ≤ k ≤ t, Kk = S1r , 0 ≤ ≤ r − 1, and k(d0 ), k(d1 ), . . . , k(d ) > k (if = 0 we have d = (k; d0 )). We set k(d ) = k. 4. d = (k; d0 , d1 , . . . , d )s where 1 ≤ k ≤ t, Kk = S1r , r ≥ 2, 1 ≤ ≤ r − 1, and k(d0 ), k(d1 ), . . . , k(d ) > k. We set k(d ) = k. We write (k; dr , d1 , . . . , d )(s) to indicate that the formal symbol s (which again stands for “sup”) may or may not appear. We require r ≥ 2 and ≥ 1 in
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the last case so as to avoid (k; d )s . Allowing this would create a redundancy, as this will evaluate to the same as d . Remark 5.8. In [Jac99] and [Jac88] the non-basic descriptions were denote (k; dr , d1 , . . . , d )(s) . That is, the first description was denoted dr (where Kk = S1r ) instead of d0 . The following definition gives the evaluation (d ; h; α) of the description. Recall that h(j) denotes the jth invariant of h (Definition 4.17). The notation h s (j) was also defined in Definition 4.17. In the following definition, the functions hi are of the form hi : dom(
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the size of the output value. Recalling the definition of the order 0 and K1 , . . . , m Kt ∈ W1 ∪ S1 . The ordering d < d on the pre-descriptions D (K1 , . . . , Kt ) is defined by reverse induction on min{k(d ), k(d )} through the following cases: I. Suppose k = k(d ) < k(d ) = k. 1. Kk = W1r . In this case we set d < d . 2. Kk = S1r . In this case d = (k ; d0 , d1 , . . . , d )(s) . We define d < d to hold iff d0 < d . II. Suppose k = k(d ) > k(d ) = k. 1. Kk = W1r . In this case we do not set d < d . 2. Kk = S1r . In this case d = (k; d0 , d1 , . . . , d )(s) . We define d < d to hold iff d < d0 or d = d0 . III. Suppose k(d ) = k(d ) = k < ∞. 1. Kk = W1r . In this case d = (k; i ), d = (k; i). We set d < d iff i < i. 2. Kk = S1r . In this case d = (k; d0 , d1 , . . . , d )(s) and d = (k; d0 , d1 , . . . , d )(s) . a. Suppose there is a least j with 0 ≤ j ≤ such that dj = dj . Then we define d < d iff dj < dj . In the remaining cases assume there is no such j. b. If < , then d < d iff d has the symbol s. c. If > then d < d iff d does not have the symbol s. d. If = , then d < d iff d has the symbol s and d does not. IV. k(d ) = k(d ) = ∞. In this case d = ·r and d = ·r . We set d < d iff r < r. We can now state the well-definedness condition, which for historical reasons we call “condition C .” The definition is also inductive, by reverse induction on k(d ). Definition 5.13. A pre-description d satisfies condition C if either d is basic, or d = (k; d0 , d1 , . . . , d )(s) is non-basic, d0 , d1 , . . . , d satisfy C , and d1 < · · · < d < d0 . Definition 5.14. Let m > 0 and K1 , . . . , Kt ∈ W1 ∪ S1 . A description (with m index m defined relative to K1 , . . . , Kt ) is a pre-description d ∈ D (K ) which m satisfies condition C . We let D (K1 , . . . , Kt ) denote this set of descriptions.
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The reader can check that if d is a description then the evaluation of d is well-defined in the sense that ∀∗ h1 if [h1 ] = [h1 ], then ∀∗ h2 if [h2 ] = [h2 ], . . . , ∀∗ ht if [ht ] = [ht ] then ∀∗ α (d ; h; α) = (d ; h ; α). This is equivalent to saying that the function space map (h1 , . . . , ht ) → (d ; h1 , . . . , ht ) = [α → (d ; h1 , . . . , ht ; α)] W1m is well-defined in the iterated ultrapower sense (Definition 5.4). Thus, the evaluation of the description is well-defined in the iterated ultrapower sense. We summarize this discussion in the following definition. Definition 5.15. Let d ∈ Dm (K1 , . . . , Kt ). Then (d ; K1 , . . . , Kt ) is the ordinal represented in the ultrapower with respect to the measure K1 by the function which assign to [h1 ] the ordinal (d ; h1 , K2 , . . . , Kt ). This, in turn, is represented with respect to the measure K2 by the function which assign to [h2 ] the ordinal (d ; h1 , h2 , K3 , . . . , Kt ). Finally, (d ; h1 , . . . , ht ) is the ordinal less than m+1 which is represented with respect to W1m by the function which assigns to α ∈ (1 )m the ordinal (d ; h; α) defined in ) is defined similarly except Definition 5.9. If g : 1 → 1 , then (g; d ; K (g; d ; h; α) = g((d ; h; α)). We next introduce the lowering operator L, as we did for the trivial descriptions. L(d ) will be the immediate predecessor of d in the above ordering on descriptions, except if d is minimal in this ordering in which case L(d ) is not defined. We will actually define approximations Lk to L, where Lk (d ) is defined for those d with k(d ) ≥ k. We will then take L = L1 . These approximations to L also correspond to the steps in the proof of the main theorem 5.17 below. Definition 5.16. Let d ∈ Dm (K1 , . . . , Kt ). We define Lk (d ), where k ∈ {1, . . . , t} ∪ {∞} and k(d ) ≥ k, by reverse induction on k(d ) through the following cases. 1. k = ∞. In this case d = ·i . If i > 1 we set L∞ (d ) = ·i−1 , and if i = 1 then d is minimal with respect to L∞ . In the remaining cases we assume 1 ≤ k ≤ t. 2. k = k(d ). We have the following sub-cases. (a) d is basic, so d = (k; i). Then Lk (d ) = (k; i − 1) if i > 1 and if i = 1 then d is minimal with respect to Lk . (b) d = (k; d0 , d1 , . . . , d ) where = r − 1. Then Lk (d ) = (dr , d1 , . . . , d )s if ≥ 1 and if = 0 then Lk (d ) = d0 . (c) d = (k; d0 , d1 , . . . , d ) where < r − 1. If Lk+1 (d0 ) is defined and satisfies Lk+1 (d0 ) > d if ≥ 1, then Lk (d ) = (k; d0 , d1 , . . . , d , Lk+1 (d0 )). Otherwise Lk (d ) = (k; d0 , d1 , . . . , d )s if ≥ 1 and if = 0 then Lk (d ) = d0 .
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(d) d = (k; d0 , d1 , . . . , d )s . If Lk+1 (d ) is defined and satisfies Lk+1 (d ) > d−1 if > 1, then Lk (d ) = (k; d0 , d1 , . . . , d−1 , Lk+1 (d )). Otherwise, Lk (d ) = (k; d0 , d1 , . . . , d−1 )s if > 1 and Lk (d ) = d0 if = 1. 3. k < k(d ) and Kk = W1r . (a) d is not minimal with respect to Lk+1 . Then Lk (d ) = Lk+1 (d ). (b) d is minimal with respect to Lk+1 (d ). Then Lk+1 (d ) = (k; r). 4. k < k(d ) and Kk = S1r . (a) d is not minimal with respect to Lk+1 (d ). Then Lk (d ) = (k; Lk+1 (d )). (b) d is minimal with respect to Lk+1 (d ). Then d is minimal with respect to Lk . We formulate the “main theorem” for level-2 descriptions as we did in Theorem 4.3 for trivial descriptions. Theorem 5.17 (Main theorem for level-2 descriptions). Suppose d ∈ ). If d is non-minimal with respect to L, Dm (K1 , . . . , Kt ) and ϑ < (d ; K If d is L-minimal then then there is a g : 1 → 1 such that ϑ < (g; L(d ); K). ϑ < 1 . We will give a brief sketch of how Theorem 5.17 is proved below. If g : 1 → 1 , then there is a real x such that the section of the Kunen tree Tx is wellfounded and g(α) < |Tx α| for all infinite α. This easily shows that ) onto ) < (L(d ); K )+ . Namely, we define a map from (L(d ); K (g; L(d ); K ) as follows. If ϑ < (L(d ); K ), then (ϑ) is defined by (g; L(d ); K h; α))|. (ϑ)([h]; α) = |(Tx (L(d ); h; α)(ϑ( ∀[h1 ], . . . , [ht ] ∀α So, we have the following immediate consequence of Theorem 5.17 ) ≤ (L(d ); K )+ . Corollary 5.18. (d ; K In fact, we will see shortly that equality holds in Corollary 5.18. Granting this for the moment, we then have the following exact computation of the ordinal represented by a level-2 description. ). Then (d ; K) = r+1 where r = |d |L is Corollary 5.19. Let d ∈ Dm (K ) given in Definithe rank of d in the ordering of the descriptions in Dm (K tion 5.16. We pause to give some examples and consequences. In the first example we compute the ultrapower jS1m (n ) of n by the measure S1m . Example 5.20. Consider the descriptions d ∈ Dn−1 (K1 ), where K1 = S1m . For each such d , (d ; K1 ) is by definition an ordinal in the ultrapower jS1m (n ). A small variation of Theorem 5.17 shows that if ϑ < jS1m (n ), then there is a
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g : 1 → 1 such that ϑ < (g; dmax ), where dmax is the maximal description in Dn−1 (K1 ) which is easily seen to be dmax = (1; ·n ). From Theorem 5.17 it follows that jS1m (n ) = +1 , where is the number of descriptions in Dn−1 (K1 ). These descriptions are all of the form (1; ·i0 , ·i1 , . . . , ·ip−1 )(s) , where 1 ≤ i1 < · · · < ip−1 < i0 ≤ n and p ≤ m (recall (s) means that s may or may not appear). For convenience here we are deviating slightly from the official definition of Dn−1 (K1 ) in that we are allowing descriptions of the form (1; ·i )s (these “extra” descriptions are just equivalent to the official descriptions ·i , and make the counting obvious). more + · · · + n−1 Clearly there are 2 n−1 such descriptions, where we regard 1 m n−1 = 0 if k > n − 1. Thus we have: k & % n−1 n−1 . + ··· + jS1m (n ) = k where k = 1 + 2 m 1 The formula of Example 5.20 when iterated allows us to compute any iterated ultrapower of the form jS m1 ◦ · · · ◦ jS1mt (n ). For the next example, 1 we do this by directly counting the corresponding descriptions. This example appeared in [Jac10]. Example 5.21. We compute jS12 ◦ jS12 (3 ). From Example 5.20, we know the answer should be jS12 ◦ jS12 (3 ) = jS12 (7 ) = 43 . The cardinals below the iterated ultrapower jS12 ◦ jS12 (3 ) correspond now to the descriptions in D2 (K1 , K2 ), where K1 = K2 = S12 . The maximal description is now dmax = (1; (2; ·2 )). The remaining descriptions can be obtained by iterating the L operation, which will list the descriptions in decreasing order. Figure 1 lists these descriptions. Thus, there are 42 descriptions in D2 (K1 , K2 ), which agrees with jS12 ◦ jS12 (3 ) = 43 . Remark 5.22. One can use the formula of Example 5.20 and the fact that jW1m (n ) = m+n to compute the number of descriptions in Dm (K1 , . . . , Kt ), in a manner similar to Example 5.21. The number of such descriptions is k, where jK1 ◦· · ·◦jKt (m+1 ) = k+1 . For example, consider D4 (S13 , S13 , W13 , S13 ) of Example 5.20. We have jS13 ◦ jS13 ◦ jW13 ◦ jS13 (5 ) = jS13 ◦ jS13 ◦ jW13 (29 ) = jS13 ◦ jS13 (32 ) = jS13 (9983 ) = 331536189167 . Thus, there are 331 536 189 166 descriptions in D4 (S13 , S13 , W13 , S13 ). We give sketches of two arguments. First we show why the lower bound holds in Corollary 5.18, and then we give a brief outline of how the proof of Theorem 5.17 goes. ) = (L(d ); K )+ it suffices to show that the conclusion of To show that (d ; K m Corollary 5.19 holds. Let d ∈ D (K ) be non-minimal, and let r = |d |L be its ). We must show that (d ; K) ≥ rank in the ordering of descriptions in Dm (K
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d = dmax = (1; (2; ·2 ))
L(d ) = (1; (2; ·2 ); (2; ·2 , ·1 ))
L (d ) = (1; (2; ·2 ); (2; ·2 , ·1 ))
L3 (d ) = (1; (2; ·2 ); (2; ·2 , ·1 )s )
L4 (d ) = (1; (2; ·2 ); (2; ·2 , ·1 )s )s
L5 (d ) = (1; (2; ·2 ); ·2 )
L6 (d ) = (1; (2; ·2 ); ·2 )s
L7 (d ) = (1; (2; ·2 ); (2; ·1 ))
L8 (d ) = (1; (2; ·2 ); (2; ·1 ))s
L9 (d ) = (1; (2; ·2 ); ·1 )
s
2
L10 (d ) = (1; (2; ·2 ); ·1 )s
L11 (d ) = (2; ·2 )
L12 (d ) = (1; (2; ·2 , ·1 ))
L13 (d ) = (1; (2; ·2 , ·1 ); (2, ·2 , ·1 )s )
L14 (d ) = (1; (2; ·2 , ·1 ); (2, ·2 , ·1 )s )s
L15 (d ) = (1; (2; ·2 , ·1 ); ·2 )
L16 (d ) = (1; (2; ·2 , ·1 ); ·2 )s
L17 (d ) = (1; (2; ·2 , ·1 ); (2; ·1 ))
L18 (d ) = (1; (2; ·2 , ·1 ); (2; ·1 ))s
L19 (d ) = (1; (2; ·2 , ·1 ); ·1 )
L20 (d ) = (1; (2; ·2 , ·1 ); ·1 )s
L21 (d ) = (2; ·2 , ·1 )
L22 (d ) = (1; (2; ·2 , ·1 )s )
L23 (d ) = (1; (2; ·2 , ·1 )s , ·2 )
L24 (d ) = (1; (2; ·2 , ·1 )s , ·2 )s
L25 (d ) = (1; (2; ·2 , ·1 )s , (2; ·1 ))
L26 (d ) = (1; (2; ·2 , ·1 )s , (2; ·1 ))s
L27 (d ) = (1; (2; ·2 , ·1 )s , ·1 )
L28 (d ) = (1; (2; ·2 , ·1 )s , ·1 )s
L29 (d ) = (2; ·2 , ·1 )s
L30 (d ) = (1; ·2 )
L31 (d ) = (1; ·2 , (2; ·1 ))
L32 (d ) = (1; ·2 , (2; ·1 ))s
L33 (d ) = (1; ·2 , ·1 )
L34 (d ) = (1; ·2 , ·1 )s
L35 (d ) = ·2
L36 (d ) = (1; (2; ·1 ))
L37 (d ) = (1; (2; ·1 ), ·1 )
L38 (d ) = (1; (2; ·1 ), ·1 )s
L39 (d ) = (2; ·1 )
L40 (d ) = (1; ·1 )
L41 (d ) = ·1
Figure 1. The descriptions in D2 (S12 , S12 ). r+1 . Note that there are r descriptions below d , say d1 < d2 < · · · < dr < d . ). Since We define an embedding from the ultrapower jW1r (1 ) into (d ; K r jW1r (1 ) = r+1 , this suffices. For f : (1 ) → 1 we define ([f]W1r ) by: ∀∗ h1 , . . . , ht ∀∗ α ∈ (1 )m ([f])(h; α) = f((d1 ; h; α), . . . , (dr ; h; α)). It is straightforward to check that is well-defined and gives an embedding. For example, to see that is well-defined, suppose [f] = [f ]. Let C ⊆ 1 be
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for all ∈ C r . Since for almost all h1 , . . . , ht = f () c.u.b. such that f() we have that all the hi have range in C , it follows that all the (di ; h; α) are in C . < · · · < (dr ; h; α) From the ordering of descriptions we also have (d1 ; h; α) ∗ ∗ (f)(h; α) = for almost all h and almost all α. This shows that ∀ h ∀ α (f )(h; α). We next outline the proof of Theorem 5.17. In general, the proof proceeds = K1 , . . . , K t ) by a sequence of partition arguments, starting with t (where K and working back to 1. At each step k we prove the analog of the statement of Theorem 5.17 for the approximation Lk of the L operation (recall that L = L1 ). After the last step, we have proved the theorem for the true L operation. We illustrate this proof by considering an example. Let d = dmax = (1; (2; ·2 )) ∈ D2 (S12 , S12 ) (see Figure 1). So, L(d ) = (1; (2; ·2 ), (2, ·2 , ·1 )). Let ϑ ∈ Ord be such that ∀∗ h ∀∗ α = (α1 , α2 ) ϑ(h; α) < (d ; h; α) = h1 (1)(h2 (1)(α2 )), where we recall that h1 (1)() = sup< (h1 (, )) and similarly for h2 (1). So, ∃ < h2 (1)(α2 ) ϑ(h; α) < h1 (, h2 (1)(α2 )), ∀∗ h ∀∗ α and so ∃ < α2 ϑ(h; α) < h1 (h2 (, α2 ), h2 (1)(α2 )). ∀∗ h ∀∗ α It follows that ϑ(h; α) < h1 (h2 (g(α1 ), α2 ), h2 (1)(α2 )). ∀∗ h1 , h2 ∃g : 1 → 1 ∀∗ α We now consider the first of the partition arguments. For every ordinal ϑ(h1 ) (this is just an ordinal, though we write it this way to be suggestive) and function h1 such that ∀∗ h2 ∃g : 1 → 1 ∀∗ α ϑ(h1 )(h2 ; α) < h1 (h2 (g(α1 ), α2 ), h2 (1)(α2 )), we consider the following partition. Fix a function [h2 ] → ϑ(h1 )(h2 ) representing ϑ(h1 ) with respect to K2 = S12 . P(h1 ) : We partition functions h2 : dom(<2 ) → 1 of the correct type according to whether ∀∗ α ϑ(h1 )(h2 ; α) < h1 (h2 (α1 + 1, α2 ), h2 (1)(α2 )). We claim that on the homogeneous side of the partition the stated property must hold. For suppose C ⊆ 1 were c.u.b. and homogeneous for the contrary side. Fix h2 : dom(<2 ) → C of the correct type such that ∃g : 1 → 1 ∀∗ α ϑ(h1 )(h2 ; α) < h1 (h2 (g(α1 ), α2 ), h2 (1)(α2 )) (the last condition holds for almost all h2 ). Fix a function g : 1 → 1 witnessing this statement, that is, ϑ(h1 )(h2 ; α) < h1 (h2 (g(α1 ), α2 ), h2 (1)(α2 )). ∀∗ α h2 :
(1)
Let D ⊆ C be c.u.b. and closed under g. Define dom(<2 ) → 1 by “sliding up” h2 along D. That is, define h2 (α, ) = h2 (ND (α), ND ()), where we recall ND () = the least element of D which is ≥ . Clearly [h2 ] = [h2 ] (they agree on D 2 ). Since h2 has the correct type, so does h2 . Finally,
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ran(h2 ) ⊆ ran(h2 ) ⊆ C . Since [h2 ] = [h2 ], we have ϑ(h2 )(h2 ) = ϑ(h1 )(h2 ). Since h2 is of the correct type with range in C , it follows from the assumed homogeneity of C for the contrary side that: ∀∗ α ϑ(h1 )(h2 ; α) ≥ h1 (h2 (α1 + 1, α2 ), h2 (1)(α2 )).
(2)
Note here that h2 (1)(α2 ) = h2 (1)(α2 ) for almost all α2 . Equations (1) and (2) contradict each other as h2 (α1 + 1, α2 ) ≥ h2 (g(α1 ), α2 ) from the definition of D and h2 . So, on the homogeneous side of the partition P(h1 ) the stated property holds. Fix C ⊆ 1 c.u.b. and homogeneous for P(h1 ). If we let g() = the least element of C greater than , then we have: ∃g : 1 → 1 ∀∗ h2 ∀∗ α ϑ(h1 )(h2 ; α) < h1 (g(h2 (α1 , α2 )), h2 (1)(α2 )). Since for almost all [h1 ] the ordinal ϑ(h1 ) satisfies our hypotheses, we therefore have: ∀∗ h1 ∃g : 1 → 1 ∀∗ h2 ∀∗ α ϑ(h1 )(h2 ; α) < h1 (g(h2 (α1 , α2 )), h2 (1)(α2 )). The net effect of the partition argument is that we have moved the ∃g : 1 → 1 quantifier to the left past the rightmost ∀∗ h2 quantifier. The second partition argument, which involves partitioning the functions h1 now, is essentially identical to the one just given, and likewise allows us to move the ∃g quantifier past the ∀∗ h1 quantifier. Thus we get: ϑ(h1 )(h2 ; α) < g(h1 (h2 (α1 , α2 ), h2 (1)(α2 ))). ∃g : 1 → 1 ∀∗ h1 , h2 ∀∗ α This verifies the conclusion of Theorem 5.17 in this case. §6. Level-3 descriptions. To compute 15 , show the strong partition property 1 , we need the level-3 descriptions. on 13 , and the weak partition property on 5 variations of the level-2 descripAs we said before, these are only fairly trivial tions we explored in the last section. Thus, in some sense the combinatorics we need is already present in the formalism of the last section. We next introduce the level-3 descriptions and show how to interpret them. Recall the definition of the canonical measure W3m from Definition 5.2. So, A ⊆ 13 has W3m measure one if there is a c.u.b. C ⊆ 13 such that for all → C of the correct type we have [f] m ∈ A. f : m+1 S1 Definition 6.1 (Level-3 pre-descriptions). Given a measure W3m and a sequence K1 , . . . , Kt (each Ki = S1mi or Ki = W1mi ), a level-3 pre-description defined relative to this sequence is an object of the form (d ) or (d )s , where ) denote the set d ∈ Dm (K1 , . . . , Kt ) is a level-2 description. We let D (W3m , K of level-3 pre-descriptions defined with respect to this sequence. We also add the one extra pre-description () which we declare to be defined relative to this sequence.
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Remark 6.2. We again write (d )(s) to denote that the symbol s may or may not appear. This occurrence of s should not be confused with those that occur within d itself. The next definition gives the interpretation of the level-3 pre-descriptions. Definition 6.3. Given f : m+1 → 13 of the correct type and h1 , . . . , ht , ), we define ((d )(s) ; f; h1 , . . . , ht ) = f((d ; h)) if and given (d )(s) ∈ D(W3n ; K the symbol s does not appear, and ((d )s ; f; h1 , . . . , ht ) = sup{f() : < (d ; h)}. We also define ((); f; h) = sup(f). If g : 13 → 13 we also define (g; (d )(s) ; f; h1 , . . . , ht ) = g(((d )(s) ; f; h1 , . . . , ht )). Note that ((d )(s) ; f; h1 , . . . , ht ) is now an ordinal below 13 . As before, in going from the pre-descriptions to the descriptions we need a well-definedness condition. satisfies condition D if for almost Definition 6.4. We say (d ) ∈ D (W3m ; K) all h1 , . . . , ht , (d ; h) is the equivalence class of a function g : (1 )m → 1 of the correct type. We say (d )(s) satisfies condition D if for almost all h1 , . . . , ht , (d ; h) is a supremum of ordinals represented by g of the correct type. Remark 6.5. This definition can be given in purely combinatorial terms, as we did for the ordering < on descriptions. Also, it is easy to see that if g : (1 )m → 1 , then [g]W1m is the supremum of ordinals of the correct type iff there is a c.u.b. C ⊆ 1 such that restricted to C the following two <m properties hold: (1) g : dom(<m ) → 1 is monotonic. That is, if α then g(α) ≤ g(). This is equivalent to saying that for some j < m, g depends almost everywhere only on (α1 , . . . , αj , αm ) and is order-preserving with respect to <j+1 . Also (2) g(α1 , . . . , αj , αm ) has uniform cofinality almost everywhere equal to αm . Definition 6.6 (Level-3 descriptions). A level-3 pre-description (d )(s) is a description if it satisfies condition D or if it is the distinguished description (). We let D(W3m , K1 , . . . , Kt ) denote the set of level-3 descriptions. As before, the descriptions give well-defined ordinals in the iterated ultrapowers as stated in the next definition. Definition 6.7. Suppose (d )(s) ∈ D(W3m ; K1 , . . . , K3 ) is a level-3 description. Then ((d )(s) ; W3m ; K1 , . . . , Kt ) is the ordinal represented in the ultrapower by W3m by the function [f]S1m → ((d )(s) ; f; K1 , . . . , Kt ), etc., and where ((d )(s) ; f; h1 , . . . , ht ) is given in Definition 6.3. We similarly define (g; (d )(s) ; W3m ; K1 , . . . , Kt ).
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is It is easy to check that if (d )(s) is a description then ((d )(s) ; W3m ; K) well-defined in the iterated ultrapower sense (the obvious extension of Defini ) < jW m ( 1 ). We see below tion 5.4). Note that by definition, ((d )(s) ; W3m ; K 3 3 that jW3m ( 13 ) = m +1 . We nextextend the L operation to level-3 descriptions. Definition 6.8 (L on level-3 descriptions). Let (d )(s) ∈ D(W3m ; K1 , . . . , K3 ). If s does not appear we set L((d )) = (d )s . When s appears we set L((d )s ) to be the first pre-description in the sequence (L(d )), (L(d ))s , (L(2) (d )), (L(2) (d ))s , . . . which satisfies condition D (i.e., is a description), where L(i) denote the ith iterate of the L operation (on level-2 descriptions). The main theorem for level-3 descriptions is now entirely analogous to that for level-1 (trivial) descriptions (Theorem 4.3) and for level-2 descriptions (Theorem 5.17). We state it next. Theorem 6.9 (Main theorem for level-3 descriptions). Let d ∈ D(W3m ; K1 , ). If d is non-minimal with respect to L . . . , Kt ) and suppose ϑ < (d ; W3m ; K 1 1 ). If d is L-minimal, then there is a g : 3 → 3 such that ϑ < (g; L(d ); W3m ; K 1 then ϑ < 3 . sketch of proof. The proof of Theorem 6.9 is a small addendum to the proof of Theorem 5.17. The non-trivial case is when d is of the form (d )s for ) we have d a level-2 description in Dm (K1 , . . . , Kt ). Given ϑ < ((d )s ; W3m ; K that ∀∗ f : m+1 → 13 ∀∗ h1 , . . . , ht ∃ < (d ; h) [ϑ(f; h) < f()]. From Theorem 5.17 we have that ∀∗ f : m+1 → 13 ∃g : 1 → 1 ∀∗ h1 , . . . , ht [ϑ(f; h) < f((g; L(d ); h))]. We assume here that the level-3 description (L(d )) satisfies condition D (otherwise we will repeat the following argument until it does). We then consider the following partition P, using the weak partition property on 13 . P: We partition functions f : m+1 → 13 of the correct type according to whether ∀∗ h [ϑ(f; h) < f((L(d ); h) + 1)]. A “sliding argument” as in the proof of Theorem 5.17 (for the example considered) shows that on the homogeneous side of the partition P the stated property holds. Fix C ⊆ 13 homogeneous for P. Let now g : 13 → 13 be of C given by g(α) = the least element of C greater than α. The homogeneity and the definition of P easily show that ∀∗ f ∀∗ h [ϑ(f; h) < g(f((L(d )); h)), which is the conclusion of Theorem 6.9. Theorem 6.9 leads to an upper bound for the ordinal (d ; W3m ; K1 , . . . , Kt ) that the level-3 description represents, as in the case of level-1 and level-2 descriptions considered earlier. There is one crucial new difference, however.
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For the level-1 and 2 descriptions, the analysis of the “g function” required only the Kunen tree. In those cases, g was a function from 1 to 1 , and the only possible cofinality below 1 was (that is, what mattered was [g]W11 ). For level-3 descriptions, g is a now a function from 13 to 13 , and what matters the three possible now is the equivalence class of g with respect to one of 1 normal measures on 3 (corresponding to the three regular cardinals , 1 , 2 below 13 ). The analysis of a function g with respect to the -cofinal normal measure uses the analog of the Kunen tree as before, but for the other cofinalities (and in general) requires a new construction, the Martin tree. The next theorem states the property of the Martin tree. The reader can refer to [Jac99] for a proof, or to [Jac10] where two somewhat different arguments are given. Theorem 6.10. There is a tree T , called the Martin tree, on × 13 such that for any g : 13 → 13 there is an x ∈ with Tx wellfounded and a c.u.b. for all α ∈ C we have g(α) < |T sup j (α)| where C ⊆ 13 such that x the supremum ranges over the ultrapower embeddings j corresponding to measures in the homogeneous tree construction on Π11 and Π12 sets. We can be a little more specific in the statement of Theorem 6.10. Namely, for α ∈ C of cofinality we need no measures at all (i.e., g(α) < |Tx α|), for cf(α) = 1 we need the measures in the homogeneous tree for a Π11 set in (which are the measures W1m ), and for cf(α) = 2 we need the measures the homogeneous tree on a Π12 set. In the cf(α) = 2 case, itis desirable to simplify the conclusion of Theorem 6.10 by replacing the more general measures in the homogeneous tree on a Π12 set by the measures in the canonical family S1m . Westate two embedding theorems which state, roughly speaking, that the canonical measures S1m dominate all the other measures (for points of cofinality 2 in one theorem). We call one the “local” embedding theorem and the other the “global” embedding theorem. We give an example of a proof of each of these results in §7. The local embedding theorem, which we state next, is the one that simplifies the conclusion of Theorem 6.10. Theorem 6.11 (Local embedding theorem). Let be a measure occurring in the homogeneous tree on a Π11 or a Π12 set. Then there is an m ∈ and a c.u.b. C ⊆ 13 such that for all α ∈ C with cf(α) = 2 we have j (α) ≤ jS1m (α). T will henceforth denote the Martin tree. From Theorem 6.10 it follows (similarly to an earlier argument, see the ) < (; L(d ); K )+ argument just before Corollary 5.18)) that (g; L(d ); W3m ; K 1 1 where : 3 → 3 is the function (α) = supm jW1m (α) if cf(α) = 1 and
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(α) = supm jS1m (α) if cf(α) = 2 (and (α) = α if cf(α) = ). So, ) ≤ (; L(d ); W m ; K) + . By countable additivity, (; L(d ); W m ; (d ; W3m ; K 3 3 m K ) = sup (j ; L(d ); W3 ; K ), where the supremum ranges over the embeddings j for the measure = W1m or = S1m . By definition of these ordinals ) = (L(d ); W m ; K , ). we have that (j ; L(d ); W3m ; K 3 m So, we have (d ; W3 ; K1 , . . . , Kt ) ≤ [supKt+1 (L(d ); W3m ; K1 , . . . , Kt , Kt+1 )]+. The new feature is now apparent; as we lower the description by the lowering operator L we must now also add a new measure Kt+1 to the sequence. More precisely, we make the following definition. Definition 6.12. We define an ordering on the sequences p = (d ; W3m ; K1 , . . . , Kt ) where d ∈ D(W3m ; K1 , . . . , Kt ) (here m is fixed, and t can vary) by taking the transitive closure of the relation (L(d ); W3m ; K1 , . . . , Kt , Kt+1 ) < (d ; W3m ; K1 , . . . , Kt ). We define the rank |p| of p in the (slightly non-standard) manner |p| = (supq
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Computing the ranks |((); W3m ; K1 )| is fairly straightforward. The anm−1 swer turns out to be ·r1 + 1, where K1 = S1r1 (details are given in [Jac99]). This gives the upper bound jW3m ( 13 ) ≤ m +1 . In particular we have: Corollary 6.16. 15 ≤ +1 . The lower-bound for 15 follows from some embedding arguments which we section. In fact, further arguments (see [JK]) show briefly indicate in the next ) are actually cardinals, and that equality that all of the ordinals ((d )(s) ; W3 ; K actually holds in the statement of Theorem 6.14. ) represented by the deThe computation of the cardinal ((d )(s) ; W3m ; K (s) m scription (d ) ∈ D(W3 ; K ) is thus reduced to a purely combinatorial rank computation. This rank can be computed in several ways. The analysis of [JK] implicitly gives a method of computing this rank. Basically, one lists all ) below (d )(s) , and for each of them an ordinal the descriptions in D(W3m ; K is written down, and these ordinals are then added. If d1 < d2 are consecutive descriptions in this list, then the ordinal written down represents the of the rank of (d2 ; W m ; K ,K ) in the supremum over measure sequences K 3 m ordering as in Definition 6.12 except any sequence (d1 ; W3 ; K , K ) is declared minimal (i.e., rank 0). Another method is to use the formulas of [Jac99] or [Jac88] for the general projective case. It was shown there that these formulas give upper-bounds for the ranks of these sequences in the ordering of Definition 6.12. In fact, with a little care these formulas will give will give the exact ranks. Rather than sketch these arguments, we illustrate with a specific example, and leave it to the reader to check the details. This example was mentioned in [Jac10]. Example 6.17. Consider (d ) ∈ D(W32 ; S12 , S12 ), where d = (1; (2; ·2 , ·1 )s , (2, ·1 ))s . This d is listed in Example 5.21 as L26 (dM ), where dM = (1; (2; ·2 )) is the maximal description defined relative to S12 , S12 . Following the method of [JK] we can give a calculation of |((d ); W32 , S12 , S12 )| as seen in Figure 2 and obtain ((d ); W32 ; S12 , S12 ) = ℵ +1 +·2+1 . The alternate way to compute the rank of this description, following [Jac99] and [Jac88], is to use formulas which are given inductively, similar to the inductive definition of the L operation (that is, by a “reverse induction” on k(d )). We again start with |((d ); W32 , S12 , S12 )| = supK3 |(L26 (dM ))s ; W32 , S12 , S12 , K3 )| + 1. After this initial step, we inductively assign to each subdescription d of d with k(d ) = i, say, an ordinal f i (d ) which represents the supremum ;K ). For the example of the Li -rank of (d ; K over measure sequences K 26 currently being considered, since L (d ) = (1; (2; ·2 , ·1 )s , (2; ·1 ))s , keeping in mind the definition of the L = L1 operation in terms of the L2 operation, we
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|((d ); W32 , S12 , S12 )| = supK3 |(L26 (dM ))s ; W32 , S12 , S12 , K3 )| + 1 ++1 = supK |((L27 (dM )); W32 , S12 , S12 , K)| +++1 = supK |((L29 (dM )); W32 , S12 , S12 , K)| + · + + + 1 = supK |((L30 (dM )); W32 , S12 , S12 , K)| + + +1 + + + 1 = supK |((L31 (dM )); W32 , S12 , S12 , K)| + + + +1 + + + 1 = supK |((L33 (dM )); W32 , S12 , S12 , K)| + + + + +1 + + + 1 = supK |((L35 (dM )); W32 , S12 , S12 , K)| + + + + + +1 + + + 1 = supK |((L36 (dM )); W32 , S12 , S12 , K)| + + + + + + +1 + + + 1 = supK |((L37 (dM )); W32 , S12 , S12 , K)| + · 2 + + + + + +1 + + + 1 = supK |((L39 (dM )); W32 , S12 , S12 , K)| )| + · 3 + + + + + +1 + + + 1 = supK |((L40 (dM )); W32 , S12 , S12 , K )| + · 4 + + + + + + +1 + + + 1 = supK |((L41 (dM )); W32 , S12 , S12 , K = +1 + · 2 + 1 Figure 2.
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$
f 1 (L26 (d )) =
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+ f 2 ((2; ·1 )).
We then have f 2 ((2; ·1 )) = f 2 ((·1 )) + f 2 ((·1 )). By inspection,'f 2 ((·1 )) = , + and so f 2 ((2; ·1 )) = · 2. Next we write f 2 ((2; ·2 , ·1 )) = 2
Thus, |((d ); W32 ; S12 , S12 )| = +1 + · 2 + 1, which agrees with the first computation. Measures on 13 . The proof of the strong partition relation on 13 is similar both in notation to that of the argument for , that is, the analysis in method and 1 of measures we gave in the proof of Theorem 4.13, and also the analysis of measures on the m we stated in Theorem 4.29.The main differences are that we must now use (generalized) level-3 descriptions and level-3 trees of uniform cofinalities, which we next introduce. Recall the notion of level-2 tree of uniform cofinalities from Definition 4.23. We call them simply level-2 trees here. We say a level-2 tree R is an immediate extension of the tree R if dom(R) is a subtree of dom(R ), dom(R ) has one extra node in it, and R dom(R) = R. We frequently assume that dom(R) is a finite subtree of < . In this case, when we speak of immediate extensions we implicitly assume that we have an embedding from dom(R) to dom(R ) allowing us to identify nodes of dom(R) with nodes of dom(R ). If s = (i1 , . . . , ik ) ∈ dom(R ) − (dom(R)), then we say that s is the new or extra node of R . A partial level-2 tree R− is a level-2 tree except that there is a single maximal node s in dom(R) such that R(s) is not defined. A level-2 tree completes the partial tree R− if dom(R) = dom(R− ) and R(s) is defined. We say the partial tree R− immediately extends the level-2 tree whose domain is dom(R− ) \ {s}, where again s is the distinguished node. If R is a partial level-2 tree of uniform cofinalities, then notice that the ordering
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immediately extends Q if |Q | = |Q| + 1. In this case, we implicitly assume that we have an injection from Q to Q allowing us to identify points of Q with points of Q . The element of Q \ [Q] will be called the new or extra element of Q . Intuitively, we think of Q as a code for the measure W1n . If Q is a level-1 tree, we let
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the Kunen tree) to generate arbitrary measures on the n . Similarly, the level-3 trees S will directly define measures MS on 13 which are essentially the measures occurring in the homogeneous tree ona Π13 set. We generate arbitrary measures on 13 by lifting these using generalized level-3 descriptions (and sections of the Martin tree). S To define M we first define an ordering <S corresponding to S. This is lexicographic ordering on sequences (i1 , 1 , . . . , ik−1 , k−1 , ik ) satisfying: 1. (i1 , . . . , ik ) ∈ dom(S). 2. If S(i1 ) = , then k = 1 (i.e., there are no ’s). If S(i1 ) = (Q, ∅) (where Q = {1}), then 1 < 1 . If S(i1 ) = (∅, R) (where R is the minimal partial level-2 tree), then 1 < 2 . 3. We inductively require that (i1 , 1 , . . . , ik−1 ) ∈ dom(<S ) and also that S(i1 , . . . , ik−1 ) = . If S(i1 , . . . , ik−1 ) = (Qk−1 , Rk−1 ), a partial level ≤ 2 tree, then there is a function f : dom(
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1. The domain of S is the tree dom(S) = {∅, (0), (0, 0), (0, 1)}. 2. S(0) is the the partial level ≤ 2 tree (∅, R0 ), where R0 is partial level-2 tree with dom(R0 ) = {∅, (0)}, R0 (∅) = (1) (the permutation of length 1), and R0 (0) is undefined. 3. S(0, 0) is the partial level ≤ 2 tree (Q0 , R1 ) where Q0 = {1}, and R1 is the level-1 tree which completes R0 by defining R1 (0, 0) = . 4. S(0, 1) = (∅, R2 ), where R2 is the partial level-2 tree extending R0 with extra node (0, 0) in its domain and with R2 (0) = (2, 1), R2 (0, 0) is undefined. The reader can check that for this S, the ordering <S has domain consisting of (0), which is maximal in the ordering, sequences (0, , 0) where < 2 and cf() = , and (0, , 1) where < 2 and cf() = 1 (the two different choices for cf() correspond to the different ways of completing R0 to a level-2 tree). If F : dom(<S ) → 13 has type S, then F (0, , 0) has uniform cofinality { < 3 : (1) = }, where cofinality 1 and F (0, , 1) has uniform (1) denotes the first invariant of (this is equivalent to saying F (0, , 1) has uniform cofinality jW11 ()). The measure MS is a measure on triples ( 0 , 0,0 , 0,1 ) from 13 . One can 0 ) < 0,1 < see that this measure concentrates on triples with 0 < 0,0 < jS11 ( jS¯ 1 ( 0 ), where S¯11 is the 1 -cofinal normal measure on 2 . 1
Similarly to Definition 4.30 we now generalize slightly the level-3 descriptions to extended level-3 descriptions which we use to lift up the measures MS . Definition 6.22. Let S be a level-3 tree of uniform cofinalities and K1 . . . . , Kt a sequence of measures each in W1 ∪ S1 . An extended level-3 description is a sequence d = i1 , d1 , . . . , dk−1 , ik where defined relative to S and K ) (for some (i1 , . . . , ik ) ∈ dom(S), each di is a level-2 description in Dm (K m), and for almost all h1 , . . . , ht , if i = (di ; h), then i1 , 1 , . . . , k−1 , ik ∈ dom(<S ). We also allow d s if for almost all h1 , . . . , ht we have k−1 is the supremum of such that i1 , 1 , . . . , , ik ∈ dom(<S ). ) in the usual If F : dom(<S ) → 13 is of type S, then we define (F ; d ; K manner via the iteratedultrapowers by the measures K1 , . . . , Kt , where (F ; d ; h1 , . . . , ht ) = F (i1 , 1 , . . . , k−1 , ik ), with the i = (di ; h) as above. Similarly, (F ; d s ; h1 , . . . , ht ) = sup F (i1 , 1 , . . . , , ik ). <k−1
The extended descriptions are ordered in the natural manner (lexicographically, using the ordering on level-2 descriptions). The necessary ingredients to generate arbitrary measures on 13 are given in the next definition, which is the analog of Definition 4.32.
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Definition 6.23. A level-3 complex is a sequence of the form C = S; x0 , . . . , xt ; d0 , . . . , dt ; K1 , . . . , Kt , where S is a level-3 tree of uniform cofinalities, xi ∈ are reals with the section Txi of the Martin tree T wellfounded, d0 , . . . , dt are extended level-3 descriptions with di defined relative to S and the measure sequence K1 , . . . , Ki . Also, d0 > d1 > · · · > dt . If C is a level-3 complex as above, F : dom(<S ) → 13 is of type S, and set < 13 , then we define the ordinal (F ; C; ) as follows. We (F ; C; ) = |(Tx0 sup j(F ; d0 ))(α1 )|, j
where α1 is represented with respect to K1 by the function which assign to h1 the value α1 (h1 ) = |(Tx1 supj j(F ; d1 ; h1 ))(α2 (h1 ))|, and α2 (h1 ) is represented with respect to the measure K2 by the function that assign to h2 the value α2 (h1 , h2 ), etc., and where αt (h1 , . . . , ht ) = |(Txt (supj j(F ; dt ; h1 , . . . , ht )) (αt+1 (h1 , . . . , ht ))|, and where αt+1 (h1 , . . . , ht ) = . In these formulas, the supj refers to the supremum over the ultrapower embeddings j for the measures in W1 ∪ S1 . Definition 6.24. Let be a measure on ϑ < 13 , and C a level-3 complex. A ⊆ 1 has V ,C measure Then V ,C is the measure on 13 defined as follows. 3 one if for almost all <ϑ there is a c.u.b. C ⊆ 13 such that for all F : dom(<S ) → C of type S we have (F ; C; ) ∈ A. The next theorem is the analysis of measures on 13 . Theorem 6.25. Let V be a measure on 13 . Then there is a measure on an that V = V ,C . ordinal ϑ < 13 and a level-3 complex C such The proof of Theorem 6.25 is similar to that for the measures on 1 (Theorem 4.9), but uses level-2 descriptions and Theorem 5.17. The proof is given in [Jac99]. We illustrate with an example which shows how we generate a typical measure on 13 . Example 6.26. Let S be the level-3 tree of uniform cofinalities from Example 6.21. We lift this to a measure on 13 . Let be the principal measure on 0. Let K1 = S12 , K2 = W11 , and let d0 be the extended level-3 description d0 = i1 = 0. Let d1 = 0, d11 ; 1s where d11 ∈ D1 (K1 ) is the level-2 description d11 = (1; ·1 ). Let d2 = 0, d21 ; 0, where d22 ∈ D1 (K1 , K2 ) is the level-2 description d22 = (1; ·1 ; (2; 1)). Finally, let x0 , x1 , x2 be reals with the sections of the Martin tree Txi well-founded. S together with the xi , the measures K1 , K2 , and the descriptions d1 , d2 , d3 define a complex C.
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Suppose F : dom(<S ) → 13 if of type S. We can view F in this case as a F (α) has uniform cofinality for α < of function from 2 to 13 such that 1 2 cofinality , and F (α) has uniform cofinality jW11 (α) for α < 2 of cofinality 1 . F represents a triple of ordinals [F ] = ( 0 , 0,0 , 0,1 ) as in Example 6.21. Note that 0 = sup(F ). Given F (actually its equivalence class [F ]), the ordinal α(F ) = (F ; C; 0) in this case is given as follows. α(F ) = |(Tx0 supj j(sup(F ))(α1 )|. α1 is represented with respect to the measure K1 by the function which assigns to h1 : dom(<2 ) → 1 of the correct type the value α1 (h1 ) = |(Tx1 supj j()) (α2 (h1 ))|, where = (F ; d1 ; h1 ) = sup{F () : < [h1 (1)]}, where we recall h1 (1) : 1 → 1 is the first invariant of h1 . α2 (h1 ) is represented with respect to K2 = W11 by the function which assigns to h2 < 1 the value α(h1 , h2 ) = |(Tx2 )(0)|, where = F ([h]) and h() = h1 (h2 (0), ) (note that h2 : 1 → 1 and h2 (0) < 1 ). The measure V C, is the measure on 13 described by: A has measure one if there is a c.u.b. C ⊆ 13 such that for allF : 2 → 13 of type S, α(F ) ∈ A. To complete the second stage of the inductive analysis we must prove the weak partition relation on 15 . The main new ingredient is that of a level This is defined similarly to how level-3 trees 4 tree of uniform cofinalities. were defined from level-2 trees. So, a level-4 tree will be a function with domain a finite tree, and to each node is assigned a partial level ≤ 3 tree. For the precise definition and proofs the reader can consult [Jac99]. The measures on ( 15 )− are described by level-4 complexes, which lift-up the measures given bythe level-4 trees of uniform cofinalities in a manner similar to that described above for level-3 trees for measures on 13 . This gives a Δ15 partition relation coding of the subsets of ( 15 )− which then gives the weak on 15 , §7. Higher Levels. We have outlined so far the first two stages of the inductive analysis of the projective ordinals. In the first stage we used trivial descriptions to compute 13 , prove the strong partition relation on 11 , and the prove the weak partition relation on 13 . In the second stage, we used next level (non-trivial) descriptions to compute 15 , prove the strong par tition relation on 13 , and prove the weak partition relation on 15 . Re Our call that in §3 we presented the general stage-n inductive hypotheses. stage-n inductive hypotheses were called I2n−1 in [Jac88] and our stage-n auxiliary hypotheses were called K2n+1 in [Jac88] (actually we have included a little more in our auxiliary hypotheses for clarity). So, at stage-n we must compute 12n+3 , prove the strong partition relation on 12n+1 and the weak partition relation on 12n+3 , as well as prove the auxiliary hypothe ses.
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The partition properties on 12n+1 and 12n+3 follow from the existence of 1 sufficiently good codings of the subsets of 2n+1 and 2n+3 (again, there is there is the small point that one must work directly with functions, rather than subsets of 12n+1 in proving the strong partition relation on 12n+1 ). As auxiliary we mentioned in §3, these codings, as well as (a), (b), (d) of the inductive hypotheses, follow from the analysis of measures on 12n+1 and 2n+3 . The arguments for analyzing these measures are essentially identical to those for the analysis of measures on 1 (Theorem 4.9), the measures on 3 (Theorem 4.29) and the measures on 13 (Theorem 6.25) sketched in this paper (the details for the measure analysis on 3 can be found in [Jac10], and for the measures on 13 and 5 in [Jac99]). In particular, one defines the notions of level-2n + 1 and level-2n + 2 trees of uniform cofinalities which generate more or less the measures in the homogeneous tree on Π12n+1 and Π12n+2 sets respectively. These define basic measures (which homogeneity measures in the homogeneous tree for the are essentially the Π12n+1 , Π12n+2 sets) which are then lifted up by descriptions to generate armeasures on 1 bitrary 2n+1 and 2n+3 . The process is entirely similar to that for level-2 and level-3 trees discussed in this paper, given the higher analogs of the descriptions (cf. Definitions 4.27, 4.33 and Definitions 6.20, 6.24). Because the general measure analysis arguments are very similar to these, they are not given in [Jac88]. Instead, [Jac88] focuses on the part of the arguments which are different (or at least need to be generalized). The main points are to (1) isolate the correct families of canonical measures (the higher level analogs of the W1m , S1m ), (2) prove the corresponding embedding theorems (which say that it is enough to define descriptions with respect to sequences of these measures), (3) give the correct next level definition of description, and (4) prove the analog of the “main theorem” for descriptions (the analog of Theorem 5.17). These points, along with a rank computation which gives the upper-bound for 12n+3 are what is proved in [Jac88]. The lower-bound for 12n+3 follows from an independent, easier argument just assuming the weak partition property on 12n+3 (this argument is self-contained can see this argument for 1 in and does not use descriptions; the reader 5 [Jac99] or [JL]). We give a very brief overview of how these generalizations take place. ,m m and S2n+1 The general families of canonical measures are denoted W2n+1 n+1 (for 1 ≤ ≤ 2 − 1). They are defined using the weak and strong partition relations on 12n+1 respectively. We will let S11,m = S1m for consistency. ,m m and W2n+1 are From the stage-n inductive hypotheses we will have that S2n−1 defined. For simplicity we assume below the strong partition property on all the 12n+1 and define all of the canonical measures. Each of these fami lies of measures will be associated to a regular cardinal below the projective
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ordinals and conversely. Each measure in one of these families will be a measure on dom() ∈ Ord except for the measures W1m which are measures on (1 )m . Recall the order <m on (1 )m was defined just before Definition 5.1. We 1 m define in a similar way the ordering <m 2n+1 on ( 2n+1 ) . That is, (α1 , . . . , αm ) <m 2n+1 (1 , . . . , m ) iff (αm , α1 , . . . , αm−1 )
2 −1,m W1m , S1m , W3m , S31,m , S32,m , S33,m , W5m , S51,m , . . . , S57,m , . . . , S2n−1 (which is the ,m m and S2n+1 as follows. order in which they are defined). We then define W2n+1 n
m Definition 7.1. W2n+1 is the measure on 12n+1 induced by the weak par2n −1,m ) → 12n+1 of the correct tition relation on 12n+1 , functions f : dom(S2n−1 2n −1,m type and the measure S2n−1 . 1,m is the measure induced by the strong partition relation on 12n+1 , S2n+1 1 functions F : dom(<m 2n+1 ) → 2n+1 of the correct type and the m-fold product of the -cofinal normal measure on 12n+1 . ,m For ≥ 2, S2n+1 is the measure induced by the strong partition relation on 1 1 1 2n+1 , functions F : 2n+1 → 2n+1 of the correct type, and the measure on 12n+1 . Here is the measure induced by the weak partition relation on 12n+1 , m. m 1 functions f : dom( ) → 2n+1 of the correct type and the the measure n 2 −1,m Finally, m is the ( − 1)st measure in the list W1m , S1m , W3m , . . . , S2n−1 (for = 2, we identify dom(W1m ) = (1 )m with the ordinal 1m by lexicographic ordering on the m-tuples). ,m m The canonical measures W11 , S11 , . . . , W2n+1 , S2n+1 dominate all of the measures arising in the homogeneous tree construction on Π11 , . . . , Π12n+2 sets are in a precise sense given by two embedding theorems which proved in [Jac88]. These are called there the “local” and “global” embedding theorems. The two embedding theorems follow from two more technical results which are proved in a simultaneous induction. Rather than state precisely these results here, we illustrate by considering a case of each of these results at the first level. Aside from illustrating the method of proof, this shows an essential difference between the natures of the two embedding results. Recall S12 is a measure on 3 corresponding to the permutation (2, 1). Let 3 S¯1 be the measure on 4 corresponding to the permutation = (3, 2, 1). In the first example we consider an instance of the local embedding theorem.
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Example 7.2. We show that there is a c.u.b. C ⊆ 13 such that for all ∈ C with cf() = 2 we have jS¯ 3 () ≤ jS12 () (it actually then follows 1 that jS¯ 3 () = jS12 ()). For sufficiently closed we define an embedding 1 : jS¯ 3 () → jS12 (). Fix a function f : 4 → , and we define a function 1 g : 3 → such that ([f]S¯ 3 ) = [g]S12 . To do this, we use the auxiliary 1 measure = S12 . Suppose h : dom(<2 ) → 1 of the correct type represents [h]W12 ∈ dom(S12 ). We represent g([h]) with respect to the measure by the function which assigns to [h ]W12 , for h : dom(<2 ) → 1 of the correct type, . the value g([h], [h ]) = f([k]W13 ) where k is defined by: k(α1 , α2 , α3 ) = h(h (α1 , α2 ), α3 ). For fixed functions h and h , k(α1 , α2 , α3 ) is well-defined for almost all α, and furthermore on a c.u.b. set we have that k is order-preserving from < to 1 . Also, k has uniform cofinality . This shows that if f and f agree almost everywhere with respect to S¯13 , then for almost all h, for almost all h we have f([k]) = f ([k]). Next observe that for almost all h that for almost all h , if [h ] = [h ] then [k(h, h )]W13 = [k(h, h )]W13 , where k(h, h ) is the k function defined using h, h and likewise for k(h, h ). Next observe that if ˜ 2 , then for almost all h we have [k(h, h )] = [k(h, ˜ h )] (this is [h]W12 = [h] W1 because almost all h are into a c.u.b. set on which h and h˜ agree). Finally, if D ⊆ 1 is c.u.b., then for almost all h and almost all h we have that h(h, h ) takes values in D for almost all α. Putting these observations together shows that is well-defined. If is closed under ultrapowers by , then it is easy to see now that is an embedding from jS¯ 3 () to jS12 (). Note here that for 1 almost all [h] that for almost all [h ] that the equivalence class [k(1)] of the first invariant of the function k = k(h, h ) is less than or equal to [h(1)]. We use also the fact that any f : dom(S¯13 ) → Ord must almost everywhere have the property that if [k1 (1)] < [k2 (1)], then f([k1 ]) ≤ f([k2 ]). For the second example we consider an instance of the global embedding theorem. Example 7.3. Let be the measure on 13 induced by the weak partition relation on 13 , function f : 3 → 13 of the correct type, and the measure S12 . measure on 1 induced Let be the by the weak partition relation on 13 , 3 1 functions f : 4 → 3 and the measure S¯13 on 4 . We show that j× ( 13 ) ≤ j ( 13 ). Note that × is the measure induced by the weak partition relation on 13 and functions f = f1 ⊕ f2 : 3 · 2 → 13 S 2 on ). of the correct type (and the measure 3 1 We again define an embedding from the first ultrapower into the second. Fix F : 13 → 13 representing [F ]× , and we define G : 13 → 13 with [G] =
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([F ]× ). Fix g : 4 → 13 , and we define G([g]S¯ 3 ). We again use an 1 auxiliary measure, this time the measure = S12 × S12 . Note that is induced by the strong partition relation on 1 and functions from dom(< ) → 1 of the correct type, where < is lexicographic ordering on tuples (α, i, ) with < α < 1 and i ∈ {0, 1}. For (1 , 2 ) = [h] ∈ dom( ) (where h : dom(< ) → 1 is of the correct type), we assign to (1 , 2 ) an ordinal ϑ(1 , 2 ) which is the equivalence class of an order-preserving function f = f1 ⊕ f2 : 3 · 2 → 4 . For such a function f we will let [f] denote the pair ([f1 ]S12 , [f2 ]S12 ). We will have that the following property is satisfied: For any A ⊆ 4 of S¯13 measure one, for almost all (1 , 2 ) we have that f is almost everywhere into A.
(∗)
(That is, f1 and f2 are both almost everywhere into A.) Granting (∗), we then represent G([g]) with respect to the measure by . the function which assigns to (1 , 2 ) = [h] the value G([g], h) = F ([g ◦ f]), where [f] = ϑ(1 , 2 ). From (∗) it is immediate that is a well-defined embedding from j× ( 13 ) to j ( 13 ). , ) and show (∗). Fix h : dom(< ) → of the It remains to define ϑ( 1 2 1 correct type representing (1 , 2 ). Let = [k] < 3 , where k : dom(<2 ) → 1 is of the correct type, and let i ∈ {0, 1}, and we define f(i, ) (we are identifying 3 ·2 with lexicographic ordering on 2×3 ). We set f(i, ) = []W13 where is defined by (α1 , α2 , α3 ) = h(α3 , i, k(α1 , α2 )). For fixed h, [] only depends on the equivalence class of k, and so f(i, ) is welldefined. Also, is order-preserving with respect to < on a c.u.b. set. Since h is order-preserving with respect to < , it follows that if (i1 , 1 )
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We will not give here the precise definition of a higher level description, nor prove the main theorem. These results are given in [Jac88]. Instead, we will make a few comments about how descriptions are defined at the next level (level-4 descriptions). This should give the reader a feel of how the general inductive step takes place. A level-4 description is defined with respect to a finite sequence K1 , . . . , Kt of canonical measures each of the form W1m , S1m , W3m , or S3,m (where 1 ≤ ≤ 3). A description will have an index associated to it, which will be a sequence I = (K¯ 1 , . . . , K¯ u ) where K¯ 1 = W3m , and K¯ i ∈ W1 ∪ S1 (there are actually two other cases, but they are similar). We let DI (K1 , . . . , Kt ) denote the descriptions defined relative to K1 , . . . , Kt with index I. ). The rough As before, there are basic and non-basic descriptions in DI (K idea is that the general level-3 descriptions give the basic level-4 descriptions. As before, the non-basic level-4 descriptions are “compositions” of these. Also as before, a description will represent an ordinal with respect to the iterated ultrapower by the measures K1 , . . . , Kt . One type of basic description is a level-3 description d defined relative (there are other types) d ∈ DI (K) to I = (K¯ 1 , . . . , K¯ u ). For h1 , . . . , ht representing ordinals in the domains of K1 , . . . , Kt , this basic description gives the ordinal (d ; h1 , . . . , ht ) which is in turn represented with respect to the measures W3m , K¯ 1 , . . . , K¯ u by the function which assign to (f; h¯ 1 , . . . , h¯ u ) the value (d ; h1 , . . . , ht ; f, h¯ 1 , . . . , h¯ u ) which we define to be (d ; f, h¯ 1 , . . . , h¯ u ) (the evaluation of the level-3 description d ). This does not depend on the functions h1 , . . . , ht . This type of basic description is analogous to one type of basic level-2 description. There we had ·r as a basic level-2 description, and ·r is essentially a lower-level (i.e., trivial) description. ) is one of the form (k; d ), An example of a non-basic description in DI (K I ¯ where d ∈ D (K ), and I = I Ku+1 where Kk = S3,m and the measure S3,m is defined using the strong partition on 13 , and functions f : dom(K¯ u+1 ) → 13 . For fixed h1 , . . . , ht , and fixed f, h¯ 2 , . . . , h¯ u , this description produces ¯ = hk ([g]) where g : dom(K¯ u+1 ) → 1 is given by: the ordinal (d ; h; f; h) 3 ¯ ¯ ¯ g([h ]) = (d ; h; f; h, h ). u+1
u+1
When Kk is of the form Kk = S31,m we allow descriptions of the form d = (k; d0 , d1 , . . . , d )(s) as before, with a similar interpretation as in the level-2 case. The proof of the main theorem (the analog of 5.17) and the rank computations are similar to the level-2 case. We refer the reader to [Jac88] for further details. §8. Concluding Remarks. The descriptions, defined relative to canonical measure sequences, describe the cardinal structure below the projective ordinals. For example, it follows from Theorem 6.9 that every cardinal κ <
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) for some level-3 description d and 5 is of the form κ = (id; d ; W3m ; K m measure sequence W3 , K . We have also seen how descriptions are used to lift-up certain basic measures (more or less the measures in the homogeneous tree construction) to generate arbitrary measures on the ordinals below the projective ordinals. However. while general embedding arguments suffice to give the lower bounds for the projective ordinals (these arguments don’t use descriptions), the arguments sketched in this paper don’t show that the ordinals represented by descriptions are cardinals. For exam are cardiple, below 5 we haven’t shown that the ordinals (id; d ; W3m ; K) nals. To show that all the descriptions represent cardinals, another argument is needed which also has independent interest. This involves producing another representation for the cardinal structure below the projective ordinals which does not involve descriptions. This allows for a simpler, self-contained presentation of the cardinal structure. The description analysis, however, is needed to show that this alternate formulation works. This alternate method is described and proved for the cardinal structure below 15 in [JK], and described not give the full details below the projective ordinals in general in [JL]. We do here, but sketch the method and indicate how it gives the cardinal structure below 15 . Following the terminology of [JL], we say an ordinal algebra is a free associative, left-distributive algebra with operations ⊕, ⊗ over a set of generators {V }<α , for some α ∈ Ord. We let Aα be the algebra with generators {V }<α . We inductively define a function o from the algebra to the ordinals as follows. We set o(V0 ) = 0. We set o(s ⊕ t) = o(s) + o(u) and o(s ⊗ u) = o(s) · o(u) (ordinal addition and multiplication). We let ht(Aα ) = sup{o(s) : s ∈ Aα }. We then set o(Vα ) = ht(Aα ). For example, for the first many generators we have o(V0 ) = 0 and o(V1 ) = 1. Since o(V1 ⊕ · · · ⊕ V1 ) = n, n
we have o(V2 ) = ht(A2 ) = . Since o(V2 ⊗ · · · ⊗ V2 ) = n , n −2
for all < , and o(V ) = we have o(V3 ) = . Similarly, o(V ) = for all ≥ . We then inductively assign measures m(V ) to each of the generators. We do not give the general definition here (it is given in [JL]), but simply give the result for the first generators. We also let ot(V ) be the ordinal on which m(V ) is a measure. We let m(V0 ) be the empty measure, with ot(V0 ) = 0
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(this is for notational convenience). Let m(V1 ) = the principal measure on {0}, and ot(V1 ) = 1. Let m(V2 ) = W11 , and let ot(V1 ) = 1 . For 3 ≤ n ∈ , let m(Vn ) = S1m−2 , and ot(Vm ) = m−1 . We extend the assignment of measures and order-types from the generators to general terms s in the algebra as follows. Let ot(s) be the order-type obtained by interpreting ⊕ by + and ⊗ by × (ordinal addition and multiplication). For example, if s = V4 ⊕ (V3 ⊗ V3 ), then ot(s) = 3 + 2 · 2 . We may identify terms s in the algebra with finite trees Ts whose nodes (except for the root node) are labeled with generators, in the same standard way as for ordinal expressions involving sums and products. So, ⊕ corresponds to splittings of a node, and ⊗ corresponds to descending in the tree. More precisely, the tree associated to s ⊕ t is the tree consisting of side-by-side copies of the trees for s and t, and the tree for s ⊗ t is the tree obtained by replacing every terminal node of the tree for t with a copy of the tree for s. For example, for the term s = V4 ⊕ (V3 ⊗ V3 ) above we have the tree: • V4
V3 V3
To every terminal node of the tree is associated a product measure obtained by taking the product of the measures corresponding to the generators as we descend along the branch to the terminal node. In fact, to each node of the tree we assign the product of the measures along the (non-maximal) branch ending with that node. Let (p) denote this product measure, for p a node of Ts . We identify the tree Ts with a subtree of < in the obvious manner. We then associate to s an ordering <s of order-type ot(s). This is lexicographic ordering on the set of tuples (i1 , α1 , . . . , ik , αk ) where (i1 , . . . , ik ) ∈ Ts and αj is in the domain of the measure associated to the node (i1 , . . . , ij ) for all j ≤ k. For the example s = V4 ⊕ (V3 ⊗ V3 ) considered above, <s has domain tuples of the form (0, α) where α < 3 together with (1, ) where < 2 , and (1, , 0, ) where , < 2 . Note that dom(<s ) has order-type ot(s). To each term s in the algebra A we assign a measure (s) on 13 as follows. Definition 8.1. (s) is the measure on 13 induced by the weak partition relation on 13 , functions f : dom(<s ) → 13 which are order-preserving and continuous (and of uniform cofinality atpoints of successor rank), and the product measures (p) associated to nodes p of Ts . For the example s we are considering, a function f : dom(<s ) → 13 rep with resents a triple of ordinals [f] = (1 , 2 , 3 ). Here 1 is represented
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respect to S12 by the map α → f(0, α). 2 is represented with respect to S11 by → f(1, ), and 3 is represented with respect to S11 × S11 by (, ) → f(1, , 0, ). Martin’s Theorem 3.8 gives that for any term s ∈ A , j(s) ( 13 ) is a cardinal. The next theorem gives our alternate representation of the cardinal structure. Theorem 8.2. The cardinals below 5 are precisely the ordinals j(s) ( 13 ) for s ∈ A . Moreover, j(s) ( 13 ) = +o(s)+1 . For the example s = V4 ⊕ (V3 ⊗ V3 ), we have j(s) ( 13 ) = 2 + ·2+1 . This alternate representation has applications, for example, it makes reading off the cofinalities of the cardinals below the projective ordinals easy. The following result from [JK] computes this below 15 . Theorem 8.3. Suppose κ = α+1 is a successor cardinal with 13 < κ < 15 . Let α = 1 + · · · + n , where > 1 ≥ · · · ≥ n , be the normal form for α. Then: • If n = 0, then cf(κ) = 14 = +2 . ordinal, then cf(κ) = • If n > 0, and is a successor ·2+1 . • If n > 0 and is a limit ordinal, then cf(κ) = +1 .
REFERENCES
Stephen Jackson [Jac88] AD and the projective ordinals, this volume, originally published in Kechris et al. [Cabal iv], pp. 117–220. [Jac90A] A new proof of the strong partition relation on 1 , Transactions of the American Mathematical Society, vol. 320 (1990), no. 2, pp. 737–745. [Jac91] Admissible Suslin cardinals in L(R), The Journal of Symbolic Logic, vol. 56 (1991), no. 1, pp. 260–275. [Jac99] A computation of 15 , vol. 140, Memoirs of the AMS, no. 670, American Mathematical Society, July 1999. [Jac08] Suslin cardinals, partition properties, homogeneity. Introduction to Part II, in Kechris et al. [Cabal I], pp. 273–313. [Jac10] Structural consequences of AD, in Kanamori and Foreman [KF10], pp. 1753–1876. Stephen Jackson and Farid Khafizov [JK] Descriptions and cardinals below 15 , in submission. Stephen Jackson and Benedikt Lowe ¨ [JL] Canonical measure assignments, in submission. Akihiro Kanamori and Matthew Foreman [KF10] Handbook of Set Theory, Springer, 2010. Alexander S. Kechris [Kec77A] AD and infinite exponent partition relations, circulated manuscript, 1977.
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[Kec78] AD and projective ordinals, this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 91–132. [Kec81A] Homogeneous trees and projective scales, this volume, originally published in Kechris et al. [Cabal ii], pp. 33–74. [Kec94] Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer, 1994. Alexander S. Kechris, Benedikt Lowe, and John R. Steel ¨ [Cabal I] Games, scales, and Suslin cardinals: the Cabal seminar, volume I, Lecture Notes in Logic, vol. 31, Cambridge University Press, 2008. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. [Cabal iii] Cabal seminar 79–81, Lecture Notes in Mathematics, no. 1019, Berlin, Springer, 1983. Alexander S. Kechris, Donald A. Martin, and John R. Steel [Cabal iv] Cabal seminar 81–85, Lecture Notes in Mathematics, no. 1333, Berlin, Springer, 1988. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. Alexander S. Kechris and W. Hugh Woodin [KW80] Generic codes for uncountable ordinals, partition properties, and elementary embeddings, circulated manuscript, 1980, reprinted in [Cabal I], p. 379–397. Donald A. Martin [Mar71B] Projective sets and cardinal numbers: some questions related to the continuum problem, this volume, originally a preprint, 1971. Donald A. Martin and John R. Steel [MS83] The extent of scales in L(R), in Kechris et al. [Cabal iii], pp. 86–96, reprinted in [Cabal I], p. 110–120. Yiannis N. Moschovakis [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. Robert M. Solovay [Sol78A] A Δ13 coding of the subsets of , this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 133–150. John R. Steel [Ste83] Scales in L(R), in Kechris et al. [Cabal iii], pp. 107–156, reprinted in [Cabal I], p. 130– 175. DEPARTMENT OF MATHEMATICS UNIVERSITY OF NORTH TEXAS P.O. BOX 311430 DENTON, TEXAS 76203-1430 UNITED STATES OF AMERICA
E-mail: [email protected]
HOMOGENEOUS TREES AND PROJECTIVE SCALES
ALEXANDER S. KECHRIS
This exposition is a sequel to [Kec78]. Its main purpose is to show how set theoretical techniques, among them infinite exponent partition relations can be used to produce homogeneous trees for projective sets. The work here is again understood as being carried completely with L(R), with the hypothesis that AD+DC holds this model. As applications, one has Kunen’s important reduction of the problem of computing 15 to the problem of computing certain ultrapowers of 13 = +1 and also a result of Martin on constructibility of . An observation on the Victoria Delfino 3rd relative to subsets +1 problem concludes the present paper. Most of the results and constructions of trees presented in §§3, 4, 5 below are due to Kunen, and go back to his [Kun71B]. On the other hand, some of the effective calculations, in §§3, 4, for the scales resulting from these trees need the recent results of Kechris and Martin [KM78] and Harrington and Kechris [HK81]. §0. Introduction. In this introductory section we collect various notational conventions and some prerequisites needed to follow this paper. 0.1. Trees. If X is a set, X is the set of all infinite sequences from X and < X the set of all finite sequences from X . If s, t ∈ <X and f ∈ X , then s ⊆ t and s ⊆ f denote the extension relation in each case. If s = (x0 , . . . , xn−1 ), we let s(i) ≡ si = xi for i < n and we put lh s = n. Sometimes we also write (x1 , . . . , nn ) for (y0 , . . . , yn−1 ), where yi = xi+1 , 0 ≤ i < n. If s = (x0 , . . . , xn−1 ) and m ≤ n, then sm = (x0 , . . . , xm−1 ). We reserve usually letters , , . . . for members of < and u, v, w, . . . for members of <Ord. As usual α, , , . . . denote reals, i.e., elements of R = . We fix a recursive 1-1 correspondence 0 , 1 , 2 , . . . between and < , such that j i ⇒ j > i and if i = lh i , then i ≤ i. Moreover we agree to take 0 = ∅, 1 = (0). The preparation of this paper was partially supported by NSF Grant MCS 76-17254 A01. The author is an A. P. Sloan Foundation Fellow. We would like to thank Y. N. Moschovakis for making a number of valuable suggestions for improving the presentation of this paper. Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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By a tree on k × , where k ∈ and ∈ Ord, we mean a set T k of (k + 1)-tuples of the form ( 1 , . . . , k , u) ∈ (<) × < , where each
i and u have all the same length and such that if ( 1 , . . . , k , u) ∈ T and n ≤ lh 1 , then ( 1 n, . . . , k n, un) ∈ T . For such a tree T and for ≤ k, we let T ( 1 , . . . , ) = {( +1 , . . . , k , u) : ( 1 , . . . , k , u) ∈ T }, T (α1 , . . . , α ) = n∈ T (α1 n, . . . , α n) and finally T ⊆ ( 1 , . . . , ) = n≤lh 1 T ( 1 n, . . . , n). By [T ] we denote the set of all infinite branches through T , i.e., [T ] = {(α1 , . . . , αk , f) : ∀n(α1 n, . . . , αk n, fn) ∈ T }. Let also p[T ] = {(α1 , . . . , αk ) : ∃f(α1 , . . . , αk , f) ∈ [T ]}. For X ⊆ , T X = {( 1 , . . . , k , u) ∈ T : u ∈ <X }. A tree T on k × is wellfounded iff [T ] = ∅. Equivalently T is wellfounded k if the Brouwer-Kleene ordering
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the class of uniform indiscernibles. Under AD, un = n for n ≤ . By a result of Solovay each ordinal < un+1 can be written in the form t L[α] (u1 ,. . . , un ), ˜ = for some term t and some real α. For each f : n[1 ] → 1 in L α L[α], n n where [1 ] = {( 0 , . . . , n−1 ) ∈ 1 : 0 < 1 < · · · < n−1 }, define ˜ 1 , . . . , un ) = t L[α] (u1 , . . . , un ), where f( 1 , . . . , n ) = t L[α] f(u ( 1 , . . . , n ) for 1 < · · · < n < 1 . Thus every ordinal < un+1 has the ˜ 1 , . . . , un ) for some f : n[1 ] → 1 in L. ˜ If now H : 1 → 1 is in L, ˜ form f(u ˜ define H˜ : u → u by H˜ (f(u1 , . . . , un )) = H ◦ f(u1 , . . . , un ). For X ⊆ 1 let also X˜ ⊆ u be defined by: ˜ 1 , . . . , un ) ∈ X˜ ⇔ ∃C ⊆ 1 (C is closed unbounded(cub) in 1 ∧ for all f(u
1 < · · · < n in C, f( 1 , . . . , n ) ∈ C ). 0.4. The work in this paper takes place in ZF+DC until otherwise specified. §1. Π11 sets; the tree S1 . 1.1. Definition of S1 . (a) Let A ⊆ R be a Π11 set of reals. Then there is a recursive tree T on × such that α ∈ A ⇔ T (α) is wellfounded. If ∈ < and lh = n define the following ordering < on {0, 1, . . . , n − 1} = n: i < j ⇔ 1. (i , j ∈ T ⊆ ( ) ∧ i < j) or def
2. (i ∈ T ⊆ ( ) ∧ j ∈ T ⊆ ( )) or 3. (i , j ∈ T ⊆ ( ) ∧ i
⊆ i ∈ T ⊆ ( ) ⇔ i ∈ T ⊆ ( ). This is of course because always i = lh i ≤ i, thus i ∈ T ⊆ ( ) ⇔ ( i , i ) ∈ T ⇔ ( i , i ) ∈ T ⇔ i ∈ T ⊆ ( ). Put now for each α ∈ R: def <α = <αn , n
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so that <α is a linear ordering of (all of) , with top element 0 again. Note that <α is just the Brouwer-Kleene ordering on T (α) with the rest of thrown in at the bottom with its natural ordering. Thus α ∈ A ⇔ T (α) is wellfounded ⇔ T (α),
S1 (A; T ) = S1 = {( , u) : ∈ <, u ∈ <1 ∧ lh = lh u ( = n, say) ∧ u : n → 1 is order preserving relative to < , i.e., for 0 ≤ i, j < n : i < j ⇔ ui < uj }. Then we obviously have that: α ∈ A ⇔ ∃f(α, f) ∈ [S1 ] ⇔ S1 (α) is not wellfounded. 1.2. Scales for Π11 sets. (a) If J is a tree on we shall say that J has an honest leftmost branch if there is a branch f ∈ [J ] such that for all branches g ∈ [J ]: ∀i, f(i) ≤ g(i). Every non-wellfounded tree J has a leftmost branch h ∈ [J ], which is by definition characterized by the property that for g ∈ [J ]: h ≤lex g, i.e., h = g ∨ ∃i[h(i) < g(i) ∧ ∀j < i(h(j) = g(j))]. But only special trees J have honest leftmost branches. We shall show that those of the form S1 (α) are among them. Indeed, let for α ∈ A: fα (i) = rank<α (i). Clearly fα ∈ [S1 (α)]. On the other hand, if g ∈ [S1 (α)], then g : → 1 is such that i <α j ⇔ g(i) < g(j), thus fα (i) = rank<α (i) ≤ g(i), and we are done. (b) Define now for α ∈ A: ϕi (α) = fα (i). Then {ϕi } is a scale on A (and thus an 1 -scale). Indeed, letting ϕ(α) ¯ = (ϕ0 (α), ϕ1 (α), . . . ) and assuming that for all n, αn ∈ A, while αn → α and
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ϕ(α ¯ n ) → g (i.e., ϕi (αn ) = g(i), for all large enough n) we conclude that (α, g) = lim(αn , ϕ(α ¯ n )) ∈ [S1 ] and moreover fα (i) = ϕi (α) ≤ g(i). Now by the usual argument (see for example Kechris and Moschovakis [KM78B]) one can show that if for α ∈ A we put i (α) = ϕ0 (α), ϕi (α), def
where , = the ordinal attached to the pair ( , ) in the lexicographical ordering of 1 × 1 , then {i } is a Π11 -scale on A. Thus we have shown that Π11 has the scale property. 1.3. Homogeneity properties of S1 . (a) Fix now ∈ <, = ∅. Let lh = n and define : n → n to be the unique permutation of n defined by: i < j ⇔ (i) < (j). In particular, (0) = n − 1. Then note that if for any A ⊆ Ord and any ∈ Ord, we let
def
[A] = the set of all increasing maps from into A,
we have ( , v) ∈ S1 ⇔ lh v = n ∧ ∃u ∈ n[1 ](v = u ◦ , i.e., v = (u (0) , u (1) , . . . , u (n−1) )). Thus def
S1 ( ) = (lh [1 ]) = {( (0) , . . . , (n−1) ) : 0 < 1 < · · · < n−1 < 1 }, so that S1 ( ) is just a “permutation” of lh [1 ]. Since (0) = n − 1 we have the following important “boundedness” property: ( 0 , . . . , n−1 ) ∈ S1 ( ) ⇔ 0 > 1 , 2 , . . . , n−1 . This will be quite useful in §2. (b) Assume now H : 1 → 1 is an increasing map, i.e., H ∈ 1[1 ]. For any u = ( 0 , . . . , n−1 ) ∈ n1 , we put also H (u) = (H ( 0 ), . . . , H ( n−1 )) and for any ( , u) ∈ n × n1 we let again H ( , u) = ( , H (u)). The claim now is that H : S1 → S1 ,
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i.e., S1 is invariant under H , so that in particular for any α: H : S1 (α) → S1 (α). This is of course because if ( , v) ∈ S1 , then for some u ∈ lh [1 ], v = u ◦ , so that H (v) = H (u) ◦ , where, since H is increasing, H (u) ∈ lh [1 ]. Thus ( , H (v)) ∈ S1 . Since ( , u) → H ( , u) clearly preserves the relation of proper extension between sequences, we have, letting D = ran(H ): S1 (α) is not wellfounded ⇒ S1 D(α) is not wellfounded. So we have shown that for each D ⊆ 1 , D uncountable, we have for all α: α ∈ A ⇔ S1 (α) is not wellfounded ⇔ S1 D(α) is not wellfounded, so that the non-wellfoundedness of S1 (α) depends only on its restriction to any uncountable subset of 1 . This too will be useful in §2. Note. The construction of S1 is due to Shoenfield [Sho61]. The homogeneity properties of S1 have been studied and used by Solovay [Sol78A], Mansfield [Man71] and Martin [Mar71B]. §2. Π12 sets; the tree S2 . Assume from now on and for the rest of this paper that for all α, α # exists. 2.1. Definition of S2 . (a) Let A ⊆ R be Π12 . Then for some Π11 set B ⊆ R × R we have α ∈ A ⇔ ¬∃B(α, ) ⇔ ¬∃∃f(α, , f) ∈ [S1 ] ⇔ S1 (α) is wellfounded, where S1 is the tree associated with B as in §1. Strictly speaking we have talked in §1 only about subsets of R but it is obvious how to modify this discussion so that it applies to Π11 subsets of any R × R × · · · × R. Thus the tree S1 associated with B will be a tree on × × 1 , so that S1 (α) is a tree on × 1 . A typical element of S1 will be a triple ( , , u), where lh = lh = lh u = n and u : n → 1 is order preserving relative to < , , which is defined as in §1.1 by replacing by , everywhere. Let now for = ∅, lh = n: S1∗ ( ) = {(i , u) : ( i , i , u) ∈ S1 ∧ 1 ≤ i ≤ n}. def
Then (S1∗ ( ),
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ALEXANDER S. KECHRIS
If S1 (α) is wellfounded, let againWα denote the wellordering S1 (α), 1 , . . . , j −1 , so that if (j , u)
Let also for each wellordering W and set A ⊆ Ord, W [A] be the set of all increasing mappings from (the domain of) W into A. As in this notation, we sometimes do not distinguish between W and its domain, when convenient. ˜ where = ∅, we will assign a tuple of ordinals To each h ∈ W [1 ] ∩ L, p (h) = ( 1 , . . . , lh ) as follows: Since h maps S1∗ ( ) into 1 , it splits into lh = n many maps h1 , . . . , hn , where dom(hi ) = S1 ( i , i ) and hi (u) = h(i , u). Now hi can be identified with the map hi : i [1 ] → 1 given by hi (v) = hi (v ◦ i ,i ), since S1 ( i , i ) = i [1 ] i ,i , as we saw in 1.3(a). Put now p (h) = (h˜ i (u1 , . . . ui ))1≤i≤n . We are now ready to define: def ˜ (h) = u) ∨ ( = u = ∅). ( , u) ∈ S2 ⇔ ∃h ∈ W [1 ] ∩ L(p
(Again S2 depends on A, B, S1 but we won’t indicate this explicitly.) We first verify that this is indeed a tree: Let ( , u) ∈ S2 and 1 ≤ n ≤ lh = ˜ such that p (h) = u. Let lh u. Put n = . We can find h ∈ W [1 ] ∩ L W ˜ h ∈ [1 ] ∩ L be defined by h = hW . Note here that S1 , W , W are ˜ Then clearly p (h ) = un, therefore ( n, un) ∈ S2 . in L, so that h ∈ L.
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(b) We shall verify now that α ∈ A ⇔ S2 (α) is not wellfounded. Indeed, if α ∈ A, S1 (α) is wellfounded, so let h : Wα → 1 be the rank function of Wα , i.e., the unique isomorphism between Wα and 1 . Clearly h ∈ L[α]. Then if for each n ≥ 1, h n = hWαn we have that pαn (h n ) ∈ S2 (αn) and also pα1 (h 1 ) ⊆ pα2 (h 2 ) ⊆ . . . , so S2 (α) is not wellfounded. Conversely, assume (1 , 2 , . . . ) is an infinite branch through S2 (α). Then for ˜ be such that pαn (nh) = (1 , . . . , n ). Then each n ≥ 1, let nh ∈ Wα n[1 ] ∩ L clearly for all i ≥ 1 and all n, m ≥ i, n# h (u , . . . , u ) = m# h (u , . . . , u ). So i
1
i
i i find Cn,m a cub subset of 1 such that for all v ∈ i [Cn,m ]: n hi (v)
i
1
i
= mhi (v).
This allows us to define the following map q from S1 C (α) into the ordinals, where i Cn,m : C = q(i , u) =
1≤i≤n,m n hi (v) = nh(i , u),
where n ≥ i and u = v ◦ αi ,i . We now claim that this is an order preserving map from S1 C (α),
S1∗ (αn), n
for n ≥ i, j, thus
h(i , u) < nh(j , w),
therefore q(i , u) < q(j , w). So S1 C (α) is wellfounded and therefore S1 (α) is wellfounded by 1.3(b), i.e., α ∈ A. 2.2. Scales for Π12 sets. We claim now that for each α ∈ A, S2 (α) has an honest leftmost branch. Indeed in the notation of 2.1(b), if h : Wα → 1 is as defined there and we let for each i > 0, hi (u) = h(i , u), for u ∈ S1 (αi , i ) and i = hi (u1 , . . . , ui ), then clearly ( 1 , 2 , . . . ) ∈ [S2 (α)]. Moreover if (1 , 2 , . . . ) is any branch of [S2 (α)], then again in the notation of 2.1(b), q is an order preserving map from S1 C (α) into 1 . Let H : 1 → C be the normal function enumerating C . Then by 1.3 H : S1 (α) → S1 C (α) and of course H preserves
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ALEXANDER S. KECHRIS
We can now define for each α ∈ A and each i ≥ 1, ϕi (α) = fα (i), where fα is the leftmost branch of S2 (α) as above. Clearly ϕ(α) ¯ = (ϕ1 (α), ϕ2 (α), . . . ) is a scale on A. We want to show actually that it is a Δ13 -scale. For this note that ϕi (α) = h (u1 , . . . , u ), i
where
hi (v)
i
= rankWα (v ◦ αi ,i ), for v ∈ [1 ]. Thus for α, ∈ A i
ϕi (α) ≤ ϕi () ⇔ rankWα (v ◦ αi ,i ) ≤ rankW (v ◦ i ,i ), for all v ∈ i [C ], where C is some cub subset of 1 ⇔ L[α, ] |= 1 (α, , v), for all v ∈ i [C ], where C ⊆ 1 is some cub set, (here i is some formula recursively determined from i), ⇔ L[α, ] |= i (α, , u1 , . . . , ui ) ⇔ α, # (i ) = 0, which is obviously Δ13 . Remark 2.1. Note that we can also describe ϕi (α) as follows: Let Sˆi = {( , , u) : lh = lh = lh u (= say, n) ∧
, ∈ < ∧ u ∈ <Ord ∧ u : n → Ord is order preserving relative to < , }. Thus Sˆ1 is the “liftup” of S1 to all ordinals. Clearly Sˆ1 1 = S1 and Sˆ1 is a definable class in L. Now by an easy indiscernibility argument we have for all i ≥ 1: ϕi (α) = fα (i) = rankSˆ1 (α),
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(i) ∀x ∈ S(h(x) > sup{h(y) : y < x}) (ii) There is { nx }x∈S with 0x < 1x < · · · → h(x), for all x ∈ S. According to a result of Martin (see [Kec78] for example) we have for any W of order type 1 and any X ⊆ W [1 ], that there is C ⊆ 1 cub such that W
C ↑ ⊆ X or W C ↑ ⊆∼ X.
This clearly defines a measure (i.e., countably additive ultrafilter) on W 1 ↑. Since W has order type 1 and since p : W [1 ] S2 ( ), this induces a measure on S2 ( ), when = ∅. For the purpose of making this measure more explicit we shall actually consider the subtree S2− of S2 defined by ( , u) ∈ S2− ⇔ ∃h ∈ W 1 ↑(p (h) = u) ∨ ( = u = ∅). It is trivial of course to check that also, α ∈ A ⇔ S2− (α) is not wellfounded. Let U¯ be the above mentioned measure on W 1 ↑, i.e., X ∈ U¯ ⇔ ∃C ⊆ 1 (C cub ∧ W C ↑ ⊆ X ), and let U be the measure on S2− ( ) induced by p , i.e., U = (p )∗ U¯ or explicitly, for A ⊆ S2− ( ): A ∈ U ⇔ ∃C ⊆ 1 [C cub ∧ p [W C ↑] ⊆ A]. We shall actually show that U is generated by the sets of the form S2− ( ) ∩
lh
C˜ ,
where C ⊆ 1 is cub. In other words, we claim that for A ⊆ S2− ( ): A ∈ U ⇔ ∃C ⊆ 1 [C cub ∧ S2− ( ) ∩
lh
C˜ ⊆ A].
For that is clearly enough to prove the following: Lemma 2.2. For any uncountable C ⊆ 1 , p [W C ↑] = S2− ( ) ∩
lh
C˜ .
Proof. (⊆): If h : W → C is order preserving, then p (h) = (h˜ (ui , . . . , u ))1≤i≤lh , i
where S2− ( )
i
hi (v) lh ∩
= h(i , v ◦ i ,i ) ∈ C, so C˜ .
h˜ i (u1 , . . . , ui )
lh
∈ C˜ , thus p (h) ∈
(⊇) Let ( 1 , . . . , n ) ∈ S2− ( ) ∩ C˜ . Then for some h ∈ W 1 ↑, p (h) = ( 1 , . . . , n ), i.e., i = h˜ i (u1 , . . . , ui ), where hi (v) = h(i , v ◦ i ,i ), for v ∈ i [1 ]. Moreover, h˜ i (u1 , . . . , ui ) ∈ C˜ , so there is D ⊆ C , D cub such that hi (v) = h(i , v ◦ i ,i ) ∈ C,
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ALEXANDER S. KECHRIS
for all v ∈ i [D]. Let H : 1 → D be the normal enumeration of D. Put g(i , u) = h(i , H (u)). Then g ∈ W [C ]. Also if gi (u) = g(i , u) and gi (v) = gi (v ◦ i ,i ), for v ∈ i [1 ], then gi (v) = hi (v) for v ∈ i [E], where E is cub and ∈ E ⇒ H ( ) = . Thus i = h˜ i (u1 , . . . , ui ) = g˜i (u1 , . . . , ui ) ∈ C . So it only remains to show that actually g ∈ W C ↑. Recalling the definition of W C ↑ before, it is clear that condition (ii) holds for g. So it is enough to verify (i), i.e., that for x ∈ S1∗ ( ) we have g(x) > sup{g(y) : y ∈ S1∗ ( ) ∧ y sup{h(j , w) : (j , w) ∈ S1∗ ( ) ∧ (j , w)
and we are done. S2− .
(b) We examine now a further homogeneity property of Let H : 1 → 1 be normal. Then if h ∈ W 1 ↑ clearly H ◦ h ∈ W 1 ↑ too (this does not happen in general if H is just increasing). Now it is easy to check that p (H ◦ h) = H˜ (p (h)). Indeed, if p (h) = ( 1 , . . . , n ), where i = h˜ i (u1 , . . . , ui ), then H˜ ( i ) = H ◦ hi (u1 , . . . , ui ) = (H ◦ h)i (u1 , . . . , ui ). Thus H˜ maps S2− into S2− (in the usual sense that if (, u) ∈ S2− , then (, H˜ (u)) ∈ S2− ), where H˜ ( 1 , . . . , n ) = (H˜ ( 1 ), . . . , H˜ ( n )). As H˜ obviously preserves the proper extension relation among sequences, we have for any cub C ⊆ 1 : α ∈ A ⇔ S2− (α) is not wellfounded ⇔ S2− C˜ (α) is not wellfounded. Remark 2.3. It is easy also to check that S2 is preserved under any H˜ , where H : 1 → 1 is just order preserving, and so for any unbounded C ⊆ 1 : α ∈ A ⇔ S2 (α) is not wellfounded ⇔ S2 C˜ (α) is not wellfounded.
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2.4. Some definability estimates for S2− . Consider the structure Q3 = u , <, {un }n< , as in Kechris-Martin [KM78]. (Recall that since we are working with AD, un = n , ∀n ≤ .) We want to show that: (i) S2− is Δ11 on Q3 . (ii) The second order relation A ⊆ S2− ( ) ∧ A ∈ U is also Δ11 on Q3 . Now (ii) follows immediately from (i) as for A ⊆ S2− ( ): A ∈ U ⇔ ∃C (C ⊆ 1 is cub ∧
lh
⇔ ∀C (C ⊆ 1 is cub ⇒
C˜ ∩ S2− ( ) ⊆ A)
lh
C˜ ∩ A = ∅).
It is needed of course to verify here that the map X → X˜ for X ⊆ 1 , is also Δ11 , or equivalently that the second order relation X ⊆ 1 ∧ ∈ X˜ is Δ11 on Q3 . This follows from the fact that for X ⊆ 1 :
∈ X˜ ⇔ ∃α∃ term t such that [t L[α] (u1 , . . . , ur(t) ) =
∧ ∃C ⊆ 1 (C cub ∧ ∀v ∈ r(t)[C ](t L[α] (v) ∈ X ))] ⇔ ∀α∀ term t[t L[α] (u1 , . . . , ur(t) ) =
⇒ ∀C ⊆ 1 (C cub ⇒ ∃v ∈ r(t)[C ](t L[α] (v) ∈ X ))]. So it is enough to prove (i). For that let us say that a sequence ( 1 , . . . , n ), where i < ui +1 , has the gap property if there are fi with f˜i (u1 , . . . , ui ) = i and there is cub D ⊆ 1 such that for all 1 ≤ i ≤ n and all v ∈ i [D]: fi (v) > sup{fj (w) : fj (w) < fi (v) ∧ w ∈ j [D]}.
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ALEXANDER S. KECHRIS
Note that that is independent of the particular choice of fi ’s which represent the i ’s. Now we have the equivalence (for = ∅) ( , ( 1 , . . . , n )) ∈ S2− ⇔ lh = n & (a) i < ui +1 ∧ cf( i ) = & (b) ( 1 , . . . , n ) has the gap property & (c) for any (all) f1 , . . . , fn such that f˜i (u1 , . . . , u1 ) = i , if we let h(i , v ◦ i ,i ) = fi (v), then for some cub set C ⊆ 1 , h(S1∗ C ( )) is order preserving relative to
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283
and that for each α ∈ A, S2+ (α) has an honest leftmost branch, say fα , and that if ϕ(α) ¯ = fα , then ϕ¯ is a Δ13 -scale on A. Our next goal will be to get an explicit description of the condition ( 1 , . . . ,
n ) ∈ S2+ ( ) in terms of the ordinals themselves instead of their representing functions. ˜ For that recall that for ( , ( 1 , . . . , n )) ∈ S2+ there must be functions fi ∈ L, i fi : [1 ] → 1 such that f˜i (u1 , . . . , ui ) = i and for some cub C ⊆ 1 and all v ∈ j [C ], v ∈ i [C ] we have (j , v ◦ j ,j ) ≺1 (i , v ◦ i ,i ) ⇒ fj (v) < fi (v ).
(∗)
Now the hypothesis of (∗) implies that j is a 1-point extension of i , so that in particular j = i + 1 and moreover if we put for simplicity j = j ,j and similarly for i , we must also have that vj (k) = v i (k) , ∀k < i or equivalently vj ◦−1 (k) = vk , ∀k < i . i
Note now that from the following commutative diagram of order preserving maps: j
j , < j ,j −−−−→ j , < ( ( ⏐ ⏐ inclusion⏐ ⏐j ◦i−1 i , < i ,i −−−−→ i , < i
we must have that j ◦ i−1 : i → j is order preserving, i.e., for some m = m( , i, j) ≤ i k if 0 ≤ k < m −1 j ◦ i (k) = k + 1 if m ≤ k < i − 1. Thus (v0 , . . . , vi −1 ) = (v0 , . . . , vˆm , . . . , vj −1 ), where vˆm signifies the fact that vm is omitted. Recall now that for each m ≥ 1 we have the following embedding jm : u → u , where
jm (un ) =
un un+1
if n < m, if n ≥ m,
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ALEXANDER S. KECHRIS
˜ 1 , . . . , ut )) = f(j ˜ m (u1 ), . . . , jm (ut )). Then an easy indiscernibility and jm (f(u argument plus the above analysis easily yields that (for = ∅) ( , ( 1 , . . . , n )) ∈ S2+ ⇔ lh = n
(∗∗)
& (i) i < ui +1 & (ii) For all 1 ≤ i, j ≤ n : j ≺1 i ⇒ j < jm( ,i,j)+1 ( i ). This is the particular form that is relevant in Martin’s proof. From this explicit form of S2+ and using again full AD one can easily check that each S2+ ( ) is a finite union of Kunen sets Atm,n , where m = max1≤i≤n i and n = lh . The notion and the notation involved here are as in Solovay [Sol78A]. Now each of these Atm,n carries a canonical measure generated by n the sets of the form C˜ ∩ Atm,n , with C ⊆ 1 cub; see again Solovay [Sol78A]. This establishes a homogeneity property of S2+ . It is relevant to notice here that S2− ( ) is exactly one of the sets Atm,n that get into S2+ ( ), so that the passage from S2+ to S2− has the effect of canonically choosing from the finitely many Kunen sets Atm,n involved in each S2+ ( ), exactly one which is then equal to S2− ( ). Although one could write a description of each S2− ( ) using the embeddings jm as in (∗∗) it would be a bit messy and not as elegant or useful as (∗∗) itself. As a final comment we mention that it would be easy to show again that S2+ has also the following homogeneity property: For all unbounded C ⊆ 1 : α ∈ A ⇔ S2+ (α) is not wellfounded ⇔ S2+ C˜ (α) is not wellfounded. Note. The construction of S2 is due to Mansfield [Man71], Martin [Mar71B] following work of Martin-Solovay [MS69]. The homogeneity properties of these trees have been studied and used by Kunen [Kun71B] and Martin. §3. Π13 sets; the tree S3 . 3.1. Definition of S3 . (a) Let now A ⊆ R be Π13 . Then for some B ∈ Π12 , α ∈ A ⇔ ¬∃B(α, ) ⇔ ¬∃∃f(α, , f) ∈ [S2− ] ⇔ S2− (α) is wellfounded. Let again for = ∅, lh = n: S2∗ ( ) = {(i , u) : ( i , i , u) ∈ S2− ∧ i < n}.
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Note that we allow here (∅, ∅) ∈ S2∗ ( ), while we have excluded it from S1∗ ( ): Again for each such , let W denote the wellordering S2∗ ( ),
0 = h 0 ,
i = [hi ]U ,i . Finally put p (h) = ( 0 , 1 , . . . , n−1 ). Then define the tree S3 by: ( , u) ∈ S3 ⇔ ∃h ∈ W [+1 ](p (h) = u) ∨ ( = u = ∅). (b) We now show that α ∈ A ⇔ S3 (α) is not wellfounded. First, if α ∈ A, then S2− (α) is wellfounded, so let h : Wα → +1 be the unique isomorphism between Wα and an initial segment of +1 , which is of course equal to α . Let hi (v) = h(i , v), for i > 0 and v ∈ S2 (αi , i ) and let i = [hi ]Uαn,i for any n > i. Let also 0 = h(∅, ∅). Then ( 0 , . . . , n−1 ) = pαn (h n ), where h n = hWαn ∈ Wαn [+1 ], so that ( 0 , . . . , n−1 ) ∈ S3 (αn), i.e., ( 0 , 1 , . . . ) ∈ S3 (α). Conversely, assume (0 , 1 , . . . ) ∈ [S3 (α)]. For each n > 0, let nh ∈ Wαn [+1 ] be such that pαn (nh) = (0 , 1 , . . . , n−1 ). Then for all n > 0, nh(∅, ∅) = 0 and for all i > 0 and all m, n > i [mhi ]Uαm,i = [nhi ]Uαn,i ;
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ALEXANDER S. KECHRIS
where Uαm,i = Uαn,i of course. By the results in §2.3, we can now find i Cn,m ⊆ 1 cub such that
i m n − i v ∈ C" n,m ∩ S2 (αi , i ) ⇒ hi (v) = hi (v).
i Cn,m . Then C ⊆ 1 is cub and the conclusion of (∗) in §2.5 i holds for all 0 < i < n, m and all v ∈ C˜ ∩ S2− (αi , i ). This allows us to define the following map q from S − C˜ (α) into the ordinals:
Let C =
0
2
q(i , v) = nhi (v), where n > i. We now claim that this is an order preserving map from S2− C˜ (α),
S2∗ (αn) n
for n > i, j, thus
h(i , u) < nh(j , w),
i.e., n
hi (u) < nhj (w),
thus q(i , u) < q(j , w). S2− C˜ (α)
is wellfounded, thus α ∈ A by 2.3(b). So (c) We note also the following two basic properties of S3 : (i) S3 is a tree on × +1 , i.e., all the ordinals occurring in it are < +1 . This is because by a result of Kunen (see [Kun71B]) if U is a measure on any set I of cardinality < +1 , then for any f : I → +1 , [f]U < +1 . (ii) Let for any measure U on a set I , i U : Ord → Ord be the embedding it generates, i.e., i U ( ) = sup{[f]U : f : I → }. = [C ]U , for C the constant function. Then we claim that if lh = n = 0: ( , ( 0 , . . . , n−1 ) ∈ S3 ⇒ i < i
U ,i
( 0 ), for i > 0.
This is because for i > 0, i = [hi ]U ,i , where h : W → +1 is order preserving and hi (v) = h(i , v), so that
0 = h(∅, ∅) > hi (v), for all i > 0, as (i , v)
U ,i
( 0 ) > [hi ]U ,i .
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287
This is analogous to a property of S1 which we established in §1, and it will be useful in §4. 3.2. Scales for Π13 sets. As usual we verify now that if α ∈ A, then S3 (α) has an honest leftmost branch. For that, in the notation of 3.1(b), if h : Wα → +1 is as defined there and we let hi and i be again as defined there, we have ( 0 , 1 , . . . ) ∈ [S3− (α)]. Moreover, if (0 , 1 , . . . ) is any branch of S3− (α) then, again in the notation of 3.1(b), q is an order preserving map from S2− C˜ (α) into +1 . Let H : 1 → C be the normal function enumerating C . Then by 2.3, H˜ : S2− (α) → S2− C˜ (α), and of course H˜ preserves
∈ D. Then H˜ (u) = u for u ∈ D, i n − therefore hi (u) ≤ hi (u) for u ∈ D˜ ∩ S2 (αn, i ), if n > i, thus [hi ]Uαn,i =
i ≤ [nhi ]Uαn,i = i for i > 0. Also 0 = h(∅, ∅) ≤ q(∅, ∅) = nh(∅, ∅) = 0 , i.e., i ≤ i , ∀i ≥ 0 and we are done. This implies now that if for each α ∈ A we put ϕi (α) = fα (i), where fα is the leftmost branch of S3 (α), then ϕ¯ = {ϕi } is an +1 -scale on A. By modifying this slightly (for reasons that will become apparent in a moment), we will obtain a Π13 -scale on A. Indeed put for α ∈ A: ¯ ϕi (α), i (α) = ϕ0 (α), α(i), where , , refers to the ordinal associated to the triple ( , , ) in the 3 lexicographical ordering of (+1 ). Now we claim that ¯ = {i } is a Π13 scale on A. For that just note that for α, ∈ A: ¯ i (α) ≤ i () ⇔ ϕ0 (α) < ϕ0 () ∨ [ϕ0 (α) = ϕ0 () ∧ α(i) ¯ < (i)] ¯ ∨ [ϕ0 (α) = ϕ0 () ∧ α(i) ¯ = (i) ∧ [ϕi (α) ≤ ϕi ()]. So if ∈ A: α ∈ A ∧ i (α) ≤ i () ⇔ [α ∈ A ∧ ϕ0 (α) < ϕ0 ()] ∨ ¯ [(α ∈ A ∧ ϕ0 (α) = ϕ0 ()) ∧ α(i) ¯ < (i)] ∨ ¯ [(α ∈ A ∧ ϕ0 (α) = ϕ0 ()) ∧ α(i) ¯ = (i) ∧ ϕi (α) ≤ ϕi ()], for which we have to calculate that it is Δ13 uniformly in . But by the results of Kechris-Martin [KM78], it is enough to show that it is Δ11 over Q3 , uniformly
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ALEXANDER S. KECHRIS
in . To check this notice that α ∈ A ∧ ϕ0 (α) ≤ ϕ0 () ⇔ There is an embedding form S2− (α), 0 is (uniformly in i, α, )
Δ11
in Q3 . Recall that for i > 0
ϕi (α) = fα (i) = [ u. rankWα (i , u)]Uαi ,i ; here the u varies over S2− (αi , i ). Since αi = i and i ≥ i , we clearly have that S2− (αi , i ) = S2− (i , i ) and Uαi ,i = Ui ,i . Then ϕi (α) ≤ ϕi () ⇔ {u ∈ S2− (αi , i ) : rankWα (i , u) ≤ rankW (i , u)} ∈ Uαi ,i . Since as above the relation rankWα (i , u) ≤ rankW (i , u) is Δ11 in Q3 , uniformly in all the parameters involved, the results in §2.4 imply that “ϕi (α) ≤ ϕi ()” is Δ11 in Q3 , uniformly in α, as above. 3.3. Homogeneity properties of S3 . The tree S3− . (a) By analogy with the work in §2.3, we define a subtree S3− of S3 as follows: ( , u) ∈ S3− ⇔ ∃h ∈ W +1 ↑(p (h) = u) ∨ = u = ∅. Again we have: α ∈ A ⇔ S3− (α) is not wellfounded. By a result of Kunen [Kun71B], we have +1 → (+1 ) , ∀, < +1 . This implies that we have the following +1 -additive measure V¯ on W +1 ↑: X ∈ V¯ ⇔ ∃C ⊆ +1 (C cub ∧ W C ↑ ⊆ X ). Since p :
W
+1 ↑ S3− ( ),
289
HOMOGENEOUS TREES AND PROJECTIVE SCALES
this induces the +1 -additive measure (p )∗ V¯ = V on S3− ( ), for = ∅. Thus for A ⊆ S3− ( ): A ∈ V ⇔ ∃C ⊆ +1 (C cub ∧ p [W C ↑] ⊆ A). We shall now try to get a more explicit form of this measure V . This will be based on the analog of Lemma 2.2. Let U be a measure on a set I of cardinality < +1 . Then, as we mentioned before, i U (+1 ) = +1 , where i U is the associated embedding generated by U . For each X ⊆ +1 , let i U (X ) be the image of X under this embedding, i.e., i U (X ) = {[f]U : {t : f(t) ∈ X } ∈ U }. Then since i U (+1 ) = +1 , clearly i U (X ) ⊆ +1 . (Caution: In general i U [X ] = {i U ( ) : ∈ X } i U (X ).) Now we have Lemma 3.2. For any unbounded C ⊆ +1 : p [W C ↑] = S3− ( ) ∩ (C × i U ,1 (C ) × i U ,2 (C ) × · · · × i U ,n−1 (C )), where lh = n > 0. Proof. First let h ∈ W C ↑. Then, if p (h) = ( 0 , . . . , n−1 ), 0 = h(∅, ∅) ∈ C , and if hi (u) = h(i , u) for i > 0, then i = [hi ]U ,i so that (since hi (u) ∈ C )
i ∈ i U ,i (C ). Thus ( 0 , . . . , n−1 ) ∈ S3− ( ) ∩ (C × i U ,1 (C ) × · · · × i U ,n−1 (C )). The proof of the converse is very similar to the proof of Lemma 2.2 and we omit the details. (Lemma 3.2) Thus the measure V on each S3− ( ) (for = ∅) is generated by the sets of the form * , + − U ,i S3 ( ) ∩ C × i (C ) 0
for C cub, C ⊆ +1 , i.e., A ∈ V ⇔ ∃C ⊆ +1 C cub ∧ A ⊇
* S3− ( )
∩
C×
+
,. i
U ,i
(C )
.
0
This bears some resemblance to the corresponding result about the generation of the measures U on S2− ( ). (b) Finally we establish the usual further homogeneity of S3− . Let H : +1 → +1 be a function. If U is a measure on I , where I has cardinality < +1 , we let i U (H ) be the image of H under i U , i.e., i U (H ) : +1 → +1 and i U (H )([f]U ) = [H ◦ f]U .
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ALEXANDER S. KECHRIS
Then if ran(H ) = C , i U (H ) : +1 i U (C ). Assume now H : +1 → +1 is order preserving and for any = ∅, if lh = n, define for u = ( 0 , . . . , n−1 ): H (u) = (H ( 0 ), i U ,1 (H )( 1 ), . . . , i U ,n−1 (H )( n−1 )). Then it is easy to check that ( , u) ∈ S3 ⇒ ( , H (u)) ∈ S3 , while if H is also normal, ( , u) ∈ S3− ⇒ ( , H (u)) ∈ S3− . In particular, if α ∈ R and we let H α = n≥1 H αn , so that H α (u) = H αn (u), where n > lh u, then H α maps S3− (α) into S3− (α). Thus if for each X ⊆ +1 , we let S3− X α = {( , ( 0 , . . . , n−1 )) ∈ S3− : 0 ∈ X ∧ for 0 < i < n,
i ∈ i Uαn,i (X )}, then for any cub C ⊆ +1 we have α ∈ A ⇔ S3− (α) is not wellfounded ⇔ S3− C α (α) is not wellfounded. (For S3 we have this for any unbounded C .) Note. The construction of S3 (and S3− ) is due to Kunen [Kun71B]. The calculation of a Π13 scale from this tree is orginally due to Martin by a different argument than the one we gave in §3.2. §4. Π14 sets; the tree S4 . 4.1. Definition of S4 . (a) Let A ⊆ R be a Π14 set of reals. Then for some B ∈ Π13 , α ∈ A ⇔ ¬∃B(α, ) ⇔ ¬∃∃f(α, , f) ∈ [S3− ] ⇔ S3− (α) is wellfounded. Again for each = ∅, lh = n, we let S3∗ ( ) = {(i , u) : ( i , i , u) ∈ S3− ∧ 1 ≤ i ≤ n}, and we define W to be S3∗ ( ),
HOMOGENEOUS TREES AND PROJECTIVE SCALES
291
and for v ∈ S3− ( i , i ) hi (v) = h(i , v). Recall that S3− ( i , i ) carries the measure V i ,i ≡ V ,i (for simplicity) and put
i = [hi ]V ,i . Finally, let p (h) = ( 1 , . . . , n ) = ([hi ]V ,i )1≤i≤n and define ( , u) ∈ S4 ⇔ ∃h ∈ W [+1 ](p (h) = u) ∨ ( = u = ∅). It is easy now to verify as in §2.1 that α ∈ A ⇔ S4 (α) is not wellfounded. Put 5 = sup{i
V ,i
(+1 ) : = ∅, 1 ≤ i ≤ lh }.
Then clearly S4 is a tree on × 5 . 4.2. Scales for Π14 sets. Again as in §2.2 we shall verify that for each α ∈ A, S4 (α) has an honest leftmost branch. For that let h : Wα → +1 be the rank function of Wα . For each i ≥ 1, let hi be the function on S3− (αi , i ) given by hi (v) = h(i , v) and let i = [hi ]Vαi ,i . Clearly ( 1 , 2 , . . . ) is a branch through S4 (α). Now let (1 , 2 , . . . ) be a branch through S4 (α). We want to show that i ≤ i , ∀i ≥ 1. Since (1 , 2 , . . . , n ) ∈ S4 (αn), let n h ∈ Wαn [+1 ] be such that pαn [nh] = (1 , . . . , n ). Let nhi (v) = nh(i , v). i Then for n, m ≥ i, [nhi ]Vαi ,i = [mhi ]Vαi ,i , so there is a cub Cn,m ⊆ +1 such / n m Uαi ,i ,j i i (Cn,m )) ∩ S3− (αi , i ). that hi (v) = hi (v) for v ∈ (Cn,m × 0<j<i i i Let C = 1
Uαi ,i ,j
(C ) for 0 < j < i },
into the ordinals: q(i , v) = nhi (v), for any n ≥ i.
292
ALEXANDER S. KECHRIS
As usual q is order preserving from S3− C α (α), 0 and j ∈ i Uαi ,i ,j (D) we have j = [f]Uαi ,i ,j , where f(x) ∈ D a.e. (mod Uαi ,i ,j ). But then H (f(x)) = f(x) a.e. (mod Uαi ,i ,j ), so j = [f]Uαi ,i ,j = [H ◦ f]Uαi ,i ,j = i Uαi ,i ,j (H )([f]Uαi ,i ,j ) = i Uαi ,i ,j (H )( j ). Similarly for j = 0. Thus i = [hi ]Vαi ,i ≤ [nhi ]Vαi ,i = i . If now for each α ∈ A, we let fα be the leftmost branch of S4 (α) and ϕi (α) = fα (i) = i = [hi ]Vαi ,i , (in the preceding notation), then we can verify again that ¯ ϕi (α) i (α) = α(i), is a Δ15 -scale on A. Indeed, for α, ∈ A: ¯ ∨ α(i) ¯ ∧ [ v. rankW (i , v)]V i (α) ≤ i () ⇔ α(i) ¯ < (i) ¯ = (i) α αi ,i ≤ [ v. rankW (i , v)]Vαi ,i ¯ ¯ ∧ ∃C ⊆ +1 [C cub ∧ ⇔ α(i) ¯ < (i) ∨ α(i) ¯ = (i) ∀( 0 , . . . , i −1 )[[(αi , ( 0 , . . . , i −1 )) ∈ S3− ∧
0 ∈ C ∧ ∀j[0 < j < i ⇒ i ∈ i Uαi ,i ,j (C )]] ⇒ rankWα (i , ( 0 , . . . , i −1 )) ≤
rankW (i , ( 0 , . . . , i −1 ))]] .
This relation can be verified to be Σ11 over the structure +1 , <, S2− , S3− , by using the results of Kechris-Martin [KM78]. By the Moschovakis Coding Lemma (see [Mos70] or Kechris [Kec78]) and the techniques of HarringtonKechris [HK81] one can verify then that every relation on reals which is Σ11 on +1 , <, S2− , S3− is Σ15 (and conversely), so “i (α) ≤ i ()” is also Σ15 . Similarly “i (α) < i ()” is Σ15 and we are done.
HOMOGENEOUS TREES AND PROJECTIVE SCALES
293
4.3. Homogeneity properties of S4 . Unfortunately not much can be said at this time about the homogeneity properties of S4 as the combinatorial property +1 → (+1 )+1 is still an open question (recall that the fact that 1 → (1 )1 is the key to establishing the homogeneity properties of S2 ). Note. The construction of S4 is due to Kunen [Kun71B]. §5. On 15 . Recall that we have defined in §4: 5 = sup{i V ,i (+1 ) : = ∅, 1 ≤ i ≤ lh }. The following result reduces the problem of computing 15 to the problem of computing these i V ,i (+1 ). Theorem 5.1 ([Kun71B]). 15 = ( 5 )+ = smallest cardinal > 5 . Proof. By the results in §4, every Π14 , and thus every Σ15 set, is 5− Suslin, i.e., it can be written in the form ∃f(α, f) ∈ S, where S is a tree on × 5 . Thus by the Kunen-Martin theorem (see Martin [Mar71B]) 15 ≤ ( 5 )+ . So it is enough to prove that 5 < 15 , i.e., that for is a measure on each , i asabove i V ,i (+1 ) < 15 . Put V ,i ≡ V. Then V S3− ( i , i ) ≡ I . Now the relation f ≺ g ⇔ f, g : I → +1 ∧ [f]V < [g]V can be easily seen to be Σ11 on the structure +1 , <, S3− . One can now use the Moschovakis Coding Lemma to code functions f : I → +1 by reals. Say ε codes fε . Then as in §4.2 one can verify that ≺ is Σ15 in the codes, i.e., the following relation is Σ15 : ε ≺∗ ⇔ ε, codes functions fε , f (resp.) from I into +1 ∧ fε ≺ f . As ≺∗ is a wellfounded relation of rank i V ,i (+1 ) (since rank≺∗ (ε) = (Theorem 5.1) rank≺ (fε ) = [fε ]V ) we have that i V ,i (+1 ) < 15 . §6. Homogeneous trees in general. We shall discuss now a general notion of homogeneity shared by the trees constructed before. We shall also formulate the type of tree construction utilized in §§2– 4 as a general transfer theorem for homogeneous trees. A tree T on × is homogeneous if for each = ∅ in < there is a measure on T ( ) with the following two properties: (i) Let for ⊇ , : T ( ) → T ( ) be the restriction map: (u) = u lh . Then ( )∗ = (i.e., for X ⊆ T ( ), X ∈ ⇔ −1 [X ] ∈ ).
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ALEXANDER S. KECHRIS
(ii) If T (α) is not wellfounded and for each n ≥ 1, Xn ⊆ T (αn) and Xn ∈ αn , then there is f ∈ with fn ∈ Xn , for all n. The basic way in which homogeneous trees have been obtained in this paper is as follows: Suppose T is a tree on × . Suppose also that there is an ordinal κ and for each = ∅ a wellordering W of order type ≤ κ and a map p :
W
[κ] T ( ),
such that
⊆ ⇒ W ⊆ W and moreover the following two conditions hold: (i) If ⊆ , then for h ∈ W [κ], p (h) lh = p (hW ). (ii) If T (α) is not wellfounded, then the union Wα = Wαn is a wellordering of order type α ≤ κ. Then granting that for each κ → (κ)· , we have that T is homogeneous. Indeed, let be the following measure on W [κ]: (X ) = 1 ⇔ ∃C ⊆ κ[C cub ∧ W C ↑ ⊆ X ]. Then let = (p )∗ . To check property (i) of homogeneity we use (i) : Indeed let X ⊆ T ( ). If (X ) = 1, then there is C ⊆ κ, C cub with W C ↑ ⊆ p −1 [X ]. If now h ∈ W C ↑, hW ∈ W C , thus p (hW ) = p (h) lh ∈ X , so p (h) ∈ −1 −1 [X ]. It follows that ( [X ]) = 1, so that ( )∗ (X ) = 1. For (ii) we use of course (ii) : Let T (α) be not wellfounded and let Xn −1 be such that αn (Xn ) = 1. Pick Cn cub in κ with Wαn Cn ↑ ⊆ pαn [Xn ]. Wα Let C = n Cn , so that C is cub in κ. Let, since α ≤ κ, h ∈ C ↑. If hn = hWαn , then hn ∈ Wαn C ↑, so that pαn (hn ) ∈ Xn . Moreover, if n < m, then pαn (hn ) is an initial segment of pαm (hm ), thus there is f ∈ with fn = pαn (hn ) so that fn ∈ Xn and we are done. It is now easy to see that S1 is an example of such a tree with κ = 1 , W = lh , < , = lh and p (u) = u ◦ . Moreover, by their construction, S2 , S3 , S4 are all of that form with W , , as given in §§2– 4. Note. Similar definitions of homogeneity apply to trees on κ × .
HOMOGENEOUS TREES AND PROJECTIVE SCALES
295
We shall now state and prove a general transfer theorem for homogeneous trees. We say below that A ⊆ R admits the tree T if A = p[T ]. Similarly for A ⊆ R × R, etc. Theorem 6.1 (Transfer Theorem for Homogeneous Trees (Kunen, Martin)). Assume B ⊆ R2 admits the homogeneous tree T (on some 2 × ). Then put α ∈ A ⇔ ¬∃B(α, ), and let be the order type of the wellordering W = (T ∗ ( ),
W
[κ] Tˆ ( )
as follows: Given h ∈ W [κ], let for 1 ≤ i ≤ lh , hi (v) = h(i , v), so that hi : T ( i , i ) → κ. Abbreviate ,i ≡ i ,i and put p (h) = ([hi ] ,i )1≤i≤lh and ( , u) ∈ Tˆ ⇔ ∃h ∈ W [κ](p (h) = u) ∨ ( = u = ∅). First note that if ⊆ and h ∈ W [κ], then p (h) lh = ([hi ] ,i )1≤i≤lh = p (hW ), so that Tˆ is indeed a tree and condition (i) before is satisfied. If now Tˆ (α) is not wellfounded, then as we will see in a moment α ∈ A so T (α) is wellfounded, therefore Wα = n Wαn is a wellordering and α ≤ κ, thus (ii) is also satisfied. Since we have assumed that κ → (κ)· we have by our preceding discussion that Tˆ is homogeneous. So it only remains to show that α ∈ A ⇔ Tˆ (α) is not wellfounded.
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ALEXANDER S. KECHRIS
The direction ⇒ is clear. To prove the other direction, assume (1 , 2 , . . . ) ∈ [Tˆ (α)]. Then for each n ≥ 1, we can find nh ∈ Wαn [κ] such that if we let n hi (v) = nh(i , v), for 1 ≤ i ≤ n, then i = [nhi ]αn,i . Since [nhi ]αn,i =
[n hi ]αn ,i for n, n ≥ i, let Zi ⊆ T (αi , i ) have αi ,i -measure 1 and be
such that nhi (v) = n hi (v), for all n, n ≥ i and all v ∈ Zi . If now α ∈ A, towards a contradiction, find with T (α, ) not wellfounded. Let for k ≥ 1, ik = k so that i1 < i2 < . . . . Let Xk = Zik ⊆ T (αk, k). By the homogeneity of T there is f ∈ [T (α, )] with fk ∈ Xk for all k ≥ 1. Then obviously (i1 , f1) >BK (i2 , f2) >BK . . . . But for each k ≥ 1, if n is large enough, then nhik (fk) = k is independent of n and also k > k+1 , so that 1 > 2 > . . . , a contradiction. (Theorem 6.1) Let now κ R be the least non-hyperprojective ordinal or equivalently the ordinal of the smallest admissible set containing the reals. Then by KechrisKleinberg-Moschovakis-Woodin [KKMW81], there are arbitrarily large < κ R with → ( ) (and also this holds for = κ R ). So it follows that every projective set admits a homogeneous tree or × for some < κ R . Moschovakis has pointed out that in the preceding theorem, it is enough to assume that B admits only a weakly homogeneous tree (to conclude again, under the appropriate assumptions, that A carries a homogeneous tree). Here a tree T on × is called weakly homogeneous if for each = ∅ there is a partition K ,i , T ( ) = i∈I
where each I is a countable set such that the following hold: (a) If ⊆ and T ( ) = i K ,i , T ( ) = j K ,j , then for every j, there is an i such that K ,j lh ≡ {v lh : v ∈ K ,j } ⊆ K ,i . (b) Each K ,i carries a measure ,i with the following property: If T (α) is not wellfounded and for each > 0, i ∈ Iαn , Xαn,i ⊆ Kαn,i and Xαn,i has αn,i -measure 1, then there is f ∈ such that for each n > 0, Xαn,i . fn ∈ i
The proof is similar to the one given before and we leave it to the reader. Notice also the simple fact that if B ⊆ R2 admits a weakly homogeneous tree, then so does C = {α : ∃B(α, )}. Note. The concepts and results in this section originate with Kunen [Kun71B] and Martin [Mar77B].
HOMOGENEOUS TREES AND PROJECTIVE SCALES
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§7. A result of Martin on subsets of 13 . Let P ⊆ R be a universal Π13 set for each A ⊆ R in Π1 , there is a of reals, i.e., assume that P ∈ Π13 and 3 n ∈ such that α ∈ A ⇔ n α = (n, α(0), α(1), . . . ) ∈ P. Let ϕ¯ = {ϕn } be a regular Π13 -scale on P—i.e., each ϕn maps P onto an initial segment of ordinals (therefore, as is well known, ran(ϕn ) = 13 = +1 ). The tree associated with this scale is defined by ¯ = {(αn, (ϕ0 (α), ϕ1 (α), . . . , ϕn−1 (α))) : α ∈ P, n ∈ }. T3 (ϕ) Thus T3 (ϕ) ¯ is a tree on × +1 . Also P = p[T3 (ϕ)] ¯ and for every α ∈ P, T3 (ϕ)(α) ¯ has an honest leftmost branch, namely ϕ(α). ¯ Note also that there is a function K : +1 → +1 such that ( , ( 0 , . . . , n−1 )) ∈ T3 (ϕ) ¯ ∧ 0 ≤ ⇒ 0 , 1 , . . . , n−1 ≤ K ( ). ¯ then for some α ⊇ , ϕ0 (α) = Indeed, if ( , ( 0 , . . . , n−1 )) ∈ T3 (ϕ),
0 , . . . , ϕn−1 (α) = n−1 . Thus max{ 0 , 1 , . . . , n−1 } ≤ sup{ϕn (α) : n ∈ def
∧ α ∈ P ∧ ϕ0 (α) ≤ } = K ( ). That K ( ) < +1 follows from the fact that for each < +1 = 13 , {α : α ∈ P ∧ ϕ0 (α) ≤ } is Δ13 , so by 1. boundedness, {ϕn (α) : n ∈ ∧α ∈ P ∧ ϕ0 (α) ≤ } is bounded below 3 1 1 If X ⊆ +1 , then we say that X is Σn in the codes or just Σn if X ∗ = {α ∈ P : ϕ0 (α) ∈ X } def
is Σ1n . One can use the results of Harrington-Kechris [HK81] to show that this is independent of the choice of P, ϕ0 , where ϕ0 is any regular Π13 -norm on a universal Π13 set P, provided that n ≥ 4. Let also S3 be the tree associated with P as in §3.1. Thus again P = p[S3 ]. The result below provides an analog of Theorem 1 in Kechris-Moschovakis [KM72]. Theorem 7.1 (Martin [Mar77B]). If X ⊆ +1 is Σ14 , then X ∈ L[S3 , T3 (ϕ)]. ¯ Proof. Say, putting T3 (ϕ) ¯ ≡ T3 , ∈ X ∗ ⇔ ∃(n0 , ∈ P) ⇔ ∃∃p(n0 , , p) ∈ [T3 ], and α ∈ P ∧ ∈ P ∧ ϕ0 (α) ≤ ϕ0 () ⇔ n1 α, ∈ P ⇔ ∃f(n1 α, , f) ∈ [S3 ],
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ALEXANDER S. KECHRIS
where as usual α, = (α(0), (0), α(1), (1), . . . ). Consider then the following game G , for < +1 : I α(0) h(0)
II (0) g(0) (0) p(0) f(0)
α(1) h(1) (1) g(1) (1) p(1) f(1) .. .
.. . α
h
g
p
f,
(where h, g, p, f ∈ +1 ), whose payoff set is defined as follows: We say that player I has played correctly up to his mth move, for m ≥ 1, if (αm, hm) ∈ T3 ∧ h(0) ≤ . Then note that also ∀i < m(h(i) ≤ K( )). We say that player II has played correctly up to his mth move, for m ≥ 1 again, if (m, gm) ∈ T3 ∧ g(0) ≤ ∧ (n0 , m, pm) ∈ T3 ∧ (n1 α, m, fm) ∈ S3 . Now player II wins iff for all m ≥ 1: Player I has played correctly up to his mth move ⇒ player II has played correctly up to his mth move. Clearly this is a closed game for player II and it is in L[S3 , T3 ], uniformly on . So it is enough (by the absoluteness of closed games) to show that
∈ X ⇔ player II has a winning strategry in G . (⇐). Say s is a winning strategy for player II in G . Let player I play (α, f) where α ∈ P, ϕ0 (α) = and h = ϕ(α). ¯ Then player I plays always correctly, so if player II, following his winning strategy s, produces (, g, , p, f) he must have played also always correctly, i.e., (, g) ∈ [T3 ] ∧ g(0) ≤
∧ (n0 , , p) ∈ [T3 ] ∧ (n1 α, , f) ∈ [S3 ], so ∈ P, ϕ0 () ≤ g(0) ≤
(as ϕ() ¯ is the honest leftmost branch of T3 ()), ∈ X ∗ and ϕ0 (α) ≤ ϕ0 (), thus ϕ0 () = ϕ0 (α) = and ∈ X . (⇒). Assume now player I has a winning strategy t in G but, towards a contradiction, that ∈ X . Then fix ∈ P with ϕ0 () = , g = ϕ() ¯ and , p so that (n0 , , p) ∈ [T3 ]. Let for each ∈ <, = ∅, V be the measure on S3 ( ) as in §3.3. Note that since +1 → (+1 ) , ∀ < +1 , ∀ < +1 , we actually have that V is +1 -additive. Of course these measures satisfy the homogeneity conditions (i), (ii) of §6. Define now inductively values α(0), h(0), α(1), h(1), . . . and sets X1 , X2 , . . . as follows (recall below that n1 α, = (n1 , α(0), (0), . . . ): First α(0), h(0) are the values called by t in I’s initial move, which is clearly correct. Now for each (f(0)) ∈ S3 ((n1 )), if player II plays (0), g(0), (0), p(0), f(0) in his first move, player I answers by t to play correctly α(1), h(1). In particular, h(1) < K ( ), so by the +1 -additivity of V(n1 ) , let X1 be in V(n1 ) and such that α(1), h(1) are always the same for (f(0)) ∈ X1 . This is our
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α(1), h(1). Then for each (f(0), f(1)) ∈ S2 ((n1 , α(0)) if player II next plays (1), g(1), (1), p(1), f(1), player I answers following t to play α(2), h(2) which, by an argument exactly as before, is the same for all (f(0), f(1)) ∈ X2 for some X2 ∈ V(n1 ,α(0)) , etc. Now as (α, h) ∈ [T3 ] and h(0) ≤ , clearly ϕ0 (α) ≤ , so ϕ0 (α) ≤ ϕ0 (), thus S3 (n1 α, ) is not wellfounded. Since Xk ∈ Vn1α,k for all k ≥ 1, we have by condition (ii) of homogeneity that there is f such that fk ∈ Xk for each k ≥ 1. If player II plays now , g, , p, f, he plays always correctly and if player I follows t he plays α, h so that he also plays correctly. But then player II won, a contradiction. (Theorem 7.1) Corollary 7.2. If X ⊆ +1 , then there is α ∈ R such that X ∈ ¯ α]. L[X3 , T3 (ϕ), Proof. By the Moschovakis’ Coding Lemma, every X ⊆ +1 is Σ14 (α) for some α ∈ R. (Corollary 7.2) §8. On the Victoria Delfino Third Problem. Let P be a universal Π13 set of reals and ϕ¯ a regular Π13 -scale on it. Let T3 (ϕ) ¯ be its associated tree as in §7. The Victoria Delfino Third Problem (see Kechris-Moschovakis (eds) [Cabal i]) is the question: Is L[T3 (ϕ)] ¯ independent of P, ϕ? ¯ ˜ 3 (ϕ)] Let also L[T ¯ = ¯ α]. Surely the independence of α∈R L[T3 (ϕ), ˜ 3 (ϕ)] L[T ¯ from P, ϕ¯ would be very strong evidence for an affirmative answer to the above problem. So the following result, despite its dependence on an unproven yet hypothesis is of interest here. Its proof uses methods of Kunen; see Kunen [Kun71D] and Kechris [Kec78]. ˜ 3 (ϕ)], ¯ Theorem 8.1. Assume +1 → (+1 )+1 . Then ℘(+1 ) ⊆ L[T ˜ 3 (ϕ)] ¯ is independent of P, ϕ. ¯ so in particular L[T Proof. The heart of the proof is the following: ˜ 3 (ϕ)] Lemma 8.2. If f : +1 → +1 , then there is g ∈ L[T ¯ such that f( ) ≤ g( ), ∀ < +1 . From that it follows that if C ⊆ +1 is cub, then there is C¯ ⊆ C , C¯ cub such ˜ 3 (ϕ)]. that C¯ ∈ L[T ¯ Indeed, let f : +1 → C be the increasing enumeration of C and let g be as in Lemma 8.2. Then if C¯ = { < +1 : is limit ∧∀ <
(g() < )}, C¯ is cub and if ∈ C¯ then ∀ < (f() ≤ g() < ), so f( ) = ∈ C , i.e., C¯ ⊆ C . Now we have Lemma 8.3. If +1 → (+1 )+1 , and for every cub C ⊆ +1 there is ˜ 3 (ϕ)], ˜ 3 (ϕ)]. ¯ C ⊆ C , C¯ cub such that C¯ ∈ L[T ¯ then ℘(+1 ) ⊆ L[T ¯
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Proof. Consider the following partition of +1[+1 ]: ˜ 3 (ϕ)]. f ∈ X ⇔ f ∈ L[T ¯ Then let C be cub such that +1C ↑ ⊆ X or +1C ↑ ⊆ X¯ . By our hypothesis, ˜ 3 (ϕ)] ˜ 3 (ϕ)]. C can be assumed to be in L[T ¯ so that we must have +1C ↑ ⊆ L[T ¯ Let { : < +1 } be the increasing enumeration of C and put H = ˜ 3 (ϕ)]. ¯ Let now A be an arbitrary {+ : < +1 }. Clearly +1[H ] ⊆ L[T unbounded subset of +1 . Let also fH be the increasing enumeration of H . Then fH [A] ⊆ H , so if g is the increasing enumeration of fH [A], ˜ 3 (ϕ)]. g ∈ L[T ¯ But ∈ A ⇔ fH ( ) ∈ fH (A) ⇔ fH ( ) ∈ ran(g), so ˜ 3 (ϕ)] A ∈ L[fH , g] ⊆ L[T ¯ and we are done. (Proof of Lemma 8.3) So we only have to prove Lemma 8.2 above. For that we will play a Solovaytype game which is a variant of a game of Kunen, see Kunen [Kun71D]. Let S be a Π12 subset of R × R, such that if S(α) ⇔ ∃S (α, ), then S ∈ Σ13 \ Π13 . Let S (α, ) ⇔ n0 α, ∈ P, so that ∃S (α, ) ⇔ ∃(n0 α, ∈ P) ⇔ ∃f∃∀k(n0 α, k, fk) ∈ T3 (ϕ) ¯ ⇔ ∃g∀m(αm, fm) ∈ U, where ((a0 , . . . , am−1 ), ( 0 , . . . , m−1 )) ∈ U ⇔ ( 0 )0 , . . . , ( m−1 )0 ∈ ∧ ¯ (n0 (a0 , ( 0 )0 , . . . , am−1 , ( m−1 )0 )m, (( 0 )1 , . . . , ( m−1 )1 )) ∈ T3 (ϕ), where → (( )0 , ( 1 )) is some simple 1-1 correspondence of +1 with (+1 ). Clearly U ∈ L[T3 (ϕ)] ¯ and if
2
S(α) ⇔ ∃S (α, ), then α ∈ S ⇔ U (α) is not wellfounded. Now we claim that U is a tree on ¯ ∈ U , then ¯ ) × , for some < +1 . Indeed in the notation above, if (a, there is a such that n0 ∈ P, = α, , a¯ ⊆ α and ϕi (n0 ) = ( i )1 for ¯ Thus (α, ) ∈ S . But S = {n0 α, : (α, ) ∈ S } is a Π12 i < m = lh a. subset of P, so (by boundedness) there is < +1 such that n0 ∈ S ⇒ ϕi (n0 ) < for all i so ( i )1 < , ∀i < m, thus there is < +1 such that
i < and we are done. Thus if U (α) is wellfounded, rank(U (α)) < +1 . Moreover, since S ∈ Δ13 , sup{rank(U (α)) : U (α) is wellfounded} = +1 .
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Otherwise for some < +1 α ∈ S ⇔ ¬(rank(U (α)) < ), therefore by Martin’s result (see Martin [Mar71B]) that Δ13 is closed under < +1 intersections and unions, S ∈ Δ13 , a contradiction. following game associated with each After these preliminaries consider the f : +1 → +1 : I w
II α
player II wins iff [w ∈ P ⇒ U (α) is wellfounded ∧ rank(U (α)) > f(ϕ0 (w))]. By a simple boundedness argument and the above remarks, player I cannot have a winning strategy in this game, so player II has a winning strategy s. Define then the following tree J on × +1 × × : ¯ ∧ (a, v) ∈ U (, u, a, v) ∈ J ⇔ (, u) ∈ T3 (ϕ) ∧ a is the result of player II playing according to s when player I plays . Clearly J ∈ L[T3 (ϕ), ¯ s]. Moreover J is wellfounded, since if (w, f, α, g) ∈ ¯ so w ∈ P, and (α, g) ∈ U , thus U (α) is not [J ] then (w, f) ∈ [T3 (ϕ)] wellfounded and also α is the result of a run in which player II follows s against player I playing w, a contradiction. Fix now < +1 . Let w ∈ P be such that ϕ0 (w) = . Let α be the result of player II playing according to s while player I plays this w. Finally let f = ϕ(w). ¯ Then the map v → (w lh v, f lh v, α lh v, v) is an embedding of U (α) into J . But notice that actually this maps U (α) into def
J( ) = {(, u, a, v) ∈ J : u(0) ≤ }. Thus rank(U (α)) ≤ rank(J( ) ). But also f( ) = f(ϕ0 (w)) < rank(U (α)), so def
f( ) < rank(J( ) ) = g( ). ¯ s], it will be enough to show that rank(J( ) ) < +1 . But As g ∈ L[T3 (ϕ), recall from §7 that for some K ( ) < +1 : w ∈ P ∧ ϕ0 (w) ≤ ⇒ ∀i[ϕi (w) ≤ K( )]. Thus if (, u, a, v) ∈ J , then u(i) ≤ K(u(0)), since (, u) ∈ T3 (ϕ), ¯ so there is w ∈ P with ⊆ w and ϕ(w) ¯ lh u = u, so ϕ0 (w) = u(0) and u(i) = ϕi (w) ≤ K (u(0)). So J( ) ⊆ J K ( ),
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thus f( ) ≤ g( ) = rank(J( ) ) ≤ rank(J K( )) < +1 , where J = {(, u, a, v) ∈ J : u ∈ <}. This completes the proof. (Lemma 8.2) (Theorem 8.1) REFERENCES
Leo A. Harrington and Alexander S. Kechris [HK81] On the determinacy of games on ordinals, Annals of Mathematical Logic, vol. 20 (1981), pp. 109–154. Alexander S. Kechris [Kec78] AD and projective ordinals, this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 91–132. Alexander S. Kechris, Eugene M. Kleinberg, Yiannis N. Moschovakis, and W. Hugh Woodin [KKMW81] The axiom of determinacy, strong partition properties, and nonsingular measures, in Kechris et al. [Cabal ii], pp. 75–99, reprinted in [Cabal I], p. 333–354. Alexander S. Kechris, Benedikt Lowe, and John R. Steel ¨ [Cabal I] Games, scales, and Suslin cardinals: the Cabal seminar, volume I, Lecture Notes in Logic, vol. 31, Cambridge University Press, 2008. Alexander S. Kechris and Donald A. Martin [KM78] On the theory of Π13 sets of reals, Bulletin of the American Mathematical Society, vol. 84 (1978), no. 1, pp. 149–151. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. Alexander S. Kechris and Yiannis N. Moschovakis [KM72] Two theorems about projective sets, Israel Journal of Mathematics, vol. 12 (1972), pp. 391– 399. [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. [KM78B] Notes on the theory of scales, in Cabal Seminar 76–77 [Cabal i], pp. 1–53, reprinted in [Cabal I], p. 28–74. Kenneth Kunen [Kun71B] On 15 , circulated note, August 1971. singular cardinals, circulated note, September 1971. [Kun71D] Some Richard Mansfield [Man71] A Souslin operation on Π12 , Israel Journal of Mathematics, vol. 9 (1971), no. 3, pp. 367– 379. Donald A. Martin [Mar71B] Projective sets and cardinal numbers: some questions related to the continuum problem, this volume, originally a preprint, 1971. [Mar77B] On subsets of 13 , circulated note, January 1977.
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Donald A. Martin and Robert M. Solovay [MS69] A basis theorem for Σ13 sets of reals, Annals of Mathematics, vol. 89 (1969), pp. 138–160. Yiannis N. Moschovakis [Mos70] Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory. Proceedings of an international colloquium held under the auspices of the Israel Academy of Sciences and Humanities, Jerusalem, 11–14 November 1968 (Y. Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam-London, 1970, pp. 24–62. Joseph R. Shoenfield [Sho61] The problem of predicativity, Essays on the foundations of mathematics (Yehoshua BarHillel, E. I. J. Poznanski, Michael O. Rabin, and Abraham Robinson, editors), Magnes Press, Jerusalem, 1961, pp. 132–139. Robert M. Solovay [Sol78A] A Δ13 coding of the subsets of , this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 133–150. DEPARTMENT OF MATHEMATICS CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA 91125 UNITED STATES OF AMERICA
E-mail: [email protected]
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ALEXANDER S. KECHRIS
This is an unpolished exposition of some work in the theory of projective ordinals under the hypothesis of definable determinacy. This is understood here as the hypothesis that every set of reals in L(R) is determined. Since the projective ordinals are absolute between the real world and L(R) we carry this study entirely within L(R). Thus we will use the full Axiom of Determinacy (AD) together with ZF+DC (DC is of course the only choice principle that is preserved under this transition to L(R)). Starting with the work of Martin, Moschovakis, and Solovay a decade ago, the exciting and unexpected possibility was discovered that one could calculate precisely the projective ordinals in terms of the aleph function. Indeed 11 = 1 and is a classical result and Martin computed that 12 = 2 , 13 = +1 1 4 = +2 (the last independently also due to Kunen). The computation of 15 and the higher 1n ’s is now the central problem of this theory. Kunen in 1971 has originateda major program towards achieving that goal, developing along the way some very important and powerful techniques. Part of his work is presented in the later sections of this survey and in Solovay’s paper [Sol78A]. We are planning to present the rest in a sequel paper, along with other more recent advances in this area. Work in descriptive set theory over the last ten years has resulted in a basically complete understanding of the analytical sets of the 3rd and 4th level of the analytical hierarchy, fully analogous to that provided by the classical effective theory for the first two. Moreover, recent results in this subject show, in our opinion, that a complete structure theory for all analytical sets at level 5 and beyond is essentially reduced to the problem of the precise calculation of the 1n ’s for n ≥ 5. Remark. Since some of the results we state below need actually only weaker forms of AD (like PD, etc.) we have put explicitly in the statements of the theorems the set theoretical assumptions which are used to establish them, beyond ZF+DC. Preparation for this paper was partially supported by NSF Grant MCS 76-17254. The author would like to thank R. M. Solovay for many interesting and helpful discussions on the topics presented in this paper. Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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§1. Definitions and the general picture. Definition 1.1. For all n ≥ 1, let 1n = sup{ : is the length of a Δ1n prewellordering of R (= )}. The following facts are known, granting AD: ... 1
1 +
1
1
4 =
3 =
5
+
+
3 +
2 +
1 +
1
1
2 =
1 =
?=
...
4
3
2
1
...
and in general for n ≥ 0, ...
; 1
2n
1 + +
2 +
1 n+
κ 2n
κ2
= 1 + ) 1 +
1 +
n ( 2
1
2n =
All the 1n are cardinals. 12n+2 =( 12n+1 )+ . + , where κ2n+1 is a cardinal of cofinality . 12n+1 = κ2n+1 1 All n are regular. (Note: without choice this does not follow from the fact that they are successor cardinals.) In fact 5) All 1n are measurable. Proofs of these and other results will be given in the sequel.
1) 2) 3) 4)
§2. For all n, 1n is a cardinal. Definition 2.1. Let F ⊆ R, ≤ a prewellordering on F , and let ϕ : F = lh(≤), be the canonical norm associated with it (i.e., α ≤ ⇔ ϕ(α) ≤ ϕ()). If f : → ℘ (R), where ℘ (R) is the power set of R, put Code(f; ≤) = {(α, ) : α ∈ F & ∈ f(ϕ(α))}. If A ⊆ n , put Code(A; ≤) = {(α1 , . . . , αn ) ∈ nF : (ϕ(α1 ), . . . , ϕ(αn )) ∈ A}. Definition 2.2. If f : → ℘ (R), we call g : → ℘ (R) a choice subfunction of f if 1) ∀ < , g() ⊆ f(). 2) ∀ < [f() = ∅ ⇒ g() = ∅].
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Theorem 2.3 (AD) (The Coding Lemma; Moschovakis [Mos70]). Let ≤ be a Δ1n prewellordering of a subset of R with length . Then every function f : → ℘ (R) has a choice subfunction g such that Code(g; ≤) is Σ1n . Proof sketch. Let G ⊆ R × R × R be universal for the Σ1n subsets of R × R; α is a code for a Σ1n set Q ⊆ R × R if Q = Gα = {(, ) : G(α, , )}. Given for all < , f : → ℘ (R), be the restriction of f to f : → ℘ (R), let , defined to be ∅ outside . Suppose there is an f with no good choice subfunction, where g is good if Code(g; ≤) is Σ1n . Let 0 ≤ be least such is limit. Consider the that f0 has no good choice subfunction. Clearly 0 following game: Player I plays α ∈ R, player II plays ∈ R, and player II wins if whenever α codes a good choice subfunction of f for some < 0 , then codes a good choice subfunction of f , where < < 0 . Case I: Player I has a winning strategy. Then for each there is an () < 0 and a good choice subfunction g() of f() , with code given by player I’s strategy applied to . Notice that sup{() : ∈ R} must be bounded below 0 (otherwise the union of the g() will be a good choice subfunction of f0 ). But since 0 was chosen least, player II can easily beat player I’s winning strategy. Hence Case I never occurs. Case II: Player II has a winning strategy. Using the recursion theorem, we can find a partial continuous function h such that for all w ∈ Field(≤) with ϕ(w) < 0 , h(w) is defined and is a Σ1n -code for a good choice subfunction g(w) of f(w) with ϕ(w) < (w) < 0. But then if g0 () = g(w) (), w∈Field(≤) ϕ(w)<0
g0 is a choice subfunction of f0 and (α, ) ∈ Code(g0 ; ≤) ⇔ ∃w[w ∈ Field(≤) & ϕ(w) < 0 & G(h(w), α, )], hence g0 is good, contradicting the choice of 0 . (Theorem 2.3) Corollary 2.4 (AD). For every A ⊆ n , Code(A; ≤) is Δ1n . Proof. Let us take n = 1 for notational simplicity. Let α0 , α1 be distinct reals, and define f : → ℘ (R) by {α0 }, if ∈ A f() = {α1 }, if ∈ A. Then the only choice subfunction of f is f itself. Hence Code(f; ≤) is Σ1n by the Coding Lemma, and α ∈ Code(A; ≤) ⇔ (α, α0 ) ∈ Code(f; ≤) ⇔ α ∈ Field(≤) & (α, α1 ) ∈ Code(f; ≤), hence Code(A; ≤) is Δ1n .
(Corollary 2.4)
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Theorem 2.5 (AD) (Moschovakis [Mos70]). For all n ≥ 1, 1n is a cardinal. Proof. If not, let f : → 1n be 1-1 and onto, where < 1n . There is a which is Δ1 . Let <∗ be defined on by prewellordering ≤ of R of length n <∗ ⇔ f() < f(). Then by the corollary, Code(<∗ ; ≤) is Δ1n . But then Code(<∗ ; ≤) is a prewell ordering on R in Δ1n with length 1n , contradiction. (Theorem 2.5) §3. The 1n ’s are successor cardinals. Definition 3.1. Let A ⊆ R. A norm on A is a map ϕ : A → Ord. The length of ϕ is the length of the prewellordering induced by ϕ on A, i.e., α ≤ϕ ⇔ ϕ(α) ≤ ϕ(). If Γ = Π1n or Σ1n and A ∈ Γ, then ϕ is a Γ-norm if the following relations are inΓ: α ≤∗ϕ ⇔ α ∈ A & [ ∈ A ∨ ϕ(α) ≤ ϕ()], α <∗ϕ ⇔ α ∈ A & [ ∈ A ∨ ϕ(α) < ϕ()]. If every set in Γ has a Γ-norm, we say that Γ has the prewellordering property. Theorem 3.2 (PD) (The Prewellordering Theorem; Martin [Mar68], Moschovakis (see [AM68])). For all n ≥ 0, Π12n+1 and Σ12n+2 have the prewellor dering property (and Σ12n+1 , Π12n+2 do nothave the prewellordering property). Definition 3.3. Let A, B ⊆ R. We say that A is reducible to B if there is a total continuous function f : R → R such that α ∈ A ⇔ f(α) ∈ B. If Γ = Σ1n or Π1n , a set A ⊆ R is called Γ-complete if A ∈ Γ and every set B ∈ Γ to A. reducible is Lemma 3.4 (AD) (Wadge’s Lemma). If A, B ⊆ R, then either A is reducible to B or B is reducible to R \ A. Proof. Consider the game in which player I plays α, player II plays and player II wins iff α ∈ A ⇔ ∈ B. (Lemma 3.4) By this lemma every set in Γ \ Δ (where Γ = Σ1n or Π1n and Δ = Δ1n ) is Γ-complete. Theorem 3.5 (PD) (Moschovakis [Mos70]). If ϕ is a Π12n+1 -norm on a 1 Π2n+1 -complete set, then lh(ϕ) = 12n+1 . Definition 3.6. A scale on a set A ⊆ R is a sequence of norms {ϕn }n∈ on A such that for every sequence {αi }i∈ of members of A, if 1) limi→∞ αi = α, and
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2) For each n there is an ordinal n such that ϕn (αi ) = n , for all large enough i, then α ∈ A and for all n, ϕn (α) ≤ n . The scale {ϕn }n∈ is a -scale if lh(ϕn ) ≤ , ∀n. If Γ = Σ1n or Π1n , we call {ϕn }n∈ a Γ-scale if the two relations S(n, α, ) ⇔ α ≤∗ϕn T (n, α, ) ⇔ α <∗ϕn are in Γ. Theorem 3.7 (PD) (The Scale Theorem, Moschovakis [Mos71]). For n ≥ 0, every Π12n+1 (Σ12n+2 ) set admits a Π12n+1 (Σ12n+2 )-scale. A tree on a set X is a set of finite sequences of members of X closed under initial segments. We will consider many times trees on × or × × , etc., where is an ordinal. Thus if T is a tree on × , its members are of the form ((k0 , 0 ), (k1 , 1 ), . . . , (kn , n )), where ki ∈ and i < for all i ≤ n. We will sometimes find it convenient to represent elements of such a T by pairs of tuples of the form ((k0 . . . kn ), ( 0 . . . n )). An (infinite) branch of tree T on X is a sequence f ∈ X such that for all n, fn ∈ T , where fn = (f(0), . . . , f(n − 1)). A branch of a tree on × is thus a sequence g ∈ ( × ), but we will represent it by the unique pair (α, f) ∈ × such that for all n, g(n) = (α(n), f(n)). If J is a tree on × put [J ] = {(α, f) : α ∈ R, f ∈ and ∀n(αn, fn) ∈ J } (i.e., [J ] is the set of branches of J ), and put p[J ] = {α ∈ R : ∃f ∈ (α, f) ∈ [J ]}. If {ϕn }n∈ is a -scale on a set A ⊆ R, the tree associated with this scale is the tree on × defined by
((k0 . . . kn ), ( 0 . . . n )) ∈ T ⇔ ∃α ∈ A such that ∀i ≤ n(α(i) = ki and ϕi (α) = i ). Claim 3.8. For A, T as above, p[T ] = A. Proof. A ⊆ p[T ] is obvious. Let α ∈ p[T ]. Find f ∈ such that (α, f) ∈ [T ]. Then for all n, (αn, fn) ∈ T , i.e., there is a sequence of reals {αn }n∈ such that ∀n(αn ∈ A) and (αn, fn) = (αn n, (ϕ0 (αn ), . . . , ϕn−1 (αn ))).
AD AND PROJECTIVE ORDINALS
Then αn → α and ϕn (αi ) = f(n) for all i > n, hence α ∈ A.
309 (Claim 3.8)
Definition 3.9. If T is a tree on × , put T (α) = {s ∈ < : (α lh(s), s) ∈ T }. Clearly for each α, T (α) is a tree on , and if T is the tree associated with a scale on A as above, then by the claim we have α ∈ A ⇔ T (α) has an infinite branch. Definition 3.10. A set A ⊆ R is -Suslin (where is an ordinal) if there is a tree T on × such that A = p[T ]. The next result is classical for Σ11 , is due to Schoenfield [Sho61] for Σ12 , to also Mansfield [Man71]) and to MoschoMartin-Solovay [MS69] for Σ13 (see vakis [Mos71] in general. Theorem 3.11 (PD). (i) For each n ≥ 0, every Σ12n+2 set is 12n+1 -Suslin. (ii) For each n ≥ 0, every Σ12n+1 set is κ2n+1 -Suslin, where κ2n+1 is a cardinal < 12n+1 . Proof. Let , : × κ ↔ κ be a coding of pairs by ordinals less than a cardinal κ, with decoding functions ( )0 and ( )1 . Then if B ⊆ R × R is κ-Suslin, let T be a tree on × × κ such that (α, ) ∈ B ⇔ ∃f ∈ κ∀n(αn, n, fn) ∈ T . Let ((k0 , 0 ), . . . , (kn , n )) ∈ T ⇔ ((k0 , , ( 0 )0 , ( 0 )1 ), . . . , (kn , ( n )0 , ( n )1 ) ∈ T. Then clearly ∃B(α, ) ⇔ α ∈ p[T ], hence {α : ∃B(α, )} is also κ-Suslin. So to prove (i) it is enough to show that every Π12n+1 set is 12n+1 -Suslin. But a Π1 -scale this is obvious by the previous results and the evident fact that 2n+1 1 1 on a Π2n+1 set must be a 2n+1 -scale. Wenow give the proofof (ii). By the closure of κ-Suslin sets under real existential quantification, it is enough to show that every Π12n set is κ-Suslin for some fixed κ < 12n+1 . The least such κ is the required κ2n+1 . Let A be a complete Π12n set, {ϕn }n∈ a Π12n+1 -scale on A. Since A is Δ12n+1 , all m. Since any -sequence lh(ϕm ) < 12n+1 for of Δ12n+1 prewellorderings can be put together to yield a new Δ12n+1 prewellordering of length at least the supremum of the lengths of the original prewellorderings, cf( 12n+1 ) > . 1 Hence there is a κ < 2n+1 such that lh(ϕm ) < κ, for all m. Hence, by passing from the scale to its associated tree, A is κ-Suslin. (Theorem 3.11) Definition 3.12. A set of reals if -Borel ( an ordinal) if it belongs to the smallest class of sets of reals containing the open sets and closed under complements and wellordered unions of length < . This class is denoted by B .
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Theorem 3.13 (Separation of κ-Suslin sets; Luzin for κ = ; see Martin [Mar71B]). If A, B ⊆ R are κ-Suslin and A ∩ B = ∅ then there is a κ + -Borel set C which separates them, i.e., A ⊆ C and C ∩ B = ∅. Proof. Let A = p[T ], B = p[S], where T and S are trees on × κ. Define a tree U on × κ × κ by (s, u, v) ∈ U ⇔ (s, u) ∈ T & (s, v) ∈ S. Since A ∩ B = ∅, U is well founded, i.e., has no infinite branches. We will define a function on U (s, u, v) → Cs,u,v ⊆ R, by induction on U , such that each Cs,u,v ∈ Bκ+ and Cs,u,v separates As,u and Bs,v , where As,u = {α ⊇ s : ∃f ⊇ u((α, f) ∈ [T ])} and Bs,v = {α ⊇ s : ∃f ⊇ v((α, f) ∈ [S])}. We can take then C = C∅,∅,∅ . Note that As,u = n, Asn,u and Bs,v = m, Bsm,v . So it is enough to define Dn, ,m, ∈ Bκ+ such that Dn, ,m, separates Asn,u from Bsm,v , since we can then take Cs,u,v = Dn, ,m, n, m,
Assume we have defined all Csn,u ,v , when (s n, u , v ) ∈ U . Case I: n = m and (s n, u , v ) ∈ U . Then take Dn, ,m, = Csn,u ,v . Case II: n = m and (s n, u , v ) ∈ U . Then either Asn,u = ∅ or Bsn,v = ∅, so they can be trivially separated. Case III: n = m. Then Asn,u and Bsn,v can be separated by disjoint open neighborhoods. (Theorem 3.13) Theorem 3.14 (Generalized Suslin Theorem; see Martin [Mar71B]). If A, R \ A are κ-Suslin, then A ∈ Bκ+ . Theorem 3.15 (AD) (⊆ Martin [Mar71B]; ⊇ Moschovakis [Mos71]). For all n ≥ 0, B 12n+1 = Δ12n+1 . 1 Proof. That Δ2n+1 ⊆ B 12n+1 follows from the fact that each Δ12n+1 set is < 1 κ2n+1 -Suslin for some κ2n+1 2n+1 (Theorem 3.11 (ii)). For the other Δ1 direction, it is enough to show that 2n+1 is closed under unions of length
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< 12n+1 . If not, let < 12n+1 be least such that for some sequence {A } < of sets, 1 Δ12n+1 uncountable cardinal, and
< A = A ∈ Δ2n+1 . Clearly is an we may assume A ⊆ A if ≤ < and A = < A if = < . Let ≤ be a Δ12n+1 prewellordering of R such that lh(≤) = . Let f : → ℘(R) be givenby f( ) = {ε : ε is a Δ12n+1 code of A }, where ε is a Δ12n+1 -code if ε = ε0 , ε1 and the Π12n+1 set coded by ε0 equals the Σ12n+1 set coded by ε1 . Denote by Δε the Δ12n+1 set coded by ε, if ε is a Δ12n+1 -code. subfunction of f such that Let g be a choice Code(g; ≤) is Σ12n+1 . Then α ∈ A ⇔ ∃[(w, ) ∈ Code(g; ≤) & α ∈ Δ ]. Hence A is Σ12n+1 . By Wadge’s Lemma, A is Σ12n+1 -complete. For α ∈ A, let (α) = the unique < such that α ∈ A +1 \ A . Claim 3.16. is a Σ12n+1 -norm (which is a contradiction since it implies property). that Σ12n+1 has the prewellordering Proof of Claim 3.16. We have α ≤∗ ⇔ ∃ < [α ∈ (A +1 \ A ) & ∈ A ] and α <∗ ⇔ ∃ < [α ∈ (A +1 \ A ) & ∈ A +1 ]. So ≤∗ and <∗ are both unions of < 12n+1 Δ12n+1 sets, hence as before they are Σ12n+1 . 3.16) (Claim (Theorem 3.15) Definition 3.17. If J is a tree on a set X and u ∈ sequences from X , then Ju = {v ∈ <X : u v ∈ J }.
<
X = set of finite
Notation. |J | < means J is wellfounded and has rank < . (Put also |∅| = −1.) ´ Theorem 3.18 (Sierpinski for κ = ). If A ⊆ R is κ-Suslin, then A ∈ Bκ++ . Proof. Let A = p[T ], where T is a tree on × κ. For each < κ + and u ∈ <κ put A u = {α : |T (α)u | < }. Then if lh(u) = n, A0u = {α : (αn, u) ∈ T }
A +1 = A u ∪ Au u A u
=
<
<
A u ,
if =
> 0.
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ALEXANDER S. KECHRIS
Thus A u ∈ Bκ+ for all u and . But α ∈ A ⇔ α ∈ p[T ] ⇔ T (α) is well founded ⇔ ∃ < κ + (|T (α)| < ) ⇔ ∃ < κ + (α ∈ A ∅ )
(Theorem 3.18)
Theorem 3.19 (Martin [Mar71B]). If A is κ-Suslin and cf(κ) > , then A ∈ Bκ + . Proof. let A = p[T ], where T is a tree on × κ. Then α ∈ A ⇔ T (α) is not well founded ⇔ ∃ < κ(T (α) is not well founded), where T = T restricted to ordinals < . Now apply 3.18. (Theorem 3.19) Theorem 3.20 (Kechris [Kec74]). For all n, 12n+1 = (κ2n+1 )+ , where κ2n+1 is a cardinal of cofinality . Proof. Let (by Theorem 3.11) κ2n+1 = least κ such that every Σ12n+1 set is κ-Suslin. If (κ2n+1 )++ ≤ 12n+1 , then every Σ12n+1 set is in B(κ2n+1 )++ ⊆ B 12n+1 = . If Δ12n+1 , a contradiction. By 3.11, κ2n+1 < 12n+1 , hence (κ2n+1 )+ = 12n+1 cf(κ2n+1 ) > , then by 3.19 every Σ12n+1 set is in B(κ2n+1 )+ = B 12n+1 = Δ12n+1 , contradiction. (Theorem 3.20) Theorem 3.21 (Kunen [Kun71C], Martin [Mar71B]). If ≺ ⊆ R × R is wellfounded and κ-Suslin, then |≺| < κ + . Proof. Let α ≺ ⇔ ∃f ∈ κ((α, , f) ∈ [T ]), where T is a tree on × × κ. Put T≺ = {(α0 , α1 , . . . , αn ) : α0 ) α1 ) · · · ) αn }. By induction one easily checks that for each α ∈ Field(≺) and each α0 , . . . , αn such that α0 ) α1 ) · · · ) αn ) α we have |α|≺ = |(α0 . . . αn , α)|T≺ . Hence |≺| ≤ |T≺ |. Let S consist of all sequences of the form s = ((s1 , t1 , u1 ), . . . , (sn , tn , un )), where si = ti+1 for all i < n and (si , ti , ui ) ∈ T , for all i ≤ n. Thus si , ti ∈ <, ui ∈ <κ. For s, s as above, define s )∗ s ⇔ lh(s) < lh(s ) and for all i < lh(s), (si , ti , ui properly extend si , ti , ui ).
AD AND PROJECTIVE ORDINALS
313
For any α, such that α ≺ let hα, be the leftmost branch of T (α, ). Now define f : T≺ → S by f(α0 . . . αn ) = ((α1 n, α0 n, hα1 ,α0 n), . . . , (αn n, αn−1 n, hαn ,αn−1 n)). Clearly f embeds in an order preserving way T≺ into (S, ≺∗ ). It only remains to show ≺∗ is wellfounded. If not, let s0 )∗ s1 )∗ s2 )∗ . . . , where sn = ((s1n , t1n , u1n ), . . . , (sknn , tknn , uknn )). Then kn → ∞, t1n → α0 , s1n = t2n → α1 , s2n = t3n → α2 , . . . , and u1n → f1 , u2n → f2 , . . . , where for all n, (αn+1 , αn , fn+1 ) ∈ [T ]. Hence α0 ) α1 ) α2 ) . . . , a contradiction. (Theorem 3.21) Theorem 3.22 (AD) (Kunen [Kun71C], Martin [Mar71B]). For all n ≥ 0, ( 12n+1 )+ = 12n+2 . Proof. If ϕ is a Π12n+1 -norm on a Π12n+1 -complete set, then lh(ϕ) = 12n+1 . Since its associatedprewellordering is Δ12n+2 , we have 12n+1 < 12n+2 . Hence 1 1 + 1 1 2n+2 ≥ ( 2n+1 ) . Since every Σ2n+2 relation is 2n+1 -Suslin, we have by that ( 1 )+ ≥ 1 . Theorem 3.21 (Theorem 3.22) 2n+1 2n+2 Theorem 3.23 (AD) (Moschovakis [Mos70] for odd n, Kechris [Kec74] for even n). For all n, 1n < 1n+1 . Proof. The theorem for n odd follows from 3.22. Suppose 12m = 12m+1 . is a , which Then 12m+1 = (κ2m+1 )+ = 12m = ( 12m−1 )+ , hence κ2m+1 = 12m−1 contradiction since cf( 12m−1 ) > . (Theorem 3.23) Theorem 3.24 (AD) (Moschovakis [Mos70], for odd n; Kunen [Kun71C], Martin [Mar71B], for all n). For all n, 1n = sup{ : is the length of a Σ1n wellfounded relation}. Proof. By Theorems 3.11, 3.21 and 3.22. (Theorem 3.24) §4. The 1n ’s are regular. Theorem 4.1 (AD) (Moschovakis [Mos70] for odd n; Kunen [Kun71C] for all n). For all n, 1n is regular. Proof. Assume not, and let f : → 1n be a cofinal map, with < 1n . Let with corresponding norm ϕ. Let ≤ be a Δ1n prewellordering of R of length g( ) = {α : α is a Σ1n code of a Σ1n well founded relation of length f( )}. of g such that Code(g ; ≤) is Σ1n . Let g be a choice subfunction
314
ALEXANDER S. KECHRIS
Let W ⊆ R × R × R be Σ1n universal, and put (α, , ) ≺ (α , , ) ⇔ [α = α , = , (α, ) ∈ Code(g ; ≤) and (, ) ∈ W ]. Clearly ≺ is Σ1n and wellfounded. But for any < , if α is such that ϕ(α) = , then for anyfixed ∈ g ( ) the map → (α, , ) embeds W into ≺. So |≺| ≥ |W | = f( ), hence |≺| = 1n , a contradiction. (Theorem 4.1) §5. The 1n ’s are measurable. Theorem 5.1 (AD) (Solovay for n = 1 (see [Sol67A]), 2; Martin [Mar71A] for odd n; Kunen [Kun71A] in general). For all n, 1n is measurable. 3 1 Proof. Let W ⊆ R be universal Σn and let S = {α : Wα is wellfounded binary relation}. For α ∈ S, let |α| = lh(Wα ). Hence 1n = sup{|α| : α ∈ S}. For A ⊆ 1n , consider the following game G A first used (for n = 1 and coding) by Solovay in his original proof that is measurable. with a different 1 Player I plays α, player II plays , and player II wins iff [∃i((α)i ∈ S or ()i ∈ S) and if i0 is the least such i, then (α)i0 ∈ S] or [∀i((α)i ∈ S and ()i ∈ S) and sup{|(α)0 |, |()0 |, |(α)1 |, |()1 |, . . . } = sup{|(α)i |, |()i |} ∈ A]. i
Here we think of a real α as coding an -sequence of reals {(α)i }i∈ , where (α)i (m) = α(pm+1 ) and pi is the ith prime. i Now define U ⊆ ℘( 1n ) by A ∈ U ⇔ player II has a winning strategy in G A . We show that U is a 1n -additive measure on 1n . Lemma 5.2. If A ∈ U and B ⊇ A then B ∈ U . Proof. Trivial. Lemma 5.3. If A, B ∈ U then A ∩ B ∈ U .
(Lemma 5.2)
AD AND PROJECTIVE ORDINALS
315
Proof. Given reals α, , let α ⊕ be a real such that (α ⊕ )2n = (α)n and (α ⊕ )2n+1 = ()n for all n. Suppose player II has a winning strategy in G A , and a winning strategy in G B . To win G A∩B , given a move α of player I, player II simultaneously builds reals , such that is the result of against α ⊕ , and is the result of against α ⊕ . Player II’s actual play is then ⊕ . If there is i0 such that ∀j ≤ i0 ((α)j ∈ S) and ( ⊕ )i0 ∈ S we are led to a contradiction. If ∀i((α)i ∈ S and ( ⊕ )i ∈ S), then supi {|(α)i |, |( ⊕ )i |} = sup{|(α ⊕ )i |, |( )i |} = (Lemma 5.3) sup{|(α ⊕ )i |, |()i |} ∈ A ∩ B. Lemma 5.4. The filter U contains no bounded sets. Proof. If A is bounded, then since sup{|α| : α ∈ S} = 1n , player I can (Lemma 5.4) easily win G A . Lemma 5.5. The filter U is an ultrafilter. Proof. We must show that A ∈ U ⇒ ( 1n \ A) ∈ U . But if player II has no winning strategy in G A , then player II canessentially follow player I’s winning 1 (Lemma 5.5) strategy in G A to win G (n \A) . Lemma 5.6. The ultrafilter U is 1n -additive. Proof. Let {A } << 1n be a sequence of < 1n members of U . It suffices to show < A = ∅. Let ≤ be a Δ1n prewellordering of R of length with associated norm ϕ. For
< let f( ) = { : is a winning strategy for player II in G A }. Let g be a choice subfunction of f such that Code(g; ≤) is Σ1n , say in Σ1n (y). Claim 5.7. For each m ≥ 0, there is a function fm : m+1S → S such that for all α 0 , . . . , α m ∈ S, for all α with (α)i = α i if i ≤ m and for all ∈ < g( ), |fm (α 0 . . . α m )| ≥ |([α])m |, where [α] = player II’s play when player I plays α and player II follows . Proof. Given α 0 , . . . , α m ∈ S, consider the following wellfounded relation: α, x, , z ≺α 0 ,...,α m α , x , , z ⇔ α = α & x = x & = & ∀i ≤ m((α)i = α i ) & (x, ) ∈ Code(g; ≤) & (z, z ) ∈ W([α])m . Then ≺α 0 ,...,α m is Σ1n (α 0 . . . α m , y) with length ≥ |([α])m | for any , α as above. Clearly one can find a continuous fm such that fm (α 0 , . . . , α m ) is a Σ1n -code 5.7) (Claim for ≺α 0 ,...,α m , proving the claim.
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ALEXANDER S. KECHRIS
Now let α 0 ∈ S and define inductively α m+1 = fm (α 0 . . . α m ). Let = sup{|α m | : m ∈ }. Given < , let player I play α such that ∀i((α)i = α i ), and let player II play using a strategy from g( ), producing a sup{|(α)i |} = . real = [α]. Then ∀i(()i ∈ S) and sup{|(α)i |, |()i |} = Since player II’s strategy was winning, ∈ A . Hence ∈ < A , proving the lemma. (Lemma 5.6) (Theorem 5.1) §6. Calculating 1n for n ≤ 4. Theorem 6.1 (Classical). 11 = 1 . Proof. Every Σ11 set is -Suslin. So every Σ11 wellfounded relation has (Theorem 6.1) length < 1 . Theorem 6.2 (AD) (Martin [Mar71B]). 12 = 2 . Proof. Obvious, since 12 = ( 11 )+ . (Theorem 6.2) Theorem 6.3 (∀α (α # exists)) (Martin-Solovay [MS69]). Every Σ13 set is -Suslin. Proof. We will show that every Π12 set admits a Δ13 -scale {ϕn }n∈ such that for all n, lh(ϕn ) < . Note that it suffices to prove that the Π12 set R# = {α # : α ∈ R} admits such a scale. Because if {ϕn∗ }n∈ is such a scale on R# , put ϕn (α) = ϕ0∗ (α # ), α # (0), ϕ1∗ (α), α # (1), . . . , ϕn∗ (α # ), α # (n), where refers to the ordinal of the 2n-tuple under the lexicographical order. Then {ϕn∗ }n∈ is a Δ13 -scale when restricted to any Π12 set A: To show this (assuming {ϕn∗ }n∈ has the right properties), let αi ∈ A, αi → α, and ϕn (αi ) = n , ∀i ≥ n. This implies that αi# → for some . Since each ϕn∗ (αi# ) is eventually constant, = α¯ # for some α. ¯ Since there is a recursive f such that f( # ) = for all , α¯ = α. To see that α ∈ A, note that for some Π12 formula ϕ α ∈ A ⇔ L[α] ϕ(α) ⇔ α # (n0 ) = 0, ¨ where n0 is a Godel number for ϕ(α) ˙ (α˙ is the constant symbol denoting α). Hence since for all i, αi ∈ A, we have αi# (n0 ) = 0, therefore α # (n0 ) = 0, i.e., α ∈ A. Finally, to see that ϕn (α) ≤ n , pick k large enough so that ∀i ≥ k, ∀p ≤ n, ϕp# (αi ) is constant and αi# (n + 1) = α # (n + 1). Then ϕn (α) = ϕ0∗ (α # ), α # (0), . . . , ϕn∗ (α # ), α # (n) ≤ ϕ0# (αk# ), αk# (0), . . . , ϕn∗ (αk# ), αk# (n) = n .
AD AND PROJECTIVE ORDINALS
317
Since the sharp operation is Δ13 , it is clear that {ϕn }n∈ is a Δ13 -scale. So we must produce a Δ13 -scale on R# whose norms have length < . Let 0 , 1 , 2 . . . be a recursive enumeration of all definable Skolem functions or terms (with variables) in the theory ZF + V=L[α] ˙ + α˙ ∈ R (where α˙ is a constant symbol) and put n = rank(n ) so that n takes only ordinal values. Say n = n (v1 , . . . , vkn ). Then define ϕn∗ (α # ) = nL[α] (1 , . . . , kn ). Note that ϕn∗ (α # ) < ϕn∗ ( # ) ⇔ nL[α] (1 , . . . , kn ) < nL[] (1 , . . . , kn ) ⇔ L[α, ] (α, , 1 , . . . , kn ) ⇔ α, # (m) = 0, where m is obtained recursively from n. Hence {ϕn∗ }n∈ is a Δ13 -scale, if it is a scale. We also have lh(ϕn∗ ) < since in fact nL[α] (1 , . . . , kn ) < kn +1 for all α (because every cardinal is an indiscernible for every L[α]). To show {ϕn∗ }n∈ is a scale, let αi# ∈ R# , αi# → and ϕn∗ (αi# ) = n for i > n. Note that ∈ R# ⇔ P() & Γ(, 1 ) is wellfounded, ¨ where P is Π01 expressing “ is a set of Godel numbers of formulas satisfying the syntactical conditions for a remarkable character relative to some real α”, and Γ(, ) is the model of ZF + V=L[α] ˙ generated by indiscernibles on the basis of . Since P(αi# ) for all i, we know that P() holds. Let I α = class of Silver indiscernibles for L[α]. Let C = i,j (I αi ,αj ∩ 1 ). Thus C is closed unbounded in 1 . Let {c } <1 be its increasing enumeration. Since for any fixed n, nL[αi ] (1 , . . . , kn ) becomes eventually constant, the same is true of nL[αi ] (c 1 , . . . , c kn ) for all 1 < · · · < kn < 1 . So define f : OrdΓ(,1 ) → Ord by f(nΓ(,1 ) (i 1 , . . . , i kn )) = eventual value of nL[αi ] (c 1 , . . . , c kn ), where I = {i : < 1 } is a generating set of indiscernibles for Γ(, 1 ). Claim 6.4. f is well defined and order preserving. Γ(,1 ) Proof. Suppose nΓ(,1 ) (i 1 , . . . , i kn ) = m (i 1 , . . . i k ), where 1 < m · · · < kn and 1 < · · · < km . Then there is a such that Γ(, 1 ) (i1 , . . . , i ) ⇔ n (i 1 , . . . , i kn ) = m (i i , . . . , i k ), where 1 < · · · < is m { 1 . . . kn , 1 . . . k n } written in increasing order.
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ALEXANDER S. KECHRIS
Thus ((v1 , . . . , v )) = 0. Since αi# → L[αi ] , the eventual # αi ((v1 , . . . , v )) is 0. Hence eventually L[αi ] (c 1 , . . . , c k ), nL[αi ] (c 1 , . . . , c kn ) = m
value of
m
so f is well defined. Similarly f is order preserving. (Claim 6.4) Hence Γ(, 1 ) is well founded, so ∈ R# . So = α # , where αi → α. Thus Γ(, 1 ) = L1 [α]. Let {i α : < 1 } be the increasing enumeration of the Silver indiscernibles of L1 [α]. Let C ∗ = { < 1 : c = i α = }. Then C ∗ is closed unbounded, and since f is order preserving, ∀c 1 < · · · < c kn in C ∗ we have nL[α] (c 1 , . . . , c kn ) = nL[α] (i α1 , . . . , i αkn ) ≤ f(nL[α] (i α1 , . . . , i nkn )) = eventual value of nL[αi ] (c 1 , . . . , c kn ). Thus nL[αi ] (1 , . . . , kn ) ≤ eventual value of nL[α] (1 , . . . , kn ) = n , i.e., ϕn∗ (α # ) ≤ n . Hence {ϕn∗ }n∈ is a scale. (Theorem 6.3) Theorem 6.5 (AD) (Martin [Mar71B]). 13 = +1 . Proof. We have 13 = κ3+ , where κ3 > is a cardinal of cofinality and such that every Σ1 set is κ -Suslin. Hence κ ≤ , κ3 is the least cardinal 3 3 3 that is the second cardinal of so κ3 = , since countable choice implies (Theorem 6.5) cofinality . Hence 13 = +1 . Corollary 6.6 (AD) (Kunen [Kun71C], Martin [Mar71B]). 14 = +2 . §7. The closed unbounded measure on 1 . Theorem 7.1 (AD) (Solovay [Sol67A] for n = 1, Moschovakis [Mos70] in general). Let P ∈ Π12n+1 \ Δ12n+1 , and let ϕ be a Π12n+1 norm on P with associated prewellordering ≤. Then for every A ⊆ 12n+1 , Code(A; ≤) ∈ 1 Π2n+1 . Proof. By the Coding Lemma we know that for all < 12n+1 , Code(A ∩ player II plays
; ≤) ∈ Δ12n+1 . Consider the following game: Player I plays w, α, and player II wins iff [w ∈ P ⇒ α is a Δ12n+1 -code of a set Δα such that Code(A ∩ (ϕ(w) + 1); ≤) ⊆ Δα ⊆ Code(A; ≤)]. If player I has a winning strategy , then { (α) : α ∈ R} = Q is a Σ11 subset of P, hence by boundedness, = sup{ϕ(w) : w ∈ Q} < 12n+1 . Soplayer II can easily beat this strategy by playing a Δ12n+1 -code of Code(A ∩ ( + 1); ≤). Hence player II has a winning strategy, and w ∈ Code(A; ≤) ⇔ w ∈ P & w ∈ Δ(w) , so Code(A; ≤) is Π12n+1 .
(Theorem 7.1)
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AD AND PROJECTIVE ORDINALS
Theorem 7.2 (AD) (Solovay [Sol67A]). For every A ⊆ 1 , ∃α ∈ R(A ∈ L[α]). Proof. Let WO = {α : α codes a wellordering of } and for α ∈ WO, let |α| = the ordinal coded by α. For A ⊆ 1 , let Code(A) = {α : |α| ∈ A}. By the above, for every A ⊆ 1 , Code(A) ∈ Π11 . We will now show (in ZF+DC) that for any A with Code(A) ∈ Σ12 , there is an α such that A ∈ L[α]. Let P∈ Σ12 be such that α ∈ Code(A) ⇔ P(0 , α) for some 0 . Then
∈ A ⇔ ∃α(P(0 , α) & |α| = ). Case I: For some , 1L[] = 1 . Then ∈ A ⇔ L[, 0 ] ∃α(P(0 , α) & |α| = ), so A ∈ L[, 0 ]. Case II: For all , 1L[] < 1 . Then if C is the notion of forcing which collapses to (for < 1 ), there are C -generic over L[0 ] sets (since (℘( ))L[0 ] is countable). Hence
∈ A ⇔ ∃α(P(0 , α) & |α| = ) ⇔ For all C -generic over L[0 ]G, L[0 , G] ∃α(P(0 , α) & |α| = ) L[0 ] ˆ ϕ(ˆ0 , ) ⇔ ∅ C
for some formula ϕ. Since forcing is definable in L[0 ], this shows that A ∈ L[0 ].
(Theorem 7.2)
Theorem 7.3 (AD) (Solovay [Sol67A]). There is a unique normal measure on 1 , namely (A) = 1 ⇔ A contains a closed unbounded set. Proof. It suffices to show that for all A ⊆ 1 , either A or 1 \ A contains a cub set. Given A ⊆ 1 , let α be such that A ∈ L[α]. Then A = L[α] (iα1 , . . . , iαk , i α1 , . . . , i αm ) for some 1 < · · · < k < 1 ≤ 1 · · · < m and some term . But then C = {iα : k < < 1 } is closed unbounded, and either C ⊆ A or C ⊆ 1 \ A (Theorem 7.3) §8. Uniform indiscernibles and the n ’s for n ≤ . Definition 8.1. An ordinal u is a uniform indiscernible if ∀α ∈ R(u ∈ I α ), where I α = {i α } ∈Ord is the class of Silver indiscernibles for L[α]. Let U = {u } ∈Ord by the increasing enumeration of the uniform indiscernibles. Clearly u1 = 1 , U is closed unbounded and every cardinal is in U . Hence u ≤ .
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Theorem 8.2 (AD) (Martin [Mar71B]). u = . Proof. In the proof of Theorem 6.3 we could have used ϕn∗ (α # ) = nL[α] (u1 , . . . , ukn ), hence Σ13 sets are u -Suslin. Hence as in Theorem 6.5 (u )+ = 13 = ( )+ hence = u .
(Theorem 8.2)
Lemma 8.3 (∀α(α # exists)). For every ordinal there is a real α, a term , and ordinals 1 < · · · < m such that = L[α] (u1 , . . . , um ). Proof. By induction on . Clear if < ℵ1 . So assume true for all < , and let ∈ U . Then for some α, ∈ I α . Thus
= L[α] (iα1 , . . . , iαm , iαm+1 , . . . , iαk ) for some term and some iα1 < · · · < iαm < < iαm+1 < · · · < iαk . Thus
= L[α] (iα1 . . . iαm , ℵ), where ℵ is a sequence of large enough cardinals. Now for all j ≤ m, iαj can be defined in some L[] using uniform indiscernibles (by induction hypothesis) hence
= L[α,] (u1 , . . . , un , ℵ), for some 1 < · · · < n and we are done.
(Lemma 8.3)
Definition 8.4. For an ordinal, let ( , α)+ = first element of I α > , ( + )α = first cardinal in L[α] > . Lemma 8.5 (∀α(α # exists)). For all , u +1 = sup(u , α)+ = sup(u + )α α∈R
α∈R
Proof. Clearly u +1 ≥ sup(u + )α ≥ sup(u + )α α∈R
#
α∈R
≥ sup(u , α) > u , +
α∈R
so it suffices to show that supα∈R (u , α)+ is a uniform indiscernible. Given ∈ R, = supα∈R (u , α)+ = supα≥T # (u , α)+ (where ≥T is Turing reducibility). (Lemma 8.5) Hence is a sup of members of I , so ∈ I . Hence ∈ U .
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Lemma 8.6 (∀α(α # exists)). If < u+1 , then there exist 1 < · · · < n ≤ , α ∈ R and a term such that
= L[α] (u1 . . . un ). Proof. We can assume inductively that u ≤ < u+1 . Then
= L[] (u1 , . . . , un−1 , u1 , . . . , uk ), where u1 < · · · < un−1 ≤ u < u1 < · · · < uk , by Lemma 8.3. Find such that
< (u , )+ < u+1 and let 1 < · · · < k be the first k cardinals above u in L[α], where α = # , # . Then 1 , . . . , k ∈ I and (u , )+ < 1 . Hence =
L[] (u1 , . . . , un−1 , 1 , . . . , k ) and since 1 , . . . , k are definable in L[α] from u , we have for some term
= L[α] (u1 , . . . , un−1 , u ).
(Lemma 8.6)
#
Theorem 8.7 (∀α(α exists)) (Solovay). For all , cf(u +1 ) = cf(u2 ). Proof. Define f : u2 → u +1 by f( L[α] (u1 )) = L[α] (u ). By indiscernibility, f is well defined and order preserving. Since supα∈R (u , α)+ = u +1 , f is cofinal. (Theorem 8.7) Theorem 8.8 (AD) (Martin [Mar71B]). For all 2 ≤ n < , cf(n ) = 2 . Proof. u = , hence n = ukn +1 for some kn .
(Theorem 8.8)
Theorem 8.9 (AD) (Kunen, Solovay). For all 1 ≤ n ≤ , un = n . Proof. Let be the closed unbounded measure on 1 . Suppose that for all n, we can prove 1 un / ∼ = un+1 . If for some k, Card(uk ) = Card(uk+1 ), then Card(uk+2 ) = Card((1uk+1 /)) = Card((1uk /)) = Card(uk+1 ) and similarly Card(uk+1 ) = Card(uk+n ) for all n. Hence u < , a contradiction. Hence it is enough to show that ∀n(1un / ∼ = un+1 ). ˜ def ˜ Lemma 8.10. Let L = L[α]. Then for all n, 1un ⊆ L. α∈R
˜ If for some α, is Proof. We prove by induction on < u that 1 ⊆ L. not a cardinal in L[α], then this is obvious by induction hypothesis. If is a cardinal in all L[α]’s, then it is a uniform indiscernible, hence has cofinality = cf(u2 ), so it is enough to show that u2 = 2 .
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ALEXANDER S. KECHRIS
To see that u2 = 2 : Clearly u2 ≤ 2 . Suppose u2 < 2 . Let A ⊆ 1 × 1 be a well ordering of 1 with order type u2 . Then for some α ∈ R, A ∈ L[α]. Hence the order type of A is < (1+ )α < u2 , contradiction. Hence u2 = 2 . (Lemma 8.10) 1 Now to complete the proof of Theorem 8.9, let f ∈ un , n ≥ 1. Then there is a real such that for some term and -almost all , f( ) = L[] ( ; u1 , . . . , un−1 ). ˜ we have by Lemma 8.6 that for some α ∈ R and for some Indeed, since f ∈ L term f( ) = L[α] ( ; u1 , . . . , un−1 , un ) < un , so for all ∈ I α ∩ 1 , f( ) = L[α] ( ; u1 , . . . , un−1 , (un−1 , α)+ ) = L[] ( ; u1 , . . . , un−1 ) for some term and = α # . Let [f] be the equivalence class of f in 1un / and define j([f] ) = L[] (u1 ; u2 , . . . , un ) < un+1 . Clearly j is well defined and order preserving by indiscernibility. Let < un+1 . Find , α such that = L[α] (u1 , . . . , un ). Now define f ∈ 1un by f( ) = L[α] ( ; u1 , . . . , un−1 ). Then j([f] ) = , so j is onto. Hence j : 1un / ∼ = un+1 . To summarize: From AD,
(Theorem 8.9)
11 = 1 = u1 , measurable 12 = 2 = u2 , measurable For n ≥ 3, n = un singular, cofinality = 2 , 13 = +1 , measurable 14 = +2 , measurable. Proposition 8.11 (AD). For n ≥ 3, 1n = u 1n . Proof. If 1n < u 1n , then since 1n is a cardinal, 1n = u for some < 1n . Now cannot be a successor (otherwise cf( 1n ) = 2 ). Hence is limit, hence 1n is singular, contradiction. (Proposition 8.11) Basic Open Problem 8.12. Compute 15 .
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§9. Back to the real world. For a moment we interrupt the development of the theory of projective ordinals in the context of ZF+DC+AD, to see what is the picture of these ordinals in a context with Choice and Projective Determinacy only. Theorem 9.1. 0) 11 = 1 . 1) 12 ≤ 2 . (Martin [Mar71B]) # (∀α(α exists)) 13 = u2 . (Martin [Mar71B]) 2) (AC + ∀α(α # exists)) 13 ≤ 3 . (Martin [Mar71B]) 1 3) (AC+PD) 4 ≤ 4 . (Kunen [Kun71C], Martin [Mar71B]) (Kunen [Kun71C], Martin [Mar71B]) 4) (PD) 12n+2≤ ( 12n+1 )+ . all n, 1 < 1 . 5) (PD) For (Kechris [Kec74], Moschovakis [Mos70]) n n+1 Proof. 1) That 12 ≤ 2 follows from the Kunen-Martin theorem and the is -Suslin. That 1 ≤ u follows from the fact that if fact that every Σ12 set 1 2 2 is a tree on × . By the proof A ∈ Σ12 (α), then A = p[T ], where T ∈ L[α] 1 of the Kunen-Martin theorem the length of a Σ12 (α) wellfounded relation is < (1+ )α < u2 , hence 12 ≤ u2 . To see that u2 ≤ 12 : For every α ∈ R and for every < (1+ )α we can find a term such that
= L[α] (i α . . . i α , 1 , ℵ), 1
n
for some 1 < · · · < n < 1 < ℵ. Coding the 1 , . . . , n by reals we can easily find a Π11 (α # ) prewellordering of reals of length > (1+ )α , so u2 = supα (1+ )α ≤ 12 . : We know that every Σ1 set is u -Suslin by the Martin2) To prove 13 ≤ 3 3 ≤ 2, cf(u ) = cf(u ), we must Hence 1 ≤ (u )+ . Since ∀n Solovay theorem. n 2 3 hence u < 3 . Hence 13 ≤ 3 . have un < 3 . But 3 is regular, 3,4) These follow from the fact that every Σ12n+2 set is 12n+1 -Suslin. a Π1 5) To prove 12n+1 < 12n+2 use the fact that 2n+1 norm on a complete 1 . Π12n+1 set has length exactly 2n+1 To prove that 1 < 1 ,prove first that 2n 2n+1 1 2n+1 = sup{ : is the length of a Σ12n+1 wellfounded relation} using the recursion theorem (see Moschovakis [Mos70]). Then let W ⊆ R × R × R be universal Σ12n , and let (α, ) ≺ (α , ) ⇔ α = α & Wα is wellfounded & W (α, , ). Then ≺ is Δ12n+1 and dominates every Σ12n wellfounded relation. Hence 12n < 9.1) 12n+1 . (Theorem
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ALEXANDER S. KECHRIS
´ Theorem 9.2 (Sierpinski for κ = ). If A is κ-Suslin, then A is the union of κ + sets in Bκ+ . Proof. Let α ∈ A ⇔ T (α) not wellfounded, T a tree on × κ. For α ∈ A let (α) = sup{|T (α)u | : u ∈ <κ & T (α)u is wellfounded} < κ + . For < κ + , let B = {α ∈ A : (α) ≤ }. Since A = <κ+ B , it is enough to show each B ∈ Bκ+ . For < κ + , α ∈ B ⇔ T (α) is not wellfounded & ∀u[T (α)u wellfounded ⇒ |T (α)u | ≤ ], therefore α ∈ B ⇔ T (α) is wellfounded ∨ ∃u[T (α)u is wellfounded & |T (α)u | > ] ⇔ |T (α)| < + 1 ∨ ∃u[|T (α)u | = + 1]. Since by the proof of Theorem 3.18 A u = {α : |T (α)u | < } ∈ Bκ+ , we are done.
(Theorem 9.2)
Theorem 9.3. ´ 1) (Sierpinski) Every Σ12 set is the union of ℵ1 Borel sets. (Martin [Mar71B]). Every Σ1 set is the union of ℵ # 2) (AC+∀α(α exists)) 2 3 Borel sets. 3) (AC+PD) (Martin [Mar71B]). Every Σ14 set is the union of ℵ3 Borel sets. Proof. 1) Every Π11 set is the union of ℵ1 Borel sets. Hence every Σ12 set is So it suffices to show that the Σ1 sets are unions of ℵ the union of ℵ1 Σ11 sets. 1 1 Borel sets. This follows from 9.2. (This proof uses AC. This can be avoided by using the uniformization theorem for Π11 sets.) 2) Every Σ13 set is u -Suslin. Since cf(un) ≤ ℵ2 for n < , we have u < ℵ3 . ℵ -Suslin. If A is Σ1 then for some tree T on × ℵ So Σ13 sets are 2 2 3 a ∈ A ⇔ T (α) not wellfounded ⇔ ∃ < ℵ2 (T (α) not wellfounded) (see 3.19). So A is the union of ℵ2 many ℵ1 -Suslin sets. By the same argument, each ℵ1 -Suslin is the union of ℵ1 many -Suslin (i.e., Σ11 ) sets and we are done. 3) Similar, using the fact that every Σ14 set is 13 -Suslin, and the fact that 13 ≤ ℵ3 . (Theorem 9.3) Basic Open Problem 9.4. Is it true that (from any reasonable hypotheses and AC): 1n ≤ ℵn , for n ≥ 5?
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§10. Infinite exponent partition relations and the singular measures . Definition 10.1. If α, , are ordinals with ≤ ≤ α, we put α → () iff for every X ⊆ [α] = {f ∈ α : f increasing} there is an H ⊆ α of order type such that either [H ] ⊆ X or [H ] ⊆ ¬X . Remark 10.2. ZFC * ¬∃κ(κ → () ). Definition 10.3. Let κ be a regular cardinal, and let be a regular cardinal < κ. The filter is the collection of all subsets of κ which contain a -closed unbounded set. (A ⊆ κ is -closed if every increasing -sequence from A has its limit in A.) Theorem 10.4 (Kleinberg [Kle70]). 1) If κ is a regular uncountable cardinal, < κ a regular cardinal, and κ → (κ) + , then is a normal measure. 2) If κ is a regular uncountable cardinal with < κ many regular cardinals below κ, and ∀ < κ(κ → (κ) ), then the normal measures on κ are exactly the for regular < κ. Proof of 2 from 1. By 1) we know each that is a normal measure. Let be another normal measure. For regular < κ let E = { < κ : cf( ) = }. Then the E ’s are pairwise disjoint, and E = { < κ : limit ordinal}. regular <κ
Since there are < κ regular cardinals below κ we can find a regular 0 < κ such that E 0 ∈ . Suppose = 0 . Then we can find a 0 -closed unbounded A such that B = κ \ A ∈ . For ∈ B ∩ E 0 , let g( ) = sup(A ∩ ). Then g( ) < for all ∈ B ∩E 0 , hence g is -a.e. constant, which contradicts (2 from 1) the unboundedness of A. So = 0 .
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ALEXANDER S. KECHRIS
Proof of 1. Assume κ → (κ) + . To show that is a normal measure, let f : κ → κ be pressing down. Consider X ⊆ + [κ] given by G ∈ X ⇔ f sup G(α) = f sup G( + α)] . α<
α<
Let H ⊆ κ be homogeneous for this partition, with Card(H ) = κ. Suppose + [H ] ⊆ ¬X . Let C be the set of limits of increasing sequences from H . Then C is -closed unbounded and for , ∈ C ,
< ⇒ f( ) = f(), i.e., f is 1-1 on C . Now we inductively define an increasing -sequence { }< of elements from C as follows: 0 = least member of C = least element of C greater than all for < which satisfies: ∀( > and ∈ C ⇒ ∀ < [f() > ]. Then if = lim< ∈ C we have f() < , hence for some < , f() < , hence ≤ +1 , a contradiction. Hence + [H ] ⊆ X . Then if < are both in C , we can find G ∈ + [H ] such that
= sup(G(α)), = sup G( + α) α<
α<
and hence f( ) = f(). So f is constant on C , proving normality for . To see now that is an ultrafilter, look at the characteristic functions of subsets of κ (which are of course pressing down). To see that is κ-additive, let {A }<<κ ⊆ and suppose < A = ∅, towards a contradiction. Then κ = < (κ \ A ), so consider least < such that ∈ A , if ≥ f( ) = 0 otherwise Then for some < , { : f( ) = } contains a -closed unbounded set, (1) i.e., (κ \ A ) = 1, a contradiction. §11. Countable exponent partition relations for 1n , n odd. We present first in an abstract form Martin’s method for proving infinite exponent partition relations from AD. It is a modification of Solovay’s technique used in the proof of Theorem 5.1. Lemma 11.1 (Martin). Let κ > be a regular cardinal, ≤ κ an ordinal. Assume:
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1) There is {C } <· , with C ⊆ R, and for each < · a map ε → f (ε) from C into κ such that if C = <· C and for ε ∈ C we let fε ( ) = f (ε) then ε → fε maps C onto · κ. 2) There are {C , } <· ,<κ such that C , ⊆ C
≤
and if : R → R is continuous and [ ≤ C ] ⊆ C then for all < · and < κ we have def
G ( , ) = sup{f ( (ε)) + 1 : ε ∈ C , } < κ. 3) If f ∈ · [κ] then there is ε ∈ C such that fε = f and ε ∈ C ,f( ) , ∀ < · . Then: AD+DC ⇒ κ → (κ) . Proof. Let A ⊆ [κ] and consider the following game: I εI
II ε II
Player II wins iff (1) ∃ < · (ε I ∈ C ∨ ε II ∈ C ) and if 0 is the least such then ε I ∈ C 0 , or (2) ∀ < · (ε I ∈ C ∧ ε II ∈ C ) and < supn {fε I ( · + n), fε II ( · + n)} >< ∈ A. Without loss of generality we can assume that player II has a winning strategy . Then by 1) [ ≤ C ] ⊆ C , ∀ < · . So by 2) above G ( , ) < κ. By the regularity of κ, let D ⊆ κ be closed unbounded such that ∈ D ⇒ ∀ < · ∀ < κ[ < ∧ < ⇒ G ( , ) < ]. Let Df = {g ∈ [D] : ∃f ∈ · [κ]∀ < (g() = sup f( · + n)}. n
We claim that Df ⊆ A, which completes the proof since then [H ] ⊆ A, where H = { + : < κ}, where { } <κ is the increasing enumeration of D. Let g ∈ Df and let f ∈ · [κ] be such that g() = supn f( · + n). Then by 3) above find ε ∈ C such that fε = f and ε ∈ C ,f( ) . Then for all < and all n ∈ :
f ·+n ( (ε)) < G ( · + n, f( · + n)) < g(), since · + n ≤ f( · + n) < g() ∈ D.
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ALEXANDER S. KECHRIS
So for all < sup{fε ( · + n), f (ε) ( · + n)} = g(), n
therefore g ∈ A.
(Lemma 11.1)
Theorem 11.2 (AD) (Martin [Mar71A]). For any n ≥ 0, 12n+1 → ( 12n+1 ) , ∀ < 1 . Proof. Fix t : · ↔ . For a real α, set α = (α)i , where t( ) = i. Let also W be a complete Π12n+1 set, ϕ a Π12n+1 -norm on W with range 12n+1 and for α ∈ W , write |α| =ϕ(α). Define now for < · , C = {α : α ∈ W } and for α ∈ C , f (α) = |α |. Finally let for < · , < 12n+1 : C , = {α : ∀ ≤ ∃ ≤ (α ∈ W ∧ |α | ≤ )}. Now obviously properties 1), 3) of Lemma 11.1 are satisfied so it is enough to verify 2). For that notice that C , ∈ Δ12n+1 , so that if is continuous then
[C , ] is Σ12n+1 . If also [C , ] ⊆ C , then {α : α ∈ [C , ]} is a Σ12n+1 , so by boundedness subset of W G ( , ) = sup{|α | + 1 : α ∈ [C , ]} < 12n+1 and we are done. (Theorem 11.2) §12. 1 → (1 )1 . Definition 12.1. For C ⊆ κ, put C f = C fκ = {f ∈ κ[C ] : ∃g ∈ κ[κ]∀ , f( ) = sup g( · + n)}. n
It is not hard to check that κ → (κ)κ ⇔ ∀X ⊆ κ[κ]∃C (C is closed unbounded on κ and C f ⊆ X or C f ⊆ ¬X ). Theorem 12.2 (AD) (Martin; see Martin-Paris [MP71]). 1 → (1 )1
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Proof. We will apply Lemma 11.1 again. Let 0 , 1 , 2 , . . . be a recursive enumeration of terms in the language of ZF + V=L[α], ˙ which take only ordinal values, as in the proof of Theorem 6.3. For < 1 put, using again the notation of 6.3: C = {ε : ε = nε = n, ε (0), ε (1), . . . & P(ε ) & the wellfounded part of Γ( + , ε ) has an ordinal of order type (denoted also by ) & nΓ( +,ε ) ( ) also belongs to the wellfounded part}. Then ε ∈ C ⇔ ∀ < 1 (ε ∈ C ) ⇔ ε = nα # , for some n, α. For ε ∈ C , let also
f (ε) = nΓ( +,ε ) ( ). Then if ε ∈ C , say ε = nα # , we have for all < 1 : fε ( ) = nL[α] ( ). Finally put for < 1 , < 1 , C , = {ε : ∀ ≤ ∃ ≤ (ε ∈ C ∧ if ε = nε , then nΓ(
+,ε )
( ) ≤ )}.
Clearly conditions 1), 3) or Lemma 11.1 are satisfied. To verify also condition 2) note first that each C , is Borel. So if is continuous [C , ] is Σ11 . If moreover [C , ] ⊆ C , then an easy boundedness argument shows that
G ( , ) < 1 and we are done. (Theorem 12.2) §13. The Martin-Paris theorem. Definition 13.1. Let κ be an uncountable cardinal, a normal measure on κ, and assume κκ/ ∼ = κ + . For each f ∈ κκ, let f(κ) = [f] (thus + κ κ = {f(κ) : f ∈ κ}). A as above is canonical if it has the following selection property: If < κ + , and { } < is a -sequence of ordinals < κ + , then there is a sequence {f } < ⊆ κκ such that f (κ) = . Note that if such a measure exists, then κ + is regular. Theorem 13.2 (AD) (Solovay). The measure on 1 is canonical. Proof. If f ∈ 11 , we can find , α such that ∀ < 1 , f() = L[α] (). It is easy to check that f(1 ) = L[α] (1 ). So in particular 1
1 / ∼ = 2 .
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ALEXANDER S. KECHRIS
Now let { } < be a sequence of ordinals less than 2 . Without loss of generality = 1 , and { : < 1 } = < 2 . Define the following prewellordering on 1 : ⇒ ≤ . By the proof of Solovay’s Theorem 7.2, find , α such that
= L[α] (1 ). For some term then,
= L[α] ( , 1 ), ∀ < 1 . (Theorem 13.2)
Take f () = L[α] ( , ).
Notation. Let κ be an uncountable cardinal carrying a canonical measure ¯ be such that . Let κ ≤ < κ + , and fix h : κ ↔ . For any < κ, let , : ¯ ↔ h[] = {h( ) : < }, with order preserving. Then consider the normal (i.e., increasing and continuous) function : κ → κ such that +1 − = ¯ (where = ())). For ∈ h[] let , = + −1 ( ). ¯
,
+1
−1 h[] Then < ∈ h[] ⇒ , < , . Thus to each < we can assign an increasing : κ → κ, (actually is defined from a point on), where () = , .
Then < ⇒ () < (). Now let f ∈ κ[κ]. For < let f [ ] () = f( ()).
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331
Thus f [ ] ∈ κ[κ]. Now let f ˜ ∈ [κ + ] be defined by f ˜ ( ) = f [ ] (κ). (Recall that for a function g ∈ κκ, g(κ) denotes the image of g in the ultrapower κκ/.) Since < ⇒ for all from a point on, () < (), we see that < ⇒ { : f [ ] () < f [ ] ()} has measure 1, so f is indeed increasing. For A ⊆ κ, let (κ[A])˜ = {f ˜ : f ∈ κ[A]} and let A∗ = {f(κ) : f ∈ κA}. Note that if A is unbounded in κ, then A∗ is unbounded in κ + . (If f ∈ κκ, define g ∈ κA by g( ) = least member of A > f( ); then g(κ) < f(κ)). Theorem 13.3 (AD) (Martin-Paris [MP71]). Let κ be an uncountable cardinal, a canonical measure on κ, κ ≤ < κ + , A ⊆ κ unbounded. Then ∗ A \ (κ + 1) ⊆ (κ[A])˜ . Corollary 13.4 (AD) (Martin-Paris [MP71]). Let κ be an uncountable cardinal carrying a canonical measure . If κ → (κ)κ , then ∀ < κ + , κ + → (κ + ) . Hence for any regular < κ + , is a normal measure on κ + . Corollary 13.5 (AD) (Martin-Paris [MP71]). 1) ∀ < 2 , 2 → (2 ) . 2) 2 has exactly two normal measures, namely , 1 . Proof of Corollary 13.4. Let X ⊆ [κ + ], κ ≤ < κ + . Put ˜X = {f ∈ [κ] : f ˜ ∈ X }. Let H ⊆ κ have cardinality κ such that, say, κ[H ] ⊆ ˜X . Then by Theorem 13.3
κ
[H ∗ \ (κ + 1)] ⊆ (κ[H ])˜ ⊆ (˜X )˜ ⊆ X.
(Corollary 13.4) Proof of Theorem 13.3. Let f ∈ [A \ (κ + 1)]. Then find {f } < ⊆ κκ such that
∗
f (κ) = f( ). We want to find G ∈ κ[A] such that ∀ < , G ˜ ( ) = f (κ),
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ALEXANDER S. KECHRIS
i.e., ∀ < , G [ ] (κ) = f (κ), i.e., ∀ < , G(, ) = f () for -almost all . For that it is enough to have for -almost all , ∀ ∈ h[], G(, ) = f (). To prove this we need the following: Lemma 13.6. There is a set C of -measure 1 such that if ∈ C : i) ∀ ∈ h[](f () ∈ A), and ii) ∀ < ∈ h[](f () < f ()). Proof of Lemma 13.6. For fixed , , let C , = { : fh( ) () ∈ A and h( ) < h() ⇒ fh( ) () < fh() ()}. Then each C , ∈ (since for all < , f (κ) ∈ A∗ ⇒ f is -equivalent to some element of κA, hence { : f () ∈ A} ∈ ). Now let C = { : ∀ , < ( ∈ C , )}. Then C has -measure 1 and has the required properties. So the proof of Lemma 13.6 is complete. (Lemma 13.6) To finish the proof of Theorem 13.3: Let 0 be large enough so that 0 ∈ h[0 ]. Now define ⎧ f () if ∈ C \ 0 , ∈ h[] and ⎪ ⎪ ⎨ f0 () > sup< G( ) G(, ) = ⎪ least member of A, greater otherwise. ⎪ ⎩ than all G() for < , Claim 13.7. For -almost all , the first definition occurs. Proof. Since is normal, for -almost all , = = sup< G(). Since f0 (κ) > κ, for -almost all , f0 () > = sup< G(), which proves the claim. (Claim 13.7) κ By the properties of C given in Lemma 13.6, G ∈ [A] and is as desired. (Theorem 13.3) Theorem 13.8 (AD) (Martin-Paris [MP71]). Let κ be an uncountable cardinal, a normal measure on κ, κ2 ∼ = κκ/. For each < κ + there is a map κ ˜ f → f sending [κ] into [κ2 ], such that if A ⊆ κ, F ∈ [A∗ \ (κ + 1)], and ∃G ∈ (κκ) such that [G( )] = F ( ), then there is an f ∈ [A] with f ˜ = F . κ
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333
Corollary 13.9. In the notation of Theorem 13.8, if κ → (κ)κ , then κ2 → (κ2 ) , ∀ < 1 . Proof of Theorem 13.8. For ≥ κ see the proof of Theorem 13.3. Assume < κ. Define a normal function on κ by +1 − = . Let , = + for < . For f ∈ κκ let f [ ] () = f(, ), and let f ˜ = {f [ ] } < . Now repeat the proof of Theorem 13.3. (Theorem 13.8) For the corollary just notice that if A ⊆ κ has cardinality κ, then A∗ \(κ +1) has order type κ2 . §14. The measure on 1n , n odd. Definition 14.1. Let κ be a cardinal. If W is a wellordering of (a subset of) κ, let for < κ HW ( ) = |W |, where W = W ∩ ( × ). Clearly HW : κ → κ. Theorem 14.2. Let κ be an uncountable cardinal, a normal measure on κ. Then for any wellordering W on κ, [HW ] = |W |. In particular, {[HW ] : W a wellordering on κ} = κ + . Proof. Notice first that {[HW ] : W a wellordering on κ} is an initial segment of ordinals. Because if F : κ → κ is such that [F ] < [HW ] , then for -almost all , F ( ) < |W |. Hence there is a map → ∗ < such that ∗ for -almost all , F ( ) = |W | (where for any wellordering W , W x = initial segment of W determined by x). So by normality, find 0 such that F ( ) = |W 0 | -almost everywhere. Hence [F ] = [HW 0 ] . To prove the theorem, it is enough to show that for wellorderings W , V on κ, [HW ] < [HV ] ⇔ |W | < |V |. ∗
If HW ( ) < HV ( ) -a.e., let F : W → V be an isomorphism for almost all , with ∗ < . Then by normality, there is 0 such that F : W → V 0 is an isomorphism -almost everywhere. For each ∈ Field(W ), let f ( ) = F () < , so by normality f ( ) = g() for -a.e. (i.e., g() is the constant value assumed -a.e. by f ). Then g : W → V 0 is an embedding, so |W | < |V |. Similarly HW ( ) ≤ HV ( ) -a.e. ⇒ |W | ≤ |V |. (Theorem 14.2)
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ALEXANDER S. KECHRIS
Corollary 14.3. Let κ be an uncountable cardinal, a normal measure on κ. Then the following are equivalent: 1) is canonical. 2) κκ/ ∼ = κ + and κ + is regular. 3) Every F : κ → κ is -a.e. equal to some HW , W a wellordering on κ, and κ + is regular. Proof. 1) ⇒ 2) by definition. 2) ⇔ 3) by Theorem 14.2. 2) ⇒ 1) Enough to show the selection property, and since κ + is regular, it is enough to show that if < κ + then we can find {f } < , where f : κ → κ is such that [f ] = . Pick a wellordering W of κ with |W | = . For each < let ∗ < κ be such that ∗
|W | = . (Corollary 14.3)
Then let f = HW ∗ .
Theorem 14.4 (AD) (Kunen [Kun71D]). For each n ≥ 0, the measure on 12n+1 is canonical. Proof. We need some lemmas first. Lemma 14.5. There is a Π12n+1 set G ⊆ R and a 12n+1 -scale {ϕm }m∈ on G such that if we put (α) =supm ϕm (α), then 1) ϕm (α) < (α), ∀α ∈ G, ∀m ∈ . 2) {(α) : α ∈ G} ∈ . 3) If A ⊆ G is Σ12n+1 , then supα∈A (α) < 12n+1 . Proof. Let W be a Σ12n+1 -complete set of reals, {m }m∈ a Π12n+1 -scale on W, W , where the range ofeach m is included in 12n+1 . Put for α ∈ ¯m (α) = 0 (α) + 1 (α) + · · · + m (α). Then {¯m }m∈ is a 12n+1 -scale on W and for all m, α ∈ W , ¯m (α) < supm ¯m (α) (we can clearly assume here without loss of generality that always m (α) > 0). Consider now G = {α : ∀i, (α)i ∈ W } and for α ∈ G, m ∈ define ϕm (α) = ¯(m)0 ((α)(m)1 ), where m → ((m)0 , (m)1 ) is a 1-1 correspondence between and ×. Clearly {ϕm }m∈ is a 12n+1 -scale on G and if (α) = supm ϕm (α) then properties 1), 3) are satisfied. (Lemma 14.5) Claim 14.6. {(α) : α ∈ G} is -closed unbounded.
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Proof. Clearly it is unbounded. Let (α 0 ) < (α 1 ) < · · · → where α 0 , α 1 , · · · ∈ G. Let α ∈ G be such that (α)i = (α (i)0 )(i)1 . Then ϕm (α) = ¯(m)0 (α(m)1 ) = ¯(n)0 ((α (m)1,0 )(m)1,1 ) < (α (m)1,0 ) < , where (m)i,j = ((m)i )j . If < , find j large enough and κ such that ϕκ (α j ) > . Then if m is such that (m)0 = (k)0 , (m)1,0 = j, (m)1,1 = (k)1 , we have ϕm (α) = ¯(k)0 ((α j )(k)1 ) > . Hence (α) = .
(Claim 14.6)
Lemma 14.7. There is a tree U on × κ2n+1 such that sup{|U (α)| : U (α) is wellfounded} = 12n+1 .
α∈R
Proof. Let S be a Σ12n+1 -complete set of reals and let U be a tree on × κ2n+1 such that p[U] = S. (Lemma 14.7) To prove now the theorem: Let F : 12n+1 → 12n+1 be given, and consider , and player II wins iff the following game: Player I plays α, player II plays α ∈ G ⇒ U () is wellfounded and |U ()| > F ((α)). If player I has a winning strategy, then by Lemma 14.5, 3) and Lemma 14.7 we get a contradiction. So assume 0 is a winning strategy for player II. Let T be the tree on × 12n+1 coming from the scale {ϕn }n∈ on G (thus G = p[T ]). Then for all α T (α) not wellfounded ⇒ U ( 0 [α]) is wellfounded and F ((α)) < |U ( 0 [α])|.
(∗)
Let [α] = ⇔ ∀n( n, αn, n) ∈ S, S a tree on × × . Let R be the tree on × × × 12n+1 × κ2n+1 defined by (s, a, b, u, v) ∈ R ⇔ (s, a, b) ∈ S & (b, v) ∈ U & (a, u) ∈ T. Then R( 0 ) is wellfounded by (∗). Suppose α ∈ G & (α) = > κ2n+1 . Let f ∈ be defined by f(n) = ϕn (α). Thus if = 0 [α], for any v ∈ U () we have (α lh(v), lh(v), f lh(v), v) ∈ R( 0 )
= {(a, b, u, v) ∈ R( 0 ) : u ∈ < }
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where lh(α0 , . . . , am−1 ) = m. Hence F ( ) < |U (α0 [α])| ≤ |R( 0 ) |. If we let W ( 0 ) be the Brouwer-Kleene wellordering of R( 0 ), viewed as a wellordering of 12n+1 (after identifying < × < × < 12n+1 × <κ2n+1 with have by the above 12n+1 ), then we F ( ) < HW ( 0 ) ( ) -a.e. Hence F ( ) = HW ( 0 ) 0 ( ) -a.e., for some 0 . So is canonical. (Theorem 14.4) §15. The measures , with > , on 1n , n odd. Lemma 15.1 (AD). There is a relation W ⊆ R × 12n+1 × 12n+1 , with the following properties: 1) If Wε ( , ) ⇔ W (ε, , ), then for every F : 12n+1 → 12n+1 there is ε0 ∈ R and 0 < 12n+1 such that Wε0 is a wellordering and F ( ) = |(Wε0 ) 0 | -a.e. In particular, sup{|Wε0 | : Wε0 is a wellordering } = 12n+2 . 2) If W , = {ε : W (ε, , )} then for each , , W , is an open set of reals. In particular for each 0 , , < 2n+1 , {ε : |(Wε ) 0 | < } ∈ Δ12n+1 . ϕ: P 3) Let P ⊆ R be a Π12n+1 -completeset and ϕ a Π12n+1 -norm on it, 1 1 1 2n+1 . Then there are Π2n+1 , Σ2n+1 relations Q, S resp. such that α, ∈ P ⇒ [W (ε, ϕ(α), ϕ()) ⇔ Q(ε, α, ) ⇔ S(ε, α, )]. Proof. By the proof of Theorem 14.4 there is a tree T on × 12n+1 such that if C : 12n+1 ↔ < 12n+1 and we put W (ε, , ) ⇔ C ( ), C () ∈ T (ε) & C ( )
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Then for α, ∈ P, Q(ε, α, ) ⇔ ∃n[(εn, C (ϕ(α))), (εn, C (ϕ()) ∈ T ∧ C (ϕ(α))
and similarly for S.
Lemma 15.2 (AD). Let n ≥ 0 be given and assume > is a regular cardinal < 12n+1 . Then there is a function F : 12n+1 → 12n+1 such that for every wellordering U on 12n+1 there is a wellordering V on 12n+1 with |V | > |U | ∩C . In particular if and a cub set C ⊆ 12n+1 such that HV ( ) < F ( ), ∀ ∈ E is a normal measure, HU < F -a.e. (Recall that E = { : cf( ) = }.) Proof. Let W be as in Lemma 15.1. Put for ∈ E F ( ) = sup{|Wε | + 1 : ε is such that ∀ < (|Wε | < )} and let F ( ) = 0 if ∈ E . If now ∈ E i.e., cf( ) = > then ∀ < (Wε is wellfounded ⇒ Wε is wellfounded), so F ( ) < 12n+1 , by boundedness and Lemma 15.1. Now given a wellordering that |U | < |Wε0 | and then find a closed unbounded C such U find ε0 such that
∈ C ⇒ ∀ < (|Wε0 | < ). Then ∈ E ∩ C ⇒ F ( ) > |Wε0 | and we are done.
(Lemma 15.2)
Theorem 15.3 (AD) (Kunen [Kun71D]). If , > , is a normal measure 1 on 12n+1 then 2n+1 12n+1 / > ( 12n+1 )+ = 12n+2 . Proof. If F is as in Lemma 15.2, then by Theorem 14.2 [F ] ≥ 12n+2 . 15.3) (Theorem From Kunen’s result (see Solovay [Sol78A]) that 13 → ( 13 ) , ∀ < 13 , it regular < < 1, follows that the conclusion of Theorem 15.3 holds for 3 1 i.e., for = 1 , 2 . It also holds for = 1 for any 2n+1 , n > 0, by the remarks following 16.1. §16. Countable exponent partition relations on 1n , n even. Theorem 16.1 (AD) (Kunen [Kun71D]). For all n ≥ 1, 12n → ( 12n ) , ∀ < 1 .
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ALEXANDER S. KECHRIS
Proof. Fix < 1 and a map t : · ↔ . Let P ⊆ R be Π12n+1 -complete, ϕ a Π12n+1 -norm on P with range 12n+1 and put |α| = ϕ(α). For any real α < · let α = (α) , where t( ) = i. and any i
1 ]. Then consider the game Fix X ⊆ [2n+1 I εI, αI
II ε II , α II
Player II wins iff 1) ∃ < · such that α I or α II ∈ P and for the least such , say 0 , α I 0 ∈ P, or I 2) ∀ < · (α I , α II ∈ P) and for some , ∈ 12n+1 × ( · ), (Wε I )|α | II or (Wε II )|α | is not well ordered and for the lexicographically least such 0 , 0 , (Wε I )|α 0 | 0 is not wellordered, or I
0
3) Both 1) and 2) fail and letting FI, () = |(Wε I )|α | | and similarly for player II, we have supn {[FI,·ϑ+n ], [FII,·+n ]}< ∈ X . I
Here W is as in Lemma 15.1 and [F ] = [F ] . Assume without loss of generality that player II has a winning strategy . Put (ε II , α II ) = [ε I , α I ]. Let then for < · ; , , < 12n+1 : II Θ( , , , ) = sup{|(W(ε II ) )|(α ) | | + 1 : ε I , α I are such that ∀ ¯ < · (α I¯ ∈ P ∧ |α I¯| < )
and for all , ≤lex , |(Wε I )|α | | < }. I
By Lemma 15.1, Θ( ; , , i) < 12n+1 so we can find a cub C such that ∈ C ⇒ ∀ < · ∀, , < (Θ( , , , ) < ). Put H = (C ∗ ∩ E ) \ ( 12n+1 ), where C ∗ = image of C under the embedding generated by the ultrapower relative to . We will show that if f ∈ H f then f ∈ X . Fix such an f and then find g ∈ · [ 12n+2 ] such that lim g( · + n) = f(), ∀ < .n < Then find ε I , α I such that Wε I is wellordered and α I ∈ P for all < · and such that [FI,·+n ] = g( · + n). Put ε II = ε II , α II = α II . Then α II ∈ P for
AD AND PROJECTIVE ORDINALS |α II | ε II
all < · and (W ) show that
339
is wellordered for all < · , so it is enough to
[FII,·+n ] < f(), ∀ < , ∀n ∈ . 12n+1
Pick F ∈
C such that [F ] = f(). Then we have to show that FII,·+n () < F () -a.e.
Let I ∈ be such that ∈ I ⇒ 1) F () > 2) F () > > sup{|α I | : < · } (for some ) 3) FI, ( ) < , ∀ < · , ∀ < 4) FI,·+n+1 () < F () 5) FI, () < FI, (), ∀ < < · . Then we claim that for ∈ I , II )|α+n | | < F (), FII,·+n () = |(Wε+n II
which completes the proof. Since F () ∈ C we have ∀ 1 < · ∀1 , 1 , 1 < F ()(Θ( 1 , 1 , 1 , 1 ) < F ()), so since < F (), < F () we only have to show that for some < F () and all , ≤lex , · + n we have FI, ( ) < . Take = max{FI,·+n+1 (), } < F (). Let , ≤lex , · + n. Then we have: Case 1. < : Then FI, ( ) < ≤ . Case 2. = and ≤ ·+n : Then FI, () < FI,·+n+1 () ≤ < F () and we are done. (Theorem 16.1) 1 1 Kechris has recently shown that for all n ≥ 2, n → ( n ) , ∀ < 2 . Thus 1 is a normal measure for all 1n , n ≥ 2. §17. The measure on 1n , n even. Theorem 17.1 (AD) (Kunen [Kun71D]). For all n ≥ 0, is a normal measure on 12n+2 and is generated by the sets of the form (C ∗ ∩ E ) \ 12n+1 , the where C ⊆ 12n+1 is closed unbounded and C ∗ is the image of C under embedding generated by the ultrapower relative to on 12n+1 . Proof. By Theorem 10.4 and §16, is a normal measure on 12n+1 . To down prove the extra statement we show that if f : 12n+2 → 12n+2 is pressing there is a set as above on which f is constant. For that consider the partition of +[ 12n+2 ] as in Theorem 10.4. Then by the proof of Theorem 16.1 there
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ALEXANDER S. KECHRIS
is a closed unbounded C ⊆ 12n+1 such that if p ∈ +[C ∗ ∩ E ] and p() > , then limn< p(n) and p(0) > 12n+1 f sup(p(n)) = f sup(p( + n)) . n
n
Let D ⊆ C be the set of all limit points of C and E ⊆ D the set of all limit points of D. Then both D, E are cub and E ∗ ∩ E ⊆ D ∗ ∩ E ⊆ C ∗ ∩ E , while every point of E ∗ is a limit point of D ∗ , which in turn is a limit point of C ∗ . So if < are in (E ∗ ∩ E ) \ 12n+1 , find 0 < 1 < · · · → < 0 < 1 < · · · → i , i ∈ D ∗ \ 12n+1 . Put p(n) = the th element of C ∗ above n and p(+n) = of C ∗ above . Then p(n) → , since ≤ p(n) ≤ , and the th element n n n+1 similarly p( + n) → . Since p ∈ +[C ∗ ∩ E ] and p() > limn p(n) = and p(0) > 12n+1 , we have f(supn (p(n))) = f() = f(supn (p( + n))) = f(), so f isconstant on (E ∗ ∩ E ) \ 12n+1 and we are done. (Theorem 17.1) Theorem 17.2 (AD) (Kunen [Kun71D]). If F : 12n+2 → 12n+2 , there is J : 12n+1 → 12n+1 such that F ( ) ≤ J ∗ ( ), for all in ( 12n+1 , 12n+2 ), where ∗ 1 1 again J : 2n+2 → 2n+2 is the image of J under the embeddinggenerated by . on 12n+1 Proof. In the notation of Theorem 16.1 consider the game I ε I ,α I
II ε II
Player II wins if 1) α I ∈ P, or I 2) α I ∈ P and either for some , (Wε I )|α | or Wε II is not a wellordering I and for the least such, say 0 , (Wε I )|α | 0 is not a wellordering or for all , I I (Wε I )|α | and Wε II are wellorderings and if fI ( ) = |(Wε I )|α | | and fII ( ) = |Wε II |, then F ([fI ]) < [fII ], where [f] = [f] . Claim 17.3. Player I does not have a winning strategy. Proof. Suppose he had one, . Then we will show that there is K : 12n+1 → 12n+1 such that if player II plays correctly so that fII is de is produced following , then [f ] < [K]. If player II then then if f fined, I I “plays fII ” such that [fII ] > sup{F () : < [K]} we immediately have a contradiction. To define K let (εI , αI ) = [ε II ]. Then let K ( ) = sup{|(WεI )|α | | + 1 : ε II is such that∀ < (|Wε II | < )}. I
Then K ( ) < 12n+1 by boundedness, since if Wε II is wellordered for all I < , then (Wε I )|α | must be wellordered by the rules of the game and the
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fact that is a winning strategy for player I. Suppose now player II “plays fII ” and find C cub in 12n+1 such that
∈ C ⇒ ∀ < (fII () < ). Then if player I produces following his strategy fI we have fI ( ) < K( ) for all ∈ C , so [fI ] < [K ] and the proof of the claim is complete. (Claim 17.3) So player II has a winning strategy . Then put for < < 12n+1 : J ( , ) = sup{|Wε II | + 1 : α I , ε I are such that |α I | < and |(Wε I )|α | | ≤ }, I
where as usual ε II = [α I , ε I ]. By boundedness again J ( , ) < 12n+1 . So put for < 12n+1 J () = sup J ( , ) < 12n+1 .
< Now we want to prove that F ( ) < J ∗ ( ), ∀ ∈ ( 12n+1 , 12n+2 ). Fix such a and find α I , ε I such that [fI ] = . Thus we have to show that F ([fI ]) < J ∗ ([fI ]). Since F ([fI ]) < [fII ] it is enough to check that [fII ] ≤ J ∗ ([fI ]), i.e., fII ( ) ≤ J (fI ( )) -a.e. Let > |α I | and fI ( ) >
(this happens -a.e.). Put = fI ( ) > . Then J (fI ( )) = J () ≥ J ( , ) > fII ( ).
(Theorem 17.2)
Theorem 17.4 (AD) (Kunen [Kun71D]). For all n ≥ 1, 2n+2 12n+2 / = ( 12n+2 )+ and cf(( 12n+2 )+ ) = 12n+2 . Proof. Let F : 12n+2 → 12n+2 . Find J : 12n+1 → 12n+1 such that for on 1 such that J ( ) =
> 12n+1 , F ( ) < J ∗ ( ). LetW be a wellordering 2n+1 HW ( ) -a.e., say J ( ) = HW ( ), ∀ ∈ I ∈ . Then J ∗ = (HW )∗ = HW ∗ on I ∗ , so F < HW ∗ on I ∗ ∩ E , thus [F ] < [HW ∗ ] = |W ∗ | < ( 12n+2 )+ . 1 So 2n+2 12n+2 / = ( 12n+2 )+ . now < 1 find f : 1 1 Given 2n+2 2n+1 → 2n+1 such that [f] = . Then ∗ 1 1 f : 2n+2 → 2n+2 . Put g( ) = [f ∗ ] < ( 12n+2 )+ . It is easy to see that g is well defined. Moreover g is cofinal by the preceding fact and Theorem 17.2. (Theorem 17.4) 1
Corollary 17.5 (AD) (Kunen [Kun71E]). For all n ≥ 1, 12n → ( 12n ) 2n . Proof. By Theorem 17.4 and Corollary 13.9. (Corollary 17.5) 1
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ALEXANDER S. KECHRIS
§18. Some singular cardinals. Let A, B be two transitive classes, i : A → B a Δ0 -elementary embedding, i.e., for every Δ0 formula ϕ, A |= ϕ(a1 , . . . , an ) ⇔ B |= ϕ(ia1 , . . . , ian ). Put i(A) = {ia : a ∈ A}. If A is closed under transitive closure, i(A) is transitive and is therefore equal to the smallest transitive class containing the range of i. Proof. If x ∈ y ∈ ia, put b = TC(a). Then A |= ∀y ∈ A∀x ∈ y(x ∈ b), so B |= ∀y ∈ ia∀x ∈ y(x ∈ ib), thus x ∈ ib. Put HWO = {a : TC(a) is wellorderable} and for each ordinal κ let Fn(HWO, κ) = κHWO ∩ HWO = {F ∈ κHWO : ran(F ) ∈ HWO}. If U is a countably complete ultrafilter on κ, let Fn(HWO, κ)/U be the usual ultrapower. By Ło´s’ theorem Fn(HWO, κ)/U |= ϕ([F1 ], . . . , [Fn ]) ⇒ { ∈ κ : HWO |= ϕ(F1 ( ), . . . , Fn ( ))} ∈ U, for ϕ ∈ Δ0 . So Fn(HWO, κ)/U is wellfounded and extensional, thus it can be collapsed to a transitive class Ult(HWO, U ). Let i U : HWO → Ult(HWO, U ) be the usual embedding which by the above is Δ0 -elementary. Also Ult(HWO, U ) = i U (HWO) ⊆ HWO. Proof. For Ult HWO, U ) = i U (HWO): If [F ] ∈ Fn(HWO, κ)/U and z = ran(F ), then { < κ : HWO |= F ( ) ∈ z} ∈ U so Fn(HWO, κ)/U |= [F ] ∈ [ → z], thus Ult(HWO, U ) |= [F ] ∈ i U (z). (We identify here [F ] with its collapse.) For i U (HWO) ⊆ HWO: Enough to show that {[G] : [G] ∈ [F ]} is wellorderable. Let < be a wellordering of TC(ran(F )). For [G], [G ] ∈ F let [G] < [G ] ⇒ {x : G(x) < G(x )} ∈ U ; then < is a wellordering on {[G] : [G] ∈ [F ]}.
Lemma 18.1 (Kunen [Kun71E]). Let κ ≤ be two cardinals such that cf( ) = κ. Let U be a countably complete uniform ultrafilter on κ. Then i U ( ) ≥ + . Proof. Fix ≤ < + . To show i U ( ) > . Let R be a wellordering of cf → . Then i U ( ) > [F ]U ≥ sup{i U ( ) : < }. of type . Let F : κ − Now |R | < , ∀ < , therefore in particular |RF ()| < , ∀ < κ. So |i U (R)[F ]U | < i U ( ). But also |i U (R)[F ]U | ≥ |i U (R){i U ( ) : < }| = (Lemma 18.1) |R| = , therefore i W ( ) > .
AD AND PROJECTIVE ORDINALS
343
Theorem 18.2 (AD) (Kunen [Kun71E]). Let n ≥ 0, 12n+1 = +1 and = +2 . Then for each k ≥ 2,
12n+2
1) cf(+k ) = +2 . 2) There is Δ0 -elementary h : Ult(HWO, U ) → HWO such that +k = sup{h( ) : < +2 }, where U = on +1 . Proof. Clearly 2) ⇒ 1). We prove now 2) by induction on k. It is trivial for k = 2, with h = identity. So assume k > 2 and it holds for all 2 ≤ k < k. If V = on +2 then by Lemma 18.1, i V (+k−1 ) ≥ +k . Case I. i V (+k ) = +k , for some 2 ≤ k < k. By induction hypothesis, let h satisfy 2) for +k , i.e., +k = sup{h( ) : < +2 }. Define h 00 : Ult(HWO, U ) → HWO by h 00 ([F ]U ) = [h ◦ i U F ]V . First we check that h 00 is well defined. Assume F = G on I ∈ U. Then i U F = i U G on i U I ∈ V, so h ◦ i U F = h ◦ i U G V-a.e. It is equally routine to check that h 00 is Δ0 -elementary. We shall now prove that sup{h 00 ( ) : < +2 } = i V (+k ) = +k , which will complete the proof in this case. First notice that h 00 ( ) < i V (+k ) if < +2 , since if [F ]U = , where F : +1 → +1 , clearly h ◦ i U F : +2 → +k . Conversely, if < i V (+k ), then for some G : +2 → +k , ≤ [G]V . Let F : +2 → +2 be given by F () = least < +2 such that G() ≤ h(). Then G ≤ h ◦ F everywhere. By Theorem 17.2 let F : +1 → +1 be such that F ≤ i U F V-a.e. Then G ≤ h ◦ i U F U -a.e., so ≤ [G]V ≤ [h ◦ i V F ]V = h 00 ([F ]V ) = h 00 ( ), where < +2 and we are done. Case II. i V (+k −1 ) < +k < i V (+k ), for some 3 ≤ k < k. Then find F : +2 → +k such that [F ]V = +k . Then F is cofinal in +k since otherwise [F ]V < i V ( ) for some < +k . But Card( ) ≤ +k −1 so Card(i V ( )) ≤ Card(i V (+k −1) ) < +k , therefore i V ( ) < +k , a contradiction. Put h 0 = i V ◦ h, where h comes from our induction hypothesis for +k . Clearly h 0 : Ult(HWO, U ) → HWO
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is Δ0 -elementary. We will show that h 0 works for +k . Notice first that sup{h 0 (α) : α < +2 } = [h+2 ]V . Indeed, h 0 ( ) = i V (h( )) < [h+2 ]V , since h+2 is cofinal in +k . On the other hand if [F ]V < [h+2 ]V , then F ( ) < h( ) V-a.e., so restricting to ’s of cofinality if we let T ( ) be the least < such that F ( ) < h(), then T ( ) is pressing down V-a.e., because h is continuous at limits of cofinality , so T ( ) = < +2 V-a.e. Then F ( ) < h() V-a.e., i.e., [f]V < i V (h()) and we are done. So it is enough to show [h+2 ]V = +k = [F ]V . Now clearly [h+2 ]V ≤ [F ]V , since F is unbounded in +k . So it suffices to prove that [F ]V = +k ≤ [h+2 ]V . For that it is again enough to show that ([h+2 ]V )+ ≥ i V (+k ). Fix < i V (+k ). Then = [H ]V , where H : +2 → +k , so ≤ [h ◦ F ]V for some F : +2 → +2 . Find HW : +1 → +1 such that [F ]V ≤ [i U HW ]V = [Hi U W ]V . Then ≤ [h ◦ Hi U W ] = [h ◦ HV ]V , where V = i U W is a wellordering on +2 . Fix g ∈ Ult(HWO, U ) such that for some I ∈ V, I ∈ Ult(HWO, U ), g( ) : HV ( ) → , ∀ ∈ I . [To see that these exist, find P( ) : |W | → , ∀ ∈ X ∈ U , so i U P( ) : |V | → , ∀ ∈ i U X ∈ V. Put g = i U P, I = i U X.] Since g ∈ Ult(HWO, U ),
∈ I ⇒ h(g( )) : h ◦ HV ( ) → h( ), so [h ◦ g]V : [h ◦ HV ]V → [h+2 ]V , thus ≤ [h ◦ HV ]V < ([h+2 ]V )+ . (Theorem 18.2) REFERENCES
John W. Addison and Yiannis N. Moschovakis [AM68] Some consequences of the axiom of definable determinateness, Proceedings of the National Academy of Sciences of the United States of America, no. 59, 1968, pp. 708–712. Alexander S. Kechris [Kec74] On projective ordinals, The Journal of Symbolic Logic, vol. 39 (1974), pp. 269–282. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978.
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Eugene M. Kleinberg [Kle70] Strong partition properties for infinite cardinals, The Journal of Symbolic Logic, vol. 35 (1970), pp. 410– 428. Kenneth Kunen [Kun71A] Measurability of 1n , circulated note, April 1971. [Kun71C] A remark on Moschovakis’ uniformization theorem, circulated note, March 1971. [Kun71D] Some singular cardinals, circulated note, September 1971. [Kun71E] Some more singular cardinals, circulated note, September 1971. Richard Mansfield [Man71] A Souslin operation on Π12 , Israel Journal of Mathematics, vol. 9 (1971), no. 3, pp. 367– 379. Donald A. Martin [Mar68] The axiom of determinateness and reduction principles in the analytical hierarchy, Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687–689. [Mar71A] Determinateness implies many cardinals are measurable, circulated note, May 1971. [Mar71B] Projective sets and cardinal numbers: some questions related to the continuum problem, this volume, originally a preprint, 1971. Donald A. Martin and Jeff B. Paris [MP71] AD ⇒ ∃ exactly 2 normal measures on 2 , circulated note, March 1971. Donald A. Martin and Robert M. Solovay [MS69] A basis theorem for Σ13 sets of reals, Annals of Mathematics, vol. 89 (1969), pp. 138–160. Yiannis N. Moschovakis [Mos70] Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory. Proceedings of an international colloquium held under the auspices of the Israel Academy of Sciences and Humanities, Jerusalem, 11–14 November 1968 (Y. Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam-London, 1970, pp. 24–62. [Mos71] Uniformization in a playful universe, Bulletin of the American Mathematical Society, vol. 77 (1971), pp. 731–736. Joseph R. Shoenfield [Sho61] The problem of predicativity, Essays on the foundations of mathematics (Yehoshua BarHillel, E. I. J. Poznanski, Michael O. Rabin, and Abraham Robinson, editors), Magnes Press, Jerusalem, 1961, pp. 132–139. Robert M. Solovay [Sol67A] Measurable cardinals and the axiom of determinateness, lecture notes prepared in connection with the Summer Institute of Axiomatic Set Theory held at UCLA, Summer 1967. [Sol78A] A Δ13 coding of the subsets of , this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 133–150. DEPARTMENT OF MATHEMATICS CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA 91125 UNITED STATES OF AMERICA
E-mail: [email protected]
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ROBERT M. SOLOVAY
§1. Introduction. We present Kunen’s [Kun71B] analysis of the measures on , from AD+DC, together with some lightface refinements that follow by mixing Kunen’s techniques with a theorem of Kechris-Martin. Our purpose in this introduction is (a) to list some applications of the Kunen method (to be presented in Section 3); (b) to give an overview of the more technical results to be proved; (c) to give some idea of the motivation behind the technicalities that follow. The principle results of type (a) are as follows (AD+DC is assumed throughout): Let < 13 . Then 13 → ( 13 ) . This should be compared with the result 1 1 of Martin that 11 → ( 11 ) 1 . It is still open whether or not 13 → ( 13 ) 3 . a highly Kunen’s proof uses detailed analysis at level 3, and it is not known how to generalize his work to odd n’s greater than 3. In particular, it is open whether 15 → ( 15 ) for all < 15 , though we have already seen in [Kec78] that 15 → ( 15 ) for countable . Note that by earlier work in [Kec78], it follows that there are exactly three normal measures on 13 (concentrating on points of cofinality 0 , 1 , and 2 respectively). (b) Our proof will be based on a Δ13 encoding of the subsets of (cf. Theorem 3.8). To state what this means, recall that the theory of sharps together with the fact that the i ’s, 1 ≤ i < are precisely the first uniform indiscernibles gives a natural Δ13 encoding of the ordinals < . We shall produce a Δ13 set, C , of codes for subsets of so that the relation “the ordinal coded by x lies in the set coded by y” is Δ13 . To get such a coding we need a concrete way of generating all subsets of . We prove that every non-empty subset of is the countable union of simple sets. Here, a simple set is a subset of some m which is the 1-1 image, by some The author would like to express his thanks to the Sherman Fairchild Distinguished Scholars Program at Caltech for its generous support during the academic year 1976–1977. Thanks also to Greg Ennis for the conscientious painstaking work of transferring a series of lectures to the printed page. Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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function constructible from a real, of one of a countable sequence of standard sets, Ajm,k . [This is slightly stronger than what we prove below (owing to a different definition of “simple”); it is not hard to prove the stronger claim with Kunen’s method.] By an ingenious reduction, (cf. Theorem 3.6) this is reduced to an analysis of measures on . Some further simple reductions (which take place in Sections 3.1 and 3.2 below) reduce the problem to the following: We are given integers m ≥ 0, k ≥ 1. Each closed unbounded subset C of 1 determines in a canonical manner a subset C˜ of m+1 (the image of C under a suitable elementary embedding of some L[z] mapping 1 into m+1 ). In k this way the sets C˜ k form a filter base for a filter F on m+1 . Our problem is to characterize the ultrafilters that extend F . The main result proved in Section 2 shows that there are only finitely many such ultrafilters. They live on disjoint sets A1m,k , . . . , Am,k (whose union is the set of k-tuples of limit ordinals less than m+1 ) and the decomposition is Δ13 in the codes. The proof is a refinement of the Martin-Paris analysis of normal measures on 2 , which was presented in [Kec78]. We replace the study of k-tuples of ordinals less than m+1 by the study of k-tuples of functions from 1m into 1 . We analyse these k-tuples carefully enough so as to be able to imitate the Martin-Paris proof. Of course, in our more general context the technicalities will be considerably greater. We emphasize that we work in ZF+AD+DC in the following. Any unexplained notation is as in [Kec78]. §2. Classification of tuples of ordinals. 2.1. W is the canonical normal measure on 1 . Wn is the product measure W × ··· × W .
n times
If A is a set, An is the n-fold Cartesian power. A[n] , for A linearly ordered, is the set of strictly increasing n-tuples from A. ˜ = {L[x] : x ∈ R}. For each x ∈ R, we have an elementary 2.2. L embedding iWn : L[x] → L[x], given by the transitive realization of the ultrapower. These maps piece together to give a map ˜ → L. ˜ iWn : L ˜ but (Note that this last map is not elementary since 1 is definable in L iWn (1 ) = 1 .)
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Note that Wn gives 1[n] measure 1 (and we usually view Wn as a measure on 1[n] ). If F : 1[n] → L[x], then F determines (via transitive realization of the ultrapower) an element [F ] ∈ L[x]. Let i : 1[n] → 1 , be given by i (α1 , . . . , αn ) = αi . Lemma 2.1. [i ] = i . iWn (1 ) = n+1 . Proof. Clearly, {α ∈ 1[n] : L[z # ] |= αi is an indiscernible for L[z]} has Wn -measure 1. Thus [i ] is a uniform indiscernible. So clearly is iWn (1 ). Since clearly [1 ] < · · · < [n ] < iWn (1 ), it suffices to show that if F : 1[n] → 1 is such that [F ] is a uniform indiscernible, then [F ] = [i ] for some i. Now F ∈ L[z] for some z ∈ R since F ⊆ L1 . By increasing the Turing degree of z, we may suppose F is definable in L[z] from z, 1 . Pick i minimal such that [F ] ≤ [i ]. (The case when [F ] > [n ] is handled similarly.) If F (α) is definable in [F ] = [i ], we are done. Otherwise, for almost all α, L[z] from z, and indiscernibles for L[z] distinct from F (α). It follows that F (α) is a.e. not an indiscernible for L[z]. Whence [F ] is not a uniform indiscernible. Corollary 2.2. Let < m+1 . Let F : 1[m] → 1 such that [F ] = . Let [m] F˜ : m+1 → m+1 be iWm (F ). Then = F˜ (1 , . . . , m ). Proof. By Ło´s’ theorem, this amounts to F (α) = F (1 (α), . . . , m (α)).
2.3. We now state the theorem which is the main goal of Section 2. Theorem 2.3 (Kunen [Kun71B]). Let m ≥ 0. Let k ≥ 1. Let Xm,k = { 1 , . . . , k : for 1 ≤ i ≤ k, i is a limit ordinal < m+1 }. We shall construct a decomposition: Xm,k = A1m,k ∪ · · · ∪ Am,k (disjoint union) with the following properties: (1) Let C be a cub subset of 1 . Let C˜ = iWm (C ). Then C˜ k ∩ Aim,k = ∅. (2) Sets of the form C˜ k ∩ Aim,k are the basis for an ultrafilter on Aim,k . (This ultrafilter is clearly countably additive since the intersection of countably many cubs is cub.) (3) Let h : 1 → 1 be normal (i.e., strictly increasing and continuous). Let ∈ Aim,k , then h˜ : m+1 → m+1 be iWm (h). If α ˜ k ) ∈ Ai . ˜ i ), . . . , h(α h(α m,k
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We shall see that properties (1) through (3) characterize the partition of Xm,k into the Aim,k ’s. 2.4. For certain applications we need that our partition is Δ13 in the codes. A code for an ordinal < is a pair z = n, x # , where x ∈ R. If z is a code as above then we put |z| = nL[x] (1 , . . . , kn ) where 0 , 1 , . . . is a recursive enumeration of all the terms in the language of L[x] (x occurring as a constant symbol) taking always ordinal values. The following result is an effective version (Δ13 replacing Δ13 ) of a theorem of Kunen [Kun71B]. Theorem 2.4. (1) Let (m, k) be the number of pieces into which Xm,k is decomposed. (cf. Theorem 2.3) Then the map m, k → (m, k) is Δ13 . (2) The following relation Φ(k, m, j, x) is Δ13 : (Here k, m, j are in , and x in R) k ≥ 1; m ≥ 0; 1 ≤ j ≤ (m, k); for 1 ≤ i ≤ k, (x)i codes a limit ordinal i < m+1 ; 1 , . . . , k ∈ Ajm,k . 2.5. Let HF be the class of hereditarily finite sets (HF = L , and there is a canonical well-ordering of HF of type .) The decomposition of Xm,k will arise in the following way. We will define a map Ψm,k : Xm,k → HF. The range of Ψm,k will be finite, say {x1 , . . . , x }, where x1 < · · · < x with respect to the canonical well-ordering of HF. Then Ajm,k = Ψ−1 m,k ({xj }). The map Ψm,k will have the following properties: (α) ran(Ψm,k ) is finite. () Ψm,k is an invariant: If h : 1 → 1 is normal, h˜ = iWm (h), α ∈ Xm,k ˜ i ) for 1 ≤ i ≤ k, then and i = h(α Ψm,k (α) = Ψm,k (). () Ψm,k is Δ13 in the codes: The following relation is Δ13 : m ≥ 0, k ≥ 1, for 1 ≤ i ≤ k (x)i codes a limit ordinal i < m+1 and = s. Ψm,k ( ) (Here s ∈ HF, x ∈ R, m, k ∈ ). 2.6. The case m = 0 is trivial but atypical. First we describe the invariant Ψ0,k : Ψ0,k ( 1 , . . . , k ) = {i, j : i < j }. It is evident that Ψ0,k has the properties (α), (), and () of 2.5.
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A moment’s thought will show that we can effectively tell which s ∈ HF lie in ran(Ψ0,k ). Thus the portion of 2.4 that relates to m = 0 is evident from () holding for Ψ0,k (uniformly in k). Finally we verify Theorem 2.3 for X0,k with respect to the decomposition induced by Ψ0,k . Let Aj0,k be a component of this decomposition. Then there is an integer r, with 1 ≤ r ≤ k, integers s1 , . . . , sr with 1 ≤ si ≤ k, and integers ni , 1 ≤ i ≤ k, with 1 ≤ ni ≤ r such that for α ∈ Aj0,k , αi = αsni ; αs1 < αs2 < · · · < αsr . By means of the map Aj0,k + Lim[r] : α1 , . . . , αn → αs1 , . . . , αsr , the claims of Theorem 2.3 for the case m = 0 reduce to the following well-known fact: sets of the form C [r] , C cub, are a basis for Wr . 2.7. We turn now to the definition of Ψm,k for m > 0. Our strategy will be as follows. We define three simpler invariant functions, Ψ1m,k , Ψ2m,k , Ψ3m,k which satisfy (α), (), () of 2.5. We then put Ψm,k (α) = Ψ1m,k (α), . . . , Ψ3m,k (α). Ψm,k will then inherit the properties (α), (), and () from the Ψim,k ’s. We now fix m ≥ 1, k ≥ 1. Let ∈ Xm,k . Pick F1 , . . . , Fk mapping 1[m] into 1 such that [Fi ] = i . Pick z ∈ R such that Fi : 1 ≤ i ≤ k is definable from z, 1 in L[z]. Let Iz be the set of canonical indiscernibles for L[z] which are less than 1 . We pick ∈ Iz[2m] . Then = {i, r, j, s : 1 ≤ i, j ≤ k; Ψ1 ( ) m,k
1 ≤ r1 < · · · < rm ≤ 2m; 1 ≤ s1 < · · · < sm ≤ 2m; Fi (r1 , . . . , rm ) < Fj (s1 , . . . , sm )}. It is necessary to see that Ψ1m,k depends only on and not on the choices of the F ’s, z, and ’s. Evidently the choice of the ’s are irrelevant so long as they are chosen in Iz . If F , z are different choices, we can find E cub so that (a) Fi and Fi agree on E [m] (1 ≤ i ≤ k), (b) E ⊆ Iz ∩ Iz . But now if we take ∈ E [2m] , we will get the same value of Ψ1 from the primed choices as from the unprimed ones. How about properties (α) through ()? Properties (α) and () are evident. ˜ i ). Since h is order-preserving, () As for property () note that [h ◦ Fi ] = h( 1 is now clear for Ψ . 2.8. Let D be a closed unbounded subset of 1 . D is the set of limit points [m] of D. Define a function HiD : D → 1 : : ∈ D [m] and Fj () < Fi ()}. HiD () = sup{Fj ()
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Evidently if E is a cub ⊆ D, [HiE ] ≤ [HiD ], since for ∈ E supremum is over a smaller set for HiE () than for HiD ().
[m]
, the
Definition 2.5. i is of type II if for some cub D, [HiD ] < [Fi ]. = {i : i is of type II}. We put Ψ2m,k ( ) As usual we must verify independence of the choice of F ’s. But if Fj and Fj agree on E [m] , E cub, and for some D, [HiD ] < [Fi ], then by the remark of the paragraph preceding Definition 2.5, [HiD∩E ] < [Fi ]. But then HiD∩E = D∩E D∩E (since the Fj ’s and Fj ’s look alike on E [m] ) so [Hi ] < [Fi ]. So i is Hi of type II with respect to the F ’s (if it is with respect to the F ’s). Property (α) is evident. Since if h is normal, it preserves the notion of limit ˜ i ), property () is clear. and non-limit, and since [h ◦ Fi ] = h( 2 Our proof of property () for Ψ will be preceded by two lemmas. F , z be as above. Let D, E be cub subsets of 1 with Lemma 2.6. Let , [m] < Fi (). Then for E ⊆ D ⊆ Iz . Suppose that for some ∈ E , HiE () [m] every ∈ D , HiD () < Fi (). Let ϑ1 , . . . , ϑn be ordinals of Iz such that = Proof. Let = HiE (). (ϑ1 , . . . , ϑn ). Pick ϑ1 , . . . , ϑn in E so that (a) ϑi ≤ ϑi (1 ≤ i ≤ n) and ϑ , are similarly ordered. (b) ϑ, (c) If α1 , α2 ∈ {0, ϑ1 , . . . , ϑn , 1 , . . . , m } with α1 < α2 , then there are at least m members of E strictly between α1 and α2 . (There is no difficulty doing this since the i ’s are limit points of E.) Let = L[z] (ϑ1 , . . . , ϑn ). By (a) ≤ . By (b) and E ⊆ Iz , < Fi (). Next select ϑ1∗ , . . . , ϑn∗ in D with , . ∗ , similarly ordered to ϑ (d) ϑ L[z]
Put (This is possible since the i ’s are in D and is similarly ordered to .) ∗ L[z] ∗ ∗
= (ϑ ). By (d) and the fact that E ⊆ D ⊆ Iz , we have < Fi (). Thus to prove HiD () < Fi () it suffices to show HiD () ≤ ∗ . Deny this towards a contradiction. Then for some ∈ D [m] , j, we have ∗
< Fj ( ) < Fi (). By (c), we can choose ∈ E [m] so that , , . ∗ , , are ordered similarly to ϑ (e) ϑ But then by (e) and E ⊆ D ⊆ Iz , (f) < Fj ( ) < Fi (). But this contradicts the fact that HiE () ≤ .
Lemma 2.7. Let D ⊆ Iz be cub. Then if i is not of type II, HiD () = Fi () [m] [m] for all ∈ D . If i is of type II, HiD () < Fi () for all ∈ D .
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Proof. Suppose that Fi () > HiD () for one ∈ D . (Note that Fi () ≥ is evident from the definitions.) Apply Lemma 2.6 with D = E. Then Fi () > HiD () for every ∈ (D )[m] . Whence [Fi ] > [HiD ], and i is of type II. This proves our first claim. Next suppose i is of type II. Then for some cub E, [HiE ] < [Fi ]. Replace E, if need be, by E ∩ D. But then Lemma 2.6 guarantees HiD () < Fi () for all [m] ∈ D (since the needed for Lemma 2.6 is guaranteed by [HiE ] < [Fi ]). [m]
HiD ()
It is now easy to prove that Ψ2 is “Δ13 in the codes”. If (x)i codes i , we can take x itself for our z. The criterion of Lemma 2.7 can be checked in L[z # ], hence recursively in z ## , hence Δ13 in z. This proves that Ψ2 satisfies (). 2.9. ˜
Definition 2.8. cf L (α) = min{cf L[z] (α) : z ∈ R}. ˜
˜
Evidently cf L (α) ≤ α. cf L (α) is regular in each L[z]. So if α is a limit ˜ ordinal cf L is either or a uniform indiscernible. ˜ It follows that cf L ( i ) = j for some j with 0 ≤ j ≤ m. We put = {i, j : i is of type II and cf( i ) = j }. Ψ3m,k ( ) Evidently Ψ3 satisfies (α). If h : 1 → 1 is normal, then so is h˜ : m+1 → ˜ i ). If m+1 . If g maps j cofinally into i , h˜ ◦ g maps j cofinally into h( ˜ ˜ ˜ i )) = cf L ( i ) and Ψ3 g is order-preserving, so is h˜ ◦ g. It follows that cf L (h( satisfies (). We need the following lemma to show that Ψ3 satisfies (). Lemma 2.9. Let z ∈ R. Let be an infinite ordinal which is regular in L[z] but not an indiscernible of L[z]. Then is cofinal with in L[z # ]. Proof. We may as well assume > . Let = sup{ < : = or is an indiscernible for L[z]}. Then ≤ ≤ . If = , then must be an indiscernible for L[z] (since the class of indiscernibles is closed and > ). This contradicts our assumption on . So < . Let ϑ1 , ϑ2 , . . . be the first indiscernibles for L[z] greater than . Then every ordinal < is definable in L[z] from ordinals ≤ and some of the ϑi ’s. We set Si = { < : is definable in L[z] from ordinals in ∪ {} ∪ {ϑ1 , . . . , ϑi }}. Then Si ∈ L[z] and in L[z] Si has power . Since is regular in L[z], i = sup(Si ) is less than . By the preceding paragraph, the i ’s are cofinal in . But clearly the sequence i is definable from in L[z # ]. #
Lemma 2.10. Let = L[z] (1 , . . . , m ), a limit ordinal. Then cf L[z ] ( ) = L˜ cf ( ).
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Proof. If cf L[z] ( ) = i , for 0 ≤ i ≤ m, then evidently i = cf L[z ] ( ) = cf ( ). If not, cf L[z] ( ) is not an indiscernible in L[z] (since it is definable in L[z] from 1 , . . . , m , but is distinct from each of 1 , . . . , m ). Whence # # ˜ by Lemma 2.9, cf L[z ] (cf L[z] ( )) = . But then clearly cf L[z ] ( ) = cf L ( ) = . 3 1 Lemma 2.10 makes it evident that Ψ is Δ3 in the codes. For again if (x)i L˜
˜
codes i , we can take x as our z and compute cf L ( i ) recursively from x ## . 2.10. Now as promised in 2.5 we put = Ψ1 ( ), Ψ2 ( ), Ψ3 ( ) Ψm,k ( ) m,k m,k m,k for ∈ Xm,k . Thus the range of Ψm,k is a finite subset of HF, say {s1 , . . . , s(m,k) }, where s1 < · · · < s(m,k) in the canonical well-ordering of HF. Let Ajm,k = Ψ−1 m,k ({sj }). The proof of part (2) of Theorem 2.4 will follow from part (1) and the fact that Ψ1 , Ψ2 , Ψ3 all satisfy property () of 2.5. To prove part (1) of Theorem 2.4 we need the Kechris-Martin Theorem. 2.11. Theorem 2.11 (Kechris-Martin [KM78]). Let A ⊆ be nonempty and Π13 in the codes. Then ∃x ∈ Δ13 (|x| ∈ A). We need the following corollaries: Lemma 2.12. Let R ⊆ n be Δ13 in the codes. (a) ¬R is Δ13 in the codes. (b) If S(1 , . . . , n−1 ) ⇔ ∃ < R( , ), then S is Δ13 in the codes. (c) If T (1 , . . . , n−1 ) ⇔ ∀ < R( , ), then T is Δ13 in the codes. Proof. (a) is obvious since the set of codes is Δ13 . (b) Let R+ (x1 , . . . , xn ) ⇔ ∀i ≤ n(xi codes an ordinal < and R(|x1 |, . . . , |xn |)). Then S + (x2 , . . . , xn ) ⇔ ∃x1 (x1 codes an ordinal and R+ (x1 , x2 , . . . , xn )). So S is Σ13 in the codes. Also, by the Kechris-Martin Theorem (2.11), S + (x2 , . . . , xn ) ⇔ ∃x1 ∈ Δ13 (x2 , . . . , xn ) [x1 codes an ordinal and R+ (x1 , . . . , xn )], so S is Π13 in the codes. (c) follows from (a) and (b). Lemma 2.13. Let R ⊆ n be Δ13 in the codes, R = ∅. Then ∃x ∈ Δ13 [(x)1 , . . . , (x)n code ordinals < & R(|(x)1 |, . . . , |(x)n |)]. Proof. By induction on n. The case n = 1 is the Kechris-Martin Theorem (2.11). If n > 1, let S(1 , . . . , n−1 ) ⇔ ∃ < R(1 , . . . , n−1 , ). By Lemma 2.12 (b), S is Δ13 in the codes, so by the induction hypothesis let x ∈ Δ13
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be such that S(|(x)1 |, . . . , |(x)n−1 |). Then { : R(|(x)1 |, . . . , |(x)n−1 |, )} is Δ13 (x) in the codes, hence Δ13 in the codes, so ∃y ∈ Δ13 with R(|(x)1 |, . . . , |(x)n−1 |, |y|). Let z ∈ Δ13 be such that (z)i = (x)i if i < n and (z)n = y. Lemma 2.14. {m, k, s : s ∈ ran Ψm,k } is Δ13 . Proof. s ∈ ran Ψm,k ⇔ ∃ x [| x | ∈ Xm,k & Ψm,k (| x |) = s]. Thus since the components of Ψm,k all satisfy condition (), this is Σ13 . But by Lemma 2.13 we can replace the existential quantifiers by ∃x ∈ Δ13 , hence it is Π13 . Proof of Theorem 2.4. Now Theorem 2.4(1) follows, since (m, k) = Card(ran Ψm,k ), so is clearly a
Δ13
function.
Theorem 2.4(2) follows easily from this. (Theorem 2.4) 1 2.12. It remains to prove that our partition Xm,k = Am,k ∪ · · · ∪ Am,k has the properties (1) and (2) of Theorem 2.3 (property (3) holds since Ψ1 , Ψ2 , Ψ3 satisfy condition ()). Let Norm = {h : 1 → 1 : h normal}. The following is a variant of Martin’s partition theorem (12.1 in [Kec78]). Lemma 2.15. Let Φ : Norm → {0, 1} be such that Φ(h) depends only on h Lim. Then ∃j ∈ {0, 1}, ∃ cub C such that if h ∈ Norm, h : 1 → C , then Φ(h) = j. Proof. Let Ψ : ℘(1 ) → {0, 1} be given by Ψ(A) = 0 iff ∃h ∈ Norm[Φ(h) = 0 and A = {h( · + ) : < 1 }]. By Martin (Theorem 12.1 in [Kec78]) ∃ cub C and a j ∈ {0, 1} such that if A ∈ C f, then Ψ(A) = j. Say j = 0. Given h ∈ Norm, h : 1 → C , let A = {h( · + ) : < 1 }. Thus A ∈ C f, so Ψ(A) = 0. Thus for some h ∈ Norm, with Φ(h ) = 0 we have A = {h ( · + ) : < 1 }. Since h, h are both increasing, we must have ∀ < 1 (h( · + ) = h ( · + )). But the closure of { · + : < 1 } is the set of limit ordinals < 1 , so since h, h are continuous they must agree at all limit ordinals. This proves the case j = 0, and the case j = 1 is easier. j 2.13. Fix m > 0, k ≥ 1, 1 ≤ j ≤ (m, k) such that if 1 , . . . , k ∈ Am,k then at least one i ≥ 1 . Note that if Ψm,k (Ajm,k ) = s we can tell using s alone whether ∃i i ≥ 1 , since if [Fi ] = i then i < 1 iff Fi is constant a.e. (Wm ), which can be determined from Ψ1m,k . So the property “∃i( i ≥ 1 )” is true of all tuples 1 , . . . , k in Ajm,k if it is true of one tuple. (Note also that if Ajm,k is a subset of 1k then we are done by the m = 0 case.)
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Lemma 2.16. ∃ G1 , . . . , Gk : 1[m] → 1 such that (i) Ψm,k ([G1 ], . . . , [Gk ]) = s k , Ψ() = s, C ⊆ 1 is cub, and ∈ C˜ , then there is a (ii) If ∈ m+1 normal h : 1 → C such that ˜ ˜ h([G . 1 ]), . . . , h([Gk ]) = Granting the lemma, we can prove Theorem 2.3 as follows: Proof of Theorem 2.3. (1) Given C ⊆ 1 cub, let h : 1 → C be normal. j ˜ ˜ ˜ ˜ ˜k Then h([G 1 ]), . . . , h([Gk ]) ∈ C . By (), h([G1 ]), . . . , h([Gk ]) ∈ Am,k , so C˜ k ∩ Ajm,k = ∅. (2) Let B ⊆ Ajm,k . We want a cub C ⊆ 1 such that either C˜ k ∩ Ajm,k ⊆ B or C˜ k ∩ Ajm,k ∩ B = ∅. Define Φ : Norm → {0, 1} by ˜ ˜ 0 if h([G 1 ]), . . . , h([Gk ]) ∈ B Φ(h) = 1 otherwise. ˜ ˜ Since [Gi ] ∈ Lim, Gi (a) ∈ Lim a.e. Hence since h([G 1 ]), . . . , h([Gk ]) = [h ◦ G1 ], . . . , [h ◦ Gk ] we have h | Lim = h | Lim ⇒ ∀i([h ◦ G1 ] = [h ◦ Gi ]), so Φ(h) = Φ(h ). Hence we can apply Lemma 2.15 to get a j ∈ {0, 1} and a cub C such that for all normal h : 1 → C , Φ(h) = j. ˜ ˜ If ∈ C˜ k ∩Ajm,k then ∃h : 1 → C normal with = h([G 1 ]), . . . , h([Gk ]). So ∈ B always (never) if Φ(h) = 0 (Φ(h) = 1) for all h : 1 → C . Thus, either C˜ k ∩ Ajm,k ⊆ B or C˜ k ∩ Ajm,k ∩ B = ∅. (3) Holds since Ψ1 , Ψ2 , Ψ3 satisfy condition () from p. 349. (Theorem 2.3) 2.14. We now prove Lemma 2.16, thus completing the proof of Kunen’s Theorem (2.3). = Proof of Lemma 2.16. We have a fixed invariant s, 1 , . . . , k with Ψ( ) s, where at least one i is ≥ 1 . Consider all m + 1 tuples i, α1 , . . . , αm such that 1 ≤ i ≤ k, α1 < · · · < αm < i . We define an equivalence relation ∼ on these tuples as follows. Given two such tuples i, α1 , . . . , αm , j, 1 , . . . , m , let 1 ≤ r1 < · · · < rm ≤ 2m, 1 ≤ s1 < · · · < sm ≤ 2m be such that r1 , . . . , rm , s1 , . . . , sm are ordered similarly to α1 , . . . , αm , 1 , . . . , m . Then put i, α ∼ j,
iff
and i, r1 , . . . , rm , j, s1 , . . . , sm ∈ / Ψ1 ( ) 1 i, s1 , . . . , sm , j, r1 , . . . , rm ∈ / Ψ ( )
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Let S be the set of ∼-equivalence classes. Linearly order S by [i, α1 , . . . , αm ] <S [j, 1 , . . . , m ] iff i, r1 , . . . , rm , j, s1 , . . . , sm ∈ Ψ1 ( ), where [ ] denotes equivalence class and r1 , . . . , rm , s1 , . . . sm are as above. Note that S, <S depends only on s not on . Let [Fi ] = i for i ≤ k, and let z ∈ R be such that F1 , . . . Fk are definable ˜ from 1 in L[z] and cf L[z] ( i ) = cf L ( i ) for i ≤ k. Let Iz be the indiscernibles for L[z] < 1 . Let h : 1 → Iz enumerate Iz in increasing order. Define h ∗ : {1, . . . , k} × 1[m] → 1 by h ∗ (i, α1 , . . . , αm ) = Fi (h(α1 ), . . . , h(αm )). Claim 2.17. ∼ is an equivalence relation. Proof. By definition of Ψ1 , h ∗ (i, α1 , . . . , αm ) = h ∗ (j, 1 , . . . , m ) iff i, α1 , . . . , αm ∼ j, 1 , . . . , m . (Claim 2.17) Claim 2.18. <S is well defined. Proof. h ∗ induces a 1-1 order preserving map from S into 1 (call this induced map h ∗ also). (Claim 2.18) Claim 2.19. S is order isomorphic to 1 . Proof. Some i is ≥ 1 , hence Fi is not constant a.e. Hence Fi [Iz[m] ] has power 1 , and h ∗ [S] ⊇ Fi [Iz[m] ]. (Claim 2.19) For [i, α1 , . . . , αm ] = x ∈ S, say x is of type I (II) if i is of type I (II) (i.e., if i ∈ / (∈) Ψ2 ( )). Claim 2.20. The type of [i, α1 , . . . , αm ] does not depend on a choice of representative. Proof. Let 1 , . . . , m ∈ Lim. Then h ∗ ([i, ]) is not a limit point of h ∗ [S] : h ∗ (j, ) < h ∗ (i, )} < h ∗ (i, ) ⇔ sup{h ∗ (j, ) ⇔ sup{Fj (h(1 ), . . . , h(m )) : Fj (h(1 ), . . . , h(m )) < Fi (h(1 ), . . . , h(m ))} < Fi (h(1 ), . . . , h(m )) −−→ −−→ −−→ −−→ ⇔ ∀ ∈ Lim[m] sup{Fj (h()) : Fj (h()) < Fi (h())} < Fi (h()) Lem. 2.7
⇔
i is of type II.
(Claim 2.20)
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Definition 2.21. Let [i, α1 , . . . , αm ] ∈ S be of type II. Then cf([i, α1 , . . . , αm ]) =
˜
if cf L ( i ) =
αj
if cf L ( i ) = j , j > 0.
˜
Claim 2.22. cf([i, α1 , . . . , αm ]) does not depend on choice of representative. Proof. Let [i, α1 , . . . , αm ] = [j, 1 , . . . , m ]. Let r1 , . . . , rm , s1 , . . . , sm be integers in 1, . . . , 2m such that r, s and α, are similarly ordered. Then by L[z] our choice of z, cf (Fi (r1 , . . . , rm )) = n for some n, 0 ≤ n ≤ 2m. Since Fi (r1 , . . . , rm ) = Fj (s1 , . . . , sm ), cf L[z] (Fj (s1 , . . . , sm )) = n . If n = 0, we are done. Otherwise, if cf L[z] (Fi (r1 , . . . , rm )) = rk and cf L[z] (Fj (s1 , . . . , sm )) = s , then rk = s , hence by the similar ordering of r, s, and α, we have αk = . So cf is well defined. (Claim 2.22) Now let T = S ∪ {x, α : x ∈ S, x of type II, ∃ ∈ Lim[m] ∃i(x = [i, ]) and α < cf(x)}. We define
or
Note that T,
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ROBERT M. SOLOVAY
Proof. Let ∈ Iz[2m] . Then = {i, r1 , . . . , rm , j, s1 , . . . , sm : Ψ1 ([G]) 1 ≤ r1 < · · · < rm ≤ 2m, 1 ≤ s1 < · · · < sm ≤ 2m, 1 ≤ i, j ≤ k & Gi (r1 , . . . , rm ) < Gj (s1 , . . . , sm )} = {i, r1 , . . . , rm , j, s1 , . . . , sm : . . . and [i, r1 , . . . , rm ] <S [j, s1 , . . . , sm ]} = Ψ ( ). 1
(Claim 2.24)
Claim 2.25. Ψ2 ([G1 ], . . . , [Gk ]) = Ψ2 ( ). Proof. We must show i is of type I w.r.t. iff i is of type I w.r.t. [G1 ], . . . [Gk ]. Then ∀ ∈ Lim[m] , Suppose i is of type I w.r.t. to . −−→ −−→ −−→ −−→ sup{Fj (h()) : Fj (h()) < Fi (h())} = Fi (h()), which by definition of h ∗ implies that ∀ ∈ Lim[m] , : h ∗ ([j, ]) < h ∗ ([i, ])} = h ∗ ([i, ]). sup{h ∗ ([j, ]) Suppose there is ∈ Lim[m] such that : [j, ] < [i, ]} < [i, ]. sup{[j, ] Then since i is not of type II, by the way S is embedded in T we must have : [j, ] < [i, ]} ≤ [k, ] < [i, ] sup{[j, ] ∈ S. This yields a contradiction by applying h ∗ . Hence for all for some [k, ] [m] ∈ Lim , : [j, ] < [i, ]} = [i, ], sup{[j, ] so i is of type I w.r.t. [G1 ], . . . [Gk ] by Lemma 2.7. then ∀ ∈ Lim[m] , Clearly (by the definition of T ), if i is of type II w.r.t. , : [j, ] < [i, ]} < [i, ], 0 < [i, ]. sup{[j, ] Hence i is of type II w.r.t. [G1 ], . . . [Gk ].
(Claim 2.25)
Claim 2.26. Ψ3 ([G1 ], . . . [Gk ]) = Ψ3 ( ). Proof. Suppose i is of type II, with cf[i, 1 , . . . , m ] = j . Then by the construction of T , [i, 1 , . . . , m ] is cofinal with j in T . The construction can be done within L, hence cf L Gi (1 , . . . , m ) = cf L (j ). Hence cf L [Gi ] = cf L [j ] = j . The case cf = 0 is similar.
(Claim 2.26)
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Hence Ψ([G1 ], . . . [Gk ]) = s as desired. This completes the proof of Lemma 2.16(i). Now to prove part (ii) of Lemma 2.16, suppose C ⊆ 1 is cub with i ∈ C˜ . We will find a normal h ∗∗∗ : 1 → C such that h˜ ∗∗∗ ([Gi ]) = i . For the remainder of the proof of the Lemma “definable in L[z]” will mean “definable from z, 1 in L[z].” ˜ We have z ∈ R with F1 , . . . , Fk definable from z, 1 and cf L[z] ( i ) = cf L ( i ). By increasing the Turing degree of z if necessary, we may assume C is also definable from 1 in L[z]. Since i ∈ C˜ , we have Fi (1 , . . . , m ) ∈ C a.e. Since C ∈ L[z], by indescernibility we get ∀1 < · · · < m Fi (h(1 ), . . . , h(m )) ∈ C . So h ∗ : S → C . The function h ∗ is definable in L[z # ]. Let x = [i, 1 , . . . , m ] be of type II, all i ∈ Lim. Let g(x) = the least map in L[z] of cf L[z] (h ∗ (x)) into h ∗ (x) continuously, order preservingly, cofinally, and with range contained in C ∩ { : > sup(h ∗ (S) ∩ h ∗ (x))}. Note that since h ∗ (x) is not a limit point of h ∗ (S), such a map exists. Now define h ∗∗ : T → C as follows: if x ∈ S, h ∗∗ (x) = h ∗ (x). For x, α ∈ T − S, g(x)(α) if cf(h ∗ (x)) = ∗∗ h (x, α) = g(x)(h(α)) otherwise. (Note that α < cf x ⇒ h(α) < cf h ∗ (x)). It is an easy exercise to prove that h ∗∗ is order preserving (using the fact that for all α, g(x)(α) > sup(h ∗ (S) ∩ h ∗ (x))). Also h ∗∗ (Gi (α1 , . . . , αm )) = h ∗∗ ([i, α1 , . . . , αm ]) = Fi (h(α1 ), . . . , h(αm )), for all α ∈ 1[m] . So since h(α) = α a.e. we have h˜ ∗∗ ([Gi ]) = i . But h ∗∗ is not normal. To get our desired normal h ∗∗∗ we need the following easy result: Fact 2.27. Let A ⊆ 1 have order type 1 , B = closure of A in 1 . Let hA , hB be the enumerations in order of A, B, respectively. Suppose is a limit ordinal with hA ( ) a limit point of A. Then hA ( ) = hB ( ) Thus let h ∗∗∗ be the enumeration of the closure of h ∗∗ (T ). Since C is closed, h ∗∗∗ : 1 → C . Now if 1 < · · · < m are limit ordinals, then [i, 1 , . . . , m ] is a limit point of T , and h ∗∗ ([i, 1 , . . . , m ]) is a limit point of h ∗∗ [T ] (since if x = [i, 1 , . . . , m ] is of type II and cf L[z] (Fi (1 , . . . , m )) = j , then cf L[z] (h ∗ (x)) = h(αj )). So by the fact, h ∗∗∗ ([i, 1 , . . . , m ]) = = . h ∗∗ ([i, 1 , . . . , m ])), so h˜ ∗∗∗ ([G]) (Lemma 2.16)
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§3. Applications. 3.1. Kunen’s theorem gives us a partition (m+1 ∩Lim)k = A1m,k ∪· · ·∪Am,k j such that each piece carries a canonical measure Vm,k . In this section we prove the following: Theorem 3.1 (Kunen [Kun71B]). Let V be a (countably additive) measure k on . Then ∃h : → m+1 for some m, k such that ˜ (1) h ∈ L. (2) h is 1-1 a.e. (V ). j (3) h∗ (V ) = Vm,k for some j. k ˜ such that g is 1-1 a.e. (V j ) and g = h −1 a.e. (4) ∃g : m+1 → , g ∈ L, m,k 3.2. By countable additivity of V we may assume V is on n for some 1 ≤ n < . We define the restricted ultrapower of Ord in the same way as the ordinary ultrapower except we only consider functions f : n → Ord which ˜ Given such an f, [f] ˜ is its image in the restricted ultrapower. lie in L. L Lemma 3.2. Let z ∈ R. Then L[z] |= ([h1 ]L˜ , . . . , [hm ]L˜ ) iff L[z] |= (h1 (), . . . , hm ()) a.e. (V ). Proof. The usual Ło´s proof works. Recall the following (Lemma 8.6 of [Kec78]).
Lemma 3.3. Let α ∈ Ord. Then there are uniform indiscernibles 1 , . . . , k ≤ α and a real z such that α is definable in L[z] from 1 , . . . , k . Proof of Theorem 3.1. Let z ∈ R and [f1 ]L˜ < · · · < [fk ]L˜ ≤ [id]L˜ uniform indiscernibles such that [id]L˜ = t L[z] ([f1 ]L˜ , . . . , [fk ]L˜ ). Let h : n → nk be h(α) = f1 (α), . . . , fk (α). We have a reverse map g : nk given by g(1 , . . . , k ) = t L[z] (1 , . . . , k ). Clearly g ◦ h = id a.e. Pick j such that Ajm,k ∈ h∗ (V ) (here of course m = n − 1). A unique such j exists since there are finitely many Ajm,k ’s, all disjoint. Let C ⊆ 1 be cub. Since [fi ] is a uniform indiscernible for i = 1, . . . , k, we have fi (α) ∈ C˜ a.e. j . (V ). Thus C˜ k ∈ h∗ (V ). So h∗ (V ) = Vm,k j That h ◦ g = id a.e. (Vm,k ) follows from the following general fact: Fact 3.4. Given (two valued) measure spaces (X, U ), (X , U ), h : X → X such that h∗ (U ) = U , and g : X → X such that g ◦ h = id a.e. (U ), then h ◦ g = id a.e. (U ). Proof of Fact. If A ∈ U and g(h(x)) = x for x ∈ A, then let B = h[A] ∈ U . On B, g is clearly equal to h −1 . (Fact) This completes the proof. (Theorem 3.1)
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3.3. Definition 3.5. A set S ⊆ is simple if there are m, j, k, C a cub subset k of 1 , and an F : m+1 → such that j (1) F is 1-1 on Am,k ∩ C˜ k (2) S = F [Ajm,k ∩ C˜ k ] ˜ (3) F ∈ L. Theorem 3.6 (Kunen [Kun71B]). If A ⊆ , then A is a countable union of simple sets. We need the following; where Θ = sup{ : is a surjective image of R}. Lemma 3.7. If I is a proper -ideal on , where < Θ, then there is a countably additive ultrafilter U on such that A ∈ I ⇒ A ∈ /U Proof. By Moschovakis [Mos70] let h : R ℘( ). So there is a h1 : R I . Let g : D → (where D = Turing degrees) be given by g(d ) = least α < α ∈ / {h(x) : x ≤T d } . Then if U0 is the Martin measure on D, g∗ (U0 ) = U is as desired. Proof of Theorem 3.6. Fix A ⊆ . Let I = {B ⊆ i< Ai : Ai is simple & (Ai ⊆ A or Ai ∩ A = ∅}. I is a -ideal. If I is not proper, then A ∈ I and we are done. So assume I is proper, towards a contradiction. By Lemma 3.7, let U be a countably additive ultrafilter such that B ∈ I ⇒ B ∈ / U . By Theorem 3.1 ˜ there are functions H : → (m+1 )k and G : (m+1 )k → , both in L, j which demonstrate that U is equivalent to Vm,k . j Case I: A ∈ U . Then H [A] ∈ Vm,k . so ∃ cub C ⊆ 1 such that H [A] ⊇ j j k ˜ ˜ Am,k ∩ C . Hence A ⊇ G[Am,k ∩ C k ] and we may assume G is 1-1 on Ajm,k ∩ C˜ k . Hence X = G[Ajm,k ∩ C˜ k ] is simple, so X ∈ I . Hence X ∈ / U,a j contradiction to H∗ (Vm,k )=U. Case II: A ∈ / U . Proof is similar to Case I. (Theorem 3.6) 3.4. Theorem 3.8 (Kunen [Kun71B], effectivized). There is a Δ13 coding of subsets of , i.e., there is a Δ13 C ⊆ R and a map C → ℘( ) taking ε to Xε , such that {Xε : ε ∈ C } = ℘( ), and the relation “w codes an ordinal < & ε ∈ Cε & |w| ∈ Xε ” is Δ13 . Proof. Let ε ∈ C∗ ⇔ ε = m, k, j, ϕ, α, & { < 1 : L[α] |= )} = F is ) ∈ ( )k+1 : L[α] |= (1 , . . . , r , , ϕ( )} = C is cub & {( , j k k ˜ a function from m+1 into which is 1-1 on Am,k ∩ C .
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For ε ∈ C∗ , let Xε∗ = F [Ajm,k ∩ C˜ k ]. Put ε ∈ C ⇔ ∀i(εi ∈ C∗ ). Let Xε = Xε∗i i
for ε ∈ C . Using the Kechris-Martin Theorem and its corollaries (cf. 2.11) it can be seen that this coding is Δ13 . 3.5. Corollary 3.9 (Kunen [Kun71B]). If U is an ultrafilter on , then i U ( 13 ) = 13 . Proof. We must show that if f : → 13 , then [f]U < 13 . It’s enough to consider f : → (since for ≤ < 31 Card( /U ) = Card( /U ). j
Furthermore, it is enough to show that i Vm,k ( ) < 13 . To see this: Define ≺ ε ⇔ , ε ∈ C & X , Xε “are” functions from Ajm,k → & [X ]V j < [Xε ]V j . m,k
m,k
This last inequality is expressed by < Xε ( ))]. there is a C ⊆ 1 cub [∀ ∈ C˜ k ∩ Ajm,k (X ( ) Thus ≺ is Σ13 , so it is bounded below 31 . 3.6.
Theorem 3.10 (Kunen [Kun71B]). For all < 13 , 13 → ( 13 ) . Proof. Assume ≥ . Fix t : · → 1-1 and onto. For < , define {, : , , ∈ Xε } if ε ∈ C (ε) = ∅ otherwise. For < · , let ε = (ε) where t( ) = . If ε “is” a well-ordering of , let |ε | be its length. Fix A ⊆ ( 13 ) ↑, and consider the game in which player I plays α, player II plays , andplayer II wins if either (1) For some < · , α or is not a w.o. of , and if 0 = the least such
, then α 0 is not a w.o., or (2) For all < · , α and are w.o.’s of , and {sup(max{|α·ϑ+n |, |·ϑ+n |})}ϑ< ∈ A.
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Assume player II has a winning strategy . Then for some < , < 13 , let Θ( , ) = sup{| [α] | + 1 : ∀ ≤ (α is a w.o. of & sup{|α | : ≤ } < }. Claim 3.11. Θ( , ) < 13 . Proof of Claim 3.11. It is easy to construct a Σ13 wellfounded relation on R of length ≥ Θ( , ). The rest of the proof is as in Lemma 11.1 of [Kec78]. (Claim 3.11) (Theorem 3.10) REFERENCES
Alexander S. Kechris [Kec78] AD and projective ordinals, this volume, originally published in Kechris and Moschovakis [Cabal i], pp. 91–132. Alexander S. Kechris and Donald A. Martin [KM78] On the theory of Π13 sets of reals, Bulletin of the American Mathematical Society, vol. 84 (1978), no. 1, pp. 149–151. Alexander S. Kechris and Yiannis N. Moschovakis [Cabal i] Cabal seminar 76–77, Lecture Notes in Mathematics, no. 689, Berlin, Springer, 1978. Kenneth Kunen [Kun71B] On 15 , circulated note, August 1971. Yiannis N. Moschovakis [Mos70] Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory. Proceedings of an international colloquium held under the auspices of the Israel Academy of Sciences and Humanities, Jerusalem, 11–14 November 1968 (Y. Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam-London, 1970, pp. 24–62. DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA BERKELEY, CALIFORNIA 94720-3840 UNITED STATES OF AMERICA
E-mail: [email protected]
AD AND THE PROJECTIVE ORDINALS
STEVE JACKSON
§1. Introduction. The purpose of this paper is to calculate an upper-bound for 12n+5 , n ≥ 0, assuming certain basic inductive assumptions concerning the projective ordinals. In a later paper, we will verify the lower bound for lower 1 2n+5 and establish the inductive assumptions at the next level. The case n = 0 appears in [Jac99], and may be obtained as a special case of the results in this paper. We assume the reader is familiar with the results in [Kec81A] and [Mar], although the reader may take as “axioms” the results needed. Indeed, the following paper is essentially self-contained, except for a knowledge of the homogeneous tree construction, which is only used indirectly. For background on the projective ordinals as well as their significance in descriptive set theory, we refer the reader to [Mos80]. We work in AD+DC throughout, although the inductive hypotheses at the lower levels suffice. §2. Definitions and preliminary results. In this section we define two families of measures; one being essentially the general measures allowed in the homogeneous tree construction, and the other a family of canonical measures. This is the only point in this paper where we use the homogeneous tree construction; the reader not familiar with it may take on faith the fact that our family captures the most general such measure. We first introduce our main inductive hypotheses: I2n+1 : 12n+1 has the strong partition relation, 12n+3 has the weak partition relation, and 12n+3 = ℵ(2n+1) , where (0) = 1 and (k + 1) = (k) (ordinal exponentiation). The author wishes to thank Tony Martin for many helpful conversations during the research for this manuscript. This work is an outgrowth of the calculations of 15 (the case n = 0 of this paper) which the author completed while working with Martin at UCLA. Aside from independently discovering many of the basic techniques and methods of use there, it was a few basic discoveries of Martin (such as [Mar]) which provided the impetus for this research. Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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K2n+3 : a) For any measure on Ξ2n+3 = the predecessor of 12n+3 , and any α < 12n+3 , j (α) < 12n+3 where j denotes the embedding from the ultrapower by the measure . b) For any g : 12n+3 → 12n+3 , and any normal measure V on 12n+3 , there 1 is a tree T on 2n+3 such that for some measure one set A with respect to V and all α∈ A, g(α) ≤ |T (sup j(α))|, where the sup ranges over embeddings j corresponding to measures on Ξ2n+3 which occur in the homogeneous tree construction on a Π12n+2 complete set. We remark that for n = 0, I2n+1 and K2n+3 (a) are well-known theorems of determinacy, and K2n+3 (b) is a theorem of Martin’s—see [Mar]. We assume I2n+1 and K2n+3 for the remainder of this paper, and establish the upper bound for 12n+5 , along with some additional auxiliary results. We introduce the family of canonical measures: n We let W1 = the n-fold product of the -cofinal normal measure on 1 . We identify the domain of W1n with an ordinal by ordering the n-tuples (α1 , . . . , αn ) by αn first, then αn−1 , etc. We define S11,n from the strong partition relation on 1 as follows: We let
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,m We let S2n+1 for 2 ≤ ≤ 2n+1 −1 be the measure induced by the strong partition relation on 12n+1 , functions F : 12n+1 → 12n+1 of the correct type, and the ,m measure ϑ2n+1 = the measure induced by the weak partition relation on 12n+1 , functions f : ϑ,m → 12n+1 of the correct type, and the measure R,m onϑ,m , 1,m m , S2n−1 , where R,m enumerates (w.r.t. ) the measures W1m , S11,m , . . . , W2n−1 2n −1,m ,m . . . , S2n−1 . That is, A has measure one w.r.t. S2n+1 ( > 1) if there is a c.u.b. subset C ⊆ 12n+1 such that for any F : 12n+1 → C of the correct type [F ]ϑ,m ∈ A. 2n+1
We frequently use the abbreviated versions of the above definitions. For = 1, we let <m denote the ordering on ( 12n+1 )m defined by (α1 , . . . , αm ) <m(1 , . . . , m ) iff (αm , α1 , . . . , αm−1 ) 1 ,m m and either S2n+1 or W2n+1 if = 1. ,m This completes the definition of the canonical family of measures R2n+1 . We now define a more general collection of measures R2n+1 = W2n+1 ∪ S2n+1 , where the measures in W2n+1 are measures on tuples of ordinals < 12n+1 , and the measures in S2n+1 are measures on tuples of ordinals < Ξ2n+3. For v ∈ R2n+1 , we let ϑv denote the set of tuples of ordinals on which v is a measure. By coding tuples, we may think of v as a measure on an ordinal, which we will also denote by ϑv , which should cause no confusion. We proceed j by induction on n, and we simultaneously define embeddings Πvv i : ϑv j → ϑv i , for certain v j , v i ∈ R. n = 0. W1 consists of the measures W1m on 1m , where a permutation m of {1, . . . , m} is associated with each such measure. We identify a measure in W1 with a measure on an ordinal by identifying (α1 , . . . , αm ) with its order-type in the ordering on these tuples where we order first by α1 , then α2 , etc., where m = (1 , 2 , . . . , m ). If V1 ∈ W1 with permutation m , and V2 ∈ W1 with permutation m+1 and the first m elements of m+1 are ordered (as integers) as the elements of m ,
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2
then we define vv1 = 12 by 12 (1 , . . . , m+1 ) = (1 , . . . , ˆk , . . . , m+1 ) where k is the last element in the permutation m+1 . S1 consists of products of basic measures in S1 , where we proceed to define the basic measures in S1 . For fixed n, we consider tuples of the form S = α1 , i1 , α2 , i2 , . . . , αm , im , where m ≤ n, α1 , . . . , αm < 1 , 1 ≤ i1 ≤ n1 , . . . , 1 ≤ im ≤ nm for some integers n1 , . . . , nm . We also assume that for each (i1 , . . . , ik ), where k < n, we have a permutation (i1 ,... ,ik ) of a size k + 1 subset of {1, . . . , n}, beginning with n, such that if (ı) extends (j), then (ı) extends (j) . We then require that for k < n, and fixed (i1 , . . . , ik ) that the ordinals (α1 , . . . , αk+1 ) are ordered as the integers in (i1 ,... ,ik ) . We let <s denote the Brouwer-Kleene ordering restricted to this set of tuples (i.e., s1 <s s2 iff s1 is less at the least point of disagreement, or if s1 extends s2 ). The measure in S1 then, is the measure on tuples (· · · , (i1 ,... ,im ) , · · · ) indexed by indices (i1 , . . . , im ), where m ≤ n as above, defined using the strong partition relation on 1 and functions F : <s → 1 of the correct type. For fixed F and (i1 , . . . , im ), the ordinal (i1 ,... ,im ) is represented with respect to the m-fold product of the -cofinal normal measure on 1 by the function k(1 , . . . , m ) = F (j1 , i1 , j2 , i2 , . . . , jm , im ), where (j1 , . . . , jm ) is the permutation of (1 , . . . , m ) ordered as (i1 ,... ,im−1 ) . We define Π on basic measures in S1 as follows: We let V1 , V2 be basic measures in S1 as above, with V1 corresponding to n1 , V2 corresponding to n2 , and n1 < n2 . We require that if (i1 , . . . , ik ) is an allowed index as in the definition of V1 , then it is allowed in V2 , and for 1 ≤ k ≤ n1 − 1 the integers 1 2 in (iV1 ,... ,ik ) and (iV1 ,... ,ik ) are ordered similarly. In this case, the tuples S1 as in the definition of V1 are tuples S2 allowed in V2 . Hence, an F2 : <s2 → 1 of the correct type induces an F1 : <s1 → 1 of the correct type. This, in 2 turn, induces the map Π21 = ΠV V 1 from ϑV 2 into ϑV 1 . It follows readily that if A ⊆ ϑV 1 has measure one w.r.t. V 1 then for almost all ϑ < ϑV 2 w.r.t. V 2 , Π21 (ϑ) ∈ A. We extend Π to products of basic measures in S1 componentwise. It also follows readily that Π extended to products has the same projection property mentioned just above. n ≥ 1. We assume W2m+1 , S2m+1 have been defined for m < n, and define W2n+1 , S2n+1 . For V1 , V2 , measures in m
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2) If (i1 , . . . , i ) ∈ dom T , then (i1 , . . . , ik ) ∈ dom T for k ≤ . 3) T associates measures to certain of the tuples (i1 , . . . , ik ), where k < n, in its domain, which we denote by v (i1 ,... ,ik ) , where v (i1 ,... ,ik ) ∈ m 1. For k = 1, we take α = 1, and allow 1(i1 ) ∈ dom
i1 , Π(i11 )
(i ,... ,ik−1 )
(α), i2 , Π(i11 ,i2 )
(α), . . . , α, ik
(j ,... ,j−1 )
(j ,... ,j−1 )
(), j2 , Π(j11 ,j2 )
(), . . . , , jk
where
)
(i ,... ,i
)
(i1 ,... ,ik )
,im+1 ) 1 Π(i11 ,... ,ik−1 = Π(i(i11 ,... ◦ Π(i1 ,... ,im+1 , where ,... ,im ) ) m)
Π(i11 ,... ,ik−1 = identity. k−1 ) T
We let < be the measure on tuples (· · · , (i1 ,... ,ik ) , · · · ) of ordinals < 12n+1 induced by the weak partition relation on 12n+1 , functions f : 1, and if k = 1 and i1 ∈ dom
2) If (i1 , . . . , in+1 ) ∈ dom T2 , then vT(i21 ,... ,in ) < vT11
.
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3) If 1i1 ∈ dom
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We let
< by a function g(· · · , (i1 ,... ,ik ) , · · · ) defined as follows: if h :
T2 It follows readily that this is well-defined on a measure one set w.r.t. S2n+1 C . 2 We take Π to be zero off this measure one set.
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This completes the definitions of the measures R2n+1 , R2n+1 , and the embeddings Π. We define an equivalence relation ∼ on the measures in R2n+1 by induction as follows: We set V1 ∼ V2 provided one of the following holds: 1,m1 1,m2 and V2 = S2n+1 1) V1 = S2n+1 m1 m2 2) V1 = W2n+1 , V2 = W2n+1 1 ,m1 2 ,m2 3) V1 = S2n+1 , V2 = S2n+1 , where 1 , 2 > 1, in which case 1 = 2 . So, V1 ∼ V2 if V1 and V2 are in the same canonical family of measures. We also let V1 < V2 mean that V1 ∼ V2 and m1 < m2 for m1 , m2 as above. ,m A useful notation is to let, for > 1, v(S2n+1 ) denote the measure v such ,m that S2n+1 is defined from the strong partition relation on 12n+1 , functions F : 12n+1 → 12n+1 of the correct type, and the measure v= the measure weak partition on 1 , functions f : ϑ → 1 induced by the v 2n+1 2n+1 of the correct type and the measure v. 2 ,m2 1 ,m1 ) ∼ v(S2n+1 ). We note that (by induction) 3 above is equivalent to v(S2n+1 We prove some basic lemmas which will be used frequently. We recall we are assuming I2n+1 throughout. Lemma 2.5. If V1 , V2 ∈ R2n+1 and V1 ∼ V2 , then cf(ϑV1 ) = cf(ϑV2 ). Proof. If not, we fix V1 ∼ V2 , such that cf(ϑV1 ) = cf(ϑV2 ). 1 ,m1 2 ,m2 We consider first the case where V1 = S2n+1 , V2 = S2n+1 , where 1 , 2 > 1 and 1 = 2 . We note that ϑV1 , ϑV2 are always limit ordinals. We fix H : ϑV1 → ϑV2 monotonicially increasing and cofinal. We consider the following partition P: We partition pairs of functions F1 , F2 : 12n+1 → 12n+1 of the correct type, where we require that for f1 : v(V1 ) → 12n+1, f2 : v(V2 ) → 12n+1 , if supa.e. f1 < supa.e. f2 , then F1 ([f1 ]v(V1 ) ) < F2 ([f2 ]v(V2 ) ) and similarly if supa.e. f2 < supa.e. f1 then F2 ([f2 ]v(V2 ) ) < F1 ([f1 ]v(V1 ) ). If supa.e. f1 = supa.e. f2 (hence one of f1 , f2 is not of the correct type a.e., since cf(ϑv(V1 ) ) = cf(ϑv(V2 ) ) by induction), we require F1 ([f1 ]) < F2 ([f2 ]). In using the strong partition relation on 12n+1 , we think of the pair F1 , F2 as being coded by correct type. We then partition such F , F (or a single function of the 1 2 more precisely the single function coding them) according to whether or not H ([F1 ]v1 ) < [F2 ]v2 , where v1 = v1 (V1 ), v2 = v2 (V2 ). We claim that on the homogeneous side of the partition, the property stated in partition P holds. We suppose not, and fix a c.u.b. C ⊆ 12n+1 homogeneous for the contrary side. We assume that C is closed under jv1 , jv2 , the ultrapowers from the measures v1 , v2 (we use K2n+1 (a) here), and C consists of limit ordinals. We fix F1 : 12n+1 → C of the correct type → C of the correct type with with F1 (α) > α for all α, and fix F2 : 12n+1 [F2 ] > H ([F1 ]) and F2 (α) > α for all α. We may also assume that F2 (α) is greater than the jv1 (α)th element of C after α.
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We let C be a c.u.b. subset of the closure points of C (= { ∈ C : for all α < , is the th element of C after α}) which is closed under F1 , F2 in the sense that for all ∈ C , if α < and f1 : ϑv1 → α then F1 ([f1 ]v1 ) < , and similarly for F2 . We then claim that there are functions F1 , F2 satisfying: 1) F1 = F1 , F2 = F2 , a.e. w.r.t. v1 and v2 respectively. 2) F1 , F2 have range a subset of C and are of the correct type. 3) F1 , F2 are ordered as in the statement of P, everywhere. We define F1 , F2 as follows: If ϑ ∈ C and cf ϑ = cf ϑv1 , then we set = F1 ([f]) for any f : ϑv1 → 12n+1 s.t. supa.e. f = ϑ. If ϑ ∈ C and cf ϑ = cf ϑv2 , then we set F2 ([f]) = F2 ([f]) for f : ϑv2 → 12n+1 with supa.e. f = ϑ. For other values of F1 and F2 we proceed inductively. That is, if F1 (ϑ) or F2 (ϑ) is not defined by the previous paragraph we proceed as follows to define it. Assume < 12n+1 and F1 (1 ) and F2 (2 ) are defined whenever 1 = [f1 ]v1 with supa.e. f1 < and likewise for 2 . Suppose ϑ = [f1 ]v1 , where supa.e. f1 = . If F1 (ϑ) is not already defined, set F1 (ϑ) = NC max sup F1 (α), sup F2 () , F1 ([f])
α
where α ranges over the ordinals less than ϑ and ranges over ordinals represented by f¯2 : ϑv2 → 12n+1 with supa.e. f¯2 < . Here NC () denotes the . For ϑ = [f ] with sup f = , if F (ϑ) th element of C greater than 2 v2 a.e. 2 2 is not already defined set F2 (ϑ) = NC max sup F1 (α), sup F2 () , α
where now α ranges over the ordinals represented by f¯1 : ϑv1 → 12n+1 with supa.e. f¯1 ≤ and ranges over the ordinals less than ϑ. We claim that F1 , F2 satisfy the above properties. Property 1 follows since F1 = F1 for all f : ϑv1 → C of the correct type, and similarly for F2 . Property 2 is immediate from the definition of F1 , F2 . Property 3 follows from the definition of F1 , F2 , the fact that F1 (α) > α, F2 (α) > α, and the definition of C . For example, for f1 : ϑv1 → C of the correct type, F1 ([f1 ]) = F1 ([f1 ]) ≥ supa.e. f1 = ∈ C , and hence F1 ([f1 ]) > > F2 ([f2 ]) for all f2 : ϑv2 → 12n+1 with supa.e. f2 < . It follows easily using the definition of ]) > > F ([f ]) as well. A similar argument shows that if C that F1 ([f 1 2 2 f2 : ϑv2 → C is of the correct type with supa.e. f2 = , then for all [f1 ] with supa.e. f1 ≤ we have F2 ([f2 ]) > F1 ([f1 ]) (here we use the fact that F2 ([f2 ]) is greater than the jv1 ()th element of C greater than ).
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From property 1, we have that [F2 ] > H ([F1 ]), which contradicts properties 2, 3 and the definition of C . Hence, on the homogeneous side of the partition, the property stated in partition P holds. We fix a c.u.b. C ⊆ 12n+1 homogeneous for P. We define F2 as follows. For α < 12n+1 represented by f2 : ϑv2 → 12n+1 with supa.e. f2 = ϑ, we let F2 (α) C greater then ϑ. Note that F is of be the · (jv1 (ϑ) + α + 1)st element of 2 the correct type with range in C . We then claim for almost all [F1 ]v1 that H ([F1 ]) < [F2 ], contradicting the fact that H is cofinal, and establishing Lemma 2.5. We fix F1 : 12n+1 → C of the correct type. To prove the claim, it is enough to find F1 , F2 satisfying: 1) F1 = F1 , F2 = F2 , a.e. w.r.t. v1 and v2 respectively. 2) F1 , F2 have range a subset of C and are of the correct type. 3) F1 , F2 are ordered as in P. The construction of F1 , F2 proceeds similarly to the above, starting with a c.u.b. C ⊆ 12n+1 contained in the closure points of C , and such that for α ∈ C and < α, if supa.e. f1 ≤ then F1 ([f1 ]) < α and likewise if supa.e. f2 ≤ then F2 ([f2 ]) < α. So, if supa.e. f1 = ∈ C with cf = ϑv1 , then we set F1 ([f1 ]) = F1 ([f1 ]) and likewise if supa.e. f2 = ∈ C with cf = ϑv2 , then F2 ([f2 ]) = F2 ([f2 ]). For other values of F1 and F2 we proceed as before. Again it is easy to check that F1 , F2 have the required properties. This completes the proof of Lemma 2.5 in the case considered. If 1 = 1, 2 > 1 (or vice-versa), the argument is similar. 2 ,m2 m If V1 = W2n+1 , V2 = S2n+1 (or vice-versa), the argument is again similar. 1 We fix an H : 2n+1 → ϑv2 cofinal, monotonically increasing, and consider the partition P: we partition pairs of functions f1 , F2 where f1 : ϑv1 → 12n+1 the n 1 is of the correct type, where W2n+1 = v2n+1 , and F2 : 12n+1 → 12n+1 of correct type, where sup f1 < inf F2 . We partition according to whether or not H ([f1 ]v1 ) < [F2 ]v2 . It follows by a similar argument that on the homogeneous side of the partition, the above property holds. By another similar argument, we contradict the cofinality of H m. If V1 ∈ m
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Lemma 2.6. For any V ∈ R2n+1 , and any ordinal , and H : ϑV → , either α w.r.t. V, or there is a there is a < such that H (α) < for almost all measure one set A w.r.t. V such that H A is dominated by a monotonically increasing function into . ,m with > 1. We fix such a Proof. We consider first the case V = S2n+1 and H . We assume the first clause fails and show the second. In particular, for all α < ϑV , we have that for cofinally many < ϑV , H (α) < H (). We then consider the partition P: we partition F1 , F2 : 12n+1 → 12n+1 of the w.r.t. v= v(V) by correct type ordered as follows: if α1 , α2 are represented f1 , f2 : ϑv → 12n+1 , and ϑ1 , ϑ2 = supa.e. f1 , f2 respectively, then if ϑ1 < ϑ2 , (α ) (here F denotes either F or F ). If ϑ > ϑ , then F1,2 (α1 ) < F1,2 2 1,2 1 2 1 2 F1,2 (α1 ) > F1,2 (α2 ). If ϑ1 = ϑ2 , then F1 (α1 ) < F2 (α2 ). We partition such pairs F1 , F2 according to whether or not H ([F1 ]v ) < H ([F2 ]v ). From our above assumption, and a sliding argument as in Lemma 2.5, it follows that on the homogeneous side of the partition, the property stated in P holds. We fix a c.u.b. subset C ⊆ 12n+1 homogeneous for P, and let C be the set of closure A to be the measure one set determined by C . For points of C . We take ∈ A represented by F : 12n+1 → C of the correct type, we define H () = sup(H ( )), where the sup ranges over represented by F : 12n+1 → C of the correct type, and < [F0 ]v , where F0 : 12n+1 → 12n+1is defined by: for α represented by f : ϑv → 12n+1 w.r.t. v, F0 (α) = supα F (α ), where w.r.t. v by f of the correct type with the sup ranges over α represented supa.e. f ≤ supa.e. f. This is well-defined in that the equivalence class of F0 depends only on the equivalence class of F with respect to v . It is clear that H A is monotonic increasing and dominates H . To show that it has the range in , and finish the proof of the claim, it is enough to establish the following claim: If F1 , F2 : 12n+1 → C are of the correct type and (F1 )0 < (F2 )0 a.e. w.r.t. v (where F0 is defined above for F : 12n+1 → 12n+1 ), then H ([F1 ]) < H ([F2 ]). to show, for such F , F , that there are To establish the claim, it is enough 1 2 F1 , F2 satisfying 1) F1 = F1 , F2 = F2 , a.e. w.r.t. v . 2) F1 , F2 have range a subset of C and are of the correct type everywhere. 3) F1 , F2 are ordered as in P. To establish this we use a sliding argument similar to, but somewhat different from, that of Lemma 2.5. We consider the partition P : we partition f : ϑv → 12n+1 of the correct type according to whether or not F2 ([f]) > (F1 )0 ([f]). It follows readily that on the homogeneous side of the partition, the property stated in P holds. We let C2 ⊆ C be a c.u.b. subset of 12n+1 homogeneous for P , where C2 is closed under jv and (F1 )0 , (F2 )0 . Welet C2 be the set of closure points of C2 . We define F1 , F2 as follows:
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1) For α represented by f : ϑv → C2 of the correct type, we set F1 (α) = F1 (α) and F2 (α) = F2 (α). 2) For α not as in 1, if α is represented by f : ϑv → 12n+1 , and ϑ = supa.e. f ∈ C2 , and f < ϑ a.e. w.r.t. v, we set F1 (α) = F1 (α2), F2 (α) = F2 (α2 ), where α2 is represented by f2 : ϑv → 12n+1 defined as follows. We let α be the least ordinal greater than α which is represented by some h : ϑv → ϑ which is monotonically increasing (it follows readily from Lemma 2.6 and induction that such α exists). We fix a monotonic h : ϑv → ϑ representing α . For < ϑv , we then set f2 () = the 2 · (( − 1) + f() + 1)st element of C after h(), where − 1 = if is a limit ordinal. It follows that α2 is well-defined. 3) For α = [f] not as in 1, ϑ ∈ C2 , and f = ϑ a.e. w.r.t. v, we set F1 (α) = NC (supα F1 (α )), where the sup ranges over α represented by f with ϑ = supa.e. f ≤ ϑ, and f < ϑ almost everywhere. Here NC () = the next element in C > . We also set F2 (α) = NC (supα F2 (α )), with α ranging over the same set. / C2 , we proceed inductively. 4) For α = [f] not as in 1, and ϑ = supa.e. f ∈ We set F1 (α) = NC (max(supα <α F1 (α ), sup F2 ())), where ranges over the ordinals represented by g : ϑv → 12n+1 with supa.e. g < ϑ. Also, ))), where now ranges over F2 (α) = NC (max(supα <α F2 (α ), sup F1 ( the ordinals represented by g with supa.e. g ≤ ϑ. It is immediate that F1 = F1 and F2 = F2 almost everywhere, that F1 , F2 have range a subset of C , and are non-normal of uniform cofinality . The fact that F1 , F0 are strictly increasing and ordered as in P follows from the definitions of C , C2 , F1 , and F2 upon consideration of several cases. The claim now follows. This establishes Lemma 2.6 in this case. The other cases are similar. (Lemma 2.6) Lemma 2.7. If V1 , V2 ∈ m≤n R2m+1 , and V1 ∼ V2 then V1 × V2 = V2 × V1 . That is, if A ⊆ ϑV1 × ϑV2 , and A has measure one with respect to V1 × V2 (that is, for almost all α ∈ ϑV1 , for almost all ∈ ϑV2 , (α, ) ∈ A), then A has measure one w.r.t. V2 × V1 (for almost all ∈ ϑV2 , for almost all α ∈ ϑV1 , (α, ) ∈ A) 1 ,m1 2 ,m2 Proof. We again consider the case V1 = S2n+1 , V2 = S2n+1 , with 1 , 2 > 1, 1 = 2 , the other cases being similar. We fix A having measure one w.r.t. V1 × V2 , and having measure zero w.r.t. V2 × V1 towards a contradiction. We consider the partition P: we partition pairs of function F1 , F2 : 12n+1 → 12n+1 of the correct type, and ordered as in the partition of Lemma 2.5, according to whether or not ([F1 ], [F2 ]) ∈ A. We similarly consider the partition P : we partition F2 , F1 : 12n+1 → 12n+1 as above, ordered as in the partition of Lemma 2.5 (with roles of F1 , F2 reversed), according to whether or not ([F1 ], [F2 ]) ∈ ¬A.
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It then follows from a sliding argument similar to that of Lemma 2.5 that on the homogeneous side of these partitions, the properties stated in P, P hold. We fix a c.u.b. C ⊆ 12n+1 homogeneous for P, P , and fix F1 , F2 : 12n+1 → C ordered as in P. Hence ([F ], [F ]) ∈ A. We then of the correct type, and 1 2 let C be the set of closure points of a c.u.b. subset of C consisting of limit ordinals, closed under jv1 , jv2 (where v1 = v(V1 ), v2 = v(V2 )), and closed under (F1 )0 , (F2 )0 (as in Lemma 2.6). It then follows from a sliding argument as in Lemma 2.5 (using the fact that cf ϑv1 = cf ϑv1 ) that there are F2 , F1 such that 1) F2 = F2 , F1 = F1 almost everywhere. 2) F2 , F1 have range a subset of C and are of the correct type. 3) F2 , F1 are ordered as in P . / A, a contradiction, which Hence it follows that ([F1 ], [F2 ]) = ([F1 ], [F2 ]) ∈ establishes Lemma 2.7. (Lemma 2.7) A simple but frequently used lemma is the following: Lemma 2.8. For any c.u.b. subset C ⊆ 12n+1 , there is a c.u.b. C ⊆ C predecessor of 1 , and any such that for any measure v on Ξ2n+1 = the 2n+1 respect to v f : ϑv → C , if f is of the correct type almost everywhere with (i.e., there is a measure one set A w.r.t. v such that fA is of the correct type), then there is an f2 : ϑv → C such that f2 = f1 almost everywhere w.r.t. v, and such that f2 is of the correct type everywhere. Proof. Given C , take C to be the set of the closure points of a c.u.b. subset of C consisting of limit ordinals. Then for f : ϑv → C , and A ⊆ ϑv of measure one such that fA is of the correct type, define f2 by: 1) for α ∈ A, set f2 (α) = f(α). 2) for α ∈ / A, set f2 (α) = NC (supα <α f2 (α )). It follows readily that f2 has the required properties. §3. A Global Embedding let denote an n − ∗ tuple with Theorem. We respect to the measures m≤n S2m+1 ∪ m≤n W2m+1 and let < denote the corresponding ordering on tuples of integers and ordinals < Ξ2n+3 . Since we are assuming I2n+1 , < 2n+3 is defined. We state the theorem of the section:
Theorem 3.1. For each n − ∗ tuple and corresponding measure < 2n+3 , < 1 there is an integer m such that the ultrapower of 2n+3 by 2n+3 is less than or m equal to the ultrapower of 12n+3 by W2n+3 . In order to prove this, we require an inductive hypothesis: B2n+1 : We let T be an n − ∗ tupleT with respect to the measures m≤n−1 S2m+1 ∪ m≤n−1 W2m+1 and let < be the corresponding ordering
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,m on tuples from Ξ2n+1 . Then there is an integer m and a measure S2n−1 , where
,m = 2n − 1 is maximal, such that K2n−1 = ϑS ,m is greater than the order type 2n−1
of
,m ,m 4) If S2n−1 satisfies the above, then so does S2n−1 for any m > m.
We have defined B2n+3 as well. We first show that the theorem follows from B2n+3 . We produce an em bedding M from the ultrapower of 12n+3 by < 2n+3 into the ultrapower by the indices in the domain of . We m W2n+3 . We let (I1 , . . . , Ik , . . . ) denote < 1 let F1 from dom(2n+3 ) to 2n+3 be given, representing [F1 ] with respect < to the measure < 2n+3 . Recall the elements of dom(2n+3 ) are tuples of the form (. . . , α (I1 ,... ,Ik ) , . . . ) where α (I1 ,...,Ik ) < 12n+3 . We define M ([F1 ]) m to be the ordinal represented with respect to W2n+3 by F2 defined as fol,m lows: given f : K2n+1 → 12n+3 (where = 2n+1 − 1 is maximal), F2 ([f]) is represented w.r.t. the measure U (the measure given by B2n+3 , this is the analog of u as in B2n+1 ) by the function which assigns to ∈ ϑU the ordinal F1 (. . . , (I1 ,... ,Ik ) , . . . ), where (I1 ,... ,Ik ) is represented w.r.t. V (I1 ,... ,Ik ) (the measures from the n − ∗ tuple ) by the function (f ◦ F )(I1 ,... ,Ik ) , where ,m F : < → K2n+1 is the map corresponding to as in B2n+3 and the superscript denotes the subfunction induced by restriction. It follows readily from B2n+3 that the embedding M is well-defined. We assume B2n+1 throughout the remainder of this section and proceed to establish B2n+3 . We fix an n − ∗ tuple , and corresponding ordering < on Ξ2n+3 . We let (I1 , . . . , Ik ), V (I1 ,... ,Ik ) denote indices and measures corresponding to , for k ≤ n, k ≤ n − 1 respectively. We have that r / T (I1 ,... ,Ik ) V (I1 ,... ,Ik ) = r rV (I1 ,... ,Ik ) , where rV (I1 ,... ,Ik ) is of the form S2n+1 rC(I ,... ,I ) , for 1 k rT (I ,... ,I )
1 k some rT (I1 , . . . , Ik ),rC(I1 , . . . , Ik ), or of the form < , or is a measure 2n+1 r r (I1 ,... ,Ik ) v ∈ m≤n R2m−1 . Here each C(I1 , . . . , Ik ) is a collection defined relative to the k − ∗ tuple rT (I 1 , . . . , Ik ) (which is k − ∗ tuple relative to the measures in m≤n−1 S2m+1 ∪ m≤n−1 W2m+1 ).
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STEVE JACKSON
From B2n+1 , there is an m and measures ru (I1 ,... ,Ik ) , corresponding to each T (I1 , . . . , Ik ) in the first case above, such that the statement of B2n+1 is r ,m satisfied by < T (I1 ,... ,Ik ) , S2n−1 (here = 2n − 1), and ru (I1 ,... ,Ik ) . We fix such an m. ,m , where = 2n+1 − 1 is maximal, and claim that B2n+3 We consider S2n+1 ,m is satisfied by < and S2n+1 . We may also take any m > m in the following. We define the measure U as in the statement B2n+3 . We let Ω be the set of tuples of ordinals and integers of the form: r
α0 , I1 , . . . , rα1(I1 ,... ,Ik ) , . . . , I2 , . . . , rα2(I1 ,... ,Ik ) , . . . , In−1 , . . . , rαn(I1 ,... ,Ik ) , . . . , In , or an initial segment of such a sequence, where: 1) α0 < 12n+1 2) The indices (I1 , . . . , Ik ) correspond to the indices in the domain of . 3) The indices (I1 , . . . , Ik ) associated with rαj correspond to the indices (I1 , . . . , Ik ) in rC(I1 , . . . , Ij ) if rV (I1 ,... ,Ij ) ∈ S2n+1 , and to the indices (i1 , . . . , ik ) ∈ dom rT (I1 , . . . , Ij ) if rV (I1 ,... ,Ij ) ∈ W2n+1 . If rV (I1 ,... ,Ij ) ∈ m 1, we define a set M (I1 , . . . , Ik ) of indexed ordinals and an ordering <M (I1 ,... ,Ik ) on them as follows: (I1 ,... ,Im ) M (I1 , . . . , Ik ) consists of indexed ordinals of the form rϑ(I where 1 ,... ,I ) (I1 , . . . , I ) is an initial segment of (I1 , . . . , Ik ), (I1 , . . . , Im ) is an index in r C (I1 , . . . , I ) and ϑ ∈ ru (I1 ,... ,I ) , for rV (I1 ,... ,Ik ) ∈ S2n+1 ; and of the form r (i1 ,... ,ik ) ϑ(I1 ,... ,I ) , where (I1 , . . . , I ) is an initial segment of (I1 , . . . , Ik ), (i1 , . . . , ik ) ∈ dom rT (I1 , . . . , I ) and ϑ is in the domain of the measure v (i1 ,... ,ik−1 ) defined by the lower-level n − ∗ tuple (for some n ) rT (I1 , . . . , I ) when rV (I1 ,... ,Ik ) ∈ W2n+1 . We also allow in the set M (I1 , . . . , Ik ) un-indexed ordinals ϑ < ,m ). ϑ(K2n−1
379
AD AND THE PROJECTIVE ORDINALS 1 (I11 ,... ,Im ) 1
2 (I12 ,... ,Im ) 2
We define r1ϑ1(I1 ,... ,I ) <M (I1 ,... ,Ik ) r2ϑ2(I1 ,... ,I ) to hold provided one of the 1 2 following cases is satisfied: 1) r1 = r2 and r1V (I1 ,... ,I1 ) , r2V (I1 ,... ,I2 ) ∈ S2n+1 . In this case, we require that r 1 ,... ,I ) there is a c.u.b. C ⊆ 12n+1 such that for F1 , F2 :
r
T (I1 ,... ,I )
T (I1 ,... ,I )
,m →
3) 4)
5) 6)
2
12n+1 , and f1 , f2 , g1 , g2 as above F1 (S1 ) < F2 (S2 ), where S1 is the se quence corresponding to g1 and (I11 , . . . , Im1 1 ), and similarly for S2 . r1 V ∈ W2n+1 and r2V ∈ S2n+1 . r1 = r2 = r and rV (I1 ,... ,I1 ) , rV (I1 ,... ,I2 ) ∈ W2n+1 . We then require that r T (I ,... ,I ) r 1 k for almost all f : < T (I1 ,... ,Ik ) → 12n+1 w.r.t. < with induced func2n+1 r r tions f1 : < T (I1 ,... ,I1 ) → 12n+1 , f2 : < T (I1 ,... ,I2 ) → 12n+1 , and subfunc (i11 ,... ,ik1 ) (i12 ,... ,ik2 ) (i11 ,... ,ik1 ) 1 2 1 tions f1 : ϑv (i1 ,... ,ik1 −1 ) → 12n+1 , and f2 , that f1 (ϑ1 ) < 2 2 (i1 ,... ,ik ) 2 (ϑ2 ). f2 r1 = r2 and r1V , r2V ∈ W2n+1 . In this case, we require r1 < r2 . ,m r1 V ∈ W2n+1 and ϑ2 ∈ K2n−1 .
,m and ϑ1 < ϑ2 . 7) ϑ1 , ϑ2 ∈ K2n−1
8)
,m V ∈ S2n+1 and ϑ2 ∈ K2n−1 . We require that there is a c.u.b. C ⊆ 12n+1 r T (I1 ,... ,I ) such that for F :
r1
1
,m almost all f : K2n−1 → 12n+1 of the correct type, with f1 , g1 as in case (1), that F (S) < f(ϑ2 ), where S corresponds to g and (I11 , . . . , Im1 1 ) as in r1 the definition of V . ,m 9) ϑ1 ∈ K2n−1 and r2V ∈ S2n+1 . Similar to (8) where we now require that f(ϑ1 ) < F (S).
We identify ϑ1(... ) , ϑ2(... ) in <M (I1 ,... ,Ik ) if neither ϑ1 <M ϑ2 or ϑ2 <M ϑ1 holds.
380
STEVE JACKSON
The measure U is a measure on tuples (. . . , (I1 ,... ,Ik ) , . . . ), indexed by indices (I1 , . . . , Ik ) in , which is induced by the strong partition relation on 12n+1 / functions H : <Ω → 12n+1 of the correct type, and the measures rru × ,m K2n−1 (I1 ,... ,Ik ) for k > 1, and 2n+1 for k= 1. Here Πr ru denotes the subproduct <M 2n+1 of V consisting of those measures in m
This completes the definition of the measure U . We proceed to define the family of embeddings F corresponding to U as in the statement of B2n+3 . We fix ∈ ϑU (the domain of the measure U ) which is represented w.r.t. the / (I1 ,... ,Ik ) by a function H : <Ω → 12n+1 of the correct type measures r ru × M 2n+1 to define F : < → (which happens for almost all w.r.t. U ). We proceed ,m (I1 ,... ,Ik ) K2n+1 . We fix an element α in the domain of < , where (I1 , . . . , Ik ) ∈ dom , and define F (α (I1 ,... ,Ik ) ) (for almost all α w.r.t. V (I1 ,... ,Ik−1 ) for k > 1). We see that α = (. . . , r , . . . , r (i1 ,... ,i ) , . . . , r (I1 ,... ,Im ) , . . . ), where the components r correspond to rV ∈ m
r
1 ,... ,I ) have functions rF : < T (I1 ,... ,Ik ) → 12n+1 and rF :
respectively. We define a function G : 12n+1 → 12n+1 represent ,m S2n−1 ,m ing F (α (I1 ,... ,Ik ) ) w.r.t. 2n+1 . We let g : K2n−1 → 12n+1 of the correct 1 < 2n+1
381
AD AND THE PROJECTIVE ORDINALS
type be given, and define G([g]). We set G([g]) = H (T ), where T = 0 , I1 , . . . , r1(I1 ,... ,Im ) , . . . , Ij , . . . , rj(I1 ,... ,Im ) , . . . , Ik is as in the definition of Ω, and the ’s are given by: 1) 0 = [g]S ,m . 2n−1 1 ,... ,Ik ) r ( ). 2) If rV (I1 ,... ,Ij ) ∈ m
1 ,... ,Im ) (ϑ) = where), and Q = g ◦ fϑ : < T (I1 ,... ,Ij ) → 12n+1 , we set rM(I j r F (S(Q, (I1 , . . . , Ij ), (I1 , . . . , Im ))), where S(· · · ) denotes the corresponding sequence as in the definition of rV . This is well-defined for almost ,m all g. We use here the fact that the ϑ ∈ K2n−1 are cofinal in the ordering <M (I1 ,... ,Ik ) . This completes the definition of F . To show that this is well-defined, we verify the following: 1 ,... ,Im ) 1) For fixed H, (r1 , . . . ,rp ), rF , and g we have that rM(I (ϑ) is wellj defined. This follows since fϑ is determined almost everywhere w.r.t. the v (i1 ,... ,ik ) , hence so is Q above, hence the sequence S above is well-defined.
2) For fixed H, (r1 , . . . ,rp ), rF , we have that the equivalence class of S
,m
,m 2n−1 F (α (I1 ,... ,Ik ) ) is well-defined with respect to 2n+1 . If g1 , g2 : K2n−1 → 12n+1 ,m agree on a measure one set A w.r.t. S2n−1 then for any (I1 , . . . , Ij ) as above, it r follows from B2n+1 that for almost all ϑ w.r.t. ru (I1 ,... ,Ij ) that fϑ : < T (I1 ,... ,Ij ) → ,m has range a.e. in A w.r.t. the measures v (i1 ,... ,ik−1 ) , where (i1 , . . . , ik ) ∈ K2n−1 r dom T (I1 , . . . , Ij ). We are assuming rV (I1 ,... ,Ik ) ∈ S2n+1 , the other cases being immediate. Hence, for Q1 = g1 ◦ fϑ , Q2 = g2 ◦ fϑ , for each (I1 , . . . , Im ) in r C(I1 , . . . , Ij ), it follows that
S(Q1 , (I1 , . . . , Ij ), (I1 , . . . , Im )) = S(Q2 , (I1 , . . . , Ij ), (I1 , . . . , Im )), 1 ,... ,Im ) 1 ,... ,Im ) and hence rM(I (1)(ϑ) = rM(I (2)(ϑ) for almost all ϑ w.r.t. j j
u , and all (I1 , . . . , Im ). Hence rj(I1 ,... ,Im ) (1) = rj(I1 ,... ,Im ) (2), and hence the sequences T1 , T2 corresponding to g1 , g2 are the same. Hence, G([g1 ]) = G([g2 ]).
r (I1 ,... ,Ij )
3) For fixed H : <Ω → 12n+1 , we have that F is well defined w.r.t. the choice of the rF representing α (I1 ,... ,Ik ) , for rV (I1 ,... ,Ik ) ∈ S2n+1 or W2n+1 . For each such r r 1 ,... ,Ik ) → 12n+1 or from < T (I1 ,... ,Ik ) → 12n+1 be given, r, we let rF1 , rF2 :
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STEVE JACKSON
representing rα (I1 ,... ,Ik ) . We let C be a c.u.b. subset of 12n+1 such that for r’s of r the first type, f : < T (I1 ,... ,Ik ) → C of the correct type, (I1 , . . . , Im ) an index r in C(I1 , . . . , Ik ), and S = S(f, (I1 , . . . , Ik ), (I1 , . . . , Im )) the corresponding ,m sequence, we have that rF1 (S) = rF2 (S). For g : K2n−1 → C of the correct type (where we also assume C is sufficiently closed w.r.t. the rF ’s so the definition of F makes sense), j ≤ k, (I1 , . . . , Im ) an index in rC(I1 , . . . , Ij ) (we consider here the case rV (I1 ,... ,Ij ) ∈ S2n+1 ) it follows that for almost all ϑ w.r.t. ru (I1 ,... ,Ij ) that Q = g ◦ fϑ has range almost everywhere in C and is of the correct type almost everywhere, since g is. By Lemma 2.8 we may assume Q has range in C and is of the correct type everywhere. Hence, for 1 ,... ,Im ) (1)(ϑ) = rF1 (S(Q, (I1 , . . . , Ij ), (I1 , . . . , Im ))) = almost all ϑ, rM(I j F2 (S(Q, (I1 , . . . , Ij ), (I1 , . . . , Im ))) = rMj(I1 ,... ,Im ) (2)(ϑ). Hence, the r’s appearing in the sequence T are the same for rF1 , rF2 in the case rV ∈ S2n+1 . The other cases are immediate. Hence the value of G([g]) is the same when computed using rF1 or rF2 .
r
M (I ,... ,I )
1 k 4) If H1 , H2 : <Ω → 12n+1 of the correct type agree a.e. w.r.t. < , 2n+1 ,m then the embeddings F1 , F2 : < → K2n+1 corresponding to H1 , H2 agree, for each fixed (I1 , . . . , Ik ) ∈ dom , almost everywhere w.r.t. V (I1 ,... ,Ik−1 ) = / r (I1 ,... ,Ik−1 ) , if k > 1, and agree if k = 1. r V
We fix a c.u.b. C ⊆ 12n+1 such that for (I1 , . . . , Ik ) ∈ dom , and the of the correct corresponding ordering <M (I1 ,... ,Ik ) , if h : <M (I1 ,... ,Ik ) → C is/ r1 type, then for all ( , . . . , rp ) ∈ A, a measure one set w.r.t. r rv, we have H1 (T ) = H2 (T ), where T = 0 , I1 , . . . , Ij , . . . , rj(I1 ,... ,Im ) , . . . , Ik is the sequence corresponding to h as in the definition of U . ,m For k = 1 we require that for g : K2n−1 → C , of the correct type that H1 ([g], I1 ) = H2 ([g], I1 ). (I1 ,... ,Im )
For each ϑ1() , ϑ2() ∈ dom <M (I1 ,... ,Ik ) , where ϑ1() = rϑ1(I1 ,... ,I 1) or ϑ1() = r (i1 ,... ,ik ) ϑ1(I1 ,... ,I ) 1
1
,m or ϑ1() = ϑ1 < K2n−1 and similarly for ϑ2() , it follows that there is a rT (I
c.u.b. Cϑ1 ,ϑ2 ⊆ 12n+1 such that for rF1 :
(I11 ,... ,Im1 )
(I12 ,... ,Im2 )
1) a) If r1V, r2V ∈ S2n+1 , and r1ϑ(I 1 ,... ,I 1 1) <M (I1 ,... ,Ik ) r2ϑ(I 2 ,... ,I 2 2) then there is 1
1
1
2
a c.u.b. Dϑ1 ,ϑ2 ⊆ 12n+1 (which depends on the rF as well) such that for ,m g : K2n−1 → D of the correct type, r1 T (I11 ,... ,I1 ) 1
fϑ1 : <
r2 T (I12 ,... ,I2 ) 2
,m → K2n−1 , fϑ2 : <
,m → K2n−1 ,
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AD AND THE PROJECTIVE ORDINALS
the corresponding embeddings from B2n+1 , and Q1 = g ◦ fϑ1 , Q2 = g ◦ fϑ2 , then r1
F (S(Q1 , (I11 , . . . , I11 ),(I11 , . . . , Im1 1 ))) < r2
F (S(Q2 , (I12 , . . . , I22 ), (I12 , . . . , Im2 2 ))).
b) similar to a) with ϑ1 >M ϑ2 . c) if neither ϑ1 <M ϑ2 nor ϑ2 <M ϑ1 then we require that r1F (· · · ) = r2 F (· · · ). () ,m , and r1ϑ1() <M ϑ2 , we proceed as in 1a) 2) a) If r1V ∈ S2n+1 , ϑ2 ∈ K2n−1 where we require that r1F () < g(ϑ2 ). b) and c) similar to b), c) of 1) above. () ,m r2 3) a) similar to 2) a), with ϑ1 ∈ K2n−1 , V ∈ S2n+1 and ϑ1 <M ϑ2() . b), c) as above. (i11 ,... ,ik1 )
4) a) If r1V, r2V ∈ W2n+1 and r1ϑ1(I 1 ,... ,I1 1 ) <M 1
r1
(i11 ,... ,ik1 ) 1
F (ϑ1 ) < r2F b), c) similar to above.
(i12 ,... ,ik2 ) 2
1
2 2 r2 (i1 ,... ,ik2 ) ϑ2(I 2 ,... ,I 2 ) , 1
then we have
2
(ϑ2 ).
,m are all less than 12n+1 , it follows readily Since ϑru(I1 ,... ,Ik ) , ϑrv(I1 ,... ,Ik ) , K2n−1 that there is a c.u.b. C2 ⊆ C such that for each ϑ1 <M ϑ2 , if the rF ’s are of the correct type into C2 representing a tuple in the product space V (I1 ,... ,Ik ) , then 1– 4 above hold. For (I1 ,... ,Im ) ,...) α (I1 ,... ,Ik ) = (. . . , r , . . . , r (i1 ,... ,ik ) , . . . , r (I 1 ,... ,Il )
where (r1 , . . . ,rp ) ∈ A, and for r such that rV ∈ S2n+1 ∪ W2n+1 the r ’s rT (I ,... ,I ) are represented by rF :
type (and ordered as in the product measure), we claim that FH1 (α (I1 ,... ,Ik ) ) = FH2 (α (I1 ,... ,Ik ) ). For such fixed (r1 , . . . ,rp ) and rF , it follows from the 12n+1 -additivity of ,m K2n−1 ,m 2n+1 that there is a c.u.b. D ⊆ 12n+1 such that for g : K2n−1 → D of the correct type, 1– 4 above hold for all appropriate ϑ1 , ϑ2 . We may also assume D ⊆ C2 , and that the least element of D > sup rF for rV ∈ W2n+1 . It then ,m follows that for g : K2n−1 → D of the correct type that the corresponding M (I1 ,... ,Ik ) 1 → 2n+1 is order-preserving. Since the rF have range in C2 , and h: < r the F are of the correct type, it also follows that h has range in C2 , and is of the correct type. Hence FH1 (α (I1 ,... ,Ik ) ) = H1 (T ) = H2 (T ) = FH2 (α (I1 ,... ,Ik ) ), where T is the sequence corresponding to h and (I1 , . . . , Ik ). For k = 1, the result is immediate. We now establish that for almost all w.r.t. U , that F is order-preserving ,m . We fix H : <Ω → 12n+1 of the correct almost everywhere from < into K2n+1
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STEVE JACKSON
type representing (which happens for almost all ). We show that F is (I11 ,... ,Ik1 )
order-preserving almost everywhere. We let α1 fix (r1 1 , . . . , rp 1 ), and r
r
T (I11 ,... ,Ik1 )
1
(I12 ,... ,Ik2 )
, α2
rT (I 1 ,... ,I 1
F1 :
1
k1
)
2
be given, and
→ 12n+1
of the correct type representing the components of α1 for rV ∈ S2n+1 ∪ W2n+1 , and similarly for α2 , which happens for almost all α1 , α2 . We assume that α1() < α2() and show that F (α1() ) < F (α2() ). We let G1 , G2 represent K
,m
2n−1 F (α1() ), F (α2() ) w.r.t. 2n+1 as in the definition of F . We show that for ,m almost all g : K2n−1 → 12n+1 of the correct type, G1 ([g]) < G2 ([g]). We consider the following cases:
Case I. k1 = 1 and k2 ≥ 1. In this case (from the definition of < ) I11 < I12 . From the definition of F and the ordering <Ω it follows that for almost all g that G1 ([g]) = H ([g], I11 ) < H ([g], I12 , . . . , rj() , . . . , ) = G2 ([g]), where the rj() appear if k2 > 1. Case II. k1 > 1 and k2 = 1. Hence I11 ≤ I12 . If I11 < I12 , we proceed as above. If I11 = I12 , the result follows since for almost all g, G1 ([g]) = H (T1 ) = H ([g], I11 , . . . , rj() ) < H ([g], I12 ) = H (T2 ), since T1 extends T2 . Case III. k1 , k2 > 1. We assume this case for the rest of the argument. We (I1 ,... ,Ik
−1 )
let α¯ 1 denote the sequence α¯ 1 = I11 , Π(I1 ) 1 (α1 ), . . . , Ik11 −1 , α, Ik11 , and similarly for α¯ 2 . We consider the following subcases (a), (b), (c). Subcase III.a. There is a least point where α¯ 1 , α¯ 2 disagree, which of the 1 2 form Iq+1 . Hence I11 = I12 = I1 , . . . , Iq1 = Iq2 = Iq and Iq+1 < Iq+1 . For fixed
l ,m g : K2n−1 → 12n+1 , we consider the sequences T1 , T2 as in the definitions of which were used in defining F (α¯ ), F (α¯ ). We claim that G1 ([g]), G2 ([g]) 1 2 1 2 for almost all g that T1 and T2 agree before the Iq+1 term. Since Iq+1 < Iq+1 and H is order-preserving, it then follows that F (α¯ 1 ) < F (α¯ 2 ). We use the following observations, where we use the notation of the definition of the (I1 ,... ,Ik
−1 )
sequence T . For q¯ ≤ q since Π(I1 ,... ,Iq¯1) that:
(I1 ,... ,Ik
−1 )
(α1 ) = Π(I1 ,... ,Iq¯2)
(α2 ), it follows
1) If (r1 (1), . . . , rp (1)), (r1 (2), . . . , rp (2)) enumerate the components of (I1 ,... ,Ik
−1 )
(I1 ,... ,Ik
−1 )
Π(I1 ,... ,Iq¯1) (α1 ) and Π(I1 ,... ,Iq¯2) (α2 ) corresponding to measures of the form r V ∈ m
AD AND THE PROJECTIVE ORDINALS
385
2) For r with rV ∈ W2n+1 , the induced functions rF1(i1 ,... ,ik ) = rF2(i1 ,... ,ik ) agree almost everywhere w.r.t. v (i1 ,... ,ik−1 ) for indices (i1 , . . . , ik ) ∈ dom rT (I1 , . . . , Iq¯ ). Hence, the rq(i¯ 1 ,... ,ik ) (as in T1 , T2 ) agree for the two sequences T1 , T2 , corresponding to rF1 , rF2 for all q¯ ≤ q. 3) For r with rV ∈ S2n+1 , there is a c.u.b. C ⊆ 12n+1 such that for ,m g : K2n−1 → C of the correct type, for all indices (I1 , . . . , Im ) in the domain of rC (I1 , . . . , Iq¯ ) we have that for almost all ϑ w.r.t. ru (I1 ,... ,Iq¯ ) that r MIq¯ 1 ,... ,Im (1)(ϑ) = rMIq¯ 1 ,... ,Im (2)(ϑ), where 1, 2 refer to the definitions relative to rF1 and rF2 respectively. Hence it follows that there is a c.u.b. C ⊆ 12n+1 ,m 1 ,... ,Im ) such that for g : K2n−1 → C , we have that T1 , T2 agree on the terms rq(I ¯ for q¯ ≤ q. Since H is order-preserving, it follows that G([g1 ]) < G([g2 ]). Subcase III.b. There is a least position where α¯ 1 , α¯ 2 disagree which of the (I11 ,... ,Ik1
form Π(I1 ,... ,Iq )1 I11
−1 )
(I12 ,... ,Ik2
(α1 ) < Π(I1 ,... ,Iq )2
I12 . . . , Iq1 (I1 ,... ,Iq )
Hence = to the product V (I11 ,... ,Ik1
then Π(I1 ,... ,Iq )1
−1 )
(α1 )(r)
−1 )
(α2 ). 2 = Iq , and if r¯ is the least component with respect / = r rV (I1 ,... ,Iq ) such that the above tuples disagree, (I12 ,... ,Ik2 −1 ) < Π(I1 ,... ,Iq )2 (α2 )(r). It then follows that there is
,m → C of the correct type, we a c.u.b. C ⊆ 12n+1 such that for g : K2n−1 have:
1) For q¯ < q, rq¯ (1) = rq¯ (2), for all r, where the ’s are elements of the sequences T1 , T2
¯ r q (1) = r q (2). 2) For r < r, 3) If r¯V (I1 ,... ,Iq ) ∈ m
(I11 ,... ,Ik1
Π(I1 ,... ,Iq )1
−1 )
(α1 )(r) in this case, and similarly for r¯q (2).
4) If r¯V (I1 ,... ,Iq ) ∈ W2m+1 , then in the fixed ordering we have on the indices (i1 , . . . , ik ) ∈ dom rT (I1 , . . . , Iq ) which we use to identify ϑr (I1 ,... ,Iq ) with an V ordinal, if (¯ı1 , . . . , ı¯k ) denotes the least index s.t. r¯ F1(¯ı1 ,... ,¯ık ) = r¯ F2(¯ı1 ,... ,¯ık ) , then the first is smaller. Hence r¯q(¯ı1 ,... ,¯ık ) (1) < r¯1(¯ı1 ,... ,¯ık ) (2), for the least index (¯ı1 , . . . , ı¯k ) where r¯1(¯ı1 ,... ,¯ık ) (1) and r¯q(¯ı1 ,... ,¯ık ) (2) disagree. 5) If r¯V (I1 ,... ,Iq ) ∈ S2n+1 , then for the least index (I 1 , . . . , Im ) in the do(I1 ,... ,Im ) (I1 ,... ,Im ) r¯ r¯ r¯ main of C(I1 , . . . , Iq ), such that Mq (1) = Mq (2) , the first is smaller. (I11 ,... ,Ik1
1 −1
Π(I1 ,... ,Iq )
)
This follows since for the least (I1 , . . . , Im ) such that (I12 ,... ,Ik2
(α1 )(I1 , . . . , Im ) = Π(I1 ,... ,Iq )2
−1 )
(α2 )(I1 , . . . , Im ), the first is
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STEVE JACKSON
smaller. Hence for the least (I1 , . . . , Im ) s.t. r¯q(I1 ,... ,Im ) (1) = r¯q(I1 ,... ,Im ) (2), the first is smaller. Hence, for such g, it follows that for the least position where T1 , T2 disagree, the first is smaller. Since H is order-preserving, G1 ([g]) < G2 ([g]). Subcase III.c. α¯ 1 is an extension of α¯ 2 . Proceeding as in (a) and (b) above, it follows that for almost all g, the sequence T1 extends T2 , so G1 ([g]) < G2 ([g]). Hence, for almost all w.r.t. U , we have that F is order-preserving almost everywhere. ,m ,m We now establish B2n+3 , We let A ⊆ K2n+1 have measure one w.r.t. S2n+1 . 1 1 We let C be a c.u.b. subset of 2n+1 such that if G : 2n+1 → C is of the correct ,m S2n−1 type, then the ordinal represented by G w.r.t. 2n+1 is in A. We let B be the measure one set w.r.t. U determined by C . We fixed ∈ B, and a function H : <Ω → C of the correct type representing with respect / (I1 ,... ,Ik ) to the measures r rv (I1 ,... ,Ik ) × <M for (I1 , . . . , Ik ) ∈ dom . For 2n+1 r (I1 ,... ,Ik ) ∈ dom < and F of the correct type representing the components α of α for the rV ∈ W2n+1 ∪ S2n+1 , (which happens for almost all α), and G S
,m
2n−1 , it follows from the definition of F that representing F (α (I1 ,... ,Ik ) ) w.r.t. 2n+1 ,m 1 for almost all g : K2n−1 → 2n+1 , that G[(g)] = H (T ), for the sequence T hence G([g]) ∈ C . as in the definition of F , and Since H is of the correct type, it follows that G is discontinuous and has uniform cofinality on a measure one set. Also, since the ordinal 0 (as in the definition of T ) is equal to [g], it follows that G is strictly increasing on a measure one set. By a sliding argument (as in Lemma 2.7), we may assume G is of the correct type and has range C everywhere. Hence F (α (I1 ,... ,Ik ) ) ∈ B. This establishes B2n+3 .
§4. A Local Embedding Theorem. Theorem 4.1. For any measure V ∈ m≤n S2m+1 ∪ m≤n W2m+1 , and regular cardinal κ < 12n+3 , there is a measure R ∈ R2n+1 (i.e., a canonical C ⊆ 1 measure) and a c.u.b. 2n+3 such that for α ∈ C with cf(α) = κ we have j denote the embeddings from the ultrapowers jV (α) ≤ jR (α). Here jV and R by the measures V and R respectively. / As in §3, we again use the notation V = r rV where each rV is basic. Also as in the proof of the global embedding theorem, we require an inductive hypothesis:
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H2n+1 : For any basic measures rV1 ∈ m≤n S2m−1 ∪ m≤n W2m−1 , for 1 ≤ /r0 r r ≤ r0 , and corresponding product measure V1 = r=1 V1 , there are r measures V2 , for 1 ≤ r ≤ r0 , in m≤n R2m−1 , a measure u on Ξ2n−1 , and a map /r0 αr → [fα ]u such that for almost all α = (α1 , . . . , αr0 ) w.r.t. V2 = r=1 V2 the ordinal [fα ]u is represented by an fα : ϑu → ϑV1 defined a.e. w.r.t. u satisfying: 1) For any A1 ∈ ϑV1 of measure one w.r.t. V1 , there is an A2 ⊆ ϑV2 of measure one w.r.t. V2 such that for all α ∈ A2 , fα has range u almost everywhere in A1 . 2) There is a measure one set A2 ⊆ ϑV2 such that for α ∈ A2 , sup fα < ϑV1 (that is there are 1 < ϑ1V1 , . . . , r0 < ϑr0 V1 s.t. for almost all w.r.t. u, if fα () = (1 , . . . , r0 ), then 1 < r0 ). 1 , . . . , r0 < 3) If rV1 , 1 ≤ r ≤ r0 are basic measures in m≤n S2m−1 ∪ m≤n W2m−1 r
V
such that the embeddings ΠrV11 are defined for 1 ≤ r ≤ r0 , then there are rV2 ∼ rV2 such that 1) and 2) hold for V1 , V2 We also that if rV2 > rV2 for 1 ≤ r ≤ r0 , then H2n+1 holds with / require r V2 = r/V2 replacing V2 (with a possibly different u, etc.). 4) If V1 = r rV1 , where each rV1 ∈ R2n−1 and r1V1 ∼ r2V1 for 1 ≤ r1 ≤ r2 ≤ r0 , then there is a V2 ∈ R2n−1 with V2 ∼ rV1 , and measure u on Ξ2n−1 such that H2n+1 1), 2), 3) above are satisfied (here V2 is now a single measure rather than a product). We require the following additional hypothesis: D2n+1 : 1) We let rV1 , rV2 , u be as in H2n+1 1), 2), 3). Then for any 1 ≤ r ≤ r0 , there is an r ≤ r0 and a measure one set A w.r.t. V2 such that if (1 , . . . , r , . . . , r0 ), (¯1 , . . . , ¯r , . . . , ¯r0 ) are in A and r < ¯r , then for almost all w.r.t. u we have that f() ()(r) < f() ¯ ()(r), these denoting the rth components of these tuples. We further require that if rV1 ≤ rV1 for 1 ≤ r ≤ r0 , are such that H2n+1 , 1), 2), 3) are also satisfied by rV1 , rV2 (for some u ) then the r ¯ as for the measures rV is the same as for the rV1 , and in fact, for , 1
V
V
1 above, for almost all w.r.t. u , (ΠV11 f() ())(r) < (ΠV1 f() ¯ ())(r).
2) If rV1 ∈ R2n+1 , r1V1 ∼ r2V1 for 1 ≤ r1 ≤ r2 ≤ r, V2 ∈ R2n−1 , and u are as in H2n+1 (4), then there is a measure one set A w.r.t. V2 such that for 1 < 2 in A we have that for almost all w.r.t. U that f1 ()(r) < f2 ()(r) for all 1 ≤ r ≤ r0 . We first establish Theorem 4.1 from H2n+3 (using also our overall inductive hypothesis K2n+3 ). We let rV1 , 1 ≤ r ≤ r0 , be basic measures in m≤n S2m+1 ∪ m≤n W2n+1 / r and V1 = r V1 . We let C be a c.u.b. subset of 12n+3 closed under jV
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STEVE JACKSON
for all measures V in m≤n S2m+1 ∪ m≤n W2n+1 . We first show that there r are measures V2 ∈ R2n+1 s.t. for all α ∈ C we have jV1 (α) ≤ jV2 (α), where / r V2 = r V2 . We let rV2 be as in H2n+3 , and fix α ∈ C . We define an embedding E from jV1 (α) into jV2 (α) as follows. If F : ϑV1 → α, then we represent E([F ]V1 ) by G : ϑV2 → α defined as follows: if = (1, . . . , r0) ∈ ϑV2 , we represent G() w.r.t. U (as in H2n+3 ) by the function F ◦ f : ϑU → α (here → [f ]U is as in H2n+3 ). It follows form H2n+3 and the definition of C that E is well-defined. / r r Now assume, by the previous paragraph, that V11 = r V1 where V1 ∈ now a regular cardinal κ < 2n+3 and proceed to show m≤n R2m+1 . We fix that for some V2 ∈ m≤n R2m+1 that jV1 (α) ≤ jV2 (α) for all α ∈ C with cf(α) Lemma 2.7, we may assume that V1 is of the form V1 = /q0 r = κ./By r0 r1 r2 r1 r2 r V × 1 r=1 r=q0 +1 V1 , where V1 ∼ V1 for 1 ≤ r1 , r2 ≤ q0 , and V1 ∼ V1 for r1 ≤ q0 and r2 > q0 , and also cf ϑ(r1V1 ) = κ./ It now follows from r0 r Lemmas 2.5, 2.6 and the definition of C that if V1 = r=q V1 and α ∈ C 0 +1 then then jV1 (α) = α. Hence, we may assume without loss of generality that / V1 = r rV1 where r1V1 ∼ r2V1 for 1 ≤ r1 , r2 ≤ r0 . We then select V2 as in H2n+3 (4), and proceed as in the previous paragraph to define an embedding from jV1 (α) into jV2 (α). For the rest of this section we assume H2n+1 and D2n+1 , and proceed to establish H2n+3 , D2n+3 . We first consider H2n+3 . We let rV1 , for each 1 ≤ r ≤ r0 be basic measures in m≤n S2m+1 ∪ / . For each r, we will define rV2 and rU1 and will take V2 = r rV2 m≤n W2m+1 / r and U = r U . Hence, in defining V2 and U we need only consider a fixed r V1 and we therefore suppress writing the presuperscript r throughout the definition. We consider the following cases I, II, III for the definition of U . Case I. V1 (= rV1 ) ∈ W2n+1 . We fix an n − ∗ tuple T and corresponding T ,m ordering on Ξ2n+1 such that V1 = < 2n+1 . We fix a measure S2n−1 s.t. the hypothesis B2n+1 from the global embedding theorem is satisfied by
S
,m
,m 2n−1 m , with measure u. We then let V2 = 2n+1 = W2n+1 , and let the U in S2n−1 H2n+3 be the u from B2n+1 .
/ dom
AD AND THE PROJECTIVE ORDINALS
389
and (3) it follows that there is a measure v˜2 s.t. for all v (n1 ,i2 ,... ,ik ) , H2n+1 (n1 ,... ,ik ) holds for v (n1 ,... ,ik ) , v˜2 and a measure , and u v˜2 = w1 × · ·· × wn where w1 , . . . , wn are basic measures in m≤n R2m−1 = m≤n S2m−1 ∪ m≤n W2m−1 . / We also let v (n1 ) = s s v (n1 ) , a product of basic measures. We let s¯ be the s such that in identifying ϑv (n1 ) with Πs ϑsv (n1 ) , we order by ϑs¯v (n1 ) most significantly. We let s¯¯ ≤ n be, from D2n+1 , the integer s.t. there is a measure one set A w.r.t. v˜2 such that for (1 , . . . , n ), (1 , . . . , n ) ∈ A and s¯¯ < s¯¯ we have that for all (n1 , i2 , . . . , ik ) that for almost all w.r.t. u (n1 ,... ,ik ) that 1 ,... ,ik ) 1 ,... ,ik ) ¯ < (Π(n ¯ (Π(n f ())(s) f( ) ())(s). (n1 ) (n1 ) We let v˜22 = v˜2 × v˜2 = (w1 × · · · × wn ) × (w1 × · · · × wn ), and let E1 , . . . , Ep 2 enumerate the components of v˜2 equivalent (w.r.t. ∼) to ws¯¯ . By H2n+1 (4), we let v2 ∈ m≤n R2n+1 be s.t. H2n−1 holds for E1 × · · · × Ep , v2 , and some ,m ,m , where v(S2n+1 ) = v2 (here is not measure u2 . We then let V2 = S2n+1 necessarily maximal). We may also take any m ≥ m in the following. We proceed to define the measure U . We define two orderings <Ω1 , and <Ω2 on tuples of ordinals and integers. The domain of <Ω1 consists of tuples of the form T = α0 , I1 , . . . , α (i1 ) , . . . , I2 , . . . , α (i1 ,i2 ) , . . . , Ik , . . . , α (i1 ,... ,ik ) , In , . . . , α (i1 ,... ,in ) or an initial segment of such, satisfying: (a) α0 ≤ 12n+1 (b) (I1 , . . . , Ik ) is an index in C. (c) the ordinals α (i1 ,... ,ik ) following Ik are indexed by indices (i1 , . . . , ik ) in C(I1 ,... ,Ik ) and occur in the same order as in C(I1 ,... ,Ik ) . (d) α (i1 ,... ,ik ) < α0 . We let <Ω1 be the Brouwer-Kleene ordering on the tuples. We let <Ω2 denote the usual ordering on 12n+1 . define A to have U measure We define the measure U as follows: We 1 one if there is a c.u.b. C2 ⊆ 2n+1 s.t. for all H2 : <Ω2 → C2 of the correct type, there is a c.u.b. C1 ⊆ 12n+1 s.t. for all H1 : <Ω1 → C1 of the correct type = (. . . , (I1 ,... ,Ik ) , . . . ) ∈ A, where for fixed (I1 , . . . , Ik ), an (I1 ,... ,Ik ) index in C, is defined through the following sequence of definitions. T (1) (I1 ,... ,Ik ) is represented w.r.t. < by a function F¯ , 2n+1
of the correct type, F¯ ([f]) is represented w.r.t. (2) for fixed f : < → v2 (as above) by a function g. T
12n+1
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STEVE JACKSON
(3) for < ϑv2 we have g() = H2 (), where is represented w.r.t. u2 (defined above) by a function h2 . (4) For ∈ ϑu2 , we represent h2 () w.r.t. D1 × · · · × Dq by a function g¯1 , where D1 , . . . , Dq enumerate the components of v˜22 not equivalent to the E1 , . . . , Ep . The function g¯1 is defined as follows. Let i1 , . . . , iq enumerate the integers a ≤ 2n where wa ∼ Di for some i, and let j1 , . . . , jp enumerate the integers a ≤ 2n where wa ∼ E1 . Then g¯1 ( i1 , . . . , iq ) = g1 ( 1 , . . . , n , 1 , . . . , n ) where { 1 , . . . , n , 1 , . . . , n } = { i1 , . . . , iq } ∪ { j1 , . . . , jp }. Here ( j1 , . . . , jp ) = f (), where → f is as in H2n+1 for E1 × · · · × Ep , v2 , and u2 . (5) g1 is defined by g1 ( 1 , . . . , n , 1 , . . . , n ) = H1 (T ), where T = α0 , I1 , . . . , α i1 , . . . , Ik , . . . , α (i1 ,... ,ik ) , . . . is defined by: (5a) α0 is represented w.r.t. u (n1 ) by a function h10 defined as follows: We let fs : ϑv (n1 ) → 12n+1 represent the least equivalence class of an a.e. mono tonically increasing function with supa.e. fs = supa.e. f (n1 ) . For ∈ ϑu (n1 ) , we set h10 () = fs (f( 1 ,... , n ) ()). Here, ( 1 , . . . , n ) → f( 1 ,... n ) is as in H2n+1 for v (n1 ) , v˜2 , and u (n1 ) . (5b) α (n1 ,i2 ,... ,ik ) is represented w.r.t. u (n1 ,i2 ,... ,ik ) by a function h1(n1 ,i2 ,... ,ik ) defined by: for ∈ ϑu (n1 ,i2 ,... ,ik ) , we set h1(n1 ,i2 ,... ,ik ) () = f( (n1 ,i2 ,... ,ik ) ) where = f( 1 ,... , n ) (), where ( 1 , . . . , n ) → f( 1 ,... n ) is as in H2n+1 for v (n1 ,i2 ,... ,ik ) , v˜2 , and u (n1 ,i2 ,... ,ik ) . (5c) α (i1 ,... ,ik ) , for i1 < n1 , is represented w.r.t. v (i1 ,... ,ik ) by the induced function f (i1 ,... ,ik ) . We show that U is well-defined in case II. For fixed H2 , H1 , and f :
12n+1 ,
1) For almost all f there is a measure one set A w.r.t. v˜22 s.t. for all tuples ( 1 , . . . , n , 1 , . . . , n ) ∈ A and T the corresponding sequence as above, (for any fixed (I1 , . . . , Ik )) we have that α0 > α (i1 ) , . . . , α (i1 ,... ,ik ) . This follows from H2n+1 (i) and (ii). 2) By Lemma 2.7, we have that there is a measure one set E w.r.t. E1 × · · · × Ep s.t. for ( j1 , . . . , jp ) ∈ E, for almost all ( i1 , . . . , iq ) w.r.t. D1 × · · · × Dq , ( 1 , . . . , n , 1 , . . . , n ) ∈ A, where, as above, ( 1 , . . . , n , 1 , . . . , n ) is the enumeration of ( i1 , . . . , iq ) ∪ ( j1 , . . . , jp ) by the subscripts. We next claim that for fixed H2 , H1 that for almost all f :
AD AND THE PROJECTIVE ORDINALS
391
H2 , H1 , f :
(I ,... ,I
)
1 k−1 U(i1 ,... , where (i1 , . . . , i ) is the index in C(I1 ,... ,Ik−1 ) occurring last (note ,i ) that this really only depends on (I1 , . . . , Ik−1 )). 1 ,... ,Ik ) For (i1 , . . . , i ) an index in C(I1 ,... ,Ik ) , we define U(i(I1 ,... as follows. We ,i ) 1 q let (i ), . . . , (i ) denote the elements of C preceding (and including)
(I1 ,... ,Ik )
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STEVE JACKSON
(i1 , . . . , i ), as well as the elements of C(I1 ) , . . . , C(I1 ,... ,Ik−1 ) , except that we ex1 ,... ,Ik ) clude (n1 ) if it occurs. We let <Ω (I be the ordering on tuples (α1 , . . . , αq ) (i1 ,... ,i ) as above. We then define A to have measure one if there is a c.u.b. C ⊆ 12n+1 1 ,... ,Ik ) such that for all H : <Ω (I → C of the correct type we have α ∈ A, where (i1 ,... ,i ) T
T 1 α is represented w.r.t. < 2n+1 by the function which assigns to f : < → 2n+1 of the correct type the ordinal H (α1 , . . . , αq ). Here, α1 = the largest of 1
q
{α (i ) , . . . , α (i ) }, α2 = the second largest of these, etc., where f represents the ordinals α (i1 ,... ,ik ) . We then let U be the product: m+2 + + m+2 U(i(I1 )1 ) U = U (I1 ) × × ··· × ⎡ ⎣
I1 i1 ∈CI1
+ (I1 ,... ,In−1 )∈C
⎤m+2
U (I1 ,... ,In ) ⎦
×
+
+
1 ,... ,In ) U(i(I1 ,... ,ik )
m+2 ,
(I1 ,... ,In )∈C (i1 ,... ,i )∈C(I1 ,... ,In )
where we order the factors in each product according to the ordering of indices in each (I1 , . . . , Ik ). This completes the definition of U (= U r ) in this case, and hence completes the definition of U . We now proceed to define the/family of embeddings F as in H2n+3 . We r0 fix = (1 , . . . , r0 ) ∈ V2 = r=1 V2r , and define F . We fix functions G1 , . . . , Gr0 representing 1 , . . . , r0 as elements in V2r . We fix a tuple ϑ = /r0 r (ϑ1 , . . . , ϑr0 ) ∈ r=1 U = U . We set F (ϑ) = (F1 (G1 , ϑ1 ), . . . , Fr0 (Gr0 , ϑr0 )) = (α1 , . . . , αr0 ), say, where it remains to define αr = Fr (Gr , ϑr ) for 1 ≤ r ≤ r0 . We again suppress writing the subscript r. Definition 4.2 (Definition of α = F(G, ϑ)). We consider the following cases:
,m Case I. V1 ∈ W2n+1 . In this case, G : S2n−1 → 12n+1 , and ϑ is in the T measure space U as in B2n+1 for T corresponding to V1 = < 2n+1 . We represent ,m α by f :
resenting ϑ = (· · · , ϑ(I1 ,... ,Ik ) , · · · ) ∈ dom(U ) with respect to < 2n+1 (note that there has been a slight change of notation from the definition of U where there we represented the elements of dom(U ) as (· · · , (I1 ,... ,Ik ) , · · · )).
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We then represent α = (· · · , α (I1 ,... ,Ik ) , · · · ) w.r.t. < 2n+1 by the function (I1 ,...,Ik ) (I) T ¯ ([f]) = G(F ([f]) for f : < → 12n+1 of the correct [f] → F type. This is well-defined. Case III. V1 ∈ S2n+1 and 1(n1 ) ∈ dom
m+2 1 ,... ,Ik ) representing the components of ϑ corresponding to the factor U (I (i1 ,... ,i ) in U , where the index (i1 , . . . , i ) may not appear. When this index does not appear note that each Hj(I1 ,...,Ik ) is actually a b-sequence of functions Hj(I1 ,...,Ik ) (), 1 ≤ ≤ b, where b is defined above (when k = 1, each Hj(I1 ) is a b-sequence of ordinals less than 12n+1 ). For fixed f : 1. We let ri = the number of indices in C(I1 ,... ,Ii ) for 1 ≤ i ≤ k. ' In general, we assume that αj has been defined for j ≤ u = q≤i (2rq + 2), ,... ,Ii ) and that αu is of the form H1 (i(I11,... (. . . , ϑ(j) , . . . ) where (i1 , . . . , i ) denotes ,i ) the last index in CI1 ,... ,Ii and (j) denotes the indices in CI1 ,... ,Ii and CI1 ,... ,Ii for i < i. Also, the ϑ(j) are represented w.r.t. v (j) by the functions induced from ,... ,Ii ) f :
αu+1 = H0(I1 ,... ,Ii+1 ) (Ii+1 )(. . . , ϑ(j) , . . . ), which makes sense since the ordering <Ω(I1 ,... ,Ii+1 ) used in defining the function 1 ,... ,Ii ) H0(I1 ,...,Ii+1 ) (Ii+1 ) is the same as the ordering <Ω (I used in defining the (i1 ,... ,i ) 1 ,...,Ii ) Hj (I for (i1 , . . . , i ) the last index in CI1 ,... ,Ii . We also set αu+2 = (i1 ,...,i )
H1(I1 ,... ,Ii+1 ) (Ii+1 )(. . . , ϑ(j) , . . . ). For 2 ≤ v ≤ ri+1 we set
1 ,... ,Ii+1 ) αu+2·v−1 = H0 (I (. . . , ϑ(j) , . . . ), (i1 ,... ,i )
where (i1 , . . . , i ) is the v − 1st index in C(I1 ,... ,Ii+1 ) and the (j)’s enumerate the indices in C(I1 ,... ,Ii+1 ) occurring before (i1 , . . . , i ) in C(I1 ,... ,Ii+1 ) along with
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the indices in C(I1 ,... ,Ii ) , i < i + 1, and occurring in the same order as in 1 ,... ,Ii+1 ) <Ω (I . We also set (i1 ,... ,i )
,... ,Ii+1 ) αu+2·v = H1 (i(I11,... (. . . , ϑ(j) , . . . ). ,i )
If i + 1 < k, we proceed as above for the last index (i1 , . . . , i ) in C(I1 ,...,Ii+1 ) as well. Namely, we set ,... ,Ii+1 ) αu+2·ri+1 +1 = H0 (i(I11,... (. . . , ϑj, . . . ), ,i )
and ,... ,Ii+1 ) (. . . , ϑj, . . . ). αu+2·ri+1 +2 = H1 (i(I11,... ,i ) 1 ,... ,Ii+1 ) If i + 1 = k, we also set αu+2·ri+1 +1 = H0 (I (. . . , ϑj, . . . ) as (u1 ,... ,i ) 1 ,... ,Ii+1 ) (. . . , ϑ(j) , . . . ), out to αm = above. We then set αu+2·ri+1 +c = Hc (I (i1 ,... ,i ) αu+2·ri+1 +(m−u−2·ri+1 ) . This completes the definition of F(G, ϑ).
We show that F(G, ϑ) is well-defined in case (3) (for the fixed function G : <m → 12n+1 ). We let H1 , . . . , Hp and H1 , . . . , Hp (for the appropriate given representing the same ϑ as in the above definition of U . integer p) be ,... ,Ik ) then Hj = Hj almost everywhere with Hence for all j ≤ p, if Hj = H (i(I11,... ,i ) ,... ,Ik ) respect to <Ω (i(I11,... . More precisely, they agree with respect to the measure ,i ) 1 ,...,Ik ) 1 ,...,Ik ) = (I on ( 12n+1 )q implicitly defined in the definition of U(i(I1 ,...,i . That (i1 ,...,i ) ) T 1 is, A has measure one if for almost all f : < → 2n+1 the corresponding 1 ,...,Ik ) tuple (α1 , . . . , αq ) of ordinals (as in the definition of U(i(I1 ,...,i ) represented ) by f lies in A. We fix a c.u.b. C ⊆ 12n+1 such that for f :
,m Case I. V1 ∈ W2n+1 . If G1 , G2 : S2n−1 → 12n+1 agree on a measure one ,m set A w.r.t. S2n−1 , then from B2n+1 , we have that for almost all ϑ, that fϑ has range in A almost everywhere. Hence, G1 ◦ fϑ = G2 ◦ fϑ agree almost
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everywhere w.r.t. the v (i1 ,... ,ik ) (and agree if k = 1). Since F(G1 , ϑ) = [G1 ◦ fϑ ] and likewise for F(G2 , ϑ), we are done. Case II. V1 ∈ S2n+1 and 1(n1 ) ∈ / dom
F2(I) as in the definition of F, then we show that for almost all f :
2
(i.e., it represents ϑ corresponding to H2 , H1 ). Recall that F (I) ([f]) is repre ,m ,m (= v(V2 )) by a function g, where for < ϑ(S2n−1 ) sented with respect to S2n−1 we have that g() = H2 (), for some as in the definition of U , and hence g() ∈ C2 ⊆ C . Since H2 is of the correct type, g is of uniform cofinality . It remains to show that g is almost everywhere strictly increasing. It follows from D2n+1 that there is an r and a measure one set A w.r.t. v˜22 s.t. if ( 1 , . . . , n , 1 , . . . , n ), (1 , . . . , n , 1 , . . . , n ) are in A, then:
(1) If r < r , there then for all indices (n1 , i2 , . . . , ik ) ∈ dom T for almost (n1 ,... ,ik ) all w.r.t. we have u (n1 ,i2 ,... ,ik ) 1 ,i2 ,... ,ik ) Π(n1 ) ◦ f( 1 ,... , n ) () (r) < Π(n ◦ f () (r), (1 ,... ,n ) (n1 ) where f( 1 ,... , n ) , f(1 ,... ,n ) denote embeddings as in H2n+1 , and r corresponds to the component of the measure v (n1 ) such that in identifying ϑv (n1 ) with an ordinal, we order first by the rth component of the product measure v (n1 ) . Hence, we have f((f( 1 ,... , n ) ())(n1 ,... ,ik ) ) < f((f(1 ,... ,n ) ())(n1 ,... ,ik ) ). (2) If r < r , then for almost all w.r.t. U (n1 ) we have (f( 1 ,... , n ) ())(r) < (f(1 ,... ,n ) ())(r), and hence fs (f( 1 ,... , n ) ()) ≤ fs (f(1 ,... ,n ) ()), where fs is (as in the definition of U ) monotonically increasing and represents the minimal equivalence class of a monotonically increasing function with supa.e. fs = supa.e. f (n1 ) .
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Hence, it follows that if g1 denotes the function as in the definition of U , . then g1 ( 1 , . . . , n , 1 , . . . , n ) < g1 (1 , . . . , n , 1 , . . . , n ) for such , From H2n+1 (i) and D2n+1 it also follows that there is a measure one set ,m A2 w.r.t. S2n−1 such that for 1 < 2 in A2 we have that for almost all w.r.t. u2 (as in the definition of U , u2 is the measure from H2n+1 for v˜22 and ,m S2n−1 ), if f1 () = ( j1 , . . . , jp ), f2 () = (j1 , . . . , jp ), then for almost all (i1 , . . . , iq ) w.r.t. D1 × · · · × Dq (as in the definition of U ), if we consider the corresponding sequences ( 1 , . . . , n , 1 , . . . , n ), (1 , . . . , n , 1 , . . . , n ) are in A and r < r , < . Hence it follows that for 1 < 2 in then , r r A2 we have g(1 ) < g(2 ), so g is almost everywhere strictly increasing. Hence F([G], ϑ) is well-defined in this case. Case III. V1 ∈ S2n+1 and 1(n1 ) ∈ dom αi = Hi ([f]), since the Hi ([f]) do not depend on f(n1 ). Here we have written Hi ([f]) to abbreviate Hi evaluated at the sequence of ordinals represented by the appropriate subfunctions of f, as in the definition of F for this case. It also follows readily that we may assume (which happens for almost all ϑ) that the H1 , . . . , Hp are chosen s.t. for almost all f, Hi ([f]) < Hi+1 ([f]). 1 ,... ,Ik ) This uses the fact that the indices j occurring in the definition of <Ω (I (i1 ,... ,i ) corresponding to Hi are a subset of those corresponding to Hi+1 . Hence it follows that α2 < · · · < αm , since if αi = Ha ([f]), ai+1 = Hb ([f]), then a < b, from the definition of F. Hence F([G], ϑ) is well-defined in this case. This completes the proof that F is well-defined in all cases. We proceed to establish H2n+3 . First we consider H2n+3 (i). / We let A have measure one with respect to V1 = r V1r . From the definition of the product measure and the strong partition relation on 12n+1 , it follows
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readily that there is a c.u.b. C ⊆ 12n+1 such that if α = (α1 , . . . , αr0 ) is represented by functions F1 , . . . , Fr0 (as in the definition of the measures V1r ) then α ∈ A provided the following hold: P1) F1 , . . . , Fr0 have range in C and are of the correct type (where Fi : <Ti → 12n+1 if V1i ∈ W2n+1 and Fi : <Ωi → 12n+1 if V1i ∈ S2n+1 ). P2) a) If i < j and V1i , V1j ∈ W2n+1 , then supa.e. Fi < inf Fj . / dom <Ti , 1(n1 ) ∈ / dom
j
measure v n1 corresponding to <Ti and similarly for j), then there is a c.u.b. C2 ⊆ 12n+1 such that if f1 : <Ti → C2 , f2 : <Ti → C2 are of with sup f = sup f , then F ([f ]) < F ([f ]). the correct type i 1 j 2 a.e. 1 a.e. 2 c) Same as b above where 1(n1 ) ∈ dom <Ti and 1(n1 ) ∈ dom
j
means f1 (n1 ) and similarly for f2 . We fix a / c.u.b. set C . We fix = (1 , . . . , r , . . . , r0 ) in the measure space V2 = r V2r represented by functions G1 , . . . , Gr , . . . , Gr0 satisfying the following (which happens for almost all w.r.t. V2 ): i) G1 , . . . , Gr0 have range in C and are of the correct type. ii) a) If i < j and V2i , V2j ∈ W2n+1 , then supa.e. Gi < inf Gj . ,m with > 1 (, m may depend on i, j) and b) If i < j and V2i , V2j ∈ S2n+1 ,m j j i ,mi cf ϑ(S2n−1 ) = cf ϑ(S2n−1 ) (this is in fact equivalent to i = j ), then i ,mi there is a c.u.b. subset C2 ⊆ 12n+1 such that for g1 : ϑ(S2n−1 ) → C2 , j ,mj g2 : ϑ(S2n−1 ) → C2 of the correct type, if supa.e. g1 ≤ supa.e. g2 , then Gi ([g1 ]) < Gj ([g2 ]). 1,m , then there is a c.u.b. C2 ⊆ 12n+1 c) If i < j and V2i , V2j ∈ S2n+1 such that for (2 , . . . , m , 1 ), (2 , . . . , m , 1 ) in C2 , if 1 ≤ 1 then G1 (2 , . . . , m , 1 ) < G2 (2 , . . . , m , 1 ). We fix G1 , . . . , Gr0 satisfying (i) and (ii). / It then follows readily that for almost all ϑ = (ϑ1 , . . . , ϑr0 ) w.r.t. U = r U r , 1 , . . . , Hr0 representing ϑ1 , . . . , ϑr0 (where H r = (Hr,1 , Hr,2 ) if where we fix H (I ,... ,I ) 1 k (n1 ) Tr (n ) r = (. . . , H ∈ / dom < and H , . . . ) if 1 1 ∈ dom <Ti ), that if 1 (i1 ,... ,i ) α = (α1 , . . . , αr0 ) = F (ϑ), where αi is represented by Fi as in the definition of F, then the following are satisfied: Claim 4.3. a) If i < j and V1i , V1j ∈ W2n+1 , then supa.e. Fi < inf Fj . / dom <Ti , then for almost all fi : <Ti → 12n+1 , b) If V1i ∈ S2n+1 and 1(n1 ) ∈ i ,mi if Fi ([fi ]) = Gi ([g]), where g : ϑ(S2n−1 ) → 12n+1 is as in the definition of F, then supa.e. g = supa.e. fi .
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c) If V1i ∈ S2n+1 and 1(n1 ) ∈ dom <Ti , then for almost all fi : <Ti → 12n+1 , Fi ([fi ]) = Gi (α2 , . . . , αm , α1 ), where α1 = sup f1 Proof. a) and c) are immediate from the definition of F and the choice of the Gi . b) follows from H2n+1 (i), (ii), the fact that measure one sets are cofinal in v n1 , and the fact that almost all fi are sufficiently closed with respect to the Hi . That is, there is a measure on set E w.r.t. E1 ×· · ·×Ep (as in the definition of U ) s.t. for ( i1 , . . . , ip ) ∈ E, for almost all ( j1 , . . . , jq ) w.r.t. D1 × · · · × Dq and ( 1 , . . . , n , 1 , . . . , n ) the enumeration of these ordinals in the order corresponding to the measures in v˜22 , we have that g1 ( 1 , . . . , n , 1 , . . . , n ) = Hi,1 (T ), where (for a fixed (I1 , . . . , Ik )), T = α0 , I1 , . . . , Ik , α (i1 ,... ,ik ) , . . . and α0 , . . . , α (i1 ,... ,ik ) < sup ran fi . This follows from H2n+1 (ii) and the fact that the range of fi may be taken as closed under ju (n1 ,i2 ,... ,ik ) . It further follows from the definition of the measures E1 , . . . , Ep and D2n+1 that there is a measure one set E w.r.t. E1 × · · · × Ep s.t. if ( i , . . . , ip ) ∈ E then sup j ,... , jq g1 ( 1 , . . . , n , 1 , . . . , k ) < sup ran fi . 1 It then follows from another application of H2n+1 (ii) that for almost all ,m w.r.t. S2n−1 (= v(V2 )) that g() is represented w.r.t. u2 (as in the definition of
,m U , u2 is the measure from H2n+1 corresponding to E1 × · · · × Ep and S2n−1 ) by a function h2 s.t. supa.e. h2 () < sup ran fi . Hence, we may assume that h2 () < sup ran fi almost everywhere. Hence supa.e. g ≤ sup ran fi . Also supa.e. g ≥ sup ran fi follows from H2n+1 (i) and the fact that measure on sets in v n1 are cofinal in ϑv n1 . This establishes (b) of Claim 4.3.
The second claim P2 above about the Fi now follows from a), b), c) of Claim 4.3 and the choice of the Gi . Hence, it remains to establish P1 above, and it is enough to establish this componentwise. We again suppress the subscript r. We consider the following cases: Case I. V1 ∈ W2n+1 . In this case, F (ϑ) is represented by f¯ :
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We show that F is order-preserving almost everywhere. We fix indices (I1 , . . . , Ik ), (J1 , . . . , J ) in the collection C , and fix functions f1 , f2 :
w
1 ,... ,Iv ) 1 ,... ,Iv ) (i1 , . . . , iw ) precedes (i1 , . . . , ip ), that H(i(I ,... ([f1 ]) = H(i(I ,... ([f2 ]), ,i ) ,i ) 1
w
1
w
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STEVE JACKSON
1 ,... ,Iv ) 1 ,... ,Iv ) since from the definition of U(i(I ,... we have that H(i(I ,... ([f]) depends ,i ) ,i )
only on the ordinals (i1
1
,... ,iw )
w
1
w
represented by f for indices (i1 , . . . , iw ) (i ,... ,i )
(i ,... ,i )
w w preceding and including (i1 , . . . , iw ) and hence 1 1 = 2 1 for f1 , f2 . Hence α2 = 2 , . . . , αk = k for k corresponding to the H preceding (I ,... ,I ) (I ,... ,I ) (i ,... ,i ) (i ,... ,i ) H0(i11,... ,ipp) . Since H0(i11,... ,ipp) is order-preserving and 1 1 p < 2 1 p , it then follows that αk+1 < k+1 . Since G is order-preserving, it then follows that G(α2 , . . . , αm , α1 ) < G(2 , . . . , m , 1 ).
Subcase III.c. α1 = 1 , and the least position where S1 , S2 disagree is of the form Ip < Jp . Proceeding as above, we have that α2 = 2 , . . . , αk = k corresponding to the functions H preceding H (I1 ,... ,Ip ) (1) used in the definition of F. It then follows that αk+1 < k+1 since αk+1 = H (I1 ,... ,Ip ) (Ip )([f]) < H (I1 ,... ,Ip ) (Jp )([f]) = k+1 , since in the b−fold product measure U (I1 ,... ,Ip ) , we may assume the functions H (I1 ,... ,Ip ) (1), . . . , H (I1 ,... ,Ip ) (b) are such that if i < j then H (I) (i)([f]) < H (I) (j)([f]). Hence, G(α2 , . . . , αm , α1 ) < G(2 , . . . , m , 1 ). Subcase III.d. α1 = 1 , and S1 extends S2 . We again have that α2 = ,... ,I ) 2 , . . . , αk = k corresponding to the H functions preceding H0 (i(I11,... , ,ip ) where (i1 , . . . , ip ) denotes the last index in C(I1 ,... ,I ) , and (I1 , . . . , I ) = (J1 , . . . , J ) here. We then have from the definition of F that αk+1 = (I1 ,... ,I ) (I1 ,... ,I ) H0(i ([f1 ]) = H0(i ([f2 ]) = k+1 as above. Also, αk+2 = 1 ,... ,ip ) 1 ,... ,ip ) (I1 ,... ,I ) (I1 ,... ,I ) ([f1 ]) < H2(i ([f2 ]) = k+2 without loss of generality. Hence H1(i 1 ,... ,ip ) 1 ,... ,ip ) G(α2 , . . . , αm , α1 ) < G(2 , . . . , m , 1 ).
This establishes P1 above in all cases. This completes the proof of H2n+3 (i). We now consider H2n+3 (ii). It is enough to establish this componentwise. We consider the following cases: Case I. V1 ∈ W2n+1 . In this case, for all w.r.t. V2 represented by a function ,m ) → 12n+1 of the correct type we have from the definition of F that G : ϑ(S2n−1 for almost all ϑ w.r.t. U that F (ϑ) = [G ◦ fϑ ], and supa.e. G ◦ fϑ ≤ supa.e. G (in fact strictly <), which immediately gives H2n+3 (ii). / dom T . We fix represented by a Case II. V1 ∈ S2n+1 and 1(n1 ) ∈ G : 12n+1 → 12n+1 of the correct type. It then follows as in the proof of T H2n+3 (i) that for almost all ϑ w.r.t. U that if F¯ represents ϑ w.r.t. < 2n+1 that for almost all f :
H2n+3 (ii).
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Case III. V1 ∈ S2n+1 and 1(n1 ) ∈ dom T . We again fix in the measure space V2 represented by a G : <m → 12n+1 of the correct type. For any ϑ in the measure space U , it follows thedefinition of F that if F represents T T 1 F (ϑ) w.r.t. < 2n+1 , then for almost all f : < → 2n+1 we have that F ([f]) = G(α2 , . . . , αm , α1 ), where α1 = sup f = f(n1 ), and α2 , . . . , αm < α1 . Hence F ([f]) < G (α1 ) ≡ supα2 ,... ,αm <α1 G(α2 , . . . , αm , α1 ) < 12n+1 . This establishes H2n+3 (ii). We now consider H2n+3 (iii). It is enough to establish this componentwise. We again consider the following cases: Case I. V1 ∈ W2n+1 . For V1 as in H2n+3 (iii) and corresponding measures S
,m
S
,m
2n−1 2n−1 , V2 = 2n+1 for some m, m (here = 2n − 1 V2 , V2 , we have that V2 = 2n+1 is maximal). From the definition of ∼, we have that V2 ∼ V2 .
/ dom
follows from the definition of Π that since ΠV11 is defined, v n1 = v 1 , a measure on ϑ(v n1 ) = ϑ1 × · · · × ϑb for some b, where in identifying ϑ(v n1 ) with an ordinal, we order first by ϑc , say, for some 1 ≤ c ≤ b (which is the same for V1 , V1 ). Hence, it follows from H2n+1 (iii) that v˜22 = (E1 × · · · × Ep )2 where Ei ∼ Ej and Ei ∈ R2n−1 for 1 ≤ i, j ≤ p and also v˜ 22 = (E1 ×· · ·×Ep )2 , where Ei ∼ Ei . Hence, it follows from H2n+1 (iv) that v2 = v(V2 ) ∼ Ei ∼ Ei ∼ v2 , and hence V2 ∼ V2 from the definition of ∼. n
Case III. V1 ∈ S2n+1 and 1(n1 ) ∈ dom
Since ΠV11 is defined, it follows that n1 ∈ dom
1,m ,m and hence V1 = S2n+1 , V1 = S2n+1 for some m. From the definition of ∼ it follows that V1 ∼ V1 .
This establishes / H2n+3 (iii)./We now consider / D2n+3 (i). For V1 = r V1r , V2 = r V2r , U = r U r as in D2n+3 , we recall that F is defined componentwise, that is, for = (1 , . . . , r0 ) in the measure space V2 , ϑ = (ϑ1 , . . . , ϑr0 ) in the measure space U we have F (ϑ) = (F1 (ϑ1 ), . . . , Fr0 (ϑr0 )) in the the measure space V1 . For any fixed r ≤ r0 , we take r = r, and consider V2r , U r . We consider the following cases. Case I. V1r ∈ W2n+1 . If 1r < 2r are represented by functions G1 , G2 : ,m ϑ(S2n−1 ) → 12n+1 of the correct type, so G1 < G2 almost everywhere, then for almost all w.r.t. u (as in B2n+1 ) [G1 ◦ f ] < [G2 ◦ f ], using B2n+1 . This establishes D2n+3 (i) in this case. Case II. V1r ∈ S2n+1 and 1(n1 ) ∈ / dom
1r
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STEVE JACKSON T
¯ ¯ w.r.t., < 2n+1 then F1 ([f]) = G1 (F ([f])) < G2 (F ([f])) = F2 ([f]), since we may assume F¯ has the property (which happens for almost all w.r.t. U ) ,m
S2n−1 that for almost all f, F¯ ([f]) is in a measure one set A w.r.t. 2n+1 on which G1 < G2 .
Case III. V1r ∈ S2n+1 and 1(n1 ) ∈ dom
,m
r
2n−1 ( = 2n − 1 is Case I. V1r ∈ W2n+1 , hence is of the form V1r = 2n+1 maximal). We let m be such that B2n+1 holds for the ordering corresponding ,mr ,m to T defined by v r = S2n−1 for 1 ≤ r ≤ r0 and S2n−1 (so T is the r0 sum of
S
,m
,mr 2n−1 the S2n−1 ). We let V2r = 2n+1 , and let U be the measure as in B2n+1 . We construct F as in H2n+3 (i)–(iii) in this case. In fact, our previous consideration of H2n+3 (i)–(iii) for the case V1 ∈ W2n+1 included this case. D2n+3 (ii) follows readily from B2n+1 . ¯r ,mr Case II. V1r ∈ S2n+1 and is of the form S2n+1 where ¯r > 1. Recall that
¯
r ,mr S2n+1 is the measure induced from the strong partition relation on 12n+1 and ¯r ,mr ¯r ,mr a measure ϑ2n+1 on 12n+1 . Also, ϑ2n+1 is the measure induced from the weak ¯ ¯ partition relation on 12n+1 and the measure Rr ,mr where the R,m enumerate 1,m 2n −1,m m , S2n−1 , . . . , S2n−1 . Since V1r1 ∼ V1r2 it the measures W1m , S11,m , . . . , W2n−1 ¯ r ,mr ¯ say. We consider the case R,m follows that ¯1 = ¯2 = · · · = ¯r0 = , = S2n−1 the other cases being similar. ,mr ,mr Since S2n−1 ∼ S2n−1 , it follows from H2n+1 (iv) that there is an m s.t. / ,mr ,m H2n+1 (iv) holds for v1 = r S2n−1 and v2 = S2n−1 for some measure u. We ¯
,m let V2 = S2n+1 . We let U be the measure on r0 tuples of ordinals (ϑ1 , . . . , ϑr0 ) defined as follows: A has measure one w.r.t. U if there is a c.u.b. C ⊆ 12n+1 such that for H : 12n+1 → C of the correct type, (ϑ1 , . . . , ϑr0 ) ∈ A where ϑr is ,mr S2n−1 represented with respect to 2n+1 by a function F¯r defined as follows. For
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AD AND THE PROJECTIVE ORDINALS
,mr f : ϑ(S2n−1 ) → 12n+1 of the correct type, F¯r ([f]) = [gr ]S ,m , where gr is 2n−1 ,m defined by: for α < ϑ(S2n−1 ), gr (α) = H ([h]), where h : ϑ(u) → 12n+1 is given by, for < ϑ(u), h() = f(fα ()(r)) and here α → fα is as in H2n+1 (iv). This is well defined by H2n+1 (iv). S ,m
2n−1 We define F as follows. If is represented w.r.t. 2n+1 by G : 12n+1 → space 1 2n+1 of the correct type, and ϑ = (ϑ1 , . . . , ϑr0 ) is in the measure U represented by F¯1 , . . . , F¯r0 , then we set F (. . . , ϑr , . . . ) = (α1 , . . . , αr0 ),
S ,mr
,mr 2n−1 where αr is represented w.r.t. 2n+1 by Fr , where for f : ϑ(S2n−1 ) → 12n+1 of ¯ the correct type, Fr ([f]) = G(Fr ([f])). This is well defined. The proofs of H2n+3 (iv) and D2n+3 (ii) now follows as in H2n+3 (i)–(iii) and D2n+3 (i) before. 1,mr Case III. V1r ∈ S2n+1 and is of the form S2n+1 . We let m = (max≤r≤ro mr )+ 1,m 1, and we set V2 = S2n+1 . We let p = r0 + m and U be the p-fold product of the -cofinal normal measure on 12n+1 . We define F as follows. For represented by G : <m → 2n+1 of the correct type, and (1 , . . . , r0 , ¯1 , . . . , ¯m ) ∈ ϑ(U ), where i , ¯i < 12n+1 , we set F (1 , . . . , ¯m ) = (α1 , . . . , αr0 ), where αr for 1 ≤ r ≤ r0 is represented by mr 1 Fr : < → 2n+1 defined as follows. For 2 < · · · < mr < 1 , we set Fr (2 , . . . , mr , 1 ) = G(r , ¯1 , . . . , ¯m−m¯ r −1 , 2 , . . . , mr , 1 ).
This is well defined for almost all 2 , . . . , mr , 1 . Since G(r1 , 2 , 1 ) < G(r2 , 2 , 1 ) for r1 < r2 (so r1 < r2 ) and any 1 , 1 , 2 , it follows readily that H2n+3 (i) is satisfied. The remaining parts of H2n+3 and D2n+3 follow readily as before. This completes the proof of H2n+3 and D2n+3 and hence of the local embedding theorem. §5. The Main Lemma. The purpose of this section is to prove a main lemma analyzing functions defined with respect to be made precise) the to (in ma sense ,m canonical measures m≤n R2m+1 = m≤n W2m+1 ∪ m≤n S2n+1 . We recall that we are still assuming I2n+1 and K2n+3 as in Section 2. We outline the methods of this section. We introduce a set D of “descriptions” which will be finitary objects which “describe” functions F : 12n+3 → L on 12n+3 with respect to the canonical measures, and a lowering operation them. We then introduce a main inductive hypothesis H2n+1 , and a main auxiliary lemma H¯ 2n+1 which analyze functions F from 12n+3 to 12n+3 in terms of D and L. We will assume H2n+1 and H¯ 2n+1 and establish H2n+3 , H¯ 2n+3 . We will also require several auxiliary definitions and conditions.
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STEVE JACKSON
Throughout this section, we will be using the notation K1 , . . . , Kt to denote a sequence of canonical measures in R2m+1 where m < n (recall R2m+1 = ,k ,m k m k W2m+1 ∪ ,k S2m+1 ) or of the form W2n+1 or S2n+1 for some , m. Given such a measure Kj , we let Kj denote the corresponding function space mea,m sure. For example, in the case Kj = S2n+1 and > 1, Kj will be the measure 1 1 on functions hj : 2n+1 → 2n+1 of the correct type induced from the strong partition relation on 12n+1 ,and if = 1, then Kj will be a measure on functions hj : <m → 12n+1 of the correct type. We use the notation h1 , . . . , ht to denote a sequence of functions in the spaces K1 , . . . , Kt . of D). We proceed to define the set D = Definition5.1 (Definition ,m D = D , for +1 ≤ ≤ 2n+1 − 1 or = −1. We as2n+1 n n ,m 2n+1 sume D2m+1 is defined for m < n and define D2n+1 . Our definition will be by a simultaneous induction in which we also define an ordering < on D2n+1 , two functions h and H associated with descriptions, conditions C , D, A and a numerical function k. We assume these notions are defined for D2m+1 , m < n. We define D2n+1 relative to a fixed sequence K1 , . . . , Kt of canonical measures as above. We denote the set of descriptions in D2n+1 which are defined relative to the sequence of measures K1 , . . . , Kt by D2n+1 (K1 , . . . , Kt ). So we ,m ,m When it causes no confusion we frequently will have D2n+1 = K D2n+1 (K). ,m = K1 , . . . , Kt for the rest of just write D2n+1 or D2n+1 . Fix such a sequence K ,m ,m ). = D2n+1 (K the definition, and we proceed to define the set D2n+1 We define basic and non-basic descriptions and subdivide these into types −1, 0, and 1; these types will correspond to = −1, = 1, and > 1 ,m ,m respectively. We will denote a general element of D2n+1 by d2n+1 . It will be an (Ia ) indexed tuple of the form d , where the index Ia is one of three forms: • Ia = (f(K¯ 1 ); K¯ 2 , . . . , K¯ a ). This will correspond to type 1, that is, > 1. • Ia = (fk ; K¯ 2 , . . . , K¯ a ). This will correspond to type 0, that is = 1. • Ia = (; K¯ 2 , . . . , K¯ a ). This will correspond to type −1, that is, = −1. The indices are viewed as formal symbols. The symbols K¯ 2 , . . . , K¯ a designate measures of the form v(Kj ) for some 1 ≤ j ≤ t, where Kj ∈ S2n+1 ,m ,m (and we recall that v(S2n+1 ) is the measure u such that S2n+1 induced from 1 1 the strong partition relation on 2n+1 , functions F : 2n+1 → 12n+1 , and the measure u2n+1 on 12n+1 ). Similarly, fk and f(K¯ 1 ) are formal symbols, where ,m K¯ 1 designates the measure v(S2n+1 ) and k is an integer. With a slight abuse of notation, we may also think of the symbol K¯ j (for j > 1) as coding a particular integer, which we denote by r(K¯ j ), between 1 and t, such that K¯ j = v(Kr(K¯ j ) ). Of course, we may have K¯ j = v(Kr1 ) = v(Kr2 ) for different integers r1 = r2 , but will assume that such a particular r(K¯ j ) is coded in the symbol K¯ j .
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Our induction in the following definitions is reverse induction on a function k, which although defined along with D2n+1 below, is actually defined outright. To define D2n+1 , we consider the following cases: Basic type −1: ¯ ¯ −1,m a) For n > 0, we allow descriptions of the form d2n+1 = d (;K2 ,... ,Ka ) , m where d = (k; ), and k is an integer 1 ≤ k ≤ t, such that Kk ∈ W2n+1 . k(d ) = k in this case. ¯ ¯ b) For n > 0, we allow d (Ia ) = d (;K2 ,... ,Ka ) = (k; d¯2 )s(Ia ) , where Ia = ¯ (K¯ 2 , . . . , K a ), s is a formal symbol which may or may not appear, and d¯2 ∈ m
,m¯ S2n−1 . Also k(d ) = k in this case. For n = 0, we allow d = (k; i), m where 1 ≤ i ≤ m, where Kk = W2n+1 . 1,m Basic type 0: We allow d = d2n+1 = (r), where r is an integer 1 ≤ r ≤ m. We set k(d ) = ∞ in this case. ¯ ¯ ¯ ,m Basic type 1: We allow d2n+1 = d (f(K1 );K2 ,... ,Ka ) , where d = (d¯2 )s , where the symbol s may or may not appear, d¯2 is defined relative to K¯ 2 , . . . , K¯ a , ¯ m¯ ¯ m¯ ,m , , m¯ where s < n, v(S2n+1 ) = K¯ 1 , and K¯ 1 = S2s+1 or W2s+1 ded¯2 ∈ D2s+1 ¯ ¯ pending on whether ≥ 1 or = −1. We set k(d ) = ∞. We also allow the distinguished description d = (). Non-Basic Descriptions: 1,mk a) If Kk (for some integer 1 ≤ k ≤ t) = S2n+1 , we allow d = d (Ia )
where d = (k; d1(Ia ) , . . . , dr(Ia ) )s where s may or may not appear, Ia is an index of one of the above forms, r ≤ mk , r > 1 if s ,m appears, d1(Ia ) , . . . , dr(Ia ) ∈ D2n+1 are defined w.r.t. K1 , . . . , Kt with k(d1(Ia ) ), . . . , k(dr(Ia ) ) > k, and d1(Ia ) > d2(Ia ) , . . . , dr(Ia ) w.r.t. the ordering < to be defined below (being defined simultaneously). We set k(d ) = k. k ,mk b) If Kk = S2n+1 where k > 1, we allow d = d (Ia ) where d = ¯ (k; d2(Ia ;Ka+1 ) )s , where the symbol s may or may not appear, K¯ a+1 = v(Kk ), d2 (with the index I(a+1) = (Ia ; K¯ a+1 )) is defined relative ,m , with k(d2 ) > k. We set k(d ) = to K1 , . . . , Kt and d2 ∈ D2n+1 k.
This completes the definition of D2n+1 . The functions h, H we will be defining in our induction will have the following properties:
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,m If d ∈ D2n+1 and satisfies condition C relative to K1 , . . . , Kt (where C is to be defined) then for almost all h1 , . . . , ht w.r.t. K1 × · · · × Kt , we have that h(d ; h1 , . . . , ht ) will be defined, and be an ordinal. The function H will have the properties:
a) If d as above with > 1 then for almost all h1 , . . . , ht as above, we have for almost all f : ϑ(K¯ 1 ) → 12n+1 of the correct type (where Ia = (f(K¯ 1 ); K¯ 2 , . . . , K¯ a ) in this case) that for almost all h¯ 2 , . . . , h¯ a w.r.t. the product of the function space measures corresponding to K¯ 2 , . . . , K¯ a that H (d ; h1 , . . . , ht ; f; h¯ 2 , . . . , h¯ t ), an ordinal, is defined. b) Same as above except = 1, in which case Ia = (fk ; K¯ 2 , . . . , K¯ a ), and we require f : k → 12n+1 (k an integer). c) For = −1, we have that for almost all h1 , . . . , ht , h¯ 2 , . . . , h¯ a , that the ordinal H (d ; h1 , . . . , ht , h¯ 2 , . . . , h¯ a ) is defined. We introduce, in our simultaneous induction, two hypothesis concerning the functions h and H . F2n+1 : If d is defined and satisfies condition C relative to K1 , . . . , Kt then for almost all h1 if h1 (1) = h1 (2) = h1 almost everywhere (representing elements of K1 ), for almost all h2 if h2 (1) = h2 (2) = h2 almost everywhere, . . . , for almost all ht and ht (1) = ht (2) = ht almost everywhere we have that h(d ; h1 (1), . . . , ht (1)) = h(d ; h1 (2), . . . , ht (2)). 2 F2n+1 : If d = d (Ia ) is defined and satisfies condition C relative to K1 , . . . , Kt , then for almost all h1 and h1 (1) = h1 (2) = h1 almost everywhere, . . . , for almost all ht and ht (1) = ht (2) = ht almost everywhere we have: i) If Ia = (f(K¯ 1 ); K¯ 2 , . . . , K¯ a ), then for almost all f : ϑ(K¯ 1 ) → 12n+1 , if [f1 ] = [f2 ] = [f] (w.r.t. K¯ 1 ) then for almost all h¯ 2 and h¯ 2 (1) = h¯ 2 (2) = h¯ 2 , . . . , for almost all h¯ a and h¯ a (1) = h¯ a (2) = h¯ a we have H (d ; h1 (1), . . . , ht (1); f1 ; h¯ 2 (1), . . . , h¯ a (1)) = H (d ; h1 (2), . . . , ht (2); f1 ; h¯ 2 (2), . . . , h¯ a (2)). ii) If Ia = (fk ; K¯ 2 , . . . , K¯ a ), then same as above with f : k → 12n+1 . iii) If Ia = (K¯ 2 , . . . , K¯ a ), then same as above except we omit the f. 2 In particular, we are assuming F2n+1 , F2n+1 for m < n. We now define conditions C and D. We first consider D. We assume d ∈ D2n+1 is defined relative to K1 , . . . , Kt , and we define when condition D holds for objects of the form d or (d )s (for d ∈ D2n+1 , (d )s is not a description, but this does not affect the definitions).
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,m ,m Definition 5.2 (Condition D). Given d or (d )s , where d = d2n+1 = D2n+1 and where d is defined relative to K1 , . . . , Kt , we require that d satisfy condition C and further: a) If > 1 (so Ia = (f(K¯ 1 ); K¯ 2 , . . . , K¯ a )), then if s does not appear we require that for almost all h1 , . . . , ht , if ϑ represents h(d ; h1 , . . . , ht ) w.r.t. ,m v2n+1 (where v = v(S2n+1 )), that there is a measure one set A w.r.t. v2n+1 restricted to which ϑ is strictly increasing of uniform cofinality . Further, if C ⊆ 12n+1 is c.u.b. then for almost all h1 , . . . , ht we have that ϑ has range almost everywhere in C . If s appears, then we require that for almost all h1 , . . . , ht that [ϑ] is the supremum of [ϑ ] for ϑ of the correct type with range a.e. in C . b) If = 1 (so Ia = (fk ; K¯ 2 , . . . , K¯ a )), then for almost all h1 , . . . , ht , if ϑ represents h(d ; h1 , . . . , ht ) w.r.t. the m-fold product of the -cofinal normal measure on 12n+1 , then we require that there is a measure one set A ϑ is (strictly) order-preserving w.r.t. <m and of uniform restricted to which cofinality , if s does not appear. In the case where s appears, we require ¯ represented by such functions. that [ϑ] is a supremum of ordinals [ϑ] ¯ ¯ c) If = −1 (so Ia = (K2 , . . . , Ka )), then for almost all h1 , . . . , ht , if ϑ m ), then there is a measure represents h(d ; h1 , . . . , ht ) w.r.t. v = v(W2n+1 one set A w.r.t. v restricted to which ϑ is of the correct type, if s does not appear. In the case where s appears, we require [ϑ] to be the supremum of ¯ represented by functions of the correct type. ordinals [ϑ]
We now define condition C. ,m ,m ∈ D2n+1 Definition 5.3 (Condition C). We again let d (Ia ) , where d = d2n+1 be given and defined relative to K1 , . . . , Kt . We let k = k(d ). We consider the following cases: d basic. 1) = −1. −1,m = (k; ). In this case d satisfies condition C. a) d2n+1 b) d = (k; d¯2 )s , where s may or may not appear. We require d¯2 or (d2 )s satisfy condition D relative to Kb1 , . . . , Kbm , K¯ 2 , . . . , K¯ a depending on whether s does not or does appear in d . 1,m 2) = 1 so d = d2n+1 = (r) where 1 ≤ r ≤ m. In this case d satisfies C. ,m 3) > 1, so d = d2n+1 = (d¯2 )s , where s may or may not appear. We require that d¯2 or (d2 )s satisfy D w.r.t. K¯ 2 , . . . , K¯ a depending on whether s does not or does appear in d . If d = (), then we define d to satisfy C. d non-basic. k ,mk ,m and k > 1, where d = d2n+1 = (k; d2(Ia ,Ka+1 ) )s , where s 1) Kk = S2n+1 may or may not appear. We require d2 to satisfy C w.r.t. K¯ 2 , . . . , K¯ a ,
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K¯ a+1 (defined by induction), and if the symbol s does not appear, we require for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a that the function g defined almost everywhere w.r.t. K¯ a+1 by g([ha+1 ]) = H (d2 ; h1 , . . . , ht ; f; h¯ 2 , . . . , h¯ a , h¯ a+1 ) is almost everywhere increasing of uniform cofi2 nality . This makes sense since H is defined and satisfies F2n+1 , by induction, for d2 . If s appears, we require that for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a that [g] is the supremum of [g ] for g of the correct type with supa.e. g = supa.e. g . 1,mk , where d = (k; d1(Ia ) , . . . , dr(Ia ) )s , r ≤ mk , and s may 2) Kk = S2n+1 or not may appear. We then require that d2(Ia ) < d3(Ia ) < · · · < dr(Ia ) < d1(Ia ) , and that for almost all h1 , . . . , ht ; f; h¯ 2 , . . . , h¯ a that Ia H (d1 ; h1 , . . . , h¯ a ), . . . , H (dr−1 ; h1 , . . . , h¯ a ) have cofinality . Also, Ia ; h1 , . . . , h¯ a ) has if s does not appear we further require that H (dr−1 cofinality . We now define the functions h and H . We first consider H . We assume d is defined and satisfies Condition C relative to K1 , . . . , Kt . ,m is of type We consider the following cases. Recall a description d = d2n+1 −1 if = −1, of type 0 if = 1 and of type 1 if > 1. 1) d basic of type −1. a) d = (k; ). For fixed h1 , . . . , ht , h¯ 2 , . . . , h¯ a where Ia = (K¯ 2 , . . . , K¯ a ) in this case, we set H (d ; h1 , . . . , ht , h¯ 2 , . . . , h¯ a ) = supa.e. hk , where mk mk mk Kk = W2n+1 , κ = κ(W2n+1 ), and hk : κ → 12n+1 (recall W2n+1 is in 1 duced by the weak partition property on 2n+1 , functions hk : κ → 12n+1 2n −1,mk and the measure S2n−1 on κ). b) d = (k; d¯2 )s , where s may or may not appear, and d¯2 or (d¯2 )s sat¯ m¯ , for isfies D relative to Kb1 , . . . , Kbu , K¯ 2 , . . . , K¯ a . Here d¯2 ∈ D2s+1 ¯ m¯ ¯ m¯ mk , , m¯ s < n, where v(W2n+1 ) = R2s+1 (= S2s+1 if ¯ > 0, and = W2s+1 if ¯ ¯ = −1). Since d2 satisfies C, we have by induction that for almost all h1 , . . . , ht , for almost all h¯ 2 , . . . , h¯ a that h(d¯2 ; hb1 , . . . , hbu , h¯ 2 , . . . , h¯ a ) mk is defined and is an ordinal in the measure space v(W2n+1 ). We set ⎧ if d = (k; d¯2 ) ⎨hk (h(d¯2 ; h¯ 2 , . . . , h¯ a , hbu )) ¯ ¯ H (d ; h1 , . . . , ht , h2 , . . . , ha ) = sup hk () if d = (k; d¯2 )s ⎩ ¯ ¯ ¯
c) For n = 0 and d = (k; i), we set H (d ; h1 , . . . , ht , h¯ 2 , . . . , h¯ a ) = hk (i). Note in this case that hk : mk → 1 , where Kk = W1mk . 2) Basic type 0. 1,m Here d = d2n+1 = (r), where r ≤ m. For fixed h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a 1 where f : m → 2n+1 , we set H (d ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = f(r).
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3) Basic type 1. ¯ ¯ ¯ Here d (Ia ) = d (f(K1 );K2 ,... ,Ka ) , where d = (d¯2 )s , where s may or may not appear and d¯2 is defined and satisfies Condition C relative to K¯ 2 , . . . , K¯ a . Hence by induction, for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , the ordinal h(d¯2 ; h¯ 2 , . . . , h¯ a ) is defined. We set ⎧ ⎨f(h(d¯2 ; h¯ 2 , . . . , h¯ a )) if d = (d¯2 ) ¯ ¯ H (d ; h1 , . . . , ht , h2 , . . . , ha ) = sup f() if d = (d¯2 )s ⎩
This makes sense since h(d¯2 ; h¯ 2 , . . . , h¯ a ) is an element of the measure space K¯ 1 . If d = (), we set H (d ; h1 , . . . , ht ; f, h¯ 2 , . . . , h¯ a ) = supa.e. f. k ,mk 4) d non-basic with Kk = S2n+1 and k > 1. Here d = (k; d2(Ia+1 ) )s , where s may or may not appear. Since d2 satisfies Condition C, for almost all h1 , . . . , ht , f; h¯ 2 , . . . , h¯ a , h¯ a+1 , H (d2 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , h¯ a+1 ) is defined. For fixed h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a we let g : ϑ(K¯ a+1 ) → 12n+1 be defined by g([h¯ a+1 ]) = H (d2 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , h¯ a+1 ), which is well 2 defined almost everywhere for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a by F2n+1 and induction. We then set ⎧ ⎨hk ([g]) if d = (k; d2(Ia+1 ) ) ¯ ¯ H (d2 ; h1 , . . . , ht , f, h2 , . . . , ha ) = sup h () if d = (k; d (Ia+1 ) )s k ⎩ 2 <[g]
1,mk 5) d non-basic with Kk = S2n+1 . Here d = (k; d1Ia , . . . , drIa )s with r ≤ mk , and s may or may not appear. Since d1Ia , . . . , drIa satisfy Condition C, we have by induction that for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a that the ordinals H (diIa ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) are defined. Also by induction and the definition of < (given below),
H (d2Ia ; · · · ) < · · · < H (drIa ; · · · ) < H (d1Ia ; · · · ). We set H (d ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = hk (H (d2Ia ; · · · ), . . . , H (drIa ; · · · ), H (d1Ia ; · · · )), if s does not appear and r = mk , H (d ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = sup{hk (H (d2Ia ; · · · ), . . . , H (drIa ; · · · ), r+1 , . . . , mk , H (d1Ia ; · · · )) : r+1 , . . . , mk < H (d1Ia ; · · · )}
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if r < mk and s does not appear, and H (d ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = Ia sup{hk (H (d2Ia , . . . ), . . . , H (dr−1 ; · · · ), r , . . . , mk , H (d1Ia ; · · · )) :
r < H (drIa ; · · · ), r+1 , . . . , mk < H (d1Ia ; . . . , )} if s appears. We now consider h. If Ia = (f(K¯1 ); K¯ 2 , . . . , K¯ a ), we represent h(d ; h1 , . . . , ht ) with respect to K¯ 1 2n+1 by a function [f] → h(d ; h1 , . . . , ht , f) for f : ϑ(K¯ 1 ) → 12n+1 of the correct type. Similarly, if Ia = (fk ; K¯ 2 , . . . , K¯ a ), we represent h(d ; h1 , . . . , ht ) w.r.t. the k-fold product of the -cofinal normal measure on 12n+1 by f → h(d ; h1 , . . . , ht , f) where now f ∈ ( 12n+1 )k . In general. we represent h(d ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ ) (where f does not appear if Ia = (K¯ 2 , . . . , K¯ a )) w.r.t. K¯ +1 by the function [h¯ +1 ] → h(d ; h1 , . . . , ht , f; h¯ 2 , . . . , h¯ , h¯ +1 ) for [h¯ +1 ] ∈ ϑ(K¯ +1 ). Finally, we set h(d ; h1 , . . . , ht , f; h¯ 2 , . . . , h¯ a ) = H (d ; , h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ). 2 2 By F2n+1 , this is well defined (where F2n+1 is established below). We now define the ordering < on the set of descriptions of the form d (Ia ) , for a fixed index Ia ; defined and satisfying Condition C relative to fixed K1 , . . . , Kt .
Definition 5.4 (Definition of ordering < on D2n+1 ). If d1 = d1(Ia ) , d2 = ,m we set d1 < d2 if for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a we d2(Ia ) ∈ D2n+1 (K), have H (d1 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < H (d2 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ). This is well defined by induction. 2 Before verifying F2n+1 and F2n+1 , we introduce a further condition. Definition 5.5 (Condition A). We let d (Ia ) be defined and satisfy Condik ,mk tion C w.r.t. K1 , . . . , Kt . If d = (k; d2(Ia+1 ) )s where Kk = S2n+1 with k > 1, and where s may or may not appear, then there is a dˆIa satisfying Condition C w.r.t. K1 , . . . , Kt , such that for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a we have H (dˆ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ k ) = sup H (d2 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ 1 , h¯ a+1 ), a.e. [h¯ a+1 ] 2 which makes sense by F2n+1 . We also require k(dˆ) > k, and all component ˆ tuples of d as well as d satisfy A.
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2 We now verify (in our simultaneous induction) F2n+1 and F2n+1 . We note 2 2 that F2n+1 implies F2n+1 , so we consider F2n+1 . We consider the following cases: 2 1) d (Ia ) basic type 1. Here d = (d2 )s , where s may or may not appear. F2n+1 s follows by induction, F2s+1 for s < n, and the fact that d2 or (d2 ) satisfies D w.r.t. K¯ 2 , . . . , K¯ a . 2) d (Ia ) basic type 0. Here d = (r) for some r ≤ m, where the index 2 follows immeIa is now of the form Ia = (fm ; K¯ 2 , . . . , K¯ a ). F2n+1 ¯ ¯ diately since H (d ; h1 , . . . , ht , fm , h2 , . . . , ha ) = fm (r) only depends on fm : m → 12n+1 . 3) d (Ia ) basic type −1. mk 2 i) d = (k; ), where Kk = W2n+1 . Since (as in the statement of F2n+1 ) hk (1) = hk (2) almost everywhere, supa.e. hk (1) = supa.e. hk (2), and the results follows. ii) d = (k; d¯2 )s , where s may or may not appear. Since d2 or (d2 )s satisfies Condition D w.r.t. Kb1 , . . . , Kbu , K¯ 2 , . . . , K¯ a , it follows from 2 induction, F2s+1 for s < n, and the definition of H that F2n+1 is satisfied. 4) d (Ia ) non-basic where d = (k; d2(Ia+1 ) )s , where s may or may not appear, 2 holds for d2 relative to the sequence and k > 1. By induction F2n+1 ¯ ¯ h1 , . . . , ht , f, h2 , . . . , ha+1 . Hence, for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , the function g : ϑ(K¯ a+1 ) → 12n+1 ) defined by ¯ g([ha+1 ]) = H (d2 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , h¯ a+1 )
is well defined almost everywhere. Since k(d2 ) > k = k(d ) it follows from the definition of H that we may assume that g has range almost everywhere k in a c.u.b. set C ⊆ 12n+1 defining a measure one set w.r.t. v2n+1 (where vk = v(Kk )) where hk (1), hk (2) agree. If the symbol s does not appear, then from the definition of Condition C we may assume that g is almost everywhere strictly increasing of uniform cofinality . Since we may assume C is sufficiently closed, we may assume g is everywhere increasing of uniform cofinality , and has range in C . Hence hk (1)([g]) = hk (2)([g]), 2 and F2n+1 follows. If the symbol s appears, [g] is the supremum of [g ] where g is of the correct type. Since C may be taken sufficiently closed, it follows readily that [g] is the sup of [g ] where g is of the correct type hav2 ing range in C . Hence sup<[g] hk (1)() = sup<[g] hk (2)(), and F2n+1 follows. 1,mk 5) d (Ia ) non-basic where d = (k; d (Ia ) , . . . , dr(Ia ) )s , and Kk = S2n+1 , where r ≤ mk and s may or may not appear. We again let C ⊆ 12n+1 be a of the c.u.b. set defining a set of measure one w.r.t. the m-fold product 1 -cofinal normal measure on 2n+1 where hk (1) and hk (2) agree. Since
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k(d1 ), . . . , k(dr ) > k, it follows that for almost all sequences h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , that the ordinals 1 = h(d1 ; h1 , . . . , ht , f, h¯ 2 ,. . . ,h¯ a ),. . . , r = H (dr ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) are in C . From the definition of Condition C we also have 1 < · · · r < 1 , and 1 , . . . , r−1 have cofinal2 now follows the definition of ity , as does r if s does not appear. F2n+1 H and the fact that C is sufficiently closed, so r is a supremum of points in C of cofinality . This completes the simultaneous induction defining D, h, H, <, conditions 2 . C, D, and A, and establishing F2n+1 , F2n+1 We require a lemma concerning the ordering <, which gives a combinatorial reformulation of the ordering. Lemma 5.6. If d1(Ia ) , d2(Ia ) are defined and satisfy conditions C and A relative to K1 , . . . , Kt , then d1 < d2 iff one of the following is satisfied: I) d1 , d2 are basic of type −1, so of the form a) d1 = (k1 ; ) or b) d1 = (k1 ; (d¯2 )1 )s , where s may or may not appear, and similarly for d2 (where d¯1 , d¯2 are integers for n = 0). There we require that one of the following is satisfied: 1) k1 < k2 . 2) k1 = k2 and d2 is of type (a) and d1 of type (b). 3) k1 = k2 , both are of type (b) and (d¯2 )1 < (d¯2 )2 , where for n = 0, by < we mean the usual ordering on the integers. 4) k1 = k2 , both are of type (b), (d¯2 )1 = (d¯2 )2 , and d1 has the symbol s and d2 does not. II) d1 , d2 basic of type 0, so d1 = (r1 ), d2 = (r2 ), and r1 < r2 . III) d1 , d2 basic of type 1, so d1 = ((d¯2 )1 )s or d1 = (), and similarly for d2 , where s may or may not appear. Then we require (d¯2 )1 < (d¯2 )2 , or (d¯2 )1 = (d¯2 )2 and d1 involves the symbol s and d2 does not, or d2 = () and d1 is of the first type. IV) At least one of d1 , d2 is non-basic. We then require that one of the following is satisfied: k ,mk 1) k(d1 ) > k(d2 ), in which case we require dˆ2 ≥ d1 if Kk(d2 ) = S2n+1 with k > 1 (here dˆ2 is as in Condition A). If k = 1, then d1 ≤ (d2 )1 , (Ia ) (Ia ) s a) where d = (k; (d2 )(I ) . 1 , (d2 )2 , . . . , (d2 )r k ,mk ˆ 2) k(d1 ) < k(d2 ), in which case we require d1 < d2 if Kk(d1 ) = S2n+1 mk with k > 1. If k = 1, then we require (d1 )1 < d2 or Kk(d1 ) = W2n+1 . 3) k(d1 ) = k(d2 ) = k, in which case one of the following is satisfied: k ,mk with k > 1, so d1 = (k; (d2 )1 )s , d2 = (k; (d2 )2 )s , i) Kk = S2n+1 where s may or may not appear in d1 , d2 . We require that (d2 )1 < (d2 )2 or (d2 )1 = (d2 )2 and d1 involves s and d2 does not.
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(Ia ) s k ,mk a) ii) Kk = S2n+1 with k = 1, so d1 = (k; (d1 )(I 1 , . . . , (dr1 )1 ) , a) s d2 = (k; (d1 )2(Ia ) , . . . , (dr2 )(I 2 ) , where s may or may not appear. We then require that: a) (d1 )1 < (d1 )2 or b) (d1 )1 = (d1 )2 and there is a p ≤ min{r1 , r2 } such that (dp )1 = (dp )2 , and for the least such p, (dp )1 < (dp )2 ; or c) (dp )1 = (dp )2 for 1 ≤ p ≤ min{r1 , r2 } and r1 < r2 and d1 involves s; or r1 > r2 and d2 does not involve s; or r1 = r2 and d1 involves s and d2 does not.
We also have: Lemma 5.7. If d1 = d2 , and d1 , d2 satisfy Conditions C and A, then d1 < d2 or d2 < d1 . Lemma 5.7 follows immediately from Lemma 5.6 and the corresponding statement for the ordering < defined from the statement of Lemma 5.6. This in turn follows readily from a consideration of cases. Before establishing Lemma 5.6 we introduce the following notation. k ,mk Definition 5.8. Suppose Kk = S2n+1 and hk represents the ordinal [hk ] ∈ dom(Kk ) Recall that if k > 1 then hk : 12n+1 → 12n+1 of the correct type k ,mk ) is a measure on represents [hk ] with respect to v2n+1 where v = v(S2n+1 k ,mk mk 1 κ(S2n+1 ). If k = 0 then hk : < → 2n+1 is order-preserving. We define hk (0), the 0th invariant of hk , as follows.If k > 1 we set k ,mk hk (0)() = sup{hk ([g]) : g is of the correct type from κ(S2n+1 ) to },
and if k = 1 we set hk (0)() = sup hk (2 , . . . , mk , ) : 2 < · · · < mk < }. To establish Lemma 5.6, it is enough to show that d1 < d2 implies d1 < d2 . We assume this for descriptions d ∈ D2s+1 for s < n, and note that for almost all h1 , . . . , ht , we may assume that: mk
mk
1 2 1) If k1 < k2 and Kk1 = W2n+1 , Kk2 = W2n+1 , then inf hk2 > sup hk1 .
k ,mk
k ,mk
1 1 2 2 2) If k1 < k2 and Kk1 = S2n+1 , Kk2 = S2n+1 , then the range of hk2 is closed under hk1 (0). From these two properties and the definition of H , a straightforward induction shows that Lemma 5.6 is satisfied. We now prove some technical lemmas concerning the family of canonical measures, which we prove in greater detail than required for this paper, and which simplify the description of conditions C and D. It is convenient to introduce an auxiliary family of canonical measures ˜ R˜ 2n+1 and embeddings Π.
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W˜ 1m : We set W˜ 1m = W1m = the m-fold product of the -cofinal normal measure on 1 , which we identify with an ordinal, as in W1m , by ordering ˜ on these by the largest component first, then next largest, etc. We define Π measures exactly as before on W1m , so Πm (α , . . . , α ) = (α 1 m m−k+1 , . . . , αm ). k 1,m 1,m ˜ ˜ S1 : We define A to have measure on w.r.t. S1 if there is a c.u.b. C ⊆ 1 such that for all h : <m+1 → C of the correct type (where <m+1 is as in the definition of S11,m+1 ), [h] ∈ A, where h : <m → C is defined by h(α2 . . . , αm , α1 ) = supαm+1 <α1 h (α2 . . . , αm+1 , α1 ). This is essentially the same as the definition of S11,m except the functions no longer have uniform cofinality . 2 ˜m ˜ 1,m2 ) → (S˜ 1,m1 ) (for m2 > m1 ) as follows: given α We define Π m1 : ϑ(S1 1 represented by h : <m2 → 1 , (induced by an h : <m2 +1 → 1 , as above), we m1 ¯ 2 ˜m → defined by let Π m1 (α) be represented by h : < ¯ 2 , . . . , αm , α1 ) = h(α 1
sup αm1 +1 ,... ,αm2
h(α2 , . . . , αm1 , αm1 +1 , . . . , αm2 , α1 ).
In general, we proceed as follows: m m : We define A to have measure one w.r.t. W˜ 2n+1 if there is a c.u.b. W˜ 2n+1 ,m+1 1 C ⊆ 2n+1 such that for all f : ϑ(S˜2n−1 ) → C of the correct type (where, as in the definition of W m , = 2n − 1 is maximal), [f] ∈ A, where 2n+1
,m f : ϑ(S˜2n−1 ) → 12n+1 is defined by f(α) = sup : Π˜ m+1 f (), where m ()=α ˜ m+1 ˜ ,m+1 ˜ ,m Π m , of course, denotes the embedding from ϑ(S2n−1 ) to ϑ(S2n−1 ). m m m 2 1 2 ˜ m from ϑ(W We define Π 2n+1 ) → ϑ(W2n+1 ), for m2 > m1 , as follows: 1 ,m2 given α represented by f : ϑ(S˜2n−1 ) → 12n+1 induced by an f as above, m ¯ ¯ 2 ˜ m (α) be represented by f : ϑ(S˜ ,m1 ) → 1 let Π 2n+1 defined by f(α) = 2n−1 1 2 ˜m ˜ ,m2 ˜ ,m1 sup : Π˜ mm2 ()=α f(), where Π m1 denotes the embedding from S2n−1 to S2n−1 . 1 ,m S˜2n−1 : If = 1, we proceed as in the definition of S˜11,m , using 12n+1 instead ˜ similarly here. of 1 . We define Π ,m For > 1, we let R˜ 2s+1 denote the ( − 1)st measure in the enumeration of the measures W˜ 1m , S˜11,m , W˜ 3m , S˜31,m , S˜32,m , . . . , etc., similarly to the definition of S ,m . We let then define A to have measure one w.r.t. S˜ ,m if there 2n+1
2n+1
is a c.u.b. C ⊆ 12n+1 such that for all F : 12n+1 → C of the correct type, ,m R˜ 2s+1 is the measure on 12n+1 from the weak partition [F¯ ] R˜ ,m ∈ A. Here 2n+1 2s+1 2n+1 ,m 1 ˜ relation on 2n+1 and functions f : ϑ(R2s+1 ) → 12n+1 which are induced ,m+1 ) → 12n+1 of the correct type (via the corresponding by an f : ϑ(R˜ 2s+1 ,m+1 ,m ˜ m+1 embedding Π for R˜ 2s+1 and R˜ 2s+1 ). Also, F¯ is given by F¯ ([f]) = m ,m+1 ˜ ˜ sup{F ([g]) : Π([g]) = [f]}, where Π([g]), for g : ϑ(R˜ 2s+1 ) → 12n+1 , is defined as above.
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,m2 ,m1 2 ˜m We define Π m1 from ϑ(S2s+1 ) to ϑ(S2s+1 ) as follows: for α represented w.r.t. ˜
,m
˜
2
,m
1
R2s+1 R2s+1 2 ˜m 2n+1 by F¯ , as above, we represent Π m1 (α) w.r.t. 2n+1 by G defined by: for g induced by an appropriate g , we set G[(g)] = supf F¯ ([f]), where the sup ,m2 ranges over f : ϑ(R˜ 2s+1 ) → 12n+1 of the appropriate type (i.e., induced by an 2 ˜m ˜ m2 f ) such that f(ϑ) < g(Π m (α)) for all ϑ (where Πm is defined by induction). 1
1
from We also extend the embeddings slightly to include embeddings Πm+1 m ,m+1 ,m R2n+1 to R˜ 2n+1 as follows: ,m+1 ,m+1 m+1 If R2n+1 = W2n+1 , and f : ϑ(S2n−1 ) → 12n+1 of the correct type represents ,m m+1 α, then we let Πm (α) be represented by f¯ : ϑ(S˜2n−1 ) → 12n+1 defined by ,m+1 ¯ f() = sup{f( ) : Π( ) = }, where Π denotes the embedding from S2n−1 ,m to S˜2n−1 , defined by induction. It follows readily that for almost all α (w.r.t. S ,m+1
2n−1 m+1 2n+1 = W2n+1 ) that Π(α) is of the appropriate type, that is, induced by a ,m+1 ˜ f : ϑ(S2n−1 ) → 12n+1 of the correct type. ,m+1 ,m+1 = S2n+1 , and F : 12n+1 → 12n+1 of the correct type represents α If R2n+1 ,m+1 R2s+1 ¯ , then we represent Πm+1 w.r.t. 2n+1 m (α) by a function F defined as follows: ,m ,m+1 1 ¯ ˜ given f : ϑ(R2s+1 ) → 2n+1 induced by an f : ϑ(R2s+1 ) → 12n+1 of the ¯ m+1 ¯ correct type, we set F¯ ([f]) = sup{F ([f] : Πm+1 m ([f]) = [f]}. Here the Πm appearing in this formula is as defined in the previous paragraph. We now state four lemmas which we prove by simultaneous induction on n:
m ) = 12n+1 → Ord monotonically inLemma 5.9. Given any H : ϑ(W2n+1 creasing almost everywhere (i.e., restricted to a measure one set), if m > 1 m then there is a measure one set A w.r.t. W2n+1 s.t. H A is increasing, or H (α) for α ∈ A depends only on Πm (α). If m = 1, then we replace the last part m−1 with: H (α) depends only on supa.e. f, for [f] = α, for n > 1, and for n = 1, with: H is constant almost everywhere. ,m Lemma 5.10. Given any H : ϑ(S2n+1 ) → Ord monotonically increasing al,m most everywhere, there is a measure one set A w.r.t. S2n+1 s.t. H A is increasing, or:
If = 1, m > 1, then H (α) depends only on Πm m−1 (α) (for α ∈ A). If = 1, m = 1, then H A is constant. If > 1, m > 1, then H A depends only on Πm m−1 (α). If > 1, m = 1, then H (α), for α represented by F : 12n+1 → 12n+1 of the correct type, depends only on F (0) (recall here Definition 5.8). ,0 5) If m = 0, let S2n+1 be the measure induced from the strong partition relation on 12n+1 , functions F : 12n+1 → 12n+1 of the correct type, and the ˜ ˜ ,1 ,1 ,1 cf(ϑ(R2s+1 )) cofinal normal measure on 12n+1 , where R2s+1 = v(S2n+1 ). 1) 2) 3) 4)
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,0 Then any H : ϑ(S2n+1 ) → Ord is either increasing almost everywhere, or constant almost everywhere. m Lemma 5.11. Let H : ϑ(W2n+1 ) = 12n+1 → Ord be pressing down almost everywhere. If m > 1, then there is a measure one set A such that for α ∈ A, m H (α) depends only on Πm−1 (α). For m = 1, H A is constant. ,m Lemma 5.12. If H : ϑ(S2n+1 ) → Ord is pressing down almost everywhere ,m then there is a measure one set A w.r.t. S2n+1 such that: 1) If = 1, m > 1, then H (α) for α ∈ A depends only on Πm m−1 (α). 2) If = 1, m = 1, then H A is constant. 3) If > 1, m > 1, then H (α) for α ∈ A depends only on Πm m−1 (α). 4) If > 1, m = 1, then H (α) for α ∈ A depends only on F (0), for F representing α.
We require two additional lemmas for the proofs, which we prove first in a separate induction: ,m ) → C of the correct Lemma 5.13. Given any C ⊆ 12n+1 , and f, g : ϑ(S2n−1 n type (where = 2 − 1 is maximal) and satisfying [f] < [g] and Πm m−1 ([f]) = ([g]), then there are f , g satisfying: Πm 2 2 m−1 1) [f2 ] = [f], [g2 ] = [g]. 2) f2 (α) < g2 (α) < f2 (α + 1) for all α, and f2 , g2 are of the correct type. 3) f2 , g2 have range in C .
If m = 1, we require f, g satisfy supa.e. f = supa.e. g, and have the same conclusion. Lemma 5.14. The same as Lemma 5.13 except we use F , G : 12n+1 → 12n+1 m of the correct type with [F ] R,m < [G] R,m , and Πm m−1 ([F ]) = Πm−1 ([G]) if 2s+1 2n+1
2s+1 2n+1
m > 1 and if m = 0, [F (0)] = [G(0)]. We have for n = 0, [F ]W1m < [G]W1m in the hypothesis. We have the same conclusion as Lemma 5.13. We first prove Lemmas 5.13 and 5.14 by induction. We first consider Lemma 5.13. We let C be given, and f, g as in Lemma 5.13. We assume m > 1, the case m = 1 being similar. We let C1 be a c.u.b. subset of 12n−1 such that for α represented by correct type, depending on whether H : 12n−1 → C1 or <m → C1 of the > 1 or = 1, we have f(α) < g(α) and sup m : Πm m−1 ()=Πm−1 (α)
f() =
sup
g().
m : Πm m−1 ()=Πm−1 (α)
We first assume > 1. We consider the following partition P: We partition functions H1 , H2 : 12n−1 → 12n−1 of the correct type with H1 (α) < H2 (α) <
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H1 (α + 1) for all α, according to whether or not f([H2 ]) > g([H1 ]), where v(S ,m )
2n−1 [H1 ], [H2 ] mean, of course, with respect to the measure 2n−1 . It follows readily that for H1 , H2 for the above type that Πm m−1 ([H1 ]) = Πm ([H ]). We claim that on the homogeneous side of the partition the 2 m−1 1 property stated in partition P holds. We suppose not, and let C2 ⊆ 2n−1 be a c.u.b. set homogeneous for the other side. We let C3 = C1 ∩C2 and C4 ⊆ C3 be contained in the closure points of C3 , that is if α ∈ C4 and , < α, then the ( · )th element of C3 after is less than α (we use this notation throughout). We fix H1 : 12n−1 → C4 of the correct type. Since C4 ⊆ C1 , it follows that m there is an H2 : 12n−1 → 12n−1 such that Πm m−1 ([H ]) = Πm−1 ([H2 ]) and f([H2 ]) > g([H1 ]). We may clearly assume that ran H2 ⊆ C3 , replacing H2 by a larger function if necessary (e.g., the function H˜ 2 (α) = the next th element of C3 after max(H2 (α), sup<α H˜ 2 ())). By Lemma 5.14 and induction, it now follows that there are functions H1 , H2 satisfying:
1) [H1 ] = [H1 ], [H2 ] = [H2 ], and hence f([H2 ]) > g([H1 ]). 2) H1 , H2 are of the correct type everywhere and ordered as in P. 3) ran H1 , H2 ⊆ C3 . This, however, contradicts C3 ⊆ C2 and the definition of C2 . Hence, on the homogeneous side of the partition, the property stated in P holds. We let C2 ⊆ C1 be homogeneous for P. We let C3 be contained in the closure points of C2 (as above), and C4 be contained in the closure points of C3 . We then define the functions f2 , g2 as follows: For α represented by H : 12n−1 → C4 of the correct type, we set f2 (α) = f(α), g2 (α) = g(α). For not of this form, we let be the least ordinal > which is represented by an H : 12n−1 → C3 of the correct type. For < α, with α as above, it is easily seenthat < α as well. We fix such an H representing , with H ⊆ C3 everywhere. We then define H : 12n−1 → C2 by H () = the ( · H ())th element of C2 after H (), where H represents . We let [H ] = . We then set f2 () = f( ), g2 () = g( ). This is well-defined. We note that is also of the correct type, since H was, and ran H ⊆ C3 . For α as above, that is, represented by H of the correct type with range in C4 , it follows readily that if < α then < α. From this, and the definitions of C1 , C2 , it now follows that f2 and g2 have the desired properties. The case = 1 is similar, using functions H : <m → 12n−1 instead. We now consider Lemma 5.14, We again consider the case m > 1, the case m = 1 being similar. We fix F , G : 12n+1 → 12n+1 of the correct type as in P: We partition functions partition Lemma 5.14. We consider the following ,m 1 f, g : ϑ(R2s+1 ) → 2n+1 of the correct type with f(α) < g(α) < f(α + 1) to whether or not F ([g]) > G([f]). everywhere, according It follows as in the proof of Lemma 5.13, using induction and Lemma 5.13, that on the homogeneous side of the partition, the property stated in P holds.
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We let C2 ⊆ C1 be homogeneous for P, where C1 ⊆ 12n+1 is a c.u.b. such ,m ) → C1 of the correct type, F ([f]) < G([f]), and that for f : ϑ(R2s+1 sup m : Πm m−1 ()=Πm−1 ([f])
F () =
sup
G().
m : Πm m−1 ()=Πm−1 ([f])
We let C3 be contained in the closure points of C2 , and C4 be contained in the closure points of C3 . We then define F2 (α) = F (α) and G2 (α) = G(α) ,m for α represented by f : ϑ(R2s+1 ) → C4 of the correct type, and for other , we defined , similarly to Lemma 5.13, and set F2 () = F ( ), G2 () = G( ). It follows that F1 , G2 have the desired properties. We now consider Lemma 5.9. We let H : 12n+1 → Ord be monotonically ,m increasing almost everywhere, say restricted to [f] for f : ϑ(S2n−1 ) → C of the correct type, where we assume n ≥ 1 here, the case n = 0 being similar. We consider the following partition P: We partition functions f, ,m ) → 12n+1 of the correct type with f(α) < g(α) < f(α + 1) g : ϑ(S2n−1 everywhere, according to whether or not H ([f]) > H ([g]) or not. We let C1 ⊆ 12n+1 be c.u.b. and homogeneous for P, and let C2 be contained in the closure points on C ∩ C1 . We then claim that C2 defines the measure one set as in the statement of Lemma 5.9. This follows from the definition of C2 and Lemma 5.13. In the event we are on the homogeneous side in which P holds, we use the trivial fact that if f, g are of the correct type with m range in C2 and Πm m−1 ([f]) < Πm−1 ([g]), then there is an f of the correct type with range in C ∩ C1 such that f < f < g almost everywhere, and m Πm m−1 ([f ]) = Πm−1 ([g]). The proof of Lemma 5.10 is similar to that of Lemma 5.9, only we partition functions F , G : 12n+1 → 12n+1 instead, and we use Lemma 5.14. Lemma 5.11. We now consider 1 We fix H : 2n+1 → Ord pressing down almost everywhere, say on a measure S ,m 2n−1 one set w.r.t. 2n+1 determined by C ⊆ 12n+1 , where we assume n ≥ 1, the case n = 0 being easier. We consider the following partition P: We partition functions f, g : ,m ϑ(S2n−1 ) → 12n+1 of the correct type with f(α) < g(α) < f(α + 1) ev erywhere, according to whether or not [f] > H ([g]). It follows readily that on the homogeneous side of the partition, the property stated in P holds. We let C1 be homogeneous for P, and let C2 be contained in the closure points of C ∩ C1 . ,m It follows if f : ϑ(S2n−1 ) → C2 is of the correct type, and f¯ is defined by ¯ > H ([f]). ¯ f(α) = the next th element of C ∩C1 after sup>α f(), then [f] We now consider the following partition P2 : We partition functions f, g : ,m ϑ(S2n−1 ) → 12n+1 of the correct type with f(α) < g(α) < f(α + 1) every to whether or not H ([f]) < H ([g]). where, according
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419
If on the homogeneous side of the partition, the property stated in P2 fails, then the lemma follows readily from Lemma 5.13. Otherwise, we let C3 ∩ C2 be homogeneous for P2 , and let C4 be contained ,m in the closure points of C3 . We fix f : ϑ(S2n−1 ) → C4 of the correct type, and ¯ let f be as above. ,m We next claim that if f2 is any function f2 : ϑ(S2n−1 ) → C3 of the correct m m ¯ > H ([f2 ]). This will give a type with Πm−1 ([f2 ]) = Πm−1 ([f]), then [f] contradiction, since it will give an order-preserving map, namely H , from the ¯ and the former is easily seen to have larger order type. set of such [f2 ] into [f], ,m To prove the claim, we fix f2 : ϑ(S2n−1 ) → 12n+1 of the correct type with m ˜ ˜ Πm m−1 ([f]) = Πm−1 ([f2 ]). It is enough to establish that f = f2 almost ˜ ˜ everywhere, where f(α) = sup>α f(), and similarly for f˜2 . We suppose ,m ˜ ˜ ) not, say f2 > f almost everywhere. We then define the function h on ϑ(S2n−1 ˜ by h(α) = by least < α such that f2 () > f(α), defined almost everywhere. Since h is pressing down, it follows from induction and Lemma 5.12 that h(α) depends only on Πm m−1 (α) if m > 1, and on F (0), for F representing m α, if m = 1. It follows that Πm m−1 ([f]) < Πm−1 ([f2 ]), contradicting our assumption. This establishes the claim, and completes the proof of Lemma 5.11. The proof of Lemma 5.12 is similar, using induction and Lemma 5.11. Finally, we remark that the above six lemmas are also true with the measure ,m ,m , W˜ 2n+1 , the proofs being essentially identical to those give above. S˜2n+1 To state the next two lemmas, we require a technical definition: ,m Definition 5.15. Given two functions f, g : ϑ(S2n−1 ) → Ord, we say that they are ordered of k-type, where 1 ≤ k ≤ m, if they satisfy the following: 1) f, g are of the correct type. ,m ) are represented by F, G of the correct type, then f(α) < 2) If α, ∈ ϑ(S2n−1 m m m g() if Πk (α) ≤ Πm k (), and f(α) > g() if Πk (α) > Πk (). For other ordinals the ordering is more or less arbitrary; we use the following: 3) If α1 < α < α2 where α1 , α2 are of the correct type (i.e., represented by m functions of the correct type) with Πm k (α1 ) = Πk (α2 ), and ≥ the sup of m m all the α of the correct type with Πk (α ) = Πk (α1 ), then f() > g(α). 4) If α2 is the least ordinal > α of the correct type, α < inf α for all α of m the correct type with Πm k (α ) = Πk (α2 ), and α ≥ sup α for all α of the m m correct type with Πk (α ) < Πk (α2 ), and also satisfies the above (for the same α2 ), then f(α) < g() if α ≤ , and f(α) > g() if α > . 5) If α2 is as in above 4, corresponding to α, and > inf α for α of the m correct type with Πm k (α ) = Πk (α2 ), then f() > g(α).
It is easily seen that this defines a unique way of ordering ran f ∪ ran g.
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STEVE JACKSON R,m
2s+1 Definition 5.16. Given F, G : ϑ(2n+1 ) = 12n+1 → Ord, we say F, G are the same 5 conditions above, ordered of k type, for 1 ≤ k ≤ m, if they satisfy ,m where α of the correct type means represented by an f : ϑ(R2s+1 ) → 12n+1 of m the correct type, and we use the corresponding Πk defined for such α. Finally, for F, G : <m → 12n+1 of the correct type, F, G are ordered of k-type if for if there is an α = (α2 , . . . , αm , α1 ), = (2 , . . . , m , 1 ) then G(α) > F () r < k such that αr = r and for the least such r we have αr > r , or αr = r for all r < k and αk ≥ k . Otherwise we require G(α) < F ().
We now state two lemmas which generalize Lemmas 5.13 and 5.14. ,m ) → C of Lemma 5.17. Given any c.u.b. C ⊆ 12n+1 and f, g : ϑ(S2n−1 m ([f]) = Π ([g]) for some 1 ≤ k ≤ m − 1, and the correct type with Πm k k m Πm ([f]) < Π ([g]), then there are f , g satisfying: 2 2 k+1 k+1 1) [f2 ] = [f], [g2 ] = [g]. 2) f2 , g2 are ordered of k + 1-type. 3) f2 , g2 have range in C . m Also, if Πm 1 ([f]) < Π1 ([g]) then f2 , g2 are ordered of 1-type if sup f = sup g, and if sup f < sup g, then inf g2 > sup f2 . R,m
2s+1 ) = Lemma 5.18. The same as Lemma 5.17 except we use F, G : ϑ(2n+1 1 m m m 2n+1 → C , and the corresponding Πk ’s. If Π1 ([F ]) < Π1 ([G]) here, we require F2 , G2 to be ordered of 1-type if [F (0)] = [G(0)], and if [F (0)] < [G(0)], then we require that F2 (α) < G2 () if supa.e. fα ≤ supa.e. f , and if supa.e. fα > supa.e. f then G2 () < F2 (α) for fα , g representing α, .
Proof of Lemma 5.17. We fix such f, g of the correct type, and suppose m m m m m Πm k ([f]) = Πk ([g]) and Πk+1 ([f]) < Πk+1 ([g]), the case Π1 ([f]) < Π1 ([g]) being similar. We consider the following partitions: P1 : We partition functions H1 , H2 : 12n−1 → 12n−1 ordered of k + 1-type for k + 1-type if = 1, according > 1, and H1 , H2 : <m → 12n−1ordered of to whether or not f([H2 ]) > g([H1 ]). P2 : We partition functions H1 , H2 as above, ordered of k + 2-type, according to whether or not g([H1 ]) > f([H2 ]). (If k = m − 1, we don’t consider P2 .) We then claim that on the homogeneous sides of these partitions that the properties stated in them hold. We consider first P1 We suppose not, and fix a c.u.b. C1 ⊆ 12n−1 homogeneous for the contrary side. We may assume C1 is contained in a c.u.b. C such that for H having range in C of the correct type, sup : Πm ()=Πm ([H ]) k k
f() =
sup : Πm ()=Πm ([H ]) k k
g().
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We fix H1 of the correct type with range in C1 . It follows that there is an H2 of the correct type with range C1 such that f([H2 ]) > g([H1 ]) and with m m m Πm k ([H2 ]) = Πk ([H1 ]), and Πk+1 ([H2 ]) > Πk+1 ([H1 ]). It then follows from Lemma 5.18 and induction that there are H1 ,H2 ordered of k + 1-type with range in C1 and [H1 ] = [H1 ], [H2 ] = [H2 ]. This contradicts the definitions of C1 . We let C1 be a c.u.b. subset of 12n−1 homogeneous for P1 . It similarly follows that there isa c.u.b. C2 ⊆ 12n−1 homogeneous for P2 . We let C3 = C1 ∩ C2 , and C4 be contained in the closure points of C3 . Let ,m A3 and A4 be the measure one sets (w.r.t. (S2n−1 )) they define. Restricted to A3 , it follows that f, g are ordered of k +1-type. For example, if α, are represented by H1 , H2 having range in C3 of the correct type, and m Πm k+1 () > Πk+1 (α), then since we may assume without loss of generality m that Πm k () = Πk (α) (since f, g are increasing), we have from Lemma 5.18 and induction that α = [H1 ], = [H2 ] for some H1 , H2 ordered of k + 1-type with range in C3 . So, since C3 ⊆ C1 we have f() > g(α). Similarly, if m m m Πm k+1 (α) = Πk+1 () and (without loss of generality), Πk+2 (α) ≤ Πk+2 () for m < k − 1, then from Lemma 5.18 and induction, and C3 ⊆ C2 we have g(α) > f(). We then define f2 , g2 as follows: For α ∈ A4 , we set f2 (α) = f(α), g2 (α) = g(α). For α ∈ / A4 , we define, by induction, f2 (α) = the th element in the range of f ∪ g which is greater that sup{f2 (), g2 ()}, where , range over the ordinals such that for any f , g ordered of k + 1-type, f () < f (α), g () < f (α). We similarly define g2 (α), where , now range over ordinals such that f () < g (α), g () < g (α), for f , g as above. It then follows readily that f2 , g2 are ordered of k + 1-type. This is immediate once it is shown that f2 , g2 are increasing, and this follows once it is shown that if α < , α ∈ / A4 , and ∈ A4 , then f2 (α) < f2 (), and similarly for g2 . This, in turn, follows from the definition of A4 and k + 1type. We use here the facts that if 1 < < 2 and 1 , 2 of the correct m type with Πm k+1 (1 ) = Πk+1 (2 ), then the definition of k + 1-type requires no value g() between f(1 ) and f(2 ), and no value f() between g(1 ) and g(2 ). Also, if f() is required to be less than inf f() for of the correct m type with Πm k+1 () = Πk+1 (0 ) for some fixed 0 , then is less than the least such . This completes the proof of Lemma 5.17. The proof of Lemma 5.18 is entirely similar, using induction and Lemma 5.17. We require two additional technical lemmas. ,m Lemma 5.19. Given H : ϑ(S2n−1 ) → Ord, there is a measure on set A restricted to which H is monotonically increasing.
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m Lemma 5.20. Same as Lemma 5.19 using H : ϑ(W2n+1 ) = 12n+1 → Ord, where n ≥ 1. ,m Proof of Lemma 5.19. We fix H : ϑ(S2n−1 ) → Ord. For each 1 ≤ k ≤ m, we consider the following partition Pk : We partition functions H1 , H2 : 12n−1 → 12n−1 (or if = 1 we have H1 , H2 : <m → 12n−1 ), ordered of k-type to whether or not H ([H ]) ≤ H ([H ]). Ifk = m, then by “ordered according 1 2 of m-type” we mean H1 (α) < H2 (α) < H1 (α + 1) for all α. For k = 0, we also consider the partition P0 , where H1 , H2 are ordered as in the last clause of Lemma 5.18 for > 1. For = 1 and α = (α2 , . . . , αm , α1 ), and if α1 ≥ 1 , then = (2 , . . . , m , 1 ), if α1 < 1 then H2 (α) < H1 () H2 (α) > H1 (). It follows from an easy well-foundedness argument that on the homogeneous side of the partition the property statedin Pk holds. We let Ck , for 0 ≤ k ≤ m, be homogeneous for Pk , and C = k Ck . If A is the measure one set determined by C , then we claim that H A is monotonically increasing. For, let α < be in A, say represented by H1 , H2 of the correct type with m range in C . We let 1 ≤ k ≤ m be maximal such that Πm k (α) = Πk (), if such a k exists. In this case, by Lemma 5.18, it follows that there are H1 , H2 ordered of k + 1-type with range in C ⊆ Ck with [H1 ] = [H1 ], [H2 ] = [H2 ]. m Hence H ([H1 ]) = H ([H1 ]) ≤ H ([H2 ]) = H ([H2 ]). If Πm 1 (α) < Π1 (), then H1 , H2 are ordered as in the last part of Lemma 5.18 (depending on whether [H1 (0)] < [H2 (0)] or not) for > 1, and for = 1 as in the paragraph above, and H ([H1 ]) = H ([H1 ]) ≤ H ([H2 ]) = H ([H2 ]) again follows. (Lemma 5.19) The proof of Lemma 5.20 is similar. We now state two lemmas which simplify conditions C and D.
Lemma 5.21. For any c.u.b. C ⊆ 12n+1 , there is a c.u.b. C ⊆ C such that ,m (for s ≤ n) for any f : ϑ(R2s−1 ) → C which is monotonically increasing, [f] ,m is the supremum of ordinals [g], for g : ϑ(R2s−1 ) → C of the correct type. Lemma 5.22. For any c.u.b. C ⊆ 12n+1 , there is a c.u.b. C ⊆ C such that for any F : 12n+1 → 12n+1 monotonically increasing almost everywhere w.r.t. ,m R2s+1 2n+1 , [F ] is the supremum of ordinals [G] for G : 12n+1 → C of the correct non-constant almost type, provided [F ] is not minimal with respect to being everywhere. We first consider Lemma 5.21. ,m Proof of Lemma 5.21. We consider the case R2s−1 = W1m , this case following directly. For a given C , we let C ⊆ C be contained in the closure points of C . ,m We fix f : ϑ(R2s−1 ) → C monotonically increasing almost everywhere, say
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on the measure one set determined by C1 ⊆ 12s−1 . Applying Lemmas 5.9 and 5.10, we conclude that either f is increasing almost everywhere, m or f(α) depends only on Πm−1 (α) almost everywhere. In the latter case, ,m−1 we view f as a function on f : ϑ(R˜ 2s−1 ) (for m > 1). That is, there ,m−1 m−1 ˜ is a f : ϑ(R2s−1 ) → C with f(α) = f m−1 (Πm m−1 (α)) almost everywhere. Applying Lemmas 5.9 and 5.10 again to f m−1 , we get that either f m−1 is increasing almost everywhere, or there is an f m−1 : ϑ(R˜ ,m−2 ) → C 2s−1
(for m > 2) such that f(α) = f m−2 (Πm m−2 (α)) almost everywhere, where ˜ m−1 (α)(Πm (α)), etc. Continuing in this manner, we con(α) ≡ Π Πm m−2 m−2 m−2 ,k clude that either for some 0 ≤ k ≤ m there is an f k : ϑ(R˜ 2s−1 ) → C (or ,m k k m f : ϑ(R2s−1 ) → C if k = m) with f(α) = f (Πk (α)) almost everywhere k (where Πm m = identity) and f is increasing almost everywhere, or else f is ,0 we mean the measure constant almost everywhere. Here, for k = 0, by S˜2s−1 ,1 1 1 on H : 2s−1 → 2s−1 induced by the measure S2n+1 on H : 12s−1 → 12s−1 of type 0 the correct and H = H (0). By W2s−1 we mean the measure (for s > 1) 1 induced by f : κ(W2s−1 ) → 12s−1 of the correct type, the weak partition is the κ-cofinal normal measure on 1 ). 1 relation on 2s−1 and sup f (this 2s−1 In the case where f is constant almost everywhere the lemma easily follows, so we assume the former. ,k ,m ) → C , where if k = m we use R2s−1 , and let We fix an f k : ϑ(R˜ 2s−1 C2 ⊆ C1 be a c.u.b. subset of 12s−1 defining a measure on A2 restricted to which f k is increasing and f(α) = f k (Πm k (α)) holds. We consider the following two cases: Case 1. There is a c.u.b. C3 ⊆ C2 defining a measure one set A3 ⊆ A2 restricted to which f k is discontinuous, that is, for α ∈ A3 we have f k (α) > ,m sup<α,∈A2 f k (). We let f : ϑ(R2s−1 ) → 12n+1 be given with [g] < [f], and we proceed to show that there is an f¯ of the correct type with range in C with ¯ < [f]. From Lemmas 5.19 and 5.20, there is a C4 ⊆ C3 defining a [g] < [f] measure one set A4 ⊆ A3 such that gA4 is monotonic increasing. We then ¯ define f¯ as follows: for α ∈ A4 , we set f(α) = the next th point in C ¯ ¯ after max{g(α), sup<α f()}. For α ∈ = the next th / A4 , we define f(α) ¯ element in C after sup<α f(). It follows readily that for C5 ⊆ the closure ¯ 5 is less than fA5 . points of C4 , that fA Case 2. There is a C3 ⊆ C2 defining a measure one set A3 ⊆ A2 restricted to which f k is continuous, that is, for α ∈ A3 we have f k (α) = ,m sup<α,∈A2 f k (). Hence, for [g] < [f], for almost all α w.r.t. R2s−1 we k m have that g(α) < f(α) = f (Πk (α)), and it follows that for almost all all k α that there is a ∈ A3 with < Πm k (α) such that g(α) < f (). Applying Lemmas 5.11 and 5.12 repeatedly, we conclude that depends only on
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Πm k−1 (α), for k > 1, and if k = 1, then is constant almost everywhere if m = 1, and for > 1, depends only on Πm 0 (α). Here Π0 (α) denotes [F (0)] for ,m ,m F representing α if R2s−1 = S2s−1 , and denotes supa.e. f for f representing ,m m α if R2s−1 = W2s−1 . If k = 0, then is constant almost everywhere. Hence, k−1 ¯ < f k−1 (where f k−1 (α) = sup : Πk ()=α f k ()) almost there is an f k−1 ,k−1 everywhere w.r.t. S˜2s−1 (in the case k > 1, or k ≥ 1 if > 1) such that for almost all α w.r.t. S ,m we have g(α) < f¯k−1 Πm (α). The result now 2s−1
k−1
follows easily from the choice of C 1 . If k = 1, = 1 or k = 0, > 1, the result similarly follows easily. This completes the proof of Lemma 5.21. (Lemma 5.21) The proof of Lemma 5.22 is similar. The last two lemmas allow us to simplify the description of conditions C ¯ and D. In condition C, if d (Ia ) is non-basic of the form d = (k; d2(Ia ;Ka+1 ) )s k ,mk where s appears, K¯ a+1 = v(Kk ), and Kk = S2n+1 , then for almost all func¯ ¯ tions h1 , . . . , ht , f, h2 , . . . , ha , we consider the function g : ϑ(K¯ a+1 ) → 12n+1 defined by g([h¯ a+1 ]) = H (d2 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a+1 ), as in the definition m of condition C. If k > 2, so K¯ a+1 is not of the form W1 , then from Lemmas 5.21 and 5.22 it follows that d satisfies condition C provided [g] is not minimal amongst all [g ] with supa.e. g = supa.e. g. Thus, for d ∈ D2n+1 with the symbol s appearing on all component tuples, condition C reduces to the following: 1) All basic component tuples of d satisfy condition C; which is just a condition on certain components tuples d¯2 ∈ D2n−1 of d . k ,mk 2) The previous definition in the case d is non-basic with Kk = S2n+1 and k = 1 or 2 3) The above non-minimality condition on g, for k > 2. ,m s As for condition D, we note the following. If > 1, then (d2n+1 ) sat,m isfies condition D if d2n+1 satisfies condition C (relative to K1 , . . . , Kt ), Ia = (f(K¯ 1 ); · · · ), and for almost all h1 , . . . , ht we have that [h(d ; h1 , . . . , ht )] K¯ 1 is not minimal subject to being nonconstant almost everywhere w.r.t. 2n+1 . ¯ This follows from Lemma 5.22 and the fact that for Ia = (f(K1 ); · · · ), and C ⊆ 12n+1 a c.u.b. set, for almost all h1 , . . . , ht we have that h(d ; h1 , . . . , ht ) is ¯1 represented w.r.t. K 2n+1 by a function F having range in C almost everywhere. This follows easily from the definition of h. These facts will be of use later. We now proceed to define the lowering operation L on D2n+1 . We first require a preliminary definition.
Definition 5.23. Given d1(Ia ) , d2(Ia+1 ) satisfying condition C relative to K1 , . . . , Kt where Ia+1 = (Ia ; K¯ a+1 ), we say condition M (d1(Ia ) , d2(Ia+1 ) ) is
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satisfied if for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , H (d1 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = sup H (d2 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a+1 ). a.e. [h¯ a+1 ]
We say M2 (d1(Ia ) , d2(Ia+1 ) ) is satisfied if M (d1 , d2 ) is satisfied and the function g : ϑ(K¯ a+1 ) → 12n+1 defined by g([h¯ a+1 ]) = H (d1 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a+1 ) is not minimal with respect to the set of [g ] for g : ϑ(K¯ a+1 ) → 12n+1 with supa.e. g = supa.e. g. We will define the operation L on objects of the form (d ) or (d )s , where d = ∈ D2n+1 , where d or (d )s satisfies condition D and d satisfies condition A relative to fixed K1 , . . . , Kt . To do this, we first define a preliminary operation ˆ ) will also satisfy conditions C Lˆ on d satisfying conditions C and A. L(d and A. We introduce a notation. For d (Ia ) satisfying condition C relative to K1 , . . . , Kt where Ia = (f(K¯ 1 ); K¯ 2 , . . . , K¯ a ) (or f = fk or does not appear) we let d (K¯ i ) for 2 ≤ i ≤ a be the integer d (K¯ i ) = j, where 1 ≤ j ≤ t such that v(Kj ) = K¯ i which was used in the construction of d . We recall that (with a slight abuse of notation), the K¯ i are tagged with integers to such Kj . The function d just recovers this integer. ˆ we will actually define a more general operation Lˆ k (d ), deIn defining L, fined for d satisfying conditions C and A relative to K1 , . . . , Kt and satisfying d (K¯ a ) < k ≤ k(d ), and also such that d is not minimal w.r.t. the ordering < restricted to the set of d with k(d ) ≥ k satisfying conditions C and A. Also, we will assume inductively that for m < n and any sequence of measures K1 , . . . , Kt in R2m+1 such that there is a description d = d (Ia ) or (d )s , for d ∈ D2m+1 , defined satisfying conditions D and A, that there is a canonically defined maximal description d˜ or (d˜)s satisfying conditions D and A. That is, d˜ ≥ d if d or (d )s satisfies conditions D and A. This ¯ m¯ such that d ∈ D,¯ m¯ . We will definition is, of course, relative to fixed m, , 2m+1 ,m is below. say explicitly what d˜2n+1 We consider the following cases: ,m d2n+1
Case I. k = ∞, so d basic of type 1 or 0. Subcase I.a. d basic of type 1, so d (Ia ) = (d¯2 )s , where s may or may not appear. If s does not appear, we let Lˆ ∞ (d ) = (d¯2 )s . If s appears and d¯2 is not minimal w.r.t. the operation L on D2n−1 (which is defined by induction), then we set Lˆ ∞ (d ) = (L(d¯2 ))(Ia ) . If s appears and d¯2 is minimal w.r.t. L on D2n−1 , then d is minimal w.r.t. Lˆ ∞ , and Lˆ ∞ (d ) is not defined.
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¯
Subcase I.b. d basic of type 0, so d (Ia ) = d (fm ;K2 ,... ,Ka ) where d = (r) and 1 ≤ r ≤ m. If r > 1, we set Lˆ ∞ (d ) = r − 1, and if r = 1 then d is minimal w.r.t. Lˆ ∞ (that is, we do not define Lˆ ∞ (d )). mk Case II. k < ∞, k = k(d ), and Kk = W2n+1 . So, d is basic of type −1 s ¯ of the form d = (k; ) or d = (k; d2 ) , where s may or may not appear, and Ia = (K¯ 2 , . . . , K¯ a ). (For n = 0, d¯2 is an integer and s does not appear).
Subcase II.a. d = (k; ). We recall that Kb1 , . . . , Kbu enumerates the sub,m m sequence of K1 , . . . , Kt of measures not of the form S2n+1 or W2n+1 (i.e., ,m those of the form R2s+1 for s < n). We let Kb(k) , . . . , Kbu enumerate those ¯
k ,m¯ k if n ≥ 1. As above, in Kk+1 , . . . , Kt . We let, in this case v(Kk ) = S2n−1 ¯k ,m¯ k let d˜ = d˜2n−1 be the maximal tuple relative to Kb(k) , . . . , Kbu , K¯ 1 , . . . , K¯ a if defined. If d˜ is not defined, we define d to be minimal with respect to Lˆ k . If d˜ is defined, we set Lˆ k (d ) = (k; d˜) or (k; d˜)s , depending on whether d˜ satisfies condition D or not. Subcase II.b. d = (k; d¯2 )s , where s may or may not appear. If n = 0, so d = (k; i), we set Lˆ k (d ) = (k; i − 1) if i > 1, and if i = 1, then d is minimal w.r.t. Lˆ k . We assume now n > 0. If s does not appear, we set Lˆ k (d ) = (k; d¯2 )s . If s appears and d¯2 is not minimal with respect to L relative to the sequence of measures Kb(k) , . . . , Kbu , K¯ 2 , . . . , K¯ a , we set Lˆ k (d ) = (k; L ((d˜2 )s )) or (k; L ((d˜2 )s ))s depending on whether L((d˜2 )s ) does not or does involve s. Here we use the notation that if L((d˜2 )s ) = (d ) or (d )s , then L ((d¯2 )s ) = d . If s appears, and d¯2 is minimal with respect to L relative to the sequence Kb(k) , . . . , K¯ a , then d is minimal with respect to Lˆ k . k ,mk Case III. k < ∞, k = k(d ), Kk = S2n+1 with k > 1. Hence d = ¯ (Ia ;Ka+1 ) s ) where s may or may not appear. We let dˆ be the tuple from (k; d2 condition A for d . ¯ Subcase III.a. If s does not appear, then Lˆ k (d ) = (k; d (Ia ;Ka+1 ) )s . 2
Subcase III.b. > 2, s appears, and d2 is not minimal w.r.t. Lˆ k+1 . We then set Lˆ k (d ) = (k; Lˆ k+1 (d2 )) if this tuple satisfies condition C and M (dˆ, Lˆ k+1 (d2 )) is satisfied. Otherwise we set Lˆ k (d ) = (k; Lˆ k+1 (d2 ))s if M2 (dˆ, Lˆ k+1 (d2 )) holds and condition C is satisfied, and if not then we set Lˆ k (d ) = dˆ. Subcase III.c. = 2, s appears, and d2 is not minimal w.r.t. Lˆ k+1 . We set Lˆ k (d ) = (k; Lˆ k+1(q) (d2 ))s , where s appears if (k; Lˆ k+1(q) (d2 )) does not satisfy condition C. Here Lˆ k+1(q) (d2 ) denotes the least iterate of Lˆ k+1 such
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that (k; Lˆ k+1(q) (d2 )) or (k; Lˆ k+1(q) (d2 ))s satisfies condition C. If no such q exists, we set Lˆ k (d ) = dˆ. Subcase III.d. Otherwise we set Lˆ k (d ) = dˆ. 1,mk Case IV. k < ∞, k = k(d ), Kk = S2n+1 . Hence d = (k; d1(Ia ) , . . . , dr(Ia ) )s where s may or may not appear, and r ≤ mk . Subcase IV.a. s appears, r > 2, and Lˆ k+1 (dr ) is not defined or Lˆ k+1 (dr ) ≤ dr−1 . We set Lˆ k (d ) = (k; d (Ia ) , . . . , d (Ia ) )s . 1
r−1
Subcase IV.b. s appears, r = 2, and Lˆ k+1 (dr ) is not defined. We set L (d ) = d1 . ˆ k
Subcase IV.c. s appears, Lˆ k+1 (dr ) is defined, and if r > 2 then Lˆ k+1 (dr ) > (Ia ) ˆ k+1 ,L (drIa ))s , where s appears if dr−1 . Then Lˆ k (d ) = (k; d1(Ia ) , . . . , dr−1 without s this tuple does not satisfy condition C. Subcase IV.d. s does not appear and r < mk . Subcase IV.d.i. Lˆ k+1 (d1 ) is defined, and if r ≥ 2 then Lˆ k+1 (d1 ) > dr . We set Lˆ k (d ) = (k; d1(Ia ) , . . . , dr(Ia ) , Lˆ k+1 (d1 ))s , where s appears if without s the tuple does not satisfy C . Subcase IV.d.ii. Lˆ k+1 (d1 ) is defined, but IV.d.i immediately above fails. We set Lˆ k (d ) = (k; d (Ia ) , . . . , dr(Ia ) )s . 1
Subcase IV.d.iii. Lˆ k+1 (d1 ) not defined. It will follow in this case from our main inductive hypothesis that r = 1, since Lˆ k+1 (d1 ) not defined implies d1 is minimal w.r.t. the ordering < on the set of d satisfying conditions C and A and with k (d ) > k, which is not the case if r > 1. In this case, we set Lˆ k+1 (d ) = d1 . Subcase IV.e. s does not appear and r = mk . We set Lˆ k (d ) = (k; d1(Ia ) , . . . , dr(Ia ) )s if mk > 1 and Lˆ k (d ) = d1 if mk = 1. mk Case V. k < ∞, k < k(d ), and Kk = W2n+1 . k+1 ˆ If d is not minimal with respect to L , then we set Lˆ k (d ) = Lˆ k+1 (d ). k+1 ˆ If d is minimal with respect to L , then we set Lˆ k (d ) = (k; ), a basic −1 description. 1,mk . Case VI. k < ∞, k < k(d ), and Kk = S2n+1
Subcase VI.a. Lˆ k+1 (d ) defined and (k; Lˆ k+1 (d )) satisfies condition C. Then we set Lˆ k (d ) = (k; Lˆ k+1 (d )). Subcase VI.b. Lˆ k+1 (d ) defined, but VI.a fails. We set Lˆ k (d ) = Lˆ k+1 (d ).
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Subcase VI.c. VI.a and VI.b fail. Lˆ k+1 (d ).
Then d is minimal with respect to
k ,mk Case VII. k < ∞, k < k(d ), and Kk = S2n+1 with k > 1.
Subcase VII.a. Lˆ k+1 (d ) defined and for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , cf H (Lˆ k+1 (d ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = cf κ(Kk ). ˜ˆ k+1 (d )(Ia ;K¯ a +1) )s , where L ˜ˆ k+1 (d ) denotes the tuple We set Lˆ k (d ) = (k; L obtained by replacing in all component tuples of d the indices (f(K¯ 1 ); K¯ 2 , . . . , K¯ a , · · · ) by (f(K¯ 1 ); K¯ 2 , . . . , K¯ a+1 , · · · ). Subcase VII.b. Lˆ k+1 (d ) defined and VII.a fails. We set Lˆ k (d ) = Lˆ k+1 (d ). Subcase VII.c. Lˆ k+1 (d ) not defined. Then d is minimal with respect to L . ˆ k
Finally for (d ) or (d )s satisfying conditions D and A, we set L(d ) = (d )s , and L(d )s = (Lˆ 1(p) (d ))s , where s appears if the tuple without s does not satisfy condition D, and Lˆ 1(p) denotes the pth iterate of Lˆ 1 . Here p is minimal such that Lˆ 1(p) (d ) or (Lˆ 1(p) (d ))s satisfies condition D. If no such p exists, then (d )s is minimal with respect to L. Lemma 5.24. If d satisfied conditions C and A relative to K1 , . . . , Kt and d (K¯ a ) < k ≤ k(d ), then Lˆ k (d ) also satisfies conditions C and A. Hence, if ˆ ) ≡ Lˆ 1 (d ) d (Ia ) satisfies conditions C and A where Ia = (f(K¯ 1 ); ), then L(d satisfies conditions C and A and in this case, if d satisfies condition D, then so does L(d ). Proof. The first statement clearly implies the second. The proof of the first statement is routine upon consideration of the cases. ¯ k ,mk For example, we consider the case d = (k; d2(Ia ;Ka+1 ) )s , where Kk = S2n+1 , k k and v(Kk ) = K¯ a+1 . We consider first Lˆ (d ) where k = k(d ). If Lˆ (d ) = (k; Lˆ k+1 (d2 )), then condition C is satisfied by definition. By induction (reverse induction on k(d )), Lˆ k+1 (d2 ) satisfies condition A. Since M (dˆ, Lˆ k+1 (d2 )) is satisfied in this case, condition A is also satisfied by Lˆ k (d ), since for Lˆ k (d ) we may take dˆ. Similarly, if Lˆ k (d ) = (k; Lˆ k+1 (d2 ))s , conditions C and A are satisfied. If Lˆ k (d ) = dˆ, conditions C and A are satisfied by definition. We consider the case with d as above and k < k(d ). If Lˆ k (d ) = Lˆ k+1 (d ), ˜ˆ k+1 (d )(Ia ;K¯ a+1 ) )s . By inducwe are done by induction. Hence Lˆ k (d ) = (k; L ˜ˆ k+1 (d ) tion, Lˆ k+1 (d ) satisfies conditions C and A. It follows readily that L also satisfies conditions C and A (if M (d1 , d2 ) is satisfied for component tuples d1 , d2 of Lˆ k+1 (d ), then M (d˜1 , d˜2 ) is also satisfied). Since for almost all
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h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , the function ˜ˆ k+1 (d ), h , . . . , h , f, h¯ , . . . , h¯ , h¯ ) g([h¯ a+1 ]) = H (L 1 t 2 a a+1 is constant almost everywhere in this case, it follows readily that condition C is satisfied, and to satisfy condition A we may take dˆ = Lˆ k+1 (d ). The remaining cases follow similarly. Finally, to complete the inductive definition of L, we define the maximal ,m relative to K1 , . . . , Kt . tuple d˜2n+1 1) = −1. In this case d˜(Ia ) = d˜() . We let 1 ≤ w ≤ t denote the largest mw integer such that Kw = W2n+1 . We let a1 , . . . , ap enumerate the integers mw ,mi ,mi and cf κ(S2n+1 ) = cf κ(W2n+1 ), i from 1 to w with Ki of the form S2n+1 mw ,mi (or of the form S11,mi if n = 0) or equivalently, v(S2n+1 ) ∼ v(W2n+1 ). () s () (K¯ 1 ) s ˜ ˜ ˜ ˜ ¯ We set d = (d1 ) where d1 = (a1 ; d2 ) where K1 = v(Ka1 ), and ¯ ¯ ¯ d˜(K1 ) = (a2 ; d˜(K1 ,K2 ) )s , with K¯ 2 = v(Ka ). We continue in this manner 2
3
2
(K¯ ,...,K¯ p−1 ) (K¯ ,...,K¯ p ) s and finally set d˜p 1 = (ap ; d˜p+11 ) where d˜p+1 = (w; ) (or = (w; mw ) if n = 0). 2) = 1. In this case d˜(Ia ) = d˜(fm ) . We let b1 , . . . , bq enumerate the (fm ; ) 1,mi measures Ki in K1 , . . . , Kt of the form S2n+1 . We set d˜∞ = (m). We let (fm ; ) ) for 1 ≤ i ≤ q (where d˜p+1 = d˜∞ ). We set d˜ = d˜1 or d˜i(fm ; ) = (bi ; d˜i+1 (d˜1 )s , depending in whether d˜1 does or does not satisfy condition D. If b1 does not exist, d˜ is not defined. ¯ 3) > 1, d˜(Ia ) = d˜(f(K1 ); ) . We let e1 , . . . , er enumerate the integers i with i ,mi i ,mi ,m ) = κ(S2n+1 ) (this is equivalent to Ki = S2n+1 with i > 1 and cf κ(S2n+1 i ,mi i ,mi ,m ¯ saying that v(S2n+1 ) ∼ v(S2n+1 ) = K1 or equivalently that cf κ(S2n+1 )= ¯ 1 );) ¯ 1 );K¯2 ) (f( K (f( K ˜ ˜ ¯ ¯ cf ϑ(K1 )). Set d = (e1 ; d ), where K2 = v(Ke1 ). Likewise, 1
¯
2
¯ ¯ ¯ ¯ d˜2(f(K1 );K2 ) = (e2 ; d˜3(f(K1 );K2 ,K3 ) ) where K¯ 3 = v(Ke2 ). We continue in this manner and finally set (f(K¯ );K¯ ,...,K¯ r−1 ) (f(K¯ 1 );K¯ 2 ,...,K¯ r−1 ,K¯ r ) s = (er , d˜∞ ) d˜r 1 2 ¯ ¯ ¯ (f(K¯ 1 );K¯ 2 ,...,K¯ r ) where d˜∞ = ()(f(K1 );K2 ,...,Kr ) . We set d˜ = d˜1 or (d˜1 )s depending on whether d˜1 does or does not satisfy condition D. If e1 does not exist, d˜ is not defined.
We now introduce some additional notation required for the proof. We fix K1 , . . . , Kt and an index Ia . We let ϑ be an ordinal. We represent ϑ w.r.t. the measure K1 by the function g. We write g(h1 ) = g([h1 ]) for [h1 ] ∈ ϑ(K1 ). Thus g is defined almost everywhere with respect to K1 . For a fixed h1 , we represent g(h1 ) w.r.t. K2 by the function g(h1 , h2 ). We emphasize that this is defined only for a fixed
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choice of g, and fixed function representing g(h1 ). Continuing, we define g(h1 , . . . , ht ). If Ia is of the form Ia = (f(K¯ 1 ); K¯ 2 , . . . , K¯ a ) then we repre¯1 sent g(h1 , . . . , ht ) with respect to K 2n+1 by the function [f] → g(h1 , . . . , ht , f) for f : ϑ(K¯ 1 ) → 12n+1 of the correct type. If Ia = (fk ; K¯ 2 , . . . , K¯ a ), then we represent g(h1 , . . . , ht ) with respect to the k-fold product to the -cofinal normal measure on 12n+1 by the function f → g(h1 , . . . , ht , f) where now f : k → 12n+1 . Continuing, we represent g(h1 , . . . , ht , f) with ¯ ¯ respect to the measures K2 , . . . , Ka to get g(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ). Again, g(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ i , h¯ i+1 ) is only defined after having chosen a specific function representing g(h1 , . . . , h¯ i ). Given d satisfying condition C and A relative to K1 , . . . , Kt , we define a sequence of component tuples d 0 , d 1 , . . . , d for some ≥ 0 as follows: (I ) We set d 0 = d . We assume d i = d i ai has been defined. (Iai+1 )
a) If d i(Iai ) = (k; d2
(Ia )
k ,mk ) where Kk = S2n+1 with k > 1, we set = i. (Ia )
1,mk and r = mk , we set b) If d i(Iai ) = (k; d1 i , . . . , dr i ) where Kk = S2n+1 = i. (Ia ) (Ia ) c) If d i = (k; d2 i+1 )s , we set d i+1 = d2 = d2 i+1 . d) If d i = (k; d1 , . . . , dr )s , we set d i+1 = dr . 1,mk e) If d i = (k; d1 , . . . , dr ) where r < mk (here Kk = S2n+1 ), we set d i+1 = (Ia )
d1 = d1 i . f) If d i is basic, we set = i. We let ki = k(d i ), so 1 ≤ ki ≤ t or ki = ∞, and k0 < k1 < · · · < k . For a fixed ordinal ϑ or description d˜i(Iai ) satisfying condition C and A w.r.t. K1 , . . . , Kt , where the indices Iai are as above, we define the ordinal h(d ; (d i → ϑ)) or h(d ; (d i → d˜i )) as follows. We represent this ordinal K¯ 1 (or the m-fold product of the with respect to the measures K1 , . . . , Kt , 2n+1 1 ¯ -cofinal normal measure on 2n+1 ), and K2 , . . . , K¯ a by H (d ; h1 , . . . , ht ; f, h¯ 2 , . . . , h¯ a ; (d i → ϑ)) or H (d ; h1 , . . . , ht ; f, h¯ 2 , . . . , h¯ a ; (d i → d˜i )) where these are defined exactly as H (d ; h1 , . . . , ht ; f, h¯ 2 , . . . , h¯ a ) except that in defining the ordinal H (d i−1(Iai−1 ) ; h1 , . . . , ht ; f, h¯ 2 , . . . , h¯ ai−1 ), the ordinal H (d i ; h1 , . . . , ht ; f, h¯ 2 , . . . , h¯ ai ) is replaced by g(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) i
in the first case and by H (d˜i ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ ai ) in the second case.
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If C is a c.u.b. subset of 12n+1 , we let NC (α) be the th element of C greater 1 than α. If g : 12n+1 → 2n+1 , we let Ng (α) be the th element in the range of g greater than α. We abbreviate Nhi (α) by Ni (α). For g : 12n+1 → 12n+1 , we let h(d ; (d i → g ◦ d˜i )) be defined as above, replacing H (d i ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ ai ) by g(H (d i ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ ai )). ,m If d = d2n+1 or (d )s is given and satisfies conditions D and A w.r.t. K1 , . . . , Kt , and g : 12n+3 → 12n+3 we define the ordinal (g; d ; K1 , . . . , Kt ). ,m We represent this with respect to 2n+3 , where = S2n+1 if ≥ 1, and = m W2n+1 if = −1. We represent by the function [f] → (g; f; d ; K1 , . . . , Kt ), for f : ϑ( ) → 12n+3 of the correct type. We represent this, in turn, w.r.t. K1 by the function[h1 ] → (g; f; d ; h1 , K1 , . . . , Kt ), for [h1 ] ∈ ϑ(K1 ). Continuing, we represent in turn with respect to K2 , . . . , Kt by the functions [h2 ] → (g; f; d ; h1 , h2 , K3 , . . . , Kt ),. . . ,[ht ] → (g; f; d ; h1 , . . . , ht ). Finally, we set (g; f; d ; h1 , . . . , ht ) = g(f(h(d ; h1 , . . . , ht ))). For (d )s , the procedure is similar, except at the end we use (g; f; (d )s ; h1 , . . . , ht ) = g(sup
Definition 5.25 (Main Inductive Hypothesis H2n+1 ). H2n+1 consists of the following assertions. ,m H2n+1 (a): We let d Ia or (d )s , where d = d2n+1 ∈ D2n+1 , be defined and ,m or = satisfying conditions D and A w.r.t. K1 , . . . , Kt . We let = S2n+1 m W2n+1 depending on whether ≥ 1 or = −1. We assume Ia = (f(K¯ 1 )) or (fm ; ) or (), and assume L(d ) is defined. We let F : 12n+3 → 12n+3 be . given and satisfy F < (id; d ; K1 , . . . , Kt ) almost everywhere w.r.t. 2n+3 1 1 Then there is a g : 2n+3 → 2n+3 such that F < (g; L(d ); K1 , . . . , Kt ) almost everywhere w.r.t. 2n+3. Similarly for (d )s . H2n+1 (b): As above, except we assume now that L(d ) is not defined. We then have that for some α < 12n+3 , F () = α for almost all w.r.t. 2n+3 . H2n+1 (c): If for almost all f: ϑ( ) → 12n+3 of the correct type, and almost all h2 , . . . , ht we have that F (f, h1 , . . . , ht ) < supa.e. f, and the maximal ,m tuple d˜2n+1 is defined, then for almost all f we have that F ([f]) < ˜ (g; f; d ; K1 , . . . , Kt ), for some g : 12n+3 → 12n+3 . H2n+1 (d): As above, except d˜ is not defined. We then have that F is constant almost everywhere.
The rest of this section is devoted to a proof of H2n+1 . We proceed by induction on n, so we assume H2m+1 for m < n. We require the following lemma:
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,m Lemma 5.26 (Cofinality Lemma). We let d (Ia ) , where d = d2n+1 ∈ D2n+1 , be defined and satisfy conditions C and A relative to K1 , . . . , Kt . We let ϑ ∈ Ord be represented by g(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) as defined previously. We assume that for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a that
g(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < H (d ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ). We then have that there is an ordinal ϑ (where is as in the definition of d 0 , d 1 , . . . , d ) such that if we represent the ordinal ϑ as before by g (h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ), then g (h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < H (d ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) almost everywhere, and for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a we have g(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < H (d ; (d → ϑ ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ). Proof. By induction on 0 ≤ i ≤ , we establish that there is a ϑi such that for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ ai ϑi (h1 , . . . , ht , f, h¯ 2 , . . . , h¯ ai ) < H (d i ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ ai ) and g(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < H (d ; (d i → ϑi ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ). For i = 0 this is true by assumption. Assuming true for i, it follows upon consideration of the cases (one of which we consider below) that there is an ordinal ϑi+1 such that ϑi+1 (h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < H (d i+1 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) i
i+1
and ϑi (h1 , . . . , ht , f, h¯ 2 , . . . , h¯ ai ) < H (d i ; (d i+1 → ϑi+1 ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ ai ) hold almost everywhere. Hence almost everywhere g(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < H (d ; (d i+1 → ϑi+1 ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) follows. k ,mk As an example, we consider the case d i = (k; d i+1 )s , where Kk = S2n+1 with k > 1. So, for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ ai we have ϑi (h1 , . . . , ht , f, h¯ 2 , . . . , h¯ ai ) < H (d i ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ ai ) = sup hk (), <[g]
where g([hai+1 ]) = H (d i+1 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ ai+1 ) almost everywhere. Hence, for almost all h1 , . . . , h¯ ai , there is a g < g almost everywhere with
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ϑi (h1 , . . . , h¯ ai ) < hk ([g ]). Equivalently, for almost all h1 , . . . , h¯ ai+1 , there is a ϑi+1 (h1 , . . . , h¯ ai+1 ) < H (d i+1 ; h1 , . . . , h¯ ai+1 ) such that ϑi (h1 , . . . , h¯ ai+1 ) < H (d ; (d i+1 → ϑi+1 ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ ai ).
The remaining cases are similar.
The main part of the proof of the main inductive hypothesis consists of establishing the main inductive lemma, which we now state: Lemma 5.27 (Main Inductive Lemma). For 1 ≤ i ≤ t or i = ∞ we consider ,m be given, where d is the following statement H¯ (i): we let d (Ia ) for d = d2n+1 defined and satisfies conditions C and A relative to K1 , . . . , Kt , and d is not minimal with respect to Lˆ i where d (K¯ a ) < i ≤ k(d ). Let ϑ be an ordinal. We assume that for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a that ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < H (d (Ia ) ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ). Then for almost all h1 , . . . , hi−1 , there is a c.u.b. Ci ⊆ 12n+1 such that for almost all hi , . . . , ht we have that ϑ(h1 , . . . , ht ) < h(Lˆ i (d ); (Lˆ i (d ) → NCi (Lˆ i (d ))); h1 , . . . , ht ). If d (Ii ) is minimal with respect to Lˆ i (i.e., Lˆ i (d ) is not defined), then we require that for almost all h1 , . . . , hi−1 , there is an αi < 12n+1 such that for almost all hi , . . . , ht we have ϑ(h1 , . . . , ht ) < αi . 5.1. Proof of the main inductive lemma. We prove the main inductive lemma by reverse induction on i. We consider the necessary cases. ,m is basic of type 0 or 1. Case I. i = ∞. Hence d = d2n+1 ¯
¯
¯
Subcase I.a. d basic of type 1, so d (Ia ) = (d2 )s(f(Ka );K2 ,... ,Ka ) , where s may ¯ m¯ ¯ m¯ , , and K¯ 1 = R2s+1 . or may not appear, d2 ∈ D2s+1 Subcase I.a.i. d (= d ) not minimal with respect to Lˆ ∞ . We let ϑ be given such that for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a we have ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < H (d ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ). We have in this case that Lˆ ∞ = L((d2 )) or L((d2 )s ) depending on whether s does not or does appear. By induction and H2n−1 (a), it follows that for almost all h1 , . . . , ht , there is a c.u.b. C ⊆ 12n+1 such that for almost all f, h¯ 2 , . . . , h¯ a we have ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < (NC ; f; L((d2 )s ); h¯ 2 , . . . , h¯ a ) = H (Lˆ ∞ (d ); (Lˆ ∞ (d ) → NC ◦ Lˆ ∞ (d )); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ), and similarly if s does not appear.
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Subcase I.a.ii. d is minimal with respect to Lˆ ∞ . Hence, the symbol s appears, and (d2 )s is minimal with respect to L. For almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < H (d ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = (f; (d2 )s ; h¯ 2 , . . . , h¯ a ). Hence it follows by induction and H2n−1 (b) that for almost all h1 , . . . , ht , there is an α < 12n+1 such that for almost all f1 , h¯ 2 , . . . , h¯ a , ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < αt . ¯
¯
Subcase I.b. d (= d ) basic type 0. Hence d (Ii ) = (k)(fm ;K2 ,... ,Ka ) , where k ≤ m. Subcase I.b.i. d not minimal w.r.t. Lˆ ∞ , so k > 1. Then for almost all h1 , . . . , ht , for almost all f : m → 12n+1 , for almost all h¯ 2 , . . . , h¯ a , we have ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < f(k). It follows readily that for almost all h1 , . . . , ht there is a c.u.b. C ⊆ 12n+1 such that for almost all f, h¯ 2 , . . . , h¯ a , ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < NC (f(k − 1)) = H (Lˆ ∞ (d ); (Lˆ ∞ (d ) → NC ◦ Lˆ ∞ (d )); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ). We use here a simple partition (of f : m → 12n+1 with an extra value inserted between f(k − 1) and f(k)) and the countable additivity of the measures ¯h2 , . . . , h¯ a . Subcase I.b.ii. d minimal w.r.t. Lˆ ∞ (d ). Similar to i). Case II. i < ∞ and i < k(d ). i ,mi Subcase II.a. Ki = S2n+1 with i > 1, and d not minimal w.r.t. Lˆ i . Hence, i+1 ˆ d is non-minimal w.r.t. L . We let ϑ be given such that for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a we have
ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < H (d (Ii ) ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ). By induction, for almost all h1 , . . . , hi , there is a c.u.b. Ci+1 ⊆ 12n+1 such that for almost all hi+1 , . . . , ht , ϑ(h1 , . . . , ht ) < h(Lˆ i+1 (d ); Lˆ i+1 (d ) → NCi+1 (Lˆ i+1 (d )); h1 , . . . , ht ). We let κ be such that for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , κ = cf H (Lˆ i+1 (d ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ).
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Subcase II.a.i. κ = cf κ(Ki ). For fixed h1 , . . . , hi−1 such that ϑ(h1 , . . . , hi−1 , Ki , . . . , Kt ) < h(d ; h1 , . . . , hi−1 , Ki , . . . , Kt ), we consider the following partition Pi (h1 , . . . , hi−1 ): we partition functions hi : 12n+1 → 12n+1 of the correct type with the extra value g(α) inserted between hi (0)(α) = sup hi ([f]) f : f<α a.e.
and the next element in the range of hi after hi (0)(α), where g has uniform cofinality , according to whether or not for almost all hi+t , . . . , ht , ϑ(h1 , . . . , ht ) < h(Lˆ i+1 (d ); Lˆ i+1 (d ) → g ◦ Lˆ i+1 (d ); h1 , . . . , ht ). It follows from a sliding argument (as in Lemma 5.18 of this section) that on the homogeneous side of the partition, the property stated in P(h1 , . . . , hi−1 ) holds. We let Ci be homogeneous for this partition. We therefore have that for almost all h1 , . . . , hi−1 there is a Ci such that for almost all hi (say with range in the closure points of Ci ), hi+1 , . . . , ht , ϑ(h1 , . . . , ht ) < h(Lˆ i+1 (d ); Lˆ i+1 (d ) → (NCi ◦ hi (0)) ◦ Lˆ i+1 (d ); h1 , . . . , ht ) = h(Lˆ i (d ); Lˆ i (d ) → NCi ◦ Lˆ i (d ); h1 , . . . , ht ). The last equality follows from Lˆ i (d ) = (i; Lˆ i+1 (d ))s and the definition of h in this case. Subcase II.a.ii. κ = cf(Ki ). For fixed h1 , . . . , hi−1 , we consider P(h1 , . . . , hi−1 ) as above, and have that on the homogeneous side of the partition, P(h1 , . . . , hi−1 ) holds. However, for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a we have that H (Lˆ i+1 (d ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = sup f :
f
hi ([f ]).
¯ ¯ 1 ,... ,ht ,f,h2 ,... ,ha ) a.e.
since for almost all α < 12n+1 of cofinality κ we have α = supf : f <α a.e. hi ([f]) almost all h , . . . , h we have as κ = cf(Ki ). Hence, for 1 t i+1 i+1 (d ); Lˆ (d ) → NCi ◦ Lˆ i+1 (d ); h1 , . . . , ht ) ϑ(h1 , . . . , ht ) < h(Lˆ = h(Lˆ i (d ); Lˆ i (d ) → NCi ◦ Lˆ i (d ); h1 , . . . , ht ) as Lˆ i (d ) = Lˆ i+1 (d ) in this case. i ,mi Subcase II.b. Ki = S2n+1 with i > 1, and d minimal with respect to Lˆ i . Hence, d is minimal with respect to Lˆ i+1 . Hence, by induction, for almost all h1 , . . . , hi , there is an αi+1 < 12n+1 such that for almost all hi+1 , . . . , ht , ϑ(h1 , . . . , ht ) < αi+1 .
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Hence, by the 12n+1 additivity of Ki , it follows that for almost all h1 , . . . , hi−1 , 1 there is an αi < 2n+1 s.t. for almost all hi , . . . , ht , ϑ(h1 , . . . , ht ) < αi . 1,mi Subcase II.c. Ki = S2n+1 and d not minimal w.r.t. Lˆ i . The proof is similar to a, where for fixed h1 , . . . , hi−1 as in a, we consider the partition P(h1 , . . . , hi−1 ): we partition hi : <mi → 12n+1 of the correct type with the extra value g(α) inserted between hi (0)(α) =
sup
hi (2 , . . . mi , α)
2 <···<mi <αi
and the next element in the range of hi (0)(α) according to whether or not for almost all hi+1 , . . . , ht , ϑ(h1 , . . . , ht ) < h(Lˆ i+1 (d ); Lˆ i+1 (d ) → g ◦ Lˆ i+1 (d ); h1 , . . . , ht ). It follows that for almost all h1 , . . . , hi−1 that on the homogeneous side of the partition, the property stated in P(h1 , . . . , hi−1 ) holds. Taking cases on whether or not cf(H (Lˆ i+1 (d ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a )) = (and so Lˆ i (d ) = (i; Lˆ i+1 (d )(Ia ) )) or Lˆ i (d ) = Lˆ i+1 (d ), the result follows as in cases II.a.i and II.a.ii above. i ,mi with i = 1, and d minimal w.r.t. Lˆ i . Similar to Subcase II.d. Ki = S2n+1 b) above. mi and d not minimal w.r.t. Lˆ i+1 . Once again, by Subcase II.e. Ki = W2n+1 induction for almost all h1 , . . . , hi there is a Ci+1 s.t. for almost all hi+1 , . . . , ht , ϑ(h1 , . . . , ht ) < h(Lˆ i+1 (d ); Lˆ i+1 (d ) → NCi+1 (Lˆ i+1 (d )); h1 , . . . , ht ). By an easy partition and sliding argument (partitioning hi followed by g : 12n+1 → 12n+1 , where inf g > sup hi ) for almost all h1 , . . . , hi−1 there is a Ci such that for almost all h , . . . , h , i
ˆ i+1
ϑ(h1 , . . . , ht ) < h(L
t
(d ); Lˆ i+1 (d ) → NCi (Lˆ i+1 (d )); h1 , . . . , ht ).
The result follows since Lˆ i (d ) = Lˆ i+1 (d ) in this case. mi and d minimal w.r.t. Lˆ i+1 . By induction, for Subcase II.f. Ki = W2n+1 almost all h1 , . . . , hi , there is an αi+1 < 12n+1 s.t. for almost all h1 , . . . , ht , ϑ(h1 , . . . , ht ) < αi+1 . For almost all h1 , . . . , hi−1 we consider Pi (h1 , . . . , hi−1 ) where we partition hi : v(Ki ) → 12n+1 of the correct type (or hi : m → 11 if n = 0) with the extra almost all h , . . . , h , value α aftersup hi according to whether or not for i+1 t ϑ(h1 , . . . , ht ) < α. On the homogeneous side, Pi holds, and, for almost all h1 , . . . , hi−1 , we let Ci be homogeneous for Pi . We then have that for almost all hi , . . . , ht that ϑ(h1 , . . . , ht ) < NCi (supa.e. hi ), and the result follows since
437
AD AND THE PROJECTIVE ORDINALS
Lˆ i (d ) = (i; ) in this case, so H (Lˆ i (d ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = sup hi . a.e.
The case n = 0 is similar. Subcase II.g. Ki ∈ m 1. ¯ Subcase III.a.i. d (Ii ) = (k; d2(Ia ;Ka+1 ) )s(Ia ) , where s appears. Recall K¯ a+1 = v(Kk ). We let ϑ be such that for almost all h1 , . . . , ht ,
ϑ(h1 , . . . , ht ) < h(d ; h1 , . . . , ht ). By the cofinality lemma, there is a ϑ2 such that for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a+1 , we have ϑ2 (h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a+1 ) < H (d2 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a+1 ) and ϑ(h1 , . . . , ht ) < h(d ; (d2 → ϑ2 ); h1 , . . . , ht ). We note that d is not minimal w.r.t. to Lˆ i in this case. We assume that d2 is not minimal with respect to Lˆ i+1 . We will show in case (iv) below that this is the case. By induction, it then follows that for almost all h1 , . . . , hk there is a Ck+1 s.t. for almost all hk+1 , . . . , ht , ϑ2 (h1 , . . . , ht ) < h(Lˆ i+1 (d2 ); Lˆ i+1 (d2 ) → NCk+1 ◦ Lˆ i+1 (d2 ); h1 , . . . , ht ), and hence ϑ(h1 , . . . , ht ) < h(d ; d2 → NCk+1 ◦ Lˆ i+1 (d2 ); h1 , . . . , ht ). We let dˆ be the tuple corresponding to d as in condition A. Subcase III.a.i.1. M (dˆ, Lˆ k+1 (d2 )) is satisfied and (k; Lˆ k+1 (d2 )) satisfies condition C. That is for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , the function g defined by g([h¯ a+1 ]) = H (Lˆ k+1 (d2 ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a+1 ) is strictly increasing of uniform cofinality . Recall in this case that Lˆ k (d ) = (k; Lˆ k+1 (d )). For almost all h1 , . . . , hk−1 , we consider the partition Pk (h1 , . . . , hk−1 ) where we partition functions hk : 12n+1 → 12n+1 of the cor h (α)and h (α + 1), rect type, with the extra value g(α) inserted between k k and g has uniform cofinality , according to whether or not for almost all hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < h(Lˆ k (d ); Lˆ k (d ) → Ng ◦ Lˆ k (d ); h1 , . . . , ht ).
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We claim that on the homogeneous side of the partition, the property stated in Pk (h1 , . . . , hk−1 ) holds. We suppose not and let Ck ⊆ 12n+1 be a c.u.b. set h : 1 2 homogeneous for the contrary side. Let Ck2 be such that for k 2n+1 → Ck of the correct type, there is a Ck+1 such that for almost all hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < h(d ; d2 → NCk+1 ◦ Lˆ k+1 (d2 ); h1 , . . . , ht ). We let Ck3 be contained in the closure points of Ck ∩ Ck2 . We fix now a function hk : 12n+1 → Ck3 of the correct type, and fix Ck+1 as above. We then get hk2 and g satisfying: k) (1) hk2 = hk almost everywhere w.r.t. v(K 2n+1 .
(2) hk2 , g are of the correct type and ordered as in Pk . (3) hk2 , g have range in Ck3 (in fact a subset of the range of hk ). (4) g([f]) > hk ([NCk+1 ◦ f]) for all f : κ(Kk ) → 12n+1 . The construction of hk2 , g follows as in the previous technical lemmas and will be omitted. We then elect hk+1 , . . . , ht such that ϑ(h1 , . . . , ht ) < h(d ; d2 → NCk+1 ◦ Lˆ k+1 (d2 ); h1 , . . . , ht ), and hence ϑ(h1 , . . . , ht ) < h(Lˆ k (d ); Lˆ k (d ) → Ng ◦ Lˆ k (d ); h1 , . . . , ht ) by (4) above; and also ϑ(h1 , . . . , hk2 , . . . , ht ) > h(Lˆ k (d ); Lˆ k (d ) → Ng ◦ Lˆ k (d ); h1 , . . . , ht ) by (2), (3). Also, since ϑ(h1 , . . . , hk ) = ϑ(h1 , . . . , hk2 ) by (1), we may assume that ϑ(h1 , . . . , hk , . . . , ht ) = ϑ(h1 , . . . , hk2 , . . . , ht ). This contradiction establishes that Pk (h1 , . . . , hk−1 ) holds. We let Ck be homogeneous for Pk , for fixed h1 , . . . , hk−1 , and let Ck2 be contained in the closure points of Ck . It then follows that for almost all hk , namely hk : 12n+1 → Ck2 of the correct type, that for almost all hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < h(Lˆ k (d ); Lˆ k (d ) → NCk ◦ Lˆ k (d ); h1 , . . . , ht ) and we are done. Subcase III.a.i.2. M2 (dˆ, Lˆ k+1 (d )) holds, case (i) fails, and (k; Lˆ k+1 (d2 ))s satisfies condition C. Recall in this case Lˆ k (d ) = (k; Lˆ k+1 (d2 ))s . For almost all h1 , . . . , hk−1 , we consider the partition Pk (h1 , . . . , hk−1 ) where we partition hk : 12n+1 → 12n+1 of the correct type with the extra value g(α) inserted sup h () and h (α) according to whether or not for almost all between k <α k hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < h(Lˆ k (d ); Lˆ k (d ) → Ng ◦ Lˆ k (d ); h1 , . . . , ht ).
AD AND THE PROJECTIVE ORDINALS
439
If Pk fails, we fix Ck homogeneous for the contrary side. We define Ck2 , Ck3 as in (i.1) above, fix hk : 12n+1 → Ck3 of the correct type, and fix Ck+1 as in (i.1) above. We then get hk2, g satisfying: k) (1) hk2 = hk almost everywhere w.r.t. v(K 2n+1 .
(2) hk2 , g are ordered as in Pk for this case. (3) hk2 , g have range in Ck3 . (4) g([f]) > hk ([NCk+1 ◦f]) for all f : κ(Kk ) → 12n+1 not of the correct type. We then proceed as in case (i.1) above. Subcase III.a.i.3. Cases (i.1) and (i.2) fail. In the case, > 2, Lˆ k (d ) = dˆ. We first note that M (dˆ, Lˆ k+1 (d2 )) is satisfied. For if not, then for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , we consider the functions g, g˜ defined almost everywhere, where g([ha+1 ]) = H (d2 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a+1 ) and g˜ : ϑ(K¯ a+1 ) → 12n+1 is minimal subject to supa.e. g˜ = supa.e. g. Hence, ˜ < [g] for almost all h1 , . . . , h¯ a , since d = (k; d2 )s satisfies condition C. [g] Hence, by induction, for almost all h1 , . . . , hk there is a Ck+1 s.t. for almost ˜ < [g2 ] where all hk+1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , [g] g2 ([h¯ a+1 ]) = NCk+1 (H (Lˆ k+1 (d2 ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a+1 )), a contradiction since supa.e. g˜ > supa.e. g2 for g2 with a range in a set closed under NCk+1 , which happens almost everywhere. Hence M (dˆ, Lˆ k+1 (d2 )) holds. We first consider the case > 2. We first claim in this case that M2 (dˆ, k+1 ˆ L (d2 )) is not satisfied. We suppose not. For almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , we consider the function g¯ where ¯ h¯ a+1 ) = H (Lˆ k+1 (d2 ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a+1 ). g( ¯ = supa.e. [g ] where g ranges over functions From our technical lemmas, [g] 1 ¯ from ϑ(Ka+1 ) into 2n+1 of the correct type. Since M2 (dˆ, Lˆ k+1 (d2 )) is sat isfied, g¯ = supa.e. [g ], where g ranges over functions from ϑ(K¯ a+1 ) into 1 2n+1 with supa.e. [g ] = supa.e. g. Hence (k; Lˆ k+1 (d2 ))s satisfies condition C, contrary to the assumption of this case. Hence M2 (dˆ, Lˆ k+1 (d2 )) fails. By induction, for almost all h1 , . . . , hk there is a Ck+1 such that for almost all hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < h(d ; d2 → NCk+1 ◦ Lˆ k+1 (d2 ); h1 , . . . , ht ). For such fixed h1 , . . . , hk−1 , we consider the partition Pk (h1 , . . . , hk−1 ) where we partition hk : 12n+1 → 12n+1 of the correct type with the extra value g(α) inserted between sup<α h k (0)() (which we recall is equal to sup{hk ([f]) :
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STEVE JACKSON
supa.e. f < α}) and the next element in the range of hk after sup<α hk (0)(), where g is of uniform cofinality , according to whether or not for almost all hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < h(dˆ; dˆ → Ng ◦ dˆ; h1 , . . . , ht ). If Pk fails, we fix Ck homogeneous for the contrary side, let Ck2 such that for hk : 12n+1 → Ck2 of the correct type there is a Ck+1 such that the above is satisfied, and let C 3 be contained in the closure points of C ∩C 2 . inequality k k k We fix hk : 12n+1 → Ck3 of the correct type and Ck+1 , and get hk2 , g satisfying: v(Kk ) (1) hk2 = hk a.e. w.r.t. 2n+1 . (2) hk2 , g are of the correct type and ordered as in Pk . (3) hk2 , g have range in Ck3 . (4) g(α) > hk ([NCk+1 ◦ f]), where f represents α w.r.t. v(Kk ). The construction of hk2 , g is similar to that in the proofs of the technical lemmas. We then have that for almost all hk+1 , . . . , ht , f, h¯ 1 , . . . , h¯ a , ¯ ϑ(hk , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < hk ([NCk+1 ◦ g]) ¯ It follows that almost ¯ = supa.e. g. with g¯ : ϑ(K¯ a+1 ) → 12n+1 as above, and [g] everywhere we have ¯ ϑ(h1 , . . . , ht , f, h¯ 1 , . . . , h¯ a ) < Ng (sup g) a.e.
= Ng (h(dˆ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a )). However, ϑ(h1 , . . . , hk2 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ). This contradiction establishes Pk . The result then follows readily as in previous cases. We now consider the case = 2. Here v = (Kk ) = the m-fold product of the normal measure on 1 = v say. We recall ϑ(v) is identified with an ordinal (= 1 ) by ordering by the largest ordinal first, then the next largest, etc. For almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , we again consider the function g : ϑ(v) → 12n+1 defined by g([h¯ a+1 ]) = H (Lˆ k+1 (d2 ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a+1 ). Here h¯ a+1 is a m-tuple of ordinals < 1 . We first assume M2 (dˆ, Lˆ k+1 (d2 )) is satisfied. For fixed h1 , . . . , hk−1 , we then consider the partition Pk where we partition hk : 12n+1 → 12n+1 of the correct type according to whether or not for almost all hk+1 , . . . ,ht , ϑ(h1 , . . . , ht ) < h((k; Lˆ k+1 (d2 )); h1 , . . . , ht ).
AD AND THE PROJECTIVE ORDINALS
441
We note that (k; Lˆ k+1 (d2 )) does not satisfy condition C, but this partition still makes sense. We then claim that on the homogeneous side of the partition the property stated in Pk holds. We suppose not, and fix Ck , Ck2 , Ck3 , hk , Ck+1 as in the previous cases. We then get hk2 satisfying: (1) hk2 = hk a.e. w.r.t. v2n+1 . (2) hk is of the correct type. (3) hk , g has range in Ck3 . (4) For α represented by gα : 1m → 12n+1 not of the correct type almost ev where here g represents α w.r.t. W m . erywhere, hk2 (α) > hk ([NCk+1 ◦ gα ]), α 1 We then proceed to a contradiction as in the previous cases, establishing that for almost all h1 , . . . , ht , ϑ(h1 , . . . , ht ) < h((k; Lˆ k+1 (d2 )); h1 , . . . , ht ). We then consider, for fixed h1 , . . . , hk−1 , the partition Pk where we partition hk : 12n+1 → 12n+1 of the correct type according to whether or not for almost , all hk+1 , . . . , h t ϑ(h1 , . . . , ht ) < h((k; Lˆ k+1 (d2 ))s ; h1 , . . . , ht ). If Pk fails, we again get Ck , Ck2 , Ck3 and hk . We then get hk2 satisfying (1)–(3) as above and (4): for α represented by gα : 1m → 12n+1 not monotonically construction is similar increasing a.e. w.r.t. W1m , sup<α hk2 () > hk (α). The to that of previous cases, using here the fact that if g : 1m → 12n+1 is not monotonically increasing almost everywhere, then [g] is not the supremum of [g ] for g of the correct type. We then proceed to a contradiction establishing that for almost all h1 , . . . , ht , ϑ(h1 , . . . , ht ) < h((k; Lˆ k+1 (d2 ))s ; h1 , . . . , ht ). If (k; Lˆ k+1(q) (d2 )) or (k; Lˆ k+1(q) (d2 ))s satisfies condition C for some q, we then repeat the above argument to establish H¯ 2n+1 (k). Hence we may assume without loss of generality that M2 (dˆ, Lˆ k+1 (d2 )) is not satisfied. Thus, Lˆ k+1 (d ) = dˆ. The result now follows as in the corresponding case for > 2. Subcase III.a.i.4. d2 minimal with respect to Lˆ k+1 . We show that this case does not occur. We show that M2 (dˆ, d2 ) is not satisfied. If it were, then define ϑ2 such that for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , ˜ ϑ2 (h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = [g] where g˜ : ϑ(K¯ a+1 ) → 12n+1 is minimal s.t. sup g˜ = H (dˆ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ). a.e.
442
STEVE JACKSON
We then have that for almost all h1 , . . . , h¯ a+1 , ϑ2 (h1 , . . . , h¯ a+1 ) < H (d2 ; h1 , . . . , h¯ a+1 ). However, for almost all h1 , . . . , hk there is no αk < 12n+1 such that for almost all hk+1 , . . . , ht , f, h¯ 2 , . . . , h¯ a we have H (dˆ; h1 , . . . , h¯ a ) < αk (this follows easily from k(dˆ) > k). This contradicts the minimality of d2 and induction. ¯
Subcase III.a.ii. d = (k; d2(Ia ;Ka+1 ) ), where s does not appear. In this case
¯ Lˆ k (d ) = (k; d2(Ia ;Ka+1 ) )s . Since d satisfies condition C, it follows readily that condition M2 (dˆ, d2 ) is satisfied. For fixed h1 , . . . , hk−1 , we consider the partition Pk (h1 , . . . , hk−1 ) where we partition hk : 12n+1 → 12n+1 of the sup h () and correct type with the extra value g(α) inserted between <α k hk (α), where g has uniform cofinality , according to whether or not for not almost all hk+1 , . . . , ht ,
ϑ(h1 , . . . , ht ) < h(Lˆ k (d ); Lˆ k (d ) → Ng ◦ Lˆ k (d ); h1 , . . . , ht ). We claim that on the homogeneous side of the partition Pk holds. If not, we let Ck be homogeneous for the contrary side, and let Ck2 be such that for hk : 12n+1 → Ck2 of the correct type, for almost all hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < H (d ; h1 , . . . , ht ). We let Ck3 be contained in the closure points of Ck ∩Ck2 . We fix hk : 12n+1 → Ck3 of the correct type. We let h˜ k : · 12n+1 → Ck ∩Ck2 exhibit that hk has uniform cofinality . Then for almost all hk+1 , . . . , ht , f, h¯ 2 , . . . , h¯ a there is an n < ˜ where g˜ is defined by such that ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < h˜ k (n, [g]), ¯ h¯ a+1 ]) = H (d2 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a+1 ). g([ By countable additivity of the measures, it follows that there is a g : 12n+1 → Ck ∩ Ck2 of the correct type with hk , g ordered as in Pk and such thatg(α) > k˜ k (n, α) for all α, where n is fixed so that the above inequality holds almost everywhere. This contradiction establishes Pk . The result then follows as in the previous cases. k ,mk Subcase III.b. Kk = S2n+1 where k = 1. (Ia )
Subcase III.b.i. d (Ia ) = (k; d1(Ia ) , . . . , dr(Ia ) )s , where 2 ≤ r ≤ mk , and s appears. By the cofinality lemma it follows that there is a ϑ2 such that for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , ϑ2 (h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < H (dr(Ia ) ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) and such that ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < H (d ; (dr → ϑ2 ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ).
AD AND THE PROJECTIVE ORDINALS
443
Subcase III.b.i.1. dr(Ia ) non-minimal with respect to Lˆ k+1 , Lˆ k+1 (dr ) > dr−1 if r > 2 and cf H (Lˆ k+1 (dr ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = almost every(Ia ) ˆ k+1 where. In this case Lˆ k (d ) = (k; d1(Ia ) , . . . , dr−1 ,L (dr(Ia ) )). By induction, for almost all h1 , . . . , hk there is a Ck+1 such that for almost all hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < h(d ; (dr → NCk+1 ◦ Lˆ k+1 (dr ), h1 , . . . , ht ). For such fixed h1 , . . . , hk−1 , we consider the partition Pk (h1 , . . . , hk−1 ): we partition hk : <m → 12n+1 of the correct type with the extra value g(α2 , . . . , αr , α1 ) inserted between hk (r)(α2 , . . . , αr , α1 ) :=
sup
hk (α2 , . . . , αr , r+1 , . . . , mk , α1 )
r+1 <···<mk <α1
and the next element in the range of hk after hk (r)(α2 , . . . , αr , α1 ), where g has uniform cofinality , according to whether or not for almost all hk , . . . , ht , ϑ(h1 , . . . , ht ) < H (Lˆ k (d ); Lˆ k (d ) → Ng ◦ Lˆ k (d ); h1 , . . . , ht ). If Pk fails, then we let Ck be homogeneous for the contrary side. We let Ck2 be such that for hk : <m → Ck2 of the correct type there is a Ck+1 such that for almost all hk+1 , . . . , ht the above inequality (with NCk+1 ) is satisfied. We let Ck3 be contained in the closure points of Ck ∪ Ck2 , and fix hk : <m → Ck3 of the correct type, and Ck+1 as above. We then get hk2 , g such that: (1) hk2 = hk a.e. w.r.t. the m-fold product of the -cofinal normal measure on 12n+1 . (2) hk2 , g are of the correct type and ordered as in Pk (h1 , . . . , hk−1 ). (3) hk2 , g have range in Ck3 . (4) g(α2 , . . . , αr , α1 ) > hk (r)(α2 , . . . , αr−1 , NCk+1 (αr ), α1 ), for almost all α2 , . . . , αr , α1 . The existence of hk2 and g follows from an easy sliding argument which we omit. We then elect hk+1 , . . . , ht such that ϑ(h1 , . . . , hk , . . . , ht ) = ϑ(h1 , . . . , hk2 , . . . , ht ), ϑ(h1 , . . . , ht ) < h(d ; dr → NCk+1 ◦ Lˆ k+1 (dr ); h1 , . . . , ht ) and ϑ(h1 , . . . , hk2 , . . . , ht ) > h(Lˆ k (d ); Lˆ k (d ) → Ng ◦ Lˆ k (d ); h1 , . . . , hk2 , . . . , ht ). This contradiction establishes Pk , and the result H¯ 2n+1 (k) then follows as in the previous cases. Subcase III.b.i.2. dr(Ia ) non-minimal w.r.t. Lˆ k+1 , Lˆ k+1 (dr ) > dr−1 if r > 2, and cf H (Lˆ k+1 (dr ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = almost everywhere. In this
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(Ia ) ˆ k+1 case, Lˆ k (d ) = (k; d1(Ia ) , . . . , dr−1 ,L (dr(Ia ) ))s . We proceed as in the previous case, and for fixed h1 , . . . , hk−1 consider Pk (h1 , . . . , hk−1 ) where we partition hk : <m → 12n+1 of the correct type with the extra value g(α2 , . . . , αr , α1 ) inserted between sup<αr hk (r)() and the next element in the range of hk after this, where g has uniform cofinality , according to whether or not for almost all hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < H (Lˆ k (d ); Lˆ k (d ) → Ng ◦ Lˆ k (d ); h1 , . . . , ht ).
If Pk fails, we proceed as in the previous case, fixing Ck , Ck2 , Ck3 , hk , Ck+1 respectively, and get hk2 and g satisfying (1)–(4) as in that case. We the proceed to get a contradiction as in that case, which establishes Pk (h1 , . . . , hk−1 ), from which H¯ 2n+1 (k) again readily follows. Subcase III.b.i.3. dr(Ia ) non-minimal with respect to Lˆ k+1 , r > 2, and either Lˆ k+1 (dr ) ≤ dr−1 or dr is minimal with respect to Lˆ k+1 . In this case, (Ia ) s ) . Actually, the second case cannot arise since Lˆ k (d ) = (k; d1(Ia ) , . . . , dr−1 d satisfies condition C so dr−1 < dr and (since k(dr−1 ) > k) for almost all h1 , . . . , hk there is no αk+1 < 12n+1 such that for almost all hk+1 , . . . , ht , h¯ 2 , . . . , h¯ a , H (dr−1 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < αk+1 . For almost all h1 , . . . , hk−1 , we consider Pk (h1 , . . . , hk−1 ) where we partition functions hk : <m → 12n+1 of the correct type with the extra value sup<αr−1 hr (r − 1)(α2 , . . . , αr−2 , , α1 ) g(α2 , . . . , αr , α1 ) inserted between and the next element in the range of hk after this, where g has uniform cofinality , according to whether or not ϑ(h1 , . . . , ht ) < h(Lˆ k (d ); Lˆ k (d ) → Ng ◦ Lˆ k (d ); h1 , . . . , ht ). If Pk fails, we fix Ck , Ck2 , Ck3 , hk , Ck+1 as in the previous case, and get hk2 and g satisfying (1)–(3) as in that case, and (4): g(α2 , . . . , αr , α1 ) > hk (α2 , . . . , αr−2 , αr−1 , NCk+1 (αr−1 ), α1 ) almost everywhere. We then elect hk+1 , . . . , ht , f, h¯ 2 , . . . , h¯ a such that ϑ(h1 , . . . , hk , . . . , ht ) = ϑ(h1 , . . . , hk2 , . . . , ht ), ϑ(h1 , . . . , hk , . . . , h¯ a ) < H (d ; dr → NCk+1 ◦ Lˆ k+1 (dr ); h1 , . . . , hk , . . . , h¯ a ) = sup{hk (H (d2 ; h1 , . . . , h¯ a ), . . . , H (dr−1 ; h1 , . . . , h¯ a ), NCk+1 (H (Lˆ k+1 (dr ); h1 , . . . , h¯ a )), r+1 , . . . , mk , H (d1 ; h1 , . . . , h¯ a )) : r+1 < · · · < m < H (d1 ; h1 , . . . , h¯ a )} k
≤ g(H (d2 ; . . . ), . . . , H (dr−1 ; . . . ), H (d1 ; . . . )) = Ng (sup{hk (r − 1)(H (d2 ; . . . ), . . . , H (dr−2 ; . . . ), r−1 , H (d1 ; . . . )) : r−1 < H (dr−1 ; h1 , . . . , hk2 , . . . , h¯ a )}) = H (Lˆ k (d ); Lˆ k (d ) → Ng ◦ Lˆ k (d ); h1 , . . . , hk2 , . . . , ht , f, h¯ 2 , . . . , h¯ a )
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and also such that ϑ(h1 , . . . , hk2 , . . . , ht ) > h(Lˆ k (d ); Lˆ k (d ) → Ng ◦ Lˆ k (d ); h1 , . . . , hk2 , . . . , ht ). This contradiction establishes Pk and the result follows as in the previous case. Subcase III.b.i.4. dr(Ia ) minimal with respect to Lˆ k+1 and r = 2. In this case we have Lˆ k (d ) = d1 . By induction, for almost all h1 , . . . , hk there is an αk+1 < 12n+1 s.t. for almost all hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < h(d ; dr → αk+1 , . . . , h1 , . . . , ht ). For almost all h1 , . . . , hk−1 , we consider the partition Pk (h1 , . . . , hk−1 ) where we partition functions hk : <m → 12n+1 of the correct type with g(α) inserted between sup<α hk (1)() and the next element in the range of hk after this, according to whether or not for almost all hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < (Lˆ k (d ); Lˆ k (d ) → Ng ◦ Lˆ k (d ); h1 , . . . , ht ). If Pk fails, we elect Ck , Ck2 , Ck3 , hk , αk+1 similarly to the previous cases. We then get hk2 and g satisfying the usual (1)–(3) and (4): g(α) >
sup
hk (αk+1 , 3 , . . . , mk , α)
3 <···<mk <α
for all α > αk+1 . We then elect hk+1 , . . . , ht , f, h¯ 2 , . . . , h¯ a such that ϑ(h1 , . . . , hk2 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < h(d ; dr → αk+1 , h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) ≤ Ng (sup{hk2 (2 , . . . , mk , 1 ) : 1 < H (d1 ; h1 , . . . , h¯ a ), 2 < · · · < mk < 1 }) = Ng (H (d1 ; h1 , . . . , hk2 , . . . , ht , f, h¯ 2 , . . . , h¯ a )) = H (Lˆ k (d ); Lˆ k (d ) → Ng ◦ Lˆ k (d ); h1 , . . . , hk2 , . . . , ht , f, h¯ 2 , . . . , h¯ a ). This contradiction establishes Pk from which the result follows. Subcase III.b.ii. d (Ia ) = (k; d1(Ia ) , . . . , dr(Ia ) )(Ia ) , where s does not appear, and r = mk . We consider the case mk > 1, the case mk = 1 being similar. In this case Lˆ k (d ) = (k; d1 , . . . , dr )s . As in the previous case, we consider Pk (h1 , . . . , hk−1 ) for almost all h1 , . . . , hk−1 , where here the extra value g(α2 , . . . , αr , α1 ) is inserted between sup<mk hk (α2 , . . . , , α1 ) and the next element in the range of hk after this. If Pk fails, we fix Ck homogeneous for the contrary side, let Ck2 be such that for hk : <m → Ck2 of the correct type, for almost all hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < h(d ; h1 , . . . , ht )
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and let Ck3 in the closure points of Ck ∪ Ck2 . We again fix hk : 12n+1 → Ck3 of the correct type, and let h˜ k (α2 , . . . , αmk , n, α1 ) : × <m → 12n+1 exhibit the n < s.t. for uniform cofinality of hk . By countable additivity there is an ¯ ¯ almost all hk+1 , . . . , ht , f, h2 , . . . , ha , ϑ(h1 , . . . , h¯ a ) < h˜ k (H (dr , . . . ), . . . , n, H (d1 , . . . )). We then get hk2 , g satisfying the usual (1)–(3) and (4): g(α2 , . . . , αmk , α1 ) > h˜ k (α2 , . . . , αmk , n, α1 ) for all α1 , . . . , αmk . The result follows as in previous cases. Subcase III.b.iii. d (Ia ) = (k; d1(Ia ) , . . . , dr(Ia ) )(Ia ) , where s does not appear, and r < mk . Subcase III.b.iii.1. d1 is not minimal w.r.t. Lˆ k+1 , Lˆ k+1 (d1 ) > dr if r ≥ 2, and cf H (Lˆ k+1 (d1 ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) = almost everywhere. In this case Lˆ k (d ) = (k; d1 , . . . , dr , Lˆ k+1 (d1 )). By the cofinality lemma, there is a ϑ2 such that for almost all h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a , ϑ2 (h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < H (d1 ; h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) and ϑ(h1 , . . . , h¯ a ) < H (Lˆ k (d ); Lˆ k+1 (d1 ) → ϑ2 ; h1 , . . . , h¯ a ). Hence, by induction for almost all h1 , . . . , ht there is Ck+1 such that for almost all hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < H (Lˆ k (d ); Lˆ k+1 (d1 ) → NCk+1 ◦ Lˆ k+1 (d1 ); h1 , . . . , ht ). For fixed h1 , . . . , hk−1 , we consider Pk (h1 , . . . , hk−1 ) where we partition hk of the correct type with g(α2 , . . . , αr+1 , α1 ) inserted between hk (r + 1)(α2 , . . . , αr+1 , α1 ) and the next element in the range of hk according to whether or not for almost all hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < H (Lˆ k (d ); Lˆ k (d ) → Ng ◦ Lˆ k (d ); h1 , . . . , ht ). If Pk fails, we fix Ck , Ck2 , Ck3 , hk , Ck+1 , as in the previous cases, and get hk2 and g satisfying the usual (1)–(3) and (4): g(α2 , . . . , αr+1 , α1 ) > hk (r + 1)(α2 , . . . , αr , NCk+1 (αr+1 ), α1 ). We then proceed as in the previous cases to establish H¯ 2n+1 (k). Subcase III.b.iii.2. d1 not minimal w.r.t. Lˆ k+1 , Lˆ k+1 (d1 ) > dr if r ≥ 2, and cf H (Lˆ k+1 (d1 ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) =
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almost everywhere. In this case Lˆ k (d ) = (k; d1 , . . . , dr , Lˆ k+1 (d1 ))s . We proceed as in (iii.1) except that in Pk (h1 , . . . , hk−1 ) we insert the function value g(α2 , . . . , αr+1 , α1 ) between sup<αr+1 hk (r + 1)(α2 , . . . , αr , , α1 ) and the next element in the range of hk after this. We proceed as in (iii.1), getting hk2 and g which satisfy (1)–(4) of that case, and then proceed to establish H¯ 2n+1 (k) as in that case. Subcase III.b.iii.3. d1 not minimal w.r.t. Lˆ k+1 and Lˆ k+1 (d1 ) ≤ dr (where r ≥ 2). In this case Lˆ k (d ) = (k; d1 , . . . , dr )s . We proceed as in the above cases, where in Pk (h1 , . . . , hk−1 ) the function value g(α2 , . . . , αr , α1 ) is inserted between sup<αr hk (r + 1)(α2 , . . . , αr−1 , , α1 ) and the next element in the range of hk . We proceed as before getting hk2 and g which satisfy the usual (1)–(3) and (4): g(α2 , . . . , αr , α1 ) > hk (r + 1)(α2 , . . . , αr , NCk+1 (αr ), α1 ). We then finish as in the previous cases. Subcase III.b.iii.4. d1 minimal w.r.t. Lˆ k+1 . It follows readily that r = 1. In this case Lˆ k (d ) = d1 . By the cofinality lemma and induction, we have that for almost all h1 , . . . , hk there is αk+1 < 12n+1 such that for almost all hk+1 , . . . , ht , ϑ(h1 , . . . , ht ) < h(d˜; d˜˜ → αk+1 ; h1 , . . . , ht ), where we define d˜ = (k; d1 , d˜˜). We assume here that mk ≥ 2, the case mk = 1 following similarly. As in the previous cases, we consider Pk (h1 , . . . , hk−1 ) where the value g(α) is inserted between sup<α hk (1)() and the next element in the range of hk . If Pk fails, we fix Ck , Ck2 , Ck3 , hk , αk+1 and get hk2 and g satisfying the usual (1)–(3) and (4): g(α) >
sup
hk (αk+1 , 3 , . . . , mk , α).
3 <···<mk <α
We then proceed as in the previous cases to establish H¯ 2n+1 (k). mk Subcase III.c. Kk = W2n+1 . In the following we assume n > 0, the case n = 0 following easily.
Subcase III.c.i d (Ia ) = (k; )(Ia ) We consider the sequence of measures Kb(k) , . . . , Kbu , K¯ 2 , . . . , K¯ a . where we recall Kb(k) , . . . , Kbu enumerates the subsequences of Kk+1 , . . . , Kt ¯k ,m¯ k of measures in m
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f, h¯ 2 , . . . , h¯ a , there is an ordinal ϑ2 = ϑ2 (hk , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < supa.e. hk such that ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < ϑ2 . It follows readily (by 12n+1 ,m m additivity of W2n+1 and S2n+1 ) that for almost all h1 , . . . , hk−1 there is an ordinal ϑ2 = ϑ2 (h1 , . . . , hk−1 ) such that for almost all hk , hb(k) , . . . , hbu , h¯ 2 , . . . , h¯ a , ϑ2 (hk , . . . , h¯ a ) < sup hk a.e.
and for almost all hk , . . . , ht , f, h¯ 2 , . . . , h¯ a , ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < ϑ2 (h1 , . . . , hk−1 )(hk , hb(k) , . . . , hbu , . . . , h¯ a ). By H¯ 2n−1 (c) it follows that for almost all h1 , . . . , hk−1 there is a c.u.b. Ck ⊆ 12n+1 such that for almost all hk , hb(k) , . . . , hbu , h¯ 2 , . . . , h¯ a , ¯k ,m¯ k ϑ2 (h1 , . . . , hk−1 )(hk , hb(k) , . . . , h¯ a ) < (NCk ; hk ; d˜2n−1 ; hb(k) , . . . , h¯ a ) = H (Lˆ k (d ); Lˆ k (d ) → NC ◦ Lˆ k (d ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ). k
This establishes H¯ 2n+1 (k) in this case. ¯k ,m¯ k Subcase III.c.i.2. the maximal tuple d˜2n−1 relative to Kb(k) , . . . , Kbu , K¯ 2 , ¯ . . . , Ka is not defined. In this case d is minimal w.r.t. Lˆ k . As above, for almost all h1 , . . . , hk−1 , there is an ordinal ϑ2 = ϑ2 (h1 , . . . , hk−1 ) such that for almost all hk , hb(k) , . . . , h¯ a ,
ϑ2 (h1 , . . . , hk−1 )(hk , hb(k) , . . . , h¯ a ) < sup hk a.e.
and for almost all hk , . . . , ht , f, h¯ 2 , . . . , h¯ a , ϑ(h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ) < ϑ2 (h1 , . . . , hk−1 )(hk , hb(k) , . . . , h¯ a ). By H¯ 2n−1 (d ) it follows that for almost all h1 , . . . , hk−1 there is an αk < 12n+1 such that for almost all hk , . . . , ht , f, h¯ 2 , . . . , h¯ a , ϑ(h1 , . . . , ht ) < ϑ2 (h1 , . . . , hk−1 )(hk , hb(k) , . . . , h¯ a ) < αk , which establishes H¯ 2n+1 (k) in this case. Subcase III.c.ii d = (k; d¯2 )s where s appears. Here d¯2 is defined relative to the sequence Kb(k) , . . . , Kbu , K¯ 2 , . . . , K¯ a and (d¯2 )s satisfies conditions D and A. Subcase III.c.ii.1. (d¯2 )s is non-minimal w.r.t. L relative to Kb(k) , . . . , Kbu , K¯ 2 , . . . , K¯ a . In this case Lˆ k (d ) = (k; L ((d¯2 )s ))s , where s appears if it appears in L((d2 )s ) (here we use the notation that if L((d2 )s ) = (d )s , then L ((d2 )s ) =
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d ). Proceeding as above, we have that for almost all h1 , . . . , hk−1 the ordinal ϑ(h1 , . . . , hk−1 ) is such that for almost all hk , hb(k) , . . . , hbu , h¯ 2 , . . . , h¯ a , ϑ(h1 , . . . , hk−1 )(hk , hb(k) , . . . , h¯ a ) < (id; hk ; (d¯2 )s ; hb(k) , . . . , hbu , h¯ 2 , . . . , h¯ a ). Hence by H2n−1 (a) it follows that for almost all h1 , . . . , hk−1 , there is Ck ⊆ 12n+1 such that for almost all hk , hb(k) , . . . , hbu , h¯ 2 , . . . , h¯ a , ϑ(h1 , . . . , h¯ a ) < (NCk ; hk ; L((d2 )s ); hb(k) , . . . , h¯ a ) = H (Lˆ k (d ); Lˆ k (d ) → NC ◦ Lˆ k (d ); h1 , . . . , ht , f, h¯ 2 , . . . , h¯ a ), k
which establishes H¯ 2n+1 (k) in this case. Subcase III.c.ii.2. (d¯2 )s is minimal w.r.t. L relative to Kb(k) , . . . , Kbu , K¯ 2 , . . . , K¯ a . In this case d is minimal w.r.t. Lˆ k . We proceed as in the previous case, using H2n−1 (b) and a simple partition and sliding argument (partitioning hk with the extra value αk+1 inserted before inf hk ). Subcase III.c.iii d = (k; d¯2 ) where s does not appear. Here Lˆ k (d ) = (k; d¯2 )s . We proceed as in case (b)(ii) above. This establishes H¯ 2n+1 (k) in all cases. ,m 5.2. H2n+1 . Now we consider H2n+1 . We fix the measure V = S2n+1 ,m
Case I. s does not appear. In this case L((d )) = (d )s We consider the partition P: We partition functions f : κ → 12n+3 of the correct type with the extra and f(α), with g of uniform cofivalue g(α) inserted between sup<α f() nality , according to whether or not F ([f]) < (Ng ; L((d )); f; K1 , . . . , Kt ). It follows readily by countable additivity of the measures K1 , . . . , Kt , that on the homogeneous side of the partition the statement in P holds. We let C be a c.u.b. subset of 12n+3 homogeneous for P. It then follows that for almost all F that F ([f]) < (NC ; L((d )); f; K1 , . . . , Kt ).
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Case II. s appears. We first claim that there is a c.u.b. C ⊆ 12n+3 such ˆ ); f; K1 , . . . , Kt ). We have that for almost all f, F ([f]) < (NC ; L(d that for almost all f, for almost all h1 , . . . , ht , there is a ϑ(h1 , . . . , ht ) < h(d ; h1 , . . . , ht ) such that F (f; h1 , . . . , ht ) < f(ϑ(h1 , . . . , ht )). Hence, by H¯ 2n+1 (1), for almost all h1 , . . . , ht there is a c.u.b. C ⊆ 12n+1 such that for almost all h1 , . . . , ht , ˆ ); L(d ˆ ) → NC ◦ L(d ˆ ); h1 , . . . , ht ). ϑ(h1 , . . . , ht ) < h(L(d We consider the partition P: We partition functions f : κ → 12n+3 of the f(α) and correct type with the extra value g(α) inserted between the values f(α+1), with g of uniform cofinality , according to whether or not F ([f]) < (Ng ; L((d )); f; K1 , . . . , Kt ). We claim that on the homogeneous side of the partition the property stated in P holds. If not, we fix a c.u.b. C 1 homogeneous for the contrary side, and let C 2 ⊆ 12n+3 be such that for f : κ → C 2 of the above. We let C 3 be contained in the correct type there is a C ⊆ 12n+1 as 1 2 closure points of C ∪ C . We fix f : κ → C 3 of the correct type, and fix a c.u.b. C ⊆ 12n+1 as above. We then get f 2 , g satisfying: (1) f 2 = falmost everywhere w.r.t. V. (2) f 2 , g are of the correct type and ordered as in P. (3) f 2 , g have range in C 3 (in fact have range a subset of the range of f) (4) If h : 12n+1 → 12n+1 (or h : <m → 1 , if n = 0) represents [h] w.r.t. ,m v(S2n+1 ) 2n+1 , then g([h]) > f([NC ◦ h]). The construction of f 2 , g is similar to that given previously and will be omitted. We then fix h1 , . . . , ht such that F (f; h1 , . . . , ht ) = F (f 2 ; h1 , . . . , ht ) < f(ϑ(h1 , . . . , ht )), and ˆ ); L(d ˆ ) → NC ◦ L; ˆ h1 , . . . , ht ), ϑ(h1 , . . . , ht ) < h(L(d and ˆ ); f 2 ; h1 , . . . , ht ) F (f 2 ; h1 , . . . , ht ) > (Ng ; L(d ˆ ); h1 , . . . , ht ))) = Ng (f 2 (h(L(d ˆ ˆ ) → NC ◦ L(d ˆ ); h1 , . . . , ht )). > f(h(L(d ); L(d This contradiction establishes P. If C ⊆ 12n+3 is homogeneous for P, it ˜ ); f; K1 , . . . , Kt ) and follows readily that for almost all f, F ([f]) < (NC ; L(d we are done with the claim. We next claim that if a description d satisfies conditions C and A relative to K1 , . . . , Kt , but (d ) does not satisfy condition D, and if there is a c.u.b. C ⊆ 12n+3 such that for almost all f, F ([f]) < (NC ; d ; f; K1 , . . . , Kt ),
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then for almost all f, F ([f]) < (id; d ; f; K1 , . . . , Kt ). To see this, consider the partition P where we partition f : κ → 12n+3 of the correct according to whether or not F ([f]) < (id; d ; f; K1 , . . . , Kt ). If P fails, we fix a c.u.b. C 1 ⊆ 12n+3 , where C 1 ⊆ C and C 1 is homogeneous for the contrary side, let C 2 ⊆ 12n+3 be such that for f : κ → C 2 of the correct type F ([f]) < (NC ; d ; f;K1 , . . . , Kt ) holds, and let C 3 be contained in the closure points of C 1 ∪ C 2 . We fix f : κ → C 3 of the correct type. ,m such that for almost all h1 , . . . , ht , We fix a measure one set A w.r.t. S2n+1 h(d ; h1 , . . . , ht ) ∈ / A (which we can do as (d ) does not satisfy condition D). We let C be a c.u.b. subset of 12n+1 such that for h : 12n+1 → C of the correct let C be contained in type (or h : <m → C if n = 0)we have [h] ∈ A. We the closure points of C . We then get f 2 satisfying: (1) For h of the correct type having range in C we have f 2 ([h]) = f([h]). (2) f 2 is of the correct type and has range in C 3 (in fact a subset of the range of f). (3) For [h] not represented by h as in (1), f 2 ([h]) > Nf (f([h])), hence f 2 ([h]) > NC (f([h])). The existence of f 2 follows from an easy sliding argument. We then fix h1 , . . . , ht such that F (f; h1 , . . . , ht ) = F (f 2 ; h1 , . . . , ht ), F (f; h1 , . . . , ht ) < (NC ; d ; f ; h1 , . . . , ht ) = NC (f(h(d ; h1 , . . . , ht ))), and F (f 2 ; h1 , . . . , ht ) > (id; d ; h1 , . . . , ht ) = f 2 (h(d ; h1 , . . . , ht )), and also h(d ; h1 , . . . , ht ) ∈ / A. From this, the last equation, and (3) above it follows that F (f 2 ; h1 , . . . , ht ) > NC (f(h(d ; h1 , . . . , ht ))). This contradicts the first two equations and establishes that on the homogeneous side the ˆ ); f; property stated in P holds. Hence, for almost all f, F ([f]) < (id; L(d K1 , . . . , Kt ). ˆ )) satisfies condition D, then our first considerations apply, and we If (L(d are done. If not, then we are in a position to repeat the argument in (i) to get ˜ ))s ; f; K1 , . . . , Kt ). If C ⊆ 12n+3 s.t. for almost all f, F ([f]) < (NC ; (L(d s ˆ )) satisfies condition D, we are done. If not an argument similar to the (L(d ˜ ))s ; f; K1 , . . . , Kt ). above establishes that for almost all f, F ([f]) < (id; (L(d The proof proceeds by considering the partition P as above, and constructing f 2 satisfying (1), (2) as above and (3): for [h] not represented by a function of the correct type, f 2 ([h]) > f() where is the least ordinal represented by a function of the correct type which is greater than [h]. We are now in a position to repeat the previous argument. Repeating this argument,
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we eventually get a c.u.b. C ⊆ 12n+3 such that for almost all f, F ([f]) < (NC ; L(d ); f; K1 , . . . , Kt ), which establishes H2n+1 (a). ,m s ) is 5.4. H2n+1 (b). We consider H2n+1 (b). We assume that (d )s = (d2n+1 minimal w.r.t. L relative to K1 , . . . , Kt . Case I. d is minimal w.r.t. Lˆ relative to K1 , . . . , Kt . We have that for almost all f, h1 , . . . , ht there is a ϑ(h1 , . . . , ht ) < h(d ; h1 , . . . , ht ) such that F (f; h1 , . . . , ht ) < f(ϑ(h1 , . . . , ht )). By H¯ 2n+1 (1), for almost all f there is α < κ such that for almost all h1 , . . . , ht , ϑ(h1 , . . . , ht ) < α. We consider the partition P where we partition f : κ → 12n+3 of the correct with the extra value (of cofinality ) inserted before inf f according to whether or not F ([f]) < . If P fails, we let C 1 be homogeneous for the contrary side, let C 2 be such that for f : κ → C 2 of the correct type F (f; h1 , . . . , ht ) < f(α) almost everywhere (where α = α(f) < κ is as above) and let C 3 = C 1 ∩ C 2 . We fix f : κ → C 3 of the correct type, let α < κ be as above, and get , f 2 satisfying: (1) f 2 = f a.e. w.r.t. V. (2) < inf f 2 , cf = , and f 2 is of the correct type. (3) , f 2 have range in C 3 . (4) > f(α). Then for almost all h1 , . . . , ht we have F (f h1 , . . . , ht ) = F (f 2 ; h1 , . . . , ht ), F (f; h1 , . . . , ht ) < f(α), and F (f 2 ; h1 , . . . , ht ) > > f(α). This contradiction establishes P. It follows readily that H2n+1 (b) is satisfied. Case II. d is not minimal w.r.t. Lˆ relative to K1 , . . . , Kt . It follows as in the proof of H2n+1 (a) that for almost all f, h1 , . . . , ht , ¯ ))s ; f; h1 , . . . , ht ), F (f; h1 , . . . , ht ) < (id; (L(d and in fact F (f; h1 , . . . , ht ) < (id; (L¯ p (d ))s ; f; h1 , . . . , ht ), ˆ We elect p minimal such that where (Lˆ p (d )) denotes the pth iterate of L. p+1 ˆ L (d ) is not defined, and then proceed as in (i). This establishes H2n+1 (b).
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453
5.5. H2n+1 (c). We consider H2n+1 (c). We assume that F is given s.t. for almost all f : κ → 12n+3 of the correct type, for almost all h1 , . . . , ht we have ,m F ([f]) < sup f, and the maximal tuple d˜ = d˜2n+1 is defined. For almost all f, h1 , . . . , ht , there is an α < κ such that F (f; h1 , . . . , ht ) < f(α). It follows readily that for almost all f, h1 , . . . , ht , there is a c.u.b. C∞ ⊆ 12n+1 such that α(f; h1 , . . . , ht ) < h(d˜∞ ; d˜∞ → NC ◦ d˜∞ ; h1 , . . . , ht ), where d˜∞ is the basic type 1 description () with index Ia = (f(K¯ 1 ); ) for n > 0 (for n = 0, d˜∞ = (m), a basic type 0 description). This follows from an easy partition argument (partitioning f : κ → 12n+3 with the extra value after sup f). We recall here that h(d˜∞ ; h1 , . . . , ht ) is represented by the function [f] → H (d˜∞ ; h1 , . . . , ht , f) = supa.e. f. We then proceed as in the proof of H¯ 2n+1 (k) to establish that for 1 ≤ k ≤ t the following holds: for almost all f, h1 , . . . , hk−1 , there is a c.u.b. Ck ⊆ 12n+1 such that for almost all h1 , . . . , ht α(h1 , . . . , ht ) < h(d˜k ; d˜k → NC ◦ d˜k ; h1 , . . . , ht ), where d˜k is as in the definition of the maximal tuple d˜ (which is equal to d˜1 ). We then proceed as in H2n+1 (a) to establish that there is a c.u.b. C ⊆ 12n+3 such that for almost all f, h1 , . . . , ht , ,m s F (f; h1 , . . . , ht ) < (NC ; (d˜2n+1 ) ; f; h1 , . . . , ht ), ,m where s appears if (d˜2n+1 ) does not satisfy condition D. This establishes H2n+1 (c). 5.6. H2n+1 (d). We consider H2n+1 (d). We again assume that for almost all f, h1 , . . . , ht ,
F (f; h1 , . . . , ht ) < sup f, a.e. ,m d˜2n+1
but that the maximal description is not defined. In this case, the integer ,m ˜ e1 as in the definition of d2n+1 does not exist (or b1 if = 1). As above, we have that for almost all f, h1 , . . . , hk−1 , there is a c.u.b. C∞ ⊆ 12n+1 such that for almost all hk , . . . , ht we have α(f, h1 , . . . , ht ) < h(d˜∞ → NC ◦ d˜∞ ; h1 , . . . , ht ) and F (f; h1 , . . . , ht ) < f(α) (where α = α(f) is as before). Here d˜∞ = (), a basic type 1 description (for n = 0, d˜∞ = (m), a basic type 0 description). We the proceed as in H¯ 2n+1 (k), and since e1 does not exist we have that for almost all f, there is a c.u.b. C1 ⊆ 12n+1 such that for almost all h1 , . . . , ht , α(f; h1 , . . . , ht ) < h(d˜∞ ; d˜∞ → NC1 ◦ d˜∞ ; h1 , . . . , ht ). We then proceed as in H2n+1 (b) to establish H2n+1 (d). This establishes H2n+1 in all cases.
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§6. The Main Theorem. We define an ordering r ((d2 )s ; K1 , . . . , Kt2 ) >r . . . , ,m where t1 ≤ t2 ≤ · · · , s may or may not not appear in any tuple, and di ∈ D2n+1 for all i. Then for any p we have that for almost all f, h1 , . . . , htp (where ,m f : ϑ(S2n+1 ) → 12n+3 ) we have that (id; (d1 )s ; f; h1 , . . . , ht1 ) > · · · > (id; (d2 )s ; f; h1 , . . . , htp ).
Letting p become arbitrarily large, we define an infinite decreasing sequence of ordinals. Hence
sup
((d¯) ;K¯ 1 ,... ,K¯ t¯ )
⎟ f((d¯)s ; K¯ 1 , . . . , K¯ t¯)⎠ + 1.
Remark 6.2. The rank function of Definition 6.1 is computed in a slightly non-standard manner so that all ranks are successor ordinals. We introduce the following notation: Definition 6.3. For any ordinal α we let E(0, α) = α and E(n + 1, α) = E(n,α) . Also, we let E(n) = E(n, 1). We now state the main theorem of this section: Theorem 6.4. Let (d )s , where s may or may not appear, satisfy condi,m ,m tions D and A relative to K1 , . . . , Kt , where d = d2n+1 . Let V = S2n+1 . V < 1 1 1 Let F : 2n+3 → 2n+3 be defined w.r.t. the measure 2n+3 on 2n+3 by ,m F ([f]) = (id; (d )s ; K1 , . . . , Kt ), for almost all f : κ(S2n+1 ) → 12n+3 . Then V in the ultrapower of 12n+3 by the measure < 2n+3 , F represents an ordinal ≤ ℵE(2n+1)+f((d )s ;K1 ,... ,Kt ) .
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455
Proof. First consider the case f((d )s ; K1 , . . . , Kt ) = 1. In this case (d )s is minimal w.r.t. L relative to K1 , . . . , Kt . By H2n+1 (b), if ϑ < (id; (d )s ; K1 , . . . , Kt ) then α < jK1 ◦ · · · ◦ jKt () for some < 12n+3 . Using our inductive hypothesis I2n+1 we have that α < 12n+3 and so (id; (d )s ; K1 , . . . , Kt ) ≤ 12n+3 = ℵE(2n+1)+1 . Next assume f((d )s ; K , . . . , K ) > 1. We may assume that (d )s is 1 t non-minimal w.r.t. L. We then have from the main inductive hypothesis (Lemma 5.25) that if F ([f]) < (id; (d )s ; f; K1 , . . . , Kt ) for almost all f, then there is a g : 12n+3 → 12n+3 such that for almost all f we have F ([f]) < (g; L((d )s ); f; K1 , . . . , Kt ). By K2n+3 , there is a tree T on 12n+3 and a c.u.b. C ⊆ 12n+3 such that for supremum ranging over embeddings j α ∈ C , g(α) < |T supj j(α)|, the from measures in m≤n R2m+1 . We fix such a T and C . Hence for almost all f we have that F ([f]) < (|T sup j|; L((d )s ); K1 , . . . , Kt ) j
< (sup j; L((d )s ); K1 , . . . , Kt )+ . j
Here “supj j” stands for the function α → supj j(α) where again the supre mum ranges over embeddings j from measures in m≤n R2m+1 . To prove the second inequality, we define an ordering on the ordinal (supj j; L((d )s ); K1 , . . . , Kt ) as follows: We set ϑ1 ϑ2 if for almost all f, h1 , . . . , ht , |T (sup j; L((d )s ); f, h1 , . . . , ht )(ϑ1 (f, h1 , . . . , ht ))| j
< |T (sup j; L((d )s ); f, h1 , . . . , ht )(ϑ2 (f, h1 , . . . , ht ))|, j
where |T α()| denotes the rank of in the tree T α (we are identifying finite tuples of ordinals with ordinals here). If now < (|T sup j|; L((d )s ); K1 , . . . , Kt ), j
then there is a ϑ = ϑ() < (sup j; L((d )s ); K1 , . . . , Kt ) j
such that for almost all f, h1 , . . . , ht , (f, h1 , . . . , ht ) = |T (sup j; L((d )s ); f, h1 , . . . , ht )(ϑ(f, h1 , . . . , ht ))|. j
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STEVE JACKSON
The map → ϑ() is order-preserving from the ordinal ((|T sup j|; L((d )s ); K1 , . . . , Kt ) j
with the usual ordering to the ordinal ((sup j; L((d )s ); K1 , . . . , Kt ) j
with the ordering . This establishes the above inequalities. It follows immediately that (id; (d )s ; K1 , . . . , Kt ) ≤ (sup j; L((d )s ); K1 , . . . , Kt )+ . j
By countable additivity, (sup j; L((d )s ); f; K1 , . . . , Kt ) = sup(j; L((d )s ); K1 , . . . , Kt ), j
j
where again over the embeddings corresponding to the supremum on j ranges measures m≤n R2m+1 . For Kt+1 in m≤n R2m+1 , it follows immediately from the definitions that (jKt+1 ; L((d )s ); f; K1 , . . . , Kt ) = (id; L((d )s ); K1 , . . . , Kt , Kt+1 ), where jKt+1 denotes the embedding from the measure Kt+1 . Putting this together we have (id; (d )s ; K1 , . . . , Kt ) ≤ [sup(id; L((d )s ); K1 , . . . , Kt , Kt+1 )]+ K + t+1 ≤ ℵE(2n+1)+supK f(L((d )s );K1 ,... ,Kt ,Kt+1 ) t+1 = ℵE(2n+1)+f((d )s ;K1 ,... ,Kt+1 ) , using induction for the second inequality and the definition of the rank function for the third. This completes the proof of the theorem. §7. A Rank Computation. The main goal of this section is to compute the bound sup
f(d ; K1 , . . . , Kt ) ≤ E(2n + 3)
d,K1 ,... ,Kt
We recall (this notation was introduced right after Definition 5.23) that for all ¯ d (Ia ) = d (f(K );) satisfying conditions C and A relative to K1 , . . . , Kt , for all the ¯ ¯ ¯ component tuples d2f(K1 ;K2 ,... ,Ka ) of d we have that the map d2 : a → {1, . . . , t} is defined. Here d2 (i) is an integer j associated to K¯ i such that v(Kj ) = K¯ i . It is easy to see that d satisfies the following: (1) d is strictly increasing. (2) v(Kd (i) ) = K¯ i .
AD AND THE PROJECTIVE ORDINALS
(3) for all component tuples d d . (4) d (K¯ a ) < k(d ).
(Ia )
457
of d (Ia ) (where Ia extends Ia ), d extends
Definition 7.1. We define t − p), etc. We also define <0p to be the ordering generated by the relations: (1) (d1 ; K1 , . . . , Kt ) <0p (d2 ; K1 , . . . , Kt ) for d1 < d2 . (2) (L˜ 1 (d1 ); K1 , . . . , Kt−p , Kt+1 , . . . , Kt ) <0p (d1 ; K1 , . . . , Kt ) for all Kt+1 , (L˜ 1 (d1 ); K1 , . . . , Kt−p , Kt+1 , Kt+2 , . . . , Kt ) <0p (d1 ; K1 , . . . , Kt−p , Kt+1 , . . . , Kt ) for all Kt+1 , Kt+2 , etc. Here t + 1, t + 2, etc. precede K1 if t = p. We let fpk (d ; K1 , . . . , Kt ), for d (a) < k < k(d ), be the rank of the tuple (d ; K1 , . . . , Kt ) with respect to t − p. Also, if p = t and k = 0, then f¯t0 (d1 ; K1 , . . . , Kt ) = f¯t0 (d1 ; Kk+1 , K1 , . . . , Kt ).
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Bpk : let d (Ia ) , k be as above, and assume d is obtained by reindexing d . That is, if Ia = (f(K¯ 1 ), K¯ 2 , . . . , K¯ a ) is the index of d and Ia = (f(K¯ 1 ), K¯ 2 , . . . , K¯ a ) is the index of d , where d (Ka ) < k(d ) = k(d ), then K¯ 2 , . . . , K¯ a is a subsequence of K1 , . . . , Ka , and d = d except that for each (Iai )
component tuple di
of d with index
Iai = (f(K¯ 1 ), K¯ 2 , . . . , K¯ a , K¯ a+1 , . . . , K¯ ai ), the corresponding component tuple di
(Ia ) i
of d has index of the form
Ia i = (f(K¯ 1 ), K¯ 2 , . . . , K¯ a , K¯ a+1 , . . . , K¯ ai ). We then require that f¯pk (d ; K1 , . . . , Kt ) =f¯pk (d ; K1 , . . . , Kt ). Rpk : suppose d1(Ia ) , d2(Ia ) are defined and satisfy conditions C and A relative to K1 , . . . , Kt , where d1 (a) < k ≤ k(d1 ) and similarly for d2 , and d2 < d1 . Then f¯k+b (d2 ; K1 , . . . , Kt−p , Kt+1 , . . . , Kt ) < f¯k (d1 ; K1 , . . . , Kt ), p
p
where b = 0 if k ≤ t − p, and b = 1 if k > t − p or k = 0. ,m Rp : if d ∈ D2n+1 , then f¯pk (d ; K1 , . . . , Kt ) < E(2n, mk ). We assume inductively that Akp , Bpk , Rpk , Rp are satisfied for d ∈ D2n−1 , and the corresponding function g¯pk on D2n−1 . Once we have defined the functions f¯pk , the following definition will define the functions f¯p . Definition 7.2. Let d (Ia ) satisfy conditions C and A with respect to the sequence K1 , . . . , Kt . We set f¯p = f¯p1 (and similarly g¯p = g¯p1 on D2n−1 ). We also introduce the following notation. Definition 7.3. We let M(K1 , . . . , Kt ) denote the sequence of measures i ,mi v(Ki ) corresponding to those Ki = S2n+1 for i > 1. Assuming f¯pk defined we will extend it slightly to a function f¯¯kp defined on objects of the form (d ) or (d )s , where d satisfies conditions C and A relative to K1 , . . . , Kt (but not necessarily condition D). We do this as follows: For s not appearing, we set f¯¯k ((d ); K , . . . , K ) = 2 · f¯¯k (d ; K , . . . , K ) + 1, p
1
t
p
1
t
and for s appearing we set f¯¯kp ((d )s ; K1 , . . . , Kt ) = 2 · f¯pk (d ; K1 , . . . , Kt ). If we consider the corresponding versions of Akp , Bpk , Rpk , for f¯¯kp , it is immediate that they are satisfied provided they are satisfied for f¯pk .
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It will be convenient for the definition of the f¯pk to introduce the following ordinal function. Definition 7.4. For m ≥ 1 and ∈ Ord, the ordinal m () is defined (recursively on m) by 1 () = 2 · and m () = (m + m−1 () + m) · + (m + m−1 ()). We have the following simple lemma. Lemma 7.5. For all m ≥ 1 and ≥ we have m () ≤ m ·2. Furthermore, if 1 ≤ k < and ≥ then (( + 1) + () + ( + 1)) · + ( + −1 () + ) · + · · · + ((k + 2) + k+1 () + (k + 2)) · + ((k + 1) + k () + (k + 1)) · ( + 1) ≤ +1 () + k + 1. Proof. Expanding the recursive definition of +1 () and stopping at the k () terms immediately gives the result. k ¯ We now proceed to define fp . We proceed by reverse induction on k. Case I. k = ∞. Hence d basic type 1 or 0. ¯
¯
Subcase I.a. d basic of type 1, so d (Ia ) = (d¯2 )s(f;K2 ,... ,Ka ) where s may or may not appear, d¯2 ∈ D2n−1 , and (d¯2 )s satisfies conditions D and A relative to K¯ 2 , . . . , K¯ a (where s appears here iff s appears in d ). Subcase I.a.i. If s appears we set f¯p∞ (d ; K1 , . . . , Kt ) = E(2n − 1) + 2 · g¯q (d¯2 ; M(K1 , . . . , Kt )) + 1. Subcase I.a.ii. If s does not appear f¯p∞ (d ; K1 , . . . , Kt ) = E(2n − 1) + 2 · g¯q (d¯2 ; M(K1 , . . . , Kt )) + 2, where q is minimal such that (K˜ w−q ) ≤ t − p, where K˜ 1 , . . . , K˜ w enumerates M(K1 , . . . , Kt ) and (v(Ki )) = i for v(Ki ) in the sequence M(K1 , . . . , Kt ). ¯
¯
Subcase I.b. d basic type 0, so d (Ia ) = (k)(fm ;K2 ,... ,Ka ) , where k ≤ m. We set f¯∞ (d ; K1 , . . . , Kt ) = E(2n − 1) + m p
Case II. k < ∞, k < k(d ), and k = t − p. k ,mk Subcase II.a. Kk = S2n+1 , with k > 1. We set $ f¯pk (d ; K1 , . . . , Kt ) = 2 · ( + 1) + 1.
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STEVE JACKSON
1,mk Subcase II.b. Kk = S2n+1 .
$
f¯pk (d ; K1 , . . . , Kt ) =
(mk + mk −1 () + mk ) + 1.
f¯pk (d ; K1 , . . . , Kt ) = E(2n, mk ) + f¯pk+1 (d ; K1 , . . . , Kt ). Subcase II.d. Kk is not of these forms. f¯pk (d ; K1 , . . . , Kt ) = f¯pk+1 (d ; K1 , . . . , Kt ). Subcase II.e. If k = 0, we set f¯pk (d ; K1 , . . . , Kt ) = f¯pk+1 (d ; K1 , . . . , Kt ). Case III. k < ∞, k < k(d ) and k = t − p. k ,mk Subcase III.a. Kk = S2n+1 , with k > 1. We set $ f¯pk (d ; K1 , . . . , Kt ) = <
Subcase III.b. Kk =
1,mk S2n+1 .
$
f¯pk (d ; K1 , . . . , Kt ) = <
Subcase III.c. Kk =
( + 1).
f¯k+1 (d ;K1 ,... ,Kt ) p
(mk + mk −1 () + mk ).
f¯k+1 (d ;K1 ,... ,Kt ) p
mk W2n+1 .
f¯pk (d ; K1 , . . . , Kt ) = E(2n, mk ) +
k+1 (d ;K ,... ,K ) f¯p t 1
.
Subcase III.d. Kk is not of these forms. f¯pk (d ; K1 , . . . , Kt ) =
k+1 (d ;K ,... ,K ) f¯p t 1
. ¯k+1 (d ;K ,... ,Kt ) 1
fp Subcase III.e. k = 0, we set f¯pk (d ; K1 , . . . , Kt ) =
.
Case IV. k < ∞, k = k(d ), and k = t − p. k ,mk Subcase IV.a. Kk = S2n+1 , with k > 1. Hence d (Ia ) = (k; d2(Ia +1) )s(Ia ) , where s may or may not appear. We set $ f¯pk (d ; K1 , . . . , Kt ) = 2 · ( + 1) + 2 · f¯pk+1 (d2 ; K1 , . . . , Kt )
+ (1 if s appears, and 2 if s does not appear). Here dˆ is as in condition A.
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461
1,mk Subcase IV.b. Kk = S2n+1 . Here d (Ia ) = (k; d1(Ia ) , . . . , dr(Ia ) )s(Ia ) , where r ≤ mk and s may or may not appear.
Subcase IV.b.i. r = mk . $
f¯pk (d ; K1 , . . . , Kt ) =
(mk + mk −1 () + mk )
+ ((mk − 1) + mk −2 (f¯pk+1 (d1 ; K1 , . . . , Kt )) + (mk − 1)) · f¯pk+1 (d2 ; K1 , . . . , Kt ) + ··· + (2 + 1 (f¯pk+1 (d1 ; K1 , . . . , Kt )) + 2) · f¯pk+1 (dr−1 ; K1 , . . . , Kt ) + 1 (f¯k+1 (dr ; K1 , . . . , Kt )) + (1 or 2) p
depending on whether s appears or not. Subcase IV.b.ii. r < mk and s appears. $
f¯pk (d ; K1 , . . . , Kt ) =
(mk + mk −1 () + mk )
+ ((mk − 1) + mk −2 (f¯pk+1 (d1 ; K1 , . . . , Kt )) + (mk − 1)) · f¯pk+1 (d2 ; K1 , . . . , Kt ) + ··· + ((mk − r + 1) + mk −r (f¯pk+1 (d1 ; K1 , . . . , Kt )) + (mk − r + 1)) · f¯pk+1 (dr ; K1 , . . . , Kt ) + 1 Subcase IV.b.iii. r < mk and s does not appear. $
f¯pk (d ; K1 , . . . , Kt ) =
(mk + mk −1 () + mk )
+ ((mk − 1) + mk −2 (f¯pk+1 (d1 ; K1 , . . . , Kt )) + (mk − 1)) · f¯pk+1 (d2 ; K1 , . . . , Kt ) + · · · + ((mk − r + 1) + m −r (f¯k+1 (d1 ; K1 , . . . , Kt )) k
p
+ (mk − r + 1)) · f¯pk+1 (dr ; K1 , . . . , Kt ) + 1 + mk −r (f¯pk+1 (d1 ; K1 , . . . , Kt )) + (mk − r + 1) mk Subcase IV.c. Kk = W2n+1
Subcase IV.c.i. d = (k; ). We set f¯pk (d ; K1 , . . . , Kt ) = E(2n, m2 ).
462
STEVE JACKSON
Subcase IV.c.ii. d = (k; d¯2 )s , where s may or may not appear. If n > 0 we set f¯pk (d ; K1 , . . . , Kt ) = E(2n − 1) + 2 · (g¯q (d¯2 ; K˜ 1 , . . . , K˜ w ) + (1 or 2) depending on whether s appears or not. Here K˜ 1 , . . . , K˜ w enumerates the measures in the sequence Kb(k) , . . . , Kbu concatenated with the sequence M(K1 , . . . , Kk−1 ). Recall that Kb(k) , . . . , Kbu enumerates the subsequence of Kk+1 , . . . , Kt consisting of the measures in m 1. Also, in this formula q is the appropriate integer such S2n+1 that adding a measure to the K1 , . . . , Kt sequences after Kt−p corresponds to adding a measure to the K˜ 1 , . . . , K˜ w sequence after K˜ w−q . More precisely, extending our previous notation slightly, let the function be such that: for Kj in the sequence K¯ b(k) , . . . , K¯ bu we have (Kj ) = j and for v(Kj ) in the sequence M(K1 , . . . , Kk−1 ) we have (v(Kj )) = j. Then q is the least integer greater than the length of M(K1 , . . . , Kk−1 ) such that (K˜ w−q ) ≤ t − p. If no such q exists then we set q = w. If n = 0, so d = (k; r) for r ≤ mk , we set f¯pk (d ; K1 , . . . , Kt ) = r. Case V. k < ∞, k = k(d ), and k = t − p. k ,mk , with k > 1. Say d = (k; d2 )s , where s may or Subcase V.a. Kk = S2n+1 may not appear. We set k+1 (d ;K ,... ,K ) $ f¯p t 2 1 ( + 1) + + (1 or 2) f¯pk (d ; K1 , . . . , Kt ) = <
k+1 (dˆ;K ,... ,K ) f¯p t 1
depending on whether s appears or not. Here dˆ is as in condition A. 1,mk Subcase V.b. Kk = S2n+1 . So, d (Ia ) = (k; d1(Ia ) , . . . , dr(Ia ) )s(Ia ) , where r ≤ mk and s may or may not appear.
Subcase V.b.i. r = mk . We set $ f¯pk (d ; K1 , . . . , Kt ) = <
(mk + mk −1 () + mk ) + (mk − 1)
f¯k+1 (d1 ;K1 ,... ,Kt ) p
+ ((mk − 1) + mk −2 (
k+1 (d ;K ,... ,K ) f¯p t 1 1
) + (mk − 1)) ·
+ ··· + (2 + 1 ( + 1 (
k+1 (d ;K ,... ,K ) f¯p t 1 1
k+1 (d ;K ,... ,K ) f¯p r 1 t
) + 2) ·
) + (1 or 2)
depending on whether s appears or not.
k+1 (d ;K ,... ,K ) f¯p t 2 1
k+1 (d ;K ,... ,K ) f¯p t 2 1
463
AD AND THE PROJECTIVE ORDINALS
Subcase V.b.ii. r < mk and s appears. Similarly to (i) above, f¯pk (d ; K1 , . . . , Kt ) is obtained from the formula in IV.b.ii by replacing terms of the form k+1 (··· ) f¯p . f¯k+1 (· · · ) by p
Subcase V.b.iii. r < mk and s does not appear. As above, using now the formula from IV.b.iii. mk Subcase V.c. Kk = W2n+1
Subcase V.c.i. d = (k; ). We set f¯pk = E(2n, mk ). Subcase V.c.ii. d = (k; d¯2 )s , where s may or may not appear. Here we use the same formula f¯pk (d ; K1 , . . . , Kt ) = E(2n − 1) + 2 · (g¯q (d¯2 ; K˜ 1 , . . . , K˜ w ) + (1 or 2) from IV.c.ii (as there, we use 1 if s appears and 2 if s does not). The measures K˜ 1 , . . . , K˜ w are as in IV.c.ii, and q is also as defined there. We now proceed to establish Akp , Bpk , and Rpk for d ∈ D2n+1 . We assume inductively these statements for d ∈ D2n−1 and the corresponding functions g¯pk . We establish these simultaneously by reverse induction on k. We first consider Akp . We consider the following cases. Case I. k = ∞ Subcase I.a. d basic of type 1. With notation as in I.a on p. 459 Akp follows from A1q for d¯2 ∈ D2n−1 , with q as in I.a on p. 459. Subcase I.b. d basic of type 0. The result is immediate. Case II. k < ∞, k < k(d ), and k = t − p. The result follows by induction, Ak+1 and the formulas for f¯pk . p Case III. k < ∞, k < k(d ), and k = t − p. We require the following easy lemma. ' Lemma 7.6. If α = for some ∈ Ord, then <α n = α for all n. By induction and Ak+1 p , we have f¯pk+1 (d ; K1 , . . . , Kt ) = f¯pk+2 (d ; K1 , . . . , Kt−p , Kt+1 , . . . , Kt ). k+1 (d ;K ,... ,K ) f¯p t 1
k+2 (d ;K ,... ,K f¯p t−p ,Kt+1 ,... ,Kt ) 1
= Hence k ¯ formulas for fp , it follows that
f¯pk (d ; K1 , . . . , Kt ) =
. From the lemma and the
k+1 (d ;K ,... ,K ) f¯p t 1
(or with a +1 added) and likewise f¯pk+1 (d ; K1 , . . . , Kt−p , Kt+1 , . . . , Kt ) =
k+2 (d ;K ,... ,K f¯p t−p ,Kt+1 ,... ,Kt ) 1
,
464
STEVE JACKSON
hence f¯pk (d ; K1 , . . . , Kt−p , Kk+1 , . . . , Kt ) = f¯pk (d ; K1 , . . . , Kt ). Case IV. k < ∞, k = k(d ), and k = t − p. The result follows by induction mk 1 0 ¯ and Ak+1 p , and in case 3 (Kk = W2n+1 ) by Aq or Aq for d2 ∈ D2n−1 . Case V. k < ∞, k = k(d ), and k = t − p. Akp then follows from the formulas for f¯pk , Ak+1 p , and Lemma 7.6. For k ¯ example, in case V.b.i of the definition of fp we have $
f¯pk (d ; K1 , . . . , Kt ) = <
+ mk −2 ( =
f¯k+1 (d1 ;K1 ,... ,Kt ) p
f¯k+1 (d1 ;K1 ,... ,Kt ) p
+ · · · + 1 (
(mk + mk −1 () + mk )
f¯k+1 (d1 ;K1 ,... ,Kt ) p
) ·
+ mk −2 (
k+1 (d ;K ,...,K ) fp r 1 t
k+1 (d ,K ,... ,K ) fp t 2 1
f¯k+1 (d1 ;K1 ,... ,Kt ) p
+ ··· ) ·
k+1 (d ,K ,... ,K ) fp t 2 1
) + (1 or 2).
and f¯pk (d ; K1 , . . . , Kt−p , Kt+1 , . . . , Kt ) = $ (mk + mk −1 () + mk )
+ ((mk − 1) + mk −2 (f¯pk+1 (d1 ; K1 , . . . , Kt−p , Kt+1 , . . . , Kt )) + (mk − 1)) · fpk+1 (d2 ; K1 , . . . , Kt ) + · · · + 1 (fpk+1 (dr ; K1 , . . . , Kt−p , Kt+1 , . . . , Kt )) + (1 or 2). t+1 ,mt+1 where t+1 > 1. From III.a We further specialize to the case Kt+1 = S2n+1 we have
f¯pk+1 (d1 ; K1 , . . . , Kt−p , Kt+1 , . . . , Kt ) =
k+2 (d ;K ,... ,K f¯p t−p ,Kt+1 ,... ,Kt ) 1 1
,
and similarly for d2 , . . . , dr . Using Lemma 7.6 it follows that f¯pk (d ; K1 , . . . , Kt ) = f¯pk (d ; K1 , . . . , Kt−p , Kt+1 , . . . , Kt ). In case V.3, we again use A0q on D2n−1 . This establishes Akp in all cases. We now consider Bpk . We again consider the same cases. Case I. k = ∞. Subcase I.a. d (Ia ) basic of type 1, so d = (d¯2 )s , where s may or may not appear, and Ia = (f(K¯ 1 ); K¯ 2 , . . . , K¯ a ).
465
AD AND THE PROJECTIVE ORDINALS
We let d be a re-indexing of d as in the statement of Bpk , so d a = ¯ ¯ ¯ (d¯2 )(f(K1 );K2 ,... ,Ka ) , where d¯2 = d¯2 , and K¯ 2 , . . . , K¯ a is a subsequence of K¯ 2 , . . . , K¯ a . The formula for f¯pk (d ), however, involves g¯q (d¯2 ) computed relative to the full sequence M(K1 , . . . , Kt ) and likewise for fpk+1 (d ). Hence f¯pk (d ; K1 , . . . , Kt ) = f¯pk (d ; K1 , . . . , Kt ). (I )
Subcase I.b. d basic of type 0. The result is immediate. Case II. k < ∞, k < k(d ), and k = t − p. and Case III. k < ∞, k < k(d ), and k = t − p. The result follows immediately from induction and Bpk+1 . Case IV. k < ∞, k = k(d ), and k = t − p and Case V. k < ∞, k = k(d ), and k = t − p. The result is also immediate from induction and Bpk+1 , where in case c.ii, we use the fact that M(K1 , . . . , Kk−1 ) is the same for both d and d . This establishes Bpk . We now consider Rpk . We let d1(Ia ) , d2(Ia ) satisfying conditions C and A relative to K1 , . . . , Kt and assume that d1 < d2 . We must establish that f¯k+b (d1 ; K1 , . . . , Kt−p , Kk+1 , . . . , Kt ) < f¯k (d2 ; K1 , . . . , Kt ), p
p
where b = 0 if k ≤ t − p and b = 1 if k > t − p or k = 0. We recall that d1 (K¯ a ) = d2 (K¯ a ) < k ≤ k(d1 ) or k(d2 ). From Akp we have that f¯pk (d2 ; K1 , . . . , Kt ) = f¯pk+b (d2 ; K1 , . . . , Kt−p , Kt+1 , . . . , Kt ). Note that for k = 0, this is part of the statement of Akp when t = p, and if p < t then fp0 (d2 ; K1 , . . . , Kt ) = fp1 (d2 ; K1 , . . . , Kt ) (by inspection of the formulas) and by A1p this is equal to fp1 (d2 ; K1 , . . . , Kt−p , Kt+1 , . . . , Kt ). So, in all cases it suffices to establish that f¯pk+b (d1 ; K1 , . . . , Kt−p , Kk+1 , . . . , Kt ) < f¯pk+b (d2 ; K1 , . . . , Kt−p , Kk+1 , . . . , Kt ). Changing notation, it suffices to show that if d1 < d2 both satisfy conditions C and A with respect to K1 , . . . , Kt then f¯k (d1 ; K1 , . . . , Kt ) < f¯k (d2 ; K1 , . . . , Kt ). p
We consider the following cases. Case I. k(d1 ) > k(d2 ). Subcase I.a. k < k(d2 ).
p
466
STEVE JACKSON
Subcase I.a.i. k = t − p. By induction, f¯pk+1 (d1 ; K1 , . . . , Kt ) < f¯pk+1 (d2 ; K1 , . . . , Kt ). From the formulas for f¯pk (specifically, those of case II), it follows that f¯pk (d1 ; K1 , . . . , Kt ) < f¯pk (d2 ; K1 , . . . , Kt ). Note that all the terms in the sum in cases II.a and II.b are positive. Subcase I.a.ii. k = t − p. By induction, f¯pk+1 (d1 ; K1 , . . . , Kt ) < f¯pk+1 (d2 ; K1 , . . . , Kt ). We have from the formulas for f¯k that f¯k (d2 ; K1 , . . . , Kt ) = p
k+1 (d ;K ,...,K ) f¯p t 2 1
, and immediately.
f¯pk (d1 ; K1 , . . . , Kt )
=
p
k+1 (d ;K ,...,K ) f¯p t 1 1
. The result follows
k ,mk where k > 1. Hence, d1 ≤ dˆ2 Subcase I.b. k = k(d2 ) and Kk = S2n+1 where dˆ2 is as in condition A. Subcase I.b.i. k = t − p. By induction, f¯pk+1 (d1 ; K1 , . . . , Kt ) ≤ f¯pk+1 (dˆ1 ; K1 , . . . , Kt ). From formulas II.a and IV.a for f¯pk , it follows that f¯pk (d1 ; K1 , . . . , Kt ) < f¯k (d2 ; K1 , . . . , Kt ), since if d2 = (k; (d2 )2 )s then f¯k+1 ((d2 )2 ;
K1 , . . . , Kt ) ≥ 1.
p
p
Subcase I.b.ii. k = t − p. By induction, f¯pk+1 (d1 ; K1 , . . . , Kt ) < f¯pk+1 (dˆ2 ; K1 , . . . , Kt ). From formula V.a for f¯pk and Lemma 7.6 we have that f¯pk (d2 ; K1 , . . . , Kt ) =
k+1 (dˆ ;K ,...,K ) f¯p t 2 1
+
k+1 ((d ) ;K ,...,K ) f¯p t 2 2 1
+ (1 or 2),
where d2 = (k; (d2 )2 )s (where s may or may not appear, and we use 1 in the formula if s appears and 2 if it does not). From formula III.a we also have f¯pk (d1 ; K1 , . . . , Kt ) =
k+1 (d ;K ,...,K ) f¯p t 1 1
k+1 (dˆ ;K ,...,K ) f¯p t 2 1
≤ < f¯k (d2 ; K1 , . . . , Kt ). p
1,mk S2n+1 .
Subcase I.c. k = k(d2 ) and Kk = Hence, d1 ≤ (d2 )1 where we use the notation d2 = (k; (d2 )1 , (d2 )2 , . . . , (d2 )r )s (where s may or may not appear). Subcase I.c.i. k = t −p. By induction, f¯pk+1 (d1 ; K1 , . . . , Kt ) ≤ f¯pk+1 ((d2 )1 ; K1 , . . . , Kt ). From formulas II.b and IV.b.i-iii for f¯pk , it follows that f¯pk (d1 ; K1 , . . . , Kt ) < f¯pk (d2 ; K1 , . . . , Kt ). Note here that the formulas from IV.b start off with the same sum as the formula for II.b, and then contain extra terms which are strictly positive. Subcase I.c.ii. k = t−p. By induction, f¯pk+1 (d1 ; K1 , . . . , Kt ) < f¯pk+1 ((d2 )1 ; K1 , . . . , Kt ). From formulas V.b.i-iii for f¯k we have that p
f¯pk (d2 ; K1 , . . . , Kt ) =
k+1 ((d ) ;K ,...,K ) f¯p t 2 1 1
+ α,
467
AD AND THE PROJECTIVE ORDINALS
where α ≥ 1.
f¯k+1 (d1 ;K1 ,...,Kt ) p
Also, from formula III.b we have f¯pk (d1 ; K1 , . . . , Kt ) =
, and the result follows.
mk Subcase I.d. k = k(d2 ) and Kk = W2n+1 . This case can not arise from the definition of <.
Case II. k(d1 ) < k(d2 ). Subcase II.a. k < k(d1 ). We proceed as in I.a.i or I.a.ii depending on whether k = t − p or k = t − p. k ,mk with k > 1. Hence, dˆ1 < d2 Subcase II.b. k = k(d1 ) and Kk = S2n+1 ˆ (where again d1 is as in condition A). Subcase II.b.i. k = t − p. By induction, f¯pk+1 (dˆ1 ; K1 , . . . , Kt ) < f¯pk+1 (d2 ; K1 , . . . , Kt ). From formula IV.a for f¯pk we have that $
f¯pk (d1 ; K1 , . . . , Kt ) =
2 · ( + 1)
+ 2 · (f¯pk+1 ((d1 )2 ; K1 , . . . , Kt )) + (1 or 2), where d1 = (k; (d1 )2 )s and s may or may not appear. From formula II.a we have ⎡ f¯pk (d2 ; K1 , . . . , Kt ) = ⎣
$
⎤ 2 · ( + 1)⎦ + 1.
Note here that if d1 = d1(Ia ) has index (Ia ), then dˆ1 also has index (Ia ) while (d1 )2 has index (Ia+1 ) = (Ia , v(Kk )). By Bpk+1 we have that f¯pk+1 (dˆ1 ; K1 , . . . , Kt ) = f¯pk+1 (dˆ1(Ia+1 ) ; K1 , . . . , Kt ) where dˆ1(Ia+1 ) is obtained from dˆ1(Ia ) by re-indexing as in B. We also have that (I ) (d1 ) a+1 ≤ dˆ(Ia+1 ) and hence by induction, 2
1
(I ) f¯pk+1 (dˆ1(Ia+1 ) ; K1 , . . . , Kt ) ≥ f¯pk+1 ((d1 )2 a+1 ; K1 , . . . , Kt ). (I ) Hence f¯pk+1 (dˆ1 ; K1 , . . . , Kt ) ≥ f¯pk+1 ((d1 )2 a+1 ; K1 , . . . , Kt ). In fact, we we have strict inequality here unless s appears in d1 (i.e., d1 = (k; (d1 )2 )s ) So we
468
STEVE JACKSON
have f¯pk (d1 ; K1 , . . . , Kt ) ≤
$
2 · ( + 1)
≤f¯pk+1 (d˜1 ;K1 ,...,Kt )
≤
$
2 · ( + 1)
<
$
2 · ( + 1) + 1
= f¯pk (d2 ; K1 , . . . , Kt ) Subcase II.b.ii. k = t − p. By induction, f¯pk+1 (dˆ1 ; K1 , . . . , Kt ) < f¯pk+1 (d2 ; K1 , . . . , Kt ). From formula V.a for f¯pk we have that $ f¯pk (d1 ; K1 , . . . , Kt ) = 2 · ( + 1) <
+ =
k+1 (dˆ ;K ,...,K ) f¯p t 1 1
k+1 ((d ) ;K ,...,K ) f¯p t 1 2 1
f¯k+1 (dˆ1 ;K1 ,...,Kt ) p
+ (1 or 2)
+
k+1 ((d ) ;K ,...,K ) f¯p t 1 2 1
+ (1 or 2).
¯k+1 (d ;K ,...,Kt ) 2 1
fp We also have from III.a that f¯pk (d2 ; K1 , . . . , Kt ) = follows from Bpk+1 , as in the previous case, that
. It also
(I ) f¯pk+1 ((d1 )2 a+1 ; K1 , . . . , Kt ) ≤ f¯pk+1 (dˆ1 ; K1 , . . . , Kt ).
Hence f¯pk (d1 ; K1 , . . . , Kt ) ≤ <
k+1 (dˆ ;K ,...,K ) f¯p t 1 1
f¯k+1 (d2 ;K1 ,...,Kt ) p
· 2 + (1 or 2) = f¯pk (d2 ; K1 , . . . , Kt ).
1,mk Subcase II.c. k = k(d1 ) and Kk = S2n+1 . Hence, if d1 = (k; (d1 )1 , (d1 )2 , . . . , (d1 )r )s (where s may or may not appear) then we have (d1 )1 < d2 . Subcase II.c.i. k = t − p. By induction, f¯pk+1 ((d1 )1 ; K1 , . . . , Kt ) < f¯pk+1 (d2 ; K1 , . . . , Kt ). From formulas IV.b.i–iii for f¯pk we have that $ f¯pk (d1 ; K1 , . . . , Kt ) = (mk + mk −1 () + mk ) + α,
where α ≤ mk −1 (f¯pk+1 ((d1 )1 ; K1 , . . . , Kt )) + mk . This follows from the formulas for f¯pk , the fact that f¯pk+1 ((d1 )i ; K1 , . . . , Kt ) < f¯pk+1 ((d1 )1 ; K1 , . . . , Kt )
AD AND THE PROJECTIVE ORDINALS
469
for all i > 1, and Lemma 7.5 which gives that ((mk − 1) + mk −2 () + (mk − 1)) · + ((mk − 2) + mk −3 () + (mk − 2)) · + · · · + ((mk − r + 1) + mk −r () + (mk − r + 1)) · + mk −r () + (mk − r + 1) ≤ mk −1 () + (mk − r + 1) ≤ mk −1 () + mk for all (where we use here = f¯pk+1 ((d1 )1 ; K1 , . . . , Kt )). We also have that $ f¯pk (d2 ; K1 , . . . , Kt ) = (mk + mk −1 () + mk ) + 1,
and hence f¯pk (d1 ; K1 , . . . , Kt ) < f¯pk (d2 ; K1 , . . . , Kt ). Subcase II.c.ii. k = t − p. As in the previous case we have by induction that f¯pk+1 ((d1 )1 ; K1 , . . . , Kt ) < f¯pk+1 (d2 ; K1 , . . . , Kt ). From formulas V.b.i–iii and Lemma 7.6 we have that $ f¯pk (d1 ; K1 , . . . , Kt ) = (mk + mk −1 () + mk ) + α <
=
k+1 ((d ) ;K ,...,K ) f¯p t 1 1 1
k+1 ((d ) ;K ,...,K ) f¯p t 1 1 1
+α
where as in the previous case we have α ≤ mk −2 ( + mk −3 (
k+1 ((d ) ;K ,...,K ) f¯p t 1 1 1
k+1 ((d ) ;K ,...,K ) f¯p t 1 1 1
+ · · · + 1 ( ≤ mk −1 (
) ·
) ·
k+1 ((d ) ;K ,...,K ) f¯p t 1 1 1
f¯k+1 ((d1 )1 ;K1 ,...,Kt ) p
k+1 ((d ) ;K ,...,K ) f¯p t 1 1 1
k+1 ((d ) ;K ,...,K ) f¯p t 1 1 1
) + mk
) + mk .
We also have from formula III.b and Lemma 7.6 that $ (mk + mk −1 () + mk ) f¯pk (d2 ; K1 , . . . , Kt ) = <
=
k+1 (d ;K ,...,K ) f¯p t 2 1
k+1 (d ;K ,...,K ) f¯p t 2 1
.
From Lemma 7.6 we have that mk −1 (
k+1 ((d ) ;K ,...,K ) f¯p t 1 1 1
) + mk <
k+1 (d ;K ,...,K ) f¯p t 2 1
470
STEVE JACKSON
and it then follows from the above equations that f¯pk (d1 ; K1 , . . . , Kt ) < f¯pk (d2 ; K1 , . . . , Kt ). mk Subcase II.d. k = k(d1 ) and Kk = W2n+1 .
Subcase II.d.i. k = t − p. From formula IV.c for f¯pk and R1 on D2n−1 we have that f¯pk (d1 ; K1 , . . . , Kt ) ≤ E(2n, mk ). Also, from formula II.c we have f¯pk (d2 ; K1 , . . . , Kt ) = E(2n, mk ) + α, where α ≥ 1 (in fact α ≥ E(2n − 1)), and hence the result follows. Subcase II.d.ii. k = t − p. The result follows as in the previous case. Case III. k(d1 ) = k(d2 ). Subcase III.a. k < k(d1 ) = k(d2 ). Subcase III.a.i. k = t − p and Subcase III.a.ii. k = t − p. We proceed as in I.a.i and I.a.ii. k ,mk with k > 1. Hence Subcase III.b. k = k(d1 ) = k(d2 ) and Kk = S2n+1 (Ia+i ) s (Ia+i ) s ¯ ¯ d1 = (k; d2,1 ) , d2 = (k; d2,2 ) , where s may or may not appear in d1 , d2 .
Subcase III.b.i. d¯2,1 < d¯2,2 . Subcase III.b.i.1. k = t − p. We have by induction that f¯pk+1 (d2,1 ; K1 , . . . , Kt ) < f¯pk+1 (d2,2 ; K1 , . . . , Kt ). Since d2,1 < d2,2 , it follows that dˆ1 ≤ dˆ2 , and hence by induction, f¯pk+1 (dˆ1 ; K1 , . . . , Kt ) < f¯pk+1 (dˆ2 ; K1 , . . . , Kt ). From formula IV.a it then follows that f¯pk (d1 ; K1 , . . . , Kt ) < f¯pk (d2 ; K1 , . . . , Kt ). Note here that if we exclude the final term of (1 or 2) from the formulas for these ordinals, then the right-hand side is at least 2 greater than the left-hand side so adding this last term maintains the inequality. Subcase III.b.i.2. k = t − p. Similar to the above case. Subcase III.b.ii. d2,1 = d2,2 and the symbol s appears in d1 and not in d2 . In this case Rpk is immediate from the formulas for f¯pk . 1,mk . Here d1 = (k; d1,1 , d2,1 , Subcase III.c. k = k(d1 ) = k(d2 ) and Kk = S2n+1 s s . . . , dr1 ,1 ) and d2 = (k; d1,2 , d2,2 , . . . , dr2 ,2 ) , where s may or may not appear in d1 , d2 .
Subcase III.c.i. d1,1 < d1,2 . Subcase III.c.i.1. k = t − p. By induction, f¯k+1 (d1,1 ; K1 , . . . , Kt ) < f¯k+1 (d1,2 ; K1 , . . . , Kt ). p
p
From the formulas of IV.b for f¯pk we have $ f¯pk (d2 ; K1 , . . . , Kt ) >
(mk + mk −1 () + mk )
471
AD AND THE PROJECTIVE ORDINALS
and
⎡
$
f¯pk (d1 ; K1 , . . . , Kt ) = ⎣
⎤ (mk + mk −1 () + mk )⎦ + α
where α ≤ mk −1 (f¯pk+1 (d1,1 ; K1 , . . . , Kt )) + mk . Hence it follows that f¯pk (d1 ; K1 , . . . , Kt ) < f¯pk (d2 ; K1 , . . . , Kt ). Subcase III.c.i.2. k = t − p. Again, by induction f¯pk+1 (d1,1 ; K1 , . . . , Kt ) < f¯pk (d1,2 ; K1 , . . . , Kt ). ¯k+1 (d
fp From the formulas of V.b we have f¯pk (d2 ; K1 , . . . , Kt ) > ¯k+1 (d
fp f¯pk (d1 ; K1 , . . . , Kt ) ≤ ( than f¯pk (d2 ; K1 , . . . , Kt ).
1,1 ;K1 ,...,Kt ) ·m)
1,2 ;K1 ,... ,Kt )
and
for some m, which is therefore less
Subcase III.c.ii. There is an r, with 2 ≤ r ≤ min{r1 , r2 }, such that di,1 = di,2 , for 1 ≤ i < r, and dr,1 < dr,2 . Subcase III.c.ii.1. k = t − p. By induction, f¯pk+1 (dr,1 ; K1 , . . . , Kt ) < f¯k+1 (dr,2 ; K1 , . . . , Kt ). For i < r, let p
i = f¯pk+1 (di,1 ; K1 , . . . , Kt ) = f¯pk+1 (di,2 ; K1 , . . . , Kt ). From the formulas IV.b for f¯pk we have that (note that the smallest of the values occurs in case IV.b.i or IV.b.ii depending on whether r = mk ) $ (mk + mk −1 () + mk ) + · · · f¯pk (d2 ; K1 , . . . , Kt ) ≥ <1
+ ((mk − r + 2) + mk −r+1 (1 ) + (mk − r + 2)) · r−1 + ((mk − r + 1) + mk −r (1 ) + (mk − r + 1)) · f¯pk+1 (dr,2 ; K1 , . . . , Kt ) + 1. We also have that (note that the largest of the values occurs in case IV.b.i or IV.b.iii) $ f¯pk (d1 ; K1 , . . . , Kt ) ≤ (mk + mk −1 () + mk ) + · · · <1
+ ((mk − r + 2) + mk −r+1 (1 ) + (mk − r + 2)) · r−1 + ((mk − r + 1) + m −r (1 ) + (mk − r + 1)) · f¯k+1 (dr,1 ; K1 , . . . , Kt ) k
+α
p
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STEVE JACKSON
where α ≤ mk −r (1 ) + (mk − r + 1). Hence ((mk − r + 1) + mk −r (1 ) + (mk − r + 1)) · f¯pk+1 (dr,1 ; K1 , . . . , Kt ) + α ≤ ((mk − r + 1) + m −r (1 ) + (mk − r + 1)) · (f¯k+1 (dr,1 ; K1 , . . . , Kt ) + 1) p
k
≤ ((mk − r + 1) + mk −r (1 ) + (mk − r + 1)) · (f¯pk+1 (dr,2 ; K1 , . . . , Kt ) and it follows that f¯pk (d1 ; K1 , . . . , Kt ) < f¯pk (d2 ; K1 , . . . , Kt ). Subcase III.c.ii.2. k = t − p. Similar to the above case. Subcase III.c.iii. r1 < r2 and di,1 = di,2 for 1 ≤ i ≤ r1 . From the definition of the ordering < on the descriptions we must have that s appears in d1 (and may or may not appear in d2 ). Subcase III.c.iii.1. k = t − p. We may in fact also assume that s appears in d2 as the formula in IV.b.iii results in a strictly larger ordinal than that of formula IV.b.ii. From formula IV.b.ii we see that f¯pk (d1 ; K1 , . . . , Kt ) is of the form ϑ + 1 while f¯pk (d2 ; K1 , . . . , Kt ) is of the form ϑ + α + 1 with α ≥ 1, and the result follows. Subcase III.c.iii.2. k = t − p. Similar to the above case. Subcase III.c.iv. r1 > r2 and di,1 = di,2 for 1 ≤ i ≤ r2 . We must have in this case that s does not appear in d2 . Subcase III.c.iv.1. k = t − p. For i ≤ r2 we again let i = f¯pk+1 (di,1 ; K1 , . . . , Kt ) = f¯pk+1 (di,2 ; K1 , . . . , Kt ). We may assume that s does not appear in d1 , as this results in the larger value for f¯pk+1 (d1 ; K1 , . . . , Kt ). From formula IV.b.iii (and IV.b.i if r1 = mk ) we have $ (mk + mk −1 () + mk ) + · · · f¯pk (d1 ; K1 , . . . , Kt ) = <1
+ ((mk − r2 + 1) + mk −r2 (1 )) + (mk − r2 + 1)) · r2 +α where from Lemma 7.5 α = ((mk − r2 ) + mk −r2 −1 (1 ) + (mk − r2 )) · f¯pk+1 (dr2 +1,1 ; K1 , . . . , Kt ) + ··· + ((mk − r1 + 1) + mk −r1 (1 ) + (mk − r1 + 1)) · (f¯pk+1 (dr1 ,1 ; K1 , . . . , Kt ) + 1) ≤ mk −r2 (1 ) + (mk − r1 + 1) < mk −r2 (1 ) + (mk − r2 + 1).
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473
We also have that $ f¯pk (d2 ; K1 , . . . , Kt ) = (mk + mk −1 () + mk ) + · · · <1
+ ((mk − r2 + 1) + mk −r2 (1 ) + (mk − r2 + 1)) · (r2 + 1). Thus, f¯pk (d1 ; K1 , . . . , Kt ) < f¯pk (d2 ; K1 , . . . , Kt ). Subcase III.c.iv.2. k = t − p. Similar to the above case. Subcase III.c.v. r1 = r2 and di,1 = di,2 for 1 ≤ i ≤ r1 . We must have that s appears in d1 and not in d2 . Rpk then follows immediately from the formulas for f¯pk . mk Subcase III.d. k = k(d1 ) = k(d2 ) and Kk = W2n+1 .
Subcase III.d.i. d1 = (k; d¯2,1 )s , where s may or may not appear, and d2 = (k; ). Then f¯pk (d2 ; K1 , . . . , Kt ) = E(2n; mk ). Also, by Rq and induction, g¯q (d¯2,1 ; Kb(k) , . . . , Kb(u) , M(K1 , . . . , Kk−1 )) < E(2n; mk ), where q, etc. refer to the sequence K1 , . . . , Kt−p , Kk+1 , . . . , Kt . Rpk then follows. Subcase III.d.ii. d1 = (k; d¯2,1 )s and d2 = (k; d¯2,2 )s , where d¯2,1 < d¯2,2 , and s may or may not appear in d1 , d2 . Let q and K¯ 1 , . . . , K¯ w = Kb(k) , . . . , Kb(u) M(K1 , . . . , Kk−1 ) be as in IV.c.ii of the definition of f¯pk corresponding to K1 , . . . , Kt . Note that q and K¯ 1 , . . . , K¯ w only depend on K1 , . . . , Kt , p, and k, and so are the same for both d1 and d2 . We then have that f¯pk (d1 ; K1 , . . . , Kt ) = E(2n − 1) + 2 · g¯q (d2,1 ; K¯ 1 , . . . , K¯ w ) + (1 or 2) and similarly for d2 . By induction, g¯q (d2,1 ; K¯ 1 , . . . , K¯ w ) < g¯q (d2,2 ; K¯ 1 , . . . , K¯ w ) and it follows that f¯pk (d1 ; K1 , . . . , Kt ) < f¯pk (d2 ; K1 , . . . , Kt ). Subcase III.d.iii. As in (ii) immediately above, where d¯2,1 = d¯2,2 . We must have that s appears in d1 and not in d2 . This case is immediate from the formulas for f¯pk (since the last term of the expression for f¯pk (d1 ; K1 , . . . , Kt ) is +1 and for f¯pk (d2 ; K1 , . . . , Kt ) is +2). Rpk has now been established in all cases. Rp follows now easily from reverse induction on k(d ), Rpk , the formulas of f¯pk , and the fact that if α < E(2n + 2, mk ) then α n < E(2n + 2, mk ) for all n. In particular it now follows from Rpk that for any (d (Ia ) )s , where s may not appear, and Ia = (f(K¯ 1 ; ), and d is defined and satisfies conditions C and A relative to K1 , . . . , Kt , that f((d )s ; K1 , . . . , Kt ) < E(2n + 3).
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§8. The upper bound for 12n+5 . We are now in a position to collect the and obtain the upper bound for 1 . We results of the previous sections 2n+5 recall we are assuming I2n+1 and K2n+3 . Theorem 8.1. 12n+5 ≤ ℵE(2n+3)+1 Proof. We recall (cf. [Kec81A]) that 12n+5 = [sup j( 12n+3 )]+ , where the the ultrapowers by the measures supremum ranges over embeddings j from in m
let F : 12n+3 → 12n+3 be given representing [F ] w.r.t. S2n+1 . From the week relationon 1 , there is a g : 1 1 partition 2n+3 2n+3 → 2n+3 such that for almost all ,m f : ϑ(S2n+1 ) → 12n+3 we have F ([f]) < g(supa.e. f). By the argument of the main theorem ofsection 6 (Theorem 6.4), it follows that F represents an ordinal [F ] ≤ [supK1 (id; f; d ; K1 )]+ , where d = () is the distinguished description. Repeating the argument gives (id; f; d ; K1 ) ≤ [supK2 (id; f; (d˜)s ; K1 , K2 )]+ , where (d˜)Ia is the maximal tuple relative to K1 , K2 (where Ia = (f(K¯ 1 ); ) and ,m K¯ 1 = v(S2n+1 )). From Theorem 6.4 and Rp (where p = 0), it follows that F represents an ordinal less than ℵE(2n+2,m) , and the result follows. §9. A Lower Bound for fp . Our goal in this section is to obtain a lower bound for a certain rank function. We first define some auxiliary measures. For each regular cardinal κ < 12n+1 , we let Mκ be defined using the strong partition relation on 12n+1 , on functions F : 12n+1 → 12n+1 of the correct type, and the normal measure 1 2n+1 concentrating in the points of cofinality κ. We let N = Mκ1 × · · · × Mκp , where κ1 , . . . , κp enumerates the regular cardinals less than 12n+1 . ,m We next modify slightly the measures S2n+1 as follows. For = 2 we ,m ,m 2,m let S˜2n+1 = S2n+1 . For = 2 we define S˜2n+1 as follows. Let k(m) = 1 + m + m(m − 1) + m(m − 1)(m − 2) + · · · + m! be the number of sequences = i1 , i2 , . . . , i , where 0 ≤ ≤ m, each 1 ≤ ij ≤ m, and the ij are all distinct. For such a fixed , we say a function f : 1m → 12n+1 is of type ( , . . . , ), if f(α1 , . . . , αm ) < f(1 , . . . , m ) whenever (αi1 , . . . , αi ) < lex i1 i if (αi , . . . , αi ) = (i , . . . , i ). It is easy to see (using and f(α) = f() 1 1 the weak partition relation on 1 ) that for any f : 1m → 12n+1 , there is a and a measure one set A w.r.t. W1m such that fA is of type. We define 2,m S˜2n+1 to be the measure on k(m) tuples of ordinals defined by: A has measure one if there is a c.u.b. C ⊆ 12n+1 such that for all F : 12n+1 → 12n+1 of the correct type (. . . , α , . . . )∈ A, where for a fixed ,α is represented
475
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with respect to V := 2n+1 = the measure on 12n+1 induced by the weak 1 partition relation on 12n+1 and functions f : 1m → 2n+1 of -type which are everywhere discontinuous and of uniform cofinality, by α ([f]) = F ([f]). /2n+1 −1 ,m m 1,m m We let B2n+1 = (S2n+1 × N × =2 S˜2n+1 ) . We modify some of the previous definitions as follows. mk We consider sequences of measures K1 , . . . , Kt , where each Kk = B2n+1 . We mk k k let K1 , . . . , Kck enumerate the measures in the product measure B2n+1 = Kk . We consider d ∈ D2n+1 to be defined relative to K1 , . . . , Kt if d is defined relative to K11 , . . . , Kc11 , . . . , K1t , . . . , Kctt . We let Kk denote the kth element of the sequence K11 , . . . , Kctt . We allow now D2m+1 to further contain descriptions of the form d (Ia ) = (k; d2(Ia ) ), where Kk = Mκ for some κ. We consider d to satisfy condition C relative to K1 , . . . , Kt if d satisfies condition C relative to K11 , . . . , Kctt , where we remove the restriction that 1,mk r > 1 if s appears for Kk = S2n+1 (so now d = (k; d1 )s will evaluate to the same ordinal as d1 ). Also, if Kk = Kκ we define (k; d2 ) to satisfy condition C, provided ∀∗ h 1 , . . . , h t f, h¯ 1 , . . . , h¯ a cf(H (d2 ; h 1 , . . . , h t , f, h¯ 1 , . . . , h¯ a )) = κ. ct
1
1
ct
2,m S˜2n+1 ,
s
If d = (k; d2 ) , where Kk = then condition condition C imposes no restriction if s appears, and if s does not appear, we define d to satisfy condition C if for almost all h11 , . . . , hctt , f, h¯ 1 , . . . , h¯ a , the function g : 1m → 12n+1 given by g([h¯ a+1 ]) = H (d2 ; h11 , . . . , hctt , f, h¯ 1 , . . . , h¯ a+1 ) is such that some measure one set A w.r.t. W m , gA is discontinuous of uniform for 1 cofinality . Conditions D and A are as before. We define modified operations LM , Lˆ M , Lˆ k M on descriptions defined and satisfying condition C relative to K1 , . . . , Kt . The definition of Lˆ k M proceeds as in the definition of Lˆ k (pp. 425– 428), with the following modifications: (cases here are numbered as in the definition of Lˆ k ; Cases I, II, and IV–VII are the same). ˆ k Case III. If s appears and d2 not minimal w.r.t. Lˆ k+1 M , then we set LM (d ) = k+1 (I ) (k; Lˆ M (d2 ) a+1 ) if this tuple satisfies condition C, and otherwise = (k; k+1 2,m Lˆ M (d2 )(Ia+1 ) )s . This case includes now the case Kk = S˜2n+1 . If d2 is k+1 k minimal with respect to Lˆ then d is minimal w.r.t. Lˆ . M
M
We also add the following cases to the definition of Lˆ k M (we continue the numbering of the cases from the definition of Lˆ k ): Case VIII. k < ∞, k = k(d ), and Kk = Mk . Hence, d (Ia ) = (k; d2(Ia ) ). ˆ k If d2 is minimal w.r.t. Lˆ k+1 M , then d is minimal w.r.t. LM . Otherwise, we set
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STEVE JACKSON
k k+1 Lˆ M (d ) = (k; Lˆ M (d2 )) if this description satisfies condition C, and otherwise k Lˆ (d ) = d2 . M
k+1 , Case IX. k < ∞, k < k(d ), and Kk = Mk . If d is minimal w.r.t. Lˆ M k k k+1 then d is minimal w.r.t. Lˆ M . Otherwise, we set Lˆ M (d ) = (k; Lˆ M (d )) if this k+1 k+1 description satisfies condition C and if not then Lˆ M (d ) = Lˆ M (d ). (Ia ) We set Lˆ M (d ) = Lˆ 1 with Ia = (f(K¯ 1 ); ). M (d ) for d If d satisfies condition C relative to K1 , . . . , Kt , it is easy to see that Lˆ M (d ) 2 also satisfies condition C. Also, property F2n+1 is still satisfied for d satisfying condition C. If d is defined and satisfies condition C relative to K1 , . . . , Kt and we are given g : 12n+3 → 12n+3 , then we define (id; f; (d )s ; K1 , . . . , Kt ) and ,m (g; f; (d )s ; K1 , . . . , Kt ) as before, where f : ϑ(S2n+1 ) → 12n+3 (here K¯ 1 = ,m v(S2n+1 )). If s does not appear here, then we require (d ) to satisfy condition D. From Lemma 5.21 of section 5, if follows that if s appears, then (d )s satisfies condition D, so these ordinals are well-defined. We will establish in part 2 that the ordinals (id; d ; K1 , . . . , Kt ), for d satisfying condition C, are all cardinals. We let <M denote the ordering on tuples (d ; K1 , . . . , Kt ), where d is defined and satisfies condition C relative to K1 , . . . , Kt , generated by the relation: (Lˆ M (d ; K1 , . . . , Kt ); K1 , . . . , Kt , Kt+1 ) <M (d ; K1 , . . . , Kt ) . for all Kt+1 We let f(d ; K1 , . . . , Kt ) denote the rank of (d ; K1 , . . . , Kt ) in the ordering <M . It then follows from the above remark (granting the unproven assertion above about the descriptions representing cardinals) that
12n+5 ≥ ℵsup f(d ;K ,... ,K ) + 1. t d,K ,... ,Kt 1 1 We now proceed to establish that supd,K ,... ,Kt f(d ; K1 , . . . , Kt ) ≥ E(2n + 1 3), yielding the lower bound. We define two auxiliary orderings
Here d (K¯ a ) < k ≤ k(d ), d satisfies condition C relative to K1 , . . . , Kt , and (q) denotes the qth iterate of Lˆ M . Lˆ M
AD AND THE PROJECTIVE ORDINALS
477
We let fq (d ; K1 , . . . , Kt ) and fqk (d ; K1 , . . . , Kt ) denote the ranks of the tuple (d ; K1 , . . . , Kt ) in these orderings. We let gq , gqk denote the corresponding rank functions on D2n−1 . We define an auxiliary function f˜qk (d ; K1 , . . . , Kt ), for 1 ≤ k ≤ c, on d satisfying condition C relative to K1 , . . . , Kt and d (K¯ a ) < k ≤ k(d ). We use the following definition. Definition 9.1. We define an ordering on tuples (α1 , αj )s , where j = 1 or 2, α2 ≤ α1 , and the symbol s may or may not appear. We define (α1 , αj )s (1 , j )s iff one of the following holds: i) α1 < 1 . ii) α1 = 1 , j1 = j2 = 2, and α2 < 2 . iii) α1 = 1 and one of the following holds: a) j1 = 1, j2 = 2, (α) s involves s, and 2 > 0. does not involve s. b) j1 = 2, j2 = 1, and () s does not. c) j1 = j2 = 1, (α) involves s and () To make a linear order we further identify (α1 , α2 ) with (α1 , α2 )s , and identify (α1 , 0) with (α1 )s . We also identify (α, α) with (α). We let ((α1 , αj )s ) denote the rank of (α1 , αj )s in the ordering with these identifications (so ((α1 , α2 )) = ((α1 , α2 )s ) and ((α1 , 0)) = r((α1 )s )). We also let (α) abbreviate ((α)). The reader will note that the ordering mimics the ordering on descriptions 1,m of the form d = (k; d1 )s or d = (k; d1 , d2 )s where Kk = S2n+1 . It can be considered a simplified version of the ordering on these descriptions. We now define f¯qk through the following cases. Case I. k(d ) = ∞ and k = c. Subcase I.a. d = d (Ia ) is basic basic type 1, so d = (d¯2 )s , where s may gq·4 (d2 ;K¯ 2 ,... ,K¯ a ) or may not appear. We set f˜qk (d ; K1 , . . . , Kt ) = + 1, where Ia = (f; K¯ 2 , . . . , K¯ a ) as usual, for gq·4 (d2 ; K¯ 2 , . . . , K¯ a ) > 0, and otherwise f˜k (d ; K , . . . , K ) = 0. q
1
t
Subcase I.b. d of basic type 0. We set f˜qk (d ; K1 , . . . , Kt ) = 0. Case II. k < c and k < k(d ). We set f˜qk (d ; K1 , . . . , Kt ) = f˜qk+1 (d ; K1 , . . . , Kt ). Case III. k < c and k = k(d ). 1,mk Subcase III.a. Kk = S2n+1 . Hence, d = (k; d1 , . . . , dr )s , where r ≤ mk and s may or may not appear. We set
f˜qk (d ; K1 , . . . , Kt ) = (((f˜qk+1 (d1 ; K1 , . . . , Kt ), f˜qk+1 (d2 ; K1 , . . . , Kt ))s )
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STEVE JACKSON
where is the function of Definition 9.1 and s appears here iff it appears in d (if r = 1 we don’t have the second argument of in the formula). k ,mk Subcase III.b. Kk = S2n+1 , with k > 1. Here d = (k; d2 )s , where s may or may not appear. We set f˜qk (d ; K1 , . . . , Kt ) = f˜qk+1 (d2 ; K1 , . . . , Kt )
Subcase III.c. d of basic type −1. We set f˜qk (d ; K1 , . . . , Kt ) = 0 Case IV. k = c and k = k(d ). As in case (III) above where we now replace gq·4 () k+1 ˜ in subcases a and b and retain subcase c. (Actually only fq () by m subcase b arises due to the definition of B2n+1 ). We introduce the following hypothesis: Rqk : for d satisfying condition C relative to K1 , . . . , Kt , where d (K¯ a ) < k < k(d ), we have fqk (d ; K1 , . . . , Kt ) ≥ f˜qk (d ; K1 , . . . , Kt ). We establish Rqk by reverse induction on k, and for fixed k by induction on the rank of the tuple with respect to <. We consider the following cases. Case I. k(d ) = ∞ and k = c. If d is basic of type 0, the result is immediate. Hence d is basic of type 1 of the form d = (d2 )s , where we may assume s appears. We have that ) + 1] fqc (d ; K1 , . . . , Kt ) ≥ sup[fqc ((L(q) (d2 ; K¯ 2 , . . . , K¯ a ))s ; K1 , . . . , Kt , Kt+1 Kt+1
≥ sup[sup fqc ((k1 ; (k2 ; · · · (ku ; (kv ; (L(q·2) (d2 ; K¯ 2 , . . . , K¯ a ))s )) · · · )); Kt+1 Kt+2
K1 , . . . , Kt , Kt+1 , Kt+2 ) + 1]
≥ sup (m) (fqc ((L(q·2) (d2 ; K¯ 2 , . . . , K¯ a ))s ; K1 , . . . , Kt )). m
1,m where k1 , . . . , ku enumerate the components of Kt+1 of the form S2n+1 and Kkv is the component of the Kt+1 of the form Mκ such that the above description satisfies condition C. Here (m) denotes the mth iterate of . The first inequality uses the definition of fqk , the second inequality uses also the , and the third inequality definition of L and the form of the measure Kt+1 1,m k k+1 uses the fact that fq (k; d ) ≥ (fq (d )) for Kk of the form S2n+1 . We also have that
fqc ((L(q·2) (d2 ; K¯ 2 , . . . , K¯ a )s ); K1 , . . . , Kt ) ≥ sup[fqc ((L(q·3) (d2 ; K¯ 2 , . . . , K¯ a ))s ; K1 , . . . , Kt , Kt+3 )] Kt+3
≥ sup[fqc ((k1 ; (k2 ; · · · (ku ; (L(q·4) (d2 ; K¯ 2 , . . . , Kt+3
)], K¯ a ))s(Ia+u ) )s(Ia+u−1 ) · · · )s(Ia+1 ) )s(Ia ) ; K1 , . . . , Kt , Kt+3
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479
k ,mk where k1 , . . . , ku now enumerate the components of Kt+3 of the form S2n+1 with k > 1. By induction this is ≥ sup[f˜qc ((L(q·4) (d2 ; K¯ 2 , . . . , K¯ a ))s(Ia+u ) ; K1 , . . . , Kt , Kt+3 )] Kt+3
where c is the value of c corresponding to the sequence K1 , . . . , Kt , Kt+3 . This is
≥ sup K¯ a+1
= ≥
[gq·4 (L(q·4) (d2 ;K¯ 2 ,... ,K¯ a );K¯ 2 ,... ,K¯ a ,K¯ a+1 )]
sup ¯ (g (L(q·4) (d2 ;K¯ 2 ,... ,K¯ a );K¯ 2 ,... ,K¯ a ,K¯ a+1 )) Ka+1 q·4
gq·4 (d2 ;K¯ 2 ,... ,K¯ a )−1
.
where − 1 = if is a limit ordinal. But, for all ordinals we have that +1 supm (m) ( ) = . Hence, fqc (d ; K1 , . . . , Kt ) ≥
gq·4 (d2 ;K¯ 2 ,... ,K¯ a )
= f˜qc (d ; K1 , . . . , Kt ).
Case II. k < c and k < k(d ). Subcase II.a. k(d ) < ∞. For d (K¯ a ) < k1 < k2 ≤ k(d ), it follows readily that fqk1 (d ; K1 , . . . , Kt ) ≥ fqk2 (d ; K1 , . . . , Kt ). Hence, fqk (d ; K1 , . . . , Kt ) ≥ fqk(d ) (d ; K1 , . . . , Kt ) ≥ f˜qk(d ) (d ; K1 , . . . , Kt ) ≥ f˜qk (d ; K1 , . . . , Kt ). The second inequality is by induction, and the third is from the definition of f˜qk . Subcase II.b. k(d ) = ∞. We have fqk (d ; K1 , . . . , Kt ) ≥ fqc (d ; K1 , . . . , Kt ) ≥ f˜qc (d ; K1 , . . . , Kt ), by case I, and f˜qc (d ; K1 , . . . , Kt ) = f˜qk (d ; K1 , . . . , Kt ) from the definition of f˜qk . Case III. k < c and k = k(d ). Subcase III.a. Kk = Mκ . Hence d = (k; d2 ). Then, fqk (d ; K1 , . . . , Kt ) ≥ fqk+1 (d2 ; K1 , . . . , Kt ) ≥ f˜qk+1 (d2 ; K1 , . . . , Kt ) = f˜qk (d ; K1 , . . . , Kt ). k ,mk Subcase III.b. Kk = S2n+1 , where k > 1. Here d (Ia ) = (k; d2(Ia+1 ) )s , where we may assume s appears.
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k+1(q) Subcase III.b.i. d2 is minimal w.r.t. Lˆ M , that is, Lˆ k+1(q) (d2 ) is not M defined. Then by induction, f˜qk+1 (d2 ; K1 , . . . , Kt ) = 0, hence f˜qk (d ; K1 , . . . , Kt ) = f˜qk+1 (d2 ; K1 , . . . , Kt ) = 0, so Rqk is immediate. k+1(q) Subcase III.b.ii. Lˆ M (d2 ) is defined. We first establish that fqk (d ; K1 , . . . , Kt ) ≥ fqk+1 (d2 ; K1 , . . . , Kt ) for d of this form by induction w.r.t. the ordering <. We have that fqk (d ; K1 , . . . , Kt ) = sup[fqk (Lˆ k(q) (d ; K1 , . . . , Kt ); K1 , . . . , Kt , Kt+1 ) + 1] M Kt+1
≥ sup[fqk ((k; Lˆ k+1(q) (d2 ; K1 , . . . , Kt ))s ; K1 , . . . , Kt+1 ) + 1] M Kt+1
≥ sup[fqk+1 (Lˆ k+1(q) (d2 ; K1 , . . . , Kt ); K1 , . . . , Kt+1 ) + 1] M Kt+1
= fqk+1 (d2 ; K1 , . . . , Kt ), k(q) the first inequality using the definition of Lˆ M (d ; K1 , . . . , Kt ) and the second inequality being by induction. We then have that
fqk (d ; K1 , . . . , Kt ) ≥ fqk+1 (d2 ; K1 , . . . , Kt ) ≥ f˜qk+1 (d2 ; K1 , . . . , Kt ) = f˜qk (d ; K1 , . . . , Kt ). 1,mk Subcase III.c. Kk = S2n+1 . Hence, d = (k; d1 , . . . , dr )s where we may assume s appears.
Subcase III.c.i. r > 2. We have that fqk (d ; K1 , . . . , Kt ) ≥ fqk ((k; d1 , d2 )s ; , Kt ) ≥ f˜qk ((k; d1 , d2 )s ; K1 , . . . , Kt ) = f˜qk (d ; K1 , . . . , Kt ). We use here the fact that d ≤ d then fqk (d ; K1 , . . . , Kt ) ≤ fqk (d ; K1 , . . . , Kt ), established by an easy induction. K1 , . . .
Subcase III.c.ii. r = 2. Subcase III.c.ii.1. Lˆ k+1(q) (d2 ; K1 , . . . , Kt ) is defined. We have that M fqk (d ; K1 , . . . , Kt ) k(q) ≥ sup fqk (Lˆ M (d ; K1 , . . . , Kt ); K1 , . . . , Kt+1 )+1 Kt+1
≥ sup fqk ((k; d1 , Lˆ k+1(q) (d2 ; K1 , . . . , Kt ))s ; K1 , . . . , Kt+1 ) + 1, M Kt+1
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by cases on whether or not r = mk . By induction, this is ≥ sup (fqk+1 (d1 ; K1 , . . . , Kt ), fqk+1 (Lˆ k+1(q) (d2 ; K1 , . . . , Kt ); M Kt+1
K1 , . . . , Kt+1 )) + 1
(d2 ; K1 , . . . , Kt ); ≥ (fqk+1 (d1 ; K1 , . . . , Kt ), sup fqk+1 (Lˆ k+1(q) M Kt+1
) + 1). K1 , . . . , Kt+1
Hence, fqk (d ; K1 , . . . , Kt ) ≥ (fqk+1 (d1 ; K1 , . . . , Kt ), fqk+1 (d2 ; K1 , . . . , Kt )) = f˜qk (d ; K1 , . . . , Kt ). k+1(q) Subcase III.c.ii.2. Lˆ M (d2 ; K1 , . . . , Kt ) is not defined. In this case
fqk (d ; K1 , . . . , Kt ) ≥ fqk ((k; d1 )s ; K1 , . . . , Kt ) ≥ ((fqk+1 (d1 ; K1 , . . . , Kt )s ) = (fqk+1 (d1 ; K1 , . . . , Kt ), 0) = f˜qk (d ; K1 , . . . , Kt ) where the second inequality follows by induction, the first equality uses the definition of , and the second equality holds since fqk+1 (d2 ; K1 , . . . , Kt ) = 0. Subcase III.c.iii. r = 1 and s appears. k+1(q) Subcase III.c.iii.1. Lˆ M (d1 ; K1 , . . . , Kt ) is defined. We have that fqk (d ; K1 , . . . , Kt ) ≥ sup[fqk (Lˆ k+1(q) (d ; K1 , . . . , Kt ); K1 , . . . , Kt+1 ) + 1] M Kt+1
k+1(q) ≥ sup[fqk (k; Lˆ M (d1 ; K1 , . . . , Kt ); K1 , . . . , Kt+1 ) + 1] Kt+1
≥ sup[(fqk+1 (Lˆ k+1(q) (d1 ; K1 , . . . , Kt ); K1 , . . . , Kt+1 )) + 1] M Kt+1
= ((fqk+1 (d1 ; K1 , . . . , Kt+1 ))s ) = f˜qk (d ; K1 , . . . , Kt )
from the definition of . k+1(q) Subcase III.c.iii.2. Lˆ M (d1 ; K1 , . . . , Kt ) is not defined. k+1 In this case f˜ (d1 ; K , . . . , K ) = 0, so Rk is immediate. q
1
t
q
Subcase III.c.iv. r = 1 and s does not appear. k+1(q) Subcase III.c.iv.1. Lˆ M (d1 ; K1 , . . . , Kt ) is defined.
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Proceeding as above, fqk (d ; K1 , . . . , Kt ) (d1 ; K1 , . . . , Kt ))s ; K1 , . . . , Kt+1 ) + 1] ≥ sup[fqk ((k, d1 , Lˆ k+1(q) M Kt+1
≥ sup[(fqk+1 (d1 ; K1 , . . . , Kt ), Kt+1
(d1 ; K1 , . . . , Kt ); K1 , . . . , Kt+1 )) + 1] fqk+1 (Lˆ k+1(q) M
≥ (fqk+1 (d1 ; K1 , . . . , Kt ), (sup fqk+1 (Lˆ k+1(q) (d1 ; K1 , . . . , Kt ); K1 , . . . , Kt+1 ) + 1)) M Kt+1
= f˜qk (d ; K1 , . . . , Kt ), where the last equality uses the definition of f˜qk and the fact that (α, α) = (α). (d1 ; K1 , . . . , Kt ) is not defined. Subcase III.c.iv.2. Lˆ k+1(q) M We have f˜qk+1 (d1 ; K1 , . . . , Kt ) = 0, so Rqk is immediate. Case IV. k = c and k = k(d ). This case is similar to the previous cases (in m fact by a trivial change in the definition of B2n+1 , so that its last component is in m
REFERENCES
Stephen Jackson [Jac99] A computation of 15 , vol. 140, Memoirs of the AMS, no. 670, American Mathematical Society, July 1999. Alexander S. Kechris [Kec81A] Homogeneous trees and projective scales, this volume, originally published in Kechris et al. [Cabal ii], pp. 33–74. Alexander S. Kechris, Donald A. Martin, and Yiannis N. Moschovakis [Cabal ii] Cabal seminar 77–79, Lecture Notes in Mathematics, no. 839, Berlin, Springer, 1981. Donald A. Martin [Mar] AD and the normal measures on 13 , unpublished.
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Yiannis N. Moschovakis [Mos80] Descriptive set theory, Studies in Logic and the Foundations of Mathematics, no. 100, North-Holland, Amsterdam, 1980. DEPARTMENT OF MATHEMATICS UNIVERSITY OF NORTH TEXAS P.O. BOX 311430 DENTON, TEXAS 76203-1430 UNITED STATES OF AMERICA
E-mail: [email protected]
PROJECTIVE SETS AND CARDINAL NUMBERS: SOME QUESTIONS RELATED TO THE CONTINUUM PROBLEM
DONALD A. MARTIN
Editorial Note (2010). This paper was originally written in 1971 and accepted to appear in the Journal for Symbolic Logic, but was never published. In 1991, a version incorporating some of the author’s corrections from the 1970s was included in a volume of papers for the author’s 50th birthday. This retyped paper is based on the 1991 version. It reflects the state of knowledge from the early 1970s; some of the conjectures made in this paper turned out to be false. For instance at the end of §1, the author conjectures that 2n+1 = ℵ·n under AD. While correct for n = 0, 1, the conjectured values turned out to be too low, as discussed in [Jac11]. §1. Introduction. A prewellordering of a set X is the relation on X induced by a function f : X → Ord, where Ord is the class of all ordinal numbers. In other words, to prewellorder X , divide X into equivalence classes and wellorder the equivalence classes. The length of a prewellordering is the order type of the associated wellordering of equivalence classes. There are prewellorderings of the continuum of every length < (2ℵ0 )+ , where α + is the least cardinal (initial ordinal) greater than the ordinal α. For each positive integer n, let 1n be the least ordinal other than 0 not the length of a Δ1n pre the continuum. (See [Sho67] for information about projective wellordering of 1 sets.) Note that every Δn prewellordering of the continuum has length < 1n . It is essentially a classical result that 11 = ℵ1 . We shall prove that 12 ≤ ℵ2 ,that, if a measurable cardinal exists, 13 ≤ ℵ3 , and (as Kunen has independently shown) that, if all projective games are determined, then for all n, we have 12n+2 ≤ ( 12n+1 )+ ), so that, in particular, 14 ≤ ℵ4 . (See [Mos70] and [Myc64] information for about infinite games and determinacy.) We conjecture that for all n, we have 1n = ℵn . Of course, even 1n ≥ ℵ2 cannot be proved without one knows how to do at present. refuting the continuum hypothesis, which no The result that measurable cardinals imply 13 ≤ ℵ3 should probably also be counted as independently due to Kunen. We earlier proved that projective determinacy implies 13 ≤ ℵ3 and Theorem 3.1 below—independently due to weaken the hypothesis. Kunen—allows one to Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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Both Kunen’s proof and ours of 12n+2 ≤ ( 12n+1 )+ under PD depend on work of Moschovakis [Mos71]. Wellorderings of the continuum might seem a more natural subject than prewellorderings of the continuum. However, a fairly plausible conjecture (and a consequence of projective determinacy) is that no projective wellordering of the continuum exists and that every projective wellordering of a subset of the continuum is countable. 1.1. As is customary in modern descriptive set theory, we work with the space of functions from the natural numbers into rather than with the reals themselves. is given the product topology, where is given the discrete topology. Call a subset of κ-Suslin if it has the form Af(n) , f∈κ n∈
where κ is a cardinal (initial von Neumann ordinal), f(n) is the sequence f0 , . . . , f(n − 1), and each Af(n) is clopen (closed and open). There are classical results to the effect that every Σ11 is ℵ0 -Suslin and every Σ12 set is cardiℵ1 -Suslin. Every ℵ0 -Suslin set is Σ11 also, but—at least if a measurable is Σ1 (although the contrary is consistent nal exists—not every ℵ1 -Suslin set 2 with ZFC; see [MS70, pp. 165–166], bearing in mind that being ℵ1 -Suslin is equivalent to being the union of ℵ1 Borel sets (3.4 below)). We shall prove, assuming the existence (MC) of a measurable cardinal, that every Σ13 set is deterℵ2 -Suslin. Recently Moschovakis [Mos71] has shown, from projective 1 minacy (PD), that every Σn set is n -Suslin for all n ≥ 1, where n+1 = 1n if n is odd. (Hence, in particular, PD implies that every Σ14 set is ℵ3 -Suslin, which we had already proved by another method.) We conjecture that Moschovakis’ result can be improved to show, from PD, that for all n, we have n = ℵn−1 . This would imply 1n ≤ ℵn for all n, by Theorem 3.1 below. We also conjecture that, for each n ≥2, not every Σ1n set is ℵn−2 -Suslin. For n = 2, this is a clas sical theorem (since otherwise every Σ12 set is Σ11 ). For each n it would follow subset B of is 2ℵ0 -Suslin. 1 from n ≥ ℵn by Theorem 3.1. Notethat every Simply let the empty set if f(0) ∈ B; Af(n) = the clopen set determined by (f(0))(n) if f(0) ∈ B, for f : → . So if 2ℵ0 = ℵn for some n, our conjecture is wrong. 1.2. If A is κ-Suslin and ℵ0 < κ < ℵ , then, as we shall show, A is the union of κ Borel sets.1 Thus, under suitable assumptions, (as in 1.1), every Σ1n+1 set is the union of ℵn Borel sets, for n = 1, 2, 3. Our conjectures imply 1 The
converse is true without the restriction κ < ℵ .
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that this generalizes to all n ≥ 1 and that not every Σ1n+1 set is the union of < ℵn Borel sets. 1.3. If α is an ordinal, an algebra of subsets of a set is an α-algebra if it is closed under unions and intersections of order type < α.2 If α is an ordinal, Bα is the α-algebra generated by the clopen subsets of . We shall mainly concerned with the algebras B 1 . We have that B 1 = B is the Borel sets, be ℵ1 n 1 so the Suslin-Kleene Theorem [Sho67, p. 185] yieldsthat B 11 = Δ11 . There is no serious hope of generalizing this result, since it is a classical result that B 12 Σ12 , and we shall prove, assuming MC, that B 13 Σ13 . Also it follows from PD via Moschovakis [Mos71] that B 1n Σ1n forn even. However, there is a peculiar fact concerning the proofs of these results: The proof for n = 3 makes strong use of the axiom of choice, whereas the proof for n = 2 uses no choice, and the general proof for even n > 2 uses only the axiom of dependent choice (DC). Without the axiom of choice, we are able to prove from MC that Δ13 ⊆ B 13 , i.e., half the natural generalization of the Suslin-Kleene Theorem for n = 3. Using only PD+DC, Moschovakis [Mos71] shows Δ1n ⊆ B 1n , for all odd (and, of course, even) n. The significance of these proofs will be made clear in the next paragraph. 1.4. In §7 we shall prove some consequences of the (false) full axiom of determinacy (AD) (see [Myc64]). Naturally we shall not use the axiom of choice (AC), but we shall use DC. We shall show that AD+DC implies that for all odd n, we have Bn1 ⊆ Δ1n . the Moschovakis theorem mentioned in the last This result, combined with section (which was proved later) gives a Generalized Suslin-Kleene Theorem: AD+DC implies that for all odd n, we have Bn1 = Δ1n . Unhappily, as we remarked in the last paragraph, this pleasant characterization of Δ1n fails in the real world. Indeed, what remains of the result we extract all use of the axiom of choice from its proof, will be Σ13 ⊆ B 13 , when enough—combined with B 13 ⊆ Δ13 , whose proof does not use DC—to prove the following curious consequence of determinacy: AD implies that ℵn is singular for all n ≥ 3. Using the “choiceless hull” of a related theorem, we show and
AD implies that 13 = ℵ+1 AD+DC implies that 14 = ℵ+2 .
2 This definition is slightly odd for α not a cardinal. The theorems listed below and in 1.4 explain why we chose this definition.
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Since results already mentioned (whose proofs use no more choice than DC)— together with Moschovakis [Mos70]—give that AD+DC implies that 12n+2 = ( 12n+1 )+ , the 1n would all be computed (assuming AD+DC) if we would compute the odd 1n , n ≥ 5. Concerning this, Kechris has used Moschovakis’ work [Mos71] to generalize partially our results on 13 : AD+DC implies that 12n+2 = ( 2n+1 )+ where 2n+1 is a cardinal cofinal with . The natural conjecture would appear to be that 2n+1 = ℵ·n . §2. Discussion of the hypotheses. 2.1. We have made several conjectures above. Some of these we hope to be consequences of large cardinal axioms and of projective determinacy. Those which contradict the continuum hypotheses, however, follow neither from known large cardinal axioms nor from, say, determinacy for sets ordinal definable from a real (except possibly through inconsistency). On the other hand it is possible, as far as we know now, that these conjectures can be refuted on the basis of the ZFC axioms alone. Since many of our theorems are of the form PD =⇒ ϕ or MC =⇒ ϕ, a few words are in order about the status of these hypotheses. Note first that MC is essentially a weaker hypothesis. Although PD does not imply MC, PD does imply the existence of inner models with many measurable cardinals (Solovay; see also [Mar77A]). Actually, in this paper we are always able to assume instead of MC only the weaker hypothesis (∀x)(∃y)(y = x # ) which is implied by MC [Sol67B] and which is equivalent [Mar77A] with a certain weak form of Det(Δ12 ): in the terminology of [Mar77A], Det(α-Π11 ) for all α < 2 . We regard both MC and PD as hypotheses for which there is considerable, though nothing remotely like conclusive, evidence. In the case of MC the evidence is mostly a priori: analogy with and reflection principles (or pseudo-reflection principles) which imply MC. In the case of PD the evidence is mostly a posteriori: its consequences (such as the prewellordering theorem [AM68, Mar68]) look right—whereas the consequences of, say, V = L sometimes do not. (Obviously we are using “a posteriori” in a way having nothing to do with sense experience.) One way to increase the evidence for PD would be to prove it from large cardinal axioms—perhaps from the existence of compact or supercompact cardinals.
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2.2. The status of full AD is quite different from that of MC and PD: It is known to be false. One may still hope that AD holds in some transitive class satisfying the axioms of set theory other than choice, perhaps even in a class containing all ordinals and all reals. If AD holds in any such class, it also holds in L(R), the minimal model of set theory containing all reals and all ordinals. ADL(R) has been conjectured by several people, including Mycielski, Solovay, and Takeuti. As Solovay perhaps first noted, it is natural to assume DC when one assumes AD, since the axiom of choice implies that L(R) |= DC. If one hopes for such a model M of determinacy, one might also hope that the model is fat enough to look in some respects like the full universe of sets. The author in fact thought at one time that one might have (ℵα )M = ℵα for α not too large. Our own results contradict this hypothesis for α = 3, since the real ℵ3 is not singular. Nevertheless, it remains possible that, for small enough α the αth regular cardinal in M is the αth regular cardinal. One reason for proving consequences of AD, is that—assuming ADL(R) or PD—such results can sometimes be turned into interesting, though slightly more complicated, theorems about the real world. For example, the assertion B 13 = Δ13 can be modified by considering only special kinds of transfinite unions and intersections, and the modified statement can be proved from, say, PD. Of course, one could argue in favor of AD+DC that many of its consequences are elegant and appealing, e.g., the Generalized Suslin-Kleene theorem. One might even argue that AD+DC is just as good as the axiom of choice. However, elegance seems less important than truth. 2.3. For obvious reasons, we wish in this paper to keep track of the axioms and hypotheses used in proving our theorems. In particular, we want to keep track of when the axiom of choice (AC) and dependent choice (DC) are used. Since we do not wish to suggest that AC is a hypothesis in the same way that PD is, or merely a formal assumption as AD is, we adopt the following conventions: In the absence of any indication to the contrary, the proof of a theorem uses only the axioms of ZF (not including AC). If AC is used, we indicate its use by writing “Theorem (AC)”. All other hypotheses (including DC) will appear simply as antecedents in conditional theorems, as “AD+DC implies that ϕ”. §3. κ-Suslin sets. In this section we prove some properties of κ-Suslin sets. In §§4–6 we shall apply the results of this section to Σ1n sets for various n. 3.1. Finite sequences and trees. As before, if f : → X then, for each n ∈ , f(n) is the sequence f(0), . . . , f(n − 1). If is a finite sequence and n = lh( ), then (0), . . . , (n − 1) are the terms of . We shall usually think of a finite sequence of n-tuples as an n-tuple of finite sequences (all of
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the same length). The collection of all finite sequences of natural numbers we call <. Let n → n be an effective bijection of onto < with lh( n ) ≤ n. By a tree on a set X we shall mean a set T of finite sequences of elements of X with the property that if ∈ T and extends , then ∈ T . What we call a “tree” is a special case of what in set theory is usually called a “tree”. If T is a tree on X1 × · · · × Xn and f1 , . . . , fj belong to X1 , . . . , Xj respectively, then Tf1 ,...,fj = {j+1 , . . . , n : f1 (k), . . . , fj (k), j+1 , . . . , n ∈ T }. (Recall our convention about sequences of n-tuples.) If T is a tree on X1 × · · · × Xn and Y ⊆ Xj , then T Y, j = {1 , . . . , n ∈ T : ran(j ) ⊆ Y }. We write T Y, j as T Y when there is no danger of confusion. If T is a tree on X1 × · · · × Xn and a1 , . . . , an ∈ X1 × · · · × Xn , then T a1 ,...,an = {1 , . . . , n : a1 ∗ 1 , . . . , an ∗ n ∈ T }, where (ai ∗ i )(0) = ai and (ai ∗ i )(k + 1) = i (k). We partially order each tree by setting 1 < 2 ⇐⇒ 1 properly extends 2 . We often think of a tree as being the partial order of its elements. Recall that a binary relation R is wellfounded if, for every non-empty subset X of the field of R, there is an x ∈ X such that no y ∈ X bears R to x. Using DC, R is wellfounded just in case there are no infinite descending chains with respect to R: i.e., there are no x0 , x1 , . . . such that x1 , x0 ∈ R, x2 , x1 ∈ R, . . . . Note that if x, x ∈ R, then x, x, . . . constitutes an infinite descending chain. (DC is not necessary if the field of R is wellordered.) Let R be any wellfounded relation. For each element x of the field of R we define an ordinal |x|R : |x|R = sup {|y|R + 1}. x,y∈R
The ordinal of a wellfounded relation R is supx {|x|R + 1}. The ordinal of a wellfounded tree T is then |Λ|T , where Λ is the empty sequence. A subset A of ()n is κ-Suslin, for κ an infinite cardinal (i.e., an infinite initial von Neumann ordinal), if there is a tree T on n × κ such that x1 , . . . , xn ∈ A ⇐⇒ Tx1 ,...,xn is not wellfounded. (This definition agrees with that given in §1.1.)
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3.2. Theorem 3.1, stated and proved in this section, will yield bounds on many of the 1n . In 1969, we proved 12 ≤ ℵ2 by an argument only slightly different from the appropriate special case of the proof of the theorem given below. The theorem itself was recently proved independently by Kunen and us. Our proof uses forcing, whereas Kunen’s is elementary. Indeed, if both proofs are given in detail, Kunen’s is almost a sub-proof of ours. We nevertheless feel that our proof may be helpful in that it can be presented without considering combinatorial details. Theorem 3.1. If R is a wellfounded relation and R is κ-Suslin, then the ordinal of R is < κ + . Proof. We first prove the case κ = ℵ0 . (Kunen’s proof does not require considering this case separately.) This case is essentially a classical theorem. Assume R is ℵ0 -Suslin (= Σ11 ) and R has ordinal ≥ ℵ1 . We show, for a contradiction, that the set W of relations r on which are wellorderings of is Σ11 : r ∈ W ⇐⇒ (∃f)[f is a function and dom(f) ⊆ and ran(f) = and (∀y)(∀n)(y ∈ dom(f) and n, f(y) ∈ r =⇒ (∃z)(z ∈ dom(f) and z, y ∈ R and (n = f(z) or n, f(z) ∈ r)))]. (If r ∈ W , let f(y) be the number, if any, whose segment in r has order type |y|R .) Furthermore, one easily sees that f may be taken to be countable, so W is Σ11 . (The choices used in picking a countable f can be made canonically.) Now we consider the case κ > ℵ0 . Let T be a tree on × κ witnessing that R is κ-Suslin. To say that there is an infinite descending chain x0 , x1 , . . . with respect to R is to say that there is an infinite sequence of descending chains in T , witnessing that x1 , x0 ∈ R, x2 , x1 ∈ R, etc. Shuffling, we produce a tree T ∗ on × κ which has an infinite descending chain just in case there is an infinite descending chain with respect to R. In addition, R is wellfounded if and only if T ∗ is wellfounded which in turn is equivalent (without DC) to the statement that T ∗ has no infinite descending chains.3 Let B be the usual complete Boolean algebra for collapsing κ onto : The forcing conditions are the finite functions with domain ⊆ and range ⊆ κ. ˇ In the Boolean-valued universe V (B) , we have (κ + )V = 1 , where x → xˇ is the canonical embedding of V in V (B) , extended to proper classes. We argue in V (B) . Let C := {x, y : Tˇx,y is not wellfounded}. Then ˇ Rˇ ⊆ C since Tˇx,ˇ y ˇ = (Tx,y ) and wellfoundedness is absolute. Furthermore, ∗ C is wellfounded, since (Tˇ ) = (T ∗ )ˇ and wellfoundedness is absolute. But Tˇ 3 Kunen
completes the proof by verifying that ordinal(T ∗ ) ≥ ordinal(R).
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is countable, so C is ℵ0 -Suslin, whence ˇ ≤ ordinal(C ) < 1 = (κ + )Vˇ . ordinal(R)
Corollary 3.2 (to proof of Theorem 3.1). If R is a wellfounded relation, and T witnesses that R is κ-Suslin, then the ordinal of R is < the first T admissible ordinal > κ. Proof. The proof is just the lightface analogue of the proof of the theorem, and we omit it. Using the proof of Theorem 3.1 (either ours or Kunen’s) and known results, one can show that 1n is the least ordinal not the ordinal of a Σ1n wellfounded relation, assuming, say, PD. This fact was noted by both Kunen and us, and Kunen has exploited it in proving that AD+DC implies that for all n, 1n is a measurable cardinal. Kunen’s proof of the theorem has the advantage 1 over ours that it yields a direct proof of this characterization of n which furthermore needs to assume only Moschovakis’s [Mos71], and soonly the determinacy hypotheses of that paper; for n = 2 no hypotheses are needed, and for n = 3 only “for all x there is a y such that y = x # ” is needed, as one can show using §5 below and [MS69]. The case of odd numbers n assuming Δ1n−1 determinacy follows from [AM68] and [Mar68] and was already known. 3.3. A subset C of separates disjoint subsets A and B of if C ⊇ A and \ C ⊇ B. The following theorem and its proof are straightforward generalizations of the classical separation theorem (Luzin) for Σ11 sets and one of its proofs [Kur58, pp. 393–395]. Theorem 3.3. If A and B are disjoint κ-Suslin sets, then A and B are separated by elements of Bκ+ . Proof. Let T A and T B witness that A and B respectively are κ-Suslin. Suppose that for each m, n ∈ and α, ∈ κ, the disjoint sets {x : x(0) = m and TxA has an infinite descending chain beginning with α} and {x : x(0) = n and TxB has an infinite descending chain beginning with } are separated by elements Cmαn of Bκ+ . Then C = Cmαn mα n
separates A and B. Hence, if A and B are not separable by an element of Bκ+ , then we can continue in this way and find functions x, y : → and g : → κ such that, for each n, f, Ax(n),f(n) = {z : z(n) = x(n) and TzA has an infinite descending chain beginning with f(n)}
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and the similarly κ+ . defined By(n),g(n) are not separated by any element of B Now suppose n Ax(n),f(n) and n By(n),g(n) have members x and y respectively. Since x ∈ A and y ∈ B and A is disjoint from B, there is an n with x(n) = y(n). But then {z : z(n) = x(n)} separates Ax(n),f(n) and By(n),g(n) . If either n Ax(n),f(n) or n By(n),g(n) is empty, then either the empty set or separates some Ax(n),f(n) from By(n),g(n) . The foregoing proof makes a superficial use of the axiom of choice. To avoid AC, observe that the tree T = {x(n), y(n), f(n), g(n) : x(n) = y(n) and x(n), f(n) ∈ T A and y(n), g(n) ∈ T B } is wellfounded, since A and B are disjoint. For non-elements of T , sets separating Ax(n),f(n) and By(n),g(n) can be defined trivially, as we have indicated. For elements of T , separating sets are defined by transfinite induction on T . Corollary 3.4. If A and \ A are κ-Suslin, then A ∈ Bκ+ . 3.4. Theorem 3.5 in this section implies Theorem 3.3 for the case κ < ℵ . However, its proof uses AC, which will be important later. Without AC, we still get a result useful in the context of full AD, and also implying Theorem 3.3 when κ is not cofinal with : Theorem 3.5 (AC). If A is κ-Suslin and ℵ0 < κ < ℵ , then A is a union of κ Borel sets. Proof. By induction on the cardinal κ. It is a classical result [Kur58, p. 391, Corollary 3] that every Σ11 set is a union of ℵ1 Borel sets. The idea of one proof is that, for each countable ordinal α, the set of x ∈ A such that the Brouwer-Kleene ordering with respect to x (see §4) has wellfounded initial segment of order type ≤ α is Borel.4 For n ≥ 1 let T on × ℵn witness that A is ℵn -Suslin. We have that A = {x : Tx is not wellfounded} = α<ℵn {x : (T α)x is not wellfounded}. By induction, each of the terms of this union is a union of ≤ κ Borel sets, so we are done. Corollary 3.6 (to the proof of Theorem 3.5). If A is κ-Suslin and κ is not cofinal with , then A ∈ Bκ+ . Proof. By the proof of the theorem, it is enough to show that every -Suslin set, < κ, belongs to Bκ+ . For this we show that, for every , each -Suslin set is an intersection of + elements of B + (It can also be shown to be a union 4 The
proof given in [Kur58] is closely related to this one.
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of + elements of B + .) By induction on α < + , we prove that, for all trees T on × , the set {x : ordinal(Tx ) < α} ∈ B + . We have that {x : ordinal(Tx ) < α} = {x : ordinal(Tx ) ≤ } <α
=
{x : x(0) = n and ordinal(Txn, ∗ ) < },
<α < n<
where x ∗ (m) = x(m + 1). (Txn, is defined in 3.1) To finish the proof, note that {x : Tx not wellfounded} = α< + {x : Tx does not have ordinal < α} for T a tree on × . 3.5. Theorem 3.7. The class of κ-Suslin sets is closed under projection. Proof. Let C = {x : (∃y)(x, y) ∈ A} with A κ-Suslin. Let T witness that A is κ-Suslin. Then x ∈ C ⇐⇒ (∃y)(Tx,y is not wellfounded) ⇐⇒ Tx is not wellfounded.
§4. Σ12 sets. That Σ11 consists exactly of ℵ0 -Suslin sets is just the normal form theorem for Σ11sets. It follows by Theorem 3.1 that 11 = ℵ1 , since 1 obviously 1 ≥ ℵ1 . To prove (the well-known fact) that every Σ12 set is ℵ1 -Suslin, it would be enough by Theorem 3.7 to show that every Π11 set is ℵ1 -Suslin. Nevertheless work can readily be applied to we shall work directly with Σ12 sets so that our the study of Σ12 sets in 5.3. 1 Let A ∈ Σ 2 . Then A is of the form A = {x : (∃y ∈ )(∀z ∈ )(∃n ∈ ) R(x(n), y(n), z(n))}. Choose such an R. Say that z(n) is secured with respect to x, y if (∃m ≤ n)R(x(m), y(m), z(n)) and unsecured otherwise. Observe that z(n) being secured with respect to x, y depends only on x(n), y(n). The Brouwer-Kleene ordering < of < (see 3.1) is defined by
< ⇐⇒ properly extends or (∃m)( (m) < (m) and (∀p < m)( (p) = (p))). Lemma 4.1 (Brouwer-Kleene). For A as before, we have x ∈ A if and only if for some y, the Brouwer-Kleene ordering of the sequences unsecured with respect to x, y is a wellordering.
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Let us define a total ordering <xy of the whole of <, for each x and y ∈ . Set ⎧ ⎨ n secured and m unsecured, or
n secured and n < m, or
n <xy m ⇐⇒ ⎩
m unsecured and n < m . In other words, we give the secured sequences the ordering induced by n → n (see 3.1) and place them before the unsecured sequences. Clearly Lemma 4.2. For A as above, we have x ∈ A if and only if there is y such that <xy is a wellordering of <. Let T be the collection of all x(n), y(n), H (n), where x, y ∈ and H ∈ ℵ1 such that H ∗ preserves order (<xy ), where H ∗ ( n ) = H (n). Then T is a tree on × × ℵ1 (Note that since lh( n ) ≤ n, H (n) depends only on x(n), y(n).) Lemma 4.3. For A and T as above, we have x ∈ A if and only if Tx is not wellfounded. Proof. We have that x ∈ A if and only if there is a y such that <xy is a wellordering which in turn is equivalent to the existence of y and H such that H ∗ preserves <xy . This is equivalent to Tx being illfounded. Theorem 4.4. Every Σ12 set is ℵ1 -Suslin. Corollary 4.5. 12 ≤ ℵ2 . Proof. Immediate from Theorem 4.4 and Theorem 3.1.
Corollary 4.6. Every Σ12 set is a union of ℵ1 Borel sets. Proof. Immediate from the theorem and Theorem 3.5. The proof appears to use AC, since that of Theorem 3.5 does. However, if T is a tree on ×α and α < 1 , then {x : Tx is not wellfounded} is canonically a union of ℵ1 Borel sets, as can be shown by the same proof as for the case α = (see 3.4). Corollary 4.7. Σ12 B 12 . 1 Proof. We have Σ2 ⊆ B 12 by Corollary 4.6, since 12 > ℵ1 . (To see 12 > ℵ1 , let x < y ⇐⇒ x codes a wellordering of and y does not code as short a wellordering. This prewellordering of has length 1 + 1 and is Π11 .) B 12 ⊆ Σ12 since Π12 ⊆ B 12 . Except for Corollary 4.5, the results above are classical. As we mentioned earlier, we proved Corollary 4.5 directly some time before we (and independently Kunen) proved Theorem 3.1.
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§5. Σ13 sets. As we have remarked, every subset of is 2ℵ0 -Suslin. In [MS69] it is shown (though only implicitly) that each Σ13 set is κ-Suslin for some κ via a tree which can be defined in not too complicated a fashion. Mansfield and the author independently noted this fact and have made use of it. Mansfield [Man71] observes, for example, that the tree in question is ordinal definable from a code for the Σ13 set and draws consequences of this. We show below that the tree essentiallydefined in §7 of [MS69] has cardinality ≤ ℵ2 , and so that every Σ13 set is ℵ2 -Suslin. Our analysis of Σ13 setshere more or less follows that of Mansfield [Man71]. of taste; we could just as well proceed as in [MS69]. We This is purely a matter depart from Mansfield in that we use the prewellordering defined in [MS69, §7] rather than using a measurable cardinal directly (since the latter approach does not yield a tree of cardinality ≤ ℵ2 .) We also depart from [Man71] and [MS69] by defining the tree in a more natural way, so that—in particular—our Lemma 5.12 does not depend upon the axiom of choice for countable sets of sets of reals. We assume familiarity with [Sol67B]. 5.1. The orderings Fn∗ . In this section we define, for each n ∈ , a wellordered set Fn∗ . In 5.2 we shall use a result of Solovay to prove that Card(Fn∗ ) ≤ ℵ2 . In 5.3 we prove that every Σ13 set is Card(Fn∗ )-Suslin. exists, then with each x ∈ is associated 5.1.1. If a measurable cardinal x a canonical proper class C of ordinal numbers with the properties: a) C x is closed and unbounded in the order topology. b) If κ is an uncountable cardinal, then Card(C x ∩ κ) = κ. c) If a1 < · · · < an , b1 < · · · < bn are elements of C x and ϕ(v1 , . . . , vn+1 ) is a formula of set theory, then L[x] |= ϕ[a1 , . . . , an , x] ⇐⇒ L[x] |= ϕ[b1 , . . . , bn , x]. d) C x ∪ {x} generates L[x]; i.e., every element of L[x] is definable in L[x] from x and elements of C x via a formula of set theory. See [Sol67B] for the proof that a unique class C x satisfying a)–d) exists. We denote the elements of C x by c1x , c2x , . . . , cαx , . . . in order of magnitude. ¨ The set x # is the set of Godel numbers of those sentences of set theory, with constants for x and each cn , n ∈ , which are true in L[x]. The set x # , if it exists (i.e., if a C x satisfying a)–d) exists), is the unique element of a certain Π12 set [Sol67B], which is empty if x # does not exist. The set C x can be defined from x # in a simple fashion.
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If ϕ(v1 , . . . , vn+k+2 ) is a formula of set theory and α1 , . . . , αk , 1 , . . . , n are ordinals with α1 < · · · < αk , set ⎧ ⎨ (L[x] |= ϕ[1 , . . . , n , cαx1 , . . . , cαxk , , x]) x x x if such a exists, hϕ (1 , . . . , n , cα1 , . . . , cαk ) = ⎩ 0 otherwise. Lemma 5.1. Assume x # exists. If (∀i ≤ n)(i < cαx1 & i < cαx ) and 1
hϕx (1 , . . . , n , cαx1 , . . . , cαxk ) < cαx1 , then hϕx (1 , . . . , n , cαx1 , . . . , cαxk ) = hϕx (1 , . . . , n , cαx , . . . , cαx ). 1
k
Proof. A sketch of the proof for the case that each i ∈ C using essentially our a) and c) is found on page 67 of [Sol67B]. The lemma can be reduced to this special case by replacing the i by their definitions as guaranteed by d) and by applying the special case to the definitions. By Lemma 5.1, we define x
hϕx (1 , . . . , n ) = hϕx (1 , . . . , n , cαx1 , . . . , cαxk ) for arbitrary sufficiently large α1 < · · · < αk . 5.1.2. If x, y ∈ , define [x, y](2n) = x(n) [x, y](2n + 1) = y(n). Lemma 5.2. Assume [x, y]# exists. Then C [x,y] ⊆ C x ∩ C y . Proof. Let α ∈ C x . Then by d) α is defined in L[x] from x and cx1 < · · · < < α < cxk+1 < · · · < cxn . By Lemma 5.1, cxk+1 , . . . , cxn may be replaced by any cx1 < · · · < cxn−k with α < cx1 . In particular, by a) they may be replaced by elements of C [x,y] ∩ C x . Similarly, each cxi , i ≤ k, is defined in L[x, y] from [x, y] and elements of C [x,y] which are ≤ cxi < α, together with elements of C [x,y] which may be chosen larger than α. Hence α is defined in L[x, y] from [x, y] and elements of C [x,y] distinct from α. By c), α ∈ C [x,y] . 5.1.3. For each set X and positive integer n, set [X ]n = {Y ⊆ X : Card(Y ) = n}. If X is an ordered set, we shall think of [X ]n as the collection of all n-tuples x1 , . . . , xn with x1 < · · · < xn . For each n, let Fn be the collection of functions f : [ℵ1 ]n → ℵ1 such that f is constructible from an element of . If f, g ∈ Fn , we say that f ∼ g if there is a closed unbounded X ⊆ ℵ1 such that cxk
f(α1 , . . . , αn ) = g(α1 , . . . , αn ) for all α1 , . . . , αn ∈ [X ]n , and f < g if there is a closed unbounded X ⊆ ℵ1 such that f(α1 , . . . , αn ) < g(α1 , . . . , αn ) for α1 , . . . , αn ∈ [X ]n .
PROJECTIVE SETS AND CARDINAL NUMBERS
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For f ∈ Fn let [f] be the equivalence class of f with respect to ∼. Let Fn∗ = {[f] : f ∈ Fn }. Partially order Fn∗ by the relation induced by <. Lemma 5.3. If for all x there is a y such that y = x # , then Fn∗ is wellordered by <. Proof. The ordering is total, since if f ∈ L[x] and g ∈ L[y] then one of the following holds for every α1 < · · · < αn ∈ C [x,y] ∩ ℵ1 with α1 bigger than the countable ordinals used to define f and g in L[x, y] from [x, y] and elements of C [x,y] : f(α1 , . . . , αn ) < g(α1 , . . . , αn ) f(α1 , . . . , αn ) > g(α1 , . . . , αn ) f(α1 , . . . , αn ) = g(α1 , . . . , αn ). If f1 > f2 > . . . then, since the intersection of countably many closed unbounded subsets of ℵ1 is closed and unbounded, there is a closed and unbounded X ⊆ ℵ1 such that α1 , . . . , αn ∈ [X ]n implies that fi (α1 , . . . , αn ) : i ∈ is an infinite descending chain. We have used DC in showing Fn∗ wellordered. This was not really necessary, since we shall later (Theorem 5.5) define, without any choices, embeddings of the Fn∗ into the ordinals. Hence we have not listed DC as a hypothesis of the lemma.5 5.2. The order type of Fn∗ . 5.2.1. The class of uniform indiscernibles is C = x∈ C x . We have that C is the class of α that are cardinals in all models L[x], or, equivalently, the class of all α that are x-admissible for all x ∈ , since “α is x # -admissible” implies that α ∈ C x , and that implies that α is a cardinal in L[x] which in turn implies that α is x-admissible. We denote the uniform indiscernibles by c1 , c2 , . . . , cα , . . . in order of magnitude. By a) and b) every uncountable cardinal belongs to C . Clearly c1 = ℵ1 . Lemma 5.4 is due to Solovay, and we present his proof of it with his permission. Solovay proved the lemma to get a bound on the number of constructible subsets of [Sol67B, p. 51]. The observation that cf(dα ) = cf(d1 ), which is used in Solovay’s proof, is due to us. We originally used it to show that Solovay’s proof of “AD implies that ℵ2 is measurable” does not extend to ℵ3 . Lemma 5.4 (Solovay). Assume that for all x there is a y such that y = x # . Then cf(cα+1 ) = cf(c2 ) for all α ≥ 1. 5 Jeff
Paris pointed out to us that DC is not really used.
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DONALD A. MARTIN
Proof. If cα = cx then cα is definable in L[x # ] from and x # . As in the proof of Lemma 5.2, if < cα we can derive a contradiction. Thus cα = ccxα .6 Set dα = supx∈ ccxα +1 . We show that dα = cα+1 and cf(dα ) = cf(d1 ). Obviously dα ≤ cα+1 . If dα < cα+1 then there is an x such that dα ∈ C x . x By a) of 5.1.1 there is a such that cx < dα < c+1 . By definition of dα there y x ≤ dα < c+1 . By Lemma 5.2, we get cc[x,y] ≥ ccx +1 > dα , is a y with cx < cα+1 α +1 a contradiction. To evaluate the cofinality of dα , note that
ccx1 +1 = ccy1 +1 ⇐⇒ ccxα +1 = ccyα +1 by c) of 5.1.1, since c x[x,y]# c
is equivalent to
+1
= c y[x,y]# c
+1
# L[[x, y]# ] |= ϕ c[x,y] , [x, y]# #
for a certain formula ϕ, and c1 and cα belong to c [x,y] . n 5.2.2. Given f ∈ Fn , we define the canonical extension of f to [Ord] . The canonical extension of f we also denote by f. If f ∈ Fn then, using Lemma 5.1, f(1 , . . . , k ) = hϕx (cαx1 , . . . , cαxj , 1 , . . . , k ) for some x, ϕ, and α1 < · · · < αj < ℵ1 . We use this same equation to define the canonical extension of f. We must show that the extension is well-defined. If hϕx (cαx1 , . . . , cαxj , 1 , . . . , k ) = hx (cy1 , . . . , cyn , 1 , . . . , k ) for some 1 < · · · < k , then we can define in L[x, y]—from [x, y], countable elements of C [x,y] and arbitrary sufficiently large elements of C [x,y] —the least k such that for some 1 < · · · < k−1 < k inequality holds. Since k < c[x,y] for some , k < cℵ[x,y] by c) of 5.1.1. 1 Theorem 5.5. Assume that for all x there is a y such that y = x # . Then the order type of Fn∗ is cn+1 . Proof. To show that the order type of Fn∗ is ≤ cn+1 , we prove that [f] < [g] ⇐⇒ f(c1 , . . . , cn ) < g(c1 , . . . , cn ) and that f(c1 , . . . , cn ) < cn+1 . 6 This
fact is not really needed in the proof which follows, but it simplifies the notation.
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Suppose for example that n = 1 and f() = hϕx () and g() = hy (). Then f(c1 ) < g(c1 ) ⇐⇒ hϕx (c1 ) < hy (c1 ) ⇐⇒ (∀α ≥ 1)(hϕx (cα[x,y] ) < hy (cα[x,y] )) ⇐⇒ [f] = [g]. The argument in the general case is merely more complicated to state, and we omit it. Since f(c1 , . . . , cn ) < cα for some α, we must have f(c1 , . . . , cn ) < cn+1 by c) of 5.1.1. To prove that the order type of Fn∗ is ≥ cn+1 , we show that every ordinal < cn+1 is f(c1 , . . . , cn ) for some f ∈ Fn . We proceed by induction on n. If < cn+1 , then by the proof of Lemma 5.4, < ccxn +1 for some x. By d) of 5.1.1 and Lemma 5.1, = hϕx (cαx1 , . . . , cαxk , cn ) for some ϕ and α1 < · · · < αk < cn . The desired result follows by induction. Corollary 5.6 (AC). Assume that for all x there is a y such that y = x # . Then the order type of Fn∗ is < ℵ3 . Proof. By Lemma 5.4. If the axiom of choice is not assumed, we get only the following result:
Corollary 5.7. Assume that for all x there is a y such that y = x # . Then the order type of Fn∗ is ≤ n+1 . Corollary 5.8. Assume that for all x there is a y such that y = x # . If any k , k ≥ 2 has cofinality = cf(c2 ), then the order type of Fn∗ is < k . These latter two corollaries will be important in drawing consequences of full determinacy. 5.3. Analysis of Σ13 sets. We wish to prove that every Σ13 set is Card( n Fn∗ ) 3.7 it is enough to prove that every Π 1 set is Card( F ∗ )Suslin. By Theorem 2 n n Suslin. 5.3.1. Let E be Π12 and let T be the tree defined from some normal form 1 Σ2 representation ofthe complement A of E as in §4. By varying the relation (see §4), we may arrange that the empty sequence is unsecured. R We note that T is homogeneous in the following sense: Lemma 5.9. For each x(n), y(n), there is a unique permutation of the sequence 0, . . . , n − 1 such that, for every : n → ℵ1 , x(n), y(n), ∈ T ⇐⇒ (∀j, k < n)((j) < (k) ⇐⇒ (j) < (k)). Furthermore (0) = n − 1.
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Proof. Immediate from the definition of T . We have (0) = n − 1 because
0 is the empty sequence, which is maximal in the Brouwer-Kleene ordering and unsecured by assumption. For convenience, we vary the definition of T (and so of Tx ) for the remainder of this section, stipulating that the empty sequence does not belong to T . If x ∈ E, Tx is wellfounded and so there is a function H : Tx → ℵ1 which preserves the tree ordering: Set H( , ) = | , |Tx . Then H( , ) < ℵ1 by the final clause of Lemma 5.9 and our temporary convention that the empty sequence is not in T . The H we have defined is, furthermore, constructible from Tx and so from a real (any real coding x and R). For fixed , to define an H : Tx → ℵ1 which is constructible from a real on all arguments , is, by Lemma 5.9, the same thing as choosing an element f of Flh( ) . Hence any function H : Tx → ℵ1 constructible from a real determines functions f 1 , f 2 , . . . with f n ∈ Flh( n ) and any such f 1 , f 2 , . . . determine an H : Tx → ℵ1 (which is constructible from a real, assuming a form of the countable axiom of choice). Using Lemma 5.9, we may extend T to a tree Tˆ on × × Ord by setting , ∈ Tˆx ⇐⇒ ((j) < (k) ⇐⇒ (j) < (k)) for every : n → Ord, with as in Lemma 5.9. Lemma 5.10. Let Y ⊆ Ord be uncountable. Then Tˆx Y is wellfounded ⇐⇒ Tx is wellfounded. Proof. If Tˆx is not wellfounded, Tˆx Y is not wellfounded for some countable Y . By homogeneity, Tˆx Z is not wellfounded for every Z of order type ≥ that of Y . Suppose we are given functions f 1 , f 2 , . . . with f n ∈ Flh( n ) . Using the canonical extensions of the f n , we can define a canonical extension Hˆ : Tˆx → Ord of the H : Tx → ℵ1 associated with the f n . ˆ Lemma 5.11. If H is order preserving, then so is H. Proof. Similar to the proof in 5.2.2 that the canonical extension of f ∈ Fn to [Ord]n is well-defined. ∗ By the proof of Theorem 5.5, an element [f] of Fk determines uniquely f[C ]k . where C is the class of uniform indiscernibles. Therefore, if we pick only [f n ] and not f n for n = 1, 2, . . . , we still define H( n , ) for all sequences from C . We are now ready to define a tree T on × n Fn∗ . Set x(n), ∈ T just ∗ ˆ Tˆ C ) in case for all 0 < m < n, we have (m) ∈ Flh( and furthermore H( m) is order preserving, where H : Tˆ → Ord is the function determined by setting
PROJECTIVE SETS AND CARDINAL NUMBERS
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[f n ] = (n), n = 1, 2, . . . . One might note that the latter condition is the same as that H(Tˆ {cn : n ∈ }) be order preserving. Lemma 5.12. x ∈ E ⇐⇒ Tx is not wellfounded. Proof. Suppose x ∈ E. Then an order preserving H : Tx → ℵ1 exists which is constructible from a real. Let f 1 , f 2 , . . . be the functions associated with H as above. Lemma 5.11 implies that [f 1 ], [f 2 ], . . . yield an infinite descending chain in Tx . On the other hand, any infinite descending chain in Tx determines a function Hˆ : Tˆx C → Ord which preserves order and so witnesses that Tx is wellfounded, by Lemma 5.10. Theorem 5.13. If for all x there is a y such that y = x # , then every Σ13 set is Card( n Fn∗ )-Suslin. Corollary 5.14 (AC). If for all x there is a y such that y = x # , then every set is ℵ2 -Suslin.
Σ13
Proof. Immediate from Corollary 5.6. Without the axiom of choice, we can still use Corollaries 5.7 and 5.8 to get:
Corollary 5.15. If for all x there is a y such that y = x # , then every Σ13 set is ℵ -Suslin. Corollary 5.16. If for all x there is a y such that y = x # and if some ℵn+1 , n ≥ 1, has cofinality differing from and the cofinality of c2 , then every Σ13 set is ℵn -Suslin. From Corollary 5.14 and Theorem 3.1, we derive 13
Corollary 5.17 (AC). If for all x there is a y such that y = x # , then ≤ ℵ3 . Without the axiom of choice, Corollary 5.15 and Theorem 3.1 yield
Corollary 5.18. If for all x there is a y such that y = x # , then 13 ≤ ℵ+1 . In §7 we show that AD implies 13 = ℵ+1 . Corollary 5.19 (AC). If for all x there is a y such that y = x # , then every Σ13 set is a union of ℵ2 Borel sets. Proof. Immediate from Corollary 5.14 and Theorem 3.5. Without AC we still get the following result: Corollary 5.20. If for all x there is a y such that y = x # and if Card(c ) is not cofinal with , then every Σ13 set belongs to B(c )+ .
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Proof. Immediate from Theorem 5.13 and Corollary 3.6.
#
Corollary 5.21. If for all x there is a y such that y = x and if Card(c ) is not cofinal with , then every Σ13 set belongs to Bc2 +1 . Proof. By Lemma 5.4, every ℵn , n ≥ 2 no larger than c must be cofinal with c2 . The result then follows from Corollary 5.20. Lemma 5.22. c < 13 . Proof. Since Fn∗ is isomorphic to cn+1 , we can code ordinals < cn+1 by elements of Fn . Elements f of Fn all have the form f(1 , . . . , n ) = hϕx (cαx1 , . . . , cαxj , 1 , . . . , n ) with the αi countable. The cαxi and x can be coded by some y. We can then code f by y and ϕ. In §4 of [MS69] it is essentially shown that the resulting prewellordering is Δ13 . Corollary 5.23. If for all x there is a y such that y = x # and if Card(c ) is not cofinal with , then Σ13 ⊆ B31 . Proof. Immediate from Corollary 5.21 and Lemma 5.22. Corollary 5.24. If for all x there is a y such that y = x # , then any two disjoint Σ13 sets are separated by elements of B 13 . Proof. Immediate from Theorem 5.13, Theorem 5.5, Lemma 5.22, and Theorem 3.3. Corollary 5.25. If for all x there is a y such that y = x # , then Δ13 ⊆ B 13 . 5.3.2. Relative to the uniform indiscernibles we can evaluate 12 exactly: Theorem 5.26. If for all x there is a y such that y = x # , then 12 = c2 . Proof. By Corollary 3.2 and by the proof of Theorem 4.4, we see that each wellfounded relation Σ12 in x has ordinal < the first x-admissible ordinal after c . ℵ1 , which is smaller than 2 If α < c2 , then α < cℵx1 +1 for some x by the proof of Lemma 5.4. Each ordinal < α is coded by hϕx (cαx1 , . . . , cαxn , cℵx1 ) for some ϕ and α1 , . . . , αn < ℵ1 . If we code by ϕ and codes for α1 , . . . , αn , the resulting prewellordering is Π11 in x # . Hence α < 12 . §6. Higher levels in the projective hierarchy. Theorem 6.1 (Moschovakis [Mos71]). Det(Δ12n )+DC implies that every 1 Σ2n+2 set is 12n -Suslin, and every Σ12n+1 set is 2n+1-Suslin, where 2n+1 < 12n+1 .
Using results of Moschovakis [Mos70], it can be shown that AD+DC implies that Σ1n is equal to the set of n -Suslin sets for all n ≥ 1, where 2m+2 = 12m+1 .
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Corollary 6.2 (Independently due to Kunen). Det(Δ12n )+DC implies that ≤ ( 12n+1 )+ . Proof. Immediate from Theorem 3.1.
12n+2
Corollary 6.3 (AC). Det(Δ12 ) implies that 14 ≤ ℵ4 . Proof. From Corollary 5.14 and Corollary 5.17, since Det(Δ12 ) implies “for all x there is a y such that y = x # ”. (See [Fri71A] or [Mar68].) Corollary 6.4. Det(Δ12 )+DC implies that 14 ≤ ℵ+2 . Proof. From Corollary 5.14 and Corollary 5.18. Corollary 6.5 (Moschovakis [Mos71]). Δ12n+1 ⊆ B 12n+1 . Proof. Immediate from Corollary 3.4.7
Det(Δ12n )+DC
implies that
1n
Corollary 6.6 (AC). If PD holds and if ≤ ℵn for all n ≥ 1, then every Σ1n+1 set is a union of ℵn Borel sets. Proof. Immediate from Theorem 3.5. §7. Consequences of the full axiom of determinacy. In this section we show that AD+AC implies that B 1n ⊆ Δ1n for all odd n (and so that B 1n ⊆ Δ1n by Moschovakis’ Corollary 4 to(his) Theorem 5). This theorem, combined with the results of §5, has surprising consequences concerning the 1n and the ℵα . We recall that PWO(Σ1n ) is the assertion that there is a map g : U → Ord, where U is a universal Σ1n set, such that there are a Σ1n relation R1 and a Π1n relation R2 with g(x) < g(y) ⇐⇒ x, y ∈ R1 and y ∈ U =⇒ (g(x) < g(y) ⇐⇒ x, y ∈ R2 ). Sep(Σ1n ) is the assertion that any two disjoint (Σ1n ) sets are separated by a Δ1n 1 ) is the assertion that, for any Σ1 sets A and B, there are disjoint set. Red(Σ n n A ∪ B = A ∪ B. Σ1n sets Aand B with A ⊆ A, B ⊆ B, and PWO(Π1 ), Red(Π1 ), and Sep(Π1 ) are similarly defined (putting “Π” for n n n
“Σ” everywhere). Obviously Red(Π1 ) =⇒ Sep(Σ1 ) and Red(Σ1 ) =⇒ Sep(Π1 ). The following n n n n fact. (Inparticular, it holds lemma is a specialcase of a well-known general with “Π” in place of “Σ”.) 7 The
case n = 1 is, of course, just a weak version of Corollary 5.25.
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Lemma 7.1. PWO(Σ1n ) =⇒ Red(Σ1n ) =⇒ ¬Sep(Σ1n ). Proof. Let U , g, R1 and R2 be as in the definition of PWO(Σ1n ). Let A = {x : y0 , x ∈ U } and B = {x : y1 , x ∈ U }. Let x ∈ A ⇐⇒ x ∈ A & g(y0 , x) < g(y1 , x); x ∈ B ⇐⇒ x ∈ B & g(y0 , x) < g(y1 , x). Using the properties of R1 and R2 , we see that A and B are Σ1n and so that they have the properties as in the definition of Red(Σ1n ). 1 1 Assume that Red(Σn ) and Sep(Σn ) both hold. Consider a Σ1n enumeration Pull the pairs apart by Red(Σ1 ) and push of all pairs of Σ1n sets. them back n 1 1 together by Sep(Σn ). This yields a universal Δn set, which an easy diagonal be impossible. argument shows to Theorem 7.2. AD implies that B 1n ⊆ Δ1n or PWO(Σ1n )). Proof. We first give the proof of a result of Wadge which shows essentially that any set in Σ1n \ Π1n is a “universal” Σ1n set. Say that A1 ≤W A2 for A1 , A2 ∈ if there is acontinuous function f : → such that f(A1 ) ⊆ A2 and f( \ A1 ) ⊆ \ A2 . Lemma 7.3 (Wadge). AD implies that for any A1 and A2 , we have either A1 ≤W A2 or A2 ≤W \ A1 . Proof of Lemma 7.3. Consider the game which player I wins if I’s play belongs to A2 if and only if player II’s play belongs to A1 . A winning strategy for player I gives a continuous function witnessing A1 ≤W A2 and a winning strategy for player II gives a function witnessing A2 ≤W \A1 . (Lemma 7.3) Note that we needed determinacy only for the game used in the proof, whose payoff set is closely related to A1 and A2 . For projective A1 and A2 , for example, only PD is needed. Returning to the proof of the theorem, assume AD and that B 1n ⊆ Δ1n and let be the least ordinal such that some union of elements of Δ1n is not Δ1n . is not Δ 1 . By our assumption, < 1n . Let Aα , α < be Δ1n sets whose union n Using the minimality of , we may assume that the family is increasing (i.e., if α < , then Aα ⊆ A ). Also by the minimality of , we know that for every α < , the set Aα = <α A is Δ1n . Let x → |x| be a surjection of onto such that the relation |x| ≤ |y| is Δ1n . This function exists, since < 1n . By [Mos70, Lemma 6], every union of < 1n Σ1n sets is Σ1n , so {x, y : x ∈ A|y| & |y| = α}) ∈ Σ1n . {x, y : x ∈ A|y| } (= α<
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By the same lemma, the relations x ∈ A|y| and x ∈ A|y| are Σ1n and we have that A = α< Aα ∈ Σ1n . Let U be a universal Σ1n set, and let h :× → be the standard homeomorphism. By Wadge’s Lemma 7.3, either h(U ) ≤W A or A ≤W \ h(U ). The latter alternative implies A ∈ Π1n contrary to hypothesis. Let f then witness h(U ) ≤W A. Define g(x) = α[fh(x) ∈ Aα ]; R1 (x1 , x2 ) ⇐⇒ (∃y)(fh(x1 ) ∈ A|y| & fh(x2 ) ∈ A|y| ) & fh(x2 ) ∈ A; R2 (x1 , x2 ) ⇐⇒ (∀y)(fh(x2 ) ∈ A|y| =⇒ fh(x1 ) ∈ A|y| ). U , g, R1 and R2 witness PWO(Σ1n ). (Theorem 7.2) Note that Theorem 7.2 holds for any reasonable class closed under projection in place of Σ1n . Corollary 7.4. If n is odd, then AD+DC implies that B 1n ⊆ Δ1n . Proof. By [AM68] and [Mar68], AD+DC gives Sep(Σ1n ), so PWO(Σ1n ) is ruled out by Lemma 7.1. Corollary 7.5 (Generalized Suslin-Kleene Theorem). If n is odd, then AD+DC implies that Δ1n = B 1n . Proof. Immediate from Corollary 7.4 and Corollary 6.5. (Moschovakis pointed out Corollary 7.5 when he proved Corollary 6.5 [Mos71].) In the case n = 3, DC was not used in proving “if for all x there is a y such that y = x # , then Δ13 ⊆ B 13 ” (Corollary 5.25). Furthermore, DC is not needed to prove Red(Π13 ) and so Sep(Σ13 ) from Det(Δ12 ), as we see as follows: get by with the weaker assumption In the proof of Red from PWO, we can that g maps U into a linearly ordered set (instead of a wellordered set such as the ordinals). In the proof of PWO(Π13 ) from Det(Δ12 ) (see, e.g., [Mar68]), a well-ordered set. Hence we have DC is used only to prove that g maps into Corollary 7.6. AD implies that Δ13 = B 13 . Corollary 7.7. AD implies that c = ℵ . Proof. Suppose c < ℵ . Then Card c = ℵn for some n ≥ 1. But AD implies that ℵn is not cofinal with , since (Friedman; Moschovakis [Mos70]) 2ℵn−1 is a surjective image of and the axiom of choice for countable sets of sets of reals follows from AD [Myc64]. The Corollary follows by Corollary 5.23 and Corollary 7.6.
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Kunen and Solovay have independently used Corollary 7.7 to show that AD =⇒ cn = ℵn for each n. Parts of Corollaries 7.8 and Corollary 7.11 below depend upon an unpublished result of Solovay. Corollary 7.8. AD implies that for all n ≥ 2, we have that ℵn has cofinality ℵ2 . Proof. The cardinal ℵ2 is regular, since Solovay (unpublished) has proved ℵ2 measurable. By Lemma 5.4 (Solovay) and the fact that each ℵn , n ≥ 1, is some cm , m ≥ 1, cf(ℵn ) = cf(ℵ2 ). Corollary 7.9. AD implies that 13 = ℵ+1 . Proof. Assume AD. That 13 ≤ ℵ+1 follows from Corollary 5.18 and the x there is a y such that y = x # . fact that AD implies that for all 1 By Lemma 5.12, 3 > c = ℵ and, by [Mos70, Theorem 8], 13 is a . cardinal, so 13 ≥ ℵ+1 Corollary 7.10. AD+DC implies that 14 = ℵ+2 . Proof. 14 > 13 follows from PWO(Π13 ). Also, 14 is a cardinal by Moscho inequality 1 ≤ ℵ is Corollary vakis [Mos70]. The 6.3. +2 4 Corollary 7.11. AD implies that the nth uncountable regular cardinal = 1n for n = 1, 2, 3. Proof. By Theorem 5.26, 12 = c2 = ℵ2 , since 12 is a cardinal [Mos70, Theorem 8]. The cardinal ℵ2 is regular by Solovay’s result mentioned in the proof of Corollary 7.7.8 The case n = 3 follows from the case n = 2, Corollaries 7.8 and 7.9, and the fact that 13 is a regular cardinal. (If 13 were singular, we would have 3.6. [Mos70] shows 1 regular Σ13 ⊆ B( 13 )+ = B 13 by the proof of Corollary 3 from AD, but he needs DC.) Kunen has recently shown that 1n is measurable for all n (we earlier showed 7.11 can be extended (assuming DC) to this for all odd n), and so Corollary cover the case n = 4. Furthermore “regular” can be replaced by “measurable” in the statement of that corollary. Given our results and the result of Kunen just alluded to, we know that ℵ1 , ℵ2 , ℵ+1 and ℵ+2 are measurable (ℵ1 and ℵ2 due to Solovay) while ℵ3 , ℵ4 , . . . are singular, assuming AD+DC. One would expect ℵ+3 , . . . to be singular, ℵ·2+1 and ℵ·2+2 to be measurable, etc. This expectation is reinforced by Kechris’ recent result that AD+DC =⇒ each odd 1n is the successor of a 8 Solovay’s
proof in fact shows that c2 is a measurable cardinal.
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cardinal cofinal with . Recall also that 1 and 2 are regular, while 3, 4, . . . are singular. REFERENCES
John W. Addison and Yiannis N. Moschovakis [AM68] Some consequences of the axiom of definable determinateness, Proceedings of the National Academy of Sciences of the United States of America, no. 59, 1968, pp. 708–712. Harvey Friedman [Fri71A] Determinateness in the low projective hierarchy, Fundamenta Mathematicae, vol. 72 (1971), pp. 79–95. Stephen Jackson [Jac11] Projective ordinals. Introduction to Part IV, 2011, this volume. Casimir Kuratowski ´ [Kur58] Topologie. Vol. I, 4`eme ed., Monografie Matematyczne, vol. 20, Panstwowe Wydawnictwo Naukowe, Warsaw, 1958. Richard Mansfield [Man71] A Souslin operation on Π12 , Israel Journal of Mathematics, vol. 9 (1971), no. 3, pp. 367– 379. Donald A. Martin [Mar68] The axiom of determinateness and reduction principles in the analytical hierarchy, Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687–689. [Mar77A] α-Π11 games, planned but unfinished paper, 1977. Donald A. Martin and Robert M. Solovay [MS69] A basis theorem for Σ13 sets of reals, Annals of Mathematics, vol. 89 (1969), pp. 138–160. [MS70] Internal Cohen extensions, Annals of Mathematical Logic, vol. 2 (1970), no. 2, pp. 143– 178. Yiannis N. Moschovakis [Mos70] Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory. Proceedings of an international colloquium held under the auspices of the Israel Academy of Sciences and Humanities, Jerusalem, 11–14 November 1968 (Y. Bar-Hillel, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam-London, 1970, pp. 24–62. [Mos71] Uniformization in a playful universe, Bulletin of the American Mathematical Society, vol. 77 (1971), pp. 731–736. Jan Mycielski [Myc64] On the axiom of determinateness, Fundamenta Mathematicae, vol. 53 (1964), pp. 205– 224. Joseph R. Shoenfield [Sho67] Mathematical logic, Addison-Wesley, 1967. Robert M. Solovay [Sol67B] A nonconstructible Δ13 set of integers, Transactions of the American Mathematical Society, vol. 127 (1967), no. 1, pp. 50–75.
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DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES, CALIFORNIA 90024 UNITED STATES OF AMERICA
E-mail: [email protected]
REGULAR CARDINALS WITHOUT THE WEAK PARTITION PROPERTY
STEVE JACKSON
§1. Introduction. We work throughout in the theory ZF+DC, though our main results will require AD as well. Recall that κ → (κ)ϑ if for all partitions P : (κ)ϑ → {0, 1} of the increasing functions from ϑ to κ into two pieces, there is a homogeneous set H ⊆ κ of size κ. That is, there is an i ∈ {0, 1} such that P restricted to (H )ϑ has constant value i. We say κ has the weak partition property, κ → κ <κ , if κ → κ ϑ for all ϑ < κ, and κ has the strong partition property if κ → κ κ . Our purpose is to prove a result which shows, assuming AD, that there are many regular cardinals without the weak partition property. In contrast, a result of Steel [Ste95] shows, assuming AD + V=L(R), that every regular cardinal below Θ is measurable. It is probably true (but not proven) that every regular Suslin cardinal has the strong partition relation, so the problem hinges on the regular cardinals between Suslin cardinals. The first two of these are the even projective ordinals 12 = ℵ2 , 14 = ℵ+2 , and a theorem gives that they have the of Martin and Paris (see corollary 13.3 of [Kec78]) weak partition relation. However, between 14 and 15 there are two additional there are 2n − 2 regular regular cardinals, ℵ·2+1 and ℵ +1 . In general, cardinals strictly between 12n and 12n+1 . Our result here implies that these cardinals do not have the weak partition property. In fact, we show exactly what exponent partition relations they satisfy. By a measure on a set (usually an ordinal) we mean a countably additive ultrafilter. Recall from AD that every ultrafilter on a set is countably additive, and so is a measure. If is a measure and f : dom( ) → Ord, we write [f] for the ordinal represented by the equivalence class of f in the ultrapower by the measure . §2. Negative Partition Results. We say a function f : ϑ → Ord has uniform cofinality if there is a f : · ϑ → Ord such that for all α < ϑ f(α) = sup<·(α+1) f (). We say f is of the correct type if f is increasing, everywhere discontinuous, and of uniform cofinality . We similarly define f Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II ¨ Edited by A. S. Kechris, B. Lowe, J. R. Steel Lecture Notes in Logic, 37 c 2011, Association for Symbolic Logic
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having uniform cofinality 1 , etc. If ≺ is a well-ordering, we say a function f : dom(≺) → Ord has the correct type if f is order-preserving with respect to ≺, everywhere discontinuous, and of uniform cofinality . We employ a useful notational convention throughout. If α ∈ Ord, and is a measure (on an ordinal), we write ∀∗ P(α()) to mean: if [f] = α, then ∀∗ P(f()). Similarly, if 1 , 2 are measures and α ∈ Ord, ∀∗ 1 ∀∗ 2 P(α(, )) abbreviates: if [f] 1 = α, then ∀∗ 1 if [g] 2 = f(), then ∀∗ 2 P(g()). The following fact is well-known. It says that the partition property can be reformulated using c.u.b. homogeneous sets provided we restrict the “type” of the function. Fact. For ϑ with ϑ = · ϑ, κ → (κ)ϑ iff for all partitions P : (κ)ϑ → {0, 1} there is an i ∈ {0, 1} and a c.u.b. C ⊆ κ such that ∀f : ϑ → C of the correct type, P(f) = i. Proof. Assume κ → (κ)ϑ , and let P : (κ)ϑ → {0, 1}. Define P : (κ)·ϑ = (κ)ϑ → {0, 1} by: P (f ) = P(f), where f(α) := sup<·(α+1) f (). If H is homogeneous for P , and C is the set of limit points of H , then C is as required. Conversely, given P : (κ)ϑ → {0, 1}, and a c.u.b. set C as in the fact, the function h(α) := the · (α + 1)st element of C enumerates a set H homogeneous for P. We will henceforth officially use the c.u.b. version of the partition property (equivalent to the original definition for ϑ = · ϑ). We fix some notation we will use for the rest of this section. We assume κ is a cardinal with the strong partition property, and < κ < κ is a regular cardinal. The c.u.b. filter restricted to points of cofinality κ defines a normal measure on κ which we denote by . Similarly, , etc., denotes the -cofinal normal measure on κ. We will assume that the c.u.b. filter restricted to points of cofinality of κ also defines a normal measure on κ, which we denote by . Let M denote the transitive collapse of the ultrapower of V by . For a measure, let j denote the embedding into the (transitive collapse of) the ultrapower of V by . We state now our main result. Theorem 2.1. Suppose κ → (κ)κ , < κ < κ is regular, let , be as above, and assume κ is closed under j . Then := j (κ) does not have the weak partition property, in fact, ( )ϑ for ϑ = [h] where h(α) = j (α). The next lemma is similar to the arguments of the Martin-Paris theorem. Lemma 2.2. Let = j (κ), and ϑ < . Fix h : κ → κ with [h] = ϑ. Let ≺ be lexicographic ordering on pairs (α, ) with α < κ, < h(α). Then → ( )ϑ iff (∗) for every g : ϑ → ( − κ) of the correct type, there is a G : dom(≺) → of the correct type with [G] = g.
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Remark 2.3. (∗) implies that P(ϑ) ⊆ M. Proof. Assume first (∗), and let P be a given partition of the functions g : ϑ → of the correct type. Let P be the partition of G : dom(≺) → κ of the correct type defined by: P (G) = P([G] ). Note that if G : dom(≺) → κ is of the correct type, then [G] : ϑ → is of the correct type. Let C ⊆ κ be c.u.b. and homogeneous for P , say for the 1 side. Let D = j (C ) ⊆ . Easily, D is c.u.b. in . Suppose g : ϑ → D is of the correct type. By (∗), let G : dom(≺) → κ be of the correct type with [G] = g. Since g has range in D, we easily have that ∀∗ α ∀ < h(α) G(α, ) ∈ C . By changing the value of G(α, ) for α off a measure one set, we may assume G has range in C . Then, P(g) = P (G) = 1, and we are done. Suppose now → ( )ϑ . First consider the partition P of functions g : ϑ → of the correct type, with P(g) = 1 iff g ∈ M and there is a function of the correct type representing g. Let D ⊆ be c.u.b. and homogeneous for P. Suppose D were homogeneous for the 0 side of P. There is a c.u.b. C ⊆ κ such that j (C ) − κ ⊆ D. To see this, partition functions f1 , f2 : κ → κ of the correct type with f1 (α) < f2 (α) < f1 (α + 1) according to whether [f2 ] > ND ([f1 ] ), where ND (α) := the least element of D greater than α. Using the fact that changing f1 , f2 off a c.u.b. set does not change [f1 ] , [f2 ] , an easy argument shows that on the homogeneous side the stated property holds. If C is homogeneous for this partition, and C = C is the set of limit points of C , then j (C ) − κ ⊆ D. Fix G : dom(≺) → C of the correct type. Then g := [G] : ϑ → D is of the correct type, a contradiction. Thus, D is homogeneous for the 1 side of P. As above, let C ⊆ κ be c.u.b. such that j (C ) − κ ⊆ D. Suppose g : ϑ → is of the correct type. Let g2 : ϑ → be defined by g2 (α) = g(α)th element of j (C ). Thus, g2 is of the correct type with range in D. So we may let G2 : dom(≺) → κ be of the correct type with [G2 ] = g2 . We may assume G2 has range everywhere in C . Since each g2 (α) is an limit of points in j (C ), we may also assume that each G2 (α, ) is a limit point of C . Let G : dom(≺) → κ be such that G2 (α, ) = G(α, )th element of C . It follows that G is of the correct type. Also, [G] = g, and we are done. Lemma 2.4. Assume κ is closed under j , and let = [h] , where h(α) = j (α). Then P() M. Proof. Let A = {α : κ < α < ∧ cf(α) = κ}. Suppose A ∈ M. Let A = [F ] , and we may assume F (α) ⊆ h(α) − α for all α < κ. Consider the partition P: we partition u : κ → κ which are increasing, everywhere discontinuous, and with u() of uniform cofinality , according to whether [u] ∈ F (sup(u)). Suppose first that on the homogeneous side of P the stated property fails. Let C ⊆ κ be homogeneous for the contrary side, and C = (C ) . For α ∈ C
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of cofinality κ, define f(α) by: ∀∗ f(α)() = the th element of C > α(). For such α and < κ, define f (α) by: ∀∗ f (α)() = the th element of C > α(). By the normality of , sup<κ f (α) = f(α). Thus, [f] has cofinality κ, and hence ∀∗ α f(α) ∈ F (α). However, ∀∗ α, f(α) = [u] where u : κ → C is increasing, discontinuous, and u() has uniform cofinality . Thus, f(α) = [u] ∈ / F (α), a contradiction. Next, suppose that on the homogeneous side of P the stated property holds. Let C be homogeneous, and C = (C ) . Fix k : κ → C increasing, discontinuous, with k(α) of uniform cofinality α. Let be defined on pairs (α, ) with < α such that k(α) = sup<α (α, ). For α ∈ C of cofinality κ closed under k, define f(α) by: ∀∗ f(α)() = k(α()). We claim that f(α) has uniform cofinality α. To see this, define for α as above, and < α, f2 (α, ) by: ∀∗ f2 (α, )() = (α(), ). If < f(α), then ∀∗ ∃ < α() () < (α(), ). Thus, we have that ∃ < α ∀∗ () < (α(), ). So, ∃ < α < f2 (α, ). Thus, f(α) = sup<α f2 (α, ), and proves the claim. It follows that cf([f] ) = κ, so [f] ∈ / A. Thus, ∀∗ α f(α) ∈ / F (α). Fix for the moment α < κ of cofinality κ closed under k. Let v : κ → α be increasing, continuous, and cofinal, so that [v] = α. Then f(α) = [u] , where u() = k(v()). Clearly u is increasing, discontinuous, cofinal in α, and has range in C . Also, u() has uniform cofinality , since u() = k(v()) = sup f(sup(u)). Easily, on the homogeneous side this holds. Let C ⊆ κ be homogeneous, and p(α) = the th element of C greater than α. Then, ∀∗ α ∀∗ f(α)() < p(α()). From the Kunen analysis (see [Kec78]), there is a wellordering W on κ such that ∀∗ < κ p() < |W |. Let W () denote W restricted to ordinals
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which are W less than . Thus, ∀∗ α < κ ∀∗ < κ ∃ < α() f(α)() < |(W α())()|. By normality of and we have: ∃ < κ ∀∗ α < κ ∀∗ < κ f(α)() < |W α()()|. Thus, if we define a well-order ≺ of κ by 1 ≺ 2 iff ∀∗ α ∀∗ |W α()(1 )| < |W α()(2 )|, we have |≺| ≥ [f] . Corollary 2.6 (ZF+DC+AD). There are measurable cardinals without the weak partition property. §3. Positive Partition Results. In this section we specialize to within the projective hierarchy. If is a regular cardinal, 12n < < 12n+1 , then according 1 1 to Theorem 2.1 ( ) 2n . We are assuming here 12n−1 → ( 12n−1 ) 2n−1 and 12n−1 is closed under j for all regular κ < 12n−1 . These arefacts from the projective hierarchy analysis (the reader may consult [Jac99] for the complete 1 details below 15 ). Thus, the best one could expect is → ( )<2n . We show in this sectionthat this is the case. The proof of the next lemma uses some of the general theory of descriptions, as developed in [Jac99], or [Jac88] for the general case. Since it is not feasible to review the general theory here, we merely sketch the proof and illustrate with a specific example for n = 2, and = ℵ +1 . We briefly recall some terminology used in the theory of descriptions. Let
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The description analysis shows that every regular cardinal with 12n−1 < < 12n+1 is of the form = j ( 12n−1 ), where is the κ-cofinal normal on 1 , for some regular κ < 1 . measure 2n−1 2n−1 Lemma 3.1 (ZF+DC+AD). Let = j ( 12n−1 ) be a regular cardinal, 12n−1 < 1 + + 1 < 12n+1 . Then for every successor cardinal 2n−1 < ≤ , cf( ) > 2n−1 . in + Furthermore, if ≤ < , then there is a well-ordering of of length M (the ultrapower of V by ). Remark 3.2. cf( + ) > 12n−1 holds for all 12n−1 < + < 12n+1 , and can and Woodin be shown without the description theory, a result of Kechris [KW80]. Proof. We consider the case n = 2, κ = 2 , so = ℵ +1 . If 13 < < ℵ +1 , then + is represented by a description. The reader can consult [Jac99] for the complete definition of description and associated terminology. We briefly recall here what the definitions amount to in the particular case we are considering. In this case one can see that + = (id; d ; S1r ; W1 ) for some integers r, , and description d . Here id : 13 → 13 is the identity func tion. In general, for g : 13 → 13 , (g; d ; S1r , W1 ) is defined to be the ordinal represented with respect to (the 2 -cofinal normal measure on 13 ) by the function which assigns to α = [f]S11 (here f will be increasing, continuous, with sup(f) = α) the value (g; f; d ; S1r , W1 ). This, in turn, is represented with respect to S1r by [h]Wr1 → (g; f; d ; h; W1 ), which is represented with respect to W1 by (1 , . . . , ) → (g; f; d ; h; 1 , . . . , ) := g(f(d ; h; 1 , . . . , )). Finally, (d ; h; 1 , . . . , ) < 2 is represented with respect to W11 by → (d ; h; 1 , . . . , )(). This last ordinal will be of the form h(j)(a1 , . . . , aj , ), or h s (j)(a1 , . . . , aj , ) (depending on d ). It is not difficult to check that this definition is well-defined. To illustrate, if + = ℵ 5 ·2+ 3 ·2+1 , then + = (id; d ; S17 , W18 ), where d is such that (d ; h, 1 , . . . , 8 )() = h1s (4)(3 , 4 , 7 , ). There is a description, denoted L(d ) in the notation of [Jac99], such that if < + , then ∃g : 13 → 13 < (g; L(d ); S1r , W1 ). The ordinal (g; L(d ); S1r ; W1 ) depends only on [g] or [g] 1 , depending on whether ∀∗S r [h] ∀∗W 1 , . . . , cf (d ; h; 1 , . . . , ) = (case 1) or 1 (case 2). Let+
1
1
ting g vary, we obtain a cofinal embedding from either ℵ+2 = j ( 13 ) or ℵ·2+1 = j 1 ( 13 ) into + . for + as above, L(d ) is such that For example, (L(d ); h; 1 , . . . , 8 )() = h(4)(3 , 4 , 6 , ). In this case, ∀∗ h ∀∗ 1 , . . . , 8 (L(d ); h; 1 , . . . , 8 ) has cofinality 1 . The map [g] 1 → (g; L(d ); S1r , W1 ) defines a cofinal map from ℵ·2+1 to + .
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Suppose now < + , and let g : 13 → 13 be such that < (g; L(d ); S1r , W1 ). that ∀∗ α < 1 g(α) < |T α|, and There is a wellfounded tree T on 13 such 3 to Kunen, and the ∀∗ α g(α) < |T supm (jW1m (α))|. The first part is due 1 second to Martin (who proved also a similar result for 2 , replacing W1m by S1m ; see [Jac99]). Fix such a tree T . T defines, in M, a wellordering of = (id; L(d ); S1r , W1 ) in case 1, or of = (supm jW1m ; L(d ); S1r , W1 ), in case 2, of length ≥ . Here supm jW1m denotes the function α → supm jW1m (α). For example, in case 2, for α = [f]S11 < 13 of cofinality 2 and sufficiently . closed, we define a wellordering <α of Ωα = (supm jW1m ; f; L(d ); S1r , W1 )(α) as follows (note that Ωα < 13 and depends only on α, not on the choice of f, and we may assume f is increasing and continuous with sup(f) = α). For 1 , 2 < Ωα , define: 1 <α 2 ↔ ∀∗ [h]W1r ∀∗ 1 , . . . , |(T Ω([h], )))(1 ([h], ))| < |(T Ω([h], ))(2 ([h], ))|. The key point is that this computation can be done entirely locally, that is, just using T α. Then [α → <α ] is the required well-ordering. Theorem 3.3 (ZF+DC+AD). Let 12n ≤ < 12n+1 be a regular cardinal. 1 Then → ( )<2n . Proof. Fix ϑ < 12n . We verify (∗) of Lemma 2.2. We first show that any g : ϑ → is in M. Let ≤ be the least cardinal such that ∃g : ϑ → with g∈ / M. Easily, ≥ 12n (using Lemma 3.1 and the fact that every subset of 13 is in M). If is a successor, then From Lemma 3.1 we may assume ran(g)is bounded in . By Lemma 3.1 again, there is a bijection between sup(g) and − in M. This produces a g : ϑ → − not in M, a contradiction. Since M is closed under sequences, is not a limit cardinal (all the projective ordinals 1n are below ℵ1 ), and we are done. Let [h] = ϑ, and ≺ be lexicographic ordering on pairs (α, ) with α < 12n−1 , < h(α). We next show that if g : ϑ → ≤ has uniform cofinality , then there is a G : dom(≺) → 12n−1 of uniform cofinality with [G] = g. Let again be a minimal counterexample, and assume g : ϑ → . Easily ≥ 12n , and is a successor cardinal. Fix G : dom(≺) → 12n−1 with at least g. Let [t] = − , and [W ] be a well-ordering of − of length [G] = 1 sup(g). We may assume that for all α < 2n−1 that W (α) is a wellordering of t(α) of length at least sup
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If F has uniform cofinality , then so does G. To see this, suppose that F : dom(≺) × → 12n−1 is such that F (α, ) = supn F (α, , n) for all , n) = sup{|| α, . Then define G (α, W (α) : < G (α, , n)}. Easily, G(α, ) = supn G (α, , n) for all α, . To see F has uniform cofinality , it suffices, by minimality of to show that f := [F ] : ϑ → − has uniform cofinality . Let g : ϑ × → induce g, that is, g() = supn g (, n). Define then f : ϑ × → − as follows. If g (, n) = [u] , then set f (, n) = [v] , where v(α) := least such that sup{| |W (α) : < } ≥ u(α), if one exists, and otherwise v(α) = 0. Easily, supn f (α, n) = f(α) for all α. Finally, if g : ϑ → ( − 12n−1 ) is of the correct type, let G : dom(≺) → 12n−1 is with [G] = g. Then for almost all α, G(α) have uniform cofinality 1 increasing and discontinuous from h(α) to 2n−1 . Changing G off a measure one set, we may assume G is of the correcttype.
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REGULAR CARDINALS WITHOUT THE WEAK PARTITION PROPERTY DEPARTMENT OF MATHEMATICS UNIVERSITY OF NORTH TEXAS P.O. BOX 311430 DENTON, TEXAS 76203-1430 UNITED STATES OF AMERICA
E-mail: [email protected]
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