Extensions of the Axiom of Determinacy 9781470472108, 9781470475659

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UNIVERSITY LECTURE SERIES VOLUME 78

Extensions of the Axiom of Determinacy Paul B. Larson

10.1090/ulect/078

Extensions of the Axiom of Determinacy

UNIVERSITY LECTURE SERIES VOLUME 78

Extensions of the Axiom of Determinacy Paul B. Larson

EDITORIAL COMMITTEE Christopher Bishop Panagiota Daskalopoulou

Robert Guralnick (Chair) Emily Riehl

2020 Mathematics Subject Classification. Primary 03E60, 03E15, 03E25, 03E45. The writing of the book was supported by NSF grants DMS-1201494 and DMS-1764320.

For additional information and updates on this book, visit www.ams.org/bookpages/ulect-78

Library of Congress Cataloging-in-Publication Data Names: Larson, Paul B. (Paul Bradley), 1970- author. Title: Extensions of the axion of determinacy / Paul B. Larson. Description: Providence, Rhode Island : American Mathematical Society, [2023] | Series: University lecture series, 1047-3998 ; volume 78 | Includes bibliographical references and index. Identifiers: LCCN 2023032439 | ISBN 9781470472108 (paperback) | 9781470475659 (ebook) Subjects: LCSH: Determinants. | Descriptive set theory. | Axiomatic set theory. | Logic, Symbolic and mathematical. | AMS: Mathematical logic and foundations – Set theory – Determinacy principles. | Mathematical logic and foundations – Set theory – Descriptive set theory. | Mathematical logic and foundations – Set theory – Axiom of choice and related propositions. | Mathematical logic and foundations – Set theory – Inner models, including constructibility, ordinal definability, and core models. Classification: LCC QA191 .L37 2023 | DDC 512.9/432–dc23/eng/20231010 LC record available at https://lccn.loc.gov/2023032439

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established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

28 27 26 25 24 23

Contents Introduction 1. Notation 2. Prerequisites 3. Forms of Choice 4. Partial orders Part 1.

vii x xii xii xii

Preliminaries

1

Chapter 1. Determinacy 1. The Axiom of Determinacy 2. Turing Determinacy

3 3 5

Chapter 2. The Wadge Hierarchy 1. Lipschitz determinacy 2. Lipschitz degrees and Wadge degrees 3. Pointclasses 4. Universal sets 5. The height of the Wadge hierarchy

9 9 12 15 17 22

Chapter 3. Coding Lemmas

25

Chapter 4. Properties of Pointclasses 1. Separation and reduction 2. The prewellordering property 3. Prewellorderings and wellfounded relations 4. Closure under wellordered unions

31 31 35 40 42

Chapter 5. Strong Partition Cardinals

45

Chapter 6. Suslin Sets and Uniformization 1. Suslin sets 2. Uniformization 3. The Solovay sequence

53 53 59 60

Part 2.

AD+

63

Chapter 7. Ordinal Determinacy

65

Chapter 8. Infinity-Borel Sets 1. ∞-Borel codes 2. Local ∞-Borel codes

71 71 76 v

vi

CONTENTS

Chapter 9. Cone Measure Ultraproducts 1. S-cones 2. Measurable cardinals from cone measures 3. The degree order on sets of ordinals 4. Pointed trees 5. Coding ultrapowers 6. Forcing with positive sets

83 85 87 88 92 93 95

Chapter 10. Vopˇenka Algebras 1. The Vopˇenka algebra 2. Codes for projections, and Uniformization 3. The Vopˇenka algebra for ∞-Borel sets

99 99 104 108

Chapter 11. Suslin Sets and Strong Codes 1. Generic codes 2. Producing strong generic codes 3. ∞-Borel representations from Uniformization 4. Closure of the Suslin cardinals

115 115 119 121 124

Chapter 12. Scales from Uniformization 1. Ordinal determinacy in the codes 2. Boundedness for ∞-Borel relations 3. Becker’s argument

127 127 132 135

Chapter 13. Real Determinacy from Scales 1. Weakly homogeneous trees 2. Normal measures on Pℵ1 (λ) 3. AD implies ADR if all sets of reals are Suslin

143 143 149 152

Questions

157

Bibliography

159

Index

163

Introduction

The Axiom of Determinacy (AD) is the statement that all length-ω integer games of perfect information are determined (see Section 1.1 for a more precise definition). If AD holds then it holds in any inner model of ZF containing the real numbers. In particular it holds in L(R), the smallest model of ZF containing the reals and the ordinals. There may be larger models of AD, however. For instance, assuming the stronger axiom ADR (which says that all length-ω real games of perfect information are determined), there are infinitely many models of the theory ZF + V =L(P(R)) containing the reals and the ordinals. Wadge’s Theorem (Theorem 2.4) shows, assuming only AD, that the models of ZF + V =L(P(R)) containing the reals and the ordinals are linearly ordered by containment. The axiom AD+ is an extension, due to W. Hugh Woodin, of the Axiom of Determinacy. Assuming AD, AD+ holds in an initial segment of the models in the linear order just described. Whether it holds in all such models, i.e., whether it is equivalent to AD, is an open question. As we discuss below, AD+ was designed to capture the theory which reflects from certain larger models of AD to smaller ones. In practice it provides a useful general setting for the study of models of determinacy. In particular, models of AD+ are the natural setting for Woodin’s Pmax forcing [52]. They also play an important role in contemporary inner model theory (see, for instance, the recent books [49] and [43, in preparation]). We define AD+ as the conjunction of three statements. 0.1 Definition. The axiom system AD+ consists of the following three statements. (1) DCR . (2) Every subset of ω ω is ∞-Borel. (3) (