Springer Monographs in Mathematics Thomas Jech Set Theory

The Third Millennium Edition, revised and expanded

123 Thomas Jech e-mail: [email protected]

The original edition was published in 1978 by Academic Press, Inc. in the series Pure and Applied Mathematics, vol. 79, with the ISBN 0-12-381950-4

The second edition was published in 1997 by Springer-Verlag under the same title in the series Perspectives in Mathematical Logic

Catalog-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Jech, Thomas J.: Set theory / Thomas Jech. - 3rd Millennium ed, rev. and expanded. - Berlin ; Heidelberg;NewYork;HongKong;London;Milan;Paris;Tokyo: Springer, 2002 (Springer monographs in mathematics) ISBN 3-540-44085-2

Mathematics Subject Classification (2000): 03Exx

ISSN 1439-7382

Corrected 4th printing 2006 ISBN-10 3-540-44085-2 Springer-Verlag Berlin Heidelberg New York ISBN-13 978-3-540-44085-7 Springer-Verlag Berlin Heidelberg New York ISBN 3-540-63048-1 2nd ed. Springer-Verlag Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 1997, 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Typesetting by the author using a Springer TEX macro package Production: LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper 46/3100YL - 5 4 3 2 1 0 For Paula, Pavel, and Susanna Preface

When I wrote the first edition in the 1970s my goal was to present the state of the art of a century old discipline that had recently undergone a revolutionary transformation. After the book was reprinted in 1997 I started contemplating a revised edition. It has soon become clear to me that in order to describe the present day set theory I would have to write a more or less new book. As a result this edition differs substantially from the 1978 book. The major difference is that the three major areas (forcing, large cardinals and descrip- tive set theory) are no longer treated as separate subjects. The progress in past quarter century has blurred the distinction between these areas: forcing has become an indispensable tool of every set theorist, while descriptive set theory has practically evolved into the study of L(R) under large cardinal assumptions. Moreover, the theory of inner models has emerged as a major part of the large cardinal theory. The book has three parts. The first part contains material that every student of set theory should learn and all results contain a detailed proof. In the second part I present the topics and techniques that I believe every set theorist should master; most proofs are included, even if some are sketchy. For the third part I selected various results that in my opinion reflect the state of the art of set theory at the turn of the millennium. I wish to express my gratitude to the following institutions that made their facilities available to me while I was writing the book: Mathematical Institute of the Czech Academy of Sciences, The Center for Theoretical Study in Prague, CRM in Barcelona, and the Rockefeller Foundation’s Bellagio Cen- ter. I am also grateful to numerous set theorists who I consulted on various subjects, and particularly to those who made invaluable comments on prelim- inary versions of the manuscript. My special thanks are to Miroslav Repick´y who converted the handwritten manuscript to LATEX. He also compiled the three indexes that the reader will find extremely helpful. Finally, and above all, I would like to thank my wife for her patience and encouragement during the writing of this book. Prague, May 2002 Thomas Jech Table of Contents

Part I. Basic Set Theory

1. Axioms of Set Theory ...... 3 Axioms of Zermelo-Fraenkel. Why Axiomatic Set Theory? Language of Set Theory, Formulas. Classes. Extensionality. Pairing. Separation Schema. Union. Power Set. Infinity. Replacement Schema. Exercises. Historical Notes.

2. Ordinal Numbers ...... 17 Linear and Partial Ordering. Well-Ordering. Ordinal Numbers. Induction and Recursion. Ordinal Arithmetic. Well-Founded Relations. Exercises. Historical Notes.

3. Cardinal Numbers ...... 27 Cardinality. Alephs. The Canonical Well-Ordering of α × α. Cofinality. Ex- ercises. Historical Notes.

4. Real Numbers ...... 37 The Cardinality of the Continuum. The Ordering of R. Suslin’s Problem. The Topology of the Real Line. Borel Sets. Lebesgue Measure. The Baire Space. Polish Spaces. Exercises. Historical Notes.

5. The Axiom of Choice and Cardinal Arithmetic ...... 47 The Axiom of Choice. Using the Axiom of Choice in Mathematics. The Count- able Axiom of Choice. Cardinal Arithmetic. Infinite Sums and Products. The Continuum Function. Cardinal Exponentiation. The Singular Cardinal Hy- pothesis. Exercises. Historical Notes.

6. The Axiom of Regularity ...... 63 The Cumulative Hierarchy of Sets. ∈-Induction. Well-Founded Relations. The Bernays-G¨odel Axiomatic Set Theory. Exercises. Historical Notes.

7. Filters, Ultrafilters and Boolean Algebras ...... 73 Filters and Ultrafilters. Ultrafilters on ω. κ-Complete Filters and Ideals. Boolean Algebras. Ideals and Filters on Boolean Algebras. Complete Boolean Algebras. Complete and Regular Subalgebras. Saturation. Distributivity of Complete Boolean Algebras. Exercises. Historical Notes. X Table of Contents

8. Stationary Sets...... 91 Closed Unbounded Sets. Mahlo Cardinals. Normal Filters. Silver’s Theo- rem. A Hierarchy of Stationary Sets. The Closed Unbounded Filter on Pκ(λ). Exercises. Historical Notes.

9. Combinatorial Set Theory ...... 107 Partition Properties. Weakly Compact Cardinals. Trees. Almost Disjoint Sets and Functions. The Tree Property and Weakly Compact Cardinals. Ramsey Cardinals. Exercises. Historical Notes.

10. Measurable Cardinals ...... 125 The Measure Problem. Measurable and Real-Valued Measurable Cardinals. Measurable Cardinals. Normal Measures. Strongly Compact and Supercom- pact Cardinals. Exercises. Historical Notes.

11. Borel and Analytic Sets ...... 139 Borel Sets. Analytic Sets. The Suslin Operation A.TheHierarchyofProjective Sets. Lebesgue Measure. The Property of Baire. Analytic Sets: Measure, Category, and the Perfect Set Property. Exercises. Historical Notes.

12. Models of Set Theory ...... 155 Review of Model Theory. G¨odel’s Theorems. Direct Limits of Models. Re- duced Products and Ultraproducts. Models of Set Theory and Relativization. Relative Consistency. Transitive Models and ∆0 Formulas. Consistency of the Axiom of Regularity. Inaccessibility of Inaccessible Cardinals. Reflection Principle. Exercises. Historical Notes.

Part II. Advanced Set Theory

13. Constructible Sets ...... 175 The Hierarchy of Constructible Sets. G¨odel Operations. Inner Models of ZF. The L´evy Hierarchy. Absoluteness of Constructibility. Consistency of the Ax- iom of Choice. Consistency of the Generalized Continuum Hypothesis. Relative Constructibility. Ordinal-Definable Sets. More on Inner Models. Exercises. Historical Notes.

14. Forcing ...... 201 Forcing Conditions and Generic Sets. Separative Quotients and Complete Boolean Algebras. Boolean-Valued Models. The Boolean-Valued Model V B . The Forcing Relation. The Forcing Theorem and the Generic Model Theorem. Consistency Proofs. Independence of the Continuum Hypothesis. Indepen- dence of the Axiom of Choice. Exercises. Historical Notes.

15. Applications of Forcing ...... 225 Cohen Reals. Adding Subsets of Regular Cardinals. The κ-Chain Condition. Distributivity. Product Forcing. Easton’s Theorem. Forcing with a Class of Conditions. The L´evy Collapse. Suslin Trees. Random Reals. Forcing with Perfect Trees. More on Generic Extensions. Symmetric Submodels of Generic Models. Exercises. Historical Notes. Table of Contents XI

16. Iterated Forcing and Martin’s Axiom ...... 267 Two-Step Iteration. Iteration with Finite Support. Martin’s Axiom. Indepen- dence of Suslin’s Hypothesis. More Applications of Martin’s Axiom. Iterated Forcing. Exercises. Historical Notes.

17. Large Cardinals ...... 285 Ultrapowers and Elementary Embeddings. Weak Compactness. Indescribabil- ity. Partitions and Models. Exercises. Historical Notes.

18. Large Cardinals and L ...... 311 Silver Indiscernibles. Models with Indiscernibles. Proof of Silver’s Theorem and 0. Elementary Embeddings of L. Jensen’s Covering Theorem. Exercises. Historical Notes.

19. Iterated Ultrapowers and L[U] ...... 339 The Model L[U]. Iterated Ultrapowers. Representation of Iterated Ultrapow- ers. Uniqueness of the Model L[D]. Indiscernibles for L[D]. General Iterations. The Mitchell Order. The Models L[U]. Exercises. Historical Notes.

20. Very Large Cardinals ...... 365 Strongly Compact Cardinals. Supercompact Cardinals. Beyond Supercom- pactness. Extenders and Strong Cardinals. Exercises. Historical Notes.

21. Large Cardinals and Forcing ...... 389 Mild Extensions. Kunen-Paris Forcing. Silver’s Forcing. Prikry Forcing. Mea- surability of ℵ1 in ZF. Exercises. Historical Notes.

22. Saturated Ideals ...... 409 Real-Valued Measurable Cardinals. Generic Ultrapowers. Precipitous Ideals. Saturated Ideals. Consistency Strength of Precipitousness. Exercises. Histor- ical Notes.

23. The Nonstationary Ideal ...... 441 Some Combinatorial Principles. Stationary Sets in Generic Extensions. Pre- cipitousness of the Nonstationary Ideal. Saturation of the Nonstationary Ideal. Reflection. Exercises. Historical Notes.

24. The Singular Cardinal Problem ...... 457 The Galvin-Hajnal Theorem. Ordinal Functions and Scales. The pcf Theory. The Structure of pcf. Transitive Generators and Localization. Shelah’s Bound on 2ℵω . Exercises. Historical Notes.

25. Descriptive Set Theory ...... 479 1 The Hierarchy of Projective Sets. Π1 Sets. Trees, Well-Founded Relations 1 and κ-Suslin Sets. Σ2 Sets. Projective Sets and Constructibility. Scales and 1 1 Uniformization. Σ2 Well-Orderings and Σ2 Well-Founded Relations. Borel Codes. Exercises. Historical Notes. XII Table of Contents

26. The Real Line ...... 511 Random and Cohen reals. Solovay Sets of Reals. The L´evy Collapse. Solo- 1 vay’s Theorem. Lebesgue Measurability of Σ2 Sets. Ramsey Sets of Reals and Mathias Forcing. Measure and Category. Exercises. Historical Notes.

Part III. Selected Topics

27. Combinatorial Principles in L ...... 545 The Fine Structure Theory. The Principle
κ. The Jensen Hierarchy. Projecta, Standard Codes and Standard Parameters. Diamond Principles. Trees in L. Canonical Functions on ω1.Exercises.HistoricalNotes.

28. More Applications of Forcing ...... 557 1 A Nonconstructible ∆3 Real. Namba Forcing. A Cohen Real Adds a Suslin Tree. Consistency of Borel’s Conjecture. κ+-Aronszajn Trees. Exercises. His- torical Notes.

29. More Combinatorial Set Theory ...... 573 Ramsey Theory. Gaps in ωω. The Open Coloring Axiom. Almost Disjoint Subsets of ω1. Functions from ω1 into ω. Exercises. Historical Notes.

30. Complete Boolean Algebras ...... 585 Measure Algebras. Cohen Algebras. Suslin Algebras. Simple Algebras. Infinite Games on Boolean Algebras. Exercises. Historical Notes.

31. Proper Forcing ...... 601 Definition and Examples. Iteration of Proper Forcing. The Proper Forcing Axiom. Applications of PFA. Exercises. Historical Notes.

32. More Descriptive Set Theory ...... 615 1 1 Π1 Equivalence Relations. Σ1 Equivalence Relations. Constructible Reals and Perfect Sets. Projective Sets and Large Cardinals. Universally Baire sets. Exercises. Historical Notes.

33. Determinacy ...... 627 Determinacy and Choice. Some Consequences of AD. AD and Large Cardinals. Projective Determinacy. Consistency of AD. Exercises. Historical Notes.

34. Supercompact Cardinals and the Real Line ...... 647 Woodin Cardinals. Semiproper Forcing. The Model L(R). Stationary Tower Forcing. Weakly Homogeneous Trees. Exercises. Historical Notes.

35. Inner Models for Large Cardinals ...... 659 The Core Model. The Covering Theorem for K. The Covering Theorem for L[U]. The Core Model for Sequences of Measures. Up to a Strong Cardinal. Inner Models for Woodin Cardinals. Exercises. Historical Notes. Table of Contents XIII

36. Forcing and Large Cardinals ...... 669 Violating GCH at a Measurable Cardinal. The Singular Cardinal Problem. Violating SCH at ℵω. Radin Forcing. Stationary Tower Forcing. Exercises. Historical Notes.

37. Martin’s Maximum ...... 681 RCS iteration of semiproper forcing. Consistency of MM. Applications of MM. Reflection Principles. Forcing Axioms. Exercises. Historical Notes.

38. More on Stationary Sets ...... 695 The Nonstationary Ideal on ℵ1. Saturation and Precipitousness. Reflection. Stationary Sets in Pκ(λ). Mutually Stationary Sets. Weak Squares. Exercises. Historical Notes.

Bibliography ...... 707

Notation ...... 733

Name Index ...... 743

Index ...... 749