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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
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