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Perspectives • In Mathematica I Log ic Editors S. Feferman W. A. Hodges M. Lerman (Managing Editor) A. J. Macintyre M. Magidor Y. N. Moschovakis Springer-Verlag Berlin Heidelberg GmbH Thomas Jech Set Theory Second Corrected Edition Springer Thomas Jech Department of Mathematics The Pennsylvania State University University Park, PA 16802 USA e-mail: [email protected] Cataloging-in-Publication Data applied for Die Deutsche Bibliothek -CIP-Einheitsaufnahme Jech, Thomas J.: Sct theory / Thomas Jech. -2., corr. ed. -Berlin; Heidelberg; New York; Barcelona ; Budapest ; Hong Kong; London; Milan; Paris ; Santa Clara; Singapore ; Tokyo : Springer, 1997 (Perspectives in mathematical logic) The original edition was published in 1978 by Academic Press, Inc. in the series Pure and Applied Mathematics, vol. 79. with the Mathematics Subject Classification (1991): 03E05, 03EIO, 03E15, 03E20, 03E25, 03E30, 03E35, 03E40, 03E45, 03E47,03E50, 03E55, 03E60, 04AI0, 04A15, 04A20, 04A25, 04A30 ISSN 0172-6641 ISBN 978-3-662-22402-1 ISBN 978-3-662-22400-7 (eBook) DOI 10.1007/978-3-662-22400-7 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 microfilms or in any other ways. 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-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Originally published by Springer-Verlag Berlin Heidelberg New York in 1997. Softcover reprint of the hardcover 2nd edition 1997 Typesetting: in LATEX by the author; pages 1-621 were reproduced from the original edition SPIN 10629652 4113143 -543210 - Printed on acid-free paper Perspectives in Mathematical Logic This series was founded in 1969 by the Omega Group consisting of R. O. Gandy, H. Hermes, A. Levy, G. H. Muller, G. E. Sacks and D. S. Scott. Initially sponsored by a grant from the Stiftung Volkswagenwerk, the series appeared under the auspices of the Heidelberger Akademie der Wissenschaften. Since 1986, Perspectives in Mathematical Logic is published under the auspices of the Association for Symbolic Logic. Mathematical Logicis a subject which is both rich and varied. Its origins lie in philosophy and the foundations of mathematics. But during the last half century it has formed deep links with algebra, geometry, analysis and other branches of mathematics. More recently it has become a central theme in theoretical computer science, and its influence in linguistics is growing fast. The books in the series differ in level. Some are introductory texts suitable for final year undergraduate or first year graduate courses, while others are specialized monographs. Some are expositions ofw ell established material, some are at the frontiers of research. Each offers an illuminating perspective for its intended audience. For Paula, Pavel, and Susanna Aus dem Paradies, das Cantor uns geschaffen, soil uns niemand vertreiben konnen. David Hilbert PREFACE The main body of this book consists of 106 numbered theorems and a dozen of examples of models of set theory. A large number of additional results is given in the exercises, which are scattered throughout the text. Most exer cises are provided with an outline of proof in square brackets [ ], and the more difficult ones are indicated by an asterisk. I am greatly indebted to all those mathematicians, too numerous to men tion by name, who in their letters, preprints, handwritten notes, lectures, seminars, and many conversations over the past decade shared with me their insight into this exciting subject. XI CONTENTS Preface xi PART I SETS Chapter 1 AXIOMATIC SET THEORY I. Axioms of Set Theory I 2. Ordinal Numbers 12 3. Cardinal Numbers 22 4. Real Numbers 29 5. The Axiom of Choice 38 6. Cardinal Arithmetic 42 7. Filters and Ideals. Closed Unbounded Sets 52 8. Singular Cardinals 61 9. The Axiom of Regularity 70 Appendix: Bernays-Godel Axiomatic Set Theory 76 Chapter 2 TRANSITIVE MODELS OF SET THEORY 10. Models of Set Theory 78 II. Transitive Models of ZF 87 12. Constructible Sets 99 13. Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis 108 14. The In Hierarchy of Classes, Relations, and Functions 114 15. Relative Constructibility and Ordinal Definability 126 PART II MORE SETS Chapter 3 FORCING AND GENERIC MODELS 16. Generic Models 137 17. Complete Boolean Algebras 144 18. Forcing and Boolean-Valued Models 159 19. Independence of the Continuum Hypothesis and the Axiom of Choice 176 20. More Generic Models 187 21. Symmetric Submodels of Generic Models 197 XIII