Introduction to Set Theory,Revised and Expanded

INTRODUCTION TO SET THEORY

PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl 1. Tafi Rutgers University New Brunswick, New Jersey

Zuhair Nashed University of &laware Newark, Delaware

EDITORIAL BOARD M. S. Baouendi University of Calijbrnia, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Anil Nerode Cornell University Dunald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi Uniwrsity of Cal~$ornia, Berkeley

Gian-Carlo Rota Massachusetts Institute of Technology

Marvin Marcus University of Calflornia, Santa Barbara

David L. Russell Virginia Polytechnic Institute and State University

W. S. Mmsey Yale University

Walter Schempp Universitat Siegen

Mark Teply University of Wisconsin, Milwa ukee

MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS 1. K. Yano, Integral Formulas in Riiannian Geometry (1970) 2. S. Kobayashi, Hyperbolic Manifdds and Momorphic Mappings (1970) 3. V. S. Vladimimv, Equations of Mathematical Physics (A. Jefkey, ed.; A. Littlewood, trans.) (1970) 4. 8. N. Pshenichnyi, Necessary Conditions for an Exfremum (L. Neustadt, translation ed.; K. Makowski, trans.) (1971) 5. L. N a k i et al., Functional Analysis and Valuation Theory (1971) 6. S. S. Passman, Infinite Group Rings (1971) 7. L. DMhft:Group Representation Theuy,Part A: Ordinary Representation Theory. Part B; M u l a r RepresentationTheuy (1971,1972) 8. W. Boothby and G. L. Weiss, eds.,Symmetric Spaces ( l 972) 9. Y. Matsushima, Differentiable Manifolds (E. T. Kobayashi, trans.) (1972) 10. L. E. Ward, Jr., Topdogy (1972) 1l. A. Babakhanian, Gohomological Methods in Group Theory (1972) 12. R. Gilmer,Multiplicative Ideal Theory (1972) 13. J. Yeh, Stochastic Pmesses and the Wiener Integral (1973) 14. J. &m-Neto, Introduction to the Theory of Distributions (1973) 15. R. Larsan, Functional Analysis (1973) 16. K. Yano and S. Ishihara, Tangent and Cotangent Bundles (1973) 17. C. Prucesi, Rings with PdynomialIdentities (1973) 18. R. Hennann, Geometry, Physics, and Systems (1973) 19. N. R. Wallach,Harmonic Analysis cm Homogeneous Spaces ( l 973) 20. J. Dieudonnd, Introduction to the Theuy of Formal Groups (1973) 21. l, Vaismen, Cohomdogy and Differential Forms (1973) 22. 8.-Y. Chen, Geometry of SubmanifoMs (1973) 23. M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973, 1975) 24. R. Larsen, Banach Algebras (1973) 25. R. 0.Kyala and A. L. Vitter; eds., Value Distribution Theory: Part A; Part B: Defiat and Bezout Estimates by Wilhelm SWl(1973) 26. K. 8. Stdarsky, Algebraic Numbers and m a n t i n e Approximation (1974) 27. A. R. Magid, The Separable GaloisTheory of Commutative Rings (1974) 28. 8. R. McDonald, Finite Rings with Identity(1974) 29. J. Satake, Linear Algebra (S. Koh et al., trans.) (1975) 30. J. S. W a n . Localization of NoncommutativeRings (1975) 31. G. Klambauer. Mathematical Analysis (1975) 32. M. K Agoston, Algebraic Topology (1976) 33. K. R. Goodearl, Ring Theory (1976) 34. L. E. Mansfield, Linear Algebra with Geomebic Applications (1976) 35. N. J. Pullman, Matrix Theory and Its Applications (1976) 36. 8. R. McDaneld. Geometric Algebra Over Local Rings (1976) 37. C. W. Groetsch, Generalized Inverses of Linear Operators (1977) 38. J. E. Kucrkowski and J. L. Gersting, Abstract Algebra (1977) 39. C. 0.Christenson and W. L. Voxman, Aspeds of Topdogy (1977) 40. M. Nagata, Field Theuy (1977) 41. R. L. Long, Algebraic Number Theuy (1977) 42. W. F. Plb#br,Integrals and Measures (1977) 43. R. L. Wwe&n and A. Z p n d , Measure and Integral (1977) 44. J. H- Curtiss, Introduction to Funcbionsof a Complex Variable (1978) 45. K. Hdmcek and T. Jech, lnhdudicm to Set Theory (1978) 46. W. S. Massey, Homdogy and c o h m b g y Theory (1978) 47. M. Marcus, Introductionto Modem Algebra (1978) 48. E. C. Young, Vector and Tensor Anatysis (1978) 49. S. 8. Nadkr, Jr., Hyperspacasof Sets (1978) 50. S. K. &gal, Topics in Group Klngs (1978) 51. A. C. M. van Row, Non-Archimedean Functional Analysis (1978) 52. L. Corwin and R. Szczadm, Calculus in Vector Spaces (1979) 53. C. Sadosky, Interpolation of Operators and Singular Integrals (1 979) 54. J. Cmin, Differential Equations (1980) 55. C. W. Gmtsch, Elements of Applicable Functional Analysis (1980)

56. 57. 58. 59.

1. Vaisman, Foundations of Three-Dimensional Eudidean Geometry (1980) H.I.Fmedan, Deterministic Mathematical Models in Population Emlogy (1980) S. B.Chae, Lebesgue Integration (1980) C. S. Rees et al., Theory and Applications of Fourier Analysis (1981) 60. L. Nachbin, Introduction to Functional Analysis (R. M. Arm, trans.) (1981) 61. G.Onech and M. Omch, Plane Algebraic Curves (1981) 62. R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical Analysis (1981) 63. W.L. Voxman and R. H. Goetschel, Advanced Calculus (1981) 64. L. J- Corwin and R. H. Szczadm, Multivariable Calculus (1982) 65. V. I. Istrdtescu, t ntroduction to Linear Operator Theory (1981) 66. R. D. Jdwinen, Finite and Infinite Dimensional Linear Spaces (1981) 67. J. K. Beem and P. E. Ehrlich, Global Lorentrian Geometry (1981) 68. D.L. Amacost, The Structure of Lcxally Compad Abelian Groups (1981) 69. J. W. h w e r and M. K. Smith, eds., Emmy Noether: A Tribute (1981) 70. K. H. Kim, Boolean Matrix Theory and Applications (1982) 71. T. W. Wmting, The Mathematical Theory of Chromatic Plane Ornaments (1982) 72. D. B.GauM, Differential Topdogy (1982) 73. R. L. Faber, Foundations of Eudidean and Non-Eudidean Geometry (1983) 74. M. Carmeli, Statistical T h q and Random Matrices (1983) 75. J. H.Camrfh et al., The Theory of Topdogical Semigroups (1983) 76. R. L. Faber, Differential Geometry and Relativity Theory (1983) 77. S.Barnett, Pdynomials and Linear Control Systems (1983) 78. G. Ka@lovsky, Commutative Group Algebras (1983) 79. F. Van Oystaeyenand A. Verschmn, Relative lnvariants of Rings (1983) 80. 1. Vaisman, A First Course In Differential Geometry (1984) 81. G. W.Swan, Applications of Optimal Contrd Theory in Biomedicine (1984) 82. T. Pet* and J. D. Randall, Transformation Groups on Manifdds (1984) 83. K. Goebel and S. Reich, Uniform Convexity, Hyperbdic Geometry. and Nonexpansive Mappings (1984) 84. T. Albu and C. Ndstdsescu, Relative Finiteness in Module Theory (1984) 85. K. Hhacek and T. Jech, Introduction to Set Theory: Second Edition (1984) 66. F. Van Oysbeyen and A. Verschoren, Relative Invariants of Rings (1984) 87. 8.R. McDmald. Linear Algebra Over Commutative Rings (1984) 88. M. Namh, Geometry of Projective Algebraic Curves (1984) 89. G. F. Webb. Theory of Nonlinear Age-Dependent Population Dynamics (1985) 90. M. R. Bremner et al., Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985) 91. A. E.Fekete, Real Linear Algebra (1985) 92. S. B.Chae, Holamorphy and Calculus in Normed Spaces (1985) 93. A. J.,e n iJ Introduction to Integral Equations with Applications (1985) 94. G.Kap'lovsky, Projective Representations of Finite Groups (1985) 95. L. Nariciand E. Beckenstein, Topdogical Vector Spaces (1985) 96. J. Weeks, The Shape of Space (1985) 97. P. R. Gribik and K. 0.Kortanek, Extremal Methods of Operations Research ( 1985) 98. J.-A. Chao and W. A. Woycrynski, eds.. Probability Theay and Harmonic Analysis (1986) 99. G. D. Cmwn et al., Abstract Algebra (1986) 00. J. H. Carmth et al., The Theory of Topological Semigroups, Vdume 2 (1986) 01. R. S.Doran and V. A. Beffi, Characterizations of C*-Algebras (1986) 02. M. W. Jeter, Mathematical Programming (1986) 03. M. Anman, A Unified Theory of Nonlinear Operator and Evolution Equations with Applications ( l 986) 04. A. V e r s c h n , Relative lnvariants of Sheaves (1987) 05. R. A. Usmani, Applied Linear Algebra (1987) M. P.Blass and J. fang, Zariski Surfaces and Differential Equations in Characteristic p > 0 (1987) 07. J. A. Reneke et al., Structured Hereditary Systems (1987) 08. H. Busemann and 8.B. Phadke, Spaces with Distinguished Geodesics (1987) 09. R.Harfe. Invertibility and Singularity for Bounded Linear Operators (1988) 10. G.S. Ladde et al., Oscillation Theory of Differential Equations with Deviating Arguments (1987) 11. L. Dudkin etal., IterativeAggregationTheory(1987) 12. T. Okuh, Differential Geometry (1987) 13. D.L. Stancl and M.L. Stancl, Real Analysis with Point-Set Topotogy (1987)

114. 115. 116. 1 17. 118. 1 19. 120. 121.

T. C. Gad, lntroduction to S t d a s t i c Differential Equations (1988) S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux (1988) H. Strade and R. Famsteiner, Modular Lie Algebras and Their Representations (1988) J. A. Huckaba, Commutative Rings with Zero Divisors (1988) W. D. Wallis, Combinaton'alDesigns (1988) W. Wipshw, Topdogical Fields (1988) G. Karpilovsky, Field Theory (1988) S. Caenepeel and F. Van Oystmyen, Brauer Groups and the Cohomdogy of Graded Rings (1989) 122. W. Kozlowski, Modular Function Spaces (1988) 123. E. Lowen-Cdebunders, Function Classes of Cauchy Continuous Maps (1989) 124. M. Pave/, Fundamentals of Pattem Recognition(1989) 125. V. Lakshmikantham et al., Stability Analysis of Nonlinear Systems (1989) 126. R. Sivaramakrishnan, The Classical Theory of Arithmetic Functions (1989) 127. N. A. Watson, Parabolic Equations on an Infinite Strip (1989) 128. K. J. Hastings, Introduction to the Mathematics of Operations Research (1989) 129. 8. Fine, Algebraic Theory of the Bianchi Groups (1989) 130. D. N. Dikranjan et al., Topdogical Groups (1989) 13l.J. C. Mowan /I, Point Set Theory ( l990) 132. P. Biler and A. Wlkowski, Problems in Mathematical Analysis (1990) 133. H. J. Sussmann. Nonlinear Controllability and Optimal Contrd (1990) 134. J.-P. Flomns et al., Elements of Bayesian Statistics (1990) 135. N. Shell, Topological Fields and Near Valuations (1990) 136. B. F. Dodin and C. F, Martin, lntroduction to Differential Geometry for Engineers (1990) 137. S. S. Hdland, Jr.. Applied Analysis by the Hilbert Space Method (1990) 138. J. Okninski, Semigroup Algebras (1990) 139. K. Zhu, Operator Theory in Function Spaces (1990) 140. G. B. Price, An Introduction to Multimplex Spaces and Functions (1991) 141. R. 8. Darst, Introduction to Linear Programming (1991) 142. P. L. Sachdev, Nonlinear Ordinary DifferentialEquations and Their Applications (1991) 143. T. Husain, Orthogonal Schauder Bases (1991) 144. J. F o m , Fundamentals of Real Analysis (1991) 145. W. C. Bmwn, MatricesandVecbr Spaces (1991) 146. M. M. Rao and L.D. Ren, Theory of Orlicz Spaces (1991) 147. J. S. Gdan and T.Head, Modules and the Structures of Rings (1991) 148. C. Small, Arithmetic of Finite Fidds (1991) 149. K. Yang, Complex Algebraic Geometry (1991) 150. D. G. Hoffman et al., Coding Theory (1991) 151. M. 0. Gonzdlez, Classical Complex Analysis (1992) 152. M. 0.Gonzdlez, Complex Analysis (1992) 153. L. W. Baggett, Functional Analysis (1992) 154. M. Sniedovich, Dynamic Programming (1992) 155. R. P. Agamal, Difference Equations and lnequalihies (1992) 156. C. Bmrinski, Biorthogonality and Its Applications to Numerical Analysis (1992) 157. C. Swattz, An Introduction to Fundional Analysis (1992) 158. S. 8. Nadler, Jr., Continuum Theory(1992) 159. M. A. Al-Gwaiz, Theory of Distributions (1992) 160. E. Peny, Geometry: Axiomatic Developments with Problem Solving (1992) 161. E. Castilh and M. R. Ruiz-Cob, Functional Equations and Mdelling in Saence and Engineering (1992) Jn i Integral and Discrete Transforms with Applications and Error Analysis 162. A. J. ,e (1992) 163. A. Charlier et al., Tensars and the Cliord Algebra (1992) 1 . P. Bilerand T. Nadzieja, Problems and Examples in DifFerentialEquations (1992) 165. E. Hansen, Global Optimization Using lnteival Analysis (1992) 166. S. Guem-DelabMm, Classical Sequences in Banach Spaces (1992) 167. Y. C. Wong. Introductory Theory of Topological Vector Spaces (1992) 1M. S. H. Kulkamiand 8. V, Limaye, Real Function Algebras (1992) 169. W. C. Bmwn, Matrices Over Commutative Rings (1993) 170. J. Loustauand M. Dillon, Linear Geometry with Computer Graphics (1993) 171. W. V. Petryshyn. Approximation-Sdvability of Nonlinear Functional and Differential Equations (1993) 172. E. C. Young, Vector and Tensor Analysis: Second Edition (1993) 173. T. A. Bick, Elementary Boundary Value Problems (1993)

174. M. Pavel, Fundamentals of Pattern Recognition: Seoond Edition(1993) 175. S. A. Albewrio et al., Nmcommutative Distributions(1993) 176. W. Fulks, Complex Variables(1993) in.M. M. Rao, Conditional Measures and Applications(1993) 178. A. Janicki and A. Wem, Simulation and Chaotic Behavior of a-Stable Stochastic P m s s e s (1994) P. NeitfaanMkiand D. Tiba, Optimal Control of Nonlinear Parabdic Systems (1994) J. Cmin, Differential Equations: Introduction and Qualitative Theory, S@

Edition

(1994)

S. HeikkMand V. LBkshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Rfferential Equations (1994) X. Mao, Exponential Stability of Stochastic Differential Equabions (1994) B. S. Thornson, Symmetric FVoperties of Real Functions (1994) J. E. Rubio, Optimirationand Nonstandard Analysis (1994) J. L. Bueso et al., Compatibility, Stability, and Sheaves (1995) A. N. Mjcheland K. Wang, Qualitative Theory of Dynamical Systems(1995) M. R. Damel, Theory of Lattice-Ordered Groups (1995) 2.Naniewicz and P. D. PanagEotopoulos, Mathematical Theory of Hemi~liational Inequalities and Applications(1 995) L. J. Cornin and R. H. Szczadw, Calculus in Vector Spaces: Second Edition(1 995) L. H. Erbe et al., Oscillah Theuy for Functional Differential Equations(1995) S. Agaian et al., Binary Polynomial Transforms and Nonlinear Digital Filters(1995) M.I.Gil', Norm Estimations for Operation-Valued Functions and Applications(1995) P. A. Gn'Ilei. Semigroups: An lntroduction to the Structure Theory (1995) S. Kichenassamy, Nonlinear Wave Equations (1996) V. F. Kmiov, Global Methods in Optimal Control Theory (1996) K. I. Beidaret al., Rings with Generalized Identities (1996) V. I.Amautov et al., Introduction to the Theory of Topological Rings and Modules

(1996) G.Sierksma, Linear and Intqpr Programming (1996) R. Lassar, Introduction to Fourier Series(1 996) V. Sima, Algorithms for Linear-QuadraticOptimization (1996) D. Redmond, Number Theory(1 996) J. K. Beem et al., Global Lmntrian Geometry: Second Edition(1 996) M. Fontana et al.. Prijfer Domains (1997) H. Tanabe, Functional Analybc Methods for Partial Differential Equations(1997) C. Q.Zhang, Integer Flows and Cyde Covers of Graphs(1997) E. Spiegeland C. J. Olhnell. Incidence Algebras(1997) 8. Jakubczyk and W. Respondek, Geometry of Feedback and Optimal Contrd(1998) T. W. Haynes et al., Fundamentals of Domination in Graphs (1998) T W. Haynes et al., Domination in Graphs: Advanced Topics(1998) L. A. D'Alotto et al., A Unified Sgnal Algebra Approach to TwDimensional Parallel Dgital Sgnal Processing (1998) F. Halter-Koch, Ideal Systems(1 998) N. K. Govil et al., Approximation Theory(1 998) R. Cms, MultivaluedLinear Operators (1998) A. A. Martynyuk. Stability by 'liapunovls ~ a t r i xFunction Method with Applications

(19981 A . ~ a v i nand i A. Yagi, Degenerate Differential Equations in Banach Spaces (1999) A. lllanes and S. Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances

(1999)

G. Kato and D. Stnrppa, Fundamentals of Algebraic Micrdocal Analysis (1999) G. X.-2. Yuan, KKM Theory and Applications in Nonlinear Analysis (1999) D. Mdmanu and N. H. Pavel. Tangency, Flow Invariance for Differential Equations, and Optimization Problems(1999) K. Hrbacek and T. Jech, Introduction to Set Theory, Third Edition(1999) G. E. Kolosov, Optimal Design of Control Systems(1 999) A. I.Prilepko et al., Methods for Solving Inverse Problems in Mathematical Physics

(1999)

INTRODUCTION TO SET THEORY Third Edition, Revised and Expanded

Karel Hrbacek The City College of the City Universijr of New York Ne W Y&, New York

Thornas Jech The &nnsylvania State University University Park, Pknnsylwania

Library of Congress Cataloging-in-Publication

Hrbacek, Karel. Introduction to set theory / Karel Hrbacek, Thornas Jech. - 3" ed., rev. and expanded. p. cm. -(Monographs and textbooks in pure and applied mathematics; 220). Includes index. ISBN 0-8247-7915-0 (alk. paper) 1. Set theory. I. Jech, Thomas J. 11. Title. 111. Series. QA248.H68 1999 5 1 1 -3'22--dc2 1 99-15458 CIP

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Copyright O 1999 by Marcel Dekker, Inc. All Rights Reserved. Ne~therthis book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing fiom the publisher. Current printing (last digit) 1 0 9 8 7 6 5 4 3 2 1

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Preface to the 3rd Edition Modem set theory has grown tremendously since the first edition of thia book was published in 1978, and even since the second edition appeared in 1984. Moreover, many ideas that were then at the forefront of research have become, by now, important tor>lain other branches of mathematics. Thua, for example, combinatorial principles, such as Principle Diamond and Martin's Axiom, are indispensable tools in general topology and abstract algebra. Non-well-founded sets turned out to be a convenient setting for semantia of artificial as well as natural languages, Nonstandard Analysis, which is grounded on structum constructed with the help of ultrdlters, has developed into an independent methodology with many exciting applications. It seems appropriate to incorporate some of the underlying set-theoretic ideas into a textbook intended as a general introduction to the aubject. We do that in the form of four new chapters (Chapters 11-14), expanding on the topics of Chapter 11 in the second edition, and containing largely new material. Chapter 11 presents fdters and ultrafiltera, develops the basic properties of closed unbounded and stationary aets, and culminates with the proof of Silver 'a Theorem. The first two aections of Chapter 12 provide an introduction to the partition calculus. The next two sections study trees and develop their relationship to Suslin'a Problem. Section 5 of Chapter 12 is an introduction to combinatorial principles. Chapter 13 is devoted to the measure problem and measurable cardinals. The topic of Chapter 14 is a fairly detailed study of well-founded and non-well-founded sets. Chapters 1-10 of the aecond edition have been thoroughly revised and reorganized. The material on rational and real numbers has been consolidated in Chapter 10, so that it does not interrupt the development of set theory proper. To preserve continuity, a wction on Dedekind cuts has been added to Chapter 4. New material (on normal forms and Goodstein sequences) has also been added to Chapter 6. A aolid basic course in set theory Bhould cover most of Chapters 1-9. This can be supplemented by additional material from Chapters 10-14, which are almost completely independent of each other (except that Section 5 in Chapter 12, and Chapter 13, refer to some concepts introduced in earlier chapters).

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Preface to the 2nd Edition The fht version of this textbook was written in Czech in spring 1968 and accepted for publication by Academia, Prague under the title h o d do teorie m n o h However, we both left Czechdovakia later that year, and the book in set never appeared. In the following y e . we taught introductory COtheory at various uaiversitiea in the United Statea, and found it difficult to select a textbook for w in these m m . Some existing books are based on the "naive" approach to set theory rather than the axiomatic one. We consider some understanding of set-theoretic "paradoxes" end of undecidable propositions (such as the Continuum Hypothesis) one of the important goals of such a CO-, but neither topic can be treated honestly with a "naiven approach. Moreover, set theory is a natural choice of a field where students can first become acquainted with an axiomatic development of a mathematical discipline. On the other hand, all currently available texts presenting set theory from an axiomatic point of view heavily stress logic and logical formalism. Most of them begin with a virtual m i n i c o w in logic. We found that students often take a course in set theory before taking one in logic. More importantly, the emphasis on formalization obscures the essence of the axiomatic method. We felt that there was a need for a book which would present axiomatic set theory more mathematically, at the level of rigor customary in other undergraduate courses for math majors. This led us to the decision to rewrite our Czech text. We kept the original general plan, but the requirements for a textbook suitable for American colleges reaulted in the production of a completely new work. We wish to stress the following featurea of the book: l. Set theory is developed axiomatically. The reasons for adopting each axiom, ars arising both fn>m intuition and mathematical practice, are carefully pointsd out. A detailed discussion is provided in "controversial" cases, such as the Axiom of Choice. 2. The treatment is not formal, Logical apparatus is kept to a minimum and logical formalism is completely avoided. 3. We show that Iudomatic set theory is powerful enough to serve as an underlying framework for mathematics by developing the beginnings of the theory of natural, rational, and real numbers in it. However, we carry the development only as far as it is useful to illustrate the general idea and to motivate settheoretic generalizations of some of these concepts (such as the ordinal numbers end operations on them). Dreary, repetitive details, such as the proofs of all

vi

PREFACE TO THE SECOND EDITION

the usual arithmetic laws,are relegated into exercises. 4. A substantial part of the book is devoted to the study of ordinal and cardinal numbers. 5. Each section ia accompanied by many exercises of w y i n g difficulty. 6. The h a l chapter is an informal outline of some recent developments in set theory and their significance for other areas of mathematics: the Axiom of Constructibility, questions of consistency and independence, and large cardinals. No prooh are given, but the exposition is sufficiently detailed to give a nonspecialist mme idea of the problem arising in the foundatiom of set theory, methods used for their solution, and their effectson mathematics in general. The fimt edition of the book has been extensively used as a textbook in undergraduate and first-year graduate courses in set theory. Our own experience, that of our many colleagues, and ~uggestionaand critickm by the reviewem led us to consider some changes and improvements for the present aecond edition. Those turned out to be much more extensive than we originally intended. As a result, the book has been substantially rewritten and expanded. We list the main new features below. l. The development of natural numbers in Chapter 3 has been greatly simplified. It is now based on the definition of the set of all natural numbers as the least inductive set, and the Principle of Induction. The introduction of transitive sets and a characterization of natural numbers as those transitive sets that are well-ordered and inversely well-ordered by E is postponed until the chapter on ordinal numbers (Chapter 7). 2. The material on integers and rational numbers (a separate chapter in the first edition) has been c o n d e d into a single section (Section l of Chapter 5). We feel that m& students learn this subject in another course (Abstract Algebra), where it properly belongs. 3. A series of new sectiona (Section 3 in Chapter 5, 6, and 9) deals with the set-theoretic properties of sets of real numbers (open, d m d , and perfect sets, etc.), and provides interesting applications of abstract B& theory to real analysis. 4. A new chapter called "Uncountable Setsn (Chapter 11) has been added. It introduces some of the concepts that are fundamental to modern set theory: ultrafilters, closed unbounded sets, trees and partitions, and large cardinals. These topics can be used to enrich the usual one-semester course (which would ordinarily cover most of the material in Chapters 1-10). 5. The study of linear orderings has been expanded and concentrated in one place (Section 4 of Chapter 4). 6. Numerous other small additions, changes, and corrections have been made throughout. 7. Finally, the discussion of the present state of set theory in Section 3 of Chapter 12 has been updated.

Content S iii

Preface to t h e T h i r d Edition Preface to t h e Second Edition 1 Sets l Introduction to

2 3 4

Sets

.........................

Prop&iw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TheAxioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary Operations on Sets . . . . . . . . . . . . . . . . . . .

2 Relations. E'unctions. and Orderings

1 2 3 4 5

Ordered Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F'unctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalences and Partitions . . . . . . . . . . . . . . . . . . . . . Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

S Natural Numbers l Introduction to Natural Numbers . . . . . . . . . . . . . . . . . . 2 Properties of Natural Numbm . . . . . . . . . . . . . . . . . . . 3 The Fkursion Theorem . . . . . . . . . . . . . . . . . . . . . . . 4 Arithmetic of Natural Numbers . . . . . . . . . . . . . . . . . . . 5 Operations and Structures . . . . . . . . . . . . . . . . . . . . . . 4 Finite. Countable. and Uncountable S e t s 1 Cardinality of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 2 FiniteSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Countable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Linear Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Complete Linear Orderings . . . . . . . . . . . . . . . . . . . . . 6 Uncountable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Cardinal N u m b e r s 1 Cardinal Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . The Cardinality of the Continuum . . . . . . . . . . . . . . . . . 2

vii

viii

CONTENTS

8 Ordinai Numbers 103 . . . . . . . . . . . . . . . . . . . . . . . . . . l Well-Ordered Sets 103 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Ordinal Numbera 107 . . . . . . . . . . . . . . . . . . . . . 3 The Axiom of Replmment 111 4 38nsfinite Induction and Recursion . . . . . . . . . . . . . . . . 114 5 Ordinal Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6 The Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7 Alephs 129 1 Initial Ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 2 Addition and Multiplication of Alephs . . . . . . . . . . . . . . . 133 8 T h e Axiom of Choice 1 The Axiom of Choice end its Equivalents . . . . 2 The Use of the Axiom of Choice in Mathematics

.........

137

137 . . . . . . . . . 144

9 Arithmetic of Cardinal Numbers 1 Infinite Sums and Products of Cardinal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Regular and Singular Cardinals . . . . . . . . . . . . . . . . . . . 3 Ekponentiation of Cardinals . . . . . . . . . . . . . . . . . . . . .

155

10 Sets of Real Numbers 1 Integers and Rational Numbers . . . . . . . . . . . . . . . . . . . 2 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Topology of the Real L i e . . . . . . . . . . . . . . . . . . . . . . 4 Sets ofRRal Numbera . . . . . . . . . . . . . . . . . . . . . . . . 5 Bore1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171

11 Filtera and Ultrafilters 1 Filters and Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 UltrsGlters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 C l d Unbounded and Stationary Sds . . . . . . . . . . . . . . 4 Silver's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .

201

155 160 164 171 175 179 188

194 201 205 208 212

12 Combinatorial Set Theory 217 1 Ramsey's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 217 2 Partition Calculus for Uncountable Cardinals . . . . . . . . . . . 221 Thea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 3 4 Suslin's Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 5 Combinatorial Principles . . . . . . . . . . . . . . . . . . . . . . . 233 13 Large Cardinab 241 1 The Measure Problem . . . . . . . . . . . . . . . . . . . . . . . . 241 2 Large Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

CONTENTS

ix

14 The Axiom of Foundation

1 2 3

251

. . . . . . . . . . . . . . . . . . . . . . . 251 . . . . . . . . . . . . . . . . . . . . . . . . . . 256 .. .... . . . . . . . . . . . . . . . .. 2W

Well-Founded Relations Wd1-Founded S& Non-Well-Founded Sds

15 The Axiomatic Set Theory l The Zermel+Fraenkel Set Theory

.

2 3

287

. .

. .

.

With Choice. . . . . . . . . . . . . . . . . . . . . . . . 267 Coasiskncy and Independence . . . . . . . . . . . . . . . . 270 The Univerae of Set Theory . . . . . . . . . . . . . . . . . 277

Bibliography Index

.. ..

.

. .

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Chapter 1

Sets Introduction to Sets The central concept of this book, that of a set, is, at least on the surface. extremely simple. A set is any collection, group, or conglomerate. So we have the set of all students registered at the City University of New York in February 1998, the set of all even natural numbers, the set of all points in the plane n exactly 2 inches distant from a given point P, the set of all pink e1ephant.s. Sets are not objects of the real world, like tables or stars; they are created by our mind, not by our hands. A heap of potatoes is not. R set of potsat.oc?s,the set of all molecules in a drop of water is not the same object ;is that drop of water. The human mind possesses the ability to abstract, to think of a variety of different objects as being bound together by some common propert,y. and thus to form a set of objects having that property. The property in question may be nothing more than the ability to think of these objects (as being) t,ogetl~er. Thus there is a set consisting of exactly the numbers 2, 7, 12, 13, 29, 34, and 11,000, although it is hard to see what binds exactly those numbers together, besides the fact that we collected them together in our mind. Georg Cantor, a German mathematician who founded set theory in a series of papers published over the last three decades of the nineteenth century, expressed it as follows: "Unter einer Menge verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten in unserer Anschauung oder unseres Denke~is (welche die Elemente von M genannt werden) zu einem ganzen." [A set is a collection into a whole of definite, distinct objects of our intuition or our thought. The objects are called elements (members) of the set.) Objects from which a given set is composed are called elements or members of that set. We also say that they belong to that set. In this book, we want to develop the theory of sets as a foundation for other mathematical disciplines. Therefore, we are not concerned with sets of people or molecules, but only with sets of mathematical objects, such as numbers. points of space, functions, or sets. Actually, the first three concepts can be defined in

CHAPTER 1 . SET'S

2

set theory as sets with particular properties, and we do that in the following chapters. So t h e only objects with which we are concerned from now on art. sets. For purposes of illustration, we talk about sets of numbers or points evt.11 hefore these notions are exactly defined. We do that, however, only ill ex:inlples. exercises arid problems, not in thc nl:tin body of theory. Sets of mnthemat,ic:al objects are, for example: 1.1 (a) (b) (c) (cl) (e)

Example T h e set of The set of T h e set of T h e set of The set of

all prime divisors of 324. a11 rlurribers divisible by 0. all continuous real-valued functioris on the interval (0.l ] . all ellipses with major axis 5 and eccentricity 3. all sets whose elements are rlatural riunlbers less than 20.

Examination of these and many other similar examples revtlals that sc?t.s with which mathematicians work are relatively simple. They i~icluclr!the set o f natural numbers and its various subsets (such as the set of all prime numbers). :ts well as sets of pairs, triples, and in general n-tuples of natural riumbers. Integcrs and rational numbers can be defirietl using only such sets. Real r~umberscall then be defined as sets or sequences of rational numbers. Mathematical analysis deals with sets of real numbers allcl functions on real numbers (sets of ordered pairs of real numbers), and in some investigations, sets of functions (11. cvrn sets of sets of functions are considered. But a working mathematician rarely eri(:ounters objects more con~plicatt?dthan that. Perhaps it is not surprising t h a t uncritical usage of "sets" renlote from "everyday experience" n ~ a ylead to contradictions. Consider for example the "set" R of all those sets which are rlot elrlncr~ts of themselves. In other words, R is a set of all sets X such that X 4 a ( E reads -;belongs to," 4 reads "does not belong to"). Let 11s now (ask whether R E R . If R R, then R is not an element of itself (because no eleme~itof R belongs t o itself), so R $ R; a contradiction. Therefore, ~~ttcessarily R 4 R. But tl~t.11 R is a set which is not an element of itself, and a11 such setasbelong t o R.P'!! conclude that R E R; again, a collt,ritdiction. Tlie argument call be briefly sulnmarized a follows: Define R by: .I. E R it' nritl o ~ i l yif X # X. Now consider X = R; by definition of R, R G R if itllci ~ 1 1 1 if~ R 4 R; a contradiction. A few comments on this argument (due to Bertrand Russell) rriight bc helpful. First, there is nothing wrong with R being a set of sets. Many sets wl~osc elements are again sets are legitimately enlployed in mathematics - set. Exanlple 1.1 - and do not lead t o contradictions. Second, it is easy t o give clx;~mpl(ts of elements of R; e.g., if X is the set of all natural numbers, then X 4 s (the set of all natural numbers is not a natural number) and so X E R. Third. it is riot so easy to give examples of sets which do not belong to R, but this is irrelevant. T h e foregoing argument would resr~ltin a contradiction even if there were 110 sets which are elements of themselves. (A plausible clindidatt! for a set wliiclh is a11 r~lenientof itself woulcl be the "st:t of all sets" V ; c:le~rlyV E C'. Howevtlr.

2. PROPERTIES

3

the "set of all sets" leads to contradictions of its own in a more subtle way see Exercises 3.3 and 3.6.) How can this contradiction be resolved? We :issunled that we have a sett R defined as the set of all sets which are not elements of t,hemselves. anci derivrd a contradiction as an immediate consequence of the definition of R. This can only mean that there is no set satisfying the definition of R. In other words. this argument proves that there exists no set wllose l~ienlberswould be precisely the sets which are not elements of themselves. The lesson contained in Russell's Paradox and other similar examples is that by merely defining a set we: clo not prove its existence (similarly as by defining a unicorn we do riot prove t,hat unicorns exist). There are properties which do not define sets; that is, it is not possible to collect all objects with those properties into one set. This observiitioll leaves set-theorists with a task of determining the properties which do dcfirle sets. Unfortunately, no way how to do this is k~lawn,and some results i n logic Theorenls discovered by K urt Giic.lt4) (especially the so-called In~omplet~eness seem to indicate that a complete answer is not cvcn possible. Therefore, we attempt a less lofty goal. We formulate some of the relatively simple properties of sets used by mathematicians as axioms, and then t a k e c a r e to check that all theorems follow logically from the axioms. Since the axioms are obviously true and the theorems logically follow from them, the theorems are also true (not necessarily obviously). We end up with a bodj. of truths about sets which includes, among other things, tlw basic properties of natural. rational, and real numbers, functions, orderings, etc.. but as far as is knowli, no contradictions. Experience has shown that practically all notio~ls~ ~ s ei tl li contemporary mathematics can be defined, and their rnatheniatical propcrti~s derived, in this axiomatic system. In this sense, the axiomatic set theory serves as a satisfactory foundation for the other branches of n~athematics. On the other hand, we do not claim that every true fact about. sets c:nn 11e derived from the axioms we present. The axiomatic system is not co~npletcin this sense, and we return to the discussion of the question of completeness in the last chapter.

Properties In the preceding section we introduced sets as collections of objects Iinving some property in common. The notion of property rnerit,s some analysis. Solne properties commonly considered in everyday life are so vague that they can hardly be admitted in a mathematical theory. Consider. for example. t h(?'-set of all the great twentieth century American novels.'' Different persons' judg~ilents as to what constitutes a great literary work differ so nluch that there is no generally accepted way how to decide whether a given book is or is not an element of the "set." For an even more startling example, consider the "set of those naturai numbers which could be written down in decimal not;ition9 (by "coul(l" we mean that someone could actually do it with paper and pencil). Clearly. 0 can be so

CHAPTER I. SETS

4

written down. If number n can be written down, then surely number n + 1 can also be written down (imagine another, somewhat faster writer, or the person capable of writing n working a little faster). Therefore, by the familiar principlt. of induction, every natural number n can be written down. But that is plainly absurd; to write down say 10lol" in decimal notation would require to follow 1 by 10'' zeros, which would take over 300 years of continuous work at a rate of a zero per second. The problem is caused by the vague meaning of "could." To avoid similar difficulties, we now describe explicitly what we mean by a property. Only clear. mathematical properties are allowed; fortunately, these properties are sufficient. for expression of all mathematical facts. Our exposition in this section is informal. Fkaders who would like to see how this topic can be studied from a more rigorous point of view can t:onsult some book on mathematical logic. The basic set-theoretic property is the membership property: " . . . is a11 element of . . . ," which we denote by E . So "X E Y" reads "X is an elen~c~lt of' Y1' or "X is a member of Y" or "X belongs to Y." The letters X and Y in these expressions are variables; they stand for (denote) unspecified, arbitrary sets. The proposition "X E Y" holds or does not. hold depending on sets (denoted by) X and Y. We sometimes say "X E Y" is a property of X and Y . The reader is surely familiar with this informal way of speech from other branches of mathematics. For example, "m is less than 71" is a property of m and n. The letters m and n are variables denoting unspecified numbers. Some m and n have this property (for example, "2 is less than 4" is true) but others do not (for example, "3 is less than 2" is false). All other set-theoretic properties can be stated in terms of membership with the help of logical means: identity, logical connectives, and q~ant~ifiers. We often speak of one and the same set in different contexts and f i ~ dit convenient to denote it by different variables. We use the identity sign "=" t o express that two variables denote the same set. So we write X = Y if X is thr. same set as Y [ X is identical unth Y or X is equal t o Y ] . In the next example, we list some obvious facts about identity: 2. l Example (a) X = X .

(b)

If X = Y, then Y

=

X.

( c ) If X = Y and Y = Z, then X = Z.

(d) If X = Y and X E 2, then Y E 2. (e) If X

=

Y and Z

E

X, then Z E Y.

(X is identical with X.) (If X and Y are identical, then Y and X are identical.) (If X is identical with Y and Y is identical with 2, then X is ider~tical with Z.) (If X and Y are identical and X belongs to Z, then Y belongs to 2.) (If X and Y are identical and Z belongs to X , then 2 belongs to Y.)

2. PROPERTIES

5

Logical connectives can be used to construct more complicated properties from simpler ones. They are expressions like "not . . . ," " . . . and . . . ," "if. . . , then . . . ," and " . . . if and only if . . . ."

2.2 Example (a) "X E Y or Y E X" is a property of X and Y. (b) "Not X E Y and not Y E X" or, in more idiomatic English, "X is not an element of Y and Y is not an element of X" is also a property of X and Y. (c) "If X = Y, then X E Z if and only if Y E Z" is a property of X , Y and Z. (d) "X is not an element of X" (or: "not X E X") is a property of X . We write X # Y instead of "not X Y" and X # Y instead of "not X = Y." Quantifiers "for all" ("for every") and "there is" ( "there exists" ) provide additional logical means. Mathematical practice shows that all mathematical facts can be expressed in the very restricted language we just described, but that this language does not allow vague expressions like the ones at, the beginning of the section. Let us look a t some examples of properties which involve quantifiers. 2.3 (a) (b) (c)

Example "There exists Y E X." "For every Y E X , there is Z such that Z E X and Z E Y." "There exists Z such that Z E X and Z # Y."

Truth or falsity of (a) obviously depends on the set (denoted by the variable) X . For example, if X is the set of all American presidents after 1789, then (a) is true; if X is the set of all American presidents before 1789, (a) becomes false. [Generally, (a) is true if X has some element and false if X is empty.] We say that (a) is a property of X or that (a) depends on the parameter X . Similarly, (b) is a property of X , and (c) is a property of X arid Y. Notice also that Y is not a parameter in (a) since it does not make sense to inquire whether (a) is true for some particular set Y; we use the letter Y in the quantifier only for convenience and could as well say, "There exists W E X ," or "There exists some element of X ." Similarly, (b) is not a property of Y, or Z , and (c) is not a property of 2. Although precise rules for determining parameters of a given property can easily be formulated, we rely on the reader's common sense, and limit ourselves to one last example. 2.4 Example (a) "YE X." (b) "There is Y f X ." (C) "For every X, there is Y E X."

Here (a) is a property of X and Y; it is true for some pairs of sets X , Y and false for others. (b) is a property of X (but not of Y), while (c) has no parameters. (c) is, therefore, either true or false (it is, in fact, false). Properties

CHAPTER l . SEI'S

6

which have no parameters (and are, therefore, either true or false) are callecl statements; all mathematical theorems are (true) statements. We son~etimeswish to refer to an arbitrary, unspecified propertry. 1%. 1 1 s ~ boldface capital letters to denote statements and properties and, if convenient.. list some or all of their parameters in parentheses. So A ( X ) stancts for any property of the parameter X , e.g.. (a). (L>), or (c) in Exa~nple2 . 3 . E ( X . Y ) is a property of parameters X and Y, e.g., (c) in Exari~ple2.3 or (a) ill Ex:lrrlplt. 2.4 or (cl)

"X E

Y o r X = Y or Y E

X.''

In general, P(X, Y,. . . , Z ) is a property whose truth or falsity deper~dsc111 parameters X, Y ,. . . , Z (and possibly others). We said repeatedly that all set-theoretic properties can be expressed in our restric:ted language, consisting of n~embershipproperty and logical means. However, as the developmerlt proc:eeds a11d rliore and more complicated t h m r t t ~ r arcb ~s proved, i t is practical to give names to various particular properties, i.e.. to Liefine new properties. A new symbol is then introduced (defined) to dt!l~ott,thcl property in question; we can view it ;is a shorthand for t.11~ explicit f'orlr~ul;itiori. For example, the property of I~einga subset is defined by 2.5

X

CY

if and or~lyif every elerrlent of X is an element of Y

"X is a subset of Y" (X

c Y) is a property of

X and Y. We can use it irl more cotnplicated formulations and, whenever desirable, rcpl;tcc? X Y by its tlefinition. For example, the explicit defirlition of

"If X

c Y ancl Y G Z, then X c 2.''

would be

"If every element of X is arl ttlt.merit of Y and every elenlent of Y is

ill1

element of 2, then every element o f X is an element of 2." It is clear that mathematics witllout definitions woulcl hcl possible. but ext:ctedingly clumsy. For another type of definition. consider the property P ( X ) : "There clxists no Y E X ." We prove in Section 3 that (a) There exists a set X such that P ( X ) (there exists a set X with no elements). (b) There exists at most one set X such that P ( X ) , i.e., if P ( X ) and P ( X f ) . then X = X' (if X has no t ~ l t ~ l ~ ~and e n t X' s ha,! no eler~lents,the11 X i i r ~ c l X' are identical). (a) and (b) together express the fact that there is a unique set X with thtb property P ( X ) . We can the11 givt? this set a name, say 0 (the ernyty set). arl available. The reader may have noticed that we did not guarantee existence of any infinite sets. This deficiency is removed in Chapter 3. Other axioms are introduced in Chapters 6 and 8. The complete list of axioms can be fourltl in Section 1 of Chapter 15. This axiomatic system was essentially formulated by Ernst Zermelo in 1908 and is often referred to as the Zemelo-Fraenkel uxiomntic system for set theory.

Exercises 3.1 Show that the set of all X such that X E A and X 4 B exists. 3.2 Replace the Axiom of Existence by the following weaker postulate: Weak Axiom of Existence Some set exists. Prove the Axiom of Existence using the Weak Axiom of Existence and the Comprehension Schema. [Hint:Let A be a set known to exist; coilsider {X E A l X # X ) . ]

CHAPTER 2 . SET'S

12

3.3 (a) Prove that a "set of all sets" does not exist. [Hint: if V is a set of all sets, consider ( X E V I X 4 X ) .] (b) Prove that for any set A there is some X 4 A. 3.4 Let A and B be sets. Show that there exists a unique set C such that. X E C if and only if either X E A and X 4 B or X E B and X 4 A. 3.5 (a) Given A, B, and C, there is a set P such that X E P if and only if x=Aor~=Bors=C. (b) Generalize to four elements. 3.6 Show that P ( X ) X is false for any X. In particular, P ( X ) # X for any X. This proves again that a "set of all sets" does not exist. /Hint: Let Y = { U € X [ u ~ u ) Y ;E P ( X ) b u t Y 4 X.] 3.7 The Axiom of Pair, the Axiom of Union, and the Axiom of Power Set can be replaced by the following weaker versions. Weak Axiom of Pair For any A and B, there is a set C such that A E C and B E C. Weak Axiom of Union For any S, there exists U such that if X E A and A E S, then X E U. Weak Axiom of Power Set For any set S, there exists P such that X c S implies X E P. Prove the Axiom of Pair, the Axiom of Union, and the Axiom of P o w r ~ Set using these weaker versions. [Hint: Use also the Co~nprehension Schema.]

c

4.

Elementary Operations on Sets

The purpose of this section is to elaborate somewhat on the notions introduced in the preceding section. In particular, we introduce simple set-theoretic: operations (union, intersection, difference, etc.) and prove some of their basic properties. The reader is certairily familiar with them to some extent and we leave out most of the details. Definition 3-10 tells us what it means that A is a subset of B (i7~cludedill B), A 5 B. The property 5 is called inclusion. It is easy to prove that, for any sets A , B, and C , (a) A 5 A. (b) If A B and B C A, then A = B. (c) If A 2 B and B G C , then A C C. For example, to verify (c) we have to prove: I f X E A, then s E C. But if X E A , then s E B, since A B. Now, X E B implies X E C, since B C C. So X E A implies X E C. If A C B and A # B, we say that A is a proper subset of B (A is prope7.ly contained in B) and write A C B. We also write B 2 A instead of A c B a11d B 2 A instead of A C B. Most of the forthcoming set-theoretic operations have been rr~entionedb e fore. The reader probably knows how they can be visualized using Venn &agrams (see Figure 1).

c

4. ELEMENTARY OPERATIONS ON SETS

Figure 1: Venn diagrams. (a) Intersection: Tile shaded part is A n B . (b) Union: The shaded part is A U B. (c) Difference: The shaded part is A - B. (d) Symmetric difference: The shaded part is A A B. (e) Distributive law: The shaded part obviously represents both A n ( B U C) and (A n B ) U (A n C). 4.1 Definition The intersection of A and B, A

n B, is the set of all

which which belong to both A and B. The union of A and B, A U B, is the set of belong in either A or B (or both). The diflerence of A and B, A - B, is the set of all X E A which do not belong to B. The s?ymmetP.ic diflerence of A and B, A A B, is defined by A A B = (A - B) U ( B - A ) . (See Examples 3.3 and 3.8 and Exercises 3.1 and 3.4 for proofs of existence and uniqueness.) X all X

As an exercise, the reader can work out proofs of some simple properties of these operations. Commutativity

Associativity

AnB=BnA AuB=BuA (AnB)nC=An(BnC) (AuB)uC=AU(BUC)

So forgetting the parentheses we can write simply A n B n C for the intersection

of sets A, B , and C. Similarly, we do not need parentheses for the rinioil arlcl for more than three sets. A n ( B U C ) = ( A n B )U ( A n C ) A u ( B n C )= ( A u B ) n ( A u C )

Distributivity

DeMorganLaws C - ( A n

B ) = (C - A) U (C - B )

C - ( A uB) = ( C - A ) n ( C - B ) Some of the properties of the difference and the symmetric difference arc

A n ( B - C ) = ( A n B )- C A - B =. O if and only if A C B AAA=@ AAB=BAA ( A A B ) A C =A A ( B A C ) Drawing Venn diagrams often helps one discover nikd prove these and sitiii1;ir relatiorlships. For example, Figure l i e ) illustrates the distributive law A n ( B LJ C ) = ( A n B ) U ( A n C ) . The rigorous proof proceeds as follows: We hwvv t o prove that the sets A n (B U C ) tirlrl ( A n B ) U ( A n C ) h:vt? t.he same eI~illt?rits. That requires us to show two facts: (a) Every element of A n ( B U C ) belongs to ( A n B ) U ( A n C ) . (b) Every element of ( A n B) U ( A n C ) belongs to A n ( B U C ) . To prove (a), let a E A ~ ( B u C )Then . a E A and also a E BUC. Therefore: either a f B or a E C. So a E A nilri a E B or a E A and a E C'. Tllis tinhiins t.hat it E A n B or a E A n C ; htantsc,finally, U E ( A n B ) U ( A n C ) . To prove (b), let a E ( A n B ) U ( A n C ) . Then a E A n B or a E A n L'. Irl the first case, a E A and a E B , so U E A and a E B U C . and a E A n ( B u C ' ) . 111the second case, U E A and U E C , SO again a E A arkcl cr. E B U C , arltl finally. U E An(3uC). U The exercises should provide slificierlt material for practicing similar ()l(.mentary arguments about sets. The union of a system of sets S was defined in the preceding section. IV(> now define the intersection S of a rlonernpty system of sets S : rr: E S if ;illil r~rllyif z E A for all A E S. The11 intersection of two sets is again ;i spcc*i;il case of the more genera1 operation: A n B = n { A ,B ) . Notice that wr do 11c1t define (78; the reason is that rLvc?rp:r belongs to all A E Q) (si11r:c there is 110 such A), so B would have to be i i set of all sets. We pustponc lilortl drtailvtl itivestigation of general unions and intersections until Chapter 2, where) ii ir~orc wit3ld y tkotatiolr becoirles available. Finally, we say that sets A i i t l r l B are dis~ointif' A n B = 8, Alorc. ge~i~riilly. S is a system of mutuully disjoint s e t s if A n B = Q) for a11 A, B E S sutnht l l i i t A # B.

n

n

n

4. ELEMENTARY OPERATIONS ON SETS

15

Exercises

4.1 Prove all the displayed formulas in this section and visualize them using 4.2

Venn diagrams. Prove: (a) A C B i f a n d o n l y i f A n B = A i f a n d o n l y i f A ~ B =B i f a n d o n l y ifA-B=@. (b) A B n C if and only if A G B and A C C. ( c ) B U C A if and only if B A and C A. (d) A - B = ( A u B ) - B = A - ( A n B ) . (e) A n B = A - ( A - B ) . (f) A - ( B - C) = ( A - B ) u ( A n C ) . (g) A = B if and only if A A B = 8. For each of the following (false) statements draw a Venn diagram in which it fails: (a) A - B= B - A . (b) A n B c A. (C) A B U C implies A G B or A C. (d) B n C A implies B C A or C A. Let A be a set; show that a "complement" of A does not exist. (The "complement" of A is the set of all s 4 A.) Let S # 8 and A be sets. (a) Set Tl = {Y E P ( A ) I Y = A n X for some X E S), and prove A n US = UT1 (generalized distributive law). (b) Set Tz = { Y E P ( A ) 1 Y = A - X for some X E S), and prove

c

4.3

c

4.4 4.5

c

c

c

c c

(generalized De Morgan laws). 4.6 Prove that r) S exists for all S # 8. Where is the assumption S # in the proof?

0 used

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Chapter 2

Relations, Functions, and Orderings 1.

Ordered Pairs

In this chapter we begin our program of developing set theory as a foundation for mathematics by showing how various general mathematical concepts, suclh as relations, functions, and orderings can be represented by sets. We begin by introducing the notion of the ordered pair. If a and b arc sets. then the unordered pair {a, b ) is a set whose elements are exactly n and b. The "order" in which a and b are put together plays no role; {a, b ) = {b.a). For many applications, we need to pair a and b in a way making possible to "read off" which set comes "first" and which comes "second.'! We denote this ordered pair of n and b by ( a , b); a is the first coordinate of the pair (a, b), b is the second coordinate. As any object of our study, the ordered pair has to be a set. It should bc defined in such a way that two ordered pairs are equal if and only if their first, coordinates are equal and their second coordinates are equal. This guarantres in particular that (a, b ) # (b, a ) if a # b. (See Exercise 1.3.) There are many ways how to define (a, b) so that the foregoing conditioll is satisfied. We give one such definition and refer the reader to Exercise 1.6 for alternative approach. 1.1 Definition (a, b) = {{a), { a , b ) ) .

If a # b, (a, b) has two elements, a singleton { a ) and an unordered pair {a, b). We find the first coordinate by looking a t the element of { a ) . The second coordinate is then the other element of {a, b). If a = b , then (ata ) = {{(L), {a, a ) ) = {{a) ) has only one element. In any case, it seems obvious that both coordinates can be uniquely "read off" from the set (a, b). WP make this statement precise in the following theorem.

CHAPTER 2. RELATIONS, FUNCTIONS, AND ORDERINGS

18

1.2 Theorem (a, b) = (a', b') if and only if a = a' and b = b'.

Proof. If a = a' and b = b', then, of course, (a, b) = { { a ) ,{a, b)) = {{a'), {a', 6')) = (a', 6'). The other implication is more intricate. Let us assume that {{a), {a,b)) = {{a'), { a f , b f ) ) .If a # b, {a) = {a') and {a,b) = {a',bt). S O, first, a = a' and then {a, b) = {a, b') implies b = b'. If a = b, { { a ) ,{a. a ) ) = {{a)). So { a ) = {a'), {a) = {a', h'}, and we get a = a' = b', so a = a' and b = b' holds in this case, too. 0 With ordered pairs at our disposal, we can defirie ordered triples ("l

b, c) = ((a,b), c),

ordered quadruples (a, b, c, d) = ((a,b, c), d), and so on. Also, we define ordered "one-tuples" (a) = a. However, the general definition of ordered n-tuples has to be postponed until Chapter 3, where natural numbers are defined. Exercises

1.1 Prove that (a, b) E P ( P ( { a ,b))) and a , b E U(a, b). More generally. if a f A and b E A, then (a, b) f P(P(A)). 1.2 Prove that (a, b), (a, b, c), and (a, b, c, d) exist for all a , b, c, and d. 1.3 Prove: If (a, b) = (6, a), then a = b. 1.4 Prove that ( a ,b, c) = (a', b', c') implies a = a', b = b', and c = c'. S t . a t ~ and prove an analogous property of quadruples. 1.5 Find a, b, and c such that ((a, b), c) # (a, (b, c)). Of course, we could use the second set to define ordered triples, with equal success. 1.6 To give an alternative definition of ordered pairs, choose two different sets C3 and A (for example, U = 0, A = (0))and define

State and prove an analogue of Theorem 1.2 for this notion of ordered pairs. Define ordered triples and quadruples.

2.

Relations

Mathematicians often study relations between mathematical objects. Relations between objects of two sorts occur most frequently; we call them binary relations. For example, let us say that a line 1 is in relation R1 with a point P if I passes through P. Then RI is a binary relation between objects called lines a l i r i objects called points. Similarly, we define a binary relation Rz between positive

2. RELATIONS

19

integers and positive integers by saying that a positive integer m is in relation R2 with a positive integer n if m divides n (without remainder). Let us now consider the relation R: between lines and points such that a line 1 is in relation R', with a point P if P lies on 1. Obviously, a line l is in relation RI to a point P exactly when 1 is in relation R\ to P. Although different properties were used to describe RI and R:, we would ordinarily consider R , and R; to be one and the same relation, i.e., RI = R;. Similarly, let a positive integer m be in relation R; with a positive integer n if n is a multiple of m. Again, the same ordered pairs (m, n) are related in R2 as in R;, and we consider to be the same relation. R2 and A binary relation is, therefore, determined by specifying all ordered pairs of objects in that relation; it does not matter by what property the set of these ordered pairs is described. We are led to the following definition. 2.1 Definition A set R is a binary relation if all elements of R are ordered pairs, i.e., if for any z E R there exist s and y such that z = (x,y).

2.2 Example The relation R2is simply the set { t ( there exist positive intege.rs m and n such that z = (m, n ) and m divides n ) . Elements of R2 are ordered pairs

It is customary to write xRy instead of (X,y) f R. We say that relation R with y if s R y holds. We now introduce some terminology associated with relations.

.X

is in

2.3 Definition Let R be a binary relation. (a) The set of all s which are in relation R with some y is called the domain of R and denoted by dom R. So dorn R = { s I there exists y such that xRy ) . don1 R is the set of all first coordinates of ordered pairs in R. (b) The set of all y such that, for some X, z is in relation R with y is called the range of R, denoted by ran R. So ran R = { y ( there exists X such that :t-Ry ). ran R is the set of all second coordinates of ordered pairs in R. Both dom R and ran R exist for any relation R. (Prove it. See Exercise 2.1). (c) The set dom R U ran R is called the field of R and is denoted by field R. (d) If field R X , we say that R is a relation in X or that R is a relation between elements of X .

c

2.4 Example Let R2 be the relation from Example 2.2.

dom R2 = {m I there exists n such that m divides n ) = the set of all positive integers

CHAPTER 2. RELATIONS, FUNCTIONS. AND ORDERINGS

20

because each positive integer m divides some n, e.g., n = m; ran R2 =

1 there exists m such

that 7n divides n ) = the set of all positive integers (71

because each positive integer

YL is

divided by some m. e.g., by m =

7 ~ ;

field R:! = dom R2 LJran R2 = the set of all positive integers;

R2 is a relation between positive integers. We next generalize Definition 2.3. 2.5 Definition (a) The image of A under R is the set of all y from the range of R related R to some element of A; i t is denoted by R [ A J So .

R[A]= { y f ran R ( there exists

X

it1

E A for which xRp).

(b) The inverse image of B under R is the set of all X from the domain of R related in R to some element of B; it js denoted R-' [B].So

-

R-'[B]

{X

E dom R ( there exists I/ E

B for which x R y ) .

2.6 Example ~ ; ' [ { 3 , 8 , 9 , 1 2 } j= { l , 2,3,4,6,8,9,12); R1[{2}] = the set of' all even positive integers. 2.7 Definition Let R be a binary relation. The inverse of R is the set

R-

L

=

{z1 z

= ((L, g )

for some z ;md y such that (y.s)E R )

2.8 Example Again let

R2 = { t 1 t = (rn,n ) , rn ancl n are positive integers, and 7n divides 7 1 ) : tbc~ii R,' = {W1 W = ( n , m ) , and ( m , n ) Rz} ~ = {WI W = ( n , m ) , m and n are positive integers, and m divides n}. In our description of Rz, we use variable m for the first coordinate and variable n for the second coordinate; we also state the property describing R2 so that the variable m is mentioned first. It, is a customary (though not necessitr!.) practice to describe R;' in the same way. All we have to do is use letter 771 instead of n , letter n instead of m. and change the wording: l

R; = { W (

W

= {WI

W

(m, n ) , n , m are positive integers, and n divides 7 7 1 ) = ( m , n ) , m , n are positive integers, and m is a multiple of n ) =

Now R2 and R;' are described in a parallel way. In this sense, the inverse o f the relation "divides" is the relation "is a multiple." The reader may notice that the symbol Rb'IB] in Definition 2.5(b) for tile inverse image of B under R now also denotes the image of B under R - ' . Fortunately, these two sets are equal.

2. RELATIONS

21

2.9 Lemma The inverse image of B under R is equal to the image of B under

R- l. Notice first that dom R = ran R-' (see Exercise 2.4). Now, a: E Proof. dom R belongs t o the inverse image of B under R if and only if for some E B, (a:, y) E R. But ( X ,y) E R if and only if (y, a:) E R - ' . Therefore, a: belongs to the inverse image of B under R if and only if for some y E B, (y, X) E R- '. i.e.. U if and only if a: belongs to the image of B under R-'. In the rest of the book we often define various relations. i.e.. sets of ordered pairs having some particular property. To simplify our notation, we introduce the following conventions. Instead of {W

)

W

= (a:, y) for some X and y such that P ( x , y ) ) ,

we simply write Y) l P(x,Y)). For example, the inverse of R could be described in this notation as { ( X . ? j ) 1 (y, X ) +E R ) . [As in the general case, use of this notation is admissible only if we prove t h a t there exists a set A such that. for all a: and y. P ( x . y) implies (5, y) f A.] {(X,

2.10 Definition Let R and S be binary relations. The composition of R and S is the relation

So R =

{(X, z )

1 there exists y for which

(a:, y) E R and (y, 2) E S ) .

So (a:, z ) E S o R means that for some y, x h y and ySz. To find objects relat~cl to X in S o R, we first find objects y related t o X in R , and then objects related t o those y in S . Notice that R is performed first and S second, but notation S o R is customary (at least in the case of functions; see Section 3). Several types of relations are of special interest. We introduce some of them in this section and others in the rest of the chapter. 2.11 Definition T h e membership relation on A is defined by

€ A = ((a,b) I a f A, ~ E Aa n, d a E h ) . The identity wlation on A is defined by 1 d =~ {(a,b) I

~ A,E

b f A, a n d a = b ) .

2.12 Definition Let A and B be sets. The set of all ordered pairs whose first coordinate is from A and whose second coordinate is from B is called the cartesian product of A and B and denoted A X B. In other words,

A

X

B = {(a,b) I a . € A a n d b ~B}.

Thus A X B is a relation in which every element of A is related to every element. of B.

CHAPTER 2. RELATIONS, FUNCTIONS, AND ORDERINGS

22

It is not completely trivial t o show that the set A X B exists. However, one gets from Exercise 1.1 that if a E A and b E B, then ( a . b ) E P ( P ( A U B ) ) . Therefore,

A x B = { ( a , b ) f P ( P ( A u B ) ) I U E A a n d b e B). Since P ( F ( A U B ) ) was proved to exist, the existence of A X B follows from the Axiom Schema of Comprehension. [To be completely explicit, we can write,

A x B= We denote A readily:

{W

X

E P(P(AU

B ) ))

W

= (a, b) for some a E A and b E B).]

A by A 2 . The cartesian product of three sets can be introduced AxBxC=(AxB)xC.

Notice that

(using an obvious extension of our notational convention). A X A denoted A 3 . We can also define ternary relations.

X

A is usually

2.13 Definition A ternary relation is a set of unordered triples. More explicitly, S is a ternary relation if for every U E S, there exist X , y, and t such that. U = ( X ,y, 2). IE S G A ~ we , say that S is a ternary relation in A . (Note t h a t a binary relation R is in A if and only if R C A'.)

We could extend the concepts of this section t o ternary relations and also define 4-ary or 17-ary relations. We postpone these matters until Section 5 in Chapter 3, where natural numbers become available, and we are able to defint* n-ary relations in general. At this stage we only define, for technical reasons. unary relations by specifying that a unary relation is any set. A unary relation i n A is any subset of A. This agrees with the genera1 conception that a unary relation in A should be a set of l-tuples of elements of A and with the definition of ( X )= a: in Section 1. Exercises

2.1 Let R be a binary relation; let A = U(UR ) . Prove that (X, E R implies s E A and y E A . Conclude from this that dom R and ran R exist. 2.2 (a) Show that R-' and S 0 R exist. [Hint: R-' (ran R ) x (dorn R). S o R (dom R) X (ran S ) . ] (b) Show that A X B X C exists. 2.3 Let R be a binary relation and A and B sets. Prove: (a) R [ AU B] = RIA] U R [ B J . (b) R [ A n B] C R[A]n R [ B J .

v)

c

c

3. FUNCTIONS

23

c

(d) Show by an example that and _> in parts (b) and (c) cannot be replaced by =, (e) Prove parts (a)-(d) with R-' instead of R. (f) R-' [R[A]]2 A n dom R and R(R- l (B]] 2 B n ran R; give examples where equality does not hold. 2.4 Let R C X X Y. Prove: (a) R[X] = ran R and R-' [Y] = dom R. (b) If a $ dom R, R[{a}] = 0; if b $ ran R, R-' [{b)]= 0. (c) dom R = ran R-'; ran R = dom R-'. (d) ( R - ' ) - l = R. (e) R-' 0 R 2 IddomR;R 0 R-' 3 IdrauR. 2.5 Let X = (0,{g)), Y = P ( X ) . Describe (a) € v , (b) I ~ Y . Determine the domain, range, and field of both relations. 2.6 Prove that for any three binary relations R, S, and T

(The operation o is associative.) 2.7 Give examples of sets X , Y, and Z such that (a) X x Y # Y x X . (b) X X (Y X Z ) # ( X X Y) X Z. ( C) X 3 # X X X 2 (i.e., (X X X ) X X # X X (X X X)]. !Hint for part (c): X = {a) .] 2.8 Prove: (a) A X B = 0 if and only if A = 0 or B = 0. (b) (Al U AS) X B = (Al X B) U (A2 X B); A X (Bl U B2) = (A X B1) U (A X 82). (c) Same as part (b), with U replaced by n, -, and A.

3. Functions Function, as understood in mathematics, is a procedure, a rule, assigning to any object a from the domain of the function a unique object b, the value of the function a t a. A function, therefore, represents a special type of relation, a relation where every object a from the domain is related to precisely one object in the range, namely, to the value of the function at a. 3.1 Definition A binary relation F is called a junction (or mapping, correspondence) if aFbl and aFbz imply bl = b2 for any a , b l , and b2. In other words, a binary relation F is a function if and only if for every a from d o m F there is exactly one b such that aFb. This unique b is called the value of F at a and is denoted F ( a ) or F,. (F(a) is not defined if a 4 dom F.] If F is a function with dom F = A and ran F C B, it is customary to use the notations

CHAPTER 2. RELATIONS, FUNCTIONS, AND ORDERINGS

24

F

:

A

4

B , ( F ( a )I a

range of the function

F

A), (F, I a E A), (Fa)afA for the function F. can then be denoted { F ( a )1 a E A) or

'rl~r

T h e Axiom of Extensionality can be applied to functions as follows. 3.2 Lemma Let F and G be functions. anti F ( x ) = G ( x ) for all X E dom F .

F

=G

2f

and only zf d o ~ F n = do~G n

We leave the proof to the reader. Sitice functions are binary relations, concepts of domain, range. image. inverse image, inverse, and composition can be applied to them. We introduce several additional definitions. 3.3 Definition Let F be a function and A and B sets. ( a ) F is a function on A if dom F = A. ( b ) F is a function into B if ran F C B. (c) F is a function onto B if ran F = B. ( d ) Tlie restriction of the function F to A is the function

F T A = { ( a , b )E F

( U E

A).

If G is a restriction of F to sorne A, we say that F is an extenszon of G. 3.4 Example Let F = { ( X , 1 / x 2 ) I X # 0, X is a real number). F is a function: If aFbl and aFb2, bl = l/u%nd bn = l / a 2 , so bl = b2. Slightly stretching our riotatioilal conventions. we can also write F = ( l/.rL X is a real number, X # 0). The value of F at X , F ( x ) , equals l / x 2 . It is ; l function otl A , where A = { X ) s is n real number and ,2: # 0 ) = dotn F. F is it function illto the set of all real nuitll~ers,but not onto the set ut' all real n u r ~ ~ l ~ r r s . If L3 = { X I X is a real number tud L > 0). then F is onto B. If C = { : I . l 0 1 ) i~rldf'-'[C] = { X 1 x 5 - 1 o r : c > 1 ) . Let us find the composition f o f : f

0

1 there is 2~ for which ( X , y ) E f and ( y , z ) C f } = { ( X . z) I there is y for which X # 0, y = 1 / x 2 ,and y # 0, z = = { ( X , z ) I X # 0 and z = X ' }

f =

=

{(X,z )

(X"

X

1/y2}

# 0).

Notice t h a t f o f is a function. This is not an accident. 3.5 Theorem Let f and y be functions. Then y o f is a functior~. g o j' i s defined at X if and only if f is defined at X and y is defined at f ( x ), 2 . e.,

dorn(g o f ) = dom f 17f-'[domg].

Also, (y o f ) ( X )

=

y ( f ( X ) ) for all

.T

E ciom(g o f ).

26

CHAPTER 2. RELATIONS, FUNCTIONS, AND ORDERINGS

Example Let f = (1/x2 ] X # 0);find f a ' . As f = { ( x , l / x " ) x # 0}, we get f - l = {(1/x2,X) 1 X # 0). f - l is not a function since ( 1 . -1) E f - l . ( 1 , l ) E f - l . Therefore, f is not one-to-one; ( 1 , l ) E f . ( - 1 , l ) E f . (b) Let g = (2x - 1 ) X real); find g - l . g is one-to-one: If 2 x l - 1 = 2x2 - 1. then 2x1 = 2x2 and X I = X ? . y = {(X,y) 1 y = 2x - 1, x real), therefore. g-l = { ( y , x ) I y = 22 - 1, z real).

As customary when describing functions, we express the second coordinate (value) in terms of the first:

Finally, it is usual to denote the first ("independent") variable x and the second ("dependent") variable y. So we change notation:

l

S-' = {(x,Y) Y =

x+l X +1 7 X, real} = (' r real).

3.10 Definition (a) Functions f and g are calIed compatible if f ( X )= y(x) for all X E dom f n dom g. (b) A set of functions F is called a compatible system of functions if any two functions f and y from F are compatible. 3.11 Lemma (a) Functions f and y are compatible if and only if f U y is a function. (6) f i n c t i o n s f und y are compatible if and only if f 1 (dom f n d o m g ) = g (dom f n domg).

We leave the easy proof to the reader, but we prove the following. 3.12 Theorem If F is a compatible system of functions. then U F is a functzun with d o m ( U F ) = U{dom f ) f E F ) . The function U F extends all f E F.

Functions from a compatible system can be pieced together t o form a single function which extends them alt. Clearly, U F is a relation; we now prove that it is a function. If Proof. (a, bl) f U F and (a, b z ) f U F, there are functions f l , f 2 E F such that ( a , bl) E f l and (a, bz) E f2. But f l and f2 are compatible, and a E dom f l n dorn f 2 ; so b l = f(a1) = f (a21 = b2. I t is trivial t o show that X f dom(U F ) if and only if X E dom f for some ffF. U 3.13 Definition Let A and B be sets. T h e set of all functions on A into B is exists; this is done in Exercise denoted B A . Of course, we have t o show that 3.9.

3. FUNCTIONS

27

It is useful to define a more general notion of product of sets in terms of functions. Let S = (S, I i f I ) be a function with domain I . The reader is probably familiar mainly with functions possessing numerical values; but for us, the values S, are arbitrary sets. We call the function (S, I i E I) an indexed system of sets, whenever we wish to stress that the values of S are sets. Now let S = (Sj1 i E I ) be an indexed system of sets. We define the product of the indexed system S as the set S = {f

If

is a function on I and f, E S, for all i

E I)

Other notations we occasionally use are

The existence of the product of any indexed system is proved in Exercise 3.9. The reader is probably curious to know how this product is related to the previously defined notions A X B and A X B X C. We return to this technical problem in Section 5 of Chapter 3. At this time, we notice only that if the indexed system S is such that Si = B for all i E I , then

The "exponentiation" of sets is related t o t'multiplication" of sets in the same way as similar operations on numbers are related. We conclude this section with two remarks concerning notation. U A and A were defined for any system of sets A (A # Q) in case of intersection). Often the system A is given as a range of some function, i.e.. of some indexed system. (See Exercise 3.8 for proof that any system A can be so presented, if desired.) We say that A is indexed by S if

n

where S is a function on I, It is then customary to write

and similarly for intersections. In the future, we use this more descriptive notation. Let f be a function on a subset of the product A X B. It is customary to denote the value of f at ( X ,y) f A X B by f( X ,y) rather than f ((X,Y)) and regard f as a function of two variables X and y.

CHAPTER 2. RELATIONS, FUNCTIONS, AND ORDERINGS

28 Exercises

3.1 Prove: If ran f C dom g, then dom(y o f ) = dorn f . 3.2 T h e functions f , : i = 1 , 2 , 3 are defined as follows:

fl

= (2x - l l X real),

f2

=

-

f3

3.3

3.4

(6 1 X > O), {l/x l z real,

X

# 0).

Describe each of the following functions, and determine their ciomaitls and ranges: f2 0 f l , f l 0 f ~ f ~, 0 . f ~f l U f3. Prove t h a t the functions f l , f.2, f 3 from Exercise 3.2 are one-to-one, atitl find the inverse functions. 111 each case, verify that dorn f , = ran( f , - l ) . ran f , = dom( f i L ) . Prove: (a) If f is invertible, f - o f = Idno,,,I , f o f - l = Id ,, f . (b) Let f be a function. If there exists a function g such t h a t y o f = IddOknf then f is invertible and f = y r ran f . If there exists function h such t h a t f 0 h = Id,,, f then f may fail t o be invertible. Prove: If f and y are one-to-one functions, y o f is also a one-to-one function, and (y 0 f ) - ' = f - ' 0 y-l. T h e images and inverse images of sets by functions have the properties exhibited in Exercise 2.3, but some of the inequalities cat1 now be replac.etl by equalities. Prove: (a) If f is a function, f l [ A n B] = f l [ A ] n f L 1 \ B ] . (b) If f is a function, f [A - B] = f [A] - f - [B]. Give an example of a fuliction f and a set A such t h a t f n A" f [ A. Show t h a t every system of sets A can be indexed by a function. [Hint: Take I = A and set S, = i for all i E A.] (a) Show that the set B A vxists. [Hint: E P(A x B).] (b) Let (S, I i E I ) be at] indexed system of sets: show t h a t S, exists. (Hint: S* 5 P ( I X UzcIS,).\ Show that unions and intersections satisfy the following general f o r ~ nof' the associative law:

'

-'

3.5

3.6

-'

3.7 3.8 3.9

-'

n,,,

nlE,

3.10

nEU 5.'

'

C E S aEC

if S is a nonempty system of nonempty sets. 3.11 Other properties of unions and intersections can be generalized similarly. De Morgan Laws:

4. EQUIVALENCES AND PARTITIONS

Distributive Laws:

3.12 Let f be a function. Then

!-'I

U Fa] aE A

=

U f-'[Fa], aEA

If f is one-to-one, then in the third formula can be replaced by 3.13 Prove the following form of the distributive law:

=.

assuming t h a t Fa,,, n Fa,bz= Q) for all U € A and bl , b2 € B , h l # b2- [Hint: Let L be the set on the left and R the set on the right. F ~ , , (c ~ )U ,, F ~ .hence ~; F ~ , ~ C( ~ , F ~ .=~ L, ) so finally, R C_ L. To prove t h a t L C R, take any n: E L. P u t ( a ,6) E f if and only if z E Falb,and prove t h a t f is a function on A into B for which X E (loEA so z E R.]

naEA

4.

naEA(u,,,

Equivalences and Partitions

Binary relations of several special types enter our considerations more frequently. 4.1 Definition Let R be a binary relation in A. (a) R is called reflexive in A if for all U A, a R u . (b) R is called spmmetric in A if for all a , b E A, uRb implies bRa. (c) R is called transitive in A if for all a , 6, c E A, u R b and bRc imply uR.cm. (d) R is called a n equivulence on A if it is reflexive, symmetric, and t,ransit,ivc. in A .

30

CHAPTER 2. RELATIONS, FUNCTIONS, AND ORDERINGS

4.2 Example

(a) Let P be the set of all people living on Earth. We say t h a t a person p is equivalent to a person q (p q ) if p and q both live in the same country. Trivially, = is reflexive, symmetric, and transitive in P. Notice that the set P can be broken into classes of mutually equivalent elements; all people living in the United States form one class, all people living in France are another class, etc. All members of the same class are mutually equivalent; members of different classes are never equivalent. T h e equivalence classes correspond exactly to different countries. (b) Define a n equivalence E on the set Z of all integers as follows: z E y if and only if X - y is divisibIe by 2. (Two numbers are equivalent if their difference is even.) T h e reader shouId verify 4.l(a)-(c). Again, the set. Z can be divided into equivalence classes under (or, as is customary to say, rnodulo) the equivalence E . In this case, there are two equivalence classes: the set of even integers and the set of odd integers. Any two even integers are equivalent; so are any two odd integers. But an even integer cannot be equivalent to an odd one. T h e situation encountered in the previous exampIes is quite general. Any equivalence on A partitions A into equivalence classes; conversely, given a suitable partition of A, there is a n equivalence on A determined by it. T h e foIIowing definitions and theorems establish this correspondence. 4.3 Definition Let E be an equivalence on A and let a E A. The equi~)ale~~cc~ class o f a rnodulo E is the set

4.4 Lemma Let a , b E A. (a) a i s equivalent to b rnodulo E if and only if [ a ]=~[bIE. (b) a is not equivalent to b rnodulo E if and only if [ a ]n~l b ] = ~ 4)-

Proof. i.e., xEa. By transitivity, x E a and ( a ) (1) Assume t h a t aEb. Let z aEb imply xEb, i.e., X E [bIE. Similarly, X E [bIE implies X E [a]E (bEa is true because E is symmetric). So [a]E = IbIE. (2) Assume t h a t [a]E = [b]E . Since E is reflexive, aEa, so a E [ a ] ~But . then a E [bIE,that is, aEb. (b) (1) Assume aEb is not true; we have to prove [ a ]f7 ~[bIE = Q). If not, there is X E [alEn IbIE; SO xEa and xEb. But then, using first symmetry and then transitivity, a E x and xEb, so aEb, a contradiction. (2) Assume finally that [a]E n [b]E = 4). If a and b were equivalent rnodulo E , aEb would hold, so a € [bIE. But also a E [aIE, implying [ u ] fl ~ [bjE # 0 , a contradiction. U

4. EQUIVALENCES AND PARTITIONS

31

4.5 Definition A system S of nonempty sets is called a partition of A if (a) S is a system of mutually disjoint sets, i.e., if C E S, D E S, and C # D, then C f~D = 8, (b) the union of S is the whole set A, i.e., US = A. 4.6 Definition Let E be an equivalence on A. The system of all equivalence classes modulo E is denoted by A/E; so A/E = { ( a J EI a E A).

4.7 Theorem Let E be an equivalence on A; then A/E is a partition of A.

Proof. Property (a) foIlows from Lemma 4.4: If ( u ]#~[bjE, then a and b are not E-equivalent, so f~ [bIE = Q). To prove (b), notice that U A/E = A because a E Notice also that no equivalence class is empty; surely a t least a E [a)& • We now show that, conversely, for each partition there is a corresponding equivalence relation. For example, the partition of people by their country of residence yields the equivalence from (a) of Example 4.2. 4.8 Definition Let S be a partition of A. The relation Es in A is defined by

Es = {(a, b) E A

X

A I there is C E S such that a

E

C and b E C).

Objects a and b are related by Es if and only if they belong to the same set from the partition S. 4.9 Theorem Let S be a partition of A; then Es is an equivalence on A.

Pmof. (a) Reflezivity. Let a E A; since A = US, there is C E S for which a E C , so ( a ,a ) E Es. (b) Symmetry. Assume aESb;then there is C E S for which a E C and b E C. Then, of course, b E C and a E C, so bEsa. (c) Transitivity. Assume aEsb and bEsc; then there are C E S and D E S such that a E C and b E C and b E D and c E D. We see that b E C n D, so C n D # 8. But S is a system of mutually disjoint sets, so C = D. Now we have a E C , c E C, and so aEsc. The next theorem should further cIarify the relationship between equivalence~and partitions. We Ieave its proof to the reader. 4.10 Theorem (a) If E is an equivalence on A and S = A/E, then Es = E. (b) If S is a partition of A and Es is the corresponding equivalence, th.en. A/Es = S.

32

CHAPTER 2. RELATIONS, FUNCTIONS, AND ORDERINGS

So equivalence relations and partitions are two different descriptions of the same "mathematical reality." Every equivaIence E determines a partition S = A / E . The equivalence Es determined by this partition S is identical with the original E. Conversely, each partition S determines an equivalence Es; when we form equivalence classes modulo Es, we recover the original partition S. When working with equivalences or partitions, it is often very convenient t o have a set which contains exactly one "representative" from each equivalence class.

4.11 Definition A set X C A is called a set of representatives for the equivalence Es (or for t h e partition S of A ) if for every C E S, X n C = { a ) for sotne a E C. Returning to Example 4.2 we see that in (a) the set X of a11 Heads of State is a set of representatives for the partition by country of residence. T h e set X = {0, l } could serve as a set of representatives for the partition o f integers into even and odd ones. Does every partition have some set of representatives'? Intuitively. the answer may seem t o be yes, but it is not possible t o prove existence of some such set for every partition on the basis of our axioms. We return t o this problem when we discuss the Axiom of Choice. For the moment we just say that. for rnany mathematically interesting equivalences a choice of a natural set of' representatives is both possible and useful. Exercises 4.1 For each of t h e following relations, determine whether they are reflexive. symmetric, or transitive: (a) Integer X is greater than integer y. (b) Integer n divides integer m. (c) z # y in the set of all natural numbers. (d) C and C in P ( A ) . (e) Q) in Q). (f) Q) in a nonempty set A. 4.2 Let f be a function on A onto B. Define a relation E in A by: uEb if and only if f (a) = f (b). (a) Show that E is an equivalence relation on A. (b) Define a function cp o r 1 .4/E onto B by p ( [ a j E ) = f (u.) (verify t.hilt ~ ( [ uE )] = cp([ul]E ) if [ a ]E = l a ' ] ~ ) . (c) Let j be the function on A onto A / E given by j ( u ) = [ u J ~ ; Show . that rp o j = f. 4.3 Let P = { ( r ,y) E R X R ] r > O), where R is the set of all real numbers. View elements of P as polar coordinates of points in the plane, and define a relation on P by: (r, y ) r- ( r ' , y ' ) if and only if r = r' and 7 - 7' is an integer multiple of 27r. Show that .- is an equivalence relation on P. Show that each equivalence class contains a unique pair (r.y ) with 0 5 y < 27r. T h e set of all such pairs is therefore a set of representativ~s for .-.

5. ORDERINGS

Orderings

5.

Orderings are another frequently encountered type of relation. 5.1 Definition A binary relation R in A is antisymm.etric if for all uRb and bRa imply a = b.

U,

b

A.

5.2 Definition A binary relation R in A which is reflexive, antisymmetric. and transitive is called a (partial) ordering of A. The pair ( A , R ) is called an ordered

set.

aRb can be read as "a is less than or equal t o b" or "6 is greater than or equal t o a" (in the ordering R). So, every element of A is less than or equal t o itself. If a is less than or equal to 6 , and, at the same time, b is less than or equal t o U , then a = b. FinaIIy, if a is less than or equal to b and b is less than or equal t o c, U has to be less than or equal to c. 5.3 E x a m p l e (a) 5 is an ordering on the set of all (natural, rational, real) numbers.

y and (b) Define the relation C A in A as follows: X C A y if and only if z X ,y E A. Then S A is an ordering of the set A. (c) Define the relation _>A in A as follows: X 2~ y if and only if z 2 y and X , y E A. Then 2~ is also a n ordering of the set A. (d) The relation ( defined by: n ( m if and only if n divides m is a n ordering of the set of all positive integers. (e) The relation IdA is an ordering of A. T h e symbols 5 or $ are often used t o denote orderings. A different description of orderings is sometimes convenient. For example. instead of the relation 5 between numbers, we might prefer to use the relation < (strictly less). Similarly, we might use C A (proper subset) instead of L A . Any ordering can be described in either one of these two mutually interchangeable ways. 5.4 Definition A relation S in A is asymmetric if aSb implies that bSa does not hold (for any a, b E A). That is, aSb and bSu can never both be true. 5.5 Definition A relation S in A is a strict ordering if it is asymmetric and

transitive. We now establish relationships between orderings and strict orderings. 5.6 T h e o r e m (a) Let R be a n ordering of A; then the relation S defined in A by

aSb i f and only i f aRb and a # b is a strict ordering of A.

CHAPTER 2. RELATIONS, FUNCTIONS, AND ORDERINGS

34

(b) Let S be a strict ordering of A; then the relation R defined in A by

aRb if and only if

aSb or a = b

is an ordering of A. We say that the strict ordering S corresponds to the ordering R and vice versa. Proof. (a) Let us show that S is asymmetric: Assume that both aSb and bSa hold for some a , b E A. Then also aRb and bRa, so a = b (because R is antisynlmetric). That contradicts the definition of aSb. We leave the verification of transitivity of S to the reader. (b) Let us show that R is antisymmetric: Assume that aRb and bRa. Becausc aSb and bSa cannot hold simultaneously ( S is asymmetric). we conclude that a = b. Reflexivity and transitivity of R are verified similarly.

5.7 Definition Let a, b E A, and let 5 be an ordering of A. We say that a and b are comparable in the ordering 5 if a < b or b 5 a. We say that a and b are incomparable if they are not comparable (i.e., if neither a 5 b nor b 5 a holds). Both definitions can be stated equivalently in terms of the corresponding strict ordering )secluellc.rl / such that ran f = X. [ H i l ~ e Use : Exercise 3.5.1

4.

Arithmetic of Natural Numbers

As an applicatiorl of the Recursioil 'Lheoretn we now show how to defiile aclcfitiori of ~ i a t u r a numbers, l and we use Induction t o prove basic properties of addition. In similar fashion, orre can introtlu 0)

5. OPERATIONS AND STRUCTURES

55

Prove the usual laws of exponents. 4.8 Verify the axioms of Peano arithmetic. The needed results can be found in the text and exercises. 4.9 For each finite sequence (kj ( 0 5 i < n ) of natural numbers define C ( k i 0 < i < n) (more usually denoted ki or C::. ki) SO that

I

C0 = 0; C ( k o }= ko; C ( k o , .. . , k,) =

5.

C{ko,. . . . kn-

1)

-I-k,

for n

> 1.

Operations and Structures

The functions +, ., etc., are usually referred to as operations. The aim of this section is to define the general concept of operation and to study its properties. Each of the aforementioned operations assigns to a pair of objects (numbers, sets) a third object of the same kind (their sum, difference, union, etc.). The order may make a difference, e.g., 7 - 2 and 2 - 7 are two different objects. We therefore submit the following definition. 5.1 Definition A binary operation on S is a function mapping a subset of S 2 into S.

Nonletter symbols such as +, X , *, A, etc., are often used to denote operations. The value (result) of the operation * at (X,y) is then denoted X * y rather than *(X,y). There are also operations, such as square root or derivative, which are applied to one object rather than to a pair of objects. We introduce the following definitions. 5.2 Definition A unary operation on S is a function mapping a subset of S into S. A ternary operation on S is a function on a subset of S 3 into S.

c

5.3 Definition Let f be a binary operation on S and A S, A is closed under the operation f if for every X,y C A such that f ( X ,y) is defined, f (X, Y) E A. One can give similar definitions in the case of unary or ternary operations.

5.4 Example (a) Let + be the operation of addition on the set of all real numbers. Then + is defined for all real numbers a and b. The set of all real numbers, as well as the set of all rational numbers and the set of all integers, are closed under +, The set of even natural numbers is closed under +, but the set of odd natural numbers is not.

CHAPTER 3. NA'l'URA L NUMBERS

56

(b) Let t be the operation of division on the set of all real numbers. + is not defined for (a,b) where b = 0. The set of all rational numbers is closccl under +, but the set of all intcgers is not. (c) Let S be a set; define binary operations Us and ns on S as follows; (i) If X,y E S and 3: U y E S. then 3: u.7 y = 3: U y. (ii) If X,y E S and x n y E S, then X nsy = .TI I y. If we take S = P ( A ) for some A, Us and ns are defined for every pair (2, Y) E S2. When developing a particular branch of mathematics, we are usually interested in sets of objects together with some relations between them and operations on them. For example, in number theory or analysis, we study the set of all (natural or real) numbers together with operations of addition and multiplication, relation of less than, etc, In geometry, we study the set of all points antl lines with relations of incidence ancl between and operation of intersectio~~. etc. We introduce the notion of structure to describe this situation abstractly. In general, a structure consists of a set A and of several relations a r ~ dopcrations on A. For example, we consider structures with two biniiry reli-itions arid two operations, say a unary operation and a binary operation. Let A be a set,. let R1 and Rz be binary relatioris in A, and let f be a unary operation ant.! g ;I binary operation on A. We ~ n a k euscl of the five-t;uple (A. R I .R 2 .f.g ) t,o clc~~iotc* the structure thus defined.

5.5 Example ( a ) Every ordered set is a struc:turc: with one binary relatioii. (b) (A, UA,n ~ A, ) is a structurtl with two binary operatiorls nrld orle binary relation. (c) Let R be the set of all real ~lurnbers. (R,+, -. X , t) is t i strur.t,urt. with four binary operations.

c

We now proceed to give a i general definition of a structure. In Chapter 2. we introduced unary ( l-ar y) , binary (2-ar y ) , and ternary (3-ar y ) relations. The rekson we did not talk about rl-itry relations for arbitrary T I is silrlplv that !v(. c.ould rlot then handle arbitrary liatural nurnbers. This obstitcle is [low ~.ert~ovt.d. and the present section gives precise definitiorls of n - t ~ ~ p l en-ary s ? relat,io~lsancl operations, and n-fold (:artesian products, for all natural numbers 71. We start with the definition of ;in ordered n-tuplt. Recall that ortlerecl pair ( a o , a l ) has been defined in Section 1 of Chapter 2 zts a set that uniciuely determines its two coorclinates, U() and u1: i.e., (no,al) = (bo,bl)

it'anclanly if

a0

= ba and u1 = b I .

We called a0 the first coordinnt,e of (uO,u1),ancl a1 its second coortlinntc. 111 analogy, an 71-tuple (ao,a l l . . . . U,,- l ) should be a set that uniquely tlcter~~lirlrs its n coorclinates, ao, u1, . . . . U , , - , ; that is, we want

(*l (IQ.

.

. . . ~ ~ - =1 ()b O , .. . D,,- l ) if ?

i i l l ~ lo111yif

uI:= L, for all L = U. . . .

, 11 -

l.

5, OPERATIONS AND STRUCTURES

57

But we already introduced a notion that satisfies (*): it is the sequence of length n , (ao,a l l . . . , un-l). The statement that (ao,. - . , a n - l ) = (bo,. . .

,bn-1)

if and only if a,

=

b, for all i = 0,. . . , n - 1

is just a reformulation of equality of functions, as in Lemma 3.2 in Chapter 2. We therefore define n-tuples as sequences of length n. For each i, D 5 i < n: ai is called the (i + 1)st coordinate of ( a o , a l , .. . , a n - l ) . SO a0 is the first coordinate, a1 is the second coordinate, . , . , an- l is the n-th coordinate. IReally, a more consistent terminology would be zeroth, first, . . . , (n - 1)st coordirlate. but the custom is to talk about the first and second coordinates of points in the plane, etc.] We notice that the only Q-tupleis the empty sequence () = 0, having no coordinates. l-t,uples are sequences of the form (ao>[i.e.,sets of the form ( (Q, n o ) )1; it usually causes no confusion if one does not disting~iishbetween a l-tuple ( a o ) and an element UO. If (A, 1 0 5 i < n) is a finite sequence of sets, then the n-fold cartesian product A,, as defined in Section 3 of Chapter 2, is just the set. of all n-tuples a = ( a o , a l , .. . ,a,- l) such that a* E Aol a1 E A I , . . . , an-l C An-1. A, = A n is the set of all n-tuples If A, = A for all i, D 5 i < n, then with all coordinates from A. We note that AO = {()) and A' can be identified with A. An n-an, relation R in A is a subset of A n . We then write R(au,a l , . . . . a, - l ) instead of ( a o , a l , .. . ,a,-l) E R. Similarly, an n - u q operation F o n A is a function on a subset of A n into A; we write F ( a o ,a , , . . . , a,- 1 ) inst,ead of F ( ( a o ,a l , . . . , a n - l ) ) . Finally, we generalize the notation already introduced in the case of pairs and triples. If P ( s o , s l , . . . , X,- 1) is a property with paramcters xo, X I , . . . , X,- 1, we write

no,,,,

no5i,n

{ ( a o , . . . , a n - l ) ( a 0 E Ao,. .. ,an-l E

A,,-1

and P(a0 , . . . , a n - l ) )

to denote the set

n

AI 1 for sorneao,... ,a,-l, a = (a0 , . . . . a n - l ) a n d P ( a o, . . . .a,,-l)).

~ < z < R

We note that l-ary relations need not be distinguished from subset.^ of A, and l-ary operations can be identified with functions on a subset of A into A. 0ary relations (0 and { ( ) ) do not have much use, but nonempty D-ary operstioils are quite useful. They are objects of the form ((0, a ) ) where a C A. We

CHAPTER 3. NATURAL NUMBERS

58

no t and xk E Y ) if such k exists; undefined otherwise.] This defines a sequence (ko,., . , km-l). When we let p, = xk, for all i < m, then Y = {yi [ i < m). The reader should verify that m < n (by induction, k, i whenever defined, so especially m - 1 < km-l 5 n - 1).

>

2.5 Theorem If X is a finite set and f is Moreover, [f[Xj[ ki such that k < n and f ( x k ) # f ( x ~for ) all j 5 i (if it exists), and y, = f ( x k i ) . Then f [X] = {po,.. . , ym-l} for some m

< n. 0

As a consequence, if (ai 1 i < n) is any finite sequence (with or without repetition), then the set ( a i 1 i < n ) is finite. All the constructions made possible by the Axioms of Comprehension w h e ~ l applied t o finite sets yield finite sets. We now show that if X is finite, then F ( X ) is finite, and if X is a finite collection of finite sets, then U X is finite. Hence addition of the Axiom of Infinity is necessary in order to obtain infinite sets. 2.6 L e m m a If X and Y are finite, then X U Y is finite. Moreover, IYI, and i f X and Y are disjoint, then J X U Y J= +IYI.

1x1+

1x1

\XU Y ( 5

1 f X = { x 0 , . . . , X , - 1 ) a n d i f Y ={yo , . . . , y , - ~ ) , w h e r e ( z o . . . . . Proof. xT1-1 ) and (yo, . . . , Y , ~ - are one-to-one finite sequences, consider the finite sequence z = ( x O , . . , xn- 1, yo, . . . , pm- 1 ) of length rt + m. (Precisely, define z = ( ~ , I O ~ i < n + m by)

72

CHAPTER 4. FINITE, COUNTABLE, AND UNCOUNTABLE SETS

+

Clearly, z maps n m onto X U Y. so X U Y is finite and [XU Y I 5 n + m bv Theorem 2.5. If X and Y are disjoint, z is one-to-one and ( X U Y I = n + m.

2.7 Theorem

If S is finite and if every X

E S is

f i n ~ t e ,then

U S is finite.

Proof.

We proceed by induction on the number of elements of S. Tht. statement is true if IS1 = 0. Thus assunze t h a t it is true for all S with IS1 = 7 1 . and let S = {Xo,. . . , X,-1, X,) be a set with n 1 elements, each X, E S being a finite set. By the induction hypothesis,; :U : X, is finite. and we hirvt~

+

us

=

( i X*)

X,,

which is, therefore, finite by Lemma 2.6. 2.8 Theorem If X i s finite, then P ( X ) i s finite.

By induction on / X ) . If J X J= 0, i.e., X = fl, then P ( X ) = (m) is finite. Assume t h a t P ( X ) is finite whenever 1x1 = n , and let Y be a set with n + 1 elements: Y = { y o,... .yn). Let X = { y o , . . . , ? j l l - l ) . We note t h a t P ( Y ) = P ( X ) U U, where U = {U ( .U C Y and y, E [ L ) . Wc, further note t h a t 1 U J = J P ( X )I because there is a one-to-one mapping of U onto P ( X ) : f ( u > = U - { ~ n )for all U E U . Hence P ( Y ) is a union of two finite sets anti. consecluently, finite. E

Proof

The final theorem of the section shows t h a t infinite sets really have more elements t h a n finite sets. 2.9 Theorem If X is infinite, then ]XI> n for all n

N.

Proof. I t suffices t o show t h a t 1x1> n for all n E N. This can be dorle by induction. Certainly 0 < (XI. Assume t h a t '2. n: then there is a one-to-one function f : n -+ X. Since X is infinite, there exists X E ( X - 1.iii1 j'). D~fiilo 9 = f U { ( n X)); , g is a one-to-one function on n + 1 into X . We1 co~icludethat 1x1> n + 1. C

1x1

To conclude this section, we briefly discuss another appr0ac.h t o finiteness. T h e following definition of finite sets does not use natural numbers: A set X is finite if and only if there exists R relation 4 such t h a t (a) + is a linear ordering of X . (b) Every nonempty subset of X has a least and a greatest element in 4 . Note t h a t this notion of finiteness agrees with the one we defined using finitr sequences: If X = {so,. . . , X,- l ), then so 4 . - . 4 X,- l describes R lincitr ordering of X with the foregoing properties. O n the other hand. if (X. + ) satisfies ( a ) and (b), we construc:t, by rec~lrsion,a sequence (h. f l . . . ) . ;ts ,

2. FINITE SETS

73

in Theorem 3.4 in Chapter 3. As in that theorem, the sequence exhausts a11 elements of X ,but the construction must come t o a halt after a finite rlunlber of steps. Otherwise, the infinite set Ifo, f l , f 2 , , . . ) has no greatest element in

(X, 4. We mention another definition of finiteness that does not involve natural numbers. We say that X is finite if every nonempty family of subsets of X has a c-maximal element; i.e., if 0 # U C P ( X ) . then there exists z E U such that for no y E U , z C y. Exercise 2.6 shows equivalence of this definition with Definition 2.1. Finally, let us consider yet another possible approach t o finiteness. It follows from Lemma 2.2 that if X is a finite set, then there is no one-to-one mapping of X onto its proper subset. On the other hand, infinite sets. such as t11e set of all natural numbers N, have one-to-one mappings onto a proper subset [e.g.? f ( n ) = n + l]. One might attempt to define finite sets as those sets which are not equipotent t o any of their proper subsets. However, it is impossible to prove equivalence of this definition with Definition 2.1 without using the Axiom of Choice (see Exercise 1.9 in Chapter 8).

Exercises

2.1 If S =

2.6

2.7

2.8

n- l

C i = o IxilIf X and Y are finite, then X X Y is finite, and / X X Y (= 1x1.( Y / . If X is finite, then )P(X)t= 2Ix1. If X and Y are finite, then X Y has lxllY elements. If = n > k = IYI,thenthenumber ofone-to-one functions f : Y + X is n - ( n - l ) . - . . . ( n - k + l ) . X is finite if and only if every nonempty system of subsets of X h ; ~ a E-maximal element. [Hint; If X is finite, 1 x1 = 11, for some 1 1 . If U C P ( X ) , Iet m be the greatest number in {IY/I Y E U). If Y E U and IY1 = m , then Y is maximal. On the other hand, if X is infinite. let U = {Y g X / Y is finite).) Use Lemma 2.6 and Exercises 2.2 and 2.4 to give easy proofs of commutativity and associativity for addition and multiplication of natural numbers, distributivity of multiplication over addition, and the usual arithmetic properties of exponentiation. [Hint: To prove, e.g., the commutativity of multiplication, pick X and Y such that ( X ( = ~ r a ,( Y J= 72. By Exercise 2.2, m . n = [X X YI, n . m = IY X XI. But X X Y and Y X X are equipotent.] If A, B are finite and X A X B, then = CoEA ku where k, = IX n ( { a ) B)I.

l US1

2.2 2.3 2.4 2.5

{Xo,. . . , Xn-l) and the elements of S are mutually disjoint, the11

1x1

c

1x1

74

CHAPTER 4. FINITE?COUNTABLE, AND UNCOUNTABLE SETS

3. Countable Sets The Axiom of Infinity provides us with an example of an infinite set - the set N of all natural numbers. In this section, we investigate the cardinality of N ; that is, we are interested in sets that are equipotent to the set N. 3.1 Definition A set S is countable if JSI= [NI.A set S is at most c o u n t a b l ~ if IS] IN).




i f 0 < p j n - I; i f p = n,

where g(k) = (p,q, (r:, . . . l rf)-'}, . . . l { 0T ~ -. .~. , ~f:-~'-l})+ The definition of ai+l is designed so as to make it transparent that if ai enumerates C,, a,+l enumerates Ci+1 (with many repetitions). By induction, U , enumerates C, for each i E N, as required. E We conclude this section with the definition of the cardinal number of countable sets. 3.15 Definition IAl = N for all countable sets A

We use the symbol No (aleph-naught) to denote the cardinal number of countable sets (i.e., the set of natural numbers, when it is used as a cardinal number). Here is a reformulation of some of the results of this section in terms of tflie new notation. (a) No > n for all n E N ; if No > )E for some cardinal number n, then n = No or K = n for some n E N (this is Corollary 3.3). (b) If ) A ] = No, 1B1 = No, then IAU Bj = No, J A X B1 = No (Tkieorenis 3.5 and 3.7). (c) If [A[= N o , then I Seq(A)(= No (Theorem 3.10). 3.16

Exercises

3.1 Let IAl( = ( A z f , lBlt = 1B21-Prove: (a) If A1 n A2 = 0, B1 n B2 = 0, then (A1 U A21 = IBI U Bzl. (b) ( A 1 X A21 = (B1X B2l. (c) I Ses(A1) 1 = I Seq(A2)I. 3.2 The union of a finite set and a countable set is countable. 3.3 If A # 0 is finite and B is countable, then A X B is countable. 3.4 If A # 0 is finite, then Seq(A) is countable. 3.5 Let A be countable. The set [AIn = { S C_ A I IS1 = n ) is countable for all n E N, n # 0.

4. LINEAR ORDERINGS

79

3.6 A sequence (s~)?&of natural numbers is eventually constant if there is no E N , s E N such that S, = S for all n no. Show that the set of eventually constant sequences of natural numbers is countable. 3.7 A sequence (S,)F=~ of natural numbers is (eventually) periodic if there are no,p E N, p 1 , such that for all n 2 no, S,+, = S,. Show that the set of all periodic sequences of natural numbers is countable. 3.8 A sequence (sn):=o of natural numbers is called a n arithmetic progression if there is d E N such that s,+l = S, d for all n E N. Prove that the set of all arithmetic progressions is countable. 3.9 For every s = (so,. . . , Sn- 1) E S e q ( N - ( Q } ) , let f (S) = p': . . - p ,S,,- ,- 1 where p, is the ith prime number. Show that f is one-to-one and use this fact to give another proof of I Seq(N)I = No. 3.10 Let [ S , C ) be a linearly ordered set and let (A, 1 n E N) be an infinite =:,- An is at most countable. sequence of finite subsets of S. Then U [Hint: For every n E N consider the unique enumeration ( a , ( k ) j k < ]A,,1) of A, in the increasing order.] 3.11 Any partition of an a t most countable set has a set of representatives.

>

>

+

4.

Linear Orderings

In the preceding section we proved countability of various familiar sets, such as the set of all integers Z and the set of all rational numbers Q. The important point we want to make is that we cannot distinguish among the sets N : Z ? and Q solely on the basis of their cardinality; nevertheless, the three sets "look" quite different (visualize them as subsets of the real line!). In order to capture this difference, we have to consider the way they are ordered. Then it becomes apparent that the ordering of N by size is quite different from the usual ordering of Z (for example, N has a least element and Z does not), and both are quite different from the usual ordering of Q (for example, between any two distinct rational numbers lie infinitely many rationals, while between any two distinct, integers lie only finitely many integers). Linear orderings are a n important tool for deeper study of properties of sets; therefore, we devote this section to the theory of linearly ordered sets, using countable sets for illustration. 4.1 Definition Linearly ordered sets (A, ( N I Proof. The function f : N -+ P ( N ) defined by f ( 7 1 ) = { n ) is on(?-to-one, so ( N1 < I P ( N )I . We prove that for every sequence (S, 1 n C N ) of stibsets of N there is some S N such that S # S, for all n E N . This shows that t.herr! is no mapping of N onto P ( N ) ,and hence (NI < I P ( N ) J . We define the set S N as follows: S = {n E N I n 4 S,). The 1111ulber n is used to distinguish S from S,: If n E S,, then lz. @ S, and if 12 $ S,,. then n.C S. In either case, S # S,,, as required. •

c

Detailed study of uncountable sets is the subject of Chapter 5 (and subsequent chapters). Here we only prove that the set 2" = {U, 1)" of all infinite sequences of U's and l's is also uncountable, and, in fact, has the same cardinality as P ( N ) and R. 6.3 Theorem lP(N)I = 1 2 ~=1 (RI.

Proof. For each S (0, l ) , as follows:

C N define the characteristic

function of S ! x.9 : N

-+

It is easy t o check t h a t the correspondence between sets and their characteristic functions is a one-to-one mapping of P ( N ) onto {0,1}". To complete the proof, we show t h a t IR( < IP(N)I and also 12N( < [RI and use the Cantor-Bernstein Theorem. (a) We have constructed real numbers as cuts in the set Q of all rational numbers. The function that assigns t o each real number 7- = (A. B) the set A C Q is a one-to-one mapping of R into P ( Q ) . Therefore (RI IP(Q)(. As JQI = ( N I , we have / P ( & ) [= IP(lV)I (Exercis(->6 . 3 ) . Hence \RI 5 IP(N)I, ( h ) To prove 12" 1 5 IRI we use the decimal representation of real nurnbers. Tlie function that assigns to each infinite sequence (a,):==, of 0's and l's the unique real number whose decimal expansiori is O.aaalae- - . is a one-to-oric mapping of 2" into R. Therefore we have (2"I 5 (RI. U


0. (i) If ~ 1 5 K:! and X1 5 XZ, then K I X 1 5 ~ 2 X2.. To make the analogy between multiplication of cardinals and multiplicatior~of numbers even more complete, let, us prove (j) K + K = ~ . K . +

If / A / = K , then 2 . r; is the cardinal of (0, I ) X A. We note that Prooj, (0, l ) X A = ((0) X A ) U ( { l ) X A), that I{O) X AI = / { l ) X AI = K , and that the two summands are disjoint. Hence 2 . K = K + K . 0 As a consequence of (j), we have (k) K + K 5 K . K , whenever K ) 2 .

1. CARDINAL ARITHMETIC

95

As in the case of addition, multiplication of infinite cardinals has some properties different from those valid for finite numbers. For example, No . No = NI, [see 3.16(b) in Chapter 41. (And the Axiom of Choice implies that K. - K = K. for all infinite cardinals.) To define exponentiation of cardinal numbers, we note that if A and B are finite sets, with a and b elements respectively, then a b is the number of all functions from B to A.

1.5 Definition

K.'

=

The definition of

K.'

1 ~ ~ where 1 , [A[ = K. and

(B1 = X.

does not depend on the choice of A and B.

1.6 Lemma If IA] = IA'I and (B1= (B'I, then IA

B

1-I

1.

Let f : A -+ A' and g : B -+ B' be one-to-one and onto. Let F : A B -+ A ~ be~ defined ' as follows: If k E A ~ let, F ( k ) = h, whew h E A ' ~ ' is such that h(g(b))= f(k(b)) for all b E B, i.e.. h = f o k o $ - l . Proof.

Then F is one-to-one and maps A B onto A' B'. It is easily seen from the definition of exponentiation that if X > 0. (1) K. J XI

holds for all X

C

S.

Proof. Let Y = P ( U S). By Cantor's Theorem, 1Y 1 > ( US ( ,and clearly ( U S 1 2 1x1 for all X E S (because X 2 U S i f X E S). U

Exercises

1 .l Prove properties (a)-(n) of cardinal arithmetic stated in the text of this section, 1.2 Show that K* = 1 and K' = K for all K . 1.3 Show that Z K = 1 for all K and 0" = 0 for all K > 0. 1.4 Prove that 5 2K'K. 1.5 If ( A ( 5 (B1 and if A # 0, then there is a mapping of B onto A. We later show, with the help of the Axiom of Choice, that the converse is also true: If there is a mapping of B onto A, then IA( 5 ( B ( . 1.6 If there is a mapping of B onto A, then 2IAI 21sl. [Hint: Given g mapping B onto A, let f (X) = g- '[X], for all X A.] 1.7 Use Cantor's Theorem to show that the "set of all sets" does not exist. 1.8 Let X be a set and let f be a one-to-one mapping of X into itself such that f \ X ] C X . Then X is infinite.


0 ) . (d) N;: = No (71 E N , 72 > 0 ) . In the present section we study t h e second rnost important infinite cardilinl number, the cardiriality of the c:oritinuum. 2"0. 'To begirl with. wc recall that 2"" is indeed the cardinality of t l ~ t set l R of all real riumbers.

+

2.1 Theorem

Proof.

1RJ= 2 N 0

This is the second half of Theorem 6 . 3 in Chapter 4 .

l1

T h e next theorem summarizes arithmetic properties of'the carciint~lityof' tlio continuum.

.

2.2 Theorem (a) n + 2N~1= N o + 2'0 = 2'1' + 2'" = 2'0 (, E N ) . (b) . 2"~' = N o - 2"" = 2N" 2H" = 2H" (, E N , > 0). (c) (2'0)" = (2"")"0 = NO = = 2'11 (71 E N , n > 0)

NOHI

Proof. (a) This follows from the obvious sequence of inequalities

by the Cantor-Bernstein Theorem. (b) Similarly, we have

(c:)

We have both 2N"

and

< ( 2K1 )

2No < - nNo < -

5 (2"")""

N N I I< 0 -

2

= 2% = 2

(2N")'"

p,, = ~ L I , C1

It is interesting t o notice that Theorem 2 . 2 , although a n e~asyc o r o l l ; ~ ~o .f, ~ laws of cardinal arithmetic and the Cantor-Bernstein Theorem, h{zs some rather unexpected consequences. For example, 2N0 . 2'" = 2'~' means that ( R X RI = IR1; however, the set R X R of all pairs of real numbers is in a one-t,o-one correspondence with the set of all poirlts in the plane (via a cartesian coordiri:ite system). Thus we see that there exists a one-to-one mapping of a straight line

2. THE CARDINALITY OF THE CONTINUUM

99

R onto a plane R X R (and similarly, onto a three-dimensional space R X R X R, etc.). These results (due to Cantor) astonished his contetnporaries; they seem rather counterintuitive, and the reader may find i t helpful to actually construct such a mapping (see Exercise 2.7). The next theorem shows that, several important sets have the cardinality of the continuum. 2.3 Theorem (a) The set of a16 (b) The set o j all (c) The set of all (dj The set of all

points in the n-dimensional space Rn has cardinality 2'" complex numbers has cardinality 2"". infinite sequences of natural numbers has cardinality 2'". injinite sequences of real numbers has cardinalitlj 2"".

Prooj. (a) [Rnl = ( 2 ' ( 1 ) ~by definition of cardinal exponentiation; ( 2 ' 0 ) ~ = 2'0 by Theorem 2.2(c). (b) Complex numbers are represented by pairs of reals (see Exercise 2.6 in Chapter 10),so the cardinality of the set of all complex numbers is /RxRI = (2H(1)2= 2Ho (c) T h e set of all infinite sequences of natural numbers is N N and lNNl = = 2"". (d) pN1= (2No)W" = 2'".

~2'

0 T h e next theorem helps t o establish further results of this sort. 2.4 Theorem I f A is a countable subset of B and ( B (= 2N". then [B-AI = 2N".

(Here we remark t h a t using the Axiom of Choice, we are able to show in general t h a t if [ A [< IB(,then IB - AI = IBI.)

Proof.

We can assume without loss of generality t h a t B = R

Let P = dom A:

P=

{X

E R

I (x,y) E

A for some

X

R.

g).

Since IAl = No, we have IPI 5 No. Thus there is xo E R such t h a t xo # P. Consequently, the set X = ( s o )X R is disjoint from A, so X (R X R) - A . Clearly, 1x1 = JRI=2'0, and we have J ( R x R ) - A1 2 2'1'. U

c

CHAPTER 5. CARDINAL NUMBERS 2.5 (a) (b) (c)

Theorem The set of all in-atzonal numbers has cardinality 2N0. The set o j all infinite sets o j natural numbers has cardinality 2N0. The set o j all one-to-one mappings oJ N onto N has cardinality 2 N o .

Prooj. (a) The set of all rationals Q is countable, hence the set R - Q of all irrational numbers has cardinality 2N0by Theorem 2.4. (b) The set of all subsets of N, P ( N ) , has cardinality 2"", and the set of all finite subsets of N is countable (see Corollary 3.11 in Chapter 4 ) , hence the set of all infinite subsets of N has the cardinality of the continuum. (c) Let P be the set of all one-to-one mappings of N onto N; as P 2 N N . clearly IPI 5 2'". Let E and 0 , respectively, be the sets of all even and N odd natural numbers. If X C E is infinite, define a mapping jx : N as follows:

-

jx(2k) = the kth element of X (k E N); jx(2k

+ 1) = the kth element of N

-

X (k

C

N).

Notice that N - X 3 0 is infinite, so jx is a one-to-one mapping of IV onto N. Moreover, it is easy to show that XI # Xzimplies jx, # f x , . We thus have a one-to-one correspondence between infinite subsets of E and certain elements of P. Since there are 2'(' infinite subsets of E by Theorem 2.5(b), we get [ P ] 2"" as needed.

>

0 Yet other similar results are provided by the next theorem. The definitions and basic properties of open sets and continuous functions can be found in Section 3 , Chapter 10. 2.6 Theorem

(a) The set o j all continuous junctions on R to R has ca~dznality2"". (b) The set o j all open sets o j reals has cardinality 2N0. Prooj. (a) We use the fact (proved in Theorem 3.1 1 , Chapter 10) that every continuous function on R is determined by its values on a dense set, in particular 63. its values a t rational arguments: If f and g are two continuous functioris on R, and if j ( q ) = g ( q ) for every rational number g , then $ = g . 'Tlius let C be the set of all continuous real-valued functions on R. Let F be ;I defined by F ( j ) = f I Q. By the fact above. F' mapping of C into is one-to-one, so (C( 1 ~ = (2N'1)N" ~ 1 = 2'(l. On the other hand. clearly jCI 2 2'" (consider the constant functions). (b) Every open set is a union of a system of open intervals with rational endpoints (see Lemma 3.14 in Chapter 10). There are N o open intervals with rational endpoints (each such interval is determined by an ordered pair of


2N11.

Proof.

The cardinal number of RR is (2"0)2*" = 2 ~ 0 . 2-~ 22'O. ~'

0

Exercises

2. Z Prove that the set of all finite sets of reals has cardinality 2N".We remark here that the set of all countable sets of reals also has cardinality 2N0, but the proof of this requires the Axiom of Choice. 2.2 A real number X is algebmic if it is a solution of some equation

where a*, . . . ,a, are integers. If X is not algebraic, it is called transcendental. Show that the set of all algebraic numbers is countable and hence the set of all transcendental numbers has cardinality ZNn. 2.3 If a linearly ordered set P has a countable dense subset, then IPI 5 2N0. 2.4 The set of all closed subsets of reals has cardinality 2"".

102

C H A P T E R 5 . C A R D I N A L NUMBERS

2.5 Show that, for n > 0, n . 22N"= No . z2'" = 2N" . 22*" = 22'" . 22'" 22Nc~ 0'2 n - 22'" Nt) = p0 p0 (2 1 - ( 1 (2 1 2.6 The cardinality of the set of all discontinuous functions is 2"". [Hint: Using Exercise 2.5, show that I R- C ~ )= 2'*" whenever JCl 2N0.J 2.7 Construct a one-to-one mapping of R X R onto R. [Hint: If a. b E (0. l ? have decimal expansions 0.alu;?a3 . . - and 0. bl b2b3 - - . rimp the ord(tret1 pair ( a ,b) onto 0.albla2b2asb3. E [0,l]. Make adjustments to avoid sequences where the digit 9 appears from some place onward.]


W . It should be pointed out that ordinal arithmetic differs substantially fro111 arithmetic of cardinals. Thus, for instance, 2" = o and w" are countable ordinals, while 2*0 = H*" is uncountable. One can use arithmetic operations to generate larger and larger ordinals:

The process can easily be continued. We define E = sup(w. d i j , One can then form E + 1, E w , E", E € , E € ' , etc.

+

Exercises

5.1 Prove the associative law (a . P) y = a . (0 . -y). 5.2 Prove the distributive law a - (0+ y ) = cr . P + a . y. 5.3 Simplify: (a) (U l) + W . (b) w + W2. ( C ) ( W + 1) . W 2 .

+

, UJ&-

,

.

.

)

5. ORDINAL ARITHMETIC

123

5.4 For every ordinal a , there is a unique limit ordinal ,8 and a unique natural number n such that a = + n. [Hint:p = sup(? 5 a 1 y is limit).] 5.5 Let a 5 p. The equation f + o = ,8 may have 0, 1, or infinitely many solutions. 5.6 Find the least a > w such that + a = cr for a11 < a. 5.7 (a) If a l , c r 2 , and p are ordinals and # 0, then a ] < a 2 if and only if


0 then by the induction hypothesis, . kz + . . . + . kn p= m

>

a

for some > . . - > lc, and finite k 2 , ., . , kn > 0, As p < w o l , we have wa2 5 p < wal and so pl > h. It follows that cw = wal .kl + w a 2 . k 2 + - .- + d r a v k n is expressed in normal form. To prove uniqueness, we first observe that if < 7 , then wo - k < d r for W'. From this it easily every finite k: this is because wa k < w @. w = do+' follows that if a = - w A kn is in normal form and y > PI! then - kl r a <w . We prove the uniqueness of normal form by induction on a. For a = 1, the expansion 1 = w0 . 1 is clearly unique. So let a = wP1 kl - . - L J ~ * I k, = wY1. e1 - - - wrTr8 - em. The preceding observation implies that Pl = 71. If we let 6 = wD1 = ~ 7 1p, = W h- k2 + . . - f W @. ~kn and a = w'2 . ez + . . . + .ern1 we have cr = 6 . k l + p = 6 - t l +a,and since p < d and a < 6, Lemma 6.3 implies that k l = el and p = a. By the induction hypothesis, the normal form for p is unique, and so m = n , = 72, . .. , = m,k2 = t2,. . . , kn = ln.If follows I7 that the normal form expansion for Q. is unique.

+ +




with bl > > b, and O < k, < a, i = l,... , n . For instance, the number 324 can be written as 44 43 4 in base 4 and 7 2 6 7 . 4 2 in base 7. A weak Goodstein sequence starting at m > 0 is a sequence mo, m1, m2,. . . of natural numbers defined as follows: First, let m0 = m , and write in base 2:

+ +

To obtain

ml,

+

+

increase the base by 1 (from 2 to 3) and then subtract 1:

CHAPTER 6. ORDINAL NUAIBERS

126

In general, to obtain m k + t from m k (as long as mk # O), write m k in base k + 2, increase the base by 1 (to k + 3 ) and subtract 1. For example, the weak Goodstein sequence starting a t m = 21 is as follows: m0

= 21 = 24

m1

= S4

+ 22 + 1

+ 32 = 90

ma=44+42-1 ~ 4 ~ + 4 - 3 + 3 = 2 7 1 m g ~ +55 . ~ 3+2=642 = 64 -16 . 3

m4

= 74

+ 1 = 1315

+ 7 . 3 = 2422

ms=84+8.2+7=4119

+ 9 . 2 + 6 = 6585

m7

= g4

m8

= 104

+ 10.2+5

= 10025

etc. Even though weak Goodstein sequences increase rapidly a t first, we have

6.5 Theorem For each m > 0 , the weak Goodstein sequence starting at Tn eventually terminates with m, = 0 jor some n.

Proof. We use the normal form for ordinals. Let m > 0 and m o , m l .m:!:. . . be the weak Goodstein sequence starting a t m. Its a t h term is written in base a -t2:

Consider the ordinal Q,

= cJb'

. kI +

'

. . + LJb., . k ,

obtailied by replacing base a + 2 by LJ in (6.6). I t is easily seen that a0 > (11 > (12 > . . - > Qa > - . . is a decreasing sequence of ordinals, necessarily finite. Therefore, there exists some T L such that a , = 0 . But clearly ~ n , (1, for ever?. U = 0 , 1 , 2 , .. . ,?L. Hence m, = 0 .


0. is the set of all finite subsets of N,. ( C ) IINa] No.

D

As a result, the Continuum Hypothesis can be reformulated as a conjecture that

zNo= N,, the least uncountable cardinal number.

142

C H A P T E R 8. T H E A X I O M O F CHOICE

1 .l1 Theorem* If f is a function and A is a set, then ( f [ A ] (. I/Al. For each b E f [ A ] ,let Xb = j - ' ( { b ) ) . Note that X b # 0 aiirl Proof. X b , n Xbr = 8 if bl # b l . Take g E n b c f l a l X b ;then g : f [ A ] + A and bl 6 2 implies g ( b l ) E Xb,, g(bl) E X b z . so g ( b l ) # g(b2). We corlclutie thitt is i i one-to-one mapping of f [ A ]into A, and consequer~tlyl f [ A }l < j Al. J

+

L ,-:

1-12Theorem* I f [ S )5 N, and, for all A E S . IAl 5 N,,. then

1Us15

h',,.

Proof. Tliis is a generalization of Theorenl 1.7. We usurne that S # V) at~cl a11 A E S art? nonempty, writct S = { A u I v < N,,) and for e w h v < H,,. c.tloosci a trilnsfinite sequence Au = (u,,(K) I K < N n ) such that A, = {a,(r;) I ti < N,,). (Cf. Exercise 1.9 in Chapter 4.) \YP define a rnapping f on N,, X N,, onto US by f (U,K ) = a , ( ~ ) .B y Theorem 1.11.

a

(the last step is Theorem 2.1 in Chapter 7 ) .

We rlext derive Zorn's Leinrrl:~,iitlother important versiorl of' the Axiorli of Choice.

1.13 Theorem The following statements are epuivc~lent: ( a ) ( T h e A x i o m of C/ioice) T/~err:e:rists a choice f u ~ ~ c t i o~ ~O tT C-I ) P T ? / .sy,st~rr/.of' sets. (6) ( T h e Well- Ordering Principle) Every set can be well-ordered. (c) (Zorn's Lemma) I f e ~ ~ e chain r y in a partially ordered set has an upper 1101lr~d. then the pa?-tiallg ordered set has a maximal element. Wt. remind the reader that :t cllain is 11 lil~earlyorderecl suhset of' 2 t p;trtiallj. orclerecl set (see Section 5 of Chapter 2 for this definition, as well as tllcl t1t:finitiolis of "ordere(l set," "upper bouilcl." ..maximal r!l~rrlent.'' iirltl otllctr. r.t)lic.cq)t s relattlrl to orderirlgs). 1.1: Proof. Equivalence of' (a) tincl (L>) follows immediately from Tht?ol.ibrr~ tllerefore, it is enough to show that (;I) inlplies (c) arld (c) irrlplies ( i t ) . ( a ) implies (c). Let ( A , 4)be n (partially) ordered set in which every chai11 11% ;in upper bound. Out sttategy is to search for a rnaximal element of ( A . $ 1 by corlstructing a 4-incre 0, there is n, E N such that jx, - a1 < E for all n 2 n,. In particular, lxnt - a1 < e and X,= E A. (b) implies (a). The usual proof proceeds as follows: Let X, = {X E A I ( X - a [ < l / n ) . By (b), X , # 8 for all n E N . Let (X, I n E N )be a sequence such that X, E X, for all n E N . Then each X, E A and (X, I n E N ) converges t o a. T h e question usually passed over is, What reasons do we have to assume t h a t any such sequence (X, I n E N ) exists? Notice that we do not give any property P ( x , y) such that P ( n , y) holds if and only if y = X, (for all n E N ) . Such a property can be exhibited in special cases (e.g., if A is open - see Exercise 2.1); however, it has been shown that the equivalence of (a) and (b) for all A c R cannot be proved from the axioms of Zermelo-Fraenkel set theory alone. Of course, if we do assume the Axiom of Choice, the fact t h a t X, # Q1 for all n E N immediately implies that X, # 0.

nnEN

2.2 E x a m p l e . C o n t i n u i t y o f a Function. The standard definition of continuity of a real-valued function of a real variable is as follows: (a) j : R -+ R is continuous at a E R if and only if for every c > 0 there is 6 > 0 such t h a t 1j ( x ) - f ( a)( < E for all X such that [X - a1 < 6. Continuity is also characterized by the following property: (b) f : R -+ R is continuous at a E R if and only if for every sequence (X, ( n E N ) which converges t o a, t h e sequence (f (X,) I n E N ) converges to f (a). I t is easy to see t h a t (a) implies (b): If (X, 1 71 E N ) converges t o a and if E > 0 is given, then first we find 6 > 0 as in (a), and because (X, 1 n N ) converges, there exists ns such t h a t [X, - a1 < 6 whenever n 716- Clearly, If (X,) - f ( a ) [ < e for all such n. If we assume t h e Axiom of Choice, then (b) also implies (a), and hence (a) and (b) are two equivalent definitions of continuity. Suppose that (a) fails, then there exists E > 0 such that for each 6 > 0 there exists an X such that Is - a1 < d

>

CHAPTER 8. THE AXIOM OF CHOICE

146

but (f(X)- f ( a ) ( 2 E . In particular, for each k = 1 , 2 , 3 , .. . , we can choose some X, such that lxk - a1 < l / k and I f (xk) - f (a)l E. T h e sequence (xk I k E N ) converges t o U , but the sequence ( f (:ck) I k E N )does not converge to f ( U ) . so (b) fails as well. U

>

As in Example 2.1, it can be shown that the equivalence of (a) arid ( l ) ) cannot be proved from the axioms of Zermelo-Fraenkel set theory alone. 2.3 Example. Basis of a Vector Space. We assume that the rc:ader is farrliliar with the notion of a vector space over a field (e.g., over the held of rcbiil numbers). Definitions and basic algebraic properties of vector spaces (:an h(. found in any text on linear algebra. A set A of vectors is linearly zndependent if no finite linen^. corribin:tt,ior~ a , v l + - . +a;u, of elements v1, . . . , U, of A with nonzero coefficients CL 1 . . . . . ( l , , from the field is equal to the zero vector. A basis of a vector space V is t t maxinlal (in the ordering by inclusion) linearly independent subset of V. One of the fundamental facts about vector spaces is m

2.4 Theorem* Every vector space has a basis.

T h e theorem is a straightforward application of Zorn 's Lenlnla: Proof. If C is a C -chain of independent subsets of the given vector space. then the union of C is also an independent set. Consequently, a maximal independent r3 set exists. lFor more details, see the special case in the next example.) This theorern cannot be proved in Zermelo-Fraenkel set theory alone, without using the Axiom of Choice. 2.5 Example. Hamel Basis. Consider the set of all real numbers as a vect,or space over the field of rational numbers. By Theorem 2.4 this vector space hiis a basis, called a Hamel basis for R. In other words, a set X C R is a Hamel basis for R if every s E R can l)e expressed in a unique way as

for some rnutually distirict X I ,. . . ,X,, E X and sorrle nonzero l.ittioli;il riull~hers r l , . . . , r,. Below we give a detailed argument that a set X with the List mentioned property exists. A set of real numbers X is called dependent if there are mutually tlistinc:t, X I , .. . ,X, E X and r l , .. . ,rn E Q such t h a t

and a t least one of the coefficients r l , . . . ,rn is not zero. A set which is not dependent is called independent. Let A be the system of all independent sets of real numbers. We use Zorn's Le111mat o show that A llas a maximal elemerit

2. THE USE OF THE AXIOM OF CHOICE IN MATHEMATICS

147

in the ordering by C; the argument is then conlpleted by showing that any &-maximal independent set is a Hamel basis. To verify the assumptions of Zorn's Lemma. corlsider A. A linearls ordered by C. Let X. = U A o ; X. is an upper bound of A. in (A. 2 ) if X. E -4. i.e., if X. is independent. But this is true: Suppose there were X I , . . . ,X, E X. and r l , . . . , r, E Q, not all zero. such that rl . . X I + . . . + r, - 2, = 0. The11 there would be X I , . . . ,X, E A. such that xl E X l , .. . .X, E X,. Since A. is linearIy ordered by C,the finite subset { X 1 , .. . , X n ) of A. would have a 2-greatest element, say X,. But then 2 1 , .. . ,X, E X,, so X, would not be independent. By Zorn's Lemma, we conclude that (A, C) has a maximal element X. It remains t o be shown that X is a Hamel basis. Suppose that X E R cannot be expressed as rl sl + - .. + r, - X, for any r1,... , r n E Q a n d x l , . . . ,X, E X . T h e n a : f X (otherwisex = l q s ) , so X U { X ) 3 X , and X U { X ) is dependent (remember t h a t X is a maximal independent set). Thus there are X I , .. . ,X, E X U { X ) and s l , . . . .S, E Q, not all zero. such t h a t sl 2 1 + - . + S, . X, = 0. Since X is independent. X E {:rl.. . . , X, ): say. X = sZ, and the corresponding coefficient S, # 0. But then

c

where X I , .. . , X,-l, xi+l, . . . , X, E X , and the coefficients are rational ~lumbers. This contradicts the assumption on X. Suppose now that some a: E R can be expressed in two ways: .I: = 7.1 . X I + m . . + rn . X n = S L + + ~k - YC;,where ~ 1 , ... ,X,. y l . . . . . ! l k E X . r l , . . . ,r,, s l , . . . , s k E Q - (0). Then +

If { X I , . . . ,X,) # {yl,. . . ,yk) (say, XI 4 { y l , . . . ,PI;)), (2.6) can be written as a combination of distinct elements from X with a t least one nonzero coefficient (namely, rl ), contradicting independence of X . We can conclude that 72 = k and X 1 = y t l , . , -, X, = y,,, for some one-to-one mapping (il, . . . , in) between indices 1 , 2 , .. . , 71. We can thus write (2.6) in the form ( r l - s , , ) . z 1 + . . . + ( r , - s , , , ) . ~ , , = 0. Since zl , . . . , X, are mutually distinct elements of X , we conclude tllat, 7.1 - szl= 0, . . . . T, - ~ i , ,= 0 , i.e., t h a t rl = s i , , . . . ,r, = St,, . These arguments show that each X E R has a unique expression in thp desired form and so X is a Hamel basis. The existence of a Hamel basis cannot be proved in Zermelo-Fraenkel theory alone; it is necessary to use the Axiom of Choice.

sc3t

2.7 Example. Additive Functions. A function f : R -. R is called additive if f ( z + v ) = f ( z )+ f (y) for all X, y E R. An example of an additive function if provided by f a where f,(z) = a X for all X E R, arltl U E R is fixed.

CHAPTER 8. THE AXIOM OF CHOICE

148

Easy computations indicate t h a t any additive function looks much like f, for some a E R. More precisely, let f be additive, and let us set f ( l ) = a. We then have f(2) = f ( l )

+ f (l)= a

2,

f (3) = f (2) + f (1)= a - 3,

+

and, by induction, f ( b ) = a . b for all b E N - (0). Since f (0) f (0) = f (0+ 0) = f (o), we get f (0) = 0. Next, f (-6) + f(b) = f (0) = 0, so f ( - b ) = -f ( h ) a (-6) for b E N. To compute f ( l / n ) , notice that

-

+

n times

consequently, f ( l / n ) = a . l / n . Continuing along these lines, we can easily prow. t h a t f (X)= a X for all rational numbers X. It is now natural t o conjecture that f (X) = a - X holds for all real numbers x; in other words, t h a t every additive function is of t h e form fa for some U E R . It turns out that this corljecturc cannot be disproved in Zermelo-Fraenkel set theory, but t h a t it is false if we assume t h e Axiom of Choice. We prove the following theorem. 2.8 Theorem* There exists a n additive junction f : R for all a E R.

Proof.

+R

such that f #

fa

Let X be a Hamel basis for R. Choose a fixed Z E X . Define rt

0

i f x = r , . x l + v v . + r i . ~ l + ~ ~ a .n d+x ~, =nT~, ~ n otherwise.

It is easy t o check that f is additive. Notice also t h a t 0 4 X and X is infinite (actually, IX I = 2*0). We have f (Z) = l (because Z = 1 - Z is the basis representation of Z), while f ( 5 )= 0 for any Z E X , 5 # Z (because Z does not occur in t h e basis representation uf 2 = 1 Z of 2). If f = fa were to hold for some a E R , we would have f (T)= 1 = a Z, showing a # 0, and, on t h e other G hand, f(Z) = 0 = a . E , showinga = O . 2.9 Example. Hahn-Banach Theorem. A function f defined on a vector space V over t h e field R of real numbers and with values in R is called a l'i.rw(~,r. functional on V if

holds for aH U,v E V and a , 0 E R. A function p defined on V and with values in R is called a sublinearfunctionul on V if p(u + v) 5 p(u) + p(v) for all U,U E V and p ( & . U) = o p(u) for all

U E

V and

Q

2 0.

2. THE USE OF THE AXIOM OF CHOICE IN MATHEMATICS

149

The following theorem, due to Hans Hahn and Stefan Banach, is one of the cornerstones of functional analysis. 2.10 Theorem* Let p be a sublinear functional on the vector space V , an.d let fo be a linear functional defined on a subspace V. of V such that fo(v) 5 p(v) for all v E Vo. Then them is a linear functional f defined on V such that f 3 fil and f ( v ) 5 p(v) for all v E V .

Proof. Let F be the set of all linear functionals g defined on some subspacc W of V and such that fo G g

g(v) 5 p(v) for all v E W.

and

c).

We obtain the desired linear functional f as a maximal element of (F, To verify the assumptions of Zorn's Lemma, consider a nonempty F. c F linearly ordered by C. If go = U Fo, go is an &-upper bound on F. provided go E F. Clearly, go is a function with values in R and go fo. Since the union of a set of subspaces of V linearly ordered by C is a subspace of V, dorn go = UgEfi, dom g is a subspace of V . To show that go is linear consider U , v E dom go and a , P E R. Then there are g , g' E F. such that U E dom g and v E dorn g'. Since F. is linearly ordered by G , we have either g C g' or g' C g. In the first case. U ,V ,cr.u+/?Uv E domg' and go(cr-U+@-v) = gf(cr-?~+ijl.?)) = a . g ' ( ~ ~ ) + / 3 . g= '(t~) cr . go(u)+ @ go(v);the second case is analogous. Finally, go(u)= g(u) < p(u) for any U E domgo and g E Fo such t h a t u E domg. This completes the check of g0 E F. By Zorn's Lemma, ( F ,2 ) has a maximal element f. It remains t o be shown that dorn f = V. We prove that dorn f C V implies that f is not maximal. Fix u E V - dorn f ; let W be the subspace of V spanned by dom f and U . Sincc2 every W E W can be uniquely expressed as W = X + a . U for some X E dorn f and cr E R?the function f , defined by

>

m

is a linear functional on W and c E R can be chosen so that

fc

3 f . The proof is complete if we show that

for all X E dom f and cr E R. The properties of f immediately guarantee (2.11) for a = 0. So we have t,o choose c so as t o satisfy two requirements: (a) For all a > 0 and a: E dom f , f ( x ) + cr c 5 p(x ~ t . U ) . ( b ) For all a > 0 and y E d o m f , f ( y ) + ( - ~ ) ~ Ir p: ( y + ( - & ) . U ) . Equivalently,

+

+

CHAPTER 8. THE AXIOM O F CHOICE

150 and then (2.12)

f(

Y )

should hold for all

-P(:Y-U)

X ,y E

c P ( . I + ' L )

do111f and cr

f ( V ) + f ( t ) = f (v

> 0. But, for all

U,

-

t E clom f .

+ t ) < y(v + t ) Ip(fJ- U ) + 142 + 4

If A = sup{ f (v) - p(v - U ) 1 .o E tlorrl f ) kind B = inf {p(t+ ZL)- f ' ( t ) I t (10111 f ) . 5 B. By choosing c such that A 5 c 5 B, we (:an make the iclerlt,itj-

we have A

--

(2.12) hold.

-_I

2.13 Example. The Measure Problem. An irrlportant problenl irl analysis is t o extend the riotion of le~lgtliof a11 interval to Inore complic-atetlsets c)f r.c.;ll ilurnbers. Ideally, one would like to have a function p defirled on P ( R ) . with

values ill 10. X ) U {m). and having the followirlg prupert,ies: 0 ) p([a, 61) = 6 - a for arly a. 6 E R. a < b.

ii) If

is a collect.ion of nlutually disjoint subsets of R. then

(This property is ci~lIedcountable additivity or a-additivity of p . ) iii) If a E R, A G R, and A + u = { x + u ( trur~slutior~ i~~uu~'ia7tce of p ) . Several additional propertks of 2.3):

)L

15

E A ) , the11 p ( A + u ) = p ( A )

follow irrlnlediately frortl U).-iii) (Exrrc:isv

iv) If A n B = QI then p(A U B) = p(A) + p(B) (finite additivity).

However, the Axiom of Choic-e implies that nlentioned properties exists:

110

function p witeh t,lw iil)o\.c.-

2. THE USE OF THE AXIOM OF CHOICE IN A4ATHEMATICS 2.14 Theorem* There is no function p : P(R) properties 0) -v).

Proof.

We define an equivalence relation

sxy

if and only if X

4

151

[O.m) U { m ) uith the

= on R by:

- y is a rational number,

and use the Axiom of Choice to obtain a set of representatives X for z . It is easy to see that

R=

(2.15)

U { X+ r I r is rational);

moreover, if q and 1. are two distinct rationals, then X + q and X + r are ciis+joillt.. We note that p ( X ) > 0: if p ( X ) = 0, then p ( X + (I) = 0 for every q E Q, i ~ l l d

a contradiction. By countable additivity, there is a closed interval la, b] such t h a t p ( X n [a.b ] ) > 0. Let Y = X n [a, b). Then

and the left-hand side is the union of infinitely Inany mutually dis.joint set,s Y + g , each of measure p ( Y + q ) = p ( Y ) > 0. Thus the left-hand sid(. of (2.16) has measure m, contrary to the fact t h a t p ( [ a ,b + 1))= 6 + 1 - U. T h e theorem ure just established demonstrates that some of the requirelne~lts on p have t o be relaxed. For the purposes of rnat.henlat,ical arlalysis it is nlost fruitful t o give up t h e condition that p is defined for all subsets of R, arlcrl recluir~ only t h a t the domain of p is closed under suitable set. operations. 2.17 Definition Let S be a nonempty set. A collection 6 P ( S ) is algebra of subsets of S if (a) @ E 6 and S E 6. (b) If X E 6 then S - X E 6. ( c ) If X , E 6 for all n, then U:=o X n E B and Xx, E 6.

ii G -

nr=q

2.18 Definition A a-additive measure on a a-algebra 6 of subsets of S is a function p : G -. 10, m ) U {m) such t h a t

ii) If { X n ) r = o is

(Z

collection of mutually disjoint sets from 6 , then

CHAPTER 8. THE AXIOM OF CHOICE The elements of 6 are called p-measurable sets. T h e reader can find some simple examples of a-algebras and a-additive measures in the exercises. In particular, P ( S ) is the largest U-algebra of subsets of S; we refer t o a measure defined on P ( S ) as a measure o n S. The theorem we proved in this example can now be reformulated as follows. 2.19 Corollary Let p be any a-additive measure on a U -algebra 6 of subsets

of R such that

(0) [a,bj E G and p ( [ a ,bj)

=

b - a , for all a , b E R, a

< b.

(iii) If A E G then A + a E G and p ( A + a ) = p(A), for all

U

E

R.

Then there exist sets o j real numbers which are nut p-measurable. In real analysis, one constructs a particular a-algebra 1137 of Lebesgue measurable sets, and a U-additive measure p on 1137, the Lebesgue measure, satisfying properties (0) and (iii) of the Corollary. So existence of Lebesgue nonmeasurable sets is a consequence of the Axiom of Choice. Robert Solovay showed that the Axiom of Choice is necessary t o prove this result. Other ways of weakening the properties 0)-iv) of p have been considered. and led to very interesting questions in set theory. For example. it is possiblo t o give up the requirement iii) (translation-invariance) and ask sinlply wllet,ht:r there exist any a-additive measures p on R such that p ( [ a ,b]) = b - a for all a , b E R, a < b. This question has deep connections with the theory of large cardinals, and we return t o it in Chapter 13. Another interesting possibility is t o give u p ii) (countable additivity) and require only iv) (finite additivity). Nontrivial finitely additive measures do exist (assuming the Axiom of Choice) and we study them further in Chapter 11. Let us now resume our discussion of various aspects of the Axiom of Choice. First of all, there are many fundamental and intuitively very acceptable results concerning countable sets and topological and measure-theoretic properties of the real line, whose proofs depend on the Axiom of Choice. We have seen two such results in Examples 2.1 and 2.2. It is hard to image how one could study even advanced calculus without being able to prove them, yet it is known that they cannot be proved in Zermelo-Fraenkel set theory. This surely constitutes some justification for the Axiom of Choice. However, closer investigation of the proofs in Examples 2.1 and 2.2 reveals that only a very limited form of tlio Axiom is needed; indeed, all of the results can still be proved if one assumes only the Axiom of Countable Choice. Axiom of Countable Choice

There exists a choice function for every countable

system of sets. It might well be that the Axiom of Countable Choice is intuitively justified, but the full Axiom of Choice is not. Such a feeling might be strengthened by realizing that the full Axiom of Choice has some counterintuitive corlsc?quent.tls.

2. THE USE OF THE AXIOM OF CHOICE IN MATHEMATICS

153

such as the existence of nonlinear additive functions from Example 2.7, or the existence of Lebesgue nonmeasurable sets from Example 2.13. Incidentally, none of these consequences follows from the Axiom of Countable Choice. In our opinion, it is applications such as Example 2.9 which mostly account for the universal acceptance of the Axiom of Choice. The Hahn-Banach Theorem, Tichonov's Theorem (A topological product of any system of compact topological spaces is compact.) and Maximal Ideal Theorem (Every ideal in a ring can be extended t o a maximal ideal.) are just a few examples of theorems of sweeping generality whose proofs require the Axiom of Choice in almost full strength; some of them are even equivalent t o it. Even though i t is true that we d o not need such general results for applications to objects of more immediate mathematical concern, such as real and complex numbers and functions, the irreplaceable role of the Axiom of Choice is t o simplify general topological and algebraic considerations which otherwise would be bogged down in irrelevant set-theoretic detail. For this pragmatic reason, we expect that the Axiom of Choice will always keep its place in set theory. Exercises

2.1 Without using the Axiom of Choice, prove t.hat the two defirlitions of closure points are equivalent if A is an open set. [Hint: X, is open, so X, n Q # 8,and Q can be well-ordered.] 2.2 Prove that every continuous additive function f is equal t o fa for some a E R. 2.3 Assume t h a t p has properties 0)-ii). Prove properties iv) and v). Also prove: vi) p(A U B) = p(A) 2.4*

2.5

2.6

2.7 2.8

+ p ( B ) - p(A n B).

vii) ~ ( U zAn) o 5 Cz=op(An). Let 6 = {X C_ S I 5 N o o r I S -XI 5 No). Prove that 6 is a a-algebra. Let C be any collection of subsets of S. Let G = C I and T is a a-algebra of subsets of S). Prove that G is a a-algebra (it is called the a-algebra generated by C). Fix a E S and define p on P ( S ) by: p ( A ) = 1 if a E A, p ( A ) = 0 if a 4 A. Show that p is a a-additive measure on S. For A 5 S let p(A) = 0 if A = 0, p(A) = m otherwise. Show that p is a a-additive measure on S. For A S let p(A) = IAl if A is finite, p(A) = m if A is infinite. p is a D -additive measure on S; it is called the counting measure on S.

1x1

n{l) c

c

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Chapter 9

Arithmetic of Cardinal Numbers 1.

Infinite Sums and Products of Cardinal Numbers

In Chapter 5 we introduced arithmetic operations on cardinal numbers. It is reasonable t o generalize these operations and define sums and products of infinitely many cardinal numbers. For instance, it is natural t o expect that

1

+ 1+

=

No

NI, times

or, more generally, X times

was defined .Sthe cardinality of The sum of two cardinal numbers I C ~and A I U A2, where A1 and A2 are disjoint sets such that A1 1 = ~1 and I A2 1 = K Z . Thus we generalize the notion of sum as follows. 1.1 Definition Let ( A i 1 i E I) be a system of mutually disjoint sets. and Ict /All = K, for all i E I. We define the sum of ( K , I i E I) by

T h e definition of C l En,I uses particular sets A, (i E I ) .In the finite case. when I = {l,2) and n1 tcz = (Ar U Azl, we have shown t h a t the choice of A1 and A2 is irrelevant. We have proved that if A', , A; is another pair of disjoint sets such that [ A \ I = n l , IAil = n2, then JA; U Ail = / A 1 LJ A2J.

+

CHAPTER 9. ARITHME;TIC OF CARDINAL NUMBERS

156

In general, one needs the Axiom of Choice in order to prove the corresponding lemma for infiuite sums. Without the Axiom of Choice, we cannot exclude t,he following possibility: There may exist two systems ( A , ( n E N ) , (A; [ n E N ) of mutually disjoint sets such that each A, and each A; has two elements, but , :U A , is not equipotent to U%oA;! For this reason, and because many subsequent considerations depend heavily on the Axiom of Choice, we use the Axiom of Choice from now on without explicitly saying so each time.

1.2 Lemma If (A, I i E I) and ( A : I z E I) are systems of mutually disjo2n.t sets such that /All = IA:l for all i E I , then /UicfA,/ = Ail. Proof, Then f =

For each i E I, choose a one-to-one mapping fz of A , onto A : .

UXEI is a oneto-one mapping of Uie,A, onto UiEfA:. D This lemma makes the definition of C,,, legitimate. Since infinite unions fL

of sets satisfy the associative law (Exercise 3.10 in Chapter 2), it follows that the infinite sums of cardinals are also associative (see Exercise 1.l ) . The operation C has other reasonable properties, like: If K., X, for all i E I, then CisrK' < C,,I X, (Exercise 1.2).However, if K, < Xi for all i E I, it does not necessarily follow that C, K, < C, X, (Exercise 1.3). If the summands are all equal, then the following holds, as in the finite case: If K~ = K for a11 z E X, then



Proof. Let I9 = ~ f ( 2 ~ "I9) ;is acardinal. T h u ~ 2 ~is 1the ~ limit o f a n increasing sequence of length 6, and it follows (see the proof of Lemma 2.3 for details) that = K" ,

r

where each K , is a cardinal smaller than 2N1p. By Konig's Theorem (where we let X, = 2'- for all u < IY), we have

and hence 2Naj < (2'1.)'.

a contradiction.

Now if

29

were less than or equal to H,, we would get

0

The inequalities (3.1), (3.2), and (3.4) are the only properties that can be proved for the operation 2N- if the cardinal N, is regular. If N, is singular, then various additional rules restraining the behavior of 2N1.are known. We prove one such theorem here (Theorem 3.5); in Chapter 11 we prove Silver's Theorem (Theorem 4.1): If H, is a singular cardinal of cofinality cf(N,) 2 N1,and if 2Nt = for all < a, then 2Na = Nocl

CHAPTER 9. ARITHMETIC OF CARDINAL NUMBERS

166

3.5 Theorem Let fi, be a singular cardinal. Let us assume that the 71al7ir o] 2 ' ~ is the same f o ~all 5 < C), say 2N' = No. Then 2N17 = N N .

For

Note that it is implicit in the theorem t h a t No is greater tlian N,. instance, if we know t h a t 2N11= Nu+:, for a11 n < w , the11 2'u =

Since N, is singular, there exists, by Lenlnia 2.3, a colle(:tion (ti, 1 Pr.ooJ i E I } of cardinals such that ti, < N, for all i E I, arid (I1 = H, is n cardilia1 less than H,, and N,, = C,,I K , . By the assumption, we have 2"* = N , j for i E I, irnd iilso 2H7 = NLj. SO

U

1% riow approach the p r o b l ~ r ~o fi evalui~tir~g N,,". whored N,, aiitl N., i~1.t. arbitrary infinite cardinals. First. we make the following obsckrvat,iori.

Proof.

Clearly,

2Hli

5 NE". Since N, 5

W l i e ~ trying i t o evaluate

2 H 1 1 ,

we also have

NE" for a > 8. we find the folluwi~lguseful.

>

3.7 Lemma Let a B and let S be the set of all subsrlts X / X i = N,j. Then IS/ = N?'.

C

SIL(:II. tlrt~t


0 such that. IT - trl < n' implies 1 f ( X ) - f ( a ) l < E . S O we found 6 > 0 sr~clithat ( X- a / < S i1ny1ic.s f ( x ) E A , i.e.! z E f [A],showing f-'[A] open. (b) iniplies (a). Let a E R and E > 0. T h e assumption (b) implies that f [(f ( a ) - E , !(a) E )] is open (and contains a). T h u s there is 6 > 0 sucli that Is-U/ < d i m p l i e s s ~f - ' [ ( f ( a ) - ~ , f ( a ) + ~ ) ] , i . ef (. ,z ) E ( f ( a ) - E . f ( u ) + ~ ) . establishing the continuity of f at a. The equivalence of (b) and (c) follows immediately from Exerciscb 3.G(b) i l l Chapter 2. L C '

-' +

In t h e rest of this section we prove some results about open aiid closecl sets. T h e following lemma is used in Chapter 5 t o determine the cardinalitv of tlie systern of a11 open sets.

CHAPTER 10. SETS OF REAL NUMBERS

184

3.14 Lemma Every open set is a union of a system of open intervals ~ 1 2 t h rational endpoints.

Proof. Let A be open and let S be the system of all open intervals with rational end points included in A. Clearly, U S G A; if a E A, then (a - 6, a + 6 ) c A for some 6 > 0 and we use the density of Q in R t o find r l , rz E Q such that a - 6 < rl < a < r2 < a + b . Now a E ( r l , r 2 ) C A , so a E U S , showing that US=A, Q We have defined closed sets as colnplements of open sets. It is useful t,o h a w a characterization of them in terms of the behavior of their points. 3.15 Definition a E R is an accumz~lationpoint of A C R if for every (i > U there is X E A , X # a, such that Jx- a[< 6 . a E R is an isolated poznt of A R if a E A and there is 6 > 0 such that Ix - a1 < 6 , X # a, implies X 4 A.

c

It is easy t o see that every element of A is either a n isolated point or ali accumulation point, and that there may be accumulation points of A that clo )lot belong t o A. 3.16 Lemma A to A.

R is closed if and only if all accumulation points of A belong

Proof. Assume that A is closed, i.e., R - A is open. If a E R - A , then there is 6 > 0 such that Ix - a / < 6 implies X E R- A, so a is not a n accumulation point of A. Thus all accumulation points of A belong to A. Conversely, assume that all accumulatiorl points of A belong t o A. If a E R - A, then it is not a n accumulation point of A, so there is 6 > 0 such tliat there is no X E A, X # a, with Ix - a1 < 6. But then lx - a1 < 6 implies X E R - A, so R - A is open and A is closed. 0 As an application, we generalize Theorem 3.4. 3.17 Theorem Any nonempty systent of closed and bounded sets with the finitc intersection property has a nonempty intersection. Let S be such a system. We note that the intersection of any PmoJ nonempty finite subsystem T of S is ii nonempty closed and bounded set. So S= T I T is a nonempty finite subsystem of S) is again a nonempty system of closed and bounded sets with the finite interse~t~ion property, and the atldit,ion;~l property that the intersection of any nonempty finite subsystem of S actunllv belongs t o S . Since obviously S = 3, it suffices t o prove that S # B. The proof closely follows that of Theorem 3.4. Let A = {inf F 1 F E S). 13'~ show that A is bounded from above. Fix G E 3; then for any F E S we L~avc. F n G E 3 and inf F 5 inf(F n G ) 5 s u p ( F f~G) 5 sup G , so A is bounded by s u p G . By the completeness property, A has a supremunl a. The proof is concluded by showing 7i E F for ally F E 3.

{n

n

n

n

3. TOPOLOGY OF THE REAL LINE

185

Suppose that E 4 F. For any 6 > 0 there is H E S such that E - 8 < inf H 5 E, by definition of E. Since H n F E S and inf H 5 inf(H ll F ) , we have also E - 6 < inf(H n F) 5 E < iz + S. It now follows from the definition of infimum that there is X E H n F for which inf(H n F) < X < E + 8. We conclude that for each 6 > 0, there exists X E F such that X # Z (we assumed that a 4 F) and (X- El < 6. But this means that E is an accum~~lation point of F, and as F is closed, Si E F , a contradiction. Cl Closed sets contain all of their accumulation points; in addition, they may or may not contain isolated points. An example of a closed set without isolat.ed points is any closed interval. [O, l] U N is an example of a closed set with (infinitely many) isolated points. Nonempty closed sets without isolated points are particularly well-behaved; they are called perfect sets. We use them in the next section as a tool for analyzing the structure of closed sets. We conclude this section with a n example showing that, their good behavior notwithstanding, perfect sets may differ quite substantially from the familiar examples, the closed intervals. 3.18 Example Cantor Set. Let S = Seq((0, l}) = UnEN{O, l } be the srt of all finite sequences of 0's and l's. We construct a system of closed intervals D = (D, I s E S) as follows:

Do = [O. l];

-

1

D(o,o) - 101 g]'

2 1

D(o.1) =

8

2 7

D(l,o) = I-, 3 -1, 9

D(1,1)=

I ] , etc.

+

In general, if D(,,,,,.. ,S,,- 1 ) = [a, bl, we let D(,,,,.. . ,S. - 1 ,O) - [a,a $ ( b - a)] and D(s, , , , , , l = [a + - a), b] be the first and last thirds of [a, b]. The existence of the system D is justified by the Recursion Theorem, although it takes some thought t o see: we actually define recursively a sequence (D n f 71 E N ) , where each D" is a system of closed intervals indexed by (0, l } " : D:, = [O, l];having defined D n , we define D"+' so that for all (so, - . . , sn- 1 ) E (0, l } " .

i(b

CHAPTER 10. SETS OF REAL NUkIBERS

l86

+

a

[ a ,a 5 ( b - a ) ] and D"+' = [a + ( h - ( 1 ) . 61 rnlwrr. (S,,,. . 1 .l} 1% bl = D: ,,.,. , S , , , and then let D = UnE D" . NOW let Fn = U{Ds I S E {O. l)" ), SO that F. = [O, l], FI [O. U .1; I]. etc. Notice that each Fn is a union of a finite system of closer1 iiltervals. alltl hence closed (see Lemma 3.12). Tlle set F = F,, is tllus also closed: it is called the Cantor set. Df For any f E 10, 1 ) We let D j = We note that D, C F i L ~ l c I D j f 0 (Theorem 3.4). Moreover, the length of the interval D / is 1/3". so the infimum of the lengths of the intervals in the system (Df ( n C N ) is zero. It follows that D j contains a unique element: Df = {clf). Conversely. for any a E F there is a unique f E (0, 1 I N such that a = dj: let f = U { s C Seq((O.1)) I a E D,). (Compare these arguments with Exercise 3.13 i l l Cl~npter2.) d = (df / f E {O. 1 I N ) is thus a ulie-to-one mapping of {U. l j N tr~lto F, and we conclude that the Cantor set has the cardinality of the (:ontinuu~r~ 2N0 ~ n + l (so,

-

,S,*-

= ~$0)

,S,,-

-

1

i]

nnEN nnEN

We prove two out of many other interesting properties of the set F. (1) The Cantor set is a perfect set.

Proof. We know that F is closed and nonempty already. so i t rernains t o prove that it hiu no isolated points. Let a E F and 6 > 0; we have t o tint! X E F, X # a , such that JJ:- a1 < 6. Take 7 1 E N fbr which 1/3n < 6 . As we know, a = d j for some f E (0, 1IN: let X = d, where g is any sequ~rlcrof 0's and l's such that g n = f / 72 i\11(1 f # g. Then a # X , but (1.1 E Dlrr2= Dvl,, which has length 1/3" < 6, so ( X - a1 < 6, as required. L3 ( 2 ) The relative complement o f the Cantor set i n (0, l] is dense in [O. l ] .

P r o Let 0 5 a < b 5 1; we show that (a, 6) contains elements not in F. Take 72 E N such that & < < ( b - U). Let k be the least natural nunlhrr fi)r k (k+l) < b. The open interval which $ 2 U ; we then have a 5 F 0 there is X E A such that ( X - a1 < 6. a E R is a closure point if and only if it is either an isolated point of A or an accumulation point of A. Let A be the set of all closure points of A. A is closed if and only if A = A . 3.15 Every open set is a union of a system of mutually disjoint open intervals. [Hint: If A is open and a f A, U { ( x , g ) I a E ( X , ? / ) C A ) is an open interval.] 3.16 Give an example of a decreasing sequence of closed sets with empty intersection. 3.17 Show that Exercises 3.11 and 3.12 remain true for any closed and bo~rndr?tl set C in place of the closed interval [a, b]. 3.18 We describe an alternative construction of the completion of (Q.

c

CHAPTER 10. SETS O F REAL NUMBERS

188

rational p > 0 there is no such that la,, - a,,,J < p whenever n 2 n.0. Let C be the set of all Cauchy sequences of rational numbers. We define an equivalence relation on C by (a,):!, = (bn)r==oif and only if for each p > 0 there is no such that [a, - b,l < p whenever n 2 no. and ;I preordering of C by {an)?=, 4 {bn)F=oif and only if for each p > 0 there is no such that a, - b, < p whenever n >_ no. Prove that is an equivalence relation on C. (a) (b) + defines a linear ordering of C/ =. ( c ) The ordered set C/x is isomorphic to R.

4.

Sets of Real Numbers

The Continuum Hypothesis postulates that every set of real numbers is either at most countable or has the cardinality of the continuum. Although the Continuum Hypothesis cannot be proved in Zermelo-Fraenkel set theory, the situation is different when one restricts attention to sets that are simple in some sense. for example, topologically. In the present section we look a t several results of this type. 4.1 Theorem Every nonempty open set of reals has cardinality 2N0.

Proof. The function tarix, familiar from calculus, is a one-to-one mapping of the open interval (-.rr/2,~/2) onto the real line R; therefore, the interval ( - ~ / 2 ,n/2) has cardinality 2'". If (a,b ) and (c, d) are two open intervals. tllth function d-C f ( X ) = c -(X - a) b- a is a one-to-one mapping of (a, b) onto (c, cl); this shows that all open intervals have cardinality 2'". Finally, every nonempty open set is the union of a nonempty system of open intervals, and thus has cardinality at least 2N0.As a subset of R, it has cardinality at most 2N"as well.

+

4.2 Theorem Every perfect set has cardznalzty 2N0. We first prove a lemma showing that every perfect set contains two disjoint perfect subsets (the "splitting" lemma). Let P denote the system of all perfect subsets of R. 4.3 Lemma There are functions Go and G1 from P into P such that, for each F E P, Go(F) G F, G I ( F ) F and Go(F) n G l ( F ) = 8.

c

Proof. We show that for every perfect set F there are rational numbers r. S, r < S, such that F f~ ( - - m , r ]and F f l [ S , +m) are both perfect. Let a = inf F if F is bounded from below (notice that in this case u E F ) and a = -m otherwise. Similarly, let P = sup F if F is bounded from abovtb (p E F) and p = +m otherwise. There are two cases to consider:

4. SETS OF REAL

NUMBERS

189

(a) ( a ,P) C F . In this case, any r, s E Q such that a < r < S < P have the desired property: for example, if a E R, then F n ( - m , r ] = [a,rj; otherwise, F n ( - m , r] = (-m, r] (and we noted that all closed intervals are perfect sets). (b) (a,0) g F. Then there exists a E ( a ,P), a # F. The set F is closed (i.e., its complement is open), and thus there is 6 > 0 such that ( a - 6, a + S) n F = 0. We note that a < a - d (because a = inf F, there exists X E F, a < s < a ) and similarly a d < P. Any r, s E Q such that a - d < r < a < .s < u + 6 have the desired property: for example, F n ( - m , r] is clearly closed and nonempty ( F n (-m, r] = F n ( - m , a ] ) . If it had an isolated point h, we could find 6' 6 such that X E (b - h/, b + 6'), X # b, implies X # F n (-m, r] = F n ( - m , a+&). But for such X, X < b+df 5 r+d' < a+d' 5 u + d , so X E ( - m , a + 6 ) . We conclude X @ F, showing that b is an isolated point of F . This contradicts the assumption that F is perfect. To complete the proof of the lemma, we fix an enumeration ((r,, S,) I n E N) of the countable set of all ordered pairs of rationals, and define G o ( F ) = F n ( - 0 0 , r,], G 1( F ) = F n [S,, m), where (rn1 S , ) is the first (in our enumeration) pair of rationals for which r, < S, and both F n ( - m , r,] and F n [S,, m ) are perfect.

+


Conti~iuumHypothesis: They are either at ~ n o s countable t or have carclinalit y 2*". This is an immediate consequence of Theorern 4.2 tincl the following: 4.5 Theorem Every uncou~~table cmlosed set. c o ~ ~ t a i na sperfccf. s~rbset.

4.6 Corollary Every closed set of reals is either at nlost cou7ttatle ~{irdinality2'".

07.

/sus thc~

L5'e give t,wo proofs of Theoreni 4.5. The first proof is quite si~nple:howevr.~. it uses the fact that the urliori of ariy couritable system of at rnost cormtable sets is at most countable. The proof of this requires the Axiom of Choice (stit* Chapter 8 ) . Our second proof does not depend on the Axioru of Choice at ;ill: it also provides a deeper analysis of the structure of closecl sets, which is of iriclcpendent interest. It uses trilllsfi~iiterecursion (see Chapter 6).

Proof of IIYteo~.em :L5. Let A I>e a set of real numbers. We call (I E R it condensation point of A if for ('very d > 0 the set of all :r E -4 sui:li t l ~ i l t 1: - ul < 6 is ullcountablc (i.e., ilclit1it.r firiitc nor c:ountable). It, is obvious f r c ~ i i i the definition that any co~~der~satior~ poi~itof A is i t r l accur1~u1:~tion poilit of' .-l. l f ' ~dc?llotethe set of all coridensaticm points of A by A". (1) A C is a closed set. By Lemma 3.16 it suffices to show that every accumulation poitlt PmoJ of A" belorlgs to A C. If a is an ticcumulation point of A" and 6 > 0, the11 then: is X E A c such that Is - a1 < 6 . Let E = 6 - Is - a1 > 0. There t i ~ . t . uncountably rnany g E A such that ( y - sl < E ; but our choice of E guilraritclos t h ~ ift J y- s.1 < E , the11 ly - a ] 5 / g - z J+ ]X- a1 < E + )X - (LJ = 6, SO there ;ire 3 ~rncourltablynlarly y E A sut'h tt~iit13 - a1 < (5. ancl (L E A C . Now let F be a closet1 set; thc.11 Fc ol>servationis

CF

(sr~caLerii~riti3.16). O u r

~ic.~st

(2) C = F - F' is a t most c:ountt~ble.

If a E C , then a is not R condensation point of F. so thew is Proof. d > 0 such that F n ( a - 6, a + h ) is a t most countable. The density of t11c rationals in the real lirle guararltees the existerice of rational numbers r , S st11.11 that u - 6 < 7 - < a < S < (L + d . This shows that for ttac:h (L E C' thtt1.c. is

4. SETS OF REAL NUMBERS

191

an open interval ( r , S ) with rational endpoints such that a E F n (r.S) and F n (r, S ) is at most countable; in other words, C 2 U{ F n (r, S ) I r, S E Q. 7- < S, and F n ( r ,S) is a t most countable). But the set on the rjght side of the Irtst. inclusion is a union of an a t most countable system of a t rnost countable sets. so it, and consequently C, too, is at most count.nble. (Here we IISP the fiirt, dependent on the Axiom of Choice.) (3) If F is an uncountable set, then F" is perft?c.t,.

We know FC is closed [observation ( l ) ]a ~ l dilorlernpty [observat.ion Proof. (2)]. It, remains t o show that F' has no isolated poir1t.s. So ;fisunle that. ( I E FC is an isolated point of F'. Then there is 6 > 0 such that lx - a1 < 6. .X # ( I , implies X $ F C . But then, for this 6 , [ X - U ) < 6 ancl .T E k' imply .r ( I or C X E F - F , which is a n a t most countable set bj- observ:ition (2). Thus tIlt.rt1 are at most countably many X E F for which lx - U I < h, contradicting the assumption a E F C .

-

Observation (3) completes the proof of Theore1114.5. For an alternative proof of Theorem 4.5, at1c1 for its own sake, wt. lwgi11 a somewhat deeper analysis of the structure of closed sctts. In gerieral. c.losc(l srts differ from perfect sets in that they may have isolated points. We first provtb 4.7 Theorem E v e q closed set has at most countubl?j ntany isolated poir~ts.

Proof. If a is an isolated point of a closed set F, t,lic?n there is 6 > 0 such t h a t the only element of F in the open interval ( U - 6: u + 6) is a. Using thc density of the rationals in the reals, we can fincl r. .s C Q such that (L - h < r- < a < s < a + 6. so that a is the only element, of F ill t,he opcrl i n t c r ~ i l( v , S) with rational endpoi~lts.The function that assigns to ctilcll open interval with rational endpoints containing a unique element of F that element :LS R value, is a mapping of all a t rrlost countable set onto the set of ill1 isolntclrl points of F. The cot~clusionfollows from Theorem 3.4 in Chapter 4. 4.8 Definition Let A C R; the derived set of A. tlerioterl by A'. is tlw sct of' all accumulation poirlts of A.

c

It is easy t o see that a set A is closed if alld only if A' A ant1 pclrfrc.t if' and only if A # 0 and A' = A. Also, the derived set is itself a closet1 set. (Set. Exercise 4.4 for these results.) Theorem 4.7 says that for any closed F. tllo s ~ t F - F' of its isolated points is a t most countable. Tllese considerations suggest a strategy for a n investigation of the cardinality of closed sets. Let F # 8 be a closed set. If F = F', F is perfect and IF1 = 2N". Otherwise, F = (F- F') U F' ant1 ( F- F ' [ < No, so it remains to determint. the ct~rc1irialit.t. of the smaller closed set F'. If F' = 0 then F = F - F' h i ~ only s isolat.ed points and JFI 5 No.

f

CHAPTER 10. SETS OF REAL NUMBERS

192

If F' is perfect, then IF'J = ZNO, so J F J= 2N0. It is possible, however, that F' again has isolated points; for an example, consider F = N U { m- !I nj..71 E N , m 2 1 , n > l}. In that case we can repeat the procedure a n d examirlr F" = ( F ) ' . If F'' = 0 or F" is perfect, we get ( F ( N o or IF1 = ' L ~ o . But F" may again have isolated points, arld the analysis must continue. We (:an t l e t i ~ l r an infinite sequence (Fn I n E N ) by recursion:


0, m

= { B C RI B =

U B, whereeach B,

E H:}

n=O

and

m

H:+, = { B c R I B =

r) Bn where each B,

E E:).

n=O

c

B, for all k E N , k > 0. It can also be shown By induction, C: C B and (with the help of t h e Axiom of Choice; we omit the proof) t h a t

so t h e hierarchy produces new Borel sets at each level. One might hope t h a t all Borel sets a r e obtained in this fashion, t h a t is, t h a t B = C: = UE, H:, but it is not so. There are sequences (B, I n E N )such t h a t B, E C:,, CB M B, does not belong to any C: or II! B, or for each n E N, but (although it is of course Borel). We see again t h e need for a continuation of the construction of the hierarchy into "transfinite." 5.3 D e f i n i t i o n For all ordinals a < w lwe define collections of sets of reals Z !! and 11° as follows: (a) C[= { B 2 R ( B is open), II - { B C R 1 B is closed). b - = { B C_ R I B = Ur=oBnwhere each B, E IIO,), (b) E,+1 ,:l = { B R I B = nr'D=, Bn where each B, E CO,). : = { B C R I B = U z o B n where each B, E IIi for some 4 < a } , ( C) X II: = { B R I B = where each Bn E X i for some D < rr) ( t t ii limit ordinal). [In view of the next lemma, one could use clause (c) for successor ordinals i i S well, instead of (b).]

n;=fiB,

c

5.4 Lemma (2)

For every a < w l , C: C H:+, and IIL G

( i i ) For every a

E

+

(iii) I f B E X:,

< w 1, and II: 2 11:+1 then R - B E IT:; z f B E l:,

then R - B E C:.

(iv] i f B, E EL for each n E N , then UF.oBn E X:; n E N , then Bn E IT:.

nEo

zf B,,

E

II: for eacl~

5. BOREL SETS

Proof. (i) Trivial: in Definition 5.3(b), let Bn = B for all n. (v) Every open set is in C: (see Exercise 5.31, and similarly, every closed set is in II!. Hence 23; C C: and I l y IT;. Also, IT: C 23; and C: C IT:. by (i). Then we proceed to prove t h a t the assertion holds for every a < p, by induction on 0.

c

(ii) Follows from (v). (iii) By induction on a . (iv) This innocuous statement requires the Countable Axiom of Choice: For =:, B,, for some countable collection each n, if B, t C:, then B, = U {B,, I m E N ) of sets B, E U,, Il;. For each n, we choose one such collection (along with its enumeration (B,, I m E N ) ) , and then

is the union of the countable collection {B,,

1 (71. m ) E N

X

N ) and thus

in C:. T h e proof for IT: is similar. I t is clear (by induction on a ) t h a t each B: and each II: consists of Borel sets. W h a t is important, however, is that this hierarchy exhausts all Borel sets.

5.5 Theorem A set of reals is a Bowl set if and only i f it belongs to C: for some a < wl (if and only if it belongs to ll: for some a < wl).

Proof.

It suffices t o show that the set

is closed under countable unions and intersections. Thus let { B , ( n E N ) be a countable collection of sets in S. We show that, e.g., ,=, Bn is in S . For each n let an be the least cr such that B, E II,. T h e set ( a , ) n E N ) is an a t most countable set of ordinals less than wl and thus (by the Countable Axiom of Choice) its supremum a = sup{an ( n E N } = U { a n I n E N } is a n ordinal less than wl. By Lemma 5.4 we have Bn E IT! for each n. Consequently, C l Bn belongs t o C:, and hence t o S.

Vrn

We note, without proof, t h a t Theorem 4.5 can be extended to all Borel sets: Every uncountable Borel set contains a perfect subset. In particular, every uncountable Borel set has cardinality 2H0. In Chapter 8 we mentioned the (Ialgebra C of Lebesgue measurable sets of reals; of course, B g C, but there

CHAPTER 10. SETS OF REAL NUMBERS

198

exist also Lebesgue measurable sets which are not Borel. In fact, one cannot, prove in Zermelo-Raenkel set theory t h a t all urlcountable Lebesgue nleasural)lc. sets have cardinality 2N". T h e construction of the Bore1 hierarchy is just a special case of ;t very ge11c.ti-il procedure. We observe t h a t the infirlitary unions and intersec:tions usecl iri it are. in some sense, generalizations of' the usual operations U arld n. t,Vo t l ~ f i i i t ~ : F is a (countably) infinitary oycratio~~ on A if F is a function on H suI>sttt of' A N (the set of all infinite sequences of elellients o f A ) into A, aiid i1itrc1tluc.t. structures with infinitary operutions. This car] bt: done forrnallv bv exteii(liil:: the definition of type (see Section 5 in Chapter 3) by allowing fi E N u { N } i l t ~ t l postulating t h a t F, is an infinitary operation wherlever f;, = N . Tlle dc~finit,ioll of' 21 set B C A being cEosed reinailis ttsserltially uncllallgeil; if f, N wt. 1.tbcluil.tt F,((u, ( i E N ) ) E i? for ali ( a , ( i E N ) such t h a t u I E i? for ;ill i E N allcl F, is defined. T h e closure 5;; of it set C C A is still t h e I e t ~ c:los~rl ~t s ~ c,ontiii~ii~ig t ;ill tllements of C. 111this terminology, the systerrl of Borel sets is just the closltre of t.he systtltii of a11 open and closed sets under the irifirlitary uniorl allcl intersec-tion. 51oi.tb precisely, we let U = (A, F l , F 2 ) w11t:re A = P ( R ) , Fl({B, I -i E N ) ) = U'1 = I ) L?, and F2((B, i E N ) )= B,, i ~ Cd = { B 2 R I B is open or B is (.k)stbrl). Then B = C. Exercise 5.9 provides another example of lt structure with infil~itaryc)l,cbl.iltiorls arld a closed subset of somc! ~nathematicaliuterest. One important general questiorl about the closure is t h a t of its c:i.trdii~iilit.~-. \jre proved the simplest result prcwicling ari a1lswt.r iri 'rlic?orcni 3.14 o f Cliilptc~~' 4. There it uras slssurned tellat C' is ;it most countable, and all operat,ioils ii1.c' fi~iitary. We flaw generalize this, first. t.o i~rbit~rary C, ii11c1 tlit:~lt o strll(.tltrcss with infinitary operittions.

-

nEo

I

5.6 Theorem Let 2l = ( A , (Ru,. . . . R ,,,-l ) , ( F u , .. . . F,- 1)) 116 U s ~ T u ( : ~ (171(i u~.~J let C -4. Let T]; be the closurc: of C. If C is finite, then z s at .moat cot~~ltahlo; i f C is 27~finite,then /C1= /Cl.

c

-

,,: C, wlrt.i.cb Proof. We proved in Theoreril 5.10 in Chapter 3 t h a t C = U CO= C and C,+* = C,UF~[c("]u. - u F , ~ _ ~ [ c [ If" -C' ]is. finite, t h c ! ~~karll ~ C', is finite, and No. If C is illfinite and ICI = N,; then wr prove by in(lu(:tioil ! t h a t lCkl = N,, for a11 k: Assume that it is true for C,; the11 IC,'J ' I Ntr' = N,,. . I IFJIC,']I < No. tint1 IC,+ll = N,,. Hence (Cl = N u . N , , = K,,.


2, then

> 2'0, and we have

IC[N4l


0. Show that d(A) = l / p . 1.8 Prove that the set (2'' 1 71 E N ) has density 0. l .g Collstxuct a set A of natural riurrlbers such that

linlsupA(n)/7i = 1 and liminf A ( r ~ ) / n= 0. n+cu

n-+m

2. ULTRAFILTERS

2.

Ultrafilters

2.1 Definition A filter U on S is an ultrafilter if for every X or S - X U.

C S, either X

E

U

The dual notion is a prime ideal:

2.2 Definition An ideal I on S is a prime ideal if for every X

C S. either

Xf IorS-XfI. 2.3 L e m m a A filter F on S is an ultrafilter if and only zf it is a maxi~naljilter on S .

Proof. If F is an ultrafilter, then it is a maximal filter, because if F' > F is a filter on S, then there is X C S in F'- F. But because F is an ultrafilter, S - X is in F , and hence in F', and we have 0 = X n (S - X ) E F', a contradi~t~ion. Conversely, let F be a filter but not an ultrafilter. There is some X S such that neither X nor S - X is in F. Let G = F U{X). We claim that U h ~ s the finite intersection property. I f X 1 , . . . , X , a r e i n F , t h e n Y = X I n . . . n X , E F , ; s n d Y n X # 0 because otherwise (S - X ) 2 Y and so S - X E F, contrary to the assumption. Hencc X l n.. . n X , n X # 0,which means that G = F U { X ) has the finite int,ers~c.tio~l property. Thus there is a filter F' on S such that F' F U { X ) . But t.hat ~ ~ l c a that ns F is not a maximal filter. S

c

>

We have seen earlier that there are principal filters that are rnaximal. 111 other words, there exist principal ultrafilters. Do there exist nonprincipal ultrafilters? Let S be an infinite set, and let F be the filter of cofir~itesubsets of S. If U is an ultrafilter and U extends F, then U cannot be principal. Thus. to fillcl a nonprincipal ultrafilter it is enough to find an ultrafilter that extellds the filter of cofinite sets. of' Conversely, if U is a nonprincipal ultrafilter, then it ext.ends t,lle filtc>~. cofinite sets because every X E U is infinite (Exercise 2.2). The next theorem states that any filter can be extended to an ultrafilt, 0 , { n I la,

- a1

<E)

E U.

First we observe t h a t if a U-lirnit exists then it is unique. For, assume that u < b and both U a n d b are U-limits of (a,)?==,. Let E = (b - a ) / 2 . Then t h t ~ sets { n ( (a, - a ( < E ) and {n ( lan - b( < E ) are disjoint and so cannot both 110 in U, If U is a principal ultrafilter, U = {A I no E A) for some no. then limr: a,, =: U,,,,for any sequence ( a n ) ~ ~ =This o . is because for every E > 0, { r 1~ Iw, - r ~ , , , , l < E ) 2 {no) U. If (a,):!, is convergent and lirn,,, a, = U , then for every nonprincipal illtrafilter U . limll a, = U. This is because for every E > 0, tilere exists sorrlr k such t h a t ( n I \a,, - a1 < E ) ( 7 1 1 ?L 2 k ) , and { n I n > k ) E U . T h e important property of ultrafilter lirnits is t h a t the U-limit exists for every bounded sequence:

>

2.

ULTRAFILTERS

207

2.8 Theorem Let U be an ultrafilter on N and let quence of real numbers. Then limu a, exists.

be a bounded se-

Proof. As is bounded, there exist numbers a. and b such that a < a, < b for all n. For every X E [a, b], let

A, whenever X 5 y. Therefore A, Clearly, A, = 0, Ab = N , and Ax Al,f U. and if A, E U a n d x 5 y , then A, E U. Now let C

= SUP{X(

A,

4

U,

U);

we claim that C = limu a,. As for every E > 0, A,-, 4 U while A,-+, E U . i d because A,+, = A,- E2 U {n I c - E < a, < c + E ) , it follows that ( 7 1 ( [a, - cl < E) E U. As an application of ultrafilter limits, we construct a nontrivial measure on

N. 2.9 Theorem There exists a measure m on N such that m(A) = d ( A ) for every set A that has density.

Proof.

Let U be a nonprincipal ultrafilter on N . For every set A

c N.

m(A) = lim 1J n where A(n) = ( A n nl. Clearly, if A has density then m(A) = d(A). Also, it is easy to verify that m(0) = 0 and m ( N ) = 1, and that A B implies m ( A ) < m ( B ) . If A and B are disjoint then (A U B)(n) = A(n) + B ( n ) , and the additivity of m follows from this property of ultrafilter limits:

c

lim(a, U

+ bn) = lima, + limb,. U U

We leave the proof, which is analogous to that for ordinary limits, as an exercise L' (Exercise 2.5).

Exercises 2.1 If U is an ultrafilter on S, then P ( S ) - U is a prime ideal. 2.2 If U is a nonprincipal ultrafilter, then every X E U is infinite. 2.3 Let U be an ultrafilter on S. Show that the collection V of sets X C S X S defined by X E V if and only if {a E S I { b E S 1 ( a , b ) E X ) E U} E U is an ultrafilter on S X S. 2.4 Let U be an ultrafilter on S and let f : S -+ T. Show that the collection V of sets X c T defined by X E V if and only if f-'[X] E U is an ultrafilter on T. 2.5 Prove that limu (a, + 6,) = limu an + limlJ b,.

CHAPTER 1 1 . FILTERS AND ULTRAFILTERS

208

Closed Unbounded and Stationary Sets

3.

In this section we introduce an important filter on :i regular uncountable carclirial - the filter generated by the closed unbounded sets. Although the results of this sectiorl car] be formulatecl and provecl for any regular uncou~lt~able c.arcliilal. n-t1 restrict our investigations to the lei~stuncountable cardirial N I . (See Exercischs 3.5-3.9.)

c

3.1 Definition A set C wl is closed unbounded if (a) C is unbounded in w l , i.e.. s u p C = w l . (b) C is closed, i.e., every increasing sequence

of ordinals in C has its suprenlurn sup{a,

I n E W ) E C.

An in~portant,albeit simple, fact about closed unbounded sets is that tl~c>?. have the finite intersection property; this is a consequence of the following: 3.2 Lemma If Cl and Cz are closed unbounded subsets of closed unbounded.

wl,

then C l n C ,

,.v

Proof. It is easy to see that Cl n C2 is closed; if a1 < a:! < . . . is a sequence in both Cl and CL, t,hen cr = sup{n, ) n W ) E Cl n C2. To see that CLnC;! is unbounded, let y < w l be arbitrary and lct us find r k in Cl n C2such that a > y. Let us construct an increasing sequence

of countable ordinals as follows: Let a 0 be the least ordinal in Cl above y. let Do be the least 3 > 00 in C2. 'l'llen a ] E C l ,P1 E C 2 ,tincl so 011. Let a be the suprernum of { o , , ) , ~ it ~ ;is also the suprernum of ( J n The ordinal a is in both C, nrld C2 and therefore cu Cl n C2.

'l'lit*~i

Inch.. 0

The set w l of all countable ordinals is closed and unbounded. (Howevc.r. here we use a consequence of the Axiom of Choice, namely the regularity of L J :~ the supremum of every countable sequence of countable ordinals is a c*ountabltb ordinal.) Another example of a closed unbounded set is the set of all limit c:ountable ordinals (Exercise 3.2). For other, less obvious examples see Exerriscbs 3 . 3 and 3.4. As the collection of all closed urlbounded subsets of wl llas the finite int,vrsection property, it generates a filter on wl: (3.3)

F = {X

c

I

L J ~

X

> C f ~ some r (:losecl unbounded C )

The filter (3.3) is called the closed unbounded filter or1 w l .

3. CLOSED UNBOURrDED AND STAT1ONAR.YSETS

209

3.4 L e m m a If {Cn)nEwis a countable collection of closed unbounded sets, then Cn is closed and unbounded. Conseguentlp, zf {Xn)nEd is a countabl~ collection of sets i n the dosed unbounded filter F , then X. E F.

nZz0

nzo n;=,

C, is closed. ?b Proof. It is easy to verify that the intersection C = prove that C is unbounded, let y < wl; we find cr E C greater than y. We note that C = n r = o D,, where for each n, D,, = COn- .nC, ; the sets D, are closed unbounded and form a decreasing sequence: Do 2 D1 2 D2 2 . . . . Let ( a , I n E W ) be the following sequence of countable ordinals: y < a0 < (11 < . - . , ancl for each n , cxn+l is the least ordinal ~ I ID, above G , , . Let rr LW the supremum of To show that a E C, we prove that CL E D, for each n. But for any n , CY is the supremum of {akI k > 72 + l ) , and all thc a k for . . . 2 Dk 2 . . . ( k > n ) . k > n + 1 are in D, because D, 2

>

Closely related to closed unbounded sets arc stationary sets. Stntionaq sets are those sets S g wl which do not belong to the ideal dual to the c:losc(l unbounded filter. A reformulation of this leads us to the followiilg definition.

C W * is

3.5 Definition A set S C, S n C is nonempty.

stationary if for every closed unbou~ldetlwt.

Clearly. every closed unbounded set is stationary, and if S is stnt,ionary arid S C_ T C wl, then T is also stationary. Later in this section we present a n example of a stationary set that does not have a closerl unbounded subset. But first we prove the following theorem. A function f with domain S C_ wl is reyressive if f ( a ) < u for all (L # 0 .

c

3.6 Theorem A set S wl is stationary i j and onlp zj r7~e7yregressi~lefunction. j : S -+ wl is constant on an unbounded set. i n fact, f has a constan.t value on. a stationary set,

This theorem is a good example of how an ul~countablealeph such as N I differs substantially from No. On uncountable cardinals there is no analogue of the function f : w -+ w defined by f(n) = n - 1 for n > 0 , f ( 0 ) = 0 , wl~ich has each value only finitely many times. A consequence of 'I'heoreni 3.6 is that. on wl! a function that satisfies f ( a )< a repeats some value ~ l n c o u n t a boften, l~ unless the domain of f is "small," that is, not stationary. One direction of the theorem states that if S C wl is not statiorlary, t11en there is a function on S such that f ( a )< a for all a # 0 , and f takes each value a t most countably many times. Such a function is corlstructed in the following example. 3.7 E x a m p l e Let A C wl be a nonstationary set; hence there is a closed unbounded C such that A n C = 0. Let f : A -+ wl be defined as follows:

f (CL)

=

sup(C n a ) ( a E A).

CHAPTER 1 1. FILTERS AND ULTRA FILTERS

210

Because f(a)E C (see Exercise 3.1) and C n A = 0, we have f ( a ) < CL. And for each y < wl, when a E A is greater than the least element of C above 7 . thcrl f ( a )> y and so f does not have the same value for uncountably many a ' s . To prove the other direction of the theorem we need 3.8 Lemma Let {Ce I set C C wl defined by

1emm:s.

< < wl } be a collection of closed unbounded sets.

a E C i f and only if a E Ccfor all

(3.9)

A

< a (a E

Thr

wl)

called the diagonal intersection of the C ( , is closed unbounded. Proof. First we prove that C is closed. So let a0 < a1 < . - . < cr, < . - . be a11 increasing sequence of elements of C and let a be its supremurn. To prove that a is in C we show that a f C( for all < a . Let. < a . Then there is sorne k such that < ~ k k and . hence < ck,, for all 71 k . Since each a , is in C, it satisfies ( 3 . 9 ) , and so for each n k we hi~vc, a , E C€. But C( is closed and so cr, the supremum of { a n ) n ~isk in , C€. Next we prove that C is unbounded. Let y < wl and let us find ( 1 E C' greater than y. We construct an increasing sequence a0 < a1 < 02 < . . . itS follows: Let a0 = y. The set C( is closed unbounded (by Lemma 3.4) i~rldso we let crl be its least element above ao. Then we let a2 be the least element of' Cc above a1, and in general,

>




n(,,,,

nECn,

7 let us prove that cr E C. The let a be the suprernum of { a n ) n E wand We are to show that a E C€ for all < a . So let < a . There is k such t l ~ t < ak, and for all n 2 k we have a,+ l E C € ,by (3.10). But cr is the suprerrlunl of { a , and hence is in C c .

Proof of Theorem 3.6. Let S be a stationary set and let f : S -+ L J ~1)e ii regressive function. For each y < wl , let A, = f - l [{y )]; we show that for some y the set A, is stationary. Assuming otherwise, no A, is stationary and so for each -j < w l there is n closed unbounded set C, such that A, n C, = 0. Let C be the diagol~al intersection of the C,, that is, (3.11)

a E C if and only if cr E C, for all

y

< rr.

If a E C,, then a pl A, and so f ( a ) # y. Thus (3.11)means that if a E C, then f ( a ) # y for all y < a ; in other words, f ( a ) is not less than a . But C is r:losecl unbounded and hence it intersects the stationary set S. But for all 0 E S, f ( t r ) C? is less than cr. A contradiction.

3, CLOSED UNBOUNDED AND STATIONARY SETS

2 11

As an application of Theorem 3.6 we construct a stationary set S G wl, whose complement is also stationary. Note however that the construction uses the Axiom of Choice. (It is known that the Axiom of Choice is indispensable in this case.) 3.12 Example A stationary set whose complement is stationary. Let C be the set of all countable limit ordinals. For each E C there exists a n increasing sequence X, = (X,, I n f W) with limit cr. For each 7 1 . we let, f, : C --+ wl be the function f n ( a ) = X,,. For each n , because f,,(a) < a on C, there cxist,s 7, such t,hat the set is stationary. Sn = {a E C I fn(o) = We claim t h a t a t least one of the sets S, has a stationary complement. If not, then each S, contains a closed unbounded set, so their irltersection does too. Hence S,, contains an ordinal a that is greater than the supremum of But in that case the sequence X, = (X,, I 12 E U ) = ( f n ( a ) I the set n E w) = (7, I n E w) does not converge to a, a c~ntradict~ion.

f)r=O

{

~

~

)

~

~

~

a

As another application of Theorem 3.6 we prove a combinatorial theorem known as the A-lemma. Although it is possible to prove the A-lemma direct,ly, the present proof illustrates the power of Theorem 3.6. 3.13 Theorem Let { A , I i E I} be an uncountable collection of finite scts. Then there exists an uncountable J C_ I and a set A such that for all distinct i , j E J , Ai n A, = A.

We may assume that I = wl, and since the union of N I finite sets Proof, has size N I ,we may also assume that all the A, are subsets of wl. Clr:arly, uncountably many A, have the same size and so we ,assume t h a t we have a collectioi~{A, I cr < wl) of subsets of L J ~ each , of size 71, for some fixed riurnber n. Let C be the set of all a < wl such t h a t maxA( < a whenever < a . The set C' is closed unbounded (compare with Exercise 3.4). For each k < 7 1 , let Sk= {cr E C 1 /A, n a [= k ) ; there is at least one k for which Sk is stationary. For each m = 1 , . . . , k , let f,(cr) = t h e m t h element of A,; we have {,,(Q) < c k on S k . By k applications of Theorem 3.6, we obtain a stationary set T 2 Sk and a set A (of size k) such that A, n a = A for all a! E T. Now when a < 0 are in T, then A, C 0 (because 0 E C) and A, n 0 = ADnp = A, and it follows that A, n Aq = A Ibecause (A, - a) n A3 B]. Thus {A, I CY E T) is an uncountable subcollection of {A, I a < wl) which satisfies t h e theorem. 0

c

-

Exercises

3.1 An unbounded set C c w l is closed if and only if for every X C C, if sup X < w l , then s u p X E C. 3.2 T h e set of all countable limit ordinals is closed unbounded.

CHAPTER 11. FILTERS AND UL'IRAFILTERS If X is a set of ordinals, then cr is a limit point of X if for every y < tr tllerc* is E X such that y < P < ck. A countable a is a limit point of X if i111cl oilly if there exists a sequence a 0 < 01 < - . . in X such that sup{cr,, I 7 1 E ~ j = ) (I. Every closed unbounded C 2 w l r:ontains all its countable limit points. 3.3 If X is an unbounded subset of w l , then the set of all cour~ta~blc! limit points of X is a closed unbounded set. If f : w l if f

(c) < cu

is ail increasing function, then wlientlver ,€ < a. -4

w l

CY

< W ] is

it

closu~ep o i r ~ tof' f'

3.4 The set of all closure points of every increasing furiction f : closed unbounded. be a regular uricountable cardinal. A set C C_ sup C = K arlcl s u p ( C n a ) E C for all a < K. Let

K

3.5 If Cl

arid

K

W , -+ ~ J J

is

is closed unbounded if'

C2 are closed unbounded, then Cl n C2 is closed unbouacli~l

The closed unbounded filter on bounded sets.

K

is the filter generated by the c1osc:ti

url-

3.6 If X < K and each C,, a < X, is closed urlbounded, then n a < x Cc?is closed unbounded.

A set S C h: is stationary if S n C # 0 for every closed unbounded

set, C' C

K

3.7 The set { a < K I U is a lirnit ordinal and c f ( a ) = w ) is statiorlary. It' h: > NI, then the set { C L < K ) cf(a) = W I ) is stationary. 3.8 If each C.',,, 0 < K. is closecl urlboundecl, then the dtt~yonal~ Y L ~ ~ ~ . . ~ . c I ( . ~ / . o I I ( 1 1 < ti I (L E C( for it11 €, < cr) is closed unbounded.

A function with dom f

c ti is ~ g ~ e s s i uifef (a)< n for. all

(1

E don1 f . (1

# 0.

c

3.9 A set S K is stationary if and only if every regressive function on S is constant on an unbounded set.

4.

Silver's Theorem

Using the techniques introduced in Sections 2 and 3 we now prove the followirlg result on the Generalized Contiriuurn Hypothesis for singular cardinals.

-

4.1 Silver's Theorem for every n. < X, 2' then zN" Nx+i .

Ij

We prove Silver's Theorem for the special case NA = N,, , using the theory of stationary subsets of N l . The full result can be proved i11 a similar way, usirig

4. SILVER'S THEOREM

213

the general theory of stationary subsets of K = cf X (see Exercise 4.1). Thus assume that 2 N 1 1= N,+l for all a < ol. One consequence of this assumption, to be used repeatedly in this section, is that N,N1 < Nu, for all a < wl (see Theorem 3.12 in Chapter 9). Let f and g be two functions on wl. The functions f and g are almost disjoint if there is some rr < w l such that f ( P ) # g ( 0 ) for all /3 2 a .

4.2 Lemma Let {A, I a < wl} be a family of sets such that [A-I every a < wl, and let F be a family of almost disjoint functions,

< N,

for

Then JFJ 5 NW,. Proof. Without loss of generality we may assume that A, 2 w,, for each a < wl. Let So be the set of all limit ordinals 0 < tr < wl. For f E F a1111 a E So, let f' ( a ) denote the least ,!3 such that f ( a ) < wg . As f * ( a ) < (I for every a E So,there exists, by Theorem 3.6, a stationary set S C Sosuch that f * is constant on S. Therefore f 1 S is a function from S into i J n . for soirle ,d < wl. Let us denote this function f 1 S by c p ( f ) . If f and g are two different functions in F, then p( f ) and ( ~ ( g are ) also different: even if their domains are the same, say S, t,hc?n f S # g I S becbaust. f and g are almost disjoint. Thus cp is a one-to-one rnapping with donlail~F. The values of p range over functions defined on a subset S of w l into some wp < W,, . Thus we have

A slight modification of Lemma 4.2 gives the following:

n,,,, A,

4.3 Lemma Let F C that the set T = { U < wl

I

be a fumilp of almost disjoint functions, sl~ch. lAal 5 N), is stationary. Then IF1 5 N,, .

Proof. In the proof of Lemma 4.2, let Sobe the stationary set So= ( n E T ( a is a limit ordinal). The rest of the proof is thc same. 0 Using Lemma 4.3, we easily get the following:

4.4 Lemma Let f be a function o n ol such that f ( 0 ) < N,+, Let F be a famzly of almost disjoznt functions on w , and let

Ff = {g E F I for some stationary set T Then (Ff(lNW,.

c u l , g(a) < f ( C L )

for all cr < w l . for all

tr

E

T).

CHAPTER I l. FILTERS AND ULTRAFILTERS

214

Proof. For a fixed stationary set T, the set { g E F I g(a) < f ( a )for all (L c T} has cardinality at most N,, , by Lemrna 4.3. Thus IF\ 2'1 - N,, = N,, .


(L. T h u s F = { f x ( X E P(w,,)) is a fanlily of a l ~ ~ i odisjoint st functions, i ~ l ~byd Le111111it

4.5, JFI < NWl+1.Therefore, 2N-1 =

G

4. SILVER'S THEOREM

Exercises

4.1 Prove Silver's Theorem for arbitrary X of uncountable cofinality. [Throughout Section 4, replace wl by K = cf X, and replace t,he sequence {N, I a < wl)by a continuous increasing sequence of cardinals {X, I a < K). Use Exercise 3 .g.]

This page intentionally left blank

Chapter 12

Combinatorial Set Theory 1.

Ramsey's Theorems

The classic example of the type of question we want to consider in this section is the well-known puzzle: Show that in any group of 6 people there are 3 who either all know each other or are strangers to each other. Wt. arc) inlplicitly assuming that the relation "X knows y" is symmetric. The argument goes follows. Consider one of these people. say ,Jot. Of t11v remaining 5 people, either there are at least 3 who know .Joe. or thcjrt. arch;it least 3 who do not know Joe. Let us assume that Peter, Paul, anc! h,l:rry krloxv Joe. If two of them, say Peter and Paul, know each other. then we havc ;3 pc'oplc who know each other (Joe, Peter, and Paul). Otherwise, Peter, Paul. and Mary are 3 people who are strangers to each other. If Peter, Paul, and hlary are 3 people who do not know Joe, the argument is similar. We now restate this problem in a inore abstract form that allows for ready generalizations. Let S be a set. For r E N, r # 0, [S]' = { X S I 1x1= r ) is the collection of all r-element subsets of S. (See Exercise 3.5 in Chapter 4 . ) Let. {A,):;: be s partition of [S]' into S classes ( S E N - (0)); i.e., [S]' = A, and A; n A j = 0 for i # j . We say that a set H S is homogeneous for the part,ition if [H]' Ai for some i; Le., if all r-element subsets of H belong to the sanw class Ai of the partition. It may help to think of the elements of the diffclrerlt c:la~sesm btling colore(1 by distinct colors. We thus have S colors (numbered 0, l , , . . , s - 1) ailcl eiic:l~ r-element subset of S is colored by one of them. Homogeneous set are nlonoch,romatic; i.e., all r-element subsets of them are colored by the same color. In this terminology, the introductory example shows that, for a set S with IS1 2 6, every partition of [SI2into two classes has n homogeneous set H with 1HI 2 3. (The classes are A. = { { X , y} E [S]" X and y know each other) and Al = {{X, y } E lsj2I X and y are stxangers to each other}.) Put, yet another way: pick a set S of a t least G points and connect t.;iibli

c

c

CHAPTER 12. COMBINATORIAL SET THEORY

218

pair of points by a line segment colored either red or blue (but not both). Tllo resulting graph then contains a monochromatic triangle (i.e., all red or all blue). Let K , X be cardinal numbers. We write K -+ (X): as a shorthand for the statement: for every set S with IS1 = ti and every partition of [S]' into S classes. there exists a homogelleous set H with [ H (> X. The negation of this st.atenl~rlt is denoted K f , (X):. We proved that 6 -+ (3);; it is an easy exercise to show that 5 f . (3);. (Sec. Exercise 1.1.) More generally, K -,(X I , . . . , X,): means: for every set S with 1 S ( = h: t i ~ i c l every partition { A i j f ~ofi [S]' into S classes there is some i and a set H c S with [H]' 2 A, and IHtI > X,. Exercises 1.2 and 1.3 exhibit some very sin~pl(, properties of these symbols. The fundamental result about partitions on finite sets is the Finite Ramst.y's Theorem. I t asserts that if S is sufficierltly large, any coloring of its T-elvmellt subsets will have monochromatic sets of the prescribed size k. 1.1 The Finite Ramsey's Theorem For any positive natural numbers k . r , s there exists a natural number n such that n -+ (k):.

The readers interested primarily in infinite sets can skip the proof wittiout. harm. We first consider s = 2 and prove that for all r,p, q E N - ( 0 ) Proof. there exists n E N for which n -+ (p, g ) ; by induction on r . Let r = 1. Then it suffices to take n = p q - 1: it is clear that if 15') = 1 1 and [S]' = S = A. U A l , we cannot have both [Ao[ p - 1 and \ A 11 5 q - l . If (Ao[ 2 p let H = Ao; otherwise, let H = A l . We now assume that the statement is true for 7. (and any p. (1) and prow it for r l. We denote the least 71 for which n + (p, q); by R(p,q ; T ) . Tlle proof now proceeds by induction on (p p). Consider a set S with /S(= t~ > 0 (as yet unspecified) and a ~)r-~rtitio[~ [S]'" = A. U A I where A. n A l = 8. The statement is true if p < 7. or rl < ,.: if p < T (q < r , respectively) we car1 take as H any subset of S with \HI = 11 (IHI = g, respectively); if p = q = 7. then either A. # O and any H E A. will do, or A I f 8 and any H E A I will do. So we 7,. (1 > 7- it1lc1 tl~cl statement is true for p', g' if p' + q' < p + g. Fix a E S, let Sa = S - (a) and define a partition {Bo,B l ) of [S"]'' 11y X E B. ( B 1respectively) , if and only if { U ) U X E A. ( A l , respectively) (for X E [Sa]'). If we choose n so large that n - 1 = R ( p l ,g'; r ) (p', g' as yet unspecified) then there will be a set Ha C Sa so that either

+

+

(ii) [H a ]" C B l and (HaI 2 g'

+


N I . Clearly [BnI2 A,,, so H = B, is the desired homogeneous set. P?-oof.

>

1x1

It is not too hard to generalize this result. First, we define the transfinite sequence of beths, the cardinal numbers obtained by iterated exponentiation.

C H A P I ' E R 12. COAdBIXA'I'ORIA L SE?' 'I'HEO R \ '

224 2,4 Definition

I*= No;

I,+]= 23-; 7~= sup(^,, I u < X) for limit X # U. A more general version of the Erdos-Rado Partition Theorem can now stated. 2.5 Theorem (In)+ -. (N,);:' iiulds

107

ull

ri E

N.

Tlie proof is by iriduction or1 n, following closely the special case n = 1 in Theorem 2.3. There are an:ilogous results for ariy infinite c;irdinnl h. see Exercise 2.5. Many results of sirnilar nature, both positive :i11(1 ricbgt~tiirc~. have been obtained by researchers in the part of set theory known as infinitary cornbinatorics. One problem iri particular h a played a very iruportii~lt.r o l t l iri the developmerit of moderri set theory. It is the question of wlittt11t.r thc3t.t~ are any uncountable cardinal numbers K. for which an analogue of Ramsey's Theorem holds. 2.6 Definition All uncountable ciirdilial Ilolds for all F , s E N - { U ) .

K

is called weakly compact if

K

-+

(K)::

2.7 Theorem Weakly compact cardinals am strongly inaccessible.

PT-uof. \Ve liave to prove that a wr:iikly conipact c:arctiii;~lr; is r.egulal. a11t1 a strorig limit cardinal. ( i ) Regularity: Assume

Define a partition of ( CL,

K.

P, where X < K and 1P,I < r; ibr ;ill v < A.

=

[K]'

by:

P ) E AO if and only if thcre is v < X sucli that

E P, n r ~ t lljl E P,,:

( a ,0)E AI otherwise. Clearly (Ao, AI ) caiiriot havcb

:i

liorriogeneous set of' ctirdilltility

(iij Strong limit property: Assurne X < K 5 2" By Exercise 2.1. 2A hence K f , (X+); and, as X+ ti, ti (K);.


with functions from X into A: if F E then {F 1 U. ( CL. < X) is brarl(:h in T. Conversely, if B is a branch in T then B is a compatible systerri of' functions and F = U B E A A . We note that all branches are cofinal. (c) More generally, if T 2 A N I subsets of T , hence more than N 1 sets X C T which our construct,ion h a s to prevent from beconling :intick~:iill..;. The crux of the matter is being able to take care of more than NI sets ill h'l steps. It is here that 0 is used. Roughly speaking, we proceed in such a way that those A E W, that are maxirnal antichains in the subtree T ( " )(:onstruc.t,etl before stage a can no longer grow -- they remain maximal antichains for tilt. rest of the construction of T. 0 implies that when X is a maximal antichcziri in T , then X n T ( ~is )a maximal antichain in T ( ~and ) X n T ( ~E W,, ) for sonie cu < w l . As X n T(") subsequently does not grow, we must have X = X n T ( ' ~ ) so in particular X is a t most countable. We now proceed with the details. One minor problem arises because 0 applies to subsets of u l ,while we pIan to construct T as a subset of Wvious one for trees. An ordered set (P,5 ) satisfies the coun.table antichnin condition if rvery ;iritichain in P is at most countable. Let ti be an infinite cardinal. We can now state Martin's Axtom for K . M a r t i n ' s A x i o m MA, If (P, 5 ) is an ordered set satisfying t h c.ouilt:iblt~ ~ antichairi condition, then, for every collectiorl C of cofi~lalsubsets of P n*it.t~ ICI . IK , there exists a C-generic set. We note that MAN,,is true by Theorem 5.9?so MAN, is the first. intcrestiiig case. We prove two of its consequences here, and consider sorne further ~xarrlpl(ls in the exercises.

CHAPTER 12. COMBINATORIAL SET THEORY

2 38

5.13 Theorem MA, implies that the intersection of any collection O of 0pc.n. dense sets i n R of (01 < K is dense.

Proof. Exactly the same as the proof of Corollary 5.10, with the refereric.t~ to Theorem 5.9 replaced by a reference to MA,. P satisfies the countable antichain condition on account of Theorem 3.2 in Chapter 10. E 5.14 Corollary MA, implies 2'" > K .

Proof. For each X E R, 0, = R - { X ) is open dense. If' 2*11 < - K the11 C3 = ( 0 , I X E R) is a collection of open dense sets in R with 101 . IK , hut 0 = 0 is not dense in R.

n

In particular, MAN, implies 2N" > NI, the negation of the Continuum HIpothesis. 5.15 Theorem MAN, implies that there are no Suslin lines.

Proof. Let T be a regular S~lslintree of height wl . We let f' = { t E T 1 {(I- F r } is uncountable). It is clear that T is a subtree of T and T satisfies t h ~ condition from Definition 5.4 (although 5? need not be regular, even if T is!); in particular, if'\ N I . For each cr < w l , p ( a ) = {y E F 1 .r 5 y for some z E is cofinal in T. Let G be C-generic for C = { ~ ( a I )a < w l } . The11 G is ii cofinal branch through p. hence through T , contradicting the assumption t,hi~t T is Suslin. 1-1

T1t


NI. Thus MA can be regarded as a ger1er;ilization of the Continuum Hypothesis. R o m Theorem 5.13 we see immediately that, if MA holds, then the intersection of any collection 0 of open dense ~111)sets of R of i01 < 2"(l is dense. MA also implies that the union of less than 2'" sets of Lebesgue measure 0 has measure 0, the Lebesgue measure is K-additivtl for all K < 2'" (not just countably ;dditive}, and many other results about R. topological spaces, and sets in general. Exercises 5.1 Show that 0 is equivalent to the statement: There exists a sequencr (IVA I cr < W ] )such that, for each a < w l , W[, C_ a&.W: is at most countable, and for any f : w l -+ wl the set { a < w l I f 1 cr E W ; ) is stationary. 5.2 Assuming 0 show that there is a Suslin tree ( T ,5 ) such that the only automorphism of (T, 5 ) is the identity mapping (a rigid Suslin tree). [Hint: Imitate the proof of Theorem 5.2; a t limit stages a make sure that no automorphism h of ?'(") such that h E 2, can be extended to itn automorphism of T ( a + l ) . ]

5. COMBINATORIAL PRINCIPLES

239

5.3 Assuming 0 show that there are 2N1 mutually nun-isomorphic norinal Suslin trees. 5.4 Assuming 0 show that there exist 2N1stationary subsets of w l such that the intersection of any two of them is at most countable. [Hint: Consider Sx = { a < w l I X n a E W,) for all X C w l . ] 5.5 Show that MA, is equivalent to the statement: If (P,5 ) is an ordered set satisfying the countable antichain condition, then for every collection C of maximal antichains with [C[ j K there exists a directed set G such that, for every C E C, there exist c E C and a E G so that c < a . 5.6 Show that there exist 2N0 subsets of W such that the intersection of any two of them is finite. [Hint: For f E {0, let A! = { f [ n 1 71 E W ) C_ W < " . Of course = 1wl.j

5.7 Assuming zN"= NI show that there exist 2Nl subsets of intersection of any two of them is at most countable.

wl

such that the

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Chapter 13

Large Cardinals 1.

The Measure Problem

Theorem 2.14 in Chapter 8 shows that there exists no a-additive translation invariant measure p on the a-algebra of all subsets of R such that p([a.h ] ) = h-(]. for every interval [ a ,b] in R. This raises the quest.ion whether t.here exists any a-additive measure on P(R), or for that matter oil P ( S ) for any infiilitc st:t S . Of course, the counting measure, which allows the value p ( S ) = X . is ii trivial example, so we formulate the problem differently. We allow a rne~asureto have only finite values, and we further require (without loss of generality) that.

p(S) = l. 1.1 Definition Let S be a nonempty set. A (nontrivial probabilistic a-additive) measure on S is a function p : P(S)+ [0, l] such that (8) p(@)= 0, p ( S ) = 1. (b) If X C Y, then p ( X ) p(Y). (c) If X and Y are disjoint, then p ( X U Y) = p ( X ) + p(Y). (d) p({a)) = O for every a E S. is a collection of mutually disjoint subsets of S, then (e) Zf


we mentioned in Chapters 9 and 13. The typical pattern is that such q l ~ ~ s t , i o i ~ s (:a11 be answered one way tfisu~i~irig an appropriate large (:rtrrli~lalexists. nrltl have an opposite answer :tssuming it does not, with the answer obtained with the help of large cardinals being preferable in the sense of being much morr, "n;ttural," "profound," or "beautiful." Tile great amount of research into these ~llikttersperformed over the last 40 years has protluced thth very rich, often very subtle and difficult theory of large cardinals, whose c?sthetic:aypcal rnakihs it difficult not to believe that, it c1t:scribes true itsyttcts of the univcbrstlof st.t t1ir:ory. We cliscuss sorrie of these results very briefly in Section 3.

2.

Consistency and Independence

111orcler to understand ~ ~ i e t ~ i ~for o tsllowing ls consisteocy and i~lclepenilcnt:t~ of t110 Continuuni Hypothesis with respt.c:t t.o Zermelo-F'raenkel axiorr~sfor set tllc-)or!.. let u s first iiivcstigate a similar, but ~rluchsimpler, probleni. In Chapter 2. defined (strictly) ordered sets ;is pairs ( A , lcb. arid the classical descriptive st2t tliclory as developed I,y Nikoli~iLuzili ii1i(1 Ilis students confirttls this to art tlxtetit. For example: it is kriowtl t h i ~ till1 arlnlytic, ancl coanalytic sets are Lebesgue tneiisurable, and that all uric:ountablt? analyt,ic: sets contain cr perfect subset. As to going further, ill ZFC with t.he Axioni of' Cotistructibility one gets discouragi~~g results: there exist C; (or I l l ) sets t.hi~t ;ire not Lebesgue measurabli?, irr~dtliere exist rincountirble c:oirnalytic ( I l l ) a.ts without a perfect subset. Another important classical result is t h a t and C : sets liaw the so-(:allc>d r ~ d u c t i u nproperty and C : and IIi sets do not have it. (A class o f sets is si~itl to have the reduction property if for all A, B E r there exist A'. B' E r stt(.ll t h a t A' A , B' C B , A' U B' = A U B , arid A' fl B' = C?.) O n e rriight expect nf,C:, Il:, etc., sets to have the reduction property, and CA, U:. et(... to fail to have it. but the Axiorrl of Constructibility leaves tlie case 71 = 1 at1 odd exception and proves instead tll;~tall C!, sets for n 2 2 li:rvt~ t h r wdiictinti property and all I l k sets, n 2 2 , fail t,o liave it.

c

g.

3. THE UNIVERSE O F SET THEORY

279

Surprisingly, a more satisfactory answer can be obtained by usirlg sonle of t h e so-called large cardinal arciams. T h e inaccessible cardinal numbers clcfint.(l in Chapter 9 provide the simplest examples of large cardinals. We remind t.he reader t h a t a cardinal number K > N o is called inaccessible if it is regular and a limit of smaller cardinal numbers. I t is shown in Section 2 of Chapter 9 that the first inaccessible cardinal (assumin g it exists) must be greater than N I , N1, . . . , N,, . . . , N N l , . . . , NNxl,. . . , NHH,, . . . , etc.. hence the adjrctive "large." I t is known t h a t the existence of inaccessible cardinals cannot be proved in ZFC; in fact, the construction of a model for ZFC in which there are iio inaccessible cardinals is rather simple. If the const,ruct,ible universe L contains no inaccessible cardinals (in the sense of L), then it is sudi a model. Otherwisc. we take t h e least inaccessible cardinal 6 in L and r:otlsider a model, set,s of which are precisely the elements of Ls, and the membership relation is the usual enc. It is quite easy to use the inaccessibility of 6 ant1 prove that all axioms of ZFr are satisfied in this model. T h e fact t h a t 6 is tile Ile~st.inaccessible rarc.linii1 ill L guarantees t h a t there are no inaccessible cardinals i l l this model. So tlxiste~~ce of inaccessible cardinals is independent of ZFC. Unlike the Continuum Hypot,hesis or the Suslin's problem, however, it is not possibIe to prove that illacc(>ssible cardinals are consistent with ZFC by way of constructing an appropriate model (doing so would contradict the celebrated Second Irlconlpleteness Theorctlil of Giidel). This means that a set theory with an inac:cessible carc1in:tl is essellt.iidly stronger than plain ZFC. T h e assumption of existence of inaccessible carc1in;lls requires a "leap of faith" (similar to that requiretl for iicr:ept;incc3o f the Asin111 of Infinity), but some intuitive justification for t . h ~ plausibility o f n~iiki~ig it (.;it1 be given. I t goes roughly as follows. Matlie~natirotltl-c)~.dt.r mathematician still cannot form the "set" of all sei:orlci-order sets wllicll are llot elements of themselves; this is a task for a third-order ~nathematician.)1%now

+

+

CHAPTER 15. THE AXIOMATIC SET THEORY claim that 0, the least second-order ordinal, is an inaccessible cardinal in t h e second-order universe. Certainly 0 > No and 0 is a Iimit of (the first-order) cardinal numbers, If ( K , I L < a) is a sequence of ordinals less than 0 having length CY < 0 ,then both CY and all K , are first-order ordinals, and the arguments we used to justify the Axiom Schema of Replacement again convince us that, a first-order mathematician should be able to coIlect { K , I L < a ) into a firstorder set. But then sup,,, K , is a first-order ordinal, so sup,,, K , < 0 , ancl 0 is regular. We can conclude that the set-theoretic universe of the second-orcler mathematician satisfies the axiom "There exists an inaccessibIe cardinal." Let us now return to descriptive set theory, in particular to the questio~l whether all countable coanalytic sets have a perfect subset. Robert Solovay discovered a profound connection between this and inaccessible cardinals: If every uncountable coar~alyticset has a perfect subset, then NI is an inaccessible cardinal in the constructible universe L. (We pointed out in Section 2 t,hat thc notion of cardinality is not iibsolute, so the "real" cardinal N I need not be '.Hl in the sense of the model L" if not all sets belong t o L.) In fact, more than this is true:

(*l

For any real number a , N I is inaccessible in L[a],

where L[a] is a model constructed just as L, but starting with Lo[a] = W IJ {a) (it is the least model of set theory containing all ordinals and the real number a ) . Conversely, from the "large cardinal axiom" (*) it follows t h a t a11 uncountable II: sets have a perfect subset. In addition, (*) further implies t h a t all u~lcountableC: sets have a perfect subset (and thus cardinality 2'") arid t h a t all Cl and Ill sets are Lebesgue measurable (but it does not imply t,hr existence of perfect subsets of uncountable Ill sets, nor measurability of C:! sets). Overall, the axiom (*) produces a mild improvement upon the results obtainable from the Axiom of Constructibility. To get better results, we havcb t o assume existence of cardinals rnuch larger than mere inaccessibles. But b(:fot.t' outlining some of them, we have to make another digression. A very important role irl modern descriptive set theory is played by a certain kind of infinitary game. Let us consider a garne between two players, I and 11. described by the following rules. A finite set M of possible moves is given. 'l'lw players choose rnoves, taking turns a r ~ deach moving n times. That is, player I begins by making a move p1 E M ; player I1 responds by making a move qt E ;If. Then it is 1's turn t o make a move p2 E M , t o which I1 responds by p l a y i ~ g (12 E M , etc. T h e resulting sequence of moves ( p l ,(11 , pz, (12, - . - , Pn, (1), is ciille(1 a play. A set of plays S is specified in advance and known to both players. Player I wins if (p],.. . , qn) E S; player I1 wins if ( p l , . . , q,) # S. Many intellectual games between two players, including checkers, chess, and go. Ciul be rrlathematically represented in this abstract form by choosing suitable A f i . 7 1 . and S (some artificial conventions have to be adopted, e.g., in order to allow fur draws). A fundamental notion in game theory is t h a t of a strategy. A strategy for player I is a rule which tells him what move to make at each of his turns.

3. THE UNIVERSE O F SET THEORY

281

depending on the moves played by both players previously. If a strategy for player I has the property that, by following it, I always wins, it is called a winning strategy for I. Similariy, one can talk about a winning strategy for 11. The basic fact about games of this type is that one of the players always has a winning strategy; in other words, the game is determined. The reason is rather simple: If there is a move p1 such t h a t for every ql there is a move p2 such that for every (12 . . . there is a move p, such that for every qn the play (pl,ql,. . . , p n , qn) E S , then obviously player I has a winning strategy. If the opposite holds, it means that For every p1 there is ql such that for every p2 there is q2 such that . . . for every p, there is q, such that the play ( p l ,q l , . . . , p,, q,)

# S.

But in this case, I1 is the player with a winning strategy. We can now modify the game by allowing each player infinitely many turns. The play now is an infinite sequence of moves: (pl, ql , pz, q 2 , . - . ). The payoff set. S is a set of infinite sequences of elements of M , and I is the winner if and only if ( p l ,ql , p2, q2, . . . ) E S. The game played according t o these rules is referred to as G.?. It is no longer obvious that such games are determined, and as u matter of fact they need not be. Using the Axiom of Choice one car] construct a payoff set S M N for any M with IMI > 2 such that neither player has a winning strategy in the game Gs.(The argument is quite similar t o the one we used t o construct an uncountable set without a perfect subset in Example 4.1 1 of Chapter 10.) T h e interesting question is whether the game Gs is determined for "simple" sets S. In order to study this question, we take the finite set A d to be a natural number m; infinite sequences of elements of m can then be viewed as expansions of real numbers in base m, and the payoff set S can be viewed as a subset of R. In this way we can talk about payoff sets being Borel, analytic, etc. A profound theorem of D. Anthony Martin states that all games with Borel payoff sets are determined. The situation a t higher levels of the projective hierarchy is similar to that discussed in connection with the question of existence of perfect subsets, but the large cardinals irivolved are much larger. To be specific, the Axiom of Constructibility can be used to construct games with analytic (E:)payoff sets that are not determined. The work of Martin and Leo Harrington showed that determinacy of all games wit h C : (or, equivalently, H:) payoff sets is equivalent t o a certain large cardinal axiom. It would take us too far to try to state this axiom here (it is known technically as "For all real numbers X , X# exists."); for our purposes it suffices t o say that this axiom implies Solovay's assumption (*), and much more (for example, it implies the existence of Mahlo cardinals and weakly compact cardinals in the constructible

CHAPTER 15. THE AXIOMATIC SET THEORY

282

universe L), and is itself a consequence of the existence of measurable cardinals. So if a measurable cardinal exists therl all analytic and coanalytic games arc) determined. T h e existence of' measurable cardinals cloes riot suffice t o prove determinacy of all games with E: (or II:) payoff sets. Thc work of Martin. John Steel, and Hugh Woodin in the early nineties showed t h a t there are 1argc. i:i~rdinals(much larger than the first mvasurable orw) whosc c?xist,t:nc*einlpIic.~i tllc Axiom of Projective Deteminacy: all galnes with projective payoff sets it1.t' determined. Conversely, the Axiom of Projective Determinacy implies ~xist~erip; i.e., it implies that Hi sets have tlie reduction property and sets do not. T h e full Axiom of Projective Deterrriinacy 11% ns its consecluences the Lebcsgut. rrieasurability of all projective sets, tlie fact that every uncountable projectivcl set has a perfect subset, and the "correct" behavior of the reduction property ( ,E , , E l . . - sets have it, E:, H i , E:, n i , . . . do not). among 111;uiy others not stated here. It is the results like these that are convincing workcm it1 descriptive set theory t h a t P D rriust be true. What emerges from the above c:onsiderations is a hierarchy postu1at.ing t.tie r.xisteiice of larger and larger r:ardiniils and providing better and better approximations t o the ultimate truth about the universe of sets. Further support for this gerit:ral picture has corn(! from the work on undecidable statcArrlentsiri ;u.itllmetic. By aritllrnetic we Inearl t h e theory of Peano arithnletic whoso iixioi 11s were given in Chapter 3; it is easy to establish t h a t Peano arithmetic is equivalent t o the theory of finite sets; i.e., the theory that is obt;iint!cl frorrl ZFC t ~ j , o ~ n i t t i ~ the l g Axiom of' Infinity (ill the resulting tllt:ory, only finite sets (:all bcl proved t o exist). It has been know11 since the fundamental work of Kurt Giiclrl in 1931 that tllere are true statements about natural rlunlbers (or finite sclts) t h a t cannot be decided (eitlier proved or clisproved) from the axiorils of Pctutlo arithmetic (or the theory of fir~itcsets). T h e statements are true because tlley can be proved in ZFC, but the use! of a t least sorrie infinite sets is c~sseriti:~lfor ttie proofs. However, Godel's exmnples of such staternents are based on logic:i~l (:onsiderations (similar to Russell's Paradox) and have no transparent matliernatic:il rneaning. Iri 1977, Jeffrey Paris discoverecl the first siniple nlcitlie~i~aticxl exarnples of such statements, and others have been found subsequently. One of' the rriost interesting is Theorem 6.7 in Chapter 6, the clainl t h a t every Gooilstein sequelice terrr~ir~ates with ii vt.illue of 0 after a firlite nurnber of steps. Our proof there uses irifiriite ordinals; tile work of Paris showed that some use of

)A:

)::

3. THE UNIVERSE OF SET THEORY infinity is necessary. Another example is a version of the Finite Ramscy's T11eorem discussed in Section 1 of Chapter 12. It is possiblc! to determine the sizt. of infinite sets needed t o prove a particular statement, and it turns out that. Ilerc, too, there is a hierarchy of theories postulating the esist,erice of larger and l:irgc?r infinite sets and providing better and better approximations t o the truth about natural numbers (or finite sets). In most cases these throries are subtheories of ZFC (so few mathematicians doubt that they are t,ruc?),but there are examples of (rather complicated) staternents of arithmetic that carinot be decided even in ZFC (but can be decided, for example, in ZFC with all inaccessible c.ardinal). so the hierarchy obtained from the study of the strength of arithmetic stat,rtnetlts merges into the hierarchy of large cardinals needed for the study of the strr?ngth of statements about real numbers arising in descriptive set theory. with ZFC being just one of the stages. Even the techniques used t o prove the rcsuIts bout arithmetic are closely related t o those used to study large cardinals; they rely heavily on such concepts as partitions, trees, and games. Most of the work discussed in this section is relatively recent, atid by no means complete. Both the theory of large cardirlals and t h e study of the undecidable statements of arithmetic are very active research arec% wllere new insights and interconnections continue to be discovered. As Godel assures us in his Incornpleteness Theorem, no axiomatic theory can decide all staternents of arithmetic o r set theory. We can thus feel corifident t h a t the eltterpristb of getting closer and ctoser t o the ultimate truth about the mathelnatical universe will continue indefinitely.

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Bibliography [l]Peter Aczel. Non-well-founded sets. Stanford University Center for the

Study of Language and Information, Stanford, CA, 1988. With a foreword by Jon Barwise [K. Jon Barwise].

[2l Keith Devlin. The joy of sets. Springer-Verlag, New York, rrecond edition, 1993. Fhdamentals of contemporary set theory.

[3l F. R Drake and D. Singh. Infamediate set theory. John Wiley & Sow Ltd., Chichester, 1996.

[4] Herbert B. Enderton. Elements of set thtmy. Academic Preaa parcourt Brace Jovanovich Publishers], New York, 1977. 151 Paul Erdb, An&& Hajnal, Attila MAtC, and Richard WO. Combinatmid set theory: portition &ons for cuniinab. North-Holland Publishing Co., Amsterdam, 1984. (61 b n a l d L. Graham, Bruce L. Rothschild, and Joel H. Spencer. hmey theory. John Wiley & SOWInc., New York, second edition, 1990. A WileyIntemcience Publication.

Naive set thtmy. Springer-Verlag, New York, 1974. [7] Paul R. Halm-. Reprint of the 1960 edition, Undergraduate I?exts in Mathematics. [8] Jam= M. Henle. An outline of set themy. Springer-Verlag, New York, 1986. [g] Thomas Jech. Set t h q . Springer-Verlag, Berlin, second edition, 1997.

[l01 Winfried J w t and Martin We-. Discovering modern set theory. I. American Mathematical Society, Pddence, RI, 1996. The M c s . [l11 Akihiro Kanamori. Thc h i g h infinite. Springer-Verfag, Berlin, 1994. Large cardinals in set theory from their beginnings. [l21 Alexander S. Kechris. Classical h M i p t i u e set theory. Springer-Verlag, New York, 1995.

286

BIBLIOGRAPHY

[l31 Kenneth Kunen. Set theory. North-Holland Publishing Co., Amsterdam, 1983. An introduction to independence prooh, Reprint of the 1980 original. [l41 Yiannia N. Mtxchovakis. Descrigtiue set fheorfl. North-Holland Publishing Co.,Amsterdam, 1980. [l51 Yiannie N. M O B C h o ~ .Notes on set theory. Springer-Verlag, New York, 1994. [l61 Judith Rnitman. Introduction to modern set thsory. John Wiley & Sow Inc., New York, 1990. A Wiey-Interscience Publication. [l71 Itobert L.Vaught. Set theory. Birkh$iwr Boston Inc., Boston, MA, second edition, 1995. An introduction.

Index absolute, 273 accumulation point, 184 addition, 52,172,176 addition of cardinal numbers, 93 addition of ordinal numbers, 119 additive function, 147 aleph, 131 aleph naught No, 78 algebraic number, 101 almost diejoint functions, 213 analytic set, 278 antichain, 225, 237 antieyrnmetric relation, 33 A r o m j n tree, 228 associativity, 13, 94, 122,159, 160 asymmetric relation, 33 at m a t countable, 74 atom, 244 atomless meaaure, 244 automorphism, 60 Axiom of Anti-Foundation, 262 Axiom of Constructibility, 273 Axiom of Countable Choice, 144,152 Axiom of Existence, 7 Axiom of Extensionality, 7 Axiom of Foundation, 259 Axiom of Infinity, 41 Axiom of Pair, 9 Axiom of Power Set, 10 Axiom of Regularity, 259 Axiom of Union, 9 Axiom of Universality, 265 Axiom Schema of Comprehension, 8 Axiom Schema of Replacement,112 axiom, 3 Bake function, 200

Banach, SteEan, 149 M e , 146 beth, 223 binary operation, 55 binary relation, 19 biiimulation, 263 Bore1 set, 194 bounded sequence, 181 bounded set, 86,161 branch, 225 Brouwer-Kleene ordering, 85 Cantor set, 185 Cantor's Theorem, 96 Cantor, Georg, 1, 90,99,101 Cantor-Bernstein Theorem, 66 cardinal number, 68, 130, 140 cardiiity, 65, 68 cartesian product, 21,57 Cauchy sequence, 181 chain, 34 characteristic function, 91 choice function, 138 dosed interval, 179 closed set, 182 closed unbounded filter, 208,212 cl& unbounded set, 208, 212 closed under operation, S5 closure, 60 closure point, 145, 212 d y t i c set, 278 cohal sequence, 143 cohdity, 163 cofhite Bet, 202 Cohen, Paul, 101,139,269,275-277 collection, 9 commutativity, 13, 94

comparable, 34 compatible, 26, 237 complete linear ordering, 87 completion, 87 complex number, 179 composition, 21 computation, 48,115 concatenation, 62 condensation point, 190 condition, 275 connectivee, 5 constant, 7,57 constructible model, 271 conatructible set, 273 continuous function, 145, 182 continuum, 92 Continuum Hypotheeie, 101,141,268 convergence, 145 coordinate, 57 countable additivity, 150 countable set, 74 counting measure, 153, 204 cumulative hierarchy, 257 cut, 88 De Morgan Laws,14 decimal exparusion, 86 decoration, 261 decreclrsing eequence, 144 Dedekind cut, 88 Dedekind finite set, 97 A-lemma, 211 denae linear ordering, 83 density, 203 derived W, 191 deecriptive set theory, 278 determined, 281 diagonal intereection, 210, 212 diagoaalization, 90 diflerence, 13 digit, 174 disjoint, 14 distributivity, 14, 94, 122, 160 division, 173, 178 domain, 19 double induction, 46

element, 1 empty quence, 46 empty set, 698 equiptent, 65 equivalence dam, 30 equivalence relation, 29 ErdbDushnik-Miller Theorem, 222 Erd&M6 Theorem, 223 expansion, 178 exponentiation, 54 exponentiation of cardinals, 95 exponentition of ordinals, 122 extension, 24 extensional relation, 254 hctorial, 48 Fibonacci eequence, 50 filter, 201 finite additivity, 204 finite character, 144 finite intemction property, 180,202 finite eequence, 46 finite set, 6 9 , n h e d point, 69 forcing, 275 function, 23 function into, 24 function on, 24 function onto, 24

Giidel, Kurt, 3, 101, 269, 271, 274, 279,282,283 GCH, 165 generic set, 236 Goodstein quence, 125,126 graph, 261 greatest element, 35 greatest lower bound, 35 Hahn, Hana, 149 Hahn-Banach Theorem, 148 Hamel h i s , 146 Harringbn, h,281 Hartog number, 130 Hausdorff'e Formula, 168 hereditarily finite, 114

INDEX Hilbert, David, 101 homogeneous set, 217 ideal, 202 identity, 4 image, 20 inaaeasible cardinal, 162,168 inclusion, 12 incomparable, 34 incompatible, 237 increasing function, 105 increasing sequence, 160,181 Induction Principle, 42,44,114,260 inductive set, 40 idmum, 35 infinitary operation, 198 infinite product of carttindn, 157 infinite sequence, 46 infinite set, 69 infinite eum of cardinals, 155 initial ordinal, 129 initial segment, 104 injective function, 25 integer, 171 integer part, 174 intersection, 7, 13, 14 inverse, 20 inverse image, 20 invertible function, 25 isolated point, 184 hmorphic structures, 59 isomorphism, 36,58 isomorphiam type, 260 Jensen'e Principle, 234 Jensen, h n a l d , 274 kcomplek filter, 246 KBnig'a Lemma,227 KBnig'e Theorem, 158 least element, 35 least upper bound, 35 Lebesgue measure, 152 length, 62 lexicographic ordering, 82

limit, 160,181 limit cardinal, 162 limit point, 212 linear functional, 148 linear ordering, 34 linearly independent, 146 linearly ordered set, 34 lower bound, 35 Mahlo cardinal, 249 mapping, 23 Martin'e Axiom, 237,238 Martin, Donald Anthony, 281,282 mcurimal element, 35 measurable cardinal, 247 meaaure, 150, 204,241 measure problem, 241 minimal element, 35 model, 270 monochromatic set, 217 monotone function, 69 Mostoweki'e Lernma, 255 multiplication, 54, 172, 177 multiplication of cardinals, 94 multiplication of ordinale, 121 n-ary operation, 57 n-ary relation, 57 n-tuple, 56, 118 natural number, 41 node, 225 nondecreasing quence, 181 normal form, 124 nonnal ultra;filter,249 o n e b o n e function, 25 open interval, 179 open set, 182 operation, 7, 55 order type, 79, 113, 260 ordered n-tuple, 56 ordered field, 178 ordered pair, 17 ordered set, 33 ordering, 33 ordinal number, 107

INDEX

palindrome, 64 parameter, 5 Paris, Jeflrey, 282 partition, 31 Peano arithmetic, 54 Peano, Giuseppe, 54 perfect set, 185 power set, 10 prime ideal, 205 principal filter, 201 principal id&, 202 Principle of Dependent Choices, 144 product of an indexed eystem, 26 projection, 62 Projective Determinacy, 282 projective set, 278 proper subset, 12 P~~P&Y 1, quantifiers, 5 Ramaey's Theorem, 218,219 range, 19 rank, 255 rational number, 173 real number, 89 Recursion Theorem, 47, 115,260 reflexive relation, 29 regressive function, 209,212 regular cardind, 161 relation, 29 restriction, 24 R d l ' s Paradax, 3 R d l , Bertrand, 2 quence, 46 set, 1 set of all seta, 2 set of representatives, 32, 139 setofeets, 2 U-additive, 150, 151, 241 U-algebra, l51 a-complete filter, 246 Silver'e Theorem, 212 singleton, 17 singular cardinal, 160

Skolem-Ewenheim Theorem, 274 fhlovay, Robert, 152, 280, 281 statement, 6 Etationary set, 209,212 Steel, John, 282 strategy, 280 strict ordering, 33 strong limit cardinal, 168 structure, !X, 58 eubaequence, 181 subaet, 6,10 successor, 40, 51 successor cardinal, 162 successor ordinal, 108 eum of linear orderings, 81 supremum, 35,109 Swlin tree, 230 Swlin's Problem, 180,230,268 Suslin, Mikhail, 230 symmetric difference, 13 symmetric relation, 29 system of sets, 9

ternary relation, 22 Tichonov's Theorem, 153 total ordering, 34 transcendental number, 101 transfinite sequence, 115 transitive doare, 256 transitive relation, 29 transitive set, 107 translation invariance, 150 tree, 225 tree property, 229, 248 trivial filter, 201 trivial ideal, 202 trivia) measure, 204 W e y ' s Lemma, 144 twwdued measure, 206,244 U-limit, 206 ultrafilter, 205 unary operation, 55 unary relation, 22 unbounded set, 161 uncountable set, 90

INDEX union, 9, 13 universe, 58 unordered pair, 9 upper bound, 35 value, 23 variables, 4 Venn diagra,ma, 12 weakly compact cardinal, 224 well-founded relation, 251 well-ordered set, 44, 104 well-ordering, 44 Well-Ordering Principle, 142 winning strategy, 281 W d n , Hugh, 282

Zermelo, Ernst, 11,138,139 ZermeleFkaenkel Set Theory, 267 Zorn'e Lemma,142