http://dx.doi.org/10.1090/gsm/018
Selected Titles in This Series
18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997 15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997 14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing 10 Barry Simon, Representations of finite and compact groups, 1996 9 Dino Lorenzini, An invitation to arithmetic geometry, 1996 8 Winfried Just and Martin Weese, Discovering modern set theory. I: The basics, 1996 7 Gerald J. Janusz, Algebraic number fields, second edition, 1996 6 Jens Carsten Jantzen, Lectures on quantum groups, 1996 5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995 4 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994 3 William W. Adams and Philippe Loustaunau, An introduction to Grobner bases, 1994 2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity, 1993 1 Ethan Akin, The general topology of dynamical systems, 1993 This page intentionally left blank Discovering Modern Set Theory. II
Set-Theoretic Tools for Every Mathematician This page intentionally left blank Graduate Studies in Mathematics
Volume 18
Discovering Modern Set Theory. II
Set-Theoretic Tools for Every Mathematician
Winfried Just Martin Weese Editorial Board James E. Humphreys (Chair) David J. Saltman David Sattinger Julius L. Shaneson
1991 Mathematics Subject Classification. Primary 04-01, 03E05, 04A20.
ABSTRACT. Short but rigorous introductions to various set-theoretic techniques that have found numerous applications outside of set theory are given. Topics covered include: trees, partition cal• culus, applications of Martin's Axiom and the O-principle, closed unbounded and stationary sets, measurable cardinals, and the use of elementary submodels. This volume is aimed at advanced graduate students and mathematical researchers specializing in areas other than set theory who want to broaden their knowledge of contemporary set theory. It can be studied independently of Volume I of the same text.
Library of Congress Cataloging-in-Publication Data Just, W. (Winfried) Discovering modern set theory / Winfried Just, Martin Weese. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339; V. 8) Includes bibliographical references and index. Contents: 1. The basics ISBN 0-8218-0266-6 (v. 1 : hard cover : alk. paper) 1. Set theory. I. Weese. Martin. II. Title. III. Series. QA248.J87 1995 511.3/22-dc20 95-44663 CIP
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected]. © 1997 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 02 01 00 99 98 97 Contents
Preface ix Notation xi
Chapter 13. Filters and Ideals in Partial Orders 1 13.1. The general concept of a filter 1 13.2. Ultraproducts 8 13.3. A first look at Boolean algebras 12 Mathographical Remarks 24
Chapter 14. Trees 27 Mathographical Remarks 48 Chapter 15. A Little Ramsey Theory 49 Mathographical Remarks 65
Chapter 16. The A-System Lemma 67
Chapter 17. Applications of the Continuum Hypothesis 71 17.1. Applications to Lebesgue measure and Baire category 71 17.2. Miscellaneous applications of CH 79 Mathographical Remarks 85
Chapter 18. From the Rasiowa-Sikorski Lemma to Martin's Axiom 87 Mathographical Remarks 94
Chapter 19. Martin's Axiom 95 19.1. MA essentials 95 19.2. MA and cardinal invariants of the continuum 102 19.3. Ultrafilters on a; 110 Mathographical Remarks 116
Chapter 20. Hausdorff Gaps 117 Mathographical Remarks 122
Chapter 21. Closed Unbounded Sets and Stationary Sets 123 21.1. Closed unbounded and stationary sets of ordinals 123 21.2. Closed unbounded and stationary subsets of [X] Chapter 22. The <>-principle 139 Mathographical Remarks 146 viii CONTENTS Chapter 23. Measurable Cardinals 147 Mathographical Remarks 157 Chapter 24. Elementary Submodels 159 24.1. Elementary facts about elementary submodels 159 24.2. Applications of elementary submodels in set theory 167 Mathographical Remarks 185 Chapter 25. Boolean Algebras 187 Mathographical Remark 205 Chapter 26. Appendix: Some General Topology 207 Index 217 Index of Symbols 223 Preface This is the second volume of a graduate course in set theory. Volume I covered the basics of modern set theory and was primarily aimed at beginning graduate students. Volume II is aimed at more advanced graduate students and research mathematicians specializing in fields other than set theory. It contains short but rigorous introductions to various set-theoretic techniques that have found applica• tions outside of set theory. Although we think of Volume II as a natural continuation of Volume I,1 each volume is sufficiently self-contained to be studied separately. The main prerequisite for Volume II is a knowledge of basic naive and axiomatic set theory.2 Moreover, some knowledge of mathematical logic and general topology is indispensible for reading this volume. A minicourse in mathematical logic was given in Chapters 5 and 6 of Volume I, and we include an appendix on general topology at the end of this volume. Our terminology is fairly standard. For the benefit of those readers who learned their basic set theory from a different source than our Volume I we include a short section on somewhat idiosyncratic notations introduced in Volume I. In particular, some of the material on mathematical logic covered in Chapter 5 is briefly reviewed. The book can be used as a text in the classroom as well as for self-study. We tried to keep the length of the text moderate. This may explain the ab• sence of many a worthy theorem from this book. Our most important criterion for inclusion of an item was frequency of use outside of pure set theory. We want to em• phasize that "item" may mean either an important concept (like "equiconsistency with the existence of a measurable cardinal"), a theorem (like Ramsey's Theorem), or a proof technique (like the craft of using Martin's Axiom). Therefore, we occa• sionally illustrate a technique by proving a somewhat marginal theorem. Of course, the "frequency of use outside set theory" is based on our subjective perceptions. At the end of most chapters there are "Mathographical Remarks." Their pur• pose is to show where the material fits in the history and literature of the subject. We hope they will provide some guidance for further reading in set theory. They should not be mistaken for "scholarly remarks" though. We did not make any effort whatsoever to trace the theorems of this book to their origins. However, each of the theorems presented here can also be found in at least one of the more specialized texts reviewed in the "Mathographical Remarks." Therefore, we do not feel guilty of severing chains of historical evidence. 1This is the reason why the present volume starts with Chapter 13. 2Possible alternatives to our Volume I are such texts as: K. Devlin, The Joy of Sets. Funda• mentals of Contemporary Set Theory, Springer Verlag, 1993; A. Levy; Basic Set Theory, Springer Verlag, 1979; or J. Roitman, Introduction to Modern Set Theory, John Wiley, 1990. ix X PREFACE Much of this book is written like a dialogue between the authors and the reader. This is intended to model the practice of creative mathematical thinking, which more often than not takes on the form of an inner dialogue in a mathematician's mind. You will quickly notice that this text contains many question marks. This reflects our conviction that in the mathematical thought process it is at least as important to have a knack for asking the right questions at the right time as it is to know some of the answers. You will benefit from this format only if you do your part and actively par• ticipate in the dialogue. This means in particular: Whenever we pose a rhetorical question, pause for a moment and ponder the question before you read our answer. Sometimes we put a little more pressure on you and call our rhetorical questions EXERCISES. Not all exercises are rhetorical questions that will be answered a few lines later. Often the completion of a proof is left as an exercise. We also may ask you to supply the entire proof of an interesting theorem, or an important example. Nevertheless, we recommend that you attempt the exercises right away, especially all the easier ones. Most of the time it will be easier to digest the ensuing text if you have worked on the exercise, even if you were unable to solve it. We often make references to solutions of exercises from earlier chapters. Some• times the new material will make an old and originally quite hard exercise seem trivial, and sometimes a new question can be answered by modifying the solution to a previous problem. Therefore it is a good idea to collect your solutions and even your failed attempts at solutions in a folder where you can look them up later. The level of difficulty of our exercises varies greatly. To help the reader save time, we rated each exercise according to what we perceive as its level of difficulty. The rating system is the same as used by American movie theatres. Everybody should attempt the exercises rated G (general audience). Beginners are encouraged to also attempt exercises rated PG (parental guidance), but may sometimes want to consult their instructor for a hint. Exercises rated R (restricted) are intended for mature audiences. The X-rated problems must not be attempted by anyone easily offended or discouraged. In Chapters 17, 18, 19, and 22 we will discuss consequences of statements that are relatively consistent with, but no provable in ZFC: the Continuum Hypothesis (abbreviated CH), Martin's Axiom (abbreviated MA), and the Diamond Principle (abbreviated 0). We will write "THEOREM n.m (CH)" in order to indicate that Theorem n.m is provable in the theory ZFC + CH rather than ZFC alone. We are greatly indebted to Mary Anne Swardson of Ohio University for reading the very first draft of this book and generously applying her red pencil to it. Special thanks are due to Ewelina Skoracka-Just for her beautiful typesetting of this volume. Notation Here is a list of somewhat ideosyncratic symbols that will be used in this volume: f\A — restriction of a function / to a subset A of its domain; f[A] — image of a set A under a function /; A — symmetric difference of two sets; a — abbreviation for (ao,... , an); Now let us review the rudiments of mathematical logic that were introduced in Chapter 5. The logical symbols of a first-order language L are A, -«, 3, =, brackets, and variable symbols V{ for every i G LJ. The symbols V, —>, <-•, V are considered abbreviations. Each language L also has nonlogical symbols: A set {r; : i G /} of relational symbols, a set {fj : j G J} of functional symbols, and a set {ck : k G K} xi Xll NOTATION of constant symbols. The sets /, J, K may be empty. The arity of r; is denoted by T0(i), the arity of ft by n(j). Here are some prominent examples of first-order languages: The language LQ of group theory has no relational symbols, one functional symbol * of arity 2, and one constant symbol e. The language Ls of set theory has no functional or constant symbols, and only one relational symbol G of arity 2. Similarly, the only nonlogical symbol of the language L< is the relational symbol < of arity 2. In principle, set theory could be developed in L< and the theory of partial orders could be expressed in Ls; the difference is purely a matter of convention. Given a language L, one defines the set TerrriL of all terms of L as the smallest set T of finite strings of symbols of L that satisfies the following conditions: (i) Vi eT for all i G u>; (ii) ckeT for all k G K; (hi) If to,...,tn-i eT, j G J, and n(j) = n, then ft(t0,....,tn-i) G T. The set ForrriL of all formulas of L is the smallest set of finite strings of symbols of L that satisfies the following conditions: (i) If s, t G Term then s = t G F; (ii) If t0, ...,tn_i G Term, i G /, and ro(i) = n, then n(t0, ...,tn_i) G F; (hi) If <^ and ^ are in F, then so are ( The set A is called the universe or underlying set of the model 21. For each i G /, Ri is a relation on A of arity ro(i) (i.e., Ri ^ Aro(^); for each j £ J, Fj is a function from ATl^ into ^4; and for each k G K, Ck is an element of A. For example, if M is a set and G denotes the membership relation restricted to elements of M, then (M, G) is a model of Ls- Now let us recall what it means for a model of L to satisfy a sentence of L. A valuation (of the variables) is a function s that assigns an element of A to each natural number. Given s, we assign to each t G Term some t5 G A as follows: < = *(i); <£ = //(*0,...,*ri(j)-l) = ^j(*0»-J*ri(j)-l)- By recursion over the length of formulas we define a relation (read as: "21 satisfies a(=flt0 = *i iff tg = tf; aKn(to,...,^(0-i) iff (^...^.^Gf^; 21 |=s -> and a K ; 21 f=s 3t^ iff there exists a valuation 5* such that 21 |=s* y? and s(k) = s*(k) forallfc^i. Let (/? G Form, let a^0,..., airi_1 € A, and let 5 be a valuation. We write 21 K In particular, if Completeness Theorem, (p is a theorem of T if and only if 21 |= (^ for all models 21 of T. This page intentionally left blank This page intentionally left blank Index Fcr-set, 207 absolute formula, 168 Gs-set, 207 accumulation point, 207 Q-embedding of a tree, 41 additivity T2-space, 210 of a measure, 147 T3-space, 210 of an ideal, 74 X-principle, 144 almost contained in, 102 ^-sequence, 144 almost disjoint families, 119 <0>-principle, 141 almost disjoint family, 82, 127 217 218 26. INDEX disjoint elements, 199 complete metric, 209 disjoint subset, 199 completely metrizable space, 209 dual, 16 completion incomparable subset, 203 of a Boolean algebra, 22 /^-complete, 19 of a l.o., 19 separated subsets, 201 cone, 134 superatomic, 197 connected space, 213 trivial, 14 continuous function, 207 weakly homogeneous, 190 continuum, 213 weakly K-homogeneous, 190 convergent sequence, 209 Boolean combination, 33 countability property, 211 Boolean order, 15 countable chain condition Boolean ring, 187 in a p.o., 7 Boolean space, 18 in a topological space, 211 Borel Conjecture, 74 countable limit, 202 Borel set, 13 countable separation property, 201 bounded subset cover, 211 of an ordinal, 123 cut, 21, 192 of ww, 76 bounding number, 76 de Morgan's Laws, 16 branch, 29 Dedekind cut, 19 definability, 172 canonical coloring, 49 dense set Cantor discontinuum, 18 in a Boolaen algebra, 19 Cantor set, 18 in a l.o., 43 Cantor space, 18 in a p.o., 7 Cantor-Bendixson characteristics, 198 in a topological space, 211 Cantor-Bendixson derivative, 197, 212 density, 211 Cantor-Bendixson height, 198 derived set, 123, 207 cardinal function, 210 diagonal intersection, 125 cardinal invariant, 74 diagonal union, 127 Cauchy sequence, 209 diameter, 209 cellularity, 199 discrete space, 18 centered set, 5, 91 discrete topology, 208 character disjoint refinement, 200 of a point, 210 dominating number, 76 of a space, 211 dual ideal, 1 characteristic function, 210 clopen set, 14, 207 elementary embedding, 24, 159 closed elementary equivalence, 160 in ON, 131 elementary submodel, 159 under a function, 124 embedding, 159 closed subset elementary, 159 of a topological space, 207 Erdos cardinal, 60 of an ordinal, 123 Erdos-Dushnik-Miller Theorem, 60 of [X] normal function regressive function on an ordinal, 124 on an ordinal, 127 on [X] strongly A-independent family, 84 weakly compact cardinal, 39, 58 strongly meager set, 78 weight, 211 subcover, 211 well-met p.o., 5 subfield, 13 submodel zero-dimensional space, 210 elementary, 159 generated by a set, 132 subspace topology, 209 substructure generated by a set, 132 subtree, 27 Suslin Hypothesis, 40 Suslin line, 45 Suslin number, 211 Suslin Problem, 47 Suslin tree, 40 symmetric difference, 187 Tarski-Vaught criterion, 161 tidy formula, 33 topological space, 207 topology, 207 topology induced by a metric, 209 totally disconnected space, 18 transversal, 127 tree, 27 bushy, 44 full binary, 28 lexicographical ordering, 42 splitting, 28 tall, 46 tree property, 36 Tychonoff's Theorem, 6 Ulam matrix, 148 Ulam measurable, 153 ultrafilter in a p.o., 4 fixed, 4 free, 4 principal, 4 normal, 151, 184 on a set, 1 principal, 1 quasi-normal, 115 selective, 110 uniform, 150 ultrapower, 11 ultraproduct, 8 unbounded in ON, 131 unbounded subset of an ordinal, 123 of ULJ, 76 of [X] Vaught relation, 196 Vaught's Theorem, 196 This page intentionally left blank Index of Symbols /, 3 ©|a, 190 _L, 3, 119, 201 21 x 53, 190 n^W8 B(a), 191 G, 11 ~/, 188 -X, 13 [a, 6), 194 E°, 13 (X)a, 201 n°, 13 EUi-Ki, 209 A-(/?)», 62 c(a), 199 A^(/3)<-,62 c(B), 199 a;-(^",114 cB, 22 <*» 75 c.cc, 7, 211 V + X,78 cFr(K), 189 CM02 Xa,210 ^,123,207 x(x,X),210 Aa<7Ca, 125 X(X), 211 Ax€x£r, 135 c/j3j dT(B) 207 Va<^a, 127 Clop(X), 14 ^> 134 c/w6, 123 0°(C), 140 CLUB(j), 123 • »_141 club, 133 0 > 142 CLUB([X]<«), 133 0*, 143 co/(I)j 79 0+> 143 aw(X), 79 *, 144 0(E), 146 D, 18 21 -< 53, 159 5, 75 21 = 53, 160 dd, 209 168 A(s,t), 42 aA6, 187 cftara(.4), 209 223 224 26. INDEX OF SYMBOLS D(K), 18 Mg, 124 d(X), 211 MG, 125 m, 147 expn(«), 55 M(x,y), 209 T*, 1 A/\ 72 Fa, 1 n(A), 198 Fa, 4 nep U, 23 F(B), 4 A4, 1, 207 4 FmcoO ), 13 nw{X), 176 fip, 2 Fiz(/), 124 OCA, 64 Fia;(F), 131 a;*, 209 F «-c.c, 7, 19, 211 rd, 209 KH, 31 r^, 208 «(A), 60 rM, 176 ker(ir), 188 r|V, 209 /C(X), 3 r(<), 208 r(e), 208 /##, 195 lub, 15 C/a, 17 L(X), 211 UltB, 17 WM, 147 MA, 90 MABa, 97 vWCT, 32 MAcBa, 97 vWTY, 32 MA(K), 90 MAProperty K, 94 WCT, 32 WTY, 32 MA(T_linked, 94 MA^.centered, 94 w(X), 211 MAtop, 97 X VK/U, 11 MA~(«), 98, 132 XV/U, 24 M, 72