Selected Titles in This Series
59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Prenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable homotopy theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental groups of compact Kahler manifolds, 1996 43 James E. Humphreys, Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Freese, Jaroslav Jezek, and J. B. Nation, Free lattices, 1995 41 Hal L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, 1995 40.3 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 3, 1998 40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 2, 1995 40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 1, 1994 39 Sigurdur Helgason, Geometric analysis on symmetric spaces, 1994 38 Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, 1993 37 Leonard Lewin, Editor, Structural properties of polylogarithms, 1991 36 John B. Conway, The theory of subnormal operators, 1991 35 Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, 1990 34 Victor Isakov, Inverse source problems, 1990 33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, 1990 32 Howard Jacobowitz, An introduction to CR structures, 1990 31 Paul J. Sally, Jr. and David A. Vogan, Jr., Editors, Representation theory and harmonic analysis on semisimple Lie groups, 1989 30 Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, 1989 29 Alan L. T. Paterson, Amenability, 1988 (See the AMS catalog for earlier titles) Consequences of the Axiom of Choice Mathematical Surveys and Monographs Volume 59
Consequences of the Axiom of Choice
Paul Howard Jean E. Rubin
AttEM^ American Mathematical Society Editorial Board Georgia M. Benkart Tudor Stefan Ratiu, Chair Peter Landweber Michael Renardy
1991 Mathematics Subject Classification. Primary 03E25, 04A25.
ABSTRACT. The book, Consequences of the Axiom of Choice, contains a comprehensive listing of statements that have been proved in the last 100 years using the axiom of choice. Also included is a thorough exposition of the relationships between these statements.
Library of Congress Cataloging-in-Publication Data Howard, Paul, 1943- Consequences of the axiom of choice / Paul Howard, Jean E. Rubin. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 59) Includes bibliographical references and index. ISBN 0-8218-0977-6 (alk. paper) 1. Axiom of choice. I. Rubin, Jean E. II. Title. III. Series: Mathematical surveys and monographs ; v. 59. QA248.H67 1998 511.3-dc21 98-18622 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 reprint-permissionOams.org.
© 1998 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 03 02 01 00 99 98
Per contracts with the authors, the American Mathematical Society is acting solely as a distributor with respect to the accompanying software. All in quiries concerning the software, including copyright, technical support, or any problems arising from the use of the software, should be addressed to the authors. Any future enhancements, upgrades, and/or corrections to the software will be the responsibility of the authors. For further information, see the authors' web site at http://www.math.purdue.edu/~jer/Papers/ conseq.html. TABLE OF CONTENTS
Introduction
Part I: Numerical List of Forms
Part II: Topical List of Forms
Part III: Models
Part IV: Notes
Part V: References for Relations Between Forms
Part VI: Bibliography
Part VII: Table 1 and Table 2 (May be constructed by using mk_book.c on the enclosed disk.)
Subject Index
Author Index
Software (Inside the back cover.)
Added at posting, November 1, 2013: The original published volume included a diskette that contained companion programs and data files. For electronic access to this information, as well as changes and additions, please see this page.
vn SUBJECT INDEX
V-compact 269 Birkhoff representation theorem <* 267 14, 76 Boolean space 247 Boolean topological space 296 Aczel's realization principle 45, boundable formula 284 127 box product 261 affine function 250 branch in a tree 243, 298 Alaoglu's theorem 22, 80 Alexander's subbase lemma 22, C*-algebra 303 129, 251 Cantor-Bernstein theorem 12, algebra of type r 250, 296 48, 88, 90, 112 algebraic closure 275 Cantor complete space 247 algebraic over a field 275 cardinal representatives 40, 91 almost finite 279 Caristi's fixed point theorem 9, almost huge cardinal 242 103, 120, 134 almost maximal ideal theorem Cech complete space 247 107 central factor of a group 244 almost well orderable sets 245 chromatic number 298 amorphous set 262, 263 class forms of AC 265 analytic hierarchy 264 closed graph theorem 57, 81 antichain 253, 258, 298 co-compact space 247 antichain in a tree 243 cofinality principle 33, 116 antichain principle 40, 108, 121 Cohen models 2, 145 Artin-Schreier theorem 53, 74 coherent local lattice 248 atomless poset 310 collectionwise Hausdorff 268, 316 atoms 175 collectionwise normal 268, 316 axiom of choice 11, 91 collectionwise normality operator 268 5-lattice 239 collectionwise separated 268 Baire property 246 compact local lattice 248 Baire space 246 compactness theorem 277 Baire's classification 28, 131 compatible 258 Banach algebra 269, 303 completely normal 267 Banach space 244 completely regular 269 Banach-Tarski paradox 60, 115, concave function 250 133 cone in a linear space 239 binary mess 293 conjunctive local lattice 248
421 422 SUBJECT INDEX consistency principle 21, 100, forms true in every FM model 112 240 convex-compact subset of a topolog Fraenkel-Mostowski (FM) models ical vector space 244 145, 175 convex domination theorem 64, Frai sse's theorem 25, 113 79 frame 248 convex function 250 frame envelope 249 countable axiom of choice 17, 91 Frattini subgroup 244 countable set 2 Frechet complete space 247 covering chain 258 Frechet space 281 Cowen-Engler lemma 22, 112 free algebra in a variety 251 free complete Boolean algebra 276 Dedekind complete partially ordered free subset 243 set 250 fully ordered group 273 Dedekind finite 278 functionally separated 268 definitions of finite 278 dense subset of a quasi-order 258 denumerable set 2 Gateaux differentiable 302 diagonal intersection 241 Gauntt's condition L(n, Z) 235 directed set 249 Gelfand algebra 248 disjoint union 315 Gelfand extreme point theorem divisible group 244 27, 64, 80, 82 double uniformization 50, 102 Gelfand-Kolmogoroff theorem 48, 81, 130 e lattice 239 general linear system 265 effectively collectionwise normal generic filter 258 268 generic point 248 effectively functionally separated Gleason cover 296 268 Goodner's theorem 48, 81, 130 effectively normal 267 effectively Tietze-Urysohn 268 Hahn-Banach theorem 33, 57, elementary Abelian group 244 64, 77, 79, 82, 301 Engelers lemma 23, 100 Hajnal's free set principle 12, 111 externally disconnected 296 Hall's theorem 42, 102 Hausdorff lemma 29, 107 filter complete space 247 height of a tree 298 filter in a quasi-order 258 Henkin principle 31, 106 finite axioms of choice 235 hereditarily countable set 10, 111 FIP extension principle 22, 107 hereditarily effectively normal 268 first category set 246 hereditarily normal 267 first countable 256 forest 253 Heyting algebra 248 forms equivalent to AC in ZF 271 highly functionally separated 268 forms false in every Cohen model highly separated 268 240 highly Tietze-Urysohn 268 forms false in every FM model HOD 274 240 hypergraph 297 SUBJECT INDEX 423
implication codes 3 monotone normality operator 268 implications true in every FM model Monteiro's theorem 25, 32, 71, 229 72 inaccessible cardinal 240 Mostowski's condition D(n,Z) incompatible elements 298 235 increasing choice function 269 Mostowski's condition K(n, Z) independence results 229 235 initial set in a variety 251 multi-algebra 260 injective Abelian group 264 injective object in a category 264 injectively boundable 285 n-divisible group 244 invariant metric 281 net 250 irreducible function 296 Nielsen property 309 irreducible set in a topological space Nielsen-Schreier theorem 14, 16, 248 37, 67 nilpotent group 244 K, colorable graph 298 norm 302 K -> (/?)« 281 normal measure 242 Konig's lemma 20, 21, 104, normal space 315 119, 240 normality operator 267 Krein-Milman theorem 35, 57, nowhere dense set 246 78, 79, 81, 82, 282 Kuratowski complete space 247 u;-tree 243 Los' theorem 55, 106 onto property 286 L1 274 open problems 229 level in a tree 243 order extension principle 32, 116 level set 309 ordered linear space 239 linear global retraction 265 ordered vector space 250 linking axiom 51, 73, 108 ordering principle 29, 115 local lattice 248 locally convex 281 locally countable 316 7r-base 310 locally finite 316 p-group 244 locally finite tree 243 para-Lindelof 316 Lowenheim-Skolem theorem 30, paracompact 257, 316 106 partially ordered group 273 Lowig's theorem 40, 78 partition principle 41, 89 partition relation 241 Martin's axiom 18, 118 patching lemma 24, 101 meager set 246 Peano algebra 296 measurable cardinal 240 perfect set 246 measure extension theorem 60, 73 permutation models 2, 145 measure on a Boolean algebra 318 point countable 316 Mersenne prime conjecture 238 point finite 316 meta-Lindelof 316 positive cone 273 metacompact 256, 316 pre-order 249 424 SUBJECT INDEX principal ideal domain 275 Sikorski's extension theorem 32, projective Abelian group 264 60, 72, 73 projective hierarchy 264 similarity type 250 projective limit 269 sober topological space 248, 252, projective object in a category 269 264 software description 5 projective set 264 Souslin space 256 projective system 269 spatial lattice 248 pseudo Ka minimal 262 special p-group 244 pseudo-intersection 310 Stone-Cech compactification theo rem 21, 129 quasi-cofinal 253 Stone representation theorem 21, quasi-order 258 71 strong antichain 259, 298 Rado's selection lemma 22, 23, strong finite intersection property 24, 40, 41, 91, 100, 101, 129 (sfip) 310 ramification lemma 12, 106, 112 ramification system 243 strongly compact cardinal 241 ramified set 253 strongly Dedekind finite 279 Ramsey cardinal 242 strongly incompatible 258 Ramsey's theorem 28, 61, 113, subhypergraph 297 115, 258 sublinear function 250, 301 Rasiowa-Sikorski axiom 62, 73, subnet 250 110 subtree 243 Rasiowa-Sikorski property 62, sup function 252 71, 270 supercompact cardinal 242 real closed field 275 superlinear function 302 regular local lattice 248 support theorem 64, 79 regular measure 242 supports 175 regular ring 248 regular ultrafilter 242 supremal set 252 Riesz theorem 9, 80 surjectively boundable 285 Robinson's valuation lemma 23, 100 T-finite 278 Rowbottom cardinal 242 T space 315 table 1, description 4 cr-compact 256 table 2, description 4 cr-discrete 316 Tarski's lemma 31, 72 a- disjoint 316 term-choice sentence 286 (j-extremally disconnected 296 Tietze-Urysohn extension theorem cr-point finite 316 sandwich theorem 64, 80 315 scattered space 247 Tietze-Urysohn operator 268 second countable 256 Tietze-Urysohn space 268 seminorm 302 Tietze-Urysohn theorem 65, 128 set mapping 243 torsion free group 244, 273 shrinkable 316 transfer theorems 284 SUBJECT INDEX 425 transferable 175 van Douwen's choice principle 44, transferable forms 239 93 tree 243, 253, 298 variety 251 Turinici's fixed point theorem 11, vector lattice 239 14, 103 vertex cover 297 two valued measure 318 Vitali equivalence classes 58, 133 Tychonoff's theorem 20, 21, 22, 42, 47, 122, 123 W-set 245 type of an algebra 260 weak* topology 244 weakly KQ, minimal 262 U space 315 weakly compact cardinal 241 ultra metric group 257 weakly Lindelof 257 ultrafilter choice function 269 wedge 239 ultranet 250 uniformly convex 302 wedge in a linear space 239 unique factorization domain 275 universal net 250 Z-valued Banach limit 250 upper semi-continuous 252 Zermelo-Fraenkel set theory 1 urelements 175 Zermelo's fixed point theorem 9, Urysohn operator 268 103, 119 Urysohn's lemma 38, 126, 315 ZF° 1 AUTHOR INDEX
Abian 9, 12, 26, 57, 75, 103, 157, 161, 175, 176, 177, 181, 104, 119, 120 189, 190, 191, 193, 198, 199, Andreka/Nemeti 10, 13, 75, 204, 206, 208, 210, 212, 235, 260 236, 245, 250, 252, 258, 260, Apter 153, 242, 258, 327, 345 263, 264, 271, 285, 298, 299, Apter/Henle 242 312, 321, 322, 324, 325, 326, Armbrust 36, 67 327, 330, 335, 336, 338, 344, Ash 54, 68, 133, 341, 368 345, 348, 349, 351, 358, 359, 361, 368 Bacsich 25, 32, 71, 72 Bleicher 40, 42, 52 , 78, 97, Baker 57, 104 186, 299, 313, 325, 336 Banach/Tarski 59, 115, 133 Borel 28, 85, 327 Banaschewski 21, 22, 23, 32, Brunner 9, 10, 12, 17, 20, 39, 57, 69, 70, 72, 74, 80, 21, 28, 30, 31, 35, 36, 40, 107, 130, 131, 248, 298, 321, 41, 42, 44, 45, 46, 47, 48, 336, 364 49, 58, 60, 80, 81, 85, 87, Banaschewski/Harting 21, 69, 88, 95, 96, 97, 98, 99, 106, 107 108, 111, 112, 113, 117, 119, Banaschewski/Moore 12, 30, 120, 121, 122, 123, 124, 125, 48, 82, 88, 89, 90, 112, 132, 126, 127, 129, 130, 131, 132, 321, 322, 324, 326, 328, 329, 134, 146, 147, 148, 149, 151, 343 154, 155, 156, 158, 159, 160, Bell 13, 32, 33, 57, 58, 72, 161, 162, 163, 164, 168, 169, 108, 125, 127, 129, 130, 131, 170, 174, 176, 177, 178, 181, 146, 296, 298, 321, 332, 375 182, 183, 184, 185, 186, 189, Bell/Fremlin 13, 69, 106 190, 191, 193, 198, 199, 200, Bell/Jellett 282, 335 201, 202, 206, 208, 210, 212, Benejam 25, 113 214, 217, 218, 219, 229, 238, Benthem 24, 76, 96, 113 239, 245, 246, 254, 255, 256, Bernstein 48, 82, 132 257, 261, 266, 271, 280, 281, Birkhoff 14, 42, 76, 128, 345 298, 299, 300, 319, 321, 322, Blair 31, 121, 134 324, 325, 326, 327, 328, 329, Blass 13, 16, 21, 22, 31, 41, 330, 334, 335, 336, 338, 339, 42, 45, 46, 48, 50, 51, 68, 340, 344, 345, 346, 347, 348, 69, 74, 76, 77, 78, 82, 89, 349, 350, 351, 352, 353, 354, 95, 96, 99, 107, 109, 112, 355, 356, 357, 360, 368, 374, 113, 114, 119, 122, 146, 152, 375, 381
427 428 AUTHOR INDEX
Brunner/Howard 44, 47, 86, Feferman 148, 157, 265, 321, 114, 193, 195, 208, 246, 323, 334, 337 328, 346, 353, 354 Feferman/Levy 28, 30, 82, Brunner/Howard/Rubin 305 131, 132, 154, 327, 329 Brunner/Rubin 230 Feigner 1, 3, 4, 29, 39, 108, Bull 50, 59, 90, 170, 241, 115, 145, 146, 148, 150, 156, 242, 360, 377 159, 160, 177, 198, 235, 265, Buskes 301, 303 319, 326, 328, 330, 332, 334, 340, 364 Cantor 19, 29, 86, 87, 328 Felgner/Flannagan 265 Caristi 9, 103, 120, 134 Feigner/Jech 40, 116, 177, Chandler 258 184, 271, 321, 336, 340, 360 Cohen 145, 146, 150, 152, Feigner/Truss 220, 326 154, 321, 324, 330, 331, 338 Figura 51, 83, 86, 90, 167, Conway 235, 237 230, 329, 361, 362 Cowen 21, 22, 23, 24, 25, Flannagan 265 26, 27, 39, 46, 100, 101, 102, Fleischer 265 104, 105, 113, 274, 297, 351 Foreman/Wehrung 341 Cowen/Emerson 26, 57, 74 Forti/Honsell 41, 89, 121, 155, 344 David 49, 114, 146, 167, 245, Fossy/Morillon 32, 34, 41, 42, 322, 352, 358 47, 77, 78, 81, 95, 96, 119, Davies 50, 132, 360 121, 130, 248, 325, 326, 332, Dawson/Howard 50, 84, 177, 344, 354 183, 204, 265, 360 Fourman 161 Day 239 Fraenkel 145, 177, 178, 179, De la Cruz/Di Prisco 19, 64, 180, 319, 320, 339 98, 99, 118, 177, 202, 218, Fraleigh 367 335, 351, 386 Dedekind 18, 87, 324 Freyd 49, 93, 161, 358 Degen 59, 69, 186, 305, 376, Fuchs 49, 68 377 Derrick/Drake 154, 333 Gaifman 10, 71 Diel 278, 279 Galvin/Komjath 14, 104, 297 Diener 13, 14, 18, 36, 75, Gardiner 24, 70, 77 76, 115, 261, 296 Gauntt 235, 236, 330 Douwen 42, 43, 44, 46, 93, Gelfand/Kolmogoroff 47, 81, 116, 122, 128, 181, 182, 190, 130 191, 298, 345, 346 Gitik 28, 42, 49, 59, 82, 83, Drake 11, 59, 90, 151, 153, 158, 261, 327, 344, 345, 358, 240, 260 377 Gleason 58, 131, 375 Easton 154, 155, 161 Goldblatt 31, 41, 72, 106, Ellentuck 257 119, 121, 134, 303, 304, 330, Erdos/Rado 11, 56, 111, 114 344 Gonzalez 57, 86, 91, 118, Federer/Jonsson 62, 69, 309 152, 183, 328 AUTHOR INDEX 429
Good/Tree 30, 44, 61, 128, Howard 1, 11, 14, 16, 22, 24, 129, 135, 146, 152, 232, 291, 40, 41, 54, 59, 62, 67, 69, 380, 381 73, 85, 101, 102, 104, 106, Good/Tree/Watson 220, 221 110, 146, 157, 162, 163, 164, Goodner 47, 81, 130 178, 179, 183, 184, 190, 191, Gratzer 14, 15, 76, 260 193, 196, 197, 204, 206, 210, Gross 53, 79, 215, 364 211, 212, 217, 218, 229, 230, 298, 299, 321, 322, 323, 326, 327, 328, 334, 339, 361, 368, Haddad/Morillon 146, 321, 377, 382 324, 325, 326, 335, 338, 368 Howard/Keremedis/Rubin/Rubin Hales 10, 71 16, 36, 37, 38, 45, 64, 96, Hall 42, 102, 345 124, 126, 127, 135, 183, 221, Halpern 11, 44, 77, 108, 177, 340, 342, 346, 347, 348, 353, 183, 313, 321, 324, 346 355, 356, 361, 368, 373, 375, Halpern/Howard 2, 17, 84, 376, 384, 386, 387 152, 188, 321, 323, 324 Howard/Keremedis/Rubin/Stanley Halpern/Levy 3, 146, 321, 15, 16, 37, 38, 48, 49, 53, 324, 325, 335, 368 64, 134, 135, 136, 177, 221, Hamel 35, 62, 63, 74, 78, 79, 336, 357, 364, 387 133, 335, 385 Howard/Rubin 42, 57, 96, 97, Harper/Rubin 33, 39, 46, 51, 176, 177, 184, 185, 186, 187, 54, 55, 108, 109, 110, 177, 190, 191, 196, 197, 206, 210, 253, 265, 329, 332, 352, 361, 212, 216, 229, 271, 330, 340, 368, 369 345, 374 Hausdorff 327 Howard/A. Rubin/J. Rubin 60, Haussler 2, 17, 48, 84, 89, 97, 265, 311 190, 191, 324 Howard/H. Rubin /J. Rubin 206, Henkin 21, 25, 27, 105 369 Hickin/Plotkin 24, 101, 274 Howard/Solski 285 Hickman 2, 10, 17, 49, 50, Howard/Spisiak 56, 88, 279, 52, 53, 54, 68, 74, 83, 88, 89, 90, 117, 118, 177, 178, 370 200, 201, 203, 206, 214, 216, Howard/Yorke 2, 13, 14, 17, 257, 258, 262, 263, 278, 323, 18, 19, 35, 38, 58, 67, 69, 358, 359, 360, 363, 365, 366 86, 87, 88, 94, 106, 108, 110, Higasikawa 30, 40, 41, 61, 183, 185, 186, 199, 278, 279, 62, 86, 89, 90, 91, 189, 190, 280, 339, 372, 376 191, 299, 329, 382 Hilbert 38, 116, 132, 338 Jech 1, 8, 9, 10, 11, 18, 19, Hodges 11, 49, 54, 56, 68, 21, 22, 27, 28, 29, 30, 34, 74, 75, 106, 109, 110, 160, 35, 37, 38, 39, 40, 41, 46, 161, 177, 205, 258, 358, 367, 48, 59, 67, 71, 74, 78, 82, 370 83, 84, 87, 88, 89, 90, 91, Hoft/Howard 31, 49, 59, 76, 92, 93, 100, 105, 107, 108, 77, 104, 118, 188, 209, 216, 111, 112, 113, 114, 116, 121, 217, 260, 330, 338, 339, 340, 122, 131, 132, 134, 145, 146, 357, 359, 377 149, 150, 151, 152, 154, 175, 430 AUTHOR INDEX
176, 177, 178, 181, 183, 184, Kolany/Wojtylak 10, 55, 56, 186, 187, 189, 190, 191, 192, 102, 105, 106, 277, 330, 334, 193, 194, 195, 198, 217, 233, 370 234, 235, 254, 259, 271, 319, Krom 20, 122, 183, 184, 197, 320, 321, 323, 324, 325, 326, 210, 212, 217, 218, 298, 345 328, 329, 330, 331, 332, 333, Krull 16, 74, 107 334, 335, 336, 337, 338, 339, Kruse 10, 115, 258 340, 341, 342, 343, 344, 345, Kunen 18, 118, 145, 267 351, 353, 364, 368 Kurepa 39, 108, 121, 340 Jech/Sochor 175, 177, 187, 192, 284, 319, 320, 333, 335, 336, 338, 341, 343, 351, 353, Lampe 39, 55, 71, 76, 77, 362, 364 91, 103, 370 Jensen 162, 170, 198, 233, Lang 255 330 Lauchli 21, 47, 51, 54, 68, Johnston 161 74, 104, 130, 176, 177, 183, Johnstone 22, 80, 122, 161, 187, 192, 205, 321, 328, 336, 252 338, 343, 355, 356, 362, 364, 366 Lebesgue 327 Kanamori/Magador 151, 153 Levy 12, 30, 32, 35, 37, 38, Kaniewski/Rogers 49, 102, 39, 82, 87, 92, 93, 94, 95, 114, 341 116, 145, 177, 183, 184, 185, Kanovei 10, 50, 58, 88, 94, 186, 194, 195, 233, 278, 279, 133, 151, 162, 169, 170, 264, 319, 320, 321, 324, 330, 334, 265, 360, 376 335, 336, 337, 339, 340, 343, Kasahara 13, 103, 120 352 Keisler 47, 121, 245 Lewis 22, 122, 129 Keisler/Tarski 42, 90, 345 Lindelof 20, 131, 325 Kelley 14, 22, 122, 129, 210, Lindenbaum/Tarski 9, 19, 41, 291, 315, 324, 386 83, 84, 89, 341, 343, 385 Kenyon 57, 104 Litman 54, 83, 329, 367 Keremedis 15, 16, 18, 32, Lolli 20, 105, 260 36, 44, 45, 60, 61, 62, 67, Los/Ryll-Nardzewski 21, 22, 69, 71, 77, 79, 81, 91, 95, 33, 77, 100, 122, 332 96, 97, 98, 99, 106, 107, 113, Lowig 40, 78 120, 123, 124, 125, 126, 127, Luxemburg 33, 34, 72, 73, 129, 130, 179, 186, 271, 272, 81, 110, 111, 332 298, 299, 310, 328, 372, 380, Luzin/Sierpihski 63, 134, 312 383 Kharazishvili 50, 78, 265, 341 Kinna/Wagner 27, 60, 90, 96, MacLane 161 97, 326 Makowski/Wisniewski 57, 104, Kleinberg 46, 99, 326, 351 199, 371 Kleinberg/Seiferas 56, 114, Malitz 258 281, 282, 371 Manka 9, 11, 12, 14, 33, Knight 49, 123 103, 108, 116, 119, 120, 134, Kolany 56, 102 252 AUTHOR INDEX 431
Mathias 8, 34, 50, 51, 53, 116, 118, 119, 121, 147, 148, 74, 82, 84, 87, 89, 90, 94, 149, 150, 163, 164, 166, 169, 95, 114, 120, 133, 150, 151, 171, 172, 173, 174, 175, 177, 154, 184, 229, 230, 235, 321, 178, 180, 182, 183, 184, 187, 322, 323, 324, 325, 326, 332, 194, 195, 202, 207, 210, 212, 333, 337, 364 218, 229, 233, 265, 270, 271, McCarten 15, 123, 290, 291 276, 282, 284, 285, 286, 287, Mirsky 341 298, 319, 320, 321, 323, 324, Monro 17, 19, 38, 48, 56, 325, 326, 328, 329, 330, 331, 84, 87, 88, 89, 91, 98, 146, 332, 334, 337, 339, 341, 343, 151, 157, 159, 167, 191, 257, 344, 347, 349, 351, 352, 353, 258, 276, 322, 323, 334, 339, 355, 356, 360, 362, 363, 370, 370, 375 372, 375 Monteiro 25, 32, 71, 72 Pineus/Solovay 52, 111, 163, G. Moore 8, 17, 19, 20, 28, 164, 229, 361, 363 29, 30, 34, 41, 62, 84, 85, Plotkin 49, 54, 68, 73, 160, 87, 89, 92, 106, 121, 128, 166, 275, 357, 367, 368 131, 132, 133, 293, 321, 323, Pogorzelski/Prucnal 25, 105 324, 325, 326, 327, 328, 329, Prikry 53, 114, 151, 364 330, 334, 341, 385 R. L. Moore 344 Morillon 26, 27, 32, 43, 44, Rado 40, 101, 343 52, 60, 61, 63, 70, 71, 73, Raisonnier 48, 117, 132, 151, 74, 80, 82, 91, 92, 93, 102, 341 107, 110, 111, 113, 116, 117, Ramsey 28, 60, 113, 115, 326 121, 123, 125, 128, 130, 131, 162, 163, 167, 267, 296, 299, Rav 15, 23, 24, 41, 52, 53, 326, 329, 330, 334, 337, 346, 68, 74, 75, 76, 100, 101, 102, 380, 381, 382, 385 109, 118, 129, 272, 273, 274, Morris 12, 51, 57, 83, 115, 343, 346, 363, 364 118, 156, 163, 169, 196, 197, Rimscha 49, 91, 259, 260, 210, 212, 217, 298, 299, 327, 324, 325 332, 362, 374 Roy den 35, 78, 81 Mostowski 32, 38, 92, 173, A. Rubin 189, 277, 292, 322, 179, 180, 181, 183, 199, 233, 323 235, 236, 270, 271, 321, 326, H. Rubin 271 329, 330, 331, 339 J. Rubin 359 Mycielski 21, 39, 105, 122 A. Rubin/J. Rubin 258 Myhill/Scott 274 H. Rubin/ J. Rubin 1, 3, 8, 15, 24, 33, 34, 35, 40, 42, Paseka 26, 70, 71, 248 52, 57, 58, 70, 73, 77, 78, Pawlikowski 376 79, 81, 82, 108, 110, 116, Pelc 30, 37, 51, 82, 89, 90, 124, 186, 265, 271, 280, 298, 251, 321, 322, 329, 330, 343, 321, 335, 340, 363 361 Pincus 20, 25, 27, 28, 29, Rubin/Scott 21, 122, 129 37, 51, 52, 57, 71, 85, 86, Rudin 47, 56, 81 88, 91, 94, 97, 110, 111, 115, Russell 29, 82 432 AUTHOR INDEX
Sacks 162, 163 Tarski 2, 9, 10, 11, 16, 17, Sageev 17, 91, 146, 152, 165, 19, 25, 30, 31, 32, 37, 38, 183, 229, 257, 321, 323, 324 71, 74, 83, 84, 87, 88, 89, Schechter 14, 22, 34, 37, 38, 94, 108, 110, 116, 120, 278, 40, 63, 78, 79, 80, 92, 101, 279, 323, 324, 330, 332, 337, 107, 112, 120, 122, 132, 249, 339 337 Taskovic 14, 103, 104 Schnare 15, 106, 123, 290, Truss 9, 10, 11, 13, 29, 32, 291 44, 46, 51, 54, 56, 57, 83, Schoenflies 17, 89, 324 84, 88, 89, 90, 92, 93, 94, Schrijver 23, 50, 51, 73, 108, 103, 114, 119, 122, 131, 132, 109, 113, 114, 340, 361 133, 148, 149, 151, 154, 155, Schroder 29, 115, 328 189, 197, 201, 208, 235, 236, Scott 21, 74 258, 263, 265, 271, 275, 278, Shannon 1, 10, 12, 18, 20, 279, 282, 321, 322, 323, 328, 39, 44, 48, 85, 86, 118, 119, 329, 330, 331, 334, 335, 337, 186, 258, 259, 298, 325, 339 340, 346, 351, 361, 363, 368, Shelah 151, 158, 168, 341, 370, 371 351 Tsukada 53, 86, 197, 333, Shoenfield 146 334, 364 Sierpinski 17, 28, 29, 30, 33, 38, 40, 41, 48, 62, 63, 83, Urysohn 38, 64, 126, 128, 338 89, 91, 92, 103, 110, 111, 116, 131, 132, 133, 323, 324, 326, 328, 329, 332, 341, 343 Vickers 248 Sikorski 32, 72 Vopenka 321 Smith 13, 112 Smullyan 57, 110, 146, 372 Wagon 59, 69, 73, 111, 151, Solovay 145, 151, 158, 162, 164, 352, 363, 376 163, 168, 281, 332, 341, 351, Whitehead 17, 112, 323 363, 371 Wisniewski 57, 94, 235, 238, Solovay/Reinhardt/Kanamori 371 151, 153 Wojtylak 60, 106, 340, 379 Specker 177 Wolk 22, 31, 39, 108, 120, Spector 31, 32, 106, 113, 121, 122, 129 120, 258, 261 Wright 56, 81, 135, 151, 281, Spisiak 278, 279 371 Spisiak/Vojtas 278, 279 W§glorz 219 Stavi 10, 55, 71, 73, 177, 369 Stern 52, 133, 151, 158, 351, 363 Young 20, 131, 325 Stewart 276 A. H. Stone 46, 122 Zermelo 9, 103, 119 M. Stone 21, 71 Zuckerman 49, 96, 179, 235, Szczepaniak 53, 114, 165, 364 238, 261, 330, 331, 340