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Home , Dana Scott, Solomon Feferman, William Lawvere

http://dx.doi.org/10.1090/pspum/013.2

AXIOMATIC PROCEEDINGS OF SYMPOSIA IN PURE

VOLUME XIII, PART II

AXIOMATIC SET THEORY

AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1974 PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

HELD AT THE UNIVERSITY OF CALIFORNIA LOS ANGELES, CALIFORNIA JULY 10-AUGUST 5, 1967

EDITED BY THOMAS J. JECH

Prepared by the American Mathematical Society with the partial support of National Science Foundation Grant GP-6698

Library of Congress Cataloging in Publication Data Symposium in Pure Mathematics, University of California, DDE Los Angeles, 1967. Axiomatic set theory. The papers in pt. 1 of the proceedings represent revised and generally more detailed versions of the lec• tures. Pt. 2 edited by T. J. Jech. Includes bibliographical references. 1. Axiomatic set theory-Congresses. I. Scott, Dana S. ed. II. Jech, Thomas J., ed. HI. Title. IV. Series. QA248.S95 1967 51l'.3 78-125172 ISBN 0-8218-0246-1 (v. 2)

AMS (MOS) subject classifications (1970). Primary 02K99; Secondary 04-00

Copyright © 1974 by the American Mathematical Society

Printed in the United States of America

All rights reserved except those granted to the United States Government. This book may not be reproduced in any form without the permission of the publishers. CONTENTS

Foreword vii Current problems in descriptive set theory 1 J. W. ADDISON Predicatively reducible systems of set theory 11 Elementary embeddings of models of set-theory and certain subtheories 3 3 HAIM GAIFMAN Set-theoretic functions for elementary syntax 103 R. O. GANDY Second-order cardinal characterizability 127 STEPHEN J. GARLAND The consistency of partial set theory without extensionality 147 P. C. GlLMORE On the existence of certain cofinal subsets of "(D 155 STEPHEN H. HECHLER Measurable cardinals and the GCH 175 RONALD BJORN JENSEN The order extension principle 179 A. R. D. MATHIAS "Embedding classical in 'intuitionistic' type theory." A correction 185 JOHN MYHILL Remarks on reflection principles, large cardinals, and elementary embeddings 189 W. N. REINHARDT Axiomatizing set theory 207 Author Index 215 Subject Index 217 Lectures delivered during the Institute 219 FOREWORD

This volume is the second (and last) part of the Proceedings of the Summer Institute on Axiomatic Set Theory held at U.C.L.A., July 10—August 5, 1967. Many of the lectures delivered during the Institute have been published in the first volume of these PROCEEDINGS, edited by Dana S. Scott. Although we were unable to obtain all the remaining manuscripts, this volume contains most of them. A small number of the contributions was meanwhile published elsewhere; the complete list of lectures is provided at the end of this volume. For several reasons, the publication of this volume was slightly delayed. I wish to thank the authors of the papers for their patience.

THOMAS J. JECH Author Index

Roman numbers refer to pages on which a reference is made to an author or a work of an author. Italic numbers refer to pages on which a complete reference to a work by the author is given. Boldface numbers indicate the first page of the articles in the book.

Ackermann, W., 192, 205 Hadamard, Jaques, 1 Addison, J. W., 1, 10, 128, 146 Hanf, W. P., 71, 101, 145, 146, 205 Aleksandrov, Pavel S., 5 Hausdorff, Felix, 2, 3, 8, 155, 173 Asser, G., 135, 146 Hechler, Stephen H., 155 Hinman, Peter G., 7 Baire, Rene, 1, 2, 3 Barnes, Robert F., Jr., 4 Jech, T., 183, 183, 198, 205 Barwise, J., 103, 126 Jensen, Roland Bjorn, 103, 104, 106, 108, 109, Bennett, J. H., 122, 126, 135, 146 126, 175, 179, 183 Benson, Guy M., 5 Johsson, B., 179, 183 Bernays, P., 195, 205 Borel, Emile, 1, 2, 3, 4, 8, 9 Kalmar, Laszlo, 4, 8, 9 Brouwer, Luitzen Egbertus Jan, 4 Kantorovic, Leonid, 6 Karp, C, 103, 108, 126 Cantor, Georg, 2 Keisler, H. J., 68, 84, 96, 100, 101 Cohen, Paul J., 10, 100, 144, 146, 155, 159, 161, Kleene, Stephen C, 4, 137, 138, 146 165, 173 Kolmogorov, Andrei N., 7, 8 Coppleston, F., 198, 205 Kreisel, G„ 17, 21, 30, 31, 32, 32, 125, 126 Kripke, S., 30, 32 Kunen, K., 37, 38, 65, 71, 72, 76, 92, 95, 96, Devlin, K. J., 104, 126 101, 131, 144, 146, 200, 201, 205

Feferman, Solomon, 11, 11, 12, 16, 17, 20, 21, La Vallee Poussin, Charles J. de, 2 30, 32,32, 116, 125, 126 Lavrent'ev, Mihail A., 3 Fitch, F. B., 149, 153 Lebesgue, Henri, 1, 3 Fraenkel, Abraham A., 8, 9 Levy, A., 103, 108, 126, 141, 146, 192, 194, 196, Friedman, H., 32, 32 205 Livenson, E. M., 6 Gaifman, Haim, 33, 36, 37, 42, 46, 95, 96, 100, Luzin, Nikolai N., 3, 4, 5, 6, 8 101 Lyndon, Roger C, 148, 149, 150, 153 Gandy, R. O., 30, 31, 32, 103, 126 Garland, Stephen J., 127 Magidor, M., 86, 88, 94, 101 Gilmore, P. C, 147, 147, 150, 153 Martin, Donald Anthony, 10 Godel, K., 10, 111, 126, 186, 187, 188 Mathias, A. R. D., 175, 179 Grzegorczyk, A., 104, 126 Montague, R. M., 143, 146

215 216 AUTHOR INDEX

Moschovakis, Yiannis N., 6, 10, 126, 128, 146 Sierpinski, Waclaw, 155, 173 Mostowski, Andrzej, 181, 183 Silver, J. H., 9, 36, 72, 101, 145, 146, 198, 205 Myhill, John, 185, 185 Smullyan, R. M., 104, 122, 126 Sochor, A., 183, 183 Pincus, D., 183, 183 Solovay, Robert M., 35, 78, 86, 94, 101, 155, Platek, R., 30, 32 173, 189 Powell, W. C, 35, 91, 94, 101, 198, 205 Suslin, Mihail Ya., 5, 6, 7, 8 Suzuki, Y., 137, 146 Reinhardt, W. N., 35, 78, 86, 94, 101, 189, 192, 198, 200, 205 Tarski, Alfred, 68, 96, 100, 101 Ritchie, R. W., 121, 126 Tennenbaum, Stanley, 155, 173 Rogers, Hartley, Jr., 2 Rowbottom, F., 36, 101 Urysohn, P. S., 5 Scholz, H., 135, 146 Scott, Dana S., 13, 71, 95, 101, 145, 146, 200, 205, 207 Vaught, R. L., 42, 101, 205 Selivanovskil, E. A., 5 Shoenfield, J. R., 71, 74, 101, 132, 136, 137, 146, Zermelo, Ernst, 8, 9 198, 205 Zykov, A. A., 130, 146 Subject Index

AJ, 128 A0 predicates, 105, 106, 108, 114-117 a-scale, 155 A0-separation, 12 Alternating sum, 3 Definability, 36, 39 Antitone, 3 general concept, 41, 42 Approximations of embeddings (y-approxima- Definable, 1 tions, < ^-approximations), 34, 78-83, 86, Defining schema, 39, 40, 41, 42 87 Descriptive set theory, 1 Axiom(s) Detail, 4 Vv la (Ra exists Av e Ra], 33, 54 OJ,128 of constructibility, 9 Direct limit, 36, 41, 46, 47 of definable determinateness, 10 Duals, 3 of extendibility, 199 of infinity, 9, 189, 199 Embeddings of measurable cardinals, 9 of intuitionistic type theory, 185 recognizing, 204-205 ordinals and the first ordinal move by, 34, 35, 67, 75, 85-87 Basic closure, 109 critical ordinals of, 67 Basic functions, 105 see also: Approximations of embeddings, Co- Basic numerals, 118 final embeddings, and "Local" conditions Bilateral, 3 Exhaustion principle, 5 Blowing up small structures, 72-75 VJ, 128 Borel hierarchy, 2 Extendible, 192, 197, 199, 202 Cardinals in models of ZC+, 95 Extendible cardinals, 35, 94, 197, 199 Cardinals, see Large cardinals various concepts of extendibility, complete Cartesian product, 105, 111 extendibility, 86, 87, 94, 95 (of structures), 40, 41 Extensionality axiom, 147 Characterizable cardinals, 128 Extension operators, 36, 39-42, 95-99 Classes (as distinguished from sets), 37, 50-54 Iterations of—see Iterations of extension Classical descriptive set theory, 2 operators Cofinal embedding, 33, 34, 36, 54-60, 79, 90, 91 y and < y-cofinality, 34, 35, 78-84 First separation principles, 7 Consistency results, 144, 147 First separation property, 7 Constructible closure, 111 , 156 Construction principle, 3 Functor, 41, 45, 49 Coreduction property, 7 Fusion, 5

S-separated, 8 General recursive functions, 2 Decomposition of embeddings, 34, 82, 83 Generalized , 131 Sm, 136 Godel embedding, 185

217 218 SUBJECT INDEX

Hausdorff hierarchy, 3 Predicatively reducible systems, 11 Hierarchies, 2,128 Projections, 5 Higher-order definability, 143 Projective hierarchy, 6 Hyperarithmetic comprehension rule, 16 Provably definite relative to 5, 18 Provably 2 H II formula, 13 Inclusion principle, 5 Pushing up ordinals, 37, 68, 69, 72 Indexing of functions, 123 Indiscernibles, 49, 50, 68, 71 R-sets, 7 Inner quantifier, 10 Ramsey cardinals, 36, 145 Intuitionistic type theory, 185 Random reals, 155 Invariant for e-extensions relative to S, 17 Recursive function theory, 2 Irreducible cover, 201, 202 Reduction property, 7 Iterations of extension operators, 36-38, 43-54, Reflection principles, 190 60-62, 64-78 Representation structures (which are ordinally coded well-founded trees in OJ), 12 Rudimentary predicates, 104,121 Kalmar hierarchy, 4 Rudimentary functions, 104

A-extendible, 94,199 S-admissible set, 30 (s), 145 Scales, 155 definitions of, via elementary embeddings, Self-extension(s), 58 35, 85-88, 189, 198 see also: Extension operator properties, 193 Separated unions, 9 in iterated extensions, 75-78 Separation Urprinciple, 8 see also: Measurable cardinals, Extendible X H 11-separation rule, 14 cardinals and Supercompactness X-formula, 13 Liftings of embeddings, 35, 36, 88-93 ^-reflection rule, 12 Limit ultrapowers, 34, 83-85 Simple predicate, 104 "Local*' conditions on elementary embeddings, Spectrum, 128 35, 86-88 Spectrum problem, 135 Luzin hierarchy, 6 Strongly compact, 202 Strong separation principle, 7 Measurable cardinal(s), 36, 68, 74, 75,145 Subtheories of set theory Models of set theory, 144 Z+, 33, 54 Z\ 55 Natural extension operator, see Extension ^finite* 55, 59 operators + ZC , 33, 78 Supercompactness, 35, 86, 87, 202 O-classes, 198, 199, 200, 201 Suslin hierarchy, 5 Operation (A), 5 Syntax, elementary, functions and predicates of, Operator, see Extension operators Operator R, 7 103-104,122,125 m Ordinal functions K (fi) and the ordinal r0» 12 Ordinal sufficiency rule, 13 Theory of definability, 1 Outer quantifier, 10 Transfinite induction rule, 16 Partial set, 147 recursion rule, 16 Persistent for 8-extensions relative to S, 17 Il-formula, 13 Undefinability, 2 Predicative predicate, 109 Uniformization problem, 1 see also: Simple predicate Predicative set theories, 115 Well-foundedness, 37,62-64, 73-75, 88-90,101 LECTURES DELIVERED DURING THE INSTITUTE

John Addison Current problems in descriptive set theory Robert Bradford Undecidability of the theory of Dedekind cardinal addition C. C. Chang Sets constructible using LKK Remarks on the foundation of set theory John E. Doner and Extended arithmetic of ordinal numbers Paul Erdos Finite and infinite combinatorial analysis Paul Erdos Problems and results on combinatorial set theory S. Feferman Predicative set theory Paul Fjelstad Set theory as algebra William Frascella Tactical configurations for infinite sets Haim Gaifman Pushing up the measurable cardinal Robin Gandy Recursive functions of sets Stephen Garland Second-order cardinal characterizability R. J. Gauntt Undefinability of cardinality

219 220 LECTURES DELIVERED DURING THE INSTITUTE

P. C. Gilmore Partial set theory Petr Hajek Bibliography of the Prague seminar on foundations of set theory (1962-1967) James Halpern The Boolean prime ideal theorem Stephen Hechler Generalized scales on number-theoretic functions Ronald Jensen Measurable cardinals and the GCH Carol Karp Some interconnections between infinitary logic and set theory H. Jerome Keisler End extensions of models of set theory On the application of Boolean-valued models to solutions of problems in Boolean algebra Saul Kripke Transfinite recursion, constructive sets, and analogues of cardinals Indescribability and the continuum F. Category-valued higher-order logic Azriel Levy The independence of the axiom of choice, the generalized continuum hypothesis y and the axiom of cons true tibility of L2 statements of set theory Azriel Levy Independence proofs by the Cohen forcing method. I Azriel Levy Independence proofs by the Cohen forcing method. II Azriel Levy The sizes of the indescribable cardinals Categorical alternatives to set theory R. Mansfield The solution of one of Ulam's problems concerning analytic rectangles A. R. D. Mathias The order-extension principle Kenneth McAloon Cohen's method applied to questions about ordinal definability LECTURES DELIVERED DURING THE INSTITUTE 221

Yiannis N. Moschovakis Predicative classes Jan Mycielski On the axiom of determinateness John Myhill Imbedding classical type theory in Hntuitionistic' type theory John Myhill and Dana Scott Ordinal definability K. Namba An axiom of strong infinity and the analytic hierarchy of ordinal numbers Katuzi Ono On Russell type paradoxes and some related problems Richard Platek Eliminating the continuum hypothesis Lawrence Pozsgay Liberal intuitionism as a basis for set theory Karel Prikry Models constructed using perfect sets Degrees of unsolvability and constructive sets William Reinhardt Conditions on natural models of set theory Gerald Sacks Degrees of nonconstructibility Dana Scott Axiomatizing set theory Dana Scott Lectures on Boolean-valued models for set theory J. R. Shoenfield Constructible sets Jack Silver On the consistency of the GCH with the existence of a measurable cardinal Jack Silver The independence ofKurepa's conjecture and two-cardinal conjectures in Robert Solovay Measurable cardinals and the axiom of determinateness Robert Solovay Real-valued measurable cardinals 111 LECTURES DELIVERED DURING THE INSTITUTE

Robert Solovay Solution of a problem ofFodor and Hajnal Robert Solovay IfZF + AC + MC is consistent, then ZF + MC + 2Ko is a real-valued meas• urable cardinal is consistent. A sketch of the proof G. L. Sward Transfinite sequences of axiom systems for set theory Gaisi Takeuti Hypotheses on power set R. L. Vaught Axiomatizability by a schema Martin Zuckerman Finite versions of the axiom of choice