THE AXIOM OF CHOICE THOMAS J. JECH State University of New York at Bufalo and The Institute for Advanced Study Princeton, New Jersey 1973 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM LONDON AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK 0 NORTH-HOLLAND PUBLISHING COMPANY - 1973 AN Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner. Library of Congress Catalog Card Number 73-15535 North-Holland ISBN for the series 0 7204 2200 0 for this volume 0 1204 2215 2 American Elsevier ISBN 0 444 10484 4 Published by: North-Holland Publishing Company - Amsterdam North-Holland Publishing Company, Ltd. - London Sole distributors for the U.S.A. and Canada: American Elsevier Publishing Company, Inc. 52 Vanderbilt Avenue New York, N.Y. 10017 PRINTED IN THE NETHERLANDS To my parents PREFACE The book was written in the long Buffalo winter of 1971-72. It is an attempt to show the place of the Axiom of Choice in contemporary mathe- matics. Most of the material covered in the book deals with independence and relative strength of various weaker versions and consequences of the Axiom of Choice. Also included are some other results that I found relevant to the subject. The selection of the topics and results is fairly comprehensive, nevertheless it is a selection and as such reflects the personal taste of the author. So does the treatment of the subject. The main tool used throughout the text is Cohen’s method of forcing. I tried to use as much similarity between sym- metric models of ZF and permutation models of ZFA as possible. The relation between these two methods is described in Chapter 6, where I pre- sent the Embedding Theorem (Jech-Sochor) and the Support Theorem (Pincus). Consistency and independence of the Axiom of Choice (due to Godel and Cohen, respectively) are presented in Chapters 3 and 5. Chapter 4 introduces the permutation models (Fraenkel, Mostowski, Specker). Chapters 2 and 10 are devoted to the use of the Axiom of Choice in mathematics. Chapter 2 contains examples of proofs using the Axiom of Choice, whereas Chapter 10 provides counterexamples in the absence of the Axiom of Choice. Chapters 7 and 8 deal with various consequences of the Axiom of Choice. The topics considered in Chapter 7 include the Prime Ideal Theorem, the Ordering Principle and the Axiom of Choice for Finite Sets, while Chapter 8 is concerned with the Countable Axiom of Choice, the Principle of De- pendent Choices and related topics. In Chapter 9 we discuss the differences between permutation models of ZFA and symmetric models of ZF. Chapter 11 is devoted to questions concerning cardinal numbers in set theory without the Axiom of Choice. Finally, in Chapter 12 we discuss VII VIII PREFACE briefly the Axiom of Determinacy and related topics. This subject has been recently in the focus of attention, and it seems that solution of the open problems presented at the end of Chapter 12 would mean an important con- tribution to the investigation of the Foundations of Mathematics. The book is more or less self-contained. Most of the theorems are accom- panied by a proof of a sort. There are, however, several instances where I elected to present a theorem without proof, and an insatiable reader will have to resort to looking up the proof in the journals. Each chapter contains a number of problems. Some of them are simply exercises, but many of the problems are actually important results which for various reasons are not included in the main text. These are in most cases provided with an outline of proof. To help the reader, I adopted the classification used by producers of French brandy: The one star problems are generally more difficult than the plain problems. Two stars are reserved for results presented without a proof (in most cases, such a result is the main theorem of a long paper). And finally, three stars indicate that the problem is open (there are still a few left). Among those whose work on the subject has had most influence on this book are (apart from Godel and Cohen) A. Levy, A. Mostowski and A. Tarski, to name a few. My special thanks are due to P. VopEnka, who is responsible for my interest in the subject and under whose guidance I wrote my 1966 thesis ‘The Axiom of Choice’. I am very much indebted to D. Pincus, for two reasons. Firstly, many of the results of his 1969 thesis found its way into this book, and secondly, his correspondence with me throughout the past five years had a significant influence on my treatment of the subject. Finally, I am grateful to U. Felgner for his valuable comments after he was so kind to read my manuscript last summer. Princeton, N.J. December 1972 CHAPTER 1 INTRODUCTION 1.1. The Axiom of Choice A mathematician of the present generation hardly considers the use of the Axiom of Choice a questionable method of proof. As a result of algebra and analysis going abstract and the development of new mathematical Is- ciplines such as set theory and topology, practically every mathematician learns about the Axiom of Choice (or at least of its most popular form, Zorn’s Lemma) in an undergraduate course. He also probably vaguely knows that there has been some controversy involving the Axiom of Choice, but it has been resolved by the logicians to a general satisfaction. There has been some controversy about the Axiom of Choice, indeed. First, let us see what the Axiom of Choice says: AXIOMOF CHOICE.For every family 9 of nonempty sets, there is u function f such that f (S)E S for each set S in the family 9. (We say that f is a choice function on F.)In some particular instances, this is self-evident. E.g., let 9 consist of sets of the form {a, b}, where a and b are real numbers. Then the function IS a choice function on 9.Or, if all the sets in 9are singletons, i.e. sets of the form {a}, then one can easily find a choice function on 9.Finally, if .F is ajkite family of nonempty sets, then there is a choice function. (To show this, one uses induction on the size of 9;naturally, a choice function exists for a family which consists of a single nonempty set.) 1 2 INTRODUCTION [CH. 1, $2 Thus the idea of having a choice function on every family of nonempty sets is not against our mathematical mind. But try to think of a choice func- tion on the family of all nonempty sets of reals! Or, let 2F be the family of all (unordered) pairs { S, T},where S and T are sets of reals. After thinking about these problems for a while, one should start to appreciate the other point of view. The Axiom of Choice is different from the ordinary principles accepted by mathematicians. And this was one of the sources of objections to the Axiom of Choice, as late as in the thirties. The other source of objections is the fact that the Axiom of Choice can be used to prove theorems which are to a certain extent ‘unpleasant’, and even theorems which are not exactly in line with our ‘common sense’ intuition. The rest of Chapter 1 is devoted to such examples. 1.2. A nonmeasurable set of real numbers The most popular example of an ‘unpleasant’ consequence of the Axiom of Choice is the existence of a set of real numbers which is not Lebesgue measurable. Let p(X) denote the Lebesgue measure of a set X of real numbers. As is well-known, p is countably additive, translation invariant and ,u( [a, b]) = b-a for every interval [a, b]. For real numbers in the interval [0, 11, let us define x-Y whenever x-y is a rational number. The relation - on [0, I] is an equiva- lence; denote by [XI the corresponding equivalence class, for each x E [0, I]. By the Axiom of Choice, we can choose one element out of each equivalence class. Thus there is a set M of real numbers, M c [0, 11, with the property that for each real number x, there exists a unique y E M and a unique rational number r such that x = y + r. If we let M,. = {y+r : y E M} for each rational number Y, then we have a partition of the set R of all real numbers into countably many disjoint sets (1.1) R = u {M,. : r rational). It follows that the set M is not measurable. Assuming that M is measurable, we get a contradiction as follows: First, p(M) = 0 is impossible, since this CH.1, $31 A PARADOXICAL DECOMPOSITION OF THE SPHERE 3 would imply p(R) = 0, using (1.1). However, p(M)> 0 is also impossible, since this would imply p([O, 11) > p(u {M, : r rational and 0 < r < l}) r rational using the fact that each Mr would have to have the same measure as M. 1.3. A paradoxical decomposition of the sphere The theorems presented here demonstrate what 'absurd' results can be expected when employing the Axiom of Choice. THEOREM1.1. A sphere S can be decomposed into disjoint sets S=AUBUCUQ such that: (i) the sets A, B, C are congruent to each other; (ii) the set B u C is congruent to each of the sets A, By C; (iii) Q is countable.