INTRODUCTION TO AXIOMATIC SET THEORY

INTRODUCTION TO AXIOMATIC SET THEORY

SYNTHESE LIBRARY MONOGRAPHS ON EPISTEMOLOGY, LOGIC, METHODOLOGY, PHILOSOPHY OF SCIENCE, SOCIOLOGY OF SCIENCE AND OF KNOWLEDGE, AND ON THE MATHEMATICAL METHODS OF SOCIAL AND BEHAVIORAL SCIENCES Editors: DONALD DAVIDSON, Rockefeller University and Princeton University J AAKKO HINTIKKA, Academy of Finland and Stanford University GABRIEL NUCHELMANS, University of Leyden WESLEY C. SALMON, Indiana University

JEAN-LOUIS KRIVINE INTRODUCTION TO AXIOMATIC SET THEORY D. REIDEL PUBLISHING COMPANY / DORDRECHT-HOLLAND

THEORIE AXIOMATIQUE DES ENSEMBLES First published by Presses Universitaires de France, Paris Translated/rom the French by David Miller Library of Congress Catalog Card Number 71-146965 ISBN-13: 978-90-277-0411-5 e-isbn-13: 978-94-010-3144-8 DOl: 10.1007/978-94-010-3144-8 All Rights Reserved Copyright 1971 by D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint of the hardcover 1st edition 1971 No part of this book may be reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publisher

INTRODUCTION This book presents the classic relative consistency proofs in set theory that are obtained by the device of 'inner models'. Three examples of such models are investigated in s VI, VII, and VIII; the most important of these, the class of constructible sets, leads to G6del's result that the axiom of choice and the continuum hypothesis are consistent with the rest of set theory [1]I. The text thus constitutes an introduction to the results of P. Cohen concerning the independence of these axioms [2], and to many other relative consistency proofs obtained later by Cohen's methods. s I and II introduce the axioms of set theory, and develop such parts of the theory as are indispensable for every relative consistency proof; the method of recursive definition on the ordinals being an important case in point. Although, more or less deliberately, no proofs have been omitted, the development here will be found to require of the reader a certain facility in naive set theory and in the axiomatic method, such as should be achieved, for example, in first year graduate work (2 e cycle de mathernatiques). The background knowledge supposed in logic is no more advanced; taken as understood are such elementary ideas of first-order predicate logic as prenex normal form, model of a system of axioms, and so on. They first come into play in IV; and though, there too, all the proofs (bar that of the reduction of an arbitrary formula to prenex normal form) are carried out, the treatment is probably too condensed for a reader previously unacquainted with the subject. Several leading ideas from model theory, not themselves used in this book, would nevertheless make the understanding of it simpler; for example, the distinction between intuitive natural numbers and the natural numbers of the universe, or between what we call formulas and what we call expressions, are easier to grasp if something is known about 1 Numbers in brackets refer to items of the Bibliography, which is to be found on p.98.

VI INTRODUCTION non-standard models for Peano arithmetic. Similarly, the remarks on p. 44 will be better understood by someone who knows the completeness theorem for predicate calculus. All these ideas can be found for example in [4] (s I, II, III) or [5] (s I, II, III). The approach of the book may appear a little odd to anyone who thinks that axiomatic set theory (as opposed to the naive theory, for which, perhaps, this is true) must be placed at the very beginning of mathematics. For the reader is by no means asked to forget that he has already learnt some mathematics; on the contrary we rely on the experience he has acquired from the study of axiomatic theories to offer him another one: the theory of binary relations which satisfy the Zermelo/Fraenkel axioms. As we progress, what distinguishes this particular theory from other axiomatic theories gradually emerges. For the concepts introduced naturally in the study of models of this theory are exactly parallel to the most fundamental mathematical concepts - natural numbers, finite sets, denumerable sets, and so on. And since standard mathematical vocabulary fails to provide two different names for each idea, we are obliged to use the everyday words also when referring to models of the Zermelo/Fraenkel axioms. The words are thereby used with a completely different meaning, the classic example of this being the 'Skolem paradox', which comes from the new sense that the word 'denumerable' takes when interpreted in a model of set theory. Eventually it becomes obvious that even the everyday senses of these words are by no means clear to us, and that we can perhaps try to sharpen them with the new tools which are developed in the study of set theory. Had this problem been posed already at the beginning of the study it would have been tempting to shirk it, by saying that mathematics is merely the business of manipulating meaningless symbols. It is an open question how much set theory can do in this field; but it seems likely that what it can do will be of interest beyond mathematical logic itself. Note: This translation incorporates a number of additions and corrections to the original French edition.

CONTENTS INTRODUCTION v I: The ZermelojFraenkel Axioms of Set Theory II: Ordinals, Cardinals III: The Axiom of Foundation IV: The Reflection Principle V: The Set of Expressions VI: Ordinal Definable Sets. Relative Consistency of the Axiom of Choice 63 VII: FraenkelfMostowski Models. Relative Consistency of the Negation of the Axiom of Choice (without the Axiom of Foundation) 70 VIII: Constructible Sets. Relative Consistency of the Generalized Continuum Hypothesis 81 BIBLIOGRAPHY 98 1 13 35 48 56