Set Theory and Its Philosophy : a Critical Introduction

This page intentionally left blank Set Theory and its Philosophy This page intentionally left blank Set Theory and its Philosophy A Critical Introduction Michael Potter 3 3 Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao˜ Paulo Shanghai Taipei Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and certain other countries Published in the United States by Oxford University Press Inc., New York c Michael Potter 2004 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2004 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organizations. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available ISBN 0–19–926973–4 (hbk.) ISBN 0–19–927041–4 (pbk.) 13579108642 Typeset by the author Printed in Great Britain on acid-free paper by T J International Ltd., Padstow, Cornwall Preface This book was written for two groups of people, philosophically informed mathematicians with a serious interest in the philosophy of their subject, and philosophers with a serious interest in mathematics. Perhaps, therefore, I should begin by paying tribute to my publisher for expressing no nervous- ness concerning the size of the likely readership. Frege (1893–1903, I, p. xii) predicted that the form of his Grundgesetze would cause him to relinquish as readers all those mathematicians who, if they bump into logical expres- sions such as ‘concept’, ‘relation’, ‘judgment’, think: metaphysica sunt, non leguntur, and likewise those philosophers who at the sight of a formula cry: mathematica sunt, non leguntur. And, as he rightly observed, ‘the number of such persons is surely not small’. As then, so now. Any book which tries to form a bridge between mathematics and philosophy risks vanishing into the gap. It is inevitable, however hard the writer strives for clarity, that the requirements of the subject matter place demands on the reader, sometimes mathematical, sometimes philosophical. This is something which anyone who wants to make a serious study of the philosophy of mathematics must simply accept. To anyone who doubts it there are two bodies of work which stand as an awful warning: the philosophical literature contains far too many articles marred by elementary technical misunderstandings,1 while mathematicians have often been tempted, especially in later life, to commit to print philosophical reflections which are either wholly vacuous or hopelessly incoherent. Both mathematicians and philosophers, then, need to accept that studying the philosophy of mathematics confronts them with challenges for which their previous training may not have prepared them. It does not follow automat- ically, of course, that one should try, as I have done here, to cater for their differing needs in a single textbook. However, my main reason for writing the book was that I wanted to explore the constant interplay that set theory seems to exemplify between technical results and philosophical reflection, and this convinced me that there was more than expositional economy to be said for trying to address both readerships simultaneously. 1I should know: I have written one myself. vi Preface In this respect the book differs from its predecessor, Sets: An Introduction (1990), which was directed much more at philosophically ignorant mathematicians than at philosophers. The similarities between the new book and the old are certainly substantial, especially in the more technical parts, but even there the changes go well beyond the cosmetic. Technically informed readers may find a quick summary helpful. At the formal level the most significant change is that I have abandoned the supposition I made in Sets that the universe of collections contains a sub- universe which has all the sets as members but is nevertheless capable of be- longing to other collections (not, of course, sets) itself. The truth is that I only included it for two reasons: to make it easy to embed category theory; and to sidestep some irritating pieces of pedantry such as the difficulty we face when we try, without going metalinguistic, to say in standard set theory that the class of all ordinals is well-ordered. Neither reason now strikes me as sufficient to compensate for the complications positing a sub-universe creates: the category theorists never thanked me for accommodating them, and if they want a sub-universe, I now think that they can posit it for themselves; while the pedantry is going to have to be faced at some point, so postponing it does no one any favours. The style of set-theoretic formalism in which the sets form only a sub- universe (sometimes in the literature named after Grothendieck) has never found much favour. (Perhaps category theory is not popular enough.) So in this respect the new book is the more orthodox. In two other respects, though, I have persisted in eccentricity: I still allow there to be individuals; and I still do not include the axiom scheme of replacement in the default theory. The reasons for these choices are discussed quite fully in the text, so here I will be brief. The first eccentricity — allowing individuals — seems to me to be something close to a philosophical necessity: as it complicates the treat- ment only very slightly, I can only recommend that mathematicians who do not see the need should regard it as a foible and humour me in it. The second — doing without replacement — was in fact a large part of my motivation to write Sets to begin with. I had whiled away my student days under the delusion that replacement is needed for the formalization of considerable amounts of mathematics, and when I discovered that this was false, I wanted to spread the word (spread it, that is to say, beyond the set theorists who knew it already). On this point my proselytizing zeal has hardly waned in the intervening dec- ade. One of the themes which I have tried to develop in the new book is the idea that set theory is a measure (not the only one, no doubt) of the degree of abstractness of mathematics, and it is at the very least a striking fact about mathematical practice, which many set theory textbooks contrive to obscure, that even before we try to reduce levels by clever use of coding, the over- whelming majority of mathematics sits comfortably inside the first couple of dozen levels of the hierarchy above the natural numbers. Preface vii The differences from standard treatments that I have described so far affect our conception of the extent of the set-theoretic hierarchy. One final differ- ence affecting not our conception of it but only the axiomatization is that the axioms I have used focus first on the notion of a level, and deduce the properties of sets (as subcollections of levels) only derivatively. The pioneer of this style of axiomatization was Scott (1974), but my earlier book used unpublished work of John Derrick to simplify his system significantly, and here I have taken the opportunity to make still more simplifications. Lecturers on the look-out for a course text may feel nervous about this. No executive, it is said, has ever been sacked for ordering IBM computers, even if they were not the best buy; by parity of reasoning, I suppose, no lecturer is ever sacked for teaching ZF. So it is worth stressing that ZU, the system which acts as a default throughout this book, is interpretable in ZF in the obvious manner: the theorems stated in this book are, word for word, theorems of ZF. So teaching from this book is not like teaching from Quine’s Mathematical Logic: you will find no self-membered sets here. Most of the exercises have very largely been taken unchanged from my earlier book. I recommend browsing them, at least: it is a worthwhile aid to understanding the text, even for those students who do not seriously attempt to do them all. It is a pleasure, in conclusion, to record my thanks: to Philip Meguire and Pierre Matet for correspondence about the weaknesses of my earlier book; to the members of the Cambridge seminar which discussed chapters of the new book in draft and exposed inadequacies of exposition; and especially to Timothy Smiley, Eric James, Peter Clark and Richard Zach, all of whom spent more time than I had any right to expect reading the text and giving me perceptive, judicious comments on it. M. D. P. This page intentionally left blank Contents I Sets 1 Introduction to Part I 3 1 Logic 6 1.1 The axiomatic method 6 1.2 The background logic 11 1.3 Schemes 13 1.4 The choice of logic 16 1.5 Definite descriptions 18 Notes 20 2 Collections 21 2.1 Collections and fusions 21 2.2 Membership 23 2.3 Russell’s paradox 25 2.4

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