Bertrand Russell and the Origins of the Set-theoretic ‘Paradoxes’ PDF
Preview Bertrand Russell and the Origins of the Set-theoretic ‘Paradoxes’
Bertrand Russell In 1907 Alejandro R. Garciadiego Bertrand Russell and the Origins of the Set-theoretic 'Paradoxes' 1992 Birkhauser Verlag Basel· Boston· Berlin Author's address Dr. A.R. Garcladlego Umversldad Naclonal Aut6noma de MexIco * Cubiculo 016 Departamento de Matematlcas Facultad de Clenclas Cludad Umversltarla 04510 Mexico, D.F. (MexIco) Library of Congress Cataloging in Publication Data Garciadiego Dantan, Alejandro Ricardo. Bertrand Russell and the origins ofthe set-theoretic 'paradoxes' / Alejandro R. Garcladlego. p.cm. Includes bibliographical references and index. ISBN-13 978-3-0348-7404-5 e-ISBN-13 978-3-0348-7402-1 001 10 1007/978-3-0348-7402-1 1. Logic, Symbolic and mathematical. 2. Set theory. 3. Paradox. 4. Russell, Bertrand, 1872-1970 Principles of mathematiCs. I. Title. QA9.G24 1992 511.3 - dc20 Deutsche Bibliothek Cataloging-in·Publication Data Garciadiego, Alejandro R.: Bertrand Russell and the origins of the set theoretic 'paradoxes' / Alejandro R. Garcladlego. - Basel; Boston; Berlin: Birkhauser, 1992 ISBN-13 978-3-0348-7404-5 This work IS subject to cOPYright. All rights are reserved, whether the whole or part ofthe material IS concerned, speCifically those of translation, reprinting, re-use of Illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage In data banks. Under § 54 of the German COPYright Law, where copies are made for other than private use a fee is payable to «Verwertungsgesellschaft Wort», Munich. © 1992 Birkhauser Verlag Basel Softcover reprint of the hardcover 1st edition 1992 Cover deSign: Albert Gomm ISBN-13 978-3-0348-7404-5 I've have put in so many enigmas and puzzles that it will keep the professors busy for centuries arguing over what I meant, and that's the only way of ensuring one's immortality. James Joyce TO LUPITA, GABRIELA & ALEJANDRO Contents Prologue by Ivor Grattan-Guinness ix Preface 1. The goal of the book xi 2. The sources xx Acknowledgements xxvii Chapter I. Antecedents 1.1 Introduction 1 1.2 Cantor's transfinite numbers 2 1. 3 Some reactions 13 Chapter II. A standard interpretation 2.1 Introduction 19 2.2 The Burali-Forti 'paradox' 21 2.3 Cantor's 'paradox' 32 2.4 Epilog 40 Chapter III. The philosophical and mathematical background to The Principles of Mathematics, 1872-1900 3.1 Introduction 41 3.2 Russell's childhood and adolescence 43 3.3 Russell as a student at Cambridge 49 3.4 Russell's An Essay on the Foundations of Geometry 56 3.5 Russell's early attempts to write a book on the principles of arithmetic 61 viii Contents Chapter IV. Russell's discovery of the 'paradoxes' 4.1 Introduction 81 4.2 The First International Congress of Philosophy, Paris 1900 83 4.3 Russell's writing of the 'first final' draft of The Principles of Mathematics, November - December 1900 87 4.4 The 'paradoxes' 100 4.5 A last effort to finish The Principles of Mathematics, April 1902 - May 1903 117 Chapter V. The 'semantic paradoxes' 5.1 Introduction 131 5.2 Berry's argument 134 5.3 The Konig - Zermelo proposition 137 5.4 Richard's contradiction 140 5.5 Polemics within the London Mathematical Society over the Well-Ordering Theorem 143 5.6 Konig's and Dixon's Arguments 146 Conclusions 151 Appendix. Correspondence 6.1 Alys Russell 155 6.2 G. E. Moore 160 6.3 David Hilbert 162 6.4 Cesare Burali-Forti 164 6.5 G. G. Berry 166 6.6 Alfred N. Whitehead 185 6.7 G. H. Hardy 188 6.8 E. H. Moore 205 List of tables 207 List of illustrations 209 Bibliography 211 Index 257 Prologue The history of the paradoxes of set theory is well known. Cantor found the one concerning the greatest cardinal in the 1890s, and soon afterwards Burali-Forti discovered the corresponding for the ordinals. Then around 1900 Russell showed that the set of all sets which do not belong to themselves led to a paradox. This finding caused a crisis, and mathematicians, philosophers and logicians struggled to free mathematics froql these infections. However, other paradoxes came to light, such as those of finite definability and of the visiting card. Like all well-known stories, this one is better known than studied. While some historians have revealed a more complicated story, Dr. Garciadiego has gone to new levels of detail to reveal the complications of the true record. They involve not only more information of the same kind but especially many corrections of our understanding. My own favourites are these: neither Cantor nor Burali-Forti thought of their results as paradoxes; Russell was their principal founder and first librari an, in that he both collected them and interpreted them as paradoxes; the status of such result as paradoxes was entwined in a melange of poorly distinguished paradoxes, contradictions, antinomies, reductio ad absur dum arguments, and strange objects like the axioms of choice which one had to accept or at least tolerate; hardly any major figure was drawn to set theory by the paradoxes; and the community of mathematicians in general gave the matter little thought. Dr. Garciadiego covers the period from the mid 1890s to the late 1900s. He begins his tale with the final form of Cantor's set theory and transfinite numbers, and he steers it through Russell's own discoveries x Prologue and reinterpretations to the emergence of the 'semantic' paradoxes (as they were normally not distinguished at the time). He adjoins a substan tial quantity of unpublished correspondence, mostly with Russell, in which the perplexities and some disagreements of the time are vividly exhibited. His book is unusually readable for an academic work and deserves to gather a substantial measure of attention-more than arose at the time, indeed, when the full consequences of the paradoxes were not yet clear. Dr. Ivor Grattan-Guinness, Middlesex Polytechnic
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