Table Of ContentBertrand 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
