Zermelo’s Axiom of Choice: Its Origins, Development, and Influence

424 Pages·1982·9.404 MB·English

by  Gregory H. Moore (auth.)

Studies
in the History of Mathematics and
Physical Sciences
8
Editor
G. 1. Toomer
Advisory Board
R. P. Boas P. 1. Davis T. Hawkins
M. 1. Klein A. E. Shapiro D. Whiteside
Ernst Zermelo as a young man
(courtesy of Mrs. Gertrud Zermelo)
Gregory H. Moore
Zermelo's Axiom of Choice
Its Origins, Development, and Influence
Springer-Verlag
New Yark Heidelberg Berlin
Gregory H. Moore
Department of Mathematics and
Institute for History and Philosophy
of Science and Technology
University of Toronto
Toronto, Ontario M5S IAI
Canada
AMS Subject Classifications (1980): 01A55, 01A60, 03-03, 03E25, 04-03, 04A25
Library of Congress Cataloging in Publication Data
Moore, Gregory H.
Zermelo's axiom of choice.
(Studies in the history of mathematics and
physical sciences; 8)
Bibliography: p.
Includes index.
l. Axiom of choice. 2. Zermelo, Ernst, 1871-1953.
I. Title. II. Series.
QA248.M59 1982 511.3 82-10429
With 1 Illustration
© 1982 by Springer-Verlag New York Inc. All rights reserved. No part of this book may
be translated or reproduced in any form without written permission from Springer
Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A.
Softcover reprint of the hardcover 1s t edition 1982
Typeset by Composition House Ltd., Salisbury, England.
987654321
ISBN-13: 978-1-4613-9480-8 e-ISBN-13: 978-1-4613-9478-5
DOl: 10.1007/978-1-4613-9478-5
To
Leslie G. Dawson
We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started
And know the place for the first time.
T. S. Eliot
"Four Quartets"
4.5,26-29
Preface
This book grew out of my interest in what is common to three disciplines:
mathematics, philosophy, and history. The origins of Zermelo's Axiom of
Choice, as well as the controversy that it engendered, certainly lie in that
intersection. Since the time of Aristotle, mathematics has been concerned
alternately with its assumptions and with the objects, such as number and
space, about which those assumptions were made. In the historical context
of Zermelo's Axiom, I have explored both the vagaries and the fertility of
this alternating concern. Though Zermelo's research has provided the focus
for this book, much of it is devoted to the problems from which his work
originated and to the later developments which, directly or indirectly, he
inspired.
A few remarks about format are in order. In this book a publication is
indicated by a date after a name; so Hilbert 1926, 178 refers to page 178 of
an article written by Hilbert, published in 1926, and listed in the bibliography.
A few anonymous articles are cited by journal, according to the list of ab
breviations before the bibliography. Sections of chapters have one decimal
point, while important propositions have two. Thus 2.5 refers to the fifth
section of Chapter 2, and 3.6.4 to the fourth proposition indented in section 6
of Chapter 3. If a theorem or axiom is stated exactly as it was originally
given by its author, it occurs in quotation marks. Otherwise, while I have
endeavored to remain close to an author's phrasing, some terminology or
notation may have been altered for readability. Unless specifically noted, any
translation from a foreign language is my own. Finally, certain symbols that
occur repeatedly in the text deserve mention: the set N of all natural num
bers 0, 1, 2, ... ; the set IR of all real numbers; the set IRn of all points in
Vlll Preface
n-dimensional Euclidean space; and the least infinite ordinal w. Natural
numbers are designated by i, j, k, m, n, ordinals by lower-case Greek letters
such as IX, 13, y, and cardinals by lower-case German letters such as m, n, p,
and q.
Burlingame, California GREGORY H. MOORE
10 January 1982
Acknowledgments
Gratitude, as the poet William Blake once wrote, is heaven itself. In this
spirit it is a pleasure to record the many archival and intellectual debts that
I have accumulated while writing this book. I am grateful to the Master and
Fellows of Trinity College, Cambridge University, for permission to publish
letters from G.H. Hardy to Bertrand Russell. Likewise, I am indebted to the
Bibliotheque at Chaux-de-Fonds in Switzerland for the right to quote from
unpublished correspondence between Louis Couturat and Russell. I also
wish to thank the Universitiits-Bibliotek of the University of Freiburg im
Breisgau for permitting quotations from Zermelo's Nachlass. To Kenneth
Blackwell and Carl Spadoni of the Russell Archives (McMaster University,
Hamilton, Canada) lowe a special debt of gratitude for archival assistance
with Russell's letters and manuscripts; the copyright on the letters from
Russell to both Couturat and Hardy, quoted with permission, rests with Res.
Lib. Ltd., McMaster University. At the University of Toronto many reference
librarians gave unstintingly of their time to aid me in obtaining materials
locally or via interlibrary loan. Finally, I am particularly grateful to the
Societe Mathematique de France for permission to publish my translation of
the "Cinq lett res " ofBaire, Borel, Hadamard, and Lebesgue (see Appendix 1).
My intellectual debts can be acknowledged with even greater pleasure. A
philosopher (Bas van Fraassen), a number of mathematicians (Solomon
Feferman, Thomas Jech, Azriel Levy, Jan Mycielski, Franklin Tall), and
several historians of mathematics (Joseph Dauben, Ivor Grattan-Guin
ness, Thomas Hawkins, Calvin Jongsma, Barbara Moss) read and comment
ed on various drafts of this book. In particular, Thomas Jech patiently an
swered innumerable technical questions about the Axiom of Choice and its
x Acknowledgments
consequences. The bibliography has benefited from suggestions by J. M. B.
Moss, and the translation in Appendix 1 from comments by Jean van
Heijenoort. I am indebted to John Addison, Leon Henkin, Robert Solovay,
and Alfred Tarski for conversations on particular historical points. After
such generous assistance from so many quarters, any remaining errors are
mme.
lowe my chief intellectual debt to the late Kenneth O. May, who first in
spired me with the potentialities inherent in the history of mathematics.
When I came to Toronto to study with him in 1970, he turned my interest
in set theory to Zermelo's life and work. Out of the first papers that I wrote
for him, this book eventually grew. While I deeply regret that he did not live
to see its completion, there is hardly a page that does not bear the imprint of
his insight and enthusiasm.
Concerning the actual production of the book, I would like to thank the
staff of Springer-Verlag New York Inc. for their unflagging patience and
cooperation. Leslie Ovens typed the manuscript, thereby turning innumer
able revisions into readable copy. I was generously aided in reading galley
proof by Joseph Dauben, Richard Epstein, Raymond Lim, and my father,
W. Harvey Moore.
Finally, on a more personal level, I wish to thank my parents, Dr. and
Mrs. W. Harvey Moore, and my friends, Eva Silberberg and Marylin
Kingston, for their continuing encouragement.
Part of my research was supported by a University Research Fellowship
from the Natural Sciences and Engineering Research Council of Canada.