Cohen Paul J - Cohen Set Theory and the Continuum Hypothesis

H ANDBOOK OF M ATHEMATICAL F UNCTIONS , Milton Abramowitz and Irene A. Stegun. (0-486-61272-4)

T ENSOR A NALYSIS ON M ANIFOLDS , Richard L. Bishop and Samuel I. Goldberg. (0-486-64039-6)

V ECTOR AND T ENSOR A NALYSIS WITH A PPLICATIONS , A. I. Borisenko and I. E. Tarapov. (0-486-63833-2)

T HE H ISTORY OF THE C ALCULUS AND I TS C ONCEPTUAL D EVELOPMENT , Carl B. Boyer. (0-486-60509-4)

T HE Q UALITATIVE T HEORY OF O RDINARY D IFFERENTIAL E QUATIONS: A N I NTRODUCTION , Fred Brauer and John A. Nohel. (0-486-65846-5)

P RINCIPLES OF S TATISTICS , M. G. Bulmer. (0-486-63760-3)

T HE T HEORY OF S PINORS , lie Cartan. (0-486-64070-1)

A DVANCED N UMBER T HEORY , Harvey Cohn. (0-486-64023-X)

S TATISTICS M ANUAL , Edwin L. Crow, Francis Davis, and Margaret Maxfield. (0-486-60599-X)

F OURIER S ERIES AND O RTHOGONAL F UNCTIONS , Harry F. Davis. (0-486-65973-9)

C OMPUTABILITY AND U NSOLVABILITY , Martin Davis. (0-486-61471-9)

A SYMPTOTIC M ETHODS IN A NALYSIS , N. G. de Bruijn. (0-486-64221-6)

T HE M ATHEMATICS OF G AMES OF S TRATEGY , Melvin Dresher. (0-486-64216-X)

A PPLIED P ARTIAL D IFFERENTIAL E QUATIONS , Paul DuChateau and David Zachmann. (0-486-41976-2)

A SYMPTOTIC E XPANSIONS , A. Erdlyi. (0-486-60318-0)

C OMPLEX V ARIABLES: H ARMONIC AND A NALYTIC F UNCTIONS , Francis J. Flanigan. (0-486-61388-7)

D IFFERENTIAL T OPOLOGY , David B. Gauld. (0-486-45021-X)

O N F ORMALLY U NDECIDABLE P ROPOSITIONS OF P RINCIPIA M ATHEMATICA AND R ELATED S YSTEMS , Kurt Gdel. (0-486-66980-7)

A H ISTORY OF G REEK M ATHEMATICS , Sir Thomas Heath. (0-486-24073-8, 0-486-24074-6) Two-volume set

P ROBABILITY: E LEMENTS OF THE M ATHEMATICAL T HEORY , C. R. Heathcote. (0-486-41149-4)

I NTRODUCTION TO N UMERICAL A NALYSIS , Francis B. Hildebrand. (0-486-65363-3)

M ETHODS OF A PPLIED M ATHEMATICS , Francis B. Hildebrand. (0-486-67002-3)

T OPOLOGY , John G. Hocking and Gail S. Young. (0-486-65676-4)

M ATHEMATICS AND L OGIC , Mark Kac and Stanislaw M. Ulam. (0-486-67085-6)

M ATHEMATICAL F OUNDATIONS OF I NFORMATION T HEORY , A. I. Khinchin. (0-486-60434-9)

A RITHMETIC R EFRESHER , A. Albert Klaf. (0-486-21241-6)

C ALCULUS R EFRESHER , A. Albert Klaf. (0-486-20370-0)

P ROBLEM B OOK IN THE T HEORY OF F UNCTIONS , Konrad Knopp. (0-486-41451-5)

I NTRODUCTORY R EAL A NALYSIS , A. N. Kolmogorov and S. V. Fomin. (0-486-61226-0)

S PECIAL F UNCTIONS AND T HEIR A PPLICATIONS , N. N. Lebedev. (0-486-60624-4)

C HANCE , L UCK AND S TATISTICS , Horace C. Levinson. (0-486-41997-5)

T ENSORS , D IFFERENTIAL F ORMS, AND V ARIATIONAL P RINCIPLES , David Lovelock and Hanno Rund. (0-486-65840-6)

S URVEY OF M ATRIX T HEORY AND M ATRIX I NEQUALITIES , Marvin Marcus and Henryk Minc. (0-486-67102-X)

A BSTRACT A LGEBRA AND S OLUTION BY R ADICALS , John E. and Margaret W. Maxfield. (0-486-67121-6)

F UNDAMENTAL C ONCEPTS OF A LGEBRA , Bruce E. Meserve. (0-486-61470-0)

F UNDAMENTAL C ONCEPTS OF G EOMETRY , Bruce E. Meserve. (0-486-63415-9)

F IFTY C HALLENGING P ROBLEMS IN P ROBABILITY WITH S OLUTIONS , Frederick Mosteller. (0-486-65355-2)

N UMBER T HEORY AND I TS H ISTORY , Oystein Ore. (0-486-65620-9)

M ATRICES AND T RANSFORMATIONS , Anthony J. Pettofrezzo. (0-486-63634-8)

T HE U MBRAL C ALCULUS , Steven Roman. (0-486-44139-3)

P ROBABILITY T HEORY: A C ONCISE C OURSE , Y. A. Rozanov. (0-486-63544-9)

L INEAR A LGEBRA , Georgi E. Shilov. (0-486-63518-X)

E SSENTIAL C ALCULUS WITH A PPLICATIONS , Richard A. Silverman. (0-486-66097-4)

I NTERPOLATION , J.F. Steffensen. (0-486-45009-0)

A C ONCISE H ISTORY OF M ATHEMATICS , Dirk J. Struik. (0-486-60255-9)

P ROBLEMS IN P ROBABILITY T HEORY , M ATHEMATICAL S TATISTICS AND T HEORY OF R ANDOM F UNCTIONS , A. A. Sveshnikov. (0-486-63717-4)

T ENSOR C ALCULUS , J. L. Synge and A. Schild. (0-486-63612-7)

M ODERN A LGEBRA : T WO V OLUMES B OUND A S O NE , B.L. Van der Waerden. (0-486-44281-0)

C ALCULUS OF V ARIATIONS WITH A PPLICATIONS TO P HYSICS AND E NGINEERING , Robert Weinstock. (0-486-63069-2)

I NTRODUCTION TO V ECTOR AND T ENSOR A NALYSIS , Robert C. Wrede. (0-486-61879-X)

D ISTRIBUTION T HEORY AND T RANSFORM A NALYSIS , A. H. Zemanian. (0-486-65479-6)

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Manufactured in the U.S.A.

Set Theory

and the

Continuum
Hypothesis

Paul J. Cohen

with a new introduction by

Martin Davis

Dover Publications, Inc.
Mineola, New York

Copyright

Copyright 1966 by W. A. Benjamin, Inc.

Copyright renewed 1994 by Paul J. Cohen

Introduction to the Dover edition copyright 2008 by Martin Davis

Foreword to the Dover edition copyright 2008 by Charles Cohen

Bibliographical Note

This Dover edition, first published in 2008, is an unabridged republication of the work first published by W. A. Benjamin, Inc., in 1966, and includes a new introduction by Martin Davis. My Interaction with Kurt Gdel: The man and his work from Kurt Gdel and the Foundations of Mathematics: Horizons of Truth, by Paul J. Cohen, is reprinted courtesy of Cambridge University Press, 2009.

Library of Congress Cataloging-in-Publication Data

Cohen, Paul J., 19342007.

Set theory and the continuum hypothesis / Paul J. Cohen. Dover ed.

p. cm.

This Dover edition, first published in 2008, is an unabridged republication of the work first published by W. A. Benjamin, Inc., in 1966, and includes a new introduction by Martin Davis.

ISBN-13: 978-0-486-46921-8

ISBN-10: 0-486-46921-2

1. Set theory. 2. Logic, Symbolic and mathematical. Continuum hypothesis. I. Title.

QA248.C614 2008

511.3'22dc22

2008042847

Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

Although Paul Cohens reputation will forever be defined by his Fields Medal winning contributions to logic and set theory, he was equally famous amongst friends and colleagues for his stunning breadth and depth of mathematical knowledge. He could not only give the proof of any great theorem you could name, but in all likelihood it would be one more elegant and beautiful than you had ever seen before. Later when you came back to him frustrated that you could not find that particular method in any textbook, he would explain that was because he had come up with it himself.

For Paul was one of the greatest auto-didacts in twentieth century mathematics; most famously of course in logic and set theory, where in two years he went from knowing almost nothing about the field to solving its most famous problem. Through this process of discovering subjects anew he came up with insightful methods that were legendary in his teaching career.

Unfortunately few of these examples survive in written form, which is why we are so pleased that this book is being reissued. Based on a course that Paul taught at Harvard shortly after his work on the continuum hypothesis, it serves as a small glimpse into an amazing mind that could strip away the technical complications of mathematics leaving its beauty clear to see. Paul used to say that his first love was really music, not math, and we think that any student of mathematics will hear the musicality of this work.