Table Of ContentFoundations 01 Mathematics
Symposium Papers Commemorating the
Sixtieth Birthday of Kurt Codel
Edited by
Jack J. Bulloff
Thomas C. Holyoke· S.W. Hahn
Springer-Verlag New York Inc. 1969
Managing Editor:
JACK J. BULLOFF
State University of New York at Albany· Albany, N. Y.
Technical Editors:
THOMAS C. HOLYOKE
Antioch College' Yellow Springs, Ohio
SAMUEL W. HAHN
Wittenberg University' Springfield, Ohio
The Photograph of Kurt GOdel is by Orren J. Turner
All rights reserved. No part of this book may be translated or reproduced in any
form without written permission from Springer-Verlag
ISBN-13: 978-3-642-86747-7 e-ISBN-13: 978-3-642-86745-3
DOl: 10.1 007/978-3-642-86745-3
© by Springer-Verlag, Berlin' Heidelberg 1969
Library of Congress Catalog Card Number 68-28757
Softcover reprint of the hardcover 1s t edition 1969
Title No. 1518
Kurt Cadel
Biographical Data
KURT GODEL was born on April 28, 1906 in Brunn (Brno), Czecho
slovakia (at that time part of the Austro-Hungarian Monarchy). He
studied mathematics and physics at the University of Vienna and took
the Ph. D. degree there in 1930. He was Dozent in mathematics at the
University of Vienna 1933-1938. He has been a member of the Institute
for Advanced Study from 1940-1953, and several times before 1940.
He joined its faculty in 1953. He became an American citizen in 1948
and received honorary degrees from Yale and Harvard in 1951 and 1952.
He is a member of the National Academy of Sciences and of the American
Philosophical Society and corecipient of the 1951 Einstein Award.
Preface
Dr. KURT GODEL'S sixtieth birthday (April 28, 1966) and the thirty
fifth anniversary of the publication of his theorems on undecidability
were celebrated during the 75th Anniversary Meeting of the Ohio Ac
ademy of Science at The Ohio State University, Columbus, on April 22,
1966. The celebration took the form of a Festschrift Symposium on a
theme supported by the late Director of The Institute for Advanced
Study at Princeton, New Jersey, Dr. J. ROBERT OPPENHEIMER: "Logic,
and Its Relations to Mathematics, Natural Science, and Philosophy."
The symposium also celebrated the founding of Section L (Mathematical
Sciences) of the Ohio Academy of Science.
Salutations to Dr. GODEL were followed by the reading of papers by
S. F. BARKER, H. B. CURRY, H. RUBIN, G. E. SACKS, and G. TAKEUTI,
and by the announcement of in-absentia papers contributed in honor of
Dr. GODEL by A. LEVY, B. MELTZER, R. M. SOLOVAY, and E. WETTE.
A short discussion of "The II Beyond Godel's I" concluded the session.
This Anniversary Volume has been prepared under the sponsorship
of Section L. Thanks are due to Mr. John H. MELVIN, Executive Secre
tary of the Ohio Academy of Science, and to Dr. ROBERT C. STEPHENSON,
Chairman of the 75th Anniversary Committee, The Ohio State University
Research Foundation, for the many things they did to make the meeting
possible. Dr. CARL KAYSEN, Director of the Institute for Advanced
Study, provided items from his files which have made for a better and
more attractive volume. Valuable suggestions were made to authors and
editors by Dr. KLAUS PETERS of Springer-Verlag. Finally, it is a pleasure
to acknowledge the helpful advice offered by the referees, by colleagues
of Dr. GODEL, and by members of the Association for Symbolic Logic.
The Editors
List of Contributors
STEPHEN F. BARKER, The Johns Hopkins University, Baltimore, Mary
land/USA.
HASKELL B. CURRY, University of Amsterdam, Amsterdam/The Nether
lands.
AzRIEL LEVY, Hebrew University, Jerusalem/Israel, and Stanford
University, Stanford, California /USA.
BERNARD MELTZER, University of Edinburgh / Scotland.
HERMAN RUBIN, Purdue University, Lafayette, Indiana/USA.
GERALD E. SACKS, Massachusetts Institute of Technology, Cambridge,
Massachusetts jUSA.
ROBERT M. SOLOVAY, The Rockefeller University, New York, New York,
and University of California, Berkeley, California/USA.
GAISI TAKEUTI, University of Illinois, Urbana, Illinois, and The Institute
for Advanced Study, Princeton, New Jersey /USA.
EDUARD WETTE, Uckerath, and Radevormwald/West Germany.
Contents
Greetings from Dr. J. ROBERT OPPENHEIMER VIII
Greetings from Dr. GODEL . . . . . . . . . VIII
Tribute to Dr. GODEL by JOHN VON NEUMANN IX
Bibliography of KURT GODEL . . . . . . . . XI
Realism as a Philosophy of Mathematics. By STEPHEN F. BARKER 1
The Undecidability of AK-Conversion. By HASKELL B. CURRY 10
The Definability of Cardinal Numbers. By AZRIEL LEVY . .. 15
The Use of Symbolic Logic in Proving Mathematical Theorems by
Means of a Digital Computer. By BERNARD MELTZER . . .. 39
A New Approach to the Foundations of Probability. By HERMAN
RUBIN. . . . . . . . . . . . . . . . . . . . 46
Measure-Theoretic Uniformity. By GERALD E. SACKS . . . . .. 51
On the Cardinality of 2~ Sets of Reals. By ROBERT M. SOLOVAY . 58
The Universe of Set Theory. By GAISI TAKEUTI . . . . . . .. 74
Definition eines (relativ vollstandigen) formalen Systems konstruk-
tiver Arithmetik. By EDUARD WETTE . . . . . . . . . .. 130
Greetings
Greetings to GOdel Symposium from Dr. J. Robert Oppenheimer
"It is an honor and a pleasure for me to help to celebrate KURT
GODEL, his anniversaries, and his great work, which has not only im
measurably deepened and enriched the understanding of the logical
structure of so much abstract and mathematical argument, but illumi
nated the role of limitation in human understanding in general. I salute
the scholars, many of high distinction, who have assembled to honor
GODEL by presenting their discoveries and their views, and I hope that
the contributions to this occasion will bring great pleasure to the man
who has inspired it."
Greetings from Dr. GOdel
"I wish to convey my greetings to the symposium, and in particular
to the speakers. I am sorry I cannot attend, but I am looking forward
with interest and pleasure to the volume in which the lectures will be
presented."
Tribute to Dr. Gijdel
JOHN VON NEUMANN
The true significance of Dr. GODEL'S work was perhaps most clearly
and succinctly summarized by the late Professor JOHN VON NEUMANN
in his remarks in March 1951, on the occasion of the presentation of
the ALBERT EINSTEIN Award to Dr. GODEL. The textual reproduction
of those remarks follows:
"KURT GODEL'S achievement in modern logic is singular and monu
mental - indeed it is more than a monument, it is a landmark which
will remain visible far in space and time. Whether anything comparable
to it has occurred in the logic of modern times may be debated. In any
case, the conceivable proxima are very, very few. The subject of logic
has certainly completely changed its nature and possibilities with
GODEL'S achievement.
GODEL'S name is associated with many important achievements in
detail, and with two absolutely decisive ones. The occasion is such that
I think I should only talk about the two latter.
The nature of the first one is easy to indicate, although its exact
technical character and execution escape an adequate characterization
without the specialized and rather intricate techniques of formal logic.
GODEL was the first man to demonstrate that certain mathematical
theorems can neither be proved nor disproved with the accepted, rigorous
methods of mathematics. In other words, he demonstrated the existence
of undecidable mathematical propositions. He proved furthermore that
a very important specific proposition belonged to this class of undecidable
problems: The question, as to whether mathematics is free of inner
contradictions. The result is remarkable in its quasi-paradoxical 'self
denial': It will never be possible to acquire with mathematical means the
certainty that mathematics does not contain contradictions. It must be
emphasized that the important point is, that this is not a philosophical
principle or a plausible intellectual attitude, but the result of a rigorous
mathematical proof of an extremely sophisticated kind.
The formulation that I gave above has coarsened the result and
obliterated some of the fine points of its rigorous formulation, but if
one is to state the theorem without having recourse to the difficult
technical language of formal logic this is, I think, the best approximation
that one can achieve.
x Tribute to Dr. Giidel
GODEL actually proved this theorem, not with respect to mathematics
only, but for all systems which permit a formalization, that is a rigorous
and exhaustive description, in terms of modern logic: For no such system
can its freedom from inner contradiction be demonstrated with the means
of the system itself.
GODEL'S second decisive result can only be stated in the terminology
of formal logic and of an important but rather abstruse modern math
ematical discipline: Set theory. Two surmised theorems of set theory, or
rather two principles, the so-called 'Principle of Choice' and the so-called
'Continuum Hypothesis' resisted for about 50 years all attempts of de
monstration. GODEL proved that neither of the two can be disproved
with mathematical means. For one of them we know that it can not be
proved either, for the other the same seems likely, although it does not
seem likely that a lesser man than GODEL will be able to prove this.
I will not attempt a detailed evaluation of these achievements, I will
limit myself to repeat: In the history oflogic, they are entirely singular.
No indemonstrability within mathematics proper had ever been rigorously
established before GODEL. The subject of logic will never again be the
same."
