On Formally Undecidable
Of Principia Mathematica
And Related Systems
R. B. BRAITHWAITE
Library of Congress Cataloging-in-Publication Data
[_ber formal unentscheidbare S_tze der Principia Mathematica
und verwandter Systeme I. English]
On formally undecidable propositions of Principia Mathematica
and related systems / Kurt Gödel; translated by B. Meltzer; introduc-
tion by R.B. Braithwaite.
Translation of a paper entitled _ber formal unentscheidbare S_tze
der Principia Mathematica und verwandter Systeme I, published 1931
in the Monatshefte fnr Mathematik und Physik, v. 38, p. 173-198.
Reprint. Originally published: New York: Basic Books, c1962.
ISBN 0-486-66980-7 (pbk.)
1. Gödel's theorem. I. Title.
Kurt Gödel's astonishing discovery and proof, published in 1931, that even
in elementary parts of arithmetic there exist propositions which cannot be proved
or disproved within the system, is one of the most important contributions to
logic since Aristotle. Any formal logical system which disposes of sufficient
means to compass the addition and multiplication of positive integers and zero
is subject to this limitation, so that one must consider this kind of
incompleteness an inherent characteristic of formal mathematics as a whole, which
was before this customarily considered the unequivocal intellectual discipline
No English translation of Gödel's paper, which occupied twenty-five pages of
the Monatshefte für Mathematik und Physik, has been generally available, and
even the original German text is not everywhere easily accessible. The argument,
which used a notation adapted from that of Whitehead and Russell's Principia
Mathematica, is a closely reasoned one and the present translationbesides
being a long overdue act of pietyshould make it more easily intelligible
and much more widely read. In the former respect the reader will be greatly aided
by the Introduction contributed by the Knightbridge Professor of Moral Philosophy
in the University of Cambridge; for this is an excellent work of scholarship in
its own right, not only pointing out the significance of Gödel's work, but
illuminating it by a paraphrase of the major part of the whole great argument.
I proposed publishing a translation after a discussion meeting on
"Gödel's Theorem and its bearing on the philosophy of science", held
in 1959 by the Edinburgh Philosophy of Science Group. I wish to thank this society
for providing the stimulus, the publishers for their ready co-operation on the
proposal, and Professor Braithwaite not only for the Introduction but also for
meticulous assistance in translation and proof-reading of a typographically
intricate text. It may be noted here that the pagination of the original article
is shown in the margins of the translation, while the footnotes retain their
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