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On Formally Undecidable
Propositions
Of Principia Mathematica
And Related Systems


KURT GÖDEL

Translated by
B. MELTZER

Introduction by
R. B. BRAITHWAITE


Library of Congress Cataloging-in-Publication Data

Gödel, Kurt.
    [_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.
    QA248.G573 1992
    511.3-dc20

TO
CHRISTOPHER FERNAU
in gratitude

PREFACE

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 par excellence.

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 translation–besides being a long overdue act of piety–should 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 original numbers.

B. MELTZER

University of Edinburgh
January, 1962


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