David Hilbert's Lectures on the Foundations of Arithmetic and Logic 1917-1933

1082 Pages·2013·6.784 MB·German-English

by  William Ewald, Wilfried Sieg (auth.), William Ewald, Wilfried Sieg (eds.)

William Ewald
Wilfried Sieg
Editors
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David Hilbert’s Lectures
on the Foundations
of Arithmetic and Logic
1917–1933
DavidHilbert’s Lectures onthe Foundations
ofMathematics andPhysics,1891–1933
GeneralEditors
WilliamEwald,MichaelHallett,UlrichMajerandWilfriedSieg
Volume1
DavidHilbert’sLecturesontheFoundationsofGeometry,
1891–1902
Volume2
DavidHilbert’sLecturesontheFoundationsofArithmeticandLogic,
1894–1917
Volume3
DavidHilbert’sLecturesontheFoundationsofArithmeticandLogic,
1917–1933
Volume4
DavidHilbert’sLecturesontheFoundationsofPhysics,1898–1914
Classical,RelativisticandStatisticalMechanics
Volume5
DavidHilbert’sLecturesontheFoundationsofPhysics,1915–1927
Relativity,QuantumTheoryandEpistemology
Volume6
DavidHilbert’sNotebooksandGeneralFoundationalLectures
William Ewald
Wilfried Sieg
Editors
Michael Hallett
Associate Editor
David Hilbert’s Lectures
on the Foundations
of Arithmetic and Logic
1917–1933
in collaboration with
Ulrich Majer and Dirk Schlimm
123
Editors
William Ewald Wilfried Sieg
Law School Department of Philosophy
University of Pennsylvania Carnegie Mellon University
Philadelphia, Pennsylvania Pittsburgh, Pennsylvania
USA USA
Ulrich Majer Dirk Schlimm
Philosophisches Seminar Department of Philosophy
Universität Göttingen McGill University
Göttingen Montréal, Québec
Germany Canada

Michael Hallett
Department of Philosophy
McGill University
Montréal, Québec
Canada
ISBN 978-3-540-20578-4 ISBN 978-3-540-69444-1 (eBook)
DOI 10.1007/978-3-540-69444-1
Springer Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013938947

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This Volume is dedicated to the memory of David Hilbert,
on the occasion of his 151st birthday.

David Hilbert. Courtesy of the Voit Collection in the Manuscript Division of
the Niedersächsische Staats- und Universitätsbibliothek, Göttingen.

Preface
ThepresentVolumeisthethirdinaseriesofsixpresentingaselectionfrom
Hilbert’s previously unpublished lecture notes on the foundations of mathe-
matics and physics during the period 1890 to 1933. The Hilbert Nachlaß
contains approximately eighty lecture notebooks, covering all aspects of his
mathematicalactivity,andspanningalmosttheentiretyofhisteachingcareer;
someareinhisownhand,otherswerecarefullyworkedoutbyhisassistantsas
official protocols of his lectures. Roughly one quarter of these notebooks deal
with foundational subjects. Hilbert’s lecture courses represent an enormous
fund of learning and invention, and embrace almost every subject common
in the mathematical sciences of his time. The notes therefore provide a re-
markablerecord,sometimesalmostfromdaytoday,ofthedevelopmentofhis
ideas, and show, in addition, his engagement with the work of other scientific
figuresofthefirstrank. ThepresentVolumetreatsHilbert’slecturesonlogic,
arithmetic and proof theory from the autumn of 1917 on. During this period
Hilbert actively resumed his research investigations into the foundations of
mathematics, and undertook his intensive collaboration with Paul Bernays,
who wrote up many of the lecture notes reproduced in this Volume. The
period covered here sees the emergence of modern mathematical logic; the
explicit posing of questions of completeness, decidability, and consistency for
logical systems; the investigations of the relative strengths of various logical
calculi; the formulation of the decision problem; the birth of proof theory
and the energetic pursuit by Hilbert and by Bernays of technical work on the
Hilbert consistency programme. These developments can here be followed in
greaterdetailthanhasbeenpossiblefromthepublishedrecordalone: onesees
thevarietyofapproaches,theshiftsinstrategy,andobtainsafullerpictureof
the motivation for Hilbert’s investigations, as well as of his intellectual rela-
tionship to the work of such contemporaries as Russell, Whitehead, Brouwer,
andWeyl. ThewidespreadpictureofHilbertasanaïve‘formalist’disappears,
tobereplacedbyamuchmoresubtleandnuancedrecordofthedevelopment
of Hilbert’s views on the philosophy of mathematics.
The structure of this Edition, the nature, location, and condition of the
Hilbert lecture notes, their provenance, and what we have been able to re-
construct of their history, are all described in the general ‘Introduction to
the Edition’, which is to be found at the beginning of Volume 1 (Hallett and
Majer 2004). That Introduction also explains in detail the criteria for the
selectionofthetexts, thewayinwhichtheywereedited, and generalmatters
of textual policy. Those matters are uniform for the entire Edition, and we
havenotrepeatedthefullaccounthere. Wedo,however,includeadescription
of the textual policies in the section ‘The Editing and Reproduction of the
Texts’; this section is intended to provide all the basic information necessary
toareadingofthetexts. ThisVolumealsoreprintsfromVolume1(inslightly
revised and expanded form) the list of Hilbert’s lecture courses; see pp. 991ff.
That these lectures are finally being published is the result of the efforts,
over two decades, of many individuals and institutions. The series as a whole
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