William Ewald Wilfried Sieg Editors ddaavviidd hhiillbbeerrtt’’ss Lectures s e r on the Foundations u t of Arithmetic and Logic c e L 1917–1933 l a n o i t a d n u o F s ’ t r e b l i H d i v 123 a D 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 © Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. 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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 Preface ThepresentVolumeisthethirdinaseriesofsixpresentingaselectionfrom Hilbert’s previously unpublished lecture notes on the foundations of mathe- matics and physics during the period 1890 to 1933.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 IX