David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917–1933 David Hilbert’s Lectures on the Foundations of Mathematics and Physics, 1891–1933 General Editors William Ewald, Michael Hallett, Ulrich Majer and Wilfried Sieg Vo lu me 1 David Hilbert’s Lectures on the Foundations of Geometry, 1891–1902 Vo lu me 2 David Hilbert’s Lectures on the Foundations of Arithmetic and Logic, 1894–1917 Vo lu me 3 David Hilbert’s Lectures on the Foundations of Arithmetic and Logic, 1917–1933 Vo lu me 4 David Hilbert’s Lectures on the Foundations of Physics, 1898–1914 Classical, Relativistic and Statistical Mechanics Vo lu me 5 David Hilbert’s Lectures on the Foundations of Physics, 1915–1927 Relativity, Quantum Theory and Epistemology Vo lu me 6 David Hilbert’s Notebooks and General Foundational Lectures 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 Montré, al Québec Germany Canada Michael Hallett Department of Philosophy McGill University Montré, al 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 Preface The present Volume is the third in a series of six presenting a selection from Hilbert’s previously unpublished lecture notes on the foundations of mathe- matics and physics during the period 1890 to 1933.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 mathematical activity, and spanning almost the entirety of his teaching career; some are in his own hand, others were carefully worked out by his assistants as 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- markable record, sometimes almost from day to day, of the development of his ideas, and show, in addition, his engagement with the work of other scientific figures of the first rank. The present Volume treats Hilbert’s lectures on logic, 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 greater detail than has been possible from the published record alone: one sees the variety of approaches, the shifts in strategy, and obtains a fuller picture of the motivation for Hilbert’s investigations, as well as of his intellectual rela- tionship to the work of such contemporaries as Russell, Whitehead, Brouwer, and Weyl. The widespread picture of Hilbert as a naïve ‘formalist’ disappears, to be replaced by a much more subtle and nuanced record of the development 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 selection of the texts, the way in which they were edited, and general matters of textual policy. Those matters are uniform for the entire Edition, and we have not repeated the full account here. We do, however, include a description 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 to a reading of the texts. This Volume also reprints from Volume 1 (in slightly 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 X Preface is under the supervision of four General Editors, William Ewald, Michael Hal- lett, Ulrich Majer and Wilfried Sieg, who bear the collective responsibility for editorial policy. For each individual volume, Volume Editors were designated to produce the final selection of texts and to write the scholarly apparatus; this work was carried out in consultation with the General Editors. The des- ignated Editors for this Volume were Ewald and Sieg, in collaboration with Majer and Dirk Schlimm, with Hallett as Associate Editor. As before, the General Editors wish to express their thanks to the Deutsche Forschungsgemeinschaft (DFG) for its generous financial support from 1993 to 2003. To edit even the mere fragment of the voluminous Hilbert Nachlaß that appears in these six volumes required a considerable institutional apparatus located in proximity to the archives in Göttingen. Without the assistance of the DFG, which enabled us to establish a permanent staff in Göttingen, the present Edition could never have been realized. Ulrich Majer, the General Editor who was constantly ‘vor Ort’, supervised the permanent staff and thus had the task of dealing with all the technical problems that an edition of this sort must inevitably face. We again acknowledge the indispensable schol- arly, editorial and technical contributions to the Edition as a whole of Ralf Haubrich, Albert Krayer and Tilman Sauer, all at one time full-time members of the permanent staff. We furthermore thank the Institut für Wissenschaftsgeschichte at the Uni- versity of Göttingen (in particular Lorraine Daston, its former director) for giving the project its first physical home and for recognizing its significance. We are also grateful to the Philosophisches Seminar at the University of Göt- tingen for space and support. Numerous other institutions and individuals provided significant support for the Edition. In Göttingen, from the first, formative stages of the project, we received encouragement and advice from the late Martin Kneser, Samuel Patterson, Günther Patzig and Helmut Rohlfing. The Mathematisches Insti- tut and the Niedersächsische Staats- und Universitätsbibliothek in Göttingen (SUB), the holders of the original Hilbert documents, granted the necessary permission for publication. At an early stage, Martin Kneser was kind enough to place at our disposal his father Hellmuth’s Mitschriften of the lectures ‘Grundlegung der Mathematik’ and ‘Grundlagen der Arithmetik’ given by Hilbert and Bernays in the Winter Semesters 1921/22 and 1922/23 respec- tively, as well as notes for some lectures given by Hilbert in February 1924; Martin Kneser also granted permission for their publication. The manuscripts now form part of an archive of Hellmuth Kneser’s papers in the Handschriften- abteilung of the SUB (signature Cod. Ms. H. Kneser), which has also kindly granted permission for their publication. Warm thanks are also due to Frau Brigitte Peterhans of Chicago for allowing us access to a polished Nachschrift by Walter Peterhans of Hilbert’s 1922/23 lectures. (The Nachschrift is described in some detail below; see pp.