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

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Springer is part of Springer Science+Business Media (www.springer.com) 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

The present Volume is the third in a series of six presenting a selection from Hilbert’s previously unpublished lecture notes on the foundations of mathe- matics and physics during the period 1890 to 1933.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. 569ff. Thanks are also due to the Folkwangmuseum in Essen, where the manuscript now resides, for permission to quote from it.) We also thank Paul Bernays’s nephew, Ludwig Bernays, for permission to reproduce Bernays’s second Habilitationschrift from 1918, which Preface XI appears as an Appendix to Chapter 1. The Universitätsarchiv of the University of Freiburg kindly granted permission to quote (on p. 291) from a postcard written in 1920 by Bernays to Zermelo, part of the Zermelo Nachlaß kept there. We are also grateful for permission to reprint the various photographs used here. For the first photograph (see p. V), we thank the Voit Collection in the Manuscript Division of the SUB. For the second photograph (p. 273) we thank the late Dorothée Fuchs of Ithaca, New York. She was born in Göttingen, the daughter of the physicist Heinrich Rausch von Traubenberg and the mathematician Marie Rosenfeld, both of whom studied with Hilbert; she later married the mathematician, Wolfgang Fuchs. The photograph was taken in the Hilberts’ garden on the occasion of her receiving a camera for her sixteenth birthday, 27 July 1937. We have been unable to ascertain the origins of the third photograph (p. 653). The two women standing behind Hilbert are said to be the family housekeepers; Hilbert’s wife is to the left. The Institute for Advanced Study in Princeton, through the offices of Harry Woolf and Phillip Griffiths, provided the Editors with a collective work- ing environment in the summer of 1997. The Alexander von Humboldt Stiftung, the Social Sciences and Human- ities Research Council of Canada, McGill University, Carnegie Mellon Uni- versity, the University of Pittsburgh, the University of Pennsylvania Research Foundation, and the John Templeton Foundation have all provided funding for travel and research, or generously supported conferences on the topics of these Volumes. Carnegie Mellon University, the Georg-August Universität, Göttingen and the Universität Bern hosted a series of conferences on Hilbert’s unpublished foundational writings. Catriona Byrne of Springer Verlag has given the Edition abundant support and advice, and has been patient with the inevitable delays; at an earlier stage of the project, similar assistance was provided both by Martin Gilchrist and Elizabeth Johnston of Oxford University Press. A large number of people have been of assistance in various technical and research capacities. For their help we thank: Volker Ahlers, Tobias Brendel, Willem Hagemann, Julia Hartmann, Nina Hehn, Arnim von Helmolt, Stefan Krämer, Pamela Klapproth, Michael Mai, Heiko Schilling, Rebecca Pates, Friedericke Schröder-Pander, Hans-Jakob Wilhelm, and many others. The technical work of Oliver Keller, ably assisted by Stefan Krämer, on the final stages of Volume 1 has also proved indispensable. The typesetting of the book by Hilbert and Ackermann reproduced in Appendix A was mainly done by Andreas Voellmer and Rachel Rudolph, to whom again many thanks. This Volume was originally set up, and the texts processed, under the supervision of Ralf Haubrich, who played an essential role in the design of the editorial apparatus; it was subsequently greatly advanced by Albert Krayer. The later stages of its preparation, organisation and presentation were largely in the hands of Dirk Schlimm, to whom we are enormously indebted.

The General Editors William Ewald, Michael Hallett, Ulrich Majer and Wilfried Sieg

Contents

Preface IX

The Editing and Reproduction of the Texts XVII

Introduction 1 1. 1899–1917: Towards Mathematical Logic...... 4 2. 1917–1920: Logic and Metamathematics...... 9 3. 1920–1922: From Logic Towards Proof Theory...... 12 4. 1922–1925: Finitist Proof Theory...... 17 5. 1925–1931: An Elementary Finiteness Theorem? ...... 21 6. Material Omitted from this Volume...... 29

Chapter 1 Lectures on the Principles of Mathematics (1917/18) 31 Introduction...... 32 ‘PrinzipienderMathematik’...... 59 TextualNotes...... 215 DescriptionoftheText...... 220 Appendix: The Bernays Habilitation Thesis of 1918 ...... 222 Introduction to the Bernays Habilitationsschrift ...... 222 ‘Beiträge zur axiomatischen Behandlung des Logik-Kalküls’ . . . . 231 TextualNotes...... 269 DescriptionoftheText...... 270

Chapter 2 Lectures on Logic (1920) 275 Introduction to the 1920 Lectures ...... 276 ‘Logik-Kalkül’...... 298 TextualNotes...... 336 DescriptionoftheText...... 339 ‘ProblemedermathematischenLogik’...... 342 TextualNotes...... 372 DescriptionoftheText...... 375 IntroductiontotheUndatedDraft...... 378 ‘Consistency Proofs for Fragments of Arithmetic (Undated Draft)’ 396 TextualNotes...... 410 DescriptionoftheText...... 413 Photographic Reproduction of a Page from the Manuscript . . 415

XIII XIV Contents

Chapter 3 Lectures on Proof Theory (1921/22 – 1923/24) 417 Introduction...... 418 ‘Grundlagen der Mathematik’ (1921/22) ...... 431 TextualNotes...... 522 DescriptionoftheText...... 527 ‘Logische Grundlagen der Mathematik’ (1922/23) ...... 528 TextualNotes...... 548 DescriptionoftheText...... 549 ‘Logische Grundlagen der Mathematik’ (1923/24) ...... 550 TextualNotes...... 563 DescriptionoftheText...... 564 Appendix: Kneser’s Mitschriften ...... 565 IntroductiontoAppendix...... 565 Kneser Mitschrift: The lectures from 1921/22 ...... 577 TextualNotes...... 595 DescriptionoftheText...... 598 Kneser Mitschrift: The lectures from 1922/23 ...... 599 TextualNotes...... 631 DescriptionoftheText...... 634 Kneser Mitschrift: The lectures from 1923/24 ...... 636 TextualNotes...... 644 DescriptionoftheText...... 645 Appendix: Bernays’s Note (c. 1923/24) ...... 646 IntroductiontoAppendix...... 646 Bernays’s Note: ‘Wf-Beweis für das logische Auswahlaxiom’ .... 647 TextualNotes...... 651 DescriptionoftheText...... 652

Chapter 4 Lectures on the Infinite (1924/25, 1931, 1933) 655 Introduction...... 656 ‘Über das Unendliche’ (1924/25) ...... 668 TextualNotes...... 757 DescriptionoftheText...... 759 Introduction to ‘Ueber die Grundlagen des Denkens’ ...... 761 ‘Ueber die Grundlagen des Denkens’ (c. 1931) ...... 765 TextualNotes...... 774 DescriptionoftheText...... 775 Photographic Reproduction of Two Pages from the Manuscript 776 Introduction to ‘Über das Unendliche’ of May 1933 ...... 778 ‘Über das Unendliche’ (23.5.1933) ...... 779 TextualNotes...... 783 DescriptionoftheText...... 785

Appendices 787 IntroductiontotheAppendices...... 788 Contents XV

Appendix A: First Edition of Hilbert-Ackermann (1928) ...... 806 Introduction...... 806 Grundzüge der theoretischen Logik ...... 809 TextualNotes...... 916 Appendix B: Hilbert’s Second Hamburg Lecture (1927) ...... 917 Introduction...... 917 ‘Die Grundlagen der Mathematik’ ...... 922 TextualNotes...... 942 Weyl:‘Diskussionsbemerkungen’...... 943 Bernays:‘ZusatzzuHilbertsVortrag’...... 946 Bernays: Letter to Weyl, January 1928 ...... 950 Appendix C: Hilbert’s Bologna Lecture (1928) ...... 954 Introduction...... 954 ‘Probleme der Grundlegung der Mathematik’ ...... 957 TextualNotes...... 966 Appendix D: Hilbert’s Third Hamburg Lecture (1930) ...... 967 Introduction...... 967 ‘Die Grundlegung der elementaren Zahlenlehre’ ...... 974 Appendix E: Hilbert’s 1931 Göttingen Lecture ...... 983 Introduction...... 983 ‘BeweisdesTertiumnondatur’...... 985 TextualNotes...... 990

Hilbert’s Lecture Courses 1886–1934 991

Bibliography 1011

Name Index 1051

Subject Index 1057

The Editing and Reproduction of the Texts

The principal task of this Edition is to present a comprehensive selection from Hilbert’s unpublished lecture notes on the foundations of mathematics and physics. Hilbert left behind a rich collection of writings covering almost the entire span of his teaching career. For the most part, these documents are in Göttingen, either in the Niedersächsische Staats- und Universitätsbibliothek or in the library of the Mathematisches Institut. A description of the archival holdings and of the principles governing the selection of texts for the Edition can be found in the Introduction to the Edition in Volume 1. At the end of the present Volume can be found a list of Hilbert’s lecture courses from 1886 to 1934; this list gives an indication both of the full range of topics covered and of the relevant extant manuscripts. This Volume includes, so far as we are aware, all the surviving, substantive lecture notes on logic and arithmetic, from the autumn of 1917 on, which were prepared under Hilbert’s supervision. A predecessor volume (Volume 2) contains the principal lecture notes up to and including the summer of 1917. For reasons explained in the Introduction, it made sense to split the two logic volumes at the point where Paul Bernays returned to Göttingen as Hilbert’s assistant. Hilbert’s lecture notes fall into three broad categories: (1) Hilbert’s own handwritten notes; (2) official Ausarbeitungen prepared by Hilbert’s assis- tants under his supervision; and (3) unofficial Mitschriften, i. e., classroom notes taken by somebody who attended the lectures or seminar in question. In general, Hilbert’s own handwritten notes—with frequent interlineations, insertions, deletions, corrections and marginal comments—present the great- est textual difficulties. Most of those texts date from the period prior to 1900. However, for later texts, and certainly for those which form the bulk of the present Volume, the situation is relatively favorable. These texts are for the most part polished typescripts (Ausarbeitungen), carefully prepared by Bernays or other close collaborators, with only occasional handwritten ad- ditions by Hilbert or the Ausarbeiter. Except for the Appendices, the texts here are given in chronological order. The chief textual difficulties occur in Hilbert’s undated and untitled draft (the last text in Chapter 2) and his un- titled notes from 1923–1924 (which we have called ‘Logische Grundlagen der Mathematik’), the last document (before the Appendices) in Chapter 3. We refer to the Introductions of the respective chapters for more detailed infor- mation. In order to clarify and date more sharply the developments reported in Chapter 3, we have reproduced as Appendices to that Chapter material

XVII XVIII The Editing and Reproduction of the Texts from Mitschriften of Hilbert’s lectures in the period 1921–1924 composed by Hellmuth Kneser. Roughly half of the Mitschrift for the 1921/22 lectures is given, whereas the other two Mitschriften are presented in full. All other texts given are complete. The texts as published in this Edition are for the most part ‘clear texts’: that is, the originals have been edited to present complete and error-free Ger- man sentences. The guiding editorial principle has been to edit the texts with a light hand: to clean up trivial slips of punctuation or spelling, but to record for the reader any change that could even conceivably be regarded as significant. The alterations to the text itself divide into two categories. (i)A few standard corrections have been carried out without any editorial indica- tion whatsoever. These ‘silent emendations’ include such things as failures to capitalize words or to insert a full stop at the end of a sentence. (ii)Aseries of other corrections has been carried out ‘quietly’, i. e., the changes, although not of any special significance, have nevertheless been recorded in the Textual Notes at the end of each document. In some cases, interventions in the text are signalled through Textual Symbols, a full list of which can be found in the last section below. All deletions, additions, corrections or replacements (of one piece of source text by another) which were deemed to be even potentially significant have been noted on the page itself. In doubtful cases, the policy was always in favour of explicit inclusion, either on the page or in the Textual Notes, so that only those alterations which were utterly of no consequence were executed ‘silently’.

1. Presentation of the Documents We have arranged the texts into chapters containing one or more docu- ments. Each chapter has been given an English title; each document or group of documents is preceded by an Introduction discussing the significance, the contents, the historical and scientific background, and, if necessary, any pe- culiarities of the text. Manuscript pagination is given in the outer page margin; page breaks in the original are indicated by vertical lines ‘|’ in the text. If a page begins with a displayed formula or a new paragraph the vertical line is omitted. The same applies to the publications reproduced in Appendices B–E. Note that page reference to all documents, whether published or unpublished, is always to the original pagination, regardless of whether the document appears in this or any other Volume in the series. The Ausarbeitungen have often been provided with Tables of Contents by their authors, although the documents themselves are usually not separated into sections. As an aid to orientation, these section headings from the Table of Contents have been inserted into the text in angled brackets. In the case of Bernays’s Habilitationschrift given in an Appendix to Chapter 1, a Table of Contents has been added, reflecting the section/sub-section headings in the text. The Editing and Reproduction of the Texts XIX

Footnotes in the source text and Hilbert’s marginal remarks, additions or replacements are rendered in a first series of footnotes separated from the main text by a short horizontal line. This first series of notes also marks uncertain or ambiguous readings. Footnotes by the editors (textual remarks as well as substantive comments) are indicated by superscribed Arabic numerals and printed at the bottom of the page in a second series of notes beneath a full horizontal line. These notes (a) point out textual features which are of immediate significance for a proper understanding of the document, including deleted pieces of text; (b) identify places, persons, and literature cited; (c) refer to differences or similarities in other versions of the document (if those exist); (d) indicate errors in the text; and (e) elaborate on the scientific and historical context. The basic principle is: everything above the line belongs to the text proper; everything below is editorial commentary or deleted text. The list of Textual Notes is attached at the end of each text; as indicated, this list records the less significant differences between the appearance of the text in the original document (including deletions, additions, substitutions, corrections to spelling) and the text as published in this Edition. The Notes are keyed to the text by page and line number; for example ‘63.7’ refers to p. 63 of the Volume, line 7. For this purpose, line numbers for the texts are printed in the inner page margin, every fifth line being counted. After the Textual Notes, there follows a description of the original doc- ument. These descriptions are schematised and include the following details about the document: (a) provenance and means of identification, if any; (b) size and appearance of the cover and/or leading page; (c) composition and pagination; (d) the original title; and (e) whatever information is avail- able on the genesis of the text. These descriptions also describe peculiarities of the document, differences between it and further copies, and so on.

2. Rules for the Constitution of Hilbert’s Annotations In this Edition, Hilbert’s own text is reproduced with especial care, but we follow a slightly more flexible policy for text by his collaborators. It should be noted that, while the Ausarbeitungen were written up by collaborators, they were usually prepared under Hilbert’s direct supervision, and many of these documents actually contain corrections or marginalia in Hilbert’s own hand. These annotations are always explicitly identified and treated with the more stringent policy. Drawings, figures and other sketches, sometimes added or supplemented by Hilbert in order to make the text more intelligible, are inserted or indicated at the appropriate place. These interventions are always noted, either directly on the page or through the Textual Notes. Additions, deletions, and substitutions as well as underlinings are exe- cuted only if they are obviously intended as an immediate correction of the Ausarbeitung, i. e., if they are made by Hilbert in connection with, or clearly shortly after, the composition of the text. In all other cases, Hilbert’s annota- XX The Editing and Reproduction of the Texts tions are reproduced in the footnotes or the Textual Notes according to their significance. Longer remarks and substantial revisions not intended as direct correc- tions, or which were obviously made well after the composition of the text, are treated precisely like Hilbert’s additional remarks in his own manuscripts. Other changes (for instance, Hilbert’s habit of putting parts of the text in brackets) are listed in the Textual Notes but not executed.

3. Rules for the Constitution of Texts Prepared by Hilbert’s Collaborators Because the Ausarbeitungen are relatively ‘clean’ compared with Hilbert’s manuscripts, a number of modifications (mainly simplifications) of the basic editorial practice were deemed desirable. Obvious errors in spelling are corrected silently; in the case of words spelt correctly but differently in different places in the document (e. g., ‘Princip’ and ‘Prinzip’) the ‘rule of overwhelming use’ has been followed, i. e., we adopt that spelling which is used in the overwhelming majority of cases. In typescripts, commas, semi-colons etc. sometimes occur too far away from the words they follow; such errors have been silently corrected. In the case of mathematical expressions, the number of dots ‘...’ continuing a formula is standardised to three. Multiplication points are either omitted altogether or printed in all instances depending on the local use. Additions, deletions, and substitutions executed by the authors of the text are for the most part executed silently. This includes handwritten addition to typescripts of special symbols and scripts not present on the typewriter keyboard. However, changes which seem substantively relevant are noted. Substantial mistakes are not corrected but pointed out in footnotes. Indentations at the beginning of a paragraph are not reproduced exactly as in the original but are standardised. The same is true for formulas, which are reproduced in a standard form. Furthermore, formulas displayed on a separate line are centred following standard practice. In addition to emphasis by underlining, typical for manuscripts, other forms of emphasis occur in typescripts, above all Sperrschrift (a standard form of emphasis in printed German, with extra spacing between the let- ters and words), and vertical lines beside the text, etc. Both underlining and Sperrschrift are reproduced by italics; any combination, like underlining and Sperrschrift together, or double underlining, is not reproduced but noted in the footnotes. The same holds for vertical lines and other forms of emphasis.

4. General Procedure The Layout The typesetting of the source text has meant that the original physical layout is not fully maintained. However, where possible the placement of marginalia is indicated and the original position of diagrams relative to the The Editing and Reproduction of the Texts XXI text has been preserved. The diagrams in the edited text have been recon- structed on the basis of the diagrams in the source text and any accompanying elucidations. Mathematical formulas are printed in italics, in accordance with standard typesetting practice, regardless of whether the formulas in the source text were underlined or not. The handwritten Sütterlin script was not infrequently used in mathematical and logical formulas, though not by Hilbert. Since Sütterlin is a handwritten formal German script, and the Fraktur font is the printed form standardly used in Hilbert’s published texts for the corresponding symbols, we have used the latter to represent any Sütterlin letters. Underlined source text is printed in italics. If an underlining does not extend over a full word or phrase but Hilbert nevertheless clearly intended to emphasize that word or phrase, the full word or phrase is rendered in italics. If a specific feature of underlining (such as underlining by an undulating line or in a different colour) is of possible significance for the text, this will be pointed out in a remark.

Footnotes, Marginal Remarks and Larger Pieces of Text As indicated above, footnotes by the author and marginal remarks are printed immediately below the main body of the text and are separated off from it by a short horizontal line. The marginal remarks are often handwritten additions by Hilbert which reflect on the text and thus cannot be integrated into the text itself. As mentioned, in some cases these remarks were clearly added years after the original text was composed. (Of course, the boundaries between later additions and insertions at the time of composition cannot be precisely drawn, and in some cases editorial judgement was called for.) ∗ ∗∗ For the footnotes, the original style of numbering, in most cases ‘ )’, ‘ )’, ∗∗∗ ‘ )’or‘1’, ‘2’, ‘3’, has been preserved. But to avoid confusion some slight ∗∗ ∗ modifications are sometimes silently introduced, e. g., ‘ )’ instead of ‘ )’. In a ∗ ∗ few cases the source text contains inconsistencies (e. g., ‘ ’inthetextand‘)’ in the footnote). These inconsistencies have likewise been silently eliminated. The marginal remarks are numbered by superscript uppercase Latin let- ters (‘A’, ‘B’, ‘C ’, etc.). These remarks are found mostly in the margins or at the top or bottom of the page, and occasionally on the blank page verso. They can often be assigned with certainty to a particular point in the text, either because of the positioning of the remark or its content, or because of an insertion sign. If such an assignment is not possible, or not possible without ambiguity, the remark is assigned by the editors to an appropriate location, and the ambiguity is recorded. Occasionally, for elucidation, a detailed de- scription of the original appearance of the page has been provided. = Sometimes symbols such as ‘| ’,‘=|’, ‘ ’, etc. appear in the source text. These symbols indicate the place where longer passages, either written on an extra sheet of paper or occuring on another page and marked by the same sym- bol, are to be inserted. Normally, the intended insertion is executed without printing the symbol, though its presence is recorded in the Textual Notes. XXII The Editing and Reproduction of the Texts

Spelling and Punctuation As a general rule, the spelling and punctuation of the source text is pre- served. There are exceptions, however. Spelling. Hilbert’s own lapses from his usual patterns of spelling, slips of the pen and grammatical mistakes are corrected and listed in the Textual Notes. Examples of such mistakes are1: ‘auf einer Geraden gelegen Punkte’ instead of ‘auf einer Geraden gelegene Punkte’, ‘eine Zweite Weise’ instead of ‘eine zweite Weise’, ‘die Nachweis’ instead of ‘der Nachweis’. Variations in, and misspelling of, proper names are uniformized respectively corrected and noted in the Textual Notes. For instance, ‘Paskal’ and ‘Herz’ are corrected to ‘Pascal’ and ‘Hertz’. Hilbert’s idiosyncrasies and variations of orthography, however, are not corrected. Examples are the use of both ‘nicht-Euklidische Geometrie’ and ‘nichteuklidische Geometrie’, ‘Variabele’ and ‘Variable’, ‘Ger- ade’ and ‘Grade’, ‘anderen’ and ‘andern’, ‘gelegene’ and ‘gelegne’. Capitalization and Punctuation. When necessary, the first word of a sen- tence is silently capitalized; any capitalization or decapitalization executed by the editors within a sentence is recorded in the Textual Notes. Full stops are silently added at the end of sentences, and commas and semi-colons are silently inserted whenever this improves readability. To illustrate, in ‘Doch Vorsicht da sie leicht irreleitet’, Hilbert omits the comma after ‘Vorsicht’; this would stand however, since the omission does not affect readability. Hilbert rarely places a comma after ‘d. h.’; for the same reason, this omission remains uncorrected. Inconsistencies in the patterns of punctuation in enumerations and similar constructions are silently corrected. For example, if the writer gives a list of formulas itemised by ‘1.’, ‘2.’ etc. but omits a full stop after one of the numerals, it is silently added. Missing full stops are also silently added to commonly used abbreviations; for example,‘Bd’, ‘z. B’, ‘etc’, ‘d. h’ and ‘wzbw’ become respectively ‘Bd.’, ‘z. B.’, ‘etc.’, ‘d. h.’, ‘w. z. b. w.’. Incor- rect occurrences of full stops, commas and semi-colons in the source text are corrected, the correction being noted in the Textual Notes; an example would be the comma in ‘so giebt es mindestens einen 3ten Punkt, der Geraden’. However, no attempt has been made to impose a consistent scheme of punctuation, and any change of punctuation which may possibly be queried is explicitly noted in the Textual Notes. For mathematical formulas, as opposed to the practice in the case of non- mathematical text, insertion of commas by the editors is noted in the Textual Notes; for example, Hilbert often writes ‘die Punkte ABCliegen’, which would appear here as ‘die Punkte A, B, C liegen’. In enumerations like ‘A, B, ... , Z’, a comma is inserted silently after the ellipsis when missing, and the number of full stops in an ellipsis has been standardized to three. Thus the frequently observed form ‘A, B, .. Z’ would become ‘A, B, ... , Z’. Common abbreviations, such as those mentioned above, are not expanded, but less common ones are; this expansion is listed in the Textual Notes. How-

1All the examples given here and in the rest of this chapter are taken from Hilbert’s own manuscripts on geometry. The Editing and Reproduction of the Texts XXIII

ever, abbreviations in references given in the source text are not expanded; some examples are ‘Math. Vereinig.’, ‘Math. Ann.’, and ‘Math. Zeitschr.’. The abbreviations ‘m’ und ‘n’ are always silently expanded to ‘mm’ and ‘nn’ respectively. Unclear handwriting. Illegible letters or words are indicated in the text by the Textual Symbols ‘?’ (one letter), ‘??’(severalletters)or‘???’ (word). In the case where a word is clearly identifiable, even though not all of its letters are legible (for instance, because they have not been completely formed), the word is silently completed. Where a letter is ambiguous (for instance, a character could be read as either ‘i’ or ‘e’), it is silently transcribed as one or the other depending on what seems to be demanded by the context. Unresolved ambiguities are recorded either through Textual Symbols or in the footnotes or in the Textual Notes.

Omissions and Repetitions Omissions of a word or a phrase or of symbols (like parentheses) have been corrected by the editors; these cases are signalled in the text by angled brackets or, in exceptional cases, recorded in the Textual Notes. Such inter- ventions are made only if the text would be grossly incomplete, incorrect or unintelligible without them. The grammatical errors caused by Repetitions of words are corrected and the corrections recorded in the Textual Notes.

Additions, Deletions and Substitutions Handwritten or typewriten corrections to the text either by addition, dele- tion or substitution are carried out. These are always reported, either directly or through the Textual Notes, in the following cases: (a) Corrections of false starts which indicate a change of intention. For example, in ‘Die Geometrie unterscheidet sich wesentlich von den rein math- ematischen Wissensgebieten wie z. B. [[Al]] Zahlentheorie, Algebra, Funktion- theorie’, it is highly likely that ‘Al’ is the beginning of the word ‘Algebra’, in which case ‘Al’ was deleted in order to put ‘Zahlentheorie’ first. (b) Corrections which enhance the precision of the text. (c) Corrections which modify the content of a sentence. Whenever the changes are significant for the understanding of the text, they are reported directly; additions are marked by means of ↑textual symbols↓, deletions in footnotes, and substitutions by a combination of both. 1 In less significant cases the differences between the published text and the 2 source text are treated quietly and recorded in the Textual Notes. The present 3 paragraph contains illustrations of the method followed for recording these 4 insignificant changes, for which purpose we have printed explicit line numbers 5 in the inner margin. (In the text themselves, numbers only appear every 6 five lines.) Note that the published text itself contains no textual symbol or 7 footnote to indicate an alteration; thus, these relatively minor alterations are 8 only to be found by consulting the Textual Notes. These Notes are normally 9 keyed to the pages and lines of the published text by means of the traditional 10 ‘lemma]’ form. For example: XXIV The Editing and Reproduction of the Texts

XXI.1: published] publ. XXI.6: five] ten XXI.6–7: symbol or footnote] symbol ↑or footnote↓ XXI.7: an alteration] [[a difference]]--↑an alteration↓ XXI.7–10: These ... form.] Added. The word or phrase before the square bracket ‘]’ repeats the reading of the published text, either in full or in abbreviated form; the text after the square bracket reports the source text and/or editorial comments. Note that frequent use is made in the Textual Notes of textual symbols to elucidate the changes made to the text. Completely insignificant corrections are silently executed. Some instances are: Instantaneous corrections so brief that they give no sense of the prelimi- nary intention. For instance, in ‘Um zu dem Gegenstück zu gelangen, bedarf [[d]] es der elementaren Sätze aus der Geometrie’, the deleted ‘d’ would be silently omitted. Corrections of misspelling or false punctuation which do not indicate a change in the sense of the passage. For instance, in ‘Beide Flächen schneiden sich in einer [[B]]--↑b↓estimmten Schraubenlinie’, the information encoded in ‘[[B]]--↑b↓’ would be omitted and the word printed would be ‘bestimmten’. Corrections of non-mathematical text due to slips where there is no signifi- cant change in meaning. For example, in ‘Die Schrauben[[linie]]fläche ent- hält alle Schraubenlinien vom nämlichen h’, the context indicates that ‘Die Schraubenlinie’ would have made no sense and that therefore the writing of ‘-linie’ was a slip. The same holds when there is a change of intention, but one of only minor importance. In the example ‘... eine Reihe bedeutender Män- ner, ... aus deren [[Untersuchun]] Händen die heutige Geometrie der Lage ... hervorging’, the change from ‘Untersuchungen’ to ‘Händen’ is merely stylistic and so ‘Untersuchun’ is silently omitted. Corrections of mathematical passages (whether composed of symbols, or of words), which do not affect the sense. One example is given by ‘folglich schneiden sich AA und BB ↑etwa in D↓’. This is rendered in Volume 1 as ‘folglich schneiden sich AA und BB,etwainD’, since Hilbert employs the letter ‘D’ in the very next sentence to denote the point of intersection in question. Another example would be the rendering of ‘Winkel ↑t↓’as ‘Winkel t’, on the grounds that although ‘Winkel’ was first written without the term ‘t’, the context makes it clear that the angle denoted elsewhere by ‘t’ is intended. Mistakes of punctuation arising from the execution of an alteration are silently corrected. For example, ‘Linie. ↑und umgekehrt↓’ would be printed as ‘Linie ↑und umgekehrt↓.’ And with ‘ML schneidet dann alle Strahlen c, d, ... k. [[Ist c der]] in C, D, ..., K’, the ‘[[Ist c der]]’ would be silently omitted. For source text reading ‘P1, P2, P3, ↑und wenn alle Punkte auf einer Seite eines Punktes A liegen↓ ... so giebt ...’, the addition would be executed after the ellipsis and not before as in the original. The Editing and Reproduction of the Texts XXV

Text Obscured by Pasting Lastly, as with all manuscripts, an attempt has been made, wherever pos- sible, to decipher text which was pasted over. Such passages are described in the Textual Notes (if very short) and in footnotes (if the passage extends over several lines).

Textual Symbols The textual symbols used in these volumes are the following: | Page break in the source text; the number of the new page is printed in the outer margin. comment Editioral insertion. ... Ellipsis. dubious Unsafe reading of words or letters (‘dubious’ is the unsafe word). ? Unreadable letter. ?? Several unreadable letters. ??? Unreadable word. ↑addition↓ Addition by the author (‘addition’ is the word added). [[deletion]] Deletion by the author (‘deletion’ is the word deleted). [[ori]]--↑sub↓ Substitution by the author (‘ori’ is replaced by ‘sub’).

William Ewald, Michael Hallett, Ulrich Majer and Wilfried Sieg

Introduction∗

The second Volume in this series concludes with Hilbert’s lectures on set theory from the Summer Semester of 1917. Near the end of those lectures, which finished around 15 August, Hilbert remarks without any elaboration that next semester ‘I hope to be able to go more deeply into the foundations of logic’. The present Volume picks up the story some six weeks later, begin- ning with the 1917/18 Winter Semester lectures ‘Prinzipien der Mathematik’, and ending with a manuscript of a lecture on the infinite from 1933, right at the end of Hilbert’s active research career. The years covered by this Vol- ume saw the most energetic phase of his collaboration with Paul Bernays, the development of axiomatic investigations of logic and arithmetic, the birth of proof theory, and the beginnings of work on the Entscheidungsproblem. These projects were pursued not only by Hilbert and Bernays themselves, but also in collaboration or interaction with, among others, Wilhelm Ackermann, Hein- rich Behmann, Gerhard Gentzen, Jacques Herbrand, Johann von Neumann, Moses Schönfinkel and Hermann Weyl. In reading the lecture notes, one should not lose sight of their connection to the wider world of Göttingen mathematics, and it is important to remem- ber that Hilbert always emphasized the connections between foundational questions, questions in the mainstream of mathematics, and questions arising from developments in the natural sciences. The period covered in this volume was a time of intellectual ferment in Göttingen. Noether and her student van der Waerden were laying the groundwork for the new abstract algebra. In physics, the discoveries of Einstein had been absorbed, and physicists like Max Born now turned their attention to the mathematical elaboration of the new quan- tum mechanics of Heisenberg and Jordan. Hilbert was deeply involved in all these developments and taught or co-taught (with Bernays, with Noether and with Debye, with Courant, and especially with Born) over two dozen seminars

∗This Introduction and several of the later Introductory Notes have made use of the detailed material and broad considerations set out in Sieg 1999 and Sieg 2002 ,which provide a continuous chronological account of the development of Hilbert’s foundational ideas during the years from the late eighteen nineties to the early nineteen thirties. The latter paper and its companion piece Sieg 1990 explore an extension of Hilbert’s Program and its connection to contemporary proof-theoretic work, but also its roots in the nineteenth century transformation of mathematics. These roots are found, in particular, in Dedekind’s work, see Sieg and Schlimm 2005 ,andalsoSieg 2009 .

1 2 Introduction and lecture courses that deal with them.1 The mathematical culture of Göt- tingen in those years was remarkably broad. Weyl moved as freely as Hilbert himself between pure mathematics, mathematical physics and foundations. Born had written up detailed notes for Hilbert’s 1904 lectures ‘Zahlbegriff und Quadratur des Kreises’ and the lectures from 1905 ‘Logische Principien des mathematischen Denkens’, and Courant had done the same for the 1910 lec- tures ‘Elemente und Prinzipienfragen der Mathematik’. Bernays attended the algebra lectures of Noether and van der Waerden; Herbrand visited Göttingen in early 1931 to interact with Bernays, but also with Emmy Noether; Hilbert commonly met Bernays and his physics assistant, Adolf Kratzer, simultane- ously, since each was expected to be able to participate in his discussions with the other. This broad background is a salient aspect of Hilbert’s intellectual life, and his lectures on logic should not be viewed as pursuing an isolated technical programme, but as being part of a much richer investigation of the foundation of the mathematical sciences. In keeping with the influential accounts in Weyl 1944 and Bernays 1935 , it is usual to divide Hilbert’s work in foundations of mathematics into two distinct phases. On this account, the first phase lasts from roughly 1899 to about 1904, when he was mostly occupied with the axiomatics of geometry and the consistency of arithmetic. The second phase is then taken to be- gin in early 1921 with talks Hilbert gave in Copenhagen and Hamburg; the contents of these talks is conveyed in the paper ‘Neubegründung der Mathe- matik’, a vigorous response to the contemporaneous writings of Brouwer and Weyl. During this second phase, Hilbert developed proof theory and pursued his quest for what he called a finitist consistency proof of arithmetic. Hilbert is charged also with having adopted a ‘formalist’ philosophy of mathematics, holding (in the words of Ramsey 1926 ) that the propositions of mathematics are ‘meaningless formulae to be manipulated according to certain arbitrary rules’, that ‘mathematical knowledge consists in knowing what formulae can be derived from what others’, and that the term ‘2’ is ‘a meaningless mark occurring in these meaningless formulae’2. This second phase allegedly cul- minates in two co-authored books. The Grundzüge der theoretischen Logik, written with Wilhelm Ackermann and published in 1928, presents mathemat- ical logic in its definitive modern form, while the two volume Grundlagen der Mathematik from 1934 and 1939, written with Paul Bernays, provides an en- cyclopædic synthesis of foundational investigations, and in particular of work in proof theory. Hilbert’s research publications on foundational matters do indeed cease with his Heidelberg talk of August 1904 (published as Hilbert 1905b), and the publication of substantive novel work does not begin until 1922. But the lecture notes tell a more complex story about the development of his thought. That development is dramatically different in important respects from the

1To gain some idea of this breadth, see ‘Hilbert’s Lecture Courses’ beginning on p. 991 below. 2See Ramsey 1926 , 339, pp. 2 and 153 respectively of the later reprintings. Introduction 3 standard account. Hilbert continued to lecture on fundamental matters in- cluding, importantly, the foundations of mathematics, throughout the ‘fallow’ period from 1904 to 1922, delivering almost every year a lecture course on foundational subjects. The standard account is, however, correct that his research can be divided into two phases. But the break occurs, not in the early 1920s, but during the summer of 1917, precisely between the lectures on set theory and the 1917/18 lectures on logic. Furthermore, there is no indication whatsoever that the new investigations were initiated as a reaction to Brouwer or Weyl. Rather, what was clearly of great importance in the new examination of logic was the detailed study of Whitehead and Russell’s Principia Mathematica. Two events during the short summer vacation of 1917 signal the begin- nings of a turn in Hilbert’s foundational research. Hilbert, as was his custom, travelled to Switzerland; he delivered his programmatic lecture ‘Axiomatisches Denken’ on 11 September to the Swiss Mathematical Society in Zürich. There he invited Paul Bernays, a promising young mathematician with strong philo- sophical interests, to return to Göttingen as his assistant for the foundations of mathematics.3 Over the next seven years, a new approach to foundational is- sues was to evolve in a remarkable series of lecture courses. Most of them were developed in cooperation with Bernays, who also bore the major responsibil- ity of writing up formal protocols of the lectures. These are: ‘Prinzipien der Mathematik’ (Winter 1917/18); ‘Logik-Kalkül’ (Winter 1920); ‘Probleme der mathematischen Logik’ (Summer 1920); ‘Grundlagen der Mathematik’ (Win- ter 1921/22); and ‘Logische Grundlagen der Mathematik’ (Winters 1922/23 and 1923/24); all of the lectures are included here. In this Introduction, we give a broad overview of the early development of Hilbert’s Programme that can be reconstructed only on the basis of these lecture notes. More details and relations to later work in the nineteen-twenties will be found in the in- troductory notes to the individual chapters. It is often said that Hilbert in the 1920s was pursuing ‘the Hilbert Pro- gramme’, which aimed at securing the foundations of analysis against the attacks of Brouwer and Weyl; that he and his students pursued ‘the’ pro- gramme with great technical skill, but that ‘it’ was finally refuted by Gödel’s incompleteness results. Such an account overlooks the complex array of con- siderations that informed Hilbert’s foundational work, the extreme flexibility of his thought, and the way his approach developed and evolved out of his other mathematical preoccupations (and in particular his work on geometry in the 1890s). In interviews he gave in 1977 (see Bernays 1977* ), Bernays points out that a great scientific research programme like Hilbert’s typically develops over a period of years, with a good many false starts. And indeed, what we find in the lecture notes is not a single, dogmatic idée fixe, but rather a collection of insights and strategies that change over time. There is a cre- ative, fluid character to the lecture notes, a groping towards solutions, and a

3See the Introduction to Chapter 1, n. 9. 4 Introduction responsiveness to new ideas. It is hoped that the documents presented here will help to reveal some of that growth and richness.

1. 1899–1917: Towards Mathematical Logic. Hilbert had long viewed the axiomatic method as holding the key to a sys- tematic organization of any sufficiently developed subject. He saw it also as the basis for metamathematical investigations of independence and complete- ness, as well as for foundational and mathematical reflections. The problem of consistency had been of central importance ever since he turned his attention to geometry and the foundations of analysis in the late 1890s. For example, Hilbert stresses in both his ‘Über den Zahlbegriff’ (Hilbert 1900a)andthe subsequent Paris address on mathematical problems (Hilbert 1900b), that a proof of consistency is necessary to underwrite the legitimacy of any axiom system and to establish the existence of its subject-matter. The nineteenth century roots of Hilbert’s work are extremely important; they link his focus on consistency to broader themes within mainstream mathematics, and re- veal the major intellectual forces that led him to the initial formulation of a syntactic consistency programme in his Heidelberg address of 1904.4 Consistency had been viewed as a semantic notion by Dedekind and by logicians in the nineteenth century more generally; in his famous monograph on the natural numbers (Dedekind 1888 ) Dedekind had quite explicitly at- tempted to show — by providing a logical model — that his notion of a simply infinite system does not contain ‘internal contradictions [innere Wider- sprüche]’ (see Dedekind’s letter to Keferstein of 1890 (Dedekind 1890* ). Hil- bert formulated consistency as a syntactic notion in ‘Über den Zahlbegriff’, and also in Grundlagen der Geometrie. That does not mean, however, that he sought to prove consistency by syntactic methods: the (relative) consistency proofs given in Grundlagen der Geometrie are all straightforwardly semantic, using arithmetic models, although information about the possibility or im- possibility of proofs is extracted from the semantic arguments. However, to prove the consistency of arithmetic itself, Hilbert thought ‘a suitable modifi- cation of known methods of inference’ was needed5; and in his Paris lecture on mathematical problems he suggested that a ‘direct method’ of proof could be found:

4Let us mention stenographically: Dedekind, consistency concerns and semantic argu- ment (Dedekind 1888 and Dedekind 1890* ); Kronecker, emphasis on a thoroughly con- structive approach; Cantor, letters to Hilbert and Dedekind on inconsistent multiplicities from 1897 and 1899; Hilbert, from semantic to syntactic arguments (Hilbert 1900b; Hil- bert 1905b). These connections are discussed by Sieg 1990 , Sieg 1997 , Sieg 2002 , Sieg and Schlimm 2005 , Ewald 2005 , Stein 1988 , Hallett 1994 , Hallett 1995 and Ferreirós 1999 . Many of the primary works are translated in Ewald 1996 ; additional important secondary literature is referred to in all of these studies. 5The phrase quoted comes from this passage of the original German: Um die Widerspruchslosigkeit der aufgestellten Axiome zu beweisen, bedarf es nur einer geeigneten Modification bekannter Schlußmethoden. (Hilbert 1900a, 184.) Introduction 5

I am convinced that it must be possible to find a direct proof for the consis- tency of the arithmetical [real number] axioms by means of a careful study and suitable modification of the known methods of reasoning in the theory of irrational numbers.6 Hilbert believed, it seems, that the genetic ‘construction’ of the real numbers, could somehow be exploited to yield a consistency proof in Dedekind’s ‘logi- cist’ style. This conjecture is supported to some extent by Hilbert’s treatment of arithmetic in contemporaneous lectures, but also by a more methodologi- cal statement from the Einleitung of the notes for the lectures on Euclidean geometry of 1898/99. He maintains there: It is important to fix precisely the starting point of our investigations: as given, we consider the laws of pure logic and in particular all of arithmetic. (On the relation between logic and arithmetic, cf. Dedekind, Was sind und was sollen die Zahlen?)7 And, clearly, for Dedekind, arithmetic is part of logic. It appears that Hilbert changed his basic attitude only after the discovery of the elementary contradiction of Russell and Zermelo. That paradox con- vinced him that there was a deep problem, that difficulties appeared already at an earlier stage, and that the issue had to be addressed in a different way. In late 1904, Hilbert sent a letter to his colleague and friend Adolf Hurwitz, in which he says: It seems that the most various parties are now again taking up the inves- tigation of the foundations of arithmetic. It has been my view for a long time that exactly the most important and most interesting questions were not settled by Dedekind and Cantor (and a fortiori not by Weierstrass and Kronecker). In order to be forced into the position of having to reflect on these matters in a systematic way, I have announced a lecture course on the ‘logical foundations of mathematical thought’ for next semester.8

6Hilbert 1900b, 265, 1104 of the English translation. In the original German, the passage reads: Ich bin nun überzeugt, daß es gelingen muß, eine direkten Beweis für die Widerspruchs- losigkeit der arithmetischen Axiome zu finden, wenn man die bekannten Schlußmethoden in der Theorie der Irrationalzahlen im Hinblick auf das bezeichnete Ziel genau durcharbeit und in geeigneter Weise modificiert. In Bernays’s review of Hilbert’s foundational work from 1935, one finds a similar remark in the discussion of these early foundational investigations: Zur Durchführung des Nachweises gedachte Hilbert mit einer geeigneten Modifikation der in der Theorie der reellen Zahlen angewandten Methoden auszukommen. (Bernays 1935 , 198–199.) 7Hilbert 1899* , 2. The German text is found in Hallett and Majer 2004 , 303: Es ist von Wichtigkeit, den Ausgangspunkt unserer Untersuchungen genau zu fixieren: Als gegeben betrachten wir die Gesetze der reinen Logik und speciell die ganze Arithmetik. (Ueber das Verhältnis zwischen Logik und Arithmetik vgl. Dedekind, Was sind und was sollen die Zahlen?) 8The letter is to be found in the Handschriftenabteilung of the Staats- und Universitäts- bibliothek of the University of Göttingen, under the signature Cod. Ms. Math. Arch. 76:324. It is undated. However, Hilbert did indeed hold a lecture course in the Summer Semester of 1905 entitled ‘Logische Principien des mathematischen Denkens’ (i. e., Hilbert 1905a* , 6 Introduction

Lecture notes from the following summer term contain more informative re- marks on Dedekind’s ‘epochal’ achievements, but pointedly insist that funda- mental difficulties remain: He [Dedekind] arrived at the view that the standpoint of considering the integers as self-evident cannot be sustained; he recognized that the difficulties Kronecker saw in the definition of irrationals arise already for the integers; furthermore, if they are removed here, they disappear there. This work [Was sind und was sollen die Zahlen?] was epochal, but it furnished nothing definitive; certain difficulties remain. As in the definition of the irrational numbers, the difficulties are here connected above all with the concept of the infinite; ...9 This insight sets the stage for Hilbert’s Heidelberg talk of 1904, a talk in which he indicates a novel way of solving the consistency problem for arith- metic, taken now in a much more restricted sense, namely, the arithmetic of the natural numbers alone. Note, however, Hilbert’s expectation that if the problems are resolved for the natural numbers, then they will also be resolved for the reals. The consistency of the Heidelberg system would guarantee, as Hilbert puts it, ‘the consistent existence of the so-called smallest infinite’10. The system, reproduced in Volume 2 of this series), and internal evidence from the letter suggests that it was written shortly before the turn of the year. These facts date it with high probabil- ity to the last few days of 1904. The letter is partially reproduced in Appendix XLIX of Dugac 1976 (and likewise dated 1904), an Appendix which contains excerpts from several letters of Hilbert to Hurwitz; the passage cited is to be found on pp. 271–272. In the original German, it reads: Die Beschäftigung mit den Grundlagen der Arithmetik wird jetzt, wie es scheint, wieder von den verschiedensten Seiten aufgenommen. Dass gerade die wichtigsten u. interessantesten Fragen von Dedekind und Cantor noch nicht (und erst recht nicht von Weierstrass und Kronecker) erledigt worden sind, ist eine Ansicht, die ich schon lange hege und, um einmal in die Notwendigkeit versetzt zu sein, darüber im Zusammenhang nachzudenken, habe ich für nächsten Sommer ein zweistündiges Colleg über die „logischen Grundlagen des math. Denkens“ angezeigt. 9The lectures were ‘Zahlbegriff und Quadratur des Kreises’. The passage cited is from the Ausarbeitung by Max Born, i. e., Hilbert 1904* , 166. In the original German, the passage reads: Er drang sich zu der Ansicht durch, dass der Standpunkt mit der Selbstverständlichkeit der ganzen Zahlen nicht aufrecht zu erhalten ist; er erkannte, dass die Schwierigkeiten, die Kronecker bei der Definition der irrationalen Zahlen sah, schon bei den ganzen Zahlen auftreten und dass, wenn sie hier beseitigt sind, sie auch dort wegfallen. Diese Ar- beit war epochemachend, aber sie lieferte doch noch nichts definitives, es bleiben gewisse Schwierigkeiten übrig. Diese bestehen hier, wie bei der Definition der irrationalen Zahlen, vor allem im Begriff des Unendlichen; ... 10The remark comes from the following passage: Indem wir die bekannten Axiome für die vollständige Induktion in die von mir gewählte Sprache übertragen, gelangen wir in ähnlicher Weise zu der Widerspruchsfreiheit der so vermehrten Axiome, d. h. zum Beweise der widerspruchsfreien Existenz des sogenannten kleinsten Unendlich (d. h. des Ordnungstypus 1, 2, 3, ...). (Hilbert 1905b, 181.) Hilbert has a footnote to the word ‘Unendlich’, which states: Vgl. meinen auf dem Internationalen Mathematiker-Kongreß zu Paris 1900 gehaltenen Vor- trag: Mathematische Probleme, 2. Die Widerspruchslosigkeit der arithmetischen Axiome [i. e., Hilbert 1900b, 264–266]. Introduction 7 properly formulated, would consist of axioms for identity and Dedekind’s re- quirements for a simply infinite system; the induction principle (which follows from Dedekind’s minimality condition) is mentioned, but neither formulated properly nor treated in the consistency argument. Using modern notation, the axioms can be stated in this way: (1) x = x (2) x = y ∧ W (x) → W (y) (3) x = y → x = y (4) x =1 . The rules, implicit in Hilbert’s description of ‘consequence’, are modus ponens together with a substitution rule allowing replacement of variables by arbitrary sign combinations. Other ‘modes of logical inferences’ are alluded to, but not stated explicitly and, consequently, not incorporated into the consistency proof. Finally, the view that the problem for the reals is resolved once matters are settled for the natural numbers is strongly reemphasized in the Heidelberg address. Hilbert claims: The existence of the totality of real numbers can be proved in a way similar to that of the existence of the smallest infinite. Indeed, the axioms which I have given for the real numbers can be expressed by precisely such formulas as the axioms just given. In the same paragraph, Hilbert continues: ... and the axioms for the totality of real numbers do not differ qualitatively in any respect from, say, the axioms necessary for the definition of the inte- gers. In the recognition of this fact lies, I believe, the real refutation of the conception of arithmetic associated with L. Kronecker ...11 For the consistency proof, Hilbert formulates the property of homogeneity: an equation a = b is called homogeneous if and only if a and b have the same number of symbol occurrences. It is easily seen, by induction on derivations, that all equations derivable from axioms (1)–(3) are homogeneous. A con- tradiction can be obtained only by establishing an unnegated instance of (4) from (1)–(3); such an instance is necessarily inhomogeneous and, thus, not provable. Hilbert comments: The considerations just sketched constitute the first case in which a direct proof of consistency has been successfully carried out for axioms, whereas the method usual in such proofs, particularly in geometry, of some suitable specialization or of the construction of examples, necessarily fails here.12

11The passages are both from Hilbert 1905b, 185. In the original German, they are: Ähnlich wie die Existenz des kleinsten Unendlich bewiesen werden kann, folgt die Existenz des Inbegriffs der reellen Zahlen: in der Tat sind die Axiome, wie ich sie für die reellen Zahlen aufgestellt habe, genau durch solche Formeln ausdrückbar, wie die bisher aufgestell- ten Axiome. and ... und die Axiome für den Inbegriff der reellen Zahlen unterscheiden sich qualitativ in keiner Hinsicht etwa von der zur Definition der ganzen Zahlen notwendigen Axiome. In der Erkenntnis dieser Tatsache liegt, wie ich meine, die sachliche Widerlegung der von L. Kron- ecker vertretenen ... Auffassung der Grundlagen der Arithmetik. 12Hilbert 1905b, 181. In the original German, the passage reads: 8 Introduction

In the Heidelberg talk, Hilbert stresses that the goal is to develop logic and mathematics simultaneously, but the actual work has significant shortcom- ings: there is no calculus for sentential logic; there is no proper treatment of quantification; and induction is not incorporated. In sum, while there is an important shift from semantic arguments to syntactic ones, the set-up is woefully inadequate as a formal framework for mathematics. The foundational import of the approach adopted by Hilbert in the Hei- delberg lecture was incisively challenged, above all by Poincaré. As is well- known, the main objection centred on the inductive character of the consis- tency proof, and on this, Poincaré criticized Hilbert’s approach severely and justly. But Poincaré brought out additional methodological shortcomings in Hilbert’s paper. His analysis shifted Hilbert’s attention, not away from foun- dational concerns per se (for these are well documented in lecture courses held throughout the period from 1905 to 1917), but rather from the specific and novel syntactic approach advocated in the Heidelberg talk. Indeed, the notes for Hilbert’s course on ‘Mengenlehre’ in the Summer Semester of 1917 and the talk ‘Axiomatisches Denken’ (Hilbert 1918a) reveal a logicist tendency in his work. Hilbert says there: Since the examination of consistency is a task that cannot be avoided, it appears necessary to axiomatize logic itself and to establish that number theory and set theory are only parts of logic.13 In this paper, Hilbert reviews the successes of the axiomatic method through- out mathematics. Turning to foundations, he praises Frege and ‘the acute mathematician and logician Russell’, saying that: One could regard the completion of this magnificent Russellian enterprise of the axiomatization of logic as the crowning achievement of the work of axiomatization as a whole. He continues, But this completion will require further work. When we consider the matter more closely we soon recognize that the question of the consistency of the integers and of sets is not one that stands alone, but that it belongs to a vast domain of difficult epistemological questions, which have a specifically mathematical tint: . . . 14

Die eben skizzierte Betrachtung bildet den ersten Fall, in dem es gelingt, den direkten Beweis für die Widerspruchslosigkeit von Axiomen zu führen, während die sonst — insbesondere in der Geometrie — für solche Nachweise übliche Methode der geeigneten Spezialisierung oder Bildung von Beispielen hier notwendig versagt. 13Hilbert 1918a, 412. The original German reads: Da aber die Prüfung der Widerspruchslosigkeit eine unabweisbare Aufgabe ist, so scheint es mir nötig, die Logik selbst zu axiomatisieren und nachzuweisen, daß Zahlentheorie, sowie Mengenlehre nur Teile der Logik sind. 14Hilbert 1918a, 412. In the original German, the passages read: In der Vollendung dieses großzügigen Russellschen Unternehmens der Axiomatisierung der Logik könnte man die Krönung des Werkes der Axiomatisierung überhaupt erblicken. And: Diese Vollendung wird indessen noch neuer und vielseitiger Arbeit bedürfen. Bei nähe- rer Überlegung erkennen wir nämlich bald, daß die Frage der Widerspruchslosigkeit bei den Introduction 9

He then gives examples of the kind of questions he has in mind, questions that were to occupy him for the next fifteen years, and that he tacitly regards as meta-systemic questions about various axiomatic systems: the problems of consistency, decidability, and provability within mathematics and logic. (See Hilbert 1918a, 412–413.) Far from being a dogmatic formalist, Hilbert appears at this time, as in the earlier phase of his foundational reflections under Dedekind’s influence, to have been (at least) tempted by the logicist programme.

2. 1917–1920: Logic and Metamathematics. In retrospect, Hilbert’s brief remarks at the end of the Zürich address can be seen as his first public announcement of the material he was to begin teaching three weeks later. No doubt the central ideas had been incubating for much longer. The detailed examination of Russell’s writings and of Principia Mathematica started around 1913, and the core idea of posing precise, meta- mathematical questions about various axiom systems goes back even further to the preparatory studies in the 1890s for his Grundlagen der Geometrie. And yet, neither in the 1917 set theory lectures, nor even in the first weeks of the 1917/18 lectures, is there any indication of the path he was planning to pursue, or of how he proposed to extend the axiomatic method into the realm of logic. Suddenly, probably in early November, his lectures change course. Until that point, his systematic presentation of geometry and the axiomatic method was not radically different from what we find in earlier years, but now he begins systematically to unveil to his students the modern conception of mathematical logic. Over the coming weeks he presents a graduated sequence of axiomatic logical calculi, starting with propositional logic, and ending with (in effect) full second-order logic based on the ramified theory of types with the Axiom of Reducibility. He carefully distinguishes questions that arise within the various axiomatic systems from questions about those systems. Indeed, these lectures mark the start of the metamathematical study of formal logical systems. They explic- itly pose an array of questions that were foreign to the logical tradition of Frege-Whitehead-Russell and also to the algebraic tradition of Boole-Peirce- Schröder. The logical and mathematical questions are driven from the very beginning by philosophical reflections on the foundations of mathematics, and it is noteworthy that every step taken in expanding the logical framework is semantically motivated. After developing first-order logic, Hilbert turns to higher-order logic as a foundation for analysis; detailed reflection on the paradoxes leads him ‘in the most natural way’ to type theory.15 The formal framework of type theory is seen, however, as too narrow for mathematics,

ganzen Zahlen und Mengen nicht eine für sich alleinstehende ist, sondern einem großen Be- reiche schwierigster erkenntnistheoretischer Fragen von spezifisch mathematischer Färbung angehört: . . . 15See the 1917/18 notes, p. 220, below p. 199. 10 Introduction because it does not, for example, allow the proper formalization of Cantor’s proof of the existence of uncountable sets. To achieve greater flexibility, Rus- sell’s Axiom of Reducibility is adopted. This broader framework is then used to develop the beginnings of analysis and, in particular, to establish the least upper-bound principle. The notes end with the remark (p. 246): Thus it is clear that the introduction of the Axiom of Reducibility is the appropriate means to turn the calculus of types into a system out of which the foundations for higher mathematics can be developed.16 The protocol for Hilbert’s 1917/18 lectures forms the basis of Chapter 1 of this Volume. In contrast to any earlier text in the history of logic, Hilbert’s 1917/18 lectures could still provide, almost a century later, a satisfactory first introduction to mathematical logic, perhaps with some modest supplementa- tion. In this sense, the lectures are an historical milestone, fully comparable in importance to Frege’s Begriffsschrift and to Whitehead and Russell’s Prin- cipia Mathematica. The treatment of the central themes in these lectures is substantially identical with the account found in Hilbert and Ackermann 1928 , a work that has hitherto been taken to lie near the end of Hilbert’s second phase of foundational research; indeed, a comparison of the two texts reveals that large parts of the 1928 book are simply taken verbatim from Bernays’s notes for the 1917/18 lectures. Hilbert’s crediting of Ackermann with co- authorship of one of the pivotal texts in the history of logic is both remark- ably generous and at the same time somewhat misleading about the extent of Ackermann’s limited role. To facilitate comparisons between these two texts, and because the first edition of the book by Hilbert and Ackermann is now hard to obtain, we have reproduced it as Appendix A to this Volume. The protocol for the 1917/18 lectures was written by Bernays, who in July of 1918 submitted his (second) Habilitationschrift, in which he analyzed the propositional logic of Principia Mathematica. Bernays’s thesis grew directly out of the Hilbert lectures and contains several important advances: a sharper formulation of the semantic completeness theorem for propositional logic (‘Ev- ery valid formula is provable and, conversely, every provable formula is valid’); a careful, model-theoretic investigation of the independence and dependence of various groups of propositional axioms; and an investigation of ways in which axioms can be replaced by rules of inference. Some of these techniques were to be used by Hilbert in subsequent lectures. Bernays’s thesis, previously unpublished17, is so intimately bound up with the developments chronicled in this Volume that we have included it as an Appendix to Chapter 1 with its own introduction. For two years after the 1917/18 lectures, Hilbert appears to have been occupied with other matters. Though he gave a popular course of lectures in

16For the original German, see below, p. 214. 17Some of the investigations were published later in Bernays 1926 , though this paper by no means captures the full richness of Bernays’s research. In particular, the completeness result is not mentioned. Introduction 11

1919 entitled ‘Natur und mathematisches Erkennen’18, his attention was fo- cused primarily on mathematical physics. The 1917/18 lectures were engaged with the logic of Principia Mathematica and do not contain any specifically proof-theoretic considerations. The names of Brouwer and Weyl are nowhere mentioned. But in the two lecture courses he delivered in 1920, Hilbert now explicitly rejects the logicist route, and begins to address the foundational views of Brouwer and Weyl. The response to Brouwer and Weyl comes in two distinct steps: first, Hilbert presents a strictly finitist version of number theory; and then, when that enterprise is shown to lead to a dead end, he takes up the suggestion, broached in the 1904 Heidelberg and 1917 Zürich addresses, of developing a theory of (syntactic) proofs. The ‘Logik-Kalkül’ lectures from the Winter Semester of 1920 recapitu- late some of the logical material from the 1917/18 lectures and then turn to a completely different topic, attempting to secure the foundations of ordinary number theory by adopting a radically constructive point of view. From 1905 onwards, Hilbert had repeatedly criticized Kronecker for not being sufficiently radical19, and pointed out that even the commutative law of arithmetic could not be justified by finitely many instances. Constructive arithmetic, as pre- sented here, is developed on a basis stricter than that which appears a little later as finitist mathematics. It is stricter because the directly meaningful part consists only of equations between specific numerals or, more generally, closed terms; the intuitive general concept of numeral is not yet assumed, and equations like x + y = y + x are given a constructive and extremely rule-based interpretation. For this interpretation, tertium non datur does not hold, as Hilbert and Bernays recognized; consequently, this approach cannot secure the foundations of classical mathematics. However, this deficiency is addressed by a second strategic step which is taken in the lectures ‘Probleme der mathematischen Logik’ held in the following semester. Here, after a careful re-examination of rival approaches to the foundations of mathematics, Hilbert unites the considerations behind a constructive foundation for number theory with the detailed formal logical work. Recall that already in his Heidelberg address of 1904, and again in his Zürich lecture of 1917, Hilbert argued for a ‘Beweistheorie’, but did not then pursue the suggestion systematically. In Section 7 of the lectures ‘Probleme der mathematischen Logik’ we find initial steps, namely a consistency proof for an extremely restricted part of elementary number theory that involves negation applied only to equations; the part of number theory considered here really is the same as that in the Heidelberg lecture of 1904. It is only at this stage that the study of ‘meaningless marks’ enters the picture, but manifestly as a programmatic technical device only, and not as the centrepiece of a philosophy of mathematics. The Ausarbeitungen of ‘Logik-Kalkül’ and ‘Probleme der mathematischen Logik’ make up the bulk of Chapter 2.

18This is Hilbert 1919* , published as Hilbert 1992 . 19See p. 425 below. 12 Introduction

At the very end of ‘Probleme der mathematischen Logik’, Hilbert formu- lates arithmetic theories going beyond the purely equational theory he had shown to be consistent. Their proof-theoretic treatment required a new per- spective and entirely different techniques. The first indications of such novel investigations are preserved, we conjecture, in an untitled manuscript in Hil- bert’s hand. This manuscript, which we have entitled ‘Consistency Proofs for Fragments of Arithmetic’, is the third document in Chapter 2. In their ‘Introductory Note’ to that document, Sieg and Tapp give ample evidence for Hilbert’s having composed this manuscript before the spring of 1921. The particular proof-theoretic considerations did not enter Hilbert’s later lectures, since they are incorrect as they stand. Sieg and Tapp, however, reconstruct the intricate proof transformations and succeed in correcting the consistency arguments, using a more complex ordering of proofs than Hilbert had used.

3. 1920–1922: From Logic Towards Proof Theory. Hilbert lectured on these topics in 1921, first (probably) in Göttingen, then in Copenhagen in the spring, and finally in Hamburg in the summer; the latter lectures were the basis for the first publication on his new foundational research, Hilbert 1922b.20 The formal theories he examines there are quasi- constructive on account of a restricted treatment of negation; nevertheless, Hilbert takes a very public and pointed stand against Weyl’s and Brouwer’s rejection of ‘classical’ mathematics. His position stirred deep interest among mathematicians and philosophers. In his celebration of the Hamburg Mathe- matisches Seminar in Behnke 1976 , Behnke gives a brief description of the impact of the Hamburg lectures in the summer of 1921:

20In a footnote to the title of his paper Hilbert 1922b (p. 157), Hilbert writes: Diese Mitteilung ist der wesentliche Inhalt der Vorträge, die ich im Frühjahr dieses Jahres in Kopenhagen auf Einladung der dortigen Mathematischen Gesellschaft und im Sommer in Hamburg auf Einladung des Mathematischen Seminars der Universität daselbst gehalten habe. Here, the ‘dieses Jahres’ clearly refers to the year the paper was composed (1921), and not the year it was published (1922). Hilbert’s lectures in Copenhagen in March of 1921 are documented in Sauer and Majer 2009 , 376–377. Hilbert’s visit to Copenhagen was occa- sioned by the award of an honorary doctorate by the University of Copenhagen, presented on 14 March; on the same day, Hilbert gave a lecture in the University’s Festsal entitled ‘Natur und mathematisches Erkennen’. (The manuscript on which this lecture was based is published in Volume 5 of this series, i. e., Sauer and Majer 2009 , 382–392.) On the two days following, Hilbert then gave lectures entitled ‘Axiomlehre und Widerspruchsfreiheit’. There are brief reports and summaries for all three lectures in the Danish press; see for example the Berlingske Tidende for 15, 16 and 18 March. As far as the lectures in Hamburg are concerned, in the letter from Bernays to Hilbert from 21 October 1921 mentioned below (p. 15) in which he reports on his own lecture at the Jena conference in September 1921, Bernays states: Man fragte mich des öfteren, wie es mit der Publikation Ihrer Hamburger Vorträge stehe. Ich wußte in dieser Hinsicht über Ihre Absichten nicht recht Bescheid. Jedenfalls würde Hecke diese Vorträge gern in der neuen Hamburger Zeitschrift drucken. This journal was precisely where Hilbert’s ‘Neubegründung’ based on these lectures was published. Introduction 13

In the summer of 1922 sic, Hilbert presented here in Hamburg his often- cited first article on the ‘Neubegründung der Mathematik’. It was a con- frontation with the writings of Hermann Weyl und L. E. J. Brouwer, who, because they had challenged the tertium non datur, and had thereby called into doubt the existence of the real numbers, had caused consternation and unrest among mathematicians. In the discussion on Hilbert’s lectures there spoke Ernst Cassirer, at that time Professor of Philosophy in Hamburg, Hein- rich Scholz from Kiel, who later was the founder of mathematical logic in Germany, Kowalewski — Königsberg, who promoted Vaihinger’s ‘as if’ phi- losophy, and several others. The discussions were livelier and more thorough than is usually the case after a mathematical talk. One understood that Hilbert had set himself the task of definitively securing mathematics against the attacks of the intuitionists.21 Only a few weeks later, Bernays gave a talk at the meeting of the Deutsche Mathematiker-Vereinigung, the German Association of Mathematicians.22 In his article (Bernays 1922a) which emerged from this lecture, Bernays presents the newly evolving ideas for a more substantive proof theory and puts them into a broad systematic and historical context. He analyzes this foundational approach and brings out very clearly the tentative character of the proposed solution. In order to provide a rigorous foundation for arithmetic (includ- ing analysis and set theory) one proceeds axiomatically and starts out with the assumption of a system of objects satisfying certain structural conditions. However, in the assumption of such a system ‘lies something so-to-speak tran- scendental for mathematics, and the question arises as to which principled position one should take [towards that assumption]’23. Bernays considers two ‘natural’ positions, positions which had been thoroughly explored in the lec- tures from 1920. The first, attributed to Frege and Russell, attempts to prove the consis- tency of arithmetic by purely logical means; this attempt is judged to be a

21Behnke 1976 , 231. In the original German, the passage reads: Im Sommer 1922 sic trug Hilbert hier seinen viel zitierten ersten Aufsatz zur Neubegrün- dung der Mathematik vor. Es war eine Entgegnung auf die Schriften von Hermann Weyl und L. E. J. Brouwer, die Unruhe unter die Mathematiker gebracht hatten, weil sie das tertium non datur bestritten und damit schon die Existenz der reellen Zahlen in Frage stellten. In den Diskussionen zu Hilberts Vorträgen sprachen Ernst Cassirer, damals Pro- fessor der Philosophie in Hamburg, Heinrich Scholz aus Kiel, der spätere Begründer der mathematischen Logik in Deutschland, Kowalewski — Königsberg, der für die Philosophie des ‘Als ob’ von Vaihinger warb, und mancher andere. Die Erörterungen waren viel reger und grundsätzlicher, als es sonst nach mathematischen Vorträgen der Fall zu sein pflegt. Man begriff, daß Hilbert sich um die entscheidende Sicherung der Mathematik gegen Angriffe der Intuitionisten wandte. Behnke’s reference to ‘Kowalewski — Königsberg’ is presumably to Christian Kowalewski, a mathematician and philosopher at the University of Königsberg, and the brother of the mathematician Gerhard Kowalewski. 22The meeting took place in Jena from 18 to 24 September 1921; Bernays’s lecture, enti- tled ‘Über Hilberts Gedanken zur Grundlegung der Arithmetik’, was held on 23 September (see Bernays 1921 ), and was published as Bernays 1922a. 23See Bernays 1922a, 10. In the original German, the passage reads: . . . liegt nun etwas für die Mathematik gleichsam transzendentes, und da entsteht die Frage, welche grundsätzliche Stellung man dazu einnehmen soll. 14 Introduction failure. The second position is associated with Kronecker, Poincaré, Brouwer and Weyl. It is seen in counterpoint to the logicist foundations of arithmetic: Since one does not succeed in establishing the logical necessity of the mathe- matical transcendental assumptions, one asks oneself, is it not possible simply to do without them.24 Thus one attempts a constructive foundation, replacing existential assump- tions by construction postulates. The methodological restrictions to which this position leads are viewed as unsatisfactory, as one is forced ‘to give up the most successful, most elegant, and most proven methods only because one does not have a foundation for them from a particular standpoint’25. From these two foundational positions, Bernays continues, Hilbert takes what is ‘positively fruitful’: from the first, the strict formalization of mathematical reasoning, and from the second, the emphasis on constructions. Hilbert does not want to give up the constructive tendency. On the contrary, he empha- sizes it in the strongest possible terms. A constructive attitude is viewed as part of the Ansatz to finding a principled position toward the transcendental assumptions; it takes into account the tendency of the exact sciences to use as far as possible only the most primitive ‘Erkenntnismittel’: From this perspective, we are going to attempt to see whether it is possible to give a foundation for these transcendental assumptions in such a way that only primitive intuitive knowledge is used.26 Bernays emphasizes that taking this perspective does not mean denying any other, stronger form of intuitive evidence. The new approach is thus taken as a tool for an alternative, constructive foundation for all of classical mathematics. The great advantage of Hilbert’s method is judged to be this: [T]he problems and difficulties that present themselves in the foundations of mathematics are transferred from the epistemological-philosophical to the properly mathematical domain.27 In this way Bernays, without great fanfare, presents an illuminating summary of four years of intense work.

24Bernays 1922a, 13. In the original German, the passage reads: Da es nicht gelingt, die mathematisch transzendenten Grundannahmen als logisch notwendig zu erweisen, so fragt man sich, ob diese Annahmen nicht überhaupt entbehrt werden können. 25Bernays 1922a, 14. In the original German, the passage reads: . . . die erfolgreichsten, elegantesten und bewährtesten Schlußweisen sollen preisgegeben wer- den, und zwar bloß deshalb, weil man von einem bestimmten Standpunkt keine Begründung für sie hat. 26Bernays 1922a, 11. In the original German, the passage reads: Unter diesem Gesichtspunkt werden wir versuchen, ob es nicht möglich ist, jene transzen- denten Annahmen in einer solchen Weise zu begründen, daß nur primitive anschauliche Erkenntnisse zur Anwendung kommen. 27Bernays 1922a, 19. In the original German, the passage reads: . . . die Probleme und Schwierigkeiten, welche sich in der Grundlegung der Mathematik bie- ten, aus dem Bereich des Erkenntnistheoretisch-philosophischen in das Gebiet des eigentlich Mathematischen übergeführt werden. Introduction 15

It is in lectures that began a few weeks later, the lectures of the Winter Semester of 1921/22, that the terms ‘Hilbertsche Beweistheorie’ and ‘finite Mathematik’ appear for the first time. In his outline of the forthcoming lectures (sent to Hilbert on 17 October) Bernays indicated that those lectures were to conclude with a treatment of proof theory: III. Further elaboration of the constructive idea: Construction of proofs, whereby the formalisation of higher inferences can be carried out, and the general problem of consistency becomes accessible. At this point the exposition of proof theory would commence.28 In the outline, Bernays refers to ‘constructive arithmetic’; but in the lec- tures themselves the terminology is changed to ‘finitist arithmetic’ and ‘fini- tist mathematics’. Although the term ‘Hilbertsche Beweistheorie’ is self- explanatory, ‘finite Mathematik’ is not. Does the new terminology signal a novel philosophical perspective, or a precise claim about the foundations of mathematics? For three reasons, neither alternative seems likely. First, the term ‘finitism’ was already in use. In 1919, Felix Bernstein pub- lished a very informative paper entitled ‘Die Mengenlehre Georg Cantors und der Finitismus’; it is one of the few papers listed at the end of the Introduc- tion to the Second Edition of Principia Mathematica in 1927.29 Bernstein describes ‘Finitismus’ as a familiar foundational stance that was opposed to set theory since the very beginnings of that theory in Cantor’s and Dede- kind’s work. Somewhat indiscriminately, Kronecker and Hermite, Borel and Poincaré, Richard and Lindelöf, and then also Brouwer and Weyl (Weyl’s work Das Kontinuum is cited, i. e., Weyl 1918 ) are viewed as members of the finitist movement. Bernstein’s paper was published before Hilbert and Ber- nays viewed ‘finite Mathematik’ as foundationally significant for proof theory. The characteristic features of finitism which Bernstein formulates are actually close to those of the ‘finite Mathematik’ emphasised by Hilbert and Bernays in the 1921/22 lectures. Hilbert himself frequently had referred to constructive demands (which he usually associated with Kronecker), and as late as 1931

28Cod. Ms. Hilbert 21. In the original German, the entire outline for the forthcoming lectures, the ‘Disposition’, reads: I. Bisherige Methoden der Beweise für Widerspruchslosigkeit oder Unabhängigkeit. A. Methode der Aufweisung. Beispiel des Aussagenkalküls in der mathem[atischen] Logik. B. Methode der Zurückführung. Beispiele: 1) Widerspruchslosigkeit der Euklidischen Geometrie 2) Unabhängigkeit des Parallelenaxioms 3) Widerspruchslosigkeit des Rechnens mit komplexen Zahlen II. Versuche der Behandlung des Problems der Widerspruchslosigkeit der Arithmetik. A. Die Zurückführung auf die Logik bietet keinen Vorteil, weil der Standpunkt der Arith- metik schon der formal allgemeinste ist. (Frege; Russell) B. Die konstruktive Arithmetik: Definition der Zahl als Zeichen von bestimmter Art. III. Die weitere Fassung des konstruktiven Gedankens: Konstruktion der Beweise, wodurch die Formalisierung der höheren Schlußweisen gelingt und das Problem der Widerspruchslo- sigkeit in allgemeiner Weise angreifbar wird. Hier würde sich dann die Ausführung der Beweistheorie anschließen. 29See Whitehead and Russell 1927 , Volume I, xlvi. Bernstein’s paper is Bernstein 1919 . 16 Introduction he remarked that Kronecker’s conception of mathematics ‘essentially coincides with our finitist mode of thought’.30 Secondly, in the lectures themselves Hilbert and Bernays give no philo- sophical explication of the term ‘finit’. Rather, they insist that intuitive con- siderations be used, and they develop elementary number theory on such a foundation; their development is a marvellously detailed presentation of the beginnings of (primitive) recursive number theory. The opposition between ‘finite Mathematik’ and ‘transfinite Mathematik’ is founded on that between ‘finite’ and ‘transfinite Logik’, since the step from ‘finite’ to ‘transfinite Mathe- matik’ is taken when the classical logical equivalences between the quantifiers are assumed to be valid even when dealing with infinite totalities. But this latter cannot be done in ‘finite Mathematik’: For the assumption of the the unrestricted validity of those equivalences amounts to thinking of totalities with infinitely many individuals as some- thing completed. (Ausarbeitung of the 1921/22 lectures, p. 2a, p. 489 below.) Thirdly, in 1932 Gödel and Gentzen proved that classical arithmetic is consistent relative to its intuitionistic version: until that time the widespread assumption had been that finitism and intuitionism were co-extensional. In- deed, as late as 1930, in his penetrating analysis of Hilbert’s proof theory, Bernays still emphasizes co-extensionality, and interprets Brouwer’s work as showing that considerable parts of analysis and set theory can be ‘given a finitist foundation’.31 These various considerations, as well as the matter-of-fact way the con- cept of finitism is introduced in the 1921/22 lectures, suggest that Hilbert and Bernays are employing a familiar, intuitive and informal concept; to that extent, attempts to pin down in the light of modern knowledge precisely what they meant by ‘finitism’ are historically misplaced. In particular, it should be observed that elementary number theory as Hilbert and Bernays develop it should not be understood as all of what they term ‘finite Mathematik’. On the contrary, a dramatic expansion is envisioned (as we will see below, at the beginning of the next section). This expansion is required, on the one hand, to develop analysis and set theory fully and, on the other hand, to recognise on the basis of finitist logic why and to what extent ‘the application of trans- finite inferences [in analysis and set theory] always leads to correct results’ (Ausarbeitung of the 1921/22 lectures, p. 4a, p. 490 below). The strategy reflects to some extent what Hilbert had done at the turn of the nineteenth and twentieth centuries, namely, shifting the Kroneckerian constructivity re- quirements from mathematics itself to metamathematics. In this way, Hilbert and Bernays try to resolve two problems simultaneously: (i) to fix the stand- point on the basis of which consistency proofs are to be carried out (‘finite Mathematik’), and (ii) to establish a broad, formal framework within which mathematics can be developed systematically.

30See Hilbert 1931a, 487, below p. 976. 31See Bernays 1930 , 350, 42 in the reprint. Introduction 17

What we find in the lecture notes is thus not a ‘programme’ whose de- tails were fully specified in advance, but rather something more flexible: they present rapidly developing and innovative mathematical work which tries to overcome the obstinate difficulties of giving consistency proofs for stronger and stronger formal theories, but it is ultimately unsuccessful. This exten- sion will be briefly sketched now; it will be presented in more detail in the Introductions to the individual Chapters. The evolution of these proof-theoretic investigations is complex, and in ad- dition to the official typescripts (Ausarbeitungen) of Hilbert’s lectures, Chap- ter 3 includes (a) the beginnings of an Ausarbeitung by Bernays of the lectures from 1922/23; (b) some handwritten notes by Hilbert himself for these lec- tures and beyond; (c) Mitschriften for the 1921/22 lectures, for the 1922/23 lectures, and for three lectures which Hilbert held in the Winter Semester of 1923/24. These Mitschriften are by the mathematician Hellmuth Kneser, and it is doubtful that Hilbert even knew of their existence. Their accuracy, readability, and indeed their fortuitous preservation, have given us enormous insight into the lectures as Hilbert (in the case of the 1922/23 lectures, assisted by Bernays) actually held them.

4. 1922–1925: Finitist Proof Theory. As was previously mentioned, the lectures for the 1921/22 Winter Semester contain for the first time the terms ‘finite Mathematik’, ‘transfinite Schluss- weisen’, ‘Hilbertsche Beweistheorie’. The third part of these lectures is en- titled ‘Die Begründung der Widerspruchsfreiheit der Arithmetik durch die neue Hilbertsche Beweistheorie’ (see p. 432, below). The clear separation of mathematical and metamathematical considerations allows Hilbert to address, finally, Poincaré’s criticism by distinguishing between contentual metamathe- matical induction and formal mathematical induction. The understanding of generality is broadened: the interpretation is no longer tied to a formal calcu- lus that permits only the establishment of free-variable statements; rather the intuitive general concept of numeral is assumed as part of the finitist stand- point. With this understanding of universal quantification, the conclusion concerning the non-validity of tertium non datur is obtained again. Hilbert writes, formulating a different consequence of the consistency considerations: Thus we see that, for a strict foundation of mathematics, the usual inference methods of analysis must not be taken as logically trivial. Rather it is precisely the task of the foundational investigation to recognize why it is that the application of transfinite inference methods as used in analysis and axiomatic set theory leads always to correct results. (Ausarbeitung of the 1921/22 lectures, p. 4a; p. 490 below.) That recognition has to be obtained on the basis of finitist logic; so Hilbert argues that we have to extend our considerations in a different direction in order to go beyond elementary number theory: We have to extend the domain of objects to be considered; i. e., we have to apply our intuitive considerations also to figures that are not number signs. 18 Introduction

Thus we have good reason to distance ourselves from the earlier dominant principle according to which each theorem of pure mathematics is in the end a statement concerning integers. This principle was viewed as expressing a fundamental methodological insight, but it has to be given up as a prejudice. (Ibid., p. 490 below.) This is a strong statement. After all, the position decisively rejected here num- bered among its supporters such distinguished mathematicians as Dirichlet, Weierstraß, Dedekind and (perhaps) Kronecker.32 What is the new extended domain of objects, and what has to be retained from the ‘fundamental method- ological insight’? As to the domain of objects, it is clear that the formulas and proofs from formal theories have to be included; as to methodological requirements, Hilbert remarks: . . . the figures we take as objects must be completely surveyable and only discrete determinations are to be considered for them. It is only under these conditions that our claims and considerations have the same reliability and evidence as is the case in intuitive number theory. (Ausarbeitung of the 1921/22 lectures, p. 5a; p. 490 below.) From this new standpoint, Hilbert exploits the formalizability of a frag- ment of number theory and proves its consistency by finitist proof-theoretic means. Here we close the gap to the published record with an almost fully developed programmatic perspective. The dialectic of the developments that emerges from these lectures given between the autumn of 1917 and the spring of 1922 is reflected, as we saw, in Bernays’s description of Hilbert’s work ‘Über Hilberts Gedanken zur Grundlegung der Arithmetik’ (Bernays 1922a); this essay makes beautifully clear the connection to the ‘existential axiomatics’ of the nineteenth century. (This connection is also strikingly made in the Nach- schrift of Hilbert’s 1922/23 lectures by Walter Peterhans. See the various extracts given below, pp. 571–576.) In Hilbert’s ‘Neubegründung’ essay (based on lectures given in the spring and summer of 1921, first in Copenhagen, then in Hamburg: see n. 20 above),

32That Dirichlet and Dedekind held to such a principle is stated in the Vorwort to De- dekind 1888 . For Kronecker’s position, see Kronecker 1887 , 338–339. Hilbert himself alludes to Weierstraß’s adherence to such a principle in his ‘Feriencurs’ for 1896, pp. 6–7; see Hallett and Majer 2004 , 154. This adherence is also stated in Hurwitz’s Mitschrift of Weierstraß’s lectures ‘Einleitung in die Theorie der analytischen Funktionen’ from 1878. Cantor (Cantor 1883a, 553) states the view as follows: Das eigentliche Material der Analysis wird, ausschliesslich, dieser Ansicht zufolge, von den endlichen, realen, ganzen, Zahlen gebildet und alle in der Arithmetik und Analysis gefunde- nen oder noch der Entdeckung harrenden Wahrheiten sollen als Beziehungen der endlichen ganzen Zahlen untereinander aufzufassen sein; . . . . Cantor distances himself from the position stated, but goes on to say that there are un- doubtedly certain advantages attached to it, and moreover . . . spricht doch für ihre Bedeutung auch der Umstand, dass zu ihren Vertretern ein Theil der verdienstvollsten Mathematikern der Gegenwart gehört. It would be an error to think that only a single, explicit principle was endorsed by all of these mathematicians, despite Cantor’s statement. For a discussion of some of the nuances, see Schappacher and Petri 2007 . Introduction 19 a general consistency result is formulated. But, as the Editors of Hilbert’s Gesammelte Abhandlungen remark, it is provable only in a restricted form (see Hilbert 1935 , p. 176, n. 2).33 This restricted result is actually established in Hilbert’s 1921/22 lectures, as is recorded in the Kneser Mitschrift;cf.the Introduction to Chapter 3, section 2. Indeed, through this Mitschrift, we can say that the proof-theoretic considerations began on 2 February 1922 and ended on 23 February. They concern a basic system of arithmetic and are ex- tended to treat definition by (primitive) recursion and proof by induction; the latter principle is formulated as a rule and restricted to quantifier-free formu- las. The part of the official Ausarbeitung which presents a finitist consistency proof was clearly written by Bernays after the term had ended, and contains a different argument; that argument pertains only to the basic system and is sketched in Hilbert 1923a. The more extended argument for a system that incorporates definition by primitive recursion and proof by quantifier-free in- duction is given in Kneser’s Mitschrift 1922/23, and is discussed below in the Introduction to Chapter 3. From a contemporary perspective, this argument shows something very important. If a formal theory contains a certain class of finitist functions, then it is necessary to appeal to a wider class of functions in the consistency proof: an evaluation function is needed to determine uniformly the numerical value of terms, and such a function is no longer in the given class. The formal system considered in the above consistency proof includes primitive recursive arithmetic, and the consistency proof goes beyond the means available in this arithmetic. At this early stage of proof theory, finitist mathematics is consequently stronger than primitive recursive arithmetic, something that remains constant throughout the development of proof theory by Hilbert and Bernays.34 In his paper of 1925 (Ackermann 1925 , 4–7), Ackermann reviews the con- sistency proof discussed above in his Section II entitled ‘The consistency proof before the addition of the transfinite axioms [Der Widerspruchsfreiheitsbe- weis vor Hinzunahme der transfiniten Axiome]’. The very title reveals the restricted significance of this result, since it concerns a theory that is by defi- nition part of finitist mathematics and thus need not be secured by a consis- tency proof. The first step which genuinely goes beyond the finitist methods involves the treatment of quantifiers. Already in Hilbert’s ‘Neubegründung’ lectures of 1921 (Hilbert 1922b), we find a brief indication of his approach; this is elaborated in the 1921/22 lectures (see the Kneser Mitschrift for these lectures, pp. 1–12 of Notebook III, below pp. 589–594). The treatment of

33The Editors of Volume 3 of Hilbert’s papers are not explicitly named, although various people are thanked by Hilbert in the Foreword ‘für ihre Mitarbeit, für ihre kommentierende Anmerkungen u. s. w.’, including the General Editor Helmut Ulm, as well as Paul Bernays and Arnold Schmidt. It is plausible to assume that Bernays was responsible for the remark mentioned here. See n. 3 and especially n. 6 to Appendix D below. 34The discussion of the strength of finitist mathematics has been taken up in recent writings; see in particular the various essays on proof theory in Tait 2005 and Zach 2003 . As to the internal developments in the Hilbert School, see Ravaglia 2003* and Sieg 2009 . 20 Introduction quantifiers is just the first of three steps which have to be taken. The second concerns the incorporation of the general induction principle for number the- ory, and the third deals with an expansion of proof-theoretic considerations to analysis. In his Leipzig address of September 1922, Hilbert sketched the consistency proof set out in the 1921/22 lectures. The really novel aspect of his proof- theoretic discussion is the treatment of quantifiers with the τ-function, the dual of the later ε-operator. The logical function τ associates with every predicate A(a) a particular object τa(A(a)) or τA for short. It satisfies the transfinite axiom A(τA) → A(a), which, according to Hilbert, states ‘if a predicate A holds for the object τA, then it holds for all objects a’. The τ-operator allows the definition of the quantifiers: (a)A(a) ↔ A(τA) (Ea)A(a) ↔ A(τ¬A) Hilbert then extends the consistency argument to the ‘first and simplest case’ going beyond the finitist system. This ‘Ansatz’ will evolve into the ε- substitution method. It is only in the Leipzig address that Hilbert gave proof theory its principled formulation and discussed its refined technical tools; there he gave a hint of the first step that has to be taken in order to ob- tain consistency proofs for comprehensive theories like full first-order number theory, namely, the treatment of quantifiers. What of the third step, the extension to analysis? In the 1917/18 lectures Hilbert had taken ramified type theory with the axiom of reducibility as the formal frame for the development of analysis. In the Leipzig address he con- siders a third-order formulation: appropriate functionals (‘Funktionenfunktio- nen’) allow him to prove (i) the least upper bound principle for sequences and sets of real numbers, and (ii) Zermelo’s Choice Principle for sets of sets of real numbers.35 He conjectures that the consistency of the additional transfinite axioms can be patterned after that for τ. He ends the paper with the remark: Through the precise execution of the basic ideas of my proof theory sketched above, the founding of analysis is completed, and the ground prepared for the founding of set theory.36

35See Hilbert 1923a, 163–165, to be found on pp. 189–191 of the republication in Hil- bert 1935 . 36Hilbert 1923a, 165. In the original German, the passage reads: Durch die genaue Ausführung der soeben skizzierten Grundgedanken meiner Beweistheorie wird die Begründung der Analysis vollendet und die der Mengenlehre angebahnt. The passage was altered for the republication in Hilbert’s Gesammelte Abhandlungen: There remains now the task of the precise execution of the basic ideas sketched above; with this execution, the founding of analysis is completed, and the ground for the founding of set theory prepared. [Es bleibt nun noch die Aufgabe einer genauen Ausführung der soeben skizzierten Grundgedanken; mit ihrer Lösung wird die Begründung der Analysis vollendet und die der Mengenlehre angebahnt sein.] (Hilbert 1935 , 191.) Introduction 21

In Supplement IV of the Second Volume of Grundlagen der Mathematik Hil- bert and Bernays give a beautiful exposition of analysis. They state: Here a formalism for the deductive treatment of analysis is presented which, leaving aside inessential differences, is the same as that given in Hilbert’s lectures on proof theory, and is also similar to the one described in Acker- mann’s dissertation.37 The concrete proof-theoretic work establishing the consistency of what we call primitive recursive arithmetic was directly continued in Ackermann’s thesis (Ackermann 1924a* ), on which Ackermann 1925 is based. It starts out, as was mentioned, with a concise review of the earlier considerations for quantifier-free systems. Hilbert had replaced the τ-symbol with the ε-symbol in 1923, and in Ackermann’s work the ε-calculus as we know it replaces the τ-calculus of 1922.38 The transfinite axiom for the ε-calculus appears in Sec- tion III, entitled ‘The consistency proof after the addition of the transfinite axioms and higher function types [Der Widerspruchsfreiheitsbeweis bei Hinzu- nahme der transfiniten Axiome und der höheren Funktionstypen]’. For number variables, the crucial axiom is just dual to the τ-axiom and is formulated as A(a) → A(εA); it yields the following definitions of the quantifiers: (Ea)A(a) ↔ A(εA) (a)A(a) ↔ A(ε¬A) The remaining transfinite axioms are adopted from Hilbert’s Leipzig paper (Hilbert 1923a). However, the ε-symbol is actually characterized as the least- number operator, and the recursion schema with number variables alone is ex- tended to a schema that also permits function variables. The proof-theoretic argument is complex.39 The connection to the mathematical development in Hilbert’s paper is finally established in Section IV entitled ‘The reach of the transfinite axioms [Die Tragweite der transfiniten Axiome]’. At first it was be- lieved that Ackermann had completed the ‘precise execution’ indicated at the end of the Leipzig talk and had thereby established not only the consistency of full classical number theory, but even that of analysis.

5. 1925–1931: An Elementary Finiteness Theorem? Ackermann’s thesis had just been published when, on 4 June 1925, Hil- bert gave a triumphant talk in Münster on his proof-theoretic programme. His lecture ‘Über das Unendliche’ (published as Hilbert 1926 ) is arguably the

37Hilbert and Bernays 1939 , 451. In the original German, the passage reads: Es sei hier ein Formalismus für die deduktive Behandlung der Analysis dargelegt, wie er — bis auf unwesentliche Unterschiede — in den Hilbertschen Vorlesungen über die Beweistheorie aufgestellt wurde und wie er ähnlich auch in der Ackermannschen Dissertation beschrieben ist. 38See, e. g., Hilbert’s lecture for 1 February 1923, reported in the Kneser Mitschrift, pp. 30ff., pp. 623ff. below. 39For analysis of this argument, see Zach 2003 and Tapp 2006* . 22 Introduction most studied of Hilbert’s papers from the 1920s; he not only gives a brilliant exposition of the methodological framework for proof theory, but he also tries to support the work by sketching the ‘fundamental ideas’ for a proof of Can- tor’s ‘Kontinuumssatz’. (The methodological framework is also discussed at greater length in the lecture course ‘Über das Unendliche’ presented in Chap- ter 4.) Hilbert applies here the method of ‘ideal elements’, already discussed by him at length in previous work, most particularly his work on the foun- dations of geometry from the 1890s, and widely used before this in geometric and algebraic practice.40 Here, Hilbert applies the method in the first place to statements: the genuinely transfinite statements are considered as ‘ideal elements’ whose coherent addition to the finitist basis has to be secured by a consistency proof. As to the ‘Kontinuumssatz’, there are attempts at tack- ling it in earlier lectures and in further manuscripts preserved in the Nachlaß. However, all this material is, without exception, very difficult to decipher and even more difficult to understand. The broad ideas that appear to have guided Hilbert in this attempt, and their relation to Gödel’s proof of the consistency of the Continuum Hypothesis relative to the other axioms of set theory, are discussed by Robert Solovay in Volume III of Gödel’s Collected Works.41 As for concrete progress in the basic proof-theoretic programme, Hilbert asserts that the consistency of the ‘arithmetic axioms’ has been established. Hilbert goes on: In fact, we can carry out this demonstration, and with it we arrive at a justification for the introduction of our ideal statements. Thus, we experience the pleasant surprise that this gives us the solution of a problem that became pressing a long time ago, namely, the problem of proving the consistency of the arithmetic axioms.42 The triumphalism displayed by Hilbert was misplaced. It turned out that the consistency proofs of Ackermann (and von Neumann) established the consis- tency only of arithmetic with quantifier-free induction; for further discussion, see the Introductions to Chapter 3 and to Appendix B below. The major document included in Chapter 4 is an Ausarbeitung of the lec- ture course ‘Über das Unendliche’ held by Hilbert in the Winter Semester of

40For discussions of Hilbert’s earlier appeal to the method of ‘ideal objects’ and ‘ideal elements’, see Majer 1993 and Hallett 2008 , 216–218. 41See Solovay 1995 , §5, esp. 121. See also Kanamori and Dreben 1996 ,§7,whichbrings out forcefully the novelty and fruitfulness of Hilbert’s central ideas, even though the proof, of course, does not succeed. Of especial interest are Gödel’s remarks in a letter of 25 June 1969 to Constance Reid; the letter is published in Volume V of the Collected Works, Gödel 2003b, 188–189. See also Gödel’s letter to van Heijenoort from 8 July 1965, Gödel 2003b, 324. 42Hilbert 1926 , 179. In the German original, the passage reads: Tatsächlich läßt sich dieser Nachweis erbringen, und damit gewinnen wir die Berechtigung zur Einführung unserer idealen Aussagen. Zugleich noch erkennen wir die freudige Überraschung, daß wir damit ein Problem lösen, das längst brennend geworden ist, nämlich das Problem, die Widerspruchsfreiheit der arith- metischen Axiome zu beweisen. The same passage is repeated almost word for word (but without the paragraph break) in Hilbert’s Hamburg address in 1927, Hilbert 1928a, 74; see below, p. 931. Introduction 23

1923/24. It bears many similarities to, and also exhibits deviations from, the famous paper which has just been discussed. The other items in Chapter 4 are a short essay (possibly prepared for a public lecture) entitled ‘Über die Grund- lagen des Denkens’ from 1931 or 1932, and a short lecture on the infinite from 1933. It is unfortunate that there are extant no lecture (or other) notes from 1925 through to the early 1930s which offer insight into Hilbert’s perspective on the further development of proof theory.43 Given this, we have chosen to include some of Hilbert’s previously published work in Appendices B–E, reprinting four papers from the period 1927 to 1931. The special circum- stances under which they were presented are discussed in the Introductions to the respective Appendices; their content is discussed briefly below, and more extensively in the general introduction to the Appendices. In so far as they are able, these Introductions serve to make clear the striking discontinuities between the papers in Appendices B and C and those in D and E. The change in Hilbert’s approach is attributed to the impact of Gödel’s Incompleteness Theorems. But first, a word about Appendix A. This Appendix reproduces Hilbert’s famous textbook on logic from 1928, the Grundzüge der theoretischen Logik, co-authored with Wilhelm Ackermann. This book does not reflect the proof- theoretic work of the 1920s; the intention was always to present that in a separate book co-authored with Bernays.44 Nevertheless, the Hilbert and Ackermann book, now somewhat hard to obtain, is a central, indeed clas- sical text in the development of mathematical logic. Moreover, because of its historical centrality, it is important to be able to compare it closely with Hilbert’s lectures of 1917/18 and the two lecture courses from 1920, and thus to see that the core content of the book is taken over from those lectures. § 6.2 of the Introduction to Chapter 1 provides a detailed comparison of the book with those earlier lectures, and the Introduction to Appendix A gives a brief synopsis of the source of the various sections of the book based on an abbreviated translation of the book’s Inhaltsverzeichnis. Appendix B reprints a paper (Hilbert 1928a) which is based on a lecture Hilbert gave in Hamburg in July of 1927. The general considerations by and large follow the presentation in ‘Über das Unendliche’ from some two years earlier, but contain also some new aspects. In particular, the lecture includes more detailed proof-theoretic considerations and a sketch of Ackermann’s new approach. More of the mathematical details of this new approach are given in Bernays’s ‘Zusatz’, and also in a letter Bernays wrote to Weyl on 5 Jan- uary 1928. Both this letter and the ‘Zusatz’ are reprinted in Appendix B, as

43A possible exception is Collatz’s protocol of Hilbert’s lectures on set theory from 1929, though see below, p. 29. 44In a letter to Weyl of 5 January 1928 (reproduced in Appendix B, below), Bernays remarks that only one chapter of the Hilbert and Bernays book had so far been drafted. However, by the beginning of 1931, a presentation of the proof-theoretic work was close to completion. See the remark of Bernays from the Foreword to Hilbert and Bernays 1934 , V–VI. 24 Introduction is the paper containing Weyl’s ‘Diskussionsbemerkungen’, which directly fol- lowed Hilbert’s Hamburg lecture. Weyl’s remarks contain deep and important philosophical insights concerning Hilbert’s general approach. Appendix C contains the first version (Hilbert 1929 ) of the paper which re- sulted from Hilbert’s address to the International Congress of Mathematicians in Bologna in September 1928. In this address, Hilbert stated that the con- sistency of full number theory had been secured by the proofs of Ackermann and von Neumann; according to Bernays in his Introductory Remarks to the Second Volume of Grundlagen der Mathematik, that belief was sustained until 1930. (See Hilbert and Bernays 1939 , VI.) A number of precise metamathe- matical problems were formulated in Hilbert’s Bologna lecture, among them the completeness question for first-order logic. As to the consistency problem for analysis, Hilbert states that it is just a matter of establishing ‘a purely arithmetic elementary finiteness theorem [ein rein arithmetischer elementarer Endlichkeitssatz]’. Hilbert’s Bologna address shows dramatically, through its broad perspec- tive and clear formulation of open problems, how far mathematical logic had been moved in roughly a decade. This is a remarkable achievement with im- pact not just on proof theory; the (in-)completeness theorems, after all, are to be seen as part of this broader foundational enterprise. These theorems give finally a negative answer to the question that serves as the title of this section. They show that there are intrinsic reasons why no striking progress had been made in proof theory in the late 1920s, even though extraordinarily talented people were working on the consistency programme. In the Preface to the first volume of Grundlagen der Mathematik, Bernays asserts that a presentation of proof-theoretic work had almost been completed in 1931. But, he continues, the publication of papers by Herbrand and Gödel produced a deeply changed situation that resulted in an extension of the scope of the work and its division into two volumes. The remaining two papers (according to Bernays’s recollection formulated in an interview of 27 August 1977) were an attempt of Hilbert’s to deal with Gödel’s results in a positive way. The interviewer raised the question: ‘How did Hilbert react when he first learned of Gödel’s proof that it is impossible to carry out a consistency proof for number theory inside number theory itself?’45 Here is Bernays’s response: Yes, yes, he was rather irritated about this. But he didn’t react merely negatively; instead he undertook extensions [of the basic system], for example in the Hamburg lecture of 1930. That isn’t the first [Hamburg lecture], nor the second, but the third. There he introduced some further considerations. And then he had a paper, which he called ‘Proof of the tertium non datur’.

45In the original German, the question reads: Wie hat Hilbert reagiert, als er von Gödels Beweis der Unmöglichkeit, einen Widerspruchs- freiheitsbeweis für die Zahlentheorie in der Zahlentheorie selbst zu führen, erfuhr? Introduction 25

There he expressed those [considerations] in somewhat further detail. So yes, indeed, he certainly also dealt with this positively.46 The lecture Bernays mentions in his remarks was actually given in Decem- ber 1930 to the Hamburg Philosophische Gesellschaft; the resulting paper was received on 21 December 1930 for publication in Mathematische Annalen. The mathematical part of the talk seems to be concerned with a version of Gödel’s first incompleteness theorem as it was presented by Gödel at the Königsberg Congress on 7 September 1930. Assuming that classical mathematics is con- sistent, Gödel argues that: One can give examples of statements (and in fact statements of the type of Goldbach’s or Fermat’s) that are in fact contentually true, but are unprovable in the formal system of classical mathematics. Therefore, if one adjoins the negation of such a proposition to the axioms of classical mathematics, one obtains a consistent system in which a contentually false proposition is provable.47 It is not implausible that Hilbert learned about Gödel’s result shortly after the Congress, possibly even in Königsberg. For details of the historical facts we know, see the discussion in Section 4 of the general Introduction to the Appendices, pp. 796–797 and also the Introductions to Appendices D and E, pp. 967–973 and 983–984 respectively. Hilbert presents the crucial and novel aspect that allows the introduction of universal axioms based on finitist argu- ments. (See Hilbert 1931a, 491–492, below, pp. 980–980.) More precisely, he adds to the formal system of number theory a finitist inference rule for closed quantifier-free or numeric formulas: If it is shown that, for any given numeral z, the formula A(z) is always a correct numeric formula, then the formula (x)A(x) canbeusedasastarting formula.48 An extension of Ackermann’s inductive argument to Hilbert’s rule establishes the consistency of the new theory, as Ackermann’s proof was taken to show

46In the original German, the passage reads: Ja, ja, er war ziemlich ärgerlich darüber. Aber er hat nicht bloss negativ reagiert, sondern er hat ja eben auch Erweiterungen vorgenommen, z. B. schon im Hamburger Vortrag von 1930. Das ist weder der erste noch der zweite, das ist der dritte. Da hatte er schon so Einiges eingeführt. Und dann hatte er eine Arbeit, die nannte er ‘Beweis des Tertium non datur’. Da hat er das noch etwas weitergehend ausgedrückt. Also, er hat sich dann doch auch positiv damit auseinandergesetzt. 47Hahn et al. 1931 , 148. The original German reads: Man kann (unter Voraussetzung der Widerspruchsfreiheit der klassischen Mathematik) sogar Beispiele für Sätze (und zwar solche von der Art des Goldbachschen oder Fermatschen) angeben, die zwar inhaltlich richtig, aber im formalen System der klassischen Mathematik unbeweisbar sind. Fügt man daher die Negation eines solchen Satzes zu den Axiomen der klassischen Mathematik hinzu, so erhält man ein widerspruchsfreies System, in dem ein inhaltlich falscher Satz beweisbar ist. The background is described in detail in Dawson’s ‘Introductory Note’ (Dawson 1986 ), and also in Dawson 1997 , 68–79. 48Hilbert 1931a, p. 491; see p. 980 below. For the subsequent introduction of the so-called ω-rule by Bernays, and his correspondence with Gödel, see the Introduction to Appendix D, pp. 969–972. 26 Introduction the consistency of full number theory. The completeness of the new theory is established as well for statements of the form indicated.49 In the final third of his paper, Hilbert vigorously defends proof theory against various objections, all of which he considers to be unjustified. These objections are numbered and with respect to the second Hilbert writes: It has been said, in criticism of my theory, that the theorems are indeed consistent, but that they are not thereby proved. Certainly they are provable, as I have shown here in simple cases. (Hilbert 1931a, 492, below p. 981.) This seems to be a reference to Gödel’s Königsberg result and to the crucial, though limited role of the mathematical result described above. However, Hilbert emphasizes then, as he had done in his second Hamburg lecture (see p. 84, below, pp. 940), that proving consistency is the essential task for proof theory: It turns out also in general (as I was convinced from the outset) that securing consistency is the essential thing in proof theory, and that the question of provability is then settled at the same time, possibly with an appropriate extension of the requirements in such a way as to preserve their finitist character. (Ibid.) It is not clear what Hilbert precisely had in mind with the last claim in this passage: it is concerned with the question of provability, but how is that set- tled once consistency is obtained? What is to be understood by ‘appropriate extension of the requirements that preserves their finitist character’? Perhaps Hilbert is thinking of finitist rules as above, but for more complex statements. The goal, it seems, is to ensure provability of a broader class of statements, while taking for granted that consistency has been, or will be, established. Hilbert’s last paper on proof-theoretic matters, ‘Beweis des Tertium non datur’, was presented to the Göttingen Gesellschaft der Wissenschaften on 17 July 1931; it is of a completely different character from the Hamburg lecture. The whole paper is deeply technical and difficult to understand. Here we merely mention that Hilbert attempts to construct an argument showing that the addition of the tertium non datur in the form (x)A(x) ∨ (∃x)A(x) to some constructive base theory extends that base theory consistently. The necessity of ‘proving’ the tertium non datur is strongly re-emphasized towards the end of the lecture: The tertium non datur occupies a distinguished position among the axioms and theorems of logic in general: for while all the other axioms and theorems can be immediately traced back without difficulty to definitions, the tertium non datur expresses a new, contentually meaningful fact that stands in need of proof. (Hilbert 1931b, 124–125, below p. 988.)

49It should be mentioned that in his 1931 paper (Herbrand 1931 ) Herbrand included Hilbert’s ‘rule’ as part of the system (Axiom Group D) for which consistency is established, and when Gödel established the consistency of classical elementary number theory rela- tive to its intuitionist version, he used Herbrand’s system including Axiom Group D, thus including Hilbert’s ‘rule’. See Gödel 1933 , 35. Introduction 27

By contrast, Hilbert had pointed out in the Hamburg lecture of 1927 that the consistency of elementary number theory, as supposedly obtained by Acker- mann and von Neumann, guarantees the admissibility of all transfinite infer- ences and in particular of tertium non datur. Hilbert insists here that the new method can be applied to the case of higher function types, so that analysis in particular would be covered. Hilbert ends the paper with an ode to the tertium non datur: Through the tertium non datur, logic attains complete harmony; its theorems assume such a simple form, and the system of its concepts is rounded off in a way that is appropriate for the importance of a discipline expressing the structure of our entire thought. (Hilbert 1931b, 125, below p. 989.) This paper was undoubtedly written against the background of a full knowledge of Gödel’s results and, it is plausible to assume, von Neumann’s conviction that there is no finitist consistency proof for classical mathemat- ics. The latter had drawn very strong and rather dramatic consequences from Gödel’s results. In a letter to Gödel of 29 November 1930, von Neumann writes: Thus, I think that your result has solved negatively the foundational ques- tion: there is no rigorous justification for classical mathematics. What sense to attribute to our hope, according to which it is de facto consistent, I do not know — but in my view that does not change the completed fact.50 It should be emphasized that Gödel himself, even as late as 25 July 1931, defended the view he had expressed at the end of his paper on incompleteness (i. e., Gödel 1931a, 197) that his results do not conflict with Hilbert’s formalist standpoint, and that there may be finitist proofs that cannot be formalized in the system of Principia Mathematica and other equally strong, or even stronger, foundational theories. He writes in a letter to Herbrand: Clearly, I do not claim either that it is certain that some finitist proofs are not formalizable in Principia Mathematica, even though intuitively I tend toward that assumption. In any case, a finitist proof not formalizable in

50See Gödel 2003b, 338–341. In the original German, the passage reads: Infolgedessen halte ich die Grundlagenfrage durch Ihr Resultat für im negativen Sinn erledigt: es gibt keine strenge Rechtfertigung für die klassische Mathematik. Welcher Sinn unserer Hoffnung, wonach sie doch de facto widerspruch[s]frei ist, dann zuzuschreiben ist, weis[s] ich nicht — aber das ändert m. E. nichts an der vollendeten Tatsache. For the whole correspondence between Gödel and von Neumann, see Gödel 2003b, 336–375, and also Sieg 2003b. At a meeting of the Vienna Circle on 15 January 1931, Gödel reports von Neumann’s view as follows: . . . if there is any sort of finitist consistency proof at all, then it can be formalized. Thus, Gödel’s proof implies the impossibility of any consistency proof. (Protokoll des Schlick Kreises for the meeting of 15 January 1931; Carnap Archives at the University of Pittsburgh.) This is quoted from Mancosu 1999b, 38; the German original can be found on p. 36. This latter reads: Wenn es einen finiten Widerspruchsbeweis überhaupt gibt, dann lässt er sich auch forma- lisieren. Also involviert der Gödelsche Beweis die Unmöglichkeit eines Widerspruchsbeweises überhaupt. 28 Introduction

Principia Mathematica would have to be quite extraordinarily complicated, and on this purely practical ground there is very little prospect of finding one; but that, in my opinion, does not alter anything about the possibility in principle.51 To overcome the ‘impossibility result’ defended by von Neumann, the re- strictions on proof-theoretic methods have to be weakened, and constructive methods stronger than finitist ones have to be allowed. That such methods were in fact available was indicated by the Gödel-Gentzen consistency proof for classical number theory relative (somewhat ironically) to its intuitionist version. That forced the recognition that finitist methods and intuitionist ones were not co-extensional as those in the Hilbert School and, in particular, von Neumann had believed. This relative consistency result pointed the direction for important proof-theoretic work that has been pursued ever since.52 The two volumes of Hilbert and Bernays’s Grundlagen der Mathematik, published in 1934 and 1939, give a superb, masterly presentation of mathe- matical logic and, in particular, of proof theory. It has been a cornerstone of much subsequent research. The part of mathematical practice that was of spe- cial concern to Hilbert and Bernays, namely, analysis, has been shown to be consistent relative to intuitionist principles. In Supplement IV of the second volume (Hilbert and Bernays 1939 ), Hilbert and Bernays developed analysis very elegantly in a form of second-order arithmetic with the full comprehen- sion principle; cf. p. 21 above. On closer examination, one can see that the 1 latter principle as well as induction is needed only for Π1-formulas. The sub- 1 system (Π1-CA)0 of second-order arithmetic encapsulates these restrictions, and this has been shown to be consistent relative to ID<ω(O), the intuitionist theory of finite constructive number classes.53 However satisfactory the result we just mentioned may be, it certainly does not support Hilbert’s claim made at the end of his final Hamburg lecture:

51Gödel to Herbrand, 25 July 1931; see Gödel 2003b, p. 23. In the original German, the passage reads: Selbstverständlich behaupte ich auch nicht, es sei sicher, daß irgendwelche finite Beweise in den Prin. Math. nicht formalisierbar sind, wenn ich auch gefühlsmäßig eher zu dieser Annahme neige. Jedenfalls müßte ein in den Princ. Math. nicht formalisierbarer finiter Beweis ganz außerordentlich kompliziert sein und es besteht aus diesem rein praktischen Grunde sehr wenig Aussicht einen zu finden, aber das ändert nach meiner Meinung nichts an der prinzipiellen Möglichkeit. For the correspondence between Gödel and Herbrand, see Gödel 2003b, 14–25, and also Sieg 2003a. 52See Bernays 1967 , p. 502. Gödel and Gentzen established their results independently and at about the same time. Gödel’s version of the result is in Gödel 1933 . Gentzen’s paper was to appear in Mathematische Annalen, but Gentzen withdrew it having learned of the Gödel paper. For more information, see p. 802, below. 53The papers Feferman 1970 and Friedman 1970 , in particular, demonstrate the reduc- 1 tion of (Π1-CA)0 to the classical theory of finite tree classes ID<ω(W); the final step to the 1 intuitionist ID<ω(O) was taken in Sieg’s Stanford thesis, Sieg 1977* .—(Π1-CA)0 is the strongest subsystem of arithmetic considered in Reverse Mathematics; see Simpson 1999 . For a glimpse at the progress that has been made in proof-theoretic investigations, see Pohlers 2009 . Introduction 29

I believe that, with proof theory, I have completely achieved what I wanted and promised: the question of the foundations of mathematics is hereby, as I believe, finally eliminated.54 Proof theory has not eliminated foundational questions. Rather, it has in- troduced, and continues to introduce, mathematical methods, posing sharply formulated problems, and philosophical issues, calling for deep reflection; to- gether, they are of the utmost significance for gaining a proper perspective on the fundamental questions.

6. Material Omitted from this Volume. This Volume reproduces the principal archival texts concerned with logic and foundations of mathematics from 1917 on, supplemented by some pub- lished works that are either now difficult to obtain, or germane to the develop- ment of Hilbert’s thought during the period in question. None of the Volumes in this series is intended to be complete, but it is nevertheless important to give a brief sketch of material omitted. After all, the list of Hilbert’s lecture courses (reproduced at the end of this Volume) mentions several courses given by Hilbert which might appear relevant to this Volume. In most cases, how- ever, there are no notes extant for these courses, though detailed Mitschriften may eventually come to light. A known exception is the lecture ‘Mengenleh- re’ given in the summer term of 1929, for which there exists a very careful protocol by Lothar Collatz, mentioned in Menzler-Trott’s book on Gentzen, Menzler-Trott 2007 , n. 8, pp. 22–23. These notes show that Hilbert’s course was an adaptation of the set theory course Hilbert gave in 1917. The final third discusses the finitist consistency programme, but this is carried out only in a very elementary way. The Collatz-Notes will be taken fully into account in Volume 2 of this series, in connection with the Göttingen notes for the 1917 lectures on set theory.) For two of the three courses on ‘Zahlbegriff und Quadratur des Kreises’ in the relevant period, those for 1924/25 and 1928/29, there do exist some hand- written notes, but these consist of only two pages and a few lines respectively, and in both cases are little more than introductory remarks. More relevantly, these notes are to be found pasted into the Ausarbeitung for the 1904 course with the same title prepared by Max Born, which is reproduced in Volume 2 of this series. This suggests (although it does not establish) that there was nothing substantially new in Hilbert’s presentation. There do exist notes as- sociated with the courses ‘Einleitung in die Philosophie auf Grund moderner Wissenschaft’ (1931/32), and ‘Grundlagen der Logik, allgemeinverständlich (für Hörer aller Fakultäten)’ (1933/34). However, in both cases, the notes consist only of a page and are at a very general level. Apart from these specific courses, there also exist notes for general lec- tures on the foundations of science which contain sections of great importance for understanding various aspects of Hilbert’s approach to the foundations

54Hilbert 1931a, 494, below p. 982. 30 Introduction of mathematics. This holds especially for the Ausarbeitungen of ‘Natur und mathematisches Erkennen’ (1919) by Paul Bernays (Hilbert 1919* )andof ‘Wissen und mathematisches Denken’ (1922/23) by Wilhelm Ackermann (Hil- bert 1922/23b* ). However, these are wide-ranging and more popular lectures, and the foundations of logic and arithmetic are not their central focus. The decision was therefore taken not to publish them in this Volume. It is intended that this Edition be completed with a final Volume 6 containing some of these more popular works as well as excerpts from Hilbert’s Notebooks. In addition to the manuscripts just cited, there exist many other items in Hilbert’s Nachlaß which contain material clearly related to the development of his approach to logic, the axiomatisation of arithmetic, the development of the ε-calculus, and consistency proofs for one or more restricted systems. Nevertheless, these notes are scattered, disconnected and often difficult to decipher, to reconstruct and to date. The decision was therefore taken to omit them from this Edition.

William Ewald and Wilfried Sieg