Jan Von Plato Gerhard Gentzen's Shorthand Notes on Logic and Foundations of Mathematics

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Jan von Plato Saved from the Cellar Gerhard Gentzen’s Shorthand Notes on Logic and Foundations of Mathematics Sources and Studies in the History of Mathematics and Physical Sciences

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Saved from the Cellar Gerhard Gentzen’s Shorthand Notes on Logic and Foundations of Mathematics

123 Jan von Plato Department of Philosophy University of Helsinki Helsinki, Finland

ISSN 2196-8810 ISSN 2196-8829 (electronic) Sources and Studies in the History of Mathematics and Physical Sciences ISBN 978-3-319-42119-3 ISBN 978-3-319-42120-9 (eBook) DOI 10.1007/978-3-319-42120-9

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This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Alles Unwichtige bzw. Ubernommene¨ aus 245 in den Keller. Hier nur ”noch einiges vielleicht Verwendbare.“

Gerhard Gentzen, addition to manuscript page BTIZ 245.10

Seite 161–198.2 (außer 170–171, welches den Hauptwertbegriﬀ behandelt)” in den Keller, betriﬀt die Ausarbeitung zur Habilschrift.“

Gerhard Gentzen, addition to manuscript page BTIZ 160d Preface

In the summer of 1939, at the time of finishing his great paper on initial segments of transfinite induction, Gerhard Gentzen started to write down ideas for a popular book on the foundations of mathematics. His somewhat unli- kely motto for the planned book was “Spannend wie ein Kriminalroman!” (“Exciting like a detective story”). Nothing definitive came of this book idea, for Gentzen was drafted to military service by the end of September of the same year. His war ended in a nervous breakdown by the beginning of 1942, with nothing at all of his plans finished after 1939, but just a life lost in 1945, at the age of 35. Two slim folders of stenographic materials in Gentzen’s hand were found in 1984. They contain notes and developments Gentzen thought could still be of some use to him, from between 1931 and 1944. In the past ten years or so, these notes and some additional manuscripts I had the fortune to find have been the object of a detective work of mine, done alongside systematic studies on proof theory. The major part of the work has consisted in the control, word for word, of the contents of transcribed manuscripts against the stenographic originals, in their interpretation in English, and in thinking of their significance against the rest of Gentzen’s results. Occasional forays into archives in Germany, Switzerland, and elsewhere have provided some additional excitement. I worked on the shorthand notes initially on the basis of transcriptions prepared by Christian Thiel. The sources of the present collection were completed through transcriptions made by Gerlinde Bach on my commission. On occasion, I filled in some gaps myself, though very slowly. I am indebted to both transcribers for what they have accomplished. I also thank Gereon Wolters for his help with the source materials. The letters of Paul Bernays are published through an agreement with Dr. Ludwig Bernays. My special thanks go to Eckart Menzler: Our knowledge of the details of Gentzen’s life is the result of the relentless efforts of his amateur histo- rianship in the proper sense of the word. I dedicate this edition to him, in thankfulness for his invaluable services to scholarship.

vii Contents

Preface ...... vii Part I: A sketch of Gentzen’s life and work 1.Overture ...... 3 2.Gentzen’syearsofstudy ...... 9 3. Dr. Gentzen’s arduous years in Nazi Germany, 1933–45 ...... 12 4.Thescientiﬁcaccomplishments ...... 16 5.Looseends ...... 50 6.Gentzen’sgenius ...... 61 Part II: An overview of the shorthand notes 1.Gentzen’sseriesofstenographicmanuscripts ...... 65 2.Theitemsinthiscollection ...... 67 3.Practicalremarksonthemanuscripts ...... 80 4. Manuscript illustrations ...... 84 The German alphabet in Latin, Sutterlin,andFrakturtype¨ ...... 86 Part III: The original writings 1. Reduction of number-theoretic problems to predicate logic ...... 99 2. Replacement of functions by predicates ...... 101 3.Correspondenceinthebeginningofmathematics ...... 112 4.Fivediﬀerentformsofnaturalcalculi ...... 113 4bis.Twofragmentsonnormalization ...... 116 5.FormalconceptionofcorrectnessinarithmeticI ...... 119 6.Investigationsintologicalinference ...... 146 7.Reductionofclassicaltointuitionisticlogic ...... 186 8.CVofthecandidateGerhardGentzen ...... 189 9.LetterstoHeyting ...... 190

ix x Contents

10.FormalconceptionofcorrectnessinarithmeticII...... 193 11.Prooftheoryofnumbertheory ...... 209 12.Consistencyofarithmetic,forpublication ...... 222 13.CorrespondencewithPaulBernays ...... 237 14.Formsoftypetheory ...... 260 15.Predicatelogic ...... 265 a. Unnaturalnesses of the ways of inference in formal logic ...... 265 b. Decidability in predicate logic ...... 266 c. Decidability and the cut theorem ...... 267 d. Natural introduction of the calculus LK ? ...... 268 e. Formulation of intuitionistic logic with symmetric sequents . . . . . 271 16.Propositionallogic ...... 275 a. Completeness of the propositional calculus LK ...... 275 b.Correctnessandcompletenessinpositivelogic ...... 276 17. Book: Mathematical Foundational Research ...... 282 BibliographyforPartsI andII ...... 305 IndexofnamesforPartsI andII ...... 313 IndexofnamesintheGentzenpapers ...... 315 Part I: A sketch of Gentzen’s life and work

1Overture 3

1. Overture

Gerhard Gentzen died on August 4, 1945, in a prison in Prague. His fellow prisoners were professors of the local German university, and there are accounts of his last days and how he was, rendered weak by lack of food, still pondering over the consistency problem of analysis. After the war, some attempts were made to find any manuscripts he might have left behind; a mythical suitcase one letter reports he had been carrying around, filled with papers with a near-proof of the consistency of analysis. Nothing was found, though, in Prague. In Göttingen, instead, there were manuscripts that were preliminary studies for published work, by the account of Arnold Schmidt. He wrote in 1948 to Gentzen’s mother that the papers would be placed and kept together with Hilbert’s papers; yet again, nothing has been found. More than thirty years later, in 1984, two slim folders of stenographic notes by Gentzen surfaced as if by a miracle, one blue, the other violet in colour. They were given by Gentzen’s sister Waltraut Student through the mediation of Prof. Hans Rohrbach to Prof. Christian Thiel of Erlangen University. Thiel had learned the “unified shorthand” as youth, like many Germans, and listed the contents, some three-hundred-odd pages, and started doing some transcriptions. For various reasons, the project slowed down gradually, and none of his transcriptions into German have reached the stage of publication. It appears from a note in the violet folder that Gentzen had left the papers in a summer place in the village of Putbus, on the island of Rugen¨ in the Baltic Sea close to his hometown Stralsund. That took place in the summer of 1944 and suggests that Gentzen bore no illusions about a future in Prague. The Putbus folders contain, by the passages that take the place of a frontispiece in this book, those parts of his manuscripts he thought could still prove to be useful. Page 245 of the series BTIZ is actually a set of over 20 pages, with the added remark on page 245.10: Alles Unwichtige ” bzw. Ubernommene¨ aus 245 in den Keller. Hier nur noch einiges vielleicht Verwendbare.“ (All that is unimportant resp. superseded from 245 in the cellar. Here just some things that could be usable.) On page 160d of the same series, he has added in October 1942: Seite 161–198.2 (außer 170–171, ” welches den Hauptwertbegriff behandelt) in den Keller, betrifft die Ausar- beitung zur Habilschrift.“ (Page 161–198.2 (except for 170–171 that treats the concept of main value) in the cellar, concerns the elaboration for the habilitation thesis.) Save for the pages that were preserved in the two folders, the Gentzen manuscripts got lost in some forgotten cellar, in the attic of the Göttingen mathematics department to be subsequently discarded, and the rest burnt. The amount of notes that Gentzen wrote was enormous. They are usually 4 Part I: A sketch of Gentzen’s life and work divided into series that bear acronyms, such as WA, WTZ, and BTIZ. These stand for Widerspruchsfreiheit Analysis, Widerspruchsfreiheit transfinite Zahlen, Beweistheorie der intuitionistischen Zahlentheorie (Consisten- cy analysis, Consistency transfinite numbers, Proof theory of intuitionistic number theory). WA runs to at least page 339, by a summary that has been preserved, whereas the last extant page is 254 of September 1943. WTZ is the series of notes in which ordinal proof theory was created, but nothing is left of it except references such as WTZ 150 and WTZ 210 in the other series. The remaining pages of BTIZ, instead, amount to some two hundred printed pages. That is less than half of what the page numbering, running to 275, would suggest for the whole of it. My work with the manuscripts began in 2005 with a stroke of luck right at the beginning, during my first visit to Erlangen in February of that year, namely, the rediscovery of a handwritten version of Gentzen’s thesis with a detailed proof of the normalization of derivations in intuitionistic natural deduction. I made a translation into English and published the proof in The Bulletin of Symbolic Logic in 2008, seventy-five years after Gentzen had written it down. The motive for my visit was that Dov Gabbay had generously invited me to write a chapter on Gentzen’s logic to the fifth volume of the Handbook of the History of Logic (von Plato 2009), and I felt that it would be important to give the readers at least some idea of the contents of the stenographic manuscripts. I had been in contact with Thiel about the manuscripts some five years earlier and he had even sent me a few samples of originals and transcriptions in that connection. My interest in the Gentzen papers comes mainly from a desire to understand better his published work. For example, there is an offhand remark in his published thesis that he uses in the proof of the cut elimination theorem a rule called Mischung (and bear with me for a paragraph and a half if this means little), rendered as mix or multicut today, “to make the proof easier.” A responsible author of a scientific text indicates by such even the possession of a proof without the multicut rule. The problem is how to permute cut up if its right premiss has been derived by a rule of left contraction. I figured out a solution on my own in 1998 and then tried to find similar in the Gentzen papers. My proof, in von Plato (2001), reduces contraction on a formula to contractions on its subformulas, but there is no such method in use in any of Gentzen’s work, published or otherwise preserved. There was another, more likely candidate for an original proof of cut elimination, but I wasn’t quite sure about it. So I read the published papers over and over again and found more and more things in them. For example, the thesis contains a sequent calculus in which half of the logical rules are replaced by “logical groundsequents” such as A & B → A and A → A ∨ B and A, A ⊃ B → B. Cut elimination can be only partial with this calculus, but it happens that the remaining cuts are innocent in the sense that they 1Overture 5 can be arranged so as not to violate the subformula property, the crucial consequence to cut elimination by which all formulas in a cut-free derivation are found as parts in the assumptions or the claim to be proved. In cut elimination for the groundsequent calculus, the cut formula is never principal in both premisses of a cut. There was thus the promising possibility that the problem with multicut, namely, that the right premiss of cut has been derived by a contraction on the cut formula, would vanish in this treatment. Some of these questions are explored in von Plato (2012, section 12). A decisive final insight on the problem came through systematic work: If possible multiplicities of formulas in sequents are erased at once at their appearance in the conclusions of rules, a cut elimination procedure quite different from Gentzen’s published one with multicut is needed. The details of this proof have been worked out in Negri and von Plato (2016): The main result is a cut elimination procedure in which cut on the original cut formula is first permuted up, followed by cuts on the components of that cut formula. I found it fascinating to ponder what Gentzen may have accomplished without telling anyone else, and applied the “hypothetico-deductive method” to textual interpretation, something I learned from Harry Wolfson’s The Philosophy of the Kalam (Harvard U.P. 1976; don’t ask me why I started reading that book): If Gentzen solved problem X in a certain way Y, then there should be some thing Z somewhere in his texts. If Z is found, it acts as a confirmation of the presence of solution Y. For the specific example of cut elimination without multicut, the evidence that the kind of solution proposed in Negri and von Plato (2016) was known to Gentzen came from two sources, the first his 1938 proof of the consistency of arithmetic: It contains a peculiar “altitude line” construction and a related cut elimination procedure with a direct connection to our proof. There is even a letter of 1938 to Paul Bernays in which Gentzen writes: “How I have obtained the consistency proof from the methods of proof in my dissertation is, I believe, now somewhat easy to see in the new version.” Secondly, the idea of modifying derivations in sequent calculus so that no multiplicities appear is found in Gentzen’s thesis, in the section in which he proves the decidability of intuitionistic propositional logic (1934–35,IV§1). The way to a proof of cut elimination for Gentzen’s calculus LI through this modification is straightforward, as detailed out in Negri and von Plato (2016). Systematic logical work is often helpful in understanding Gentzen’s accomplishments. There is behind the polished articles an enormous amount of detailed results and profound insights. Moreover, such historically motivated systematic work can lead to valuable new results on the proof systems of logic and on their application to arithmetic. It remains to be seen to what extent this turns out to be the case for the two unknowns of the stenographic notes, the big series BTIZ and WA. The Putbus notes give an idea of how Gentzen worked. He would start 6 Part I: A sketch of Gentzen’s life and work with a theme with an absolute independence of mind. Thus, having decided to try his hands on the consistency problem of arithmetic and analysis early in 1932, he would first clarify the nature of mathematical reasoning as it appears in practice. This starting point led soon to the abandonment of the prevailing axiomatic logical tradition. By September, he had found what is now the standard system of natural deduction, mainly by trying out all the possibilities, and had the first ideas about a proof of normalization. At that point, his notes for a series called D,possiblyforDeduktion or Dissertation, run to the eighties. Each page is actually a four-page sheet, numbered, say, 87.1 to 87.4. In October 1932, the series D got rebaptized into INH that stands, in translation, for something like The formal conception of the notion of contentful correctness in number theory. Relation to proof of consistency. Two things emerge from this 36-page bound and covered manuscript, the only one of its kind among the Gentzen papers: The idea of the subformula property comes from an attempt at a semantical explanation of intuitionistic logic that should be, as one says now, compositional. The first idea for a consistency proof of arithmetic is an extension of the subformula property to a formal system of arithmetic. Gentzen managed to prove the normalization theorem first for a fragment of predicate logic with conjunction, negation, and the universal quantifier. Success with the & ¬∀-fragment led soon, in January 1933, to the realization that the classical law of double negation elimination is redundant for that fragment if all atomic formulas are double-negated at the outset. It also led to the mentioned very detailed proof of normalization for the full system of intuitionistic natural deduction. The former result made it to a stage of proofs of an article in which classical arithmetic is reduced to the intuitionistic one (Gentzen 1933). Also the idea to prove the consistency of intuitionistic arithmetic by transfinite induction appears in INH but was putasideforthetimebeing. No description of Gentzen’s work methods exists. From some sources, as in Menzler-Trott (2007, p. 30), it seems he could figure out things in his head, out of touch with the world. When finished with a train of thought, he would write it down. That is the impression one gets from many of the series that have a couple of pages of notes with intervals of one or two days. In one case, the series WA, there is a secondary series titled WAV in which the V stands for Veröffentlichung, publication, with detailed section titles and outlines of their contents. These notes, running to eighty pages and beyond, were then used for writing the manuscript of Gentzen’s famous proof of the consistency of arithmetic, published as a 73-page article in 1936. At their best, the Gentzen notes clarify the development of his consistency program as it appears from his published work, as with the normalization idea. Here is another example: The consistency proof of the 1936-article was 1Overture 7 preceded by another one that used at a crucial point, as Gödel was quick to point out in the fall of 1935, Brouwer’s principle of bar induction. From WAV, we find a reference to a first such proof that used the intuitionistic sequent calculus LI, and an outline of a similar proof with the classical calculus LK, referred to as “the second proof.” Thus, the proof in the 1935- manuscript within yet another logical calculus, preserved in the form of galley proofs, is a third proof, and the published one by transfinite induction afourth. It becomes clear from the manuscripts that many aspects of Gentzen’s published papers stem from concerns about presentation rather than int- rinsic logical reasons. For example, the 1936 paper uses classical logic even if it is not needed, the likely reason being that Gentzen did not want to give the impression that his result somehow depended on the acceptance of intuitionistic logic. Secondly, the paper uses a special notation for logical derivations, what is known as natural deduction in sequent calculus style, by which derivation trees are avoided. An inspection of the literature shows that such two-dimensional proof patterns were a novelty at the time. Third, it avoids the use of the classical sequent calculus LK with its multiple succedent sequents Γ → Δ in which Δ stands for a finite number of cases under assumptions Γ instead of the traditional single conclusion of a mathematical claim, as in Γ → C. Finally, the paper introduces a special decimal notation for ordinals. When in 1938 Gentzen turned into using the standard notation for ordinals, he was very careful to point out that even if the notation comes from set theory, “all definitions and proofs in this section are completely ‘finite’ and in this respect even of a particularly elementary kind” (1938, pp. 37–38). Thus, with the decimal notation of 1936 he most likely wanted to avoid the impression that his consistency proof depended on set-theoretic concepts. Except for an eleven-page summary of the consistency problem of analysis, dated February 1945 and preserved through Hans Rohrbach, and the longhand manuscript of Gentzen’s thesis, the Putbus notes are the only ones that have remained. The pre-war notes kept in Göttingen have been discarded decades ago. There are in addition some letters by Gentzen, those to Paul Bernays and Arend Heyting with a detailed scientific content and therefore included in this collection. Many more can be found in Menzler-Trott’s Gentzen biography. The stenographic notes together with the thesis manuscript can be divided into three: The series WA,theseriesBTIZ, and the rest. Each of these is grosso modo about two hundred pages in standard print. The present collection contains the English translations of the third group of notes and the mentioned letters. The other two series come in parts, with a first attempt at the consistency of analysis in the second half of 1938, followed by a more extensive series in 1943 and a brief summary in February 1945. 8 Part I: A sketch of Gentzen’s life and work

A similar first attempt at the proof theory of intuitionistic arithmetic from the summer of 1939 is preserved, followed by a shorter period of thoughts in the spring of 1942 and a sustained attempt in 1943. There are gaps, for example, WA starts in August 1938 from page 77, and the series BTIZ in May 1939 from page 133. The stenographic notes often complement the published papers, by offe- ring alternatives. The more unified series are typically inconclusive, loose ends in Gentzen’s work, such as the attempt at a semantical decision procedure for intuitionistic propositional logic in the series AL,datedOctober 1942. One has to get used to the nature of the texts. They are for the most part clearly written, but meant for the writer’s eyes, who would remember their context. Phrases can be incomplete, a stenographic mark unreadable, and there are abrupt starts, loose ends, and connections that have to be figured out by the reader, if at all possible, and so on. My attitude is to take what I can understand and hope to get more out with a next reading. Bits and pieces here and there help, from other texts, from the published papers, and from works by others. The English translations have been produced in two stages, the first the deciphering of the stenographic original, the second a translation into Eng- lish. In the latter, the idea has been that the thought behind should be decisive. My method is this: Having had my elementary school training in German, at the Deutsche Schule Helsinki, that language became the one that dominated my verbal thinking for a good number of decisive years of my childhood. Taking in the German text, I fancy seeing in my mind’s eye precisely the thought behind the words, and try to reproduce it in English; it is good to keep in mind that the original German need not be anything like handsome writing and that turning complicated German sentence constructions around, into typical sentence structures in English, may somewhat distort the contents. With the correspondence included in this collection, there are no exact matches to the different forms of addressing and saluting a person. I have given literal translations. German is full of little words that convey some meaning such as a grade of hesitation but that may give an exaggerated feeling if translated into English: eben, doch, also, etc. As the guiding principle has been to always give precedence to thought over words, I have not followed any uniform policy with such words. A part of the transcriptions had been prepared by Thiel, but the greater part is work commissioned by myself from Gerlinde Bach, a retired academic secretary in Munich who is fluent with the stenography and who even knows the logical symbolism. The translations into English have been made with the transcriptions and the originals at hand. My own experience is that a sufficient amount of concentrated effort leads to the ability to read 2 Gentzen’s years of study 9 the stenographic script against its transcription, to control the correctness of the latter word for word. Errors remain, mainly because the originals are sometimes unclear. On the other hand, as the contents are logical and mathematical, it is instead usually clear what reading makes sense in a context. The texts have many cancelled passages. Often these are just wrong beginnings, formulated at once in an improved form, and usually not indicated in the translations. At other times I have judged the contents of a cancelled passage to merit being translated. The guiding idea has always been to reproduce the train of thought of Gentzen and not distracted by cancelled false ends, for this is no complete German edition, such as may have been Thiel’s objective.

2. Gentzen’s years of study

Gerhard Karl Erich Gentzen was born on November 24, 1909, in the northern German town of Greifswald by the Baltic Sea. He seems to have been a quiet and reserved person with a unique talent for mathematics. Craig Smorynski (2007) describes in an appendix to Eckart Menzler’s Gentzen-biography, Lo- gic’s Lost Genius, the first preserved expression of Gentzen’s mathematical talent, namely a little theorem about triangles and intersection points in the Euclidean plane that he found at the age of thirteen. Gentzen’s university studies began in 1928 under the guidance of the ma- thematician Hellmuth Kneser in Greifswald, but as was the habit of the times, he shifted between different places. In the Summer Semester of 1929, he was following Hilbert’s lectures titled Mengenlehre (set theory) in Göttingen. Beautifully finished notes of the course by Gentzen’s friend Lothar Collatz have been preserved; the topics include ordinal numbers and their arithmetic, and the second number class (i.e., constructive ordinals). A final part discusses paradoxes, the language of first-order logic, and the problem of consistency of arithmetic. In the fall of 1930, Gentzen was enrolled in Berlin, where Johann von Neumann gave a course on Hilbert’s proof theory, titled Beweistheorie.He had heard in Königsberg in September about Gödel’s discovery of the incompleteness of arithmetic and decided to explain this work in his course. Contemporary accounts tell of a tremendous excitement that these developments aroused among the Berlin mathematicians. Carl Hempel, later a very famous philosopher, was one of the participants, and his recollection (Hempel 2000, pp. 13–14), even evidenced by contemporary correspondence for which see Mancosu (1999), is as follows:

I took a course there with von Neumann which dealt with Hil- bert’s attempt to prove the consistency of classical mathematics by ﬁnitary means. I recall that in the middle of the course von 10 Part I: A sketch of Gentzen’s life and work

Neumann came in one day and announced that he had just received a paper from... Kurt Gödel who showed that the objectives which Hilbert had in mind and on which I had heard Hilbert’s course in Göttingen could not be achieved at all. Von Neumann, therefore, dropped the pursuit of this subject and devoted the rest of the course to the presentation of Gödel’s results. The finding evoked an enormous excitement. One who was present is Jacques Herbrand, who wrote a short paper on incompleteness in the spring of 1931. There are in addition letters he wrote late in 1930 that contain the basic ideas of Gödel’s proof, as well as von Neumann’s independent discovery of the second incompleteness theorem, by which the consistency of arithmetic is among the unprovable propositions. Von Neumann got in fact a copy of Gödel’s paper with the second theorem only in early 1931. I think it likely that Gentzen followed von Neumann’s lectures, as perhaps suggested on p. 143 below, or at least shared some of the atmosphere around these foundational matters. A letter to Hellmuth Kneser of 2 July 1930 clearly indicates his interest (Menzler-Trott, p. 24): IhavestudiedthebookTheoretische Logik by Hilbert and Acker- mann, as I would like to come to a greater clarity on the foundations of mathematics. Now I will attempt to acquire the recent treatments of Hilbert on these matters.

What could Gentzen have learned from von Neumann, next to G¨odel’s two theorems? There is, ﬁrst of all, von Neumann’s great paper of 1927, Zur Hil- bertschen Beweistheorie, (On Hilbertian proof theory), with the intriguing footnote (note 9, p. 38): The possibility lies close of substituting the logical axioms by the logical ways of inference that rely on the axioms. There would be in place of a single syllogistic rule of inference several ones, by which the (rather arbitrarily built) group of axioms I would disappear. We have refrained from such a construction, because it departs too radically from the usual one. The axioms are those of classical propositional logic with implication and negation as the primitive notions, and the rule is implication elimination, so here is one possible origin of Gentzen’s calculus of natural deduction. A rule system that corresponds to von Neumann’s axioms for logic is studied to some extent in von Plato (2014). It turns out that if a system has only rules and no axioms, the addition of a rule is required that in terms of natural deduction is implication introduction. In terms of axiomatic logic, the additional rule is called “the deduction theorem.” 2 Gentzen’s years of study 11

Von Neumann’s paper on Hilbert’s proof theory is known for its proof of the consistency of arithmetic when the principle of induction is restricted to free-variable formulas. It is referred to by Gentzen in his dissertation that gives a similar result, with a completely different proof. More could be learned from von Neumann: A long paper of his of 1928 discusses transfinite induction and three versions of set theory, one of them intuitionistic set theory. Further, it is known that von Neumann knew well and had by 1929 even applied the bar theorem, a fundamental principle of intuitionistic mathematics. By the summer of 1931, Gentzen had already done work of his own in logic: At the suggestion of Paul Bernays, he had studied the logical theory of “sentence systems” of Paul Hertz. This was in Göttingen, the leading center for mathematics and exact science worldwide at the time. In connection with the preparation of Gentzen’s Collected Papers, Bernays suggested in a letter of 14 May 1968 to the editor Manfred Szabo that the work of Hertz be explained, and then added: It would be worth mentioning that Gentzen showed here his great mathematical ability, bringing to a complete resolution a difficult problem that had remained open in the theory of sentence systems. During the fall of 1931, Gentzen worked on rather detailed logical questions, especially the translation of arithmetic into pure predicate logic: The arithmetical operations are translated into relations such as σ(x, y, z)(“thesum of x and y is z”). This was the way in which Gentzen in his thesis was able to reduce the consistency of arithmetic without the full induction principle to the cut elimination theorem of pure predicate logic. In early February 1932, Gentzen sent his work on Hertz systems to the Mathematische Annalen, then started a systematic work the central aim of which was to find consistency proofs for arithmetic and analysis. The dean of mathematics in Göttingen was David Hilbert whose main interest then was in the foundations of mathematics. Hilbert’s closest colla- borator in that field was the Professor extraordinarius Bernays, a native of Switzerland. The latter was writing the volume Grundlagen der Mathematik (Foundations of mathematics) that was planned to contain a full statement of the famous program Hilbert had set up, namely, to secure the foundation of mathematics through the steps of formalization of the language of mathematics and of mathematical proofs, through a proof of the consistency and completeness of the formalization, as well as through a study of the Entscheidungsproblem, the question whether a mathematical theory accepts a method for deciding theoremhood. The high hopes of Hilbert’s program were shattered by the incompleteness theorem of Kurt Gödel. Bernays, who 12 Part I: A sketch of Gentzen’s life and work soon was to become Gentzen’s teacher, had to start over in 1931 the writing of the magnum opus. Mathematical logic and the foundations of mathematics became the fo- cus of Gentzen’s studies, and he finished in May 1933 a thesis with the title Untersuchungen uber¨ das logische Schliessen, (Investigations into logical inference). Meanwhile, Gentzen’s teacher Bernays had been fired in April as a “non-Aryan,” on the basis of the racial laws of the Nazi government of Hitler. These events may have contributed to the haste with which Gent- zen finished the doctorate. He sought financing for continued studies on the problem of the consistency of analysis, but he also took an exam that gave him the right to teach at high schools. It was a rather serious undertaking that took a lot of his time, from the summer of 1933 until November of that year. He had to hand in two written works, one the published paper on Hertz systems, another a manuscript, subsequently lost, on the application of the recently discovered quantum-mechanical tunneling effect to cosmic radiation. After the doctorate, Gentzen lived on small scholarships in Stralsund. In the second part of 1935, he was nominated assistant to David Hilbert. Hilbert’s years of research were bygone at this point, and Gentzen was able to concentrate on his own research. Even before the assistant’s position, he had finished his original proof of the consistency of arithmetic, then changed it around the turn of the year 1935–36 into the well-known proof based on transfinite induction. After this success, published in 1936, Gentzen fell in a serious depression that required treatment, but regained his powers in stages in 1938. In the summer of 1939, he finished his Habilitationsschrift, a considerable span of time from his doctorate, in part caused by his health problems, in part by the general circumstances of a country headed for war.

3. Dr. Gentzen’s arduous years in Nazi Germany, 1933–45

Some days before the state exam of November 1933, Gentzen applied for membership in the SA in G¨ottingen, the paramilitary Nazi troops, and explained this as something “urgently recommended from several quarters” in a letter to Bernays. In the exam, he had to sign a document by which he swore that “no circumstances are known to me that would justify the assumption that I stem from non-Aryan parents or grandparents; especially, none of my parents or grandparents have belonged to the Jewish religion at any time.” Gentzen was at this time living in Stralsund and there is no record of any participation on his part in any of the G¨ottingen SA-activities, to the extent that his membership was unknown there in 1935. That year, he applied for the assistantship with Hilbert, and got it towards the end of the year, despite reports from the Nazi teachers’ union by which he had contacts with someone in Jerusalem, clearly Abraham Fraenkel, and thereby 3 Dr. Gentzen’s arduous years in Nazi Germany, 1933–45 13

“had shown his loyalty to the Chosen People.” At this time and later, Her- mann Weyl attempted to bring Gentzen to the Institute for Advanced Study in Princeton, but failed to get the financing needed. He had perhaps a special interest in the matter, because Gentzen had found a consistency proof of Weyl’s system of predicative analysis in the 1918 book Das Kontinuum. In 1937, Heinrich Scholz wanted to take Gentzen to the international philosophy conference in Paris, and it seems that the trip was possible only under a Nazi party membership (of a “helpless opportunist,” as Menzler- Trott writes, p. 77). It turned out to be the last time Gentzen and Bernays met, but the two continued their correspondence until the war. For Gentzen, it meant contacts with the expelled Jewish professor Bernays, an obvious risk to his own conditions in Göttingen. Bernays in turn must have had some understanding for Gentzen’s decisions. Contacts other than occasional letters were greatly limited as the 1930s was coming to its disastrous end; for example, Gentzen was invited together with Wilhelm Ackermann to a conference in Zurich in 1938 but neither managed to participate. In September 1939, when Hitler and Stalin began their war against the rest of Europe, Gentzen was called to military service, and what seems a last encounter with another logician took place in December 1939 when Gödel gave a guest lecture in Göttingen and Gentzen got a leave from his service station in nearby Brunswick to attend it. Gentzen gives the impression of having been in practice blind towards the Nazi regime and the conditions it set on life in the 1930s, academic and otherwise. Thus, Menzler reports the following (p. 260): “According to Bernays, Gentzen is supposed to have dismissed his reports on rioting against Jewish colleagues with the words: The government wouldn’t allow such a thing!” In a letter to Hellmuth Kneser that involved a possible job, of 16 February 1935, he writes about “the recommendation of Professor Bernays” among others and ends the letter with a “Heil Hitler!” (see Menzler-Trott, p. 55). There were strict rules about such salutes in public places: A failed Hitler salute could lead to the loss of one’s university job (see Allert 2008). By 1941, his readiness to use such salutes seems to have come to an end. Kneser had written to him and the reply, from May 1941, begins with: “I thank you for your letter, which to me seemed almost like news from another, past world.” Then he continued with his dreams of a post-war Germany with several chairs devoted to logic and foundations (Menzler-Trott, p. 135). His own “respected Professor Bernays,” the central figure in foundational research, robbed of his position and chased out of Göttingen, had no role in these plans.1 The same totally blind attitude is seen in his notes of October 1942, on completeness in intuitionistic propositional logic in the series AL

1 Incidentally, I had the occasion to ask in 2008 Dr. Ludwig Bernays why Paul Bernays didn’t request the restitution of his position after the war: “Oh, uncle Paul would never have done such a thing!” 14 Part I: A sketch of Gentzen’s life and work

(page 137), where the work of Wajsberg on Heyting’s propositional logic is referred to with great emphasis. It does not seem to have crossed Gentzen’s mind that in a world governed by the Nazis, no such praise could ever be published. – Poor Mordchaj Wajsberg who had already lost his life in some unknown Nazi concentration camp by the time of Gentzen’s writing. It is perhaps all too easy to exercise moral judgment decades later. Alex- ander Soifer writes in his The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators (p. 480) about Van der Waer- den that “one’s response to living under tyranny can only be to leave, to die, or to compromise.” One who left was Peter Thullen, a Catholic youth by his own words “totally absorbed by my mathematical research” at the time of Hitler’s rise to power, as reported in Reinhard Siegmund-Schultze’s pionee- ring study of the fate of mathematicians in the Third Reich, Mathematicians Fleeing from Nazi Germany (p. 395). Thullen’s diary contains the following from 27 May 1933, one day after Gentzen had handed in his doctoral thesis (p. 403): One could read in the papers today that a bank employee had been punished because he had refused to do the Hitler salute while the Horst-Wessel song was being sung. I guess this will happen to me sooner or later, too. Obliged to watch what is happening, yet refusing to go along, seeing the way noble ideals and all that is good are increasingly being replaced by brutality, meanness, vacuity and the cowardice of petty bourgeois, all this drains one’s energy and generates a feeling of impotent rage. Two months later, after a visit to Munster,¨ he had come to his conclusion (diary entry of 23 July, p. 408): I must admit that in Munster¨ even I was tempted to give up resistance, at least outwardly. However, I just could not bring myself even to raise my arm, which felt as if it were weighed down by lead, for the German salute or–worse still–to end my letters with the obligatory “Heil Hitler,” in the name of the man who ruined our Germany. The research grant I obtained in 1932 to go to Rome now seemed providential. It would allow me to gain some distance and to observe events from outside. It was clear to me that I would not return to Germany as long as Hitler remained in power. Whatever we now think of Gentzen’s compromises, let us keep in mind that he seems not to have had a choice with things such as Hitler salutes and that he, too, applied repeatedly for a scholarship, to go to Princeton. Eckart Menzler had the fortunate idea of writing already in the 1980s to people who had known Gentzen with the results reported in his biography. 3 Dr. Gentzen’s arduous years in Nazi Germany, 1933–45 15

One missive was Egon Mohr who reported back about his friend Gentzen among others that “it pleased him greatly when he could acquire a copy of the second edition of Oskar Perron’s Irrazionalzahlen in 1943 in a Prague bookstore.” The edition came out in 1939 and ridiculed the racist ideas of Ludwig Bieberbach about a German-Aryan mathematics (Menzler-Trott, pp. 241–242). Mohr himself got a death sentence for having listened to for- eign radio stations in early 1944, but was saved because of delays in the execution. Gentzen’s Habilitationsschrift was finished in 1939, when he entered military service in Brunswick, not far from Göttingen, in the end of September. He was given the task to observe the British airplanes that flew over and to listen to their radio traffic; not anything one with a mere school training would be able to do. Perhaps he had practised English with people such as Saunders MacLane with whom he studied in Göttingen. Moreover, Gent- zen’s mother Melanie Gentzen (born Bilharz) was born and lived the first years of her childhood in St. Louis. The task saved him from front service. Even so, he was by the end of 1941 in a bad shape and suffered a nervous breakdown. After some months of convalescence, he took up again scientific work. In 1943, Gentzen was called to teach at the German University of Prague, and to obtain the position, he had to deliver a lecture. One report about the lecture mentions the “slowness while lecturing of Dr. Gentzen caused by nervous disease” (Menzler-Trott, p. 237). The period in Prague ended with an arrest in May 1945 and imprisonment with the other German professors that had remained. The separate German university was created towards the end of the 19th century, through the division of the university founded in the 14th century into a Czech and a German unit. The position of the latter became by the 1930s a question of nationalistic contend by the Czech majority. The German occupation of 1938 led to the closing of the Czech university in November 1939. Gentzen, when accepting the position in Prague, must have been aware of this situation. Gentzen figures as one of the mathematicians in Maximilian Pinl’s article suite Kollegen in einer dunklen Zeit, Colleagues in a dark era (Pinl 1969– 76). Pinl was a Czech and a professor of the German University of Prague, arrested for half a year by the Gestapo in 1939–40 and forbidden any teaching activity in Germany, including occupied Czechoslovakia. This is how his series begins: One generation has passed since the troopers of apocalypse roa- red around a great part of the world. The time has at last come to look back at the caesura [complete break] of the years 1933– 45 in the development of German mathematical science and to recollect the colleagues of these dark years the innocent victims 16 Part I: A sketch of Gentzen’s life and work

of which they were. Gentzen appears in the fourth part among the German mathematicians in Prague. Pinl describes first briefly the history of the German university, then begins with (Pinl 1974, pp. 173–74): Members of the mathematical institute of the German University of Prague and of the mathematical teaching unit of the German technical university had to lament the loss of colleagues Peter G. Bergmann, Lipman Bers, Ludwig Berwald, Philipp Frank, Walter Fröhlich, Paul Georg Funk, Gerhard Gentzen, Paul Kohn, Paul Kuhn, Heinrich Löwig, Karl Löwner, Ernst Mohr, Georg Pick, Max Pinl, Artur Winternitz. Each person is then described in detail. Of Gentzen, we read a general account of his career and then the following (p. 173): Against the advice of the author of this report, he did not want to leave his position voluntarily in the Easter of 1945. It was taken from him when the pendulum of bloody terror of the past years hit back. He was put in a post-war forced-labour camp together with everyone else employed at the university. He did not measure up to the physical hardship and died on 4 August 1945. He hoped until the end to be able to prove the consistency of analysis, after the successful proof of the consistency of number theory, and dreamt of the founding of an institute of mathematical logic and foundational research. The actual end was ghastlier than Pinl’s report would suggest: The prisoners had to repair the cobbled streets that had been damaged, and passers-by threw stones on them. One stone cut tendons in Gentzen’s hand; unable to work, he was deprived of food and died of starvation in a prison cell.

4. The scientific accomplishments

Gentzen’s scientific results begin with his paper on what are known as Hertz systems, with research conducted in 1931 and published in 1932. The last work published during his life was the 1943 work on transfinite induction. These are the two fixed points of his achievement, and there do not seem to be any great results hidden in the stenographic manuscripts. There is, instead, one result Gentzen left unpublished that stands on a par with his most remarkable achievements, namely the proof of normalization for intuitionistic natural deduction in the longhand thesis manuscript of 1932–33. For the rest, Gentzen’s central achievements can be read from his publications, and the manuscript sources serve mainly to enrich the understanding of 4 The scientific accomplishments 17 the published papers. In the account of his achievements that follows, I have used in part my earlier reports, especially the Handbook-article of 2009, and the review Gentzen’s proof systems of 2012.

1. Hertz systems Gentzen’s first research accomplishment came in the summer of 1931, at the age of 21: Bernays had given him the task of studying an open problem in a logical theory another extraordinarius at Göttingen, the mathematical physicist turned into a logician Paul Hertz had developed. Hertz, otherwise known for his contribution to the foundations of statistical mechanics, put up in the 1920s a general theory of “systems of sentences” (Hertz 1923, 1929, identically titled). A sentence is an expression of the form a1,...,an → b. It has several interpretations: the circumstances given by a1,...,an bring about b, objects with the properties a1,...,an have also the property b,etc. The logical interpretation is that from the propositions a1,...,an,proposition b follows. The arrow thus represents the notion of logical consequence. Hertz systems have two forms of rules of inference, the first with zero premisses: Table 1. The rules of Hertz systems.

a1,...,an → ai for 1 i n → → → a11 ,...,a1n b1 ... am1 ,...,amk bm b1,...,bm,c1,...,cl c → a11 ,...,a1n ,...,am1 ,...,amk ,c1,...,cl c Hertz called the former kind of rules “immediate inferences” and the latter “syllogisms.” If a collection of sentences S is closed with respect to these rules, it forms a sentence system. A collection of sentences T is an axiom system for a sentence system S if each sentence in S is derivable from the ones in T . One of the principal questions for Hertz was the existence of independent axiom systems, with independence defined in the obvious way, as underivability by the rules. Bernays had followed and clearly also sustained the work of Hertz, and he suggested that Gentzen work with the problem. The result was Gentzen’s first paper, titled, in English translation, On the existence of independent axiom systems for infinitary sentence systems.Heshowedbya counterexample that there is an infinite sentence system for which there is no axiom system with independent axioms. Gentzen noticed in the course of his work, conducted in the summer of 1931, that the second form of Hertz’ rules can be replaced by one in which there is just one term b as a middle term of syllogism: Table 2. Gentzen’s cut rule in Hertz systems.

a1,...,am, → bb,c1,...,cn → c a1,...,am,c1,...,cn → c 18 Part I: A sketch of Gentzen’s life and work

He called this form of rule cut (Schnitt), and the first form of Hertz’ rules thinning (Verdunnung).¨ The “sentences” of the calculus of Hertz became the “sequents” of Gentzen’s later work. The calculus of Hertz was one of the two main sources of Gentzen’s sequent calculus. The other source was his development of the calculus of natural deduction. The rules of the latter, when adapted to the notation of sequents, led to sequent calculus proper. The background of sequent calculus in the work of Hertz and in Gentzen’s first paper is discussed at length in Bernays (1965). Gentzen’s work on Hertz systems can be seen as an early contribution to logic programming. The sentences are the same as program clauses, and derivations consist of cuts with the terms in the clauses. Gentzen’s work is described from this point of view in Peter Schroeder-Heister’s paper (2002). The main results contain a normal form for derivations that amounts, in modern terms, “to the completeness of propositional SLD-resolution in logic programming” (ibid., p. 246). Gentzen uses in his proof of the existence of a normal form a semantical notion of “consequence” (Folgerung), instead of a direct method in which a derivation is transformed into normal form through the permutation of cuts, with the motivation that it “gives important additional results, namely the soundness and completeness of our ways of inference,” and just mentions the possibility of a direct proof (p. 337). The normal form and completeness proofs of Gentzen are studied in great detail in Moriconi (2015), with a very detailed direct proof of the normal form theorem in what is the deepest account of Gentzen’s first work so far.

2. A logical formulation of the decision problem of arithmetic A two-page stenographic note, the earliest of all and dated September 1931, contains a translation of elementary arithmetic with the standard operations of sum and product into a relational language in which three-place relations σ(x, y, z)andπ(x, y, v)representthesumz and product v, respectively, of x and y. The eﬀect is that the whole of arithmetic is expressed in the language of pure predicate logic. On this basis, Gentzen clearly thought that one should not expect predicate logic to be decidable. This little piece begins the present collection of notes. Its title amounts to Reduction of number- theoretic problems to the decision problem of the lower functional calculus and bears the motto: A result in the small is worth more than no overview at large. The earliest systematic series is from the same period, with the signum VOR, the second item in this collection. It contains work on the replacement of functions by relations, in the same way in which the previous short note treated sums and products. Gentzen used this formulation in his doctoral 4 The scientiﬁc accomplishments 19 thesis, in the proof of the consistency of arithmetic without the induction principle.

3. Interpretation of Peano arithmetic in Heyting arithmetic When intuitionistic logic was in its infancy in the late 1920s, a lively debate arose on its proper formalization. It was conducted on the pages of the Bulletin of the Royal Belgian Academy. Valeri Glivenko contributed to this discussion in 1929 by a result by which classically provable negative formulas of propositional logic are already intuitionistically provable. The origins of the idea go back to Andrei Kolmogorov’s paper of 1925; In it, Kolmogorov gave a double-negation interpretation of classical logic in intuitionistic logic. For each formula A,letA∗ stand for the formula obtained by prefixing each subformula of A with a double negation. If A is provable classically, then A∗ is provable intuitionistically. Kolmogorov’s paper was written in Russian and remained unknown to logicians outside Moscow, except for its indirect influence through Glivenko (1929). Gödel found in 1932 an interpretation of classical Peano arithmetic in the intuitionistic arithmetic of Heyting. Gentzen arrived at a slightly differently formulated interpretation by early 1933. His translation was as follows: Table 3. The Gödel-Gentzen translation. 1. a = b∗ ; a = b 2. (¬A)∗ ; ¬A∗ 3. (A&B)∗ ; A∗&B∗ 4. (A ⊃ B)∗ ; A∗ ⊃ B∗ 5. (∀xA)∗ ; ∀xA∗ 6. (A ∨ B)∗ ; ¬(¬A∗&¬B∗) 7. (∃xA)∗ ; ¬∀x¬A∗ Gentzen tells to add double-negations to equations if these contain variables. Gödel’s translation removes also implications A ⊃ B,bytranslating them into ¬(A∗&¬B∗). The crucial points of the translation are disjunction and existence. The former is translated into the intuitionistically weaker ¬(¬A∗ & ¬B∗), and the same for existence that is translated into ¬∀x¬A∗. Even though there is a difference in the translations Gödel and Gentzen defined, the translation theorem that they established was the same: A formula A is a theorem of classical Peano arithmetic if and only if A∗ is a theorem of intuitionistic Heyting arithmetic. Gentzen’s work was planned to be a chapter in his doctoral thesis, but by February 1933, he had submitted it as a separate paper to the Mathemati- sche Annalen, the leading journal of mathematics of the day. Gödel heard 20 Part I: A sketch of Gentzen’s life and work of Gentzen’s discovery through Heyting, with Gödel writing in a letter of 16 May that he had found the translation and that “it should have been known in Göttingen since at least June 1932.” As a reason for the latter, Gödel wrote that at that time he had “reported on it in the Menger collo- quium, to wit, in the presence of O. Veblen who shortly afterwards went to Göttingen.” (See the Gödel-Heyting correspondence in volume V of Gödel’s Collected Works.) We know from Menger’s later recollections, based on mi- nutes of the meetings, that Veblen got very enthusiastic about the lecture, and invited Gödel to visit the newly established Institute for Advanced Stu- dy in Princeton. However, there is no reason to think that Gentzen would haveknownofGödel’s discovery. Gentzen’s reaction to the simultaneous discovery was to withdraw his paper from publication. The paper was preserved in the form of galley proofs and appeared in an English translation in 1969, in Manfred Szabo’s edition of Gentzen’s Collected Papers,thenin the original German in 1974. The importance of the Gödel-Gentzen translation was that it clearly showed the consistency of arithmetic to be within the reach of intuitionistic arguments. This was the first goal Gentzen had set to himself early in 1932. As a special case of the Gödel-Gentzen translation, with all atomic formulas double-negated, a translation of classical predicate logic into intuitionistic predicate logic is obtained. Clarification of the role of negation in intuitionistic arithmetic: Gentzen’s paper of 1933 has a section 6.2 that has been added in the end: He had sent the original paper to Heyting and responded in a letter of 25 February 1933 to Heyting’s assessment, meant for the journal Mathematische Annalen.In that connection, he also explained the contents of his section 6.2, by which it could not have been included in the original version. Gentzen’s point is that one can “avoid entirely the negation in intuitionistic arithmetic,” as well as the rule of falsity elimination, by defining negation as ¬A ⊃⊂.A ⊃ 1 =1. Now the axiom A ⊃ (B ⊃ A) gives the instance 1 =1 ⊃¬B,andin particular, 1 =1 ⊃¬¬x = y. As Gentzen had earlier shown equalities to be decidable, we have ¬¬x = y ⊃ x = y, and in consequence 1 =1 ⊃ x = y. From the derivability of falsity elimination for atomic formulas follows now easily the derivability of 1 =1 ⊃ A for any A, by induction on the length of the formula A. A three-page note of January 1933, item 7 below with the title Reduction of classical to intuitionistic logic, goes much further than the Godel-Gentzen¨ translation: It contains a translation from derivations in classical natural deduction for predicate logic to derivations in the ⊃,¬, ∀-fragmentofwhatis today called minimal logic, i.e., natural deduction with these logical operations, but without the rule of falsity elimination. One can see from the note how Gentzen arrived at the double negation 4 The scientific accomplishments 21 translation: In his attempts to prove normalization for natural deduction, he treated first the fragment without ∨ and ∃. Looking at the derivations in the fragment, he realized the following (here ( ) is the universal quantifier and V rule ⊥E, falsity elimination): A proposition of the classical predicate calculus can be written equivalently with only ⊃, (),andF, with the elementary propositions occurring only negated, i.e., with ⊃F. And this proposition is classically and intuitionistically of the same value (i.e., correct or incorrect in both systems). By the subformula theorem it is therefore provable, if at all, already with the inferences for ⊃, (),andV. Next he found that steps of indirect inference to a formula A in the fragment can be reduced to the immediate subformulas of A, so, in the end to atomic formulas, and if these are negated at the outset, even falsity elimination or rule V can be dispensed with. Gentzen’s last preserved letter to Bernays, of 16 June 1939, is a reply to a missing letter by Bernays. It contains cryptic remarks about a double- negation translation in which a formula and each of its subformulas are prefixed with ¬¬. It is noted that number-theoretic axioms and even the comprehension axiom scheme remain correct under the interpretation. A letter from Bernays to Gödel of 28 September 1939 tells (Gödel 2003,p. 126): I wanted to inform you that Mr. Gentzen has recently found out that the method of interpretation of the classical propositional calculus within the intuitionistic one can be extended from the number-theoretic formalism easily to the simple theory of types. He adds that no intuitionistic interpretation is achieved, because impredicative definitions remain. Gödel in his reply of 29 December writes that “the result seems to be of no special interest because the constructivity of the concepts used is problematic” (ibid., p. 130).

4. Natural deduction After Gentzen had finished the preparation of his Hertz-paper, he turned in early 1932 into the problem of the consistency of arithmetic and analysis, and wrote in December 1932 to his professor in Greifswald Hellmuth Kneser: I have set as my specific task to find a proof of the consistency of logical deduction in arithmetic... The task becomes a purely mathematical problem through the formalization of logical deduction. The proof of consistency has been so far carried out only for special cases, for example, the arithmetic of the integers 22 Part I: A sketch of Gentzen’s life and work

without the rule of complete induction. I would like to proceed further at this point and to clear at least arithmetic with complete induction. I am working on this since almost a year and hope to finish soon, and would then present this work as my dissertation (with Prof. Bernays). Progress was remarkably fast: The fourth item in this collection is a three- page note on “Five different forms of natural calculi.” This note gives a clear impression of Gentzen’s mode of working, of going through all the possibilities of formulating logical calculi and choosing ones that work best in a given situation. I have discussed in detail the discovery of natural deduction in my (2012, section 5). A note of 1931 in this collection adds a little piece: In the series VOR, item 2 below, there appears the idea of inference under assumptions. It is called “indirect inference,” and at the end of page VOR 13, his reasoning is as follows: Given A ∨¬A,ifB follows from A, then one can write B ∨¬A which is the same as A → B. “So the principle is: If B is provable from A,thenonecanwritedownA → B.” The first part of the series INH, written in October-November 1932, contains discussions about the semantical explanation of intuitionistic logic, about normalization, and about consistency proofs. There are no definitive results, but there are clear insights on, say, “where the Gödel point is hiding.” Gentzen saw that a notion of correctness in arithmetic cannot be defined by arithmetical means. Some time towards the end of 1932, it seems, Gentzen gave to Bernays a sketch for a doctoral thesis. This is item 6 below, written, in contrast to most of the other materials, in longhand and not stemming from the two folders, but from the Bernays archive in Zurich. It contains a definitive formulation of the system of natural deduction, and mentions normalization as a conjecture proved so far for a “partial calculus” with just conjunction, negation, and universal quantification. At some point, the word conjecture has been cancelled and a detailed proof of normalization added. As described above in section 3 on the interpretation of classical arithmetic, the double negation translation from classical Peano arithmetic to intuitionistic Heyting arithmetic was also included in the manuscript, but became then a separate article. Other parts of the manuscript contain the earliest formulations of sequent calculi and proofs of the equivalence of axiomatic, natural, and sequent formulations of logic. Derivation trees: Gentzen’s thesis manuscript begins with a “foundation and setting up of the calculus N1I ”: The way logical inference is formalized in Russell, Hilbert, Hey- ting (for intuitionistic inference) among others is rather far apart from the way of inference as it is practised in reality (say, in number-theoretic proofs). Notable formal advantages are won 4 The scientific accomplishments 23

thereby. I shall for once set up a formalism (“calculus N1I ”) that comes as close as possible to actual inference. One can assume that such a calculus has certain advantages and I think I can maintain, on the basis of my further results, that this is the case. Next comes the basic idea of inference from assumptions: Natural deduction, in contrast [to axiomatic logic], does not in general start from logical axioms, but from assumptions.... In- ference proceeds further from these. The result is made independent of the assumptions through a later inference. Natural deduction is not just a system of rules, but it is founded on much more general consideration of how any system of proofs from assumptions has to be organized: A proof consists of a number of propositions in an arrangement in tree form. Its look is as follows, in an example:

A1 A2 A6 A11 A3 A4 A5 A10 A7 A8 A9

The tree form was in practice a novelty in Gentzen: The arrangement of formal deductions had been invariably a linear succession of formulas, with some device such as a numbering of the formulas, so that one could refer to formulas deduced earlier. The first one, apparently, to depart from this form and to present proofs in tree form had been Paul Hertz in his work of 1923, as suggested by Schroeder-Heister (2002). Gentzen’s first paper (1932) treated these Hertz-systems and introduced tree derivations that stem from a rule of cut that has two premisses. The work of Hertz was not widely read, by which Gentzen’s work can be considered the one that made the use of the tree form for formal derivations generally known. He himself continues from the display of the above tree figure with the following remarks: I deviate somewhat from actual reasoning by the requirement of an arrangement in tree form, because 1. there is in actual reasoning a linear succession of propositions and 2. it is common in continuing further to use repeatedly a result already won, whereas the tree form allows to use in continuation only once a proposition already proved. Both deviations are obviously inessential, but they make it instead easier to conceive the notion of aproof. 24 Part I: A sketch of Gentzen’s life and work

Each of the horizontal lines represents an inference. The proposition directly below the line is the conclusion of the inference, and those directly above the line its premisses. There can further belong assumptions to an inference, these are certain uppermost propositions of the proof that stand above the conclusion of the inference (indicated as belonging to the inference through a common numbering). (I.e. not necessarily directly above, but in general: uppermost propositions of “proof threads” that pass through the inference line.) The last point is just an explanation of how a step of inference can close assumptions, indicated by a numbering next to the inference line and above the closed assumptions. The explanation continues with: All uppermost propositions of a proof are assumptions, and each belongs to exactly one inference. All propositions that stand under an assumption (i.e.,inthe proof thread determined by the assumption), but still above the inference line to which the assumption belongs, are said to depend on the assumption. The assumption itself included (so the inference makes the propositions following it, beginning with its conclusion, independent of the assumption in question.) Gentzen met Stanislaw Ja´skowski in Munster¨ in June 1936 and became acquainted with the linear version of natural deduction of the latter, Ja´skowski (1934). As Gentzen emphasizes in the above passage, the tree form displays the open assumptions on which a given formula in a derivation depends, as those topmost formulas of the upward threads that begin from the given formula but are not yet closed at the point in which the formula is found. This full display of the deductive dependences in a proof is a crucial property by which parts of derivations can be taken in isolation and composed in new ways. The method of permutation of parts of derivations is the most essential method in all of Gentzen’s central results: normalization for natural deduction, cut elimination for sequent calculus, and consistency proofs for arithmetic. The permutable parts of proofs are the subtrees determined by a given formula. All of this is lost in the linear variant of natural deduction, because the dependence of formula occurrences on assumptions is muddled and the permutation of parts of proof blocked. It is no accident that no results on the structure of formal derivations were won in the linear variant comparable to normalization and the subformula property. It has instead become the preferred approach in ﬁrst introductions to natural deduction, where the tree form is too abstract as a starting point. 4 The scientiﬁc accomplishments 25

Gentzen’s lost proof of normalization for intuitionistic natural deduction: The copy of the early handwritten manuscript of Gentzen’s thesis that I found in February 2005 contained, as the greatest surprise, a detailed proof of the normalization theorem of intuitionistic predicate logic. The chapter on normalization is a later addition to the manuscript, distinguished by a standard handwriting style that can be read without greater diﬃculties. The other parts are written in what is called Sutterlin¨ or also “deutsche Schrift,” a form of handwriting taught at schools at Gentzen’s time that is quite hard to read. There are also notational and terminological changes as compared to the ﬁrst chapter, such as the use of horseshoe implication in place of the earlier arrow. The following rules of the calculus of natural deduction for intuitionistic logic come from the introductory chapter, with the notation changed into that of chapter III. AI, AE stand for And-Introduction and And-Elimination, for Gentzen’s UE, UB (Und-Einfuhrung¨ and Und- Beseitigung), and similarly for Or, Universal, Existence,andFollow: AI AE OI OE [A] [B] AB A & B A & B A B A ∨ B C C A & B A B A ∨ B A ∨ B C ————————————————————————————————– UI UE EI EE [Dy] Dy (x)Dx Da (Ex)Dx C (x)Dx Da (Ex)Dx C ————————————————————————————————– FI FE V [A] B AA⊃ B A ⊃ B B A

Rule V, for “completion step” (Vervollst¨andigungsschritt), is from a marginal remark that indicates a change with the negation rules: In chapter III, negation is deﬁned through falsum and only rule V is kept of the earlier negation rules. Rules UI and EE have the standard variable restrictions. Square brackets indicate any number 0 of closed assumption formulas. De- rivations are such that implication introductions discharge in the end any remaining open assumptions. The same is true of Gentzen’s printed thesis, even if his system of natural deduction is usually presented as one in which completed derivations can have open assumptions. Thus, in Gentzen’s calculus, all formulas in a normal derivation are subformulas of the endformula that is also the longest formula in the derivation. 26 Part I: A sketch of Gentzen’s life and work

Contrary to the other extant parts of Gentzen’s thesis manuscript, the detailed proof of normalization for intuitionistic natural deduction is what he would have described as “publikationsreif verfasst,” ready for publication. The text is, even if ready for publication, not easy to read. Gentzen had developed his own style of writing in which the proof of normalization is presented by laborious verbal arguments, with a minimum of formal notation. The same style can be seen, even though to a lesser extent, in the published thesis and in, say, the proof of a crucial lemma 3.4.3 in Gentzen’s 1938 proof of the consistency of arithmetic. Gentzen had arrived at the system of natural deduction by September 1932, as in item 4. The idea of a normal form appears also at this time, and some indications of the problems with normalization can be seen in the accidentally preserved page and a half that is included here as an addition to the Five different forms,aswellasinitem5.Alostseriesofnotes with the signum HUG¨ , for “Hugel”¨ or “hillock” in English, seems to have contained the working-out of details of normalization. The proof in the manuscript for the thesis is very detailed, especially in questions such as the change of variables, called “purification of derivations,” to the effect that steps of normalization of derivations can be made without violating variable restrictions. The main idea appears already in the ∨, ∃-free fragment: An introduction followed by the corresponding elimination gives rise to what is called today a detour convertibility, and its elimination removes the “hillock formula,” a local peak of maximal length, from the derivation. The said fragment was of interest because it arose from the study of a classical system of natural deduction. Thus, Prawitz (1965) proves first a normalization theorem for classical logic without ∨, ∃. Gentzen gives a proof directly for the full language of intuitionistic logic. The main obstacle is when a disjunction or existence elimination separates a step of introduction from the corresponding elimination. Gentzen gives what are now called permutative conversions by which the order of elimination rules can be permuted until a detour convertibility is reached. The resulting proof is the same in substance as the standard proof today, say in Troelstra and Schwichtenberg’s text book Basic Proof Theory.

5. Sequent calculi Gentzen abandoned his idea of proving the consistency of arithmetic through a subformula result for a formulation in terms of natural deduction, for the reason that it is not possible to delimit the induction formulas in a derivation in advance. He also failed in proving a normalization theorem for classical natural deduction and was led to believe that there is no such result. His way out of these two failures was to turn into the development of sequent calculus for pure logic, both intuitionistic and classical, and to leave arithmetic aside for the time being. This was perhaps in February 1933, as suggested by a 4 The scientific accomplishments 27 letter to Heyting of 25 February 1933 (item 9 below). He worked out soon the Hauptsatz or cut elimination theorem, formulated so that it applied both to intuitionistic and classical logic, and all the rest of his thesis by 26 May when he handed it in. The real significance of Gentzen’s Hauptsatz is contained in its easy corollary, the subformula property. There had been attempts at separating the role of the different logical operations within the axiomatic logical tradition, similarly to the separation of the basic notions of geometry, but all axioms had to contain an implication or a negation. With sequent calculus, logical reasoning reached for the first time a pure form in which only those principles need be used that correspond to the logical forms found in the proposition to be proved; a consequence of the subformula property of cut-free derivations. Gentzen’s thesis Untersuchungen uber¨ das logische Schliessen got published with some delay in two parts in 1934 and 1935. It contained many new results found through the use of sequent calculus, such as a decision procedure for intuitionistic propositional logic. He had obviously many more results that were not included in the thesis, including remarkable results about intuitionistic predicate logic: Another letter to Heyting, of 23 January 1934, contains that he had achieved, through “a quite general theorem about intuitionistic and classical propositional and predicate logic [the cut elimination theorem] . . . the intuitionistic unprovability of simple formulas of predicate logic, such as (x)¬¬Ax. ⊃¬¬(x)Ax.” This result is now called the underivability of the “double-negation shift.” A related result that Gentzen must have cleared is the intuitionistic unprovability of ¬¬(x)(Ax∨¬Ax); both had been shown earlier through counterexamples based on the intuitionistic theory of real numbers as found in Brouwer (1928) and Heyting (1930)that Gentzen had studied in detail. He obtained his unprovability results syntac- tically from an analysis of cut-free derivations in sequent calculus, thereby showing that these results belong to pure intuitionistic logic, rather than de- pending on the intuitionistic theory of real numbers. The letters to Heyting in item 9 contain announcements of results that are not found elsewhere in Gentzen’s work. One trace of these letters is found in Heyting’s little book Mathematische Grundlagenforschung: Intuitionismus, Beweistheorie of 1934 that mainly discusses intuitionism and formalism. It contains a reference to Gentzen’s thesis, “to appear soon,” and mentions that it contains a decision method for intuitionistic propositional logic, with the disjunction property as a consequence (p. 17). The only other result explained is the “seemingly paradoxical theorem” by which ∀xA(x) is provable in intuitionistic logic, if ∃xA(x) is (p. 18). This observation is contained in Gentzen’s last known letter to Heyting, of 16 April 1934, where it is first stated as an answer to Heyting’s question that the disjunction property “follows in fact from my theorem,” with the addition that the theorem should lead similarly to the 28 Part I: A sketch of Gentzen’s life and work result Heyting announces in his book, a result not included in Gentzen’s thesis. Heyting’s knowledge of Gentzen’s thesis was at this point based only on their correspondence. The sequent calculi for classical and intuitionistic logic and their subformula properties, in particular, displayed the complete command over the structure of logical proofs that Gentzen had achieved in a very short time, though he did not waste much time in explaining these matters. For example, it is an easy exercise in sequent calculus to show that the standard classical connections between the connectives and quantifiers are un- derivable in intuitionistic logic, such as the purely classical implications ¬(A & ¬B) ⊃ (A ⊃ B)and¬∀x¬A(x) ⊃∃xA(x). Gentzen’s thesis clears this matter with a single phrase: He notes first that in a system of classical natural deduction, implication can be defined in terms of negation and conjunction and then adds just that “nothing of the kind holds for the calculus NI ” (p. 194). The underivability results came in fact through the intuitionistic sequent calculus LI, shown equivalent to the natural calculus NI through a translation of derivations that is given in the last part of the thesis. Semantic proofs of the independence of the intuitionistic connectives got published some five years later, by Wajsberg in 1938 and McKinsey in 1939. Gentzen’s method of syntactic proof of underivability in sequent calculus became a research topic with Kleene and is treated in detail in his Introduction to Metamathematics of 1952 (§ 80, pp. 479–492, titled Decision procedures, intuitionistic unprovability). Another early appreciation of Gentzen’s method is in Curry’s booklet A Theory of Formal Deducibility (1950), based on lectures as early as 1937. Kleene had obtained similar underivability results for the case of intuitionistic arithmetic by his “realizability method” already in his (1945), such as the unprovability of the double negation shift. Strangely enough, Heyting (1946) in commenting on Kleene’s results mentions Gentzen’s method, but seems to have forgotten Gentzen’s letter with the specific example of the double negation shift, and does not pursue the matter in any way. There is a good reason for Gentzen’s caution regarding the independence of the intuitionistic connectives and quantifiers: The underivability results do not establish undefinability, because there could be some other, more complicated formulas that define one connective or quantifier in terms of the others. Gentzen would have been still rather far from the exclusion of such possibilities, which requires, for example, consideration of the disjunction property under disjunction-free assumptions, as in Harrop (1960). Proofs of independence of the intuitionistic connectives and quantifiers by the methods of proof theory were given five years later, in Prawitz (1965, pp. 59–62). I have not seen these proofs carried through in sequent calculus, but they are beautiful advanced exercises in what are known as contraction-free sequent calculi, for which latter see Negri and von Plato (2001). 4 The scientific accomplishments 29

Gentzen continued to work on systems of sequent calculus even after the thesis. For example, item 15E, apparently from 1943 and titled Formulati- on of intuitionistic logic with symmetric sequents, contains an intuitionistic calculus with several formulas in the succedent part, similarly to the classical symmetric sequent calculus LK, but with precisely the right restrictions for rules of right implication, negation, and universal quantification. The multi-succedent intuitionistic calculus was re-invented, naturally with no knowledge of Gentzen’s anticipation, by Maehara in 1954. Even Curry’s early paper (1939) that investigates translations between sequent calculus and axiomatic logic notes in the end that except for the right negation and implication rules, the commas at right of symmetric sequents can be read as disjunctions, a remark limited to propositional logic. There is a manuscript with the signum WKRd from 1944 that contains an overview of variants of sequent calculi that Gentzen hoped could help in finding consistency proofs for arithmetic within an intuitionistic sequent calculus, the topic of the series BTIZ, and for analysis, the topic of the series WA. Whether Gentzen’s briefness contributed to it or not, misunderstandings about the basic properties of intuitionistic logic prevailed for a long time. One example is Wittgenstein who, judging by the publications of his pupil R. L. Goodstein, thought the inference from ¬∃x¬A(x)to∀xA(x)intuitionistically legitimate. The converse implication is intuitionistically provable, so with the claimed inference, the universal quantifier could be defined by the existential one; Instead, this particular argument against intuitionism and for the “strict finitism” of Wittgenstein and Goodstein is just fallacious: In Goodstein (1951, p. 49), written under Wittgenstein’s influence around 1940, it is stated that “some constructivist writers maintain that. . . a ‘reduction’ proof of universality is acceptable.” In Goodstein (1958, p. 300), we find again that Brouwer rejects indirect existence proofs, here ¬(∀x)¬P (x) → (∃x)P (x), “whilst retaining the converse implication ¬(∃x)¬P (x) → (∀x)P (x).” In other words, if (∃x)¬P (x) turns out impossible, a reduction gives (∀x)P (x); certainly not anything Brouwer or any other constructivist thinker would have endorsed. Heyting was initially enthusiastic about natural deduction, as is shown by his series of Dutch papers from 1935 on (see von Plato 2012, pp. 331–2 for details). Further, Gentzen’s intuitionistic sequent calculus opened up a whole new method for studying the properties of intuitionistic logic that had remained on an intuitive level, with the exception of the unprovability of excluded middle and the law of double negation. For example, a letter of Heyting’s of 26 November 1932 to Gödel contains the remark that “I haven’t been able to prove purely formally even that ¬a ∨¬¬a is unprovable” (see Gödel’s Collected Works, p. 66). With sequent calculus, one obtains easily the disjunction property of intuitionistic logic, as Gentzen tells in a letter of 30 Part I: A sketch of Gentzen’s life and work

16 April 1934 to Heyting: “I can answer affirmatively to your question. It follows in fact from my theorem that if A ∨ B is intuitionistically provable, either A or B is intuitionistically provable, and not just for propositional logic, but for the whole of predicate logic.” The unprovability of the weak law of excluded middle, as in Heyting’s letter to Gödel, is an almost immediate consequence. The enormous step ahead in the study of intuitionistic logic brought by sequent calculus is seen clearly in the work of Ingebrigt Johansson. He had found Heyting’s 1930 paper and wrote to him about his misgivings concerning the ex falso-axiom A & ¬A ⊃ B, “the magic works of a contradiction” as he called it. Johansson had made a very careful study of which results in Heyting’s article depend on this axiom; just a few indeed, and in particular, the axiom is interderivable with (A ∨ B)&¬B ⊃ A which latter he was not prepared to give up. In his answer, of 7 September 1935, Heyting recom- mends Johansson to study Gentzen’s work. It took Johansson about two months to clear all things up and to write the well-known article about minimal logic that appeared in 1936. Here is his reasoning: By the disjunction property, if A ∨ B is a theorem, either A or B is a theorem. If even ¬B is a theorem, the former must be the case, because predicate logic is consistent. Although Johansson does not have a name for it, he had shown the modus tollendo ponens-rule to be admissible, i.e., a rule the conclusion of which is derivable whenever its premisses are derivable, and this substitute of the full rule was enough for him. Johansson’s further remarks contain: the ex falso- axiom holds in minimal logic for negative propositions, as in A & ¬A ⊃¬B, and it is not needed if the law of excluded middle is admitted. It is difficult to understand why Heyting later completely ignored Gent- zen’s new methods by which all such open questions can be answered in a methodical way. The Heyting-Johansson correspondence shows clearly that Johansson had a far better command of intuitionistic logic than Heyting. The latter was moreover bound to the Brouwerian position by which mathematical proof comes from an extra-linguistic intuition, as in Heyting’s letter of 9 November 1935: We have the mathematical intuition at our disposal that allows us to build up the whole of mathematics without the help of any logic whatsoever. Theorems such as ‘π is a transcendental number’ surface from an empty system of axioms. Johansson’s answer to this particular point is that “I don’t understand youifbythisyoumean (π is transcendental). Whereas, if you mean (axioms for numbers ) ⊃ (π is transcendental), then I understand you fully.” His letter ends with the following words: Even if intuition is the same with all human beings, the meaning of words and expressions is in any case different. I can appreciate 4 The scientific accomplishments 31

the thoughts of another person only if I know the formal rules by which he uses those words.

By the above, it may be an expression of Heyting’s misgivings about formal logic that his book of 1956, Intuitionism: An Introduction does not mention any of Gentzen’s work even once. Some three decades got wasted before Gentzen’s proof systems, natural deduction in particular, became a regular part of the trade of intuitionistic logic and mathematics. Even a brief look at books by Heyting’s students and successors, such as Metamathematical Investigation of Intuitionistic Arithmetic and Analysis (Troelstra 1973)or Constructivism in Mathematics (Troelstra and van Dalen 1988), displays the close ties between natural deduction and intuitionism. Sequent calculi are equally central in Basic Proof Theory (Troelstra and Schwichtenberg 1996).

6. Type theory and the consistency of Weyl’s predicative analysis After his state exam in November 1933, Gentzen turned again into the proof theory of arithmetic and analysis. He seems to have tried out diﬀerent ideas ﬁrst: There are some scattered notes on these topics in the second part of the series INH, here given as item 10, including some of the basic ideas of ordinal proof theory. In the spring of 1934, he was working on type theory (second-order arithmetic) and wrote to Bernays:

To begin with, I treated type theory and carried through a consistency proof for it, one that is connected to the well-known simple proof of consistency for predicate logic. Then I added the mathematical axioms by which, naturally, the real diﬃculty just begins. It turned out that the consistency of mathematics is equivalent to the carrying over of the Hauptsatz of my dissertation from predicate logic to type theory.

The first result mentioned here is contained in the article Gentzen (1936b), Die Widerspruchsfreiheit der Stufenlogik (Consistency of type theory). Also from the spring of 1934 stems a proof of the consistency of Hermann Weyl’s system of predicative analysis. Very little is known about the proof: One letter from Bernays to Weyl tells that Gentzen was not able to reproduce it without his notes in 1937, when he met Bernays in Paris (in Menzler- Trott 2007,p.82).EvenGödel was aware of the proof, for he writes in the “Zilsel lecture” of 1938 that “Gentzen had the consistency of number theory, probably also of Weyl’s Kontinuum”(inGödel 1938, p. 108). The result seems to have been a byproduct of the attempts at producing a proof of the consistency of arithmetic, thus, not a strong result about analysis. Jean Cavaillès has in his book Méthode axiomatique et formalisme arather clear indication of what Gentzen’s consistency proof for Weyl’s system of 32 Part I: A sketch of Gentzen’s life and work analysis was. First he explains the proof of consistency of arithmetic quite at length (1938, pp. 157–162), then adds (p. 162): The method of the consistency proof for arithmetic extends without modification to those mathematical theories in which the predicates and functions are decidable or calculable in finitary terms: so for the constructive part of analysis. Gentzen wrote on 11 December 1935 to Bernays about the discussions he had had with Cavaillès who was visiting Göttingen at the time (see Menzler- Trott 2007, p. 64). The stenographic notes contain disappointingly little about type theory, just some five pages of notes of a general nature, given as item 14 below; perhaps that just indicates that Gentzen expected no results of great importance to be gained from type theory.

7. The original proof of the consistency of arithmetic The first consistency proof of arithmetic was finished around October 1934: The earliest preserved source on it is in item 12, the series WAV that stands for Consistency of arithmetic, for publication. The central idea of this first proof is a “reduction procedure” for sequents Γ → C in the calculus LI + CI (complete induction) by which it is shown that if C has some falsifiable consequences, Γ as well contains some falsifiable assumptions. Therefore, whenever nothing in Γ is falsifiable, also C is not falsifiable. A crucial Hilfs- satz (lemma) shows that such reducibility of sequents is maintained under cut. The details of the reduction procedure are explained in von Plato (2015). The problematic passage of the consistency proof is the termination of the reduction procedure in a finite number of steps. Pages 78–80 of WAV contain a remarkable three-page outline of what Gentzen calls “the second proof of consistency (LK consistency proof).” Contrary to the first proof, he uses multisuccedent sequents Γ → Δ. The result to be proved is that if Γ → Δ is derivable in LK + CI , it reduces to a sequent Γ∗ → Δ∗ in which either some formula in Γ∗ is false or some formula in Δ∗ true. Gentzen’s 1935 manuscript: Gentzen prepared in the spring of 1935 a long manuscript that he sent to Bernays chapter after chapter, titled Die Wi- derspruchsfreiheit der reinen Zahlentheorie. It uses a calculus of natural deduction for classical logic, with the notation of sequent calculus that displays the open assumptions a formula depends on at the left of a sequent arrow, thus, a third variant of the consistency proof. There has been an extensive correspondence between Gentzen and Ber- nays about the 1935 consistency proof, as well as some letters between Gent- zen and Weyl, and Gentzen and Van der Waerden. On this topic, only the 4 The scientific accomplishments 33 letters of Gentzen to Bernays have been preserved; they are found here in item 13. The first of these letters, dated June 23, 1935, was sent from Gent- zen’s hometown Stralsund and included the “final part” of the consistency paper. It went then on to discuss the suggestions made by Bernays and notes, among others, that the existence property of arithmetic follows for formulas ∃xA(x), “in case A(x) is not transfinite,” i.e., has no quantifiers. Towards the end Gentzen writes that he wanted to discuss in the final chapter transfinite ordinal numbers and their relation to reduction procedures and construction procedures, and then continues: “In the end, these things did not seem ripe for a presentation yet but could perhaps find place in a later separate publication.” A second letter written three weeks later, 14 July, contains passages re- miniscent of Brouwer’s explanation of bar induction in his (1924), where the connection to transfinite induction is also made. Gentzen is well aware that a new type of proof is about to surface here, but it is not known what he knew about bar induction. There is a reason to think that he had encountered it: In Heyting (1930, p. 65), the example of failure of the “double negation shift,” as discussed above in Section 5, is shown by the use of Brouwer’s notion of choice sequences, and the second part of the proof of failure uses the bar theorem with explicit reference to Brouwer’s 1924 paper. The first mentioning in print of Gentzen’s original consistency proof seems to be a remark in Bernays’ 1935 account of Hilbert’s researches on the foundations of arithmetic (p. 216): During the printing of this account, G. Gentzen has brought forth a demonstration of consistency of the full number-theoretic formalism (to be published soon in the Math. Ann.), by a method that corresponds throughout to the basic requirements of the finitary standpoint. Thereby the assumption mentioned above about the extent of finitary methods (p. 212) finds its refutation. The passage is somewhat ambiguous: Bernays writes earlier that the “finitary standpoint at the basis of Hilbert’s proof theory could not lead to a proof of the consistency of the usual number-theoretic formalism” (pp. 212–213). This view stems clearly from Gödel’s theorem, by which, writes Bernays, any suggested proof of consistency must contain an explanation of why it is not representable in the arithmetic formalism itself. Now, on the other hand, he seems to suggest that Gentzen’s proof has refuted the said limitation. The true nature of Gentzen’s proof was an object of a long exchange, as we shall soon see. Bernays had the submitted proof with him when he sailed to New York in September 1935. On board was Gödel; His position as the king of logicians was reflected in his status on board, in the first class. I have seen a postcard in the Bernays collection of the ETH-Zurich in which Gödel requests a meeting with Bernays, for the fired professor had to travel in a tourist class and 34 Part I: A sketch of Gentzen’s life and work could not just like that go and meet Gödel. During the Fall Term, the two commented on Gentzen’s proof, but only the answers of the latter have been preserved. They contain some information, though in a form that is often bound to frustrate the reader, such as the following passage from a letter of 11 December 1935:

The possible changes indicated by G¨odel were known to me, but are in fact inapplicable from the ﬁnite standpoint because of their impredicative character.

Gentzen answered to the criticisms by changing the 1935 consistency proof into one that uses the now generally known transfinite induction principle, with essential changes of large parts of the manuscript sent to the journal in February 1936. By good luck, the proof originally submitted for publication was preserved by Bernays in the form of galleys. They were published in English translation in the Szabo edition of Gentzen’s papers in 1969,and in the German original in 1974. Even if Bernays kept the proofs for forty years, they have been lost in connection with the 1974 publication in the Archiv fur¨ mathematische Logik und Grundlagenforschung (later Archive for Mathematical Logic). Weyl gave his copy of Gentzen’s manuscript to Stephen Kleene who, by his own telling, got a job from Wisconsin and gave the copy back after only two days. That was very unfortunate for the development of proof theory and foundational study in the U.S. It took another fifteen years before Kleene took up Gentzen’s work, in an article about sequent calculus (Kleene 1952a), and in the Introduction to Metamathematics. Criticisms of the 1935 proof: Bernays (1970) recalled that the main point of criticism was Gentzen’s implicit use of the fan theorem, a principle of Brouwer’s intuitionistic mathematics, by which, if all branches of a finitely branching tree are finite, the tree consists of a finite number of nodes. The same is explained in his prefatory words to the publication of Gentzen (1935) in 1974 (p. 97):

A methodical objection was made against the original proof, namely that it used implicitly a principle usually described today as the “fan theorem,” by which each branching figure that branches only finitely at each point and in which each thread ends after a finite number of component parts, can on the whole have only a finite extension.

The fan theorem is a special case of the bar theorem in which latter the branchings are denumerably infinite. These terminologies are much later than the results, but it is still a bit strange that Bernays explicitly describes the finite branching, when Gentzen’s proof clearly has denumerable branching. The 4 The scientific accomplishments 35 classical contrapositive of the fan theorem is König’s lemma: If a finitely branching tree has an infinity of nodes, it has at least one infinite branch. The existence of such a branch is based on classical logic. As we saw, Bernays writes that Gentzen’s use of bar induction was “implicit.” A detailed proof of Gentzen’s Hilfssatz can be given by the use of bar induction; It makes Gentzen’s proof of termination of reduction crystal clear (Siders and von Plato 2015). A missing theorem about branching trees? There is a letter of 17 July 1936 in which Gentzen, in reply to remarks by Bernays, makes the conjecture:

I consider it improbable that one could prove the theorem about the branching ﬁgure through TI up to ε0. I would rather suppose that one needs the whole II number class.

What this bar-theorem-like result is that in Gentzen’s view bears the weight of the whole class of constructive ordinals, is not revealed in detail. However, there is a letter of Gentzen’s of 4 November 1935 that contains a description of what he calls “the essence of the somewhat peculiar inductive inference” in the Hilfssatz, namely, why the reduction procedure should terminate:

Aproposition∀xF (x) is proved if each of the infinitely many special cases F (ν) is proved. Let each of these again be equivalent to a proposition ∀xFν(x), each special case Fν(μ)ofthese propositions again equivalent to a proposition ∀xFν,μ(x), etc. Let the following be known: Each arbitrary series of specializations ∀xF (x),F(ν) ⊃⊂ ∀xFν(x),Fν(μ) ⊃⊂ ∀xFν,μ(x),... ends after a finite number of components in a formula Fνμ...(), the correctness of which is known.Tobeprovednow:∀xF (x)iscor- rect. To this end, I infer as follows: The correctness of ∀xF (x)is secured if F (ν) holds for whichever arbitrarily chosen ν.Soletus assume that we had chosen a specific number ν, and it remains just to prove F (ν). This is ⊃⊂ ∀xFν(x). Now I infer just as before, namely, that to show that this proposition holds, it suffices to take whichever arbitrarily determined special case, say Fν(μ), etc. This chain of inferences must end after a finite number of steps, because each arbitrary sequence ∀xF (x),F(ν),Fν (μ),... had to be finite. Thereby ∀xF (x)isproved.

He says that this is “an analogy” that should be compared to “the image of the branching sequence of line segments.” The latter can be depicted as follows, with Gentzen’s example: 36 Part I: A sketch of Gentzen’s life and work

Fν,μ,...()

c ...... # ... c # c # . .

c ...... # ... c # c #

Fν(μ) ≡∀xFν,μ(x) c ...... # ... c # c #

F (ν) ≡∀xFν(x) c ...... # ... c # c # ∀xF (x) There is no bound on how many universal quantifiers can occur in a formula, and therefore a denumerably branching tree of any finite height can occur. “What do you think, now, about this way of inference? Shouldn’t it be finite?” These are his questions to Bernays, but he adds at once the parenthetical remark: If one turns the proof into an indirect one, i.e., begins like this: Assume that ∀xF (x) does not hold, then there is a counterexample ν so that F (ν) does not hold, so neither ⊃⊂ ∀xFν(x), etc, then the tertium non datur enters. Now we can read the suggestion as the choice of a path in a reduction tree that has a denumerable branching at each node. If there is at least one sequence of choices such that the topformula Fνμ...() gets falsified, we have established ¬Fνμ...(). If not, i.e., if no counterexample was found, proceeding all the way down to the root of the reduction tree we get that the assumption that ∀xF (x) does not hold is false, and a classical step of double negation elimination (the tertium non datur)gives∀xF (x). Gentzen is well aware that a new type of proof is about to surface here. One reason for expecting something new is, naturally, that the proof must go beyond principles that can be justified in elementary arithmetic. We can collect together the elements here: The essence of Gentzen’s bar induction argument: A classical indirect form of induction on a well-founded tree, as suggested by Gentzen, would be: Assume the root element in the tree has a property (here, a counterexample). Next show that if stage n has the property, there exists at least one case at stage 4 The scientific accomplishments 37 n + 1 with the property. Now, as in König’s lemma, there exists an infinite sequence of stages in which the property is found. On the other hand, it was shown that along each sequence, the property fails at some stage. Therefore the initial assumption, here the existence of a counterexample, is impossible. Bar induction, in analogy to the fan theorem and König’s lemma, is a constructive version of the above: If each case at stage n+1 has the opposite property (here: provability), even the previous stage n has the property. It is quite remarkable that Gentzen had seen through this matter. Brouwer’s choice sequences, bar theorem, and transfinite induction: In Brou- wer (1924) to which Gentzen refers in his (1938a), the bar theorem is called “the main theorem on well-ordered sets.” The additional remarks in Brouwer (1924a) make quite explicit the associated principle of transfinite induction on well-founded trees the bar theorem rests on (p. 645). The theorem was known to Gödel, and also to von Neumann who also was in Princeton at that time and must have heard discussions about Gentzen’s result. Von Neumann’s knowledge of Brouwer’s “fundamental theorem on finite sets” is shown by his letter to Brouwer, from April 1929, that contains a constructive proof of the existence of a winning strategy in chess by the fan theorem (see van Dalen 2011). Incidentally, von Neumann’s 1927 paper on transfinite induction in set theory mentions three set theories: naive set theory, intuitionistic set theory, and formal set theory, by which one can presume that he had studied Brouwer’s papers. Kreisel had had extensive discussions with Bernays about Gentzen’s original proof, and he writes (1987, p. 173) that “Gödel and von Neumann criticized the original–posthumously published–version.” There is a more general principle behind the fan and the bar theorem; In Kreisel (1976,p. 201) we find stated that both Gödel and von Neumann “naturally knew the theory of choice sequences that Brouwer had developed systematically, and especially the problematic assumption (of which Brouwer was particularly proud), namely that all functions F with arbitrary choice sequences of natural numbers as arguments and natural numbers as values. . . can be produced inductively. The best-known corollary is the fan theorem.” Correspondence with Bernays reveals some of the crucial details of Gent- zen’s original proof, not found in the printed version or in the galley proofs that were preserved, as in a letter of 14 July 1935:

I have written in fact nonsense on pp. 75–76; I held my eye on an older form of the notion of reduction, in which the reduction steps are uniquely determined. The passages could be corrected more or less as follows: At 15.21, reducibility should be replaced by: “There is a number ν so that for each series Rν of ν numbers, a series of at most ν sequents can be given such that the ﬁrst one is Sq, and each of these is formed from the preceding 38 Part I: A sketch of Gentzen’s life and work

one through a reduction step, and the last one has endform, and further, the possible choices are determined through the associated numbers from the series Rν.” Correspondingly under 15.23: “For each infinite series R of numbers, a finite series of sequents can be given, the first of which . . . ” as before. – I have, however, cancelled these passages completely, because they are not fully necessary; perhaps I could give sometime later complete proofs to both theorems in a special publication. Here it is stated quite clearly that the choice sequences in steps of reduction, represented as sequences of natural numbers, lead to endform in a finite number of steps. Gentzen’s description of the reduction procedure brings him close to Brouwer’s way of describing bar induction, in his 1924 formulation of the bar theorem. Here M is the set of choice sequences over which a function with values in N is given: If to each element of a set M a natural number β is associated, M is decomposed by this association into a well-ordered species S of subsets Mα, such that each of these is determined by a finite initial segment of choices. To each element of the same Mα is associated the same natural number βα. In Gentzen, M consists of the collection of reduction sequences of sequents and the choices to single reduction steps. Bar induction in Gentzen’s original proof seems not to have been noticed by Brouwer or Heyting. The latter took, with good reasons, the denumerable ordinals, on which the proof by transfinite induction rests, to be a creation of Brouwer’s, and mentions in his (1958) in a good tone each of Gentzen’s three papers on the topic (1936, 1938, 1943). Gödel’s notes on choice sequences in Gentzen’s original proof: The fate of Gentzen’s original proof was that it was simply put aside, just like Gentzen had put aside his detailed proof of normalization for natural deduction, the former saved only because Bernays had kept the galley proofs, the latter only because he had kept Gentzen’s handwritten notes. Gentzen’s use of induction on well-founded trees had been saved also in another sense, the extent of which is yet to be fully determined: Namely, as shown by the titles of topics in Gödel’s stenographic notes in his Arbeitshefte,thereareatleast 150 pages of work of his on Gentzen’s proof, with such suggestive titles as Principal lemma of Gentzen’s consistency proof with choice sequences (Ar- beitsheft 11, p. 28). In the earlier Arbeitsheft 4 (p. 39), there is the title Gentzen with choice sequences. The proof ends on p. 50 with: “Theorem. Induction Principle. [(n)A(Φn)] ⊃ A(Φ) ). A(const.) ⊃ (Φ)A(Φ).” The meaning is that if from the assumption that every one-step continuation Φn of a reduction sequence Φ has the property A it follows that Φ has the property 4 The scientific accomplishments 39

A, then from the base case A(const.) follows that all reduction sequences have the property A.

8. Ordinal proof theory During and after the criticisms by Bernays and Gödel, seconded by von Neu- mann and possibly even Weyl (as suggested by a letter of Weyl’s for which see Menzler-Trott 2007, p. 58), Gentzen laid the foundation of today’s ordinal proof theory: It can be seen clearly from his letters how this topic emerged in a few months’ time, with the consequence that the semantical explanation of sequents through a notion of reducibility and the consistency proof by induction on well-founded trees receded in the background. By his (1938), after having closed his new proof of consistency by a presentation of transfinite induction, he writes that he puts no specific weight on the notion of reducibility of derivable sequents and ends up with what seems almost a contradiction in terms: “I resorted to it at the time as one argument against radical intuitionism.” This paper was the second part of an issue of Heinrich Scholz’ publication series on logic and foundations. The first part was Gentzen’s essay on The present situation in mathematical foundational research. There he contrasts the lessons from intuitionism, presumably those of Brouwer’s (1928) insights, against “radical intuitionism, that rejects as senseless everything in mathematics that does not correspond to the constructive point of view.” Gentzen became a Brouwerian intuitionist in 1932 but then found by 1936 that Brouwer’s constructive ordinals codify intuitionistic principles in more conventional terms, it seems. Gentzen’s letters to Bernays from 4 November 1935 to 17 July 1936 in item 13 below describe to some extent the way to ordinal proof theory. Earlier attempts in this direction can be found in INH, items 5 and 9. In the first of these letters, the connection between the original proof and transfinite induction is explained to some extent. An obvious source for Gentzen’s idea of using constructive ordinals is Hilbert’s 1925 article Uber¨ das Unendliche. It is one of the few papers Gent- zen mentions in his printed work, and Hilbert’s discussion and presentation of ordinals up to ε0 is extensive. Hilbert’s idea was to prove the continuum hypothesis in this framework. He states among others that transfinite induction could be replaced by ordinary induction in arithmetic, something Gentzen proved to be impossible, and that he can prove the continuum hypothesis! Bernays (1935a) is an overview of the state of proof theory prior to Gentzen’s consistency proof. It contains an incredibly clear anticipation of Gentzen’s consistency proof of arithmetic. In the search of “intuitionistic proofs that cannot be formalized in N” (Hilbert’s formalization of first- order arithmetic), he notes that the consistency of arithmetic is provable “in a formalism N∗ that one obtains from N by the adjoining of certain 40 Part I: A sketch of Gentzen’s life and work non-elementary recursive definitions,” backed up by an example. Then (p. 90): It follows that this sort of recursive definitions surpass the formalism N. On the other hand, such a recursive definition intervenes also in the formal deduction of the principle of transfinite induction applied to an order of the type of ordinal limn αn,or

αk α0 =1,αk+1 = ω (k =0, 1....) .

This type of order can be realized for the integers by an order

a ≺ b

that is deﬁnable by elementary recursions. And the said principle is expressed, for this order, by the formula

(x){(y)(y ≺ x −→ A(y)) −→ A(x)}−→(x)A(x)

Bernays suggests that this principle is not provable in the standard arithmetic formalism but it is instead provable intuitionistically. Therefore (p. 91): It is apparent that the special case of the principle of transfinite induction considered is already an example of a theorem that is provable by intuitionistic mathematics, but not deducible in N. Consequently one would propose, in concordance with the theorem of Gödel, that one finds an intuitionistic proof of the consistency of the formalism N in which the only part non-formalizable in N is the application of the said principle of transfinite induction. He adds that the proposal is “just one possibility at the moment.” The specific example of recursive definition Bernays chose was not strong enough, as Skolem (1937, p. 439) pointed out. Bernays therefore modified his example in the second volume of the Grundlagen (p. 340). More details on ordinal proof theory are found in subsection 11 below.

9. Summing up of several inductions into one Gentzen’s last paper from his hand was a short piece of the fall of 1944, just three pages. It was published in 1954 together with the editorial statement by which “Gentzen had dedicated this little article to Heinrich Scholz on his 60th birthday 17 December 1944.” The result of the paper is that a formal derivation in arithmetic can be so transformed that only one step of inductive inference occurs in it. It expresses an insight that Gentzen arrived at 4 The scientiﬁc accomplishments 41 early on, around the turn of 1934–35, namely: the number of steps of inductive inference is not a crucial parameter in the proof-theoretical analysis of formal arithmetic derivations, but rather the length of induction formulas. In the series BZ,forBeweistheorie der Zahlentheorie (Proof theory of number theory, item 11), this result is attempted, with a negative outcome on page 6. Then there is an addition: “Yes. Page 6.2!” A letter to Bernays of 3 March 1936 mentions the “little result.” At the request of the latter, Gentzen sent a detailed proof in a letter of 13 August 1936. These letters are found in item 13 of this collection.

10. Yet another consistency proof The changes in the original consistency proof led to a presentation that mixed elements from different proofs. Gentzen was not satisfied with the resulting published version (1936) and produced yet another proof of consistency within the classical sequent calculus LK, this time with an explicit ordinal assignment in standard notation for ordinals, and published in 1938 under the title Neue Fassung des Widerspruchsfreiheitsbeweises fur¨ die reine Zahlentheorie, abbreviated as NEU (New conception of the proof of consistency for pure number theory). The architecture of this paper is very clear. The earlier proof contained a reduction procedure for arbitrary derivable sequents, but NEU considers only derivations of the empty sequent → that is the expression of incon- sistency in sequent calculus. As a consequence, the reduction procedure is closely analogous to cut elimination in LK , as we shall see. As a preparation to the reduction, free variables in the derivation of → other than eigenvariables are replaced by any numbers, say 1. Next the endpiece of the derivation is defined: It is the subtree with → as a root that includes all derivation branches up to conclusions of logical rules, or if none is reached, up to topsequents that are then groundsequents, either logical ones of the form D → D or arithmetical ones that express instances of the basic laws of arithmetic other than the induction principle. No complete list is given, but one list of such laws is in Gentzen (1933). The leading ideas of the reduction of derivations of → are two: First, if there is a step of proof by induction CI in the endpiece, it must be a numerical induction, i.e., one in which the conclusion is a natural number, for a free variable in the conclusion is excluded by the preparation explained above. The rule of induction is written as: F (a), Γ → Θ,F(a) CI F (0), Γ → Θ,F(t) In a numerical induction, the term t in the succedent of the conclusion is some number n, and the conclusion F 0), Γ → Θ,F(n)canbederived through a series of substitutions and structural inferences: In the premiss 42 Part I: A sketch of Gentzen’s life and work

F (a), Γ → Θ,F(a), a is an eigenvariable so that a substitution of a by anumberm in its derivation gives a derivation of F (m), Γ → Θ,F(m). The structural inferences are cuts and possibly repeated contractions and exchanges, the latter two indicated by Str* in the following reduction:

F (0), Γ → Θ,F(1) F (1), Γ → Θ,F(2) Cut F (0), Γ, Γ → Θ, Θ,F(2) Str* F (0), Γ → Θ,F(2) F (2), Γ → Θ,F(3) Cut F (0), Γ, Γ → Θ, Θ,F(3) . . F (0), Γ → Θ,F(n − 1) F (n − 1), Γ → Θ,F(n) Cut F (0), Γ, Γ → Θ, Θ,F(n) Str* F (0), Γ → Θ,F(n)

After numerical inductions have been removed, the endpiece has as essential rules only contractions and cuts (weakenings and exchanges can be dispensed with). A crucial lemma 3.4.3 shows that there is at least one cut formula in the endpiece such that it is the principal formula of a right resp. left logical rule in two steps that delimit the endpiece from above. The proof of the lemma is a long and verbose argument that I had never been able to formulate to my satisfaction. There is a letter of Bernays, written 13 June 1938 after he had read the page proofs of Gentzen’s article, in which he expresses dissatisfaction with the proof of the lemma and suggests another one. Gentzen, in turn, admits in a letter of 22 June that the proof is not “tight.” Perhaps symptomatically, Takeuti’s book Proof Theory gives a different proof (pp. 105–6). After discussions about the proof with Dag Prawitz, he presented in correspondence in May 2012 the following delightful reading of Gentzen’s proof. In the proof, a cluster of formulas stands for Gentzen’s Formelbund, i.e., all the occurrences of a cut formula up from the premisses of cut to the topsequents of the endpiece, even multiplied if there are contractions. The following is Prawitz verbatim, except that the word ending is changed into endpiece that I have used even earlier as a translation of Gentzen’s Endstuck:¨ Prawitz’ proof of Gentzen’s lemma 3.4.3: A suitable cut is a cut such that it holds for the cluster of formulas determined by the cut formula that both its left and right side has an uppermost formula that stands in a sequent as the principal formula of a logical inference and belongs to the endpiece of the derivation. Gentzen considers a collection of threads T in the endpiece: t belongs to T whenever its first element is a sequent in the endpiece that stands as the conclusion of a logical inference and its last element is the endsequent. The property P of sequents that stand in such threads is defined by: S has the property P whenever there is formula F in S such that the cluster associated 4 The scientific accomplishments 43 with F has an uppermost formula that stands above F or is F and stands as principal formula of a logical inference. The following assertions (made more or less by Gentzen) hold obviously: (1) The first sequent in a thread t in T has the property P . (2) The last sequent in a thread t in T lacks the property P .

(3) There is a sequent S1 in each thread t in T such that S1 has the property P , but the sequent S3 immediately below S1 lacks the property P .Inthis case, S1 is the premiss of a cut, and more precisely, there is a unique formula F that is responsible for S1 having the property P and is the cut formula. Gentzen argues for the following assertion:

(4) In case (3), the other premiss S2 of the cut belongs to a thread t in T . The argument is that F is logically compound and as such it can belong to S2 only if there is a sequent above S2 that is the conclusion of a logical inference, but then this sequent is either the ﬁrst element of a thread t in T or stands above such an element. Consider now all cuts such that the conclusion lacks the property P but one of the premisses has the property P . Choose an uppermost one, that is, a cut of this kind such that there is no cut of this kind standing above (one of its premisses). There must be such a cut, and for this cut C it holds:

(5) Both premisses S1 and S2 of C have the property P , but the conclusion S3 lacks the property.

The unique formulas F1 and F2 that are responsible for S1 and S2 having the property P are by definition such that the associated cluster has an uppermost formula that stands above Fi or is Fi and stands as principal formula of a logical inference. But, as Gentzen says, the cut is then a suitable cut. QED. Any two cuts can be permuted, so a cut with the cut formula that has been singled out by the lemma can be shifted up until its two premisses have been derived by the right and left rules in which the formula is principal. Now we have exactly the situation of cut elimination for the logical calculus LK in which, after some permutations, cuts on a formula are reduced to cuts on its immediate subformulas. The search of suitable cuts, followed by the reduction of these cuts to ones on shorter formulas, expands little by little the endpiece upwards in the derivation of → . The permutability of cuts is lost if the rule of multicut is used. Therefore Gentzen could not simply extend his earlier proof of cut elimination for the classical sequent calculus LK to arithmetic. Here is the hidden reason for the complications of the 1938 proof, especially the altitude line construction mentioned already in the introductory section above (for details see Negri 44 Part I: A sketch of Gentzen’s life and work and von Plato 2016). Next in NEU comes the proof of termination of the reduction procedure. This is a straightforward thing, even if complicated in details: Gentzen gives an ordinal assignment to derivations and shows that each reduction step lowers the ordinal. The assignment of ordinals to derivations is adjusted in great detail so that the result follows even under the said construction that apparently complicates derivations, instead of simplifying them. A subsequent work is promised in which the nature of the transfinite induction in the proof is to be treated, but it never appeared nor are there any notes left for such a piece. The 1938 paper suggests towards the end that a consistency proof of analysis could be expected. Gentzen worked intensely on the problem from the summer on, with a set of 38 well-organized stenographic pages preserved, between page numbers WA77 and WA180. However, by the end of the year, the attempt came to an end and Gentzen started to work on the proof theory of intuitionistic arithmetic that soon led to his next and, as it turned out, last great paper, the Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der Zahlentheorie. I shall discuss WA below, in Section 5.3. Gentzen’s new proof had two direct early descendants: The first one was a consistency proof within axiomatic logic, by Laszlo Kalmár. It is preserved as a 21-page typewritten manuscript in the Bernays collection, and the essential contents are also found in the second edition of the second volume of the Grundlagen der Mathematik. In a letter of 4 October 1938,Kalmár explains some of his views about his proof to Bernays. He writes that his proof simplifies the one in NEU:

At the same time, the original (Hilbertian, I believe) basic idea of the consistency proof comes out quite clearly (one has with Gentzen the feeling that the basic idea stands in the possibility of an intuitionistic interpretation of formal number theory, the one given by Gentzen and G¨odel; This idea recedes entirely in the background in my formulation). What I describe as the “basic idea of the consistency proof” is an approach without Hilbert’s ε-symbol; if I remember right, I learnt this approach from you in G¨ottingen in 1929. . .

I emphasize that I consider Gentzen’s proof, especially in the new formulation in Scholz’ Forschungen as quite clear and understandable; Moreover, Gentzen’s natural calculus (the “NK” as well as the “LK”) is most sympathetic to me. I can think, though, of a reader who is unwilling to learn a new calculus for 4 The scientiﬁc accomplishments 45

the purpose of the consistency proof, but would prefer to see the proof within the old calculus. A second proof of Gentzen’s result was given by an unwilling Wilhelm Acker- mann, after repeated pleadings on the part of Bernays. Ackermann recast Gentzen’s proof in the terms of Hilbert’s proof theory of the 1920s, namely what is called the ε-subsitution method, with the article Zur Widerspruchs- freiheit der Zahlentheorie (On the consistency of number theory) published in 1940. Haskell Curry, a student of Hilbert’s, has written (1977,p.25): The German writers tend to shy away from the Gentzen techni- que and to devise ways of modifying the ordinary formulations so as to obtain its advantages without its formal machinery. This is much the same as if one attempted to develop group theory without introducing the abstract group operation. . . 11. Provable and unprovable cases of transfinite induction Gentzen finished in the summer of 1939 his Habilitationsschrift the title of which is Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie (Provability and unprovability of initial segments of transfinite induction in pure number theory). It was submitted as a paper to Mathematische Annalen in July 1942, after Gentzen had recovered from the nervous breakdown that was caused by the conditions of life in military service (for which see even the last page of this collection). The paper was published in a second-to-last issue in 1943, before publication began again several years after the war. It has turned out recently that the Habilitationsschrift is extracted from the much longer series BTIZ that is preserved from page 133 to page 274. There is a gap from page 161 to 198, but a brief summary note page numbered 160d tells the following: The gap contains an elaboration of the materials for the Habilitationsschrift. I shall turn to the unpublished parts in Section 5.2. All but the first four pages of the original Habilitationsschrift have been preserved in the form of a stencil copy, but by pure accident: The typescript with corrections, deletions, and additions done in ink for the printed version was used for writing down the series BTIZ in 1943. It is likely that Gentzen had it from the printer’s together with the proofs of the article. The BTIZ- pages are written on a stapled set, from the back of the last page on and with each page ripped off as it got full. Incredibly, a review of the paper was published in the US during the war, in The Journal of Symbolic Logic of 1944. The reviewer was Bernays who had moved to Switzerland in 1934. He worked as a bridge between European and American logicians through the late thirties and the war: I have seen 46 Part I: A sketch of Gentzen’s life and work a letter from Bernays to Georg Henrik von Wright who wrote reviews for the journal in Helsinki during the war. Bernays asks for second copies of the reviews, because the originals “were destroyed in an air attack.” Despite Bernays’ efforts, the methods of Gentzen’s 1943 paper, or the earlier 1938 paper, did not become widely used: Both works remained by and large unread in the 1940s and early 1950s, except for a chosen few such as Schutte,¨ Kreisel, and Takeuti. Kleene’s Introduction to Metamathematics had covered the logical aspects of sequent calculi, but no accessible introduction to the proof theory of arithmetic was available before Takeuti’s Proof Theory of 1975. Even today, one can find whole books devoted to Gödel’s incompleteness theorem, both popular and scientific, that do not even mention Gentzen, the person who has contributed most to solving the problems of consistency and incompleteness of arithmetic, not to talk about explaining his fundamental discoveries about incompleteness. Gentzen’s idea in his last paper was to extend the domain of natural numbers by constructive ordinals and to formulate transfinite induction as a separate rule of proof within the extended system. The basis was, as in 1938, his classical sequent calculus LK of the doctoral thesis of 1933. He then proceeded to prove that transfinite induction up to ordinals below ε0 is an admissible rule of the sequent calculus for arithmetic with ordinary induction. This observation, apparently due in some form to Bernays, was made already in 1936. As a second and main result, Gentzen showed that transfinite induction up to ε0 is not provable by ordinary arithmetic, even if it is expressible in arithmetic. By the unprovability of transfinite induction, arithmetic is consistent, because anything would be provable in an inconsistent system. Secondly, arithmetic is incomplete: Transfinite induction expressed as an arithmetic formula is a true but unprovable sentence. As Gentzen points out, transfinite induction is a mathematical principle, whereas Gödel’s original proof of incompleteness had produced a “metamathematical” principle that was coded into arithmetic through Gödel numbering. Thus, it was often held to be an artificial construction that has no serious bearing on “real mathematics.” Ordinal numbers below ε0 are defined by the inductive clauses that 0 is an ordinal below ε0 and that α + β and ωα are ordinals below ε0 whenever 0 ωα α and β are. The notation is ω0 ≡ 0,ω1 ≡ 1 ≡ ω ,ω2 ≡ ω,...,ωα+1 ≡ ω if α ≺ ω,andωα ≡ 0ifω α, the last using the basic relations α ≺ β and α β. The place of natural numbers is taken by the finite ordinals:thoseα for which α ≺ ω. The successor function for natural numbers is represented as sn ≡ n + ω0 ≡ n+ 1. Ordinals for which α ≺ ω is not the case are transfinite 4 The scientific accomplishments 47 ordinals. The rule of induction is as follows: y ≺ ω, A(y), Γ → Θ,A(sy) Ind t ≺ ω, A(0), Γ → Θ,A(t) In the rule, y is the eigenvariable. The only difference with Gentzen’s earlier rule of induction is that the finiteness conditions y ≺ ω and t ≺ ω have been added in the assumption parts. Theruleoftransfinite induction is 0 ≺ y, ∀z(z ≺ y ⊃ A(z)), Γ → Θ,A(y) TI A(0), Γ → Θ,A(t) Also here, y is the eigenvariable of the rule. If t is a numerical term (an ordinal less than ε0), application of TI gives a “derivation of transfinite induction up to the ordinal t.” Gentzen shows first that rule TI is not needed below ε0. To this purpose, he introduces a one-place “property parameter” P.WhenP(0) → P(t)is the endsequent of a derivation, we have “derivability of transfinite induction up to t.” Gentzen proves that if t is a numerical term, then P(0) → P(t) is derivable from the premiss 0 ≺ t, ∀y(y ≺ t ⊃P(y)) → P(t)ofruleTI in the calculus LK+Ind. Thus, initial segments of transfinite induction can be eliminated, analogously to the elimination of standard arithmetic induction up to a given number n, as in the 1938 consistency proof explained in Section 4.10 above. The main result is that transfinite induction up to ε0 is not derivable in the calculus LK+Ind, unlike inductions to lesser ordinals. The proof is based on the reduction procedure of the 1938 proof of consistency of arithmetic. First the notion of an endpiece is extended to LK+TI. In 1938, the endpiece ended with an empty sequent; now instead, the endsequent has the form P(0) → P(t1),...,P(tn). The next step is to assign ordinals to derivations of such endsequents. A crucial lemma states that in a derivation of an endsequent, the least of the ordinals ti cannot be greater than the ordinal of the derivation, or else the derivation can be reduced. The ordinals of derivations are less than ε0, so it follows that transfinite induction up to ε0 is not derivable. A more precise result can be given: In a derivation of P (0) → P (t), the maximum length n of induction formulas in the derivation has the following connection to t: t ≺ ωn+3. There is a contemporary assessment of Gentzen’s paper by Wilhelm Acker- mann (1940a) who, together with Heinrich Scholz (1940), was the examiner of the Habilitationsschrift. One of the main points of the former was that one could dispense with the explicit introduction of transfinite ordinals in the formalism: 48 Part I: A sketch of Gentzen’s life and work

One could then formulate transfinite induction as a purely number-theoretic statement in which no symbols for ordinal numbers occur. The demonstration of unprovability of this statement could follow in exactly the way given by the author. The fact would be expressed, even more clearly, that the incompleteness of methods of proof concerns the pure system of number theory, with no additions at all. Ackermann’s paper of the same year gives a consistency proof through the ε-substitution method, a monstrous undertaking in comparison to Gentzen’s 1938 lucid proof within sequent calculus. Ackermann notes that “the natural numbers can be ordered after every order type of the second number class,” with the consequence that “natural numbers themselves, in such an order, represent the transfinite ordinal numbers needed” (p. 179). A large part of the paper is devoted to the definition of such an order. Bernays gives a similar example already in the 1939 second volume of the Grundlagen der Mathematik, p. 361. Gentzen’s last paper was a milestone in the second main direction of proof theory in addition to structural proof theory, namely ordinal proof theory. The proof-theoretical ordinal of a theory is defined as the least ordinal that is needed for proving its consistency, just as with ε0 for Peano arithmetic. The central aim in ordinal proof theory has been to determine the proof-theoretical ordinals of ever stronger subsystems of second-order arithmetic. Among the highlights of this development is the determination of the “ordinal of predicativity” around 1964, independently of each other, by Schutte¨ and Solomon Feferman. The present state is marked by ordinal 1 analysis of the principle of Π2-comprehension in the work of Toshiyasu Arai and Michael Rathjen. These developments are reviewed in Rathjen (1995) and (1999).

12. The foundations of mathematics Gentzen was no great friend of abstract philosophy, but he took in his papers standpoints in what are questions of the philosophy of logic and mathematics. In his early work, he endorsed fully the four “insights” expressed in Brouwer’s 1928 Intuitionistische Betrachtungen uber¨ den Formalismus.His view was less radical than that of Brouwer, in that he suspended judgment on classical mathematics. The translation of classical to intuitionistic arithmetic of 1933, especially, must have been a useful lesson in this respect: It showed that classical reasoning in arithmetic is not more “unreliable” than intuitionistic reasoning. What was lacking in the former was a clear explanation of the meaning, of the inhaltliche Deutung,asitissoconcisely expressed in German. Gentzen is usually counted among the formalist school of Hilbert, but 4 The scientific accomplishments 49 he was more of an intuitionist than a formalist. In a short piece from the fall of 1931 on the formation of abstract mathematical concepts, item 3 below, he describes Hilbert’s formalism as “artificial.” The natural numbers as well as other mathematical notions are “concepts of thought that have no reality” but just receive a sufficient meaning through their “repeated practical applicability.” At one point, on page WA 80, Gentzen uses for his position the invented word Innentheoretiker, something like “internal theorist,” in contrast to a finitist. At another place, on page 9 of the series WAdd of 1944, he describes in third person “the task of the constructivist.” On the other hand, he took on several occasions distance from “radical intuitionism.” Gentzen’s original proof of the consistency of arithmetic was based on intuitionistic ideas and methods, and even its modification was based on the intuitionistic theory of constructive ordinals. He believed that one could give a constructive justification to analysis in a similar vein, by proceeding further in the hierarchy of ordinals, describing the difference between arithmetic and analysis as (1938, p. 44): I could not really say at what “point” the constructively indubi- table ends and the continuation of transfinite induction becomes dubitable. I believe rather that the relation between the indubitability of the whole domain needed for the proof of consistency and its first beginnings, say up to ω2, is not different from the relation between the indubitability of a hundred-page numerical calculation and a calculation of a few lines. Gentzen was firmly convinced that the notion of actual infinity will never receive an acceptable justification. He followed Skolem in taking set-theoretical infinity to be a relative, even fictional concept. Thus, he doubted whether set theory will ever find a justification. In Gentzen, foundational standpoints were a means to an end: Their role was in making it clear what one was aiming at. An example is offered by his attempts at proving a normalization theorem and the consistency of arithmetic as a corollary in the fall of 1932. In the series INH, he is asking what one accomplishes with a consistency theorem, and the aim in seeking an answer is to understand better the significance of formal work. Gödel’s incompleteness theorems were clearly the leitmotif of Gentzen’s work. For some, incompleteness was a failure that meant the end of the foundational enterprise. It is interesting to note what Gentzen’s general attitude was in this respect – his view is found in the final reflections of the 1936 consistency proof (§17.1): Practical number theory need not be bounded by any formal limitations, it can always be extended by new kinds of concepts, 50 Part I: A sketch of Gentzen’s life and work

perhaps even by new kinds of ways of inference. How is it, then, with the proof of consistency? The thing is that one must extend, with each surpassing of the boundaries so far, even the consistency proof to cover what has been added. The consistency proof is already organized so that this is possible without diﬃculty in the widest measure.

5. Loose ends

It is fair to say that Gentzen was mostly able to pursue his research program until the outbreak of World War II. Nazism cut a lot his potential scientific contacts and rendered him lonely. However, his delicate mental balance was a greater factor: Medical reports tell of periods of “mental slackness and da- zedness” (see Menzler-Trott 2007, p. 231). Gentzen appears to be the fragile genius who can lose his drive and fall into lengthy periods of depression. After one such period of depression that occurred between 1936 and 1938 had passed, Gentzen was able to finish his “new proof” of consistency of arithmetic (Gentzen 1938). He then concentrated on extending the new proof to analysis. Judging from his steady progress with the series WA in the fall of 1938, nothing, not even the Kristallnacht of 9 November 1938, seems to have made a difference. The Finnish student Oiva Ketonen arrived to Göttingen on the day of the violence and was shocked, both by the destruction and the attitudes of some of the academics. Some time later, he wrote in a frustrated tone to his professor about Gentzen’s inaccessibility. To me he told that Gentzen was active at the time with transfinite induction, that is, the topic of his Habilitationsschrift that he managed to finish before his military service began at the end of September 1939 (see also the detailed account of von Boguslawski 2011). The notes for the series WA go on until December 1938. Apparently, Gentzen then began the notes for the series BTIZ (Prooftheoryofintui- tionistic arithmetic), as the first preserved page is number 133 from May 1939. Recent transcriptions from this series have shown that the habilitation thesis, the Initial segments of transfinite induction-paper published in 1943, was part of it, with the basic ideas ready by late June, and a detailed plan of the thesis written on pages 161–198 of the series BTIZ. Gentzen’s unfinished work, as it appears from the stenographic notes, was on four topics. They are: first, his attempt at a semantical decision method for intuitionistic propositional logic. This is item 16B of the present collection. The second unfinished piece of work is the proof theory of intuitionistic arithmetic, and the third the great problem of the consistency of analysis. Finally, he planned to write a popular book on the foundational research in mathematics. The notes for the planned book form item 17 with which this collection ends. They are at the same time an addition to Gent- zen’s views on the philosophy of mathematics, a theme he discusses here 5 Loose ends 51 and there in his papers, most notably in the first and last parts of the 1936 article on the consistency of arithmetic and the 1938 article The present situation in mathematical foundational research. From Gentzen’s biography and the shorthand manuscripts, written with the notation of month and year he used, the following picture emerges of his time since the military service began: The first preserved notes are from VIII–IX.40, on the book on foundations. Next there are the military hospital notes for the series BTIZ, IV.42, followed by IX.42 by notes for the book andbyX.42bytheAL-pages 137–139. Now follows an intensive period III– IX.43, with the first two months used for producing the continuous BTIZ- pages 248–275, and the rest for WA-pages between 183 and 254, not all preserved. After the last-mentioned work on WA, Gentzen started lecturing at the mathematical institute of the German university of Prague. I shall describe in this Section in turn the four unfinished topics from Gentzen’s manuscripts, to end up with what is known about three lost items.

1. A handy decision procedure for intuitionistic propositional logic This set of notes comes from the series AL, with the extant pages 133– 142, four of them blank. The pages that deal with the decision procedure are AL137–139, dated X.42. As a background, let us note that Gentzen had solved the decision problem of intuitionistic propositional logic in his doctoral thesis: It contains the intuitionistic sequent calculus LI and a proof of cut elimination that gives the subformula property of derivations in LI as a result. There is a further result by which the number of copies of a formula in the antecedent of any sequent in LI can be limited to at most three, by which there is a bound on proof search. Under the title Richtigkeitsbegriﬀ und Vollst¨andigkeitsnachweis fur¨ die positive Aussagenlogik (The concept of correctness and demonstration of completeness for positive propositional logic), Gentzen introduces a semantics in which correctness is determined in stages. Thus, (A ⊃ B) ⊃ C is correct if C is correct whenever A ⊃ B is. The latter, in turn, is correct if B is correct whenever A is. On the whole, we have: (A ⊃ B) ⊃ C is correct if C is correct whenever the correctness of B follows from that of A. Gentzen gives Peirce’s law ((A ⊃ B) ⊃ A) ⊃ A as an example with which correctness fails. Let’s try: Assume the antecedent correct, i.e., assume that the correctness of A follows if the correctness of B follows from that of A. These conditions in no way entail the correctness of A. For a more systematic approach, a system of numberings is introduced to keep track of the conditions of correctness in iterated implications. In the end, a failure is found in the approach, with the interesting intuitionistically provable formula ((B ⊃ A) ⊃ (A ⊃ B)) ⊃ (A ⊃ B). There is in Gentzen’s last known letter, written to Paul Lorenzen on 12 November 1944 and by chance in my possession in original, the following 52 Part I: A sketch of Gentzen’s life and work advice:

Allow me to give you a well-meant advice: Don’t begin with so diﬃcult problems, but with, say, the propositional calculus, as I have done; It is especially suited for learning to think by oneself and for becoming used to the logical concepts. The road to the predicate calculus is a lot stonier! A couple of nice problems are, for example: a proof of the completeness of my “structural rules of inference,” or: to give a handy decision method for the intuitionistic propositional logic.

There is in the book Schmidt (1960) a section with the self-same title (hand- liches Entscheidungsverfahren) as in Gentzen’s letter, but the contents are unrelated to Gentzen’s attempt in the series AL.

2. Proof theory of intuitionistic arithmetic After Gentzen had put his attempt at proving the consistency of analysis aside in December 1938, he wanted to carry through a consistency proof of intuitionistic arithmetic within sequent calculus. There are clear similarities in the methods of the WA-series of the fall of 1938 and the BTIZ-series of the following spring. By the end of June 1939 he had concluded, however, that his attempt had not led to success (BTIZ160d):

It seems to me almost as if, say, the proofofunprovabilityup to ε0 in number theory can be carried through only with the LK- formulation, I see no way to arrive at it from the intuitionistic formulation!

Despite the failure, Gentzen had arrived at precious results that he outlines in June: namely, as indicated in the second of the quotations at the beginning of this book, the gap in the series BTIZ, from page 161 to page 198.2, contains an elaboration of the habilitation thesis. Four pages, 169–172, have been kept, because they contain an idea that is not found in the habilitation thesis. On page 170, we read:

23.VI.39 Now then: §3. The general scheme of proofs of unprovability. The question is about showing the unprovability of: TI up to certain numbers in partial systems of pure number theory. Procedure similar to the consistency proof, indeed, one must prove anyway the consistency of the partial system in question at the same, for were it inconsistent, one could prove everything, even the TI. 5 Loose ends 53

The passage is followed by a detailed description of the main proof idea in the habilitation thesis. Indeed, the text of §3 of the published thesis follows closely the sketch in BTIZ.TheseriesBTIZ contains, though, at least three more paragraphs in comparison to the published habilitation thesis. §4on page 171 is titled “The unprovability of TI to ε0 in pure number theory,” with the addition “resp. demonstration of unprovability through the use of the valuation previously applied in the consistency proof.” §5appears three times, on page 199 as “The domain without ⊃ (and ¬)inLI and LK treated by the method of main thread,” developed fully on pages 199–210. On page 207 we find §5 a second time, with the title “The train of thought.” As mentioned, the short page 160 tells that pages 161 to 198.2 contain the elaboration for the habilitation thesis, but the text read originally “page 161–210,” the latter then cancelled and 198.2 written above. Thus, there seems to have been a serious plan of including the materials of pages 199– 210 in the thesis, cancelled at some later occasion. The title of §5isgiven on page 222 as “Sharpened formulation,” the same as the title of §6 on page 225. Finally, on page 229 there is §7 with the title “Main valuation.” Why these attempts at carrying the result on transfinite induction through in LI instead of LK failed, is still unclear. The details of the series BTIZ are as follows: There was a first, sustained and continuous push with BTIZ that begins with page 133. The next page 134 is dated 19.V.39. All pages from there on up to page 160 are preserved, with the date 27.V. on page 155. Now there is the gap, with just the four extant pages 169–172 (page 170 dated 23.VI.39). Next follows a continuous suite, from 199 (11.VII.39) to 244 (242 dated 23.VII.). The last-mentioned page has the comment: “My thoughts don’t seem to be moving anymore,” and page 243 continues: “End with it for the time being.” In sum, there are 80 stenographic pages between 19.V.39 and 23.VII.39 in what could be called BTIZ-I. They are continuous, except for the gap 161–198 that, by grace, is covered by the habilitation thesis. The typed thesis must have been prepared in August and September, before Gentzen’s military service began. A second installment of BTIZ begins under Gentzen’s convalescence in a military hospital, with thoughts written on little slips of paper and numbered from 245.1 to 245.22, the extant pages being 245.9–10, 15–16, and 21–22. These are dated IV.42 and described as “Lazarettideen.” There are a few additional pages, numbered 246.1–2 and 247.1–3, with the dates IV.42 for 246 and IX.42 for 247. The former is an impressive table of 12 reduction schemes for transfinite induction. This ensemble of pages can be designated as BTIZ-II. A third attempt at the proof theory of intuitionistic arithmetic, BTIZ- III, takes place between III.43 and VII.43, with 28 consecutive stenographic pages numbered 248–275, all of them preserved. 54 Part I: A sketch of Gentzen’s life and work

In sum, the preserved pages of BTIZ are 133–160, 169–172, and 199–275. The material should amount to some two hundred printed pages. The main difficulty with these pages is that they build on earlier developments, the lost pages BTIZ 1–132, that are occasionally referred to in the extant pages. Why was it important for Gentzen to find a consistency proof within an intuitionistic sequent calculus? First, work on BTIZ took place in direct continuation to the Habilitationsschrift. There Gentzen makes it clear that not all of the treatment of transfinite induction is carried through intuitionistically. Thus, the main motivation of BTIZ seems to be that a consistency proof within intuitionistic sequent calculus would be a prerequisite for a thoroughly intuitionistic treatment of transfinite induction. The particular problem Gentzen was wrestling with in BTIZ concerned contraction, the analogue in sequent calculus of a multiple use of an assumption that gets closed in natural deduction. When a cut is eliminated, contraction is a problem, as discussed above in I.1. When in this situation a cut or analogously a non-normality is resolved, there occurs typically a multiplication of some part of a derivation. The problem is to find a measure of the complexity of derivations that does not increase with such multiplication. One solution is offered in Siders (2015). This is not the place to detail out the possible reasons for the thrice- failed proof theory of intuitionistic arithmetic, but one thing can be said: There is a very neatly written series from August 1944, with the signum WKRd that stands for something like: Widerspruchsfreiheit, Kalkule¨ und Reduktionen, Durchsicht (Consistency, overview of calculi and reductions). It is a clear indication of Gentzen’s belief, namely: a suitable formulation of a logical calculus should bring success in the proof theory of intuitionistic arithmetic, in the intuitionistic treatment of transfinite induction, and also in the consistency problem of analysis. Unfortunately, a similar overview WTZd about transfinite induction, placed originally with the Putbus notes, got removed at some point.

3. Consistency of analysis The series WA contains what is left of Gentzen’s attempts at proving the consistency of analysis. The first preserved collection of notes has been written after the 1938 new proof of the consistency of arithmetic was finished in May. His article (1938) with the new proof ends with some discussion of the extension of transfinite induction beyond the ordinal ε0: In the proof of consistency [of arithmetic] and its eventual exten- sions, the question is anyway about an initial part, a “section” of the II number class, even if it is already one that goes relatively far. For a proof of the consistency of analysis, it would probably 5 Loose ends 55

be an initial part that goes considerably further.

This idea seems to have been a rather widely endorsed one. For example, Gödel expressed in his stenographic notebooks repeatedly his belief in the possibility of a proof of the consistency of analysis through an extension of Gentzen’s transfinite induction. What is left of the series WA has been written in two stretches: A first one is from 11.VIII.38 to 8.XII.38, with the pages 77–80 (11.VIII.), 111–116 (29.IX.), 121–128 (1.–11.X.), 135–146 (–8.XI.), 169–172 (22.XI.–3.XII.), and 177–180 (8.XII.); altogether 38 pages. Pages 77–80 of WA contain a suggestion for a realizability interpretation of intuitionistic arithmetic, where functionals over ordinal numbers act as realizers. Gentzen’s letter to Bernays of 17 July 1936 mentions that he is studying the system of Church, i.e., λ-calculus, “because of the relations to transfinite numbers.” He considers in the preserved pages only implication introduction and elimination and a rule of induction. To the assumptions are assigned some variables of appropriate functional type. Whenever there is an implication introduction, there is a functional abstraction over some function Γ, written with the bound variable underlined as in η Γ(η). Whenever there is an implication elimination with an argument β, a corresponding functional application Γ(β) is written:

η Γ(η) It means: the expression that follows considered as a function of η.Wehavenaturally(η Γ(η))β =Γ(β). The type results by itself.

Whenever there is a step of the induction rule, an omega-power of the term is taken. An example derivation on page WA78 has the associated ordinal function terms written with a blue pencil next to each formula in the natural derivation. Gentzen’s aim was, naturally, to show that the reduction of a derivation leads to a reduction of the associated ordinal term. Gentzen’s ordinal realizability for natural arithmetic derivations comes quite close to the Curry-Howard correspondence of typed λ-calculus. A second series of notes with the signum WA follows between May and September 1943. The pages that have remained are 183–184 (11.V.43), 187– 197, 198.01, 198.02, 198.5–198.11 (9.VI.–1.VIII.), 209–222 (3.VIII.–15.IX.), and 225–254 (–27.IX.43); altogether 56 pages. The first page of this second series of WA is 183, with the title Allgemei- ne Gedanken. Analogien und vermutete Zusammenhänge (General thoughts. Analogies and presumed connections). The text proper begins with the bo- xed date 11.V.43. : “Promenade thoughts in connection with WTZ 150.” The extant pages of the series WA amount to some two hundred printed pages. There is in addition to the WA-pages an 11-page stencil copy of a typewritten manuscript with the signum WAdd that stands for Consistency 56 Part I: A sketch of Gentzen’s life and work of analysis, overview (or summary of overview, for the double-d). The title is From WA the most modern part and the date II.45. It is the last known writing of Gentzen’s, a stenographic overview of the notes of the consistency problem of analysis, from where the preserved notes end, so from page 255 on and “cleared through page 339” as it states. The text was transcribed around 1948 by Hellmuth Kneser and H. Urban (presumably a pseudonym for Hans Rohrbach, see Menzler-Trott 2007, p. 277). The original stenography is lost. Rohrbach showed the transcription to some people such as Bernays, but nothing definitive could be said on its basis. Counting to page 339, the WA-notes would amount to a length of some five hundred printed pages. There were in addition the pages of the series WTZ, at least 210 pages on consistency proofs by transfinite induction that are often referred to in WA, but nothing is left of it to help prospective readers. Menzler-Trott’s biography contains descriptions of the attempts at reco- vering Gentzen’s papers from the time in Prague (pp. 263–5). Maximilian Pinl writes in a letter that he had been able to ascertain “that Mr. Gent- zen had left behind a luggage with precious manuscripts.” A letter from Heinrich Scholz to Gentzen’s mother reports of the account of Pinl in the following words: “Your son up to his separation from him had been of an exceptional mental activity and occupied himself without break with the consistency of analysis.” How likely is it that Gentzen would have found a proof of consistency? The summary of February 1945 gives no indication of such a proof and after that month, the time window to Gentzen’s arrest and imprisonment on 7 May is very narrow. It remains to be seen what further transcription and study of the series WA leads to in this respect.

4. Foundations and philosophy of mathematics As said, Gentzen’s idea was to write a popular book on ‘mathematical foundational research,’ with the motto cited above, Spannend wie ein Kriminal- roman!, exciting like a detective story. There are mainly fragmentary notes for the book. Some of these complement things known from elsewhere, say the remarks on normalization and arithmetic. The notes for the series BG,“Buchuber¨ die Grundlagenforschung,” are writtenonallsortsofslipsofpaperoverseveralyears.Theremarksbeginin 1939 with a list of contents and some outlines, but most of them stem from April and September 1942. These fragmentary notes amount altogether to twenty-odd pages and are found here as the last item. The plan of April 1939 has the following chapters:

Chapter I: (The basic concepts of proof theory)

§1. The structure of mathematical propositions.

§2. The formalization of mathematical propositions. 5 Loose ends 57

The ways of inference of mathematics Formalization of these (Resp. in general a chapter on: reduction of the basic concepts etc. Examples of formalized proofs. Even difficult ones. Remark: Any illiterate can control the correctness). (Proof methods, axioms.) Equivalence of functions and predicates. Such that. Chapter II: Mathematical logic. Here the most important theorems. Such as: prenex normal form. (Possibly) my dissertation Hauptsatz. As well as: double-negation theorem in a general formulation. 1. part: propositional logic. (Possibly: the propositional calculus. Here what is calculus-like). 2. part: predicate logic. Possibly two chapters. Chapter III: (Axiomatics.) (The axioms, ways of inference and methods of concept formation of specific parts of mathematics.) Herein: the structure of theories. Recursive theories, equivalence between theories. The form of analysis. The form of geometries. Set theory. Form of proof theory. Possibly a chapter IV on: “The incompleteness theorems of proof theory.” Theorem of Gödel, possibly Skolem, (Church), Tarski. Chapter V: The problem of infinity The later notes relate usually to this structure, either through a chapter number or by topic. In the end, there are miscellaneous notes from before the book plan, and on the reverse of the last page the following unrelated cancelled text, written during Gentzen’s convalescence in 1942: have not caused any essential recovery have caused a certain, though not yet essential recovery. only a modest Considering the unfortunate fact that it is objectively difficult to demonstrate an illness of this kind, and that I have to refer to good faith in several points of my statement, I hereby assure with emphasis that I have given all the information according to my best knowledge and conscience. Gerhard Gentzen. 58 Part I: A sketch of Gentzen’s life and work

5. Three lost items (A) A treatise on quantum-mechanical tunneling: After the finished doctoral thesis, Gentzen prepared for an exam that would give him the license to teach at high schools. Gentzen’s application to the exam is dated 26 May, 1933. The date had to do with his thesis, the manuscript of which must have been ready and handed in at that date. The doctoral thesis was one of the two essays that had to be prepared by the candidates before the examination proper. The other essay had the title Elektronenbahnen in axialsymmetrischen Feldern unter Anwendung auf kosmische Probleme (Electron trajectories in axially symmetric fields in relation to applications in cosmic problems). Thus, the “scientific examination” for a high school teacher’s position seems to have been a rather demanding undertaking. The exam itself was conducted on 13–16 November in Göttingen and qualified Gentzen to teach mathematics, physics, and applied mathematics. His wish was to continue research, but there was no financing for that. A teacher’s job would work as a security. The documents about the examination have been found in 2009, at the Niedersächsische Hauptstaatsarchiv in Hannover. It appears that Gentzen, in need of copies of his thesis manuscript for other applications, obtained a permission to change it to an offprint of his first article, on Hertz systems. In his announcement to the examination of 26 May 1933, he asked to have the second work from “the theory of electricity.” The manuscript on physics is not found among the papers, but there is a report prepared by Hertha Sponer who acted as the physics examiner. A form dated 12 July 1933 was sent to Gentzen and gave the title of the physics essay, with Sponer’s name under it, the essay to be delivered by 13 September. Sponer had obtained the habilitation (permission to teach at a university) in 1932, as one of the three first women in physics in Germany the other two being Lise Meitner and Hedwig Kohn. She was nominated extraordinary professor the year after. Quantum-mechanical tunneling was explained in 1928 in a work by Ralph Fowler and Lothar Nordheim, the latter Hilbert’s physics assistant in Göttin- gen. Electron microscopy is based on this phenomenon. The title of Gent- zen’s essay suggests quantum-mechanical tunneling, but what the intended application to cosmic radiation was, I have not been able to find out. Whe- ther Nordheim initially had a role in Gentzen’s examination paper is not known, but being Jewish, he was anyway excluded from any such role by the spring of 1933 and instead emigrated the year after. Sponer’s specialty in the early 1930s was the theory of molecular spectra treated by methods of quantum mechanics, on which she published a two- volume book in 1935–36. Unable to pursue as a woman a scientific career in Nazi Germany, she emigrated in 1934 first to Norway and then to the United States. No obvious direct link to Gentzen’s manuscript can be made from 5 Loose ends 59 her published papers that became very few after the Nazi takeover. In her evaluation, she writes that “the candidate proved to be quite familiar with the mathematical presentation, whereas the purely physical side remains somewhat distant.” A separate statement is as follows: The work Electron trajectories in axially symmetric fields in relation to applications in cosmic problems has been carried through by Mr. Gentzen in a delightful way. He has gone through the literature with a very good understanding and good critique. These works are written in part in a way that is little transparent, and so he had to give a simpler and more understandable presentation. Since he was able to do this well and since the work contains even independent considerations, it can obtain the predicate “With Distinction.” H. Sponer

(B) Gentzen’s lost lecture on Kepler’s laws: In 1943, Gentzen gave a lecture on Kepler’s laws, to prove his ability as a teacher for the German university of Prague. Hans Rohrbach’s evaluation of this lecture contains (Menzler- Trott, p. 235): He began his presentation with a reference to the local connections of his topic, since Kepler had worked many years in Prague and the necessary material for the development of the laws of the planets was present in the observations of Tycho de Brahe. His historical overview of the eﬀorts of the human spirit since antiquity to describe the movements of the planets made its connection here. He closed part 1 of the lecture with an interesting demonstration that the theory of epicycles of Ptolemy is mathematically equivalent to Kepler’s First Law. Part 2 of the lecture was dedicated to the theoretical derivation of Kepler’s Laws from Newton’s general law of gravitation. It is known from Menzler’s biography that Gentzen had a great interest in astronomy as a youth. Had he perhaps made in the second part of his lecture the same work as Feynman, as found in the book Feynman’s Lost Lecture: The Motion of Planets Around the Sun, namely a geometrical derivation of Kepler’s laws? (C) A report on the V-2 missile project: Next to mathematical lectures, Gentzen’s tasks in Prague also included work in the calculation oﬃce of Hans Rohrbach at the mathematics institute, intended by the latter to save him from front service. Mysteriously, there is among the Putbus notes a large 60 Part I: A sketch of Gentzen’s life and work folded millimeter paper, the grid measuring 450 by 300 mm, with what seem like ballistic curves drawn on the millimeter side, numbered 79, and a table of values and calculations and shorthand text on the reverse white page 80. It appears that Gentzen’s time from late 1943 to the summer of 1944 was mostly taken by this work. Menzler-Trott’s biography tells (p. 238–243) that the computations were done for the V-2 rocket development at the ill-famed Peenemunde¨ rocket test site and the concentration camp rocket production site at Mittelbau-Dora. Some three thousand V-2 rockets were targeted between September 1944 and March 1945 against civil population mainly in Britain, though they led to more victims, an incredible twenty thousand, at the production site; still, Rohrbach’s letters, even in the 1980s, show no regret at having participated in and advanced this Nazi war crime and at having made Gentzen do the same (as on p. 240):

I could, through a corresponding proposal, bring Gentzen to my mathematical institute in Prague, where I had already begun to carry out computational works for the V-weapon manufacture in Peenemunde.¨ I could release Gentzen for this and transfer to him the direction of the workgroup, which I had already formed. At the time these were young girls from the upper classes of the higher schools, who with certain mathematical-statistical com- puting works, to which they would be trained, must carry out assistance work, also a sort of military action [eine Art Kriegs- einsatz, contribution to the war]. Gentzen performed his task well and worked out good results for Peenemunde;¨ the problems would be posed from there.

The two manuscript pages 79 and 80 are clearly written for some extensive report. What the latter is and whether there were more authors is unknown. If a document was finished, it is likely that it is preserved somewhere; in the spring of 1945, there was a race between the Soviets and western Allied forces to recover the remaining V-2 rockets and paraphernalia, personnel, and documents pertaining to it. The V-2 rocket was the first one to reach outer space, an altitude up to over 200 km, with an enormous military potential in connection with the development of the atomic bomb. An operation of the American forces brought Wernher von Braun and many other Nazi warfare criminals to the United States in 1945, together with over a hundred rockets for test launches. An account of the race from the Soviet side, under whose control Peenemunde¨ and Dora finally fell, is found in Boris Chertok’s Rockets and People: Creating a Rocket Industry.Agreatnumber of German rocket scientists and technicians got moved to the Soviet Union and continued the development of ballistic missiles. Michael Neufeld’s The Rocket and the Reich: Peenemunde¨ and the Coming of the Ballistic Missile Era gives a comprehensive presentation of this initial phase of rocketry. 6 Gentzen’s genius 61

6. Gentzen’s genius

Eckart Menzler’s Gentzen biography bears the title Logic’s Lost Genius.In a review that Bill Tait wrote, he plays on the double meaning of that phrase (Tait 2010, pp. 274–75): Certainly logic lost a genius with Gentzen’s early death in the sense that, whatever he might have made of the consistency problem for analysis, he was certain to have contributed much more of value to logic had he lived longer and regained his health. . . Is the meaning of the title rather (or also) that Gentzen, perhaps inhisefforttosolvehiscareerproblem,losthimself?...Ilikethe ambiguity. There is a clear sense in which Gentzen’s work bears the hallmark of genius: he began with simple, basic things and made no mistakes on the way. I wrote in my review Gentzen’s proof systems (2012, p. 314): “It is of course some kind of an arithmetic necessity that simple new things are few in number and therefore hard to find. Whenever there is such an idea, success follows if things are in the right order from the start.” To ask what the logical form of actual inference in mathematics is, is just such a simple idea. In a short time, from about February to September 1932, Gentzen had distilled out of his question a pure form of logical inference, and even a single error could have led him completely astray. The normalization result for intuitionistic natural deduction followed in a few months’ time, then the interpretation of classical arithmetic in intuitionistic arithmetic in early 1933, the development of sequent calculus with the cut elimination theorem and a detailed written account in the doctoral thesis by May, the last in a period of some three months. Things went similarly with some other problems: The consistency of arithmetic by transfinite induction up to ε0 got figured out in some four months around 1935–36, with a clear view of the basic ideas of ordinal proof theory as witnessed by the correspondence with Bernays, and finally the great work on transfinite induction in the Habilitationsschrift that was done between early 1939 and June. I have mentioned occasionally aspects of Gentzen’s published papers that appear to be motivated by reasons of exposition. He wanted to keep his papers accessible even outside the circle of the few fellow logicians. His research had a similar vein: Results were one thing, what one presented in research papers another. This much at least becomes clear from the manuscripts in this collection and in the two remaining series WA and BTIZ.Someof the unpublished results were things he just left behind, such as the proof of normalization for natural deduction, others topics he planned to discuss on a suitable occasion, of which there were just a few. Where could one present 62 Part I: A sketch of Gentzen’s life and work one’s results, where to publish them? Two papers came out in 1938 in Hein- rich Scholz’ series Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, a publication made possible through Scholz’ cunning (see Menzler-Trott 2007, pp. 85–86). Gentzen was able to close himself from the rest of the world. His shorthand notes are confined strictly to logic and foundations, with an occasional personal note of joy and optimism in the style of “Hurrah! Es stimmt!” or disappointment at things not working out the right way. He was able to fantasize, from 1939 on well into the 1940s, about writing a popular book on foundational research. Gentzen had in 1945 behind him a period of three years from the convalescence of the spring of 1942. Apparently, no conclusive results came from these years, but just loose ends. Had his brilliant mind gone broken, or were the problems too hard? In the evaluation of his test lecture for the position as Docent in Prague, Rohrbach included the remark (see Menzler-Trott, p. 237): “This judgement is not affected by the current slowness while lecturing of Dr. Gentzen caused by nervous disease.” I hope that these Gentzen manuscripts and others to come can lead to deepened insights into his way of looking at logic and foundations of mathematics, at times in a form that is well developed, at other times as just flashes into the anatomy of a unique mind’s workings. Part II: An overview of the shorthand notes

1 Gentzen’s series of stenographic manuscripts 65

1. Gentzen’s series of stenographic manuscripts

Except for the early thesis manuscript, the correspondence with Bernays and Heyting, and the summary of the series WA of February 1945, all the source materials are written in the unified shorthand. The last-mentioned item was originally written in shorthand, but it got transcribed and typewritten by 1948 (see Menzler-Trott 2007, p. 277). The original has not been found. Here follows an alphabetical list of series of notes I have found mentioned in Gentzen’s manuscripts. Then I summarize the contents of the items in this collection. The number of different stenographic series from which pages have been preserved is about 12. There are in addition loose pages and notes, with no signum, and about 9 series to which references can be found in the preserved pages but of which nothing is left. The pages are often written on four-page sheets, with the sheets numbered consecutively, and pages in a sheet in the style of 96.1 to 96.4. At times, pages are written at the back of proofs or manuscripts for other articles, or on any piece of paper that was at hand, such as a “Luftwachezettel,” a form used for reporting the allied airplanes that flew over Brunswick where Gentzen was stationed between September 1939 and his nervous breakdown around the turn of 1941–42. Alphabetical list of Gentzen’s series of stenographic notes: 1. AL. This stands for Aussagenlogik, propositional logic. The preserved pages are 133 (II.37), 137–139 (X.42), and 141–142 (IV.43). In the beginning of the first item in this collection, Gentzen mentions “my calculus D (in the book about the propositional calculus etc.)” that may be the initial part of AL. 2. Allerlei Bemerkungen. This series of “miscellaneous remarks” is mentioned in SLF 17, but next to the passages there dated as X.33–V.34, nothing is left. 3. BG. This stands for Buch uber¨ Grundlagenforschung, the book on mathematical foundational research that Gentzen fancied composing from the spring or summer of 1939 on. It is described in Section I.5.4 above. 4. BTIZ. This stands for Beweistheorie der intuitionistischen Zahlentheorie, proof theory of intuitionistic number theory, as described in Section I.5.2 above. 5. BZ. This stands for Beweistheorie der Zahlentheorie, proof theory of number theory. The extant pages are 1–6 (VIII.34–XI.34) and 9–12 (I.35– III.35). 6. D. This could stand for Deduktion, as the first page of the first item in 66 Part II: An overview of the shorthand notes this collection mentions “my calculus D (in the book about the propositional calculus etc).” D got changed into INH in October 1932, at page 92.2. The note on Five different forms of natural calculi of 23 September 1932 begins as the series PLF1, a later addition. In the corner of the page, references to pages D85.3 and DB131 have been added. The remaining pages of the series D are: 92–96, each a four-page sheet, and 97.3–4. 7. DB. The signum DB seems to indicate a later continuation of the series D, probably after the first installment of INH had been finished. As mentioned under D, PLF 1 contains the reference DB 131. A reference to page D 138 has been added later on page 17 of INH, likely a DB-page. 8. Funktionenkalkul¨ . References from the fall of 1931 to pages 20 and 25 of this lost series are found in VOR. 9. GRO.TheseriesINH, pp. 26 and 30, refers to pages 27–28 of this otherwise unknown series and BZ to page 28. The last page of the handwritten thesis manuscript contains reduction schemes that suggest a possible reading of the letters GR as Gipfelreduktion, reduction of peaks. 10. HUG¨ . This series refers obviously to the normalization procedure, in which local peaks (Hugel¨ ) get removed, but nothing is left of it. 11. INH. The abbreviation stands for Die formale Erfassung des Begriffs der inhaltlichen Richtigkeit in der reinen Zahlentheorie, Verhältnis zum Widerspuchsfreiheitsbeweis, The formal conception of the notion of contentful correctness in pure number theory, relation to proof of consistency. This 36 page series has been bound together with covers, the only one of its kind among the preserved stenographic notes. 12. MK. This series is mentioned in WA, pages 252.1 and 254.1, and in WAdd that refers to MK 62–64. Gentzen speaks in this context of “Mo- dellherleitungen,” example derivations, and the letter K seems to indicate a rule of inference that corresponds to a comprehension axiom, the latter the topic of pages WA 177–180 of December 1938. 13. PLF. Also abbreviated as PL, the former for Formen der Prädika- tenlogik and the latter for just Prädikatenlogik. This is a collection of various short texts as detailed out in Section II.2.14 below. 14. RED. This is a renaming and possible addition to pages 97.1–4 of the series D. The extant pages are RED 3–4, placed in between pages of INH. 15. SLF. This stands for Formen der Stufenlogik, or Forms of type theory (“logic of types,” literally). The pages are few, from SLF15 to SLF18. 16. SW.TheseriesSLF has a reference to SW that should stand for Stufenlogik Widerspruchsfreiheit, consistency of type theory, the topic of Gentzen’s paper published in 1936. Nothing is left of this series. 2 The items in this collection 67

17. VOR. I think this stands for Vorstudien or a related word, preliminary studies. The extant pages are 13, 14.1–14.4, and 14–20. 18. WA. The signum WA stands for Widerspruchsfreiheit Analysis, though the A could in the earlier stages also stand for Arithmetik. A description is found in Section I.5.3. WAdd is an overview of the series WA from page 255 to 339, written by Gentzen in shorthand in February 1945 and preserved in a 13-page typewritten transcript. 19. WAV. This signum is connected to the early pages of WA, and here the A stands for Arithmetik, and the whole for something like Widerspruchs- freiheit Arithmetik Veröffentlichung, consistency of arithmetic publication. Ten pages are left of this series, up to page 86. 20. WKR. This stands for Widerspruchsfreiheit, Kalkule¨ und Reduktionen, Consistency, calculi and reductions. There is an eight pages long, very clearly written overview of the series that stems from August 1944, with the signum WKRd.TheseriesWKR itself has had at least 210 pages. There is in addition a reference to a series WKRLG on page 265 of BTIZ, clearly an account of the groundsequent calculi LIG and LKG. 21. WTZ. This stands for Widerspruchsfreiheit transfinite Zahlen.Thereis a note of VIII.44 with the signum WTZd cancelled, so there has been an extensive series WTZ up to at least 210 pages, as well as an overview of it. Not a single page of WTZ, the series that signals the birth of ordinal proof theory, has remained.

2. The items in this collection

The following summaries are intended to explain to some extent the central contents of each of the items. For one who likes to discover things on one’s own, it is best to turn directly to the original papers. The list as well as the papers in this collection follow an approximate chronological order of earliest appearance, approximate because the series overlap, have gaps, and so on.

1. Reduction of number-theoretic problems to predicate logic: 1931 The ﬁrst item bears the long title Reduction of number-theoretic problems to the decision problem of the lower functional calculus.Itisanearlyseparate observation from the fall of 1931 that relates directly to the series VOR: The decision problem of arithmetic reduces to that of pure predicate logic by the translation from a language with arithmetic operations into one with relations. Gentzen’s conclusion was the one Church and Turing proved to be the case in 1936, namely that predicate logic is not a decidable theory, 68 Part II: An overview of the shorthand notes for if it were, also arithmetic would be decidable. Contrary to, e.g., Jacques Herbrand, Gentzen did not think that likely.

2. Replacement of functions by predicates: late 1931 The topic of this early series is the expression of mathematical operations in the language of pure logic, in particular, the expression of arithmetic with additional relations such as σ(x, y, z)(“thesumofx and y is z”). The systematic development starts from page 14 that has the title Replacement of functions by relations. The extant pages of the series VOR are 13, 14, 14.1–14.4, and 15 to 20. VOR proper seems to begin with page 14. The pages 14.1, 14.2, and 14.3 have been written earlier, in September 1931, and added with a new numeration in between pages 14 and 15. It is my guess that VOR stands for Vorstudien, preliminary studies that Gentzen conducted on logical and mathematical notions before starting with his consistency program in early 1932. References are made in VOR to pages 20 and 25 of a series called Funktionenkalkul¨ , calculus of functions, but no pages of such a series have survived. Page 13 seems to precede the main contents of VOR proper without being connected to the rest. Similarly, the reverse of page 14.4 is cancelled; it contains axioms for natural numbers, namely the equality axioms, the schematic replacement axiom for equals, and uniqueness axioms for functions. The reverse of page 14.3 is the unrelated short text about the origins of mathematical concepts, item 3 below. The pages of VOR pertinent to the theme of functions and relations are here given in approximate chronological order, first the pages 14.1–14.3 with the title Functions instead of relations with which page 14.1 begins, then the pages 14–20 with the title Replacement of functions by predicates, from page 14. The insertion of pages 14.1–14.3 between 14 and 15 would create a discontinuity in the text, in the middle of a phrase. Gentzen uses the word Zählausdruck (‘counting expression’) for an expression of predicate logic, or equivalently first-order expression, a common practice of the time. I have used the latter in translation. The topic of turning functions into relations is seen already in item 1, and it is also used in the printed doctoral thesis written in 1933, when the classical sequent calculus LK is extended by arithmetic, the rule of induction excluded. The problem of consistency is easy, because the formulation is in terms of relations rather than functions, i.e., within pure predicate logic to which the cut elimination theorem can be applied. The series BZ, pages 9–11 from January to March 1935, refers to VOR 14– 19 as “old investigations” and gives comparisons with the treatment of the description operation ι.Infact,thepreservedVOR-pages are found between pages 10 and 11 of BZ. 2 The items in this collection 69

3. Correspondence in the beginning of mathematics: end of 1931 The topic of this one-page text is how “primitive man began to exercise mathematics.” It relates Gentzen’s view of the nature of the basic concepts of mathematics, natural numbers in particular, to those of Russell and Hilbert. He finds that they are sufficiently clear in their own right, and that “it is not necessary to define them, as sets of the respective objects in reality that have been made to correspond to them, as in Russell, or as the signs that have been introduced for them, as in Hilbert.”

4. Five different forms of natural calculi: IX.32 This remarkable short document contains the rules of natural deduction. The text has lots of references to alternative formulations, up to dozens of different conditions on formal proofs that Gentzen had explored. A final form of the rules can be found in the thesis manuscript written towards the end of 1932, item 6 below. The Five different forms is here followed by remarks that are found at the beginning of the manuscript INH. The second page has the date 9.X.32. This short piece is saved only because INH proper happens to begin on page 92.2 of the four-page sheet 92 of the series D. Also the Five different forms refers to pages of D, up to page 84, and it is dated 23.IX.32. The preserved page and a half of D, namely 92.1 to 92.2, have been written just two weeks later, and Gentzen is trying to figure out how to deal with the rule of negation introduction, RA in his notation, and with induction, VI in his notation (for Vollständige Induktion). There are many references to earlier pages of D, and transfinite induction is also mentioned. This was a stage in which the development of natural deduction was still tied to the attempt at proving the consistency of arithmetic.

4bis. Two fragments on normalization: IX–X.32 The previous item begins at the middle of a page. Its top has a short text on normalization, quite inconclusive but of interest nevertheless. A longer similar passage is found in the beginning of INH as described above. It gives some indication of Gentzen’s way to the normalization of derivations in natural deduction.

5. Formal conception of correctness in arithmetic I: X.32 Gentzen’s manuscript with the signum INH, from the blue folder, is titled Die formale Erfassung des Begriffs der inhaltlichen Richtigkeit in der reinen Zahlentheorie, Verhältnis zum Widerspuchsfreiheitsbeweis. My translation is: “The formal conception of the notion of contentful correctness in pure number theory, relation to proof of consistency.” The stenographic manuscript has about legal size pages (Kanzleiformat in German), 36 in number, 70 Part II: An overview of the shorthand notes bound together with what are now somewhat rusted metal clips. It is the only such single bound piece in the Gentzen papers. INH has been written between October 1932 and January 1935. The first 20 pages are from between early October and the 10th of November 1932. They record, usually with a couple of days’ intervals, the progress of his thinking. Then follow a few pages dated “should be about February 33” and titled “Ultraintuitionistic criticism of the ways of inference. Finitism.” The next entries are in April and June 1933. The rest is written in the second half of 1934, especially October and November, as well as a final part in January 1935. The main question of INH is what the sense of a consistency proof for arithmetic could be. By April 1933, Gentzen had come to very clear ideas about three kinds of possible consistency proofs: 1. A “purely-formal” one (rein-formal), 2. A “semi-contentful” one (halbinhaltlich), 3. A “contentful” one (inhaltlich, sometimes even ganz inhaltlich). By April 1933, Gentzen was sure that a formal proof requires a transfinite induction, something he had surmised the previous fall. The second proof was to use an ordinary double induction, and the same for the third. Gentzen reasoned initially as follows, on the basis of a contentful notion of correctness: The axioms of arithmetic are obviously correct. The rules of logic maintain correctness. Therefore all things provable are correct. He then asked, well knowing that such a proof cannot be presented within arithmetic: “Where is the Gödel-point?” Gentzen had fixed by 23 September 1932 the system of natural deduction, a rule of induction included, and clearly saw that formal derivations by the logical rules can be normalized. The detailed proof of normalization was produced later, perhaps by early 1933. All derivations ended with the closing of the open assumptions by implication introductions, even an aspect of his printed thesis that has gone largely unnoticed. Then, the conclusion of the derivation builds up into a longest formula that has all the formulas in the derivation as subformulas, provided that there are no “inner hillocks.” If this held for arithmetic, its consistency would be an immediate consequence: Take a presumed derivation of 1 = 2, and you see that it cannot have a normal derivation. Gentzen could not get hold of the multiplication of parts of derivations in the reduction steps connected to instances of the rule of induction. Even detour conversions on implication proved difficult, but there the derivation grew broader in conversion. With the rule of induction, it grew in height when a step of induction was replaced by a numerical induction up to some number n. He thought of using the maximum number of nested inductions as the main parameter, and the overall number of inductions as a secondary parameter, but saw on the other hand that a proof should not be formalizable within Peano arithmetic. 2 The items in this collection 71

Gentzen tried first the reduction (i.e., normalization) of derivations for what he called direct derivations. These are derivations with no assumptions, but just instances of the arithmetic axioms as topformulas. The rules would be those for conjunction, universal quantification, and induction. Only the last one has assumptions that have to be explained away. One idea was to use “levels” such that there is a ground level, say the base case of an uppermost induction in a derivation. Reduction of the induction removes the assumptions and the subderivation to the uppermost induction becomes direct. None of his attempts worked properly, and could not work. Of the three possible proofs, the significance of the first, purely-formal one was not clear. What would be accomplished with it? The third, contentful one was trivial, so the second, semi-contentful one remained. The main ideawastodefineanotionofcorrectness for formulas, and derivations, based on the notion of reducibility. Correctness is a, as we would now say, semantic notion, and the idea of a proof of consistency would be that formal derivability produces results correct in the defined sense. The difference to a purely-formal proof by transfinite induction is clear now: The latter is a purely syntactic proof. The central point in the second proof was the definition of correctness. This notion is well known from Gentzen’s original proof of consistency as submitted for publication on 11 August 1935. There a reduction procedure for sequents is given. These have the form Γ → C. The sequent is reduced to simpler forms starting from C. At a certain point, it happens that there is just an equation at right. Assume it turned out false. Now the task is to reduce the antecedent part so that it also turns out to contain a false equation. Then, in whatever way one succeeds in producing a false equation at right, it should be possible to produce a false equation at left. Suppose this succeeds. Then, if all formulas in the antecedent are correct, so is the formula in the succedent. The proof is finished by showing that derivable sequents are of the said kind, “reduce to endform” as Gentzen said. The reduction procedure is an attempt at producing a counterexample to Γ → C, even if Gentzen does not use that terminology. It inspired Gödel’s Dialectica- interpretation. Gödel had studied Gentzen’s original proof in the fall of 1935 in Princeton. Some letters indicate that Gentzen had the consistency proof ready by the end of 1934. By that time, INH had been superseded by another series with the signum WTZ, obviously for something like Widerspruchsfreiheit transfinite Zahlen (Consistency transfinite numbers). The late 1934 parts of INH contain already plans and outlines and details for a paper to be published that can be compared to corresponding parts of the 1935 paper. Then, in the other direction, one can understand parts of INH by studying the 1935 paper. There is another series in the Gentzen papers that is useful for the evalua- 72 Part II: An overview of the shorthand notes tion of the latter parts of INH.IthasthesignumBZ,forBeweistheorie der Zahlentheorie (Proof theory of number theory), and extends from August 1934 to March 1935. Nine out of eleven pages have been preserved. These refer to another series WAV of which ten pages are extant. The title of the latter should stand for Widerspruchsfreiheit der Arithmetik Veröffentlichung (Consistency of arithmetic publication). A large part of BZ hastodowith the production of a prenex normal form for formulas in sequents, and the same holds for WAV.

6. Investigations into logical inference: 1932–33 This item is an early manuscript version of Gentzen’s doctoral thesis Un- tersuchungen uber¨ das logische Schliessen. I found a photocopy of the manuscript during a visit to Erlangen in February 2005. It was among the documents about Gentzen that Christian Thiel has gathered. An accompanying letter indicated that it had been sent to him by Eckart Menzler in 1988 and that the original belongs to the Paul Bernays collection at the ETH-Zurich. From their web-pages one finds that there is indeed such a document, numbered Hs. 974:271 and given as 41 pages in length and with the, somewhat misleading, indication that the first 12 pages have been handed in as a dissertation. One also recognizes the photographic reproduction of part of a page from Gentzen’s manuscript on p. xiii of Szabo’s edition of Gentzen’s Collected Papers (North-Holland, 1969). Page 9 of the manuscript contains the standard detour conversion schemes of natural deduction. The introductory chapter closes with a summary of Gentzen’s main results in the planned thesis, ending with: “The consistency of arithmetic will be proved.” The next part consists of a long, detailed proof of normalization for intuitionistic natural deduction. There follow short summaries of an intuitionistic sequent calculus, with remarks that relate to attempts at proving the consistency of arithmetic, a multisuccedent classical calculus, and a final part on translations between the new calculi and axiomatic logic. The composition of the thesis underwent changes that are indicated by the order of the material and by marginal additions. The manuscript is written in the “Sutterlin¨ Schrift,” an obsolete form of handwriting that is not easy to read, but the 14 pages of the proof of normalization are a later addition written in a less formal hand he had in other occasions used for the printer. The proof is found in English translation in The Bulletin of Symbolic Logic, vol. 14 (2008), pp. 245–257. A comparison to the printed thesis is useful because of additions and deletions, as with the inversion principle in chapter I, §5 where the manuscript clearly states that the reducibility of derivations through a match of I-andE-rules aims at decidability and consistency results, thus revealing Gentzen’s original idea for a consistency proof of arithmetic. I have made such comparisons in my Gentzen’s proof systems: byproducts in a work of 2 The items in this collection 73 genius (2012). More on this topic is found in Negri and von Plato (2015). Gentzen’s original plan for the thesis seems to have been the following: First set up the calculus of natural deduction, a rule of induction included (chapter I). Next show that the calculus is equivalent to an axiomatic calculus (II). Then show normalization for intuitionistic natural deduction (III). In the next step, show that classical arithmetic, and especially the question of its consistency, reduces to intuitionistic arithmetic (IV). Finally, extend the subformula property of natural deduction to the system of intuitionistic arithmetic, to prove the consistency of the latter (V). Gentzen’s grand plan for a doctoral thesis failed, as he found out some time around the turn of the year 1932–33, because the system of arithmetic does not have, and cannot have, a subformula property. He took out the chapter on the translation of classical into intuitionistic arithmetic, sent it first to Heyting, then to Mathematische Annalen in mid-March 1933, and developed classical and intuitionistic sequent calculi for pure logic to be presented as a doctoral thesis. This latter cannot have taken more than a few months, because the thesis was handed in on 26 May 1933. The finished thesis has not been preserved in Göttingen or elsewhere, but the published version should be the same, save for possible minor corrections. In addition to the first introductory chapter that presents the system of natural deduction and the third on normalization, the handwritten manuscript contains the very first sketches towards an intuitionistic sequent calculus, a brief section on a multisuccedent classical sequent calculus, and translations between natural deduction and an axiomatic system of logic, the topic of the originally planned chapter II. These translations are different from those in the published thesis in which Gentzen started from an axiomatic calculus in which all the standard connectives and quantifiers are present, as in Hilbert (1931). Axioms are given there separately for each connective, so that an intuitionistic calculus can be obtained by leaving out the law of double negation. Derivations in such a calculus are first translated into intuitionistic natural deduction, then to sequent calculus, then to the axiomatic calculus, to make a full circle. The last of these translations is rather complicated. Afterwards the analogous translations for the corresponding classical calculi are outlined. In the manuscript, the axiomatic calculus is that of Hilbert and Acker- mann (1928) that contains only disjunction, negation, and universal quantification. Gentzen gives a classical system of natural deduction, called N2, that has rules for conjunction, negation, and universal quantification. Con- trary to the published thesis, a direct translation of derivations in N2 to derivations in axiomatic logic is given. This translation seems to be at the origin of the discovery of sequent calculus, but no trace of it can be seen in the published thesis. The translation is presented in detail in von Plato (2014). 74 Part II: An overview of the shorthand notes

Letters from Gentzen to Heyting from the winter and spring of 1933 show that the latter had received a manuscript with the translation from classical to intuitionistic arithmetic in January or early February 1933. Heyting was corresponding at the same time also with Gödel, and a letter shows that Gödel reacted rather strongly to the news that someone else had done the thing (see Gödel’s Collected Works, vol. V, p. 70). In the end, Gentzen withdrew his paper from publication, but Bernays kept a set of galley proofs through which the work was preserved and finally published in English translation in 1969 and in the German original in 1974. A page of corrections in the manuscript of the thesis is left where there once was a planned fourth chapter, all corrections done in the proofs. It can be gathered that the text of the article came directly from that chapter. As said, the chapter on normalization is a later addition to the manuscript, distinguished by a standard handwriting style that can be read without greater difficulties. It contains also notational and terminological changes as compared to the first chapter, such as the use of horseshoe implication in place of the earlier arrow.

7. Reduction of classical to intuitionistic logic: I.33 This remarkable short note contains a proof transformation of derivations in classical natural deduction for arithmetic to ones in arithmetic with minimal logic in the fragment with ⊃, ¬,and∀. The idea is that double negation elimination reduces to components in the fragment until it reaches equations for which it holds intuitionistically. The procedure is presented very clearly in Bernays (1935a), so it can be concluded that Gentzen had explained the transformation to him. The note suggests, incidentally, that the normalization theorem was proved by January 1933, at least for the fragment in question.

8. CV of candidate Gerhard Gentzen: V.33 Gentzen wrote a short CV to accompany his doctoral thesis, dated 26 May 1933, that contains useful remarks about his early work, including that he has “been occupied with this ﬁeld in detail independently.” It is found together with the documents about his state exam of November 1933, at the Nieders¨achsische Hauptstaatsarchiv in Hanover (Hann. 144, Acc. 65/85, Nr. 2289).

9. Letters to Heyting: II.33–IV.34 There are four known letters from Gentzen to Heyting, beginning with one of 25 February 1933 that concerns Gentzen (1933), the manuscript on the interpretation of classical arithmetic in intuitionistic arithmetic. The second letter announces the withdrawal of this paper from publication. Gentzen’s 2 The items in this collection 75 last letter is from 16 April 1934, and there is in addition a postcard signed by Gentzen and Heinrich Scholz, sent during Gentzen’s visit to Scholz in Munster,¨ 28 June 1936. There is a logical thread in the correspondence that can be felt, even in the absence of replies on Heyting’s part: In the end of the second letter, Gentzen mentions that he has a decision procedure for intuitionistic propositional logic. In the next letter, he explains to Heyting that the decision procedure comes from “a quite general theorem about intuitionistic and classical propositional and predicate logic,” which we know to be the cut elimination theorem. Then he indicates an application to underivability results in predicate logic, clearly with Heyting’s (1930) article in mind. Gentzen’s last letter answers “aﬃrmatively to your question,” namely, that the disjunction property of intuitionistic logic follows from Gentzen’s theorem. The letters that Gentzen sent to Heyting contain announcements of results that are not always found elsewhere in his work, most importantly, underivability results in intuitionistic predicate calculus, especially the double negation shift that was earlier obtained through counterexamples that use intuitionistic real numbers, as in Brouwer (1928) and Heyting (1930). These are discussed in some detail in Section I.4.5 above. The letters are found in the Heyting collection at the Rijksarchief in Haarlem. The Heyting-Johansson correspondence discussed above in Section I.5isalsofoundthere.

10. Formal conception of correctness in arithmetic II: II.33–X.34 This is the second part of INH, the details of which have been described above.

11. Proof theory of number theory: VIII.34–III.35 There are ten extant pages of the series BZ,forBeweistheorie der Zahlen- theorie. These pages contain mainly a conversion of derivations in the classical sequent calculus LK so that all formulas are in prenex normal form. The extant pages BZ are found in the violet folder after the series INH. They are BZ 1–6 (VIII.34–XI.34) and 9–12 (I.35–III.35). The series is not a record of continuous work. The first page has the dates VIII.34 and X.34. The latter date records work on the substitution of existential quantifiers by functions which is one of the central concerns of the series VOR.Theextant pages of VOR are, in fact, found between pp. 10 and 11 of BZ. Pages 1–4 of BZ are dedicated to the production of sequent calculus derivations in which all formulas have a prenex normal form. It refers to WAV for a detailed treatment and the pertinent pages of the latter are preserved in between sheets of BZ. Letters by Gentzen indicate that he had obtained a first proof of the consistency of arithmetic by about October 1934, if not even in the preceding 76 Part II: An overview of the shorthand notes spring.

12. Consistency of arithmetic, for publication: X.34 Gentzen’s stenographic manuscripts contain ten pages of a series WAV, written around October 1934. The signum stands for something like Wider- spruchsfreiheit der Arithmetik, Veröffentlichung (consistency of arithmetic, publication), and the contents consist of preparatory notes for the writing of the long paper Gentzen submitted for publication in August 1935, with a major revision submitted in February 1936. The preserved pages are 55–56, 77–80, and 83–86. These are quite polished and readable and amount to 15 printed pages. The series BZ contains material similar to WAV 55–56, 77, and 83–86, namely, the transformation of derivations in the classical sequent calculus LK into ones in which all formulas appear in prenex normal form. The idea was to delimit the “transfinite part” of arithmetic into the quantifier inferences, separated and appearing below a decidable and finitistic propositional part in the style of Gentzen’s midsequent theorem. In the end, Gentzen chose to use in the paper he wrote in the spring of 1935 a natural calculus in sequent calculus style, mainly for reasons of exposition, to make the work accessible to a reader without logical training. The most remarkable part of WAV are the pages 78–80. They give a three-page outline for a consistency proof of arithmetic, similar to the 1935 proof that is based on the notion of reduction of sequents, but done for a classical sequent calculus. It is referred to as “the second proof of correctness (LK consistency proof)” (p. 78), the first being a proof with a single- succedent calculus rather similar to the 1935 version. The crucial point of the proof is that reducibility is maintained under the application of the mix rule, i.e., the multicut rule of the doctoral dissertation. The dissertation contained a form of sequent calculus that Gentzen used often, even if it was not explicitly singled out as such, namely what he called a calculus with Grundsequenzen (groundsequent calculus). It can be given in a single-succedent form, designated LIG, and a “symmetric” multisuccedent form, designated LKG. The idea is to replace half of the logical rules by logical groundsequents by which derivations can start. There is even a variant in which all classical propositional rules are substituted by groundsequents, as on p. 55 of WAV. By the “sharpened Hauptsatz,” cuts can be permuted to such sequents, but not completely eliminated. However, the cuts are harmless because they maintain the subformula property. Cut elimination is somewhat simplified by the use of groundsequents: It never happens that a cut formula is principal in both premisses of cut. 13. Correspondence with Paul Bernays: IV.34–VII.39 The correspondence with Bernays consists of 17 letters, 13 of them by Gent- 2 The items in this collection 77 zen, and is at its most significant with the letters that concern Gentzen’s original proof of the consistency of arithmetic. Bernays was in Princeton at this time, and his part of the correspondence of this period has not been preserved; The reader has to make inferences and guesses about a missing half of the letters. It seems that Gentzen understood quite well the remarks made by Bernays, and also by Gödel and von Neumann who were in Princeton at the same time. At one point, on 4 November, Gentzen sent a four-page tightly and very orderly written letter to Princeton where Bernays, possibly with the help of Gödel, had taken up the central problem of Gentzen’s proof, as can be gathered from Gentzen’s answer: I have considered all these aspects already myself, including the geometrical image of branching line segments. You are quite right that the finiteness of even a single reduction path for the sequent Γ, Δ → C can get grounded on the finiteness of a whole series of different reduction paths for D, Δ → C. But this does nothing for my proof idea! The letters contain many interesting details about Gentzen’s later work, up to the summer of 1939. For this period, four letters on the part of Bernays have been preserved, even if some others by both clearly have disappeared. In a letter of 13 June 1938, Bernays expresses dissatisfaction with the proof of Gentzen’s lemma 3.4.3 of the 1938 proof of consistency, and suggests an alternative, but at this stage, only minor changes were possible in the paper that was in print. One of the ideas of Bernays, from a letter dated 9 May 1938, is to look at the reduction procedure of sequents of Gentzen’s original 1935 proof as a non-formal calculus with an infinitistic rule and with consistency as cut elimination for the said calculus, in the style known otherwise from Kurt Schutte’s¨ work (1951). There is also discussion, in connection with the preparation of the second volume of the Grundlagen der Mathe- matik in September–November 1937, about Hilbert’s ε-substitution method that Bernays endorsed and Gentzen instead found “not worth the effort.” Gentzen’s last preserved letter to Bernays is from 16 June 1939. It is an answer to a lost letter on the part of Bernays, dated 31 May 1939. The contents of the letter appear somewhat mysterious, but Bernays refers to this exchange in a letter to Gödel of 28 September the same year (see Gödel 2003, p. 126): He tells that Gentzen has extended the double-negation interpretation to simple type theory. In Gentzen’s letter, instead of a translation of the logical language into the fragment with &, ¬,and∀, double negations are added in front of each subformula of a formula in the way of the Kolmo- gorov translation, known to some extent to Gentzen at this point. The effect is that the law of double negation, ¬¬A ⊃ A,isprovableforeachA,even for the comprehension axiom as Gentzen notes (in the formulation of page 360 of his 1936 paper on the consistency of type theory). Gödel replies to 78 Part II: An overview of the shorthand notes

Bernays in late December with the remark that the intuitionistic interpretation is of no special interest (ibid, p. 130). Here G¨odel’s opinion is based on his conviction that intuitionistic logic fails to be properly constructive, witness his 1941 article “In what sense is intuitionistic logic constructive?” A certain human touch is found in Bernays’ worries about Gentzen’s delicate psychological balance, with Gentzen telling about periods of depression – always writing to his “most respected Professor Bernays,” deposed many years earlier by the Nazis, and Bernays responding to his “dear Mr. Gent- zen.”

14. The forms of type theory: VII.34–X.37 The signum SLF stands for the series Formen der Stufenlogik or Stufenlo- gische Formen, forms of type theory. The preserved pages are 15 to 18, the ﬁrst with the date VII.34 and some additional comments dated IX.35. That is one month after Gentzen had submitted his paper Die Widerspruchsfrei- heit der Stufenlogik (The consistency of type theory) for publication. The upper part of the ﬁrst preserved page is a short unrelated text. The series proper seems to start with the title General thoughts about the axioms of type theory on page 15. There are two brief notes that come with this series, one a reproduction of Carnap’s formulation of the axiom of choice, another a remark by Bernays, noted in 1933, by which the axiom of choice is needed in analysis, for example to show that a bounded set has a point of convergence.

15. Predicate logic: 1935–IV.43 The violet folder contains a few pages from the series PLF,forPrädikatenlo- gische Formen, and some additional short texts on different aspects of predicate logic: Pages 1–2 and 6–9 belong to PLF, the first two pages containing the Five different forms of natural calculi, item 4 above. Pages 6–9 are a single sheet, with page 6 a discussion about the role of disjunction, item 15A, and page 7 blank. Item 15B, about decidability in predicate logic of February 1937, is on a separate piece of paper, with item 1 of September 1931 on the other side. Item 15C on decidability and the cut theorem of April 1938 is written at the back of item 7 of January 1933. Pages 8 and 9 contain the attempt at a “natural introduction of the calculus LK,” i.e., the multisuccedent sequent calculus. It ends with a reference to the multisuccedent sequent calculus for intuitionistic predicate logic that is found in the series AL, here placed under 15E. 15A. Unnaturalnesses of inferences in formal logic: 1935 The topic of this note is the perplexity created by the introduction rule for disjunction: from A to infer A ∨ B. Why would one weaken the claim in such a way? One answer Gentzen gives is that there can be a disjunctive 2 The items in this collection 79 assumption, as in A ∨ B.IfA leads to C and B to D,thenA ∨ B leads to C ∨ D. The discussion ends with: “The apparently ‘natural’ conception of ∨ touches on many impurities that are left aside in the conception used in formal logic. Here A ∨ B means simply: At least one of the propositions holds. Thereby the inference OI is obvious.” 15B. Decidability in predicate logic: II.37 Gentzen asks first what it would mean if, say, the negation of Fermat’s theorem were provable from the axioms of elementary arithmetic: There would be a counterexample that follows by logical inferences and no induction. Gentzen then mentions his attempts at solving special cases of the decision problem in number theory by cut elimination. Of decidability in the cases of existential propositions, he notes that such “is understandably not possible, because there need not obtain any connection between the simplicity of the axioms and the length of construction of an existing object with a determinate property. . . In any case, one does not know anything about this. With the number-theoretic axioms also presupposed, one cannot know whether by ever new computations, a number arises in the end with a determinate computable property. This can take an eternity.” 15C. Decidability and the cut theorem: IV.38 This item has been added in the bottom part of the second page of item 7. It contains, similarly to the preceding one, brief comments on Gentzen’s hope, and subsequent failure by and large, to obtain decidability results in predicate logic through the cut elimination theorem. 15D. Natural introduction of the calculus LK ?: IV.38 This item occupies pages 8 and 9 of the series PLF. The topic is to introduce the classical multisuccedent sequent calculus LK in a pedagogical way. The result is not particularly satisfying; as Gentzen notes: “A natural introduction of LK is possible to some extent, along the above path, but then one cannot yet recognize its advantage. I.e., its advantage does not depend on its being accessible in a natural way, but that is a more arbitrary side issue.” 15E. Formulation of intuitionistic logic with symmetric sequents: IV.43 Pages 141–142 of the series AL from IV.43 contain the details of a multisuccedent intuitionistic sequent calculus, the quantifiers included, with exactly the right kind of restrictions in rules for R⊃,R¬,andR∀. The multisuccedent intuitionistic sequent calculus was reinvented by Maehara in 1954.

16. Propositional logic: II.37–X.42 Very little is left of the series AL, a mere six pages between AL 133 and 80 Part II: An overview of the shorthand notes

142. There is a short note on the classical propositional calculus LK ,and a substantial attempt at a decision method for intuitionistic propositional logic, and the multisuccedent intuitionistic sequent calculus for predicate logic on pages 141–142, placed in the previous item. 16A. Completeness of the propositional calculus LK: II.37 On page 133 of AL is found a text with the date II.37 that is about the completeness of the propositional calculus of LK “after a reading of Kalmár.” The work in question is obviously Kalmár’s (1935) article. Kalmár shows that if any unprovable formula is added to classical propositional logic, the calculus becomes inconsistent. He gives, among others, a very careful formulation of the deduction theorem. Gentzen’s sequent calculus would be a natural setting for most of Kalmár’s results, as Kalmár acknowledges. 16B. Correctness and completeness for positive propositional logic: X.42 Pages 137–139 contain the attempt at a decision procedure dated X.42, and describedindetailinI.5.1above.

17. Foundational research in mathematics: IV.39–42 This item is described in some detail in Section I.5.4 above. At the back of the last page of these notes, Gentzen had sketched a statement about his illness in 1942.

3. Practical remarks on the manuscripts

The stenographic manuscripts are found in two folders, one blue, the other violet. Each cover has a glued slip of paper, the blue one with the text Buch: Die mathematische Grundlagenforschung“ and the violet one with Aussa- ” genlogik, cancelled with a light line. The violet folder contains in addition the name “Dr. Gerhard Gentzen” and the following text on a smaller paper: VIII.44. This folder contains what has still some value from PL, AL, Consistency of number theory, BZ. Not duplicated. For Putbus. As well as WKRd, [cancelled: WTZd] WA earlier. The contents are as follows: The blue folder: Book: Mathematical Foundational Research (BG) Proof theory of intuitionistic number theory (BTIZ) 3 Practical remarks on the manuscripts 81

The violet folder: The forms of type theory (SLF) Consistency of analysis (WA) Propositional logic (AL) The forms of predicate logic (PLF) Formal conception of the notion of correctness in number theory (INH) Proof theory of number theory (BZ and WAV ) The pages are not in a numerical order, but mixed somewhat for reasons of continuity of argument, as in the case of BZ and WAV. Stenographic writing has been used for various purposes. The main use has been to write down what was said, say in a dictation or lecture. Thus, the idea is just to reproduce a spoken language in written form. Stenographic writing for one’s own purposes must have another reason. With Gentzen’s research notes, the increase in writing speed in itself should not have been a consideration of importance. Rather, it seems that once the habit of writing in shorthand is acquired, the slowness of longhand writing would be felt as a disturbing element in the rendering of thoughts in a written form. Another aspect of stenography is that longhand writing takes much more space. With all the long series of manuscripts that Gentzen wrote, compactness may have played a role. He seems to have kept the piles of papers in a cellar, as indicated for example on page 245 of the series BTIZ. The idea of shorthand writing is old, and such systems were used already in Antiquity. In the cursive form of shorthand, words are written by and large as continuous lines. It was developed by Franz Gabelsberger, and his system of writing was the prevailing one in the German-speaking world from its creation in the 1830s to 1924. Logicians such as Paul Bernays and Kurt G¨odel used it constantly for notes and for the preparation of lecture series, say. Another competing shorthand notation, developed from Gabelsberger, was the Stolze-Schrey system, used for example by Rudolf Carnap and Hermann Weyl. Many variants of Gabelsberger shorthand exist in languages other than German, even one in Finnish as I happen to remember from my childhood: The mail brought every now and then to our home a –very thin!– bulletin of which I could read only the title, the rest was a mystery: Pikakirjoitus- lehti (Finnish for Shorthand Bulletin). Sweden has its own system, but the Swedish-language minority of Finland has retained the original Gabelsberger writing, perhaps a unique relict in the world. It is upheld by a tiny society with the name F¨oreningen Gabelsberger, and the main task is to record what is said in Swedish at parliamentary sessions. 82 Part II: An overview of the shorthand notes

There were too many variants of shorthand in Germany, and therefore a reformation was forced on the competing schools of stenography that led to a compromise result in 1924. The multitude of systems was replaced by a “unified shorthand” (einheitliche Kurzschrift) that retained most of the consonant writing of Gabelsberger but changed the mode of indicating the vowels. The shorthand is still taught at least in Germany, and used by secretaries and journalists among others. It is difficult to say if there is more than a casual connection, the brevity of expression, between interest in symbolic logic and in shorthand, say some common ideal of rationality. Gentzen was obviously completely at home with them. It is known that his mother had taught shorthand writing, both German and English. In addition to shorthand, Gentzen wrote in three different styles plus typewriting. One was the formal handwriting that was part of the school education, called Sutterlin-script,¨ as in the handwritten thesis manuscript. A sample can be found in the frontmatter of Szabo’s edition of Gentzen’s Collected Works that contains a photographic reproduction of part of page 2 of the manuscript. Sometimes marginal remarks are written in a rather childish-looking and clumsy, but otherwise readable hand. When writing for a printer, Gentzen seems to have used either a typewritten manuscript, or a very readable though somewhat irregular handwriting as in the chapter on normalization that was later added to the thesis manuscript. Even here, formulas are usually written with the Sutterlin¨ alphabet. The writing of formal expressions uses also in shorthand mainly the Sutterlin.¨ A printer would set such in Fraktur; well, there are hardly any printers left as the typesetting task has been relegated to us authors, and I have followed what was once the standard practice. A special aspect of logic is the use of metasymbols, Mitteilungszeichen in German, or “signs for communication.” Gentzen’s thesis manuscript, item 6 below, begins with the following explanation:

§1. Short explanation of some ways of notation. I shall distinguish between (cf. H.A. [Hilbert-Ackermann] pages 51–52): Object variables, variables for short. E.g. a,b,c... Objects. Speciﬁc objects, e.g. 1, 2, 3 ... Predicate variables, e.g. A( ,,),G(),P. Predicates. (In H.A.: logical functions) Speciﬁc predicates, e.g. <,=

Gentzen writes the letters in a Latin style, sometimes in block letters, at other times in cursive. The example proofs that follow in the introductory chapter are also written in a Latin style, first with X, Y, Z as propositional letters, then F as a predicate letter. When Gentzen comes to explaining the 3 Practical remarks on the manuscripts 83 tree structure of derivations, he starts writing formulas in Sutterlin,¨ rendered as Fraktur in a corresponding passage of the printed version of the thesis. In the printed thesis, we have the following divisions and notation (p. 179): There are signs that are divided into “determinate” ones such as signs for numbers, arithmetic and logical operations, and specific predicates and relations. Parentheses, commas, and the sequent arrow are “auxiliary signs.” The signs for object and predicate variables are as in the handwritten version. Finally, “German and Greek letters serve us as ‘signs for communication,’ so they are not signs of formalized logic, but variables for its consideration.” The Sutterlin¨ handwriting, designed on commission by Ludwig Sutterlin¨ and taken into use in 1915, is much harder to read than the printed Fraktur; Even if printing does not do justice to the cursive nature of the former, it will give some idea of the difference, as in and Fraktur. In the former, many lower case letters are hard to distinguish, as in and , meinen and nennen. Even some capitals are very close to each other, such as that stands for capital I, not J, the latter being .1 In Fraktur, letters are close to each other in just a few cases, as with capital C and E that are printed as C and E, respectively. The main thing to keep an eye on are the variables x, y, z that are printed in Fraktur as x, y,andz,fortheSutterlin¨ , ,and . There is at the end of this introduction a list of letters in today’s standard German alphabet, in Sutterlin,¨ and in Fraktur. At places, Gentzen’s writing of a cursive Latin F and Sutterlin¨ come very close to each other, but in the end, no difficulty is created if these are mixed. In deciphering formulas in the shorthand notes and in deciding how they should be printed, the handwritten thesis and its printed version are the best guide. For example, in the series WA, there is a function that seems to be denoted by sc, but it could also be an upper case Sutterlin¨ that would be printed as X, written very small, or even a Greek κ.Atonepoint, Gentzen changed the notation, writing an upper case letter on top of the s. The same letter is found in the word Schlussen¨ on the first page of chapter III of the thesis manuscript, the one that does not use Sutterlin.¨ Then, by the context of use, this Sc stands for something like “Schnitt-charakter.” The manuscripts have lots of underlined passages. It could be that this is in part a side-effect of stenography, meant to highlight topics and concepts that perhaps would not otherwise be as easy to detect as in longhand writing and in a printed text. On the other hand, even Gentzen’s published papers use a lot of emphasis in italics that may seem as exaggerated as here.The manuscripts have also often titles and other passages highlighted by a co- loured underlining, usually red, blue, or green. The scans I have used have a good resolution and show minute details of the manuscripts. Copies of these scans are kept at the Philosophisches Archiv der Universität Konstanz.

1 For similar typographical reasons, Gentzen’s intuitionistic calculi NI and LI have been almost always rendered erroneously as NJ and LJ in English. 84 Part II: An overview of the shorthand notes

The manuscripts have what are obviously later additions, sometimes seen from an added date. At other times, Gentzen indicated later additions by freely drawn triangles that I have reproduced like this . Most manuscript pages contain series and page numberings in a thick red pencil. Sometimes there is a letter d, apparently for “Durchsicht,” overview, written with a green pencil. Yet another usage is a thick vertical black line in the center of a page or passage. It could be a deletion, but it could also just indicate that Gentzen had read the passage and summarized it somewhere else. Shorter cancellations are usually clear, several light horizontal lines drawn over the text. There are also parentheses out of the normal usage. They may highlight a side issue. Finally, there are vertical marginal lines of various types, say, a double violet line. What all these emendations to the plain text mean is not easy to say. I have in each case read the text and made my own decisions.

4. Manuscript illustrations

The illustrations begin with a list of the German alphabet in three different forms, followed by a scan of a table from the book Lebendige Sprache: Ein Wegweiser in die Muttersprache by Hans Schorer and Albert Wiechmann (Verlar Moritz Diesterweg, 10th printing 1959). This table comes much closer to written Sutterlin¨ than the type LATEXoffers.Thebookismyown copy that I used at the Deutsche Schule Helsinki, found among old papers in my childhood home several years after my rediscovery of Gentzen’s thesis manuscript in 2005. By about 1960, the teaching of Sutterlin¨ had been abandoned since two decades, but older people used it, as on page 91 of Lebendige Sprache that contains a “letter from aunt Gertrud” printed in a Sutterlin¨ emulation, so children would have to learn to read it. There follow various illustrations of Gentzen’s manuscripts: The first of these illustrations, on page 88, is from the series WAV,lower part of page 78 from late 1934. It is written with a sharp pencil in a very clear hand. The bottom of the paper is somewhat damaged and a word in the middle of the line not clearly readable. What is said in the sentence is clear anyway and can be read from the facing page. Page 90 is from the series BTIZ, page 245.10 written in April 1942. The text is unified shorthand written with a dull pencil on a cardboard-like paper. The bottom has a later addition, as indicated by the triangles.TheGerman transcription is given on the facing page. Page 92 is from the handwritten version of Gentzen’s thesis, written in the Sutterlin-script¨ possibly towards the end of 1932. There is a facing transcription. Page 94 is also from the handwritten thesis manuscript, from the chapter on normalization some time in early 1933. It is written in a readable hand- 4 Manuscript illustrations 85 writing that Gentzen seems to have used when not writing in shorthand or in the formal Sutterlin-script.¨ A transcription would be hardly needed. 86 Part II: An overview of the shorthand notes

The German alphabet in Latin, Sutterlin,¨ and Fraktur type

The following table gives the alphabet in today’s upper case standard Latin (Lat), Sutterlin¨ ( ),Fraktur(Fr), and lower case Latin (lat), Sutterlin¨ ( ),Fraktur(fr). There are two variants of lower case s in Sutterlin,¨ one as in the above , and a second that is used in the end of words, as well as a symbol for double s, similar to today’s German ß. Letters with umlauts have two short vertical lines resp. two dots above.

Lat Fr lat fr

A A a a B B b b C C c c D D d d E E e e F F f f G G g g H H h h I I i i J J j j K K k k L L l l M M m m N N n n O O o o P P p p Q Q q q R R r r S S s s T T t t U U u u V V v v W W w w X X x x Y Y y y Z Z z z 4 Manuscript illustrations 87

The Sutterlin¨ alphabet was still taught at the Deutsche Schule Helsinki around 1960. This illustration with the title Buchstabenbilder and beautiful handwritten Sutterlin¨ letters is from the author’s personal copy of the book of readings Lebendige Sprache. Ein Wegweiser in die Muttersprache (part 2 for the third and fourth school-years), written by Hans Schorer and Albert Wiechmann and published by Verlag Moritz Diesterweg (tenth printing, 1959). 88 Part II: An overview of the shorthand notes

Lower part of page 88 from the series WAV. 4 Manuscript illustrations 89

9. Richtigkeits §(7–). Der 2. (Widerspruchsfreiheits) beweis. LK WfBeweis

Nach dem 3. Ersetzungssatz können wir uns beschränken auf den symmetri- schen Kalkul¨ ohne transfinites Vorkommen von &, ∨, ¬ (⊃ kommt uberhaupt¨ nicht vor.) Der Richtigkeitsbeweis kann auch ohne besondere Schwierigkeiten auf die Zulassbarkeit dieser Verknupfungen¨ ausgedehnt werden, der 3. Ersetzungs- satz stellt keinen wesentlichen Teil desselben dar. Doch wird das seit Vorweg- nahme der Widerspruchsfreiheitsbeweis selbst [cancelled: klarer] ubersicht-¨ licher, mehr auf das Wesentliche beschränkt. Der ganze Beweis verläuft analog zum 1. Richtigkeitsbeweis, formal [cancelled: noch] einfacher als jener. Wir beginnen wieder mit der Erklärung eines Begriffs reduzierbar“ fur¨ Sequenzen. ” [Cancelled: Dieser stimmt fur¨ Sequenzen mit einer Hinterformel mit dem alten uberein.¨ Ist also nur eine Verallgemeinerung derselben.] (nein! Durch die Vermehrungsmöglichkeit von Hinterformeln wesentlicher Unterschied!) Eine Sequenz mit freie Variable ist reduzierbar, wenn jede durch Einsetzen beliebiger Zahlen fur¨ die freie variable entstehende Sequenz und anschlie- ßendes Ersetzen aller Terme durch ihre Werte in dem nachfolgenden Sinn reduzierbar ist. Reduzierbarkeit fur¨ Sequenzen ohne freie Variable: Zunächst Reduktion einer Sequenz“: ” Freie Reduktionen: Man ersetzt eine Hinterformel der Gestalt ∀xF x [written above: oder Vorderformel ∃xF x] durch Fν mit beliebig gewähltem ν, Zahl. [To the right of previous: Etwa gleich nur fur¨ Sequenzen mit transfiniten Bestandteilen erlaubt. So dass finite Sequenzen ohne weiteres ⊃ nur reduzierbar sind, wenn sie richtig sind.] Gebundene Reduktionen: Man ersetzt eine Hinterformel ∃xF x beziehungs- weise Vorderformel ∀xF x durch Fν mit einem bestimmten, irgendwoher ge- gebenen ν. Termreduktionen: Ein Term ohne freie Variable wird durch den betref- fenden Zahlenwert, nach Definitionen verdunnt¨ [?], ersetzt. (Nach oben verlegt, wohl nötig wegen z.B. EE, AB. 90 Part II: An overview of the shorthand notes

Page 245.10 from the series BTIZ. 4 Manuscript illustrations 91

Zur Frage der Vermehrungen 245.10 Bemerkung: In diesen TI-Herleitungen kommt jede Annahme nur einmal vor. Das bleibt auch bei Reduktionen zwangsläufig so. Es ist auch keine wesentliche Beschränkung, denn [cancelled: die VI-Annahmen stecken ja in Wahrheit ω-fach] Annahme 1stecktjainWahrheitω-fach in dem VI-Ergebnis. (Sie wird freilich erst so weit vermehrt, wenn sie offenes Land geworden ist, d.h. nicht als Annahme, sondern als Tatsache dann.) Ich glaube, dies stimmt, und man kann es als normativ ansehen, braucht sich also nicht durch mehrfaches Vorkommen von Annahmen irrefuhren¨ zu las- sen, d.h. dieses und die dadurch bedingte Vermehrung,wiebei⊃-Reduktion nicht als das Wesentliche hierbei anzusehen! (Echte Vermehrungen treten freilich bei VI-Reduktion auf.) Das Wesentliche an der Besonderheit der ⊃- Reduktion is nicht die Vermehrung“, sondern die Durcheinanderschachte- ” lung von Teilen, die vorher in ganz anderer Reihenfolge zueinander standen. 3 Hiernach Satzteil [?]: Naturliche¨ TI-Herleitung bis ωω -Reduktionsstellen“ ” Alles Unwichtige bzw. Ubernommene¨ aus 245 in den Keller. Hier nur noch einiges vielleicht Verwendbare. 92 Part II: An overview of the shorthand notes

Part of page 12 from the early handwritten version of Gentzen’s thesis. 4 Manuscript illustrations 93

IV.−−−−−−−−− Vermutet wird der Satz: Wenn eine logische Aussage im Kalkul¨ N1I beweisbar ist, so gibt es einen solchen Beweis fur¨ sie, in welchem nur Teilaussagen von ihr vorkommen, allenfalls mit anderen Variabeln. Der Beweis hierfur¨ ist fertig [cancelled: noch nicht fertig ausser fur¨ einen Teilkalkul¨ N2I, welcher von N1I nur die Schlusse¨ UE,UB,AE, AB,R enthält, aber durch folgende Tatsache Bedeutung besitzt: Die Frage der Beweisbarkeit einer logischen Aussage im klassischen en- ” geren Funktionenkalkul¨ lässt sich zuruckf¨ uhren¨ auf die Frage der Beweisbar- keit einer (anderen) logischen Aussage im Kalkul¨ N1I.(dieshängt mit III zusammen). End of cancellation] V. Die Widerspruchsfreiheit der Arithmetik wird bewiesen; dabei wird der Begriff der unendlichen Folge von naturlichen¨ Zahlen benutzt, ferner an einer Stelle der Satz von ausgeschlossenen dritten. Der Beweis ist also nicht intuitionistisch. Vielleicht lässt sich doch tertium non datur wegschaffen. 94 Part II: An overview of the shorthand notes

Part of page 13 from the early handwritten version of Gentzen’s thesis. Dissertation, III. Abschnitt 95

Dissertation, III. Abschnitt G. Gentzen

Ubersicht.¨ Wir wollen zeigen, dass sich jede NI-Herleitung auf eine ziem- lich einfache Form bringen lässt. Diese wird z.B. folgende Eigenschaft ha- ben: Jede Formel der Herleitung ist Teilformel der Endformel, allerdings mit anderen Gegenstandsvariabeln. Der Grundgedanke der Umformung ist: In einer beliebigen Herleitung können zunächst alle möglichen Formeln vorkommen. Man betrachtet nun unter denjenigen Formeln, welche nicht Teil- formeln der Endformel sind, eine längste. Diese wird i.allg. die Unterformel einer Einfuhrung“¨ ihres äussersten Zeichens und die Haupt-oberformel einer ” Beseitigung“ desselben sein. Alsdann lässt sie sich aber durch Reduktion ” wegschaffen. (vergl. I. Abschn. §5.) Einige Ausnahmefälle (bei den Schlussen¨ OB,SB,V) erfordern besondere Behandlung, doch liegen die Verhältnisse da nicht wesentlich anders. Part III: The original writings

1 Reduction of number-theoretic problems to predicate logic 99

1. Reduction of number-theoretic problems to predicate logic

A result in the small is worth more than no overview at large. —————————————————————————————————

IX.31 Reduction of number-theoretic problems to the decision problem of the lower functional calculus. Roman numbers denote the corresponding formulas from my calculus D (in the book about the propositional calculus etc.). (Each time equipped throughout with the every-sign.) [Gentzen’s notation is: γ equality, ϕ succession, and σ sum.] (Eγ)(Eϕ)(Eσ)(E1)[I & II & III & ... & VII& XII & ... & XV I & (x)(y)(z)(σxyz → σyxz)&(a)(Eb)(ϕab& ϕa1)] This should be the claim of the commutativity of addition, written as a formula of the functional calculus. For the decision problem of the lower functional calculus, the question would thus be one of the satisfiability of this formula, with the existence signs for the relations ϕ, σ, γ left out. (Resp. the question of universal validity, in which there is in place of the underlined &a → ? Maybe not, see below. What is the relation of the induction axiom to this?) Precise consideration: If commutativity holds, the formula is satisfiable, in a denumerable domain. Namely, one defines in this, and one can without limitation take the domain of natural numbers, the relations γ,ϕ,σ as equality, succession, the sum relation. Number 1 is taken as 1, then all that stands in thesquarebracketsissatisfied. If the formula is satisfiable, the validity of commutativity follows like this: There is a satisfiability in a denumerable domain. To the 1 that there is, is associated the number 1. Then the following holds for this satisfiability: (a)(Eb)ϕab.Totheb that is associated to 1 as a one lets correspond the number 2 etc. It would be thinkable here that one comes at some time back to an element that was already there. On the one hand, this does not disturb the considerations that follow, on the other hand, it cannot even occur at all, because otherwise the axioms would not be satisfied. (If a were “followed” by an earlier b,then“b−1” would be “equal” to a (b =/ 1 because of (a)ϕa1) etc backwards, in the end an element c of the sequence “equal” to 1, there would then follow after the element c−1 in addition to c also 1, contradiction 100 Part III: The original writings with (a)ϕa1.) I posit now between the numbers that are associated to each of the elements the relations ϕ, γ, σ, just as for the elements that correspond to them. These have to be succession, equality, and the relation of sum, because they satisfy their axioms. They satisfy, further, commutativity that is thereby proved. Universal validity: Let a formula with → (naturally without the existential sign for the relations) be universally valid, in a denumerable domain. Then one takes as the domain the natural numbers and as relations equality, succession, and the relation of sum. The formula is satisfied for arbitrary relations, therefore also for these. Because all that is before the arrow holds for these, also commutativity must hold. The converse: Let commutativity hold. Then the formula need not be universally valid! I.e., there can be relations in a denumerable domain that do not satisfy the formula. The reason is that commutativity is provable only with the induction axiom, and this one does not have to hold. Take, say, the domain of natural numbers and that of even numbers in addition. Then commutativity still holds under the normal definition of addition (which should be avoidable through an appropriate definition), nevertheless one sees with other theorems, such as [page ends] 2 Replacement of functions by predicates 101

2. Replacement of functions by predicates

This item has additional pages 14.1–4 that I have ordered so that the whole makes sense. Page 13, with the number 4 written with a green pencil in the upper right corner, has been placed between pages 14.1–4. Page 14.2 ends with the remark “Continuation of these thoughts see sheet 4 of the green kind” so page 13 is placed there. Page 17 has a similar “green” number 5. The bottom of page 13 contains Gentzen’s peculiar early use of the expression “indirect proof” for proof under assumptions, at the same time a passage that is suggestive of the implication introduction rule of natural deduction.

VOR14.1

Functions instead of relations. Introduced as another way of writing. For A(a, b, c) ∨ B(d, e) ∨ ...F (r,s,t,...) ∨ H(x,y,a,e,...,s,...) one can write H(x, y, A(· ,b,c),B(d, ·),...,F(r, · ,t,...),...) and correspondingly in the general case. as already indicated by the dots...

Detailed execution: Explanation: a predicate in which there is exactly one argument place or a dot, whereas the rest are filled by variables or the individual 1, shall be called a “function”; a predicate, the argument places of which have exactly one filled by a dot, the rest by variables, 1, or functions, shall be also called a “function.” a function is, for example: γ(· ,σ(1,a,· )) Explanation: variables, 1, and functions together are called “terms.” A predicate, the argument places of which are all filled by terms, has the meaning of a certain disjunction of elementary propositions that can be always produced from it by a rule to be given shortly. The following has to be taken into account: The representation is unique only up to the renaming of certain variables. If the predicate stands in itself for a sentence, that is no concern, because renaming gives anyway provable sentences. If instead it is a part of a certain propositional connection, the following rule has to be followed in the restitution, so that all the variables introduced arbitrarily therein are to be chosen as different from each other and from those already occurring. Hereby the logical equivalence of all the sentences that arise through restitution is in every case secured. The rule is, for one single predicate A, as follows: One substitutes in the same variables (different from each other and those 102 Part III: The original writings that already occur) to all argument places filled by functions and obtains A; the functions are certain predicates with a dot in an argument place, one writes these predicates negated and connected by ∨ after A and places in each instead of the dot the variable that was placed in the place of the function in the predicate A. Should there appear in the still remaining predicates still functions, one reduces further in the same way.

VOR14.2 Example: a + b · c a · b Written precisely:1 κ(σ(a, π(b, c, · ), · ),π(a, b, · )) ∨ γ(σ(a, π(b, c, · ), · ),π(a, b, · )) Reduction to relations: κ(d, e) ∨ σ(a, π(b, c, · ),d) ∨ π(a, b, e) ∨ γ(f,g) ∨ σ(a, π(b, c, · ),f) ∨ π(a, b, g) κ(d, e)∨σ(a, h, d)∨π(b, c, h)∨π(a, b, e)∨γ(f,g)∨σ(a, i, f)∨π(b, c, i)∨ π(a, b, g)

IX.31. The dot corresponds naturally to the Hilbertian ε. One could require within a theory that such functions can be put up only if they really “exist” by the axioms (or by theorems provable from these). Continuation of these thoughts see sheet 4 of the green kind. = VOR14 ﬀ.

VOR13 “green” 4 Continuation. 4.) B is the abstraction of an expression, variable x.One applies the rule of deletion considered, namely with the deletion of the universal sign, one leaves the x in place as a variable element; with common [gemeinsam] deletion, one writes for an x a normal functional of the form: first there is, for the letter “x,” the letter that corresponds to individual elements (it is easy to make such an assignment), thereafter the variables (possibly there are none) that were free in B, in the order of succession of their first occurrence in B. (In a linear writing of B.) Hereby the uniqueness of the procedure is guaranteed. 5.) B is a disjunction of two expressions. B = C∨D (=, ∨ areheremetasigns) [Mitteilungszeichen]. a) C is a conjunction of two expressions. One applies the “distributive law.” b) C is the abstraction of an expression. Use of the first bracket rule. c) C is the negation of an expression that is not a basic expression or a predicate [Relativ]. C is then treated as B in 3.) (So case a) therein becomes left out. In case d) the general tertium non datur becomes effective here.)

1 [Here the Greek letters κ and γ stand for kleiner and gleich, less and equal.] 2 Replacement of functions by predicates 103

d) C is a disjunction of two expressions, C = E ∨ F. Then one puts for B: C ∨ (E ∨ F) (associative.) Except when B is already a disjunction of unities, then it remains unchanged. e) C is unity, B not a disjunction of unities, i.e., basic proposition or predicate or negated basic proposition or negated predicate. (This is clearly all that remains.) α) D is not unity. The one substitutes B by D ∨ C (commutative.) β)AlsoD is unity. Then B remains unchanged. Known things. Skolem normal form —————————– The procedure offers uniquely a “normal form” of the expression. One con- vinces oneself that this one is provable from the expression, and the other way around. (The first part can be false within a fixed theory, renaming will be necessary.) What shape does the normal form have? It presents a conjunction of disjunctions of unities. —————————– Concerning indirect proof. I would like to arrive at having even in this one only correct sentences. Something like: The starting point is a disjunction: A ∨ A.IfnowB follows from A, one writes down B ∨ A,etc.Thatisin order, and is it sufficient? B ∨ A isthesameasA → B. So the principle is: If B is provable from A, then one can write down A → B.

VOR14.3 (notebook 30) end of 1931 Is the concept of a function referred back to equality? Must a function be unique? It is not absolutely necessary to require uniqueness. For one can understand in case of R(· y)anyx that stands in the relation R to y. A proposition to that effect would be conceived as holding for each such x. I.e., S( R(· y))wouldmean:(x)(R(xy) → S(x)) One could perhaps even include that there should exist such an x.Inthat case, & (y)(Ex)R(xy) would have to be added to the right. Compare to Hilbert’s ε with which the latter is dropped. (Russell’s ι is unique.) With the many-valued conception of a function, it is not the case that, say, R(· 1) = R(· 1) I.e., two externally quite equal expressions need not be “equal.” So one has 104 Part III: The original writings to take an expression such as R(· 1) as still a variable, even if R and 1 can be quite fixed. If one requires that the same value is always assigned to equal function symbols in equal conditions, then R(· 1) = R(· 1) continues to hold. This should be appropriate, because it corresponds to the usual formulation. In number theory, there are also many-valued functions, e.g., “a divisor of,” “a common multiple of.” Now one avoids writing this down. Though it is not always easy to avoid it, e.g., with the infinity of primes. n!+1oritsprime divisor delivers what is desired. It is unnatural to specialize the divisor to, say, the smallest one. VOR14.4 Can I dispense with equality after the introduction of a symbol for: “there is exactly one”? [The question and the following paragraph are cancelled.] The introduction of equality in logic seems baffling, because it is the only individual relation. On the other hand, it has to be taken into account that it is the only relation that can be defined completely generally, without consideration of the special “elements.” The concept of a “same thing” holds even for each object as a completely clear stipulation. (In contrast to, say, “the thing that is greater by 1.”) about 1931 Can one write x for y when f(x, y) holds? This seems to me allowed only when the uniqueness of y has been ascertained. Otherwise one would have to say: x denotes some element that follows x, so a restricted variable. Then x = x for example is incorrect. This, though, should be doable because the equality between variables is anyway always senseless. (?) A theorem such as the following would then be possible: If two numbers arise from x through succession twice and they are equal, then the middle term is the same in both. How should this be written? It would be like: y = x & z = y & u = x & v = u & v = z → u = y The same difficulties seem to me to lie at hand with the Hilbertian ε,even that many-valued. [The reverse of this page has the cancelled text:] About 1931 Lower case Latin letters denote natural numbers. Axioms of equality. (x)(x = x) (x)(y)(x = y) → (y = x) (x)(y)(z)(x = y)&(y = z) → (x = z) 2 Replacement of functions by predicates 105

(x)(y)[(x = y) → (A(x) → A(y))] Axioms for natural numbers [Z for number, f for succession]. Z(1) [(x)(Ey) f(x, y)] & (x)(y)(z)((f(x, y)&f(x, z) → (y = z)) (Ex) f(x, 1) (x)(y)(u)(v)((f(x, u)&f(y, v)&(u = v) → (x = y)) or: [f(x, u)&f(y, u)] → (x = y)?

VOR14

XI., XII. 31.

First-order form of a complex. Replacement of functions by predicates.

Examples. 1) A(x, δy) ≡ (Eu)[(Δ(r, y) ∼ r = u)&A(x, u)] One could require for all values, instead of just for y that could even be a constant here, the existence of δ. One can leave out the equality, then the thing looks like this: Eu[Δ(u, y)&A(x, u)] This means naturally something diﬀerent from the previous. It all depends on what one wants to understand by a function, this is a question of deﬁnition. 2) a + b = b + a ≡ (Ec)(Ed)[Σ(abc)&Σ(bad)&Γ(cd)] This doesn’t quite catch the sense of the left side, because it doesn’t present at the same the uniqueness of sum. Even if this were not the case, one could read the left side also as: (Ew)Σ(uvw)&(Σ(abc)&Σ(bad) → Γ(cd)) This is: There is always a “sum” and for all c that are equal to a + b,aswell as for all d that are equal to b + a it holds that c is “equal” to d. There are various possibilities in the determination of the concept of a function: single-valued —— many-valued always existing —— not necessarily always existing In the case of many-valuedness of functions, the function symbol denotes a variable, and moreover one that is restricted to certain values. It could then, no doubt, be taken under a for-all or an existence-sign: (E(ω1))A(ω1)

To be read as (Eu)(Ω(u1)&A(u1)) 106 Part III: The original writings

The expression (E(ω1)) would have a sense already on its own, namely (Eu)Ω(u1). (If functions should always exist, that would be a triviality.)

VOR15 Here the concept of a predicate seems to be primary to the concept of a function. Though one can have in several cases the opposite conception, e.g., in the introduction of the elements of a recursive theory in which, e.g., the “element that arises through writing two elements one after the other” is a thoroughly intuitive function built up before any predicates. Here the signs are conceived as objects; the uniqueness of functions is a mathematical identity. 3) General case. Let an expression be given. One writes to begin with down all functions that appear as existing ones. This means that when + occurs, one writes down: (x)(Ey)σ(xyz) The conjunction is built from this one. There comes in addition the expression itself, reformulated in the following way: Example a: (a + b>c) ∨ (a + bc) ∨ (dc) ∨ (Ed)(f)(a + fc)(y)(z)(σafy& σdcz → yd)] Transformation:2 (x)(σacx→ xd)] I am for 1., because it is easier to formulate in a general form, and because it says the same as 2. (admittedly only with unique functions, but these are what we want to assume.) We say in general: In an expression, each relation is modified separately. The expression as a whole has no effect on this modification. So is suffices now to consider the modification of relations. 4. example: a + b · c

2 [This line and the next are followed by a large question mark.] 2 Replacement of functions by predicates 107

VOR16 calculus at hand. 5. example: a + b propositional inference (some attention!) Substitution of an element <—> the same Substitution of a functional <—> 6. example: In the relation R xy, δuis substituted for y. So this becomes (z)(Duz → R xz) ≡ (z)(Duz∨ R xz) Now even (u)(Ez)Duz holds. [End of cancellation.]

VOR17 “green” 5 How is R(xδu) to be transformed? (z)(Duz → R xz)or(z)(Duz → R xz)? 108 Part III: The original writings

By the rule that has been given, obviously the second holds. But is this correct? It reads as: (Ez)(R xz& Duz) x and u are ﬁxed for us. Then there is exactly one z such that Duz holds. So we can say equally well: (z)(Duz → R xz) for all z, for which Duz holds, R xz holds. That means: The two readings are equivalent! ———————————————————————————– Let δ occur in an expression A as the only function, of the form δu,u free. Can one then write, instead of A(δu), (x)(Dux → A(x)) ? This is precisely the procedure in which one sees A as a predicate. How does one arrive from this form to the usual one? Are they both certainly equivalent? [The following is cancelled.] Should one move from the ﬁrst-order propositions directly to their reduced forms, that is, ones with only single existential propositions, and one general proposition? Example: A(δu) ∨ B(δu)&C(δu)&D(v)=A (x)(Dux∨ Ax) ∨ (y)(Duy∨ Bx)&... ∨ ∨ ∨ (x)(y)(Dux Duy Ax Bx)&... ≡ Dux& Duy ∼ x = y & Dux (x)(y)(x = y ∨ Dux∨ Ax ∨ Bx) ∼ (x)(Dux∨ Ax ∨ Bx)&(z)(Duz∨ Cz)

VOR18 (x)(z)(Dux∨ Ax ∨ Bx.&Duz∨ Cz) Claim: This is ∼ (x)(Dux∨ Ax ∨ Bx.& Dux∨ Cx) This follows from the previous one, namely for x = z. Conversely: From the lower one follows (x)(Dux∨ Ax), and also the second. So, all of it correct. [End of cancellation.] ———————————————————– 2 Replacement of functions by predicates 109

The introduction of new predicates in mathematics and logic. Many-valued functions.

In an inference that removes an existence-sign, a new predicate is introduced (compare Funktionenkalkul¨ 20.) Namely like this: Given the expression (x)(Ey)(Axy → By) for example. One introduces: Cxy≡ Axy → By. Then one can infer (x)(Ey)Cxy, this together with Cxy∼ Axy → By is “equivalent” to the previous expression. (Even if not in the logical sense, as is well known.) Further, one can as is known (Gödel) substitute the second proposition by Cxy → (Axy → By). It hangs contentwise on: The predicate Axy → By has by assumption the property that there is for every x a y that satisfies it. But there can be several such y. It is now sufficient to introduce instead a predicate with which there is to each x just one y, but one that has to be always among the previous y. In the introduction of designating functions [Angabefunktionen], uniqueness is even obligatory. It makes no difference at all how a y is chosen from those at hand. The predicate Cxy, though, is not uniquely determined, analogously to the designating function. The justification of its introduction is seen from the fact that, e.g., Axy → By has the desired properties, so the exist such predicates. That there exist even ones with (x)(Gy)Cxy [(Gy) likely for Gibt y], should not be so easy to see formally. How does one replicate such in logic? It is not a normal inference, the second line does not “follow” from the first one. One could take it as the addition of an axiom: Cxy∼ (Axy → By)oronetakesCxyjust as an abbreviation of Axy → By! The last one is clearly the most convenient one, as long as one doesn’t require the uniqueness of y(x). In the latter case, one should need a sort of axiom of choice: (Skolem!) Whether it is preferable here to allow many-valued functions? (A single-valued function cannot give one more than a many-valued, because its choice is arbitrary.)

VOR19 There are on page 25 of “Funktionenkalkul”¨ considerations on this very same question. How is it with the difficulty that arises there concerning the introduction of many-valued functions? Let us have: (Ea)(Ec)A(ac) Formulation, let there be from the outside: Let the many-valued function α designate all such a.Itmeansthat(Ec)A(αc) holds for every such α.The c that exists can clearly be different for different α. So one is constrained to take it as a function of α: γ(α). Then we have A(α, γ(α)) for arbitrary α from its domain of values, and arbitrary γ from the domain of the α that is 110 Part III: The original writings associated to this γ. The α(x) depend on an α with a free variable x,thenγ(α(x)) is an implicit function of x. A(α(x),γ(α(x))). One should, now, give always the same value to α(x) in all places where it occurs, and as a consequence, one must not write γ(α(x)) as only a function of x alone, even if it is! Compare sheet 4,5, example. There only the first transformation works now. ———————————————————— Essentially a one-one correspondence sought between first-order propositions and complexes. (That is to say up to possible renamings of signs.) Thereupon the equality of theories in both systems has to be shown, and here transformative inferences from one to other have to be renounced. Example. (x)(Ey) Axy Complex: A(x, (vx)) The way back by the scheme: (u)(Ev) Yuv&(x)(v)(Yxv→ Axv) Provable from this one: (x)(Ey) Axy Though not the converse direction, because Y is something completely new. Or one makes the way back in the old style (step of introduction of the existence-sign): (x)(Ev) Axv. It is, though, questionable whether this functions always so well. Should be the case whenever the complex is obtained from a first-order proposition. But not with, for example, a + b = b + a. There remains in the universal case only the new procedure.

VOR20 [The whole page has been cancelled] We consider a system of inference of propositions. This will consist of certain propositions A1 ...An, and certain propositions B1 ...Bm that are concluded from the A in a determined way. All intermediate steps in proofs must be carried through! We shall define a law of assignment by which there is assigned to each of these expressions a first-order proposition. (It will depend with the A only ontheirshape,withtheB though also on the kind of associated proof, A B as the case may be!) Let these first-order propositions be 1 ... m.Ithas then to be shown that whenever any inference among the expressions took S S 1 ... S S → T place, of the form T ,then 1 & ...& is a generally valid first-order proposition. The assignment is as follows: First one proceeds as on sheet 4. (I don’t see the 2 Replacement of functions by predicates 111 freedom in the notation for bound variables and in the order of members in a disjunction as a difference in the first-order propositions concerned.) Each first-order proposition is to be added conjunctively. Existential propositions in an occurring predicate that arises from functions. No uniqueness in these. The procedure is even to be modified because of many-valuedness, see above. A A A Thereupon there remain unaltered the 1 ... n that arise from .The- re will possibly be a further modification with the remaining ones, to wit, whenever there occur functions in the B that have not yet occurred in A. Each of these must have arisen through: (these not allowed with thinning and substitution). 1. The renaming rule or 2. elimination of the existence- sign. With 1, one proceeds as if there had been no renaming. With 2, let therebearisen(as a first-order proposition)(Ex)A(u1 ...uσ,x)andthe- refore A(u1 ...uσ,ξ(u1 ...uσ)) in the first-order proposition, instead of a predicate X(u1 ...uσ,v). One substitutes for this now all over the predicate A(u1 ...uσ,v). Thereby the claim has been now proved. ——————————————————— One can make also out of one-place predicates functions, what are known as “constants.” Even so, substitutions are allowed for these, namely of expressions that, as one knows, fulfil the predicate. For example, one can consider all variables of number theory as such constants relative to the predicate “to be a number”! Maybe one will allow such substitutions in general, for arbitrary functionals. Namely, of special values resp. classes of values for many-valued functions. The role of variables in polynomials, and “propositions”! Is this an element? And what? 112 Part III: The original writings

3. Correspondence in the beginning of mathematics

(Notebook 29) End of 1931

It should be the case that primitive man began to exercise mathematics with quite concrete objects at hand. (The abacus with its beads, lines drawn in sand.) He observed relations between these (among others: rows of beads, intersections on lines) and ascertained facts. He then determined that other objects, found in another place, allowed to establish the same relations, and gave the same results. With the assumption that there was something in common between these objects and those before, namely, the possibility of a one-to-one correspondence, of the objects and the relations. This knowledge gave rise to a first abstraction: The results are freed of the specific things, one sees them as statements about certain concepts of thought that have no reality and sees their meaning only in the ever repeated practical applicability. To make these concepts of thought communi- cable, words had to be agreed upon, and signs for written communication. (Geometry. The numbers.) Connected to the formation of these abstract concepts was the addition of their arbitrary combinations. These were analogous to those in nature, though not anymore made necessary by nature. My view here is that these “concepts” have been sufficiently clarified by the above. So it is not necessary to define them, as sets of the respective objects in reality that have been made to correspond to them, as in Russell, or as the signs that have been introduced for them, as in Hilbert. Both appear so artificial, namely dependent on circumstances that should have no effect on the concepts at all. (The existence of infinitely many things in the world, the nature of the signs.) The next correspondence is between “equal” signs in different locations. (Mathematical identity.) See p. 11.8. 4 Five different forms of natural calculi 113

4. Five different forms of natural calculi

23.IX.32 Inference schemes:

AI AE OI OE FI FE A B A ...... ABA&B A&B A B CA∨ B C B AA→ B A&B A B A ∨ B A ∨ B C A → B B

UI UE EI EE RA REND CI Pa A ...... Pa xPx Ph ExPx B ¬B ¬¬A P1 Pa xPx Ph ExPx Pa ¬A A Ph

NEB W D RA2 H

A ¬A . . . . B ¬B B ¬B A ¬A ¬A ¬. A&¬AA∨¬A A B

Formal shape of proof: Each proposition conclusion of at most one inference, no circle in inference. S Tree form: Each proposition except for the endproposition a premiss in exactly one inference. Association of assumptions as known, assumption can appear in an arbitrary number, even not at all. But only as an initial proposition above its place of drawing [the associated inference]. N Net form: Each proposition except for the endproposition premiss in at least one inference, eventually several. Association of assumptions: An assumption can appear at most once, as an initial proposition, and then all threads down from it must go through the place of drawing [the inference]. Variable conditions. Condition 6: The h of UE, EI, CI can be a number, the a of UI, EE, CI not. Condition 2: The a of a UI, CI must not occur in their conclusion, the one of an EE not in its premiss. “Eigenvariable.” 114 Part III: The original writings

Both of these conditions hold in all cases, they could be taken as part of the inference schemes. Condition 11: All threads through propositions with a free all-variable must go down through their UI. Analogous for CI variables. Condition 10: There is a linear order of propositions in a proof, in which each conclusion comes after its premisses, and in which no free existence variable appears before its EE.

PLF 2

Condition 30: In a UI, there must not occur any existence variable that depends on the all-variables. Concept of dependence after 78.4 bottom. Condition 27: The free variable of an EE must not be related with any free variables occurring in other EE’s, UI ’s, CI ’s. Condition 33: In A, B of a FE; A, B, C of an OE; A of an RA; Pa, Ph of a CI, A of NEB or RA2, no free variable must occur the eigenscheme of which stands in an indirect upper part of this inference under its assumptions. Bb 4:1 The eigenvariable of an EE, UI, or CI in an indirect part of proof must not be equal to a variable in the (main) assumption or in the endproposition of this part of proof. VC 5: Equal free variables must be related. Condition 34: There is to each free variable an eigenvariable equal to it. Division of the inference schemes: Constructive2 (C ) AI to EE. Intuitionistic (I ): AI to RA, H.Classical(K ): AI to REND. Important forms of proof: ↓ that is D NNA: natural net proof general form, i.e., net form with conditions 6.2; 11, 10, 30, 33. (sheet 78.3–79.) N1: Tree form, with conditions 6.2; 27, Bb 4. (Bb 1, Bb 3, Bb 4.) (sheet 66.3.) N2: Tree form, inference forms AI,AE,UI,UE,RA and REND, with condition 6.2; Bb 4 (only the special case of condition 32.) (sheet 80.2) NN2: Net form, inference forms AI,AE,UI,UE,RA and REND,with condition 6.2; 11. (sheet 84.1)

1 [Probably for Beweisbedingung, proof condition.] 2 The German original transcribes, in Christian Thiel’s judgment, into the somewhat unusual construktiw, with an initial c instead of k so as not to clash with klassisch. Note that one has to add rule H to (C )toget(I ). 4 Five diﬀerent forms of natural calculi 115

NO: Tree form, with condition 6.2; 10, 30, 33. Corresponds to NNA, condition 11 and superﬂuous because of the tree form. For complete “generality” there is still missing the admission of several EE’s with the same variable (now blocked by condition 10.)

The satisfaction of variable condition 5 can be always achieved through the puriﬁcation of the proof, I shall presuppose it in general (as needed). The same with condition 34. (Types higher than 1 are so far not allowed.) For a puriﬁed proof holds: the eigenvariable of an EE does not occur in its upper part, the one of UI, CI only there. Each free variable is equal to exactly one eigenvariable. (Theorems 4, 5)3 The former follows from variable condition 5, condition 2 and condition 33, resp. Bb 4; in the case of net form already from condition 11 and 10. The second follows then from condition 34, as well as 1, 10, 30 with NNA, NO, condition 27 in the case of N1, without further ado with N2, NN2.

3 These theorems are found in the handwritten manuscript of Gentzen’s thesis, item 6 below, section titled The puriﬁcation of a proof towards the end. The triangles indicate that the passage is a later addition. 116 Part III: The original writings

4bis. Two fragments on normalization

The first of the following texts is from the top of the first page of the note Five different forms of natural calculi, September 1932, and with the page number PLF 1. The second text is from the series D, page 92.1 and the upper part of page 92.2. It is a preliminary work for a normalization proof. The titles in both are taken from underlined passages in the main texts.

PLF 1 Overview of all possible reductions (D 85.3, DB 131)

Series PLF: The forms of predicate logic One should not grab the hill [Berg] lying highest up, but the one buried deepest. One will want to normalize, to begin with, the small parts of a proof, say the indirect pieces [parts from assumptions to the steps in which they are closed], before one undertakes big rearrangements with them. This seems to be in the nature of things as well as reasonable. Ipreferagain the family tree form. One will reduce a maximum only when the anterior parts in question contain no maximum anymore. (Compare 69.2) Simpliﬁcations in the sense that the number of maxima diminishes! This, though, is not quite accurate. I need an overview of all possible reductions in hills, eventually further ones. (Return to a systematic work procedure from an erratic one.) [Here begins the text of Five diﬀerent forms of natural calculi,item4above.]

D 92.1 Recursive assignment of values. An RA without a further one in its anterior land is trivial to ﬁnd. [Cancelled: I assign values to its eigenthreads in a way that can be ﬁxed. Thereby possible passages through RA at left have assignable values.] How is it with the eigenpassage left through this RA! The eigenthreads that belong to it could obtain values only when one knows the increase of values in this passage. There are dependences between the RA’s of the proof. Namely, the value of an RA depends on the NI ’s [negation introductions] of its right premiss. Whether these dependences are circular? If the chain of relatedness passes through no assumption, the dependent RA must stand under the other ones, 4bis. Two fragments on normalization 117 i.e., the latter in a right thread of the former. (Not necessarily under its assumption.) (The simplest circle [loop in a derivation] thinkable can be seen as impossible through an index consideration. Perhaps in general? Attempt: See 89.3 bottom etc.) The example 82.3 top not dangerous? × D A γ α A ¬D D ¬D× I put, say, for A , = : ¬D We have already the circle. Would correspond on 70.2 bottom to: A = C → D. A α For A there would appear:

× C C → D D FE × C → D FI

Both of the C → D are related by the red1 detour. Therefore a circle is constituted when one wants to give a value to the left-passage of this FE, because it belongs at the same time to the eigenthread of the associated FI . The reductions work nevertheless, because this [thread?] is doubled and thereby the circular relatedness dissolves. An assignment of values is anyway not possible simply after the above scheme. Maybe one has to do as on page 73.4. Maybe even simply a consideration of the actual reductions in their order of occurrence helps. Say if one substitutes all variables by 1 etc.

9.X. 32 Thelineofreduction:This leads in the proof considered [Urbeweis], proceeding from the first peak, always to the next peak in the reduction, i.e., to the proposition (these can be, though, several) that is like the next peak. In the case of a unique determination of a peak, also the line is unique, up to the multiplication in the cases mentioned. Then the line will be made to branch. The line of reduction can pass through the same proposition repeatedly, namely when different copies of this become peaks one after the other in the course of time. Whether the line of reduction has to pass each proposition? This would not yet mean a simplification. Assume it has been shown that one comes at some point to a CI-peak [complete induction] and that the number of reductions to that point can be estimated. Then one can assign to the proof, as a value of its form after

1 No colour coding is visible here, so perhaps it is on page 70.2. 118 Part III: The original writings these reductions, the number of the reductions. This value would obviously diminish always by one until the CI -reduction. The question is, naturally, if the value of the form of proof before the CI -reduction can be defined so that it does diminish with CI -reductions. Thus, there is to be compared the form of proof before the first CI -reduction and before the second CI -reduction, as well as an anterior part of the latter to be determined. Let us take for now the case that no CI contains further ones in its eigenland [parts of derivation by a rule that depend on an assumption closed by the rule or that contain the eigenvariable of the rule]. This already is very difficult. D 92.2 A new idea: Is it possible to perform appropriate reductions so that one takes the longest proposition, or a proposition that is of the highest value according to some other assignment that is invariant under reductions, and eliminates it in all its places of occurrence, without multiplying propositions of the same value? The assignment of values will go into the transfinite with CI ’s. The question is naturally to be answered in the affirmative, if a simplification or reduction is at all possible. The reduction will have to be doable so that a UI, RA, CI supersedes in value all that lies in its eigenland. We don’t come, though, into the transfinite by this. It has to happen already somewhere else in the investigation[?]. 5 Formal conception of correctness in arithmetic I 119

5. The formal conception of the notion of contentful correctness in pure number theory, relations to proof of consistency I

This document is a 36-page bound notebook. The ﬁrst page has the numbering D92 with INH 1 added in strong red pencil. Page 92.1 and the top of page 92.2 have been cancelled by a single vertical line. They concern normalization and are found at the end of the previous item. The rest of page 92.2, at the same time INH 2, has been similarly cancelled. It bears the date 14.X. and the title Contentful correctness in intuitionistic proofs that relates directly to INH,evenifthemanuscriptINH begins properly on page 3, with the date 19.X.

14.X. Contentful correctness in intuitionistic proofs [Cancellation begins] (One deﬁnes contentful correctness like this: The mathematical axioms are correct. A&B is correct when A is correct and B is correct, A∨B when at least one of them is correct, Ax when for each number substitution for x this correct, the same with xAx,Aa when a number can be given so that Aa holds, the same for ExAx, A → B when from the correctness of A that of B can be concluded, ¬A when from A a contradiction can be concluded. It is to be shown now that the result of a proof is correct. This happens by CI on the length of proof. How is it in the case of FE: Let A and A → B be already correct, claim, then also B is. This seems to follow from the deﬁnition of correctness at once. The case of CI: We have P 1andx.P x → Px and conclude Pz.)

16.X. I explain the concept “B is provable from V1 ...Vν” as follows recursively:1 (More correctly: “A consequence of”) C&D is provable from the V’s when and only when C is and D is, C ∨ D is provable from the V’s when and only when C is or D is,2 C → D is provable from the V’s when and only when D is provable from the V’s and C, ←− is this still recursive? xPx is provable from the V’s when and only when Pν is provable from the V’s for arbitrary numbers ν, ExPx is provable from the V’s when and only when there is a number μ

1 The manuscript has upper case V in Sutterlin¨ script in the list of assumptions, likely for Vermutung. 2 A blatant error: For a counterexample, let V be D ∨ C, and similarly for the case of existence. 120 Part III: The original writings such that Pμ is provable from the V’s. ¬C is provable from the V’s when and only when the contradiction 1 =1is provable from the V’s and C. I.e., ¬C ∼ C → O. Nothing new. (For example, ¬.A&¬A is provable thus: From A&¬A follows A, A → O, O.) There appear still diﬃculties when from a condition something is derived through B.3 To what extent are such “contentful explanations” still formal? Why is a consistency proof through a coarse contentful explanation, A&B correct when A correct and B correct, A → B correct when from the correctness of A the one of B follows, xAxwhen Aν correct for all numbers, ¬A correct when A not correct, after G¨odel not formal? Does it contain a bridging inference [Bruckenschluss]?¨ One infers: The logical axioms are correct, the mathematical axioms are correct, inference scheme and substitution give correct from correct, therefore all things provable are correct. If, now, A&¬A were provable, it would be correct, but that cannot be. If A is correct, A cannot at the same time be not correct. All of it looks like inference in the covering theory [Dachtheorie]. [End of cancelled passage]

INH 3 (D 92.3) 19.X. I seek a clarification to the questions: How does a formal proof of correctness resp. consistency differ from a contentful one, how is it that the former is not, in the case of certain ways of inference, (after Gödel) not even possible with the help of these ways themselves, is there then something like a bridging inference at hand with the contentful proof, how great is its certainty, how are the connections with the Gödelian proof, what is the role of the mathematical axioms? Preliminary considerations. It is best if I consider, to start with, a theory with just mathematical axioms that have to be formulated as ways of inference. Then CI in addition. Only then do I add &, ∨, further all, there is. Later then →. Finally ¬ intuitionistic, then REND. The theory with only mathematical axioms. Logicalsigns,&, ∨, → etc. must not appear. I formulate the mathematical axioms as ways of inference, for example: x = y x = yy= z y = x x = z 1=1 3 The text has what looks like a large typewriter style letter B, likely for Beseitigung, i.e., elimination. 5 Formal conception of correctness in arithmetic I 121

1 = 1 holds without conditions, therefore nothing can be concluded from it. So this is an axiom in a strict sense. The same with: x = xx =1. What is the role of substitution of numbers for variables? [Added: One can speak of axiom schemes and only after substitution by numbers of axioms. ] In itself, a theory without these is already thinkable, say with: 1=1 2=2 1=1 2=2 3=3 Now 3 = 3 is provable in this theory. Though, such theories are worthless. To get further, one needs the (all)variables, and so that even this theory would not be equally trivial, the substitution of numbers for variables and in the end also of variables for variables. Isn’t this already “logical inference,” just what I wanted to avoid? One can also say: The gaining of 2 = 2 from 1 = 1=1 1and2=2 is already logical inference! [Added: “Structural inference.” ] What remains is just that I avoid the logical signs, but not the logical inferences. Reasonably, one can introduce existential variables just like all-variables, equally without the use of logical signs. Then one can infer like this: 1=1, x = x a = a a = a . One has to pay attention in further execution on dependences, “function” concepts, variable conditions. Because the variables contain actually the concepts all and there exists, it appears natural to add as ﬁrst logical signs A and S [All and Sein,for universality and existence], even before &, ∨. To start with, I consider a theory without logical signs, but with free all- variables and a general rule of substitution. For example, with

x = y x = y 1=1 x =1 1= x x = y x = y The two predicates = and = don’t have anything to do with each other to start with, but if one considers them contradictories, this system of ways of inference resp. of axioms has the properties of completeness (there obtains for each pair of numbers = or =), as well as independence, which latter is not important for us. How are, now, “completeness” and “consistent in itself” to be deﬁned formally and

INH 4 (D 92.4) proved to be useful for this system, how does one further prove at all the consistency in the case of an arbitrary number of uses of the allowed ways of inference, resp. instead of consistency rather contentful correctness, i.e., the accordance of numerical results with the ones ﬁxed that are reachable right 122 Part III: The original writings at the beginning? Speciﬁcally, these proofs are to be formalized precisely and the inferences used to be determined.

20.X. Intuitive mathematics would not begin with the mentioned axiom system, but with the following one: I explain 1 = 1, 2=2, 3 = 3, etc. ν = ν ... as well as 1 =2 , 1 =3 ,... 2 =3 , 2 =4 ,... further 2 =1 , 3 =1 ,... 3 =2 , 4 =2 ,... Let in general = hold between diﬀerent number signs, = between the same. Thereby, then, the formal =, = has been reduced back to the intuitive sameness resp. diﬀerence. Thereupon one could put up the above axioms, and to prove them from these prescriptions. One cannot do so logically, because one does not have here any equivalent of intuitive sameness. (At most an axiomatically prescribable, by which, however, nothing in respect of sense is improved.) Completeness and consistency in itself result from themselves in the intuitive procedure, with intuitive sameness even from the same concepts. Consistency of inference on the contrary not. Here one is, rather, powerless and can only say: correct things must come out from correct inference. I consider again the formal procedure. The completeness of the above axioms can be taken as a theorem of the theory itself, x, y. x = y ∨ x = y. As such, it is to be proved by ∨, all, CI . Namely like this: 1 y. f = y ∨ f = y UE f = g ∨ f = g OE f = g f = g Ax Ax f = g f = g =1 OI OI f Ax f = g ∨ f = g f = g ∨ f = g 1 = OI 1=1Ax k Ax =1∨ =1 OI OI f f 1=1∨ 1 =1 1= ∨ 1 = CI k k CI f = h ∨ f = h 1=i ∨ 1 = i UI UI y. f = y ∨ f = y y. 1=y ∨ 1 = y CI1 y. x = y ∨ x = y UI x, y. x = y ∨ x = y

Note well that all five axioms were used. The consistency of the axioms, is not, however, provable in the theory itself. The concept of an immediate numerical consequence of the axioms is important. I can define as such roughly those consequences that are obtained through applications of the axiom-inferences that are connected to immediate number substitutions. (Constructive definition, not recursive!) Completeness and consistency of the numerical consequences can be summarized in the claims: For each pair of numbers a, b, exactly one of the two propositions a = b, a = b is an immediate consequence of the axioms. 5 Formal conception of correctness in arithmetic I 123

A formal proof for this: The derivation of an immediate numerical consequence has the form: a1 δ1 b1 a2 δ2 b2 . . aν δν bν Here a, b are numbers, δ =or=. ν number, ν ≥ 1. Single lines represent inferences by axioms, the uppermost formula is an axiom, the lowest a numerical consequence.

INH 5 (D 93.1) Let a, b be any two numbers.

To be shown that there is exactly one such derivation with aν = a, bν = b. First of all that there is one: CI on a.Fora =1:CI on b. 1 = 1 is a derivation with b =1.1= x gives a derivation for an arbitrary b. Finished. Let it have been proved for a, b. To be shown for a,b. CI on b.Fora, 1 x =1 gives at once a derivation. For a,b there arises from the derivation for a, b immediately a derivation through the application of one of the inference- axioms. Hereby we are ﬁnished. One sees that the proof is quite analogous to the one in the theory itself, 92.4 middle. Now the consistency, i.e., that there can be no derivation for a = b and at the same time for a = b. I assume there are two such derivations. I use a double CI that I can comfortably formulate as follows: Let a already be that constant number for which there is a b with this property, and b already the smallest one with this property, for a.Leta or b or both be equal to 1. Then the derivations can both consist of only one formula, because application of an axiom-inference gives always two numbers with . What follows trivial, 1=1,x =1 , 1 = x are consistent. Let then both a and b not be equal to 1. Say a = c,b = d.Thenthe derivation has in the end a mathematical inference, because mathematical axioms have always a 1. This mathematical inference could be only: c = d c = d c = d resp. so there were already derivations for c = d, c = d so with c

4 Gentzen’s footnote: Better: d

As elements of the metatheory there are the proofs of the theory. At first I need the “immediate inferences.” These are to be defined constructively as follows: 1 = 1 is a derivation, x =1 , 1 = x with an arbitrary number in them are derivations, from a derivation becomes through the addition of x δy,incasexδy is the last formula of the previous one, a new derivation. Formulas are figures of the form x = y, x = y with numbers for x, y. The definition is intuitive, not formal. Its formal conception is not difficult to produce.

The theory with mathematical axioms and CI. I deﬁne as “correct” purely formally: those numerical propositions that can be concluded from the axioms through immediate derivation. Further those propositions with free variables that result, with all substitutions of numbers for the variables, in correct numerical propositions. I call a proof with the initial propositions A1 ...Aν and the endproposition B correct if, whenever all those numerical substitutions in the A’s for which these result in correct numerical propositions, also B results in a correct numerical proposition under substitutions by the same numbers for the same variables, resp. arbitrary numbers for possible other variables. I prove now the correctness of all proofs through CI according to the number of inferences like this: A proof with just one formula is trivially correct.

INH 6 (D 93.2) Now I consider a proof that is, except for the last inference, already correct. If the last inference is a mathematical inference, the thing is in order. For if the premiss is correct under a substitution with numbers, then also the conclusion is correct under the substitution of the same numbers for the same variables. (Because an immediate derivation can be produced for it from the premiss, through the addition of the same inference.) Let now the last inference be a CI . To be shown that if the assumption formulas are correct under a substitution for the variables, also the CI result is. So let there be at hand any substitution for variables under which the assumption formulas are correct. Then P 1 is already correct. Phto be shown correct. Now h is already as a result of the substitution a number ν,or already from the beginning, resp. otherwise now to be substituted by an arbitrary number ν. Let this have happened. Then we have further: The right anterior part is by the inductive hypothesis a correct proof, so therefore, in case Px is correct under a substitution for a number for x,alsoPx is correct. Now I set for x 1tostartwith.P 1 is correct, therefore also P 2. Now I set 2 for x,etc.IntheendPν is correct, finished. I have naturally used a CI with the latter proof. 5 Formal conception of correctness in arithmetic I 125 a) The proof as a whole does not make the impression of a bridging inference. It seems to me to be a quite normal proof of the mathematical theory. Nevertheless, it seems to be of a nature different from the “purely formal” proofs of correctness. Such a proof would require a transfinite induction that could be at best bypassed in the previous case in an artificial way, say through an estimate [Abschätzung]. b) The above proof shows the correctness of the result of every proof through a double application of CI . How is that possible if the “proof” contains repeatedly CI ’s? So there should be after all a bridging inference at hand here. Otherwise one would have to assume that every correctness proposition canbeprovedthroughtwoCI ’s, which is nonsensical. This is still to be investigated. [Concerning] a) further to be investigated is: Does the above proof use inferences that are avoided in the “purely formal” proof, or how is the apparent difference in character otherwise to be explained? c) Whether the Gödelian result depends essentially on the consideration of consistency instead of correctness?? 21.X. c) About the last point: Consistency is a much more formal property than “correctness.” The latter requires, for its formal explanation, in the first place detailed definitions, separately for each logical sign. With suitable prescriptions about correctness, the craziest results of proofs could be “correct.” The proof of consistency states insofar much more than the proof of such “correctness.” A proof of correctness states something only when correctness is defined reasonably, i.e., in correspondence with the true sense of the signs. Maybe it is not quite so. Namely, if one defines the correctness of numerical results of proof reasonably, i.e., as a correspondence with what is obtainable immediately from the mathematical axioms, it is then in any case a quite valuable result to be able to show that only correct numerical propositions are provable. And this one can show only if one adapts a reasonable meaning for the logical signs. And this proposition, that numerical results of proofs are always “correct,” is moreover a rather simple formal proposition, similarly to consistency. The latter follows from it, as soon as 0 one has Heyting’s inference A, the former follows from the latter if every numerical proposition is immediately decidable from the axioms, as is usually the case.

INH 7 (D 93.3) Can the contentful correctness of numerical results, now, be formally proved for the whole of intuitionistic resp. classical (N2 +CI )? Where is the Gödel- point hiding? On 92.2 bottom, I could gain no clarity here. Best if I investigate first the situation with the above simple theories. 126 Part III: The original writings b) About the question whether each proposition is provable by CI use two times etc: The metatheoretical proof shows only the correctness of results of proofs according to formal definitions of “correctness.” For example, it shows for a numerical result such as 4 = 4: There is a formal derivation of the formula 4 = 4 from the axiom formulas. If, however, one wants to conclude that the proposition “4 is equal to 4” is contentfully correct, one has to make a bridging inference. Hereby this point is clarified. It is naturally most interesting that one can apparently sometimes carry through proofs more simply through a bridging inference. a) Now to the question: Through what does the purely formal proof of correctness differ from the semi-contentful? I place them one against the other, first for the theory without CI , with the all-sign and all-variables. A proof in this theory is a number of formulas that follow one another. A formula has, e.g., the form x. x = g. Numerical formulas are clear. Two formulas in a proof that follow each other form an inference, namely mathematical axiom inference5 or UI, UE. The initial formula is a mathematical axiom, i.e., 1 = 1 or x. x =1or x. 1 = x (resp. a =1or1 = a with whatever number for a). A normal proof for a numerical proposition consists of only numerical propositions and mathematical inferences. (= earlier: “immediate derivation.”) To be proved: There is to every proof of a numerical proposition a normal proof of the same proposition. Formalized: x.. Pr x & Np res x → Ey.Noy & res x = res y Proof Numerical Normal proof proposition result [numerical] -(function) Formal proof: Let there be given a proof with a numerical result. Pr x & Np res x. Let the length of the proof be ν. Abbreviation x= ν. x. Eμ x = μ is known Precisely: x= νEE I apply CI on ν. ν = 1 is trivial, for we have: Pr x & x=1. → No x Therefore No x & res x = res x AI Ey.Noy & res x = res y EI

5 Steps such as those Gentzen indicated by Ax in the big example derivation on page INH 4. 5 Formal conception of correctness in arithmetic I 127

Inductive inference: Let the claim have been proved for all ν .Tobe shown for . The assumption is x= . If the proof x contains no UI it is, if we assume it puriﬁed, already numerical. If the proof contains a UI ,lettheformulau be the conclusion of the lowest UI of the proof. It must, then, be shown by CI that there is such a UI . Now the UI is followed by a number-UE, one puts for the variable of the UI all over this number, then one leaves out the UI and has then a proof that is shorter than x by two formulas. The formalization would require a somewhat extended carrying through of the proof.

INH 8 (D 93.4) Contentful proof: I declare a formula with variables (free or bound) to be “correct” whenever, for all substitutions of numbers for the variables, there results a numerical formula for which there is a normal proof. Formally: Cox = Fo x & y..Foy & Sbx,y. → Ez.Noz & res z = y Formula y results from x by number substitution It will be proved now, by CI on the length of x,thateachproofx has as a result a correct formula. Claim: x. Pr x → Coresx This is trivial for length 1. Inductive inference: Let the claim have been proved for all ν .Tobe shown for . The proof has by the inductive hypothesis a correct result until the next to last formula. Then follows a mathematical inference or UI, UE.Inthecase of a mathematical inference, there is a normal proof of a formula obtained from the premiss through number substitution, and a normal proof can be produced at once for the formula that results from the conclusion through a number substitution, by the addition of the same mathematical inference. This holds for arbitrary number substitutions in the conclusion, so the latter is “correct.” In the case of UI ,variableUE, the correctness of the conclusion is immediately equivalent to the one of the premiss, in the case of number UE a trivial consequence of the correctness of the premiss. The essential diﬀerence in the comparison of both proofs seems to me to be: The semi-contentful proof makes a CI over a rather complicated proposition. It contains Coresx, and this predicate becomes in complicated cases ever more complicated. The formal proof makes a CI over Ey.Noy & res x = res y, even this a proposition with logical signs, but of a simpler nature, also in more complicated cases. 128 Part III: The original writings

It seems to me further that in the semicontentful proof, all logical inferences allowed in the formal theory appear contentfully in the eigenthreads of the CI . For example, if one allowed CI , there would appear here a CI .Inthe formal proof, instead, it does not seem to be so, or at least this is not really apparent. Whether there can appear with more complicated formal proofs also a CI in the eigenthreads of the main CI ? The Gödel-point lies perhaps therein that the property “correct” [left parenthesis added] is in the general case admittedly constructively defined, but not recursively, even not reducible to recursive properties. So it is [right parenthesis added] not at all definable in Gödel’s formalism! It seems to me that correctness can be defined formally for propositions to a maximal length, but not for an arbitrary length. It is different with the predicate “provable,” even this is naturally not recursive, but reducible to the recursive, namely Ex.xBew y as with Gödel. Can, then, the consistency of a system of arithmetic that allows constructive definitions be proved from itself? The constructive definition of a numerical predicate would look like this: The predicate holds for certain numbers, furthermore, one obtains in general from a number for which it holds, resp. several such numbers, by a constructive procedure fixed in advance a new number for which the predicate holds.

INH 9 (D 94.1) The constructive definition seems to be highly admissible. For it has in a trivial way the property of self-consistency, simply because one never admits that a predicate does not hold, but always only that it holds. In any case, the contradictory predicate cannot be defined in an analogous way, at least not in the general case. It is apparently provable only with difficulty that a predicate does not hold for a number. (It almost seems as if such were provable only metamathematically!) (True, one can say in the above case about “correctness” that 1 = 1 is not “correct,” but this prescription is metamathematical, true, one can demonstrate that there is no immediate derivation of 1 = 1, nevertheless, that a further constructive definition of correctness cannot even make arise 1 = 1 is, equally true, intuitively trivial because the propositions grow always longer, but it is not anymore a fact of the metatheory in which one works, but of its metatheory, of the meta- metatheory.) The constructive definition of correctness exceeds even the framework delineated above. The reason is that whether a predicate holds for a new element can depend on whether it holds for all elements of a certain type, and it can 5 Formal conception of correctness in arithmetic I 129 be even more complicated. With ¬, for example, it depends on whether the predicate fails to hold for another element! One could introduce “correctness of level ν,” something that applies to propositions with ν signs and is defined as follows: For propositions without logical signs: There is a normal proof for these, a formal definition. For propositions with ν signs: If the outermost is &, the proposition is ν-correct in case both of the single parts are already correct in their levels, etc. As already indicated, one can arrive at a determinate level given in advance along the formal way. But up to an arbitrary level only with the meta- metatheory! Compare transfinite induction! One will be able to show in the meta-metatheory: The metatheory can prove the correctness of proofs up to a length ν. It seems to me that there is another way in which the “correctness”-definition is not constructive in the above sense: One can define without worries also incorrectness, simply when the conditions for correctness are not satisfied. This goes through consistently, even if the later propositions grow always in length. That something is correct or incorrect would be a genuine, transfinite tertium non datur-proposition. I believe I can now see clearly why a consistency proof cannot be formalized through the giving of a coarse contentful meaning. To wit, because the meaning is not formalizable, and this naturally always: in the usual formalism, e.g., of Gödel. There remains of the questions on 92.3 top really only the one about the connection to Gödel’s work. Therefore, there is a need to see clearly why it is that not just a consistency proof is not possible to which meaning can be given coarsely in the formalism, but no proof at all. Even the role of complete induction (resp. transfinite induction) in this is still unclear. It should therefore be good to place against each other the formal and the contentful proofs of consistency, for the case of mathematical axioms + CI . The concept of “correctness” is here only in its beginnings as regards its complexity, because it is used in addition to numerical propositions only to ones with free variables. 22.X. Therefore: back to the theory with mathematical axioms + CI . The concept of correctness for numerical propositions, Ey.Noy & res y = z, is (notwithstanding the same symbol in this conception) recursive and decidable. For one can determine recursively in a well-known way, on the basis of the recursion scheme for the predicate in question (e.g., =), whether it holds (= is correct, normally provable) for given numbers. (So there was, with the formal proof on 93.3 bottom, only a CI on a decidable proposition.) The concept of correctness for arbitrary propositions with free variables, 130 Part III: The original writings

93.1 bottom, is not anymore decidable, even if it should be formalizable, namely: Cox = Fo x & y..Noy & Sbx,y. → Coy corresp. to 93.4 top Proposition numerical y results from x by correct for = Formula proposition number substitution numerical propositions All of the predicates Fo,No,Sb,Co are decidable.

INH 10 (D 94.2) Now I consider the proof of correctness of 93.1 bottom – 93.2 top. The concept of “correctness” of a proof is formalizable, even if not decidable. There is a CI applied over it. The inductive inference runs in the case of CI later essentially as follows: The proof for P 1 is correct, the proof of Px from Px is correct for arbitrary x. Now the correctness of Ph for an arbitrary number ν for h follows by CI on ν like this: P 1 is correct. If Pμ is correct, then by the right anterior part also Pμ. Finished. The CI is here over CoPν, a decidable proposition. It is, though, essential that it lies in the eigenthread of the main CI . (Because this uses essentially its assumption: P 1 is correct & the proof of Px from Px is “correct.”) (This should hang somehow together with the fact that the main CI goes over a proposition that is not decidable.) Now to the purely formal proof. It will be shown that when a proof has a numerical result, the result is correct. There will be a CI on the “simplicity” of the proof, a natural number associated to it. The CI proceeds over a decidable proposition. However, precisely stated it is the following: If the result is numerical, it is correct for all results up to index ν. This, now, is just finite so that decidability is maintained, for each fixed ν. When, however, simplicity is defined so that “all proofs up to the index ν” are not finite in number, and this is indeed the case with a transfinite assignment of values, the proposition is not anymore finite, and not anymore decidable! [Cancelled: If I give, say, to the occurrence of a CI the value ω,toaCI in the eigenthread of another ω2,etc,(only indicated.)] 23.X. Realization of the purely formal proof:6 One shows now through ordinary inferences, i.e., without CI :Thereisto

6 Cancelled: I assign values as indicated: A CI without further ones in its eigenthread counts ω,aCI with CI ’s in its eigenthread counts: the highest value of these CI ’s, times ω. This definition of values is recursive. The propositions under the last CI are counted with 1 for each. 5 Formal conception of correctness in arithmetic I 131 each proof with a numerical result a proof with a lower value and the same result. Namely, one shows existence of a peak, this peak can be reduced. The assignment of values follows according to 88.3 bottom ff. So, the value ofaproofisasystemoftransfinitenumbersoftheform:apolynomialinω with natural coefficients. (To be replaced by ωa1 ++ωaν .) The main inference can be seen as a transfinite induction over a decidable proposition, namely the proposition: The numerical result is correct. (That proposition is put as a condition with a single inductive inference: The condition holds for all proofs of a lesser value than a fixed one, belongs to the essence of transfinite induction and is not itself its proposition [the proposition over which the induction proceeds].) How does one arrive from the transfinite induction (in our special case) to the two-fold complete induction of the semi-contentful proof? (Consider the proposition: the result is, if numerical, correct for all proofs up to a certain value a. This proposition results from a decidable one through an all-sign instantiation. The passage to the above one seems not to be that simple.) It could perhaps be said more generally: The lowering of transfinite values takes place in our special case only in a very special, distinguished way. One could give on this basis a justification for the transformation of the transfinite induction into two usual inductions, over formalizable

INH 11 (D 94.3) propositions; Whereas such a transformation perhaps is not possible in general. A transformation into two complete inductions should instead be possible in general, in which however one proceeds over a non-formalizable proposition. For this is how the semi-contentful proof of correctness for quite arbitrary ways of inference proceeds. (It would naturally be also thinkable that even this transformation is possible only under a special form of the transfinite induction, one that perhaps is sufficient for a purely formal proof in such a case, but not with a completely wild transfinite induction.) There must obtain, in my opinion, some kind of a connection between the informal element in the non-formalizable definition of “correctness” and the non-formalizable (?) transfinite induction. For both of them seem to make possible a non-formalizable proof of consistency. To be noticed: The transfinite induction below ω2 can be transformed into two complete inductions, one of which goes over a generalized decidable proposition, in case the transfinite induction itself was over what is decidable. There seems to be something at hand here that corresponds precisely to the above case. If the above purely formal proof, as on 8, were realizable with an assignment of values below ω2, the analogy would be produced in full. 132 Part III: The original writings

[Added: It obtains already as it is, because also the transﬁnite induction below ωω is formalizable through two CI ’s.]

INH 12 (D 94.4) 31.X. G¨ottingen Attempt at a proof of consistency of REND under the condition of the contentful correctness of intuitionistic inference. Proof with one REND [for Reduktion von Negation Doppelt]. First case: Let the REND stand in the direct [part of a proof with no assumptions] and α ¬¬ contain no free variables. Then the proof can be written as follows: A A γ B Here α and γ are already in themselves single proofs. If B were inconsistent, A would not be correct by RA,so¬A, ¬¬A provable already without REND. This is a trivial reformulation of our old reduction. α ¬¬Pg Now instead when, say, REND contains a free variable: Pg γ Qg xQx δ B Were B a contradiction, then xQx intuitionistically false, ¬xQx proved. Let Ey ¬Qy hold, i.e., EE ¬Qf.Then¬Pf holds intuitionistically, but also at the same time ¬¬Pf. Contradiction, therefore ¬Ey ¬Qy. This together with ¬xQx should give an intuitionistic contradiction. Or not?7

INH 13 (D 95.1) 1.XI.32 The duality &, ∨, correct, false. Page 86.4 middle. The position of the inference H , there only obscurely surmised, can be justiﬁed fully through the duality: There can appear where a proof starts besides assumptions: Fundamentally correct propositions axioms. There can appear where a proof ends besides the end-proposition: Funda- mentally false propositions, anti-axioms, say like 1 =1 , 2 = 3. One could prohibit such beginnings and endings by allowing the following inferences: 7 The page is cut from here, with a last addition: Taken out ↓ “dual counterpart of the CI ,” does not belong here at all. 5 Formal conception of correctness in arithmetic I 133

One can infer things correct from anything (especially the conditions). One can infer everything (especially the end-proposition) from what is false. The latter corresponds to the inference H. The fact that there is at hand a proof, what is really a metatheoretical proposition, can be written down as follows: When A is a condition, B the end-proposition: A → B.Whenthere is no condition: B (B is correct.) When there is no end-proposition ¬A (A is false.) A proof without condition and without end-proposition would be a proof of a contradiction. One can write for B also 1 = 1 → B or C → B,for¬A: A → 1 =1or A → O. (It is perhaps better if I write for C, O instead R, F : correct, false. (Not W true, because W means already contradiction.))8 The RA appears by now as a dual counterpart of a normal proof. One obtains, proceeding ahead, from the positive axioms further correct propositions. One obtains, proceeding backwards, from the negative axioms further false propositions. The all- and exists-signs as well as ascending and descending CI (94.3) can be, naturally, incorporated in the duality. The position of the → sign (by which one can express and ¬) is of a special kind. It is a sign of a metatheoretical kind. Still one works in practice within the theory with the → sign and with → signs placed one above the other, which requires a precise justification. It is here that the difficulties with the proof of consistency begin. [Added: All superseded resp. cleared through the LK -calculus. ] Dual reversal of the proof. Each proposition is replaced by its dual opposite. (As when one shifts the negation inside.) So, one puts first ¬ in front and transforms then ¬.A&B into ¬A ∨¬B, ¬.A ∨ B into ¬A&¬B, ¬xPx into Ex¬Px,¬ExPx into x ¬Px, [cancelled: ¬.]A → B into ¬B →¬A . . . still unclear. ⏐ ⏐ ⏐ A & A → B ⏐¬A. ∨ .¬.A → B ← Counterpart to FE: B ¬B Justificati- on of this inference: probably not possible without RA. Maybe: ¬.¬B → ¬A? Yes, if one had ¬A ∨¬¬A! But does it hold intuitionistically? ⏐ R ⏐ A A ⏐ γ γ ⏐ Counterpart to FI: ⏐ B B ⏐ A → B → F A → B ¬A → B This must be carried through. Were there really to each proof a dual, a lot 8 Gentzen’s choice of letters, R for richtig and W for wahr, cannot be adapted to English. 134 Part III: The original writings would be revealed thereby about intuitionistic inference.

INH 14 (D 95.2) Precise investigation of the dual transformation of an intuitionistic proof. I consider first a proof without → , ¬, , CI .Ireplace&by∨, ∨ by &, all by there is and the other way around. To what point do I now have, proceeding upwards from the bottom, a correct proof? AI and OE, AE and OI , UI and EE, UE and EI have gone over reciprocally into each other. I shall assume, something I consider very probable, that the variable conditions are in order. Then, apparently, all inferences are in order. Now the initial and end-propositions. The old condition is obviously, in the dual form into which it has been transformed, now the result, the old result in dual form the condition. But how is it with the axiomatically true initial propositions and false end-propositions? To keep things in order in the first place with numerical propositions, I must add still the following change: The predicates in the elementary propositions of the whole proof are replaced by their contradictories, e.g., 2=3 by 2 =3 , (x.)x>1by(Ex.)x ≯ 1, etc. The dual proposition is by now, after this reshaping, precisely the contrary of the proposition as formed according to HA, with negation shifted innermost. If I request that the contrary so built to an axiomatically true (resp. false) proposition be determined likewise as axiomatically false (resp. true), the new proof is fully in order. If the old proof was, for example, an RA of ExPx, the new one is a proof of x ¬Px.WereitanRA of xPx, the new one would be a proof of Ex.¬Px. The former can be done also by an intuitionistic inference, the latter not. Here one would have a very interesting intuitionistic fact that, however, has a value only if an extension to → and CI succeeds. Incorporation of CI with the help of the dual counterpart seems not difficult. The → .Forthesign→ , FI and FE, a dual counterpart seems to be lacking. I tried to create such artificially on the bottom of the previous page, with the help of the other signs. Let us assume for once a sign as a dual to → . Then, the following B A & A → B inferences should hold for it: A ∨ .A B that corresponds to B A γ as well as correspondingly ΔA ΔB Here Δ denotes the B : ΔB A → B Δγ ΔA dual operation. I.e, C D But how can a single proposition deliver a D γ C 5 Formal conception of correctness in arithmetic I 135

A γ whole proof? Maybe one prefers to look at B as being separated out, a thing by which the justification of the correctness of A → B is originally given, and possibly with a dual justification the falsity of C D. (As on the previous page.) A → γ Just as A B is the proposition that is correct on the basis of B,soshall D γ C D be the proposition that is false on the basis of C .Thequestion, now, is, if the sign can be explained by the other signs, wherein both ways of inference would be intuitionistically clean. The forms that can be considered are ¬.D → C, ¬.¬D ∨ C, D & ¬C, and others that are classically equivalent.9 D The greater difficult seems to lie in making the inference C ∨ .C D into a correct one. For this inference to be correct, must be as weak as possible. To become refuted, it has to be as strong as possible. One asks oneself if there is a compromise.

INH 15 (D 95.3) D 2.XI. The inference C ∨ .C D leads to the explanation of C D:If C holds, it must be false. If nothing is known about whether C holds (not: if C does not hold, because it gives intuitionistically a tertium), D must be claimed. A → B means analogously: If A holds, B must be claimed. If nothing is known about whether A holds, nothing must be claimed.10 D γ How is it, now, with the elimination of C D by C ? A contradiction must follow here. If C holds, one has it at once, because C D is then already false. If nothing is known about whether C holds, then D holds. By that further C. Thereby one has again the contradiction. It seems still on the whole somewhat particular. The main question is whether the “explanation” of C D is really intuitionistically acceptable. One can also ask oneself if the elimination of C D really brings back that from which C D arose initially, something that is to be required of constructi- D CC D ve inferences. This should be the case. For D is correct, one can δ C D δ “reduce” it to C . 9 Cancelled: The proof by A → B would, it seems, refute all (?) of these. 10 Added above after “nothing must”: A → B must be correct (?). 136 Part III: The original writings

According to this, the symbol seems to be correct. It would be very pleasant if it could be expressed with the other signs, because only that would make possible a secure foundation. [Cancelled: ¬C & .. C → O. → D] It doesn’t seem to me to be the case

Consideration of the special case: ¬ in place of → . ¬ A α A is introduced through a proof: α (as A through A ) Eliminated through A ¬A or ¬A ¬¬A ! Dual reversal: 1 =1 Were A = xPx, for example, the dual would be: ExP/x. ¬A has to be transformed, let us say, into a ExP/xsuch that 1. either this or the previous holds, 2. from the latter, together with α , that is, with a proof of ExP/x ExP/x, follows O. ¬ExP/x would accomplish this classically, but then the former inference would be a REND! One would read ∼B intuitionistically like this: If B holds, it must be false. If nothing is known about whether B holds, it must be correct. I.e., it must state: Nothing is known about whether B holds. That is an allusive negation β and it seems intuitionistically incorrect. Nevertheless, together with B it gives : Because B holds, ∼B is false. Therefore O. It seems now to me as if REND were necessary for a full realization of the dual transformation. Then it loses its value, because it was meant to help justify precisely REND.Alsothesign becomes herewith quite problematic. For the latter case is just a special case of the above.

INH 16 (D 95.4) I have to investigate the contentful correctness (resp. consistency) of intuitionistic mathematics. The (transfinite) negation is superfluous, because substitutable through → . I see as appropriate divisions of this theory: in the first place &, ∨, all, there is. Then CI in addition. Then in the end → . The CI is a, so to say, metatheoretical inference that is added. The → is even a metatheoretical sign that is not mixed in the theory itself. I see herein the ground for problems with the sign → . These problems should be cleared if possible. The proof of consistency, especially, has to be always considered through a contentful interpretation and assessed for its value. Especially whether there is here a particular leap with → .

3.XI The → plays a special role in the deﬁnition of correctness, because correctness is always reduced to the correctness of smaller propositions with the other signs. This does not happen with → . The correctness of A → B can be conceived as the existence of a proof of B from A. However, there is a circle in this conception once the proof operates in its turn with → . 5 Formal conception of correctness in arithmetic I 137

Maybe one has to do a recursion of a theory to one closest below (of which the former is the metatheory). Cf. 92.2 middle. More or less like this: we have first a theory without → .Nowweexplain → → A 1 B as follows: There is a proof in the old theory, i.e., without , AA→ B for B from A. The inference B is perfectly in order, it seems. → Then we can explain further: let A 2 B mean: there is a proof (possibly) → with 1 for B from A. Etc. One can agree in the end to leave out the index everywhere. But can this be done without difficulty? What conditions obtain for the indices? Next. With an FI , it is required that the index of the new → is greater than all the indices within the frontland. Nothing is required with an FE. Is it possible to effect such a numbering within an arbitrary proof? Related symbols must naturally have the same index. It seems to be difficult only when an → is related to one within the front-part of an FI . This should, however, not be possible. One more observation: On the basis → of the meaning of A ν B, namely that there is a proof of B from A in the ν − 1th theory, one can say that B is now proved even in the ν − 1th theory, provided that only this theory occurs in the proof of A, not disturbed by → → the occurrence of ν in A ν B. However, it seems that this possibility is not needed at all. (?) See below ↓ The situation here seems to me not yet quite clear. One could generalize this type of definition of correctness: A thing is declared as correct if there is a proof of a lower order for it. If one said simply: has a proof, then one could not revoke this definition in the justification of the forms of inference, because the definition is already a part of the “proof.” Though it should be possible to give a foundation for the forms of inference in this way. I presume there is a close connection between this procedure and the one of the “reduction of the longest sign.” The longest sign has the highest level, the steps of inference in its neighbor are of the highest level. They are justified by adding as already correct those of lower levels (shorter propositions). This is to be carried through precisely. Finally, one has to be careful with CI that would be an obstacle in the reduction of the longest sign. A difference: a proof of lower order need not be a proof with shorter propositions throughout. (Cf. above with → ). Finally, to secure correctness, I need something like: for a numerical result of an arbitrary proof, such as 1 = 1&2 = 2, it must be the case that this & is chosen to be of the lowest level, i.e., that both parts are provable without logical signs. I need for this the possibility of lowering the level in the way it obtains above for FE for example. (Let us consider 138 Part III: The original writings

INH 17 (D 96.1) the following meaning: Correctness means that there is a proof in which there occur only shorter propositions (and which ends with the I of the outermost sign of the proposition). I must set aside CI now. I have to show now that each inference leads from correct propositions again to correct propositions. It goes like this: it is clear with an I , for one puts before the premisses their “short-proofs,” that exist by assumption, and has hereby a short-proof for the conclusion. The procedure is the same with an AA→ B → E.SayFE: B . Since there is in front of A B its short-proof, there is an FI directly before: A γ B A → B Nothing comes of it yet, for A can be longer than B.)

DB137 The analogy to 87.1, though, should be producible somehow. And then one has to investigate how far one comes with correctness deﬁni- tionsofthetype:“Thereisaproofofacertainkindfortheproposition.” Such deﬁnitions seem to suit the intuitionistic standpoint.

4.XI A noticeable difference between a formal and a semi-contentful proof : As is known, there occur in the reduction of CI and → certain pieces repeatedly, next to each other, one under the other, etc. There would be considerable difficulties in this connection in the formal proof, because the pieces need not be reducible at the same time, but can depend on each other. In the semi-contentful proof, instead, one infers simply: The single piece is as a smaller one already correct, hence its composition is “correct” in whatever multiplicity. With CI for example, the ν-fold writing one under the other of Px . . Px. The formal proof requires with CI a transfinite assignment of values. It is impossible to see the assignment as natural. Therefore the formal proof appears to necessarily be unnatural. What does it depend on? Iconsiderthe essence of the purely-formal proof . Especially what inferences it contains. The main thing about it is that for a proof with a numerical result, a normal proof of the same result is associated. (The transformation of a proof of 5 Formal conception of correctness in arithmetic I 139 contradiction in a shortest possible proof of contradiction is essentially the same.) This association must happen in an intuitionistically unobjectionable way. I ask what the intuitionistic inferences are that are needed. Now, the association is defined through a system of mathematical axioms for them. Like recursively. As far as it goes in general, which however would not be the case. So the association itself needs no inferences. Neither presumably any logical signs, if one allows beside mathematical axioms also mathematical inference schemes. But: That the definition is allowed, when for example it is recursive, has to be justified somehow through inferences! Let us take for example the case: to each proof is associated the one that arises through the reduction of a given peak. Let’s presume that the latter is always smaller. Then one infers by CI ; that a fully reduced proof can be associated, namely the one that remains when the reduction is made over and over again. In formulas, say: x = red y, x is the proof that arises from y through reduction. It is prescribed that the following holds: x = red y → le x < le y. le = length of the proof. Here the sign → occurs. It should be avoidable by giving simply a proof x = red y . . le x < le y What inferences does this proof use? Maybe just mathematical. Namely, because red and le are explained mathematically recursively. To present the association of a minimal proof to x, nothing seems to remain over except CI [cancelled: and the existence sign!] Like this: Claim: (x)(Ey)No z & res x = res y The & should be avoidable

INH 18 (D 96.2) by a recursive introduction of the whole predicate. The there is: [small circled arrow points down (to the cancelled passage below?)] The proof goes now more or less like on 93.3 bottom. [Cancelled: It is essential that the existence sign be used. The reason is that one has to prove metatheoretically the existence of a certain function, (namely the function that associates to each proof of a numerical result a normal proof of the same result).] Certainty depends even with completely normal recursive definitions, in fact, on a metatheoretical consideration in which the concept of existence plays arole. The existence sign above should, though, not be necessary. [small circled ar- 140 Part III: The original writings row points down] Because, as already remarked above, the whole existential proposition is recursively definable. The pure-formal proof is particularly intuitive. Therein should lie the advantage one so willingly grants to it. And therein should also lie that one sees it as a foundation of the forms of inference, the latter conceived to be formal. Whereas one has the feeling of a tautological foundation with the semi-formal proof .Eventherecursive concepts that lie at the basis of the purely-formal proof are intuitive. Therein lies the ground that they are seen assosecure,eveniftheirlogical foundation is by no means simple. But is it, now, reasonable to reduce formal inference to intuition? Here intuition is, in the end, of a dubitable nature. Also the feeling of the correctness of formal inference that one commonly has depends on intuition, but here on the intuitiveness of the inferences themselves, of their sense, not on intuitive properties of the formal proof as they are considered in a purely-formal proof of consistency. ? This difference should be the basis for the following: No purely-formal consistency proof will succeed with the same intuitiveness as the correctness of the forms of inference, seen contentfully, possesses. So, the justification of intuitionistic inference by a formal proof of correctness is actually senseless. There is in it admittedly a reduction of something that is in itself in a way intuitively clear into another intuitiveness the formal justification of which in turn is not simpler than that of the formalism to be justified. So, there is no gain either intuitively or formally. ? Things may be differently with REND. Perhaps an investigation of the position of the intuitive, recursive etc definition of predicates in a recursive theory (arithmetic), in relation to the contestability of forms of inference, could be quite in place. The result should be that that definitions that are intuitively very clear get justified by inferences that are not at all simple. Anyway, supposedly the transfinite ND is not used. All intuitionistic inferences, instead. (And meta-inferences.) So, the proof of consistency of the REND cannot be carried through in practice in the way it was planned so far, in the purely-formal way. One would obtain at the same the correctness of the intuitionistic inferences on this way that, as was said, is not at all appropriate for such. It would be more reasonable to assume both. Or, say, to assume the definition of correctness in the intuitionistic sense as an allowed definition, which makes possible without difficulty a proof of correctness of the intuitionistic inferences. On 94.4 there was an attempt in the first-mentioned direction that 5 Formal conception of correctness in arithmetic I 141

INH 19 (D 96.3) however did not lead so far to the objective. There are, though, other possibilities on this way.

5.XI The theory with only mathematical axioms, UI, UE, CI, and only single-threaded proofs. Single-threadedness shall state: The mathematical axiom inferences are with one premiss. CI is to be written as: P 1 Px . . Px Ph so that there is formally one thread. I want to consider with this theory once more, side by side, the semi- contentful and the purely-formal proof of correctness. Especially, to see if the latter can be brought closer to the former in naturalness, or if it appears always artificial. Semi-contentful proof of correctness: the definition of correct goes as on 93.1 bottom, about. There are, though, bound universal variables in addition. I have to modify the definition of the correctness of an indirect part of proof as follows: Px γ Px is correct if (because of the free variables, it would be wrong to have: if from the correctness of Px as a proposition that of Px follows.) for every substitution of numbers for the free variable of Px, such that the proposition Px is correct (wherein the latter correctness is not decidable, because there can appear bound universal variables), also the proposition Px is correct, as long as one substitutes for the same variables the same numbers, (for other free variables arbitrary numbers ∼∼ this is superfluous because of the definition of correctness of a proposition ∼∼). I ask myself now if this definition of correctness is still formalizable. This should be the case. Though, one has to distinguish between correctnesses, but since the number of levels is finite, it works. The first correctness is that of numerical propositions. The second of arbitrary propositions, in their definition the first is involved. The third is the correctness of proofs, herein the second correctness is involved. The first is decidable. The second not any more, and the third not at all. In the third definition, there appear all- and nested above it → -signs 142 Part III: The original writings over decidable propositions. (Exactly as in the formal expression for the correctness of a proof, namely: Px → Px.) (An all-sign over the whole because of free variables.) The proof of correctness proceeds now easily with the help of the definition. Cf. 93.2 top [added: 93.4] and 94.2 top. The CI in the justification of the “CI ” is over the correctness of the second level, that is, over a proposition that is not within the decidable ones. (In contrast to before when no bound universal variables occurred together with CI .) Thereby it occurs in the eigenthread of the main CI . Now to the purely-formal proof. Cf. 94.2 middle. The aim is: to simplify a given proof with a numerical result step by step, until there is a numerical proof for the same result. The simplifications are served by reductions of all-hillocks and CI ’s. The CI -reduction has, it seems, to necessarily take place as usual. The problem is to see simplifications in it. I would like to get along with an assignment of values that is as simple as possible. Single-threadedness makes the thing easier. The following conception seems possible: One thinks of each indirect part one below the other ω times. With nested indirects quite analogously. Thereby the thing is already intuitively pretty clear!? One infers like this: one considers in the first place the CI ’s of the whole proof that stand one below the other, not nested. The conclusions of which, then, all belong to the direct. If one succeeds in showing that the lowest of these together with all CI ’s contained in its indirect part can be reduced, one is (through a CI )

INH 20 (D 96.4) already finished. Now the reductions grab always the lowest CI ,thatis pleasant to know. Now there is still to prove that a single CI in the presence of others [other CI ’s ?] can in itself be completely reduced. It stands firm that the number of nestings is lowered by one through its first reduction. There is after this reduction again a system of CI ’s partly not nested. It is clear that I come to the goal through a transfinite assignment of values. However, no result with a finite number of CI ’s would be in sight. Therefore I will try a proof with a finite number of CI ’s through the use of the fact that one grabs always the lowest CI .

6.XI.32 I doubt that it works. The following method of proof should be the most intuitive: One uses the 5 Formal conception of correctness in arithmetic I 143 property: “A proof is transformed through a finite number of prescribed reductions into a single proposition (resp. similar, say a normal proof).” How is its with its formalizability? Formalizable is naturally: a proof arises from another through one reduction.However,howisitwith:aproofarises from another through ν reductions? This is an iteration of the previous. Supposedly this is still, by von Neumann, Gödel, still formalizable. Then naturally also: there is a ν such that a proof is resolved through ν reductions. I.e., the proposition considered. I didn’t believe in its formalizability on 61.2– 3. To wit, because of the iteration. The thing should be naturally investigated before a decision is taken. But I don’t care about it right now. One arrives now to the goal with the said property: One shows that it obtains for all proofs through a double CI , namely, according to the highest number of nested occurrences of CI in the proof and the overall number of their occurrences. Cf. 54.1 middle. The very first procedure. Is there, now, an analogy between the property of “correctness” and of the “finite reducibility,” as well as between the two inductions at hand that take place over these properties in the two proofs? correctness uses transfinite “for all,” reducibility transfinite “there is.” The first-mentioned seems very probable. For one can define for proofs with an arbitrary end-result something like finite reducibility. Thereby the result is also, as an end-peak, reduced in a known way. It is, now, of decisive importance to know: Can the notion of finite reducibility give us a replacement for the notion of correctness, i.e., make possible something like a proof that stands between the semi-intuitionistic and the purely-formal one? [Cancelled from here on] I call a proof reducible if it collapses into a proposition after a finite number of reductions. Herein also initial and end-peak A γ reductions are allowed as reductions. Is it now the case that if B and A γ B B γ γ C are reducible, also C is reducible, similarly to correctness? One should think Yes! For only such reductions take place as are anyway in a single proof, even possibly in both at the same time, if reduction takes place in the middle (B in the first place). And the effect of such a middle reductions is for such a single proof as for a previous boundary reduction. Just that certain earlier liberties that were sometimes at hand disappear now and become necessary, makes no difference, because the notion of reducible shall state: reducible, however one proceeds with the remaining cases. Is here the key to the whole proof of consistency? To the above: If in a reduction in a lower part the upper one is multiplied, 144 Part III: The original writings so what? Does it have any effect? Further: attaching to itself, etc? The idea of the above treatment of CI is to be carried over to such cases. Second attached sheet: RED1. [The pages of RED1–2, at the same time pages of D97.1–2, are missing. The text continues in the middle of a sentence of page RED3, i.e., D97.3.]

D 97.3 (RED 3) ...andtherebyallthatisessential. One must, though, not deﬁne the eigenland of a free variable as its domain of occurrence. For this can reach in, say, the front-piece of a FI . Then the whole of it has to be counted. Further, the question presents itself if one has to count to the eigenland of a FI the whole front-piece, not only the eigenthreads. This seems necessary to me. For a inner FI -orUI -eigenland, for example, could very well reach into an area that does not belong to the eigenthreads. FI forms indeed an insurmountable border for all eigenlands the eigeninference of which is in the anterior part of the FI .

xAx EyBy Aa Ba AI Aa & Ba

It is, instead impossible to require the same of a UI, as the above [neben- stehende in original] example shows. One could anyway count its eigenland only up to the EE. This should be the best solution. It should be analogous with an OE,andinfact:IfitisfollowedbyaUE, its eigenland must ﬁnish here. In the remaining threads, if they end with an axiomatically false proposition and there is otherwise nothing on the way, it must reach to this [proposition]. It should be very simple to formulate the proof of existence of peaks through the consideration of eigenlands. For the consideration of “simpliﬁcations,” the nesting of eigenlands seems to me to be very useful. Cf. the consideration of the nesting of CI’s, i.e., of their eigenlands, on 96.4. Possibly there settles in also with the other reductions a multiplication of the next to each other instead of a multiplication of the nesting. [In margin: The nesting of eigenlands seems to me to be the decisive structure of the proof.] An important piece of knowledge that leads to the sequent form. To cover also other cases 5 Formal conception of correctness in arithmetic I 145

INH 20.1 (D 97.4)(RED 4) one could equip the → sign in contact propositions with a higher assignment. 8.XI.32 Contrast between correctness and reducibility in the case: mathematical axioms + UI, UE, CI, see 96.3–4: INH 19–20 In the case considered, a CI finds place. Say above P 1=yA1y. One infers through correctness: P 1 is correct (i.e., for all y, A1y of the first level is correct), further the proof of Px from Px,soalsoPν for any ν,byCI.The correctness of indirect proof [here: proof under assumptions] states: For all x,ifPx of the second level is correct, also Px of the second level is correct. That is, for all x:ifforally, Axy of the first level is correct, then so also for all y, Axy of the first level os correct. With reducibility one has: The proof of P 1 is reducible, the one for Px from Px also. The proof as a whole has to be shown reducible for arbitrary ν. One could want to use CI, analogously to the previous case, when one has: The proof of P 1 is reducible, and one has to show: If the proof for Px is reducible, then also the one for Px, for arbitrary x. It would suffice to have: If the proof for Px is reducible, and also the one from Px to Px, then the proof for Px composed from the two is equally reducible. However, precisely here lies the difficulty. With correctness, there would correspond to this point: if Px is correct, and the proof of Px from Px is correct, then Px is correct. Here the same thing appears to be completely trivial. It seems to me, in any case, that the difficulty has nothing to do with CI, but it lies in the nature of the implication, so the → sign. It should be possible to remove easily the recognized difficulty in single-threaded proofs, through a formal investigation. The same seems to be true in general for proofs without →. More or less with the considerations at the bottom of the previous page. It would, further, be interesting if one could succeed with still formalizable means. Therefore the question concerns mainly the nature of the definition of reducibility. 10.XI [Cancelled: ((Theory with meta-axioms, &, ∨, (), (E ), →, cf. 87.1 92.2 middle. 95.4–96.1 top.))] 146 Part III: The original writings

6. Investigations into Logical Inference

by Gerhard Gentzen.

Delimitation of the field of research: The object of the following considerations is logical inference, and I shall specifically restrict myself to the domain of what is known as the “lower functional calculus” (cf. Hilbert-Ackermann, Die Grundzuge¨ d. Theoret. Lo- gik, cited in the following as H.-A.). To this calculus are eventually added specific mathematical axioms and ways of inference (for example those of number theory).

§1. Short explanation of some ways of notation. I shall distinguish between (cf. H.A. pages 51–52): Object variables, variables for short. E.g. a,b,c... Objects. Specific objects, e.g. 1, 2, 3 ... Predicate variables, e.g. A( ,,),G(),P. Predicates. (In H.A.: logical functions) Specific predicates, e.g. <,= (I take the propositional variables of H.A. as predicate variables without argument places.) Inductive definition of a proposition (“expression” in H.A.): 1. A predicate the argument places of which have been filled in with objects is a proposition. (I shall call it specifically: an atomic proposition.) 2. From 2 propositions a new one is formed through composition with & , ∨, → . 3. From a proposition a new one is formed through the prefixing of ¬ (negation), or of (x)resp.(Ex): Here x is a variable that occurred free in the proposition (i.e., not yet in an all- or exists-sign). Parentheses resp. points are used in the known manner to ensure that the construction can be seen in a unique way. Example: (x)... x < 3 → ..¬ x =4. ∨ (Ey) y>a I shall call outermost the last logical sign &, ∨, → , (), (E) applied in the construction of a proposition according to 2.,3. Each proposition that is not an atomic proposition has exactly one outermost sign. The propositions that appear in the construction of a proposition according to 1,2,3, the proposition itself included, are its subpropositions. I shall call propositions without specific objects and predicates logical propositions. In the parts I and II that follow, only logical propositions are treated. 6 Investigations into Logical Inference 147

ULS 2

I Part. The inferential calculus N1I. §2. Foundation and setting up of the calculus N1I. The way logical inference is formalized in Russell, Hilbert, Heyting (for intuitionistic inference) among others is rather far apart from the way of inference as it is practised in reality (say, in number-theoretic proofs). Notable formal advantages are won thereby. I shall for once set up a formalism (“calculus N1I ”) that comes as close as possible to actual inference. One can assume that such a calculus has certain advantages and I think I can maintain, on the basis of my further results, that this is the case. Whole sequences of inferences are often passed by as they have become self-evident through long acquaintance. This shall not be repeated in the formalization that follows, but each individual step of a “proof” shall be represented. The point is to think out the basic components of logical inference. To illustrate what I understand by “natural” inference, I shall give next 3 examples of correct logical propositions together with a way through which one would in reality conclude their correctness. 1. Example: X ∨ Y.& .X ∨ Z.. → ..X ∨ .Y & Z (H.A. formula 20.) One would argue contentfully as follows: Let X ∨ Y and X ∨ Z hold. The ﬁrst one admits the two possibilities: X holds resp. Y holds. If X holds, then X ∨ .Y & Z follows at once. if Y holds, there remain because of X ∨ Z still the 2 possibilities X holds resp. Z holds. If X holds, then X ∨.Y & Z follows as before. If Z holds, then Y & Z holds, therefore as well X ∨.Y & Z. Hereby the above proposition is seen as correct. (Moreover, in an intuitionistically acceptable way, as will be the case until further notice.) The above train of thought could be represented formally as follows: Assumption: X ∨ Y.& .X ∨ Z X ∨ Y Division into cases: XY X ∨ .Y & ZX∨ Z XZ2. division into cases Y & Z X ∨ .Y & ZX∨ .Y & Z Result: X ∨ Y.& .X ∨ Z.. → ..X ∨ .Y & Z This will be, in a slightly modiﬁed form (see p. 7.), a proof in the calculus N1I to be set up. 148 Part III: The original writings

ULS 3 2. Example: (Ex)(y)Fxy. → .(y)(Ex)Fxy (H.A. formula 36.) The contentful argument goes as follows: Let there be an x such that Fxy holds for all y.Leta be such an x. Then for all y: Fay.Letnowg be an arbitrary object. Then Fag holds. So there is an x, (namely a) such that Fxg holds. g was arbitrary, so this holds for all objects, i.e.: There is for all y an x such that Fxy holds. Hereby the claim follows. Formal representation of this train of thought: Assumption: (Ex)(y)Fxy, (y)Fay Fag (Ex)Fxg (y)(Ex)Fxy Result: (Ex)(y)Fxy. → .(y)(Ex)Fxy 3. Example: ¬(Ex)Fx. → .(y)¬Fy is to be shown intuitionistically correct. One will reason as follows: it is assumed that there be no x for which Fx holds, and from this it shall be concluded: ¬Fy holds for all y.Letnowg be any object for which Fg holds. It then follows: There is an x for which Fx holds (namely g is one such). This contradicts the assumption ¬(Ex)Fx. Therefore Fg does not hold. Here g was completely arbitrary, i.e.: ¬Fy holds for all y. This was to be shown. Formal representation of this train of thought. Assumption:1 ¬(Ex)Fx further assumption: Fg (Ex)Fx ————————— ¬Fg (y)¬Fy Result: ¬(Ex)Fx → (y)¬Fy

1A freely drawn arrow places the assumption as the ﬁrst premiss above the inference line. 6 Investigations into Logical Inference 149

Proofs of the kind made in these three examples shall now ﬁnd their place in an exactly deﬁned calculus. (I am certainly aware that also the proofs of these propositions in the H.A.-formalism arose, at bottom, from such “natural insights of correctness.” I shall be concerned in §9 exactly with the transformation of N1I- proofs into proofs of that kind.)

§3 Setting-up of the calculus N1I. A calculus for “natural intuitionistic proofs” shall now be given. It will be shown in part II that this calculus is equivalent to what is the standard formalism so far and that I shall designate as “logistic.” The most essential external diﬀerence with “logistic” proofs is the following:

ULS 4 The correct propositions are derived in the latter from a series of logical axioms through a few ways of inference. Natural deduction, in contrast, does not in general start from logical axioms, but from assumptions, (see the 3 examples above). Inference proceeds further from these. The result is made independent of the assumptions through a later inference. – These preliminary remarks will make the following deﬁnitions appear meaningful. Presentation of an N1I-proof. (Examples in §4.) A proof consists of a number of propositions in an arrangement in tree form. So it looks as follows, for example:

A1 A2 A6 A11 A3 A4 A5 A10 A7 A8 A9

(I deviate somewhat from actual reasoning by the requirement of an arrangement in tree form, because 1. there is in actual reasoning a linear succession of propositions and 2. it is common in continuing further to use repeatedly a result already won, whereas the tree form allows to use in continuation only once a proposition already proved. Both deviations are obviously inessential, but they make instead easier to conceive the notion of a proof.) Each of the horizontal lines represents an inference. The proposition directly below the line is the conclusion of the inference, and those directly above the line its premisses. There can further belong assumptions to an inference, these are certain uppermost propositions of the proof that stand above the conclusion of the inference (indicated as belonging to the inference 150 Part III: The original writings through a common numbering). (I.e. not necessarily directly above, but in general: uppermost propositions of “proof threads” that pass through the inference line. All uppermost propositions of a proof are assumptions, and each belongs to exactly one inference. All propositions that stand under an assumption (i.e., in the proof thread determined by the assumption), but still above the inference line to which the assumption belongs, are said to depend on the assumption. The assumption itself included (so the inference makes the propositions following it, beginning with its conclusion, independent of the assumption in question.)

ULS 5 Presentation of the allowed inferences. An inference is formed from one of the following inference schemes through the substitution of arbitrary propositions for A, B, C, and for (x)Px resp. (Ex)Px a proposition with an all- or exists-sign as the outermost sign, for Pa the same proposition as for Px, though with a substitution of a variable a in place of x. (The same variable can be substituted for x and a,anda variable that occurred already elsewhere in Px can also be substituted for a. This, however, will be excluded for the inferences UI and EE through the “variable condition” that follows below, but it remains allowed for UE and EI .) The proposition that stands under the line denotes the conclusion of the inference, and the propositions that stand directly above it its premisses; The propositions that stand in square brackets above some of the latter have the following signiﬁcance: Propositions of this form can be assigned as assumptions to the inference in an arbitrary number, even not at all, (but all of the same form). They have to be then uppermost propositions of the proof above the pertinent premiss (comp. p. 4) resp. to be the premiss itself. The inference schemes. The nomenclature AI etc. shall mean: And- (A), Or- (O), Follows- (F ), All- (U ) and Exists- (E)-sign- “introduction” (I ) resp. “elimination” (E); further T : tautological inference, R: reductio ad absurdum, V : completion inference [Vervollst¨andigungsschluss]. More details below (§5.) AI AE OI OE [A] [B] AB A & B A & B A B A ∨ B C C A & B A B A ∨ B A ∨ B C ————————————————————————————————— 6 Investigations into Logical Inference 151

UI UE EI EE [Pa] Pa (x)Px Pa (Ex)Px C (x)Px Pa (Ex)Px C ————————————————————————————————— FI FE T R V 2 [A] [A] [A] B AA→ B A B ¬B A ¬A A → B B A ¬A B —————————————————————————————————

The inferences that arise from the schemes UI and EE have to satisfy the following “variable condition”: The variable a of the inference UI resp. EE must not occur in any other proposition of the inference except in Pa,and furthermore [cancelled in margin: Also for the inference FI a restriction is needed.]

ULS 6 in no assumption the conclusion of the inference depends on. The definition of a proof is hereby finished. [Cancelled: (Another, essentially equivalent version of the variable condition is the following: The variable a of UI must occur only above and in the premiss of the UI, otherwise nowhere in the whole proof; the variable a of the EE must occur only above the premiss C of the EE, otherwise nowhere in the whole proof.)] Ishallgiveclarifications to the inference schemes, by explaining for some of them their contentful meaning. I try to make it evident that the calculus really presents “natural deduction.” FI : This inference says in words: If B has been proved through the use of the assumption A, also the following holds, this time without the assumption: From A follows B. It makes no difference if A has been used for the derivation of B repeatedly or even not at all. – There can have naturally been further assumptions from which then also A → B still remains dependent. OE:Ifwehave:A ∨ B, and a result C follows from the assumption that A holds, and the result C follows also from the assumption that B holds, then C holds in any case, i.e., this time independently of the two assumptions. (Again, naturally not independently of eventual further assumptions made under the derivation of the 3 conditions.) UI :IfPa has been proved for an “arbitrary” a,then(x)Px follows. The condition that a be “completely arbitrary” can be expressed precisely as follows: Pa must not depend on any assumptions that contain the variable

2 T, R,andV have been crossed out. Margin gives rule V in the form A and states: R remains. 152 Part III: The original writings a. And this is precisely the part of the above “variable condition” that is pertinent to the inference UI . EE: One has (Ex)Px. Then one says: Let a be such an object for which P holds, i.e., one assumes: Let Pa hold. If one has proved on this basis a proposition C that does not contain a anymore and that does not depend on any assumption that contains a,thenC is provable independently of the assumption Pa. Hereby is expressed precisely the part of the “variable condition” pertinent to EE. (There is a certain analogy between OE and EE.) ULS 73 The remaining inferences should be easy to explain.4

§4. Writing of the 3 examples from §2 as proofs in the calculus N1I . 1 2 3 2 Y Z AI 1.) ∨ & ∨ & 3 1 X Y. .X Z AE X OI Y Z OI ∨ & ∨ ∨ ∨ & ∨ & X Y. .X Z AE X X Z X .Y Z X .Y Z OE 2 ∨ ∨ & ∨ & X Y X .Y Z X .Y Z OE 1 ∨ & X .Y Z FI 3 X ∨ Y.& .X ∨ Z.. → ..X ∨ .Y & Z

(The arrangement in tree form should appear somewhat artiﬁcial with this example, as, for example, the successive distinction into cases X, Y after X ∨ Y has been established, goes lost in it.) 2.) 1 (y)Fay UE Fag EI 2 (Ex)Fxg UI (Ex)(y)Fxy (y)(Ex)Fxy EE 1 (y)(Ex)Fxy FI 2 (Ex)(y)Fxy. → .(y)(Ex)Fxy

(Here, also, the assumption of the EE would in a linear arrangement follow naturally after the left premiss, as happened above in the example in §2.) 1 3.) Fg 2 EI (Ex)Fx ¬(Ex)Fx R 1 ¬Fg UI (y)¬Fy FI 2 ¬(Ex)Fx. → .(y)¬Fy

3 The following has been cancelled from the top of p. 7: R: If from an assumption A follows on the one hand B, and on the other ¬B,thenA is wrong, i.e., ¬A holds. 4Added in Bernays’ hand: V :seebelow. 6 Investigations into Logical Inference 153

§5 Considerations on the calculus N1I.

The calculus has several formal defects. The following advantages stand against them: 1. The far-reaching ﬁt with actual inference, in fact also my starting point above. 2. Proofs of correct logical propositions are in this calculus almost throughout shorter (and truly closer at hand) than those in the logistic calculi. This shorter length depends essentially on the fact that one and the same proposition appears (as a part of other propositions) usually a number of times in logistic proofs, whereas this happens only to a small extent in the calculus N1I . (Comp. the transformation of N1I -proofs into logistic proofs that follows further below.) 3. The notation applied above for the single inferences (AI ,etc.)makes us recognize the presence of a remarkable order. To each of the logical signs & , ∨, → , (), (E)

ULS 8 (I shall consider negation later separately) belongs exactly one inference that “introduces” the sign (as an outermost sign of a proposition), and one that “eliminates” it; The doubling of AE and OI presents a purely external deviation without consequence. The “introductions” present, so to say, the “deﬁnitions” of the signs in question, and the “eliminations” are actually just consequences thereof, expressed more or less as follows: In the elimination of a sign, the proposition the outermost sign of which is in question, must “be used only as what it means on the basis of the introduction of this sign.” An example will clarify what is meant hereby: The proposition A → B could be introduced when a derivation of B from the assumption A was at hand. If one now wants to use A → B further with the elimination of the sign → (uses for the construction of longer propositions, such as A → B. ∨ C (OI ), are naturally also possible), one can do it straightaway so that one concludes at once B from A that has been proved (FE). For A → B documents the existence of a derivation of B from A. Note well: It is not necessary to rely on a “contentful sense” of the sign → . I think one could show, by making precise this idea, that the E-inferences are, through certain conditions, unique consequences of the respective I - inferences. I shall limit myself to the indication a consequence of this connection that can be established purely formally. It will form the basis of later investigations into decidability and consistency. It goes as follows: If in an N1I -proof an introduction (I ) of a sign is followed immediately by its elimination (E), the proposition with the sign in question (as its 154 Part III: The original writings outermost sign) can be eliminated through a simple “reduction” of the proof. These reductions proceed after the following schemes: (α,β,...,ε,ζ denote the further lines of the proof, in a way that can be easily seen. Square brackets mean that the respective part of the proof is to be written as many times as there occurred the respective assumption before the reduction.) ULS 9 & ∨ α β α A [B] A B A [ ] α γ δ A A B AI A ∨ B OI C C & α γ A AE A C OE C ε becomes: ε ε into: ε (it is quite analogous with the other form of AE resp. OI .) () (E ) a α( ) α [Pb] Pa Pa b α UI EI γ( ) Pa (x)Px b (Ex)Px C a UE α( ) γ( ) Pb Pb C EE C ε into: ε ε into: ε →

[A] γ B α α FI A A A → B γ B FE B ε into: ε It requires some considerations to see to it that a correct proof is in fact produced in each case. I shall refrain from the exact realization, because I will not make any use of these facts, but rather present them for the sake of intuitiveness. — [Cancelled: The inference T is trivial, (it can be left out altogether).] The negation. (Inferences R and V.)5 This falls out somewhat from the frame of the rest of the logical signs. A negation can be “introduced” only if there is a negated proposition already at hand. The reductio ad absurdum (R) can be seen as a simultaneous introduction of a ¬ (namely of ¬A) and an elimination of another (of ¬B). Namely, if R appears twice in succession (and moreover so that the conclusion of the 1. right is the premiss of the 2.), a “reduction” with the

5 A marginal note indicates for the rest of this §:tobemodiﬁed. 6 Investigations into Logical Inference 155 elimination of the middle proposition is as well possible after the following scheme: ⎡ ⎤ ⎡ ⎤ 2 ⎢ 2 ⎥ ⎢ ⎥ 1 2 1 2 ⎢ C ⎥ ⎢ C ⎥ , , ⎣ ⎦ ⎣ ⎦ 2 A C A C 3 ζ 3 ζ C C γ δ A , C A , ζ B ¬B γ δ A ¬A R1 B ¬B 2 3 ¬C R ¬C R ε becomes: ε

Inference V can be seen as a further “negation elimination.” If a ¬ is introduced through R and then immediately eliminated through V, the proposition in question can

ULS 10 well be reduced, after the following scheme: A [A] [ ] α α γ δ A A ¬B α B γ δ A ¬A R B ¬B C V C V ε becomes: ε

One can “deﬁne” negation (as until now, only the intuitionistic negation is meant) in the following way, say: One introduces asinglefalse proposition F and explains ¬A as: A → F . Now the inference R canbeexpressedas follows with the help of FE and FI : 1 1 1 1 [A] [A] [A] [A] γ δ B B → F γ δ FE B ¬B F R1 FI 1 In place of ¬A enters: A → F

F Inference V can now be replaced by B. Its correctness does not follow from the “definition” but must be specifically postulated. (It can be justified contentfully as follows, say: Since F is a false proposition, it cannot in any case do any harm to stipulate that every proposition follows from it.)

§6. The “tertium non datur.” If I add to the calculus N1I the logical axiom A∨¬A (tertium non datur), a complete classical calculus “N1K ” is obtained. (As concerns “completeness,” see §7.) When a logical axiom is allowed, a small modiﬁcation is required in the deﬁnition of “proof” of §3: I must now allow as uppermost propositions of a proof, in addition to assumptions, also propositions of the form A ∨¬A. 156 Part III: The original writings

It appears to me that the special role of the tertium non datur, among all the other ways of inference, stands out especially in this calculus. Not, however, because the tertium non datur is, in the chosen presentation, the only “logical axiom,” because one could on the one hand take in its place ¬¬A an inference,say A (analogously to Hilbert and Heyting), and on the other hand also replace the inferences of the calculus N1I in part by logical axioms. Anyway, the point of view of “naturalness” should direct us here, and I have chosen from this point the formalization that has

ULS 11 taken place. Let us therefore disregard this external distinction, let us consider in place ¬¬A of the “axiom of excluded middle” simply the inference A ,say.Thislies obviously outside the frame of N1I -inferences, because it presents a new negation elimination. Its permissibility does not appear in any way from the kind of negation introduction by inference R, say in a way that would make possible a “reduction” of the above kind. Namely, a double negation would be “introduced” by R as follows:

[¬A] [¬A] γ δ B ¬B ¬¬A

¬¬A If one infers further A , it cannot be seen how an elimination of the proposition ¬¬A through reduction could be possible. The considerations in this and the preceding § can perhaps be of use in giving a positive characterization of intuitionistic inferences.

—————– Summary of my further results.

II. A proof of the equivalence of the calculus N1I and the “logistic calculi” of intuitionistic inference in Hilbert, Heyting, Glivenko.6 (Here also the construction of mediating calculi.) III. By “arithmetic” is understood the theory of natural numbers, say on the basis of the axioms of Peano (that is, with complete induction), it will then be proved: If intuitionistic arithmetic (that is, arithmetic with the ways of inference of the lower functional calculus) is consistent, also classical arithmetic is

6Margin has: In the dissertation Part II. 6 Investigations into Logical Inference 157 consistent. (I.e., the previous with the tertium non datur added.) (The proof is intuitionistic.)7 Further: To every arithmetic proposition (e.g.,anytheoremofnumber theory) A, a proposition A can be given such that A is

ULS 12 classically provable (by the means of arithmetic) if and only if A is intui- tionisticaly provable. One can, so to say, “interpret intuitionistically” the theorems of arithmetic. IV. [Cancelled: Conjectured] theorem: If a logical proposition is provable in the calculus N1I, there is a proof for it in which only subpropositions of it appear, eventually with different variables.8 The proof for this is finished.9 V. The consistency of arithmetic will be proved; The concept of an infinite sequence of natural numbers will be used in it, further in one place the law of excluded middle. So the proof is not intuitionistic. Maybe the tertium non datur can still be removed.

7Margin has in Bernays’ hand: Even finite in a strict sense. Eckart Menzler-Trott has pointed out that the handwriting of this marginal note is that of Bernays. Page 33 of this manuscript is a list of corrections to the withdrawn publication Gentzen (1933). The second-to-last sentence of its introduction reads: Die Beweise dieser und der ubrigen¨ Sätze werden intuitionistisch gefuhrt.¨ A correction indicates that the original text was: intuitionistisch (sogar “finit”) gefuhrt.¨ Thus, Bernays had taken over Gentzen’s original wording, which shows that he had read the manuscript part that later became the paper Gentzen (1933). 8Margin has: For the dissertation Part III. 9The following until point V cancelled: not yet finished except for a partial calculus N2I that maintains from N1I only the inferences AI,AE,UI,UE,R, but gains importance by the following fact: The question of the provability of a logical proposition in the classical “lower functional calculus” can be reduced [cancelled in margin: i.e., is equivalent to] the question of the provability of (another) logical proposition in the calculus N1I . (This goes together with III)] 158 Part III: The original writings

Dissertation, III Chapter.10

G. Gentzen.

Overview: We shall show that each NI-derivation can be brought into a rather simple form. It will have, e.g., the following property: Each formula in the derivation is a subformula of the endformula, eventually with different individual variables. The basic idea of the transformation is: In an arbitrary derivation, any formulas can appear in the first place. One considers now, among those formulas that are not subformulas of the endformula, a longest one. It will be, in general, the lower formula of an “introduction” of its outermost sign and the main upper formula of an “elimination” of the same sign. But then it can be subsequently eliminated through a reduction. (Comp. I Chapter §5.) Certain exceptions (with the inferences OE, EE, V ) require a special treatment, but the situation is not essentially different.

§1. Some new notations. As concerns an NI -derivation we define:11 1. 1. For the inference figures except V : 1. 1 1. One formula is the main formula,namelyA & B with AI, AE, A∨B with OI, OE,...(Ex)Dx with EI, EE.12 1. 1 2. There appear, further, subformulas of these, possibly with different variables, these are A, B, Dy in the schemes. We shall call each such formula a neighbor of the main formula (and conversely, symmetrically.) 1. 1 3. There appear further, with OE and EE,formulasC.Weshallsay that the lower formula C is bonded with both of the two upper formulas

10Top of page has: 13 IV Sheet. Right corner has in a small box: I. Only odd page numbers appear. Margin has: Changes of notation against Part I. new: old: NI = N1I Derivation, =Proof Proof ﬁgure⎫ ⎬ Formula, = Proposition Propositional formula,⎭ Expression 11 The manuscript has an elaborate numerical notation system, especially for cases in the normalization proof. These are in the form of marginal additions that have been reworked into the text in a way identical to the additions in Gentzen (1933) and other papers of the time. 12 The margin continues the list of new and old notations, with ⊃ for →, D for P,and y for a. 6 Investigations into Logical Inference 159

C (with OE), resp. with the one in EE. (A symmetric relation.) The two upper formulas in OE are,then,notbondedwitheachother. 1. 2. A bond is a sequence of (formally equal) proposition formulas that stand one under another in a derivation thread, such that each formula is bonded with the one standing under it (and therefore also with the one above it) but so that the bond cannot be continued in the same way either upwards or downwards. 1. 2 1. (The concept of a “bond” serves as a generalization of the concept of “a formula in a derivation.” Namely, such systems of C’s can be considered in practice as one formula.) We let a bond consist of just one formula. Then we have: Each proposition formula of a derivation belongs to a bond. We shall apply also to bonds

ULS 14 certain of modes speech such as subformula, outermost sign, and mean then an arbitrary formula of the bond. We shall designate the uppermost formula of a bond as its representant. The latter determines the bond uniquely. (The downmost would not, because of OE.) Two bonds are neighbors if one formula of one bond is a neighbor of one formula of the other bond. Of two bonds that are neighbors, one is a subformula of the other, possibly with diﬀerent variables. (For this holds in general for formulas that are neighbors.) 1. 3. A hillock 13 is a bond the neighboring bonds of which are subformulas of it (eventually with diﬀerent variables). The grade of a formula is the number of logical signs in it. A hillock can also be characterized in the following way: Its neighbors have a lesser grade. A bond that has the greatest grade found in a derivation is always a hillock. Therefore there is no derivation without a hillock. An inner hillock is a hillock to which the endformula does not belong. 1. 4. The main theorem [Hauptsatz] of this chapter is the following hillock theorem: An arbitrary NI -derivation, the endformula of which does not contain any free variables, can be transformed into an NI -derivation of the same formula without inner hillocks. (In this case the bond of the endformula is the only hillock.) All further results of this chapter, e.g., the more intuitive subformula theorem (that

13 Added: Corresponds more or less to the notion of a “peak,” the latter is more specific. 160 Part III: The original writings says less) become easy consequences of the main theorem. 1. 5. The concept of a “main hillock.”14 We shall define a relation of “higher” between certain formulas of a derivation: When two formulas stand in the same thread, the one that is upper is higher. If in an inference figure the main formula is an upper formula, and if there are further upper formulas (OE, FE, EE), the latter, as well as everything that stands above them, are higher than the main formula as well as everything that stands above it. We carry over the relation higher into bonds by calling a bond higher than another one if its representant is higher than the representant of the other bond.

ULS 15 A main hillock is an inner hillock such that there is (in the derivation) no inner hillock of a greater grade, and no higher one of the same grade. Each derivation that has an inner hillock has also a main hillock. (Because the relation of higher has no cycles.)

§2. Proof of the hillock theorem. It is sufficient to show: An NI -derivation with an inner hillock can be so transformed, with the endformula maintained, that a main hillock is eliminated and thereby the number of inner hillocks of the greatest grade decreased, without the appearance of inner hillocks of a greater grade. Let us, then, choose a specific main hillock for reduction. The transformation of the derivation happens in four steps. The fourth step gives the reduction proper. The others are preparatory to it. 2. 1. The first step. Renaming of individual variables. (Purification of the derivation.) 2. 1 1. The following shall be achieved: 2. 1 1 1. Each individual variable shall occur as only bound or only free. 2. 1 1 2. Each individual variable shall be the eigenvariable15 (y)ofatmost one inference figure (UI or EE). 2. 1 1 3. The eigenvariable of a UI resp. EE must appear only above the inference line of the UI resp. only above the upper formula C of the EE. (We call this part of the derivation an “anterior part of the UI resp. EE.”)

14 Marginal remark: The hillock to be reduced ﬁrst is one. 15 Marginal remark gives old notation a, new notation y. 6 Investigations into Logical Inference 161

2. 1 1. All of this is achieved through the following renaming: 2. 1 2 1. If an individual variable occurs in the derivation both as free and as bound, it will be substituted everywhere where it occurs free by one and the same variable that does not occur yet anywhere. 2. 1 2 2. The eigenvariable of a UI resp. EE will be substituted in the whole anterior part of this inference by a variable that does not occur yet anywhere. 2. 1 3. We have to convince ourselves now that 2. 1 3 1. The derivation has turned into a derivation with the same endformula. 2. 1 3 2. The number of inner hillocks of the greatest grade has not been multiplied and no new inner hillock of the greatest grade has appeared.

ULS 16 2. 1 3 3. The above three [one in 1 and two in 2] requirements are satisfied in the new derivation. 2. 1 3 4. About 2. 1 3 1: We divide this question into three parts and shall later always lay the same division as a basis: The following three requirements are to be satisfied for a figure to represent a derivation with a prescribed endformula. 1. The figure has tree form and has the right endformula, the “inference figures” are correct, i.e., producible from the inference schemes through a permissible instantiation. (We shall not count the assumption formulas that are associated to inference figures as parts of the figures.) 2. The uppermost formulas of a figure can be associated to the inference figures as assumption formulas in such a way that each uppermost formula belongs to exactly one inference figure and has the form and position that is allowed by the schemes. 3. The variable condition is satisfied in the figure. The variable condition is now (different from earlier): The eigenvariable of a UI or EE must appear among the formulas of an inference figure only in the formula Dy, and in no assumption formula in the anterior part (2.113) of the inference figure on which the lower formula of the inference figure depends. 2. 1 4. Working-out of 2. 1 3 1–2. 1 3 3: 2. 1 4 1. (2. 1 3 1.) Division as already said: 2. 1 4 1. 1. The endformula is obviously unchanged, because it could not contain any free variables. The inference figures are changed, repeatedly, in at most the way that a variable is substituted through a new variable all over (in the inference figure), or all over in places in which it occurred free. 162 Part III: The original writings

(This holds because of the variable condition also for a UI resp. EE the eigenvariable of which was renamed.) All inference figures remain correct here. 2. 1 4 1. 2. The association of assumption formulas to the inference figures is left as it was. All things are now in order, save that it is established that the same substitution finds place in an assumption formula as in its inference figure. This could be doubted in renaming 2. 1 2 2 in the following case: The assumption formula contains an eigenvariable of a UI resp. EE and stands in its anterior part, whereas the inference figure concerned is neither in this part, nor is it the UI resp. EE itself. But then it has to stand below the inference line of the UI resp. EE, i.e., its lower formula depends on the said assumption and we have a contradiction with the variable conditions, 2. part.

ULS 17 2. 1 4 1. 3. The variable condition is trivially satisfied in the new inference figure, because the new eigenvariable of a UI resp. EE occurs only in such places in which the old one already occurred.16 2. 1 4 2. (2. 1 3 2) Neighboring bonded formulas are always affected by the same substitution, each bond has become a bond, each hillock a hillock of the same grade, each non-hillock a non-hillock. 2. 1 4 3. (2. 1 3 3) Conditions 2. 1 1 1 and 2. 1 1 3 are obviously satisfied in the new derivation. 2. 1 1 2 remains: Let there be a variable in a formula that is the eigenvariable of two different inference figures. Both inference lines stand below this formula because of 2. 1 1 3. We consider the lower of the two inference figures. If it were a UI , the eigenvariable would appear in its upper formula, contrary to 2. 1 1 3 applied to the upper inference figure. If it were an EE, the eigenvariable would appear in an associated assumption formula, contrary to either 2. 1 1 3 or to the variable condition applied in both cases to the upper inference figure. 2. 2. Second step. This vanishes in the case that the main hillock is already a bond that consists of a single formula (which is the normal case). In the case that it is not, the second step serves to diminish the main hillock by one formula. We consider the two lowest formulas of the hillock, and let both have the form C. Thus, the derivation looks as follows: (We keep an eye on the arrangement of the upper formulas of an inference figure, not necessarily on

16 Right corner of page has in a small box: II. 6 Investigations into Logical Inference 163 the inference schemes.)

Γ Γ Π |C Δ |C Δ C IEE IOE |C Θ |C Θ D II D II Λ resp.: Λ In these, the two formulas C with a stroke denote the said formulas, D the lower formula of the inference ﬁgure the upper formula of which is the lower of the two. Γ, Δ, Θ, Λ, Π represent the remaining parts of the derivations. D (These can be empty.) For example, Λ denotes the part that consists of all formulas that do not stand above D, Γ the part that consists of all formulas above the C in question, Δ and Θ denote possible further upper formulas

ULS 18 that belong to the inference ﬁgure, together with whatever stands above them. The derivation is now transformed into the following form:

Γ Γ Π |C Θ |C Θ C Θ (II) (II) (II) D Δ I D Δ D I D (EE) D (OE) Λ resp.: Λ 2. 2 1 We have to show again that (comp. 2. 1 3) 2. 2 1 1. the derivation turns into a derivation with the same endformula. 2. 2 1 2. the number of inner hillocks of the greatest grade does not increase and none of a greater grade appear. 2. 2 1 3. the bond of C’s remains a main hillock and is diminished by one formula. 2. 2 2 1 (2. 2 1 1) Division as in 2. 1 3: 2. 2 2 1. 1. The new figure has tree form and has the old endformula. How is it with the inference figures? II is not changed by the transformation, it just appears possibly twice. In I , D has taken the place of C, together with an instance of EE resp. OE. All of the other inference figures have remained completely unaltered, though those in Θ can appear twice after the transformation. 2. 2 2 1. 2. The uppermost formulas remain associated to the same inference figures as assumption formulas as before. In the case that the part Θ was doubled, the same associations that held earlier within Θ are naturally kept within each part. 164 Part III: The original writings

The form of assumption formulas is, now as before, suitable to their inference figures, because neither has been changed. (The change of C into D in I changes obviously nothing here.) It must be studied carefully that their position relative to each other satisfies the requirements. D If the inference figure belongs to Λ, its assumption formulas that stand above D can have changed places and been multiplied (Θ), but this doesn’t harm anything because they stand now as before above D and are uppermost formulas, and nothing is prescribed of their number and further arrangement. Γ Π If the inference figure belongs to C, C,Δ, Θ,

ULS 19 the position of the assumption formulas has remained unchanged. There remain the inference figures I and II to be considered. Assumption formulas that belong to inference figure I stand in Γ and Π, therefore clearly also after the transformation in a correct relation to the inference figure. Possible assumption formulas associated to II can, as we shall soon show, stand only in Θ. Therefore also these stand in the right place after the transformation. The following shows that there cannot be any associated assumption formulas above the upper formula C of II : There can be assumption formulas, on the basis of the inference schemes, over the upper formula C of the inference figures OE, EE as well as over the C of a FI . C cannot be of the first kind, because it is the lowest formula of a bond. (Otherwise the bond would be extensible further down.) C cannot be of the second kind, because it belongs to a hillock, whereas the B of a FI is a neighbor of the A ⊃ B and A ⊃ B is not a subformula of B. 2. 2 2 1. 3. The variable condition is manifestly satisfied for every UI resp. Γ Π EE that belong to one of the parts C, C,Δ, Θ. For it states then something about only the conditions within this part, and these have remained completely unchanged. The variable condition remains satisfied also for each UI resp. EE that belongs to the part D.(ForD depends only on assumption formulas of the same form as before.) Inference figure II cannot be a UI , because C would then be a neighboring subformula (possibly with a different variable) of D, against the definition of a hillock. It can be, instead, an EE. Then Θ is its anterior part (because D and C are not allowed to be bonded). The variable condition is now obviously satisfied, because inference figure II together with its anterior part Θ appears in the new figure completely unchanged. Also inference figure I requires some attention in case it is an EE.True, its anterior part Δ remains unchanged, but it would be thinkable that the 6 Investigations into Logical Inference 165 eigenvariable of the EE appeared in D. This is, however, impossible by 2.113. 2. 2 2 2. (2. 2 1 2) Is it possible that there appear inner hillocks that have the same or greater grade than C? As of next we consider those formulas of the above schemes that are Γ designated by C, D.TheC of the part C belongs to the main hillock. (see Π 2.223.)IftheC of C belongs to a hillock after the change, this was the case Π also before. (Because the neighborhood relations in C and Π have

ULS 20 not been changed, and there were none for C in I .) Thus, there is no new hillock here. How is it with D? C is, as the lowest formula of a hillock in Π, the main formula. C as a lower formula can be either17 aneighborofB,and consequently a subformula of C, so of a lesser grade, or else (if II is one of OE, EE) bonded with the lowest formula of Θ. Now, Θ is just the part that was possibly doubled in the change. I.e.: If there appeared, through a transformation, at all a hillock of the same or greater grade than that of C, its representative formula (1. 2 1) must belong to Θ. (The parts Γ, Δ, Π, Θ contain after the change the same hillocks that were there before.) And there must have been a hillock of the same grade in the same place in Θ already before the change. (It is seen similarly as for C that this holds also for D.) We must now take into consideration the fact that the bond C was a main hillock. By the deﬁnition of “higher,” all formulas of Θ were (before the transformation) higher than C and the formulas above C,forC was, as noticed, the main formula of inference ﬁgure II . Thus, had there occurred in Θ before the change a representative formula of a hillock of the same or greater grade as that of C, the bond of C would not have been a main hillock and we would have a contradiction. 2. 2 2 3. (2. 2 1 3) We have already established that there is after the change no hillock of a greater grade than that of C. Before the change, the entire part Θ as well as parts of Γ and Λ were higher than C. (though not Δ and Π.) After the change it is exactly the same parts. (With the doubling of Θ only one part Θ.) And no formula can be a representative of a hillock in these parts if it was not such already earlier. (comp. 2. 2 2 2.) Therefore the formula C has in fact remained a main hillock and further manifestly diminished by one formula. 2. 2 3. There is at the end of step 2 [2. 2] still a repetition of step 1 (2. 1). Observe that what was achieved by step 2 (2. 2 1 3) remains (on the basis of

17[Margin has in a clumsy hand: Case FE?] 166 Part III: The original writings

2.142).(2.211and2.212remainnaturally,becauseof2.131and2.132.)

ULS 21 2. 3. Third step.18 If after performing step 2 the main hillock still consists of more than one formula, step 2 is repeated until it consists of one single formula. It is essential for this that the properties (2. 1 1) of the derivation, achieved through step 1 on the basis of 2. 2 1, are maintained each time. 2. 4. Fourth step. The proper reduction of the main hillock. Now the main hillock consists of one formula in the derivation. It is the lower formula of an inference figure and the upper formula of another inference figure. For: It cannot be the endformula (as an inner hillock) and it cannot be an assumption formula, because each assumption formula is a neighbor of the main formula of the associated inference figure, and a subformula of it (possibly with a different variable). We have further: The hillock is a main formula in both of the inference figures to which it belongs, provided that the inference figure is not V , for which there is not even any definition of a concept of main formula. (Because there appear in an inference figure, except for the main formula and except for V , only subformulas that are neighbors of the main formula, and formulas (C in OE, EE) that are bonded with others.) We shall next distinguish the two cases: 1. The main hillock does not belong to any inference figure V. 2. The main hillock does belong to an inference figure V. 2. 4 1. First case. By what was said before, the main hillock is the lower formula and upper formula of two inference figures, and in both cases the main formula. Then obviously exactly the following five cases are possible: The two inference figures are AI, AE; OI, OE; FI, FE; UI, UE;orEI, EE. Thus, the outermost sign of the hillock is eliminated right after its introduction. It is for this reason that the hillock can be removed, and this happens after the following five reduction schemes, corresponding to the five cases: (Only one possibility of the two in AE and OI is treated, the other one is cleared in a completely analogous way) We lay down for each case the form of the given derivation and next to it the new form that appears through reduction.

18 Right upper corner of page has in a small box: III. 6 Investigations into Logical Inference 167

ULS 22

& ∨⊃ [A]

Γ Δ Γ [A] [B] Θ Γ A B A Γ B A Γ Θ Λ A Γ |A & B A |A ∨ B C C Θ A |A ⊃ B Θ C B A Ω C B Ω Ω Ω Ω Ω ————————————————————————————————— () (E )

y Γ( ) Γ [Dy] Dy Γ(z) Dz Θ(y) Γ | x Dx Dz Dz ( ) |(Ex)Dx C z Ω Θ( ) Dz C C Ω Ω Ω 19The formula with a stroke denotes the hillock. Γ, Δ, Θ, Λ, Ωdenotethe [A] Θ other parts of the derivations. They can be empty. C, for example, means: C together with all the formulas that stand above it, [A] denotes assumption formulas of the form A that belong to the inference figure the upper formula of which is C. (It is possible that C itself is such a formula, then Θ is empty and C identical with [A].) Γ In a scheme for a modified derivation, A means that the part Γ is to be written above each of the mentioned assumption formulas A,sothatΓ appears as many times as the assumption formula appeared (possibly not at all). Where there stands Γ(y) in a scheme of the old derivation, and Γ(z)in the new one, the following is meant (and the same with Θ(y), Θ(z)): Γ(z) denotes that part of derivation that is obtained from Γ(y)wheneverywhere in Γ(y)thevariabley is substituted by z. We have to show that 1. the new figure represents again a derivation, with the same endformula. 2. the number of hillocks of the greatest grade diminishes in the reduction and no hillocks of a greater grade appear. 2. 4 1 1. Division as in 2. 1 3 4. 2. 4 1 1. 1. We have a tree form. The endformula is unchanged (also if Ω is empty).

19 Margin has: These correspond to precisely the schemes in the I chapter, §5. 168 Part III: The original writings

The following holds for the inference ﬁgures: They are in general taken over from the old derivation. An exception is Γ(z) made, though, by the inference ﬁgures in Dz with universal reduction and [Dz] Θ(z) in C with existence

ULS 23 reduction. These have come from the inference figures of the old derivation through a substitution of z for y.ItwasspecifiedtobesoforΓ, Θ, it holds for Dz on the following basis: By the rule for substitution in the inference schemes, I chapter, both of Dy and Dz come from Dx through the substitution of y resp. z for x. By the variable condition, y does not occur in Dx,(z can occur there). It follows that one obtains Dz through subsitution of z for y in Dy. — Our claim holds as well for C, because y does not occur in C on the basis of the variable condition. An inference figure remains in general correct if a variable is substituted by another one that did not occur bound in it previously. (This holds because of 2. 1 1 1.) 2. 4 1 1. 2. The assumption formulas. [B] Δ Λ 2. 4 1 1. 2 1) B is left out in & -reduction, as well as C in ∨-reduction, and as a consequence certain assumption formulas are left out. This, however, has no effect. Γ Γ(y) Γ 2. 4 1 1. 2 2) The assumption formulas in A, Dy , Dz, belong (in all reductions in which there is a part denoted by Γ) to inference figures that A C B Dz stand in this part itself or in Ω resp. Ω, Ω, Ω . They are associated after the reduction to “the same” inference figures. This is clear and in order if Γ(y) we have case 2. (Except for universal reduction in which Dy is modified through the substitution of z. But we have, because of the variable conditi- Γ(y) on: An assumption formula in Dy , the inference figure of which belongs to Dz Ω , cannot contain y. Therefore it is not at all affected by the substitution by z.) Γ Γ(y) Γ In case 1, that is, when the inference figure belongs to A, Dy , Dz itself, the matter is in order with &- and universal reduction. (A substitution of z for y in an assumption formula and its inference figure does no harm, because z did not occur bound (2. 1 1 1).) 6 Investigations into Logical Inference 169

With &, ∨, and existence reduction, a multiplication (or also a disappearance) of Γ can occur. The association of assumptions is ﬁxed in each part in itself in the way it was done earlier in the single part. [A] [A] Θ Θ 2. 4 1 1. 2 3) We consider now assumption formulas in the parts C , B , [Dy] Θ(y) C of the old derivation.

ULS 24 These are, in the first place, those assumption formulas that belong to the inference figures OE, FI, EE removed by the reduction. These were denoted by [A], [D] in the schemes. The parts Γ are attached to these in the reduction, so they have not become, say, assumption formulas without associated inference figures. Further assumption formulas in the parts considered can belong to infe- C B rence figures in the parts themselves or to Ωresp.Ω. The association can in both cases be carried over without further ado to the new figure. Existence reduction requires some attention, because y is here substituted by z.Butit suffices to present the same considerations as with universal reduction under 2. 4 1 1. 22. 2. 4 1 1. 2 4) Now there remain still those assumption formulas to be considered that belong to Ω. Also their inference figures must belong to Ω, and the associations are obviously kept under the reduction. 2. 4 1 1. 3. The variable conditions. 2. 4 1 1. 3 1) Let us consider next A, O and F -reductions. There appear in the new figure only such inference figures as there appeared already in the old figure. Therefore the first part of the variable conditions remains satisfied. The lower formulas of the new figure can, instead, depend on assumption formulas of which the corresponding ones did not depend in the old proof figure, in case of a UI resp. EE. This is ob- [A] [A] Θ Θ viously possible only if the UI resp. EE stands in C resp. B and the new assumption formulas stand in Γ. The variable condition could be violated if the eigenvariable of the UI resp. EE appeared in such an assumption. This is, however, not possible, because it cannot appear in Γ at all, in accordance with 2. 1 1 2. 2. 4 1 1. 3 2) We shall now consider the U -andE-reductions. If in inference figures UI, EE of the new figure the eigenvariable is not z, essentially the above considerations are valid. If an inference figure has z as an eigenvariable, it must have had z as an 170 Part III: The original writings eigenvariable already in the old proof figure. (Because y could, by 2. 1 1 2, be the eigenvariable of exactly that one UI resp. EE that was removed by the reduction.) Dz An inference figure with the eigenvariable z should, now, belong to Ω C resp. Ω (by 2. 1 1 3).

ULS 25 So it remains itself unchanged in the reduction.20 The first part of the variable conditions is secured by this. Now, instead, the lower formulas of the UI resp. EE can depend on assumption formulas. These would belong to inference figures standing further down, i.e., to those in Ω. However, such assumption formulas have remained completely unchanged in the reduction, as was already established in 2. 4 1 1. 2 2–4. 2.4112.(Thehillock.) The substitution of z for y has with U -andE-reduction no effect on the number and grade of the hillock. (Because the concepts of neighborhood, boundedness, bond, subformula with possibly different variables, hillock, grade, are independent of such substitution.) Therefore we do not need to pay attention to substitution in considering these reductions. The formulas A, B, Dz in the new derivation have all a lesser grade than the reduced main hillock, for they are its subformulas (with possibly other variables). Thus, there remain to be considered only those hillocks of the new derivation that belong entirely to one of the parts Γ, Θ, Ω, as well as those to which a C belongs. Such a hillock must have occurred already before the reduction at the corresponding place. (Observe that the loss of assumption formulas that can take place through the reduction cannot make a bond into a hillock if it wasn’t one before. For an assumption formula is always a subformula of its neighboring formulas (possibly with a different variable.) The hillock in Θ and Ω, as well as in Γ with A-andU -reduction, further the (possible) hillocks containing C, correspond one-to-one to hillocks of the same grade in the old derivation. With the parts Γ in O-, F -andE-reduction, one can say only: To each new hillock there is an old one of the same grade; But there is no one-to-one correspondence, because Γ can have been multiplied by the reduction. We have in any case already: No hillocks of a greater grade than the one the main hillock had has

20 Right upper corner of page has in a small box: IV. 6 Investigations into Logical Inference 171 appeared. We shall be ﬁnished if we succeed in showing that there appeared in the case of O-, F -andE-reductions in part Γ of the old

ULS 26 derivation at most hillocks of a lesser grade than the main hillock. To this end, we have to turn back to the definition of a main hillock: Namely, the formulas in Γ are all higher (in the old derivation) than the main hillock. This can be read off at once from the definition of higher and the reduction schemes. — The main hillock becomes removed and no new hillocks of its grade appear. Therefore the number of hillocks of the greatest grade is diminished. 2. 4 2. Second case. The main hillock belongs to a V -inference figure. It is then the lower formula of the inference. (For if it is the upper formula ,it is in addition a lower formula of a V -inference figure, because is not the main formula of any inference figure.) We distinguish six cases according to whether the hillock is an upper formula of AE, OE, FE, UE, EE,orV . (Further ones are not possible, because the main formula in the I -inference figures is a lower formula.)

ULS 27 [Written on the middle of the next manuscript page:] Subformula theorem from the hillock theorem: There is, among the formulas that are not subformulas of the endformula, one of a greatest grade. This is an inner hillock. Now one applies the hillock theorem. 172 Part III: The original writings

ULS 28 [Written in large by Bernays across the next page: Sheets of Mr. Gentzen] 6 Investigations into Logical Inference 173

ULS 29

Calculus NL3, “natural-logistic” parallel calculus. (without negation.) Recursive explanation of the concept: “Proof.” (The construction of a proof is not, as so far, undertaken from top downwards, i.e., from the conditions to the ﬁnal result, but from inside out. New propositions are added partly to below, partly to above, and such parts of proof are also “hung one beside the other.”) I explain the concept: “Proof of a proposition B from the propositions 21 A1,...Aν.” I shall use the following symbolic writing when one such is at hand: A1,...Aν ⊃ B to be read as: B is provable from A1,...Aν.Thisisa “sentence” in the sense of P. Hertz, I shall use the word “sentence” [Satz] only in this sense. IdesignatebyΓ, Δ, Θ, Λ complexes of propositions, they can be empty. Now the recursive explanation of “proof”: I) A is a proof of A from A. “natural” writing of this “proof”: symbolic writing for its formation: AA⊃ A x Px (tautological sentence) ( ) Ph x Px 1) Ph is a proof of from ( ) .

natural writing: symb. wr. for the form. of the proof: x Px ( ) x Px ⊃ Ph UE) Ph ( )

The following proofs are formed analogously: Ph EI) Ph ⊃ (Ex)Px (Ex)Px

A & B A & B A B ⊃ AAB ⊃ B AE) A B & &

A B A ⊃ A ∨ BB⊃ A ∨ B OI) A ∨ B A ∨ B

AA→ B A A → B ⊃ B FE) B ,

21Margin has: The order and [changed into: though not the] multiplicity of the A is irrelevant. The A could be empty; B not. 174 Part III: The original writings

ULS 30

2) A proof of Pa from A1 ...Aν gives (through the joining of (x)Px,to below) rise to a proof of (x)Px from A1 ...Aν.HereA must not occur in A1 ...Aν, (x)Px. natural writing: symbolic writing: (let Γ = A1 ...Aν)

A1 ...Aν A1 ...Aν . . Pa Γ ⊃ Pa UI) (x)Px Γ ⊃ (x)Px 22 a must not occur in A1 ...Aν, (x)Px The following expansions of proofs are analogously possible:23 A proof of B from Γ, [Pa] gives rise to a proof of B from Γ, (Ex)Px. a must not occur in Γ, B. Γ(Ex)Px Γ[Pa] . . B Γ[Pa] ⊃ B EE) B Γ(Ex)Px ⊃ B a must not occur free in Γ, B. A proof of A from Γ and a proof of B from Δ give rise to a proof of A & B from ΓΔ. Γ Δ Γ Δ . . . . A B Γ ⊃ A Δ ⊃ B AI) A & B ΓΔ⊃ A & B Further analogously: Γ A ∨ B Δ [A]Γ [B]Δ . . ··. . CC [A]Γ ⊃ C [B]Δ ⊃ C OE) C (A ∨ B)Γ Δ ⊃ C

22 A circle is drawn around the subderivation from A1 ...Aν to Pa. The same goes for the corresponding subderivations in all derivations in the left column of I. 23Margin has: Let [A]meanA ...A, i.e., A occurs in an arbitrary number, even “not at all” is allowed. 6 Investigations into Logical Inference 175

Γ [A]Γ . . B [A]Γ ⊃ B FI) A → B Γ ⊃ A → B

ULS 31 Γ P1 Γ[Pa] . . Pa Γ[Pa] ⊃ Pa CI) Ph Γ P1 ⊃ Ph II) Joining of a condition (Thinning) Γ Γ A . . . . Γ ⊃ B from B becomes B Γ A ⊃ B

Hanging one beside the other (Cut) Γ Δ A . . . . from A and B Γ ⊃ A Δ A ⊃ B becomes24 ΓΔ⊃ B Γ . . Δ A . . B

——————————————————

The logistic form of proof arises from the natural one as follows: Parts of proof that stand one after the other in the construction of the natural proof are written symbolically one below the other, as was shown in the right column above. A logistic proof obviously has the tree form. In the upper ends there are “axioms” of the form

A ⊃ A, (x)Px ⊃ Ph, Ph ⊃ (Ex)Px, A & B ⊃ A, A & B ⊃ B,

24The derivation of A from Γ after composition has an extra smaller A between Γ and A. 176 Part III: The original writings

There then follow the “inference schemes” of the form

Γ ⊃ (x)Px Γ ⊃ A Δ A ⊃ B Γ ⊃ Pa ... ΓΔ⊃ B

I shall call the axioms in I: Structural axioms, In 1: Axioms for the logical signs (namely, in order, all, there is, &, ∨, →). The inferences in II: Structural inferences, In 2: Inferences for the logical signs, and the induction inference. There is to each logical sign an axiom and an inference scheme. Another division is into

ULS 32 an introduction (I ) and an elimination (E) of a sign, which lies close at hand in the natural form of proof. Inference AI can as well be conceived as an axiom, A, B ⊃ A & B.Itwas not done because of symmetry. All logical-sign-axioms can be replaced by inference schemes, namely, for example:

Γ ⊃ A & B A ∨ B Γ ⊃ C ⊃ x Px x Px ⊃ C Γ ( ) Γ(E ) Γ ⊃ A A Γ ⊃ C Γ ⊃ A → B Γ ⊃ Ph Γ Ph ⊃ C B B A Γ ⊃ B

I shall show as an example the equivalence with the axiom in the case of the all-inference: The inference follows from the axioms like this:

Axiom Γ ⊃ (x)Px (x)Px ⊃ Ph Γ ⊃ Ph Cut

The axiom follows from the inference like this:

Str. Axiom (x)Px ⊃ (x)Px (x)Px ⊃ Ph

These inference schemes are to a large extent analogous (inversions) of those in 2). I use the axioms instead of the inference schemes because they are more convenient in the search of peaks. 6 Investigations into Logical Inference 177

Peak search. I consider the following “endpiece” of a logistic proof: I proceed from the Axiom endsentence upwards, in each thread, until I come to a beginning of a thread or (for the ﬁrst time) to a conclusion of an inference 2). I include it, whereas its premisses with their proofs are left out. This endpiece contains of the inferences only the structural ones. I can assume without essential restriction: It contains no structural axioms and no thinning, further no free variables. Then only cuts remain as inferences. As uppermost sentences sign-axioms and conclusions of sign-inferences.25 The proposition that belongs to the sign, underlined in red above, I shall call the “head” of the sentence in question.26 I shall now consider, for improved intuitiveness, the natural form of this proof-piece.

See the drawing −→

To each uppermost sentence of the piece corresponds a natural proof of the form: among others, with a head-proposition ( ). To a cut corresponds the “hanging together,” so that overall, a ﬁgure in tree form is produced. (The single “points of contact” are the “cut propositions.”) I shall now specify: Where 2 heads touch each other, there is an ×× “inner−head.” Where a head is not a contact proposition, but an initial or × ﬁnal proposition, there is a “boundaryhead.” Theexistenceofanyoneheadcannowbeseentopologically.

ULS 33 [Page 33 of the manuscript consists of a list of changes to a chapter of the original text, now absent, that Gentzen submitted for publication on 15 March 1933 under the title Uber¨ das Verh¨altnis zwischen intuitionistischer und klassischer Arithmetik. The changes are incorporated in the submitted text that has been preserved in the form of galley proofs.]

ULS 34 [Blank]

25Added in margin: If there is a CI-conclusion there is a “CI-peak” (in which case h is a number) and there is nothing further to consider. 26 The red underlinings are for the main formulas in the logical rules of sequent calculus, the right column of the above “recursive explanations” of proofs. 178 Part III: The original writings

ULS 35

Calculus LDK. (Logistic-dualistic.) (classical.) A sentence is written:

A1,...,Aμ ⊃ B1,...Bν in short: Γ ⊃ Δ The A, B are propositions. The meaning of the sentence is: If A1,...,Aμ together hold, then at least one of the propositions B1,... Bν holds. (Formally: A1 & ... & Aμ. ⊃ .B1 ∨ ...∨ Bν, can be written also as a “disjunction”: A1 ∨ ...∨ Aμ ∨ B1 ∨ Bν .) Order and multiplicity of the A,resp.theB make no diﬀerence. Both can be empty. Structure-axiom: A ⊃ A Structure-inferences: Thinning: Γ ⊃ Δ ΘΓ⊃ ΔΛ

Cut Γ ⊃ Δ AAΘ ⊃ Λ ΓΘ⊃ ΔΛ Logical sign-axioms and inferences: 1. The propositional calculus. Axioms:

& ∨

A & B ⊃ AA⊃ A ∨ B A & B ⊃ BB⊃ A ∨ B A, B ⊃ A & BA∨ B ⊃ A, B ¬ A, ¬A ⊃⊃A, ¬A (Law of contradiction) (Law of excluded middle) Remarks: ⊃ A means contentfully: A is correct, A ⊃ means: A is wrong. (A ⊃ is equivalent to ⊃¬A.) The sentence: ⊃ with an empty antecedent and succedent means “the contradiction.” (It is equivalent to ⊃ A & ¬A.) The sign → is to be deﬁned through &, ¬ or ∨, ¬. 6 Investigations into Logical Inference 179

2. The predicate calculus. Axioms:

() (E)

(x)Px ⊃ Ph Ph ⊃ (Ex)Px

ULS 36 Inferences: () (E) Γ ⊃ Pa Pa ⊃ Γ Γ ⊃ (x)Px (Ex)Px ⊃ Γ

a must not occur in the conclusion.

[The page ends with a few lines written in stenographic notation. These have been deciphered by Christian Thiel: I have proved that thinning and cut form in the propositional calculus a complete system of inferences. Since both hold also intuitionistically, it should follow: Propositions brought in the form of “sentences” are at the same time intuitionistically and classically correct or incorrect.] 180 Part III: The original writings

ULS 37

The N2-Formalism.27 Inference schemes: AI AE UI UE AB A & B A & B Pa (x)Px A & B A B (x)Px Ph RAA TND CI Pa A A ...... B ¬B ¬¬A P1 Pa ¬A A Ph

1.) a:afreevariable h: a free variable or a number. 2.) The a of a UI, CI must not occur in the conclusion. ⎛ ⎞ AA ⎜ . . ⎟ 3.) The a in a UI, CI that appears in the anterior part of a RAA ⎝ . . ⎠ B ¬B must not occur in the RAA-assumption A. Analogously for the right anterior part of a CI and its assumption Pa.

Equivalence with the H.A.-formalism. I. A proof in HA can be reproduced in N2. I deﬁne: A ∨ B as ¬.¬A & ¬B,(Ex)Px as ¬(x)¬Px, A → B as ¬.A & ¬B In the replacement of ∨, →,Ex by the deﬁning expression, the axioms of HA read: a) ¬...¬.¬A & ¬A.. & .¬A b) ¬..A & ¬¬.¬A & ¬B c) ¬...¬.¬A & ¬B.. & ..¬¬.¬B & ¬A d) ¬{¬.A & ¬B.. & ¬¬...¬.¬C & ¬A.. & ..¬¬.¬C & ¬B} e) ¬.(x)Px & ¬Ph f) ¬.Ph & ¬¬(x)¬Px

27This part was originally planned to be the second one and must have been written before the parts on normalization and sequent calculus. 6 Investigations into Logical Inference 181

AllofthesecanbeprovedinN2, as an example, I shall prove b: 1 A & ¬¬.¬A & ¬B AE 1 ¬¬.¬A & ¬B TND A & ¬¬.¬A & ¬B ¬A & ¬B A AE ¬A AE RAA 1 Axiom b

ULS 38 (I have gone through also the rest of the proofs.) The inference schemes of HA can be reproduced in N2: A ¬.A & ¬B “Inference scheme”: It reads as B Reproduction in N2: 1 A ¬B A & ¬B ¬.A & ¬B ¬¬B RAA 1 B TND ¬.A ¬Pa All-scheme: & ¬.A & ¬(x)Px Reproduction in N2: 1 A ¬ x Px & ( ) 2 A AE ¬Pa AI A & ¬Pa ¬.A & ¬Pa RAA 2 1 ¬¬Pa A & ¬(x)Px Pa TND AE UI ¬(x)Px (x)Px RAA 1 ¬.A & ¬(x)Px UI fulﬁls the variable condition, because a does not occur in ¬.A & ¬(x)Px. Existence scheme: Analogous. Induction axiom: 1 P1&.(x)¬.Px & ¬Px.. & ¬Ph AE P1&(x)¬.Px & ¬Px 3 2 AE Pa ¬Pa (x)¬.Px & ¬Px 1 AE UE P1&.(x)¬.Px & ¬Px.. & ¬Ph Pa & ¬Pa ¬.Pa & ¬Pa AE RAA2 P1&.(x)¬.Px & ¬Px ¬¬Pa 1 AE TND P1 Pa P1&.(x)¬.Px & ¬Px.. & ¬Ph CI 3 AE Ph ¬Ph RAA 1 ¬...P1&.(x)¬.Px & ¬Px.. & ¬Ph This is the induction axiom, written in &, ¬, (x). 182 Part III: The original writings

—————————————————————

II.AproofinN2 can be reproduced in HA. It is ﬁrst transformed as follows: One ﬁnds for each proposition A of the N2-proof all those assumptions above it the respective RAA’s (CI’s) of which still stand under A.Theseare B1,...Bϕ. Then one substitutes A by B1 & ... & Bϕ → A. If A itself is an assumption, A → A takes its place.

ULS 39 All initial propositions are by now correct in HA. For mathematical axioms have remained thus, and assumptions have become A → A. The inferences read now as:

AI AE UE TND D → AE→ B C → A & B C → (x)Px C → ¬¬A D & E → A & B C → A C → Ph C → A (B) These are correct HA-inferences. Some more attention is required by UI, RAA, CI:

UI C → Pa This is a regular all-scheme, because by condition 2, C → (x)Px a does not occur in (x)Px, and because of condition 3, neither in C.

RAA CI D & A → BE& A → ¬B D → P1 E & Pa → Pa D & E → ¬A D & E → Ph Also these inferences are correct in HA.WithCI, one can infer: E & Pa → Pa E → .Pa → Pa E → .(x)Px → Px

The all-scheme is correct,28 as before, because of conditions 2,3. ——————————————–

The puriﬁcation of a proof.

1. Those free variables that are not necessarily, for the sake of the correctness of the inference, equal (“related”), are substituted by diﬀerent free variables.

28The last formula should read: E → .(x).Px → Px 6 Investigations into Logical Inference 183

2. Those free variables that do not occur as the a of a UI or CI are substituted by the number 1. Then one has a purified proof for which the following theorems hold: Theorem 2. [Added: All] propositions with a free variable are connected [zusammenhängend] if one takes the conclusion of an inference to be directly connected to the premisses (also these with each other) and to eventual assumptions that belong to the inference. This follows from the 1. step of purification. Theorem 4. The a of a UI, CI occurs only in the anterior part of the UI resp. the right anterior part of the CI.

ULS 40 This follows from theorem 2 and the variable conditions 2,3. Theorem 5. Each free variable is the a (the “eigenvariable”) of exactly one UI resp. CI (of its “eigeninference”). This follows from theorem 4 and the structure of proofs.

The concept of a peak:

I designate as a peak a sentence [Satz] of the direct part of proof (i.e.,not in the anterior part of a RAA or the right anterior part of a CI )ofthe following kind: An assumption [Voraussetzung] that is the A & B of an AE or the (x)Px of anumber-UE, the A & B of an AE the (x)Px of a number-UE a sentence that is or and also of an AI and also of a UI or a conclusion of a CI or a TND, the endsentence, if it is a conclusion of a RAA.

Peak existence theorem.

Theorem 6. A purified proof pair for a contradiction [added: that does not consist of just two propositions] in which there occurs no assumption of the form ¬C that is not a unit sentence (i.e., C without variables and logical signs), has a peak. Proof: Let the contradiction be A, ¬A. Let next the proof of ¬A consist of not only this sentence. I then proceed upwards from ¬A in the proof until the first conclusion of UI, AI, RAA, CI, TND, [Added: or assumption] that I encounter. The way is unique because only inferences AE, UE are passed 184 Part III: The original writings by. The proposition encountered is already a peak. For it (and all under it) contains no free variables and is in a direct part of proof, because no UI, CI, RAA occurs below it (theorem 4 important). If it is in addition the endproposition ¬A, it cannot be a UI, AI conclusion, it is then a peak according to the definition. If it is not an endproposition, it is followed by an AE or a number-UE. Then it cannot be an RAA-conclusion; and an AI conclusion only if AE follows, UI conclusion only if UE follows. There is then also in these cases a peak, and in the remaining cases as well, when the sentence is a conclusion of CI or TND or an assumption. If the proof of ¬A consists of only ¬A,thenA is an elementary sentence. One then finds in the proof of A apeakinthesamewayasabove.The sentence that one encounters is not A, except when A is a conclusion of CI or TND.ThenA is a peak. The rest goes as above.

ULS 41

The reductions.

Reduction schemes:29 VAGR 30 VUGR CIGR Pa α γ(a) x Px P Pa A & B ( ) 1 A A Pν Pν P(ν) ε into ε ε into ε ε

AGR UGR into:

α P1 γ(1) P2 γ(2) a α( ) . α β P(a) . A B P(ν − 1) A B (x)Px − & α α(ν) γ(ν 1) A A P(ν) P(ν) P(ν) ε into ε ε into ε ε

29 The schemes are delineated in boxes by freely drawn horizontal and vertical lines. 30 V stands for Voraussetzung, assumption, and AGR for And-Ground-Reduction,and similarly for the rest. 6 Investigations into Logical Inference 185

RAAGR

× × × × A ...A A ...A α α γ δ α α A A B ¬B A ...A ... α RAA× γ δ A ¬A into B ¬B

TNDGR α ◦ ¬¬B B β γ B β γ (¬)A [¬]A α RAA◦ (¬)A [¬]A into ¬¬B ¬B

There is to each kind of peak a reduction. Each of these reductions takes over a proof pair of a contradiction again into such: (In the case of RAAGR, B must not contain any free variables, theorem 4).31 Considerations that show this are the same as before.

31 Cancelled: these can be substituted in puriﬁcation by 1 (by theorem 4). 186 Part III: The original writings

7. Reduction of classical to intuitionistic logic

This note is written on a separate sheet. It contains a reduction of derivations in Gentzen’s calculus of natural deduction for classical predicate logic into what is today called minimal logic, with just implication and universal quantiﬁcation and a parametric atomic formula that plays the role of ⊥.

Decision in classical predicate calculus reducible to decision in intuitio- ∀ nistic calculus with only ⊃ and ()? should be about the beginning of 1933

To begin with we have: A proposition of the classical PC can be written equivalently with only ⊃, (),andF, with the elementary propositions occurring only negated, i.e., with ⊃F. And this proposition is classically and intuitionistically of the same value. (i.e., correct or incorrect in both systems.) By the subformula theorem it is therefore provable, if at all, already with the inferences for ⊃, (),andV.1 I would just like to have that one does not need the inference V. This should be possible. It is done more or less as follows: One puts before each proposition and subproposition always ¬¬. Then one has to redo the inferences. This is still a bit unclear. Precisely: Let there be given an N2K proof2 for a proposition E with ∧, ¬, (). Let E not contain the propositional variable F. First transformation: Each A ∧ B is substituted by ¬.A ⊃¬B The inferences AI and AE become new ones, modiﬁed as follows:

1 A A ⊃¬B AI into: B ¬B FE 1 RA ¬.A ⊃¬B

1 2 A ¬A 2 ¬B RA ¬B FI1 FI AE into: A⊃¬B ¬.A⊃¬B resp: A⊃¬B ¬.A ⊃¬B ¬¬A RA2 ¬¬B RA2 A REND B REND All of the rest remains in order. Second transformation: Each ¬A is replaced by A ⊃F.

1 Rule ⊥E. 2 N2K and the rules below are deﬁned in ULS. 7 Reduction of classical to intuitionistic logic 187

The inferences RA and REND are aﬀected. They are replaced by:

1 1 [A] [A] B B ⊃F RA: F FE A ⊃F ⊃F FI1 REND: . A ⊃F A “DN (law of double negation) Third transformation: Each elementary proposition A is replaced by A ⊃ F. ⊃F. The proof remains naturally correct. The inferences DN, though, shall be bypassed completely. This happens in steps as follows: Let the A of DN have already the form B ⊃ C. Then the DN is replaced in the ﬁrst place as follows:

1 3 B B ⊃ C 2 C FE C ⊃F F FE FI 3 B ⊃ C. ⊃F B ⊃ C. ⊃F: ⊃F F FE FI 2 C ⊃F. ⊃F C DN for C B ⊃ C FI 1

Let the A of DN have the form (x)Px. Then the DN is replaced in the ﬁrst place as follows:

2 x Px ( ) 1 Pa UE Pa ⊃F F FE FI 2 (x)Px ⊃F (x)Px ⊃F. ⊃F F FE FI 1 Pa ⊃F. ⊃F Pa DN for Pa UI (x)Px

One chooses as a a variable that does not occur in the entire proof. Then the variable condition is satisﬁed. Let the A of DN be previously an elementary proposition. Then DN is replaced as follows: We have A = D ⊃F. ⊃F.

1 2 D ⊃F D ⊃F. ⊃F F FE FI 2 (D ⊃F. ⊃F) ⊃F (D ⊃F. ⊃F) ⊃F. ⊃F F FE FI 1 D ⊃F. ⊃F 188 Part III: The original writings

It is obvious that the inference DN can be completely eliminated by these steps. Result: If the original proposition is provable, also the proposition emerging through the three transformations is provable in the calculus FI, FE, UI, UE.3 Conversely: If the latter provability holds, then the original proposition is classically provable. For the latter proof is classically correct, when one substitutes G∧¬Gfor F everywhere in it. Its result is through this substitution equivalent to E.ForA ⊃ (G∧¬G) is equivalent to ¬A from which all follows. In sum: It is in fact suﬃcient to consider the decision problem in the calculus FI, FE, UI, UE,inshort:inNFA.4 (The proposition F plays no special role anymore, because it does not appear in the inference schemes anymore. It is now nothing more than any propositional variable.)5

3 The calculus of natural deduction for the minimal logic of ⊃, ⊥, ∀. 4 This should be NFU in translation, for “natural,” “follows,” and “universal” that correspond to the German Naturlich¨ , Folge,andAll, but I decided not to change the name of the calculus in translation. 5 The text ends with a cancelled Attempt at decision in NFA that consists of the four rules of inference FI, FE, UI, UE with the subformula relations in them indicated by arrows. 8 CV of the candidate Gerhard Gentzen. 189

8. CV of the candidate Gerhard Gentzen.

Göttingen, 26.V. 1933. I, Gerhard Karl Erich Gentzen, was born on 24 November 1909 in Greifswald, as the son of the attorney Hans Gentzen and his wife Melanie Gentzen, born Bilharz. I am a Prussian citizen and of Evangelic-Lutheran confession. After about two years of elementary school, I went to the school in Bergen on Rugen,¨ stayed there until sixth grade, and frequented then the humanistic gymnasium in Stralsund in 1920–28. I took the maturity exam there in Easter 1928. Right thereafter I studied for two semesters in Greifswald, then two semesters in Göttingen, one in Munich, one in Berlin and since then, from Easter 1931 on, again until now in Göttingen. I have never interrupted my studies and am therefore at the time at my 11. semester. I am since Easter 1929 a member of the Studienstiftung des Deutschen Volkes. My main subject of study was from the beginning mathematics, but I have also taken part in physical and astronomical lectures and exercises, as well as philosophical lectures and physical training. Within mathematics, my interests tended in the course of time more and more towards the more abstract and general questions, especially those of mathematical logic and proof theory. I have been occupied with this field in detail independently. Then I wrote the enclosed work “Uber¨ die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsysteme” that was published through the mediation of Professor Bernays in the Mathematische Annalen. I am occupied since about Christmas 1931 in the first place with the problem of the consistency of arithmetic. I wrote in connection with this occupation a smaller work “Uber¨ das verhältnis zwischen intuitionistischer und klassischer Arithmetik,” this is at present in press with the Mathema- tische Annalen. My considerations about the consistency of arithmetic were inevitably connected with a study of those properties of logical inference that can be conceived mathematically. I have summed up the most essential results obtained here in a work with the title “Untersuchungen uber¨ das logische Schliessen”; I have handed this in at the Mathematical-Scientific Faculty of the University of Göttingen on 26 May 1933, for the obtainment of the doctoral degree. Professor Weyl has offered to evaluate this work. In the field of physics, my principal interest is in the theoretical questions, and here, then, I have been occupied mainly with the theory of electricity. My work in the field of mathematical logic and foundational research has naturally brought me into contact with the related philosophical problems. Gerhard Gentzen. 190 Part III: The original writings

9. Letters to Heyting

1. Gentzen to Heyting 25 February 1933

Most respected Doctor! Göttingen, 25.II.33 Schillerstr. 7 I thank you most obligingly for your letter of 18.II. and for the evaluation of my work. I was very glad to see that your point of view is largely in accordance with mine. I will be glad to add for the publication of the work, following your suggestion, a remark about the relation to Hilbert’s idea of a metamathematical foundation of arithmetic. I think one could say about it, in accordance with your remarks, something like: A proof of consistency by finitary means (what you call sentences of the first and second type [Stufe]) has not succeeded so far, so this original aim of Hilbert’s has not been reached. If instead one accepts the intuitionistic standpoint as one that is already secure, i.e., as a consistent one, then the consistency of classical arithmetic is also secured by my result. If one wanted to satisfy Hilbert’s requirements, there would still remain the proof of the consistency of intuitionistic arithmetic. This, however, is not possible even by the formal apparatus of classical arithmetic, on the basis of Gödel’s result together with my proof. Nevertheless, I am inclined to consider possible and even desirable a consistency proof for intuitionistic arithmetic from a standpoint that is even more evident. I hope to be able to undertake investigations on this next year. A further small addition to my work is the following remark that does not appear yet, to my knowledge, in the literature on intuitionism: One can avoid entirely the negation in intuitionistic arithmetic, by declaring ¬A to be an abbreviation for A ⊃ 1 = 1, and by considering = a basic predicate for which is put up the axiom: x = y ⊃⊂ ¬ x = y (so that ¬ x = y means just x = y ⊃ 1 = 1). This conception of negation is correct, because ¬A ⊃⊂.A ⊃ 1 = 1 is intuitionistically provable. Thereu- pon axiom 2.12 11 (reductio ad absurdum) becomes provable, the second negation axiom (2.12 10) becomes equivalent to 1 =1 ⊃ B.Onecantake this to be an arithmetical axiom, but one can also leave it out and to remain content with the fact that 1 =1 ⊃ A is always provable, as long as A contains no propositional variables. (It follows like this: To begin with, we have 1 =1 ⊃.B ⊃ 1 =1,so1 =1 ⊃¬B. Especially: 1 =1 ⊃¬¬x = y, therefore (2.148) 1 =1 ⊃ x = y. Similarly 1 =1 ⊃ x

2. Gentzen to Heyting 28 November 1933

Most respected Doctor. G¨ottingen, 28.11.33

I want to inform you that my paper about the relation between intuitionistic and classical arithmetic, the one for which you had the kindness to act as a referee, will not appear in print. I found out later that Mr G¨odel had arrived at very closely the same result already before me. (The work in question is, as is probably known to you, G¨odel’s “Zur intuitionistischen Arithmetik und Zahlentheorie” in the “Ergebnisse eines mathem. Kolloquiums,” issue 4). You may be interested to know that I have found a decision procedure for the intuitionistic propositional calculus (it was probably lacking so far). The article in question is in press, when it is ready, I’ll naturally send you acopy. with devoted regards

Your Gerhard Gentzen

3. Gentzen to Heyting 23 January 1934

Most respected Doctor! G¨ottingen, 23.1.34

Many thanks for your postcard of 9.1. My dissertation will appear in the Ma- them. Zeitschrift under the title “Untersuchungen uber¨ das logische Schlies- sen.” I prove therein a quite general theorem about intuitionistic and classical propositional and predicate logic. The decision procedure for intuitionistic propositional logic results as a simple application of this theorem. One can also show with it the intuitionistic unprovability of simple formulas of predicate logic, such as (x)¬¬Ax. ⊃¬¬(x)Ax. I have not studied how far, in the end, one could go. I am now working with the proof of the consistency of analysis that has been since 2 years my real aim. with devoted regards

Your Gerhard Gentzen

4. Gentzen to Heyting 16 April 1934

Most respected Doctor! G¨ottingen, 16.IV.34

I can answer aﬃrmatively to your question. It follows in fact from my theorem that if A ∨ B is intuitionistically provable, either A or B is intuitioni- 192 Part III: The original writings stically provable, and not just for propositional logic, but for the whole of predicate logic. I have given this consequence also in my dissertation. The article is unfortunately still not in press, because it was lying for a longer time at the Notgemeinschaft der Deutschen Wissenschaft [a stipendiary fund]. If I am not wrong, the theorem leads even to the following similar consequence: If (Ex)Fx is intuitionistically provable (Fx an arbitrary formula with no [other] free variables), then also (x)Fx is intuitionistically provable. This does not hold in classical logic, as for example the formula (Ex)(¬Ax ∨ (y)Ay)shows.

with best regards your most devoted

Gerhard Gentzen 10 Formal conception of correctness in arithmetic II 193

10. The formal conception of the notion of contentful correctness in pure number theory, relations to proof of consistency II

This second part of INH has been written between February 1933 and October 1934. The ﬁrst page is 21 that faces directly the last page 20 of the notes of the fall of 1932. Pages D 97.3 and INH 20.1 stand loosely between the two parts of INH and are found at the end of the ﬁrst part.

INH 21 supposedly about II.33

Ultraintuitionistic criticism of the ways of inference. Finitism. The finitary standpoint: it is at first not allowed to infer with transfinite propositions as with finite. Definition of &, ∨, for all, and there is: AB A B A B A & B holds if A holds and B holds. Therefore and & & A & B A B are allowed inferences. (x)Px holds if Pa obtains for all natural numbers a. How such a thing can Pa be recognized is not important just now. So we have the inferences (x)Px (x)Px .HerePν means that Pa obtains for arbitrary natural numbers in Pν place of a. A∨B A B A B holds if or is recognized as holding. So: A ∨ B A ∨ B. Nothing can be inferred from A ∨ B. One has to instead go back to its appearance. The same holds for (Ex)Px. It follows from Pν. (“judgment-abstraction”.) Direct inference is already possible by the means so far. No indirect, one that starts from assumptions, though. One needs from the beginning certain basic facts, axioms. One needs further also, to be able to prove universal propositions, the “variables.” In addition, one has to be able to infer from (x)Px: Pa.Soa denotes “just any arbitrary natural number.” One can then operate further with this arbitrary number. x Px (E ) b One will admit that also the EE Pb is not dubitable if one looks at as a number the value of which is fixed in itself, but say cannot be remembered at the moment, or simply must not be given because of convenience. Though A . . even that should be defended. One says, say, that one has two proofs C and 194 Part III: The original writings

B . . C : One of them is certainly meaningful and its “condition” really holds, so also C is correct, because regularly proved. The other part is completely meaningless. One just does not know, or does not give it, which of the two. Complete induction: Here there is an assumption. Though one that has been so to say proved. Pa . . P1 Pa Ph One says, indeed: the proof of Pa from Pa proves in the first place, thinking of a as 1, P2fromP1 that has already been proved, then P3, etc. It is in any case obvious that when a fixed b is given, an ordinary proof without CI and without assumptions (unless there are otherwise some left) can be produced. And this is all one wants to know. The admissibility of an inference depends in general always on this type of thoughts. So with, e.g., EE above: for it is sufficient to control afterwards which number actually “existed,” then one puts this for b and has an ordinary proof without EE. Therefore: The belief in the security of an inference depends always on the idea of its possibility of reduction. Also propositions such as (x)(Ey) ... are possible, proved through CI ,in which the search of a y for a given x requires the unfolding of the CI .Then one does not have the existent at all that directly at hand, say, if one wants to continue the inference. But still, one has it in principle, can find it through a finite work (!).

INH 22 The theory so far and its metatheoretical foundation. The considerations so far to display the correctness of the inferences seem to me in fact necessary and standard. They are, though, nothing but metatheory. The operation with proofs plays a role. So, when one ascertains the correctness of simple inferences, one poses in fact already metatheoretical considerations. [Margin has: From here on less important. What follows after.] I had once earlier the thought that a metatheoretical consideration would not be suited to the knowledge of correctness. Here, nevertheless, it seems to be. Or does the above metatheory contain elements that cannot be taken together or changed essentially in the whole of the following presentation? 10 Formal conception of correctness in arithmetic II 195

A purely formal consideration of correctness should perhaps take an entire proof as a whole, to ﬁnd its peaks and to reduce them, as well as to show that this leads to the aim in the end. This consideration deviates, however, considerably from those above. It was the “purely-formal” proof of correctness of my earlier description, whereas the above give something like “semi-formal” proofs of correctness. I.e., they work with the concept of “contentful correctness” of a proposition. To what extent can the above considerations be formalized? Will there then be inferences in use that are not yet contained in our system? If so, how come is it that these considerations are still suitable for conﬁrming the correctness of our system? A simple example:

I consider a system with the mathematical axioms A1 ..Aν and the inferences AI, AE. (Let the mathematical axioms be without logical signs.) The correctness of inference states here: The arbitrarily continued process of inference results in, if a proposition without &, then one of the axioms. Proofinwords:ByCI . Trivial for no inferences. Let it be shown up to ν.Let now the ν-th inference give a proposition without &. To be shown that it belongs to the axioms. The inference before it is AE, for otherwise it would contain &. I call the premiss of the AE its anterior-proposition. I proceed in this way backwards until a proposition the inference of which was not AE anymore, but AI . [Added above: Peak.] There must be one such because there were only axioms in the beginning and these did not contain any &. Next it will be shown that this AI was already the premiss of the AE.Soit was already there. By this one is essentially ﬁnished. One can continue more or less like that. Or one prescribes in advance that if there is an inference from a proposition, it must be used in its earliest place of occurrence. Now, the whole thing is quite trivial. What concerns me is the formal proof, and it must be carried with the simplest possible inferences.

INH 23 The formal proof: I begin with an even simpler case, namely one axiom A and only the inference T A | A. To be shown that the theory A | A | A | A .. can in the end have, with arbitrary length, only A. The object of the metatheory are the proofs of the theory. These are single- threaded and consist of propositions. Like in the recursive production procedure: From a proof a new one is formed when a proposition is added. A proof is, however, correct only when its ﬁrst proposition is A and when each proposition in it is equal to the preceding one. Claim: Each correct proof has A as the end-proposition. 196 Part III: The original writings

The question is mainly that the metatheoretical proof operates with only ﬁnite propositions. At least almost so. It is to be determined where that is superseded. Should be so far only in the result of its CI , its end-proposition, one that can be still written with free variables, without the all-sign.

IV.33 Proof of consistency of the whole of the above theory. 1. purely- formal, 2. semi-contentful, 3. with the concept of reducibility. Ways of inference to be compared, transfinite values. 1. Purely-formal. Let there be a derivation of 1 = 1. Peak: naturally at hand. (To be shown finitely.) Simplification, assignment of values to threads. Values polynomials in ω.ForCI as known. Should not be difficult for OE.(In the passage through the middle [premiss with the disjunction] the greatest eigenthread-value is added.) Assignment of values non-circular, because no →. The value of the derivation as a whole to be written as ωP1(0) ++ωPν (0), P a polynomial in ω. ω Inferences used: Transfinite induction up to ω(ω ), resp. standard induction over transfinite propositions, not yet clearly known. Is the transfinite induction intuitively clearer than its justification through a standard induction over transfinite propositions? Something like a meta- meta-justification? 2. Semi-contentful. Two times CI over the concept of correctness. Is this still formalizable transfinitely or not anymore? Maybe not anymore in general. (But maybe still the meta-meta-justification!, in each specific case.) One can produce the HA normal form of formulas. Correctness of the matrix can be shown recursively, for fixed variables. It doesn’t go further, though, because the prefix can be wild. Without ∃ still clearly formalizable. With parts of proofs even → is added. 3. With the concept of reducibility. Even here two times CI (for the formal CI and for the length of derivation), this time over the concept of reducibility. Its formalizability should stand similar to the formalizability of correctness. Maybe it is still formalizable also in this case (say freedom of choice prescribed in advance or so, on the basis of specific properties of the reductions), but the proof of the “peak theorem” is more difficult in turn. Should still be investigated. 10 Formal conception of correctness in arithmetic II 197

General thoughts about the proof of consistency. 1

The idea as a whole: Each proof has a (transfinite) value. Consistency of a system of proofs can be shown only through a proof that has a higher value than all of these. Therefore the theorem of Gödel. This is to be attempted to be proved. The values to be determined in this connection. Further then whether there are proofs of a higher value but nevertheless higher relia- bility. (Transfinite induction, meta-meta-justification.) Connection between assignment of values and reducibility, contentful correctness etc. taken over to WTZ

INH 24 A proof of consistency for a system would prove among others its own consistency. One applies it to itself, then its own consistency comes out. To investigate: Is transfinite induction at some point not anymore formalizable, how far is the definition of greater than between transfinite numbers still formalizable? Specifically also: Isn’t a formalization with a corresponding contentful consideration still possible under ωω? This is to be investigated in connection with D54.1, D73. I would tend to affirm it. The formalization according to Bernays seems, however, not to be this one. (VI 33) The possibility of a transfinite assignment of values seems almost sure, more or less on the basis of the reducibility theorem. Let us take the following consideration: Proofs that become continuously smaller with reduction are assigned values according to the number of their sequents. Those that became such in one step are taken only later. These are assigned values correspondingly etc. One should just be able to classify each proof directly in a correct way. The best should be to begin with simple calculi. The [cancelled: idea of ] reducibility and the [cancelled: idea of ] non-circularity are the ones on which the proof of consistency depends. The first one is taken into account within contentful thinking, the second appears in the ho- rizon only within more acute considerations. Therefore also the paradoxes of set theory, where there are circular proofs. (Even if these are reducible, i.e., allow one reduction, then again another, etc, this procedure does not lead to the aim in the end [or: finitely], the latter because of the non-circularity.) Non-circularity2 = possibility of assignment of values. Reducibility = possibility of a reduction step.

1 A vertical line is drawn over the rest of page 23 and also page 24. Both lines end with the text: taken over to WTZ. 2 An arrow points to this word from the addition: An error. There is with the paradoxes already nonreducibility. 198 Part III: The original writings

Let the assignment of values of the proof be known. How is the assignment to be done for a proof that arises from it through the addition of a speciﬁc inference ﬁgure? taken over to WTZ

INH 25 X.34. Nagging doubts. About the value of the consistency proof. The contrast between the formal and the semi-contentful correctness proof: The formal one has greater diﬃculties with the “simpliﬁcation.” The semi- contentful one avoids these. And precisely because it has already summarized formal facts in a skilled way. [Margin: Comp. INH 17 middle, INH 18 middle.] Thereby it approaches the contentful proof itself, which latter takes all possible ways of summarizings. My concept of reducibility is supposed to deliver to some extent the formal contents of a contentful summarizing of individual facts. Let us compare for once the formal and the contentful proof in the case of thetheorywithonly∀, &, and ⊃.[Margin:SeeINH 19–20 middle!] Here, already, the contentful proof has a gain in that it uses propositions with ⊃, and these must be considered harmless, for they state just a meta-metatheoretical connection. What can the formal proof do in such a situation? There seems to remain to it nothing left but to perform through detours the kind of summarizings as the semi-contentful. Or?

Formal, semi-contentful, reducibility-proof:

The proof with the concept of reducibility is a kind of middle thing between the other two, to be looked at more or less as follows: It runs with & ∀∨∃ in the same way as the semi-contentful proof, in that it still admits here the contentfully corresponding. It deviates instead with ⊃ and goes over to a formal proof, in that it wants to avoid the contentful ⊃.[Margin: More correctly said: The finitary meanings are with & ∀∨∃still almost the same as with the being-in-itself, with ⊃ it is considerably [?] less so! ] The above raises the impression: One begins with the semi-contentful. In proceeding ahead, this becomes more and more unwieldy, because one apparently presupposes almost exactly the same as one wants to prove. With ⊃, finally, one decides to jump off and is saved into the formal, by which the proof becomes complicated. Devices of a peculiar kind are later needed to save it, let’s say, the super-ordering. [Margin: Well this is an exaggeration. ] One can be irritated by the fact that all purely-formal ways use in the end, 10 Formal conception of correctness in arithmetic II 199 by their foundation through the super-ordering, again the same means as the semi-contentful proofs. [Margin: I would not want to hope that with my proof. ] (I disregard ¬ because the situation is clearly different. I consider only intuitionistic mathematics.)

INH 26 The semi-contentful method with ⊃.

INH 27 ∀, ∃ with which the same is possible. Certainly, the question of whether A ⊃ B holds [cancelled: is reducible] does not reduce back to that for A or B, but to a relation between both of these. How, then, is it in the end with ∀, &, ∨?WithA & B one comes to two single holdings [cancelled: reducibilities]. With ∀ even worse. The avoidance of the “in-itself” is achieved through the word “demonstrable” [nachweisbar]. Correctly conceived, there is no definitional circle. What does “demonstrated” mean? This is somewhat embarrassing. Though, it is the same with ∀, ∃.Alreadywith∀, what is it, to have demonstrated Fx correct for an arbitrary x?[Margin: That is the crux of this ⊃-meaning. It is, surely, different with ∀ etc.]Orwith∃, what is it that an example is given? Let us assume that the number in question is written down. Then there is still the question, what is it that the number satisfies the proposition F a,saywhenF is not finite anymore, yet also already when it is finite! One works always with “possibilities,” the basic point of which is dubitable. What relation is there between the semi-contentful proof and the reduction procedure in the case of derivations of a contradiction? Can the finiteness of the latter be proved from the former. In the same way as the reducibility of derivations of a contradiction follows from the general reducibility theorem. One should make the bridge from the concept reducible to the reductions. From reducible for formulas to reductions of derivations. For the reduction of a derivation is something so natural that all of this univocally clear. The concept “reducible” in my reducibility theorem, that is, where I assume a jump with ⊃, is constructed from it [=reduction], and thereby that concept should be natural, and not artificial with respect to the difference of ∀∃.And⊃. How is it with this thing? It needs clarification. The concept reducible for derivations is on the whole something really fundamental. Especially for derivations of a contradiction. Every semi-contentful 10 Formal conception of correctness in arithmetic II 201 method should in fact deliver such [a concept], for otherwise they would not be natural to the same extent!

INH 28 How is it then with the theory without ⊃? It seems to me that already here the semi-contentful and the reducibility proofs depart. E.g., with ∃: Semi-contentfully: an example is “given,” reducibility instead: The example comes out in the prescribed reduction procedure. Therefore: One must distinguish between the semi-contentful proof that associates to each formulas resp. sequent a semi-contentful concept of correctness, and the proof by the concept of reducibility that works with reductions of a derivation. This one leads over to the purely-formal proof that considers only the reductions of a derivation of a contradiction. The method of adapting the concept of reducibility from derivations to sequents led to an apparent identification of reducible and what is semi- contentfully correct, so it seems to be a blurring of the situation, a degra- dation of the reducibility proof. [Margin: I don’t mean that ] It remains undecided whether also the reducibility proof presents just a delusion, i.e., an apparent improvement. Something like: Let ∃xF x be derived. It was: Reducibility provides us with a procedure for the finding of x. Hereby something would be accomplished also for the in- tuitionists that was so far missing. But: The demonstration of reducibility requires means that possibly present exactly the same as appears in the proof of ∃xF x, namely, the demonstration of an existence without an immediate specification of the procedure of production, only indirect ones over summarized propositions etc. [Margin: This should be partly correct, but is of no consequence for consistency ] In sum, one could assume: All kinds of proofs of consistency have in common that they already use the ways of inference that they want to prove as consistent! Sad but true! [Margin: I admit this for the theory without ⊃ but not with ⊃.See29 bottom.] And quite understandable if one thinks that a proof of correctness even proves the correctness of the formula in question once more, so cannot very well be simpler than the considered derivation of the formula, except when this had by chance been led through detours. What remains? 202 Part III: The original writings

INH 29 X.34 Solid ground to be regained ∀xPx must be explained in a close connection to the ways of inference, especially the I. As also with ∃ even more so. For P finite to begin with: How Pa is recognized for an arbitrary a makes no difference to us. We see that something like this works, apparently with great certainty, and that we accept. What is constructive lies therein that one says: ∀xPx can be stated (and ⊃ should hold) as soon as Pa has been shown. Clearer with ∃.Buta∀xPx without having been proved is senseless. The- refore, e.g., not actually allowable as an assumption! FI , ⊃ problems. ∃ is actually the most innocuous sign. The complications grow properly with the nesting of one above the other. Already ∃ below ∀ ! ∀x∃yFxy. Can one simply retain the usual explanation here? Must one not speak (like Weyl) about the givenness of a procedure? Plan for section III [of Gentzen 1936]: First, after the finite, the in-itself conception in the infinite that corresponds just to the one in the finite. We reject this conception. Then the development of the intuitionistic point of view in words, as in INH 21–22, (possibly the translation theorem of Gödel to be given at the same.) Then criticism that goes deeper, as still to be developed.

About INH 27 middle: Is my reduction procedure, then, natural? The idea of contentful correctness brings along, when doing inferences, a knowledge of the possibility of certain reduction procedures, say in the case of ∃, about the possibility of constructing examples, and this is sufficient for the idea. My idea of reduction, instead, requires a determinate, generally valid procedure for arbitrary derivations that is quite universally applicable and that grants their reduction. Is it natural to require such a thing? Isn’t the previous conception about bringing along the knowledge of certain possibilities of reduction more natural? In any case, the former need not follow from the latter. Sure, the semi- contentful conducting of a proof gives also the result: the endformula is reducible. But this doesn’t state now that just the prescribed reduction procedure, precisely, does the job. The passage from the concept reducible for derivations into reducible for single sequents would already really be a passage from the reduction procedure to the semi-contentful method! It is good that I have thereby found a further part in between the sequence contentful . . . purely-formal. It leads to 10 Formal conception of correctness in arithmetic II 203 further considerations about the connections between the different methods. Compare page 28 top middle. Overview all of page 28. Meaning of ∀xPx:Wehaveaprocedure by which we construct, for each given ν, a derivation for Pν that is already recognized as allowed. There is hidden in “each” already the metatheoretical ∀. However, nothing more can be given than a confirmation that the formal inferences are in accordance with a constructive sense. It is not proved that this giving of sense is constructive, but it is seen in words. Hereby any proper proving is set aside. Namely, because the difference between in-itself and constructive is not conceived formally. But just recognized after its sense. And this is how it must be, this is then the last, extra-mathematical basis. This seems to me to be the correct conception overall for the theory without ⊃. A substantial proof of consistency is here quite impossible, just a demonstration of the constructivity of the ways of inference. I say also that no reduction back is achieved with the theory without ⊃, but just a confirmation. [Margin has: Comp. WAV 58 bottom –59.] No reduction back to something simpler, but a confirmation of constructivity.

INH 30 Main question: What value does my proof of consistency still have for the theory with ⊃? I will admit with ⊃ that one can consider it more dubitable than ∀∃& ∨. Not must. Thence, the proof of consistency itself should avoid ⊃ when possible, or at least those of its uses that are “dubitable” under just this conception. (Let’s say: Uses of ⊃ in the antecedent part of an implication, as in (A ⊃ B) ⊃ C.) Is anything at all gained in this respect? This question is treated on page . . . One achievement of the proof of consistency by the reducibility of derivations that can be brought up is the following. Important for proof theory:Saythat ∃xAx has been proved, and it gives at once a reduction procedure that allows to calculate the example. Clearly, this has no implications for consistency, by what was said about it above (previous page). But it does have for proof theory as such.

Is there or is there not a circle in the intuitionistic ⊃-explanation? Important passages for this so far: Against the circle: INH 26 bottom and 27 top. For the circle: WAV 42, GRO 27–28 top. INH 16, lines 7 to 17, line 9. 204 Part III: The original writings

Also INH 21 is important, cf. that there ⊃ is not brought up, but it is shown that the situation is different with &∀∨∃. In general: (WAV 2, lines 5–6.) I am again inclined to admit the circle. The procedure of giving sense is, after all, this one: The giving of sense to the propositions and the demonstration of correctness of the inferences go hand in hand. The latter is: Agreement of these with the sense given to the propositions. (The same is the case also with the proof of consistency, with the concept of reducibility!) It should be the case that the giving of sense is in general based on the existence of derivations. For example ∀xF x with F finite means: There is for each number ν a finite derivation of Fν.TherebyalsoCI is in order. The possibility to see this for each number depends on the following. Either the derivation is the same for each number, just substitute the number at one place (UI ), or it grows according to a prescribed scheme (by CI ) with growing numbers. ∀∀ and ∀F with F finite are cleared analogously. ∃xF x with F finite means: There is a finite derivation of an Fν with a determinate ν. (“There is” means always that one knows it, can produce it. So one knows also the ν, it is contained therein.) One should be now able to generalize at the same: ∀xF x means, because F is arbitrary though already explained: There is for each number ν a derivation of Fν that is already allowed. Analogously for ∃. But now, what does “already allowed” mean? Here one should have, after all, considered already derivations.

INH 31 So, a foundation through the giving of sense should succeed only as follows. One considers any one derivation and gives relating to it in turn a ﬁnite sense to the single propositions that appear in the derivation. Thereby one also demonstrates that the inference ﬁgures that occur are in accordance with the given sense. To give sense to a proposition means: To associate to it a proposition one is prepared to accept as (allowed and) precisely determined in its sense. What sense should one, now, quite generally give to an assumption in a derivation? The derivation could begin with such. (With FI , otherwise not!). It should be possible to do this with the “secure” assumptions (OE, EE, CI ). But with “wild” assumptions, i.e., FI -assumptions? 10 Formal conception of correctness in arithmetic II 205

One could have doubts about assumptions in general. Perhaps only the way of giving sense of the first consistency proof would help here. One must in any case give expressly an explanation of when an indirect part of a derivation can be seen as “correct.” One is tempted to say: The part is correct if it demonstrates by ways of inference already recognized as correct: When the assumption A holds, also the result B holds. But: The concept of holding does already assume the fact that the proposition in question was derivable! Or rather: It states that there “exists” a correct derivation. It is, however, impossible to put such a precondition when one is about to define the concept of a “correct derivation!” [Cancelled: The matter would now be in order.] All the same, the matter can be saved with OE, EE, CI by the following. The derivation at hand is there already and has been shown correct in advance. Hereby even the assumption in question is a secure assumption. [Margin: Comp. page 21.] I.e., not a proper assumption, but a proposition already derived in itself, just that a substitution of a variable (EE, CI ) or the choice of a possibility (OE) is still needed. (With CI there appears metatheoretically a CI .) We come now also to the possibility of allowing wild assumptions under certain restrictions: Namely, when one has already finished the definition of a correct derivation without wild assumptions, one can then afterwards undertake an extension to allow also derivations with wild assumptions. These latter, however, must state only: That the assumption A holds means: Let there be a derivation of the previous type for it. If one incorporated the derivations of the new type, one would commit the circle. [Margin: Example: proof of (A ⊃ B) ⊃ C.] Hereby with have, in full clarity, the intuitionistic circle. [Margin: “The circulus vitiosus in intuitionistic number theory!”] I would wish to see even more markedly: 1. Why the circle does not appear already with ∀∃, 2. what train of thought led to the rejection of the circle, and whether this could somehow be recognized. [Margin: More on this p. 34 middle.] About 1.) Certainly, also with ∀∃ the giving of sense is founded on the existence of a derivation in which ∀ and ∃ can occur, even more nested. These have been, however, provided with a sense in advance and the derivation in question has consequently been shown correct in advance.With⊃,thisis achieved properly only by my first consistency proof.

INH 32 About 2.) The rejection of the circle should result only if one does not base 206 Part III: The original writings the giving of sense on the existence of certain derivations, but on a kind of “demonstratedness” (page 26 bottom). But what else can this mean than the existence of a derivation? If it mustn’t be an “in-itself” concept! Say the example: (A ⊃ B) ⊃ C. What sense should one give to a proposition like that? (E.g., (A ⊃ B&¬B) ⊃¬A) One can require that A, B,andC have been already provided with a sense. I wouldn’t be able to say anything proper here. One explanation of the circle: Let (A ⊃ B) ⊃ C have been derived. Let us assume it were possible that the whole had again the form A ⊃ B, then one could infer: (A ⊃ B) ⊃ C A ⊃ B (A ⊃ B) ⊃ C C This would be apparently circular, because A ⊃ B is here proved through the use of an assumption: “Let A ⊃ B have been proved,” but this is quite dubitable. Clearly, nothing of the kind can occur in this form. Deﬁnitely, however, in the form in which here occurs at some point in the derivation of A ⊃ B an assumption A ⊃ B that disappears later. By which, sure enough, the possibility of a self-confrontation disappears. Anyway, one can have the impression that there is indeed some kind of a circle, and that this fails in some way to have an eﬀect by accidental circumstances. To be sure, the matter remains somewhat hazy. The decision to give sense on the basis of the existence of a derivation requires that the following is not anymore allowed: A giving of sense such as: “∀xF x holds whenever the F , for which a sense has been already given, holds for each ν.” Instead only: “∀xF x holds when, for each ν,aderivation of Fν recognized as correct is known to be at hand.” Just as with ∃,here we also have: A ν must be known.

INH 33 X.34 The proof of correctness through a ﬁnite giving of sense for the theory without ⊃ and ¬. Basics: There follows hand in hand a giving of sense of the formulas and a proof of correctness of the derivations. The giving of sense to a formula follows under the consideration of “existence of a derivation.” 10 Formal conception of correctness in arithmetic II 207

“A general procedure for giving sense” given in advance: ∀xF x holds (and has thereby a sense) when there exists constructively for each ν a derivation for Fν recognized already as correct. ∃xF x holds (and has thereby a sense) when there exists constructively a determinate ν such that a derivation for Fν recognized already as correct exists constructively. & ∨ easily correspondingly. This giving of sense does not yet confer in advance so to say a sense to all formulas. For it has not at all been determined yet what a correct derivation is. One can provide a sense for formulas only stepwise, through the obtaining of correct derivations. Even the concept of a “correct derivation” requires an explanation: It states, more or less, that the ways of inference in a derivation must be in accordance with the sense of the formulas. (Comp. INH 31 top.) All of this becomes even more clear in a formal execution. Both explanations, the sense of formulas and the correctness of derivations, depend on each other. It would therefore be circular to consider them as deﬁnitions in advance. They work simply as guidelines for the stepwise procedure for giving sense and demonstrating correctness, to be pursued in what follows. (Resp. one has to convince oneself with the procedure that one is in accordance with both of these two explanations. For one makes later always use of this accordance.)

1.) Theory with mathematical axioms and AI, AE, UI, UE, OI, OE. I.e., without assumptions. One goes through the derivation at hand simply beginning from the top. Mathematical axioms “hold” according to a special proof. In passing ahead by one inference figure, the holding is taken over. Some attention for free variables with UI. I am still missing the accordance of the giving of sense with the finite sense of finite propositions. For example, how do we know that 1 = 1 cannot be derived? One cannot, after all, found something simply on the existence of a “correct” derivation. Or rather: One must know more about this derivation. Well, the explanation of sense does have to begin like this: ∀xF x with F finite holds when there is,foreachFν,afinite derivation. (Page 30 bottom). So what does ∀x∀yFxy now mean? (Really explained according to build-up.) For one is not allowed to say: It holds for each y: 208 Part III: The original writings

∀xF x in the previous sense. (For “it holds” would be an “in itself”). But say: it is shown for each y that ∀xF x holds in the previous sense?

INH 34 This, though, does not work! One has to add to it the meaning of: it “has been shown.” (Otherwise one could do the same with ⊃, with: It has been shown that when A holds, also B holds.) So one needs, after all, the “correctness of derivations.” My “reducibility” concept seems to me to be the best one, applied to formulas. By considering “reducible” as a knowledge of the procedure ... one should be able to avoid the diﬃculties here. (The ∀∃ nesting for example, and nesting on the whole.)

Perhaps there is, after all, no circle with ⊃, or better, things are as follows: One presupposes already with the previous signs a kind of “in itself” sense. This is admittedly somehow of a constructive kind. Though with ⊃ one can at most say: The in-itself concept is here essentially more complex than with the previous signs, the difference taken relatively,notabsolutely.Thereisa relatively strong difference and, since a proof of consistency is possible here, this [in-itself concept] is most recommendable. To diminish the preconditions. For there occurs, admittedly, say also by ∀ taken contentfully, an “in itself”- every. The justification by the “existence of a derivation” does not, after all, seem to be a real justification. Cannot even at all. “Correctness” presupposes again contentful knowledge about the sense of ∀, and this is as much an “in- itself”-sense, if you wish.

It is singular how I oscillate between the conceptions. Each time I am at one side, it appears to me to be really nicely illuminating and the other one dubious!

[Margin: Date ? ] Certainly, ∀ and ∃ are led back to a contentful every and there is, though not to the really dubitable “in itself”-every or there is. There is with ∃ even a clear diﬀerence. The question is: In what way does the in-itself-∀ diﬀer from the constructive ∀, even if they obey the same ways of inference? And with ∃? The constructive explanation leads back to something that is “conceptually simpler.” In the case of ⊃, one can debate whether there is something simpler at hand. Even the “idea of reducibility” is important. 11 Proof theory of number theory 209

11. Proof theory of number theory

BZ 1 VIII.34 Transformation of derivations. Attempt at an introduction of the HA-normal form (prenex normal form). (HA page 63, 64.) The basis is: Calculus LK+CI and mathematical axioms. The latter ones shall already have the HA normal form. The sign ⊃ is forbidden. I change the derivation to begin with as follows: 1. Each DS-formula [HS Formel,one in a sequent in a derivation, in a Herleitungssequenz] will be transformed into its normal form (always intended as HA-form). Now the single inference ﬁgures have to be made correct. D → D remains correct. CI remains correct: Pa | Pa P1 | Ph

For all P’sarechangedinthesameway. Structural inference figures: remain correct, because whole formulas are always changed in the same way. Inference figures for the logical signs:1 those for ∀ and ∃ remain correct. &, ∨: AE: A & B → A.LetA become A∗.NowA∗ & B∗ → A∗ holds anyway. Further, A∗ & B∗ is equivalent to (A & B)∗ as is known. Then also (A & B)∗ → A∗ holds. This is further derivable so that all DS-formulas have normal form, by the “sharpened Hauptsatz” of my thesis. The same consideration applies now also to OI,aswellasAI, OE in the forms A, B → A & BA∨ B → A, B . For also A∗ ∨ B∗ is equivalent to (A ∨ B)∗. (HA page 64) Now the inference figures for ¬. (These must require a special treatment. ? For example:

∀xFx| ∀xFx| |¬∀xFx would become: |∃x ¬Fx

F a ﬁnitary predicate). How would it be in full analogy to the above: I consider A, ¬A → and → A, ¬A.

1 Gentzen is here using a calculus that has groundsequents for the connectives and just two rules for the quantiﬁers. 210 Part III: The original writings

Can’t one say simply : The normal form is equivalent to what was there. Consequently the sequent remains logically correct. Consequently it is, by the sharpened Hauptsatz, provable through a derivation with only normal forms!2 This holds for all the cases treated, also the above ones with &, ∨. Consequences: The propositional connectives ⊃∨¬⊃do not occur anymore transﬁnitely in the derivation.3 I.e., not placed on top of ∀ and ∃. Consequences for the proof of consistency: See GRO 28.

X.34 Replacement of ∃ signs through functions: It should be provable in the theory without ⊃, ¬ that one can do without the existence signs, and especially to replace by functions. Wherein the recursive mode of definition appears where there was a CI for the proof of a ∀x∃y proposition. This does not seem to me to be of special value, because the recursive definitions are as dubious as the existence inferences. (I.e., both equally little dubious.) What happens when ⊃ and ¬ are added, is unknown. Nothing of the above kind should go through, already because of the completely different sense of the propositions, as, say: ∃xAx ⊃∃xBx.

BZ2

X.34 The change into prenex normal form, carried through in detail. For publication. [Written above: Originally ↓ .] More in WAV 56. Is here 4 1). A single step in the transformation of one formula goes like this: (concerning a subformula in it) Goal: to bring the propositional connectives over the quantifiers inward. One can choose a place at which these hit against each other. So: ¬∀xFx becomes ∃x¬Fx ¬∃xFx becomes ∀x¬Fx A & ∀xFx becomes ∀y(A & Fy) correspondingly ∀xFx & A y the next variable that occurs, to avoid collisions with possible y in A compare HA page 64 bottom. Similarly for ∨ with ∀. 2 Gentzen uses here as an argument the completeness of the calculus in the same way as in proofs of closure with respect to the cut rule, instead of cut elimination through proof transformation. 3 Gentzen uses Hilbert’s terminology in which quantifiers are “transfinite” operations. 4 The two “triangulated” additions indicate that the text that follows is the original version, that of WAV a further development. This passage suggests that the V in WAV stands for Veröffentlichung, publication, with WA standing for Widerspruchsfreiheit der Arithmetik. 11 Proof theory of number theory 211

Now &, ∨ with ∃: Similarly. One can say simply: In place of & there can be ∨,inplaceof∀ also ∃, all things remain the same. 4 cases [Written below 4: 8.] altogether. It is clearly seen that the procedure leads to prenex normal form. (One could, say, assign to each formula a number, its “distance from prenex normal form,” defined as follows: One counts for each propositional connective the ∀ and ∃ that stand inside it and sums all this. This number becomes smaller with each step so that it must become 0 in the end, and this is the case only with the prenex normal form.) (Besides, the order of the steps makes no difference, as is easily seen, so the prenex normal form is uniquely determined. (Up to the choice of the bound variable “y,” which is not essential.) → No. Not quite unique. E.g., ∀xA & ∀yB can become ∀x∀y A & B or ∀y∀x A & B! Easy to lift through a prescription. Though not necessary. For the publication: give an example. 2) Now the application to the proof theory of number theory: The claim is: A derivation can be so transformed that only prenex normal forms of formulas appear as DS-formulas. The endsequent is hereby transformed so that its formulas are replaced by prenex normal forms. So it remains classically equivalent. 1. transformation: Each formula is transformed into prenex normal form. All inference figures, i.e., the structural inference figures as well as UI, EE and CI go over into inference figures of the same kind. (If one chose the y in upper and lower sequents appropriately in agreement, which can be always done, not only as long as the derivation had a tree form.) There remain still the groundsequents. here plausibly as above [written above: page 1]. (With the sharpened Hauptsatz even proved already.) 2.) A more precise proof would go as follows: E.g. → A, ¬A.Onegetsin the transformation first → A∗, ¬A∗.LetA∗ be prenex. Now ¬ should still ∗ ∗ 5 be pushed inside A .LetA be ∀x1 ... ∀xνB. B without ∀, ∃.Nowthe groundsequent is → B, ¬B. We show by an example how it goes: Let A∗ = ∀x∃y∀zB.Then → A, ¬A becomes: → ∀x∃y∀zB(x, y, z), ¬∀u∃v∀wB(u, v, w) with u, v, w altogether new variables, not earlier in B. Derivation of this one

5∃ written below both ∀’s, to indicate an analogous formula with existential quantiﬁers. 212 Part III: The original writings as follows:6 EI → B(r, s, t), ¬B(r, s, t) UI Care for the variable condition! → , ∀v∃w... EI → ∀zB(r, s, z), UI → ∃y∀zB(r, y, z), → , ∀v∃w... etc.

The also in print! Facilitates! The same with NI,onlythatUE,EE have inference ﬁgures. 1.) Quite the same with & and ∨ inferences. Example: A, B → A & B. Becomes: E.g., ∀x∃y∀zA×xyz, ∃u∀vB×uv, → ∀r∃l somehow in order. The ∀mA×rst & B×rm

BZ3 This is derived as follows: Starting with

abc de abc de A× , B× → A× & B× UE again altogether new variables ∀vB UI ∀m etc, always the × inference figure without variable condition first, then the one with variable condition. So it goes always, no matter what the order is. The order of the inference figures follows the order of the prefixes of the A× & B×. It is unique.

XI.34 A sort of analogue of the previous in the intuitionistic calculus: Transformation of derivations so that there appear just a few, simple ways of inference: Reduction so that intuitionistic ∀, ¬ are the only transfinite As is known, reducible to intuitionistic &, ∀, ¬. (By first reducing classically to &, ∀, ¬, then using the provability of ¬¬A → A for such formulas, to eliminate the TND.) Can one reduce further into intuitionistic ∀, ¬ as the only transfinite operations? 6An unfinished derivation is written, then cancelled and the following added: The general procedure is completely clear. 11 Proof theory of number theory 213

By producing first, in the classical, prenex normal forms, then replacing ∃ by ¬∀, one obtains classically equivalent formulas. The question is just if derivability by the accompanying ways of inference is maintained. Investigation about this: (I lay as a basis the calculus LI + ¬¬A → A and CI. All DS-formulas are replaced by prenex normal form and further ∃ with ¬∀. Better an even more immediate passage in which for example an outermost ¬ remains directly outermost. Now? We have as on page 2: Structural inference figures, structural groundsequents, CI, UI, UE remain of the same kind. Same with the ¬ inference figures and groundsequents (¬¬A → A), because the ¬ remains outermost. Even better should be: I’ll lay at once as a basis the calculus with just &, ∀, ¬. In any case still with ¬¬A → A Production of the “normal form”: The &’s are to be shifted inward. A & ∀xFx becomes ∀y(A & Fy). Similarly ∀xFx & A. A & ¬B becomes: I can assume without limitation that B is equal to ¬C or to ∀xFx.ForifB contained yet another transfinite &, this latter would be shifted inward first. In case B = ¬C,weobtainA & C.IncaseB = ∀xFx, we obtain: (A & ∃x¬Fx, ∃y(A & ¬Fy)) ¬∀y¬(A & ¬Fy) continuation obvious. In this way each & is brought inward step by step, into what is finitary. Now the transformation of a derivation: All DS-formulas are brought to normal form. Correct remain: CI, ∀,and¬ inferences, structural inferences. There remain to be investigated only the & inferences. These can be taken as groundsequents: A & B → A, A & B → B; A, B → A & B. Can I now just infer simply: The transformed sequents are classically correct, and because they do not contain any ∨, ∃, they are also intuitionistically correct, that is, derivable in LI ?

BZ4 Should be all right. Though, since an essentially mathematical derivation is in question, I will not be able to use the cut theorem [the word Haupt is changed into Schnitt]. I must therefore still show in detail that the derivation in question can be carried through with only formulas in normal form. It seems to me pretty sure now that all of this holds. I’ll try still with A, B → A & B. [Cancelled: I proceed as follows: I begin with A, B → A & B and transform the formulas step after step into normal form, the new sequent is derived at each step.] Senseless. For the main thing is: The derivation must have normal form of formulas! Next: Transformations inside A and B can be made right in the beginning, at the same time on both sides. Then we get 214 Part III: The original writings

A∗, B∗ → A∗ & B∗. Now there occurs & at right. I leave out the ∗. [Cancelled to 3.] Cases: B & ∀xFx; ¬¬C; ¬∀xFx:(IfA, correspondingly.) 1) We have A, ∀xFx → ∀y(A & Fy) I produce A, Fz → A & Fz ———————–UE, UI ———————– the above The &-groundsequent is handled in the same way. 2) We have A, ¬¬C → A & C I produce ¬¬C → CA, C → A & C ... Cut 3)[End of canceling] I should be able to infer as follows: Let the claim be proved for a sequent the transformation of which requires one step less! E.g., in the case of B = ∀xFx: Let the claim have been proved for the sequent A, Fz → A & Fz.Letthis be in transformation: A, Fz → (A & Fz)∗ So let this be already correctly derivable. Obviously A & B reads in transformation ∀y(A & Fy)∗.Sothe above works, in fact, one adds UE and UI.InthecaseofB = ¬¬C,let A, C → A & C be already cleared, it becomes A, C → (A & C)∗.Letalso A & B read when transformed, there first appearing A & C,as(A & C)∗.We get from A, C → (A & C)∗ now A, B → (A & C)∗ [above B stands ¬¬C]by ¬¬C → C. This is a groundsequent, and it has normal form, because C (like B [below B stands A]) has normal form. 3. Case: B = ¬∀xFx.FromA, B → A & B we get A, B → ¬∀y¬ (A & ¬Fy)∗.LetA, ¬Fy → A & ¬Fz have been already cleared, so that A, ¬Fy → (A & ¬Fz)∗ is already derivable. By ∀y¬(y)∗ → ¬(z)∗ and the above one gets through RA A, ¬Fy → ¬∀y¬(y)∗, finished. I hope that all of this is already in order, it is still a bit sketchy. To eliminate in addition ¬¬A → A, it has to be shown that in its derivation, the normal form of formulas can be always maintained. (One could presumably eliminate ¬¬A → A also in advance, then write in the above case of A & ¬¬B not A & B, but ¬¬(A & B).) This should be possible. (Is made even simpler by not having & occur transfinitely in A.) Hereby the reduction to theory with only intuitionistic ∀ and ¬ as transfinite operations would be achieved. In detail in WAV 83–86, the next. 11 Proof theory of number theory 215

BZ5

XI.34 Transformation of CI’s,intooneswith a single anterior part In connection with WTZ 15 bottom. Calculus LK +CI as a foundation. It results from considerations on the assignment of values (WTZ 15) that vi(a, ν) [apparently for vollst¨andige Induktion, complete induction] is the same in value as something like (Value: ak(a, ω, ν +1)) sc(a, ω, ν +1) If a corresponding transformation of derivation can be found, this should happen more or less as follows: Some CI is substituted, through a cut, one anterior part of which contains again a CI , though this time of the most elementary kind, no matter how complicated the previous one was. The CI is: α Fa → Fa α F1 → Fh → ∀x(Fx⊃ Fx)

It seems to me that this results simply in the trivial formulation:

etc UIA α Fb, → Fb CI → ∀x(Fx⊃ Fx) F1, ∀x(Fx ⊃ Fx) → Fh F1 → Fh

The CI has now, in fact, above it quite a trivial derivation, more or less because the essential ∀x(Fx ⊃ Fx) is pulled through it and only later proved with all the difficulties! Does there now reappear, with “reduction,” again the former form? No! Not with the CI .Firsth has to be rendered a number and the CI destroyed, later, when one has cuts all the way through, the normal order of things reappears again. So: For the consistency proof, one can easily restrict oneself to CI that are not nested one above the other, indeed, even more specifically, altogether on CI ’s of the above form. Like using the induction axiom insteadofthe inference. That much, more or less, should be cleared? See 6.1. exact realization. It seems to me that this consideration leads at the same time over to the grounds for the harmlessness of CI in analysis. All that is decisive resides in the cut, precisely because cut is what reorgani- zes the rest, so that it becomes harmless. Specifically, to transform the CI ’s into quite simple CI ’s. 216 Part III: The original writings

In the theory without ⊃, the above will not yet go through, therefore this theory is even more harmless. Therefore the complications through ⊃. The reformulation should have no importance for the proof theory of number theory.

Uniﬁcation of complete inductions into a single one??

IformulateeachCI as an axiom: ∀x(Fx ⊃ Fx) ⊃∀y(F1 ⊃ Fy) There can still appear free variables therein. To be able to pull the axiom through down, I have to generalize it (compare the “deduction theorem.”) Thereupon the pulling down shall be possible. One writes on top each time A → A, the induction axiom A with ∀ in the head for the free variable, and pulls A through. Now the endsequent is:

A1,...Aμ, B1..Bν → C B the mathematical axioms, A the CI ’s. C endresult.

F1 ..Fμ the F of the single A.

BZ6 [Cancelled: Can one now replace the totality of the A by:

∀u1 ...∀u ∀xF1x & ...& F The u the freely occurring variables in F, each named differently.] How is it then. Two completely independent CI ’s can, after all, be united: Fa → Fa G → Ga F1 → ∀xFx and G1 → ∀xGx into:7 Fa → Fa Ga → Ga AIS,AIA Fa & Ga → Fa & Ga CI F1, G1 → & F1&G1 → ∀x Fx & Gx ———————————————— F1, G1 → ∀x Fx & ∀xGx This is admittedly weaker than what there was earlier, one doesn’t have anymore F1 → ∀xFx alone. The whole of it shouldn’t go through anymore if the CI ’s are in some way nested. Think just something like having ∀xFx = G1 ! It is already completely beyond. 7 The example is unfinished. Parentheses are missing after the quantifier in the conclusion of CI, the sequent F1, G1 → ∀x(Fx & Gx). The example is completed by taking this conclusion of CI and a cut with the derivable sequent ∀x(Fx & Gx) → ∀xFx & ∀xGx, to obtain F1, G1 → ∀xFx & ∀xGx 11 Proof theory of number theory 217

Therefore: A uniﬁcation into a single CI does not seem possible. Might it just go in some way? The considerations about assignment of values suggest it! Yes. Page 6.2. [Margin: Continuation page 6.2! ] Limitation of cuts. The derivation of the sequent A induction axioms, B mathematical axioms

A1 ...Aμ, B1 ...Bν → C can naturally be kept cut free, by the cut theorem. There will then appear cuts with the reduction of the CI ’s.8 Another formulation, with simpler CI ’s and therefore again cuts, arises as follows: . . . . . A1 A1,...A ... → C . μ . A2 A2 ... → C . . . . Aμ . B1 ...Bν → C Should work in LK and also in LI .

Each of the A is derived through the use of a CI inference ﬁgure and a few further steps of inference (in the end UI’s for the free variables.) Or remains standing as an induction axiom. &-binding of the A, compare 6.1, and can be carried through trivially in the L-calculus, each side with AIA resp. AIS, so that only one cut remains. This means for the transﬁnite assignment of values (WTZ 15) that the overall value can be taken in the form:

... sc(ω, sc(ω, sc(ω, μ, ν1),ν2),ν3) ... (should be ak(ω, ω, ))

μ the ﬁnite value of the derivation of A1 ...Aμ ...Bν → C,νthe grade of the cut formulas.

BZ9

I.35. The concept of “the one such that” in pure number theory.9 For this can be written, as is known: ∃x(A(x)&∀y[A(y) ⊃ x = y])

8 Margin has the addition: The B are the mathematical axioms, can remain. .It replaces the cancelled addition: Mathematical axioms!? They should also be taken along in the inference. 9 top of page has: Building of functions from predicates (“the one such that,” “some one such that,” eliminability [?] again etc.) 218 Part III: The original writings

Introduction according to Bernays: One can introduce on the basis of ∃xA(x) and ∀x∀y(Ax & Ay ⊃ x = y) as a new expression for an object (term) ιxA(x), and one has at the same: A(ιxA(x)). (It requires a new sign, can keep for now to ι.) A can contain free variables, by which ι represents a function.

The reinterpretation of formulas with ι. Old investigations: VOR 14–19. An example: ∃xAxa ∀x∀y(Axa & Aya ⊃ x = y) one introduces a ιxAxa = δa for short. We have A(δa,a). That is, ∀zA(δz,z) Now there appears something like: ∀xB(δx,x)&C(δ2) ⊃∀y[D(δy) ⊃ E(δy)] This has to be interpreted by a formula (or sequent) without δ. One would add, to start with: ∀u∃xAxu & ∀u∀x∀y(Axu & Ayu ⊃ x=y) I.e., the use of a ι involves its existence and uniqueness for all argument values. The old method was wider: [Cancelled: Each elementary form with δab contains, brought forward with ⊃: VOR 16, marked. There emerges with the example: ∀x(∀u(Aux⊃Bux)) & ∀u(Au2⊃Cu)⊃∀y[∀u(Auy ⊃Du)⊃∀u(Auy ⊃Eu)] Is this interpretation in order? One should conduct a general demonstration that derivations remain correct. I choose as the calculus the number- theoretic calculus of my treatise on consistency. Let a derivation be given, and it is freed of ι-terms through the above interpretation, other additions do not take place. To be shown that it has remained correct, logical groundsequents have remained logical groundsequents, mathematical are [un?]changed because mathematical axioms do not yet contain any ι-terms. By the ι-introduction prescription, there can further have appeared as sequents that hold: Γ → A(ιxA(x)). These read after the change: [Cancelled: Γ → ] a bit difficult. I put this still back. It seems already to be a main case. Now the rules of inference: A form is adjusted in the same way in different places. Therefore the only difficult ones are the inferences with a t of their own, i.e., UE, EI, CI . → ∀xFx t UE: Earlier → Ft so, shall be always a term that contains no ι. Needless to say, t contains only free variables. 11 Proof theory of number theory 219

If I consider F as elementary and t as the only term with ι, it would read as follows when transformed:

∀u1∀u2 ...(... & ... & ...⊃ Fs)wheres is a term without ι. Can one, now, derive the same from ∀xFx? Apparently trivially: One obtains Fs, therefore also ... & ... & ...⊃ Fs, further the same [?] with ∀ through UI. This should be in order. Then the determination of the, so far, arbitrary u1,u2 ...on ι-terms that are distorted with the formulas that are connected through & and pulled ahead, yes, more precise [weak original here]. It is important that they are in accordance with the terms that read the same within F! This, though, is a secondary

BZ10 thing, belongs to the task of deriving again the original Ft from the auxiliary formula ∀u1 ..., that is doubtless the main problem. One will have a very general equivalence of derivability to be demonstrated, one that clears all the remaining cases (similarly to the examples in VOR 15, 17.) Two are the concerns here: First, the equivalence of the choice in the head or aft of a new connective (as long as this one doesn’t bind an associated variable), secondly the equivalence of the choice for a speciﬁc term with multiple occurrence multiply or just once. There can be here, with the above formula Fs, obviously just one single choice for the multiply occurring s, because it has to maintain the same reading all over, whereas in the shift of the choice inwards, on elementary formulas, each single place has its own choice. Comparison for “Fifth example” of VOR 16 top. The best way to proceed should be to always choose the same parts of terms within a domain from which the choices are to be made. So: The proof procedure is to be shaped as follows: 1) Deﬁnition of the choice that belongs to a ι, in front of a formula in which no variables subordinate to ι are bound. 2) Demonstration of the equivalence of formulas in which the choice, further in or out, takes place. This is the main task. Could anyway be a rather tedious one. 3) Clearing out of the rules of inference, should be trivial at this point. To attend that the t is always without bound variables in the case of an eigen-t, i.e., that the equivalence is usable without further ado. The same with the A(ιxA(x)) that arise with the introduction, for these read with the global choice: Au ⊃ Au. 220 Part III: The original writings

I think the whole of it can be carried through without special diﬃculties. Use of the unicity conditions: Existence is needed when a ι-term disappears, e.g., with EI. Diﬀerently from UE, the chosen members have to be eliminated here, the sum total [?] of existential statements for the ι. Uniqueness is needed with the proof of equivalence, when several formally equal terms have to be resumed together; should be.

The reduction of a ι in the proof of reducibility. Reduction [?] not eliminability of ι by the above method to prove in advance, then one can take into consideration in the reducibility (consistency) proof ι-formulas after direct reduction. Like: One considers a ι-term only when it contains no bound variables anymore. Then it contains no free variables, either, on the basis of the reduction. (It can contain nested ι.) Let one such ι-term be reduced like this: The A(u) that belongs to it is chosen, and u for the written one. u is a free variable, so to be substituted immediately by an arbitrary number! It will be seen later if this number is an example, i.e., this need not show itself. In this it comes close to Ackermann’s ε-substitution method! Should be compared some time! So is the ι-reduction thereby already in order? The reduction of a ι in an antecedent formula is still lacking. The previous goes only for succedent formulas. (Supposing it is at all correct.) When is such at all needed?

III.35. The unity formulas state reducibility of the sequents ∀x¬A(x) → 1=2 for arbitrary values of the free variables that occur.

BZ11 Reducibility of C → C. There occurs a ι-term reduction. This takes place first at the back. Say A(u), C(t) → C(u) is formed. An arbitrary number enters in place of u. What is the sense of a ι-term in the antecedent? E.g., ¬B(t). Is ∀u[A(u) ⊃ ¬B(u)] the same as ¬∀u(A(u) ⊃ B(u))? The latter states ∃u¬(A(u) ⊃ B(u)), i.e., ∃u(A(u)&¬B(u)). Because there is just one u such that A(u), it is this one, so ∀u[A(u) ⊃¬B(u)] holds at once, in fact! The unicity of both parts should, however, be used in an essential way here. Reduction in the anterior part must therefore take place like this: In the 11 Proof theory of number theory 221 head of [?] B(t), Γ → 1=2. B(t) states ∀u[A(u) ⊃ B(u)], i.e., ∀u¬(A(u)&¬B(u)) in the writing without ⊃. What if one now simply substitutes B hereby ? (Seems that there is nothing to do further ahead, because the reduction of ∀ is still quite pointwise.) If the procedure goes up from the back, then C → C should be trivial to reduce, as so far. Admittedly a somewhat modified concept of simplicity, instead of the grade of C,theι-terms are to be assigned values. It should go. There arises then with each single rule of inference the equivalent to point 2, previous page. Iprefernot to bring this in the consistency proof. The simplification should not be substantial against the procedure of the previous page. And the same stands already in Bernays, and is just a side issue for consistency. 222 Part III: The original writings

12. Consistency of arithmetic, for publication

WAV55 2.1 Connectives. & introduction: A, B → A & B & elimination A & B → A .. ∨ introduction: [left unwritten] ∀ elimination ∀xFx → Fh Fh →∃xFx 2,3 ¬ elimination: ¬ introduction A, ¬A →→A, ¬A law of contradiction law of excluded middle 3. Complete induction now also [cancelled: symmetrically] with arbitrary succedent formulas: Remarks: Negation does not have its special position anymore, but there belongs to it, too, an introduction and an elimination inference. The new formulation of the forms of inference is equal to the old one. In the sense: [missing] The proof is not difficult, shall not be carried through. (Reference to my dissertation where this more or less stands.) [Added in margin: Possibly some prominent cases to be carried through, such as derivations of both of the old ¬ inferences in the new system.] The mathematical axioms and concepts are not changed. [Added: But yes: ⊃ must be substituted by ∨, ¬ ! For otherwise the ⊃ inferences are missing.] 2.) I want to show now that the transfinite use of the connectives &, ∨, ⊃ and ¬ can be avoided in the present formulation, in the same way as this took place in the previous paragraph with the connectives ¬ and ∨, ∃. Hereby number theory is reduced back to a theory in which only the connectives ∀ and ∃ remain used transfinitely, for which the consistency is proved in 3. HA normal form [Added at right: Prenex normal form. Hereby the possibility: Transfinite & ∨⊃¬can be assigned a “sense,” only ∀∃replaced by transfinite, otherwise finite connectives.] The basic idea in the elimination of the transfinite &, ∨,and¬ (⊃ is, as already noted, substitutable by ∨ and ¬) is nothing new: Each formula can be brought in the in-itself-logic [An–Sich–Logik]intowhat is known as the “prenex normal form,” it is a form in which the connectives 1 The whole page has been cancelled. 2 Added below: variable condition the inference 3 Added at right: perhaps more practical as inference figure 12 Consistency of arithmetic, for publication 223

∀ and ∃ occur only in the beginning, that is to say, not in the scope of a connective &, ∨ or ¬.4 The formal form of a formula is in the in-itself-sense equivalent to the original formula.5 The procedure, as applied to an arbitrary formula of our number theory, is the following: (Also after HA p. . .) (The prenex presentation means apparently for such a formula, by the definition, that there is only a finite occurrence of the connectives &, ∨ and ¬.) Thequestionistomovestepbystepthe∀ and ∃ that occur in the formula in its beginning, whereby the formula goes always over into a formula with an equivalent meaning (in the in-itself-sense, exactly defined: according to valuation table, for arbitrary finite domain of objects, easy to ascertain through contentful consideration, or also quite exactly on the basis of the valuation table that this is the case, thought of for arbitrary finite domains, and by the way, not important for us, because anyway sufficiently confirmed through the proof of the substitution theorem that follows.)

WAV56 Production of the prenex normal form in LK, carried out for publication The single step of transformation is done like this: One finds in the formula a place in which there is a ∀ or an ∃ immediately under a&, ∨ or ¬. The subformula in question has one of the following forms, we give at the same the new form into which it is transformed : ¬∀xFx becomes ∃x¬Fx ¬∃xFx into ∀x¬Fx A & ∀xFx becomes ∀y(A&Fy) Here a variable is to be chosen as y that does not occur in the formula. ∀xFx&A becomes ∀y(Fy&A) The remaining cases are treated just as the last two ones. They are obtained when there stands ∨ instead of & and finally also ∃ instead of ∀. Through such a step, each time a ∀ or an ∃ is shifted above a &, ∨ or ¬.It is easily seen that by a sufficient number of such steps, an arbitrary formula can be brought to “prenex normal form.” (The normal form is not [cancelled: quite] unique, but such is not needed for our purposes.) An example: The formula: ¬∀x∃y(x = y ∨∀zz

4 Margin: The same can be carried through for the forms of inference, the lemma. 5 Margin has cancelled: “Third substitution theorem.” 224 Part III: The original writings

It can be brought to the “prenex normal form” through the given procedure: [Margin: to be checked.] ∃t∀u∃v[¬(t = u ∨ v

6 Gentzen’s footnote: Observation: One could use here also the “sharpened Hauptsatz,” number . . of my dissertation. 12 Consistency of arithmetic, for publication 225 inevitably appear in their derivation. This is all the same just a plausibility consideration, there follows now the really exact demonstration. After the proof of the third substitution theorem: The transformation into prenex normal form can be again seen as a kind of giving of sense,thistime for all transfinite &, ∨ and ¬. The sense of these connectives is reduced back to the one of the finite connectives &, ∨, ⊃ and ¬, as well as the transfinite ∀ and ∃. One should bear in mind that ∀ and ∃ have now the in-itself-sense, not something like the sense given to these connectives in § ..,andthatfor this reason, not too much is gained by the [cancelled: proof of consistency] proof of correctness of number theory. The ways of inference ∀ and ∃ in their present form contain all the essential difficulties that were formerly divided among all of the ways of inference (for example, of the transfinitely applied law of excluded middle, only apparently vanished now. The inferences UI and EE in their present form are not intuitionistically acceptable.) (to be checked.) 10

WAV77

10 1. The groundsequents for & and ∨. I treat an example, to start with: Let an AI have become in the transformation to prenex normal form: ∃zA(z), ∃u∀vB(u, v) →∃l∃t∀m[A(t)&B(lm)] Let A and B contain no ∀ and ∃ anymore. This is derived as follows: (Let a, b, c, d, e be variables that haven’t yet oc- curredinthesesequents.)7

7 The symbols a, b have been cancelled from the list and the corresponding variables x, y from the formula Axyz in the antecedent and the variables r, s in the formula A rst in the succedent of the sequent to be derived, and correspondingly in the derivation. 226 Part III: The original writings

8 These words have been put in parentheses with the following parenthetical addition: where the transformation rule stands for ¬. 12 Consistency of arithmetic, for publication 227

→ B rst , ¬ B rst ————————— EI ∃ww ————————— UI ∀zz ————————— EI UI EI UI By this one, the generality should be sufficiently clear: One begins as in the previous case and introduces in order each time a ∀ resp. an ∃ of one formula and an ∃ resp. ∀ [the symbol is not inverted but an A] of the other formula. Here EI must precede UI if the case was one of a groundsequent → A, ¬A, whereas the inference figures EE and UE are needed with A, ¬A →,inwhich case UE must take precedence (because of the variable condition). Hereby the proof of the theorem is finished. One sees clearly in the proof the

WAV78 advantages of the symmetry properties of the new formalism, these make it possible to treat all of the connectives &, ∨, ¬ in an analogous way, as well as ∀ and ∃ analogously to each other. [Cancelled from top of page to this point.]

9. correctness §(7–). The second proof of (consistency). LK consistency proof

By the third substitution theorem, we can limit ourselves to the symmetric calculus without transfinite occurrences of &, ∨, ¬ (⊃ does not occur at all.) The proof of correctness can be also extended without special difficulties so that these connectives are allowed, the third substitution theorem presents no essential part of it. But the consistency proof itself becomes more surveyable through this preparation, more concentrated on what is essential. The whole proof runs analogously to the first proof of correctness, formally simpler than it. We begin again with an explanation of a concept of “reducible” for sequents. [Cancelled: This coincides with the old one for sequents with one succedent formula. So it is just a generalization of the same. (No! Essentially different because of the possibility of multiplication of succedent formulas!)] A sequent with free variables is reducible when each sequent that arises from the substitution of arbitrary numbers for the free variables and the ensuing 228 Part III: The original writings substitution of all terms through their values is reducible in the following sense. Reducibility for sequents without free variables: First, “reduction of a sequent”: Free reductions: A succedent formula of the form ∀xFx or an antecedent formula ∃xFx is substituted by Fν with an arbitrarily chosen ν, a number. Allowed just now only for sequents with transfinite components. So that finite sequents without further ⊃ are reducible only if they are correct. Bound reductions: A succedent formula ∃xFx resp. antecedent formula ∀xFx is substituted by Fν with a determinate ν given somewhere. Term reductions: A term without free variables is substituted by the respective numerical value as determined [?] by the definitions. (Shifted to above, should be needed because of, e.g., EI, AE).

WAV79

Structural reductions: Deletion of any formula as well as addition of an antecedent formula that is there already as an antecedent formula, or the addition of a succedent formula that is there already as a succedent formula. A sequent is called reducible when a procedure is known that allows one to always bring the sequent to a “groundform” through finitely many steps of reduction, in whatever way the number ν is chosen in the case of a free reduction. Groundforms are: 1. [Cancelled: Logical groundform] Sequents of the form A → A, A arbitrary formula [cancelled: without free variables]. 2. [Cancelled: Mathematical groundform] A sequent with just finitary formulas, without free variables, correct also by the definition of correctness. (Applicable to formulas, applicable to sequents to be determined.) (Remark: One can, incidentally, always reduce the first form to the second one, not essential.) [Marginal remark: E.g., 1 = 1 → 1 = 1 can be at the same time a mathematical and a logical groundform.]

Proof of reducibility of an arbitrary sequent derivable in the calculus. 1.) The initial sequents of a derivation: All groundsequents of the inferential calculus have already the groundform [cancelled: and are reducible]. (Be- cause they cannot contain anything transﬁnite.) All mathematical axioms 12 Consistency of arithmetic, for publication 229

(written as sequents) are easily seen to be reducible, as in the first consistency proof. 2.) Structural inference figures except for mix: trivial. 3.) We consider a ∀ or ∃ inference figure, let the upper sequent have already been shown reducible. Then the lower sequent is also trivially reducible, one substitutes for its free variable something, the same in the upper sequent, then one reduces so that the present form of the upper sequent is obtained (by the substitution of a proposition, resp. of an arbitrary a,andthesame above.) 4.) Mix. Let the reducibility of both upper sequents be already shown. That for the lower sequent to be shown. We do a complete induction after the grade of the mix. That is now: The number of ∀ and ∃ attheheadofthe mix formula M. Γ → Δ(M)Θ(M) → Λ Γ, Θ∗ → Δ∗, Λ 1) Let the grade be 0. Then M is a numerical formula, so decidable whether correct or false. If one reduces the left upper sequent by the known pertinent procedure, the upper sequent is thereby also reduced by leaving Θ∗ and Λ at once away. Γ and Δ∗ modified as above. (If M is correct, correspondingly at right.) To be detailed out. It works. A modification of M in the upper sequent is excluded, only multiplication or deletion. Finally, if there is a groundform above, also the lower sequent has assumed a groundform. Because M is false, the sequent that corresponds to the upper sequent through the deletion of M among the succedent formulas must be correct, if the whole sequent was correct.

WAV80 2) Let the grade be >0, proved up to − 1. Each of the upper sequents are reduced separately and their modifications taken over to the lower sequent. This goes as long as an M is modified on both sides. (If M disappears completely from one side, one obtains from the lower sequent the pertinent upper sequent through deletion and reduces further as this one alone, finished.) Let the “critical case” be at hand. Then one has always on one side a bounded case of the substituted ν, on the other freedom of choice. One chooses so that the same ν is substituted. Let the upper sequents immediately before a modification of an M be: Γ → Δ, M, ΠΣ, M, Θ → Λ 230 Part III: The original writings

They turned in their next reduction into: Γ → Δ, M∗, ΠΣ, M∗, Θ → Λ The lower sequent is: Γ, Σ×, Θ× → Δ×, Π×, Λ We write now down the figure: Γ → Δ, M∗, ΠΣ, M, Θ → Λ Γ → Δ, M, ΠΣ, M∗, Θ → Λ Γ, Σ×, Θ× → Δ×, M∗, Π×, Λ Γ, Σ×, M∗, Θ× → Δ×, Π×, Λ Γ, Σ×, Θ× → Δ×, Π×, Λ Here the old lower sequent is a lower sequent of a mix with mix formula M∗, of grade − 1. So we are finished if we can show that both upper sequents of this mix are reducible. We consider one of them, the situation is perfectly symmetric for the other. Say the left one, this one presents itself again as a lower sequent of a mix with the mix formula M and the upper sequents are the old ones, just that one of them is reduced by one further step. So there remains to be shown: When both of these upper sequents are reducible, also the lower sequent is reducible. This claim is of the same kind as the original one, and it is treated in the same way. This continuous leading back of one claim to another must, now, come to an end in a finite number of steps (quite independently of which claim among the two is considered!), for at least one of the two upper sequents is reduced each time, and both are reducible by assumption. (This somewhat peculiar inference in subjected to detailed criticism in Sec- tion IV.) CI (page 81, WAV ) trivial.

WAV83 Elimination of a transfinitely used &. (after BZ 3-4) Each formula with such a & is transformed as follows: One takes first an innermost &, that is, one under which no transfinite & occurs. Let now the right part of & be transfinite: A & B is at hand. [Margin: “subformula” not usable, to be avoided] One transforms: A & ∀xF(x)into:∀y(A & F(y)) all B equal .. A & ¬¬C into: A & C A & ¬∀xF(x)into:¬∀y¬(A & ¬F(y)) If the right part is finite, one proceeds correspondingly at left, better to write down precisely: It means a new choice of y! 12 Consistency of arithmetic, for publication 231

It is easy to see that all & signs are in the end “shifted into the finite” by just such steps, further to consider that the contentful sense of the formula is not changed. This consideration is not necessary, but it makes plausible that the transformation really is possible also with the forms (rules) of inference maintained correct.9 Further: Uniqueness of the transformation. How is it with y?10 ((Possibly shortly on the way how in [?] example.)) Now the derivation: Only the & rules of inference and their places of application should be essentially changed.11 ((&-introduction: It is assumed that the possibility of transformation has been already shown for the case that the transformation of A & B requires one step less. (Eventually to be formulated with the grade of the &.) Γ → A, Δ → B give Γ, Δ → A & B has become: Γ∗ → A∗, Δ∗ → B∗ give Γ∗, Δ∗ → (A&B)∗ [margin: explanation of ∗] I distinguish: B has the form ∀xF(x), ¬C, ¬∀xF(x), as well as C & D (When B finite, A transfinite, all things correspondingly.) 1.) (A & ∀xF(x))∗ is: (∀y(A&F(y)))∗ [margin: UE there now.] The inference figure: Γ → A, Δ → Fa give Γ → A & Fa [margin: a a new free variable.] is after the inductive hypothesis already cleared, that is, one can already reproduce the corresponding consequence: Γ∗ → A∗ and Δ∗ → (Fa)∗ give Γ → (A & Fa)∗ Now the above is transformed like this: Δ∗ → (∀xFx)∗ gives, because (∀xFx)∗ is equal to ∀x(Fx)∗, through ∀ elimination Δ∗ → (Fa)∗, from this one can already obtain Γ → (A&Fa)∗, further through ∀-introduction (variable condition in order because a new) Γ →∀y((A & Fy)∗) is the same as Γ → [∀y(A & F(y))]∗ 2.)

9 Marginal remark: To show more closely how this is concluded by CI , with “ordinal number” or so. Second marginal remark: Important for understanding to give the formulation at least for the somewhat more diﬃcult third case! 10 Marginal remark: Possibly same in meaning with the concept or so, for formulas and sequents (here also exchange of antecedent formulas eventually to be brought in)? 11 The rest of the page is cancelled. 232 Part III: The original writings

WAV84 Maybe the whole of it is better like this: A UE.LetΓ→ A & B,Γ→ B be given. One transforms little by little. To begin with: Let, say, B be equal to ∀xF(x). One makes with a UE: Γ →∀y(A & Fy), gives Γ → A & Fa, → A → Fa →∀xF x gives Γ resp. Γ ,fromthatΓ ( ). B So the first sequent is somewhat transformed. (There is no sense to derive it from the previous one, because it requires UE of the same order.) The transformation is in the direction of the prescribed global transformation. There must follow a further transformation of the new UE of a lower order. How is it with the idea: This procedure is continued precisely so, that is, there appear continuously new interruptions in the derivation, and in the end, that is, when no transfinite &-inferences appear anymore, all formulas are brought to normal form. Is a correct derivation thereby formed? One should suppose so, for all points of interruption are again pasted together by the last operation. Would this already hold? One can consider provisionally correct all inferences that arise from correct ones, when one treats whatever formulas with an arbitrary required number of &-shifts. These become in the end again clearly correct, as long as they were not &-inferences. And these have, then, also disappeared. So perhaps it works. A question: What happens with the case of Γ → A & B with B equal to C&D? Now, I transform into: Γ → A & B∗, where only a single step need happen with B, this leads then to Γ → A resp. Γ → B∗, the latter is already a transformed one of Γ → B,inorder.ThenewUE has a lower order, as follows by some consideration. [Marginal remark: In good correspondence with the procedure in the case of the &-shift itself; the inner one to be treated first.] So, altogether. There is a step by step transformation of the derivation, in which there are provisionally admitted as new rules of inference: Going over to a sequent produced by a &-shift, or the other way around. A global ordering number of the &-inferences to be cleared is diminished with each step. (The sum of the orderings of all the & that occur in their A & B.) This means in the end of all things the disappearance of all &-inferences with a transfinite &. Thereafter the final transformation, all things in order. 12 Consistency of arithmetic, for publication 233

WAV85 In detail: The derivation is first transformed so that transfinite &-inferences disappear. I allow here for the time being as new rules of inference: The going over from one sequent to another when the formulas in the sequents differ from each other by &-shifts. The transformation on this stage happens in single steps of the following kind: The global ordering of &-inferences, given as the sum of the transfinite orderings of their &, diminishes with each single step. Take any transfinite &-inference. Main formula A & B.LetnowB be transfinite. (A transfinite, B finite is treated just correspondingly.)12 Let B contain still transfinite &. Then I shift this into the finite, just as with the B of A & B. Here the inference figure remains of the same kind, but contains a lesser ordering (to think about). And the connection to further previous or following inference figures is to be established again with the “new rule of inference.” Let now B contain no transfinite &. Then I distinguish three cases: B equal to ∀xF(x), ¬¬C, ¬∀xF(x). 1. case. B equal to ∀xF(x). The inference figure can be a &-introduction and one of the two kinds of &-elimination. a) &-introduction. It is13 Γ → A and Δ →∀xF(x)giveΓ, Δ → A & ∀xF(x) There appears from the latter through a &-shift the sequent Γ, Δ →∀y(A & F(y)). This one results from the two others as follows: Δ →∀xF(x)givesΔ→ F(a)(a a new free variable), together with Γ → A:Γ, Δ → A & F(a), further Γ, Δ →∀y(A&F(y)) From this one the old sequent Γ, Δ → A & ∀xF(x), is obtained by the new rule of inference. Hereby the derivation is again composed, and the ordering number diminished. 12 Cancelled: I distinguish five cases: B equal to C&D, ∀xFx¬¬C, ¬∀xFx, ¬(C&D)(No further ones come in question, it seems.) 13 The right margin has at about this level the following text: It would not be sufficient to, say, anticipate: Something like: A formula is inferentially equivalent (herlei- tungsäquivalent) to one that arises through &-shifts. The thing has to be done as it is done here. 234 Part III: The original writings b) A &-elimination: Γ → A & ∀xFx, Γ → A (resp. Γ →∀xFx) is modified as follows: Γ → A&∀xF x gives by the new rule of inference Γ →∀y(A&F(y)), Γ → A&Fa (a new free variable), through &-elimination of a lower ordering there arises Γ → A (resp. Γ → Fa from which at once Γ →∀F). 2. case. B equal to ¬¬C. A &-introduction is: Γ → A and Δ →¬¬C give Γ, Δ → A & ¬¬C. This is turned into: Δ → C, Γ, Δ → A & C, from this one Γ, Δ → A & ¬¬C by the new rule of inference. A &-elimination is: Γ → A & ¬¬C gives Γ → A (resp. Γ →¬¬C). This is turned into: Γ → A & C, Γ → A (resp. Γ → C, from this one with the help of ¬C →¬C Γ →¬¬C). 3. case. B equal to ¬∀xF(x). Another &-introduction: Γ → A, Δ →¬∀xF(x) gives Γ, Δ→A&¬∀xF(x) that becomes: ∀y¬(A & ¬Fy)→∀y¬(A & ¬Fy) gives ∀y¬(A & ¬Fy) →¬(A & ¬Fa), (a new free variable), Γ → A and ¬Fa →¬Fa give Γ, ¬Fa → A & ¬Fa. One obtains with the previous sentence Γ, ∀y¬(A & ¬Fy) →¬¬Fa further Γ, ∀y¬(A & ¬Fy) → Fa

WAV86 Γ, ∀y ¬ .. →∀xFx.WithΔ→¬∀xFx there results Γ, Δ →¬∀y¬(A&¬Fy), this one gives with the new rule of inference again → A ¬∀xF x Γ, Δ & ( ). B A &-elimination: Γ → A & ¬∀xF(x)givesΓ→ A (resp. Γ →¬∀xF(x)) This is transformed into: There results by the new rule of inference Γ →¬∀y¬(A & ¬F(y)) A & ¬Fa → A & ¬Fa gives14 A & ¬Fa → A (resp. A&¬Fa →¬Fa) With ¬A →¬A (resp. ∀xFx →∀xFx, ∀xFx → Fa) there results ¬A →¬(A & ¬Fa), therefrom

14 Added above: a newfreevariable 12 Consistency of arithmetic, for publication 235

¬A →∀y¬(A & ¬Fy)(resp.∀xFx →¬(A & ¬F),)15 With the addition of Γ →¬∀y¬(A & ¬F(y)) there results Γ →¬¬A and from that one Γ → A (resp. Γ →¬∀xFx). Finished. If through this procedure the deletion of all transfinite &-inferences is achieved, there follows now a transformation of all formulas in the derivation after their formulas without transfinite &. Here those sequents that arose by the new rule of inference from each other, obviously go over to identical ones, so that the new rule of inference is no more applied. One has a regular derivation that satisfies everything. The result of 1 is, in its turn, not disturbed, because no transfinite connectives ∨, ∃, ⊃ have entered anywhere. ————————————————————————————————16 Concerning the reducibility proof. It must be required that the free variable is first substituted. Otherwise bad with (UI ), CI . Execution in 14.2 [subsection in Gentzen 1935)] should be best in the beginning. Therein to be mentioned, that a plays no role in UE. The terms? Because of CI for example, one must have the permission to begin with term substitutions in the tail. But this can be the case also e.g. with UE in the upper sequent! Possibly terms, as well as free variables in general removed in advance [?]? Works only with the rules of inference for & and ¬, both in the same way [?]. Perhaps it’s not worth it. The inferences for ∀ and CI have particularities with the free variables and terms. Even t cannot be removed in advance, after the substitution of all terms in F(t), the rule of inference may under circumstances fail to remain one! (Should be that a variable x bound in F(x) now becomes t, i.e., ν and a term can be substituted by another one.) Here the lemma to be used? ∀xF(x) → F(t) could take the place of UE. No gain, however. The multiplication of antecedent formulas. Cannot be avoided. (See the example Γ → A × Γ A → =Γ→, now there is a reduction in Γ to the right, pulling through down.) To work with equality of sense does not seem bad to me.

15 Written below: therefrom ∀xFx →∀y¬(A & ¬Fy) 16 The following to the end of the page is cancelled. 236 Part III: The original writings

Then, though, one has to be careful with the conception of reduction, namely that it gives the same result for sequents that have an equal sense. If one doesn’t want to require normal form. Especially with the reduction steps on antecedent formulas. How, e.g., does A & B occur repeatedly in the antecedent? I would suggest something like the shifting through of A as a step of reduction. Or: As a formal step of reduction, it is allowed to go over to a sequent with the same sense? Stupid. Then rather: To show that it makes no diﬀerence to the reduction which one of the forms with an equal sense one takes, But is it correct? Say, if one substitutes ∀xFx, ∀xFx by ∀xFx, ∀yFy that has the same sense? Stupid things! One should formulate it all so that there appears no diﬀerence for what is equal in sense. 13 Correspondence with Paul Bernays 237

13. Correspondence with Paul Bernays

There are 13 known letters from Gentzen to Bernays, all in the Bernays collection of the ETH. They are written between April 1934 and July 1939. Letters from Bernays to Gentzen have been preserved in four cases as copies or drafts that Bernays often kept. The altogether 17 letters in the Bernays collection are numbered Hs. 975:1649 to Hs. 975:1665. Here Hs. stands for Handschriftensammlung, the signum used for the collection of manuscripts of the ETH. No letters to Gentzen with a scientiﬁc content have been preserved directly, but only as occasional copies kept by a sender.

1. Gentzen to Bernays 11 April 1934 (Hs975:1649) This letter is a large two-sided card addressed to “Herrn Professor Dr. P. Bernays in Berlin-Wilmersdorf Holsteinische Str. 38.” The sender is indicated as “G. Gentzen Stralsund Schillerstrasse 35.” It is known that Bernays resided with his sister in Berlin at the time, before taking himself to safety in Switzerland. Gentzen, in turn, had left Göttingen after his doctoral thesis was accepted in the summer of 1933 and lived in his home town Stralsund. The letter shows that the result of the second of Gentzen’s 1936 publications, the consistency of the simple theory of types, was done by the spring of 1934. There is a letter from Bernays to Weyl in 1937 by which Gentzen had also found a proof of the consistency of Weyl’s system of predicative analysis at the same time. Most importantly, the letter indicates that Gentzen thought he had found a proof of the consistency of arithmetic by transfinite induction. Gentzen writes to his fired Jewish professor that he had entered the SA, the Nazi paramilitary troops, explaining that it was “urgently advised to me from several directions.” This took place some days before the state exam of November 1933, in which Gentzen got the license to teach at lyceums. At the exam, he had to sign a printed form that contained the following:

I hereby dutifully assure: No circumstances are known to me, despite careful probing, that would justify the assumption that I would descend from non-Aryan parents or grandparents. Espe- cially, none of my parents or grandparents have at any time ad- hered to the Jewish religion. I am aware that if this clariﬁcation is not true, it will lead to an exclusion from the examination or later impeachment of the nomination to a public oﬃce.

It is not known who Gentzen’s advisors were, but to pass the exam, the membership seems to have been needed. No record shows that he would 238 Part III: The original writings have participated in any SA-activities, and his SA-membership was not known in G¨ottingen in 1935 when he became Hilbert’s assistant there.

Respected Professor! Stralsund, 11.4.34 Even if there is no special reason to it, I would for once like to make myself heard. The Notgemeinschaft seems to have no hurry in clearing my application for a scholarship, at least I have had no notice from them. To an inquiry on my part about six weeks ago I got the affirmation that my documents are still being examined. I have also made an application with the school, for security, but even there, no decision has been made so far. I have also entered into the SA, because it was urgently advised to me from several directions. – For the rest, I have cultivated my research on consistency considerably: To begin with, I treated type theory and carried through a consistency proof for it, one that is connected to the well-known simple proof of consistency for predicate logic. Then I added the mathematical axioms by which, naturally, the real difficulty just begins. It turned out that the consistency of mathematics is equivalent to the carrying over of the Hauptsatz of my dissertation from predicate logic to type theory. I hope to have soon a consistency proof ready “by force.” It then remains to transform the proof so that only allowed ways of inference are used. I hope to arrive at that, analogously to just arithmetic, by the use of transfinite numbers. – How far are you with your book? Will you come again to Göttingen in the Summer Semester? With best regards, Your G. Gentzen.

2. Gentzen to Bernays 23 June 1935 (Hs975:1650)

This letter, as well as the two following ones of 14 July and 4 November 1935, concern Gentzen’s long manuscript, about a hundred pages and titled Die Widerspruchsfreiheit der reinen Zahlentheorie, in which the original proof of the consistency of arithmetic is given. Bernays had the manuscript with him when he sailed in the company of Gödel to New York in September 1935, for a stay at the Princeton institute. Criticisms by Bernays, Gödel, and possibly von Neumann made Gentzen change the original proof into one that used transfinite induction. The original proof was saved for posterity in the form of galley proofs that Bernays kept and that got published first in an English translation in Szabo’s edition of Gentzen’s Collected Papers in 1969, then in German in 1974 in the Archiv fur¨ mathematische Logik,later Archive for Mathematical Logic. An account is found in Bernays (1970), and a more brief explanation in Bernays’ foreword to the 1974 publication. The role of bar induction in Gentzen’s original consistency proof is discussed in detail in Section I.4.7 above. 13 Correspondence with Paul Bernays 239

Stralsund, 23.6.35.

Respected Professor!

I send enclosed the final part of my “Widerspruchsfreiheit der reinen Zah- lentheorie.” I thank you so much for the remarks that concern the main part. I have taken over the suggested improvements of the textual details in their substance in the final text. – I thought to answer your question concerning the process 13.6 2 with the very same example that you indicate in your card (as the second one). See also 15.2. – It is in fact so that on the basis of my proof, from the derivability of ∃xA(x)itfollows,whenA(x)is not transfinite, that a ν can be given such that A(ν). Nothing of the kind holds, however, even for ∃x∀yA(x, y). – A change of the calculus, through the addition of cut etc., would certainly lead to some formal simplification of the consistency proof; I have, even so, deviated from this formulation (from which I originally parted), to keep the rules of inference as natural as possible. I wanted to meet by this those readers who have not been trained in advance in formal logic. – As to the terminology: I have introduced the word “finite” where it is appropriate. I would be reluctant to change the designations “number sign” [Zahlzeichen] and “elementary formula”; with a “cipher” [Ziffer], one intends in the first place a single cipher, by which reason this word can be misunderstood it seems to me, and “elementary formula” is an expression already used by diverse authors, even if it is, as I admit, somewhat circumstantial. – As to 13.5 3 to 13.6 4: It is correct that 13.6 1 and the addition in 13.6 4 are in the end dispensable. They contain, nevertheless, advantages for the consistency proof. I want to change this point, namely, to leave out 13.5 3 and 13.6 1 altogether, but instead provide 13.6 2 and 13.6 3 with the same addition as 13.6 4. – I am not able to decide on renunciating part I, in the first place because I had hoped to attract through my article the interest of those readers who have so far not had to do with foundational questions. The individual questions that I take up, and they all seem to me important for an understanding of the consistency proof, have not been treated in this way in their context in the literature. – I shall cite your Grundlagen-book when referring to the ι-terms (6.3), with an indication of your elimination-theorem. The present part V has become shorter than I had initially intended. I wanted to bring forth something about transfinite ordinal numbers and their connection to reduction procedures, and construction procedures in general, but in the end, these things did not really seem to be ripe for presentation, but should perhaps find their place later in a separate publication. – It would interest me a lot if you could pass me also your judgment and eventual individual remarks about part V. Would you by chance know if Professor Weyl is going to be in Europe this summer again, or is already? I would want to send to him, as well, 240 Part III: The original writings a stencil copy of my article. – I hope to receive soon a decision about the scholarship from the Notgemeinschaft. Should it be granted to me, I plan to travel to Munster¨ in the fall. I shall then attack again the consistency of analysis. with best wishes Your Gerhard Gentzen.

3. Gentzen to Bernays 14 July 1935 (Hs975:1651)

Stralsund, 14.7.35. Respected Professor! I thank you for your letter. I include here the final formulation of part IV of my work, as well as a couple of modified pages from part III that are related to IV. All the changes are just little ones, they don’t concern what is essential. – About your remarks: To what point one wants to talk about “follows”intheproofofconsistency,isintheendamatterofterminology. I mean hereby the “follows” that finds its appearance in formalization as ⊃, whereas there occurs for example with complete induction in my formalization only → , F(a) → F(a+1). But I don’t count this as a proper “follows,” therefore also the separate mentioning of complete induction (10.5). – I have written in fact nonsense on pp. 75–76; I held my eye on an older form of the notion of reduction, in which the reduction steps are uniquely determined. The passages could be corrected more or less as follows: At 15.21, reducibility should be replaced by: “There is a number ν so that for each series Rν of ν numbers, a series of at most ν sequents can be given such that the first one is Sq, and each of them is formed from the preceding one through a reduction step, and the last one has endform, and further, the possible choices are determined through the associated numbers from the series Rν.” Correspondingly under 15.23: “For each infinite series R of numbers, a finite series of sequents can be given such that the first one . . . ” as before. – I have, however, cancelled these passages completely, because they are not fully necessary; perhaps I could give sometime later complete proofs to both theorems in a special publication. I have reflected on your question about the connection with my dissertation; I see no way to arrive at the goal along the way indicated. For, 1.) induction on the concept of “grade” cannot be taken over to the system F ∗, because the F(A)thatentersforF(X) in the reduction of a ∀-sign (Diss. III 3.11 3.3 3) can contain arbitrarily many more symbols for connectives (in A) than before; 2.) the concept of a maximal number of ∀-inference figures in any thread in a derivation cannot be set forth without further ado as a substitute, as can be seen, say, from the example that I have included on a 13 Correspondence with Paul Bernays 241 separate sheet. – On the whole, it does not seem altogether sure that the assumed theorem really holds; for the calculus F is intuitionistic,whereas F ∗ contains even the possibility of a “circulus vitiosus,” say in the case in which there is a proposition about all propositions, and in this again a proposition can be substituted (UIA∗)1 that refers to all propositions. If one prohibited with UIA∗ the occurrence of bound variables in formula A,it should be possible to carry through the desired proof, because the notion of grade would then be again usable; herein each ∀ counts as ω. It is the same state of affairs that results from the attempt at carrying the procedure at hand over to the consistency proof of analysis. If one takes the LK-calculus instead of LI, the corresponding theorem, with unlimited UIA∗, should be easily provable. The proof of ι-eliminability appeared to me of old as something that can be made shorter. I figured once an approach that really doesn’t cause much trouble; maybe I’ll find some time the occasion to compare it with yours. – I got from the Notgemeinschaft, now “Deutsche Forschungsgemeinschaft,” the notice that I have been granted the scholarship, first until the end of this year, and with the prospect of a continuation. with best wishes Your Gerhard Gentzen.

4. Gentzen to Bernays 4 November 1935 (Hs975:1652)

Göttingen, 4 Nov. 35. Feuerschanzengr. 18.III Most respected Professor! Many thanks for your letter that I received today. I believe that your worries about my consistency proof are unfounded. The trains of thought that you present in your letter are not new to me; I have considered all these aspects already myself, including the geometrical image of branching line segments. You are quite right that the finiteness of even a single reduction path for the sequent Γ, Δ → C can get grounded on the finiteness of a whole series of different reduction paths for D, Δ → C. But this does nothing for my proof idea! I have to admit that the passage 14.6 3 can be misunderstood, in that it could give the impression that there was a uniquely determined sequence of reductions on the sequent D, Δ → C. I came to think about this passage only a few days ago while doing the proof-reading, namely whether I should add a corresponding remark, but didn’t do it. I have now decided to add the following sentences (and sent today a corresponding message to Prof.

1 Original has AEA, likely for All-Einfuhrung-Antezedent¨ , universal introduction antecedent. 242 Part III: The original writings

Blumenthal, because I had sent him back the proof sheets already a couple of days ago):2 ‘Let the following be remarked to avoid misunderstandings: The type of reduction of D, Δ → C into D, Δ∗ → C∗ can eventually depend on a choice (14.6 2 1) that takes place in the reduction of the mix-sequent Γ, Δ → C. The same holds of each further step of reducing back, and, it can be added, the new mix-sequent Γ, Δ∗ → C∗ etc. need in no way always be the reduced one of the preceding sequent (14.6 2 3). So, the number of steps of proof can be very different, according to the result of the individual choices; the only thing that is certain is that it is in every case finite.Toprovea claim for every possible choice, it is sufficient to prove it for one specific, arbitrary choice. Therefore it is sufficient in the entire proof to keep an eye on just one single specific sequence of reductions of the sequent D, Δ → C, and thereby on just one single specific finite series of steps of proof.’ I would like to maintain that there is nothing to object against this type of proof idea. To carry it out a bit: In the proof of the proposition: “The mix-sequent of Γ → D and D, Δ → C is reducible,” there remains in general the task of proving a claim of the form “The mix-sequent of Γ → D and of a reduct of D, Δ → C is reducible.” The inductive hypothesis–induction on [the number of logical operations in the mix formula D]–has no say here, it is taken for a proposition already proved and therefore valid. The reduct of D, Δ → C results in a unique way, except for the case 14.6 2 1 with a freedom of choice. The way to reason in this case is as follows: The reducibility of Γ, Δ → C is satisfied when the remaining sequent is reducible for each arbitrary choice (in the first reduction step). To show that, one makes somehow a completely arbitrary choice and obtains again a single specific reduced sequent Γ, Δ∗ → C∗ and thereby also uniquely D, Δ∗ → C∗. There remains consequently in each case the task of showing the reducibility of the mix-sequent of Γ → D and a single reduct of D, Δ → C;Itisnow treated further in precisely the same way. It makes no difference if the new mix-sequent is not a reduct of Γ, Δ → C, by which the possible freedom of choice in its reduction does not belong to the choice prescription for Γ, Δ → C. There is, to wit, a complete change of the task to be performed at each step. If I put for once aside all inessential auxiliary work, the essence of the somewhat peculiar inductive inference 14.6 3 seems to me to be represented by the following analogy (comp. the image of the branching sequence of line segments): A proposition ∀xF(x) is proved if each of the infinitely many special cases F(ν) is proved. Let each of these again be equivalent to a proposition ∀xFν(x), each special case Fν(μ) of these propositions again equivalent to aproposition∀xFν,μ(x), etc. Let the following be known: Each arbitrary

2 The addition was meant for the end of part IV of Gentzen’s article, p. 112 of the printing in Archiv fur¨ mathematische Logik in 1974. 13 Correspondence with Paul Bernays 243 series of specializations ∀xF(x), F(ν) ⊃⊂ ∀xFν(x), Fν(μ) ⊃⊂ ∀xFν,μ(x),... ends after a finite number of components in a formula Fνμ...(), the correctness of which is known.Tobeprovednow:∀xF(x) is correct. To this end, I infer as follows: The correctness of ∀xF(x) is secured if F(ν)holds for whichever arbitrarily chosen ν. So let us assume that we had chosen a specific number ν, and it remains just to prove F(ν). This is ⊃⊂ ∀xFν(x). Now I infer just as before, namely, that to show that this proposition holds, it suffices to take whichever arbitrarily determined special case, say Fν(μ), etc. This chain of inferences must end after a finite number of steps, because each arbitrary sequence ∀xF(x), F(ν), Fν(μ),... had to be finite. Thereby ∀xF(x) is proved. What do you think, now, about this way of inference? Shouldn’t it be finite? (To be sure, if one turns the proof into an indirect one, i.e., begins like this: Assume that ∀xF(x) does not hold, then there is a counterexample ν so that F(ν) does not hold, so neither ⊃⊂ ∀xFν(x), etc, then the tertium non datur enters.) The proposition about line segments that you presented could be proved by this inference. – The inference has also a relation to transfinite induction. This is the connection with my earlier use of transfinite induction in the consistency proof. Transfinite ordinal numbers are the best intuitive image or general measure for the extraordinary nested chaining of mutual dependences of sequents in respect of their reducibility. Since, however, you don’t seem so far to have said anything concerning the recognition of the finiteness of the forms of inference, I have left them out of the consistency proof; also because there would be still one thing and another to clarify. It may interest you that I needed for the ordering of sequents in the number-theoretical formalism something like the number ak(ω, ω, ω), that is limν→∞ ak(ω, ω, ν), as an upper limit. The enormous chaining of dependences makes it really inevitable that one needs these extraordinarily chained inductions. It does not surprise me at all that you could not come to an end with the “resolution” of the -induction, for there one stands undeniably in front of a chaos of branchings. I have, to be sure, tried myself all that is possible to bring out more clearly the true way in which the reduction procedure for Γ, Δ → C is produced in the Hilfssatz, but I did not succeed in this. This would be the best possible point at which my proof procedure could be attacked, and to wit, for the reason that the concept “finite” in the proof of the Hilfssatz is, sure, conceived somewhat broadly. Can you, please, refer to this point with everyone who would like to criticize the proof, where I would find it, anyway, nicer if one carried through an improvement in the sense of a further finitization (see 15.1 1) instead of attacking the present conception from the point of view of rigorous finiteness. I would be very interested to hear your counter-arguments to these considerations; The exchange of thoughts over the ocean is unfortunately hindered by the great lapses of time. 244 Part III: The original writings

A devoted salute –also to Professor Weyl whom you should see frequently– Your Gerhard Gentzen

5. Gentzen to Bernays 11 December 1935 (Hs975:1653)

This letter, as well as the following ones of 15 January, 3 March, and 17 July 1936 concern the modiﬁcation of Gentzen’s original proof of consistency, into one that uses transﬁnite induction up to the ordinal ε0.

Most respected Professor! Many thanks for your letter. I must admit that the critical inference in my consistency proof is not satisfactory. I have therefore decided to rework thoroughly the proof. I hope to succeed in overcoming completely the difficulty. Publication is drawn back until further. The modification suggested by Gödel was known to me, but it is in fact not usable from a finitary point of view, because of its impredicative character. There is hardly any interest for foundational questions here in Göttingen. It is only with the Frenchman Cavaillès who is here for a couple of weeks that I have entertained myself a few times. I should presumably get the assistant’s position with Hilbert that has become vacant because Mr. Schmidt left. Hilbert told me to send you his wishes, namely, he asks you to write on occasion how it is with you. with best regards and wishes for Christmas Your Gerhard Gentzen

6. Gentzen to Bernays 15 January 1936 (Hs975:1654)

This letter mentions two formulations of the proof of consistency by transﬁ- nite induction. It seems that formulation A would be the one that uses the standard notation for ordinal numbers, as in footnote 21 of Gentzen (1936, p. 555). Formulation B would seem to be the special decimal notation for ordinals that Gentzen develops in the same paper. Formulation A was included as a separate attachment in the letter, but it has not been preserved.

Göttingen, 15. I. 36. Most respected Professor! I have undertaken in the past weeks a complete reworking of my consistency proof and believe that the critical point has now taken an essentially more pleasant form. It would interest me a lot if you could give your judgment 13 Correspondence with Paul Bernays 245 about this, and possibly to report on it again to Mr. Blumenthal. I send you therefore as an attachment the part of the proof that contains in a concise form (and quite independent of the rest) those inferences that are essential for the judgment. The remaining part of the proof is not yet ready in all details, but it seems to me as good as certain that it is in order: the inferences used in it are in any case altogether elementary and certainly finitary. The question is only about assigning to every derivation a transfinite ordinal number of the kind defined in the part attached – Sν-numbers – and to show that a derivation that leads to a contradiction can be transformed through a step of reduction into a derivation that also proves a contradiction and with a lesser ordinal number. Then the consistency of arbitrary derivations follows by an application of transfinite induction, this is the essential inference, one that cannot be taken to be provable in formalized number theory, and the proof of which forms the content of the attached notes. I would like to add to these the following remarks: There are 2 parallel formulations. Formulation B is the decisive one, as the more elegant one. I have attached formulation A because I had even that one ready, and because theassociationoftheSν-numbers to the usual way of writing of the first transfinite ordinal numbers can be carried through more easily. Hereby A has independent interest. I considered even a third formulation in which, as compared to B, even the requirement was put aside by which the Sν- numbers used in the construction of a Sν+1-number have to be distinct;this form is even more convenient in applications, though somewhat intricate for the proof of the main theorem, that of transfinite induction. About the formalizability of the main theorem (formulation B) the following holds, as far as I have come to think: First of all, it is directly representable in analysis, provided that one substitutes the concept of “accessibility of a” through the concept (as is usual with complete induction): ∀F[∀y(∀x(x < y ⊃ Fx) ⊃ Fy) ⊃ Fa] Then the proof becomes “impredicative” through the entering of the concept “for all properties.” This circumstance cannot, though, be brought to bear against the original form, because the introduction of the concept of “for all properties” represents an obvious distortion of the original intuitive concept of “continuation” etc. One can arrive further, through a reformulation of this analysis-proof, at the formalizability in pure number theory of the same [“continuation”] for each fixed Sν, certainly again by methods that are not finitarily admissible (namely ⊃-nesting). Herein the F that is needed for the formulation of the main theorem functions merely as a free formula variable. – This result appears to me as thoroughly satisfying, in that it shows that the consistency proof goes, as it were, as little beyond the framework of formalized number theory as possible. (Admittedly, it cannot be proved as of yet directly that 246 Part III: The original writings this is so.) I haven’t worked though these trains of thought in detail yet, because they have no immediate significance for the one and only essential question, namely whether the (attached) proof satisfies the requirements posed on a consistency proof. I have obtained recently the position as an assistant to Hilbert. with best wishes Your Gerhard Gentzen.

7. Gentzen to Bernays 3 March 1936 (Hs975:1655)

G¨ottingen, 3. III. 36. Feuerschanzengraben 18 Most respected Professor!

8. Gentzen to Bernays 17 July 1936 (Hs975:1656)

Stralsund, 17.VII.36. Schillerstr. 35. Most respected Professor!

I thank you for your letter and the sending of your Zurich lectures. I had read them already in Göttingen, though only after I had finished my work on consistency. It was of special interest to me that you had already set your eye on the possibility of carrying through the proof of consistency through transfinite induction up to ε0. Now to the single remarks in your letter: I consider it improbable that one could prove the theorem about the branching figure through TI up to ε0.I would rather suppose that one needs the whole II number class. – To consider TI in the descending way even I have tried, but I have ascertained, as you, that the inductive proof will not succeed then. – I have in fact used the word “Numerus” by error in a way that is not the usual one. Unfortunately it is too late now for a change. – It is certain that one does not need the free choices for a mere proof of consistency. This doesn’t come forth in the present form as clearly as in an earlier one from which I departed. The entrance of the concept at 14.2 5 3 is in my view only apparent; it can be avoided completely if one defines the concept of a reduction step in advance only for derivations with the endsequent → 1 = 2, or shall we say, to not to have to do with hypothetical derivations, for derivations with the endsequent → M where M is a minimal formula. The definition cannot be then built up in the earlier way inductively, but it is still not all too difficult to give. You find it “somewhat weird” that in the proof of consistency, one has to consider only derivations for which all reduction steps are uniquely determined, so that their number is always a quite determinate natural number. But this is just how it is. If one could use these numbers as the assigned ordinal numbers (and prove that they diminish in reduction) one would not need to use at all TI, but ordinary CI. But it is not possible to produce such an assignment in the usual “calculable” way. On the contrary, one can calculate the number of reduction steps only by actually carrying through the reduction; then one does now it. But that this calculational procedure is finite, that can be proved only through TI. These are the apparently rather paradoxical circumstances. As concerns the approximate delimitation of the stand taken in connection with the TI-proof, it is admittedly difficult to say something about it. I would say that it consists in a very strong use of the concept of “potential.” I want to wait still with a more precise investigation of this point, until it is seen how much is needed for the handling of analysis. I am especially interested in the system of Church because of the relations 13 Correspondence with Paul Bernays 249 to transfinite numbers, and want to study it more closely. – Now the story with the chain inference: indeed, I have not meant this in the way you have understood, and admit that my presentation is prone to misunderstanding. What is meant is that one writes the antecedent formulas of a lower sequent one after the other, and during this process, when one arrives at a specific formula, the decision is allowed of leaving out the formula among those already written down; if this happens because the formula occurs already among those written down, it means that the identical formula in question cannot be deleted later on its own, but that the determination had been taken earlier for this that it had to remain. The modified conception suggested by you is in fact exactly the same as the one I had meant. – The result that concerns CI is achieved so elementarily that it deserves in no way a separate publication. I will be able to communicate to you the proof from Göttingen where I return in the beginning of August; at present I don’t have my notes about this at hand. I came to the result by an admittedly strange deviation, namely because one needs in the proof of consistency arbitrarily high ordinal numbers (below ε0) already for derivations with just one CI ; this led me to the conjecture that one could get through in general always with just one CI. The same point of view has proved to be at times a heuristic aid in the finding of similar connections. In this way I have been led to various simple normal forms for number-theoretic derivations. I have presented my thoughts in talks in Göttingen in May, and in Leipzig and Munster¨ in June. – Do you know more about the destiny of Gödel who, as I heard, should be in a psychiatric hospital? with best wishes Your Gerhard Gentzen.

9. Gentzen to Bernays 13 August 1936 (Hs975:1657)

Gentzen gives in this letter a proof that formal derivations in arithmetic can be so transformed that they contain just one step of inductive inference. Even if he mentioned in the previous letter of 17 July 1936 that “the result that concerns CI is achieved so elementarily that it deserves in no way a separate publication,” he nevertheless prepared later a three-page paper, one that got published in 1954. This publication contains the editorial statement by which “Gentzen had dedicated this little article to Heinrich Scholz on his 60th birthday 17 December 1944.” The paper contains the details of the ﬁrst of the two ways of handling more than two inductions mentioned towards the end of the letter. 250 Part III: The original writings

G¨ottingen, 13.VIII.36. Feuerschanzenstr. 18 Most respected Professor!

I heard through Prof. Hilbert that you would perhaps pass through Göttingen in September. He asks to tell that he would presumably not be back from his summer trip before the 20th of September. If you want to meet him, you must come towards the end of the month. I myself would be very glad if I could speak to you once again. – I add here also the promised proof that one can summarize all complete inductions in a number-theoretic derivation into one: Calculus: as in my consistency proof, §5. For complete induction I take, though, not the inference figure formulation, but the one that is easily shown equivalent: the admission of groundsequents of the form: → [F(1) & ∀x(F(x) ⊃ F(x + 1))] ⊃∀yF(y) Let there now be given a derivation with 2 complete inductions. I.e. two such groundsequents with F1 resp. F2. Iformnow(c, d newfreevariables):(d =1⊃ F1(c)) & (d =2⊃ F2(c)). I write for this the abbreviation H(c). Now there results by precisely one complete induction: → [H(1) & ∀x(H(x) ⊃ H(x + 1))] ⊃∀yH(y) Both of the previous CI-groundsequents can be derived from this one without the use of any further complete induction. For F1 for example (for F2 quite similarly): One puts 1 for d (through ∀-introduction and elimination.) Hereby H(c) becomes (1 = 1 ⊃ F1(c)) & (1 = 2 ⊃ F2(c)). This is by (propositional) logic equivalent to F1(c). (Mathematical axiom formulas give: → 1=1and → ¬ 1=2.) So, such an expression can be transformed by purely logical inference figures, at any place, into an equivalent one. Hereby one obtains, as claimed, the corresponding previous groundsequent for F1. Hereby all is proved. More than two complete inductions can be handled quite analogously as one. (Or one can also apply the procedure for two over and over again.) If one brings the derivation in the end in a tree form,thesingle CI- groundsequent will then admittedly occur multiply. This, however, is inessential, it can be avoided by special arrangements. In sum, one can say: The number of complete inductions in a derivation is not characteristic for it. The number of connective symbols in an induction formula (F) instead is in a certain way characteristic of the “complexity” of a derivation. Nothing more specific can hardly be said about it at present. 13 Correspondence with Paul Bernays 251

with best wishes Your Gerhard Gentzen.

10. Gentzen to Bernays 30 September 1937 (Hs975:1659) The main topic of this letter concerns the proof reading for the second volume of the Grundlagen der Mathematik that Bernays was writing. The matter is continued in subsequent letters, until 1938, with discussions about the -substitution method among others.

G¨ottingen, 30.IX.37. Most respected Professor!

Please excuse me that I write to you only now about your delivery. I was occupied in the past weeks with the working out of a talk about “the present situation in foundational research” that I gave a week ago at the meeting of the D. M. V. [Deutsche Mathematiker-Vereinigung, German mathematical society] and that I plan to publish later [Gentzen 1938a]. Concerning your manuscript, I would like to say in general that it appears to me very understandably written, to be easily readable also for a beginner, I find. A bit too broad at places to my taste; I mean, some repetitions could be spared, e.g., around pages 78–83. The proof of the first -theorem has, admittedly, reached a pleasant simplicity; it bespeaks of a deeply grounded connection in the essence of the diverse ways of proof – and for not expecting a further simplification – that there obtain certain external analogies to the proofoftheHauptsatz of my dissertation, especially through the double induction (ω2) by range and grade, wherein the two concepts would have to be exchanged. The great effort required for the proof with the equality axiom included (pp. 74–104) stands in an unpleasant contrast to the first proof. Whether this can be simplified I cannot see through. I would suggest to leave out this proof altogether; its result does not seem worth the effort to me. If I have understood right, this theorem is there only because of the -symbol that should be properly (p. 16 bottom) just an auxiliary for a more convenient treatment of questions that arise independently of the symbol. For all that one needs in practice is contained in the result formulated on p. 73. It should be that the special difficulties of the proof that follows arise only because one wants to have the as unique, so that one can apply the general equality axiom also to -terms; hereby one betakes exactly those advantages that the indeterminate had. A few stylistic remarks follow: Expressions such as “des genaueren” (p. 1 middle, 50 middle, 63 bottom), “des näheren” (p. 30 middle), “von Erheb- lichkeit” (p. 38 bottom) should be better replaced by “genauer” etc. The 252 Part III: The original writings sense of the sentence “Mit der Einfuhrung¨ ...on p. 17 middle is a bit unclear. P. 23, third to last line: Nämlich was [indicated as transposed]. I will be glad if my remarks are of any use to you. I can ascertain for the first time, in the past weeks, a clear improvement in my state of health. I am already making careful attempts at an engagement with the consistency proof. with best wishes Your G. Gentzen.

[The letter is followed by a list of eight smaller corrections with page numbers given, all written in Gabelsberger.]

11. Bernays to Gentzen 15 November 1937 (Hs975:1658)

This letter is a carbon copy of a typewritten original.

Zurich,¨ 15 Nov. 37 Besenrain Str. 30. Dear Mr. Gentzen!

The response to your letter occasioned by the reading of the part of my draft I had sent, has been considerably delayed. (You must have received the provisional postcard I sent you.) I have – as I already wrote – used most of the diverse remarks you made. I give you further down (on page 3) the modifications adopted. As concerns especially the breadth of the exposition in section 2.a), I too observed it in a further reading. It seems, though, more advantageous to me to make the shortening at an earlier place, instead of pages 78–83. I believe that the impression of length comes especially from the fact that the theorem about the representability [Vertretbarkeit] of the general axiom of equality on pp. 75/76 is formulated in such detail, whereas a shorter rendition would suffice here, because the theorem had been used a little earlier. I don’t find the demonstration of the possibility of incorporating the general equality axiom in the first -theorem quite that unrewarding. Admitted- ly, its meaning does not quite become evident in the part so far. However, the more general theorem about the eliminability of the -symbol and of ι-formulas from derivations that for the rest proceed with the means of the predicate calculus and the axiom of equality, and the endformulas of which do not contain any -symbol, will be proved afterwards, in §11, with the help of this theorem. The admissibility of the introduction of occupation functions [?, Belegungsfunktionen] for the treatment of decision problems in 13 Correspondence with Paul Bernays 253 predicate calculus extended by equality axioms, for example, will be proved through this theorem. The extension of the general equality axiom even to the -symbol corresponds, moreover, definitely to the use of the -symbol in the sense of the set-theoretic principle of choice. A requirement that goes even further than the one formalized through the formulas of -equality is used in set theory, namely the one that is given by the formula scheme

(x)(A(x) ∼ B(x)) → xA(x)=xB(x) You find this in, for example, the new work by Ackermann (in the newest Annalen issue) as axiom (II.4). I would, to be sure, greet even a simplification of the proof in 2.b). I have worked quite a lot with this and had it already in different versions. At least this proof is still simpler than the Ackermannian proof for the consistency of -formulas that comes into consideration in the next part and that is connected to the first Hilbertian approach. Mr. Ackermann had indicated this proof to me in its time on less than two quarto pages, but the precise carrying through is quite tedious to fashion. I asked myself, in fact, whether I should abstain from a presentation of this proof in extenso. But it seemed to me desirable to arrive at complete clarity as regards this proof, because errors stood about the scope of this proof, and because there were repeated discussions about it in the literature – for the approach that hinges on this proof formed for a long time the main method of proof theory. The reader, on the other hand, who is not especially interested in this question, is advised that he can jump over a lengthy section. I send you now the rest of §10, pages 105–177, and would be glad to hear again your opinion and remarks about these. Hoping that you are feeling well, with warmest regards, Your P. B.

[Page 3 of the letter consists of a list of corrections to the manuscript.]

12. Bernays to Gentzen 3 January 1938 (Hs975:1660)

This letter, like the previous one, is a carbon copy of a typewritten original.

Zurich,¨ 3 Jan. 38, Besenrain Str. 30. Dear Mr. Gentzen!

Many thanks for your card of December. I was especially glad to hear that you have now carried through your proof of consistency in a simpler and more profound form, and I would be very interested to come to know it in 254 Part III: The original writings this new formulation. Would you be so kind as to have me sent a copy of the proofs when it is that far. - The reformulation of your consistency proof that you have carried through will presumably come in handy in your continuing work in this direction. §11 of the foundations book is since already two months ﬁnished, but not yet typewritten at the time. The topic is not, by the way, the G¨odelian results (they will be in turn only in §12) but mainly Herbrand’s theorem and the decision problem. As soon as the writing has been done, I’ll send it to you. I wish you will feel well and have success in your research in the coming year.

With best regards,

Your P. B.

13. Bernays to Gentzen 9 May 1938 (Hs975:1661)

This letter is again a carbon copy of a typewritten original. Bernays suggests here a reading of Gentzen’s reduction procedure as an inﬁnitary derivation, and idea worked through in Schutte¨ (1951).

Zurich,¨ 9 May 1938. Besenrain Str. 30. Dear Mr. Gentzen!

I send you in this same mail the remaining – what has become rather long –partof§12. The draft of the foundations book should be hereby finished, except for a couple of appendices. I was very sorry to hear, from your postcard of February, that you were again so severely hindered from work. I very much hope that this unpleasant state has passed in the meanwhile. Have you now been able to finish the new formulation of your consistency proof for number theory? - When coming to discuss your proof in the last part of §12 I settled, as you will see, to present the basic idea, or rather actually only the disposition of this proof; Still, I treated more in detail the special case of transfinite induction used there. The proof that I give here is, even disregarding the form of the attire, somewhat different from your work. I admit that your proof procedure is more appealing as regards evidence, whereas the organization that I chose should be somewhat more elementary, especially because the concept of “accessibility” is introduced only in relation to a specific proposition and can therefore be explicitly defined. 13 Correspondence with Paul Bernays 255

It occurred to me during the preoccupation with your proof that the reducibility of a formula can (by inverting the sequence of steps) be conceived as a certain special kind of provability. The result of your proof assumes here a form by which each formula derivable in the formalism you consider, can also be obtained deductively in a certain special way – in which admittedly a rule3 of content is needed that is not purely formal: If it can be shown that A(z) is derivable for every numeral z, thereby the formula (z) A(z)counts as derived. In this way, the method of your consistency proof appears as an extension of the result of your dissertation. And I harbor the conjecture that this should be the way you arrived at the proof. The dispute about the Finslerian objection seems to me not yet quite satisfying. I made do on this point in §12 just so provisionally. I hope that you are in a position to read through §§11 and 12 soon. (Maybe you are already acquainted with §11). The publisher wants to begin now with the printing of §9 and 10.

Pleaseletmeknowsomedayhowitiswithyou.

With best regards,

Your P. B.

You receive naturally the proofs by mail. - Incidentally, Ackermann has declared himself ready to participate in the reading of proofs.

14. Gentzen to Bernays 12 May 1938 (Hs975:1662)

G¨ottingen, 12.V.38. Most respected Professor!

Many thanks for your letter and the manuscript. I am doing again in the main better as far as my health is concerned; The new conception of the consistency proof is ready and will be printed now, I have indicated that proofs be sent also to you. Ihavereadyour§§11–12 brieﬂy, the ending somewhat more precisely. Some quotations are not quite right (the work of Finsler; my dissertation Math. Zeitschrift not Annalen, p. 38 of §11 and p. 231 of §12). – A work of Kolmogorov’s, “Sur le principe de tertium non datur” (1925), not understandable to me in detail because written in Russian, gives me the impression as if the theorem proved by G¨odel and then by me, on ¬¬A → A,isalrea- dy proved in there in its essentials, so that priority should belong to K.; a

3 Added in margin in Gabelsberger: One that occurs, for example, also in the Hilbertian article of the year 1932 in the Göttinger Nachrichten. 256 Part III: The original writings similar thing is said in the big bibliography of J. Symb. Logic [vol. 1, No. 4 of 1936] when this Russian work is detailed. – In that case, the theorem would have three authors! Your treatment of transfinite induction in connection with my consistency proof seems to me to make for a really welcome addition to publications of mine, insofar as I don’t bring in transfinite induction from the start in my new conception, but refer to the old one, and especially because you prove the representability of the proof of transfinite induction up to a number below ε0 within the number-theoretic formalism itself, something I just mention as a fact without a proof. Incidentally, I have left out the decimal expansions in the new conception and chosen the set-theoretical formulation with powers of ω.Ithinkthat you should get the proofs early enough to be able to incorporate on the basis of the new conception those changes in your presentation that seem appropriate to you. Maybe it would be preferable to refer also in your book to the ω-presentation in place of the binary expansions. It is certain that one can derive consequences about derivability in certain normal forms from my proof; I haven’t followed these possibilities so much yet. The special case of an endformula ∀xF (x), F without ∀, ∃ appears to me especially interesting. Something like the following must come out, namely that such a formula, if at all derivable in the system, is derivable by no other transfinite inferences that a single transfinite induction, to below ε0 I believe that it can now be seen rather clearly, in the new conception, how I reached the consistency proof, starting from the proof method of my dissertation. I have elucidated there the train of thought of the proof in detail. The Finslerian objection should persist at bottom in cases not as simple as the one treated by us; Something can be done against it, I believe, only when one considers more general domains of higher ordinal numbers. That remains for later. with most devoted wishes Your G. Gentzen.

15. Bernays to Gentzen 13 June 1938 (Hs975:1663)

This typewritten letter has deletions and additions, including the date and place, but gives still the impression of a ﬁnished letter. It seems that a last page is missing, because the letter is not signed. The main topic, as well as Gentzen’s answer in the next letter, concerns Gentzen’s 1938 new proof of the consistency of arithmetic. Bernays suggests a sharper formulation of the proof of Gentzen’s lemma 3.4.3, discussed above in Section I.4.10, but the paper was already in print and Gentzen left it at that. 13 Correspondence with Paul Bernays 257

Zurich,¨ 13 June 1938. Besenrain Str. 30. Dear Mr. Gentzen!

I received the proofs of your new treatise and took myself at once into reading. [Cancelled: I find that your new conception is a great step ahead in comparison to the earlier one that for the main part of the consistency proof was hardly accessible to even one reader, whereas the new train of thought is really transparent and comprehensible.] [Added in margin: It is particularly nice that you are again operating with a formalism that can be surveyed and that the ordinal number of a derivation is now defined in such a simple way. – It is interesting, in comparison with your dissertation, that whereas there cuts are eliminated in succession, your present procedure comes to the same by eliminating in succession the logical inference figures.] You have left apart the “reducibility,” and it means a certain weakening of the result, but on the other hand, a factor of difficulty in the consideration of the methodical “indubitability” is thereby lifted. Incidentally, one can formulate the result of your proof procedure also positively: Each sequent of grade 0 derivable in the formalism considered has the property that it gives a correct sequent with each substitution of variables through numerals (number symbols). It could perhaps be good to indicate by a remark, that the indirect proof procedure can be equally well turned into a positive one, – wherein the transformations of derivations considered relate to factually possible derivations. Your presentation of the proof seems to me to be most surveyable. I would just find it desirable if you formulated the considerations on page 15 line 19 and ff. [lemma 3.4.3, page 33 in Gentzen 1938] somewhat more sharply, something that could be done in the following way for example: Consider those sequents in the endpiece that satisfy both of the following conditions: 1. At least one thread runs from the sequent backwards to the lower sequent of a logical inference figure. 2. None of the principal formulas of the logical inference figures, from the lower sequent of which a thread (within the endpiece) leads to the sequent in question, belongs to the same formula bond as a formula of the sequent in question. The empty endsequent fulfils in any case both of these conditions. Consi- der further those ends of threads (going until the endsequent) that consist of only such sequents as satisfy conditions 1 and 2, and let among these one of greatest possible length be taken. 258 Part III: The original writings

Let S be the initial sequent of this end of thread, then it is easy to see the following: S is the lower sequent of a cut, and the cut formulas have a grade of at least 1; There runs therefore at least one thread backwards from each upper sequent to the lower sequent of a logical inference ﬁgure. Moreover, for at least one of these threads, the principal formula of the corresponding logical inference ﬁgure must belong to the same formula bond as the cut formula of the upper sequent considered, because the end of thread considered (from S to the end of thread) would not otherwise possess the properties required.

[There follows a list of ﬁve suggested corrections after which the letter ends abruptly, suggesting a missing third page.]

16. Gentzen to Bernays 22 June 1938 (Hs975:1664)

G¨ottingen, 22.VI. 38. Most respected Professor!

Many thanks for your reading through of my proofs; I have eﬀected your suggestions for stylistic improvements, as long as they did not cause bigger changes in typesetting. I have myself turned abundantly around the proof on p. 15, the “peak existence proof” in my old nomenclature, because it seemed somewhat unhandy to me. In the end, the form chosen in the text resulted as the relatively most intuitive to me, even if it is not formulated in a formally tight way. – I have even considered repeatedly a direct expression of the proof as a whole, but there is nothing good really to be made, because my reduction steps are now deﬁned for just the hypothetical derivations of a contradiction, not for really possible derivations. Otherwise I would have needed to generalize the reduction steps correspondingly. Of the proofs of your book, I have read through galleys 1–192. Some misprints and stylistic improvements in addition to those given already for the manuscript: [a list of corrections follows] ——– Is it known to you that there is a new work on the decision problem (further reductions) by Pepis (Fund. math. 30, 1938)?

with most devoted wishes Your G. Gentzen.

17. Gentzen to Bernays 16 July 1939 (Hs975:1665)

This letter is a reply to questions in a lost letter by Bernays dated May 31, possibly about some perished work of Gentzen’s on a topic that relates to 13 Correspondence with Paul Bernays 259 the comprehension axiom as intimated by the letter. As noted in Section I.4.3, a passage in the letter from Bernays to G¨odel of 28 September 1939 suggests that the topic is Gentzen’s recent extension of a double-negation interpretation to simple type theory. The comprehension axiom is treated at some length about half a year earlier, on pages 177–180 of the series WA, December 1938. G¨ottingen, 16.VI. 39. Most respected Professor!

Many thanks for your letter of 31.V. I would like to remark the following concerning your question about ¬¬A → A: I use the following “reinterpretation” of all formulas in derivations: Each prime formula is prefixed with ¬¬, similarly after each use of a logical connective sign (not necessary with ¬) prefixing with ¬¬. A reinterpretation of ∨ and ∃ is not necessary. (The other approach, ∨ and ∃ and possibly even ⊃ replaced by &, ∀, ¬ seems to fail with the comprehension axiom, also if one adds the prefixing of prime formulas with ¬¬.) Thereupon it results that all logical ways of inference resp. axioms remain correct, i.e., derivable from the old ones, and even without the use of ¬¬A → A. The same holds further for number-theoretic axioms, complete induction, equality axiom scheme, and the comprehension axiom scheme (say in the formulation of my paper on type theory). A carrying over to the usual axiom systems of set theory poses probably no difficulties. Now, you expressed doubts for the case in which the tertium non datur could appear as an implicit component of a definitional scheme. I cannot, however, think that there could be a difficulty here. For: One can transform the recursive definitions in analysis (after Dedekind, if I am not wrong) in- to explicit definitions. But then the tertium non datur does not occur as a component in a definition, but in an inference in a proof (in a demonstration of existence for the definition, more or less), so it falls under the treatment of the ways of inference. Admittedly, a ι-scheme for the introduction of functions cannot likely be mastered by the above ¬¬ procedure (at least not directly), but one can choose a calculus without functions, with just predicates. That means one more “reinterpretation” for a calculus with functions. with most devoted wishes Your G. Gentzen. 260 Part III: The original writings

14. Forms of type theory

The ﬁrst preserved page of the few ones that remain on type theory has on top a fragment of some previous issue, followed by a cancelled note on the axiom of choice. These have been preserved because of the longer note of October 1935 on the axioms of type theory that begins at the bottom of the page.

SLF15 It remains to be shown that the formula obtained, ∀xF∗x ⊃∀u(u ⊃⊂ h. ⊃ F∗(u for h)) where h is an argument is correct. One has here to pay attention to: From the left side follows ﬁrst F∗(u everywhere for h). i.e., where u is to be placed for h as an argument as well as an operator. Then one has to bring in h with the help of u ⊃⊂ h in the operator places, to obtain F∗(u for h) where h is an argument In every case, a more precise execution would be needed.

VII.34. The axiom of choice [Cancellation begins.] Nonsense. It is a question of the assignment ? Contentfully: There is to each set of sets that are non-empty a set that has one element from each of them. [Cancelled: Observe: “exactly” is needed. Otherwise the whole of it is trivial, one just takes the union of all sets. That is easy to represent formally. (Therefore the formulation in SW 3is nonsense.)] “Exactly one” cannot be requested. Example the sets (1), (2), (1,2). But the thing is trivial in this way, one takes the union set. The question should be in the main of the existence of a one-one correspondence between elements of the choice set and the single sets. I don’t know yet the circumstances suﬃciently. It should though, be possible to derive the axiom of choice from logical basic principles of a simple kind. It would be a ﬁrst requirement. [Cancellation ends.]

X.35. Carnap has, indeed, a conception that corresponds to mine, without the correspondence. (Even so, I wouldn’t like to abandon it.) But the existence of the correspondence should be provable.TherelationR(g, M) is simply deﬁned: Let A be the choice set that exists, then we have R(g, M)=“g is the (or one) common element of A and M,” ﬁnished! 14 Forms of type theory 261

General thoughts about the axioms of type theory Type theory should if possible represent, just like predicate logic, all the correct logical inferences that belong to a certain domain of formulas. With predicate logic, these resulted in a relatively unique way, to each concept in turn there belonged evident inferences. These remain the same for the connectives & ∨∀ ∃ ⊃ ¬,with∀ and ∃ at least analogous. (Further rules of substitution are already somewhat involved.) Now, however, there enters in addition the concept of an arbitrary set resp. property. (Disregarding ones with several places that should most likely behave analogously, though they could oﬀer something extra.) The analogy to the connectives: Introduction and elimination as corresponding to the formation of a set from elements and the removal of an element should not help. A related logical operation is the formation of a set from elements. The axioms of comprehension and choice belong here. The ca- sualness of the latter especially calls for an explanation. More speciﬁcally, a general

SLF16 entry or approach should be found for it. The axiomatizations of set theory should be studied for this. In general, the formation of sets proceeds through the composition of elements. It is diﬃcult to indicate which elements should belong to the set. One can assume other sets to be already at hand. Intuitively, one can make sort of: say a set of sets is collected together into a new set from all of their objects. NB: The concept of a “set” is much more intuitive than that of “property.” Maybe it would be better to take it as a basis also in the formalism? Further examples: A given set is collected together by the intersection of a set of sets. From a given set, an arbitrary subset is chosen. From a given set with more than two distinct elements (requires the concept of equality, can be even paraphrased), exactly two elements are chosen and united into a new set. Finally choice after the “axiom of choice”: A set is formed from a set of non- empty disjoint sets such that it has exactly one element in common with each. This kind of formation does not at all fall out of the intuitive boundaries of all these formations! 262 Part III: The original writings

It can also be combined with formations of other kinds. For the formal conception, it seems now that one manages with quite a special formulation of the two “axioms” mentioned. This is a really strange fact even with the comprehension axiom. It should contain here a new, special kind of completeness of the propositional connectives and quantiﬁers & ∀¬. Namely when concepts such as “intersection” etc. are built with these, even “subset”! There is in “subset” already hidden a kind of concept of choice. The choice of a single element can be rewritten through a suitable property, by which the axiom of comprehension can be used:

Set a1 (∃x0 a1 x0) [written above: not union?]. I form a set b1 that consists of exactly one element of this set:

b1 u0 ⊃⊂ ∀x0(b1 x0 ⊃ x0 = u0)&a1 u0 It should be noted that the intuitive type of formation and formal execution are here thoroughly apart, even more than with intersection, subset,inwhich also noticeable! About equality: It is quite justified to take it as belonging to logic. (Especially the metamathematical equality!) And also not, a point of contest that cannot be decided. One should on the whole think about a determinate domain of objects in logic. A finite domain in the first place, then all concepts about sets are deli- mited in advance, all inferences testable (admittedly only for a determinate number). Admittedly, all things must hold even for infinite sets, of which one has only a somewhat vague image. Where one could even contest the correctness of inferences; for this area is outside all experience, is in any case properly just imagined, resp. a paraphrase of the factual, finitary. (Basic idea of the proofs of consistency.)

SLF17 Just collecting together denumerable sets, one would be beyond the ﬁnite and in an area that is still half-way transparent. Though one would not be allowed to form the set of all sets of type 1 (real numbers!). Hereby there arise limitations on the comprehension axiom that are quite extra-logical.

Formal construction of the set of all sets of type 1: b2 u1 ⊃⊂ u1 = u1 or the like. Again an example of the sharp distinction between what is intuitive and a formal construction! Or b2 u1 ⊃⊂ . The whole of type theory without a domain of objects sways very much in the air. Does not have much independent sense. The view of the logicists puts all things on their head, for they take logic as primary and the domain of objects is only then developed from it. I can’t feel myself at home with 14 Forms of type theory 263 this. The logicist view becomes, admittedly, a bit clearer when one thinks that it does in fact take a domain of objets as primary, namely something like the set of all things. The natural numbers, for example, are secondary insofar as even in the natural conception, the primary concept is that of a (finite) set, whereas numbers are introduced only afterwards for the counting of these sets. What is it that gives origin to the enormous difference between type theory and type-free inconsistent general logic [Gesamtlogik]? Type theory is valid for arbitrary domains of objects. Type-free logic requires to a certain extent an infinite domain of objects, because it contains inferences that are conditioned on such a domain. Certainly, it is sufficient that the domain of type 0 is finite here, infinity obtains only at higher types, resp. in the type-free concept of an arbitrary predicate. So, the difference is here not yet that distinct. Even type theory has, in all types together, an infinity of objects, it’s just that it doesn’t make any propositions about all these together. The modern logicists attempt to obtain the infinitely many objects from logic itself (without the axiom of infinity) without getting thereby even the paradoxes. I take over from “Allerlei Bemerkungen” (Miscellaneous remarks) the following considerations from the time: X.33–V.34. on the axiom of choice: I would like to claim: This axiom is a logical self-evidence by itself, i.e., that it has to be formulated axiomatically, is just a sign of the insufficiency of the logical calculus so far to represent all that is logical! Attention to be paid to: it stands as valid to choose an element from one set. One shows: There are elements in the set (as far as it is not empty) and one concludes then simply: let a be such an element! Thereby the choice has happened. (This conception is at the basis of the existence-scheme. With my ∃E in the natural calculus even more distinctly.) One should simply allow for something similar going on, to start with especially: There is a set of predicates (classes), there are to each predicate objects that have it, then one is allowed to introduce the set of such objects! (Even to be formulated with just predicates, without the concept of a set.) — In a formulation with predicates: One introduces a certain predicate, namely the predicate: to be a chosen element. More precisely: one can show in advance: for each predicate, an element can be chosen to which it applies. Thereupon one forms for a collection of predicates, characterized by a predicate of a higher type that applies to them, the new predicate: to be a chosen element. This one defines the choice class. 264 Part III: The original writings

XI.35. The axiom of choice has definitely its justification in the axiomatics of set theory, as a specifically formulated axiom of set-existence, for here even other logically self-evidently existing sets (like intersection etc.) are expressly axiomatically required. It is, however, different in logic, in which the special position is quite unnatural.

SLF18 VIII.36. I recommend some time: to investigate the completeness of type theory! Analogous to G¨odel’s investigation of the completeness of predicate logic. Compare Skolem on its normal form, carrying over to type theory with Carnap, meaning of formulas in analysis. The role of the axiom of choice would be decisively illuminated by this. One asks if the question can be at all treated in a somewhat meaningful way, because we have to do here with “in-themselves” non-denumerable cardinalities (continuum of type 1.)

IX.36. Bernays objects quite correctly that there is involved in the completeness of type theory the completeness of all consequences of, say, the axiom of infinity, one that can be even introduced formally, i.e., written down each time as an antecedent of an implication. Herewith all the difficulties of number theory enter, e.g., the Gödelian incompleteness. Therefore: Nothing to do for the time being in this matter. Even more simply: The undecidable problems of first-order predicate calculus can be formulated in the next type through ∃P , and they are here undecidable sentences.

X.37. About the axiom of choice see both of the partial conceptions in HA [Hilbert-Ackermann] second edition. 15 Predicate logic 265 15. Predicate logic

The notes on predicate logic bear the signum PF or PLF (Formen der Prädikatenlogik. The first three are short notes, the last two more substantial pieces. Even the early note on five different forms of natural calculi, item 4 above, had been placed among these. Item 15E is found in the series AL on propositional logic but treats also the quantifiers and is referred to in the end of item 15D. It is found here right after 15D.

15A. Unnaturalnesses of the ways of inference in formal logic.

PLF6 1935? Unnaturalness of the formulation of the ways of inference is not meant, for these are rather arbitrary, but of the inferences in themselves. (Compare the objections of Hölder,König, and others. Also: Formal implication, etc.) A A → A ∨ B A ⊃ A ∨ B Let the example be: The OI. (or A ∨ B or or Γ → A or whatever.) Γ → A ∨ B This inference is not in accordance with the natural impression. One will say, if A holds, then of the two possibilities A and B just A, but there is no sense in saying that A ∨ B holds. One will instead accept A ∨ B in the following case: When out of two possibilities that obtain (say A, ¬A)atone time A follows, at another time B, then one is entitled to say: A ∨ B. A → CB→ D Formally something like: A ∨ B → C ∨ D In this way, ∨ can bring something (as with ¬!). But how does a first ∨ appear? Say through REND [reduction of double negation]. Or through mathematical axioms that already contain ∨. But is this kind of ∨ already always sufficient? To be noted: A ∨ proposition has sense in the first place in the form: ∀x(Ax∨Bx). As it is in fact the case that for certain x A holds, for others B. For example ∀x(x =1∨ x>1). If this is proved by, say CI [complete induction], one begins indeed with 1=1,so1=1∨ 1 > 1! Or in what way? No, one would say instead contentfully: For the number, 1 = 1 holds. One does not say, though: 1 = 1 ∨ 1 > 1 holds, but just: Precisely 1 = 1 holds. So this leads always to impurities. Should one then accept a ∨ only when, as with ∀x(Ax ∨ Bx), both possibilities actually occur. But one cannot even 266 Part III: The original writings conclude in this way A3 ∨ B3. For here again, just one possibility holds. Maybe one just doesn’t know which. Should the acceptance of ∨ be made dependent on whether one knows something, that is even more stupid. Result: The apparently “natural” conception of ∨ touches on many impurities that are left aside in the conception used in formal logic. Here A ∨ B means simply: At least one of the propositions holds. Thereby the inference OI is obvious. Another case that is similar: Is A ⊃ B correct, when B is correct and A arbitrary? As well as: Is A ⊃ B correct, when A is false and B arbitrary?

15B. Decidability in predicate logic

II.37. Let A be the axioms for [Vg?], =, +, ·, exponentiation, F Fermat’s theorem. This is an elementary ∀-proposition! I consider for provability the logical formula Vg = [?], formula variable): ¬ A ∨¬F ⊃⊂ A ⊃¬F. If Fermat’s theorem is correct, this formula [A ⊃¬F] does not hold generally. For it would then present a counterexample within the domain of the natural numbers. If on the other hand the formula does not hold generally (will say is not provable, after Gödel), is the Fermat theorem then correct? Its not holding generally means that there are predicates for which it does not hold. Therefore A must hold for these predicates, and F at the same. But it follows from this that Fermat’s theorem is correct, and the other way around. By associating such predicates always to the natural numbers in question. So, we have further that the following holds: “A ⊃¬F is provable” is equivalent to: “Fermat’s theorem does not hold.” Well, finally the thing clears up, and it is quite trivial: Fermat’s theorem does not hold means: there is a counterexample. This fact can then, for the example has to be given in numbers, be concluded from the axioms by purely predicate logical inferences, without CI naturally. It becomes completely clear and understandable by the above why the decision problem for predicate logic is coupled to practically unsolvable difficulties! Only finitely many axioms are needed for this, representable in predicate logic, namely, as many as suffice to carry through the technical decision for each combination of numbers. I thought of solving the decision problem (with the help of my Hauptsatz) for questions such as: intuitionistic provability of ∀x∃y A(x, y) → ∀x∃y B(x, y), in which A, B are without ∀, ∃. However, I don’t think that this case would already be equivalent to the one mentioned, i.e., that it would already contain a decision of Fermat’s theorem. For one should be able to 15 Predicate logic 267 compress the axioms into the first form without an essential difference, the Fermat theorem into the second, and the proof of the counter-existence would naturally also be intuitionistic. One will be able to show, however, that decision is possible for a ∀- proposition as antecedent formula, as one can have no use at all of the part of the premiss that expresses existence. This theorem states admittedly (if it holds) not much anymore. It seems to follow from my Hauptsatz and even quite independently through a model consideration. So, a worthless thing. A decision in the case of existential propositions is understandably not possible, because there need not obtain any connection between the simplicity of the axioms and the length of construction of an existing object with a determinate property. In any case, one does not know anything about this. With the number-theoretic axioms also presupposed, one cannot know whether by ever new computations, a number arises in the end with a determinate computable property. This can take an eternity. Example the case of Fermat. There could be possibilities of decision with axioms of a certain simple shape say the case of Pressburger. But this does not given one much.

15C. Decidability and the cut theorem

IV.38 More on the question of decidability in connection with the cut theorem. I believed on and off to obtain specific decidabilities, e.g., the equivalence of two formulas. Such things, though, naturally contain the general decidability. I must admit that I thought one could somehow pass by such generalities. But how should they be excluded? Calculus LK without ⊃¬: Contains perhaps already the general decidability in it, i.e., is equivalent to it. Namely: Let arbitrary formulas be brought to prenex normal form, here no ⊃, just only ¬ on atomic formulas. Can that perhaps be replaced by specific formula variables with added special axioms (that could be connected to the formulas) such as Fa∨ Fa, TND. Should be carefully considered. Decidability should in any case be reachable for formulas without ⊃, ¬,ones in which “generalities” are excluded in a sense, because there are no generally valid formulas of this kind. The difficulty is that ∃-decomposition can produce a, fa,thenffa fffa etc. One does not know how many specific applications one needs in reality. One reaches admittedly the goal in exceptional cases, through specific considerations that suit the case, as in the case treatedbySkoleminhiswork. 268 Part III: The original writings

15D. Natural introduction of the calculus LK?1

PLF8 IV.38. A step-by-step path can be taken for something like a textbook or a lecture for beginners. First just & and ∀. Here just sequents with precisely one formula as a succedent formula. That is to say: No notion of a sequent needed at all. Results are obtained from “axioms.” Something to be made anyway with this. Then addition of ∨ and ∃.Sayfirst∨. Here one needs, to use ∨,thedivision into cases in natural inference. And the problem arises quite naturally: How do I represent this formally? There appear for the first time propositions that are not in themselves correct propositions. But only conditionally. I.e., several cases present themselves one beside the other. The most natural presentation: a sequent without antecedent formulas, with A ∨ B several succedent formulas ! ∨–elimination: A, B Then put: To make a difference with respect to a proposition as such, we want to equip an asserted proposition, i.e., the one designated as holding in → A ∨ B aproof,with→. Correspondingly the division into cases, so: → A, B Allofitquite naturally! Now the carrying over of all inference figures A, B&C from single formulas to sequents! For example, now even holds in A, B the natural: further treatment of one case while the other remains intact. One can introduce already here contraction and exchange as structure– “inference figures” (better: structure–modifications). The first one should be even necessary! Belongs to the nature of the thing. For example, the finishing of a case distinction through the same result on both sides!: → A, B . . → C, C → C

Put here: These are formal necessities of our formalization. Sure, the same happens in a natural proof, though without special notice. F(t) Then ∃ and the free existential variables.First∃–introduction: This ∃xF(x) → Γ, F(t), Δ is for sequents: This is the general formulation, now. → Γ, ∃xF(x), Δ

1Addition: See what the basic thing is, on page 9 down middle. 15 Predicate logic 269

Our purpose is, to be sure, to give a formal image of actual reasoning. ∃xF(x) Now ∃–elimination: e a there–is–variable (a all–variable). F(e) Clearly one has to take care of the variable conditions that are revealed by the natural usage. [Added: Compare PLF 1–2, calculus NNA for example. A clearer presentation possible.] Division into cases leads to certain complications, but this not bad. Not different from the natural proof that is always possible! It would be good to have here an example proof, with considerable nesting of many cases and variables. Special role of &–introduction as the, so far, only inference with 2 premisses. Net form of derivation is more natural than tree form? All of this seems more involved than writing directly the finished calculus in its entirety. It is a tertium! The picture as whole is, admittedly, not really understandable for a beginner, before he has gone all the way through each detail! Before that, he would not know the least what it all means contentfully, in its single details. And only then can he work with it on his own. One can already develop some beginnings of number theory with the ways of inference at hand. It will be good here to add ¬ for prime formulas, together with REND [reduction of double negation]: → A, ¬A.Thelaw of contradiction is not necessary, as one does not have any negation over connectives yet. Entirely appropriate! The latter negation is anyway almost superfluous. ¬ for prime formulas goes instead for a positive proposition. (As is known, even so conceivable, = instead of ¬ =) [Cancelled: It is remarkable that everything so far is finite! Irrespective of the case distinctions, ∨ is used only constructively.] Nonsense. The whole of it is not finite, for example, ∀–introduction with additional formulas is not intuitionistically acceptable [cancelled: if I got it right.] If one wants to stay within the finite, one has to restrict appropriately the inference figures that have additional formulas. It fits badly to the calculus as a whole. Or: Example: → a =1,a=1 → ∃x(x =1),a=1 → ∃x(x =1), ∀x(x =1)

That is not intuitionistically correct. But is it still natural? No.Itcontains in fact more than stands there. Namely a transﬁnite REND. Following the natural feeling, this kind of inference should rather be forbidden through 270 Part III: The original writings a variable condition! Like this: a is a free variable for the whole division into cases. Then it must not be generalized within the same for one case. Even if it has already disappeared in the other case. The situation is instead to be conceived exactly as with the OE-formulation with assumptions! The variable condition is to be formulated correspondingly.

PLF9 So: It is throughout natural, and the feeling is correct, when the calculus is put up as a finite one; it works through a suitable variable condition that is indeed natural! Then the addition of ⊃: Reference can be made here to the limits of the finite. Say, to start with by accepting ⊃ without nesting.(Or,say,onlyfor prime formulas.) Here assumptions appear as a new form of structure. There are so far in the whole calculus no inference figures for antecedent formulas (disregarding naturally structure–modifications here) and no cut. So far all in a natural calculus. Some forms of LK can be given later, as artificial but on the other hand unifying forms. The special nature of the organization of introduction and elimination inferences results in a simplification of the variable conditions into a single condition and at the same in the turning of cut into a necessity. About the division into cases: Whether even here, in the end, the formulation with assumptions Γ → A ∨ B ΔA → C ΘB → C ΓΔΘ → C is the most natural one? For in actual reasoning, one does always so that one chooses one case alone and works it through. (Repeatedly in a demonstration of impossibility, and how is it with its representation in the calculi, in regard of naturalness?) In sequents with [several] succedent formulas, the thing does not find such a clear expression. For example, when free variables appear as new ones in one case but don’t concern at all the other cases. And when different consequences are drawn from the basic assumption of one case. It works, admittedly, even then; one has just to add each time the side cases. (As one has net form to start with.) If one wants to refrain from writing only valid sequents, one comes back to the natural calculus, this stands closest to reality, but is still somewhat unpractical for the formulations. Though because of this, the sequent is really natural: One has in an actual proof one case and follows it, keeping all the time in mind the case as a whole and its basic assumption.Theposition of this assumption of a case is lost with the sequent. The assumption itself disappears already in the first conclusion from it. One has to go back again 15 Predicate logic 271 to the first sequent in further conclusions and to use the net form in a decisive way. There is no such defect in the calculus NL [as in item 6] in the assumption formulation.Sothis one is by the same more natural. In general: As long as one remains within the finite,whichisthe natural in the first place, the calculus with symmetric sequents is still unnatural.If one wants to have it natural, one has to liken it to the formulation with assumptions through special conditions (variable condition for UI !). It obtains its proper meaning only in connection with the non-finite, especially with REND, negation more generally. Here it becomes simple and deviates from a natural one. To wit, because the “classical” is not natural! So: A natural introduction of LK is possible to some extent, along the above path, but then one cannot yet recognize its advantage. I.e., its advantage does not depend on its being accessible in a natural way, but that is a more arbitrary side issue. That is the main point. Compare also AL 141–142.

IV. 43. Formulation of intuitionistic predicate logic with symmetric instead of single succedent sequents. See AL 142.

15E. Formulation of intuitionistic logic with symmetric sequents

AL 141 IV.43. Use of symmetric sequents in positive (intuitionistic) logic.2 1. Propositional logic with just & and ∨. Here, as is well known, classical (truth value-) propositional logic and positive propositional logic are in agreement regarding “correct sequences.” It results from the agreement that one can use here also for positive logic symmetric sequents as in LK, with the advantage of formal symmetry. A proof that this classical calculus is correct in the sense of positive logic: It is simplest if we use groundsequents for the connectives: A & B → AA→ A ∨ B A, B → A & BA∨ B → A, B A & B → BB→ A ∨ B There come in addition the LK-structural inference ﬁgures as well as the groundsequents D → D. The proof goes now like this: Each sequent in such a derivation is to be changed so that all succedent formulas are bound by ∨.(Thereis,bytheway,

2 Added: (Compare PLF 8–9.) 272 Part III: The original writings no empty succedent, no empty antecedent either.) Now every groundsequent is trivially positively correct. It remains to show that this is maintained with the structural inference ﬁgures. Trivial for weakening, exchange, contraction in the head. For weakening at the back: To be reproduced with the help of OI . Exchange at the back: Also clearly positively correct. Contraction at the back: Also clearly positively correct. Formal execution for contraction: There is given: Γ → ((Δ ∨ D) ∨ D) ∨ Θ (Δ, Θ empty easy special treatment.) Γ → (Δ ∨ D) ∨ Θ This is turned into: trivial. Cut: Let there be: Γ → D ∨ ΘΔ, D → Λ Γ, Δ → Θ ∨ Λ This is turned into: Δ, D → Λ OI OI Δ, D → Θ ∨ Λ Θ → Θ ∨ Λ OE (first weakening) Γ → D ∨ Θ Δ, D ∨ Θ → Θ ∨ Λ Cut ﬁnished. Γ, Δ → Θ ∨ Λ I have naturally laid as a foundation for positive logic the calculus LI. Hereby the equivalence of the classical and positive calculi has been proved for the domain of just & and ∨ alone. Therefore one can conveniently formulate also the positive calculus as the classical one with symmetric sequents. So, to represent the “division into cases” without special assumptions. Admittedly, a liberation from assumptions altogether has not been achieved hereby, in that the sequential representation contains, seen naturally, assumptions all over. That is also unavoidable, because there are no correct formulas without assumptions at all in this domain. There remains, though, as a gain in any case: The possibility to produce a duality theorem for correct &, ∨-sequents in a positive logic. (Same for the classical as well.)

AL 142 2. Addition of ⊃ and ¬ (as well as ∀ and ∃). Theorem: Any addition of ⊃, ¬,andalso∀ and ∃, yields correctly the associated intuitionistic logic as long as one keeps to the admission of symmetric sequents in general, however so that the LK inference figures that belong to these with an occurrence in the succedent , are tied to sequents with at most one succedent formula. Then: The calculus in question is not too narrow, because it contains within itself the LI -style calculus with the general limitation to at most one succedent formula. On the other hand, it is intuitionistically correct, for: If 15 Predicate logic 273 one substitutes all sequents with ones with at most one succedent formula, as in the proof on page 141, the inference figures there treated remain correct as shown (an empty succedent should not cause any difficulty). The same with the new inference figures for the connectives in the case of no occurrence in the succedent, trivial. Those with a succedent occurrence remain unchanged because of the formulation of the limitation. The said limitation is also in general necessary for these connectives: For ⊃, e.g.: An FE can read A, A ⊃ B → B and remain so. A FI for positive logic must read: Γ, A → B Γ → A ⊃ B without additional succedent formulas. This means that a FI within a division into cases must be treated separately, as given for the case alone, under its case assumption. Any other formulation would not be positively correct. Correspondingly for ¬: Γ, A → NE A, ¬A → NI: Γ →¬A Here the admission of several succedent formulas would make the TND at once derivable, as known: A → A →¬A, A A similar thing holds for ∀:EvenUI, Γ → Fa Γ →∀xFx is as is known intuitionistically incorrect with an extra succedent formula. (Example PLF 8 bottom.) It should be said that no limitation remains for ∃ even by the above theorem, if one chooses EI in the groundsequent formulation. No limitation is required for EE, because nothing could happen in the succedent. EE would need no limitation even in the form of an inference figure, because the following holds already positively: Whenever A ⊃ B,thenalsoA ∨ C ⊃ B ∨ C.Clear. The same holds for the formulation as inferences of each of the & and ∨-inference figures: Even these can be admitted without limitations intuitionistically for symmetric sequents, proof as before. (Should anyway follow already from the proof on page 141, considering the equivalence that arises in LK between inference figures and groundsequents.) The result as a whole: One can obtain intuitionistic logic (and each partial calculus with whatever chosen connectives) from the calculus LK resp. from a formulation with equivalent groundsequents in place of the inference figures for the connectives already with lesser restrictions than the requirement that each sequent must contain at most one succedent formula, namely with the following restrictive condition: Arbitrary (symmetric) sequents are 274 Part III: The original writings admitted in a derivation, however, in the inference figures FI, UI, and NI (the groundsequent formulation of NI is not admitted), at most one succedent formula must occur in each sequent. For example, the logic of &, ∨,and∃ is classically and positively equivalent. That does not mean much. There are no correct formulas here, only correct sequents. And these have just little content. Should be fully decidable the whole domain. 16 Propositional logic 275

16. Propositional logic

Of the series AL,forAussagenlogik, i.e., propositional logic, pages 133 to 142 have been preserved in the violet folder. The pages 134–136 are blank except for the number 136. The ﬁrst date is II.37, page 133 on the completeness of the propositional calculus LK, then follow three dense pages 137–139 on a semantics and decision procedure for intuitionistic propositional logic dated X.42. The remaining pages 141–142 are dated IV.43 and give a multisuccedent sequent calculus for intuitionistic logic with the quantiﬁers included, found as the previous item 15E. One of Gentzen’s early notes from 1932 mentions “my book about propositional logic.” What is left of the series AL may very well have begun with such a notebook.

16A. Completeness of the propositional calculus LK

AL 133

II.37 The completeness of the propositional calculus LK. (After a reading of Kalm´ar.)

This can be shown easily, analogously to Kalmár’s procedure. (Axiomati- sierbarkeit des Aussagenkalkuls,¨ LV [likely for Literturverzeichnis, list of references]) Namely: One derives first the desired formula from some “distribution of truth values” over its formula variables. I.e., One begins with A → A for each formula variable. And now one builds up the formula according to the computation of truth values. If A = , it remains in the antecedent until the inference, A in the succedent will be handled. If A = , the other way around. The LK-rules are directly tailored to this procedure. In the end one has the desired formula, under the assumption of an arbitrary distribution of truth values. I.e.: The distribution considered manifests itself in the presence of formulas in the antecedent () and succedent ( assumed.). All these additional formulas are removed by successive cutting. Note well: The special position of ⊃ and ¬ is completely removed in the procedure, contrary to Kalmár. This corresponds to the special character of the symmetric calculus. One can, by the way, somewhat laboriously substitute the cutting by ulterior composition (linking together) and thereby also to prove the cut theorem for propositional logic. (Not much gained in comparison to a direct proof of the same.) This proof of completeness should be the overall simplest and most natural one. It is appropriate to renounce ⊃ (and ⊃⊂), to take these as defined. 276 Part III: The original writings

But &, ∨, ¬ all together to be included. As is on the whole appropriate with LK . But inclusion of ⊃ (and ⊃⊂) gives no particular diﬃculties. (Only an unsymmetric treatment.)

16B. Correctness and completeness in positive logic

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X.42 Concept of correctness and demonstration of completeness for the positive propositional logic. Not possible by the usual methods of truth values matrices, as is known (after G¨odel, should be). Still I think it possible more or less like this: For the concept of correctness is deﬁned according to ⊃ -degree. For ⊃ -degree 0, i.e., formulas without ⊃: Here there are no correct formulas as is well known.

Now ⊃ -degree 1, i.e., complexes A0 ⊃B0: for example [parentheses added] A &(B ∨ C) ⊃ A ∨ B (a correct formula) I distinguish three—— two truth values: and undetermined 0. Then the following holds for & and ∨: (exactly as classical with 0 = ) & = ∨ = &0=0 ∨ 0= 0& =0 0∨ = 0&0=0 0∨ 0=0 and I define: A0 ⊃B0 is “correct” (no truth value is given here) when and only when B0 = for each combination of truth values to the propositional variables for which A0 = . (The concept should be here still quite in accordance with the classical one, and the same for derivable = correct formulas. Should be differently already with the next ⊃ -degree.) I think it is so. But now for example: (A0 ⊃B0) ∨C ?Now,thisiseasy:A∨B is in general correct when one of the two is correct, A & B when both are. So, reduction to decision on subformulas. E.g., A ⊃ B.∨ A is classically correct, now false resp. 0. ⊃ -degree 2. The real difficulties begin only here. I consider the ⊃ -normal A0 ⊃B0 . ⊃ . C0 ⊃D0. A correct example: A ⊃ B.⊃ .. A & C.⊃ .B∨ D Attempt at a definition of correctness: I consider all possible truth value distributions. I exclude among these the ones for which A0 = , but instead 16 Propositional logic 277

B0 =0.IfC0 ⊃D0 is correct in the above sense, considering only the remaining combinations, the whole formula is correct. Is it already right so? An attempt with the example: The combinations with A = and B = 0 are excluded. There remain to be considered all those combinations of truth values for which A & C = , except the ones excluded. I.e., there has to be A = and C = ,aswellasB = corresponding to the exclusion, therefore also B ∨ D is = ,inorder.

Nice. But now, for example: (E0 ∨ . A0 ⊃B0) ⊃ (C0 ⊃D0). It seems to be necessary to deﬁne here: This is correct when both of the single formulas E0 ⊃ (C0 ⊃D0)andA0 ⊃B0 . ⊃ . C0 ⊃D0 are correct. Other cases correspondingly. Even to be formulated still more generally, after the contentful sense. It seems to me that the thing as a whole can be done. Let’s try again:

⊃ -degree 3. A1 ⊃B1. ⊃ .C1 ⊃D1 (⊃ -normal to begin with.)

It is more or less like this: One investigates whether C1 ⊃D1 holds under the additional knowledge that B1 holds automatically together with the holding of A1. There would remain to be investigated later the extension of the results to predicate logic. (Proof of completeness?) It speaks of itself that the concept of correctness for propositional logic must deliver a (new) decision procedure, and an indirect proof of my Hauptsatz, the latter so that derivability along a minimal way, without detours, follows from “correctness” in the demonstration of completeness. (More or less like in my Hertz-paper.) One danger is that the whole definition sound all too much the same as the one of detour-free derivability. Such lies in part in the nature of the matter. Still, the point of departure above is essentially different. (To be investigated especially in relation to this question the example A∨B ⊃C.Whetherit is really necessary to resolve this in the way of ∨E for the definition of correctness? Surely one can make a more direct connection to the ∨-sense.) Cf. HB II, Supplement III, and above all Wajsberg, Untersuchungen uber¨ den Aussagenkalkul¨ von A. Heyting.

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Considerations (up to ⊃ -degree 3). (A ⊃ B. ⊃ A) ⊃ A is classically correct,———————– intuitionistically positively not. (By HB I.) A positively correct one is for example: A ⊃ (A ⊃ B.⊃ .B ⊃ A) 278 Part III: The original writings

How does the decision proceed in the positive? One has four truth distributions: A : 00 B : 0 0 1234 A is characterized by distributions 1,2 B by 1,3 A ⊃ B is characterized how? Now, it states that those distributions for which A is correct permit only ones for which B is correct.

A0 ⊃ B0 in general: But now, it is just quite simply a blocking of truth distributions, namely those that make A0 = ,B0 = 0. (The accordance with the classical is here still appropriate) £ Written for example:£ A ⊃ B blocks distribution 2, : ¢—2 ¡ For B ⊃ A to be written: ¢—3 ¡ A0 ⊃ B0 is correct if it blocks no distribution; but A0 nevertheless satisfiable, this it seems to be requested. (Holds automatically for 0-degree). £How now A ⊃ B.⊃ .B ⊃ A? It is a condition between two blockings, ¢—2 ¡→ —3 . So when is such a formula “correct”? I take the example: A ⊃ B.⊃ .A& C ⊃ B. The distributions are: A 0 B 1234 1234 C called:1234 5678 £ £ £ A ⊃ B is now = —–¢34¡the whole of it is —–¢34¡ → ¢—3 ¡, and further £ A & C ⊃ B = ¢—3 ¡ is correct because all that is blocked at right is also blocked at left. £ £ This is how it should be in general: —¢– ¡→ —¢– ¡ is correct when all distributions blocked at right are also blocked at left. If nothing is blocked at right, it is also correct, as it should be. If everything is blocked at left, the result is also: correct. This, though, can be excluded by definition, one can require that not all can be blocked at left. This not so important for the time being. Yet another example: A ⊃ B ⊃ (B ⊃ C.⊃ .A⊃ C)wehave £ £ £ A ⊃ B = —–¢34¡ B ⊃ C = ¢—–26¡ A ⊃ C = ¢—–24¡ (3 and 4 are blocked, 2 and 6 are blocked, so also 2 and 4 are blocked) £ £ £ £ The condition ¢—–26¡→ ¢—–24¡[written below: eq ¢—–26¡→ ¢—4 ¡]isitselfnot correct, because 4 at£ right is blocked, but not at left. 6, though, is correct under the condition —–¢34¡, because also 4 is blocked by it. 16 Propositional logic 279 £ The classically correct£ formula£ above: A ⊃ B£= —–¢34£ ¡ A =£ correspondingly. £ So A ⊃ B ⊃ A = ¢—2 ¡→ ¢—3 ¡ this contains ¢—2 ¡→ ¢3and¡ ¢—2 ¡→ ¢4¡ but A cannot be obtained from these. An attempt: Reduction of decision£ to ⊃ -degree£ one lower:£ I set with£ two propositional variables A and B: ¢—1 ¡ = a£ ¢—2 ¡£= b ¢—3 ¡ = c ¢—4 ¡ = d Now A ⊃ B = b, B ⊃ A = c, A isolated = ¢—3 ¡& ¢—4 ¡, B isolated = c & d Hereby the two examples above become: (b ⊃ c & d) ⊃ c & d resp. c & d ⊃ (b ⊃ c) The latter remains clearly correct. Whether this holds in general? It does seem very daring to me. £ £ £ £Correspondingly £ £ for the transitivity example: one has ¢—3 ¡ & ¢—4 ¡⊃ ( ¢—2 ¡& ¢—6 ¡⊃ ¢—2 ¡ & ¢—4 ¡) in order. Attempt with: A ⊃ B ⊃ [A ⊃ (B ⊃ C ⊃ C)] We have first: 3 & 4 ⊃ [5&6&7&8 ⊃ (2 & 6 ⊃ 2&4&6&8)],Thisisap- parently correct. It should continue to go like that. The approach: For the general proof: I consider again the case of two propositional variables, and in the first place just the combinations given above (so not for example A∨B and similar ones.) The idea is that one looks at all truth distributions as independent of each other, i.e., no aprioriconnection between such as would be given by the TND in the classical. It seems first somewhat paradoxical that one operates, nevertheless, with truth values as in the classical. But perhaps it is still appropriate. Therefore: Let a formula be positively derivable, it is to be shown that also its transform is. (The converse should be easier.) I.e., the natural derivation (without hillocks) of the formula is to be transformed correspondingly. I undertake for the whole derivation the same substitution: A ⊃ B by b, B ⊃ C by c, in all other kinds of places of occurrence A by c & d, B by b & d. Which inferential connections became herewith false? Apparently just ⊃ I and ⊃ E with A ⊃ B resp. B ⊃ A.(WhenA ∨ B is allowed in isolation therecomealready∨I and ∨E in addition, with the same with ⊃ above even more.) An ⊃ E becomes for example: AA⊃ B c & db into B b & d in order. Now instead a ⊃ I: [A] [c & d] . . . . B b & d A ⊃ B into b It is essential that a, b, c, d are indecomposable propositional variables. Then the matter is admittedly illuminating from the contentful point of view. 280 Part III: The original writings

Formally: One obtains ﬁrst by &E b. The claim is now that B is independent of the assumption c & d.

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That would not hold if for example another assumption of the form c ⊃ b entered that was used for the derivation of b from c by ⊃ E.Thereis,however, no occurrence of ⊃ between the a, b, c, d. So? One can certainly make the derivation of b free of hillocks by the hillock theorem. Let us consider in LI , it is more common. The sequent: Γ,c& d → b derived without cut. The claim states that one could get rid of c & d throughout, to strike it out. (Contentfully: Because no dependences have been posited between the a, b, c, d (such as through TND classically), one of these is not deducible through an engagement of the others essentially used, but at most apparently so.) That should be possible even formally. Let’s see: Example: 1 2 1 2 A A ⊃ B c & d b B 1 b & d 2 A ⊃ B b Here c & d is not in fact used for the production of b, and it becomes removed in hillock reduction. But now to the example: 1 1 A ⊃ 2 2 ⊃ E ⊃ ⊃ ⊃ c & d B A (B A) (A B) 1 c c ⊃ b 1 ⊃E A A ⊃ B c & d b ⊃E B ⊃I 1 b & d 1 A ⊃ B b Here the assumption c & d is evidently really needed for the production of b, so the thing is not correct! Now? Itwouldmeaneventhatb should be deducible from c ⊃ b ! This leads to nonsense. The counterexample reads: (B ⊃ A.⊃ .A⊃ B) ⊃ .A⊃ B This is positively correct, as the given natural deduction shows. But our procedure is not applicable to it. For it yields: (c ⊃ b) ⊃ b,whichisnot. Hereby the attempt has failed. Will something remain of the attempt? The example is interesting, it can even be done in an ⊃ -normal way, like: 16 Propositional logic 281

(B ⊃ A.⊃ .A⊃ B) ⊃ (A ⊃ A.⊃ .A⊃ B) The example is on the whole important already for the whole train of thought of the previous page. The approach with the “blockings” becomes already£ questionable £ through£ it, or does it? For it does give the formulation [ ¢—3 ¡⊃ ¢—2 ¡] ⊃ ¢—2 ¡. The correctness of the formulas is, however, not anymore recognizable here. Admittedly, one can proceed with the recognition so that one resolves the consequent and asks oneself what the things are that the condition A draws with itself on the basis of the givenness of the antecedent. It draws almost automatically— 3 with itself, so obviously after the antecedent also— 2 . The decision procedure resembles in this form greatly the procedure based on the hillock theorem with its construction of all the possible few derivations together and the attempt to arrive in the end at the desired result by proceeding from the top down. So apparently one should manage to make it. The essential thing here is just whether it is possible at all to operate in some way with truth values as a basis. By the way: Pure ⊃ -formulas, if they are written as sequents as can be always done, are in any case positively correct whenever the succedent formula (of grade 0!) can be derived from the antecedent formulas by just ⊃ -eliminations, including weakening as in the counterexample. This should stand already in Hilbert-Bernays. It gives without further ado an illuminating “deﬁnition of correctness” for such, without truth values. The thing stands in a precise way in HB II pages 426–427. So it is indeed a bit more complicated. Admittedly, one seems to need also ⊃ E, not just in the end, but as an inference ﬁgure in the derivation.

IV. 43. As a substitute to the artificial attempts at definitions of correctness there must serve in positive logic the fact that the forms of inference for the connectives are reducible. This is what is proper, in the essence of positive logic. It can be formulated without difficulty in a precise way. Here is the way in which intuitionistic = positive logic is demarcated against the TND. 282 Part III: The original writings

17. Book: Mathematical Foundational Research

The notes for Gentzen’s planned book on foundational research begin with a list of contents written in the spring of 1939, with ﬁve chapters. Additions have been inserted here and there between the pages, identiﬁed by a later date. Some details about chapters I–III are found on pages BG9–10, August 1939. Shorter remarks from 1942 have been added in between. After BG10, there follow remarks from 1942–43 that can be usually placed in the context of chapters I–III, either by a chapter number or a comparison of the topics listed in BG1. From page BG5 on there are rare remarks from August and September 1940, perhaps written during a leave from military service. Work with the book begins again in April 1942, under Gentzen’s convalescence, and continues in September 1942 with ideas for chapters III and IV, the latter on incompleteness. After these, there follow remarks that are mostly written before the clear plan of a book in 1939.

BG1

Ca. IV.39. The book: “Mathematical Foundational Research.” Motto: Exciting like a detective story! Division. Plan: In the beginning: Introduction and overview. Then: Chapter I: (The basic concepts of proof theory) §1. The structure of mathematical propositions. §2. The formalization of mathematical propositions. The ways of inference of mathematics Formalization of these (Resp. in general a chapter on: reduction of the basic concepts etc. Ex- amples of formalized proofs. Even diﬃcult ones. Remark: Any illiterate can control the correctness). (Proof methods, axioms.) Equivalence of functions and predicates. Such that. Chapter II: Mathematical logic. Here the most important theorems. Such as: prenex normal form. (Possi- bly) my dissertation Hauptsatz. As well as: double-negation theorem in a general formulation. 17 Book: Mathematical Foundational Research 283

1. part: propositional logic. (Possibly: the propositional calculus. Here what is calculus-like). 2. part: predicate logic. Possibly two chapters. Chapter III: (Axiomatics.) (The axioms, ways of inference and methods of concept formation of specific parts of mathematics.) Herein: the structure of theories. Recursive theories, equivalence between theories. The form of analysis. The form of geometries. Set theory. Form of proof theory. (Possibly a chapter IV on: “The incompleteness theorems of proof theory.” Theorem of Gödel, possibly Skolem, (Church), Tarski. Chapter V: The problem of infinity BG2 [empty] BG3 Introduction and overview What is “mathematical foundational research” and what is its aim? Possibly also a preface above all, the aim so that one who reads it buys the book! BG4 [empty] [The BG-numbering becomes interrupted here and there follow notes, often on slips of paper, with a title such as “For the book” and a letter d that seems to indicate Durchsicht (overview)]

IX.42. d For the book About the basic form of all propositions of mathematics as “first-order propositions” [Zählaussagen] (better word to replace that): Each proposition about a state of affairs is a positing of a relation between objects, that transpires. In the widest sense of relation and object. (Asi- de are left propositions about possibilities, “one can” – they are not factual but subjective-human. They appear, though, in finitism-mathematics). That much of the basic building blocks of a proposition shall naturally become represented in single detail as comes out of the proposition in its instantiation in inferences (not grossly: Cons Nu, for example; number theory is consistent, shall [?] here already all equations [?] a mathematical example to be chosen.) Therefore, above all, one must give explicit expression to the logical connectives (be they “relations” in the end). Hereby one obtains the normal form of first-order propositions. It is required that the inferences are completely represented through the logical connective-inferences. Now, things stand in fact so that these are the logical ways of inference and that no other ways of inference occur in mathematics, resp. can all be reduced to the logical ways of inference. (This in the end an experimental fact.) (And the 284 Part III: The original writings

CI in addition. It is in fact justified in the most natural way possible, even a logical consequence of the definition of natural numbers as “accessible,” say, from 1 through a continuing addition of one.) Then the general form of a mathematical theory, as one that builds on a system of axioms through logical inferences. Here the axiom system can be denumerably infinite after a scheme. Even this transpires. For the whole theory is, in the end, a system of sentences of a kind in which the scheme for obtaining new theorems is given through the formalized ways of inference. “Concept formation” falls under this conception, with the axiom scheme. This general pattern of a mathematical theory is to be placed at the basis of the general theorems about equivalence. (Even to require a certain kind of recursion for the axioms scheme? That, though, means limitation against the thinkable, even if not against what occurs in practice.) IV.42. d Concerning Skolem’s theorem; concerning the most general form of an axiom system: The axioms can be taken as “first-order propositions,” the system of axioms is denumerable, namely because it can obviously contain a prescription for the continuous new construction of axioms. These remain first-order propositions. The difficulty I was suspecting earlier: axioms with a variable F for example, and their position is not at all at hand. To be sure, CI for example, cannot be conceived as a first-order proposition, but as a denumerable sequence of such, i.e., a prescription for the production of such. Nothing strange about that. Rather, each proposition can be written as a first-order proposition, naturally with a corresponding interpretation of the concepts. This interpretation, in turn, can be established through axioms, and a prescription for the production of axioms is clearly still something meaningful, practically thinkable. Over and above that, one cannot imagine a further conception of the notion of an axiom system in an ordinary sense. So, each theory consists of a finite number of propositions, and the creation of new such has to follow some rule formulated sometime; hereby the whole theory is an axiom system of the usual kind, clear!

The creation of new propositions in a theory can naturally be undecidable; but one can by all means allow it for the axioms; Skolem’s theorem remains always valid. There is hardly a point to be disputed in the whole argument. The nucleus to be underlined in every case, may be diﬃcult, though. Eventually to be thought out more precisely. 17 Book: Mathematical Foundational Research 285

BG 9 d VIII.39 Book. A mathematical theory (example number theory, Euclidean geometry, to- pology) is seen provisionally as follows: a system of sentences dependent on each other in the way: certain sentences are assumed in advance, called “axioms,” further ones derived from these by certain inferences. One essential thing to be added: Deﬁnitions by which new concepts of the most varied sort are introduced. Introductions of concepts are also thinkable that are not deﬁnitions, so let us say more generally: concept formations. Inferences brought together into proofs, therein occur also propositions. . . . Further: In- stead of “theorems,” we want to talk more generally about “propositions” so that also propositions that are in between in proofs, usually not indicated as “theorems,” can occur. Further, also especially false propositions that naturally are not “theorems” in the usual sense, but can occur in proofs as an auxiliary; an assumption, for example, that is refuted in the further course of a proof.

More concisely! Will be necessary §1. Structure and formalization of mathematical propositions. We begin with the notion of a proposition as it appears in mathematical theories. We wish to determine first what kind of concepts, even more basic, the components of a “proposition” are. Let us choose at random some examples of “propositions”: sum of angles in a triangle two rights. 4 + 5 = 3. The sine function is everywhere differentiable. If x>3theny + 8 not prime. There is no greatest prime number. In accordance with our purpose, we shall leave the “correctness” of a proposition completely at side. The set of natural numbers is [text ends] We determine as the building blocks of propositions, through the consideration of most varied examples: 1. Objects of the mathematical theory in question. And indeed proper names of such. As well as indefinite signs (x, y) for denoting arbitrary objects. “Variables.” 2. Functions: . . . (“operations”: ...) example 4 + 5. 3. “Predicates.” We denote as such concepts of the following kind: ... examples ... 4. Logical connectives. Examples : (& ⊃)also¬ “connective,” in a somewhat slanted way. We want to take also the concepts of all and there is under this collective notion. With this we are done. How come, about that later. As next, now: We want still to formulate precisely in what way a proposition can be built up from such concepts. Advisable here is: the formalization. It means nothing more than: symbols instead of words of the language, for uniqueness etc. Specific signs for the logical connectives: ... Arbitrary combinations of signs allowed for the other concepts, naturally not to be mixed with each other. To be fixed 286 Part III: The original writings for a theory at a time. Now: the concepts term, formula. Formalization of the above examples. E.g., Diff (sine), the question of the further decomposability of the concepts to be noticed. Concerning the “completeness ” of the formalized concept of a proposition. Completeness is to be assumed. Plausible. There are, not counting turns of phrase that can be brought without much trouble into a normal form (essential exceptions, like examples) that maintains sense, especially difficult cases. In these, there appear concepts that are in the first place of a completely different kind, such as: “On the basis of the formulas ... the value of the symbol (a|b) of arbitrary arguments can be calculated. There is something completely new in the “can be calculated.” Reformulation, nevertheless,as: “There is” a “computational instruction” by which it can be formulated, as long as one keeps the predicate “to be a computational instruction” as, say, indecomposed. We shall come back to such problems of completeness later. (Here just to make curious about the answer!) Other sentence forms that fall out for example: “When one is” “a figure turns” or: a decimal number shifts somehow or something, “one obtains” such and such. Or “if one distributes, in some way, n numbers in n dra- wers.” Here it is a question of concepts about action. These can, however, be eliminated, indeed, one will look at the forms freed of them as mathematically more pure. As concerns the diminishing of the number of basic concepts: possible, e.g., functions replaceable by predicates. Will go into that later when in a position to support such a kind of claim by a formal proof. (Awake curiosity!) A delightful task, the reader to be hinted about it: to bring mathematical sentences into a formalizable form. Examples as exercises. “There are three ∃ ∃ ∃ distinct numbers...”: x y zDist (x, y, z) resp. x = y & x = z & y = z such that ... “there are infinitely many prime numbers” “the power of the set of all natural numbers is the same as the power of the set of rational numbers” “∃ exactly one...” Representation: ∀x(Nat x ⊃⊂ xN )&...⊃ Pot N = Pot R ↓ ↓ ↓ ↓ predicate predicate function function BG 10 d §2. Kind and formalization of the logical ways of inference that occur in mathematical proofs. “Intuitive” inferences to be mentioned. 17 Book: Mathematical Foundational Research 287

New plan of arrangement 1. The normal form of mathematical propositions. Here without functions, just with predicates. To be mentioned that this form is assumed to be complete. That there appear first several ways of expression of quite a different kind. Examples to be mentioned and to be said: The direct formalization of these as well as the reduction to normal form is given later, once we have means to demonstrate that the reduction doesn’t change anything essential in respect of sense and utilization of the proposition. 2. As before: The logical ways of inference. 3. Possibly already: The propositional calculus and the logical Hauptsatz. Then 4.: Functions, the one such, sets, proofs of equivalence. Beginning of text 4.: We wish to deal now with the “descriptions of objects” that occur often in mathematical propositions. Is to say: An object is often not designated in a proposition by its proper name resp. by a variable, but by more complicated ways of expression such as: the greatest number with this and this property. A the-one-such-that-example. The sum of the numbers 2 and 4, 2+4. There occur further, for example: the set of all natural numbers between 100 and 1000, or the set of all rational numbers. Even these sets are objects. Objects are described in all these examples as functions of something, be it other objects of the same kind, or other objects of a subordinated kind (example of sets), or even a property, the one such that; also: An example with sets of the kind: The set of all numbers with the property ... We want to fix formal representations for these cases, so that we have the possibility to formalize directly such propositions. And we wish to demonstrate precisely in this connection how one can still bring such propositions back to the normal form, with no loss as a consequence, as we shall show exactly. It is in general often motivated in proof-theoretical investigations to limit oneself at the outset to the normal form, to make the matter thereby simpler. (II chapter: Basic concepts.) Here, equivalence of function, predicate, set. Especially also ι replaceable by a predicate. III chapter. Possibly before 2. (Possibly together with 2): Some formal logic. Hauptsatz. This possibly right after §2. Better for the reader!

IX.42. For the book For III d The Hauptsatz of predicate logic To say in this connection that this theorem proved in different forms. Rather: that different formulations by different ways of proof, with basically the same 288 Part III: The original writings nucleus of the theorem, more or less. [Cancelled: First] by Herbrand as the théorème fondamental, by [cancelled: Bernays ?] Hilbert and Ackermann, also Bernays! ε-term, by the author in the form at hand (resp. modified). The nucleus of this theorem should be: Absence of detours, i.e.: A logical theorem can be concluded in logic through just that part of the logical concepts, and inferences that belong to them, as appear in the theorem. There is no need to take detours through, say, more complicated concepts. (Against what often happens in mathematics, proofs of theorems through a completely different field say, like theorems about the natural numbers through analysis etc) Not to be titled as Hauptsatz of predicate logic, that is too assertive. Rather Hauptsatz about the normalization of chains of logical inference, or corresponding to its meaning.

IX.42. For the book About the equivalence of functions, predicates, sets: the real numbers as an example. The same as also predicates: binary fractions: predicates for natural numbers. Then make notice that predicates are not usually objects in mathematics, but only functions and sets.

IX.42. For the book. Forthesectiononthelogicalwaysofinference:d The natural calculus to be taken. No lengthy derivation as an example, but just short single forms of inference within the single ways of inference. Say like in my dissertation, only from mathematics. Later perhaps a longer example. From positive logic perhaps even to limit the domain without ⊃. To indicate with every introduction and elimination the reduction of the corresponding hillock (maybe wasn’t done in the dissertation?) From here carried over to the Hauptsatz. Hauptsatz, absence of detours. The idea as in the intuitive case: [a drawing suggests the detour conversion with implication] Direct way, detour, transformation of the detour into the shortest one through single steps, deletion of the outermost detour.

IV.43 The positive logical ways of inference are to be developed in reference to the idea of their reducibility. That is their natural rooting.

IV.43 The idea of the hillock theorem is connected to the reducibility of a single 17 Book: Mathematical Foundational Research 289 hillock, as well as to the consistency of logical inference: The 0-grade stuff that is in the end obtained from 0-grade stuff by introduction of connectives and -re- eliminations cannot result in anything that is not contained in the initial material. When this is proved, it turns out that a proof of the existence of peaks as well as above all a proof of simplification in peak reduction are additionally needed to accompany the idea of reducibility. The same holds for the hillock theorem. Now, the reduction of connectives has to be presented in the same way for the hillock theorem as for the consistency proof;forboth are the same thing. As well as an assignment of values in the same way that applies to the hillock theorem and the consistency proof. (Positive logic and without CI clearly in the first place) In addition the idea of altitude layering1 and power assignment2 already. Later, for the complete proof of consistency, there comes in addition transfiniteness only through CI. Even if I can’t bring all of this in the book (still, to be indicated in brief traits), it is quite independently a promising idea: to assimilate the proof of the hillock theorem to the proof of consistency, possible for both LK and NEU, but also for LI.Thereduction of connectives is the same in both. It should be possible to take it with the cut theorem in LK in the same way as in NEU.InLI perhaps rather like in the dissertation. It should be possible to adopt the assignment of values, the one that guarantees simplification despite multiplication, from the consistency proof to a proof of the hillock theorem, of course, here for finite numbers. BG5 Various thoughts concerning the book d For chapter I. The title something like: “The (forms of) propositions and ways of inference of mathematics.” The concept of a “predicate” has no place here. One formulates the ways of inference appropriately completely independently of the structure of the propositions, simply like this, for example: Γ → A&B Γ → A Here A and B are “formulas,” i.e., “proposition-expressions,” with nothing assumed about their structure. As well as Γ → Fa Γ → ∀xFx Here we have naturally also the concept of a variable. Obviously it has to range over a certain domain of objects. In 1 This notion appears in the series BTIZ; it means that cuts in a derivation are permuted so that no cut is followed in the same thread of derivation by a cut on a longer formula. 2 Ordinals α and β are assigned to the premisses of a cut in a derivation so that the conclusion has the ordinal αβ . 290 Part III: The original writings

Γ → ∀xFx Γ → Ft there is in addition the notion of a term = sign for an object. It is, though, analogous to the notion of a propositional expression. Whether the term has been built by functions, by ι,byε, is of no consequence. As to the sorts of objects, I can easily allow, from the beginning, several sorts,thenformulate UI, UE as follows: the expressions for objects, including the free and bound variables, here, must belong to the same sort. This is practical and no fuss about that! A division into diﬀerent sorts of propositions is not required. (Even if one uses in practice with &): propositions that belong in a certain sense together. Writing of assumptions,asinJa´skowski say: numbering of these, say A1,A2, A2.1 and writing before the posterior formula: 1, 2.1 A & B. No sequent writing so far. And for the example derivations. Natural calculus, without the imposed form of a tree. By the way, I have worked even in my natural calculus with a numbering of the assumptions! It is just one more step to the corresponding writing freed of the tree form. Example: Formulation of the FI:

Aν) A . . (..ν..) B (.. ..) A ⊃ B or just in words

An example of a derivation: The sum of angles in a triangle equal to 2R. Known theorems: ∀P, g ¬(P on g) ⊃∃h(P on h & h g) and further. [Par- entheses should be added to put all occurrences of P and g under the scope of the universal quantiﬁers.]

C Z εδ Z h γ Z Z Z Z Z Z αβZ ABc BG6 UE with c and C: ¬(C on c) ⊃ .. We have, because a triangle: ¬ C on c (= AB prolonged) so FE: ∃h(C on h) & h c). Now EE. I.e, formally: Assumption 1) C on h & h c, h the existential variable. 17 Book: Mathematical Foundational Research 291

AE 1 C on h and 1 h c I assume now the theorem that corresponding angles in parallels are equal and write directly the insight that stems from that: I obtain from 1 h c, with assumption 1) maintained: 1 β = δ,1 α = ε. We have further ε + γ + δ =2R, here it is best to write at once ≡ in place of =, and to take as a known theorem: α ≡ β ⊃ α + γ etc. Through the substitution of equals for equals there obtains: 1 α + β + γ =2R h does not occur here anymore, so one can finish the EE. Attention: The dependence of ε and γ on h did not get expressed. So better to put: ∠(b, h) and ∠(a, h). Becomes a bit circumstantial. About the structure of the propositions: The basic concepts: Object, function, predicate, propositional connectives. With these, everything can be represented. To notice in addition: Functions, for example, can be objects (examples from mathematics, “theory of functions”). This can be accommo- dated, for they are even the “objects of the theory” and even if they are at the same to be applied to other objects, it can be easily written otherwise. A secondexample:I cansay:Theequation...hasa root,writingatthesa- me time the equation, that is, not considering it as an object. Whereas when I say: All equations of a certain kind have a root (fundamental theorem of algebra) I have to, because I applied “all” to equations, make the equations into objects. The equations correspond to the functions of the previous example: For an equation is actually a proposition, i.e., an indeterminate, with free variables. (Even though it is not seen as a proposition in practice, but as a formal expression.) So, one makes also propositions into expressions or rather into empty proposition-forms, with free variables. It is easy to lift formally, in its turn, the functioning next to each other of such forms as objects on the one hand and as propositions on the other. Here to read Lewis-Langford chapter I, on Peano’s thoughts about the representability of propositions. UI= all-introduction, UE= all-elimination, FI= follows-introduction. BG5 [?] VIII.40 Book d Predicate, function, “determinate” set. A domain of objects given. (to be thought of as) finite. Infinite especially only later. So everything here in general independent of the number of objects, holds also for “infinite,” in a certain sense. The concept of a predicate as known. The concept of a (unique) function: a function with n arguments: association by which there is an object associated to every n-tuple of objects. Arguments, “function 292 Part III: The original writings value.” A close connection between predicates and functions: to a function (n)a predicate with n + 1 arguments can be associated: applies when the n +1st object is precisely the value of the function for the n first. Does not apply when it is not. It is clear that this predicate can serve as a substitute of the function. It is especially clear that when the predicate is given, also the function is given (likewise the other way around). We shall show later that this substitutability of functions through predicates is possible, through a proof for the domain of all logical ways of inference. Conversion: Incidentally, predicates also substitutable through functions. Contentfully: We give to an n-place predicate a function of n places that has, for those n arguments for which the predicate holds, a certain specified function value. For other arguments it has another value. So that even here the predicate is given whenever the function is given. The concept of a “set” of objects of a domain plays as well a role in mathematics. Its sense is clear. There are even for it similar relations to the concepts of a predicate and function. Namely: We give to a set a one-place predicate in the following way: Let the predicate hold for those objects that belong to the set and not hold for the rest. There corresponds to each one-place predicate in a one-one way a set. We don’t want to go into further connections [added: the above substitution of predicates by functions to be mentioned here] that can be equally easily established, say between the concepts of a set and a predicate, they are less important in practice. About: the formal part: We want to give expression to the new concepts in the formulation of propositions, and to further state how the propositions are short forms, to be brought into an “equivalent” casting with only predicates [gap?] stated above, i.e., to bring into main normal form, and finally to prove in a sense to be made precise this equivalence with respect to ways of inference.

About: Minimalization. Besides the substitution of sets and functions by predicates, resp. sets in higher types, also all sorts of curiosities to be mentioned: Sheﬀer stroke, inclusion (anda ˆ),possiblywithaproofofG¨odel’s TND-theorem. Then: The methods of minimalization of the researcher into decision problems. Restriction of the number and positions of the predicates (apropos, concerning notation: a predicate has a number of arguments, a predicate symbol has a number of empty positions.) Herbrand to be reread for all this! His “reductions of the decision problem.” 17 Book: Mathematical Foundational Research 293

——————————————————————————————– IX.40. §: Predicates, functions, sets as objects of a higher type. One can consider predicates, functions, sets themselves again as objects of the theory, next to the objects of the ground domain. I.e., to apply ∀ ...∃ on such, as well as to consider functions of such, sets of such. It means in any case the addition of signs for objects of a further “sort.” Determinate predicates, functions, sets can be introduced over these analogously to those over ground objects. “Mixed ones” in addition, for example predicates with one argument position for ground objects and another for the new sort. We treat three cases of addition of sets, predicates, functions, each over ground objects, alone, as new objects, each separately at a choice (they are widely analogous). The beginning with sets as the simplest and also especially practically important case: Variables for ground objects, say with index 0, for sets with index 1. Specific functions, predicates, and sets to be equipped with specification of the range of the argument positions already at their definition. We want to allow always just one of the two sorts of objects as ranges, not both mixed [added: or just allow it]. It means no essential restriction. (Also practically quite common, e.g., functions that are “extended” from natural numbers to real numbers, a “logical monstrosity” according to Russell or Frege. Said limitation not essential, because one can then let the ground objects be represented by the 1-objects (as Russell does), in the case of sets just simply as sets with one object.) The symbolˆfor sets also under variable within A,ofbothsorts. That is, precisely, set formation of the most general kind. Variable v naturally only for ground objects, because we build sets of only such. (The repeated treatment makes here the impression as if it were better to perhaps unite this § with the previous one. This, though, is not good because precisely here is made the division into theories of type one and type two. I would like to have afterwards the possibility to allow, with the recursive theories, the methods of formation of specific sets and functions. Not true. To wait.) The formalism without the ˆ -symbol: Formally simpler, though not that much in correspondence with practice, it is another way to formalization that uses a rewriting of the propositions as compared to the natural form. Therefore it spares the symbol ˆ, uses in place of a schema that belongs to it a new one, the “comprehension axiom.” Third way: UE and EI formulated differently; procedure with sets!

Equivalence Kuratowski ω-ascent. Arbitrary types, levels. Example of types in analysis. 294 Part III: The original writings

Definitions, partly theory-specific. Generally: immediate abbreviation. Defi- nitions with order [?] proof, definitions through abstraction. Definitions: + for negative numbers for example.

So the formal part: 1. [added: better 2.] Functions. These appear normally [added: e.g., the sine function is differentiable] in propositions of the form: the value of the function ... for the arguments ... . Examples. These are already routinely fully formalized in mathematics. We keep these formaliza- tions, so we set: the function sign given, the number of arguments, empty positions for the entry of the arguments. The notion of a term. The concept of a formula changed accordingly: instead of “object”: “term.” Keep an eye on free variables and bound variables. Examples. Formal substitution of the function sign through a predicate sign: associated signs P f... Then: A proof that nothing is lost in the ways of inference b the substitution. Not only to substitute ι. But functions in general. Example +. Existence. Other forms of introduction e.g.: Definition by other functions. Recursive definition. Occurs already in the axioms. Still others should be thinkable. Normal form of the introduction of new functions in a theory: ∀t1 ...tn∃yA(ϕ1 ...y)(A as a free variable of course). ∀ϕ1 ...ϕn∃y∀zA(ϕ1 ...y)&A(ϕ1 ...z) ⊃ y = z Uniqueness.

Thereafter a new function f,wehave:∀ϕ1 ...A(ϕ1 ...ϕn) f(ϕ1 ...ϕn) for f the symbol ιy A(... ,y). “Description.” Practice: a new sign. But here ι better, to keep the context always clear. Some attention to the description of objects! I.e., ι for a singular object. Perhaps an existential variable. 2. [added: better 1.] Sets: Such appear normally in propositions of the form: the object ... is an element of the set ... . We write gM. So the concept of a formula: There are in addition signs for sets (for determinate sets) in the same way as predicate signs, function signs. gM makes a formula. It is to be added to the concept of a prime formula, for the rest all remains as is. Formal substitution of signs for sets by signs for predicates: Associated sign PM ,representationPM (g)forgM.[Addedabove: no predicate, because M is not considered here as a sign for objects.] Normal form for the introduction of determinate sets in a theory: Common in words: We consider the “set of objects for which this and this proposition holds.” Formally: Let A(v) be the formula for the proposition in question, v the (only) free variable therein. Designate now the set byv ˆ[A(v)]. And need the scheme that belongs to this concept: gvAˆ (v) ⊃⊂ A(g) for arbitrary ... A etc. 17 Book: Mathematical Foundational Research 295

A further §: to review here: sets, functions, predicates as objects, etc. Possibly to show once more equivalence here? BG7 39. Various thoughts about the book. For the chapter on mathematical logic For the end: specific half-logical notions, such as =, ,possibly1(=ι). Different levels with quite easy, because I have already many-sorted! Finite set theory as a model. To notice that predicate logic, and this further one, is not yet in every sense “the logic”: rather, it is a logic of statements about states of affairs. There is lacking, for example, the treatment of statements such as: A occurs probably. Even finitary mathematics has its own logic, for example [?] at least its quite specific notions. The ways of inference remain, naturally, “the logical ways of inference.” A concept such as “one can.” “One can give a number with a property.” (Possibly this can be reduced to the form of a state of affairs). “The mathematics of “one can”.” Proof of the double negation theorem through the cut theorem. LK ,orNI ? To manage with the “theorem of freedom of detours” if possible. (To be tried, result in each thread to be first eliminated, then introduced. That is how it should be. Cf. the HUG¨ series.) Natural deduction, a ¬¬elimination-theorem is logically connected to the hillock theorem for intuitionistic. Completeness of predicate logic: With the inference figures so transparent, it should be possible to prove in some elementary way completeness, though only in a proper sense. How is it with: completeness for each finite set of objects? In general, to put into use arbitrary finite models. As the natural, finitary domain of application of the ways of inference. Then the extension into the infinite is something new. Above all, logic in itself has not yet anything to do with the difficulties about the infinite, that should come out clear! The possibilities to “diminish” the general logical basic notions. Possibly a special chapter, or the end of the chapter on logic. Here: “the one which” is replaced. Sheffer stroke. ε, sets, functions resumed. Functions to be replaced by predicates resp. sets Derivations to be transformed into a reduced form Rendering intuitive of sorts of objects. One single. I should not have to go into concept formation. Equally little to specific basic predicates of mathematical theories! To reduce their number does not 296 Part III: The original writings belong here. It can happen in connection with the presentation of specific mathematical systems, i.e., the logicist presentation of mathematics as a whole would be brought there. BG8 For III the book. For recursive theories: Objects as combinations of signs. Examples thereof. Then specifically for pure number theory: Production of “axioms” in connection with the definition of functions and predicates, here: recursion schemes to be given. Examples, formalized. To make notice: One could take even “practical axioms,” clear theorems in a greater number. As well as: Incompleteness of the recursion scheme in a certain respect, here though nothing right to be done against. Then: One on “set-theoretic treatment of pure number theory.” Here: The concept of a set and equality are counted into logic, the general theory [?] of sets is taken as given (comprehension). Characterization of the natural numbers through the axioms of Peano. Another way: (Dedekind) Frege Russell. “Definition” of the natural numbers through the validity of CI . To notice here the displeasure of such a procedure in consequence of the dubitability of set theory, the circularity. Nevertheless no entrance into “philosophy.” Elegance of the method to be admitted.

For the book: Reflections. about IV.42. Why is a + b (in natural numbers) = b + a ? Not at all simple. Here lies the beginning of inference. An intuitive-logical mixture, the concept of a finite set also plays a role. With the natural justification of this sentence, of course. The formal-logical is instead already artificial.Tosaysomething about the natural here should be appropriate. Even if penetrating in it further is inessential, for one wants to proceed ahead. The concept of a set goes before that of number, cardinality, in natural order. For “cardinality” assumes already a set to be counted. Admittedly finite sets, of course. Main theorem: The cardinality of a set is independent of the succession of the elements in counting. Proof by CI possible, should be. (Care of course with respect to what one is already assuming.) about IV.42. Sequent writing in practice: ....|..|...||... resp. , cf. Frege-Russellian use of instead of , , → this sign! clearer 17 Book: Mathematical Foundational Research 297

The commas are too little conspicuous. That is clear. No need to fear mixing with the sign a b (the Sheffer stroke) (and a b), by drawing the sequent stroke longer through. ——————————————————————————————- “Contradictions in logic?” No. I mean, the concept of logic has been taken in too broad a sense. Concept, set, and property are involved in the Russell- antinomy, but they have not been made sufficiently precise to be counted in such a universal form as a part of “logic.” Sure, logic can talk about sets, but about sets, assumed to be given, of objects of a given kind. The universal concept covers the real world, that is not a matter of logic. (The Russellian antinomy in mathematical set theory is of a different kind, here connected to the concept of infinity. The logical form is not mathematical, because it operates with concepts too undetermined for mathematics.)

IX.42. For the book: The general concept of a recursive theory: to III. Objects with a finite designation. Finitely many single signs have been given. This is no limitation, against those people who would perhaps always like to introduce new signs: One can naturally, as for example with 1, 1, 1 etc, create arbitrarily many different expressions through limited single signs, ones that can represent those new signs just as well. Next the “theorem of equivalence for recursive theories” is proved, by which they can all be mapped on another in a certain sense so that they are logically equivalent. Prototype pure number theory. “Arithmetizability.” Proof theory as a recursive theory. Ways of inference of recursive theories.: The CI added to the logical ones. Possibly to mention the possibility of extension by TI , more about that in connection with the infinity results.

X. Better description: “Theory with a domain of objects built up recursively.” For one needs the concept of recursive proving in number theory (Skolem) even in a speciﬁc, already common sense (a ﬁnitary development of theories.)

Thetypeoftheoryofa,sotosay,purelylogicalkindsuchasgeometry and algebra is also to be described. As well as the passages from these to the purely-logical. Also set theory in addition! Typical for such theories is: Objects are not independent things distinct from each other in advance but are left undetermined. Axioms assumed, consequences drawn. The concept of a model, isomorphism etc. A particularity of theories: Ways of inference: the logical ones, though usual- 298 Part III: The original writings ly one adds the natural numbers (with CI ) without special mention into the theory. This is quite in order, these count equally as among the basic stock of all of logic, not as an element alien to the theory (as, say, real numbers would be). It should not be necessary to make this addition in set theory, in that one can obtain within the theory the natural numbers as objects of the theory. Then: The addition of the concept of a set considered as an equally logical element, in geometry, should be, and in algebra. Seen logically, the question is about delimiting the means of proof in advance, and therefore all of this has to be especially noticed. (Clear that here should be the axiomatics of set theory with just predicate logic in addition, or?). Algebra especially unorderly [?]. Parts such as the theory of polynomials should be recursive theories. Parts “purely logical,” not properly axiomatic, but just drawing conclusions from assumptions (so these assumptions are no axioms in the proper sense) Then parts of set theory added.

XI.42. For the book: to III. In the theme: “The introduction of new objects, in a theory,” indication of questionable methods “postulation,” mentioning of the Frege quote about “theft,” thereafter that the, still today, usual introduction of imaginary numbers occurs in books. (The introduction confusion in Landau as an example for myself, not to be mentioned.) The conception of logicism (Frege) that requires a direct deﬁnition that pertains to the world of things. Instead the usual more free of the mathema- tician who wants to create on one’s own the numbers, apparently in order, if it goes in terms of simple points of view (naturally not: postulation). Con- ception of the association of (natural) numbers to the real things, no matter if the numbers are signs or created mental concepts. In general with each “deﬁnition through abstraction” the same comparison of conceptions.

Explicitly about the theme: “Deﬁnitions.” Direct deﬁnitions (designation to be chosen according to old authors resp. philosophical logicians).

To try withDf. = : Variables as auxiliary signs in functions and predicates, resulting from the usual writing of the latter. Examples from mathematics: tan x = sin x Df. cos x Prime number x,say. Another attempt: Deﬁnition depends on whether a proposition holds (say: lim of a sequence, presupposes its own convergence.) Of a diﬀerent kind the 17 Book: Mathematical Foundational Research 299

Legendre symbol, perhaps not anymore to be called a direct definition. Definition of 0 by its introduction as a sign. Questionable? To be reflected. Then the definitions through abstraction. Implicit definitions. To be naturally read: Dubislav “Die Definition,” Frege, Scholz, Hölder perhaps and others.

IX.42. d IV. Ideas for a direct proof of the Gödelian incompleteness theorem resp. for a clearer overview of the train of thought of his proof. One obtains the incompleteness of functions trivially by the diagonal procedure. Now the same is to be applied to the theorems. All derivable universal sentences can be brought into an ordering, this can be formulated as a number-theoretic function: Hν is the ν-th derivation in one such; Sν the corresponding sentence. Can the diagonal procedure be now somehow applied to produce a new correct universal sentence from this series, one that is not identical to any one of them? It should be possible to introduce didactically the Gödelian proof, step by step. Possibly for the book. (Could one obtain a procedure essentially different of the one of Gödel for the production of such a sentence?? I attempt: How is it with the sentence: “each universal sentence that is not in this series, is unprovable in the formalism of pure number theory.” This sentence is correct, is a number-theoretic universal sentence. But perhaps it could belong to the series. So it doesn’t go simpler than with Gödel. Cf. my earlier ideas.

IX.42. d For the book. For IV. I place at the peak of the incompleteness theorems (functions, grade, Church- terms) the diagonal procedure, their common idea. Thence as the way to overcome them the idea of transfinite ordinal numbers, even that developed from the diagonal procedure, say connected to the procedure for functions. Wherein do the incompletenesses lie? It is, that the means of formalization are so strong that they allow the formulation without trouble of the diagonal procedure; therefore what can be formulated always more than what can be “cleared” by limited means. About the theorem of Church concerning the decision problem: Does not the diagonal procedure enter here, in the way that: if one poses, quite generally, 300 Part III: The original writings as an act of decision a finite figure, so that the acts of decision are denumerable; can one now already construct through the diagonal procedure such an undecidability theorem? Or does one have to, like Church, make the decision procedure more precise? Compare on Church Hilbert-Bernays II Supplement II.

IX.42. Concerning the Gödelian incompleteness theorem: Let one take from my consistency proof the result by which each number- theoretically derivable “pure universal sentence” is derivable by one TI with no further transfinite inferences. One tries now to construct correct sentences that are derivable by only one TI , always up to a determinate ordinal number, but not by any lower one. (Consider the examples of consistency statements for which this holds, admittedly still without proof for parts of number theory.) Something like that should be somehow possible. A thought, to be attempted on occasion. Might still be too difficult. One should consider at the same time the corresponding thing in relation to the definition of number-theoretic functions, recursive ones, by a definitional scheme that can be characterized through transfinite ordinal numbers. (Example of the concepts of the reducibility result in the consistency proof.) Relations to the initial segments (ω, ω2) to be studied thoroughly! They must still be transparent here.

About Skolem’s theorem: d See also: The folder: “Book: Mathematical Foundational Research” As well as: issue 4 of the Forschungen zur Logik, private copy, marginal remarks at my talk.

VI.36 d The situation concerning Skolem’s paradox Fraenkel (Zehn Vorlesungen) thinks that impredicative processes cannot be taken into account. [Added: he errs] Still it is very unclear. See also Kauf- mann. Then: Skolem Uber¨ einige Grundlagenfragen .... 1929 important. Ber- nays ([Hilbert’s] Abhandlungen III 200 remark 3) takes escape in the fact that an axioms system need not, unlike the one by von Neumann, consist of first-order propositions. This seems to be at bottom the same objection as the one of Fraenkel. Still: it is essential that there is no comprehension axiom, this contains the impredicativity, is also what is essential in higher types. It seems to me that Skolem nevertheless believes to cover all systems that appear in practice. (Kaufmann adheres to Skolem, despite Fraenkel, in that he rejects the impredicative. That is certainly not enough.) Very important: Carnap, Die Antinomien, final part. Here the clearest repre- 17 Book: Mathematical Foundational Research 301 sentation of the meaning of Skolem’s theorem. The salvation of “set theory” really weak. Carnap seems to accept the theorem of Skolem for all axiomatizations of set theory. Also real numbers are naturally affected. (See especially Kaufmann). The specific meaning of their “non-denumerability” that Carnap adds says nothing in practice. Or nothing more than that non- denumerability has of course also constructively a meaning, as is known, though not in any higher cardinality, that is just a figure of speech. 16, 1930 Some time to treat for the collection: Kaufmann, Skolem, Carnap. Von Neu- mann: (Crelle 154 II part: very interesting and clear! However, nothing really new.

should be V.34. 3 Naive mathematics, mathematics in its beginning, thinks ﬁnitely. It goes only later more and more over into the in-itself conception. Finite – intuition against: in-itself – logical. Example: Addition is in the ﬁrst place an operation with numbers. The sum is, however, something existent in itself for the logical conception.

X.35. The analogy of the constructions: set of all sets and greatest natural number is undoubtedly noticeable. It is just that our usual mathematical thinking is already used and adapted to the second thing, to the ﬁrst one not yet similarly. How is the program of consistency proofs related to it? Perhaps one can say that the solution of the diﬃculty with the greatest natural number has not yet been complete, that the proofs of consistency take the situation in their entirety once again and more profoundly under attack.

should be IX.42. On Skolem’s theorem: Concerning model of analysis: Do the entire formally deﬁnable real numbers, functions, etc, already make up for a Skolemian model of analysis? To be deliberated! Should not turn out to be that simple, compare G¨odel’s continuum theorem work.

II.39. The super-theoretical character of the concept of truth:4

3 Top of page has the following seemingly unrelated passage heavily cancelled: it is a question of the certainty of the inferences. To this eﬀect, system 3 will form the last instance, after there had been a passage from 1 to 2 for a more convenient conception. 4 Added: (Cf. Tarski in Studia Phil. 1, page 311, remark 41.) 302 Part III: The original writings

I consider a formula given in prenex normal form. I will define “truth” like this, “recursively”: It is defined for elementary parts. ∀xF x is true whenever Fx is true for an arbitrary x; ∃xF x is true whenever there is an x for which Fx is true. The type of x is the same as contentfully in the theory. For example that of a natural number, if the theory is pure number theory. Formalization of this definition now: Auxiliary concept: A formula A is correct for a system of numbers g [likely for Gegenstand] substituted for the free variables of A: written: gRA.Now we have: “A (without free variables) is [cancelled: correct] true means: For all relations R, such that the following condition is fulfilled, holds ∀gRA. The condition is: gR ∀xF x ⊃⊂ [g+xRF x] in a sense easily understood: g+x means the substitution of the same objects as in g for the corresponding variables, and the additional object x for the variable x. & gR ∃xF x ⊃⊂ ∃x[g+xRF x] & the initial case clear. (That is naturally the classic Dedekindian way to transform a recursive definition into a direct one.) Finished! All things are clear now: It is essential that the definition of truth contains, as a super-theoretical means, a bound variable over a relation between the totality of the objects that appear in the theory on the one side, and on the other hand the formulas, i.e., practically: natural numbers. Hereby one goes by a minimum beyond the theory. It would perhaps seem natural if one had a one-place predicate, for arbitrary objects of the theory, bound. It is obtained easily from: as soon as the natural numbers themselves belong to the theory, one can hit A in a relation to g and has such a predicate, finished.

IV.42. On the practical way of writing of natural derivations in mathematics: Concerning the dependence on assumptions: One makes in practice assumptions with variables. (I.e., ﬁxed sentences. One does not assume that they are known normally to be correct or false.) One says: Let a be a number for which F (a) holds. Then one continues and does not need to take notice of the assumption all the time, rather, it is in itself clear that each proposition in which a occurs (and only those) depends on this assumption. This holds, of course for a-free assumptions only when there exists at least one number for which F (a)holds.However,suchtrickeriesplayin practice no essential role. For if one wants to derive something that does not have a, this is sensible only [if?] one wants to derive it even just from ∃xF x.And 17 Book: Mathematical Foundational Research 303 then one introduces a at once as an existential variable. So one can say formally: It should be prohibited to derive without a from propositions that contain a. Such variables conditioned by assumptions have to be naturally carefully distinguished by notation from universal variables. True, these are introduced in mathematics quite similarly: Let a be an arbitrary number, one then says. Hereby one distinguishes the sign a as a free variable. This can be also incorporated in the formalism, say, one writes then Frvar x,in which it is only required that a has not yet been used.

***

The reverse of the previous page contains the following unrelated cancelled text, written during Gentzen’s convalescence in 1942: have not caused any essential recovery have caused a certain, though not yet essential recovery. only a modest Considering the unfortunate fact that it is objectively diﬃcult to demonstrate an illness of this kind, and that I have to refer to good faith in several points of my statement, I hereby assure with emphasis that I have given all the information according to my best knowledge and conscience. Gerhard Gentzen. Bibliography for Parts I and II

Ackermann, W. (1924) Begrundung¨ des tertium non datur“ mittels der Hilbert- ” schen Theorie der Widerspruchsfreiheit. Mathematische Annalen, vol. 93, pp. 1–36. Ackermann, W. (1940) Zur Widerspruchsfreiheit der Zahlentheorie. Mathematische Annalen, vol. 117, pp. 162–194. Ackermann, W. (1940a) Zur Arbeit von G. Gentzen: Beweisbarkeit und Unbeweis- barkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie. Typewritten manuscript with the imprint “Logistisches Seminar der Universität Munster¨ i./W. Prof. Scholz.” Allert, T. (2008) The Hitler Salute. Metropolitan Books. Bernays, P. (1935a) Quelques points essentiels de la métamathématique. L’Enseig- nement Mathématique, vol. 34, pp. 70–95. Bernays, P. (1935b) Hilberts Untersuchungen uber¨ die Grundlagen der Arithmetik. In Hilbert’s Gesammelte Abhandlungen, vol. 3, pp. 196–216. Bernays, P. (1944) Review of Gentzen (1943). The Journal of Symbolic Logic,vol. 9, pp. 70–72. Bernays, P. (1965) Betrachtungen zum Sequenzen-kalkul. Contributions to Logic and Methodology in Honor of J. M. Bochenski, pp. 1–44, North-Holland. Bernays, P. (1970) On the original Gentzen consistency proof for number theory. In J. Myhill et al., eds, Intuitionism and Proof Theory, pp. 409–417, North-Holland. von Boguslawski, M. (2011) Proofs, Paradoxes, and Probabilities: The Logical Turn of Philosophy in Finland. Doctoral dissertation, University of Helsinki. Brouwer, L. (1924) Beweis, dass jede volle Funktion gleichmässig stetig ist. Ko- ninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Sciences, vol. 27, pp. 189–193. Brouwer, L. (1924a) Bemerkungen zum Beweise der gleichmässigen Stetigkeit voller Funktionen. Ibid., pp. 644–646. Brouwer, L. (1926) Zur Begrundung¨ de intuitionistischen Mathematik. III. Mathe- matische Annalen, vol. 96, pp. 451–488. Brouwer, L. (1928) Intuitionistische Betrachtungen uber¨ den Formalismus. Sitzungs- berichte der Preussischen Akademie der Wissenschaften, pp. 48–52. Cavaillès, J. (1938) Méthode axiomatique et formalisme. As republished in Oeuvres completés de Philosophie des sciences, Hermann, Paris 1994. Chertok, B. (2006) Rockets and People: Creating a Rocket Industry. The NASA History Series, Washington.

© Springer International Publishing Switzerland 2017 305 J. von Plato, Saved from the Cellar, Sources and Studies in the History of Mathematics and Physical Sciences, DOI 10.1007/978-3-319-42120-9 306 Bibliography for Parts I and II

Church, A. (1936) A note on the Entscheidungsproblem. The Journal of Symbolic Logic, vol. 1, pp. 101–102. Curry, H. (1939) A note on the reduction of Gentzen’s sequent calculus LJ. Bulletin of the American Mathematical Society, vol. 45, pp. 288–293. Curry, H. (1950) A Theory of Formal Deducibility. Notre Dame Mathematical Lec- tures no. 6. Curry, H. (1977) Foundations of Mathematical Logic. Dover reprint of the 1963 original edition. van Dalen, D. (2011) The Selected Correspondence of L.E.J. Brouwer. Springer. Feferman, S. (1964) Systems of predicative analysis, The Journal of Symbolic Logic, vol. 29, pp. 1–30. Gentzen, G. (1932) Uber¨ die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsysteme. Mathematische Annalen, vol. 107, pp. 329–250. Gentzen, G. (1933) Uber¨ das Verhältnis zwischen intuitionistischer und klassischer Arithmetik. Submitted for publication March 15, 1933 but withdrawn, published in Archiv fur¨ mathematische Logik, vol. 16 (1974), pp. 119–132. Gentzen, G. (1934–35) Untersuchungen uber¨ das logische Schliessen. Mathematische Zeitschrift, vol. 39, pp. 176–210 and 405–431. Gentzen, G. (1935) Der erste Widerspruchsfreiheitsbeweis fur¨ die klassische Zah- lentheorie. First printed in Archiv fur¨ mathematische Logik, vol. 16 (1974), pp. 97–118. Gentzen, G. (1936) Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathemati- sche Annalen, vol. 112, pp. 493–565. Gentzen, G. (1936a) Der Unendlichkeitsbegriff in der Mathematik. Semester- Berichte Munster,¨ WS 1936/37, pp. 65–80. Available in English translation in Menzler-Trott (2007), pp. 343–350. Gentzen, G. (1936b) Die Widerspruchsfreiheit der Stufenlogik. Mathematische Zeit- schrift, vol. 41, pp. 357–366. Gentzen, G. (1938a) Die gegenwärtige Lage in der mathematischen Grundlagenfor- schung. Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, vol. 4, pp. 1–18. Gentzen, G. (1938) Neue Fassung des Widerspruchsfreiheitsbeweises fur¨ die reine Zahlentheorie. Forschungen zur Logik und zur Grundlegung der exakten Wissen- schaften, vol. 4, pp. 19–44. Gentzen, G. (1943) Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie. Mathematische Annalen,vol. 120, pp. 140–161. Gentzen, G. (1954) Zusammenfassung von mehreren vollständigen Induktionen zu einer einzigen. Archiv fur¨ Mathematische Logik und Grundlagenforschung,vol.5, pp. 81–83. Bibliography for Parts I and II 307

Gentzen, G. (1964–65) Investigations into logical deduction. American Philosophical Quarterly, vol. 1, pp. 288–306 and vol. 2, pp. 204–218. Gentzen, G. (1969) The Collected Papers of Gerhard Gentzen, ed. M. Szabo. Gentzen, G. (2008) The normalization of derivations. The Bulletin of Symbolic Logic, vol. 14, pp. 245–257. Glivenko, V. (1929) Sur quelques points de la logique de M. Brouwer. Academie Royale de Belgique, Bulletin de la Classe des Sciences, vol. 15, pp. 183–188. Gödel, K. (1931) Uber¨ formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte fur¨ Mathematik und Physik, vol. 38, pp. 173–198. Also in Gödel 1986. Gödel, K. (1933) Zur intuitionistischen Arithmetik und Zahlentheorie. As reprinted in Gödel (1986), pp. 286–295. Gödel, K. (1938) Lecture at Zilsel’s. In Gödel (1995), pp. 86–113. Gödel, K. (1941) In what sense is intuitionistic logic constructive? In Gödel (1995), pp. 189–200. Gödel, K. (1958) Uber¨ eine noch nicht benutzte Erweiterung des finiten Standpunk- tes. Dialectica, vol. 12, pp. 280–287. Also in Gödel 1986. Gödel, K. (1986, 1995, 2003) Collected Works, vols. 1, 3, and 4. Oxford U. P. Goodstein, R.L. (1939) Mathematical systems. Mind, vol. 48, pp. 58–73. Goodstein, R.L. (1951) Constructive Formalism. Leicester U.P. Goodstein, R.L. (1958) On the nature of mathematical systems. Dialectica, vol. 12, pp. 296–316. Harrop, R. (1960) Concerning formulas of the type A → B ∨ C, A → (Ex)B(x)in intuitionistic formal systems. The Journal of Symbolic Logic, vol. 25, pp. 27–32. van Heijenoort, J., ed, (1967) From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931, Harvard University Press. Hempel, C. (2000) An intellectual autobiography. In Science, Explanation, and Rationality, ed. J. Fetzer, pp. 3–35. Herbrand, J. (1931) Sur la non-contradiction de l’arithmétique. Journal fur¨ die reine und angewandte Mathematik, vol. 166, pp. 1–8. English translation in Van Heijenoort. Hertz, P. (1923) Uber¨ Axiomensysteme fur¨ beliebige Satzsysteme. Teil II. Mathe- matische Annalen, vol. 89, pp. 76–102. Hertz, P. (1929) Uber¨ Axiomensysteme fur¨ beliebige Satzsysteme. Mathematische Annalen, vol. 101, pp. 457–514. Heyting, A. (1930) Die formalen Regeln der intuitionistischen Logik. Sitzungs- berichte der Preussischen Akademie von Wissenschaften, Physikalisch-mathematische Klasse, pp. 42–56. 308 Bibliography for Parts I and II

Heyting, A. (1931) Die intuitionistische Grundlegung der Mathematik. Erkenntnis, vol. 2, pp. 106–115. Heyting, A. (1934) Mathematische Grundlagenforschung: Intuitionismus, Beweis- theorie. Springer. Heyting, A. (1935) Intuitionistische wiskunde. Mathematica B, vol. 4, pp. 72–83. Heyting, A. (1946) On weakened quantification. The Journal of Symbolic Logic,vol. 11, pp. 119–121. Heyting, A. (1956) Intuitionism: An Introduction. North-Holland. Heyting, A. (1958) Blick von der intuitionistischen Warte. Dialectica, vol. 12, pp. 332–344. Hilbert, D. (1925) Uber¨ das Unendliche. Mathematische Annalen, vol. 95, pp. 161– 190. Hilbert, D. (1931) Die Grundlegung der elementaren Zahlenlehre. Mathematische Annalen, vol. 104, pp. 485–494. Hilbert, D. (1931a) Beweis des tertium non datur. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse, pp. 120–125. Hilbert, D. and W. Ackermann (1928) Grundzuge¨ der theoretischen Logik. Springer. Hilbert, D. and P. Bernays (1934, 1939) Grundlagen der Mathematik I–II. Springer. Ja´skowski, S. (1934) On the rules of supposition in formal logic, as reprinted in S. McCall, ed, Polish Logic 1920–1939, pp. 232–258, Oxford U. P. 1967. Johansson, I. (1936) Der Minimalkalkul,¨ ein reduzierter intuitionistischer Forma- lismus. Compositio Mathematica, vol. 4, pp. 119–136. Kalmár, L. (1935) Uber¨ die Axiomatisierbarkeit des Aussagenkalkuls.¨ Acta Scien- tiarium Mathematicarum, vol. 7, pp. 222–243. Kalmár, L. (1938) Uber¨ Gentzens Beweis fur¨ die Widerspruchsfreiheit der reinen Zahlentheorie. Manuscript Hs974:105 of the Bernays collection of the ETH, 21 pages. Kleene, S. (1945) On the interpretation of intuitionistic number theory. The Journal of Symbolic Logic, vol. 10, pp. 109–124. Kleene, S. (1952) Introduction to Metamathematics, North-Holland. Kleene, S. (1952a) Permutability of inferences in Gentzen’s calculi LK and LJ. Memoirs of the American Mathematical Society, vol. 10, pp. 1–26. Kolmogorov, A. (1925) On the principle of excluded middle. Translation of Russian original, in van Heijenoort (1967), pp. 416–437. Kreisel, G. (1976) Wie die Beweistheorie zu ihren Ordinalzahlen kam und kommt? Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 78, pp. 177–223. Kreisel, G. (1987) Gödel’s excursions into intuitionistic logic. In P. Weingartner and L. Schmetterer, eds, Gödel Remembered, pp. 65–186, Bibliopolis, Naples. Bibliography for Parts I and II 309

Maehara, S. (1954) Eine Darstellung der intuitionistische Logik in der klassischen. Nagoya Journal of Mathematics, vol. 7, pp. 46–64. Mancosu, P. (1999) Between Berlin and Vienna: The immediate reception of Gödel’s incompleteness theorems. History and Philosophy of Logic vol. 20, pp. 33–45. Menzler-Trott, E. (2001) Gentzen Problem: mathematische Logik im nationalsozia- listischen Deutschland. Birkhäuser. Menzler-Trott, E. (2007) Logic’s Lost Genius: The Life of Gerhard Gentzen.Ame- rican Mathematical Society. Moriconi, E. (2015) Early structural reasoning. Gentzen 1932. The Review of Sym- bolic Logic, vol. 8, pp. 662–679. Negri, S. and J. von Plato (2001) Structural Proof Theory. Cambridge. Negri, S. and J. von Plato (2015) Meaning in use. In Dag Prawitz on Meaning and Proofs, ed. H. Wansing, pp. 239–257. Springer Trends in Logic Series, 2015. Negri, S. and J. von Plato (2016) Cut elimination in sequent calculi with implicit contraction, with a conjecture on the origin of Gentzen’s altitude line construction. In D. Probst and P. Schuster, eds, Concepts of Proof in Mathematics, Philosophy, and Computer Science, pp. 269–290. De Gruyter. Neufeld, M. (1995) The Rocket and the Reich: Peenemunde¨ and the Coming of the Ballistic Missile Era. Smithsonian Books, Washington. von Neumann, J. (1927) Zur Hilbertschen Beweistheorie. Mathematische Zeitschrift, vol. 26, pp. 1–46. von Neumann, J. (1928) Uber¨ die Definition durch transfinite Induktion und ver- wandte Fragen der allgemeinen Mengenlehre. Mathematische Annalen, vol. 99, pp. 373–391. Pinl, M. (1969–76) Kollegen in einer dunklen Zeit. Jahresbericht der Deutschen Mathematiker–Vereinigung, vol. 71 (1969), pp. 167–228. Part II, vol. 72 (1971), pp. 165–189. Part III, vol. 73 (1972), pp. 153–208. Part IV, vol. 75 (1974), pp. 166–208. Part V, vol. 77 (1976), pp. 161–164. von Plato, J. (2001) A proof of Gentzen’s Hauptsatz without multicut. Archive for Mathematical Logic, vol. 40, pp. 9–18. von Plato, J. (2008) Gentzen’s proof of normalization for natural deduction. The Bulletin of Symbolic Logic, vol. 14, pp. 240–244. von Plato, J. (2009) Gentzen’s logic. Handbook of the History of Logic,vol.5,pp. 667–721. von Plato, J. (2012) Gentzen’s proof systems: byproducts in a program of genius. The Bulletin of Symbolic Logic, vol. 18, pp. 313–367. von Plato, J. (2014) From axiomatic logic to natural deduction. Studia Logica,vol. 102, pp. 1167–1184. 310 Bibliography for Parts I and II von Plato, J. (2015) From Hauptsatz to Hilfssatz. In R. Kahle, and M. Rathjen, eds, Gentzen’s Centenary: The Quest for Consistency, pp. 89–126, Springer. von Plato, J. and A. Siders (2012) Normal derivability in classical logic. The Review of Symbolic Logic, vol. 5, pp. 205–211. Prawitz, D. (1965) Natural Deduction: A Proof-Theoretical Study. Almqvist & Wicksell. Rathjen, M. (1995) Recent advances in ordinal analysis. The Bulletin of Symbolic Logic, vol. 1, pp. 468–485. Rathjen, M. (1999) The realm of ordinal analysis. In S. Wainer and J. Truss, eds, Sets and Proofs, pp. 219–279, Cambridge University Press. Schmidt, H. (1960) Mathematische Gesetze der Logik. Springer. Scholz, H. (1940) Urteil uber¨ die Habilitationsschrift Gentzen “Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlen- theorie.” Typewritten manuscript with the date 24 February 1940 and the imprint “Logistisches Seminar der Universität Munster¨ i./W. Prof. Scholz.” Schroeder-Heister, P. (2002) Resolution and the origins of structural reasoning: early proof-theoretic ideas of Hertz and Gentzen. The Bulletin of Symbolic Logic, vol. 8, pp. 246–265. Schutte,¨ K. (1951) Beweistheoretische Erfassung der unendlichen Induktion in der Zahlentheorie. Mathematische Annalen, vol. 122, 369–389. Siders, A. (2015) A direct Gentzen-style consistency proof for Heyting arithmetic. In R. Kahle, and M. Rathjen, eds, Gentzen’s Centenary: The Quest of Consistency, pp. 177–211. Siders, A. and J. von Plato (2015) Bar induction in the proof of termination of Gent- zen’s reduction procedure. In R. Kahle, and M. Rathjen, eds, Gentzen’s Centenary: The Quest for Consistency, pp. 127–130, Springer. Siegmund-Schultze, R. (2009) Mathematicians Fleeing from Nazi Germany: Indivi- dual Fates and Global Impact. Princeton U. P. Skolem, T. (1937) Uber¨ die Zuruckf¨ uhrbarkeit¨ einiger durch Rekursionen definierter Relationen auf “arithmetische.” As reprinted in Skolem’s Selected Works in Logic, pp. 425–440, 1970. Smorynski, C. (2007) Gentzen and geometry. In Menzler-Trott (2007), pp. 281–288. Soifer, A. (2009) The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators. Springer. Tait, W. (2005) Gödel’s reformulation of Gentzen’s first consistency proof of arithmetic for arithmetic: the no-counterexample interpretation. The Bulletin of Sym- bolic Logic, vol. 11, pp. 225–238. Tait, W. (2010) Review of Menzler-Trott 2007. The Bulletin of Symbolic Logic,vol. 16, pp. 270–275. Bibliography for Parts I and II 311

Takeuti, G. (1975) Proof Theory. North-Holland. Troelstra, A. (1973) Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Lecture Notes in Mathematics, vol. 344, Springer. Troelstra, A. and D. van Dalen (1988) Constructivism in Mathematics. 2 vols., North-Holland. Troelstra, A. and H. Schwichtenberg (1996) Basic Proof Theory. Second edition 2000, Cambridge U. P. Turing, A. (1936) On computable numbers, with an application to the Entschei- dungsproblem. Proceedings of the London Mathematical Society , vol. 42, pp. 230– 265. Wajsberg, M. (1938) Untersuchungen uber¨ den Aussagenkalkul¨ von A. Heyting. Wiadomsci matematyczne, vol. 46, pp. 45–101. Wolfson, H. (1976) The Philosophy of the Kalam. Harvard U.P. Index of names for parts I and II

A Gentzen, M., 15 Ackermann, W., 10, 13, 45, 47, 48, 73 Glivenko, V., 19 Allert, T., 13 Gödel, K., 7, 9–11, 13, 19–22, 29–31, Arai, T., 48 33, 34, 37–40, 44, 46, 49, 55, 57, 70, 71, 74, 77, 78, 81 B Goodstein, R., 29 Bach, G., ix, 8 Bergmann, P., 16 H Bernays, L., ix, 13 Harrop, R., 28 Bernays, P., ix, 5, 7, 11–13, 17, 18, 21, Hempel, C., 9 22, 31–42, 44–46, 48, 55, 56, Herbrand, J., 10, 68 61, 65, 72, 74, 76–78, 81 Hertz, P., 11, 12, 16–18, 20, 23, 58 Bers, L., 16 Heyting, A., 7, 14, 19–22, 27–31, 33, 38, Berwald, L., 16 65, 73–75 Bieberbach, L., 15 Hilbert, D., 3, 9–12, 22, 33, 39, 44, 45, von Braun, W., 60 48, 49, 58, 69, 73, 77 Brouwer, L., 7, 27, 29, 30, 33, 34, Hitler, A., 12–14 37–39, 48, 75 Howard, W., 55 von Boguslawski, M., 50 J C Ja´skowski, S., 24 Carnap, R., 78, 81 Johansson, I., 30, 75 Cavaillès, J., 31, 32 Chertok, B., 60 K Church, A., 55, 57, 67 Kalmár, L., 44, 80 Collatz, L., 9 Ketonen, O., 50 Curry, H., 28, 29, 45 Kleene, S., 28, 34, 46 Kneser, H., 9, 10, 13, 21, 22, 56 D Kohn, H., 58 van Dalen, D., 31, 37 Kohn, P., 16 Kolmogorov, A., 19, 77 F Kreisel, G., 37, 46 Feferman, S., 48 Kuhn, P., 16 Feynman, R., 59 Fowler, R., 58 L Fraenkel, A., 12 Lorenzen, P., 51 Frank, P., 16 Löwig, H., 16 Fröhlich, W., 16 Löwner, K., 16 Funk, P., 16 M G MacLane, S., 15 Gabbay, D., 4 Maehara, S., 29, 79 Gabelsberger, F., 81, 82 Mancosu, P., 9 © Springer International Publishing Switzerland 2017 313 J. von Plato, Saved from the Cellar, Sources and Studies in the History of Mathematics and Physical Sciences, DOI 10.1007/978-3-319-42120-9 314 Index of names for parts I and II

McKinsey, J., 28 Siegmund-Schultze, R., 14 Meitner, L., 58 Skolem, T., 40, 49, 57 Menger, K., 20 Smorynski, C., 9 Menzler-Trott, E., ix, 6, 7, 9, 10, 13–15, Soifer, A., 14 31, 32, 39, 50, 56, 59, 60, 62, Sponer, H., 58, 59 65, 72 Stalin, J., 13 Mohr, E., 15, 16 Student, W., 3 Moriconi, E., 18 Sutterlin,¨ L., 25, 72, 82–84, 86, 87 Szabo, M., 11, 20, 34, 72, 82 N Negri, S., 5, 28, 43, 73 T Neufeld, M., 60 Tait, W., 61 von Neumann, J., 9–11, 37, 39, 77 Takeuti, G., 42, 46, Nordheim, L., 58 Tarski, A., 57 Thiel, C., ix, 3, 4, 8, 9, 72 P Thullen, P., 14 Peano, G., 6, 19, 22, 48, 70 Troelstra, A., 26, 31 Perron, O., 15 Turing, A., 67 Pick, G., 16 Pinl, M., 15, 16, 56 U Prawitz, D., 26, 28, 42 Urban, H., 56

R V Rathjen, M., 48 Veblen, O., 20 Rohrbach, H., 3, 7, 56, 59, 60, 62 Russell, B., 22, 69 W van der Waerden, B., 14, 32 S Wajsberg, M., 14, 28 Schmidt, A., 3, 52 Weyl, H., 13, 31–32, 34, 39, 81 Scholz, H., 13, 39, 40, 44, 47, 56, 62, 75 Wiechmann, A., 84, 87 Schorer, H., 84, 87 Winternitz, A., 16 Schroeder-Heister, P., 18, 23 Wittgenstein, L., 29 Schutte,¨ K., 46, 48, 77 Wolfson, H., 5 Schwichtenberg, H., 26, 31 Wolters, G., ix Siders, A., 35, 54 von Wright, G., 46 Index of names in the Gentzen papers

A J Ackermann, W., 146, 220, 253, 255, 288 Ja´skowski, S., 290

B K Bernays, P., 189, 197, 218, 221, Kalmár, L., 275 237–259, 264, 281, 288, 300 Kaufmann, F., 300, 301 Blumenthal, M., 242, 245, 246 Kolmogorov, A., 255 König, J., 265 C Kuratowski, K., 293 Carnap, R., 260, 264, 300, 301 L Cavaillès, J., 244 Landau, E., 298 Church, A., 248, 283, 299, 300 Langford, C., 291 Lewis, C., 291 D Dedekind, R., 259, 296 N Dubislav, W., 299 von Neumann, J., 143, 238, 300, 301

F P Fermat, P., 266, 267 Peano, G., 156, 291, 296 Finsler, P., 255, 256 Pepis, S., 258 Fraenkel, A., 300 Presburger, M., 267 Frege, G., 293, 296, 298, 299 R G Russell, B., 103, 112, 147, 293, 296, 297 Glivenko, V., 156 S Gödel, K., 109, 120, 125, 128, 129, 143, Schmidt, A., 244 190, 191, 197, 202, 244, 246, Scholz, H., 249, 299 247, 249, 254, 255, 264, 266, Sheffer, H., 292, 295, 297 276, 283, 292, 299–301 Skolem, T., 103, 109, 247, 264, 267, 283, 284, 297, 300, 301 H Herbrand, J., 254, 288, 292 T Hertz, P., 277 Tarski, A., 283, 301 Heyting, A., 125, 147, 156, 190–192, 277 Hilbert, D., 103, 112, 146, 147, 156, 190, W 244, 246, 250, 281, 288, 300 Wajsberg, M., 277 Hölder, O., 265, 299 Weyl, H., 189, 202, 239, 244