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Gentzen’s Consistency Proof in Context

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Gentzen's Centenary

Abstract

Gentzen’s celebrated consistency proof—or proofs, to distinguish the different variations he gave1—of Peano Arithmetic in terms of transfinite induction up to the ordinal2 \(\varepsilon _{0}\) can be considered as the birth of modern proof theory.

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Notes

  1. 1.

    Cf., e.g., [13, 87, 105], and [114] as well as [97] in this volume.

  2. 2.

    For the ordinal \(\varepsilon _{0}\) see, for instance, [58] in this volume.

  3. 3.

    See, for instance, [90] in this volume.

  4. 4.

    See, for instance, [15] and [87] in this volume.

  5. 5.

    For the development of Hilbert’s programme(s), cf. e.g., [98].

  6. 6.

    See Fraenkel [28, p. 154]:

    This is the point of view of Hilbert, who, therefore, picks up himself the methodical starting point of his intuitionist opponents—but for the purpose to deny their thesis; one could almost characterize him as an intuitionist.

    (German original: “Dies etwa ist der Standpunkt Hilberts, der somit den methodischen Ausgangspunkt seiner intuitionistischen Gegner — allerdings zum Zweck der Bestreitung ihrer Thesen — selbst aufnimmt; man könnte ihn geradezu als Intuitionisten bezeichnen.”) Van Dalen adds to this citation [112, p. 309]: “Although the inner circle of experts in the area (e.g. Bernays, Weyl, von Neumann, Brouwer) had reached the same conclusion some time before, it was Fraenkel who put it on record.” See also footnote 18.

  7. 7.

    German original [32, p. 6]: “Die folgerichtigste Art der Abgrenzung ist die durch den ‘intuitionistischen’ Standpunkt […] gegebene.”

  8. 8.

    We may leave it open here whether Hilbert himself was advocating such a conservativity. The issue of conservativity can be considered, of course, without reference to historic figures.

  9. 9.

    It is reported in the Schütte school that this was also immediately recognized in Göttingen.

  10. 10.

    But one may note the puzzling lack of understanding of Russell, expressed in a letter to Leon Henkin of 1 April 1963, cf. [18, p. 89ff].

  11. 11.

    Hilbert and Bernays [55, p. VII]. German original: “Jenes Ergebnis zeigt in der Tat auch nur, daß man für die weitergehenden Widerspruchsfreiheitsbeweise den finiten Standpunkt in einer schäferen Weise ausnutzen muß, […].”

  12. 12.

    German original [1, p.1f]: “Besonders interessiert hat mich der neue meta-mathematische Standpunkt, den Sie jetzt einnehmen und der durch die Gödelsche Arbeit veranlaßt worden ist.” The letter was written after Ackermann visited Göttingen, but didn’t meet Hilbert and spoke only with Arnold Schmidt, who informed him about “everything” going on in Göttingen.

  13. 13.

    Detlefsen, [19] in this volume, however, points out that there are some fundamental differences between Gentzen’s own philosophical view and Hilbert’s view.

  14. 14.

    In German: “Überschreitung des bisherigen methodischen Standpunkts der Beweistheorie”.

  15. 15.

    German original, [38, p. 94]:

    1. 4.

      Wie also erweitern? (Erweiterung nötig.) Drei Wege \([\![\) sind\(]\!]\) bisher bekannt:

      1. 1.

        Höhere Typen von Funktionen (Funktionen \([\![\) von\(]\!]\) Funktionen von Zahlen, etc.)

      2. 2.

        Modalitätslogischer Weg (Einführung einer Absurdität auf Allsätze angewendet und eines “Folgerns”).

      3. 3.

        Transfinite Induktion, d.h., es wird der Schluß durch Induktion für gewisse konkret definierte Ordinalzahlen der zweiten Klasse hinzugefügt.

  16. 16.

    See, for instance, [75] in this volume.

  17. 17.

    German original: “Ein solcher Widerspruchsfreiheitsbeweis wäre nun wieder ein mathematischer Beweis, in dem gewisse Schlüsse und Begriffsbildungen verwendet würden. Diese müssen als sicher (insbesondere als widerspruchsfrei) bereits vorausgesetzt werden. Ein ‘absoluter Widerspruchsfreiheitsbeweis’ ist also nicht möglich. Ein Widerpruchsfreiheitsbeweis kann lediglich die Richtigkeit gewisser Schlußweisen auf die Richtigkeit anderer Schlußwiesen zurückführen. Man wird also verlangen müssen, daß in einem Widerspruchsfreiheitsbeweis nur solche Schlußweisen der Theorie, deren Widerspruchsfreiheit man beweist, als erheblich sicherer gelten können.”

    Similarly in [32]:

    In order to carry out a consistency proof, we naturally already require certain techniques of proof whose reliability must be presupposed and can no longer be justified along these lines. An absolute consistency proof, i.e., a proof which is free from presuppositions is of course impossible. [102, p. 237].

    German original: “Um einen Widerspruchsfreiheitsbeweis zu führen, braucht man natürlich bereits gewisse mathematische Beweismittel, deren Unbedenklichkeit man voraussetzen muß und auf diesem Wege schließlich nicht weiter begründen kann. Ein absoluter, d. h. voraussetzungsloser Widerspruchsfreiheitsbeweis ist selbstverständlich unmöglich.”

  18. 18.

    “Concerning the use of the word intuitionistic […], it should be noted that according to Bernays [[11, p. 502]], the prevailing view in the Hilbert school at the beginning of the 1930s equated finitism with intuitionism.” [24, p. 117]. See also footnote 6 above.

  19. 19.

    This paper was submitted in 1933, but withdrawn by Gentzen when he became known about Gödel’s paper. An English translation appeared in print in 1969, [102, #2], the German version of the Galley proofs, kept by Paul Bernays, was published only in 1974.

  20. 20.

    German original [33, p. 131]: “Wenn man die intuitionistische Arithmetik als widerspruchsfrei hinnimmt, so ist […] auch die Widerspruchsfreiheit der klassischen Arithmetik gesichert.”

  21. 21.

    An informal presentation of the main idea of the proof is given, for instance, by Takeuti in [120, p. 128ff].

  22. 22.

    A well-known proof theorists presumably heard the second joke from Kreisel but confused a “y” with an “i” attributing it—with reference to Kreisel—to “un grand mathématicien français” [35, p. 520, fn. 14]; this confusion is confirmed in [36, pp. 9 and 33] where André Weil is mentioned by name (without reference to Kreisel).

  23. 23.

    Smullyan [100, p. 56] illustrates very well this point in connection with Gödel’s (first) incompleteness result, stressing that Gödel, by using ω-consistency, makes a much weaker assumption than correctness. The pointlessness of consisteny proofs by semantic methods was well stated by Shoenfield [96, p. 214]:

    The consistency proof for P by means of the standard model […] does not even increase our understanding of P, since nothing goes into it which we did not put into P in the first place.

  24. 24.

    For sure, Weyl will have known exactly what’s going on here, and probably also classified his remark only as a joke.

  25. 25.

    See [84], cited in [98, p. 7].

  26. 26.

    See, for instance, [10, p. 203]. This separation might have been suggested by Brouwer to Hilbert in 1909, cf. [112, p. 302]. Sieg [98, p. 27] writes: “Hilbert claims in [[50]], that Poincaré arrived at ‘his mistaken conviction by not distinguishing these two methods of induction, which are of entirely different kinds’ and feels that ‘[u]nder these circumstances Poincaré had to reject my theory, which, incidentally, existed at that time only in its completely inadequate early stages’.”

  27. 27.

    It is defensible that Hilbert took Poincaré’s critics more serious than, for instance, Brouwer’s, cf. [61, 62]; but since Poincaré died already in 1912, Hilbert had lost him as discussion partner at the time his programme was worked out.

  28. 28.

    This supplement, added to the second edition of [53] and published in 1970, also presents a consistency proof of Kalmár, based on an unpublished manuscript of 1938.

  29. 29.

    Cf. Bernays in [53, p. VII]:

    Currently, W. Ackermann is developing his earlier consistency proof—by use of a sort of transfinite induction as used by Gentzen—in a way that it obtains validity for the full numbertheoretic formalism.

    German original: “Gegenwärtig ist W. Ackermann dabei, seinen früheren (…) Widerspruchsfreiheitsbeweis durch Anwendung der transfiniten Induktion in der Art, wie sie von Gentzen benutzt wird, so auszugestalten, daß er für den vollen zahlentheoretischen Formalismus Gültigkeit erhält.”

    Von Plato writes in [115, end of I.4.10]: “A second proof of Gentzen’s result was given by an unwilling Wilhelm Ackermann, after repeated pleadings on the part of Bernays.”

  30. 30.

    In the continuation of the citation, the mentioned fine structure is illustrated by the result about provably total functions of PA which one can obtain from Gentzen’s work.

  31. 31.

    See, for instance, the talk on transfinite numbers given by Poincaré in Göttingen in 1909 in the presence of Hilbert, included in [85] and translated by Ewald in [22, 22.G] (reprinted in [62]).

  32. 32.

    Szabo [102, p. viii] refers to the memories of a friend of Gentzen in the prison: “He once confided in me that he was really quite contented since now he had at last time to think about a consistency proof for analysis. He was in fact fully convinced that he would succeed in carrying out such a proof.”

  33. 33.

    Here, one can turn Hilbert’s programme upside down and use interpretations of new intuitionistic principles to justify them on classical grounds; see, for instance, [27, p. 340]. I also remember a proof theorist, making good use of such principles, but calling them—trained in classical Mathematics and therefore believing in the standard notion of mathematical truth—“totally wrong” (as translation of the German “grob falsch”).

  34. 34.

    Kreisel, in [67, p. 344], sketches also an extension of “Gödel’s old translation” of a system for classical Analysis to a specific intuitionistic reformulation of Analysis, involving the general Comprehension Axiom, which “provides an intuitionistic consistency proof of classical analysis”. He himself classifies this result as “philosophically […] not significant at all”, except for “a reduction to intuitionistic methods of proof”—which he judges a “technical” property. In the Discussion of this proof he reminds the reader to look for alternatives:

    Quite naively, this easy proof in no way reduces the interest of a more detailed proof theoretic reduction […]; just as Gödel’s original intuitionistic consistency proof for classical arithmetic Z did not make Gentzen’s reduction superfluous.

  35. 35.

    In a discussion of these proofs, Kreisel writes [67, p. 349, footnote 16]: “[I]n terms of consistency proofs, Tait’s argument would only have proved the consistency of classical analysis in third order arithmetic!”

  36. 36.

    I remember a proof-theorist classifying such a normalization proof as simply “circular.”

  37. 37.

    The worst-case scenario was experienced by Martin-Löf, when he realized that the normalization proof of his first (inconsistent) type theory was carried out in an inconsistent metatheory (see Setzer’s contribution in this volume [95]).

  38. 38.

    For a thorough discussion of Spector’s proof see [26] in this volume. Oliva and Powell [80], also in this volume, discuss some spin-offs we can get from proof-theoretic analyses in the neighborhood of Spector’s approach.

  39. 39.

    See [81, 82, 94, 108] for comprehensive presentations of the background of the respective developments.

  40. 40.

    See, for instance, [14] and [57] in this volume.

  41. 41.

    This was exemplified, in particular, by Kripke-Platek set theory, cf. e.g., [56, 81].

  42. 42.

    In the further course of the discussion, Feferman expresses some doubts about current advances in ordinal analysis with respect to the given rationale [23, p. 80]:

    Even if one succeeds in reducing the system \((\Pi _{2}^{1}\mbox{ -}\mathsf{CA}) \pm \mathsf{BI}\) to a constructive system (whether evidently so or not), one can hardly expect that doing so will appreciably increase one’s belief in its consistency (if one has any doubts about that in the first place) in view of the difficulty of checking the extremely complicated technical work needed for its ordinal analysis.

  43. 43.

    This is, admittedly, in sharp contrast to the early times of axiomatic set theory, where Poincaré, for instance, expressed his doubts about Zermelo’s axiomatization of set theory in the following words, cf. [43, p. 540]:

    But even though he has closed his sheepfold carefully, I am not sure that he has not set the wolf to mind the sheep.

  44. 44.

    Of course, this prediction is embedded in a thorough discussion which gives arguments for this claim. But one may note that Woodin speaks here about the discovery not about the existence of an inconsistency.

  45. 45.

    The argument for the intuitive model of ZF is compared with the situation for Quine’s New Foundation where the lack of such an intuitive model gives reason to look for a (relative) consistency proof.

  46. 46.

    Conveyed by Girard in French [35, p. 525]: “Les doutes quant à la cohérence sont plus douteux que la cohérence elle-même.”

  47. 47.

    German original: “Was beweisbar ist, soll in der Wissenschaft nicht ohne Beweis geglaubt werden.” cited and translated in [20, p. 97].

  48. 48.

    See, for instance, [12]: “Historically speaking, it is of course quite untrue that mathematics is free from contradiction” and later “[Contradictions] occur in the daily work of every mathematician, beginner or master of his craft, as the result of more or less easily detected mistakes, […]”

  49. 49.

    This example is taken from [20, p. 59].

  50. 50.

    German original: “Unter einer ‘Menge’ verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten m unserer Anschauung oder unseres Denkens (welche die ‘Elemente’ von M genannt werden) zu einem Ganzen.” [16, p. 282]. The translation is from [44, p. 33].

  51. 51.

    In German: “consistente Vielheiten,” letter to Hilbert from May 5th, 1899, [17, letter 160]; as “finished set” (“fertige Menge”) already in a letter from December 2nd, 1897, [17, p. 390].

  52. 52.

    In [47] he writes, [113, p. 131]:

    G. Cantor sensed the contradiction just mentioned and expressed this awareness by differentiating between “consistent” and “inconsistent” sets. But, since in my opinion he does not provide a precise criterion for this distinction, I must characterize his conception on this point as one that still leaves latitude for subjective judgment and therefore affords no objective certainty.

    In German (cited in [17, S. 436]): “G. Cantor hat den genannten Widerspruch empfunden und diesem Empfinden dadurch Ausdruck verliehen, daß er ‘konsistente’ und ‘nichtkonsistente’ Mengen unterscheidet. Indem er aber meiner Meinung nach für diese Unterscheidung kein scharfes Kriterium aufstellt, muß ich seine Auffassung über diesen Punkt als eine solche bezeichnen, die dem subjektiven Ermessen noch Spielraum läßt und daher keine objektive Sicherheit gewährt.” An even stronger statement against Cantor’s approach can be found in a lecture note from 1917, [48], cf. [59, 60].

  53. 53.

    Although this axiomatization has the flaw that its justification is extrinsic where philosophers would prefer to have an intrinsic one, cf. e.g., the discussion in [73].

  54. 54.

    One may note that Cantor’s criterion for a “finished set” also requires a consistency proof, but somehow locally for the particular construction only. However, as far as we know, Cantor only took note of the criterion in the negative cases, to dismiss a set construction when it was shown to be inconsistent.

  55. 55.

    For instance, Aczel’s Frege Structures, [3].

  56. 56.

    The situation becomes philosophically even more doubtful when such a justification depends, in addition, on the approval of a “Master”. In this respect, Lorenzen complained about Brouwer [70]:

    Unfortunately, the explanation which Brouwer himself offers for this phenomenon [that some Mathematicians consider the ‘tertium non datur’ as unreliable] is an esoteric issue: only one who listened the Master himself understands him.

    (German original: “Unglücklicherweise ist die Erklärung, die Brouwer selbst für dieses Phänomen anbietet, eine esoterische Angelegenheit: nur, wer den Meister selber hörte, versteht ihn.”)

  57. 57.

    A complementary view on this issue is given by Setzer [95] in this volume.

  58. 58.

    This claim can be substantiated by the fact that it was not possible for any of the systems to modify it in a way that the original aims of the authors would be preserved.

  59. 59.

    A thorough discussion of this issue can be found in [25].

  60. 60.

    German original [32, p. 6]: “So scheint es mir nicht ganz ausgeschlossen, daß auch in der klassischen Analysis mögliche Widersprüche verborgen sein können. Daß man bis jetzt keine entdeckt hat, besagt nicht viel, wenn man bedenkt, daß der Mathematiker in praxi immer mit einem verhältnismäßig geringen Teil der an sich logisch möglichen mannigfachen Komplizierungen der Begriffsbildung auskommt.”

  61. 61.

    German original [32, p. 7]: “doch steht der praktisch vor allem wichtige Beweis für die Analysis noch aus.”

  62. 62.

    Cf, e.g., [83] and [91] in this volume.

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Acknowledgements

Research supported by the Portuguese Science Foundation, FCT, through the projects Hilbert’s Legacy in the Philosophy of Mathematics, PTCD/FIL-FCI/109991/2009, The Notion of Mathematical Proof, PTDC/MHC-FIL/5363/2012, and the Centro de Matemática e Aplicações, UID/MAT/00297/2013; and by the project Método axiomática e teoria de categorias of the cooperation Portugal/France in the Programa PESSOA – 2015/2016.

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Kahle, R. (2015). Gentzen’s Consistency Proof in Context. In: Kahle, R., Rathjen, M. (eds) Gentzen's Centenary. Springer, Cham. https://doi.org/10.1007/978-3-319-10103-3_1

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