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Reinhard Kahle, Michael Rathjen (eds.): “Gentzen’s Centenary, The Quest for Consistency”

Springer, 2015, 561 pp.

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Notes

  1. Dedekind’s definition assumed the existence of the system of all systems, i.e., a universal set. Cantor pointed out in letters to Dedekind and Hilbert in 1899 that this multiplicity is “inconsistent”; he had discussed the distinction between consistent and inconsistent multiplicities already in earlier letters.

  2. [8], pp. 11–13. From a modern perspective one might call a particular system that falls under an abstract notion or satisfies its characteristic conditions a “model”. In [26] Hilbert gave the standard analytic model for Euclidean geometry, but he constructed also a geometric model for a complete ordered field via his “Streckenrechnung”; one could not make a more convincing case of the irrelevance of the nature of the mathematical objects from such a structuralist perspective.

  3. The notes of these stunning lectures were written by Paul Bernays who had joined Hilbert that very term as assistant for the foundations of mathematics; they literally form the core of the book Grundzüge der theoretischen Logik that was published only in 1928 with Hilbert and Wilhelm Ackermann as authors. All of the lecture notes from 1917–18 onward as well as the Grundzüge can be found in [7].

  4. Neither in these lectures nor in their publications does one find a precise articulation of finitist mathematics. It seems, that it is taken to be “familiar” as described, for example, in Felix Bernstein’s article [2]. Bernays and members of the wider Hilbert circle, like Jacques Herbrand and Johan von Neumann, considered finitist and intuitionist mathematics as co-extensional.

  5. Eckhard Menzler-Trott’s Gentzen biography, Logic’s Lost Genius—The life of Gerhard Gentzen, was published in 2007 by the American and the London Mathematical Society. It is full of rich, detailed information that is crucial also for those who do not share Menzler-Trott’s perspective on the intellectual context of Gentzen’s work, for example, the connection to Hilbert and Bernays’ proof theory.

  6. Ironically, Gödel in his attempt to reach that very goal was led to a version of his first incompleteness theorem in August of 1930.

  7. Gentzen’s discovery in [10] was simultaneous with that of Gödel in [17], who also reduced classical number theory to its intuitionist version. After learning of Gödel’s publication Gentzen withdrew his paper; it was posthumously published in 1974.

  8. The emergence of this proof and the conceptual difficulties Gentzen faced (and ingeniously overcame) are analyzed in my paper In the shadow of incompleteness: Hilbert and Gentzen [35]. The starting-point for Gentzen’s work is Hilbert’s last paper [32] in which Hilbert takes an extended foundational standpoint for giving consistency proofs: he argues for the finitist correctness of a constructive theory that includes intuitionist number theory and tries to prove that adding instances of tertium non datur does not lead to contradictions. Hilbert’s paper is a bridge between his earlier proof theory and Gentzen’s work. That is in contrast to the perspectives articulated in two contributions to this volume, namely, Michael Detlefsen’s Gentzen’s Anti-Formalist Views and von Plato’s From Hauptsatz to Hilfssatz.

  9. To articulate what a “natural” or “canonical” system of ordinal notations is, that is a significant methodological problem for contemporary proof theory.

  10. Mints was also, as the editors emphasize, “one of the leading executors of Gentzen’s legacy”. He died on 30 May 2014. In deep respect, this volume is dedicated to his memory.

  11. This part of second-order number theory is standardly denoted by \((\Pi^{1}_{1}\mbox{-CA})_{0}\). It has both an ordinal analysis and a reduction to a thoroughly constructive theory, namely, the theory of the finite constructive number classes; these results were already established in [3].

  12. That, in turn, is the starting point of the automated search for humanly intelligible proofs or, what Gowers calls, human centered automatic theorem proving. See, for example, the 2016 interview presented in the Notices of the American Mathematical Society 63 (9), 1026–1028.

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  1. Department of Philosophy, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Wilfried Sieg

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Sieg, W. Reinhard Kahle, Michael Rathjen (eds.): “Gentzen’s Centenary, The Quest for Consistency”. Jahresber. Dtsch. Math. Ver. 119, 201–211 (2017). https://doi.org/10.1365/s13291-017-0163-8

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