On the Intuitionistic Background of Gentzen's 1935 and 1936 Consistency Proofs and Their Philosophical Aspects

Abstract
Gentzen's three consistency proofs for elementary number theory have a common aim that originates from Hilbert's Program, namely, the aim to justify the application of classical reasoning to quantified propositions in elementary number theory. In addition to this common aim, Gentzen gave a “finitist” interpretation to every number-theoretic proposition with his 1935 and 1936 consistency proofs. In the present paper, we investigate the relationship of this interpretation with intuitionism in terms of the debate between the Hilbert School and the Brouwer School over the significance of consistency proofs. First, we argue that the interpretation had the role of responding to a Brouwer-style objection against the significance of consistency proofs. Second, we propose a way of understanding Gentzen's response to this objection from an intuitionist perspective.
Keywords proof theory  intuitionism  foundations of mathematics  philosophy of logic  history of logic
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DOI 10.4288/jafpos.27.0_1
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References found in this work BETA

A Semantics of Evidence for Classical Arithmetic.Thierry Coquand - 1995 - Journal of Symbolic Logic 60 (1):325-337.
Notation Systems for Infinitary Derivations.Wilfried Buchholz - 1991 - Archive for Mathematical Logic 30 (5-6):277-296.
The Collected Papers of Gerhard Gentzen.M. E. Szabo - 1971 - Journal of Philosophy 68 (8):238-265.

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