Abstract
The purpose of this paper is to discuss Gentzen’s first consistency proof for arithmetic, the ‘galley proof’ (published for the first time in the Collected Papers, Amsterdam, 1969), by analyzing its methods of proof. Opposing positions on this subject have been taken by Bernays (1970) and Kreisel (1971). According to Kreisel’s analysis, which uses a result of Tait, it turns out that the ‘galley proof’ does not involve methods of proof exceeding transfinite induction up to ε0.
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Bibliography
Bernays, P. (1970), ‘On the originals Gentzen consistency proof’, in Intuitionism and Proof Theory. Amsterdam, North-Holland. pp. 409–17.
Gentzen, G. (1969), Collected Papers, ed. by M.E. Szabo. Amsterdam, North Holland.
Kreisel, G. (1971), ‘Book review of Gentzen’s Collected Papers,’ Journal of Philosophy, 68, 238–65.
Kreisel, G. (1976), ‘Wie die Beweistheorie zu ihren Ordinalzahlen kam und kommt’, Jahresbericht der Deutschen Mathematiker-Vereinigung 78, 177–223.
Tait,W.W. (1961), ‘Nested recursion,’ Mathematische Annalen 143, 236–50.
Tait, W.W. (1965), ‘Functional defined by transfinite recursion’, Journal of Symbolic Logic 30, 155–192.
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Negri, M. (1980). Constructive Sequent Reduction in Gentzen’s First Consistency Proof for Arithmetic. In: Dalla Chiara, M.L. (eds) Italian Studies in the Philosophy of Science. Boston Studies in the Philosophy of Science, vol 47. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8937-5_10
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DOI: https://doi.org/10.1007/978-94-009-8937-5_10
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