Hilbert and the Problem of Clarifying the Infinite

Stenlund, Sören

doi:10.1007/978-1-4020-8926-8_20

Cited by 1 publication

(1 citation statement)

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“…Tait (1981Tait ( , 2002 argues that finitism is completely exhausted by PRA, though he explicitly notes that he is more concerned with the technical meaning of the term, separated as much as possible from the historical and philosophical aspects of the concept. Stenlund (2009) argues that the technical and epistemological aspects of Hilbert's finitism cannot be so easily separated, as he takes Hilbert's epistemological aims to be primary. This is because the finite can be the object of intuition in a strongly Kantian sense, 4 and thus can be taken as a firm philosophical basis for mathematics.…”

Section: Finitismmentioning

confidence: 99%

Takeuti’s Well-Ordering Proof: Finitistically Fine?

Darnell

Thomas-Bolduc

2018

Proceedings of the Canadian Society for History and Philosophy of Mathematics/ Société Canadienne D’histoire Et De Philosophie

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If it could be shown that one of Gentzen's consistency proofs for pure number theory could be shown to be finitistically acceptable, an important part of Hilbert's program would be vindicated. This paper focuses on whether the transfinite induction on ordinal notations needed for Gentzen's second proof can be finitistically justified. In particular, the focus is on Takeuti's purportedly finitistically acceptable proof of the well-ordering of ordinal notations in Cantor normal form. The paper begins with a historically informed discussion of finitism and its limits, before introducing Gentzen and Takeuti's respective proofs. The rest of the paper is dedicated to investigating the finitistic acceptability of Takeuti's proof, including a small but important fix to that proof. That discussion strongly suggests that there is a philosophically interesting finitist standpoint that Takeuti's proof, and therefore Gentzen's proof, conforms to.

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Section: Finitismmentioning

confidence: 99%