Finding a Fit Among Philosophical Finitisms

Eamon Darnell () and Aaron Thomas-Bolduc ()
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Eamon Darnell: University of Toronto
Aaron Thomas-Bolduc: Independent Researcher

A chapter in Handbook of the History and Philosophy of Mathematical Practice, 2024, pp 443-461 from Springer

Abstract: Abstract David Hilbert sought to secure the epistemic foundations of mathematics by providing consistency proofs of axiomatized mathematical theories from within the finite standpoint. This standpoint requires concrete constructions without reference to completed infinities. In 1938, Gerhardt Gentzen proved the consistency of first-order Peano Arithmetic relying on the well-ordering of certain ordinal notations. This was thought by Gentzen and Paul Bernays to be finitistically acceptable. However, a finitistically acceptable proof of the relevant well-ordering was not available until Gaisi Takeuti’s proof in 1978. Beginning with the proposition that Takeuti’s proof conforms to the finite standpoint, this chapter explores the limits of a finitism based on the methods of Gentzen and Takeuti. It is argued that the upper limit is the Feferman-Schütte ordinal, and thus that finitism may be extended to strong subsystems of second-order arithmetic-systems strong enough to provide a theory of the ordinals for which Gentzen employed notations.

Keywords: Consistency; Constructive mathematics; Finitism; Gentzen; Hilbert’s program; Ordinal notations; Predicativism; Proof theory; Takeuti; Transfinite induction (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/978-3-031-40846-5_95

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