Takeuti’s Well-Ordering Proof: Finitistically Fine?

Abstract

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.

Takeuti's Well-ordering Proof

“…That proof, and Takeuti's finitistic position and related consistency proofs, have recently been the subject of significant renewed interest. Darnell & Thomas-Bolduc [6] discuss whether and from which conceptions of finitism Takeuti's well-ordering proof is finitistically acceptable. They argue that Takeuti's proof conforms to what Takeuti himself terms the "Hilbert-Gentzen finitist standpoint", but that the finitistic acceptability of the proof ultimately depends on the philosophical motivations behind a given finitist standpoint.…”

“… Akiyoshi and Arana indicate that they intend to further explore Takeuti's philosophical commitments in that regard and we are hopeful that such investigations will contribute to both our understanding of Takeuti's thought, as well as the space of modern constructivist positions more generally.2 It appears that[2] also contains a reconstruction or reformulation of Takeuti's wellordering proof using modern techniques, however that paper is written in Japanese (which we are regrettably unable to read), and approaches the reconstruction from a different perspective than the present paper.3 This section has been adapted from §3 of Darnell & Thomas-Bolduc[6]. For an introduction to these methods and proofs requiring little background in proof theory, see[14].Australasian Journal of Logic (19:1) 2022, Article no.…”

“…See[6] for further discussion of this addition to Takeuti's proof 16. That is, (S n ) is a decreasing sequence constructed by taking all and only the n-major parts of the ordinals in (S n ) and removing multiple occurrences of identical n-major parts.Australasian Journal of Logic (19:1) 2022, Article no.…”