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Takeuti's Well-Ordering Proof: Finitistically Fine?

Darnell, Eamon and Thomas-Bolduc, Aaron (2018) Takeuti's Well-Ordering Proof: Finitistically Fine? [Preprint]

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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.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
Darnell, Eamoneamon.darnell@mail.toronto.ca
Thomas-Bolduc, Aaronathomasb@ucalgary.ca0000-0001-9955-2371
Keywords: Takeuti, Hilbert, finitism, constructivism, proof theory
Subjects: Specific Sciences > Mathematics > History of Philosophy
Specific Sciences > Mathematics > History
Specific Sciences > Mathematics > Logic
Depositing User: Aaron Thomas-Bolduc
Date Deposited: 17 Oct 2018 15:58
Last Modified: 17 Oct 2018 15:58
Item ID: 15160
DOI or Unique Handle: https://doi.org/10.1007/978-3-319-90983-7_11
Subjects: Specific Sciences > Mathematics > History of Philosophy
Specific Sciences > Mathematics > History
Specific Sciences > Mathematics > Logic
Date: October 2018
URI: https://philsci-archive-dev.library.pitt.edu/id/eprint/15160

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