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 | |||||||||
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| Keywords: | Takeuti, Hilbert, finitism, constructivism, proof theory | |||||||||
| Subjects: | Specific Sciences > Mathematics > History of Philosophy Specific Sciences > Mathematics > History Specific Sciences > Mathematics > Logic |
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| 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 |
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| Date: | October 2018 | |||||||||
| URI: | https://philsci-archive.pitt.edu/id/eprint/15160 |
Available Versions of this Item
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Takeuti's Well-Ordering Proof: Finitistically Fine? (deposited 04 Jan 2018 23:55)
- Takeuti's Well-Ordering Proof: Finitistically Fine? (deposited 17 Oct 2018 15:58) [Currently Displayed]
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