In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics the Cshpm 2017 Annual Meeting in Toronto, Ontario. Birkhäuser. pp. 167-180 (2018)

Aaron Thomas-Bolduc
University of Calgary
If 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.
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.1007/978-3-319-90983-7_11
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

Our Archive

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 50,488
External links

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library

References found in this work BETA

No references found.

Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles

Takeuti’s Well-Ordering Proof: Finitistically Fine?Eamon Darnell & Aaron Thomas-Bolduc - 2018 - In Amy Ackerberg-Hastings, Marion W. Alexander, Zoe Ashton, Christopher Baltus, Phil Bériault, Daniel J. Curtin, Eamon Darnell, Craig Fraser, Roger Godard, William W. Hackborn, Duncan J. Melville, Valérie Lynn Therrien, Aaron Thomas-Bolduc & R. S. D. Thomas (eds.), Research in History and Philosophy of Mathematics: The Cshpm 2017 Annual Meeting in Toronto, Ontario. Springer Verlag. pp. 167-180.
Takeuti's Proof Theory in the Context of the Kyoto School.Andrew Arana - 2019 - Jahrbuch Für Philosophie Das Tetsugaku-Ronso 46:1-17.
Consistency Proof Via Pointwise Induction.Toshiyasu Arai - 1998 - Archive for Mathematical Logic 37 (3):149-165.
Proof Theory for Theories of Ordinals—I: Recursively Mahlo Ordinals.Toshiyasu Arai - 2003 - Annals of Pure and Applied Logic 122 (1-3):1-85.
Proof Theory and Set Theory.Gaisi Takeuti - 1985 - Synthese 62 (2):255 - 263.
Decoding Gentzen's Notation.Luca Bellotti - 2018 - History and Philosophy of Logic 39 (3):270-288.
Determinate Logic and the Axiom of Choice.J. P. Aguilera - 2020 - Annals of Pure and Applied Logic 171 (2):102745.
[Omnibus Review].Dag Prawitz - 1991 - Journal of Symbolic Logic 56 (3):1094-1096.
Omnibus Review. [REVIEW]Dag Prawitz - 1991 - Journal of Symbolic Logic 56 (3):1094-1096.
Sur la théorie des démonstrations.Y. Gauthier - 1981 - Philosophiques 8 (2):273-285.


Added to PP index

Total views

Recent downloads (6 months)

How can I increase my downloads?


Sorry, there are not enough data points to plot this chart.

My notes