Abstract
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.
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Notes
- 1.
- 2.
See Zach (2006) for a thorough introduction to Hilbert’s program.
- 3.
Primitive recursive arithmetic contains the usual recursive definitions of 0, +, ×, and successor, as well as all other primitive recursive functions, and the quantifier free induction schema.
- 4.
See Sieg (2009) for an interesting look at Kant’s (and other preceding figures’) influence on Hilbert.
- 5.
Quoted in Zach (1998, Fn. 16).
- 6.
The proof-theoretic ordinal of PRA is ω ω, so Ackermann certainly went beyond PRA.
- 7.
An ordinal is accessible, roughly, if it can be reached from below. See Section 6. Compare to the concept of an inaccessible cardinal for which there is a strong sense in which such cardinals cannot be reached from below.
- 8.
It is likely much of what follows will apply equally well to the 1936 proof given a finitistically acceptable translation between the ordinal notation systems.
- 9.
Gentzen includes only “1,” but Takeuti makes use of this obvious notational extension so we have included it here for completeness.
- 10.
Note that it may be that β i = β i+1 = … = β i+n for some i, n > 0.
- 11.
- 12.
A regular proof in one in which all of the non-eigen variables have been replaced with 0s and the eigenvariables have been replaced with appropriate arithmetic terms.
- 13.
Likewise for other inferences, though those cases are more simple.
- 14.
We plan to publish a full reconstruction of Takeuti’s well-ordering proof in the near future.
- 15.
Takeuti takes this assumption to be uncontroversial because he sees it as an obvious consequence of his definitions of ordinals and the relations: “= ”, “+ ” and “< ” on the ordinals (1987 pp. 90–91).
- 16.
All content in this section is from or adapted from: Takeuti (1987, p. 93).
- 17.
All subsequent eliminators (and their associated terminology) are analogous to the 1-eliminator.
- 18.
That is the last term with a 1-major part.
- 19.
The content in this section is from or adapted from Takeuti (1987, p. 93).
- 20.
Where (C1) is amended such that S 0 is changed to S 1 and \(S^\prime _0\) is changed to \(S^\prime _1\).
- 21.
Where (C2) is an appropriate analogue of (C1).
- 22.
But see Incurvati (2005)
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Acknowledgements
Special thanks to Richard Zach who inspired our interest in this topic, and has provided invaluable comments on earlier drafts. Thanks as well to audiences in Philadelphia and Toronto.
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Darnell, E., Thomas-Bolduc, A. (2018). Takeuti’s Well-Ordering Proof: Finitistically Fine?. In: Zack, M., Schlimm, D. (eds) Research in History and Philosophy of Mathematics. Proceedings of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-90983-7_11
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