Takeuti’s Well-Ordering Proof: Finitistically Fine?

  • Eamon DarnellEmail author
  • Aaron Thomas-Bolduc
Conference paper
Part of the Proceedings of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques book series (PCSHPM)


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.



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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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of TorontoTorontoCanada
  2. 2.University of CalgaryCalgaryCanada

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