
Publication details
Publisher: Birkhäuser
Place: Basel
Year: 2018
Pages: 167-180
ISBN (Hardback): 9783319908557
Full citation:
, "Takeuti's well-ordering proof", in: Research in history and philosophy of mathematics, Basel, Birkhäuser, 2018


Takeuti's well-ordering proof
finitistically fine?
pp. 167-180
in: Maria Zack, Dirk Schlimm (eds), Research in history and philosophy of mathematics, Basel, Birkhäuser, 2018Abstract
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.
Publication details
Publisher: Birkhäuser
Place: Basel
Year: 2018
Pages: 167-180
ISBN (Hardback): 9783319908557
Full citation:
, "Takeuti's well-ordering proof", in: Research in history and philosophy of mathematics, Basel, Birkhäuser, 2018