FOM: FoM: Zermelo and Gentzen/Infinitary proofs

[A.P. Hazen]

  The reference to Gentzen is (obviously) to his use of transfinite
induction to prove the  consistency of elementary number theory.  The
legitimacy of this as  a proof had been a topic of philosophical
controversy fromthe beginning: cf. Tarski's quip that Gentzen  had
increased his confidence in the consistency of elementaryt number theory
"by an epsilon".
  One early PUBLISHED discussion is in a review article by Max Black in
"Mind" vol. 49 (1940), pp. 239-248.  Another, which I have  not read but
which is mentioned in Kleene's "Introduction to Metamathematics"
(discussion at end of sec. 81) is Bernays's "Sur les questions
méthodologiques actuelles de la théorie hilbertienne de la démonstration,"
in Gonseth, ed., "Les entretiens de Zurich sur les fondements et la méthode
des sciences  mathématiques" (1938). The discussion was  not all in print,
of course: I first heard of the issue (when I was a student in the 1960s)
in conversation, and got the impression it was part of the folklore of the
field.
---
Allen Hazen
Philosophy Department
U. of Melbourne