Gentzen’s original consistency proof and the Bar Theorem

The story of Gentzen’s original consistency proof for first-order number theory (Gentzen 1974), as told by Paul Bernays (Gentzen 1974), (Bernays 1970), (Gödel 2003, Letter 69, pp. 76-79), is now familiar: Gentzen sent it off to Mathematische Annalen in August of 1935 and then withdrew it in December after receiving criticism and, in particular, the criticism that the proof used the Fan Theorem, a criticism that, as the references just cited seem to indicate, Bernays endorsed or initiated at the time but later rejected. That particular criticism is transparently false, but the argument of the paper remains nevertheless invalid from a constructive standpoint. In a letter to Bernays dated November 4, 1935, Gentzen protested this evaluation; but then, in another letter to him dated December 11, 1935, he admits that the “critical inference in my consistency proof is defective.” The defect in question involves the application of proof by induction to certain trees, the ‘reduction trees’ for sequents (see below and §1), of which it is only given that they are well-founded. No doubt because of his desire to reason ‘finitistically’, Gentzen nowhere in his paper explicitly speaks of reduction trees, only of reduction rules that would generate such trees; but the requirement of wellfoundedness, that every path taken in accordance with the rule terminates, of course makes implicit reference to the tree. Gentzen attempted to avoid the induction; but as he ultimately conceded, the attempt was unsatisfactory.