Gentzen’s Original Consistency Proof and the Bar Theorem

Abstract

The story of Gentzen’s original consistency proof for first-order number theory [9], as told by Paul Bernays [1, 9], [11, 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’.