  It was unclear to Gödel and Bernays what the proof assumed; they thought it assumed what is known in intuitionistic mathematics as the fan theorem, but this was false. Termination of Gentzen's reduction procedure can be proved instead by induction on well-founded trees (“bar-induction”), a principle that was used by Gentzen on intuitive grounds. Anyway, the result of the criticism was that Gentzen changed without further ado the proof into a third proof that uses the now famous principle of transfinite induction up to the first epsilon-number. This induction was presented through a coding that used decimal numbers. The concrete result of the changes for Gentzen's paper published in 1936 was not good, however: the logical calculus was changed midway in an article of seventy-odd pages that became very difficult to read. Therefore Gentzen gave yet another, by the present count fourth proof of the consistency of arithmetic in 1938 (at the Bernays archives of the ETH Zurich), this time based on the classical sequent calculus LK of 1933.  As mentioned, correspondence with Bernays indicates that he thereby returned to the proof method that had led to success in 1934. The use of transfinite induction is made clearly visible in the 1938 paper through an ordinal notation. Such induction principles on Cantor's “second number class” are discussed in detail in Hilbert's 1925 lecture “Über das Unendliche” (“On the infinite”, published 1926), a paper to which Gentzen referred.