 The belief that Ackermann and von Neumann had established the consistency of elementary arithmetic was expressed as late as December 1930 by Hilbert in his third Hamburg talk (Hilbert 1931a) and by Bernays in April 1931 in a letter to Gödel (see Gödel 2003: 98–103). Bernays asserts there that he has “repeatedly considered [Ackermann’s modified proof] and viewed it as correct”. He continues, referring to Gödel’s incompleteness results,