Von Neumann’s Consistency Proof

Abstract

We consider the consistency proof for a weak fragment of arithmetic published by von Neumann in 1927. This proof is rather neglected in the literature on the history of consistency proofs in the Hilbert school. We explain von Neumann’s proof and argue that it fills a gap between Hilbert’s consistency proofs for the so-called elementary calculus of free variables with a successor and a predecessor function and Ackermann’s consistency proof for second-order primitive recursive arithmetic. In particular, von Neumann’s proof is the first rigorous proof of the consistency of an axiomatization of the first-order theory of a successor function.

Some Unpublished Letters by Gödel and von Neumann in the Fraenkel Archive

“…11 Here von Neumann presumably has in mind Hilbert's consistency proof of logic in the ε-calculus inHilbert (1923). SeeZach (2003Zach ( , 2004) on Hilbert's consistency proofs and the relation to vonNeumann (1927), andBellotti (2016) on von Neumann's consistency proof specifically.…”