Beneš’s Partial Model of NF: An Old Result Revisited

Abstract

A paper by Beneš, published in 1954, was an attempt to prove the consistency of $\mathsf{NF}$ (Quine’s set theory “New Foundations”) via a partial model of Hailperin’s finite axiomatization of $\mathsf{NF}$. Here, I offer an analysis of Beneš’s proof in a De Giorgi-style setting for set theory. This approach leads to an abstract version of Beneš’s theorem that emphasizes the monotone and invariant content of the axioms proved to be consistent, in a sense of monotony and invariance that this paper intends to state rigorously and to help clarify. Moreover, some tentative speculation will be made about possible developments of the topic in the following two directions: (1) which set theories can be proved to be consistent via Beneš-like constructions, and (2) how can we elaborate on Beneš’s model to get a consistency proof for full $\mathsf{NF}$?