On the consistency of NF via Fuzzy Forcing

Abstract

In this paper, we present a proof of the consistency of the New Foundations set theory ($\mathit{NF}$). $\mathit{NF}$'s main idea is to permit very large sets (including the Universal Set) by restricting set formation to stratified formulas, thereby avoiding the classic set-theoretic paradoxes. Our proof employs a new forcing method incorporating concepts from fuzzy logic. A brief outline of the proof can be as follows: (1) We extend $ZF$ to Fuzzy $\mathit{ZF}$ with a membership function $\mu$ over $D=\mathbb{Q} \cap [0,1]$; (2) we define Fuzzy $\mathit{NF}$ as $\Sigma$, and (3) we derive a crisp $\mathrm{N}$ model of $NF$. Our proof does not depend on Holmes' Tangled Type Theory ($\mathit{TTT}$). It establishes that if $\mathit{ZF}$ is consistent, then $\mathit{NF}$ is also consistent. It achieves that via the chain $\mathit{ZF} \rightarrow$ Fuzzy $\mathit{ZF} \rightarrow \Sigma \rightarrow \mathit{NF}$. The method presented in this paper offers a novel perspective connecting fuzzy logic with classic set theory.