Paraconsistent models of Zermelo-Fraenkel set theory

TL;DR

This paper develops Fidel-structures valued models for ZF set theory, demonstrating that all axioms hold in these models by leveraging paraconsistent models of Leibniz law, thus extending set theory into a paraconsistent framework.

Contribution

It introduces Fidel-structures valued models for ZF set theory and proves their validity, highlighting the role of paraconsistent Leibniz law models in this construction.

Findings

All ZF axioms are valid in Fidel-structures valued models.

Existence of paraconsistent Leibniz law models is crucial for modeling ZF.

Discussion on algebraic paraconsistent models of negation law.

Abstract

In this paper, we build Fidel-structures valued models following the methodology developed for Heyting-valued models; recall that Fidel structures are not algebras in the universal algebra sense. Taking models that verify Leibniz law, we are able to prove that all set-theoretic axioms of ZF are valid over these models. The proof is strongly based on the existence of paraconsistent models of Leibniz law. In this setting, the difficulty of having algebraic paraconsistent models of law for formulas with negation using the standard interpretation map is discussed, showing that the existence of models of Leibniz law is essential to getting models for ZF.