Paraconsistent Models of Zermelo-Fraenkel Set Theory

Abstract

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