Leibniz's law and paraconsistent models of ZFC

TL;DR

This paper constructs full models for paraconsistent set theories based on Fidel semantics, extending classical ZFC models to handle inconsistency without trivialization.

Contribution

It introduces a method to build paraconsistent models of ZFC using Fidel semantics, adapting classical proof techniques to paraconsistent contexts.

Findings

Models are built over Fidel semantics for PSTs.

Paraconsistent models of ZFC are constructed.

Proof techniques from classical set theory are adapted.

Abstract

In this paper, we present full models for some Paraconsistent Set Theories (PSTs). These models are built over Fidel semantics where they are specific first-order structures in the sense of Model Theory. These structures are known as F-structures in the literature and they are not algebras in the universal algebra sense. We demonstrate how is possible to present paraconsistent models for ZFC for any of PSTs studied in this paper, by adapting the proofs given on the celebrated John Lane Bell's books; in general, we adapt the proofs in the mentioned book throughout the work.