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Leibniz's law and paraconsistent models of ZFC

Aldo Figallo-Orellano

TL;DR

It is demonstrated 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.

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.

Leibniz's law and paraconsistent models of ZFC

TL;DR

It is demonstrated 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.

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.
Paper Structure (13 sections, 12 equations)

This paper contains 13 sections, 12 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $F$-structures for N4
  4. $F$-structures for $C_\omega$ and $C^2_\omega$
  5. Zermelo-Fraenkel Set Theory: Boolean and Heyting valued models
  6. Some properties of Leibniz's law
  7. Validating axioms
  8. Nelson's Set Theories
  9. $ZF_{\bf N3}$ Set Theory
  10. Paraconsistent empty set
  11. $ZFC^2_\omega$ and $ZFC_\omega$ Set Theories
  12. Paraconsistent models
  13. Conclusions and future works