Research on Set Theory Based on Paraconsistent Logic

Shi Jing

doi:10.11648/j.ijp.20200802.13

Abstract

Different from ZF axiomatic set theory, the paraconsistent set theory has changed the basic logic of set theory and selected paraconsistent logic which can accommodate or deal with contradictions, it effectively avoids the whole theory falling into a non-trivial dilemma when there are contradictions in set theory. In this paper, we first review the history and current situation of the praconsistent set theory; then, we give three kinds of paraconsistent logic which can be used to construct the praconsistent set theory among many kinds of paraconsistent logics. And then, we analyze the differences of methods of the paraconsistent set theory with strong or weak structure of paraconsistent logic and get different paraconsistent set theory. Finally, we verify that paraconsistent set theory is a new method to solve the paradox of set theor. The development of paraconsistent set theory can solve the difficulties in the development of set theory in a unique way, which is not only the extension of the application of paraconsistent logic, but also the new form and new trend of the development of set theory.

Highlights

Cantor established the naive set theory with the principles of comprehension and extension
We first review the history and research status of the paraconsistent set theory; we give three kinds of paraconsistent logics which can be used to construct the paraconsistent set theory; we analyze the different characteristics of the paraconsistent set theory constructed by strong or weak paraconsistent logic; we verify that the paraconsistent set theory is a new method to solve the paradox of the set theory
Naive set theory is the foundation of modern mathematics and an important branch of modern logic

Summary

Introduction

Cantor established the naive set theory with the principles of comprehension and extension. The establishment of Cantor’s naive set theory (hereafter referred to as the set theory) has brought new vitality to mathematics. It has produced two fields of modern mathematics: meta-mathematics and structural mathematics, which are completely different from the classical mathematics. ZF axiomatic set theory is a formal system formed by adding non-logical axioms about the basic properties of sets to the classic logic with “=” and “∈”. In the classical axiomatic set theory system ZFC, subset axiom is a kind of restriction to the comprehension principle of the set theory, which excludes Russell paradox in the set theory. We first review the history and research status of the paraconsistent set theory; we give three kinds of paraconsistent logics which can be used to construct the paraconsistent set theory; we analyze the different characteristics of the paraconsistent set theory constructed by strong or weak paraconsistent logic; we verify that the paraconsistent set theory is a new method to solve the paradox of the set theory

The History and Research Status of Paraconsistent Set Theory

Three Kinds of Logics That Can Be Used to Construct Paraconsistent Set Theory

The Strength and Weakness of Paraconsistent Logic

A New Solution to the Paradox of Set Theory Pradox

Conclusion