Paraconsistent set theory by predicating on consistency

Abstract

This paper intends to contribute to the debate about the uses of paraconsistent reasoning in the foundations of set theory, by means of employing the logics of formal inconsistency (LFIs) and by considering consistent and inconsistent sentences, as well as consistent and inconsistent sets. We establish the basis for new paraconsistent set-theories (such as ZFmbC and ZFCil) under this perspective and establish their non-triviality, provided that ZF is consistent. By recalling how George Cantor himself, in his efforts towards founding set theory more than a century ago, not only used a form of 'inconsistent sets' in his mathematical reasoning, but regarded contradictions as beneficial, we argue that Cantor's handling of inconsistent collections can be related to ours.

Paraconsistent Set Theory

“…3 On the other hand, a theory in a logic L is trivial if ϕ for every formula ϕ; otherwise, it is non-trivial. In case ¬ is explosive in L , then is inconsistent if and only if it is trivial.…”

“…It is to be noted that Cantor had already proved by 1891 his now famous "Cantor's Theorem", which shows that the cardinal number of the set P(S) of all subsets of a given set S is different to the cardinal number of S. Cantor's argument shows directly that, if the universe is a set, Russell's paradox obtains (see, for instance, [2] for a short proof). This chapter is based on the ideas of the paper [3] and in the references therein.…”

Linear models and set theory

“…In the next pages we will show that the set theories developed in [5] are not valid in any linear model. This will therefore motivate the rejection of these theories.…”

“…Consequently, a Zermelian approach proposes axiomatic systems as close as possible to classical ZFC, but where the logical connective receive, instead, a paraconsistent interpretation. Example of these theories may be found in [8], [9], and [5]. Although both approaches have their pros and cons, we consider the Zermelian approach closer to set-theoretical practice.…”

“…In this paper we investigate wether linear models are suitable structures for the set theories developed in [5]. The reason for asking this question is twofold.…”