Paraconsistent set theory by predicating on consistency

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. 1 A new look at antinomic sets Ever since the discovery of the paradoxes, the history of contemporary set theory has centered around attempts to rescue Cantor’s naive theory from triviality, traditionally by placing the blame on the Principle of (unrestricted) Abstraction (or Principle of Comprehension). Unrestricted abstraction (which allows sets to be defined by arbitrary conditions) plus the axiom of extensionality, and plus the laws of the underlying logic where the theory of sets is expressed, leads to a contradiction when a weird collection such as the Russell set or similar constructions are defined. The problem is not the weird collections by themselves – set-theorists are used to strange objects like large cardinals, measurable cardinals and the like, and, in fact, hypothesizing on large cardinals enables us to investigate the capabilities of possible extensions of ZFC. The problem is that some weird sets, such as Russell’s, entail a contradiction, and in classical logic a contradiction entails everything. One way of escaping this mathematical Armagedon is to consider weaker forms of separation, by patching in the Principle of Comprehension, but this