Remarks on da costa's paraconsistent set theories

Abstract

In this paper we analyse da Costa's paraconsistent set theories, i.e., the set theories constructed over da Costa's paraconsistent logics C=n, 1 ≤ n ≤ ω. The main results presented here are the following. In any da Costa paraconsistent set theory of type NF the axiom schema of abstraction must be formulated exactly as in NF; for, in the contrary, some paradoxes are derivable that invalidate the theory. In any da Costa paraconsistent set theory with Russell's set [Formula Matemática] UUR is the universal set. In any da Costa paraconsistent set theory the existence of Russell's set is incompatible with a general (for all sets) formulation of the axiom schemata of separation and replacement.