ZF and the axiom of choice in some paraconsistent set theories

In this paper, we present set theories based upon the paraconsistent logic Pac. We describe two different techniques to construct models of such set theories. The first of these is an adaptation of one used to construct classical models of positive comprehension. The properties of the models obtained in that way give rise to a natural paraconsistent set theory which is presented here. The status of the axiom of choice in that theory is also discussed. The second leads to show that any classical universe of set theory (e.g. a model of ZF) can be extended to a paraconsistent one, via a term model construction using an adapted bisimulation technique.