In this paper the existence of natural models for a paraconsistent version of naive set theory is discussed. These stand apart from the previous attempts due to the presence of some non-monotonic ingredients in the comprehension scheme they fulfill. Particularly, it is proved here that allowing the equality relation in formulae defining sets, within an extensional universe, compels the use of nonmonotonic operators. By reviewing the preceding attempts, we show how our models can naturally be obtained as fixed points of some functor acting on a suitable category (stressing the use of fixed-point arguments in obtaining such alternative semantics).

It is proved in this paper that the positive abstraction scheme is consistent with extensionality only if one drops equality out of the language. The theory obtained is then compared with GPK, a well-known set theory based on an extended positive comprehension scheme.

In this paper, we present set theories based upon the paraconsistent logic P ac. 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.

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