Paradefinite Zermelo-Fraenkel Set Theory: A Theory of Inconsistent and Incomplete Sets
Hrafn Valtýr Oddsson

Abstract:
A paradefinite logic is a logic that is both paraconsistent and paracomplete. In this thesis, we present a set theory in a four-valued paradefinite logic that can be viewed as the result of enriching the standard von Neumann universe for ZF C with various non-classical sets.
Our approach differs from most previous attempts at paraconsistent or paracomplete set theory in that we do not chase increasingly general comprehension principles. Rather, we prioritise an intuitive treatment of non-classical sets so as to make our set theory accessible to the classical mathematician who is used to working in classical ZF C. Moreover, as we work in a paradefinite logic, we provide a unified account of paraconsistent and paracomplete set theory.
We provide a natural model of our set theory starting from classical ZF C. We also show that within our theory, we can construct a class that acts as a model of classical ZF C. This allows us to translate back and forth between our theory and classical ZF C. Finally, we will generalize the construction of Boolean-valued models for classical set theory to obtain algebra-valued models of our theory.