Santiago Jockwich Martinez

This paper advances the study of so-called “iterative” paraconsistent set theories. Unlike a naive set theory, which validates both Unrestricted Comprehension and Extensionality, an iterative paraconsistent set theory uphold the axioms of Zermelo–Fraenkel set theory. The principal advantage of the iterative approach is that it yields set theories that are highly mathematically expressive. It has recently been conjectured that the mathematical expressiveness of certain iterative paraconsistent set theories may even rival that of classical set theory. However, until now the status of the Axiom of Choice in this setting has remained open. We show that iterative paraconsistent set theories are compatible with the Axiom of Choice. Finally, in the conclusion, we contrast this result with the failure of Choice in the intuitionistic setting.

In this paper, we explore the possibility of constructing algebra-valued models of set theory based on Priest's Logic of Paradox. We show that we can build a non-classical model of ZFC which has as internal logic Priest's Logic of Paradox and validates Leibniz's law of indiscernibility of identicals. This is achieved by modifying the interpretation map for $\in$ and $=$ in our algebra-valued model. We end by comparing our model constructions to Priest's model-theoretic strategy and point out that we have a tradeoff between desirable model-theoretic properties and the validity of ZFC and its theorems.