Algebra-valued models for LP-set theory

Abstract

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.