Non-classical Models of ZF

Abstract

This paper contributes to the generalization of lattice-valued models of set theory to non-classical contexts. First, we show that there are infinitely many complete bounded distributive lattices, which are neither Boolean nor Heyting algebra, but are able to validate the negation-free fragment of $$\mathsf {ZF}$$ . Then, we build lattice-valued models of full $$\mathsf {ZF}$$ , whose internal logic is weaker than intuitionistic logic. We conclude by using these models to give an independence proof of the Foundation axiom from $$\mathsf {ZF}$$ .