On non-classical models of ZFC

In this talk we present the recent developments in the study of non-classical models of ZFC. We will show that there are algebras that are neither Boolean, nor Heyting, but that still give rise to models of ZFC. This result is obtained by using an algebra-valued construction similar to that of the Boolean-valued models. We also show that by a suitable modification of the interpretation of equality and belongingness, we can significantly extend the class of algebra-valued models of ZFC. We also present an application of these constructions, showing the independence of CH from non-classical set theories.