Models and independence in non-classical ZF.

In this talk we show that the boolean-valued construction of models of set theory can be applied to algebras whose underlying logic is non-classical. We start by presenting the first results in this area by Lowe and Tarafder (2015) and  some recent extensions of their  work. We then show that there are algebra-valued models of ZF, whose underlying logic is neither classical, nor intuitionistic. In the end we will show how to extend independence results for classical ZF to a non-classical setting, showing, as an example, the independence of the Continuum Hypothesis.