Boolean-valued class forcing

Abstract

We show that the Boolean algebras approach to class forcing can be carried out in the theory Kelley-Morse plus the Choice Scheme (KM + CC) using hyperclass Boolean completions of class partial orders. Applying the Boolean algebras approach, we show that every intermediate model between a model V |= KM + CC and one of its class forcing extensions is itself a class forcing extension if and only if it is simple generated by the classes of V together with a single new class. We show that a model of KM + CC and its class forcing extension may have non-simple intermediate models, and thus, the Intermediate Model Theorem can fail for models of KM + CC.