Boolean valued models of set theory

Abstract

Given a first-order language L, a Boolean-valued model (BVM) for Lis a gener- alization of the classical notion of a Tarskian L-structure, in which formulas have truth values ranging over a Boolean algebra. In this work, we provide a short and self-contained presentation of the theory of BVMs. After that, we specialize on Boolean valued models of (the language of) set theory. These turn out to be a natural setting to develop the technique of forcing, introduced by Paul Cohen to prove the celebrated independence of the Continuum Hypothesis. In the final part of this work, we apply this framework to the study of the current literature in inner model theory, showing that the Boolean valued framework is particularly well-suited to produce general models into which any model of a sufficiently rich fragment of the theory of an initial segment of the universe can be embedded.