• boolean valued models
• forcing axioms
• saturation

This dissertation will focus on Boolean-valued models, giving some insight into the theory of Boolean ultrapowers, and developing the connection with forcing axioms and absoluteness results. This study will be divided into three chapters.

The first chapter provides the basic material to understand the subsequent work.

Boolean-valued models are well known in set theory for independence results and the development of forcing. In the second chapter of this dissertation, Boolean-valued models are studied from a general point of view. In particular, we give the definition of B-valued model for an arbitrary first-order signature, and we study Boolean ultrapowers as a general model-theoretic technique.

A more ambitious third chapter develops the connection with forcing axioms and absoluteness results. From a philosophical point of view, forcing axioms are very appealing. Not only do they imply that the Continuum Hypothesis is false, but also they are particularly successful in deciding many independent statements in mathematics. First, we give an interesting formulation of bounded forcing axioms in terms of absoluteness. Furthermore, we prove that the Axiom of Choice is a “global” forcing axiom, and that also some large cardinal axioms are in fact natural generalizations of forcing axioms.