Studies in the Structural Theory of Forcing

Abstract

This thesis can be viewed as a "meta-theoretical" study of certain issues of forcing and some of the applications arising from this structural study. Chapter 1 presents our notation, terminology and some facts needed in later Chapters. Chapter 2 analyzes the notion of genericity versus that of pre-genericity and shows why the latter does not suffice to capture the intended goal. Chapter 3 gives a simplified definition of the forcing relation and shows how the fundamental theorems of forcing effortlessly follow from that definition. Chapter 4 re-addresses the issue of the Boolean completion of a preorder from the viewpoint of "identifying the discernibles". The prototype case of ≤* is systematically studied, thereby preparing for its applications in later Chapters. Chapter 5 presents a conceptual analysis of iterated forcing, and thus provides a precise technical argument why a forcing iteration exactly corresponds to an ascending sequence of complete Boolean algebras. Chapter 6 characterizes the situation of MP ⊆ MQ, primarily by the "witness mappings" and gives applications of this characterization. Characterizations of P ~ Q, P ≤ Q are also discussed. Chapter 7 gives more applications of the techniques and results in previous Chapters in giving forcing proofs of forcing-free facts like Kripke's embedding theorem and Tarski's generalized Bernstein-Schroder theorem.