A talk at the conference on Set Theory and Higher-Order Logic: Foundational Issues and Mathematical Developments, following the summer school, at the University of London, Birkbeck, July 1-6, 2011.
What are the most general principles relating forceability and truth? As in Solovay’s celebrated analysis of provability, this question and its answer are naturally formulated in modal logic. Specifically, we dene that a set theoretic assertion phi is forceable or possible if it holds in some forcing extension, and necessary if it holds in all forcing extensions. Under this forcing interpretation, the provably valid principles of forcing are exactly those in the modal theory known as S4.2. In this talk, I shall discuss this result and recent advances in this area. This is joint work with Benedikt Loewe.