Consistency Strengths of Modified Maximality Principles

The Maximality Principle mp is a scheme which states that if a sentence of the language of zfc is true in some forcing extension V , and remains true in any further forcing extension of V , then it is true in all forcing extensions of V . A modified maximality principle mpΓ arises when considering forcing with a particular class Γ of forcing notions. A parametrized form of such a principle, mpΓ(X), considers formulas taking parameters; to avoid inconsistency such parameters must be restricted to a specific set X which depends on the forcing class Γ being considered. A stronger necessary form of such a principle, 2mpΓ(X), occurs when it continues to be true in all Γ forcing extensions. This study uses iterated forcing, modal logic, and other techniques to establish consistency strengths for various modified maximality principles restricted to various forcing classes, including ccc, cohen, coll (the forcing notions that collapse ordinals to ω), < κ directed closed forcing notions, etc., both with and without parameter sets. Necessary forms of these principles are also considered.