On a class of maximality principles

Abstract

We study various classes of maximality principles, $\rm{MP}(\kappa,\Gamma)$, introduced by J.D. Hamkins, where $\Gamma$ defines a class of forcing posets and $\kappa$ is a cardinal. We explore the consistency strength and the relationship of $\textsf{MP}(\kappa,\Gamma)$ with various forcing axioms when $\kappa \in\{\omega,\omega_1\}$. In particular, we give a characterization of bounded forcing axioms for a class of forcings $\Gamma$ in terms of maximality principles MP$(\omega_1,\Gamma)$ for $\Sigma_1$ formulas. A significant part of the paper is devoted to studying the principle MP$(\kappa,\Gamma)$ where $\kappa\in\{\omega,\omega_1\}$ and $\Gamma$ defines the class of stationary set preserving forcings. We show that MP$(\kappa,\Gamma)$ has high consistency strength; on the other hand, if $\Gamma$ defines the class of proper forcings or semi-proper forcings, then by Hamkins, it is shown that MP$(\kappa,\Gamma)$ is consistent relative to $V=L$.