On a class of maximality principles

Abstract

We study various classes of maximality principles, $\mathrm {MP}(\kappa ,\Gamma )$, introduced by Hamkins (J Symb Log 68(2):527–550, 2003), where $\Gamma$ defines a class of forcing posets and $\kappa$ is an infinite 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 (2003), MP$(\kappa ,\Gamma )$ is consistent relative to $V=L$.