On a class of maximality principles

We study various classes of maximality principles, $\mathrm{M}\mathrm{P}(\kappa ,\mathrm{\Gamma })$, introduced by J.D. Hamkins, where $\mathrm{\Gamma }$ defines a class of forcing posets and $\kappa$ is a cardinal. We explore the consistency strength and the relationship of $\mathsf{\text{MP}}(\kappa ,\mathrm{\Gamma })$ 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 $\mathrm{\Gamma }$ in terms of maximality principles MP$({\omega }_1,\mathrm{\Gamma })$ for ${\mathrm{\Sigma }}_1$ formulas. A significant part of the paper is devoted to studying the principle MP$(\kappa ,\mathrm{\Gamma })$ where $\kappa \in \{\omega ,{\omega }_1\}$ and $\mathrm{\Gamma }$ defines the class of stationary set preserving forcings. We show that MP$(\kappa ,\mathrm{\Gamma })$ has high consistency strength; on the other hand, if $\mathrm{\Gamma }$ defines the class of proper forcings or semi-proper forcings, then by Hamkins, it is shown that MP$(\kappa ,\mathrm{\Gamma })$ is consistent relative to $V=L$.