On a class of maximality principles

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 \(V=L\).