On a Class of Maximality Principles

We study various classes of maximality principles, MP(κ,Γ), introduced by J.D. Hamkins in [6], where Γ defines a class of forcing posets and κ is an infinite cardinal. We explore the consistency strength and the relationship of MP(κ,Γ) with various forcing axioms when κ ∈ {ω, ω1}. In particular, we give a characterization of bounded forcing axioms for a class of forcings Γ in terms of maximality principles MP(ω1,Γ) for Σ1 formulas. A significant part of the paper is devoted to studying the principle MP(κ,Γ) where κ ∈ {ω, ω1} and Γ defines the class of stationary set preserving forcings. We show that MP(κ,Γ) has high consistency strength; on the other hand, if Γ defines the class of proper forcings or semi-proper forcings, then by [6], MP(κ,Γ) is consistent relative to V = L.