On a class of maximality principles

TL;DR

This paper investigates various maximality principles related to forcing axioms, characterizing their consistency strength and relationships, especially focusing on stationary set preserving, proper, and semi-proper forcings.

Contribution

It provides a characterization of bounded forcing axioms via maximality principles and analyzes the consistency strength of MP principles for different classes of forcings.

Findings

MP(κ,Γ) has high consistency strength for stationary set preserving forcings.

MP(κ,Γ) is consistent relative to V=L for proper and semi-proper forcings.

Characterization of bounded forcing axioms in terms of MP(ω_1,Γ) for Σ_1 formulas.

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 $V=L$.