On a class of maximality principles

摘要: We study various classes of maximality principles, $\rm{MP}(\kappa,\Gamma)$, introduced by J.D. Hamkins, where $\Gamma$ defines a class forcing posets and $\kappa$ is cardinal. explore the consistency strength relationship $\textsf{MP}(\kappa,\Gamma)$ with axioms when $\kappa \in\{\omega,\omega_1\}$. In particular, we give characterization bounded for forcings in terms principles MP$(\omega_1,\Gamma)$ $\Sigma_1$ formulas. A significant part paper devoted to studying principle MP$(\kappa,\Gamma)$ $\kappa\in\{\omega,\omega_1\}$ stationary set preserving forcings. show that has high strength; on other hand, if proper or semi-proper forcings, then it shown consistent relative $V=L$.