Separating Maximality Principles

Abstract

We investigate fragments of generic absoluteness principles known as Maximality Principles. We determine the consistency strength of $\Sigma_n$-$\mathsf{MP}(\mathbb R)$ and $\Pi_n$-$\mathsf{MP}(\mathbb R)$, the boldface Maximality Principle restricted respectively to $\Sigma_n$- and $\Pi_n$-formulas. Further, we show that no implication between $\Sigma_n$-$\mathsf{MP}(\mathbb R)$ and $\Pi_n$-$\mathsf{MP}(\mathbb R)$ is provable in $\mathsf{ZFC}$. We also establish the consistency, relative to a Woodin cardinal, of the Maximality Principle for $\omega_1$-preserving posets with countable ordinal parameters and prove its consistency strength is bounded below by a Ramsey cardinal. Finally, we resolve questions of Ikegami-Trang and Goodman by separating the Maximality Principle for stationary set preserving posets restricted to $\Sigma_2$-formulas from $\mathsf{MM}^{++}$ in the presence of large cardinals.