Separating Maximality Principles

Abstract

We investigate fragments of generic absoluteness principles known as Maximality Principles. We determine the consistency strength of $\hat{I}{\pounds }_n$-$\mathsf{MP}(\mathbb{R})$ and ${\hat{I}}_n$-$\mathsf{MP}(\mathbb{R})$, the boldface Maximality Principle restricted respectively to $\hat{I}{\pounds }_n$- and ${\hat{I}}_n$-formulas. Further, we show that no implication between $\hat{I}{\pounds }_n$-$\mathsf{MP}(\mathbb{R})$ and ${\hat{I}}_n$-$\mathsf{MP}(\mathbb{R})$ is provable in $\mathsf{ZFC}$. We also establish the consistency, relative to a Woodin cardinal, of the Maximality Principle for $\ddot{I}{‰}_1$-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 $\hat{I}{\pounds }_2$-formulas from ${\mathsf{MM}}^{++}$ in the presence of large cardinals.

29 pages