Separating Maximality Principles

TL;DR

This paper analyzes the consistency and implications of various fragments of Maximality Principles in set theory, establishing their relative strength and separating different principles using large cardinal assumptions.

Contribution

It determines the consistency strength of specific Maximality Principles and separates their logical implications, advancing understanding of their hierarchy and interactions.

Findings

Consistency strength of $oldsymbol{ m ext{MP}}(oldsymbol{ m extbf{R}})$ fragments established.

No provable implication between $oldsymbol{ m ext{MP}}(oldsymbol{ m extbf{R}})$ variants in ZFC.

Separation of Maximality Principles for stationary set preserving posets from $ ext{MM}^{++}$ under large cardinal assumptions.

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.