Bounded Martin's Maximum with Many Witnesses

TL;DR

This paper introduces a strengthened version of Bounded Martin's Maximum, explores its implications for determinacy and its consistency with certain models, revealing nuanced relationships between forcing axioms and determinacy principles.

Contribution

It defines a new strengthening of Bounded Martin's Maximum and analyzes its consistency and implications for global projective determinacy and models like P_max.

Findings

The strengthened principle implies Global Projective Determinacy.

It does not hold in the P_max model for BMM.

A restricted version holds in the BMM model, but is not a consequence of BMM.

Abstract

We study a strengthening of Bounded Martin's Maximum which asserts that if a \Sigma_1 fact holds of \omega_2^V in a stationary set preserving extension then it holds in V for a stationary set of ordinals less than \omega_2. We show that this principle implies Global Projective Determinacy, and therefore does not hold in the \mathbb{P}_{max} model for \mathsf{BMM}, but that the restriction of this principle to forcings which render \omega_2^V countably cofinal does hold in the \mathsf{BMM} model, though it is not a consequence of \mathsf{BMM}.