The *-variation of the Banach-Mazur game and forcing axioms

TL;DR

This paper introduces a new property of posets based on a variation of the Banach-Mazur game, demonstrating its preservation of PFA under forcing and exploring its relation to other closure properties.

Contribution

It defines a strengthened strategic closedness property for posets using a modified Banach-Mazur game and shows its implications for forcing axioms and set-theoretic principles.

Findings

PFA is preserved under forcing with posets having this new property.

The property differs significantly from (+1)-operational closedness.

Reproduces Magidor's theorem on PFA and square principles.

Abstract

We introduce a property of posets which strengthens (\omega_1+1)-strategic closedness. This property is defined using a variation of the Banach-Mazur game on posets, where the first player chooses a countable set of conditions instead of a single condition at each turn. We prove PFA is preserved under any forcing over a poset with this property. As an application we reproduce a proof of Magidor's theorem about the consistency of PFA with some weak variations of the square principles. We also argue how different this property is from (\omega_1+1)-operational closedness, which we introduced in our previous work, by observing which portions of MA^+(\omega_1-closed) are preserved or destroyed under forcing over posets with either property.