Marginalia to a Theorem of Asper\'o and Schindler

TL;DR

This paper provides a game-theoretic framework to characterize when certain models of infinitary propositional formulas can be extended by specific types of forcing posets, and uses this to reprove a significant theorem linking $ ext{MM}^{++}$ to Woodin's axiom $(*)$.

Contribution

It introduces a novel game-theoretic approach to analyze models of infinitary propositional formulas and offers a new proof of Aspero and Schindler's theorem.

Findings

Characterization of models extendable by proper, semiproper, and stationary-set-preserving posets.

A general sufficient condition for the existence of such posets.

A new proof that $ ext{MM}^{++}$ implies Woodin's axiom $(*)$.

Abstract

We give a game-theoretic characterization of when a model of an infinitary propositional formula can be added by a proper, semiproper, and stationary-set-preserving poset. In the latter case, we also give a general sufficient condition for the existence of such a poset. We use this condition to give a somewhat different proof of the theorem of Asper\'o and Schindler, which states that $\mathsf{MM}^{++}$ implies Woodin's axiom $(*)$.