Marginalia to a Theorem of Asperó and Schindler

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 Asperó and Schindler, which states that ${\mathsf{M}\mathsf{M}}^{++}$ implies Woodin’s axiom $(\ast )$.