Another proof that $\mathsf{MM}^{++}$ implies Woodin's axiom $(*)$

Abstract

Let $\mathsf{MM}^{++}(\kappa)$ state that the forcing axiom $\mathsf{MM}^{++}$ can be instantiated only for stationary set preserving posets of size at most $\kappa$. We give a detailed account of Asper\`o and Schindler's proof that $\mathsf{MM}^{++}(\kappa)+$there are class many Woodin cardinals implies Woodin's axiom $(*)$ if $\Diamond_\kappa$ holds and $\kappa>\aleph_2$. Our presentation takes advantage of the notion of consistency property: specifically we rephrase Asper\`o and Schindler's forcing as a specific instantiation of the notion of ``consistency property'' used by Makkai, Keisler, Mansfield and others in the study of infinitary logics. We also reorganize the order of presentation of the various parts of the proof. Taken aside these variations, our account is quite close to the original proof of Asper\`o and Schindler.