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

TL;DR

This paper provides a detailed account and reorganization of Asperò and Schindler's proof that certain forcing axioms imply Woodin's axiom (*) under specific large cardinal and combinatorial assumptions.

Contribution

It reformulates Asperò and Schindler's proof using the notion of consistency property and reorganizes the presentation for clarity.

Findings

$ ext{MM}^{++}( ext{kappa})$ plus many Woodin cardinals imply $(*)$ under $ ext{Diamond}_ ext{kappa}$

Rephrases the proof using the concept of consistency property

Provides a clearer, reorganized presentation of the original proof

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.