${\mathsf{MM}}^{++}$ implies $(\ast )$

Abstract

We show that Martin's Maximum${}^{++}$ implies Woodin's ${\mathbb{P}}_{\max }$ axiom $(\ast )$. This answers a question from the 1990's and amalgamates two prominent axioms of set theory which were both known to imply that there are ${\aleph }_2$ many real numbers.

48 pages. To appear in Annals of Mathematics