Martin’s Maximum${}^{++}$ implies Woodin’s axiom $(*)$

Abstract

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