Forcing "${\mathrm{NS}}_{{\omega }_1}$ is ${\omega }_1$-dense" From Large Cardinals

Abstract

We answer a question of Woodin by showing that assuming an inaccessible cardinal $\kappa$ which is a limit of $<\kappa$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb{P}}\models ‘‘{\mathrm{NS}}_{{\omega }_1}\text{ is }{\omega }_1\text{-dense}"$. We also introduce a new forcing axiom $\mathrm{QM}$, show it is consistent assuming a supercompact limit of supercompact cardinals and prove that it implies ${\mathbb{Q}}_{\max }\text{-}(\ast )$. Consequently, $\mathrm{QM}$ implies ``${\mathrm{NS}}_{{\omega }_1}$ is ${\omega }_1$-dense".