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 $\mathbb{ Q}_{ \mathrm{ max}}\text{ -}(*)$. Consequently, $\mathrm{ QM}$ implies ``$\mathrm{ NS}_{ \omega_1}$ is $\omega_1$-dense".
Categories: math.LO