<i>Forcing “</i>$\mathrm {NS}_{\omega _1}$<i>is</i>$\omega _1$<i>-Dense” from Large Cardinals</i>

Abstract We answer a question of Woodin [3] by showing that “ $\mathrm {NS}_{\omega _1}$ is $\omega _1$ -dense” holds in a stationary set preserving extension of any universe with a cardinal $\kappa $ which is a limit of ${&lt;}\kappa $ -supercompact cardinals. We introduce a new forcing axiom $\mathrm {Q}$ -Maximum, prove it consistent from a supercompact limit of supercompact cardinals, and show that it implies the version of Woodin’s $(*)$ -axiom for $\mathbb Q_{\mathrm {max}}$ . It follows that $\mathrm {Q}$ -Maximum implies “ $\mathrm {NS}_{\omega _1}$ is $\omega _1$ -dense.” Along the way we produce a number of other new instances of Asperó–Schindler’s $\mathrm {MM}^{++}\Rightarrow (*)$ (see [1]). To force $\mathrm {Q}$ -Maximum, we develop a method which allows for iterating $\omega _1$ -preserving forcings which may destroy stationary sets, without collapsing $\omega _1$ . We isolate a new regularity property for $\omega _1$ -preserving forcings called respectfulness which lies at the heart of the resulting iteration theorem. In the second part, we show that the $\kappa $ -mantle, i.e., the intersection of all grounds which extend to V via forcing of size ${&lt;}\kappa $ , may fail to be a model of $\mathrm {AC}$ for various types of $\kappa $ . Most importantly, it can be arranged that $\kappa $ is a Mahlo cardinal. This answers a question of Usuba [2]. Abstract prepared by Andreas Lietz E-mail : [email protected] . URL : https://andreas-lietz.github.io/resources/PDFs/AJourneyGuidedByThe Stars.pdf .