Set Theory Seminar

October 21
Andreas Lietz, University of Münster
Forcing '$\mathrm{NS}_{\omega_1}$ is $\omega_1$-dense' from Large Cardinals - A Journey guided by the Stars

An ideal $I$ on $\omega_1$ is $\omega_1$-dense if $(\mathcal{P}(\omega_1)/I)^+$ has a dense subset of size $\omega_1$. We prove, assuming large cardinals, that there is a semiproper forcing $\mathbb{P}$ so that $$V^\mathbb{P}\models`\mathrm{NS}_{\omega_1}\text{ is }\omega_1\text{-dense}\textrm '.$$ This answers a question of Woodin positively. Our general strategy is based on the observation that replacing the role of $\mathbb{P}_{\mathrm{max}}$ in Woodin's axiom $(*)$ by $\mathbb{Q}_{\mathrm{max}}$ results in an axiom $\mathbb{Q}_{\mathrm{max}}-(*)$ which implies $`\mathrm{NS}_{\omega_1}\text{ is }\omega_1\text{-dense}\textrm '$.
We proceed in three steps: First we define and motivate a new forcing axiom $\mathrm{QM}$ and then modify the Asperó-Schindler proof of $`\mathrm{MM}^{++}\Rightarrow(*)\textrm '$ to show $`\mathrm{QM}\Rightarrow\mathbb{Q}_{\mathrm{max}}-(*)\textrm '$. Finally, assuming a supercompact limit of supercompact cardinals exists, we construct a semiproper partial order forcing $\mathrm{QM}$. This last step involves proving two new iteration theorems both of which allow for forcings killing stationary sets.

Video