Talk held by Monroe Eskew (KGRC) at the KGRC seminar on 2018-10-18.
Abstract: Using ideas from Foreman-Magidor-Shelah, one can force from a Woodin cardinal to show it is consistent that the nonstationary ideal on $\omega_1$ is saturated while the quotient boolean algebra is rigid. The key is to apply Martin’s Axiom to the almost-disjoint coding forcing to see how it interacts with a generic elementary embedding. This strategy requires the continuum hypothesis to fail. Towards showing the consistency of rigid ideals with GCH, the speaker investigated other coding strategies: stationary coding (with Brent Cody), a rigid version of the Levy collapse, and ladder-system coding (in recent work with Paul Larson). We have some equiconsistencies about rigid ideals on $\omega_1$ and $\omega_2$, as well as some global possibilities from very large cardinals. Some natural questions remain about $\omega_1$ and successors of singulars.