Comparing forcing approaches to dense ideals

Abstract

We analyze some posets involved in forcing constructions for dense ideals, showing that the Anonymous Collapse and the Dual Shioya Collapse are equivalent for collapsing a large cardinal to ${\omega }_2$. We also give a somewhat simplified construction of a normal ideal $I$ on ${\omega }_2$ such that $\mathcal{P}({\omega }_2)/I\sim \mathrm{Col}({\omega }_1,{\omega }_2)$.

arXiv commentSubmitted as part of the proceedings of the meeting, "Recent developments in axiomatic set theory", at the Research Institute for Mathematical Sciences (RIMS), Kyoto University, October 2024