November 22
Alejandro Poveda,
Harvard University
Identity crises phenomena between the first supercompact cardinal and Vopěnka's Principle
We will report on some recent results on the large cardinal hierarchy between the first supercompact cardinal and Vopěnka's Principle. We present various consistency results as well as a conjecture as for how the large-cardinal hierarchy of $\text{Ultimate}$-$L$ looks like at these latitudes. The main result will be the consistency with very large cardinals of a new Kimchi-Magidor configuration; namely, we will present a model where every supercompact cardinal is supercompact with inaccessible target points. This answers a question of Bagaria and Magidor. This configuration is a consequence of a new axiom (named $\mathcal A$) which regards the mutual relationship between superstrong and tall cardinals. Time permitting we shall discuss the interplay between $\mathcal A$ and $\text{Ultimate}$-$L$ and propose a few open questions.