${\rm AD}_{\mathbb R}$ is a strengthening of the determinacy axiom that states that all games on the real numbers are determined. It is a Theorem of Solovay that under ${\rm ZF}+{\rm AD}_{\mathbb R}$ there is a fine, countably complete and normal filter on $P_{\omega_1}(\mathbb R)$, so $\omega_1$ is $\mathbb R$-supercompact. The exact consistency strength of the theory ${\rm ZF}+ {\rm AD}+$“$\omega_1$ is $\mathbb{R}$-supercompact” is, however, weaker than the one of ${\rm ZF}+{\rm AD}_{\mathbb R}$.
One central interest of Inner Model Theory is to construct/find canonical models for theories extending ${\rm ZF}$. A natural question is, then, whether there is a canonical model for the theory ${\rm ZF}+ {\rm AD}+$“$\omega_1$ is $\mathbb{R}$-supercompact”.
In this talk, we will discuss the consistency strength and minimal models of this theory. We will discuss the proof of the uniqueness of minimal models of this theory, under various appropriate hypotheses. And time permitting we will discuss the proof of the result that under ${\rm ZFC}$ there is at most one minimal model of this theory. This is joint work with Nam Trang.

By an old result of Mathias, there are no mad families in the Solovay model constructed by the Levy collapse of a Mahlo cardinal. By a recent result of Törnquist, the same is true in the classical model of Solovay as well. In a recent paper, we show that ZF+DC+”there are no mad families” is actually equiconsistent with ZFC. I’ll present the ideas behind the proof in the first part of the talk.

In the second part of the talk, I’ll discuss the definability of maximal eventually different families and maximal cofinitary groups. In sharp contrast with mad families, it turns out that Borel MED families and MCGs can be constructed in ZF. Finally, I’ll present a general problem in Borel combinatorics whose solution should explain the above difference between mad and maximal eventually different families, and I’ll show how large cardinals must be involved in such a solution.

This is joint work with Saharon Shelah.

We’ll discuss problems arising when trying to apply CMI in models where even the weakest forms of choice might fail. We’ll show how to deal with these problems in the particular case of a model in which all uncountable cardinals are singular.