6 problems

Woodin's HOD conjecture. … mathrm{ZFC}+text{“there is a supercompact cardinal”} $$ proves the HOD hypothesis.

Assume ${\sf{AD^+}}+V=L(\wp(\mathbb{R}))$. Define $(\eta_\alpha:\alpha\leq\Omega)$ by setting $\eta_0=\theta_0$; if $\eta_\alpha<\Theta$ and…

Assume that $AD_{\mathbb{R}}+V=L(\mathcal{P}(\mathbb{R}))$ and that HOD analysis is successful, so that $\operatorname{HOD}$ is defined and the Solovay sequence consists of the rel…

Woodin's conjecture. “Club-many” cannot be replaced by “all” in the assertion that $\alpha^+$ can be measurable in ${\rm HOD}$ below a supercompact cardinal, even if “measurable” i…

Cummings–Friedman–Golshani conjecture. For club-many cardinals $\alpha$ below a supercompact cardinal, $\alpha^+$ can be measurable in ${\rm HOD}$.

Let $V$ be a model of ${\rm ZFC}$ and let $\alpha\leq\mathop{\rm ORD}$. In another model $\overline{V}$ of ${\rm ZFC}$, write $({\mathord{\rm M}}^\alpha)^{\overline{V}}$ and…