8 problems

Let $({\mathcal{P}},\Psi)$ be a hod pair such that … Let $\kappa$ be the least strong cardinal which reflects the class of strong cardinals, and let…

Let $obreaktextsf{AD}^+$ denote the strengthened axiom of determinacy. For a function $F$ as in the paper's Theorem n*x, call $F$ pe-bad when, for a cone of reals $x$, the sup…

Let $obreaktextsf{AD}^+$ denote the strengthened axiom of determinacy, and let $obreaktextsf{NLE}$ denote the no-long-extender assertion. Let $obreaktextsf{HPC}$ denote hod-pair…

Let $LSA$ abbreviate the assertion that $AD^+$ holds and there is a largest Suslin cardinal belonging to the Solovay sequence. Let $\vec{E}$ be an extender sequence and write…

Steel's translation conjecture.

Capturing of hod pairs conjecture. There is a $\Sigma$-iterate $\mathcal Q$ of $\mathcal P$ such that, if $\Lambda$ is the strategy of $\mathcal Q$ induced by $\Sigma$, then…

Generation of closed pointclasses conjecture. If there is a Suslin cardinal $\kappa>w(\Gamma)$ and $L(\Gamma,\mathbb R)\vDash MC$, then for some hod pair $(\mathcal P,\Sigma)$,

HOD conjecture. Under these assumptions, $V_\Theta^{\operatorname{HOD}}$ is a shortening of a hod mouse. Moreover, if $\Gamma\subsetneq\mathcal P(\mathbb R)$ and…