9 problems

Ultraexacting-above-extendible rank-Berkeley model conjecture. There is a set model of ZF with a rank-Berkeley cardinal.

Ultraexacting-above-extendible consistency conjecture. If ZFC holds, $c\text{?}$ is ultraexacting, and $c\text{?}<c\text{?}$ is extendible, then there is a set model of ZF with a r…

Steel's vague conjecture. There is an extension $H$ axiomatized by large cardinal hypotheses such that

Natural consistency-strength conjecture. Any natural extension of $\mathsf{ZFC}$ is either equiconsistent with $\mathsf{ZFC}$ or equiconsistent with $\mathsf{ZFC}+A$, where $A$ is…

Let $\mathfrak F$ be the class of ideals on $\operatorname{\mathbb{N}}$ described above, and consider the theory obtained by adjoining to ${\mathsf{ZF}}$ the assertion that there a…

Let $cmathsf{ZFR}$ denote the set-theoretic theory under discussion, and let $cmathsf{DC}$ denote the principle of dependent choice. Consistency-strength conjecture. The theory…

Let $\textsf{LSA}$ denote the theory “$\textsf{AD}^+ + \theta_{\beta+1}+\theta_\beta$ is the largest Suslin cardinal.” Here $\operatorname{Con}(T)$ and…

Divergent-models consistency-strength conjecture. The exact consistency strength of divergent models of $AD^+$ is the consistency strength of a Woodin limit of Woodin cardinals.

LST-hod mouse equiconsistency conjecture. The following theories are equiconsistent: