5 problems

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

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 $cmathsf{ZFR}$ 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…

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