3 problems

Menas's conjecture. Not every strongly compact cardinal has a function with the Menas property. Menas proved that every strongly compact limit of strongly compact cardinals has suc…

Let $U$ and $W$ be countably complete ultrafilters. Write $j_U:V\to M_U$ and $j_W:V\to M_W$ for their ultrapower embeddings, and let $j_{j_W(U)}$ and $j_{j_U(W)}$ denote the corres…

Extender Axiom refutation. If there is a cardinal $\kappa$ that is $2^\kappa$-supercompact, then the Extender Axiom is false. This is a consistency obstruction for the proposed gen…