Unsolved Problems

There is no set whose cardinality is strictly between that of the integers and the real numbers....

Does the generalized continuum hypothesis below a strongly compact cardinal imply it everywhere?...

Does the generalized continuum hypothesis entail the diamond principle $\diamondsuit(E_{\text{cf}(\lambda)}^{\lambda^+})$ for every singular cardinal ...

Does there exist an ultimate core model containing all large cardinals?...

If there is a proper class of Woodin cardinals, does Ω-logic satisfy an analogue of Gödel's completeness theorem?...

Does the consistency of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?...

Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?...