Unsolved Problems

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

If $\kappa$ is a singular strong limit cardinal, then $2^\kappa = \kappa^+$....

Is every abelian group $A$ such that $\text{Ext}^1(A, \mathbb{Z}) = 0$ a free abelian group?...

Does the partition principle (PP) imply the axiom of choice (AC)?...

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 the generalized continuum hypothesis imply the existence of an $\aleph_2$-Suslin tree?...

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?...

Does there exist a Jónsson algebra on $\aleph_\omega$?...

Is the open coloring axiom (OCA) consistent with $2^{\aleph_0} > \aleph_2$?...

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