Woodin's Ω-conjecture
if there is a proper class of Woodin cardinals, then Ω-logic satisfies an analogue of Gödel's completeness theorem.
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- Does the partition principle (PP) imply the axiom of choice (AC)?
- (Woodin) Does the generalized continuum hypothesis below a strongly compact cardinal imply the generalized continuum hypothesis everywhere?
- Does the generalized continuum hypothesis entail {\diamondsuit(E^{\lambda^+}_{\operatorname{cf}(\lambda)}}) for every singular cardinal…
- Does the generalized continuum hypothesis imply the existence of an ℵ2-Suslin tree?
- If ℵω is a strong limit cardinal, is 2^{\aleph_\omega} ω4, was obtained by Shelah using his PCF theory.
- The problem of finding the ultimate core model, one that contains all large cardinals.
- Does the consistency of the existence of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?
- Does there exist a Jónsson algebra on ℵω?