Recent Advances in Core Model Theory

Recent Advances in Core Model Theory

Open problems (under construction)

  1. Determine the consistency strength of "every real has a sharp + u2 = ω2" where u2 is the second uniform indiscernible.

  2. Assume PD. Then there is a largest countable Π13 set of reals called C3 . It also follows from PD that for each n < ω, there is a minimal proper class model with n Woodin cardinals called Mn . Is it true that C3 is the set of reals that are Δ13 equivalent to a mastercode of M2?

  3. Working in ZFC, how large can ΘL(R) be?

  4. Assume AD+ and that there is no countable mouse with a superstrong cardinal. Let x be an ordinal definable real. Does there exist a countable mouse M with x ε M ?

  5. Assume that the M1# operation is total on sets. Suppose that M2 does not exist. Then K exists and is closed under the M1# operation. Therefore K is Σ13 correct. Is K is Σ14 correct?

  6. Assume that the M1# operation is total. Let N be the least M1# closed model. Suppose that there is a Π13 singleton not in N. Let {x} be the least such singleton. Does N# exist and is there a Δ13 isomorphism between x and N#?

  7. (Two part question.)

    1. Let M be a countable transitive set. Suppose that there exists α and an elementary embedding of M into Vα . Is M (ω1+ 1)-iterable?

    2. Let T be a countable iteration tree of limit length on V with extenders Eξ and models Mξ. Assume that Eξ is countably closed in Mξ for all ξ < lh(T). Does T have a cofinal wellfounded branch?

      • This is an instance of CBH.

      • Countably closed means that Mξ satisfies "the ultrapower by Eξ is closed under ω sequences".

      • Woodin showed that CBH is false.

      • Extender models satisfy UBH.

  8. Let L[E] be an extender model such that every countable premouse that embeds into a level of L[E] is (ω1+1)-iterable. In terms of large cardinal axioms, characterize those L[E] successor cardinals λ with the property that every stationary subset of λ∩cof(ω) reflects.

  9. What is the consistency strength of "λ is a singular cardinal and weak square fails at λ"?

  10. Assume K exists. Let j be an elementary embedding from V to a transitive class M. Let i be the restriction of j to K. Then i is an elementary embedding from V to j(K) = KM. Does i arise from an iteration of K?

  11. (Two part problem)

    1. Rate the consistency strength of the following statement. Let I be a simply definable σ-ideal. (E.g., the ideal of countable sets, null sets, meager sets, etc.) Then the statement "Every Σ12 (projective) I-positive set has a Borel I-positive subset" holds in every generic extension.

    2. Assume 0# does not exist. Is it possible to add a real x by forcing such that RL[x] is I-positive?

  12. Assume that

    Prove there is an inner model with aleph2 many measurable cardinals.

  13. Investigate the ZFC model HODV[G] where G is V-generic over Coll(ω, < OR). In particular, does CH hold in this model?

  14. Assume 0-Pistol does not exist. Suppose κ is Mahlo and Diamondκ (Sing) fails.

    1. Must κ be a measurable cardinal in K?

    2. In addition, suppose that GCH holds below κ. Is there an inner model with a strong cardinal?

    3. Can GCH hold?

  15. (Two part question.)

    1. Assume there is no proper class inner model with a Woodin cardinal. Must there exist a set iterable extender model with the weak covering property?

    2. Assume ZFC + NSω1 is ω2 saturated. Is there an inner model with a Woodin cardinal?

  16. Let M be the minimal fully iterable extender model with a Woodin cardinal κ that is a limit of Woodin cardinals. Let D be the derived model of M below κ. Is ΘD regular in D?

  17. Determine the consistency strength of incompatible models of AD+, by which we mean that there are A and B such that L(A, R) and L(B,R) satisfy AD+ but L(A, B, R) does not satisfy AD.

  18. Let Θ = ΘL(R) and δ be the least Woodin cardinal of Mω.

    1. Is HODL(R)∩VΘ a normal iterate of Mω∩Vδ? (Neeman has evidence towards a negative answer.)

    2. If not, is there a normal iterate Q of HODL(R) such that the iteration map fixes Θ and Q∩VΘ is a normal iterate of every countable iterate of Mω∩Vδ?

  19. Assume V = L(R) and AD holds. Let Γ be a Π11 like scaled pointclass. (I.e., closed under ∀R and not self-dual.) Let Δ be the corresponding self-dual pointclass. Let δ be the supremum of the lengths of Δ prewellorderings. Is Γ closed under unions of length less than δ?