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.

• Steel and Welch showed that a strong cardinal is a lower bound. It is not known how to use the hypothesis u2 = ω2 to build models with more large cardinals than this even if granted that K exists.

• An upper bound is the existence of δ < μ where δ is a Woodin cardinal and μ is a measurable cardinal.

• A related problem is to determine the consistency strength of "every real has a sharp + every subset of ω1 in L(R) is constructible from a real".

• References:

• Steel, J. R., and Welch., P. D., Σ13 absoluteness and the second uniform indiscernible, Israel J. Math. 104 (1998) 157-190

• Woodin, W. H., The axiom of determinacy, forcing axioms and the nonstationary ideal, DeGruyter series in logic and its applications, vol. 1, 1999

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?

• C2 is the set of reals in M0 = L .

• C1 is the set of reals that are Δ11 equivalent to a mastercode of M0 = L .

• C2n is the set of reals in M2n .

• The general question is how this pattern continues at the odd levels.

• References:

• Kechris, A., The theory of countable analytic sets, Trans. Amer. Math. Soc. 202 (1975) 259-297

• Steel, J. R., Projectively well-ordered inner models, Ann. Pure Appl. Logic 74 (1995) 77-104

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

• Clearly ω1 < ΘL(R) .

• It is consistent that ω2 < ΘL(R) . For example, if u2 = ω2 .

• But what about ω3 < ΘL(R)?

• (Variant)   Assume there are arbitrarily large Woodin cardinals. Is it possible that there is a universally Baire well-ordering of type ω3?

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 ?

• Woodin obtained a positive answer under the stronger assumption that and that there is no countable mouse for the ADR hypothesis.

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?

• The following is a related theorem of Steel. Assume that the sharp operation is total on sets. Suppose that M1 does not exist. Then K exists and is closed under the sharp operation. Suppose that there is a measurable cardinal. Then K is Σ13 correct.

• It is conjectured that a measurable cardinal is not needed for Steel's Σ13 correctness theorem.

• Steel's correctness theorem extends earlier results of Jensen, Magidor and Mitchell.

• Reference:

• Steel, J. R., The core model iterability problem, Springer, 1996

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#?

• This would give Σ14 correctness for K in the case that K does not go beyond N.

• The second clause in the above conclusion is an instance of problem 2.

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.

• The iterability hypothesis is enough to conclude that all the standard condensation lemmas apply to L[E].

• A variant of the question asks about λ∩cof(<κ)

• Schimmerling has written a report (PDF) with results on this question.

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

• An upper bound is the existence of a cardinal κ that is κ-strongly compact.

• An upper bound for the failure of square at alephω is the existence of a measurable subcompact cardinal. This points to a possible difference between square and weak square.

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?

• Schindler proves instances of this in his paper, Iterates of the core model.

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

• V=W[r] for some real r,

• V and W have the same cofinalities,

• CH holds in W, and

• the continuum is aleph2 in V.

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

• Shelah showed that under this hypothesis, there is an inner model with a measurable cardinal.

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?

• Starting with a model with a measurable cardinal κ with o(κ)=κ++ + ε, Woodin produced a model with a Mahlo cardinal κ for which Diamondκ fails.

• Zeman showed that if κ is a Mahlo cardinal and Diamondκ(Sing) fails, then for all λ < κ there exists δ < κ such that oK(δ) > λ.

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.

• Neeman and Woodin showed that a Woodin limit of Woodin cardinals is an upper bound.

• Woodin showed that ADR + DC is a lower bound.

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?

• For Π13, this is a result of Kechris and Martin.

• For Π12n+5, this is a result of Jackson.

• Is there an inner model M of L[0#], an M-definable poset P ⊂ M and an M-generic filter G over P such that

• 0# is not an element of M,

• 0# is an element of M[G], and

• (M[G], ε, G) is a model of ZFC?