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 M_n . 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 .
     
     
   • C_2n is the set of reals in M_2n .
     
     
   • 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 Θ^L(R) be?
   
   
   • Clearly ω₁ < Θ^L(R) .
     
     
   • It is consistent that ω₂ < Θ^L(R) . For example, if u₂ = ω₂ .
     
     
   • But what about ω₃ < Θ^L(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 AD_R 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 V_α . Is M (ω₁+ 1)-iterable?
      
      
   2. Let T be a countable iteration tree of limit length on V with extenders E_ξ and models M_ξ. Assume that E_ξ is countably closed in M_ξ for all ξ < lh(T). Does T have a cofinal wellfounded branch?
      
      
      • This is an instance of CBH.
        
        
      • Countably closed means that M_ξ satisfies "the ultrapower by E_ξ 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^+ω-strongly compact.
     
     
   • An upper bound for the failure of square at aleph_ω 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) = K^M. 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 R^L[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 HOD^V[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 Diamond_κ (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 Diamond_κ fails.
      
      
    • Zeman showed that if κ is a Mahlo cardinal and Diamond_κ(Sing) fails, then for all λ < κ there exists δ < κ such that o^K(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 + NS_ω1 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 Θ^D 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 AD_R + DC is a lower bound.
      
      
18. Let Θ = Θ^L(R) and δ be the least Woodin cardinal of M_ω.
    
    
    1. Is HOD^L(R)∩V_Θ a normal iterate of M_ω∩V_δ? (Neeman has evidence towards a negative answer.)
       
       
    2. If not, is there a normal iterate Q of HOD^L(R) such that the iteration map fixes Θ and Q∩V_Θ is a normal iterate of every countable iterate of M_ω∩V_δ?
       
       
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 Π¹_2n+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?