Covering properties of derived models
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1 University of California, Irvine June 16, 2015
2 Outline Background Inaccessible limits of Woodin cardinals Weakly compact limits of Woodin cardinals
3 Let L denote Gödel s constructible universe. Weak covering If 0 does not exist, then L is close to V in terms of cardinals and cofinalities: 1. If κ is a singular cardinal, then (κ + ) L = κ +. (Jensen) 2. If κ ℵ 2 is regular, then cf ( (κ + ) L) κ. (Jensen) 3. If κ is weakly compact, then (κ + ) L = κ +. (Kunen) In cases (2) and (3), we can get parallel results with a model of determinacy (a derived model at κ) in place of L, and a strong axiom of determinacy (AD R ) in place of 0.
4 Definition The Axiom of Determinacy, AD, says that for every ω-length two-player game of perfect information on the integers, one of the two players has a winning strategy. Theorem (Woodin) The following theories are equiconsistent: 1. ZFC + there are infinitely many Woodin cardinals 2. ZF + AD. We will need some details of the forward direction.
5 Theorem (Woodin) Let κ be a limit of Woodin cardinals, let G be a V -generic filter over Col(ω, <κ), and define R G = α<κ R V [G α]. Then L(R G ) = AD. From a slightly stronger hypothesis, Woodin obtained AD in the L(R) of V itself.
6 For the rest of the talk: Fix a limit κ of Woodin cardinals Fix a V -generic filter G Col(ω, <κ) Define R G = α<κ RV [G α]. If κ is regular (hence inaccessible) then κ = ω V [G] 1, and R G = RV [G].
7 Let s look for models of AD larger than L(R G ). First we consider a symmetric model: Definition V (R G) = HOD V [G] V R G {R G }. Whether or not κ is regular, we have κ = ω V (R G ) 1. R G = RV (R G ).
8 AC fails in V (R G ): we cannot choose a surjection ω α for every α < κ. If κ is regular (hence inaccessible) in V, then in V (R G ) every set of reals is Lebesgue measurable and DC holds. (Solovay) AD fails in V (R G ).
9 Theorem (Woodin) In V (R G ), there is a largest (under ) pointclass Γ such that L(Γ, R G) = AD +. (AD + is a strengthening of AD that holds in L(R G )). Definition The derived model of V at κ by G is D(V, κ, G) = L(Γ, R G) for the largest pointclass Γ as above.
10 can satisfy stronger determinacy axioms than L(R G ), such as AD R. (Just as higher core models can satisfy stronger large cardinal axioms than L, such as the existence of 0.) Definition AD R is determinacy for games on R (instead of N.)
11 Recall that if 0 does not exist, then L is close to V. Question If AD R does not hold in the derived model D(V, κ, G), then is D(V, κ, G) close to V (R G )? The relevant cardinalities and cofinalities are in the vicinity of κ and κ +. We could say close to V instead of close to V (R G ) because the correspondence between cardinals and cofinalities of V and V (R G ) is straightforward.
12 A caveat in formulating close to V for derived models: In D(V, κ, G) there is a surjection R G ω 2 (by AD, using the Moschovakis coding lemma.) In V (R G ) there is no surjection R G ω 2 Because κ is ω 1 in D(V, κ, G) and V (R G ), it follows that: (κ + ) D(V,κ,G) < κ +. This also shows that V (R G ) does not satisfy AD.
13 So it seems (κ + ) D(V,κ,G) is not the relevant thing to look at. Definition Θ is the least ordinal that is not a surjective image of R (i.e. the successor of R in the sense of surjections.) If AC holds, then Θ = c +. If AD holds, then Θ is inaccessible by the coding lemma (in particular Θ > ω 2 ). Look at Θ D(V,κ,G) instead of (κ + ) D(V,κ,G).
14 Θ D(V,κ,G) κ +. If AD R holds in D(V, κ, G) then Θ D(V,κ,G) < κ +. (Using the fact P(R) D(V,κ,G) = Hom G.) If AD R fails in D(V, κ, G) then in general we may have Θ D(V,κ,G) < κ + or Θ D(V,κ,G) = κ + ; in specific cases we will be able to say more. Analogy: Θ D(V,κ,G) (κ + ) L AD R fails 0 does not exist
15 Inaccessible limits of Woodin cardinals Weakly compact limits of Woodin cardinals Theorem (W.) Let κ be an inaccessible limit of Woodin cardinals. Let G be a V -generic filter over Col(ω, <κ). If AD R fails in D(V, κ, G), then cf(θ D(V,κ,G) ) κ. An equivalent conclusion is that D(V, κ, G) is closed under ω-sequences of sets of reals in V (R G ). If AD R holds in D(V, κ, G) then cf(θ D(V,κ,G) ) = κ, but for trivial reasons.
16 Inaccessible limits of Woodin cardinals Weakly compact limits of Woodin cardinals Proof sketch: We want to show that cf(θ D(V,κ,G) ) κ. If not, assume WLOG that cf(θ D(V,κ,G) ) = ω in V. Take hull X H κ + with X κ = κ < κ and X ω X. Consider π : M = X, the uncollapse map. Extend to ˆπ : M[Ḡ] H κ +[G] where Ḡ = G κ. Set D = D(M, κ, Ḡ) and D = D(H κ +, κ, G). ˆπ[ D] is Wadge-cofinal in D (cofinality is small.)
17 Inaccessible limits of Woodin cardinals Weakly compact limits of Woodin cardinals Proof sketch (continued): In D(V, κ, G), if AD R fails, then there is a Suslin set of reals p[t ] whose complement is not Suslin. Assume WLOG that T V. Using that ˆπ[ D] is Wadge-cofinal in D, show the hull is T -full: every subset of R in L(T, Ḡ R ) is in D. Ḡ Use T -fullness and ˆπ to get a tree T in V (R G ) such that T and T project to complements, a contradiction.
18 Inaccessible limits of Woodin cardinals Weakly compact limits of Woodin cardinals So if κ is an inaccessible limit of Woodin cardinals and AD R fails in D(V, κ, G) then either 1. Θ D(V,κ,G) = κ +, or 2. cf(θ D(V,κ,G) ) = κ. Both cases are possible. Case 1 holds if κ is weakly compact (as we will see.) Can get case 2 from case 1 by forcing with Col(κ, κ + ).
19 Inaccessible limits of Woodin cardinals Weakly compact limits of Woodin cardinals Theorem (W.) Let κ be a weakly compact limit of Woodin cardinals. Let G be a V -generic filter over Col(ω, <κ). If AD R fails in D(V, κ, G), then Θ D(V,κ,G) = κ +. The hypothesis is consistent: AD R has higher consistency strength than a weakly compact limit of Woodin cardinals.
20 Inaccessible limits of Woodin cardinals Weakly compact limits of Woodin cardinals We can force a failure of covering for the derived model. This does not typically preserve weak compactness. But: Corollary If κ is a Col(κ, κ + )-indestructibly weakly compact limit of Woodin cardinals and G is a V -generic filter over Col(ω, <κ), then D(V, κ, G) = AD R. A better relative consistency result comes from Jensen Schimmerling Schindler Steel, Stacking mice.
21 Can we get weak covering in the singular case? Question Let κ be a singular limit of Woodin cardinals. If AD R fails in D(V, κ, G), then must Θ D(V,κ,G) = κ +? This would result in incompactness: Proposition (W.) Let κ be a singular limit of Woodin cardinals. If Θ D(V,κ,G) = κ +, then κ holds after some small forcing. (The small forcing is only needed if D(V, κ, G) = LSA; perhaps not even then.)
22 In the inaccessible case, where we do have weak covering, does this result in incompactness? (Note κ is trivial at an inaccessible.) Question Let κ be an inaccessible limit of Woodin cardinals. If AD R fails in D(V, κ, G), then In the case cf(θ D(V,λ,G) ) = κ, must (κ) hold? In the case Θ D(V,λ,G) = κ +, must (κ + ) hold?
23 Recall that if AD R holds, then we have Θ D(V,κ,G) < κ +. Can we still get some kind of weak covering? Question Let κ be a limit of Woodin cardinals. Assume that κ is singular, or κ is weakly compact. Assume that AD R holds in D(V, κ, G) (and maybe that some stronger determinacy axiom fails.) Is the successor of Θ D(V,κ,G) in HOD D(V,κ,G) equal to κ +?
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