On the consistency strength of the inner model hypothesis

Sy-David Friedman (KGRC, Vienna)^{∗} Philip Welch (Bristol) Hugh Woodin (Berkeley)

10 August 2006

The inner model hypothesis (IMH) and the strong inner model hypothesis (SIMH) were introduced in [5]. In this article we establish some upper and lower bounds for their consistency strength.

We repeat the statement of the IMH. A sentence in the language of set theory is internally consistent iff it holds in some (not necessarily proper) inner model. The meaning of internal consistency depends on what inner models exist: If we enlarge the universe, it is possible that more statements become internally consistent. The inner model hypothesis asserts that the universe has been maximised with respect to internal consistency:

The Inner model hypothesis (IMH): If a statement φ without parameters holds in an inner model of some outer model of V (i.e., in some model compatible with V), then it already holds in some inner model of V.

Equivalently: If φ is internally consistent in some outer model of V then it is already internally consistent in V. This is formalised as follows. Regard V as a model of Gödel-Bernays class theory, endowed with countably many sets and classes. Suppose that V^{∗} is another such model, with the same ordinals as V. Then V^{∗} is an outer model of V (V is an inner model of V^{∗}) iff the sets of V^{∗} include the sets of V and the classes of V^{∗} include the classes of V. V^{∗} is compatible with V iff V and V^{∗} have a common outer model.

∗The first author was supported by FWF Grants P16334-NO5 and P16790-NO4.

The IMH implies absoluteness for sentences which are Σ_{2} over H(ω_{1}) (equivalently, for sentences which are Σ_{3}^{1} in the sense of descriptive set theory). This is because by Lévy-Shoenfield absoluteness, such a sentence is true iff it is true in some inner model.

Theorem 1 ([5]) The inner model hypothesis implies that for some real R, ZFC fails in L_{α}[R] for all ordinals α. In particular, there are no inaccessible cardinals, the reals are not closed under # and the singular cardinal hypothesis holds.

Theorem 2 The IMH implies that there is an inner model with measurable cardinals of arbitrarily large Mitchell order.

Proof. Assume not and let K denote Mitchell’s core model for sequences of measures (see [6]). Let δ be the maximum of ω_{1} and the supremum of the Mitchell orders of measurable cardinals in K. By Mitchell’s Covering Theorem for K we have:

(∗) cof (α) ≥ δ, α regular in K → cof (α) = card (α).

Now normally iterate K by applying each measure of order 0 exactly once, i.e., if K_{i} is the i-th iterate of K, K_{i+1} is formed by applying the measure of order 0 in K_{i} at κ_{i}, the i-th measurable cardinal of K. Let σ : K → K′ be the resulting iteration map. Then:

Lemma 3 (∗) holds with K replaced by K′.

Proof. It suffices to show by induction on i that (∗)_{i} holds, where (∗)_{i} is (∗) with K replaced by K_{i}.

Base case: (∗)_{0} is just (∗).

Successor case: Suppose that (∗)_{i} holds and that α is K_{i+1}-regular with cofinality at least δ. Let π_{i,i+1} : K_{i} → K_{i+1} be the ultrapower map resulting from applying the measure of order 0 at κ_{i}.

We may assume that α is greater than κ_{i}, else α is K_{i}-regular and we are done by induction. If α is at most π_{i,i+1}(κ_{i}) then α has the same cardinality as κ_{i}^{+} of K_{i}, and, as K_{i+1} and K_{i} contain the same κ_{i}-sequences of ordinals, the same cofinality as κ_{i}^{+} of K_{i}. So we are again done by induction.

Now suppose that α is greater than π_{i,i+1}(κ_{i}). Represent α in K_{i+1}, the ultrapower of K_{i}, by [f] where f : κ_{i} → Ord. We may assume that f is either constant or increasing, and also that f(γ) is K_{i}-regular and greater than κ_{i} for all γ < κ_{i}. If f is constant then α = π_{i,i+1}(ᾱ) for some ᾱ which is regular in K_{i} and greater than κ_{i}; but then ᾱ is a fixed point of π_{i,i+1} so α = ᾱ and we are done by induction. So assume that f is increasing.

Now the K_{i}-cofinality of α is at least the supremum μ of the f(γ)’s, as we can everywhere-dominate any set in K_{i} of f(γ)-many functions from κ_{i} into ∏_{γ′>γ} f(γ′) by a single such function in K_{i}. As μ is K_{i}-singular, the K_{i}-cofinality of α is in fact greater than μ. And the K_{i}-cardinality of α is μ^{κ} = μ^{+} of K_{i}. It follows that α and μ^{+} of K_{i} have the same cofinality and the same cardinality, so we are done by induction.

Limit case: Suppose that i is a limit and α is K_{i}-regular with cofinality at least δ. For large enough j < i we can write α as π_{j,i}(α_{j}), where π_{j,i} is the natural embedding of K_{j} into K_{i}. Let κ^{∗} denote the supremum of the κ_{j}, j < i.

We may assume that each α_{j} is at least κ^{∗}, as if α_{j} is less than κ_{k}, j < k < i, it follows from the fact that κ_{k} is a fixed point of π_{j,k} that α equals π_{j,k}(α_{j}), which is regular in K_{k}, and so we are done by induction. But if α_{j} is at least κ^{∗} then α_{j} is a fixed point of π_{j,i} and therefore α = α_{j}; so we are again done by induction. □

Lemma 4 If λ is a cardinal then cof ^{K′}(λ) is not measurable in K′.

Proof. If κ = cof ^{K}(λ) is not measurable in K then λ is a fixed point of the map σ and therefore the result follows by elementarity. Otherwise, let σ_{0} : K → K′_{0} be the iteration map that results from applying only the order 0 measures at cardinals less than κ. Then λ = σ_{0}(λ) has cofinality σ_{0}(κ) = κ in K′_{0}. As the ultrapower K′_{1} of K′_{0} by the order 0 measure at κ contains all κ sequences of ordinals that belong to K′_{0}, it follows that λ has cofinality κ in K′_{1} and therefore also in K′. As κ is not measurable in K′ we are done. □

Now after [2] define a function d : Ord → ω as follows. Fix a lightface K′- definable global □-sequence ⟨C_{α} | α singular in K′⟩: C_{α} is closed unbounded in α with ordertype less than α for each K′-singular α and Cᾱ = C_{α} ∩ ᾱ whenever ᾱ is a limit point of C_{α}. If α is not K′-singular then d(α) = 0. Otherwise define: α_{0} = α α_{1} = ot (C_{α_{0}}) α_{2} = ot (C_{α_{1}}) ⋯ α_{n+1} = ot (C_{α_{n}}),

as long as α_{n} is K′-singular, and let d(α) be the least n such that α_{n} is not K′-singular. α_{d(α)} is the K′-cofinality of α.

Lemma 5 (Main Lemma, after [2]) For each n there is a ZFC-preserving class forcing P_{n} that adds a CUB class C_{n} of singular cardinals such that for all α in C_{n} of cofinality at least δ, d(α) is at least n.

Proof. We use the following.

Lemma 6 Suppose k < m, α ≥ δ, α is regular and C is a closed set of ordertype α^{+m} +1, consisting of ordinals ≥ α^{+m} (where α^{+0} = α, α^{+(p+1)} = (α^{+p})^{+}). Then (C ∩ {β | d(β) ≥ k + 1}) ∪ Cof (< δ) has a closed subset of ordertype α^{+(m−k−1)} + 1.

Proof. The proof is by induction on k, using Lemma 3.

Suppose k = 0. Let β be the α^{+(m−1)}-st element of C. Then β is K′-singular since its cofinality (= α^{+(m−1)}) is at least δ and less than its cardinality (≥ α^{+m}). Similarly, each element of Lim (C ∩β) of cofinality ≥ δ is K′-singular and therefore Lim (C∩β) is a closed subset of (C∩{β | d(β) ≥ 1})∪Cof (< δ) of ordertype α^{+(m−1)} + 1, as desired.

Suppose that the Lemma holds for k and let m+1 > k+1, C a closed set of ordertype α^{+(m+1)} +1 consisting of ordinals ≥ α^{+(m+1)}. Then μ = max C is K′-singular, as its cofinality is at least δ and less than its cardinality. Let β be the (α^{+m} + α^{+m})-th element of C ∩ C_{μ}. β is K′-singular as its cofinality is at least δ and less than its cardinality. Let β̄ be the α^{+m}-th element of C. Then C̄ = {ot C_{γ} | γ ∈ C ∩ Lim C_{β} ∩ [β̄, β]} is a closed set of ordertype α^{+m} + 1 consisting of ordinals ≥ α^{+m}. By induction there is a closed D̄ contained in (C̄ ∩ {γ | d(γ) ≥ k + 1}) ∪ Cof (< δ) of ordertype α^{+(m−k−1)} + 1. But then D = {γ ∈ C ∩ Lim C_{β} | ot C_{γ} ∈ D̄} is a closed subset of (C ∩ {γ | d(γ) ≥ k + 2}) ∪ Cof (< δ) of ordertype α^{+(m−k−1)} + 1. As m − k − 1 = (m + 1) − (k + 1) − 1, we are done. □ (Lemma 6)

Lemma 5 now follows: Let P_{n} consist of closed sets c of singular cardinals such that α ∈ c, cof (α) ≥ δ → d(α) ≥ n,

ordered by end-extension. Lemma 6 implies that this forcing is κ-distributive for every cardinal κ. □

Now for each n there is an outer model of V containing a real R_{n} such that in L[R_{n}]:

(∗)_{R_{n}} R_{n} codes a CUB class C_{R_{n}} of singular cardinals and an iterable, universal extender model K′_{R_{n}} such that a. d_{R_{n}}(α) ≥ n for α in C_{R_{n}} of sufficiently large cofinality, where d_{R_{n}}(α) is defined in K′_{R_{n}} just like d(α) is defined in K′. b. α ∈ C_{R_{n}} → cof (α) in K′_{R_{n}} is not measurable in K′_{R_{n}}.

This is because we can use Lemma 5 to force a CUB class C_{n} of singular cardinals such that d(α) ≥ n for all α in C_{n} of sufficiently large cofinality, and then L-code the model ⟨V, C_{n}, K′⟩ by a real R_{n}. The extender model K′ is universal in the extension as successors of strong limit cardinals are not collapsed and therefore weak covering holds relative to K′ in the extension at all such cardinals of sufficiently large cofinality.

Applying the IMH, there are such reals R_{n} in V. As each R_{n} codes a CUB class of singular cardinals, the K of L[R_{n}] is universal and therefore so is the K_{R_{n}} arising from (∗)_{R_{n}}. Now co-iterate the K_{R_{n}}’s to a single K^{∗}, resulting in embeddings π_{n} : K_{R_{n}} → K^{∗}. As singular cardinals in C_{R_{n}} of sufficiently large cofinality are fixed by π_{n} (as their K_{R_{n}}-cofinality is not measurable in K_{R_{n}}), it follows that there is a single γ belonging to all of the C_{R_{n}}’s which is fixed by all of the π_{n}’s. But then d^{∗}(γ) ≥ n for each n, where d^{∗}(γ) is defined relative to K^{∗} just like d(γ) was defined relative to K′. This is a contradiction. □

For each real x let M_{x}, if it exists, be the minimum transitive set model of ZF C containing x. Thus M_{x} has the form L_{μ}[x] for some countable ordinal μ = μ(x). If d is a Turing degree we write M_{d}, μ(d) for M_{x}, μ(x) (x in d).

Theorem 7 Assume the existence of a Woodin cardinal with an inaccessible above. Then the IMH is consistent. Moreover for all d in a cone of Turing degrees, M_{d} exists and satisfies the IMH.

Proof. First we prove the consistency of the IMH by showing that M_{d} satisfies the IMH for some Turing degree d in a forcing extension of V.

Let κ be Woodin with an inaccessible above in V. Let G be generic over V for the Lévy collapse of κ to ω. Work now in V [G]. Σ_{2}^{1} determinacy holds and, as there is still an inaccessible, M_{d} exists for each Turing degree d. It follows that the theory of (M_{d}, ∈) is constant on a cone of Turing degrees d. Let d be a Turing degree such that the theory of (M_{e}, ∈) is constant for Turing degrees e at least that of d.

We claim that M_{d}, endowed with its definable classes, witnesses the IMH. Indeed, suppose that φ is a sentence true in some model M of height μ(d) compatible with M_{d}. By Jensen coding there is a real y such that d is recursive in y, μ(y) = μ(d) and M is a definable inner model of M_{y}. Let e be the Turing degree of y. Then for some formula ψ, M_{e} satisfies the sentence

The inner model defined by ψ (with some choice of parameters) satisfies φ.

It follows that there is an inner model of M_{d} which satisfies φ, as desired. This proves the consistency of the IMH.

To say that a countable M, together with its countable collection of definable classes, satisfies IMH is simply a Π_{1}^{1}-statement with a real coding M as parameter, since one only needs to quantify over outer models of M of height M ∩ Ord. Thus the assertion that there exists a Turing degree d such that M_{d} (with its definable classes) satisfies IMH is a Σ_{2}^{1}-statement and hence absolute. So the existence of a Woodin cardinal with an inaccessible above implies that such an M_{d} exists in V (and indeed in L).

To prove the stronger statement that in V, M_{d} satisfies the IMH for a cone of d’s, one argues as follows. Say that a set of reals X is absolutely Δ_{2}^{1} iff there is a pair of Σ_{2}^{1} formulas φ(x), ψ(x) such that X consists of all solutions to φ(x) in V and φ is equivalent to the negation of ψ both in V and all of its forcing extensions.

Claim. Assume that there is a Woodin cardinal. Then determinacy holds for absolutely Δ_{2}^{1} sets.

Proof of Claim. As before let G be generic for the Lévy collapse of the Woodin cardinal to ω. Then Σ_{2}^{1} determinacy holds in V [G]. By the Moschovakis Third Periodicity theorem, ([7] Theorem 6E.1), if X is Σ_{2}^{1} in V [G] there is a definable winning strategy in V [G] for one of the players in the game G_{X}. By the homogeneity of the Lévy collapse, it follows that absolutely Δ_{2}^{1} sets are determined in V. This proves the Claim.

As there is an inaccessible in V, M_{d} exists for each Turing degree d in V. Now it follows from the Claim that in V, for any sentence φ, either for a cone of Turing degrees d,

M_{d} ⊧ φ

or for a cone of Turing degrees d,

M_{d} ⊧ ¬φ,

since the relevant games are absolutely Δ_{2}^{1}. Therefore in V the theory of (M_{d}, ∈) is constant for a cone of Turing degrees d. We can then apply the argument used earlier in V [G] to conclude that also in V, M_{d} satisfies IMH for d in a cone of Turing degrees. □

Parameters and the strong inner model hypothesis

How can we introduce parameters into the inner model hypothesis? The following result shows that inconsistencies arise without strong restrictions on the type of parameters allowed.

Proposition 8 ([5]) The inner model hypothesis with arbitrary ordinal parameters or with arbitrary real parameters is inconsistent.

So instead we consider absolute parameters, as in [4]. For any set x, the hereditary cardinality of x, denoted hcard (x), is the cardinality of the transitive closure of x. If V^{∗} is an outer model of V, then a parameter p is absolute between V and V^{∗} iff V and V^{∗} have the same cardinals ≤ hcard (p) and some parameter-free formula has p as its unique solution in both V and V^{∗}.

Inner model hypothesis with locally absolute parameters Suppose that p is absolute between V and V^{∗} and φ is a first-order sentence with parameter p which holds in an inner model of V^{∗}. Then φ holds in an inner model of V.

For a singular cardinal κ, a □_{κ} sequence is a sequence of the form ⟨C_{α} | α < κ^{+}, α limit⟩ such that each C_{α} has ordertype less than κ and for ᾱ in Lim C_{α}, Cᾱ = C_{α} ∩ ᾱ. Definable □_{κ} is the assertion that there exists a □_{κ} sequence which is definable over H(κ^{+}) with parameter κ. We will be interested in the special case κ = ℶ_{ω}, in which case the parameter κ is superfluous.

Theorem 9 The inner model hypothesis with locally absolute parameters is inconsistent.

Proof. We first show that definable □_{κ} fails, where κ is ℶ_{ω}. Let ⟨C_{α} | α < κ^{+}, α limit⟩ be a □_{κ} sequence definable over H(κ^{+}) without parameters. For each n let S_{n} consist of all limit α < κ^{+} such that the ordertype of C_{α} is greater than ℶ_{n}.

Claim. Let P_{n} be the forcing that adds a CUB subset of S_{n} using closed bounded subsets of S_{n} as conditions, ordered by end extension. Then P_{n} is κ^{+} distributive, i.e., does not add κ-sequences.

Proof of Claim. It is enough to show that P_{n} is ℶ_{m+1}^{+} distributive for each m < ω. Assume that m is greater than n. Suppose that p is a condition and ⟨D_{i} | i < ℶ_{m+1}^{+}⟩ are dense. Let ⟨M_{i} | i < κ^{+}⟩ be a chain of size κ elementary submodels of some large H(θ) such that M_{0} contains κ ∪ {⟨C_{α} | α < κ^{+}, α limit⟩, p} and for each i < κ^{+}, ⟨M_{j} | j ≤ i⟩ is an element of M_{i+1}. Let κ_{i} be M_{i} ∩ κ^{+} and C the set of such κ_{i}’s. Then a final segment D of C ∩ Lim C_{γ} is contained in S_{n}, where γ = κ_{ℶ_{m+1}^{+}}. Write D as ⟨κ_{α_{i}} | i < ordertype D⟩. We can then choose a descending sequence ⟨p_{i} | i < ℶ_{m+1}^{+}⟩ of conditions below p such that p_{i+1} meets D_{i} and belongs to M_{κ_{α_{i}}+1} for each i. Then the greatest lower bound of this sequence meets each D_{i}. This proves the Claim.

It follows that for each n the forcing P_{n} does not alter H(κ^{+}). By the inner model hypothesis with locally absolute parameters S_{n} has a CUB subset C_{n} in V for each n. But this is a contradiction, as the intersection of the C_{n}’s is empty.

Now we refine the above argument. As not every real has a #, there exist reals R such that κ^{+} equals κ^{+} of L[R], where κ is ℶ_{ω}. Let X be the set of such reals and for each R in X let ⟨C_{α}^{R} | α < κ^{+}, α limit⟩ be the L[R]-least □_{κ} sequence. Now for limit α < κ^{+}, define C_{α}^{∗} to be the intersection of the C_{α}^{R}, R ∈ X. Then ⟨C_{α}^{∗} | α < κ^{+}, α limit⟩ is definable in H(κ^{+}) without parameters and has the properties of a □_{κ} sequence with the sole exception that C_{α}^{∗} is only guaranteed to be unbounded in α if α has cofinality greater than 2^{ℵ_{0}}. Now repeat the above argument using ⟨C_{α}^{∗} | α < κ^{+}, α limit⟩ in place of ⟨C_{α} | α < κ^{+}, α limit⟩, to obtain a contradiction. □

To obtain the strong inner model hypothesis, we require more absoluteness. We say that the parameter p is (globally) absolute iff there is a parameter-free formula which has p as its unique solution in all outer models of V which have the same cardinals ≤ hcard (p) as V.

Strong inner model hypothesis (SIMH) Suppose that p is absolute, V^{∗} is an outer model of V with the same cardinals ≤ hcard (p) as V and φ is a first-order sentence with parameter p which holds in an inner model of V^{∗}. Then φ holds in an inner model of V.

Remark. If above we assume that the sentence φ holds not just in an inner model of V^{∗} but in V^{∗} itself, then in the conclusion we may demand that in an inner model of V witnessing φ, p is definable via the same formula ψ witnessing the absoluteness of p. (This inner model may, however, fail to have the same cardinals ≤ hcard (p) as V.) This is because we can replace the sentence φ by: “φ holds and p is defined by ψ”.

Theorem 10 ([5]) Assume the SIMH. Then CH is false. In fact, 2^{ℵ_{0}} cannot be absolute and therefore cannot be ℵ_{α} for any ordinal α which is countable in L.

Theorem 11 The SIMH implies the existence of an inner model with a strong cardinal.

Proof. Assume not, and let K be the core model below a strong cardinal (see [8]). As in the proof of Theorem 2, we let K′ denote the iterate of K obtained by applying each order 0 measure exactly once. Then by Lemma 4, if λ is a cardinal then the K′-cofinality of λ is not measurable in K′. And by weak covering relative to K, if λ is a singular cardinal, then λ^{+} is computed correctly in K (i.e., (λ^{+})^{K} = λ^{+}).

Lemma 12 For any singular cardinal λ, λ^{+} is computed correctly in K′.

Proof of Lemma 12. This is clear if the K-cofinality of λ is not measurable in K, for then λ is a fixed point of the iteration from K to K′ and (λ^{+})^{K′} = (λ^{+})^{K} = λ^{+}. Otherwise let ⟨K_{i} | i ∈ Ord⟩ result from the iteration of K to K′ and choose i so that the ultrapower map σ_{i} : K_{i} → K_{i+1} applies the order 0 measure at κ = cof ^{K_{i}}(λ). If ⟨λ_{j} | j < κ⟩ is a continuous and increasing sequence in K_{i} with supremeum λ, then λ^{+} of K_{i+1} is represented in the ultrapower of K_{i} by ⟨λ_{j}^{+} | j < κ⟩. In K_{i}, the product of the λ_{j}^{+}’s contains a subset of size (λ^{+})^{K_{i}}, consisting of functions well-ordered by dominance on a final segment of κ. It follows that (λ^{+})^{K_{i+1}} has cardinality λ^{+} and therefore K_{i+1} computes λ^{+} correctly. As λ^{+} is a fixed point of the remaining iteration from K_{i+1} to K′, it follows that K′ computes λ^{+} correctly. This proves Lemma 12.

We say that λ is a cut point of K′ iff no extender on the K′ sequence with critical point less than λ has length at least λ (i.e., λ is not overlapped in K′). As we have assumed that there is no strong cardinal in K′, there is a closed unbounded class of cut points of K′.

Let ⟨λ_{n} | n ∈ ω⟩ be the first ω-many limit cardinals of V which are cut points of K′, and let λ_{ω} be their supremum. Then each λ_{n}, and of course λ_{ω}, has cofinality ω.

Lemma 13 Each λ_{n}, and λ_{ω} as well, is an absolute parameter.

Proof of Lemma 13. We first show that λ_{0} is absolute. Let V^{∗} be an outer model of V with the same cardinals as V up to λ_{0}. Note that for some real R in V, no L_{α}[R] satisfies ZFC and therefore R^{#} does not exist in V^{∗}. It follows that for any singular cardinal λ of V^{∗}, λ is singular in V and λ^{+} is computed correctly in V. In particular, V^{∗} and V have the same cardinals, and the same singular cardinals, up to λ_{0}^{+}.

It follows by Lemma 12 that for any singular cardinal λ of V^{∗}, λ^{+} is computed correctly in both K′ and (K^{∗})′, where (K^{∗})′ denotes the K′ of V^{∗}, obtained from K^{∗}, the K of V^{∗}, by applying each order 0 measure exactly once. Note that K is universal in V^{∗}, and therefore is a simple iterate of K^{∗}. It follows that K′ is a simple iterate of (K^{∗})′, obtained by lifting the iteration map from K^{∗} to K along the iteration from K^{∗} to (K^{∗})′.

Claim. The iteration from (K^{∗})′ to K′ fixes singular cardinals of V^{∗} which are cut points either of K′ or of (K^{∗})′.

Proof of Claim. Let λ be a singular cardinal of V^{∗}. If λ is a cut point of (K^{∗})′ then as λ has non-measurable cofinality in (K^{∗})′, λ is fixed by the iteration. If λ is a cut point of K′ then as λ has non-measurable cofinality in (K^{∗})′, λ can only move if an extender overlapping λ is applied. As by assumption λ is not overlapped in K′ it must be that the least extender overlapping λ was applied. But then λ^{+} is not computed correctly in the resulting ultrapower and therefore not computed correctly in K′, the result of the iteration, contradicting Lemma 12. This proves the Claim.

It follows from the Claim that λ_{0} is the least limit cardinal of V^{∗} which is a cut point of (K^{∗})′. As V^{∗} is an arbitrary outer model of V with the same cardinals as V up to λ_{0}, we have shown that λ_{0} is an absolute parameter. The same argument shows that each λ_{n} is absolute, and therefore so is λ_{ω}, the supremum of the first ω limit cardinals which are cut points of K′. This proves Lemma 13.

Now let ⟨C_{α} | α < λ_{ω}^{+}, α limit⟩ be the least □_{λ_{ω}} sequence of K′; this is also a □_{λ_{ω}} sequence in V, as (λ_{ω}^{+})^{K′} = λ_{ω}^{+}. As in the proof of Theorem 9, there are generic extensions of V preserving H(λ_{ω}^{+}) which add CUB subsets to each S_{n} = {α < λ_{ω}^{+} | ordertype C_{α} > λ_{n}}. It follows from the strong inner model hypothesis (and the Remark immediately following its statement) that for each n there is an inner model M_{n}, with the correct λ_{ω}^{+} and λ_{n}, in which S_{n}^{M_{n}} contains a CUB subset C_{n}, where S_{n}^{M_{n}} is defined using the least □_{λ_{ω}} sequence of (K′)^{M_{n}}. The latter may of course differ from the least □_{λ_{ω}} sequence of K′. However as λ_{ω}^{+} is computed correctly in each (K′)^{M_{n}} and λ_{ω} is a cut point of non-measurable cofinality in each (K′)^{M_{n}}, it follows that the (K′ | λ_{ω}^{+})^{M_{n}}’s compare to a common K′′ of height λ_{ω}^{+} with all ordinals in some CUB subset C of λ_{ω}^{+} as closure points. But if α is such a closure point in the intersection of the C_{n}’s and α_{n} is the image of α under the comparison embedding of (K′ | λ_{ω}^{+})^{M_{n}} into K′′, then C_{α_{n}} as defined in K′′ contains elements cofinal in α and therefore C_{α} as defined in K′′, an initial segment of C_{α_{n}}, has ordertype at least that of C_{α} as defined in (K′ | λ_{ω}^{+})^{M_{n}}. It follows that C_{α} as defined in K′′ has ordertype greater than λ_{n} for each n, which is a contradiction. □

Remarks. (a) It is likely that Theorem 11 can be improved to obtain an inner model with a Woodin cardinal. But it is not possible to obtain an iterable inner model with a Woodin cardinal and an inaccessible above it (unless the SIMH is inconsistent): Otherwise every real would be generic for Woodin’s extender algebra defined in an iterate of such an inner model, implying that for every real R there is an inaccessible in L[R]; this contradicts Theorem 1.

(b) David Asperó and the first author observed that the consistency of the SIMH for the parameter ω_{1} follows as in the proof of Theorem 7 from that of a Woodin cardinal with an inaccessible above. In particular this yields the consistency of the natural extension of Lévy absoluteness asserting Σ_{1} absoluteness with parameter ω_{1} for arbitrary ω_{1}-preserving extensions. (c) For any finite set of absolute parameters, the version of the SIMH where V^{∗} is required to be a set-generic extension of V is consistent for those parameters.

Question. Is the strong inner model hypothesis consistent relative to large cardinals?

References

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[6] Mitchell, W. An introduction to inner models and large cardinals, a chapter for the Handbook of Set Theory.

[7] Moschovakis, Y. Descriptive set Theory, North-Holland Series in Logic, Amsterdam, 1980.

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