The Journal of Symbolic Logic Volume 00, Number 0, XXX 0000

STRONG COMPACTNESS AND STATIONARY SETS

JOHN KRUEGER

Abstract. We construct a model in which there is a strongly compact cardinal κ such that the set S(κ, κ^{+}) = {a ∈ P_{κ}κ^{+} : o.t.(a) = (a ∩ κ)^{+}} is non-stationary.

If κ is a κ^{+}-supercompact cardinal then the set S(κ, κ^{+}) = {a ∈ P_{κ}κ^{+} : o.t.(a) = (a ∩ κ)^{+}} is stationary. A natural question to ask is whether the κ^{+}- strong compactness of a cardinal κ implies the stationarity of S(κ, κ^{+}). In the present paper we solve this problem by constructing a model in which there is a strongly compact cardinal κ such that S(κ, κ^{+}) is non-stationary.

We use two main techniques in our consistency proof. First, we use our forcing poset from [6] for destroying stationary subsets of P_{κ}κ^{+} using partial square sequences. Secondly, we use a Magidor iteration of Prikry forcing.

The method of argument we present has applications to other problems. For example, in the final section we construct a model in which there is a strongly compact cardinal κ with 2^{(κ^{+})} = κ^{+++}, but for a club of α < κ, 2^{(α^{+})} = α^{++}.

The contents of the paper are as follows. In Section 1 we outline preliminaries and notation. Section 2 describes our method of using partial square sequences to destroy stationary sets. Section 3 provides some additional background material on large cardinals and forcing. Section 4 gives a summary of the consistency proof. In Section 5 we complete the first step of the consistency proof by iterating the partial square poset. In Section 6 we finish the proof using a Magidor iteration of Prikry forcing. In Section 7 we give another application of our method to prove a consistency result concerning strong compactness and GCH.

I would like to thank Moti Gitik for discussing this material with me.

§1. Preliminaries. We assume that the reader is familiar with iterated forcing, Prikry forcing, and supercompact cardinals; see [1], [2], and [5].

If κ is regular and κ ⊆ X, define P_{κ}X as the set of a ⊆ X such that |a| < κ and a ∩ κ ∈ κ. A set C ⊆ P_{κ}X is club if it is closed under unions of ⊆-increasing sequences with length less than κ, and for all a in P_{κ}X there is b ∈ C with a ⊆ b. A set S ⊆ P_{κ}X is stationary if for every club C, S ∩ C is non-empty.

The expression “θ ≫ κ” indicates that θ > 2^{2^{|H(κ)|}}. If θ ≫ κ and A is a structure with underlying set H(θ), then the collection of elementary substructures of A in P_{κ}H(θ) is a club set.

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This work was supported by FWF project number P16790-N04.

An ultrafilter U on P_{κ}X is normal if for every function f : P_{κ}X → X which is regressive (i.e. f(a) ∈ a for all a) there is a set A in U such that f ↾ A is constant. We assume our ultrafilters are non-principal and fine; that is, {a} ∉ U for all a, and for any i in X, the set {a ∈ P_{κ}X : i ∈ a} is in U.

Suppose that κ ≤ λ are cardinals. Then κ is λ-supercompact if there exists a normal ultrafilter on P_{κ}λ, or equivalently, there exists an elementary embedding j : V → M, where M is a transitive inner model of ZFC, such that crit(j) = κ, j(κ) > λ, and ^{λ}M ⊆ M. We say that κ is λ-strongly compact if there exists a κ-complete ultrafilter on P_{κ}λ, or equivalently, there is an elementary embedding j : V → M, where M is a transitive inner model of ZFC, such that crit(j) = κ, j(κ) > λ, and there exists a set a in M with |a| < j(κ) such that j“λ ⊆ a. A cardinal κ is supercompact (respectively, strongly compact) if for all λ > κ, κ is λ-supercompact (respectively, λ-strongly compact).

If a is a set of ordinals then o.t.(a) denotes the order type of a. If o.t.(a) is a limit ordinal, cf(a) denotes cf(o.t.(a)). Note that cf(a) = cf(sup a).

If θ ≫ κ and N ≺ H(θ) is in P_{κ}H(θ), then N ∩ κ^{+} contains as a member any limit point below sup(N ∩ κ^{+}) which has cofinality different from cf(N ∩ κ).

If M ⊆ N are transitive models of set theory with the same ordinals and λ is a cardinal, we say that M is λ-closed in N if ^{λ}M ∩ N ⊆ M. The model M is λ-closed in N iff ^{λ}On ∩ N ⊆ M.

We will use the phrase forcing poset to indicate any ordering ⟨ℙ, ≤⟩ whose relation ≤ is reflexive and transitive. For a detailed discussion of the author’s conventions concerning forcing posets, separative posets, and Boolean algebras, see [7].

If ℙ is a forcing poset and p is a condition in ℙ, we write ℙ/p for the forcing poset with underlying set {q ∈ ℙ : q ≤ p}, ordered the same as in ℙ.

Suppose φ is a statement in the forcing language for a poset ℙ. We write ^{0}φ for φ and ^{1}φ for ¬φ.

If λ is an ordinal, a canonical ℙ-name for a subset of λ is a name of the form {⟨p, α̌⟩ : p ∈ A_{α}, α < λ}, where each A_{α} is an antichain. If p forces that Ẋ is a subset of λ, then there is a canonical name Ẏ for a subset of λ such that p forces that Ẋ = Ẏ.

Suppose that G is generic for a forcing poset ℙ and φ is a statement in the forcing language. If each p in G has an extension q ≤ p which forces φ, then φ holds in V [G].

A forcing poset ℙ is λ-strategically closed for an ordinal λ if there is a strategy for Player II in the following game: Player I starts the game by playing a condition p_{1} in ℙ. Player II responds with a condition p_{2} ≤ p_{1}. The game continues in this manner, each player choosing a condition below the previous one, with Player I playing at odd stages and Player II at even successor stages. At limit stages Player II plays a condition below all the conditions played so far. Player II wins if it is able to play a condition at all stages below λ.

For a regular cardinal α and an ordinal β, the forcing poset Add(α, β) consists of conditions p which are partial functions α × β → 2 with size less than α, ordered by q ≤ p if p ⊆ q. The poset Add(α, β) is α-closed and (α^{<α})^{+}-c.c. If G is generic for Add(α, β) then G codes a sequence of β many new subsets of α.

A triple ⟨ℚ, ≤, ≤^{∗}⟩ is called a Prikry type forcing poset if ⟨ℚ, ≤⟩ and ⟨ℚ, ≤^{∗}⟩ are forcing posets, q ≤^{∗} p implies q ≤ p, and ℚ satisfies the Prikry property: for any φ in the forcing language for ⟨ℚ, ≤⟩ and for any p in ℚ, there exists q ≤^{∗} p such that q decides φ. If α is a cardinal we say that ℚ is α-weakly closed if ⟨ℚ, ≤^{∗}⟩ is α-closed. We say that ℚ satisfies the direct extension property if whenever q, r ≤^{∗} p, there is s ≤^{∗} q, r. When we say that we force with a Prikry type forcing poset ℚ, we always mean that we force with ⟨ℚ, ≤⟩.

§2. Stationary sets and partial squares. Suppose that κ is a weakly inaccessible cardinal. Then there is a club set of a in P_{κ}κ^{+} such that o.t.(a) ≤ (a∩κ)^{+} (see [7]). Define S(κ, κ^{+}) as the set of a in P_{κ}κ^{+} such that o.t.(a) = (a ∩ κ)^{+}.

In [6] we introduced the idea of a partial square sequence and showed how to use such a sequence to destroy certain stationary subsets of S(κ, κ^{+}). This method will play a crucial role in the proof of the main theorem.

Let B ⊆ κ^{+} be a set of limit ordinals. We say that □_{κ}^{B} holds if there exists a partial square sequence ⟨c_{α} : α ∈ B⟩ satisfying:

(1) c_{α} is club in α,

(2) if cf(α) < κ then o.t.(c_{α}) < κ,

(3) if β is in lim(c_{α}) ∩ B then c_{β} = c_{α} ∩ β.

It is easy to force a partial square sequence using a generalization of Jensen’s poset for adding a square sequence. Suppose that B ⊆ κ^{+} is an unbounded set of limit ordinals. Define ℙ_{B} as follows. A condition in ℙ_{B} is a sequence ⟨c_{α} : α ∈ B ∩ (β + 1)⟩, for some β < κ^{+}, satisfying (1), (2), and (3) above. We let q ≤ p if p is an initial segment of q.

Note that ℙ_{B} has size no larger than 2^{κ}. The proof of the next lemma is basically the same as the proof of the corresponding fact for Jensen’s poset for adding a square.

Lemma 2.1. The poset ℙ_{B} is (κ + 1)-strategically closed.

Proof. We describe a strategy by considering a run of the game. Suppose ⟨p_{i} : i < β⟩ is a run of the game up to stage β and it is Player II’s turn. For each i < β let γ_{i} be the least ordinal γ so that p_{i} is of the form ⟨c_{α} : α ∈ B ∩ (γ + 1)⟩. First assume β = α + 1. Let ξ be the least element of B larger than γ_{α}, and define p_{β} = p_{α} ∪{⟨ξ, c_{ξ}⟩}, where c_{ξ} is some club subset of ξ with order type cf(ξ) and min(c_{ξ}) > γ_{α}.

Suppose that β is a limit ordinal. Let γ_{β} = ⋃_{i<β} γ_{i}. If γ_{β} is not in B, then let p_{β} = ⋃{p_{i} : i < β}. If γ_{β} is in B, then define p_{β} as ⋃{p_{i} : i < β} ∪ {⟨γ_{β}, c_{γ_{β}}⟩}, where c_{γ_{β}} = {γ_{i} : i < β}. If β < κ then c_{γ_{β}} has order type less than κ. Suppose that γ is a limit point of c_{γ_{β}} in B. Then γ = γ_{i} for some limit ordinal i and c_{γ} = {γ_{j} : j < i} = c_{γ_{β}} ∩ γ. ⊣

A standard argument shows the lemma implies that ℙ_{B} does not add any subsets to κ. The lemma also implies that ℙ_{B} adds a partial square sequence with domain B.

Now let us show how to use this forcing poset to destroy stationary subsets of P_{κ}κ^{+}. Let B ⊆ κ^{+} be an unbounded set of limit ordinals and μ_{0}, μ_{1} distinct regular cardinals less than κ such that B contains all its limit points with cofinality either μ_{0} or μ_{1}.

Lemma 2.2. The poset ℙ_{B} destroys the stationarity of the set S = {a ∈ S(κ, κ^{+}) : sup(a) ∈ B}.

Proof. Let G be generic for ℙ_{B} and let ⟨c_{β} : β ∈ B⟩ be the generic partial square sequence given by G.

Suppose for a contradiction that S is stationary in V [G]. Fix θ ≫ κ^{+} regular. Then there is N in P_{κ}H(θ) such that N ∩ κ^{+} ∈ S, N ∩ κ is a cardinal, μ_{0} and μ_{1} are in N, and N is an elementary substructure of

⟨H(θ), ∈, ⟨c_{β} : β ∈ B⟩⟩.

Let a = N ∩ κ^{+}, β = sup(a), and κ_{a} = a ∩ κ. Since a is in S, the ordinal β is in B. Note that cf(β) = κ_{a}^{+}. In particular, cf(β) > μ_{0}, μ_{1}.

Let μ be one of μ_{0} or μ_{1} which is different from cf(κ_{a}). Then a is closed under suprema of bounded subsets with order type μ. By the choice of B and the (κ + 1)-strategic closure of ℙ_{B}, the set B is also closed under suprema of subsets with order type μ. By elementarity, β is a limit point of B. So a ∩ B is a stationary subset of β.

Since cf(β) = κ_{a}^{+}, o.t.(c_{β}) ≥ κ_{a}^{+}. So there exists γ in a ∩ B ∩ lim(c_{β}) such that o.t.(c_{β} ∩ γ) ≥ κ_{a}. Then c_{γ} = c_{β} ∩ γ has order type at least κ_{a}. But by elementarity, o.t.(c_{γ}) is in N ∩ κ = κ_{a}, which is a contradiction. ⊣

Note that since ℙ_{B} is (κ + 1)-distributive, the set S in Lemma 2.2 satisfies the same definition in V [G] as it does in V.

The poset ℙ_{B} is simpler and easier to work with than previous posets for destroying stationary subsets of P_{κ}κ^{+}, as found in [4] and [7]. For more applications of the partial square poset see [6].

§3. Background on large cardinals and forcing. In this section we give a summary of the background material on large cardinals and forcing which we use in our consistency proof. None of the results in this section are due to the author.

The following lemma is the main tool for extending elementary embeddings.

Lemma 3.1. Suppose that j : M → N is an elementary embedding between transitive models of set theory, ℙ is a forcing poset in M, G is generic for ℙ over M, and H is generic for j(ℙ) over N.

Then j can be lifted to j : M[G] → N[H] such that j(G) = H iff j“G ⊆ H. In particular, j can be lifted if there exists a condition s in H such that s ≤ j(p) for all p in G.

Proof. If j“G ⊆ H then the map j(ẋ^{G}) = j(ẋ)^{H} is well-defined and satisfies the required properties. ⊣

We will use Silver’s notation and refer to a condition s in j(ℙ) such that s ≤ j(p) for all p in G as a master condition.

Lemma 3.2. Suppose that U is a normal ultrafilter on a cardinal κ and j : V → M = Ult(V, U). Let ℙ be a forcing poset and let G be a generic filter for ℙ over V. Suppose that in V [G] there is a generic filter H for j(ℙ) over M such that j“G ⊆ H. Lift j to j : V [G] → M[H].

In V [G] define U^{∗} by letting X ∈ U^{∗} iff X ⊆ κ and κ ∈ j(X). Then U^{∗} is a normal ultrafilter extending U and M[H] = Ult(V [G], U^{∗}).

Proof. We omit the standard argument that U^{∗} is a normal ultrafilter extending U. Since the only isomorphism between two transitive models of set theory is the identity mapping , it suffices to prove that M[H] and Ult(V [G], U^{∗}) are isomorphic.

Define k : M[H] → Ult(V [G], U^{∗}) as follows. Let a be in M[H]. Then there is a j(ℙ)-name ẋ in M such that ẋ^{H} = a. Since ẋ is in M, we can choose f : κ → V such that [f] = ẋ. Moreover, choose f so that f(α) is a ℙ-name for all α. In V [G] define g : κ → V [G] by letting g(α) = f(α)^{G}. Now let k(a) = [g].

First we show that k is well-defined. Suppose that a = ẋ^{H} = ẏ^{H}, where [fẋ] = ẋ and [fẏ] = ẏ. Then there exists a condition q in H = j(G) which forces over M that ẋ = ẏ. In M let q = [h] and write h(α) = p_{α}. Then there exists a set A in U such that for all α in A, p_{α} forces that fẋ(α) = fẏ(α). In M[H], q is in j(G). But j(h)(κ) = q in M and hence in M[H]. By definition of U^{∗}, there is A^{∗} ⊆ A in U^{∗} such that for all α in A^{∗}, h(α) = p_{α} is in G. Then for all α in A^{∗}, fẋ(α)^{G} = fẏ(α)^{G}. Define gẋ and gẏ by gẋ(α) = fẋ(α)^{G} and gẏ(α) = fẏ(α)^{G}. Then for all α in A^{∗}, gẋ(α) = gẏ(α), so [gẋ] = [gẏ]. This proves that k is well-defined.

A similar argument shows that k is injective and that a ∈ b iff k(a) ∈ k(b). To show that k is surjective, fix [h] in Ult(V [G], U^{∗}). Let ḣ be a ℙ-name for h, and for each α let ḃ_{α} be a ℙ-name for ḣ(α). Then [α ↦ ḃ_{α}] is a j(ℙ)-name in M. Let a = [α ↦ ḃ_{α}]^{H}, which is in M[H]. Using the definition of k, it is straightforward to check that k(a) = [h]. ⊣

A standard way to extend an elementary embedding is to apply strategic closure to build a generic filter. Suppose that M ⊆ N are transitive models of set theory and λ is an N-cardinal. Let ℙ be a forcing poset in M and let p be in ℙ. Suppose that N models that ℙ is λ-strategically closed and has no more than λ many maximal antichains in M. Enumerate all maximal antichains in M as ⟨A_{i} : i < λ⟩. Applying strategic closure we can inductively define a decreasing sequence ⟨p_{i} : i < λ⟩ so that p_{0} = p and p_{i+1} is below some member of A_{i}. This sequence of conditions generates a generic filter H for ℙ over M which contains p.

The following two lemmas show how to verify closure of generic extensions.

Lemma 3.3. Suppose that M ⊆ N are transitive models of set theory with the same ordinals and λ is a regular uncountable cardinal in N such that M is < λclosed in N. If ℙ is a forcing poset in M which is λ-c.c. in N and G is generic for ℙ over N, then M[G] is < λ-closed in N[G].

Proof. We prove that ^{<λ}On ∩ N[G] ⊆ M[G]. Suppose that p is in G and p forces over N that ḟ : β → On for some β < λ. For each α < β let A_{α} be a maximal antichain below p contained in the dense set of conditions which decide the value of ḟ(α). Let X_{α} be the set of pairs ⟨q, γ⟩ such that q ∈ A_{α} and q forces over N that ḟ(α) = γ. Then |A_{α}| = |X_{α}| < λ, and so ⟨A_{α}, X_{α} : α < β⟩ is in M. Define a name ġ in M by letting ġ(α) be the unique γ so that there is q in Ġ ∩ A_{α} such that ⟨q, γ⟩ is in X_{α}. Clearly p forces that ġ = ḟ, and ġ^{G} is in M[G]. ⊣

Lemma 3.4. Suppose M ⊆ N are transitive models of set theory with the same ordinals and λ is a regular cardinal in N such that M is λ-closed in N. If ℙ is a forcing poset in M and G is a generic filter for ℙ over M lying in N, then M[G] is λ-closed in N.

Proof. In N we have ^{λ}On ⊆ M ⊆ M[G]. ⊣

Lemma 3.5. Suppose that κ is a measurable cardinal, ℙ is a forcing poset with size less than κ, and U is a normal ultrafilter on κ. Then ℙ forces that the filter U^{∗} generated by U is a normal ultrafilter.

Proof. Let j : V → M = Ult(V, U). Let G be generic for ℙ over V. Since crit(j) = κ, j(ℙ) = j“ℙ and j“G is generic for j(ℙ) over M. Extend j to j : V [G] → M[j“G]. Define U′ by letting X ∈ U′ iff X ⊆ κ and κ ∈ j(X). Then U′ is a normal ultrafilter containing U.

Let U^{∗} be the filter generated by U in V [G]. Clearly U^{∗} ⊆ U′. To show that U^{∗} = U′ we show that U^{∗} is an ultrafilter. Suppose that p is in G and p forces that Ȧ ⊆ κ. Define a map f : κ → (ℙ/p × 2) in V by letting f(α) be some pair ⟨q, i⟩ such that q ⊩ ^{i}(α ∈ Ȧ). Since |ℙ| < κ, there is B in U and ⟨q, i⟩ such that f(α) = ⟨q, i⟩ for all α in B. But then q forces that B is a subset of Ȧ if i = 0, and is a subset of the complement of Ȧ if i = 1. So U^{∗} is an ultrafilter. ⊣

§4. Outline of the main theorem. We outline the argument for the construction of a model in which there is a strongly compact cardinal κ such that S(κ, κ^{+}) is non-stationary.

Start with a model V in which there is a supercompact cardinal κ and GCH holds. Let A be the set of α < κ such that α is measurable.

The basic idea of the proof is to add a club subset to P_{κ}κ^{+} which makes almost all a in S(κ, κ^{+}) satisfy that a ∩ κ is in A, while at the same time destroying the stationarity of A.

We construct the model in two steps. In the first step we add a club to P_{κ}κ^{+} by forcing a partial square sequence. Since we will want to preserve the supercompactness of κ, we iterate such partial square posets for α in A ∪ {κ} using an Easton support iteration.

Let W be a generic extension of V by this iteration. Then in W, each α in A is still a measurable cardinal. In the second step of the proof we use a Magidor iteration to add a Prikry sequence to each α in A. We show that the Magidor iteration destroys the stationarity of A and preserves the strong compactness of κ. A key point is that by an absoluteness argument, the Magidor iteration will not change the properties of the partial square sequence added in the first stage.

§5. Iterating partial squares. In this section we complete the first step of the consistency proof as outlined above.

Start with a model V in which GCH holds and there is a supercompact cardinal κ. Also assume that there is no inaccessible cardinal above κ. Let A be the set of measurable cardinals less than κ. For each α in A let U_{α} be a normal ultrafilter on α such that A ∩ α is not in U_{α}.

Define an Easton support forcing iteration ℙ_{κ} ∗ ℚ_{κ} by induction as follows. Suppose that ℙ_{α} is defined for a fixed α ≤ κ. If α is not in A ∪ {κ} then let ℚ_{α} be trivial. Suppose that α is in A ∪ {κ}.

Let G_{α} be generic for ℙ_{α} over V. In V [G_{α}] let B_{α} be the set of limit ordinals β < α^{+} such that cf(β) is less than α and cf(β) is not equal to μ^{+} for any μ in A ∩ α.

Now let ℚ_{α} be the forcing poset from Section 2 which adds a sequence ⟨c_{β} : β ∈ B_{α}⟩ such that:

(1) c_{β} is club in β,

(2) cf(c_{β}) < α,

(3) if γ ∈ lim(c_{β}) ∩ B_{α}, then c_{γ} = c_{β} ∩ γ. The poset ℙ_{α} forces that ℚ_{α} is (α + 1)-strategically closed and has size α^{+}.

This completes the definition of the iteration.

Suppose that V_{1} is a generic extension of V by ℙ_{κ} ∗ ℚ_{κ}. We show that for almost all a in S(κ, κ^{+}), a ∩ κ is in A. Applying Lemma 2.2, let C be a club subset of P_{κ}κ^{+} in V_{1} such that for all a in C ∩ S(κ, κ^{+}), a ∩ κ is a cardinal and sup(a) has cofinality equal to μ^{+} for some μ in A. Consider a in C ∩ S(κ, κ^{+}). Then cf(sup(a)) = (a ∩ κ)^{+}. Since a ∩ κ is a cardinal, it follows that a ∩ κ is in A.

Proposition 5.1. For all α in A, ℙ_{κ} ∗ ℚ_{κ} forces that there is a normal ultrafilter U_{α}^{∗} on α such that U_{α} ⊆ U_{α}^{∗}.

Proof. Write ℙ_{κ} ∗ ℚ_{κ} = ℙ_{α} ∗ ℚ_{α} ∗ ℙ_{α,κ} ∗ ℚ_{κ}. Let G_{α} be generic for ℙ_{α} over V. Since ℚ_{α} ∗ℙ_{α,κ} ∗ℚ_{κ} does not add subsets to α over V [G_{α}], it suffices to prove that U_{α} can be lifted in V [G_{α}].

If α is not a limit point of A then |ℙ_{α}| < α, and U_{α} can be lifted by Lemma 3.5. Suppose that α is a limit point of A.

In V let j_{α} : V → M_{α} = Ult(V, U_{α}). Since α is not in j_{α}(A ∩ α), j_{α}(ℙ_{α}) factors as ℙ_{α} ∗ ℙ_{tail} where ℙ_{tail} is forced to be (min(j_{α}(A ∩ α) \ (α + 1)) + 1)- strategically closed. The poset ℙ_{α} has size α and is α-c.c., so has no more than α-many antichains. Therefore ℙ_{tail} has no more than j_{α}(α)-many antichains in M_{α}[G_{α}]. But j_{α}(α) has size α^{+} in V [G_{α}]. So we can enumerate the set of maximal antichains of ℙ_{tail} in M_{α}[G_{α}] as ⟨A_{i} : i < α^{+}⟩. Since M_{α}[G_{α}] is αclosed, ℙ_{tail} is α^{+}-strategically closed in V [G_{α}]. So we can construct a generic G_{tail} for ℙ_{tail} over M_{α}[G_{α}].

Now lift j_{α} to j_{α} : V [G_{α}] → M_{α}[G_{α} ∗ G_{tail}]. Define U_{α}^{∗} by letting X ∈ U_{α}^{∗} iff X ⊆ α and α ∈ j_{α}(X). Then U_{α}^{∗} is a normal ultrafilter on α with U_{α} ⊆ U_{α}^{∗}. ⊣

Proposition 5.2. The poset ℙ_{κ} ∗ ℚ_{κ} forces that κ is supercompact. In fact, for all λ > κ, ℙ_{κ} ∗ ℚ_{κ} forces that there is a normal ultrafilter on P_{κ}λ which concentrates on A.

Proof. Fix λ > κ^{+} regular. Let U be a normal ultrafilter on P_{κ}λ and let j : V → M = Ult(V, U). By the closure of M, κ is measurable in M, so κ is in j(A). Write j(ℙ_{κ}) = ℙ_{κ} ∗ ℚ_{κ} ∗ ℙ_{tail}.

Let G_{κ}∗G be generic for ℙ_{κ}∗ℚ_{κ}. Since we assumed that there is no inaccessible above κ, min(j(A)\(κ+1)) > λ^{+}. So ℙ_{tail} is forced to be λ^{+}-strategically closed. Since ℙ_{κ} ∗ ℚ_{κ} is κ^{++}-c.c., M[G_{κ} ∗ G] is λ-closed in V [G_{κ} ∗ G]. Therefore ℙ_{tail} is λ^{+}-strategically closed in V [G_{κ} ∗ G]. In M[G_{κ} ∗ G], ℙ_{tail} has j(κ) many maximal antichains. In V [G_{κ} ∗ G], |j(κ)| = κ^{λ} = λ^{+}. So we can construct a generic filter G_{tail} for ℙ_{tail} over M[G_{κ} ∗ G]. Extend j to j : V [G_{κ}] → M[G_{κ} ∗ G ∗ G_{tail}].

We construct a master condition. For each p in G there is β < κ^{+} such that p = ⟨c_{α} : α ∈ B_{κ}∩(β+1)⟩. So j(p) = ⟨d_{α} : α ∈ j(B_{κ})∩j(β+1)⟩. Let s = ⋃ j“G. Then s = ⟨d_{α} : α ∈ j(B_{κ}) ∩ sup(j“κ^{+})⟩. The sequence s is a condition in j(ℚ_{κ}) iff the domain of s is equal to j(B_{κ}) ∩ (sup(j“κ^{+}) + 1), that is, iff sup(j“κ^{+}) is not in j(B_{κ}). But sup(j“κ^{+}) has cofinality equal to the successor of κ, and κ is a member of j(A). So by the definition of B_{κ}, sup(j“κ^{+}) is not in j(B_{κ}).

The poset j(ℚ_{κ}) has j(κ^{++}) many maximal antichains in M[G_{κ} ∗ G ∗ G_{tail}]. But j(κ^{++}) has size λ^{+}. In M[G_{κ} ∗ G ∗ G_{tail}], j(ℚ_{κ}) is j(κ)-strategically closed. But M[G_{κ} ∗ G ∗ G_{tail}] is closed under λ-sequences and |j(κ)| = λ^{+}. So in V [G_{κ} ∗ G], j(ℚ_{κ}) is λ^{+}-strategically closed. So we can construct a generic H for j(ℚ_{κ}) which contains the master condition s. Now extend j to j : V [G_{κ} ∗ G] → M[G_{κ} ∗ G ∗ G_{tail} ∗ H].

Define U^{∗} by letting X ∈ U^{∗} iff X ⊆ P_{κ}λ and j“λ ∈ j(X). Standard arguments show that U^{∗} is a normal ultrafilter extending U in V [G_{κ} ∗ G]. ⊣

§6. Magidor iteration. Let W be a generic extension of V by the poset ℙ_{κ} ∗ℚ_{κ} from the last section. Then in W, κ remains supercompact and for every α in A there is a normal ultrafilter U_{α}^{∗} on α such that A ∩ α is not in A.

We use a Magidor iteration to iterate Prikry forcing over α in A using the ultrafilters U_{α}^{∗}. Define 𝕄_{κ} by recursion as follows. Our recursion hypothesis is that for each α in A, 𝕄_{α} forces that there is a normal ultrafilter U_{α}^{∗∗} on α which extends U_{α}^{∗}.

Suppose that 𝕄_{α} is defined for some α < κ. If α is not in A then force with the trivial poset at stage α. If α is in A, then let 𝕄_{α+1} = 𝕄_{α} ∗ ℙℝ(U_{α}^{∗∗}), where ℙℝ(U_{α}^{∗∗}) is a name for the Prikry forcing defined from the ultrafilter U_{α}^{∗∗} which exists by the recursion hypothesis. Recall that Prikry forcing on α is a Prikry type forcing poset which satisfies the direct extension property and is α-weakly closed.

Suppose that α is a limit ordinal and 𝕄_{β} is defined for all β < α using Prikry forcing at stages in A∩α. Define 𝕄_{α} as follows. A condition in 𝕄_{α} is a function p with domain α such that for all β < α, p ↾ β is in 𝕄_{β}; moreover, there exists a finite set a_{p} ⊆ α such that β is in α \ a_{p} iff p ↾ β forces that p(β) ≤^{∗} 1. We let q ≤ p if q ↾ β ≤ p ↾ β for all β < α. Let q ≤^{∗} p if for all β < α, q ↾ β ⊩ q(β) ≤^{∗} p(β).

Such an iteration satisfies the Prikry property; see [8] or [3] for a proof.

Note that 𝕄_{α} satisfies the direct extension property and is min(A)-weakly closed.

Lemma 6.1. If α is strongly inaccessible then 𝕄_{α} is α^{+}-c.c.

Proof. For each β < α, |𝕄_{β}| < α. By the direct extension property, if p and q are conditions such that a_{p} = a_{q} and p ↾ a_{p} = q ↾ a_{q}, then p and q are compatible. There are α many possibilities for a_{p}, and given a_{p}, there are less than α many possibilities for p ↾ a_{p}. ⊣

Suppose that α is in A. We verify the recursion hypothesis by showing that 𝕄_{α} forces that U_{α}^{∗} can be lifted.

If α is not a limit point of A then |𝕄_{α}| < α and all normal ultrafilters on α can be lifted by Lemma 3.5. Suppose that α is a limit point of A.

Proposition 6.2. Suppose that G_{α} is generic for 𝕄_{α} over W. Then in W [G_{α}] there is a normal ultrafilter U_{α}^{∗∗} such that U_{α}^{∗} ⊆ U_{α}^{∗∗}.

Proof. Let j_{α} : W → M_{α} = Ult(W, U_{α}^{∗}). Since α is not in j_{α}(A), write j_{α}(𝕄_{α}) = 𝕄_{α} ∗ 𝕄_{tail} where 𝕄_{tail} is forced to be α^{+}-weakly closed.

In W [G_{α}] define U_{α}^{∗∗} by letting X ∈ U_{α}^{∗∗} iff X ⊆ α and there is an 𝕄_{α}-name Ẋ for X and a condition q ≤^{∗} 1 in 𝕄_{tail} such that q forces over M_{α}[G_{α}] that α ∈ j_{α}(Ẋ).

First we show that the definition does not depend on the choice of Ẋ. Suppose that X = Ẋ^{G_{α}} = Ẏ^{G_{α}}. By the Prikry property there exists qẊ ≤^{∗} 1 in 𝕄_{tail} which decides the statement α ∈ j_{α}(Ẋ) and qẎ ≤^{∗} 1 which decides α ∈ j_{α}(Ẏ). We show that qẊ and qẎ decide their respective statements the same way.

Suppose for a contradiction that qẊ ⊩ α ∈ j_{α}(Ẋ) and qẎ ⊩ α ∉ j_{α}(Ẏ). Fix p in G_{α} which forces that Ẋ = Ẏ. Then j_{α}(p) forces that j_{α}(Ẋ) \ j_{α}(Ẏ) is empty. Since a_{p} ⊆ α is finite, j_{α}(a_{p}) = a_{p}. Therefore j_{α}(p) = p r where r ≤^{∗} 1 in 𝕄_{tail}. Note that r forces over M_{α}[G_{α}] that j_{α}(Ẋ) \ j_{α}(Ẏ) is empty. By the direct extension property, there is s which directly extends r, qẊ, and qẎ. Then s forces α ∈ j_{α}(Ẋ) \ j_{α}(Ẏ), contradicting the fact that s ≤ r.

Similar arguments show that U_{α}^{∗∗} is a uniform ultrafilter extending U_{α}^{∗}. To show that it is normal, let f : α → α be a regressive function. Fix an 𝕄_{α}-name ḟ for f and p in G_{α} which forces that ḟ is regressive. Then j_{α}(p) forces that j_{α}(ḟ) : j_{α}(α) → j_{α}(α) is regressive. Write j_{α}(p) = p r where r ≤^{∗} 1 in 𝕄_{tail}. Then r forces over M_{α}[G_{α}] that j_{α}(ḟ) is regressive. Using the α^{+}-weak closure of 𝕄_{tail} in M_{α}[G_{α}], define a ≤^{∗}-decreasing sequence ⟨q_{i} : i ≤ α⟩, where q_{0} = r and q_{i+1} decides the statement j_{α}(ḟ)(α) = i. Then q_{α} decides all such statements, so there must be some β < α such that q_{α} ⊩ j_{α}(ḟ)(α) = β. Let Ẋ be an 𝕄_{α}-name such that p forces that Ẋ = {γ < α : ḟ(γ) = β}. Since q_{α} ≤ r, q_{α} forces that α ∈ j_{α}(Ẋ). But q_{α} ≤^{∗} 1, so Ẋ^{G_{α}} is in U_{α}^{∗∗}. ⊣

This completes the definition of 𝕄_{κ}.

Proposition 6.3. The poset 𝕄_{κ} forces that κ is strongly compact.

Proof. Let λ > κ be regular. In W let U be a normal ultrafilter on P_{κ}λ which concentrates on A. Let j : W → M = Ult(W, U). Since κ is in j(A), j(𝕄_{κ}) = 𝕄_{κ} ∗ ℙℝ(U_{κ}^{∗∗}) ∗ 𝕄_{tail}. Note that ℙℝ(U_{κ}^{∗∗}) ∗ 𝕄_{tail} is forced to be κweakly closed.

Let G_{κ} be generic for 𝕄_{κ}. In W [G_{κ}] define U^{∗} by letting X ∈ U^{∗} iff there is an 𝕄_{κ}-name Ẋ for X and q ≤^{∗} 1 in ℙℝ(U_{κ}^{∗∗}) ∗ 𝕄_{tail} such that q forces over M[G_{κ}] that j“λ ∈ j(Ẋ).

The proof that U^{∗} is a fine ultrafilter on P_{κ}λ is similar to the proof of Proposition 6.2. We show that U^{∗} is κ-complete. Suppose that p is in G_{κ} and p forces that ⟨Ẋ_{i} : i < β⟩ is a partition of P_{κ}λ for some β < κ. Then j(p) forces that j(⟨Ẋ_{i} : i < β⟩) = ⟨j(Ẋ_{i}) : i < β⟩ is a partition of j(P_{κ}λ). Since j(a_{p}) = a_{p}, j(p) = p r where r ≤^{∗} 1 in ℙℝ(U_{κ}^{∗∗}) ∗ 𝕄_{tail}. Applying weak closure, find q ≤^{∗} r in ℙℝ(U_{κ}^{∗∗}) ∗ 𝕄_{tail} which decides the statements j“λ ∈ j(Ẋ_{i}) for i < β. Since r forces that ⟨j(Ẋ_{i}) : i < β⟩ is a partition of j(P_{κ}λ) and j“λ is in j(P_{κ}λ), there must be some i < β such that q forces that j“λ ∈ j(Ẋ_{i}). Then Ẋ_{i}^{G_{κ}} is in U^{∗}. ⊣

Lemma 6.4. If μ is in A, then 𝕄_{κ} forces that both μ and μ^{+} remain cardinals.

Proof. Write 𝕄_{κ} = 𝕄_{μ} ∗ℙℝ(U_{μ}^{∗∗})∗𝕄_{tail}. Then 𝕄_{μ} preserves both μ and μ^{+} because it preserves the measurability of μ and is μ^{+}-c.c. Prikry forcing preserves μ and μ^{+} since it does not add bounded subsets to μ is μ^{+}-c.c. Finally, 𝕄_{tail} is forced to be min(A \ (μ + 1))-weakly closed so does not add subsets to μ^{+}. ⊣

We now turn to the proof of the main theorem.

Let G_{𝕄} be generic for 𝕄_{κ} over W. Then κ is strongly compact in W [G_{𝕄}]. For each α in A, let c_{α} be the Prikry sequence added by ℙℝ(U_{α}^{∗∗}).

Proposition 6.5. Define C_{𝕄} in W [G_{𝕄}] as the set of β < κ such that for all α ≥ β in A, min(c_{α}) ≥ β. Then C_{𝕄} is a club subset of κ disjoint from A.

Proof. If α is in A, then min(c_{α}) < α, so α is not in C_{𝕄}. Suppose that δ < κ and ⟨β_{i} : i < δ⟩ is an increasing sequence from C_{𝕄}. Let β = sup_{i<δ} β_{i}. If α ≥ β is in A, then for each i < δ, min(c_{α}) ≥ β_{i}. So min(c_{α}) ≥ sup_{i<δ} β_{i} = β.

To show that C_{𝕄} is unbounded in κ, let β < κ and let p be some condition in G_{𝕄}. Fix ξ in κ\A larger than max(a_{p}). Define q by letting q ↾ (ξ+1) = p ↾ (ξ+1), and for α in A \ ξ, let q(α) be a name for p(α) \ ξ. Then q forces that ξ is in C_{𝕄}. ⊣

Suppose for a contradiction that S(κ, κ^{+}) is stationary in W [G_{𝕄}]. Let ⟨c_{α} : α ∈ B_{κ}⟩ be the partial square sequence given by the generic for ℚ_{κ} in W. Recall that in W, B_{κ} is the set of limit ordinals α < κ^{+} with cofinality less than κ such that the cofinality of α is not equal to μ^{+} for any μ in A.

In W [G_{𝕄}] fix θ ≫ κ^{+} regular. Since S(κ, κ^{+}) is stationary, there exists N in P_{κ}H(θ) such that N ∩ κ^{+} is in S(κ, κ^{+}), N ∩ κ is a limit cardinal, and N is an elementary substructure of the model

⟨H(θ), ∈, C_{𝕄}, ⟨c_{α} : α ∈ B_{κ}⟩⟩.

Let a = N ∩κ^{+}, β = sup(a), and κ_{a} = a∩κ. Note that κ_{a} is in C_{𝕄} and therefore κ_{a} is not in A.

Claim 6.6. The ordinal β is in B_{κ}.

Proof. If β is not in B_{κ}, then in W the cofinality of β is μ^{+} for some μ in A. In W [G_{𝕄}], μ^{+} remains a cardinal and so the cofinality of β is μ^{+}. But the cofinality of β in W [G_{𝕄}] is equal to cf(a) = κ_{a}^{+}. Since κ_{a} and μ are cardinals, μ = κ_{a}. This is a contradiction since μ is in A but κ_{a} is not in A. ⊣

Since β is in B_{κ}, c_{β} is defined and is a club subset of β with order type less than κ. By elementarity, B_{κ} is unbounded in β. Fix some uncountable regular cardinal δ < κ_{a} which is different from cf(κ_{a}) and is not the successor of a cardinal in A. Then a is closed under suprema of subsets with order type δ.

Claim 6.7. The set B_{κ} is closed under suprema of subsets with order type δ.

Proof. Let γ be the supremum of a subset of B_{κ} with order type δ. Then the cofinality of γ in W [G_{𝕄}] is δ. If γ is not in B_{κ}, then there is μ in A such that cf^{W} (γ) = μ^{+W}. Since μ^{+W} is preserved, in W [G_{𝕄}] we have cf(γ) = μ^{+}. So μ^{+} = δ, which contradicts the choice of δ. ⊣

It follows that a ∩ B is closed under suprema of subsets with order type δ, and in particular, a ∩ B is a stationary subset of α. Since c_{β} is unbounded in β, o.t.(c_{β}) ≥ cf(β) = κ_{a}^{+}. So there is γ in a∩B ∩lim(c_{β}) such that o.t.(c_{β} ∩γ) ≥ κ_{a}. But c_{β} ∩γ = c_{γ}. Since γ is in N, c_{γ} is in N, and so also o.t.(c_{γ}) is in N ∩κ = κ_{a}. This contradicts that o.t.(c_{γ}) ≥ κ_{a}.

§7. Strong Compactness and GCH. If κ is supercompact and GCH holds for cardinals below κ, then GCH holds everywhere. Woodin asked whether the same is true for a strongly compact cardinal. In this section we prove a consistency result related to Woodin’s problem. Using an argument similar to the proof of the main theorem above, we construct a model in which κ is strongly compact, 2^{(κ^{+})} = κ^{+++}, but for a club of α below κ, 2^{(α^{+})} = α^{++}.

Start with a model V in which κ is supercompact and GCH holds. Let A be the set of measurable cardinals below κ.

Define an Easton support iteration ℙ_{κ} ∗ ℚ_{κ} as follows. Suppose that ℙ_{α} is defined for some α ≤ κ. If α is not in A ∪ {κ} then let ℚ_{α} be trivial. If α is in A ∪ {κ} then let ℚ_{α} be a name for the poset Add(α^{+}, α^{+++}).

Arguments similar to those in Section 5 show that ℙ_{κ} ∗ ℚ_{κ} preserves the measurability of each α in A and normal ultrafilters on P_{κ}λ which concentrate on A can be lifted. Let W be a generic extension of V by ℙ_{κ} ∗ℚ_{κ}. Then for all α in A ∪ {κ}, 2^{(α^{+})} = α^{+++}, and for any cardinal α not in A ∪ {κ}, 2^{(α^{+})} = α^{++}.

Let 𝕄_{κ} be the Magidor iteration of Prikry forcings defined over α in A. Let G_{𝕄} be generic for 𝕄 over W. Exactly as in Section 6, κ is strongly compact in W [G_{𝕄}] and there is a club set C_{𝕄} in κ disjoint from A. Let C be the club of limit cardinals in C_{𝕄}. Then 2^{(κ^{+})} = κ^{+++}, but for all α in C, 2^{(α^{+})} = α^{++}.

REFERENCES

[1] J. Baumgartner, Iterated forcing, Surveys in set theory, Cambridge Univ. Press, 1983.

[2] J. Cummings, Iterated forcing and elementary embeddings, Preprint.

[3] M. Gitik, Introduction to prikry type forcing notions, Preprint.

[4] , Nonsplitting subset of P_{κ}κ^{+}, J. Symbolic Logic, vol. 50 (1985), no. 4, pp. 881– 894.

[5] A. Kanamori, The higher infinite, Springer, 1994.

[6] J. Krueger, Adding clubs with square, Preprint.

[7] , Destroying stationary sets, Preprint.

[8] M. Magidor, How large is the first strongly compact cardinal?, Annals of Mathematical Logic, vol. 10 (1976), pp. 33–57.

KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC

UNIVERSITY OF VIENNA WÄHRINGERSTRASSE 25 1090 VIENNA, AUSTRIA

E-mail: jkrueger@logic.univie.ac.at

URL: http://www.logic.univie.ac.at/∼jkrueger