ADDING CLUBS WITH SQUARE

JOHN KRUEGER

Abstract. We present a technique for destroying stationary subsets of P_{κ}κ^{+} using partial square sequences. We combine this method with Gitik’s poset for changing the cofinality of a cardinal without adding bounded sets to prove a variety of consistency results concerning saturated ideals and the set S(κ, κ^{+}).

In this paper we continue our study of consistency results concerning the set S(κ, κ^{+}) = {a ∈ P_{κ}κ^{+} : o.t.(a) = (a ∩ κ)^{+}} from [5] and [6]. We present a method for destroying the stationarity of certain subsets of P_{κ}κ^{+}, where κ is inaccessible, using partial square sequences.

This method of destroying stationary sets has a variety of applications. We prove several results which support the general theme that the structure of S(κ, κ^{+}) can vary greatly depending on the particular model considered. For example, if κ is supercompact and μ < κ is regular, then there exists a generic extension in which S(κ, κ^{+}) is stationary and for almost all a in S(κ, κ^{+}), a∩κ is singular with cofinality μ. On the other hand, it is relatively consistent that for almost all a in S(κ, κ^{+}), a ∩ κ is measurable with Mitchell order (a ∩ κ)^{++}. We also construct a model in which GCH holds and there is a stationary set S ⊆ P_{κ}κ^{+} such that NS ↾ S is saturated and for all regular μ < κ, {a ∈ S : cf(a ∩ κ) = μ} is stationary.

This material is related to our previous papers [5] and [6]. However, no prior acquaintance with these papers is required by the reader. Most of our consistency results use Easton support Prikry iterations. The theorems in this paper rely on Gitik’s technique from [3] for changing the cofinality of a cardinal without adding bounded sets. We present Gitik’s poset in complete detail.

The contents of the paper are as follows. In Section 1 we outline preliminaries and notation. Section 2 provides additional background on forcing posets and large cardinals. Section 3 describes Easton support Prikry iterations. In Section 4 we present our method for destroying stationary sets using partial square sequences. In Section 5 we construct a model in which almost all a in S(κ, κ^{+}) satisfy that a∩κ is a measurable cardinal with Mitchell order (a ∩ κ)^{++}. Section 6 describes Gitik’s forcing poset for changing a cofinality without adding bounded sets. In Section 7 we show how to construct models in which we can control the cofinality of a ∩ κ for almost all a in S(κ, κ^{+}). In Section 8 we construct a model with a stationary set S ⊆ P_{κ}κ^{+} such that NS ↾ S is saturated, GCH holds, and for every regular μ less than κ, the set {a ∈ S : cf(a ∩ κ) = μ} is stationary.

1. Preliminaries

We assume that the reader is familiar with iterated forcing, supercompact cardinals, and Prikry forcing; see [1] and [4].

If κ is a regular uncountable cardinal and κ ⊆ X, define P_{κ}X = {a ⊆ X : |a| < κ, a ∩ κ ∈ κ}. A subset of P_{κ}X is club if it is closed under unions of ⊆-increasing sequences of length less than κ and is cofinal in P_{κ}X. A set is stationary if it intersects every club. If A is a subset of a club set C with size less than κ and is directed, i.e. for all a and b in C, there is c in C with a ∪ b ⊆ c, then ⋃ A is in C.

The ideal of non-stationary subsets of P_{κ}λ for λ ≥ κ is denoted by NS_{κ,λ} or NS. If S is stationary then NS_{κ,λ} ↾ S denotes the ideal generated by the elements of NS_{κ,λ} along with the complement of S. An ideal on P_{κ}λ is fine if it contains the set {a ∈ P_{κ}λ : ξ ∉ a} for every ξ < λ. A function f : P_{κ}X → X is regressive if f(a) is in a for all a. An ideal I is normal if for every set S ⊆ P_{κ}λ not in I and for every regressive function f : S → λ, there is an i < λ such that the set {a ∈ S : f(a) = i} is not in I.

If I is an ideal on P_{κ}λ, the set I^{∗} = {P_{κ}λ \ A : A ∈ I} is the dual filter of I. The collection of I-positive sets I^{+} = {A ⊆ P_{κ}λ : A ∉ I} is a forcing poset ordered by A ≤ B if A \ B is in I.

An ideal I is μ-saturated if I^{+} is μ-c.c. If I = NS ↾ S for some stationary set S, then I is μ-saturated iff there is no family {S_{i} : i < μ} of stationary subsets of S such that S_{i} ∩ S_{j} is non-stationary for i < j. We say that an ideal I on P_{κ}λ is saturated if I is λ^{+}-saturated.

If I is an ideal on P_{κ}λ, the forcing poset I^{+} adds a generic set U which is an ultrafilter on the Boolean algebra (P(P_{κ}λ))^{V}. If M is the ultrapower of V by U in the generic extension, M is called the generic ultrapower and the ultrapower map j : V → M is the generic elementary embedding. If I is saturated then the generic ultrapower M is well-founded and ^{λ}M ⊆ M.

If M ⊆ N are inner models of set theory 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.

A cardinal κ is λ-supercompact if there is a normal fine ultrafilter on P_{κ}λ, or equivalently, there is an elementary embedding j : V → M, where M is a transitive inner model, such that crit(j) = κ, j(κ) > λ, and M is λ-closed. If U_{0} and U_{1} are normal fine ultrafilters on P_{κ}λ, we write U_{0} ◁ U_{1} if U_{0} is in the ultrapower of V by U_{1}. This ordering, called the Mitchell ordering, is transitive and well-founded.

When we mention an ultrafilter on P_{κ}λ we always assume that it is non-principal and fine. If U is a ultrafilter on P_{κ}λ, we write Ult(V, U) for the transitive collapse of the ultrapower of V by U. Suppose that U is normal and j : V → M = Ult(V, U). Then M is λ^{<κ}-closed, and for any function f : P_{κ}λ → V, [f] is equal to j(f)(j“λ).

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

Suppose that α is a limit ordinal which is equal to the ordinal exponent ω^{γ} for some γ ≥ 1. Then any set X ⊆ α which is cobounded in α has order type α.

The expression θ ≫ κ means that θ is larger than 2^{2^{|H(κ)|}}. If 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. If N is such an elementary substructure, then N ∩ κ^{+} is closed under suprema of bounded subsets with order type different from cf(N ∩ κ).

If F : λ^{<ω} → λ is a partial function and A ⊆ λ, we say that F is Jonsson for A if A is closed under F, and whenever B ⊊ A is closed under F, it follows that |B| < |A|. If κ is a regular cardinal then there exists a Jonsson function for κ^{+}.

If A is a set of ordinals, then A is an Easton set if for any strongly inaccessible cardinal β, A ∩ β is bounded below β.

We use the phrase forcing poset to indicate any ordering ⟨ℙ, ≤⟩ which is reflexive and transitive. We do not require that ℙ be antisymmetric or separative. We usually assume that ℙ has a maximum element 1; this should be clear from context.

Suppose that κ is regular and ℙ is a forcing poset. We say that ℙ is < κdistributive if whenever {D_{i} : i < β} is a family of dense open subsets of ℙ and β is less than κ, then ⋂ D_{i} is dense open. Equivalently, ℙ is < κ-distributive if forcing with ℙ does not add any new sequences of ordinals with order type less than κ.

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. If ℙ is λ+1-strategically closed, then ℙ does not add any subsets to λ.

Suppose that ℙ is a forcing poset and λ is a cardinal. A canonical name for a subset of λ is a ℙ-name of the form

{⟨p, α̌⟩ : p ∈ A_{α}, α < λ}

where A_{α} is an antichain for each α < λ. If p forces that Ȧ is a subset of λ, then there is a canonical name Ḃ for a subset of λ such that p forces that Ȧ = Ḃ. If Ẋ is a ℙ-name and ℙ forces that ḟ : λ → Ẋ is a bijection, then a canonical name for a subset of Ẋ is a name of the form

{⟨p, ḟ(α)⟩ : p ∈ A_{α}, α < λ}

where each A_{α} is an antichain. If p forces that Ȧ is a subset of Ẋ, then there is a canonical name Ḃ for a subset of Ẋ such that p forces that Ȧ = Ḃ.

Suppose that λ is a regular uncountable cardinal and ℙ is a forcing poset which is λ-c.c. and is a subset of H(λ). Let G be generic for ℙ over V. If a is in H(λ)^{V [G]}, then there is a ℙ-name ẋ in H(λ)^{V} such that ẋ^{G} = a. Therefore H(λ)^{V [G]} = H(λ)^{V} [G].

If H is a subset of a forcing poset ℙ, we say that H generates J if J is the set of q in ℙ such that there is p in H with p ≤ q.

Suppose that ℙ is a forcing poset and ⟨U_{i} : i < β⟩ is a ◁-increasing sequence of normal ultrafilters on P_{κ}λ. We say that ℙ forces that the sequence can be lifted if ℙ forces that there are normal ultrafilters U_{i} ⊆ U_{i}^{∗} for i < β such that U_{i}^{∗} ◁ U_{j}^{∗} for i < j.

We say that forcing posets ℙ and ℚ are equivalent if they have the same generic extensions. This is only an intuitive definition, not a formal one, since generic filters for non-trivial posets do not exist in the universe. To prove that forcing posets ℙ and ℚ are equivalent in this informal sense, we show how to construct a generic filter for one poset, given a generic filter for the other. The following lemma provides a sufficient condition for the equivalence of forcing posets.

Lemma 1. Suppose that ℙ and ℚ are forcing posets, D is a dense subset of ℙ, and E is a dense subset of ℚ. Assume there is a surjective mapping i : D → E satisfying:

(1) q ≤ p implies i(q) ≤ i(p),

(2) If p and q are incompatible, then i(p) and i(q) are incompatible. Then ℙ and ℚ are equivalent.

A mapping π : ℚ → ℙ between forcing posets is a projection mapping if it satisfies:

(1) q ≤ p implies π(q) ≤ π(p),

(2) π(1) = 1,

(3) if p ≤ π(q) then there is r ≤ q such that π(r) ≤ p.

Conditions (2) and (3) imply that π“ℚ is dense in ℙ. If G is generic for ℚ over V, then π“G generates a generic filter for ℙ over V.

Suppose that π : ℚ → ℙ is a projection mapping. Let G be generic for ℙ over V. In V [G], define a poset ℚ/ℙ as follows. A condition in ℚ/ℙ is any q in ℚ such that π(q) ∈ G. Let q ≤ p in ℚ/ℙ iff q ≤ p in ℚ. Now let ℚ/ℙ be a ℙ-name for this poset. Define k : ℚ → ℙ ∗ (ℚ/ℙ) by letting k(q) = π(q) ∗ q̌. Then k satisfies:

(1) q ≤ p implies k(q) ≤ k(p),

(2) If p and q are incompatible, then k(p) and k(q) are incompatible,

(3) k“ℚ is dense in ℙ ∗ (ℚ/ℙ). So by Lemma 1, ℚ and ℙ ∗ (ℚ/ℙ) are equivalent.

Lemma 2. Suppose that π : ℚ → ℙ is a projection mapping and μ is a regular uncountable cardinal. Assume that ℚ is μ-c.c. Then:

(1) ℙ is μ-c.c.,

(2) ℙ forces that ℚ/ℙ is μ-c.c.

Proof. Since ℚ is μ-c.c., so is ℙ ∗ (ℚ/ℙ). If A is an antichain in ℙ, then the family {p ∗ 1 : p ∈ A} is an antichain in ℙ ∗ (ℚ/ℙ). So ℙ is μ-c.c. Suppose that p forces that ⟨q̇_{i} : i < ξ⟩ is an antichain in ℚ/ℙ. Then p forces that q̇_{i} is incompatible with q̇_{j} for i less than j. It follows that the family {p ∗ q̇_{i} : i < ξ} is an antichain in ℙ ∗ (ℚ/ℙ), and so ξ is less than μ. Therefore ℙ forces that ℚ/ℙ is μ-c.c. □

A Prikry type forcing poset is a triple ⟨ℚ, ≤, ≤^{∗}⟩ such that ⟨ℚ, ≤⟩ and ⟨ℚ, ≤^{∗}⟩ are forcing posets, q ≤^{∗} p implies q ≤ p, and ℚ satisfies the Prikry property: if p is a condition in ℚ and φ is a statement in the forcing language for ⟨ℚ, ≤⟩, then there is q ≤^{∗} p such that q decides φ. If q ≤^{∗} p we say that q is a direct extension or a direct refinement of p.

We say that ℚ is α-weakly closed if ⟨ℚ, ≤^{∗}⟩ is α-closed, and α-weakly strategically closed if ⟨ℚ, ≤^{∗}⟩ is α-strategically closed. The poset ℚ has the direct extension property if whenever q, r ≤^{∗} p, there is s ≤^{∗} q, r.

2. Background on Forcing and Large Cardinals

In this section we review some additional material on large cardinals and forcing. Suppose that κ ≤ λ_{0} ≤ λ_{1} and U is a normal ultrafilter on P_{κ}λ_{1}. Define U ↾ λ_{0}

by letting X be in U ↾ λ_{0} iff X ⊆ P_{κ}λ_{0} and the set {a ∈ P_{κ}λ_{1} : a ∩ λ_{0} ∈ X} is in U. Equivalently, if j : V → M = Ult(V, U), X is in U ↾ λ_{0} iff j“λ_{0} ∈ j(X). The set U ↾ λ_{0} is a normal ultrafilter on P_{κ}λ_{0}.

Lemma 3. Suppose that κ ≤ λ_{0} < 2^{λ_{0}^{<κ}} ≤ λ_{1} are cardinals and U is a normal ultrafilter on P_{κ}λ_{1}. Let j : V → M = Ult(V, U) and i : V → N = Ult(V, U ↾ λ_{0}). Then there is an elementary embedding k : N → M such that j = k ∘ i and crit(k) = (2^{λ_{0}^{<κ}})^{+N}.

Proof. Define k as follows. Let a be in N and let f : P_{κ}λ_{0} → V be a function such that [f] = a. Define k(a) = j(f)(j“λ_{0}). It is easy to check that k is a well-defined elementary embedding and j = k ∘ i.

If β ≤ λ_{0}, then β = o.t.(i“β) = o.t.(i“λ_{0} ∩ i(β)). So β is represented by the function a ↦ o.t.(a ∩ β) in N. By the definition of k, k(β) = o.t.(j“λ_{0} ∩ j(β)) = o.t.(j“β) = β in M. So crit(k) > λ_{0}.

Since N is λ_{0}^{<κ}-closed and crit(k) > λ_{0}, k(P_{κ}λ_{0}) = P_{κ}λ_{0} and k(P(P_{κ}λ_{0})) = P(P_{κ}λ_{0}). If a is in P_{κ}λ_{0}, then k(a) is in k(P_{κ}λ_{0}) = P_{κ}λ_{0}, and α ∈ k(a) iff k(α) ∈ k(a) iff α ∈ a. Therefore k ↾ P_{κ}λ_{0} is the identity. The same argument shows that k ↾ P(P_{κ}λ_{0}) is the identity.

We prove by induction that for all β less than (2^{λ_{0}^{<κ}})^{+N}, k(β) = β. Fix β and suppose k(α) = α for all α less than β. Since β is less than (2^{λ_{0}^{<κ}})^{+N}, there is a surjective function s : P(P_{κ}λ_{0})) → β in N. By elementarity, k(s) is a surjection of k(P(P_{κ}λ_{0})) = P(P_{κ}λ_{0}) onto k(β). If k(β) > β, then there is A in P(P_{κ}λ) such that β = k(s)(A) = k(s)(k(A)) = k(s(A)) = s(A) < β, which is absurd.

Now (2^{λ^{<κ}})^{+N} < i(κ) < (2^{λ^{<κ}})^{+} = (2^{λ^{<κ}})^{+M}, and k((2^{λ^{<κ}})^{+N}) = (2^{λ^{<κ}})^{+M}. So crit(k) = (2^{λ^{<κ}})^{+N}. □

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

Lemma 4. 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] iff j“G ⊆ H. In this case, 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. The mapping 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 5. 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 G a generic filter for ℙ over V. Suppose that in V [G] there 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 any isomorphism between transitive models of set theory is the identity, 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 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 g is well-defined. Suppose that a = ẋ^{H} = ẏ^{H}, where [f_{x}] = ẋ and [f_{y}] = 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 f_{x}(α) = f_{y}(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^{∗}, f_{x}(α)^{G} = f_{y}(α)^{G}. Define g_{x} and g_{y} by g_{x}(α) = f_{x}(α)^{G} and g_{y}(α) = f_{y}(α)^{G}. Then for all α in A^{∗}, g_{x}(α) = g_{y}(α), so [g_{x}] = [g_{y}]. 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^{∗}). For each α let ḃ_{α} be a ℙ-name for h(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 models of set theory and λ is an N-cardinal. Let ℙ be a forcing poset in M and p 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 obtain closure of generic extensions.

Lemma 6. Suppose that M ⊆ N are models of set theory and λ is a regular uncountable cardinal in N such that M is < λ-closed in M. 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 forces over N that ḟ : β → On for some β < λ. For each α < β let A_{α} be a maximal antichain 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 is in M[Ġ]. □

Lemma 7. Suppose M ⊆ N are models of set theory, λ is a regular cardinal in N such that M is λ-closed in N. If ℙ ∈ M is a forcing poset, G ∈ N is a generic filter for ℙ over M, then M[G] is λ-closed in N.

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

We need some facts about the Mitchell ordering.

Lemma 8. Let U_{0} and U_{1} be normal ultrafilters on P_{κ}λ. For each a in P_{κ}λ, let π_{a} : a → o.t.(a) be the unique order preserving bijection. Then U_{0} ◁ U_{1} iff there exists a function f : P_{κ}λ → V_{κ} such that:

(1) {a ∈ P_{κ}λ : f(a) is a normal ultrafilter on (P_{a∩κ} o.t.(a))} is in U_{1},

(2) For every X ⊆ P_{κ}λ, X ∈ U_{0} iff the set of a in P_{κ}λ such that

X_{a} = {b : (∃c ∈ (P_{a∩κ}a) ∩ X) (π_{a}“c = b)} ∈ f(a)

is in U_{1}.

Proof. Let j : V → M = Ult(V, U_{1}). Let [f] = U_{0} in M. In M, [a ↦ a] = j“λ, [a ↦ a ∩ κ] = κ, and [a ↦ o.t.(a)] = λ. Clearly (1) holds. For (2), check that j(a ↦ X_{a})(j“λ) = X, and therefore [a ↦ X_{a}] = X in M. So X ∈ U_{0} iff M ⊧ X ∈ U_{0} iff {a ∈ P_{κ}λ : X_{a} ∈ f(a)} is in U_{1}. The converse is similar. □

Corollary 1. Suppose that U_{0} and U_{1} are normal ultrafilters on P_{κ}λ and U_{0} ◁ U_{1}. Let ℙ be a forcing poset which does not add subsets to λ^{<κ}. Then ℙ forces that U_{0}, U_{1} are normal ultrafilters on P_{κ}λ and U_{0} ◁ U_{1}.

Corollary 2. Suppose that M is a transitive model of set theory which is λ^{<κ}- closed in V. If U and W are normal ultrafilters on P_{κ}λ which are in M, then U ◁ W iff M models that U ◁ W.

Lemma 9. Suppose that U_{0} ◁ U_{1} are normal ultrafilters on P_{κ}λ. Let j_{0} : V → M_{0} = Ult(V, U_{0}), j_{1} : V → M_{1} = Ult(V, U_{1}), and j_{1,0} : M_{1} → N_{0} = Ult(M_{1}, U_{0}). Then the following statements are true:

(1) If f ∈ ^{P_{κ}λ}M_{1} then [f]_{M_{0}} = [f]_{N_{0}}, so N_{0} ⊆ M_{0},

(2) j_{0} ↾ M_{1} = j_{1,0}. Suppose that ℙ is a forcing poset in M_{1}, G is generic for ℙ over V, H is generic for j_{0}(ℙ) = j_{1,0}(ℙ) over M_{0}, and j_{0}“G = j_{1,0}“G ⊆ H. Extend j_{0} and j_{1,0} to j_{0} : V [G] → M_{0}[H] and j_{1,0} : M_{1}[G] → N_{0}[H]. Then j_{0} ↾ M_{1}[G] = j_{1,0}.

Proof. Let [f]_{M_{0}} be given such that f is in ^{P_{κ}λ}M_{1}, and assume that (1) holds for all functions g where [g]_{M_{0}} has rank less than the rank of [f]_{M_{0}}. We prove that [f]_{M_{0}} = [f]_{N_{0}}. Note that f is in M_{1} by the closure of M_{1}, so [f]_{N_{0}} is defined. Suppose that x is in [f]_{M_{0}}. Since f is in ^{P_{κ}λ}M_{1}, there is g is in ^{P_{κ}λ}M_{1} and A in U_{0} such that [g]_{M_{0}} = x and g(a) ∈ f(a) for all a in A. By induction, [g]_{N_{0}} = x, and clearly [g]_{N_{0}} is in [f]_{N_{0}}. So [f]_{M_{0}} ⊆ [f]_{N_{0}}. The other direction is similar.

To prove (2), let x be in M_{1}. Then j_{0}(x) = [f]_{M_{0}} where f(a) = x for all a in P_{κ}λ. Since f is in ^{P_{κ}λ}M_{1}, by (1) we have that [f]_{M_{0}} = [f]_{N_{0}}, and clearly [f]_{N_{0}} = j_{1,0}(x).

The fact that j_{0} ↾ M_{1}[G] = j_{1,0} follows from (2) and the definition of the extended mappings given in the proof of Lemma 4. □

Lemma 10. Suppose that κ is a measurable cardinal, ℙ is a forcing poset with size less than κ, and ⟨U_{i} : i < β⟩ is a ◁-increasing sequence of normal ultrafilters on κ. Then ℙ forces that ⟨U_{i} : i < β⟩ can be lifted.

Proof. Without loss of generality we assume that ℙ is in V_{κ}. For i < β let j_{i} : V → M_{i} = Ult(V, U_{i}) and k_{i} : M_{i+1} → N_{i} = Ult(M_{i+1}, U_{i}). Then j_{i}(ℙ) = k_{i}(ℙ) = ℙ. By Lemma 9, j_{i} ↾ M_{i+1} = k_{i}.

Let G be generic for ℙ over V. Then j“G = G and we can lift each j_{i} and k_{i} to j_{i} : V [G] → M_{i}[G] and k_{i} : M_{i+1}[G] → N_{i}[G]. By Lemma 6 each M_{i}[G] is κ-closed in V [G]. By Lemma 9, j_{i} ↾ M_{i+1}[G] = k_{i}.

Define U_{i}^{∗} by letting X ∈ U_{i}^{∗} iff κ ∈ j_{i}(X). Then U_{i}^{∗} is a normal ultrafilter on κ extending U_{i}, and by Lemma 5 Ult(V [G], U_{i}^{∗}) = M_{i}[G]. Moreover, since P(κ) ⊆ M_{i+1}[G] and j_{i} ↾ M_{i+1}[G] = k_{i}, M_{i+1}[G] can compute U_{i}^{∗} using the same definition. So U_{i}^{∗} is in M_{i+1}[G].

Fix i < j. Then U_{i}^{∗} is in M_{i+1}[G] ⊆ M_{j}[G] = Ult(V [G], U_{j}^{∗}). So U_{i}^{∗} ◁ U_{j}^{∗}. □

3. Iterations of Prikry type forcing posets

In this section we present the basics of Easton support Prikry iterations which we use in our consistency proofs.

Magidor [7] showed how to iterate Prikry forcing over different cardinals using conditions with full support. To overcome the difficulty in extending an elementary embedding with such an iteration, Gitik [3] developed a method for iterating Prikry type forcing posets using conditions with Easton support.

An Easton support Prikry iteration is an iterated forcing

⟨ℙ_{α}, ℚ_{α} : α < κ⟩,

for some ordinal κ, satisfying the following properties:

(1) There exists a set A ⊆ κ consisting of strongly inaccessible cardinals such that if ℚ_{α} is non-trivial, then α is in A,

(2) ℙ_{α} forces that |ℚ_{α}| < min(A \ (α + 1)),

(3) ℙ_{α} forces that ⟨ℚ_{α}, ≤, ≤^{∗}⟩ is a Prikry type forcing poset,

(4) ℙ_{α+1} = ℙ_{α} ∗ ℚ_{α},

(5) If α is a limit ordinal, then ℙ_{α} consists of functions p with domain an Easton subset of α ∩ A such that p ↾ β is in ℙ_{β} for all β less than α,

(6) If α is a limit ordinal and p, q are in ℙ_{α}, then q ≤ p if q ↾ β ≤ p ↾ β for β less than α, and there is a finite set a such that for all β in dom(p) \ a, q ↾ β ⊩ q(β) ≤^{∗} p(β).

(7) If α is a limit ordinal, q ≤^{∗} p in ℙ_{α} if q ≤ p and the finite set a in (6) is empty; i.e. for all β in supp(p), q ↾ β ⊩ q(β) ≤^{∗} p(β),

(8) q ḃ ≤^{∗} p ȧ in ℙ_{β+1} if q ≤^{∗} p in ℙ_{β} and q ⊩ ḃ ≤^{∗} ȧ.

In (3) the Prikry poset ℚ_{α} will usually not add bounded subsets to α. To iterate Prikry posets which do add bounded sets, such as Radin forcing, one can use the Magidor iteration; see [6] for an example.

If p is a condition in ℙ_{α}, the support of p, denoted by supp(p), is the domain of p as a function.

If β is less than α, then ℙ_{α} factors into ℙ_{β} ∗ ℚ_{β} ∗ ℙ_{β,α}. Suppose that ℚ_{γ} is forced to be γ-weakly strategically closed for all γ greater than β; then ⟨ℙ_{β,α}, ≤^{∗}⟩ is forced to be min(A \ (β + 1))-weakly strategically closed. It will follow from Proposition 2 that ℙ_{β,α} does not add any bounded subsets to min(A \ (β + 1)).

The following result follows from the usual proof of the corresponding fact for Easton support iterations.

Proposition 1. If |ℙ_{β}| < α for all β less than α and α is a Mahlo cardinal, then ℙ_{α} is α-c.c.

We need the following lemma in order to prove that ℙ_{α} satisfies the Prikry property.

Lemma 11. Suppose that β is less than α, and factor ℙ_{α} = ℙ_{β} ∗ℚ_{β} ∗ℙ_{β,α}. Assume that p is in ℙ_{β} and p forces that ȧ is in ℚ_{β} ∗ ℙ_{β,α}. Then there is q in the ground model such that p q is a condition in ℙ_{α} and p forces that q ≤^{∗} ȧ in ℚ_{β} ∗ ℙ_{β,α}.

Proof. First we define the support of q, which we call x. Let γ be in x iff there is r ≤ p such that r forces that γ is in the support of ȧ. We prove that x is an Easton set. Let ξ be a strongly inaccessible cardinal, which we can assume to be larger than min(A \ (β + 1)). Then |ℙ_{β}| < ξ. Now p forces that supp(ȧ) ∩ ξ is bounded below ξ. Let Y be the set of δ such that there is r ≤ p which forces that δ is the least ordinal larger than all the elements of supp(ȧ) ∩ ξ. Since ℙ_{β} has size less than ξ, sup(Y) is below ξ, and x ∩ ξ ⊆ sup(Y).

Now define q by induction. Suppose that γ is in x and q ↾ γ is already defined so that p (q ↾ γ) is in ℙ_{γ}. Then let q(γ) be a ℙ_{γ}-name which p (q ↾ γ) forces is equal to ȧ(γ), provided that γ is in supp(ȧ), and otherwise is some arbitrary element of ℚ_{γ}.

Suppose that G is generic for ℙ_{β} and contains p. Let a = ȧ^{G}. Then in V [G], supp(a) ⊆ x = supp(q), and if γ is in the support of a, then q(γ) = a(γ). So p forces that q ≤^{∗} a in ℚ_{β} ∗ ℙ_{β,α}. □

Proposition 2. The iteration ⟨ℙ_{α}, ≤, ≤^{∗}⟩ is a Prikry type forcing poset.

Proof. We prove by induction on α that ℙ_{α} satisfies the Prikry property. Suppose that β is less than α and ℙ_{β} satisfies the Prikry property. We prove the same is true for ℙ_{β+1} = ℙ_{β} ∗ ℚ_{β}. Let φ be a statement in the forcing language and let p ȧ be a condition. Fix a ℙ_{β}-name ḃ such that p forces ḃ ≤^{∗} ȧ and ḃ decides φ. Now apply the induction hypothesis to obtain q ≤^{∗} p in ℙ_{β} which decides which way that ḃ decides φ. Then q ḃ ≤^{∗} p ȧ and q ḃ decides φ.

Suppose that α is a limit ordinal and for all β less than α, ℙ_{β} satisfies the Prikry property. Let p be a condition in ℙ_{α} and φ a statement in the forcing language.

First assume that the support of p is bounded below α. Then there is β less than α such that p is in ℙ_{β}. Write ℙ_{α} = ℙ_{β} ∗ ℚ_{β} ∗ ℙ_{β,α}. Let ȧ be a ℙ_{β}-name for a condition in ℚ_{β} ∗ ℙ_{β,α} which decides φ. Apply the induction hypothesis to obtain p_{0} ≤^{∗} p which decides which way that ȧ decides φ. By Lemma 11, choose q so that p_{0} q is a condition in ℙ_{α} and p_{0} forces that q ≤^{∗} ȧ. Then p_{0} q is a direct extension of p which decides φ.

Now assume that the support of p is unbounded in α. Suppose for a contradiction that p does not have a direct extension which decides φ. We construct by induction a condition q = ⟨q(γ) : γ ∈ supp(p)⟩ which is a direct refinement of p and has the same support as p.

Suppose that γ is in supp(p) and q ↾ γ is defined. Then (q ↾ γ) p(γ) is in ℙ_{γ+1} and forces that p \ (γ + 1) is in ℙ_{γ,α}. Choose a ℙ_{γ}-name q(γ) such that q ↾ γ forces that q(γ) ≤^{∗} p(γ) and q(γ) decides the following statement: there is a direct extension of p \ (γ + 1) in ℙ_{γ,α} which decides φ. This completes the definition of q.

Claim 1. For all γ less than α, q ↾ γ forces that there is no direct extension of p \ γ in ℚ_{γ} ∗ ℙ_{γ,α} which decides φ.

Proof. We prove the claim by induction. By assumption, there is no direct extension of p which decides φ. So the claim holds for γ less than min(A).

Suppose that γ ≥ min(A) and the claim holds for all γ^{∗} less than γ. First assume that γ = γ_{0} +1. Consider the case when γ_{0} is not in supp(p). If the claim fails then without loss of generality there is r ċ ≤ q ↾ γ and ȧ such that r ċ ⊩ ȧ ≤^{∗} p \ γ and ȧ forces φ. But then since γ_{0} is not in supp(p), r forces that ċ ȧ is a direct extension of p \ γ_{0} which forces φ. This contradicts the induction hypothesis.

Suppose now that γ_{0} is in supp(p). Then q ↾ γ_{0} forces that q(γ_{0}) decides whether there is a direct extension of p \ γ which decides φ. So if the claim fails, there is r ≤ q ↾ γ_{0} and ḃ such that

r ⊩ q(γ_{0}) ⊩ ḃ ≤^{∗} p \ γ, ḃ decides φ.

It follows that r forces that q(γ_{0}) ḃ is a direct extension of p \ γ_{0} which decides φ, contradicting the induction hypothesis.

Now suppose that γ is a limit ordinal. If the claim fails then without loss of generality there is r ≤ q ↾ γ and ḃ such that r forces that ḃ ≤^{∗} p \ γ and ḃ forces φ. Let a be a finite subset of γ such that for all ξ in supp(q ↾ γ) \ a, r ⊩ r(ξ) ≤^{∗} q(ξ). Since γ is a limit ordinal, there is γ^{∗} less than γ such that a is a subset of γ^{∗}.

Therefore r ↾ γ^{∗} forces that (r \ γ^{∗}) ḃ is a direct extension of p \ γ^{∗} in ℙ_{γ^{∗},α} which forces φ. This contradicts the induction hypothesis that q ↾ γ^{∗} forces there is no such direct extension. □

Now choose r ≤ q in ℙ_{α} which decides φ. Let a be the set of ξ in supp(q) such that r ↾ ξ forces that r(ξ) is a non-direct extension of q. Since the support of p is unbounded in α, there is γ in supp(q) such that a ⊆ γ. Then r ↾ γ is a condition below q \ γ which forces that r \ γ is a direct extension of p \ γ which decides φ. This contradicts the fact that q ↾ γ forces there is no such direct extension. □

If ℙ is an α-strategically closed forcing poset, then the triple ⟨ℙ, ≤, ≤^{∗}⟩, where ≤^{∗}=≤, is a Prikry type forcing poset which is α-weakly strategically closed. So we can define Easton support Prikry iterations by combining usual Prikry forcings with strategically closed posets. Gitik showed that in some cases it is possible to use distributive posets as well, by turning them into Prikry type forcing posets using a projection mapping from a strongly compact Prikry forcing; see [5] for details.

4. Adding Clubs with Square

Suppose for the remainder of this section that κ is a weakly inaccessible cardinal. We present a method for destroying certain stationary subsets of P_{κ}κ^{+}.

Lemma 12. There is a club set of a in P_{κ}κ^{+} such that o.t.(a) ≤ (a ∩ κ)^{+}.

Proof. Fix θ ≫ κ regular and let N be an elementary substructure of ⟨H(θ), ∈⟩ in P_{κ}H(θ). Let a = N ∩ κ^{+}. We claim that o.t.(a) ≤ (a ∩ κ)^{+}.

We show that for all β in a, |a ∩ β| ≤ |a ∩ κ|. This is clear if β < κ. Suppose that β ≥ κ is in a. Fix a bijection f_{β} : κ → β in N. By elementarity, for all j in a ∩ β, f_{β}^{−1}(j) is in a ∩ κ. Therefore f_{β} ↾ a ∩ κ is a bijection of a ∩ κ onto a ∩ β, and so |a ∩ β| = |a ∩ κ|. Since each initial segment of a has size at most |a ∩ κ|, clearly o.t.(a) ≤ (a ∩ κ)^{+}. □

Define S(κ, κ^{+}) as the set of a in P_{κ}κ^{+} such that o.t.(a) = (a ∩ κ)^{+}.

Proposition 3. If κ is κ^{+}-supercompact then S(κ, κ^{+}) is stationary.

Proof. Fix j : V → M with crit(j) = κ, j(κ) > κ^{+}, and M is κ^{+}-closed. Then j“κ^{+} is in M. Note that j“κ^{+} is in j(S(κ, κ^{+})). Suppose that C is a club subset of P_{κ}κ^{+} in V. Then j“C is in M and is a directed subset of j(C) with size less than j(κ). So ⋃ j“C = j“κ^{+} is in j(C). Therefore j(S(κ, κ^{+})) ∩ j(C) is non-empty. By elementarity, S(κ, κ^{+}) ∩ C is non-empty. □

If κ is subcompact, which is a weaker assumption than κ^{+}-supercompactness, then S(κ, κ^{+}) is stationary. In [6] we prove that the strong compactness of κ does not imply that S(κ, κ^{+}) is stationary.

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.

Define a forcing poset ℙ_{B} which adds a □_{κ}^{B}-sequence as follows. This poset is the obvious generalization of Jensen’s poset for adding a □_{κ}-sequence. A condition in ℙ_{B} is a sequence p = ⟨p_{α} : α ∈ B ∩ (γ + 1)⟩ for some γ < κ^{+} satisfying (1), (2), and (3) above for all α in B ∩ (γ + 1). Let q ≤ p if p is an initial segment of q.

Proposition 4. The forcing poset ℙ_{B} is κ + 1-strategically closed.

Proof. We describe a strategy by considering a run of the game. Suppose ⟨p_{i} : 0 < i < β⟩ is the run of the game up to stage β, and it is Player II’s turn. For each 0 < i < β let γ_{i} be the least ordinal γ such 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 any club subset of ξ with order type cf(ξ) and min c_{ξ} > γ_{α}.

Suppose now that β is a limit ordinal. Let γ_{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. Now 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. □

It follows that ℙ_{B} does not add subsets to κ. Since ℙ_{B} has size 2^{κ}, if 2^{κ} = κ^{+} then ℙ_{B} preserves all cardinalities and cofinalities.

We show how to use this forcing poset to destroy stationary sets.

Theorem 1. Suppose that B is a subset of κ^{+} consisting of limit ordinals and there are distinct regular cardinals δ_{0} and δ_{1} below κ such that B contains all its limit points with cofinality δ_{0} or δ_{1}. Then if □_{κ}^{B} holds, the set

S = {a ∈ S(κ, κ^{+}) : sup a ∈ B}

is non-stationary.

Proof. Let ⟨c_{β} : β ∈ B⟩ be a partial square sequence witnessing that □_{κ}^{B} holds. Suppose for a contradiction that S is stationary.

Fix θ ≫ κ regular. Then there exists N in P_{κ}H(θ) such that N ∩ κ^{+} is in S, N ∩ κ is larger than δ_{0} and δ_{1}, and

N ≺ ⟨H(θ), ∈, ⟨c_{β} : β ∈ B⟩⟩.

Let a = N ∩ κ^{+}, β = sup a, and κ_{a} = a ∩ κ. Since a is in S, β is in B. So c_{β} is defined. Note that o.t.(a) = κ_{a}^{+} so cf(β) = κ_{a}^{+}.

By elementarity, B is unbounded in β. Let δ be one of δ_{0} or δ_{1} which is different from cf(κ_{a}). Then B and a are both closed under suprema of subsets with order type δ. Therefore B ∩ a is a stationary subset of β.

Since o.t.(c_{β}) ≥ cf(β) = κ_{a}^{+}, there is γ in a ∩ B ∩ lim(c_{β}) such that o.t.(c_{β} ∩ γ) ≥ κ_{a}. But c_{γ} = c_{β} ∩ γ, and by elementarity, o.t.(c_{γ}) is in N ∩ κ = κ_{a}. This is a contradiction since o.t.(c_{γ}) ≥ κ_{a}. □

Now let A be any subset of κ^{+} consisting of limit ordinals. Let B be the closure of A under suprema of subsets with order type ω or ω_{1}. We claim that

{a ∈ S(κ, κ^{+}) : sup a ∈ A} = {a ∈ S(κ, κ^{+}) : sup a ∈ B}

modulo clubs.

Clearly if sup a is in A then sup a is in B. Suppose that a is in S(κ, κ^{+}), sup a = β is in B, and κ_{a} = a ∩ κ is a limit cardinal. Then cf(β) = κ_{a}^{+}, which is different from ω and ω_{1}. But B can be written as A ∪ X where X consists of ordinals with cofinality ω or ω_{1}. So β is in A.

Proposition 5. The forcing poset ℙ_{B} preserves cardinals and cofinalities less than or equal to κ^{+}, does not add subsets to κ, and destroys the stationarity of the set S = {a ∈ S(κ, κ^{+}) : sup a ∈ A}.

Proof. The poset ℙ_{B} adds a partial square sequence ⟨c_{β} : β ∈ B⟩. Since ℙ_{B} does not add subsets to κ, after forcing with ℙ_{A} the set B is still the closure of A under suprema of subsets with order type ω or ω_{1}. Now apply Theorem 1. □

All of the stationary subsets of S(κ, κ^{+}) which we will consider have the form

{a ∈ S(κ, κ^{+}) : sup a ∈ A}

for some A ⊆ κ^{+}. In [5] we constructed a model in which all stationary subsets of S(κ, κ^{+}) have this form. Namely, fix a Jonsson function F : κ^{+<ω} → κ^{+}. Using the forcing poset from [5], force to make almost all a in S(κ, κ^{+}) satisfy that F is Jonsson for a. Then the function a ↦ sup a is injective on a club set intersected with S(κ, κ^{+}). So if S ⊆ S(κ, κ^{+}) is stationary, then S is equal modulo clubs to the set

{a ∈ S(κ, κ^{+}) : sup a ∈ A}

where A = {β < κ^{+} : ∃a ∈ S sup a = β}. The stationarity of S(κ, κ^{+}) can be preserved by preparing the ground model.

Lemma 13. Suppose that S is a stationary subset of S(κ, κ^{+}) such that for some X ⊆ κ,

S = {a ∈ S(κ, κ^{+}) : a ∩ κ ∈ X}.

Then there is A ⊆ κ^{+} such that

S = {a ∈ S(κ, κ^{+}) : sup a ∈ A}.

modulo clubs.

Proof. Let C be the club set of a in P_{κ}κ^{+} such that a∩κ is a limit cardinal. Define A as the set of α less than κ^{+} such that cf(α) = μ^{+} for some limit cardinal μ in X. If a is in S∩C, then cf(sup a) = o.t.(a) = (a∩κ)^{+}. But a∩κ is a limit cardinal in X, so sup a is in A. On the other hand, suppose that a is in S(κ, κ^{+})∩C and sup a is in A. Then cf(sup a) = μ^{+} for some limit cardinal μ in X. But cf(sup a) = (a ∩ κ)^{+}, so a ∩ κ = μ, and therefore a is in S. □

Note: The method presented here for destroying stationary sets using partial square sequences is simpler and easier to use than previous posets for adding clubs, as found in [2] and [5]. The Radin forcing required by these other posets is eliminated. Also the fact that the partial square poset does not add subsets to κ is very useful, as we see in the following sections.

5. The Structure of S(κ, κ^{+})

In this section and in Section 7 we prove several consistency results which contribute to the general theme that the structure of S(κ, κ^{+}), unlike its complement, is largely independent of ZFC. In the present section we construct a model in which S(κ, κ^{+}) is stationary and for almost all a in S(κ, κ^{+}), a∩κ is a measurable cardinal with Mitchell order (a ∩ κ)^{++}.

Let V be a model of set theory in which κ is κ^{++}-supercompact and GCH holds. Let A be the set of α less than κ such that α is α^{+}-supercompact.

Proposition 6. If α is in A ∪ {κ}, then there exists a ◁-increasing sequence ⟨U(α, i) : i < α^{++}⟩ of normal ultrafilters on α such that A ∩ α is not in U(α, i) for each i.

Proof. Let W be a normal ultrafilter on P_{α}α^{+} which is minimal in the Mitchell ordering and let j : V → M = Ult(V, W). Then α is not α^{+}-supercompact in M, so α is not in j(A ∩ α). Let W ↾ α be the projection of W to α, i.e. X is in W ↾ α iff X ⊆ α and α ∈ j(X). Note that A ∩ α is not in W ↾ α. Let i : V → N = Ult(V, W ↾ α). By Lemma 3, there exists an elementary embedding k : N → M such that j = k ∘ i and crit(k) = α^{++N}.

If U⃗ is a ◁-increasing sequence of ultrafilters on α in N with length (α^{++})^{N}, then k(U⃗) is a ◁-increasing sequence of ultrafilters on k(α) = α with length k(α^{++N}) = α^{++}. So it suffices to prove there exists such a sequence in N.

It suffices to prove that in N, whenever β is less than α^{++N} and U⃗ is a ◁- increasing sequence of ultrafilters on α, then there is an ultrafilter not containing A which is above the sequence. Suppose for a contradiction that this claim fails. Let β < α^{++N} be the least ordinal such that there exists a sequence U⃗ with length β, but there is no ultrafilter above it which does not contain A. Since crit(k) = (α^{++})^{N}, k(U⃗) = U⃗, and by elementarity there is no ultrafilter above U⃗ in M which does not contain A. This is a contradiction, since W ↾ α is in M. □

Define an Easton support iteration ⟨ℙ_{α}, ℚ_{α} : α < κ⟩ as follows. Suppose that ℙ_{α} is defined for some α less than κ. If α is not in A then let ℚ_{α} be trivial. Suppose that α is in A.

Let G_{α} be generic for ℙ_{α} over V. In V [G_{α}], define B_{α} as the set of β less than α^{+} such that cf(β) ≠ μ^{+} for any μ in A ∩ α. Clearly B_{α} is closed under suprema of subsets with order type ω and ω_{1}. So the forcing poset ℙ_{B_{α}} for adding a □_{α}^{B_{α}}- sequence forces that almost all a in S(α, α^{+}) satisfy that cf(sup a) = μ^{+} for some μ in A ∩ α. But cf(sup a) = (a ∩ α)^{+}, so μ = a ∩ α is in A. Let ℚ_{α} be a name for ℙ_{B_{α}}.

This completes the definition of ℙ_{κ}. Let B = B_{κ}. We force with ℙ_{κ} ∗ ℙ_{B}. By standard Easton support iteration arguments, this poset preserves all cardinals, cofinalities, and GCH. It also forces that almost all a in S(κ, κ^{+}) satisfy that a ∩ κ is in A. So to complete the proof, we show that S(κ, κ^{+}) remains stationary and for each α in A, the sequence of ultrafilters on α can be lifted.

Lemma 14. For all α in A, the poset ℙ_{κ} ∗ ℙ_{B} forces that the sequence ⟨U(α, i) : i < α^{++}⟩ can be lifted.

Proof. Write ℙ_{κ} ∗ ℙ_{B} = ℙ_{α} ∗ ℙ_{B_{α}} ∗ ℙ_{tail} ∗ ℙ_{B}. The key point is that ℙ_{B_{α}} is forced to be α + 1-strategically closed, and therefore ℙ_{B_{α}} ∗ ℙ_{tail} ∗ ℙ_{B} does not add subsets to α. So by Corollary 1, it suffices to prove that ℙ_{α} forces that the sequence can be lifted.

If α is not a limit point of A ∩ α, then |ℙ_{α}| < α, so the claim holds by Lemma 10.

Suppose that α is a limit point of A ∩ α. In V fix ultrapower maps j_{i} : V → M_{i} = Ult(V, U(α, i)) and k_{i} : M_{i+1} → N_{i} = Ult(M_{i+1}, U(α, i)). By Lemma 9, j_{i} ↾ M_{i+1} = k_{i}. We define a normal ultrafilter U(α, i)^{∗} extending U(α, i) in M_{i+1}[G_{α}].

Write j_{i}(ℙ_{α}) = ℙ_{α} ∗ℙ_{tail}. We claim that k_{i}(ℙ_{α}) can also be factored as ℙ_{α} ∗ℙ_{tail}. For each β in α \ A let ℙ_{α} = ℙ_{β} ∗ ℙ_{β,α}. Since M_{i+1} is α-closed, the sequence ℙ⃗ = ⟨ℙ_{β,α} : β ∈ α \ A⟩ is in M_{i+1}. So j_{i}(ℙ⃗) = k_{i}(ℙ⃗). Since ℙ_{tail} = j_{i}(ℙ⃗)(α), k_{i}(ℙ_{α}) = ℙ_{α} ∗ ℙ_{tail}.

Similarly, for each β in α\A let A_{β} be a ℙ_{β}-name for the set of maximal antichains in ℙ_{β,α}. Since ℙ_{α} is α-c.c. and M_{i+1} is α-closed, the sequence A⃗ = ⟨A_{β} : β ∈ α\A⟩ is in M_{i+1}, and j_{i}(A⃗) = k_{i}(A⃗). In particular, A = j_{i}(A⃗)(α) is a ℙ_{α}-name for the collection of maximal antichains in ℙ_{tail}.

In M_{i+1}[G_{α}] the size of k_{i}(α) is α^{+}, so we can enumerate A as ⟨A_{i} : i < α^{+}⟩. In V [G_{α}] this sequence also enumerates all of the maximal antichains of ℙ_{tail} in M_{i}[G_{α}]. Since N_{i}[G_{α}] is α-closed, ℙ_{tail} is α^{+}-strategically closed in M_{i+1}[G_{α}]. So we can construct a generic H_{i} for ℙ_{tail} over N_{i}[G_{α}]. Note that H_{i} is also a generic for ℙ_{tail} over M_{i}[G_{α}].

Now lift j_{i} and k_{i} to j_{i} : V [G_{α}] → M_{i}[G_{α} ∗H_{i}] and k_{i} : M_{i+1}[G_{α}] → N_{i}[G_{α} ∗H_{i}]. Note that j_{i} ↾ M_{i+1}[G_{α}] = k_{i}. Define U(α, i)^{∗} by letting X ∈ U(α, i)^{∗} iff X ⊆ α and α ∈ j_{i}(X). Since P(α) ⊆ M_{i+1}[G_{α}] and k_{i} = j_{i} ↾ M_{i+1}[G_{α}], the ultrafilter U(α, i)^{∗} can be correctly computed in M_{i+1}[G_{α}].

This defines U(α, i)^{∗} for i < α^{++}. Suppose that i < j. Then U(α, i)^{∗} is in M_{i+1}[G_{α}] ⊆ M_{j}[G_{α}] ⊆ M_{j}[G_{α} ∗ H_{j}] = Ult(V [G], U(α, j)^{∗}). So U(α, i)^{∗} ◁ U(α, j)^{∗}.

□

To prove that ℙ_{κ} ∗ ℙ_{B} preserves the stationarity of S(κ, κ^{+}), it suffices to prove that κ remains κ^{++}-supercompact. Let U be a normal ultrafilter on P_{κ}κ^{++} and let j : V → M = Ult(V, U). Note that κ is in j(A). Let G_{κ} ∗ G_{B} be generic for ℙ_{κ} ∗ ℙ_{B} over V. Write j(ℙ_{κ} ∗ ℙ_{B}) = ℙ_{κ} ∗ ℙ_{B} ∗ ℙ_{tail} ∗ j(ℙ_{B}). In M[G_{κ} ∗ G_{B}], ℙ_{tail} is κ^{+3}-strategically closed. By Lemma 6, M[G_{κ} ∗ G_{B}] is κ^{++}-closed in V [G_{κ} ∗ G_{B}]. Therefore ℙ_{tail} is κ^{+3}-strategically closed in V [G_{κ} ∗ G_{B}]. In M[G_{κ} ∗ G_{B}], ℙ_{tail} has j(κ)-many maximal antichains and |j(κ)| = κ^{+3}. So we can construct a generic filter G_{tail} for ℙ_{tail} over M[G_{κ} ∗ G_{B}]. Extend j to j : V [G_{κ}] → M[G_{κ} ∗ G_{B} ∗ G_{tail}].

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

The poset j(ℙ_{B}) has j(κ^{++}) many maximal antichains. But j(κ^{++}) has size κ^{+3}. In M[G_{κ} ∗ G_{B} ∗ G_{tail}], j(ℙ_{B}) is j(κ)-strategically closed. By Lemma 7, M[G_{κ} ∗ G_{ℚ} ∗ G_{tail}] is κ^{++}-closed and |j(κ)| = κ^{+3}. So in V [G_{κ} ∗ G], j(ℙ_{B}) is κ^{+3}-strategically closed. So we can construct a generic H for j(ℙ_{B}) 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}κ^{++} and j“κ^{++} ∈ j(X). Standard arguments show that U^{∗} is a normal ultrafilter on P_{κ}κ^{++} in V [G_{κ} ∗ G_{B}].

6. Changing Cofinalities

In this section we present Gitik’s poset for changing a cofinality without adding bounded sets which we will use in the consistency results of the following sections. If V is the core model and ℙ is a forcing poset which changes the cofinality of a regular cardinal κ to an uncountable value without collapsing cardinals, then ℙ adds bounded subsets to κ. Therefore we are required to prepare the ground model before defining the poset.

Our exposition differs from that of [3].

Let V be a model of set theory in which κ is a strongly inaccessible cardinal. Assume that there is a function

o : A ∪ {κ} → κ,

where A ⊆ κ is an unbounded set of strongly inaccessible cardinals and o(α) < α for all α in A ∪ {κ}. The function o will be a relativized version of the Mitchell order function in a particular inner model.

Assume that for each α in A, there is a fixed set

b_{α} = b_{α}^{∗} ∪ {α},

which satisfies the following properties:

(1) If o(α) = 0 then b_{α}^{∗} = ∅ and b_{α} = {α},

(2) If o(α) > 0 then b_{α}^{∗} is closed and unbounded in α, and the order type of b_{α}^{∗} is equal to the ordinal exponent ω^{o(α)},

(3) If γ is in b_{α}^{∗}, then o(γ) < o(α),

(4) If γ is in b_{α}^{∗} and for all β in b_{α} ∩ γ, o(β) < o(γ), then b_{γ}^{∗} = b_{α} ∩ γ,

(5) If γ is in b_{α}^{∗} and (4) fails for γ, then there is a maximum ordinal γ′ in b_{α} ∩ γ such that o(γ′) ≥ o(γ), and b_{γ}^{∗} = b_{α} ∩ (γ′, γ). Note that by (5), if γ is in b_{α}^{∗} and o(γ) = 0, then γ is not a limit point of b_{α}.

For each i less than o(κ) let A_{i} denote the set of α in A with o(α) = i.

We will define a forcing poset ℙ(κ, o(κ)) which adds a generic set b_{κ} = b_{κ}^{∗} ∪ {κ} which itself satisfies conditions (1) through (5). We will need to state a few additional properties of the ground model, but first we provide some analysis of the sets b_{α}.

Lemma 15. Suppose that α is in A and o(α) = γ + 1 for some γ ≥ 1. Enumerate b_{α} in increasing order as ⟨β_{i} : 1 ≤ i < ω^{γ} ·ω⟩. Then for any i < ω^{γ} ·ω, o(β_{i}) = γ iff there is 1 ≤ n < ω such that i = ω^{γ} · n. If o(β_{i}) = γ, then b_{β_{i}}^{∗} = {β_{j} : ω^{γ} · (n − 1) < j < i}.

Proof. By (3), the maximum possible value of o(β_{i}) is γ, and therefore the largest possible order type of b_{β_{i}}^{∗} is ω^{γ}. By (4) and (5) we can write b_{β_{i}}^{∗} = {β_{j} : k ≤ j < i} for some k < i. Suppose that i = ω^{γ} · n for some n ≥ 1. Then any co-bounded subset of {β_{j} : 1 ≤ j < i} has order type at least ω^{γ}. Therefore o.t.(b_{β_{i}}^{∗}) = ω^{γ} and o(β_{i}) = γ. Conversely, suppose that ω^{γ} · (n − 1) < i < ω^{γ} · n for some n ≥ 1. Since o(β_{ω^{γ}·(n−1)}) is either undefined (if n = 1) or else is equal to γ by what we just proved, (5) implies that b_{β_{i}}^{∗} is contained in the interval {β_{j} : ω^{γ} · (n − 1) < j < i}, which has order type less than ω^{γ}. So o(β_{i}) < γ.

The second statement follows immediately from (4) and (5). □

Lemma 16. Suppose that α is in A and o(α) is a limit ordinal. Enumerate b_{α} in increasing order as ⟨β_{i} : 1 ≤ i < ω^{o(α)}⟩. Fix 1 ≤ i < ω^{o(α)} and suppose that i = ω^{γ} for some 1 ≤ γ < o(α). Then o(β_{i}) = γ, and b_{β_{i}} = {β_{j} : 1 ≤ j ≤ i}.

Proof. Since b_{α} is closed and i is a limit ordinal, β_{i} is a limit point of b_{α} and therefore o(β_{i}) > 0. By (4) and (5) there exists k such that b_{β_{i}}^{∗} = {β_{j} : k ≤ j ≤ i}. But for any choice of k, this set has order type ω^{γ}. So o.t.(b_{β_{i}}^{∗}) = ω^{γ} and therefore o(β_{i}) = γ. If k > 1, then by (4) and (5) there is l < i such that o(β_{l}) ≥ γ. This is impossible since any bounded subset of {β_{j} : 1 ≤ j < ω^{γ}} has order type less than ω^{γ}. □

Fix an ordinal β ≤ o(κ). A sequence ⟨α_{0}, …, α_{n}⟩ is said to be β-coherent for κ if it is a finite increasing sequence from A such that o(α_{i}) < β for each i, and it satisfies the following property: For each 1 ≤ m ≤ n, let m′ be the least index so that for all m′ ≤ k < m, o(α_{k}) < o(α_{m}). Then ⋃{b_{α_{k}} : m′ ≤ k < m} is an initial segment of b_{α_{m}}. If m′ does not exist, or in other words, o(α_{m−1}) ≥ o(α_{m}), then the minimal element of b_{α_{m}} is greater than α_{m−1}.

Suppose that γ < β ≤ o(κ) and t = ⟨α_{0}, …, α_{n}⟩ is β-coherent. Define t(γ) = ⟨α_{i}, …, α_{n}⟩, where i is the minimal index such that for each i ≤ k ≤ n, o(α_{k}) < γ. If there is no such i, i.e. o(α_{n}) ≥ γ, then let t(γ) be the empty sequence. Note that t(γ) is γ-coherent.

Lemma 17. Suppose that t = ⟨α_{0}, …, α_{n}⟩ is β-coherent and i ≤ n. If j < i and o(α_{j}) ≥ o(α_{i}), then min b_{α_{i}} > α_{j}.

Proof. We prove the statement by induction on i. It is vacuously true for i = 0. Suppose that i is greater than 0 and there exists j < i such that o(α_{j}) ≥ o(α_{i}). Let j be the maximal such index. If j = i − 1, then min b_{α_{i}} > α_{j} by the definition of βcoherence. Otherwise let i′ = j + 1. Then for all i′ ≤ k < i, o(α_{k}) < o(α_{i}) ≤ o(α_{j}). So ⋃{b_{α_{k}} : i′ ≤ k < i} is an initial segment of b_{α_{i}}. By induction, min b_{α_{k}} > α_{j} for all such k, so min b_{α_{i}} > α_{j}. □

If t is an o(κ)-coherent sequence, let b_{t} denote ⋃{b_{α} : α ∈ t}.

Lemma 18. Suppose that t = ⟨α_{0}, …, α_{n}⟩ is β-coherent and α is an ordinal in b_{t} \ t. Let i_{0} be the minimal index such that α is less than α_{i_{0}}, and let i_{1} be the minimal index such that α is in b_{α_{i_{1}}} . Then i_{0} = i_{1}.

In particular, for each i, ⋃{b_{α_{k}} : k ≤ i} is an initial segment of b_{t}.

Proof. Since α is in b_{α_{i_{1}}} \ t, α is less than α_{i_{1}}. So α_{i_{0}} ≤ α_{i_{1}}. Suppose for a contradiction that α_{i_{0}} < α_{i_{1}}. Let i′_{1} be the least index such that for all i′_{1} ≤ k < i_{1}, o(α_{k}) < o(α_{i_{1}}). Since α < α_{i_{0}} < α_{i_{1}} and α is in b_{α_{i_{1}}} , by Lemma 17 i′_{1} exists and i′_{1} ≤ i_{0}. Since α < α_{i_{0}} and ⋃{b_{α_{k}} : i′_{1} ≤ k < i_{1}} is an initial segment of b_{α_{1}}, there exists k < i_{1} such that α is in b_{α_{k}}. This contradicts the minimality of i_{1}. □

Lemma 19. Suppose that t is β-coherent and γ is in t. If o(ξ) < o(γ) for all ξ in b_{t} ∩ γ, then b_{γ}^{∗} = b_{t} ∩ γ. Otherwise there is a maximal γ′ in b_{t} ∩ γ such that o(γ′) ≥ o(γ). In this case, b_{γ}^{∗} = b_{t} ∩ (γ′, γ).

Proof. Suppose that o(ξ) < o(γ) for all ξ in b_{t} ∩ γ. In particular, o(ξ) < o(γ) for all ξ in t ∩ γ. So ⋃{b_{ξ} : ξ ∈ t ∩ γ} is an initial segment of b_{γ}. This fact along with Lemma 18 imply that b_{t} ∩ γ = b_{γ}^{∗}. Otherwise let γ′ be the maximal ordinal in t ∩ γ such that o(γ′) ≥ o(γ). Then ⋃{b_{ξ} : ξ ∈ t ∩ (γ′, γ)} is an initial segment of b_{γ}. It follows from Lemmas 17 and 18 that b_{t} ∩ (γ′, γ) = b_{γ}^{∗}. □

Now we complete the description of the ground model. We assume that there is a family

{ U(κ, β, t) : β < o(κ), t is β-coherent }

of κ-complete ultrafilters on κ satisfying the following conditions:

(1) A_{β} = {α < κ : o(α) = β} is in U(κ, β, t),

(2) If s and t are β-coherent sequences and b_{s} = b_{t}, then U(κ, β, s) = U(κ, β, t),

(3) If t is β-coherent and γ is less than β, then the set of α less than κ such that t α is β-coherent is in the ultrafilter U(κ, γ, t(γ)).

Fix β ≤ o(κ). We describe a forcing poset ℙ(κ, β).

A set T is said to be a β-tree if T is a collection of β-coherent sequences satisfying:

(1) ⟨T, ⊴⟩ is a tree ordering, where u ⊴ v iff u is an initial segment of v. In other words, if v is in T and u ⊴ v, then u is in T,

(2) The tree T has a trunk t such that t is in T and for all u in T, either u ⊴ v or v ⊴ u,

(3) For any u in T, if t ⊴ u where t is the trunk of T, then the set of α less than κ such that u α is in T can be partitioned into a disjoint family

{ Suc(T, u, γ) : γ < β },

where Suc(T, u, γ) is a subset of A_{γ} and is in U(κ, γ, u(γ)).

If T is a β-tree with trunk t, u is in T, and t ⊴ u, then let T_{u} denote the tree consisting of those v in T such that either u ⊴ v or v ⊴ u. Note that T_{u} is a β-tree with trunk u.

If β = 0 then let ℙ(κ, β) be the trivial poset. Define b_{κ}^{∗} = ∅ and b_{κ} = {κ}. Suppose that β is greater than 0.

A condition in ℙ(κ, β) is a pair ⟨t, T ⟩ such that T is a β-tree with trunk t. If ⟨t, T ⟩ and ⟨s, S⟩ are conditions, we let ⟨t, T ⟩ ≤ ⟨s, S⟩ if:

(1) there exists t^{∗} in S such that b_{t^{∗}} = b_{t},

(2) for each u in T, the sequence t^{∗} (u \ (max t^{∗} + 1)) is in S. If ⟨t, T ⟩ ≤ ⟨s, S⟩ and s = t, then let ⟨t, T ⟩ ≤^{∗} ⟨s, S⟩.

The assumptions about the family of ultrafilters imply that β-trees exist, and therefore ℙ(κ, β) is a non-trivial forcing poset.

The proof that ℙ(κ, o(κ)) satisfies the Prikry property will depend on certain details about the preparation forcing; see Proposition 10. For the rest of this section we will just assume that ⟨ℙ(κ, o(κ)), ≤, ≤^{∗}⟩ is a Prikry type forcing poset.

If T_{0} and T_{1} are β-trees with the same trunk t, then T_{0} ∩ T_{1} is also a β-tree with trunk t. In fact, suppose that {⟨t, T_{i}⟩ : i < ξ} is a family of conditions in ℙ(κ, β), where ξ is less than κ. By the κ-completeness of the ultrafilters, ⋂ T_{i} is a β-tree with trunk t. So ⟨t, ⋂ T_{i}⟩ is a condition which directly extends each ⟨t, T_{i}⟩. It follows that ℙ(κ, o(κ)) is κ-weakly closed and satisfies the direct extension property. Therefore ℙ(κ, o(κ)) does not add bounded subsets to κ.

Proposition 7. The poset ℙ(κ, β) is κ^{+}-c.c.

Proof. Any two conditions with the same trunk are compatible. There are only κ many possibilities for the trunk. □

So ℙ(κ, o(κ)) preserves all cardinals.

Lemma 20. Suppose that ⟨t, T ⟩ ≤ ⟨s, S⟩ in ℙ(κ, β). Then there is t^{∗} in S and T^{∗} ⊆ S such that ⟨t, T ⟩ is equivalent to ⟨t^{∗}, T^{∗}⟩.

Proof. Fix t^{∗} in S witnessing that ⟨t, T ⟩ ≤ ⟨s, S⟩. Since b_{t} = b_{t^{∗}}, t and t^{∗} have the same maximal element α. Define

T^{∗} = {t^{∗} (u \ α + 1) : u ∈ T }.

Note that T^{∗} ⊆ S. The relation ⟨t, T ⟩ ≤ ⟨t^{∗}, T^{∗}⟩ is witnessed by t^{∗}, and ⟨t^{∗}, T^{∗}⟩ ≤ ⟨t, T ⟩ is witnessed by t. □

Lemma 21. Suppose that ⟨s, S⟩ is a condition in ℙ(κ, β) and α is in b_{s}. Then there is a condition ⟨t, T ⟩ such that b_{t} = b_{s}, ⟨t, T ⟩ is equivalent to ⟨s, S⟩, and α is in t.

Proof. Let ⟨s, S⟩ be a condition and suppose that α is in b_{s}. Write s = ⟨α_{0}, …, α_{n}⟩. If α is in s then we are done. Suppose that α is in b_{s} \ s. Fix i ≤ n so that α_{i} is the least ordinal in s larger than α. By Lemma 18, α_{i} is also the least ordinal in s for which α is in b_{α_{i}}. We will prove the existence of ⟨t, T ⟩ by induction on o(α_{i}).

Since α is in b_{α_{i}}^{∗}, o(α_{i}) > 0. Suppose that o(α_{i}) = 1, so that b_{α_{i}}^{∗} has order type ω. Let β_{0}, …, β_{m} enumerate in increasing order the ordinals in b_{α_{i}} ∩ α, and which are larger than α_{i−1} if i > 0. Define

t = ⟨α_{0}, …, α_{i−1}, β_{0}, …, β_{m}, α, α_{i}, …, α_{n}⟩.

Then t is β-coherent and b_{t} = b_{s}. Define

T = {t (u \ (α_{n} + 1)) : u ∈ S}.

Then ⟨t, T ⟩ is a condition, ⟨t, T ⟩ ≤ ⟨s, S⟩ is witnessed by s, and ⟨s, S⟩ ≤ ⟨t, T ⟩ is witnessed by t.

Suppose that o(α_{i}) = γ, where γ > 1, and the claim holds whenever o(α_{i}) < γ. Enumerate b_{α_{i}} in increasing order as ⟨β_{i} : 1 ≤ i < ω^{γ}⟩.

Suppose that γ is a successor ordinal, and let γ = γ_{0} + 1. Then b_{α_{i}} has order type ω^{γ} = ω^{γ_{0}} · ω. Fix 1 ≤ n < ω minimal so that α is in b_{α_{i}} ∩ (β_{ω^{γ_{0}} ·n} + 1). Let ξ_{0}, …, ξ_{m} enumerate the ordinals in b_{α_{i}} \ (α_{i−1} + 1) of the form β_{ω^{γ_{0}} ·k}, where 1 ≤ k ≤ n. By Lemma 15, these are exactly the ordinals in b_{α_{i}} \ (α_{i−1} + 1) which have order γ_{0}. Define

s^{∗} = ⟨α_{0}, …, α_{i−1}, ξ_{0}, …, ξ_{m}, α_{i}, …, α_{n}⟩,

which is β-coherent. Also b_{s} = b_{s^{∗}}. Define

S^{∗} = {s^{∗} (u \ (α_{n} + 1)) : u ∈ S}.

Then ⟨s^{∗}, S^{∗}⟩ is equivalent to ⟨s, S⟩. If α is in s^{∗} we are done. Otherwise, the least ordinal in s^{∗} which is larger than α is now ξ_{n}, and o(ξ_{n}) = γ_{0}. By induction, there is ⟨t, T ⟩ equivalent to ⟨s^{∗}, S^{∗}⟩ such that α is in t and b_{t} = b_{s^{∗}}.

Now suppose that γ is a limit ordinal. Then ω^{γ} = sup{ω^{j} : j < γ}. Fix j < γ minimal so that α is in b_{α_{i}} ∩ (β_{ω^{j}} + 1). By Lemma 16, o(β_{ω^{j}}) = j, and b_{β_{ω^{j}}} = {β_{l} : 1 ≤ l ≤ ω^{i}}. Define

s^{∗} = ⟨α_{0}, …, α_{i−1}, β_{ω^{j}}, α_{i}, …, α_{n}⟩.

Then s^{∗} is β-coherent and b_{s^{∗}} = b_{s}. Define S^{∗} as in the last case. Then ⟨s^{∗}, S^{∗}⟩ is equivalent to ⟨s, S⟩. If α is in s^{∗} we are done. Otherwise, the least ordinal above α in s^{∗} is β_{ω^{j}}, which has order less than γ. By induction, there is a condition ⟨t, T ⟩ which is equivalent to ⟨s^{∗}, S^{∗}⟩ such that α is in t and b_{t} = b_{s^{∗}}. □

Lemma 22. Suppose that ⟨s, S⟩ and ⟨t, T ⟩ are conditions in ℙ(κ, β) such that b_{s} = b_{t}. Then ⟨s, S⟩ and ⟨t, T ⟩ are compatible.

Proof. Note that s and t have the same maximal element α. Define

S_{0} = {v ∈ S : if s ⊴ v then t (v \ (α + 1)) ∈ T }.

Then ⟨s, S_{0}⟩ is a condition below ⟨s, S⟩. Apply Lemma 21 finitely many times to obtain a condition ⟨u, U⟩ which is equivalent to ⟨s, S_{0}⟩ such that s ∪ t ⊆ u and b_{u} = b_{s}. By definition of S_{0}, ⟨u, U⟩ is also below ⟨t, T ⟩, as witnessed by t. □

Now we analyze the generic object for ℙ(κ, o(κ)). Let ḃ_{κ} and ḃ_{κ}^{∗} be ℙ(κ, o(κ))- names such that

⊩ ḃ_{κ}^{∗} = {α ∈ κ : ∃⟨t, T ⟩ ∈ Ġ α ∈ t}.

and

⊩ ḃ_{κ} = ḃ_{κ}^{∗} ∪ {κ},

Lemma 23. The poset ℙ(κ, o(κ)) forces that b_{κ}^{∗} is equal to ⋃{b_{t} : ∃T ⟨t, T ⟩ ∈ Ġ}.

Proof. This is immediate from Lemma 21. □

Lemma 24. Let ⟨s, S⟩ be a condition in ℙ(κ, o(κ)). Then ⟨s, S⟩ forces that b_{s} is an initial segment of ḃ_{κ}.

Proof. By Lemma 23, ⟨s, S⟩ forces that b_{s} is a subset of ḃ_{κ}. Suppose that ⟨t, T ⟩ is below ⟨s, S⟩ and α is in t ∩ max s. Fix t^{∗} in S such that b_{t^{∗}} = b_{t}. Then α is in b_{t^{∗}}. Since s ⊴ t^{∗} and α is less than max s, by Lemma 18 it follows that α is in b_{s}. Therefore ⟨s, S⟩ forces that b_{s} is an initial segment of ḃ_{κ}.

□

We show that the generic set b_{κ} satisfies the properties (1) through (5) which we stated for the b_{α}’s.

Lemma 25. The poset ℙ(κ, o(κ)) forces that ḃ_{κ}^{∗} is closed and unbounded in κ.

Proof. By an easy density argument, ḃ_{κ}^{∗} is unbounded in κ. To show it is club, suppose that ⟨s, S⟩ forces that β is a limit point of ḃ_{κ}^{∗}. Let ⟨t, T ⟩ ≤ ⟨s, S⟩ such that max t > β. Then ⟨t, T ⟩ forces that ḃ_{κ} ∩ (max t + 1) = b_{t}, which is closed. So β is in b_{t} ⊆ ḃ_{κ}. □

Let G be generic for ℙ(κ, o(κ)). If α is in b_{κ}^{∗}, then α is in t for some ⟨t, T ⟩ in G. But t is o(κ)-coherent, so o(α) < o(κ).

Lemma 26. Let γ be in b_{κ}^{∗}. If o(β) < o(γ) for all β in b_{κ} ∩ γ, then b_{γ}^{∗} = b_{α} ∩ γ.

Otherwise, there is a maximal ordinal γ′ in b_{α} below γ such that o(γ′) ≥ o(γ). In this case, b_{γ}^{∗} = b_{α} ∩ (γ′, γ).

Proof. If γ is in b_{κ}^{∗}, then there is a condition ⟨s, S⟩ in G such that γ is in s. Since b_{s} is an initial segment of b_{κ}, the statement follows from Lemma 19. □

Proposition 8. The order type of b_{κ}^{∗} is ω^{o(κ)}. In particular, if o(κ) is a regular uncountable cardinal, then o.t.(b_{κ}^{∗}) = o(κ).

Proof. Suppose that o(κ) = 1. Then an o(κ)-coherent sequence is just a finite increasing sequence from the set {α ∈ A : o(α) = 0}, and for any such sequence t, b_{t} = t. It follows that ℙ(κ, o(κ)) forces that every initial segment of b_{κ} is finite. So b_{κ}^{∗} has order type ω.

Suppose that o(κ) = γ + 1. Then the maximum possible order an ordinal in an o(κ)-coherent sequence is γ. For any finite n, there is a dense set of conditions ⟨t, T ⟩ such that t has the form t = ⟨α_{0}, …, α_{m}, β_{0}, …, β_{n−1}⟩, where for each 0 ≤ i < n, o(β_{i}) = γ. Then b_{t} has order type at least ω^{γ} · n. Therefore b_{κ}^{∗} is forced to have order type at least ω^{γ} · ω = ω^{γ+1}. Suppose for a contradiction that there is a condition ⟨s, S⟩ in G which forces that the order type of ḃ_{κ}^{∗} is larger than ω^{γ+1}. Then there is a condition ⟨t, T ⟩ in G and an ordinal β less than κ such that max t is larger than β and ⟨t, T ⟩ forces that the order type of b_{κ} ∩ β is at least ω^{γ+1}. So the order type of b_{t} is at least ω^{γ+1}. But b_{t} is a finite union of sets with order type at most ω^{γ}, so we have a contradiction.

Finally, suppose that o(κ) is a limit ordinal. Then for each i < o(κ), there is a dense set of conditions ⟨t, T ⟩ such that there is α in t with o(α) = i. For any such condition ⟨t, T ⟩, b_{t} has order type at least ω^{i}. It follows that b_{κ}^{∗} has order type at least ω^{o(κ)}. By the same argument as in the last case, b_{κ}^{∗} cannot have an order type larger than ω^{o(κ)}. □

7. The Structure of S(κ, κ^{+}), Continued

Let V be a model of set theory in which κ is a κ^{+}-supercompact cardinal and GCH holds. Let μ be a regular cardinal less than κ. We construct a generic extension of V in which κ is strongly inaccessible, S(κ, κ^{+}) is stationary, and for almost all a in S(κ, κ^{+}), a ∩ κ is a singular cardinal with cofinality μ.

For the remainder of the section fix a normal ultrafilter U on P_{κ}κ^{+} and let j_{U} : V → M = Ult(V, U). Let U ↾ κ be the projection of U to κ; i.e., X ∈ U ↾ κ iff X ⊆ κ and κ ∈ j_{U}(X). Let i : V → N = Ult(V, U ↾ κ). Applying Lemma 3, fix an elementary embedding k : N → M such that crit(k) = κ^{++N} and j_{U} = k ∘ i.

Proposition 9. There exists a ◁-increasing sequence ⟨U(κ, i) : i ≤ μ⟩ of normal ultrafilters on κ such that U(κ, μ) = U ↾ κ.

Proof. It suffices to prove that there exists a ◁-increasing sequence of normal ultrafilters on κ with length μ in the model N = Ult(V, U ↾ κ). To prove this claim it is enough to show that any ◁-increasing sequence of ultrafilters in N with length less than μ has an upper bound in N.

If the claim fails then let α be the least ordinal less than μ such that in N, there exists a ◁-increasing sequence U⃗ with length α with no upper bound. By elementarity, the same statement holds for k(U⃗) = U⃗ in M. This is a contradiction, since U ↾ κ is above every ultrafilter in U⃗. □

Lemma 27. There is a sequence ⟨X_{i} : i ≤ μ⟩ such that each X_{i} is in U(κ, i) and each X_{i} ∩ X_{j} is empty.

Proof. For distinct i and j let X_{i}(j) be a set in U(κ, i)\U(κ, j). Let X_{i}^{∗} = ⋂{X_{i}(j) : j ≤ μ, j ≠ i}. Then for all j ≠ i, X_{i}^{∗} ∈ U(κ, i) \ U(κ, j). Now let X_{i} = X_{i}^{∗} \ ⋃{X_{j}^{∗} : j ≠ i}. □

For each i ≤ μ let j_{i}^{κ} : V → N_{i}^{κ} = Ult(V, U(κ, i)). For i < j ≤ μ let f_{i}^{j} : κ → V_{κ} be a function such that [f_{i}^{j}] = U(κ, i) in N_{j}^{κ}. Define a sequence ⟨A_{j} : j ≤ μ⟩ by induction, using the following recursion: A_{j} is the set of α less than κ such that

(1) α is in X_{j},

(2) α > μ,

(3) For i < j, f_{i}^{j}(α) is a normal ultrafilter on α and A_{i} ∩ α is in f_{i}^{j}(α),

(4) For i_{0} < i_{1} < j, f_{i_{0}}^{j}(α) ◁ f_{i_{1}}^{j}(α),

(5) For i_{0} < i_{1} < j, [f_{i_{0}}^{i_{1}} ↾ α] in Ult(V, f_{i_{1}}^{j}(α)) is equal to f_{i_{0}}^{j}(α). The reader can check by induction that A_{j} is in U(κ, j), using Lemma 9 (1) to prove (5). In particular, κ is in j_{U}(A_{μ}).

Let A be the disjoint union ⋃{A_{j} : j ≤ μ}. If α is in A, let o(α) denote the unique β such that α is in A_{β}. The function o is a relativized version of the Mitchell order function. If α is in A, then for each i < o(α) let U(α, i) = f_{i}^{o(α)}(α) and let j_{i}^{α} : V → N_{i}^{α} = Ult(V, U(α, i)). By (3), for i < o(α), A_{i}∩α is in f_{i}^{o(α)}(α) = U(α, i), and therefore α is in j_{i}^{α}(A_{i}). It follows that j_{i}^{α}(o)(α) = i in N_{i}^{α}. By (5), for each i_{0} < i_{1} < o(α), U(α, i_{0}) is in N_{i_{1}}^{α}, so U(α, i_{0}) ◁ U(α, i_{1}).

Now we are ready to construct our model. Fix a well ordering <_{κ} of V_{κ} such that for every inaccessible α less than κ, <_{κ} ∩ (V_{α} × V_{α}) is a well-ordering of V_{α}. We define an Easton support Prikry iteration ⟨ℙ_{i}, ℚ_{i} : i < κ⟩ by induction.

Suppose that ℙ_{α} is defined. Assume as an induction hypothesis that for all γ less than α, ℚ_{γ} is trivial unless γ is in A ∩ α. If γ is in A then ℙ_{γ} forces that ℚ_{γ} = ℙ_{B_{γ}} ∗ ℙ(γ, o(γ)), where ℙ_{B_{γ}} is a γ + 1-strategically closed forcing poset and ℙ(γ, o(γ)) is defined from a family of ultrafilters

{ U(γ, β, t) : β < o(γ), t is a β-coherent sequence }.

Let G_{α} be generic for ℙ_{α} over V. If α is not in A then let ℚ_{α} be trivial. Suppose that α is in A. In V [G_{α}] define B_{α} as the set of β less than α^{+} such that cf(β) is not the successor of a singular cardinal with cofinality μ. Note that B_{α} is closed under suprema of subsets with order type ω and ω_{1}. Therefore the forcing poset ℙ_{B_{α}} for adding a □_{α}^{B_{α}}-sequence forces that for almost all a in S(α, α^{+}), a ∩ α is a singular cardinal with cofinality μ. We consider ℙ_{B_{α}} as a Prikry type forcing poset with ≤^{∗}=≤.

Let G^{α} be generic for ℙ_{B_{α}} over V [G_{α}]. If o(α) = 0 then ℙ(α, o(α)) is the trivial poset and there is nothing to prove. Suppose that o(α) > 0. We define in V [G_{α}∗G^{α}] a family of α-complete ultrafilters

{ U(α, β, t) : β < o(α), t is a β-coherent sequence }

which satisfies the requirements (1), (2), and (3) described in the last section. We also assume as an induction hypothesis that the family of ultrafilters defined at any previous stage γ satisfies the same definition as at stage α.

In V [G_{α} ∗ G^{α}] fix β less than o(α) and a β-coherent sequence t. We define U(α, β, t). Consider j_{β}^{α} : V → N_{β}^{α}. Since j_{β}^{α}(o)(α) = β, by the induction hypotheses we can write

j_{β}^{α}(ℙ_{α}) = ℙ_{α} ∗ ℙ_{B_{α}} ∗ ℙ(α, β) ∗ ℙ_{α,j_{β}^{α}(α)}.

By closure, <_{α}=<_{κ} ∩ (V_{α} × V_{α}) is in N_{β}^{α}. In order to define U(α, β, t) we need to consider all subsets of α in V [G_{α}∗G^{α}]. However, ℙ_{B_{α}} does not add subsets to α over V [G_{α}]. So it suffices to consider only ℙ_{α}-names for subsets of α. Let ⟨Ẋ_{i} : i < α^{+}⟩ be the j_{β}^{α}(<_{α})-least enumeration of all canonical ℙ_{α}-names for subsets of α.

We apply the α^{+}-weak strategic closure of ℙ_{α,j_{β}^{α}(α)} to define a ≤^{∗}-decreasing sequence of names ⟨q̇_{i}, q̇_{i}^{∗} : i < α^{+}⟩ for conditions in ℙ_{α,j_{β}^{α}(α)} as follows. Consider a run of the game. Let q̇_{0} denote 1. Given q̇_{i}, let q̇_{i}^{∗} be the j_{β}^{α}(<_{α})-least name for a direct extension of q̇_{i} which decides the statement α ∈ j_{β}^{α}(Ẋ_{i}). Let q̇_{i+1} be a name for Player II’s response according to its strategy. Also Player II plays according to its strategy at limit stages.

Now define U(α, β, t) by letting X be in U(α, β, t) iff there is an index i such that Ẋ_{i}^{G_{α}} = X and there exists a β-tree T such that

N_{β}^{α}[G_{α} ∗ G^{α}] ⊧ ⟨t, T ⟩ ⊩ q̇_{i+1} ⊩ α ∈ j_{β}^{α}(Ẋ_{i}).

Lemma 28. Suppose that β < β^{∗} < o(α) and t is β-coherent. Then U(α, β, t) is in N_{β^{∗}}^{α}[G_{α} ∗ G^{α}] and satisfies the same definition as it does in V [G_{α} ∗ G^{α}].

Proof. Let k_{β,β^{∗}}^{α} : N_{β^{∗}}^{α} → Ult(N_{β^{∗}}^{α}, U(α, β)). By Lemma 9 (2), k_{β,β^{∗}}^{α} = j_{β}^{α} ↾ N_{β^{∗}}^{α}. So the lemma follows immediately from the definition of U(α, β, t). □

Lemma 29. The set U(α, β, t) is an α-complete ultrafilter on α which extends U(α, β).

Proof. We prove that the definition of a set X being in U(α, β, t) is independent of the index of its name. Suppose that X = Ẋ_{i}^{G_{α}} = Ẋ_{j}^{G_{α}}. Fix p in G_{α} which forces that Ẋ_{i} = Ẋ_{j}. Then j_{β}^{α}(p) = p forces over N_{β}^{α} that j_{β}^{α}(Ẋ_{i}) = j_{β}^{α}(Ẋ_{j}). Therefore

N_{β}^{α}[G_{α} ∗ G^{α}] ⊧ ℙ(α, β) ⊩ ℙ_{α,j_{β}^{α}(α)} ⊩ (α ∈ j_{β}^{α}(Ẋ_{i}) iff α ∈ j_{β}^{α}(Ẋ_{j})).

Suppose that there is T such that

N_{β}^{α}[G_{α} ∗ G^{α}] ⊧ ⟨t, T ⟩ ⊩ q̇_{i+1} ⊩ α ∈ j_{β}^{α}(Ẋ_{i}).

Since q̇_{j+1} is forced to decide the statement α ∈ j_{β}^{α}(Ẋ_{j}), it follows that

N_{β}^{α}[G_{α} ∗ G^{α}] ⊧ ⟨t, T ⟩ ⊩ q̇_{j+1} ⊩ α ∈ j_{β}^{α}(Ẋ_{j}).

Similar arguments show that U(α, β, t) is an ultrafilter which extends U(α, β).

Let us prove that U(α, β, t) is α-complete. Suppose that p is in G_{α} and p forces that χ : β → P(α) is a partition of α into β many sets, for some β less than α. Then j_{β}^{α}(p) = p forces over N_{β}^{α} that j_{β}^{α}(χ) is a partition of j_{β}^{α}(α) into β many sets. For each i less than β let ξ_{i} be an index such that p forces that χ(i) = Ẋ_{ξ_{i}}. For each i less than β apply the Prikry property of ℙ(α, β) in N_{β}^{α}[G_{α} ∗ G^{α}] to obtain a β-tree T_{i} such that ⟨t, T_{i}⟩ decides which way that q̇_{ξ_{i}+1} decides the statement α ∈ j_{β}^{α}(Ẋ_{ξ_{i}}). Then ⟨t, ⋂ T_{i}⟩ is a condition which directly refines each ⟨t, T_{i}⟩. Since j_{β}^{α}(χ) is forced to be a partition of j_{β}^{α}(α), there must be some index j such that ⟨t, ⋂ T_{i}⟩ forces that q̇_{ξ_{j}+1} forces α ∈ j_{β}^{α}(Ẋ_{ξ_{j}}). □

Lemma 30. Suppose that s and t are β-coherent sequences and b_{s} = b_{t}. Then U(α, β, s) = U(α, β, t).

Proof. It suffices to prove that U(α, β, s) ⊆ U(α, β, t), since they are ultrafilters. So let X be in U(α, β, s), and fix i such that X = Ẋ_{i}^{G_{α}}. Then there is S such that

N_{β}^{α}[G_{α} ∗ G^{α}] ⊧ ⟨s, S⟩ ⊩ q̇_{i+1} ⊩ α ∈ j_{β}^{α}(Ẋ_{i}).

Apply the Prikry property to find ⟨t, T ⟩ which decides whether q̇_{i+1} ⊩ α ∈ j_{β}^{α}(Ẋ_{i}). By Lemma 22, ⟨s, S⟩ and ⟨t, T ⟩ are compatible, so they must decide the statement the same way. So A is in U(α, β, t). □

Lemma 31. Suppose that t is β-coherent and γ is less than β. Then the set of ξ in α such that t ξ is β-coherent is in U(α, γ, t(γ)).

Proof. Let X be the set of ξ less than α such that t ξ is β-coherent. Fix i so that X = Ẋ_{i}^{G_{α}} and let p be a condition in G_{α} which forces that Ẋ_{i} satisfies the definition of X. Since j_{γ}^{α}(p) = p, it follows that

N_{γ}^{α}[G_{α} ∗ G^{α}] ⊧ ℙ(α, γ) ∗ ℙ_{α,j_{γ}^{α}(α)} ⊩ j_{γ}^{α}(Ẋ_{i}) = {ξ < j_{γ}^{α}(α) : t ξ is β-coherent}.

Fix any γ-tree T with trunk t(γ). Then by Lemma 24,

N_{γ}^{α}[G_{α} ∗ G^{α}] ⊧ ⟨t(γ), T ⟩ ⊩ b_{t(γ)} is an initial segment of ḃ_{α}.

Now j_{γ}^{α}(o)(α) = γ and every ξ in t(γ) has order less than γ. It follows that

N_{β}^{α}[G_{α} ∗ G^{α}] ⊩ ⟨t(γ), T ⟩ ⊩ ℙ_{α,j_{γ}^{α}(α)} ⊩ t(γ) α is γ-coherent.

So X is in U(α, γ, t(γ)). □

Now define ℙ(α, o(α)) in V [G_{α} ∗ G^{α}] using this family of ultrafilters. We show that ℙ(α, o(α)) satisfies the Prikry property.

Proposition 10. Suppose that φ is a statement in the forcing language for ℙ(α, o(α)) and ⟨t, T ⟩ is a condition. Then there is a direct extension of ⟨t, T ⟩ which decides φ.

Proof. Let φ be a statement in the forcing language and fix a condition ⟨t, T ⟩. Let φ^{0} denote φ and let φ^{1} denote ¬φ.

We define a sequence ⟨T_{n}^{∗} : n < ω⟩ such that each T_{n}^{∗} ⊆ T is an o(α)-tree with trunk t, and T_{n+1}^{∗} ⊆ T_{n}^{∗} for each n. Let T_{n}^{∗} = T for n ≤ |t|.

Suppose that n ≥ |t| and T_{n}^{∗} is defined. We define T_{n+1}^{∗}. If u is in T_{n}^{∗} and |u| ≤ n, then let u be in T_{n+1}^{∗}. Now for each u in T_{n}^{∗} with |u| = n we define (T_{n+1}^{∗})_{u}. Fix γ less than o(α). For each ξ in Suc(T_{n}^{∗}, u, γ) define n_{ξ} = l for l < 2 if there exists a tree T_{ξ} ⊆ T_{n}^{∗} with trunk u ξ such that ⟨u ξ, T_{ξ}⟩ forces φ^{l}. If there is no such tree T_{ξ}, then let n_{ξ} = 2. Now fix l ≤ 2 and X_{u,γ} in U(α, γ, u(γ)) such that n_{ξ} = l for all ξ in X_{u,γ}. If l = 2 then define

(T_{n+1}^{∗})_{u} = {u ξ v ∈ T_{n}^{∗} : ξ ∈ X_{u,γ}, γ < o(α)}

If l < 2 then define

(T_{n+1}^{∗})_{u} = {u ξ v ∈ T_{ξ} : ξ ∈ X_{u,γ}, γ < o(α)}.

Clearly T_{n+1}^{∗} is an o(α)-tree with trunk t.

Now let T^{∗} = ⋂ T_{n}^{∗}. Then ⟨t, T^{∗}⟩ is a direct refinement of ⟨t, T ⟩. The choice of T^{∗} guarantees that the following property holds: Suppose that u is in T^{∗} such that t ⊴ u, γ is less than o(α), and there is an o(α)-tree S ⊆ T^{∗} such that for some ξ in Suc(T^{∗}, u, γ), ⟨u ξ, S⟩ forces φ^{l}. Then for all ξ in Suc(T^{∗}, u, γ), ⟨u ξ, T_{u ξ}^{∗}⟩ forces φ^{l}.

Claim 2. Suppose that ⟨t u ξ, T_{t u ξ}^{∗}⟩ forces φ^{l} for some l < 2. Then for all γ ≥ o(ξ) and any β in Suc(T^{∗}, t u, γ), ⟨t u β, T_{t u β}^{∗}⟩ forces φ^{l}.

Proof. It suffices to prove the statement is true for some β in Suc(T^{∗}, t u, γ). We already know that this is true when γ = o(ξ). Suppose that γ is larger than o(ξ).

We claim that there is an ordinal β^{∗} in the set Suc(T^{∗}, t u, o(ξ)) such that

Suc(T^{∗}, t u, γ) ∩ Suc(T^{∗}, t u β^{∗}, γ) ≠ ∅.

It suffices to prove that there is an ordinal β^{∗} in Suc(T^{∗}, t u, o(ξ)) such that Suc(T^{∗}, t u, γ) is in U(α, γ, (t u β^{∗})(γ)).

Fix i so that Suc(T^{∗}, t u, γ) = Ẋ_{i}^{G_{α}}. Since Suc(T^{∗}, t u, γ) is in U(α, γ, (t u)(γ)), by the definition of this ultrafilter in V [G_{α} ∗ G^{α}] there is S such that

N_{γ}^{α}[G_{α} ∗ G^{α}] ⊧ ⟨(t u)(γ), S⟩ ⊩ q̇_{i+1} ⊩ α ∈ j_{γ}^{α}(Ẋ_{i}).

Since γ is greater than o(ξ), (t u)(o(ξ)) = (t u)(γ)(o(ξ)). It follows that the set Suc(S, (t u)(γ), o(ξ)) is in U(α, o(ξ), (t u)(o(ξ))). So fix an ordinal β^{∗} which is in the set

Suc(S, (t u)(γ), o(ξ)) ∩ Suc(T^{∗}, t u, o(ξ)).

Now ⟨(t u)(γ) β^{∗}, S_{(t u)(γ) β^{∗}}⟩ is below ⟨(t u)(γ), S⟩, therefore

N_{γ}^{α}[G_{α} ∗ G^{α}] ⊧ ⟨(t u)(γ) β^{∗}, S_{(t u)(γ) β^{∗}}⟩ ⊩ q̇_{i+1} ⊩ α ∈ j_{γ}^{α}(Ẋ_{i}).

Therefore Suc(T^{∗}, t u, γ) is in U(α, γ, (t u)(γ) β^{∗})) = U(α, γ, (t u β^{∗})(γ)), as desired.

Now choose β in Suc(T^{∗}, t u, γ) ∩ Suc(T^{∗}, t u β^{∗}, γ). Since o(β^{∗}) = o(ξ), ⟨t u β^{∗}, T_{t u β^{∗}}^{∗}⟩ forces φ^{l}. Moreover, ⟨t u β^{∗} β, T_{t u β^{∗} β}^{∗}⟩ is below this condition, so it also forces φ^{l}. Since o(β) > o(β^{∗}), β^{∗} is in b_{β}. So b_{t u β} = b_{t u β^{∗} β}. Therefore for all δ less than o(α), U(α, δ, (t u β)(δ)) = U(α, δ, (t u β^{∗} β)(δ)). It follows that S = {v \ {β^{∗}} : v ∈ T_{t u β^{∗} β}^{∗}} is an o(α)-tree. Also the conditions ⟨t u β^{∗} β, T_{t u β^{∗} β}^{∗}⟩ and ⟨t u β, S⟩ are equivalent. Therefore ⟨t u β, S∩T_{t u β}^{∗}⟩ forces φ^{l}. By the choice of T^{∗}, ⟨t u β, T_{t u β}^{∗}⟩ forces φ^{l}.

□

Suppose for a contradiction that ⟨t, T^{∗}⟩ does not have a direct extension which decides φ. Using Lemma 20, choose a condition ⟨t u ξ, T_{0}⟩ below ⟨t, T^{∗}⟩ which decides φ such that t u ξ is in T^{∗}, T_{0} ⊆ T^{∗}, and |u| is minimal. Fix l < 2 such that ⟨t u ξ, T_{0}⟩ forces φ^{l}. By Claim 2 and the choice of T^{∗}, for all γ ≥ o(ξ) and β in Suc(T^{∗}, t u, γ), ⟨t u β, T_{t u β}^{∗}⟩ forces φ^{l}.

We will prove that the condition ⟨t u, T_{t u}^{∗}⟩ forces φ^{l}, which contradicts the minimality of u. By Lemma 20 it suffices to prove that for all γ less than o(α) and for all β in Suc(T^{∗}, t u, γ), ⟨t u β, T_{t u β}^{∗}⟩ forces φ^{l}. We already know by Claim 2 that this is true for γ ≥ o(ξ). So by the choice of T^{∗} it is enough to show that for all γ less than o(ξ) there is β in Suc(T^{∗}, t u, γ) such that ⟨t u β, T_{t u β}^{∗}⟩ forces φ^{l}.

Fix i so that Suc(T^{∗}, t u, o(ξ)) = Ẋ_{i}^{G_{α}}. Since the set Suc(T^{∗}, t u, o(ξ)) is in U(α, o(ξ), (t u)(o(ξ))), there exists an o(ξ)-tree S such that

N_{o(ξ)}^{α}[G_{α} ∗ G^{α}] ⊧ ⟨(t u)(o(ξ)), S⟩ ⊩ q̇_{i+1} ⊩ α ∈ j_{o(ξ)}^{α}(Ẋ_{i}).

Define T_{1} as the set of sequences of the form t u ⟨α_{0}, …, α_{n}⟩ which are in T^{∗} and satisfy the following property: Let i ≤ n be the maximal index such that for all j ≤ i, o(α_{j}) is less than o(ξ); then (t u)(o(ξ)) ⟨α_{0}, …, α_{i}⟩ is in S.

Fix some ordinal β in the set Suc(T_{1}, t u, γ). We claim that ⟨t u β, (T_{1})_{t u β}⟩ forces φ^{l}. If this is true, then by the choice of T^{∗} the condition ⟨t u β, T_{t u β}^{∗}⟩ forces φ^{l} and we are done. If the claim fails, then by the choice of T^{∗} there exists a condition

⟨t u β ⟨α_{0}, …, α_{n}⟩, T_{t u β ⟨α_{0},...,α_{n}⟩}^{∗}⟩

which forces the negation of φ^{l}, where t u β ⟨α_{0}, …, α_{n}⟩ is in T_{1}. Let i ≤ n be maximal so that o(α_{j}) < o(ξ) for all j ≤ i. Then by the definition of T_{1}, the sequence (t u)(o(ξ)) β ⟨α_{0}, …, α_{i}⟩ is in S. Therefore N_{o(ξ)}^{α}[G_{α} ∗ G^{α}] models that

⟨(t u)(o(ξ)) β ⟨α_{0}, …, α_{i}⟩, S_{(t u)(o(ξ)) β ⟨α_{0},...,α_{i}⟩}⟩ ⊩ q̇_{i+1} ⊩ α ∈ j_{o(ξ)}^{α}(Ẋ_{i}).

But (t u)(o(ξ)) β ⟨α_{0}, …, α_{i}⟩ = (t u β ⟨α_{0}, …, α_{i}⟩)(o(ξ)). It follows that Suc(T^{∗}, t u, o(ξ)) is in the ultrafilter

U(α, o(ξ), (t u β ⟨α_{0}, …, α_{i}⟩)(o(ξ))).

So we can choose an ordinal β^{∗} in the set

Suc(T^{∗}, t u, o(ξ)) ∩ Suc(T^{∗}, t u β ⟨α_{0}, …, α_{i}⟩, o(ξ)).

Since o(β^{∗}) = o(ξ), it follows that b_{t u β ⟨α_{0},...,α_{i}⟩ β^{∗}} = b_{t u β^{∗}}. Define T_{2} as the set of sequences of the form

t u β ⟨α_{0}, …, α_{i}⟩ β^{∗} v

such that t u β^{∗} v is in T^{∗}. Then the condition

⟨t u β ⟨α_{0}, …, α_{i}⟩ β^{∗}, T_{2}⟩

is equivalent to the condition

⟨t u β^{∗}, T_{t u β^{∗}}^{∗}⟩.

Since o(ξ) = o(β^{∗}), these conditions force φ^{l}. Therefore the condition

⟨t u β ⟨α_{0}, …, α_{i}⟩ β^{∗}, T_{2} ∩ T_{t u β ⟨α_{0},...,α_{i}⟩ β^{∗}}^{∗}⟩

also forces φ^{l}. By choice of T^{∗}, the condition

⟨t u β ⟨α_{0}, …, α_{i}⟩ β^{∗}, T_{t u β ⟨α_{0},...,α_{i}⟩ β^{∗}}^{∗}⟩

forces ϕ^{l}. By Claim 2, the same holds if we replace β^{∗} by any δ in the set Suc(T^{∗}, t u β ⟨α_{0}, …, α_{i}⟩, γ^{∗}), where γ^{∗} ≥ o(ξ). But this is a contradiction since we cannot replace it with α_{i+1}, since this condition has an extension which forces the negation of φ^{l}. □

This completes the definition of ℙ_{κ}. The poset ℙ_{κ} preserves all cardinals and preserves GCH. Now we force with the poset ℙ_{κ} ∗ ℙ_{B}, where B is a name for the set of β less than κ^{+} such that the cofinality of β is not equal to the successor of a singular cardinal with cofinality μ. After forcing with ℙ_{κ} ∗ ℙ_{B}, κ remains strongly inaccessible and for almost all a in S(κ, κ^{+}), a ∩ κ is a singular cardinal with cofinality μ. We show that S(κ, κ^{+}) remains stationary.

Recall that U is a normal ultrafilter on P_{κ}κ^{+} in V and j_{U} : V → M = Ult(V, U). The sequence ⟨U(κ, i) : i ≤ μ⟩ is a ◁-increasing sequence of ultrafilters on κ and U(κ, μ) is equal to U ↾ κ, i.e. X is in U(κ, μ) iff X ⊆ κ and κ ∈ j_{U}(X). In particular, κ ∈ j_{U}(A_{μ}). So j_{U}(o)(κ) = μ. Therefore

j_{U}(ℙ_{κ}) = ℙ_{κ} ∗ ℙ_{B} ∗ ℙ(κ, μ) ∗ ℙ_{tail}.

.

Let G_{κ} ∗ G_{B} be generic for ℙ_{κ} ∗ ℙ_{B} over V. Let G(κ, μ) ∗ G_{tail} be generic for ℙ(κ, μ)∗ℙ_{tail} over V [G_{κ}∗G_{B}]. Let H = G_{κ}∗G_{B} ∗G(κ, μ)∗G_{tail}. Since j_{U}“G_{κ} = G_{κ}, in V [H] we can lift j_{U} to

j_{U} : V [G_{κ}] → M[H].

Now j_{U} ↾ H(κ^{+})^{V} is in M. For any x in H(κ^{+})^{V [G_{κ}]}, define j^{∗}(x) = ((j_{U} ↾ H(κ^{+})^{V})(ẋ))^{H}, where ẋ is a name in H(κ^{+})^{V} and ẋ^{G_{κ}} = x. Then in V [H], j^{∗} = j_{U} ↾ H(κ^{+})^{V [G_{κ}]} and j^{∗} is in M[H]. It follows that j_{U} ↾ ℙ_{B} is in M[H], and therefore also ⋃ j_{U}“G_{B} is in M[H].

Consider p in G_{B}, and write p = ⟨c_{α} : α ∈ B ∩ (β + 1)⟩ for some β less than κ^{+}. Then j_{U}(p) = ⟨d_{α} : α ∈ j(B) ∩ (β + 1)⟩. It follows that ⋃ j“G_{B} = ⟨d_{α} : α ∈ j(B) ∩ sup j“κ^{+}⟩. So ⋃ j“G_{B} is a master condition in j_{U}(ℙ_{B}) iff sup j“κ^{+} is not in j(B), which is true iff sup j“κ^{+} has cofinality equal to the successor of a singular cardinal with cofinality μ. But cf(sup j“κ^{+}) = κ^{+} and κ has cofinality μ in M[H]. So ⋃ j_{U}“G_{B} is a master condition.

Let g be a generic filter for j_{U}(ℙ_{B}) over V [H] which contains the master condition ⋃ j_{U}“G_{B}. Then in V [H ∗ g] we can lift j_{U} to

j_{U} : V [G_{κ} ∗ G_{B}] → M[H ∗ g].

Now we prove that S(κ, κ^{+}) is stationary in V [G_{κ} ∗ G_{B}]. Let C be a club subset of P_{κ}κ^{+} in V [G_{κ} ∗ G_{B}]. Since ℙ_{B} is κ + 1-strategically closed, C is a subset of H(κ^{+})^{V [G_{κ}]}. By Lemma 6, C is in M[G_{κ} ∗ G_{B}]. Therefore j_{U} ↾ C is in M[H] and so j_{U}“C is in M[H]. Now in M[H ∗ g], j_{U}“C is a directed subset of the club set j_{U}(C) with size less than j_{U}(κ). Therefore ⋃ j_{U}“C = j“κ^{+} is in j_{U}(C). But j“κ^{+} is in j(S(κ, κ^{+})), so j_{U}(C) ∩ j(S(κ, κ^{+})) is non-empty. By elementarity, S(κ, κ^{+}) ∩ C is non-empty. So S(κ, κ^{+}) is stationary.

Note: There are variations of this consistency proof to arrange a set of possible cofinalities for a∩κ, instead of a single cofinality. To obtain such models, modify the definition of the B_{α}’s and replace the ultrafilter U above with a coherent sequence of supercompact ultrafilters (see next section).

8. Saturated Ideals and GCH

In this section we construct a model in which κ is strongly inaccessible, GCH holds, and there is a stationary set S ⊆ P_{κ}κ^{+} such that NS ↾ S is saturated. Moreover, we will arrange that for all regular cardinals μ less than κ, the set

{a ∈ S : cf(a ∩ κ) = μ}

is stationary.

Previously Gitik [2] constructed a model in which NS ↾ S is κ^{+}-saturated for a set S ⊆ P_{κ}κ^{+}, where κ is inaccessible. Unlike the partial square forcing poset, Gitik’s poset for destroying stationary sets adds subsets to κ. Therefore 2^{κ} = κ^{++} in his model. It is an open question whether the GCH is consistent with the existence of a stationary set S such that NS ↾ S is κ^{+}-saturated.

Before constructing our model we introduce one of the basic forcing posets, which is an iteration of the partial square poset over the same cardinal. Let V be a model of set theory in which κ is strongly inaccessible and GCH holds. We define a κ-support forcing iteration ℙ^{κ} with length κ^{++}, which we denote by

⟨ℙ_{i}^{κ}, ℚ_{i}^{κ} : i < κ^{++}⟩.

The following induction hypotheses will be maintained:

(1) ℙ_{α}^{κ} is κ + 1-strategically closed,

(2) Let D_{α}^{κ} be the set of conditions p in ℙ_{α}^{κ} such that for all β in the support of p, there is x in V such that p(β) = x̌. Then D_{α}^{κ} is dense in ℙ_{α}^{κ}, and if α < κ^{++} then |D_{α}^{κ}| ≤ κ^{+}.

Fix a surjective function f : κ^{++} → κ^{++} × κ^{++} such that

∀α, i, j < κ^{++} (f(α) = ⟨i, j⟩ ⟶ i ≤ α),

and each f^{−1}(⟨i, j⟩) is unbounded in κ^{++}. Suppose that ℙ_{α}^{κ} is defined for a fixed α less than κ^{++} and satisfies the induction hypotheses. Let ⟨Ḃ_{i}^{α} : i < κ^{++}⟩ enumerate all canonical D_{α}^{κ}-names Ḃ for a subset of κ^{+} such that ℙ_{α}^{κ} forces that Ḃ is closed under suprema of subsets with order type ω or ω_{1}. Write f(α) = ⟨i, j⟩. Then i ≤ α, so consider the name Ḃ_{j}^{i} which was defined at stage i. Let ℚ_{α} be a ℙ_{α}^{κ}-name for

the κ+1-strategically closed forcing poset ℙḂ_{j}^{i} which adds a □Ḃ_{j}^{i} ; κ-sequence. Clearly ℙ_{α+1} = ℙ_{α} ∗ ℚ_{α} satisfies the induction hypotheses.

Now suppose that β ≤ κ^{++} is a limit ordinal and ℙ_{α}^{κ} is defined for all α less than β. Define ℙ_{β}^{κ} as the set of conditions p with support of size no larger than κ such that p ↾ α is in ℙ_{α}^{κ} for all α less than β.

Proposition 11. The iteration ℙ_{β}^{κ} is κ + 1-strategically closed.

Proof. For each α less than β, let σ_{α} be a ℙ_{α}^{κ}-name for a strategy for Player II in the game for the poset ℚ_{α}^{κ}. We describe a strategy for Player II in ℙ_{β}^{κ} by considering a run of the game ⟨p_{i} : 0 < i ≤ κ⟩.

Suppose that i < κ and Player I has just played condition p_{i}. Define p_{i+1} as the condition with the same support as p_{i} such that for all α in the support, p_{i+1}(α) is a ℙ_{α}^{κ}-name for the condition obtained by applying σ_{α} to the sequence of conditions played so far on coordinate α. An easy induction shows that for all α less than β, p_{i+1} ↾ α is a condition in ℙ_{α}^{κ} below p_{i} ↾ α. So p_{i+1} is a condition in ℙ_{β}^{κ} below p_{i}.

Suppose that i ≤ κ is a limit ordinal and ⟨p_{j} : 0 < j < i⟩ is defined. Let a be equal to ⋃{supp(p_{j}) : 0 < j < i}. Since i ≤ κ, |a| ≤ κ. Define p_{i} as the condition with support a such that for all α in a, p_{i}(α) is a name for the condition obtained by applying σ_{α} to the sequence of conditions played thus far on coordinate α. As in the last case, p_{i} is a condition and extends each p_{j} for 0 < j < i. □

Proposition 12. The set D_{β}^{κ} is dense in ℙ_{β}^{κ}, and if β < κ^{++} then |D_{β}^{κ}| ≤ κ^{+}.

Proof. Suppose first that cf(β) ≥ κ^{+}. Then ℙ_{β}^{κ} = ⋃{ℙ_{i}^{κ} : i < β} and D_{β}^{κ} = ⋃{D_{i}^{κ} : i < β}. So in this case the statement follows from the induction hypotheses.

Suppose that cf(β) ≤ κ. Let p be a condition in ℙ_{β}^{κ}. We construct a decreasing sequence ⟨p_{n} : n < ω⟩. Let p_{0} = p.

Assume that n < ω and p_{n} is defined. Since ℙ_{β}^{κ} is κ + 1-strategically closed, it forces that the sequence

⟨p_{n}(α)Ġ : α ∈ supp(p_{n})⟩

is in the ground model. So there is a condition p_{n+1} ≤ p_{n} and a sequence

⟨s_{n}^{α} : α ∈ supp(p_{n})⟩

in the ground model such that p_{n+1} forces that š_{n}^{α} = p_{n}(α) for all α in supp(p_{n}).

Now define q as follows. The support of q is ⋃{supp(p_{n}) : n < ω}. Fix α in supp(q) and let n_{α} be the least n < ω such that α is in supp(p_{n}). For m ≥ n_{α}, s_{m}^{α} is defined and is a sequence of clubs. If s_{m}^{α} is eventually constant, define s_{α} as this constant value. Otherwise let ξ_{α} be the suprema of the set {sup dom(s_{m}^{α}) : m ≥ n_{α}}, and choose c_{α} as some cofinal subset of ξ_{α} with order type ω.

Let B_{α} be a ℙ_{α}^{κ}-name for the set such that ℚ_{α}^{κ} = ℙ_{B_{α}}. Since cf(ξ_{α}) = ω and B_{α} is ω-closed, ξ_{α} is forced to be in B_{α}. Define

s_{α} = ⋃{s_{m}^{α} : m ≥ n_{α}} ∪ ⟨ξ_{α}, c_{α}⟩.

Now define q by letting q(α) = š_{α} for all α in supp(q). Then q is in D_{β}^{κ} and q ≤ p.

Any condition in D_{β}^{κ} is coded by its support and a set in ^{≤κ}H(κ^{+}). So |D_{β}^{κ}| ≤ κ^{+}. □

This completes the definition of ℙ^{κ}. Let D^{κ} be the set D_{κ^{++}}^{κ}, which is dense in ℙ^{κ}. Note that D^{κ} = ⋃{D_{i}^{κ} : i < κ^{++}}.

Proposition 13. The iteration ℙ^{κ} is κ^{++}-c.c.

Proof. It suffices to prove that D^{κ} is κ^{++}-c.c. Suppose that ⟨p_{i} : i < κ^{++}⟩ is a sequence of conditions in D^{κ}. By the Δ-system lemma, we can assume that there is a set d such that supp(p_{i}) ∩ supp(p_{j}) = d for i < j. Since there are at most κ^{+} many possibilities for p_{i} ↾ d, we assume p_{i} ↾ d = p_{j} ↾ d for i < j.

Fix i < j and let q = p_{i} ∪p_{j}. It is easy to check by induction that q is a condition below p_{i} and p_{j}. □

Now we begin to construct our model. Let V be a model of set theory in which κ is κ^{++}-supercompact and GCH holds.

First we construct a coherent sequence of supercompact ultrafilters.

Proposition 14. There exists a ◁-increasing sequence ⟨U(κ, i) : i < κ⟩ of normal ultrafilters on P_{κ}κ^{+} satisfying the following properties:

(1) for i < j, U(κ, i) ↾ κ ◁ U(κ, j) ↾ κ,

(2) For each i let j_{i} : V → Ult(V, U(κ, i)). Then for α < β there exists a function f_{α}^{β} : κ → V_{κ} such that j_{β}(f_{α}^{β})(κ) = U(κ, α).

Proof. It suffices to prove that for any ◁-increasing sequence U⃗ of normal ultrafilters on P_{κ}κ^{+} with length less than κ, there exists a normal ultrafilter U on P_{κ}κ^{+} satisfying the following properties:

(a) for each U_{i} in U⃗, U_{i} ◁ U and U_{i} ↾ κ ◁ U ↾ κ,

(b) Let j_{U} : V → Ult(V, U). Then for each U_{i} in U⃗, there is a function f_{i} : κ → V_{κ} such that j_{U}(f_{i})(κ) = U_{i}. If the claim holds then we can construct the desired sequence by induction.

Suppose for a contradiction that the claim fails. Let α be the least ordinal less than κ such that there exists a sequence U⃗ with length α for which no such U exists. Let W be a normal ultrafilter on P_{κ}κ^{++} and let j : V → M = Ult(V, W). By closure, the above statement about α holds in M.

Let U = W ↾ κ^{+}. Let j_{0} : V → M_{0} = Ult(V, U ↾ κ) and j_{1} : V → M_{1} = Ult(V, U). Apply Lemma 3 to obtain k_{0} : M_{0} → M_{1} with crit(k_{0}) = κ^{++M_{0}} and k_{1} : M_{1} → M with crit(k_{1}) = κ^{+3M_{1}}.

By elementarity, there exists a sequence ⟨U_{i}^{∗} : i < α⟩ which is a witness for the statement about α in M_{0}. For each i let U_{i} = k_{0}(U_{i}^{∗}). Then k_{0}(⟨U_{i}^{∗} : i < α⟩) = ⟨U_{i} : i < α⟩ is a witness for the statement about α in M_{1}. By the closure of M_{1}, this sequence really is a ◁-increasing sequence of ultrafilters on P_{κ}κ^{+}. Moreover, U is above each U_{i}. By the closure of M_{0}, U_{i}^{∗} ↾ κ as computed in M_{0} is exactly U_{i} ↾ κ. Therefore U_{i} ↾ κ◁U ↾ κ for each i. For each i fix f_{i} : κ → V_{κ} such that [f_{i}] = U_{i}^{∗} in M_{0}. By the definition of k_{0} as given in the proof of Lemma 3, k_{0}([f_{i}]) = j_{1}(f_{i})(κ). So j_{1}(f_{i})(κ) = k_{0}([f_{i}]) = k_{0}(U_{i}^{∗}) = U_{i}.

By elementarity k_{1}(⟨U_{i} : i < α⟩) = ⟨U_{i} : i < α⟩ satisfies the statement about α in M. But this sequence and U satisfy (a) and (b) in M, so we have a contradiction.

□

Fix ⟨U(κ, i) : i < κ⟩ as in Proposition 14.

Lemma 32. There exists a sequence ⟨X_{i} : i < κ⟩ of subsets of κ such that X_{i} is in U(κ, i) ↾ κ, and for i ≠ j, X_{i} ∩ X_{j} is empty.

Proof. For distinct i and j, let X_{i}(j) be a set in (U(κ, i) ↾ κ) \ (U(κ, j) ↾ κ). Let X_{i}(i) = κ. Define X_{i}^{∗} = △{X_{i}(j) : j < κ}, which is in U(κ, i) ↾ κ. If i and j are distinct, then X_{i}^{∗} is not in U(κ, j) ↾ κ, since X_{i}^{∗} \ (j + 1) ⊆ X_{i}(j). Now define X_{i} = X_{i}^{∗} \ ⋃{X_{j}^{∗} : j < i}. □

For each i < κ, let j_{i}^{κ} : V → N_{i}^{κ} = Ult(V, U(κ, i)). For α < β < κ, fix f_{α}^{β} : κ → V_{κ} such that j_{β}^{κ}(f_{α}^{β})(κ) = U(κ, α). Define a sequence ⟨A_{j} : j < κ⟩ by induction, using the following recursion: A_{j} is the set of α less than κ satisfying

(1) α is in X_{j} and α > j,

(2) For i < j, f_{i}^{j}(α) is a normal ultrafilter on P_{α}α^{+} such that A_{i} ∩ α is in f_{i}^{j}(α) ↾ α,

(3) For i_{0} < i_{1} < j, f_{i_{0}}^{j}(α) ◁ f_{i_{1}}^{j}(α),

(4) Let i_{0} < i_{1} < j and k : V → Ult(V, f_{i_{1}}^{j}(α)). Then k(f_{i_{0}}^{i_{1}} ↾ α)(α) = f_{i_{0}}^{j}(α). For each j, A_{j} is in U(κ, j) ↾ κ, as can be proved by induction, using Lemma 9 (2) to prove (4).

Let A be equal to ⋃{A_{i} : i < κ}. For each α in A, define o(α) as the unique i such that α is in A_{i}. If α is in A, then for each i < o(α) let U(α, i) = f_{i}^{o(α)}(α) and let j_{i}^{α} : V → N_{i}^{α} = Ult(V, U(α, i)). By (2), for i < o(α), A_{i} ∩ α is in f_{i}^{o(α)}(α) ↾ α = U(α, i) ↾ α, and therefore α is in j_{i}^{α}(A_{i}). It follows that j_{i}^{α}(o)(α) = i in N_{i}^{α}. By (4), for each i_{0} < i_{1} < o(α), U(α, i_{0}) is in N_{i_{1}}^{α}, so U(α, i_{0}) ◁ U(α, i_{1}).

Fix <_{κ} a well-ordering of V_{κ} such that for any inaccessible α less than κ, <_{α}=<_{κ} ∩ (V_{α} × V_{α}) is a well-ordering of V_{α}.

We define an Easton support Prikry iteration ⟨ℙ_{i}, ℚ_{i} : i < κ⟩.

Suppose that ℙ_{α} is defined. Assume as an induction hypothesis that for all γ less than α, ℚ_{γ} is trivial unless γ is in A ∩ α. If γ is in A then ℙ_{γ} forces that ℚ_{γ} = ℙ^{γ} ∗ ℙ(γ, o(γ)), where ℙ^{γ} is a γ + 1-strategically closed forcing poset and ℙ(γ, o(γ)) is defined from a family of ultrafilters

{ U(γ, β, t) : β < o(γ), t is a β-coherent sequence }.

Let G_{α} be generic for ℙ_{α} over V. If α is not in A then let ℚ_{α} be trivial. Suppose that α is in A. In V [G_{α}] let ℙ^{α} be the iteration of partial square posets which was defined earlier in this section. We consider ℙ^{α} as a Prikry type forcing poset by letting ≤^{∗}=≤. Let G^{α} be generic for ℙ^{α} over V [G_{α}].

If o(α) = 0 then ℙ(α, o(α)) is the trivial poset and there is nothing to prove. Suppose that o(α) > 0. We define in V [G_{α} ∗ G^{α}] a family of α-complete ultrafilters

{ U(α, β, t) : β < o(α), t is a β-coherent sequence }

which satisfies the requirements (1), (2), and (3) described in the last section. We also assume as an induction hypothesis that the family of ultrafilters defined at any previous stage γ satisfies the same definition as at stage α.

In V [G_{α} ∗ G^{α}] fix β less than o(α) and a β-coherent sequence t. We define U(α, β, t). Consider j_{β}^{α} : V → N_{β}^{α}. Since j_{β}^{α}(o)(α) = β, by the induction hypotheses we can write

j_{β}^{α}(ℙ_{α}) = ℙ_{α} ∗ ℙ^{α} ∗ ℙ(α, β) ∗ ℙ_{α,j_{β}^{α}(α)}.

By closure, <_{α}=<_{κ} ∩ (V_{α} × V_{α}) is in N_{β}^{α}. In order to define U(α, β, t) we need to consider all subsets of α in V [G_{α} ∗G^{α}]. However, ℙ^{α} does not add subsets to α over V [G_{α}]. So it suffices to consider only ℙ_{α}-names for subsets of α. Let ⟨Ẋ_{i} : i < α^{+}⟩ be the j_{β}^{α}(<_{α})-least enumeration of all canonical ℙ_{α}-names for subsets of α.

By the α^{++}-weak strategic closure of ℙ_{α,j_{β}^{α}(α)}, there exists a name q̇ for a condition in ℙ_{α,j_{β}^{α}(α)} which decides the statement α ∈ j_{β}^{α}(Ẋ_{i}) for each i < α^{+}. Let q̇ be the j_{β}^{α}(<_{α})-least such name.

Define U(α, β, t) by letting X be in U(α, β, t) if there exists an index i such that X = Ẋ_{i}^{G_{α}} and a β-tree T such that

N_{β}^{α}[G_{α} ∗ G^{α}] ⊧ ⟨t, T ⟩ ⊩ q̇ ⊩ α ∈ j_{β}^{α}(Ẋ_{i}).

Clearly Lemmas 28 through 31 hold for these ultrafilters. Let ℙ(α, o(α)) be the forcing poset defined from this family in V [G_{α} ∗ G^{α}]. By Proposition 10, ℙ(α, o(α)) satisfies the Prikry property.

This completes the definition of ℙ_{κ}. By standard Easton support arguments, ℙ_{κ} preserves GCH.

In V let F : κ^{+<ω} → κ^{+} be a Jonsson function for κ^{+}. So if X ⊆ κ^{+} has size κ^{+} and is closed under F, then X = κ^{+}. Define S^{∗} as the set of a in S(κ, κ^{+}) such that a∩κ is strongly inaccessible, F is Jonsson for a, and a is closed under suprema of bounded subsets with order type less than a ∩ κ.

Lemma 33. If U is any normal ultrafilter on P_{κ}κ^{+}, then S^{∗} is in U.

Proof. Let j : V → M = Ult(V, U). Then j“κ^{+} is in j(S(κ, κ^{+})) and j“κ^{+} ∩j(κ) = κ is strongly inaccessible in M. Since the critical point of j is κ, j“κ^{+} is closed under suprema of subsets with order type less than κ = j“κ^{+} ∩ j(κ).

To show that j(F) is Jonsson for j“κ^{+}, suppose that X is a subset of j“κ^{+} which is closed under j(F) and has size equal to |j“κ^{+}| = κ^{+}. Then there is Y ⊆ κ^{+} with size κ^{+} such that j“Y = X. Since X is closed under j(F), Y is closed under F. The fact that F is a Jonsson function implies that Y = κ^{+}, and therefore X = j“κ^{+}. It follows that j“κ^{+} is in j(S^{∗}), so S^{∗} is in U. □

For each i less than κ, let

S_{i} = {a ∈ S^{∗} : a ∩ κ ∈ A_{i}}.

Then S_{i} is in U(κ, i). Define

S = ⋃{S_{i} : i < κ}.

Lemma 34. The function a ↦ sup a is injective on S. So if T is a subset of S, there is X ⊆ κ^{+} such that T = {a ∈ S : sup a ∈ X}.

Proof. Suppose that a and b are distinct elements in S with the same supremum. Since a and b are closed under suprema of subsets with order type ω, the set a ∩ b is unbounded in sup a. Also cf(sup a) = o.t.(a) = (a ∩ κ)^{+}. It follows that |a ∩ b| = |a| = |b|. But a ∩ b is closed under F and is a subset of a and b. Since F is Jonsson for a and b, a = a ∩ b = b.

For the second statement, let X = {sup a : a ∈ T }. Since a ↦ sup a is injective, T = {a ∈ S : sup a ∈ X}. □

Let G_{κ} be a generic filter for ℙ_{κ} over V. In V [G_{κ}] we construct a subiteration ℙ_{B}^{κ} of ℙ^{κ} such that ℙ_{B}^{κ} forces that NS ↾ S is saturated, and for all regular μ less than κ, the set {a ∈ S : cf(a ∩ κ) = μ} is stationary.

In V [G_{κ}] let ⟨Ẏ_{i} : i < κ^{++}⟩ be a sequence of D^{κ}-names such that for all i < κ^{++}, ℙ_{i}^{κ} forces that ℚ̇_{i}^{κ} is the forcing poset ℙẎ_{i} for adding a □Ẏ_{i} ; κ-sequence.

For each i < κ write

j_{i}^{κ}(ℙ_{κ} ∗ ℙ^{κ}) = ℙ_{κ} ∗ ℙ^{κ} ∗ ℙ(κ, i) ∗ ℙ_{tail}^{i} ∗ j_{i}^{κ}(ℙ^{κ}).

We introduce some notation for generic filters for these posets. We use G^{κ}, G(κ, i), and G_{tail}^{i} to denote generic filters of D^{κ}, ℙ(κ, i), and ℙ_{tail}^{i} respectively (or sometimes for the names of such filters). For each α less than κ^{++}, let G_{α}^{κ} = G^{κ} ↾ α, i.e. the generic for D_{α}^{κ} given by G^{κ}. Let H_{α}^{κ} be the generic for ℙ_{Y_{α}} given by G^{κ} over V [G_{κ} ∗ G_{α}^{κ}].

We construct by induction an iteration

⟨ℙ_{B_{α}}^{κ}, ℚ_{B_{α}}^{κ} : α < κ^{++}⟩

and functions

⟨π_{α} : α < κ^{++}⟩

such that for each α, π_{α} : D_{α}^{κ} → ℙ_{B_{α}}^{κ} is a projection mapping satisfying: (1) For α < β and p in D_{β}^{κ}, π_{β}(p) ↾ α = π_{α}(p ↾ α),

(2) D_{B_{α}}^{κ} = D_{α}^{κ} ∩ ℙ_{B_{α}}^{κ} is dense in ℙ_{B_{α}}^{κ} and π_{α} ↾ D_{B_{α}}^{κ} is the identity,

(3) For each α, B_{α+1} is a ℙ_{B_{α}}^{κ}-name for a subset of α + 1,

(4) ℙ_{B_{α}}^{κ} is κ + 1-strategically closed.

In addition we construct for each i less than κ a ≤^{∗}-descending sequence of ℙ^{κ} ∗ ℙ(κ, i)-names

⟨q_{α}^{i}(0), q_{α}^{i}(1) : α < κ^{++}⟩

for conditions in ℙ_{tail}^{i}, and a descending sequence of ℙ^{κ} ∗ ℙ(κ, i) ∗ ℙ_{tail}^{i}-names

⟨s_{α}^{i}(0), s_{α}^{i}(1) : α < κ^{++}⟩,

where s_{α}^{i}(0) and s_{α}^{i}(1) are forced to be conditions in j_{i}^{κ}(ℙ_{B_{α}}^{κ}) such that for all p in G_{α}^{κ}, s_{α}^{i}(0) ≤ j_{i}^{κ}(π_{α}(p)) (i.e., s_{α}^{i}(0) is a master condition). Both of the sequences will constitute a run of the game in their respective posets, with the 0 condition interpreted as Player II’s play and the 1 condition as Player I’s play.

Let B_{0} = ∅ and for each i less than κ let q_{0}^{i}(0) = 1 and s_{0}^{i}(1) = 1. Fix α < κ^{++} and suppose that ℙ_{B_{α}}^{κ} and π_{α} are defined, and for all i less than κ, q_{α}^{i}(0) and s_{α}^{i}(0) are defined.

We consider several cases. First suppose that Ẏ_{α} is not a D_{B_{α}}^{κ}-name. In this case let B_{α+1} be a ℙ_{B_{α}}^{κ}-name for B_{α}. So ℙ_{B_{α}}^{κ} forces that α is not in B_{α+1}. Let ℙ_{B_{α+1}}^{κ} be equal to ℙ_{B_{α}}^{κ} ∗ ⟨0⟩. Clearly properties (1) through (4) hold.

Fix i less than κ. Let s_{α}^{i}(1) = s_{α}^{i}(0) and q_{α}^{i}(1) = q_{α}^{i}(0). Let q_{α+1}^{i}(0) be a name for Player II’s response to the run of the game so far in ℙ_{tail}^{i}. Define s_{α+1}^{i}(0) as follows. The support of s_{α+1}^{i}(0) is supp(s_{α}^{i}(1)). Fix γ in supp(s_{α}^{i}(1)). Consider the run of the game in coordinate γ which begins when Player I plays s_{β}^{i}(1)(γ) for the first value of β for which this term is defined. Define s_{α+1}^{i}(0)(γ) as Player II’s response to this run of the game according to its strategy.

Now consider the case when Ẏ_{α} is a D_{B_{α}}^{κ}-name. Fix i less than κ. Since s_{α}^{i}(0) is a master condition, s_{α}^{i}(0) forces that j_{i}^{κ} can be lifted to

j_{i}^{κ} : V [G_{κ} ∗ G_{B_{α}}^{κ}] → N_{i}^{κ}[G_{κ} ∗ G^{κ} ∗ G(κ, i) ∗ G_{tail}^{i} ∗ g_{α}^{i}]

where G_{B_{α}}^{κ} is the filter generated by π_{α}“(G_{α}^{κ}) and g_{α}^{i} is a name for the generic for j_{i}^{κ}(ℙ_{B_{α}}^{κ}). Choose s_{α}^{i}(1) ≤ s_{α}^{i} in j_{i}^{κ}(ℙ_{B_{α}}^{κ}) which decides the statement

sup j_{i}^{κ}“κ^{+} ∈ j_{i}^{κ}(Ẏ_{α}).

Choose q_{α}^{i}(1) as a name for a condition directly extending q_{α}^{i}(0) which decides which way that s_{α}^{i}(1) decides the given statement. Let q_{α+1}^{i}(0) be a name for Player II’s response to the run of the game in ℙ_{tail}^{i}.

Let B_{α+1} be a ℙ_{B_{α}}^{κ}-name which is forced to be equal to B_{α} unless for all i less than κ,

N_{i}^{κ}[G_{κ} ∗ G_{B_{α}}^{κ}] ⊧ (ℙ^{κ}/ℙ_{B_{α}}^{κ}) ∗ ℙ(κ, i) ⊩ q_{α+1}^{i}(0) ⊩ s_{α}^{i}(1) ⊩ sup j_{i}^{κ}“κ^{+} ∉ j_{i}^{κ}(Ẏ_{α}), in which case B_{α+1} = B_{α} ∪ {α}. Define ℙ_{B_{α+1}}^{κ} as ℙ_{B_{α}}^{κ} ∗ ℚ_{B_{α}}^{κ}, where ℚ_{B_{α}}^{κ} is forced to be trivial if α is not in B_{α+1} and is equal to ℙẎ_{α} if α is in B_{α+1}.

Note that if α is in B_{α+1}, then for all i less than κ the condition s_{α}^{i}(1) forces that sup j_{i}^{κ}“κ^{+} ∉ j_{i}^{κ}(Ẏ_{α}). Therefore ⋃ j_{i}^{κ}“H_{α}^{κ} is a master condition in j_{i}^{κ}(ℙẎ_{α}). Define s_{α+1}^{i}(0) ↾ j_{i}^{κ}(α) exactly as s_{α+1}^{i}(0) was defined in the last case by using the strategy at each coordinate in the support. If α is not in B_{α+1} then let s_{α+1}^{i}(0) = s_{α+1}^{i}(0) ↾ j_{i}^{κ}(α). If α is in B_{α+1} then let

s_{α+1}^{i}(0) = (s_{α+1}^{i}(0) ↾ j_{i}^{κ}(α)) ∪ ⟨j_{i}^{κ}(α), ⋃ j_{i}^{κ}“H_{α}^{κ}⟩.

Note that by the induction hypotheses and the comments above, s_{α+1}^{i}(0) is a master condition in k_{i}(ℙ_{B_{α+1}}^{κ}).

Lemma 35. The poset ℙ_{B_{α+1}}^{κ} is κ + 1-strategically closed and D_{B_{α+1}}^{κ} is dense in ℙ_{B_{α+1}}^{κ}.

Proof. Straightforward. □

Lemma 36. There is a projection mapping π_{α+1} : D_{α+1}^{κ} → ℙ_{B_{α+1}}^{κ} such that: (a) π_{α+1} ↾ D_{B_{α+1}}^{κ} is the identity,

(b) For all q in D_{α+1}^{κ} and β < α + 1, π_{α+1}(q) ↾ β = π_{β}(q ↾ β).

Proof. For each set x in V [G_{κ}] let ȧ_{x} be a ℙ_{B_{α}}^{κ}-name which is forced to be equal to x̌ if α is in B_{α+1} and is 1 otherwise. Define π_{α+1}(p x̌) = π_{α}(q) ȧ_{x}. Statements (a) and (b) follow from induction hypotheses (1) and (2). □

This completes the definition for successor stages. Suppose that δ ≤ κ^{++} is a limit ordinal and ℙ_{B_{α}}^{κ} is defined for α < δ. Define ℙ_{B_{δ}}^{κ} as the set of functions p with domain a subset of δ with size no larger than κ such that p ↾ α is in ℙ_{B_{α}}^{κ} for α < δ. Let B_{δ} be a ℙ_{B_{δ}}^{κ}-name for ⋃{B_{α} : α < δ}.

Lemma 37. The poset ℙ_{B_{δ}}^{κ} is κ + 1-strategically closed and D_{B_{δ}}^{κ} = D_{δ}^{κ} ∩ ℙ_{B_{δ}}^{κ} is dense in ℙ_{B_{δ}}^{κ}.

Proof. The proof is identical to Propositions 11 and 12. □

Fix i less than κ. Define q_{δ}^{i}(0) as a name for Player II’s response to the run of the game defined thus far in ℙ_{tail}^{i}. Define s_{δ}^{i}(0) as follows. The support of s_{δ}^{i}(0) is equal to ⋃{supp(s_{α}^{i}(0)) : α < δ}. If γ is in supp(s_{δ}^{i}(0)), define s_{δ}^{i}(γ) by applying the strategy to the run of the game determined on coordinate α.

Lemma 38. There is a projection mapping π_{δ} : D_{δ}^{κ} → ℙ_{B_{δ}}^{κ} such that:

(a) π_{δ} ↾ D_{B_{δ}}^{κ} is the identity,

(b) for all p in D_{δ}^{κ} and α < δ, π_{δ}(p) ↾ α = π_{α}(p ↾ α).

Proof. Define π_{δ}(p) = ⋃{π_{α}(p ↾ α) : α < δ}. The induction hypotheses (1) and (2) imply that the definition of π_{δ} makes sense and (a) and (b) hold. We show that π_{δ} is a projection mapping.

Obviously π_{δ}(1) = 1 and q ≤ p implies π_{δ}(q) ≤ π_{δ}(p). Suppose that q ≤ π_{δ}(p) in ℙ_{B_{δ}}^{κ}. We find r ≤ p in D_{δ}^{κ} such that π_{δ}(r) ≤ q. If cf(δ) ≥ κ^{+} then the supports of p and q are bounded below δ, so r exists by the induction hypotheses.

Suppose that cf(δ) ≤ κ. Let ⟨α_{i} : i < cf(δ)⟩ be increasing and unbounded in δ, with α_{0} = 0. We define ⟨r_{i}, r_{i}^{∗} : i < cf(δ)⟩ so that r_{i} is in D_{α_{i}}^{κ} and is below p ↾ α_{i}.

Let r_{0} = ∅. Suppose that i < cf(δ) and r_{i} is defined such that π_{α_{i}}(r_{i}) ≤ q ↾ α_{i}. Since q ≤ π_{δ}(p),

π_{α_{i}}(r_{i}) (q ↾ [α_{i}, α_{i+1})) ≤ π_{α_{i+1}}(r_{i} (p ↾ [α_{i}, α_{i+1})))

in ℙ_{B_{α_{i+1}}}^{κ}. Apply the fact that π_{α_{i+1}} is a projection mapping to obtain r_{i}^{∗} ≤ r_{i} (p ↾ [α_{i}, α_{i+1})) in D_{α_{i+1}}^{κ} such that

π_{α_{i+1}}(r_{i}^{∗}) ≤ π_{α_{i}}(r_{i}) (q ↾ [α_{i}, α_{i+1})).

Now define r_{i+1} as follows. The support of r_{i+1} is the same as the support of r_{i}^{∗}. For α in the support of r_{i+1}, we consider the following run of the game. Player I begins by playing condition r_{j}^{∗}(α) for the first value of j for which this term is defined. Player II responds according to its strategy with r_{j+1}(α) and at limit stages. The last play is r_{i}^{∗}(α) defined above. Let r_{i+1}(α) be the condition obtained by applying the strategy to this run of the game.

Suppose δ_{0} ≤ δ is a limit ordinal. Define r_{δ_{0}} with support ⋃{supp(r_{i}) : i < δ_{0}}, such that for each α in this support, r_{δ_{0}}(α) is obtained by applying the strategy to the run of the game at coordinate α.

Let r be an extension of r_{δ} in D_{B_{δ}}^{κ}. Then r ≤ p and π_{δ}(r) ≤ q. □

This completes the definition of ℙ_{B}^{κ}. By Lemma 2, ℙ_{B}^{κ} and ℙ^{κ}/ℙ_{B}^{κ} are κ^{++}-c.c. Suppose that G_{B}^{κ} is generic for ℙ_{B}^{κ} over V [G_{κ}].

Proposition 15. In V [G_{κ} ∗ G_{B}^{κ}], NS ↾ S is saturated.

Proof. For each set X ⊆ κ^{+}, i < κ, and α < κ^{++}, let Φ(X, i, α) be the statement

N_{i}^{κ}[G_{κ} ∗ G_{B}^{κ}] ⊧ (ℙ^{κ}/ℙ_{B}^{κ}) ∗ ℙ(κ, i) ⊩ q_{α+1}^{i}(0) ⊩ s_{α}^{i}(1) ⊩ sup j_{i}^{κ}“κ^{+} ∉ j_{i}^{κ}(X).

Let T be a stationary subset of S. If a is in T, then a ∩ κ is in A and o(a ∩ κ) is less than a ∩ κ. Therefore the map a ↦ o(a ∩ κ) is regressive. By Fodor’s Lemma, there exists an index i less than κ such that T ∩ S_{i} is stationary. So in order to prove that NS ↾ S is saturated, it suffices to prove that NS ↾ S_{i} is saturated for all i less than κ.

Fix i less than κ and let T be a stationary subset of S_{i}. Fix X ⊆ κ^{+} such that

T = {a ∈ S : sup a ∈ X}.

We claim that T is non-stationary iff there is α less than κ^{++} such that Φ(X, i, α) holds.

Suppose that T is disjoint from a club set C. Since ℙ_{B}^{κ} is κ^{++}, there is α < κ^{++} such that C and T are in V [G_{κ} ∗ G_{B_{α}}^{κ}]. Fix p s in G_{κ} ∗ G_{B_{α}}^{κ} which forces that Ṫ = {a ∈ S : sup a ∈ Ẋ} and Ċ ∩ Ṫ is empty, where Ṫ, Ẋ, and Ċ are names for T, X, and C. Since s_{α}^{i}(0) is a master condition,

N_{κ}^{i}[G_{κ} ∗ G_{B_{α}}^{κ}] ⊧ (ℙ^{κ}/ℙ_{B_{α}}^{κ}) ∗ ℙ(κ, i) ∗ ℙ_{tail}^{i} ⊩ s_{α}^{i}(0) ⊩ j_{i}^{κ}(Ċ) ∩ j_{i}^{κ}(Ṫ) = ∅.

But it is also forced that j_{i}^{κ}“κ^{+} = ⋃ j_{i}^{κ}“Ċ ∈ j_{i}^{κ}(Ċ), so j_{i}^{κ}“κ^{+} is not in j_{i}^{κ}(T), i.e. sup j_{i}^{κ}“κ^{+} is not in j_{i}^{κ}(X). Therefore Φ(X, i, α) holds.

Suppose on the other hand that there exists α such that Φ(X, i, α) holds. Let Y be the closure of X under suprema of subsets with order type ω or ω_{1}. Since sup j_{i}^{κ}“κ^{+} does not have cofinality ω or ω_{1}, Φ(X, i, α) implies Φ(Y, i, α). The poset ℙ_{B}^{κ} is κ^{++}-c.c, so there exists γ less than κ^{++} such that Y is in V [G_{κ} ∗ G_{B_{γ}}^{κ}]. Moreover, we can choose γ so that there is a canonical D_{B_{γ}}^{κ} -name Ẏ for Y and p in G_{B_{γ}}^{κ} which forces Φ(Ẏ , i, α) and the other properties of Y stated above. Then there is β < κ^{++} greater than α and γ such that Ẏ = Ẏ_{β} and for all ξ less than κ,

N_{ξ}^{κ}[G_{κ} ∗ G_{B_{β}}^{κ}] ⊧ (ℙ^{κ}/ℙ_{B_{β}}^{κ}) ∗ ℙ(κ, ξ) ⊩ q_{β+1}^{ξ}(0) ⊩ s_{β}^{ξ}(1) ⊩ sup j_{ξ}^{κ}“κ^{+} ∉ j_{ξ}^{κ}(Ẏ_{β}).

So β is in B, and therefore G_{B}^{κ} adds a □_{κ}^{Y} -sequence and destroys the stationarity of the set {a ∈ S(κ, κ^{+}) : sup a ∈ Y }. But T is contained in this set, so it is non-stationary in V [G_{κ} ∗ G_{B}^{κ}].

Suppose for a contradiction that NS ↾ S_{i} is not saturated. Let ⟨T_{X_{α}} : α < κ^{++}⟩ be a sequence of stationary subsets of S_{i} such that T_{X_{α}} ∩ T_{X_{β}} is non-stationary for α < β. Since T_{X_{α}} is stationary, there is p_{α} in (ℙ^{κ}/ℙ_{B}^{κ}) ∗ ℙ(κ, i) and γ_{α} such that

N_{i}^{κ}[G_{κ} ∗ G_{B}^{κ}] ⊧ p_{α} ⊩ q_{γ_{α}+1}^{i}(0) ⊩ s_{γ_{α}}^{i}(1) ⊩ sup j_{i}^{κ}“κ^{+} ∈ j_{i}^{κ}(X_{α}).

Then p_{α} and p_{β} are incompatible for α < β. For if q ≤ p_{α}, p_{β}, then letting γ = max{γ_{α}, γ_{β}},

N_{i}^{κ}[G_{κ} ∗ G_{B}^{κ}] ⊧ q ⊩ q_{γ+1}^{i}(0) ⊩ s_{γ}^{i}(1) ⊩ sup j_{i}^{κ}“κ^{+} ∈ j_{i}^{κ}(X_{α} ∩ X_{β}),

and therefore T_{X_{α}} ∩ T_{X_{β}} is stationary. We have a contradiction since the poset (ℙ^{κ}/ℙ_{B}^{κ}) ∗ ℙ(κ, i) is κ^{++}-c.c. □

Proposition 16. In V [G_{κ} ∗ G_{B}^{κ}] let μ be a regular cardinal less than κ. Then the set

{a ∈ S : cf(a ∩ κ) = μ}

is stationary.

Proof. If a is in S_{μ} = {a ∈ S : o(a ∩ κ) = μ}, then cf(a ∩ κ) = μ. Let C be a club subset of P_{κ}κ^{+} in V [G_{κ} ∗ G_{B}^{κ}]. Since ℙ_{B}^{κ} is κ^{++}-c.c., there is γ less than κ^{++} such that C is in V [G_{κ} ∗ G_{B_{γ}}^{κ}].

Let H be generic for j_{μ}^{κ}(ℙ_{κ})/(ℙ_{κ} ∗ ℙ_{B}^{κ}) over V [G_{κ} ∗ G_{B}^{κ}] and let h be generic for j_{μ}^{κ}(ℙ_{B_{γ}}^{κ}) over V [G_{κ} ∗ G_{B}^{κ} ∗ H] which contains the condition s_{γ}^{μ}(0). Since s_{γ}^{μ}(0) is a master condition, in V [G_{κ} ∗ G_{B}^{κ} ∗ H ∗ h] we can lift j_{μ}^{κ} to

j_{μ}^{κ} : V [G_{κ} ∗ G_{B_{γ}}^{κ}] → N_{μ}^{κ}[G_{κ} ∗ G_{B}^{κ} ∗ H ∗ h].

Recall that j_{μ}^{κ}“κ^{+} is in j_{μ}^{κ}(S_{μ}). Also j_{μ}^{κ}“κ^{+} = ⋃ j_{μ}^{κ}“C is in j_{μ}^{κ}(C). By elementarity, S_{μ} ∩ C is non-empty. □

References

[1] J. Baumgartner. Iterated forcing. In Surveys in Set Theory. Cambridge Univ. Press, 1983.

[2] M. Gitik. Nonsplitting subset of P_{κ}κ^{+}. J. Symbolic Logic, 50(4):881–894, 1985.

[3] M. Gitik. Changing cofinalities and the nonstationary ideal. Israel J. Math, 56(3):280–314, 1986.

[4] A. Kanamori. The Higher Infinite. Springer, 1994.

[5] J. Krueger. Destroying stationary sets. To appear.

[6] J. Krueger. Strong compactness and stationary sets. To appear.

[7] M. Magidor. How large is the first strongly compact cardinal? Annals of Mathematical Logic, 10(1):33–57, 1976.

Institute for Formal Logic, University of Vienna, Währingerstrasse 25, 1090 Vienna, Austria

   E-mail address: jkrueger@logic.univie.ac.at
   URL: http://www.logic.univie.ac.at/∼jkrueger