NAMBA FORCING AND NO GOOD SCALE

JOHN KRUEGER

Abstract. We develop a version of Namba forcing which is useful for constructing models with no good scale on ℵ_{ω}. A model is produced in which □_{ℵ_{n}} holds for all finite n ≥ 1, but there is no good scale on ℵ_{ω}; this strengthens a theorem of Cummings, Foreman, and Magidor [3] on the non-compactness of square.

Cummings, Foreman, and Magidor [4] proved that if the square principle □_{ℵ_{n}} holds for all finite n ≥ 1, then there exists a “square-like” sequence on ℵ_{ω+1} which exhibits uniformity at ordinals of uncountable cofinality. But in [3] they showed that the existence of squares on cardinals less than ℵ_{ω} fails to imply the existence of a square sequence on ℵ_{ω+1}. Specifically, they constructed a model in which □_{ℵ_{n}} holds for all n ≥ 1, but every stationary subset of ℵ_{ω+1} ∩ cof(ℵ_{0}) reflects to an ordinal in ℵ_{ω+1} of cofinality ℵ_{1}. Since □_{ℵ_{ω}} implies that every stationary subset of ℵ_{ω+1} contains a non-reflecting stationary subset, □_{ℵ_{ω}} fails in this model.

Stationary set reflection is consistent with weaker forms of square. For example, in [2] it is proven that the weak square principle □_{ℵ_{ω}}^{∗} is consistent with the statement that every family of ℵ_{1} many stationary subsets of ℵ_{ω+1} ∩ cof(ℵ_{0}) reflects simultaneously to some ordinal in ℵ_{ω+1} ∩ cof(ℵ_{1}). This raises the question whether the square principle holding below ℵ_{ω} implies any form of weak square principle on ℵ_{ω+1}.

In this paper we rework the forcing construction of [3] to construct a model which satisfies that □_{ℵ_{n}} holds for all finite n ≥ 1, but there does not exist a good scale on ℵ_{ω}. Thus the existence of squares below ℵ_{ω} does not imply even the weakest square principles on ℵ_{ω+1}, such as □_{ℵ_{ω}}^{∗} and the approachability property AP_{ℵ_{ω}}. To obtain this model we develop a version of Namba forcing which is useful for constructing models with no good scale.

Section 1 reviews some basic facts about good scales. Sections 2–5 develop the version of Namba forcing which we will use. In Section 6 we define a forcing iteration of Namba forcing to obtain a model with no good scale on ℵ_{ω}. Section 7 proves the main theorem of the paper; we construct a model which satisfies that □_{ℵ_{n}} holds for all finite n ≥ 1, but there is no good scale on ℵ_{ω}.

1. Good Scales

The weakest square principle on ℵ_{ω+1} is the existence of a good scale. In this section we review notation and basic ideas regarding this concept.

For an infinite set 𝔞 ⊆ ℵ_{0}, we consider the product ∏_{n∈𝔞} ℵ_{n}, ordered by eventual domination with respect to 𝔞. For functions f, g ∈ ∏_{n∈𝔞} ℵ_{n}, define f <_{𝔞}^{∗} g if there exists n < ω such that for all k ≥ n in 𝔞, f(k) < g(k). The relation ≤_{𝔞}^{∗} is defined similarly. We also define f <_{𝔞} g if for all n ∈ 𝔞, f(n) < g(n), and similarly with ≤_{𝔞}.

2010 Mathematical Subject Classification: 03E35, 03E04.

Key words and phrases: good scale, Namba forcing.

A sequence ⟨f_{i} : i < ℵ_{ω+1}⟩ of functions in ∏_{n∈𝔞} ℵ_{n} is a scale if it is <_{𝔞}^{∗}-increasing, and for every function g in ∏_{n∈𝔞} ℵ_{n}, there is i < ℵ_{ω+1} such that g <_{𝔞}^{∗} f_{i}.

Any family of ℵ_{ω} many functions in ∏_{n∈𝔞} ℵ_{n} is eventually dominated by some function in ∏_{n∈𝔞} ℵ_{n}. Thus if 2^{ℵ_{ω}} = ℵ_{ω+1}, then there exists a scale in ∏_{n∈𝔞} ℵ_{n}. On the other hand, Shelah [11] proved that there always exists an infinite set 𝔞 for which there is a scale in ∏_{n∈𝔞} ℵ_{n}.

Definition 1.1. Let 𝔞 be an infinite subset of ℵ_{0}, and suppose ⟨f_{i} : i < ℵ_{ω+1}⟩ is a scale in ∏_{n∈𝔞} ℵ_{n}. An ordinal α < ℵ_{ω+1} is said to be good with respect to the scale if there exists a set A cofinal in α and some n such that for any k ≥ n in 𝔞, the sequence ⟨f_{i}(k) : i ∈ A⟩ is strictly increasing. The scale is said to be good if there exists a club C ⊆ ℵ_{ω+1} such that every α in C is good.

Most square-like principles on ℵ_{ω+1} imply that every scale is good. For example, this is true of the weak square principle □_{ℵ_{ω}}^{∗} and the approachability property AP_{ℵ_{ω}}. More generally, if a stationary set A ⊆ ℵ_{ω+1} is in the approachability ideal I[ℵ_{ω+1}], then for any scale, modulo clubs almost every point in A is good with respect to the scale. (See [1].)

Let ⟨f_{i} : i < α⟩ be a <_{𝔞}^{∗}-increasing sequence in ∏_{n∈𝔞} ℵ_{n}, where α is a limit ordinal. A function h ∈ ∏_{n∈𝔞} ℵ_{n} is an exact upper bound of the sequence if (i) for all i < α, f_{i} <_{𝔞}^{∗} h, and (ii) if g <_{𝔞}^{∗} h then there exists i < α such that g <_{𝔞}^{∗} f_{i}. It is straightforward to show that any two exact upper bounds of the same sequence are eventually equal.

The only result we will need about good ordinals is the next standard proposition. We include a proof for the convenience of the reader.

Proposition 1.2. Let 𝔞 be an infinite subset of ℵ_{0} and suppose ⟨f_{i} : i < ℵ_{ω+1}⟩ is a scale in ∏_{n∈𝔞} ℵ_{n}. Then for any ordinal α < ℵ_{ω+1} of uncountable cofinality, the following are equivalent:

(1) α is good with respect to the scale;

(2) for every set A cofinal in α, there is a set B ⊆ A cofinal in α and some n such that for all k ≥ n in 𝔞, ⟨f_{i}(k) : i ∈ B⟩ is strictly increasing;

(3) there exists an exact upper bound h of the sequence ⟨f_{i} : i < α⟩ such that cf(h(n)) = cf(α) for all large enough n ∈ 𝔞.

Proof. (2 ⇒ 1) Immediate. (1 ⇒ 2) Fix a set X cofinal in α and n such that for all k ≥ n in 𝔞, ⟨f_{i}(k) : i ∈ X⟩ is strictly increasing. Let A be cofinal in α. Then we can easily define by induction increasing sequences ⟨α_{i} : i < cf(α)⟩ and ⟨β_{i} : i < cf(α)⟩ of ordinals in X and A respectively such that for all i, α_{i} < β_{i} < α_{i+1}. For each i < cf(α), fix m_{i} < ω such that for all k ≥ m_{i} in 𝔞, f_{α_{i}}(k) < f_{β_{i}}(k) < f_{α_{i+1}}(k). As cf(α) is uncountable, there exists a set D cofinal in cf(α) and m such that for all i in D, m_{i} = m. Let B = {β_{i} : i ∈ D} and let l = max{n, m}. Then B ⊆ A is cofinal in α. For all k ≥ l in 𝔞 and β_{i} < β_{j} in B, f_{β_{i}}(k) < f_{α_{i+1}}(k) ≤ f_{α_{j}}(k) < f_{β_{j}}(k), where the first and last inequality follow from the fact that k ≥ l ≥ m_{i} = m_{j}, and the second follows from the fact that k ≥ l ≥ n and α_{i+1} and α_{j} are in X.

Let cf(α) = ℵ_{m}. (1 ⇒ 3) Fix A ⊆ α with order type ℵ_{m} and n > m such that for all k ≥ n in 𝔞, ⟨f_{i}(k) : i ∈ A⟩ is strictly increasing. For k ∈ 𝔞, define h(k) = 0 if k < n, and h(k) = sup_{i∈A} f_{i}(k) if k ≥ n. Clearly h is an upper bound of ⟨f_{i} : i < ℵ_{m}⟩. Suppose g <_{𝔞}^{∗} h. Fix l > n such that for all k ≥ l in 𝔞, g(k) < h(k). Then for each k ≥ l in 𝔞, there is i_{k} ∈ A such that g(k) < f_{i_{k}}(k). Fix i^{∗} > sup_{k} i_{k} in A. Then g <_{𝔞}^{∗} f_{i^{∗}}.

(3 ⇒ 1) Suppose that h is an exact upper bound such that cf(h(k)) = ℵ_{m} for all k ≥ n in 𝔞. Choose ⟨γ_{i}^{k} : i < ℵ_{m}⟩ increasing and cofinal in h(k) for all k ≥ n in 𝔞. For i < ℵ_{m} and k ∈ 𝔞, define g_{i}(k) = 0 if k < n and g_{i}(k) = γ_{i}^{k} if k ≥ n. It is straightforward to define by induction increasing sequences ⟨ξ_{i} : i < ℵ_{m}⟩ and ⟨α_{i} : i < ℵ_{m}⟩ cofinal in ℵ_{m} and α respectively such that g_{ξ_{i}} <_{𝔞}^{∗} f_{α_{i}} <_{𝔞}^{∗} g_{ξ_{i+1}} for all i < ℵ_{m}. Then there is an unbounded set B ⊆ ℵ_{m} and l such that for all i ∈ B and k ≥ l in 𝔞, g_{ξ_{i}}(k) < f_{α_{i}}(k) < g_{ξ_{i+1}}(k). The set {α_{i} : i ∈ B} and l witness that α is good. Namely, for i < j in B and k ≥ l in 𝔞, f_{α_{i}}(k) < g_{ξ_{i+1}}(k) ≤ g_{ξ_{j}}(k) < f_{α_{j}}(k). □

2. Namba forcing

A tree is a set of finite sequences, closed under initial segments. The elements of a tree are called nodes of the tree. Note that the union or the intersection of any number of trees is a tree. For finite sequences η and ν, we write η⊴ν to express that η is an initial segment of ν, and η ◁ ν to express that η is a proper initial segment of ν. We say that η and ν are comparable if η ⊴ ν or ν ⊴ η; otherwise they are incomparable. Note that if η and ν are both initial segments of some finite sequence, then they are comparable. A subset of a tree is an antichain if its elements are pairwise incomparable, and is a maximal antichain if it is an antichain and every node of the tree is comparable with one of its elements.

Let T be a tree. We write Suc_{T} (η) for the set {x : η x ∈ T }. A node η is a splitting node if |Suc_{T} (η)| > 1, and is maximal if Suc_{T} (η) = ∅. A splitting node η is an n-splitting node if there are exactly n many splitting nodes of T which are proper initial segments of η. There is at most one 0-splitting node of T, which if it exists is referred to as the stem of T, and denoted by stem(T). In that case, if η is a splitting node of T, then stem(T) ⊴ η. We let T_{η} denote the set of ν in T which are comparable with η. Note that T_{η} is a tree and is a subset of T.

A function b with domain ω is a branch of T if for all n, b ↾ n ∈ T. The set of branches of T is denoted by [T]. An antichain J ⊆ T is a front if every branch of T has an initial segment which is in J. The statement “J is a front of T ” is absolute between transitive models of set theory. In a tree with no maximal nodes, every front is a maximal antichain. We will mainly be interested in trees with no maximal nodes such that every node can be extended to a splitting node. In such a tree, for every n < ω the set of n-splitting nodes is a front.

Now we define the version of Namba forcing which we will use. Fix a surjection d : ω → ω \{0, 1} such that for each k ≥ 2: (i) there are infinitely many n such that d(n) = k, and (ii) if n is least such that d(n) = k, then for all m < n, d(m) < k. For example, we can let d be an enumeration of the sequence 2, 2, 3, 2, 3, 4, …

Definition 2.1. Let ℙ be the forcing poset whose conditions are trees S satisfying: (1) if η α is in S, then α ∈ ℵ_{d(lh(η))};

(2) if η is a splitting node of S, then Suc_{S}(η) is an unbounded subset of ℵ_{d(lh(η))};

(3) for each η in S and k ≥ 2, there is a splitting node ν of S above η such that d(lh(ν)) = k.

Let T ≤ S if T ⊆ S. Let T ≤^{∗} S if T ≤ S and every η in T which is a splitting node of S is also a splitting node of T.

If η is a splitting node of S and k = d(lh(η)), then Suc_{S}(η) is an unbounded subset of ℵ_{k}, and we say that η splits in ℵ_{k}.

Note that if T is in ℙ and η ∈ T, then T_{η} is in ℙ and T_{η} ≤ T. If η ∈ T ≤ S, then T_{η} ≤ S_{η}. Also note that if T ≤ S then stem(S) ⊴ stem(T). It follows that if S and T have incomparable stems, then S and T are incompatible conditions.

Let S, T ∈ ℙ. For n < ω, define T ≤_{n} S if T ≤ S and every n-splitting node of S is an n-splitting node of T. This is equivalent to saying that T ≤ S and T and S have the same n-splitting nodes. Note that ≤_{n} is a partial ordering, and if m < n then T ≤_{n} S implies T ≤_{m} S.

Lemma 2.2. Let S be in ℙ and n < ω. Let A be the set of nodes of S of the form η α, where η is an n-splitting node of S. Then A has the same cardinality as the set of n + 1-splitting nodes.

Proof. Every n + 1-splitting node extends a unique node of the form η α, where η is an n-splitting node of S. Conversely, every such node η α extends uniquely to an n + 1-splitting node. □

We will construct conditions using the methods of amalgamation and fusion sequences. Generally speaking, amalgamation is a way to construct a condition by taking the union of a set of subtrees of a condition. Of course, taking such a union will not always produce a condition. The next lemma, whose proof we leave to the interested reader, describes the specific kind of amalgamation we will use.

Lemma 2.3. Let S be in ℙ. Assume that either B is a singleton consisting of a single splitting node of S, or B is the set of all n-splitting nodes of S for some n < ω. For each η in B let Suc(η) be an unbounded subset of Suc_{S}(η). Let A = {η α : η ∈ B, α ∈ Suc(η)}. For each ξ ∈ A, let U(ξ) ≤ S_{ξ}. Then T = ⋃{U(ξ) : ξ ∈ A} is a condition, T ≤ S, and {U(ξ) : ξ ∈ A} is a maximal antichain below T. In the case that B is the set of all n-splitting nodes of S, T ≤_{n} S.

A sequence ⟨S_{n} : n < ω⟩ of conditions in ℙ is a fusion sequence if S_{n+1} ≤_{n} S_{n} for all n.

Lemma 2.4. Let ⟨S_{n} : n < ω⟩ be a fusion sequence. For each n < ω, let J_{n} be the set of n-splitting nodes of S_{n}. Let T = ⋂_{n} S_{n}. Then:

(1) T = ⋃_{n}{ξ : ∃η ∈ J_{n} ξ ⊴ η};

(2) for all n < ω, S_{n} and T have the same n-splitting nodes; so if T ∈ ℙ, then T ≤_{n} S_{n} for all n;

(3) T satisfies Definition 2.1(1,2).

The proof is straightforward.

In general, the intersection of a fusion sequence might not be in ℙ, since it could fail Definition 2.1(3). To guarantee that the intersection is a condition, we will construct our fusion sequences carefully, as we now describe.

Notation 2.5. Let ℙ_{0} be the suborder of ℙ consisting of conditions S such that for all n < ω, every n-splitting node of S splits in ℵ_{d(n)}.

Lemma 2.6. Let ⟨S_{n} : n < ω⟩ be a fusion sequence such that for all n < ω, every n-splitting node of S_{n} splits in ℵ_{d(n)}. Then T = ⋂_{n} S_{n} is in ℙ_{0}.

Proof. By Lemma 2.4(2), if T is in ℙ, then T is in ℙ_{0}. By Lemma 2.4(3), it suffices to verify that T satisfies Definition 2.1(3). Consider η in T and let k ≥ 2. Fix n > lh(η) such that d(n) = k. Now η is in S_{n}, so we can find an n-splitting node ν of S_{n} which is comparable to η; since n > lh(η), η ⊴ ν. Then ν is an n-splitting node of S_{n}, so d(lh(ν)) = d(n) = k. By Lemma 2.4(2), ν splits in T. □

We describe now in general terms the procedure for defining fusion sequences which we will use in what follows. We start with a condition S, and define by induction a fusion sequence ⟨S_{n} : n < ω⟩. To ensure that the intersection of the sequence is a condition, we arrange that for each n < ω, every n-splitting node of S_{n} splits in ℵ_{d(n)}.

To begin, choose a node η of S which splits in ℵ_{d(0)}. Let S_{0} = S_{η}. Note that the 0-splitting node of S_{0} is η. Assume that S_{n} is given, and satisfies that every n-splitting node of S_{n} splits in ℵ_{d(n)}. We would like to define S_{n+1} which has the same n-splitting nodes as S_{n}, and whose n + 1-splitting nodes split in ℵ_{d(n+1)}. For each n-splitting node η of S_{n}, we can choose a set Suc(η) which is an unbounded subset of Suc_{S_{n}}(η). Let A_{n} be the set of nodes of S_{n} of the form η α, where η is an n-splitting node of S_{n} and α ∈ Suc(η).

For each ν in A_{n}, we can choose a condition U(ν) ≤ (S_{n})_{ν} whose stem, which extends ν, splits in ℵ_{d(n+1)}. Then we let S_{n+1} = ⋃{U(ν) : ν ∈ A_{n}}. By Lemma 2.3, S_{n+1} is a condition, S_{n+1} ≤_{n} S_{n}, and {U(ν) : ν ∈ A_{n}} is a maximal antichain below S_{n+1}. Note that the n + 1-splitting nodes of S_{n+1} are the stems of the conditions U(ν) for ν in A_{n}, and they split in ℵ_{d(n+1)}.

Let T = ⋂ S_{n}. Then T is in ℙ_{0} and satisfies that T ≤_{n} S_{n} for all n < ω. Moreover, for each n < ω, {U(ν) : ν ∈ A_{n}} is a maximal antichain below T. Note that the construction of T depends on two kinds of choices, namely, on the successor sets Suc(η) for n-splitting nodes η of S_{n}, and on the conditions U(ν) for ν in A_{n}.

3. Properties of Namba Forcing

In this section we establish some basic properties of the Namba forcing ℙ introduced in the previous section. The proofs are standard; they rely on Namba forcing type methods which were introduced by Namba [10] and Shelah [12].

Let ḃ be a name for ⋃ ⋂ Ġ. If T ≤ S, then stem(S) ⊴ stem(T); so if S and T are compatible conditions, then stem(S) and stem(T) are comparable. It follows that ⋂ G = {ξ : ∃S ∈ Ġ (ξ = stem(S))}, and ḃ is a function with domain ω such that for all n < ω, there is S in G such that ḃ ↾ n = stem(S).

Note that T forces that stem(T) is an initial segment of ḃ. Hence if η α⊴stem(T), then T ⊩ ḃ(lh(η)) = α.

Lemma 3.1. Suppose G is a generic filter for ℙ, and S ∈ G. Then for all n < ω, S_{b↾n} is in G.

Proof. It suffices to show that S_{b↾n} is comparable with each condition in G. So let T be in G. Fix U ≤ S, T in G; by a density argument we can assume that n < lh(stem(U)). Then b ↾ n ⊴ stem(U), and U = U_{b↾n} ≤ S_{b↾n}, T. □

Lemma 3.2. For each k ≥ 2, ℙ forces that the set {ḃ(n) : d(n) = k} is unbounded in ℵ_{k}. In particular, ℙ ⊩ cf(ℵ_{k}) = ℵ_{0}.

Proof. Let α < ℵ_{k} be given, and let S ∈ ℙ. We construct a condition T ≤ S by applying Lemma 2.3. Fix η in S which splits in ℵ_{k}; then d(lh(η)) = k. Let Suc(η) = Suc_{S}(η) \ α. Let A = {η β : β ∈ Suc(η)}. For each β in Suc(η), let U(η β) = S_{η β}. Let T = ⋃{U(ξ) : ξ ∈ A}. Then T ∈ ℙ and T ≤ S. Since U(η β) ⊩ ḃ(lh(η)) = β ≥ α for all η β ∈ A, T ⊩ ḃ(lh(η)) ≥ α. □

Proposition 3.3. Let λ > ℵ_{ω} be regular. Then ℙ ⊩ cf(λ) > ℵ_{0}.

Proof. Suppose S forces that ḟ : ω → λ is a function. We define a fusion sequence ⟨S_{n} : n < ω⟩. Fix η in S which splits in ℵ_{d(0)}, and let S_{0} = S_{η}. Assume S_{n} is given. Let A_{n} be the set of nodes of the form η α, where η is an n-splitting node of S_{n} and α ∈ Suc_{S_{n}}(η). For each η α in A_{n}, choose a condition V (η α) ≤ (S_{n})_{η α} and an ordinal β_{n}(η α) < λ such that V (η α) ⊩ ḟ(n) = β_{n}(η α). Choose ν in V (η α) which splits in ℵ_{d(n+1)}, and let U(η α) = V (η α)_{ν}. Define S_{n+1} = ⋃{U(ξ) : ξ ∈ A_{n}}.

Let T = ⋂_{n} S_{n}. For each n, A_{n} is a subset of ℵ_{ω}^{<ω}, and thus has size at most ℵ_{ω}. So if we let β_{n} = ⋃{β_{n}(ξ) + 1 : ξ ∈ A_{n}}, then β_{n} < λ. Now let β = ⋃_{n} β_{n}, which is less than λ. We claim that T ⊩ ḟ(n) < β for all n. For suppose that U ≤ T decides ḟ(n). Then since {U(ξ) : ξ ∈ A_{n}} is a maximal antichain below T, for some ξ in A_{n}, U ⊩ ḟ(n) = β_{n}(ξ) < β. □

Proposition 3.4. The forcing poset ℙ preserves stationary subsets of ℵ_{1}.

Proof. Let A be a stationary subset of ℵ_{1}, and we show that ℙ preserves the stationarity of A. So let S ⊩ Ċ ⊆ ℵ_{1} is a club, and we find T ≤ S and δ ∈ A such that T ⊩ δ ∈ Ċ.

For each δ < ℵ_{1} we define a two-player game G_{δ} as follows. Let η_{0} be the lexicographically least node in S which splits in ℵ_{d(0)}. Player I begins the game by choosing X_{0} which is a bounded subset of ℵ_{d(0)}, and an ordinal β_{0} < δ. Player II responds by choosing an ordinal α_{0} in Suc_{S}(η_{0}) \ X_{0}, a condition U_{0} ≤ S_{η_{0} α_{0}} whose stem splits in ℵ_{d(1)}, and an ordinal ξ_{0} < ℵ_{1} such that U_{0} ⊩ ξ_{0} ∈ Ċ \ β_{0}.

Now suppose n > 0 and stage n−1 of the game is complete, which has resulted in η_{n−1}, X_{n−1}, β_{n−1}, α_{n−1}, U_{n−1}, and ξ_{n−1}. Let η_{n} = stem(U_{n−1}). We assume as an induction hypothesis that η_{n} splits in ℵ_{d(n)} in U_{n−1}, and η_{n−1} α_{n−1} ⊴ η_{n}. Player I chooses a set X_{n} which is a bounded subset of ℵ_{d(n)}, and an ordinal β_{n} < δ. Player II responds by choosing an ordinal α_{n} in Suc_{U_{n−1}}(η_{n}) \ X_{n}, a condition U_{n} ≤ (U_{n−1})_{η_{n} α_{n}} whose stem splits in ℵ_{d(n+1)}, and an ordinal ξ_{n} < ℵ_{1} such that U_{n} ⊩ ξ_{n} ∈ Ċ \ β_{n}.

For later purposes, we will refer to the node η_{n} described above as the n-th node of the partial run of the game. If n = 0 then the 0-th node of a partial run is just the lexicographically least node of S which splits in ℵ_{d(0)}. If n > 0 then the n-th node is determined after stage n − 1 of the partial run of the game has completed, and is the stem of the last condition played by Player II and splits in ℵ_{d(n)}.

A run of the game produces a sequence of ordinals ⟨ξ_{n} : n < ℵ_{0}⟩ as described above. Player II wins if sup_{n} ξ_{n} ≤ δ. This game is open for Player I. So by the Gale-Stewart Theorem, one of the players has a winning strategy.

We claim that the set of δ < ℵ_{1} for which Player II has a winning strategy in the game G_{δ} contains a club. Suppose for a contradiction that there exists a stationary set Y ⊆ ℵ_{1} such that for all ζ in Y, Player I has a winning strategy σ_{ζ} in the game G_{ζ}. Fix a regular cardinal θ much larger than ℙ and Ċ, and let N be a countable elementary substructure of H(θ) such that N ∩ ℵ_{1} ∈ Y and which contains as elements the sets ℙ, S, Ċ, Y, and ⟨σ_{ζ} : ζ ∈ Y ⟩. Let δ = N ∩ ℵ_{1}.

We consider simultaneously runs of the games G_{ζ} for each ζ ∈ Y, where Player I’s moves are according to G_{ζ} and Player II’s moves are in N. Suppose it is stage n of the games. For simplicity assume n > 0; the case n = 0 is basically the same except with slightly different notation. We assume Player II’s previous moves are in N and are the same for all ζ. Then η_{n} is in N by elementarity and is the stem of the last condition U_{n−1} played by Player II. For each ζ in Y apply σ_{ζ} to the moves in the G_{ζ} run of the game to obtain Player I’s next move β_{n}^{ζ} and X_{n}^{ζ}, where β_{n}^{ζ} < ζ and X_{n}^{ζ} is a bounded subset of ℵ_{d(n)}. Since Player II’s moves are in N, by elementarity the sequence ⟨X_{n}^{ζ} : ζ ∈ Y ⟩ is in N. Let X_{n} = ⋃_{ζ} X_{n}^{ζ}. Then X_{n} is in N and is a bounded subset of ℵ_{d(n)} since ℵ_{d(n)} > ℵ_{1} = |Y |.

Choose some ordinal α_{n} in Suc_{U_{n−1}}(η_{n})\X_{n} which is in N. Extend (U_{n−1})_{η_{n} α_{n}} to a condition U_{n} which decides some ξ_{n} to be in Ċ \ β_{n}^{δ} and whose stem splits in ℵ_{d(n+1)}. By elementarity we can choose U_{n} and ξ_{n} in N. Player II now plays α_{n}, U_{n}, and ξ_{n} as his move in all runs of the game. It might happen that for some ζ in Y, Player II’s move violates the rules of the game. However it does not in the game G_{δ} (and in fact, for stationarily many ζ). But we consider the strategies to be total functions so we can continue in any case.

Now note that Player II wins the run of the game G_{δ}, since each ξ_{n} is in N ∩ℵ_{1} = δ. This is a contradiction since Player I used the strategy σ_{δ}.

Now we are ready to find T ≤ S which forces that A ∩ Ċ is non-empty. Fix a club D ⊆ ℵ_{1} such that for every δ in D, there is a strategy τ_{δ} for Player II in the game G_{δ}. Since A is stationary, we can fix δ in D ∩ A. Let ⟨δ_{n} : n < ω⟩ be increasing and cofinal in δ.

We define a fusion sequence ⟨S_{n} : n < ω⟩. Let η_{0} be the lexicographically least node in S which splits in ℵ_{d(0)}, and let S_{0} = S_{η_{0}}. Assume now that S_{n} is defined. As an induction hypothesis we assume that every n-splitting node η of S_{n} is the n-th node of a partial run of the game G_{δ} in which stage n − 1 has been completed (or η = η_{0} if n = 0) in which Player II used the strategy τ_{δ} and Player I played the ordinals δ_{0}, …, δ_{n−1}.

Consider an n-splitting node η of S_{n}. If n = 0 then η = η_{0} and Suc_{S}(η) = Suc_{S_{0}}(η). If n > 0 then η is the n-th node of a partial run of the game with stage n − 1 having been completed. Let U_{n−1} be the condition played by Player II in the previous move, so that η = stem(U_{n−1}). As an induction hypothesis assume that (S_{n})_{η} = U_{n−1}. Let Suc(η) be the set of α such that for some move of Player I at stage n whose ordinal part is δ_{n}, τ_{δ} instructs Player II to respond with α. So Suc(η) is a subset of Suc_{S_{n}}(η). We claim that Suc(η) is unbounded in ℵ_{d(n)}. If not, then Player I can play Suc(η) and δ_{n}, and τ_{δ} instructs Player II to play an ordinal in Suc(η) which is a contradiction.

For each η α with α in Suc(η), let U(η α) ≤ (U_{n−1})_{η α} = (S_{n})_{η α} and ξ(η α) be the condition and ordinal which τ_{δ} instructs Player II to play along with α. Let A_{n} be the set of nodes of the form η α, where η is an n-splitting node of S_{n} and α ∈ Suc(η). Let S_{n+1} = ⋃{U(ν) : ν ∈ A_{n}}.

Let T = ⋂_{n} S_{n}. Then T is in ℙ and T ≤ S. We claim that T forces δ is in Ċ. It suffices to show that for each n, T forces Ċ ∩ [δ_{n}, δ] is non-empty. But {U(ν) : ν ∈ A_{n}} is a maximal antichain below T. For each ν in A_{n}, U(ν) forces ξ(ν) is in Ċ \ δ_{n}. Since ξ(ν) was part of a move played by Player II according to τ_{δ}, ξ(ν) ≤ δ. Thus each U(ν) forces Ċ ∩ [δ_{n}, δ] is non-empty, and therefore so does T. □

4. The Bounding Property

We now establish the bounding property of the forcing poset ℙ which will be useful for constructing models with no good scale.

Recall that the surjection d : ω → ω \ {0, 1}, which we used in the definition of ℙ, was chosen to satisfy that whenever n is least such that d(n) = k, then d(m) < k for all m < n. The next lemma is the reason for this requirement.

Lemma 4.1. Let k > 2 be given, and suppose that n is least such that d(n) = k, so that n > 0. Let S be in ℙ, and assume that for all m < n, every m-splitting node of S splits in ℵ_{d(m)}. Let A be the set of nodes of S of the form η α, where η is an n − 1-splitting node of S. Then |A| < ℵ_{k}.

Proof. We prove by induction on m ≤ n that S has fewer than ℵ_{k} many m-splitting nodes. First S has one 0-splitting node. Suppose the claim is true for a fixed m which is less than n. Then there are fewer than ℵ_{k} many m-splitting nodes of S by the induction hypothesis, and for each m-splitting node η, |Suc_{S}(η)| = ℵ_{d(m)} < ℵ_{k}, since d(m) < k. So the set of nodes in S of the form η α, where η is an m-splitting node, has cardinality less than ℵ_{k}. By Lemma 2.2, S has fewer than ℵ_{k} many m + 1-splitting nodes. Finally, the case m = n − 1 gives the result. □

Proposition 4.2. Every function in ∏_{n≥2} ℵ_{n}^{V} in a generic extension by ℙ is bounded by some function in ∏_{n≥2} ℵ_{n}^{V} in the ground model.

Proof. Suppose S ⊩ ġ ∈ ∏_{n≥2} ℵ_{n}^{V}. We will define a condition T ≤ S and a sequence ⟨ζ_{k} : 2 ≤ k < ω⟩ such that each ζ_{k} is in ℵ_{k} and T ⊩ ġ(k) < ζ_{k}. By Lemma 3.2, for each k ≥ 2 we can fix a name ṁ_{k} such that S ⊩ d(ṁ_{k}) = k ∧ ġ(k) < ḃ(ṁ_{k}).

We define a fusion sequence ⟨S_{n} : n < ω⟩. Let B be the set of n such that for some k ≥ 2, n is the least integer such that d(n) = k. The definition of S_{n} will depend on whether or not n is in B. For n = 0, d(0) = 2 and 0 ∈ B. Fix S′ ≤ S which decides ṁ_{2} as some integer m_{2}. Then d(m_{2}) = 2. Pick a node η of S′ which splits in ℵ_{2} such that lh(η) > m_{2}. Now let S_{0} = S′_{η}. Note that

S_{0} ⊩ ġ(2) < ḃ(ṁ_{2}) = ḃ(m_{2}) = η(m_{2}) = stem(S_{0})(m_{2}).

Let ζ_{2} = stem(S_{0})(m_{2}). Then S_{0} ⊩ ġ(2) < ζ_{2}, and ζ_{2} < ℵ_{2}.

Now suppose that S_{n} is defined. In the first case, assume that n + 1 is not in B. Let A_{n} be the set of nodes of S_{n} of the form η α, where η is an n-splitting node of S_{n}. For each η α in A_{n}, choose some ν extending η α which splits in ℵ_{d(n+1)} in S_{n}. Let U(η α) = (S_{n})_{ν}. Now let S_{n+1} = ⋃{U(ξ) : ξ ∈ A_{n}}.

In the second case, assume that n+1 is in B. Fix k such that d(n+1) = k and for all m < n+1, d(m) ≠ k. Let A_{n} be the set of nodes of S_{n} of the form η α, where η is an n-splitting node. Consider η α in A_{n}. Extend (S_{n})_{η α} to a condition V (η α) which decides the name ṁ_{k} as m_{k}(η α). Then d(m_{k}(η α)) = k. Now choose a node ν in V (η α) which splits in ℵ_{k} and whose length is greater than m_{k}(η α). Let U(η α) = V (η α)_{ν}. Note that U(η α) ⊩ ġ(k) < ḃ(ṁ_{k}) = ḃ(m_{k}(η α)) = ν(m_{k}(η α)) = stem(U(η α))(m_{k}(η α)). Now let S_{n+1} = ⋃{U(ξ) : ξ ∈ A_{n}}.

By Lemma 4.1, A_{n} has size less than ℵ_{k}. So letting

ζ_{k} = sup{stem(U(ξ))(m_{k}(ξ)) : ξ ∈ A_{n}},

ζ_{k} < ℵ_{k}. For each ξ in A_{n}, U(ξ) ⊩ ġ(k) < ζ_{k}. Since {U(ξ) : ξ ∈ A_{n}} is a maximal antichain below S_{n+1}, S_{n+1} ⊩ ġ(k) < ζ_{k}.

This completes the construction of the fusion sequence. Let T = ⋂_{n} S_{n}. Define h in ∏_{n≥2} ℵ_{n} by letting h(k) = ζ_{k}. We claim that T forces that ġ is dominated by h. So let k ≥ 2 be given. Let n be the least integer such that d(n) = k. As shown above, S_{n} ⊩ ġ(k) < ζ_{k} = h(k). Hence T ⊩ ġ(k) < h(k). □

5. The 𝕊-condition

We review the 𝕊-condition of Shelah [12], and prove that ℙ satisfies this property. This will ensure that we can iterate ℙ with revised countable support while not adding reals.

The following concepts are due to Shelah [12].

Definition 5.1. Let 𝕊 be a family of regular cardinals larger than ℵ_{1}. An 𝕊-tree is a pair (T, H), where T is a tree and H : T → 𝕊 is a function, satisfying:

(1) if η α is in T, then α ∈ H(η);

(2) if η is a splitting node, then Suc_{T} (η) is an unbounded subset of H(η);

(3) for all η in T, there is a splitting node in T above η. Given 𝕀-trees (T, H) and (S, G), let (T, H) ≤ (S, G) if T ⊆ S and H = G ↾ T. Let (T, H) ≤^{∗} (S, G) if T ≤ S and every η in T which is a splitting node of S is also a splitting node of T.

The forcing poset ℙ can be described in the terms just introduced. Let 𝕊 = {ℵ_{n} : 2 ≤ n < ω}. Define H : ^{<ω}ℵ_{ω} → 𝕊 by letting H(η) = ℵ_{d(lh(η))}. Then a tree T is in ℙ iff (T, H ↾ T) is an 𝕊-tree such that for every η in T and k ≥ 2, there is a splitting node ν of T above η such that H(ν) = ℵ_{k}.

Let 𝕊 be a family of regular cardinals larger than ℵ_{1}. We define when a forcing poset ℚ satisfies the 𝕊-condition. This property is described in terms of the existence of a winning strategy for Player II in a certain two-player game. At the end of a run of the game, the players will have constructed an 𝕊-tree (T, H) and a function f : T → ℚ, satisfying that η ⊴ ν implies f(ν) ≤ f(η).

Suppose it is stage n of the game. If n > 0, assume that we are given a tree T_{n}^{II}, with no cofinal branches, resulting from the previous play. If n = 0, then let T_{0}^{II} = {∅}. In either case, let J_{n}^{II} be the set of maximal nodes of T_{n}^{II}. Player I chooses (1) for each η in J_{n}^{II}, a finite sequence ν with η ⊴ ν, (2) for each ξ with η ⊴ ξ ◁ ν, a cardinal H(ξ) ∈ 𝕊 such that ν(lh(ξ)) ∈ H(ξ), and (3) for each ξ with η ◁ξ ⊴ν (as well as ξ = η = ∅ if n = 0), a condition f(ξ) in ℚ such that η ⊴ζ ⊴ξ ⊴ν implies f(ξ) ≤ f(ζ). This completes Player I’s move. Let J_{n+1}^{I} be the set of such finite sequences ν chosen by Player I, and let T_{n+1}^{I} be the tree {ξ : ∃ν ∈ J_{n+1}^{I} ξ ⊴ν}.

Player II responds by playing, for each η in J_{n+1}^{I}, (a) a cardinal H(η) ∈ 𝕊, (b) a set Suc(η) which is an unbounded subset of H(η), and (c) for each α in Suc(η), a condition f(η α) in ℚ such that f(η α) ≤ f(η). Let J_{n+1}^{II} = {η α : η ∈ J_{n+1}^{I}, α ∈ Suc(η)}, and let T_{n+1}^{II} = {ξ : ∃ν ∈ J_{n+1}^{II} ξ⊴ν}. Note that J_{n+1}^{II} is the set of maximal nodes of T_{n+1}^{II}.

At the end of the run of the game, an 𝕊-tree (T, H) is determined, where T = ⋃_{n} T_{n+1}^{II}, as well as a function f : T → ℚ such that η ⊴ ν implies f(ν) ≤ f(η). The n-splitting nodes of T are exactly the nodes in J_{n+1}^{I}, and for each η in J_{n+1}^{I}, Suc_{T} (η) = Suc(η).

Player II wins the run of the game if for every 𝕊-tree (U, H ↾ U) ≤^{∗} (T, H), there exists a condition q ∈ ℚ such that

q ⊩ ∃c ∈ [U] ∀n < ω (c ↾ n ∈ Ġ).

We say that ℚ satisfies the 𝕊-condition if Player II has a winning strategy such that at stage n of a run of the game, for every η in J_{n+1}^{I}, Player II’s choice of H(η), Suc(η), and the values f(η α) for α ∈ Suc(η), depends only on (i) the node η, (ii) which initial segments of η are in J_{1}^{I} ∪ ⋯ ∪ J_{n}^{I}, and (iii) the values of f on the initial segments of η.

We will use the following theorem.

Theorem 5.2 ([12]). Assume CH. Let ℚ be a forcing poset which satisfies the 𝕊condition, for some non-empty family 𝕊 of regular cardinals larger than ℵ_{1}. Then ℚ does not add reals.

Let 𝕊 be a family of regular cardinals larger than ℵ_{1}. If ℚ is an ℵ_{1}-closed forcing poset, then ℚ satisfies the 𝕊-condition. In fact, Player II can play any moves which obey the rules of the game. After a run of the game is complete, a tree (T, H) and a function f : T → ℚ are constructed. If U ≤^{∗} T, then choose any branch b of [U], and let q be a lower bound of {f(b ↾ n) : n < ω}. Then q forces f(b ↾ n) is in Ġ for all n < ω.

The proof of the next proposition is based on the proof of Shelah [12] that Namba forcing satisfies the 𝕊-condition.

Proposition 5.3. Let 𝕊 be a family of regular cardinals larger than ℵ_{1} such that for all 2 ≤ n < ω, ℵ_{n} is in 𝕊. Then ℙ satisfies the 𝕊-condition.

Proof. We describe a winning strategy for Player II. Suppose it is Player II’s turn to move at stage n of the game. In particular, we are given a set of nodes J_{n+1}^{I}, and values f(η) for all η in J_{n+1}^{I}. Let η be in J_{n+1}^{I}. Player II’s move must be determined only from (i) the node η, (ii) which initial segments of η are in J_{1}^{I} ∪⋯∪J_{n}^{I}, and (iii) the values of f on the initial segments of η. Note that the integer n is determined by (i) and (ii).

Let H(η) = ℵ_{d(n)}. Choose from the Namba condition f(η) a node ν_{η} which splits in ℵ_{d(n)}. Note that stem(f(η)) ⊴ ν_{η}. Let Suc(η) = Suc_{f(η)}(ν_{η}). For each α in Suc(η), let f(η α) = f(η)_{ν_{η} α}. In particular, ν_{η} α ⊴ stem(f(η α)). This completes the description of Player II’s strategy.

We consider a run of the game in which Player II uses the strategy just described. This run of the game produces an 𝕊-tree (T, H), a function f : T → ℙ, and a mapping η ↦ ν_{η} defined on ⋃_{n} J_{n+1}^{I}.

We claim that for η, ξ ∈ ⋃_{n} J_{n+1}^{I},

η ⊴ ξ ⟺ ν_{η} ⊴ ν_{ξ}.

Let η and ξ be given, where η ∈ J_{n+1}^{I} and ξ ∈ J_{m+1}^{I}. Assume η ⊴ ξ. If η = ξ, we are done, so assume η ◁ ξ. Then clearly n < m. Let α = ξ(lh(η)), so that η α ⊴ ξ. Then we have

ν_{η} ◁ ν_{η} α ⊴ stem(f(η α)) ⊴ stem(f(ξ)) ⊴ ν_{ξ},

where the second relation holds because f(η α) = f(η)_{ν_{η} α}, the third relation holds because f(ξ) ≤ f(η α), and the last relation holds because ν_{ξ} is a splitting node of f(ξ). Thus ν_{η} ◁ ν_{ξ}.

Conversely, assume ¬ η ⊴ ξ. If ξ ◁ η, then ν_{ξ} ◁ ν_{η} as we just showed; therefore ¬ ν_{η} ⊴ ν_{ξ}. So assume that η and ξ are incomparable, and we will prove that ν_{η} and ν_{ξ} are incomparable. Let k < n be the largest integer such that for some σ in J_{k+1}^{I}, σ ⊴η, ξ. Note that k exists since the 0-splitting node of T in J_{1}^{I} is an initial segment of η and ξ. It follows that there are distinct α and β in Suc_{T} (σ) = Suc_{f(σ)}(ν_{σ}) such that σ α ⊴ η and σ β ⊴ ξ. Then

ν_{σ} α ⊴ stem(f(σ α)) ⊴ stem(f(η)) ⊴ ν_{η},

so ν_{σ} α ⊴ ν_{η}. Likewise, ν_{σ} β ⊴ ν_{ξ}. It follows that ν_{η} and ν_{ξ} are incomparable. This completes the proof of the claim.

Now we prove that Player II has won the game. Consider (U, H ↾ U) ≤^{∗} (T, H). Note that the n-splitting nodes of U are exactly the nodes in J_{n+1}^{I} ∩ U. We define a fusion sequence ⟨S_{n} : n < ω⟩ so that the n-splitting nodes of S_{n} are exactly the nodes of the form ν_{η}, where η is in J_{n+1}^{I} ∩ U, and moreover, they split in ℵ_{d(n)}. It follows that S = ⋂_{n} S_{n} will be a condition.

Let S_{0} = f(η)_{ν_{η}}, where η is the unique node in J_{1}^{I} = J_{1}^{I} ∩ U. Since ν_{η} splits in ℵ_{d(0)} in f(η), ν_{η} is the 0-splitting node of S_{0} and splits in ℵ_{d(0)}. Now assume that S_{n} is defined, so that the n-splitting nodes of S_{n} are exactly the nodes of the form ν_{η}, where η ∈ J_{n+1}^{I} ∩ U, and they split in ℵ_{d(n)}. Assume moreover that for each η in J_{n+1}^{I} ∩ U, (S_{n})_{ν_{η}} = f(η)_{ν_{η}}.

We define S_{n+1} by amalgamation. Consider an n-splitting node ν_{η} of S_{n}, where η is in J_{n+1}^{I} ∩ U. By the choice of Player II’s strategy, Suc_{T} (η) = Suc_{f(η)}(ν_{η}) = Suc_{S_{n}}(ν_{η}). Since U ≤^{∗} T, Suc_{U}(η) is an unbounded subset of Suc_{S_{n}}(ν_{η}). Let A_{n} be the set of nodes of S_{n} of the form ν_{η} α, where η ∈ J_{n+1}^{I} ∩ U and α ∈ Suc_{U}(η).

Consider ν_{η} α in A_{n}. The node η α is in U and extends uniquely to some ξ in J_{n+2}^{I} ∩ U. Then ν_{ξ} is a splitting node of f(ξ). Define U(ν_{η} α) = f(ξ)_{ν_{ξ}}. Note that stem(U(ν_{η} α)) = ν_{ξ}. Now by the choice of f,

ν_{η} α ⊴ stem(f(η α)) ⊴ stem(f(ξ)) ⊴ ν_{ξ}.

We need to see that U(ν_{η} α) ≤ (S_{n})_{ν_{η} α}. But

U(ν_{η} α) = f(ξ)_{ν_{ξ}} ≤ f(ξ)_{ν_{η} α} ≤ f(η)_{ν_{η} α} = (S_{n})_{ν_{η} α},

where the last equality holds since (S_{n})_{ν_{η}} = f(η)_{ν_{η}}. Now let S_{n+1} = ⋃{U(ζ) : ζ ∈ A_{n}}. Then the n + 1-splitting nodes of S_{n+1} are exactly the stems of the U(ζ)’s, which are of the form ν_{ξ} for ξ ∈ J_{n+2}^{I} ∩ U, and they split in ℵ_{d(n+1)}.

Let S = ⋂_{n} S_{n}. Let G be a generic filter on ℙ which contains S, and let b = ḃ^{G}. For each n, there is an initial segment of b which is an n-splitting node of S of the form ν_{η_{n}}, for a unique η_{n} which is an n-splitting node of U. Note that for n < m, ν_{η_{n}} ⊴ ν_{η_{m}}, which implies η_{n} ⊴ η_{m}. Let c = ⋃_{n} η_{n}, which is a branch of U.

We show that whenever ξ is an initial segment of c, f(ξ) ∈ G. It suffices to show this when ξ = η_{n} for some n. So consider η_{n}. Then ν_{η_{n}} is an initial segment of b. Also ν_{η_{n}} is an n-splitting node of S_{n} and (S_{n})_{ν_{ηn}} = f(η_{n})_{ν_{ηn}}. Now ν_{η_{n}} is in S, and S ≤ S_{n}. So

(S)_{ν_{ηn}} ≤ (S_{n})_{ν_{ηn}} = f(η_{n})_{ν_{ηn}} ≤ f(η_{n}).

But ν_{η_{n}} is an initial segment of b and S ∈ G, hence (S)_{ν_{ηn}} is in G by Lemma 3.1. Therefore f(η_{n}) is in G. □

6. No Good Scales

We now produce a model by an iteration of Namba forcing in which there is no good scale on ℵ_{ω}. Specifically, the model we produce satisfies that for every infinite set 𝔞 ⊆ ω and any scale ⟨f_{i} : i < ℵ_{ω+1}⟩ in ∏_{n∈𝔞} ℵ_{n}, there are stationarily many α in ℵ_{ω+1} ∩ cof(ℵ_{1}) for which there exists an exact upper bound h for ⟨f_{i} : i < α⟩ such that cf(h(n)) = ℵ_{0} for all n ∈ 𝔞. By Proposition 1.2 and the uniqueness of exact upper bounds, this implies that there is no good scale on ℵ_{ω}.

We construct the model using a revised countable support forcing iteration satisfying the 𝕊-condition, for a suitable family of regular cardinals 𝕊. This will ensure that no reals are added and GCH holds in the final model. If we did not care about adding reals then we could use the semiproper forcing iteration theorem, as is done in the next section. The iteration theorem we will use is stated next; it comes from [12] and [8].

Theorem 6.1. Let ⟨ℙ_{i}, ℚ̇_{j} : i ≤ α, j < α⟩ be a revised countable support forcing iteration, and for each i < α, let 𝕊_{i} be a family of regular cardinals larger than ℵ_{1}. Suppose that for all β < α:

(1) ℙ_{β} forces that ℚ̇_{β} satisfies the 𝕊_{β}-condition;

(2) ℙ_{β} forces that every ξ in 𝕊_{β} is a regular cardinal greater than or equal to ℵ_{2};

(3) (if β is not a limit ordinal, or cf(β) < β, or there is γ < β with |ℙ_{γ}| ≥ β) there exists λ and μ such that (i) λ^{<μ} = λ, (ii) if ξ is in ⋃{𝕊_{i} : β ≤ i < α}, then ξ ≥ λ^{+}, and (iii) if ξ is in ⋃{𝕊_{j} : j < β}, then ξ < μ;

(4) (if cf(β) = β and for all γ < β, |ℙ_{γ}| < β) for all ξ in 𝕊_{β}, ξ ≥ β. Then ℙ_{α} satisfies the ⋃{𝕊_{i} : i < α}-condition. Moreover, if α is strongly inaccessible and for all β < α, |ℙ_{β}| < α, then ℙ_{α} is α-c.c.

We start with a ground model in which GCH holds and κ is a supercompact cardinal. We define a revised countable support forcing iteration

⟨ℙ_{i}, ℚ̇_{j} : i ≤ κ, j < κ⟩.

by recursion. We also define families of cardinals {𝕊_{i} : i < κ} satisfying the assumptions of Theorem 6.1. We will maintain that each 𝕊_{i} is countable, and |ℙ_{i}| and every cardinal in 𝕊_{i} is less than the least inaccessible greater than i.

Let ℙ_{0} be the trivial forcing. If δ ≤ κ is a limit ordinal and ℙ_{i} is defined for all i < δ, let ℙ_{δ} be the revised countable support limit of ⟨ℙ_{i} : i < δ⟩. Note that the assumptions of Theorem 6.1 hold for ℙ_{δ}.

Assume ℙ_{β} is defined, where β < κ. Also assume {𝕊_{i} : i < β} is defined as in Theorem 6.1. Then ℙ_{β} does not add reals, and in particular, it preserves ℵ_{1}.

The definition of ℚ̇_{β} and 𝕊_{β} depends on two cases:

Case 1: β is not strongly inaccessible.

Case 2: β is strongly inaccessible.

First consider Case 1. Let ℚ̇_{β} be a ℙ_{β}-name for Coll(ℵ_{1}, (|β|^{+ω+1})^{V}), and let ℙ_{β+1} = ℙ_{β} ∗ ℚ̇_{β}. Let λ = μ be the least regular uncountable cardinal larger than |ℙ_{β}| and all the cardinals in ⋃{𝕊_{i} : i < β}. By GCH, λ^{<μ} = λ; also λ^{+} is below the least inaccessible greater than β. Let 𝕊_{β} = {λ^{+}}. Then the assumptions of Theorem 6.1 hold for ℙ_{β+1}.

Suppose we are in Case 2. Then for all γ < β, |ℙ_{γ}| < β. By Theorem 6.1, ℙ_{β} is β-c.c. By Case 1 occurring cofinally often below β, ℙ_{β} ⊩ β = ℵ_{2}. Also by induction, ⋃{𝕊_{i} : i < β} ⊆ β. Let ℚ̇_{β} be a ℙ_{β}-name for the Namba forcing ℙ described in the previous sections. Let 𝕊_{β} = {β^{+n} : n < ω}. The assumptions of Theorem 6.1 hold for ℙ_{β+1}.

This completes the definition of ℙ_{κ}. It follows by Theorem 6.1 that ℙ_{κ} does not add reals and is κ-c.c. So ℙ_{κ} collapses κ to become ℵ_{2}. Also standard arguments show that ℙ_{κ} forces 2^{μ} = μ^{+} for all cardinals μ > ℵ_{1}, so ℙ_{κ} forces GCH.

Let G be a generic filter on ℙ_{κ}. We prove that in V [G] there is no good scale on ℵ_{ω}. Note that κ^{+n} = ℵ_{n+2} for n < ω, and κ^{+i} = ℵ_{i} for i ≥ ω. We will abbreviate ℵ_{i}^{V [G]} with ℵ_{i} in what follows.

The first step towards proving that there is no good scale is to extend an elementary embedding. In the ground model V, let j : V → M be an elementary embedding with critical point κ such that j(κ) > κ^{+ω+1} and M is closed under κ^{+ω+1}-sequences.

Standard arguments using the closure of M show that j(ℙ_{κ}) factors in M as

j(ℙ_{κ}) = ℙ_{κ} ∗ ℙ ∗ Coll(ℵ_{1}, (κ^{+ω+1})^{V}) ∗ ℙ_{κ+2,j(κ)},

where ℙ is a ℙ_{κ}-name for the Namba forcing discussed in the previous sections.

Let H ∗ I ∗ J be a generic filter for ℙ ∗ Coll(ℵ_{1}, (κ^{+ω+1})^{V}) ∗ ℙ_{κ+2,j(κ)} over V [G]. Let K = G ∗ H ∗ I ∗ J. Since j[G] = G ⊆ G ∗ H ∗ I ∗ J = K, in V [K] we can extend j to j : V [G] → M[K] such that j(G) = K.

We prove that there is no good scale on ℵ_{ω} in V [G]. Working in this model, fix an infinite set 𝔞 ⊆ ω \ {0, 1} and suppose ⟨f_{i} : i < ℵ_{ω+1}⟩ is a scale in ∏_{n∈𝔞} ℵ_{n}. We will prove that this scale is not good by showing that there are stationarily many α in ℵ_{ω+1} ∩ cof(ℵ_{1}) for which there exists an exact upper bound h for ⟨f_{i} : i < α⟩ such that cf(h(n)) = ℵ_{0} for all n ∈ 𝔞. Fix a club subset C of ℵ_{ω+1}.

Let j(⟨f_{i} : i < ℵ_{ω+1}⟩) = ⟨f_{i}^{j} : i < j(ℵ_{ω+1})⟩. By the elementarity of j, it suffices to show that in M[K] there is an ordinal α in j(C) ∩ cof(ℵ_{1}) for which there exists an exact upper bound h for ⟨f_{i}^{j} : i < α⟩ such that cf(h(n)) = ℵ_{0} for all n ∈ 𝔞 = j(𝔞).

Let α = sup(j[ℵ_{ω+1}]). We claim that α is in j(C) ∩ cof(ℵ_{1}) in M[K]. By Proposition 3.3, the ordinal ℵ_{ω+1} is collapsed to have size and cofinality ℵ_{1} by ℙ ∗ Coll(ℵ_{1}, (κ^{+ω+1})^{V}). Also ℙ_{κ+2,j(κ)} preserves ℵ_{1}. The set j[ℵ_{ω+1}] is in M by the closure of M. It follows that j[ℵ_{ω+1}] is bounded below j(ℵ_{ω+1}) = (ℵ_{ω+1})^{M[K]}, and α has cofinality ℵ_{1} in M[K]. Since j[C] is a cofinal subset of α contained in j(C), α is in j(C).

Define h(n) = sup(j[ℵ_{n}]) for all n ∈ 𝔞. Since cf(ℵ_{n}) = ℵ_{0} and j[ℵ_{n}] ∈ M[K], cf(h(n)) = ℵ_{0} in M[K] and h(n) ∈ j(ℵ_{n}) = ℵ_{n}^{M[K]}. Hence h ∈ ∏_{n∈𝔞} ℵ_{n}^{M[K]}. So it suffices to show that h is an exact upper bound of ⟨f_{i}^{j} : i < α⟩ in M[K].

To see that h is an upper bound, let i < α and we show f_{i}^{j} <_{𝔞}^{∗} h. Since j[ℵ_{ω+1}] is cofinal in α, fix β < ℵ_{ω+1} such that i < j(β). Then f_{i}^{j} <_{𝔞}^{∗} f_{j(β)}^{j} = j(f_{β}). For each n ∈ 𝔞, j(f_{β})(n) = j(f_{β}(n)) ∈ j[ℵ_{n}], so j(f_{β})(n) < h(n). Thus f_{j(β)}^{j} <_{𝔞} h, and therefore f_{i}^{j} <_{𝔞}^{∗} h.

Now let g be a function in M[K] such that g <_{𝔞} h. Then for all n ∈ 𝔞, g(n) < h(n) = sup(j[ℵ_{n}]), so we can fix γ_{n} < ℵ_{n} such that g(n) < j(γ_{n}). Let g_{0} be the function in ∏_{n∈𝔞} ℵ_{n} defined by letting g_{0}(n) = γ_{n} for all n ∈ 𝔞. Then for all n ∈ 𝔞, g(n) < j(g_{0}(n)).

The function g_{0} is a countable subset of ℵ_{ω} in M[K] = M[G ∗ H ∗ I ∗ J], and ℵ_{ω} has cardinality ℵ_{1} in the model M[G ∗ H]. Since Coll(ℵ_{1}, (κ^{+ω+1})^{V}) ∗ ℙ_{κ+2,j(κ)} does not add reals, and thus not countable subsets of ℵ_{1}, g_{0} is in M[G ∗ H]. Now ℙ satisfies the bounding property by Proposition 4.2. So in M[G] there is a function g_{1} in ∏_{n∈𝔞} ℵ_{n} such that g_{0}(n) < g_{1}(n) for all n ∈ 𝔞. Since ⟨f_{i} : i < ℵ_{ω+1}⟩ is a scale in M[G], we can find β < ℵ_{ω+1} such that g_{1} <_{𝔞}^{∗} f_{β}. Fix n such that for all k ≥ n in 𝔞, g_{1}(k) < f_{β}(k).

For all k ∈ 𝔞 greater than or equal to n,

g(k) < j(g_{0}(k)) < j(g_{1}(k)) < j(f_{β}(k)) = f_{j(β)}^{j}(k).

So g <_{𝔞}^{∗} f_{j(β)}^{j} and j(β) < α. This proves that h is an exact upper bound as desired.

Remark: Several models have been constructed previously which satisfy the failure of the existence of a good scale on ℵ_{ω}. Foreman and Magidor [6] showed that the Chang’s conjecture (ℵ_{ω+1}, ℵ_{ω}) ↠ (ℵ_{1}, ℵ_{0}) implies that there is no good scale on ℵ_{ω}. Magidor [1] showed that the same conclusion follows from Martin’s Maximum. Finally, Cummings, Foreman, and Magidor [5] proved that a model of Shelah [9] in which the approachability property AP_{ℵ_{ω}} fails also satisfies that there is no good scale.

7. The Non-Compactness of Square

We now prove the main result of the paper, and produce a model in which □_{ℵ_{n}} holds for all 1 ≤ n < ω, but there is no good scale on ℵ_{ω}.

First we recall the forcing poset from [3] which forces that the square principle □_{ℵ_{n}} holds for all 1 ≤ n < ω. This poset is a forcing iteration of length ω with full support,

⟨ℙ_{n}, ℚ̇_{m} : n ≤ ω, m < ω⟩

where for each n < ω, ℙ_{n} forces that ℚ̇_{n} is the standard forcing poset for adding a □_{ℵ_{n+1}}-sequence with initial segments. For each n < ω, ℚ̇_{n} is forced to be ℵ_{1}- closed and (ℵ_{n+1}) + 1-strategically closed. Standard arguments then show that ℙ_{ω} is ℵ_{1}-closed, and preserves all cardinals and cofinalities. Also, if GCH holds in the ground model, then GCH holds after forcing with ℙ_{ω}. In what follows we will write ℙ(ω) for the iteration ℙ_{ω} just described. Note that since ℙ(ω) is ℵ_{1}-closed, it preserves stationary subsets of ℵ_{1}.

We will use the following version of the semiproper forcing iteration theorem [12].

Theorem 7.1. Let ⟨ℙ_{i}, ℚ̇_{j} : i ≤ α, j < α⟩ be a revised countable support forcing iteration, where α is a limit ordinal. Assume that for all i < α, ℙ_{i} forces that ℚ̇_{i} is semiproper, and there is n < ω such that ℙ_{i+n} ⊩ |ℙ_{i}| ≤ ℵ_{1}. Then ℙ_{α} is semiproper. Moreover, if α is a Mahlo cardinal and for all i < α, |ℙ_{i}| < α, then ℙ_{α} is α-c.c.

We will also use Lemma 3 from [7], which states that a certain kind of forcing iteration forces that for any forcing poset ℚ, ℚ is semiproper iff ℚ preserves stationary subsets of ℵ_{1}.

Let V be a model satisfying GCH in which κ is a supercompact cardinal. We define in V a revised countable support forcing iteration

⟨ℙ_{i}, ℚ̇_{j} : i ≤ κ, j < κ⟩.

Fix a Laver function l : κ → V_{κ}.

Let ℙ_{0} be the trivial forcing. If δ ≤ κ is a limit ordinal and ℙ_{i} is defined for all i < δ, let ℙ_{δ} be the revised countable support limit of ⟨ℙ_{i} : i < δ⟩.

Assume that ℙ_{α} is defined. The definition of ℚ̇_{α} splits into several cases.

Case A: α is a Mahlo cardinal, and for all β < α, |ℙ_{β}| < α.

Case B: not Case A.

If Case B holds, let ℚ̇_{α} be a ℙ_{α}-name for the collapse Coll(ℵ_{1}, |ℙ_{α}|^{V}). Let ℙ_{α+1} = ℙ_{α} ∗ ℚ̇_{α}.

Assume Case A holds. Then ℙ_{α} is α-c.c. and collapses α to become ℵ_{2}.

There are three subcases.

Case A(1): l(α) is a ℙ_{α}-name for a semiproper forcing poset of the form

ℙ(ω) ∗ ℙ ∗ Coll(ℵ_{1}, (α^{+ω+1})^{V}).

Case A(2): l(α) is a ℙ_{α}-name for a forcing poset of the form Coll(ℵ_{1}, μ), where μ is a regular cardinal larger than α.

Case A(3): Neither A(1) nor A(2).

In Case A(1) and A(2), we let ℚ̇_{α} be the name l(α). In Case A(3), we let ℚ̇_{α} be a ℙ_{α}-name for the collapse Coll(ℵ_{1}, ℵ_{2}).

This completes the definition of ℙ_{κ}. Note that ℙ_{κ} is semiproper, κ-c.c., and collapses κ to become ℵ_{2}. By Lemma 3 of [7], ℙ_{κ} forces that for every forcing poset ℚ, ℚ is semiproper iff ℚ preserves stationary subsets of ℵ_{1}. Since ℙ_{κ} forces that the forcing poset

ℙ(ω) ∗ ℙ ∗ Coll(ℵ_{1}, (κ^{+ω+1})^{V})

is a three-step iteration of forcing posets each of which preserves stationary subsets of ℵ_{1}, it forces that this poset is semiproper.

Fix an elementary embedding j : V → M in the ground model with critical point κ such that M^{κ^{+ω+2}} ⊆ M and j(l)(κ) is a ℙ_{κ}-name for ℙ(ω) ∗ ℙ ∗ Coll(ℵ_{1}, (κ^{+ω+1})^{V}). Then for any generic filter G_{κ} for ℙ_{κ}, M[G_{κ}]^{κ^{+ω+2}} ⊆ M[G_{κ}]. So M[G_{κ}] has enough closure to satisfy that the poset ℙ(ω)∗ℙ∗Coll(ℵ_{1}, (κ^{+ω+1})^{V}) is semiproper. Therefore by the definition of the forcing iteration, j(ℙ_{κ}) factors as

j(ℙ_{κ}) = ℙ_{κ} ∗ ℙ(ω) ∗ ℙ ∗ Coll(ℵ_{1}, (κ^{+ω+1})^{V}) ∗ ℙ_{κ+1,j(κ)}.

Let G_{κ} ∗G(ω) be a generic filter for ℙ_{κ} ∗ℙ(ω) over V, and let W = V [G_{κ} ∗G(ω)]. Then in W, κ = ℵ_{2} and □_{ℵ_{n}} holds for all 1 ≤ n < ω. We will show that in the model W, there is no good scale on ℵ_{ω}.

Note that in W, κ^{+n} = ℵ_{n+2} for n < ω and κ^{+i} = ℵ_{i} for i ≥ ω. We will abbreviate ℵ_{i}^{W} with ℵ_{i}.

The first step is to generically extend the elementary embedding. This is done exactly as in [3]. Let H ∗ I ∗ J be a generic filter for

ℙ ∗ Coll(ℵ_{1}, (κ^{+ω+1})^{V}) ∗ ℙ_{κ+1,j(κ)}

over the model V [G_{κ}][G(ω)]. Let K = G_{κ} ∗G(ω)∗H ∗I ∗J. Since j[G_{κ}] = G_{κ} ⊆ K, in the model V [K] we can extend j to

j : V [G_{κ}] → M[K]

such that j(G_{κ}) = K.

Note that in M[G_{κ}][G(ω)][H], ℵ_{n} has cofinality ℵ_{0} for all 2 ≤ n < ω. Thus cf(ℵ_{n}) = ℵ_{0} in M[K]. Also cf(ℵ_{ω+1}) = ℵ_{1} in M[K].

As usual, M[G_{κ}]^{κ^{+ω+1}} ⊆ M[G_{κ}]. Also H(κ^{+ω+1})^{V [G_{κ}]} = H(κ^{+ω+1})^{V} [G_{κ}], and j(ȧ^{G_{κ}}) = j(ȧ)^{K} for any ℙ_{κ}-name ȧ in V. Since j ↾ H(κ^{+ω+1})^{V} is in M and K ∈ M[K], j ↾ H(κ^{+ω+1})^{V [G_{κ}]} is in M[K]. In particular, j ↾ G(ω) is in M[K].

In M[K], j[G(ω)] is a set of conditions in j(ℙ(ω)). We define a lower bound q of this set by inductively defining q ↾ n for all n < ω. So let n < ω be given, and assume that q ↾ n is defined and is a lower bound of {j(p) ↾ n : p ∈ G(ω)}.

We claim that

q ↾ n ⊩ ⋃{dom(z(n)) : z ∈ j[G(ω)]} = sup(j[ℵ_{n+2}]).

Consider a condition s in G(ω). Then by genericity we can find t ≤ s in G(ω) such that t ↾ n decides the value of dom(s(n)), which is some ordinal α < ℵ_{n+2}. Then j(t) ↾ n forces that the domain of j(s)(n) is equal to j(α), and j(α) < sup(j[ℵ_{n+2}]). Conversely if γ < ℵ_{n+2}, then by genericity there is some t in G(ω) such that t ↾ n forces that the domain of t(n) is at least γ. Then j(t) ↾ n forces that the domain of j(t)(n) is at least j(γ). Since q ↾ n ≤ j(t) ↾ n for all t ∈ G(ω), this proves the claim.

Note that q ↾ n forces that any two conditions in {j(p)(n) : p ∈ G(ω)} are compatible, and hence one is an initial segment of the other by the definition of the ordering on j(ℚ̇_{n}). It follows that q ↾ n forces that ⋃{j(p)(n) : p ∈ G(ω)} is a function with domain sup(j[ℵ_{n+2}]). We can find a name q(n) which is forced to extend this function by attaching to the ordinal sup(j[ℵ_{n+2}]) a set of order type ω which is cofinal in sup(j[ℵ_{n+2}]). Then q(n) is forced to be a condition, since the coherence requirement is trivially satisfied as this set has no limit points. Define q(n + 1) = (q ↾ n) q(n).

Now let L be a generic filter on j(ℙ(ω)) over V [K] which contains q. Since q is a lower bound of j[G(ω)], j[G(ω)] ⊆ L. Hence we can extend j in the model V [K][L] to

j : V [G_{κ}][G(ω)] → M[K][L].

such that j(G_{κ} ∗ G(ω)) = K ∗ L.

We work in the model W = V [G_{κ} ∗ G(ω)] and show that there is no good scale on ℵ_{ω}. Let 𝔞 be an infinite subset of ω \ {0, 1}, and suppose that ⟨f_{i} : i < ℵ_{ω+1}⟩ is a scale in ∏_{n∈𝔞} ℵ_{n}. Let C be a club subset of ℵ_{ω+1}. We show that there exists an ordinal in C ∩ cof(ℵ_{1}) which is not good for the scale.

Let j(⟨f_{i} : i < ℵ_{ω+1}⟩) = ⟨f_{i}^{j} : i < j(ℵ_{ω+1})⟩. By the elementarity of j, it suffices to show that in M[K][L] there is an ordinal in j(C) ∩ cof(ℵ_{1}) which is not good for the scale ⟨f_{i}^{j} : i < j(ℵ_{ω+1})⟩.

Let α = sup(j[ℵ_{ω+1}]). We claim that α is in j(C)∩cof(ℵ_{1}). First note that ℵ_{ω+1} has size and cofinality ℵ_{1} in the model M[G_{κ}][G(ω)][H][I]. As ℙ_{κ+1,j(κ)} ∗j(ℙ(ω)) is semiproper over this model, ℵ_{ω+1} still has size and cofinality ℵ_{1} in M[K][L]. Since j ↾ ℵ_{ω+1} is in M, M[K][L] models that α has cofinality ℵ_{1}. Finally, as j[C] is a cofinal subset of α contained in j(C), α is a limit point of j(C) and hence is in j(C).

In the model M[G_{κ}][G(ω)][H][I], choose a set X cofinal in ℵ_{ω+1} with order type ℵ_{1}. Also for each 2 ≤ n < ω, choose a set S_{n} cofinal in ℵ_{n} with order type ω. Since ℙ satisfies the bounding property and Coll(ℵ_{1}, ℵ_{ω+1}) is ℵ_{1}-closed, every function in ∏_{n∈𝔞} S_{n} in M[G_{κ}][G(ω)][H][I] is bounded by some function in the scale ⟨f_{i} : i < ℵ_{ω+1}⟩. Conversely, every function in the scale is clearly bounded by some function in ∏_{n∈𝔞} S_{n}.

It follows that in M[G_{κ}][G(ω)][H][I] there exists an increasing sequence ⟨β_{i} : i < ℵ_{1}⟩ of ordinals in X and a sequence ⟨g_{i} : i < ℵ_{1}⟩ of functions in ∏_{n∈𝔞} S_{n} such that for all i < ℵ_{1},

f_{β_{i}} <_{𝔞} g_{i} <_{𝔞}^{∗} f_{β_{i+1}}.

Suppose for a contradiction that the ordinal α is good in M[K][L]. Since {j(β_{i}) : i < ℵ_{1}} is cofinal in α, by Proposition 1.2 there is a set D ⊆ ℵ_{1} of size ℵ_{1} and some n < ω such that for all k ≥ n in 𝔞, ⟨f_{j(β_{i})}^{j}(k) : i ∈ D⟩ is strictly increasing. Now for each i ∈ D, f_{j(β_{i})}^{j}(k) = j(f_{β_{i}}(k)). Hence by elementarity, ⟨f_{β_{i}}(k) : i ∈ D⟩ is strictly increasing.

We define for each k ≥ n in 𝔞 an ordinal γ_{k} in S_{k}. For each i ∈ D, let γ_{k}^{i} be the least ordinal in S_{k} such that f_{β_{i}}(k) < γ_{k}^{i}. Then for i < j in D, f_{β_{i}}(k) < f_{β_{j}}(k), and therefore γ_{k}^{i} ≤ γ_{k}^{j}. But S_{k} has order type ω, so the sequence ⟨γ_{k}^{i} : i ∈ D⟩ is eventually constant. Let γ_{k} be the eventual constant value, and let i_{k} < ℵ_{1} be such that γ_{k}^{i} = γ_{k} for all i > i_{k} in D.

Let i^{∗} = sup({i_{k} + 1 : k ≥ n, k ∈ 𝔞}). Fix i < j in D larger than i^{∗}. Then

f_{β_{i}} <_{𝔞} g_{i} <_{𝔞}^{∗} f_{β_{j}}.

Choose some k ≥ 2 in 𝔞 so that

f_{β_{i}}(k) < g_{i}(k) < f_{β_{j}}(k).

Recall that g_{i} is in ∏_{n∈𝔞} S_{n}. Since γ_{k} is the least ordinal in S_{k} such that f_{β_{i}}(k) < γ_{k}, it follows that γ_{k} ≤ g_{i}(k). But also f_{β_{j}}(k) < γ_{k}, and hence f_{β_{j}}(k) < g_{i}(k), which is a contradiction. So indeed α is not good.

Comment: As in [3], the model W satisfies that every stationary subset of ℵ_{ω+1} ∩ cof(ℵ_{0}) reflects to an ordinal in ℵ_{ω+1} ∩ cof(ℵ_{1}).

References

[1] J. Cummings. Notes on singular cardinal combinatorics. Notre Dame J. Formal Logic, 46(3):251–282, 2005.

[2] J. Cummings, M. Foreman, and M. Magidor. Squares, scales, and stationary reflection. J. Math. Log., 1(1):35–98, 2001.

[3] J. Cummings, M. Foreman, and M. Magidor. The non-compactness of square. J. Symbolic Logic, 68(2):637–643, 2003.

[4] J. Cummings, M. Foreman, and M. Magidor. Canonical structure in the universe of set theory. I. Ann. Pure Appl. Logic, 129(1-3):211–243, 2004.

[5] J. Cummings, M. Foreman, and M. Magidor. Canonical structure in the universe of set theory. II. Ann. Pure Appl. Logic, 142(1-3):55–75, 2006.

[6] M. Foreman and M. Magidor. A very weak square principle. J. Symbolic Logic, 62(1):175–196, 1997.

[7] M. Foreman, M. Magidor, and S. Shelah. Martin’s maximum, saturated ideals, and nonregular ultrafilters. I. Ann. of Math. (2), 127(2):1–47, 1988.

[8] M. Gitik and S. Shelah. On the 𝕀-condition. Israel J. Math, 48(2-3):148–158, 1984.

[9] A. Hajnal, I. Juász, and S. Shelah. Splitting strongly almost disjoint families. Trans. Amer. Math. Soc., 295(1):369–387, 1986.

[10] K. Namba. Independence proof of (ω, ω_{α})-distributivity law in complete Boolean algebras. Comment. Math. Univ. St. Paul., 19:1–12, 1970.

[11] S. Shelah. Cardinal Arithmetic, volume 29 of Oxford Logic Guides. The Clarendon Press, Oxford University Press, New York, second edition, 1994.

[12] S. Shelah. Proper and Improper Forcing. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, second edition, 1998.

Department of Mathematics, University of North Texas

E-mail address: jkrueger@unt.edu