THE NON-COMPACTNESS OF SQUARE.

JAMES CUMMINGS, MATTHEW FOREMAN, AND MENACHEM MAGIDOR

1. Introduction

This note gives proves two theorems. The first is that it is consistent to have □_{ω_{n}} for every n, but not have □_{ℵ_{ω}}. This is done by carefully collapsing a supercompact cardinal and adding square sequences to each ω_{n}. The crux of the proof is that in the resulting model every stationary subset of ℵ_{ω+1} ∩cof(ω) reflects to an ordinal of cofinality ω_{1}, that is to say it has stationary intersection with such an ordinal.

This result contrasts with compactness properties of square shown in [3]. In that paper it is shown that if one has square at every ω_{n}, then there is a square type sequence on the points of cofinality ω_{k}, k > 1 in ℵ_{ω+1}. In particular at points of cofinality greater than ω_{1} there is a strongly non-reflecting stationary set of points of countable cofinality.

The second result answers a question of Džamonja, by showing that there can be no squarelike sequence above a supercompact cardinal, where “squarelike” means that one replaces the requirement that the cofinal sets be closed and unbounded by the requirement that they be stationary at all points of uncountable cofinality.

2. Some Lemmas

In this section we define a forcing notion and show some lemmas. The forcing notion is a standard style of Namba forcing and the lemmas are standard. We prove them here for the benefit of the reader.

Let n → (n_{0}, n_{1}) be a bijection from ω to ω × (ω\{0, 1}). We say that a tree is standard for our partial ordering iff

(1) T ⊂ (ℵ_{ω})^{<ω}

(2) For all σ ∈ T and n ∈ dom(σ), σ(n) ∈ ω_{n_{1}}

(3) There is a maximal σ ∈ T (called the stem of T) such that for all τ ∈ T, σ ⊂ τ or τ ⊂ σ.

(4) For all τ extending the stem σ, if n ∈ dom(τ), n ⊃ dom(σ) then |{α : τ ↾ n^{⌢}α ∈ T }| = ω_{n_{1}}

The first author was partially supported by DMS-0070549. The second author was partially supported by DMS-0101155. The second and third authors were partially supported by BSF grant 445860-21136.

2 JAMES CUMMINGS, MATTHEW FOREMAN, AND MENACHEM MAGIDOR

Our partial ordering P will consist of standard trees with the ordering of inclusion.

It is easy to verify the following facts:

• |P| = | ∏_{n∈ω} 2^{ω_{n}}|.

• If G ⊂ P is generic then ⋃ ⋂ G is a function f : ω → ℵ_{ω} such that for all n, f(n) ∈ ω_{n_{1}} and for all m, {f(n) : n_{1} = m} is cofinal in ω_{m}.

We will say that a sequence of standard trees ⟨T_{n} : n ∈ ω⟩ is a fusion sequence if T_{n+1} ⊆ T_{n} for all n, all the trees have the same stem σ, and for every n and every τ ∈ T_{n} with dom(τ) ≤ dom(σ) + n + 1 we have τ ∈ T_{n+1}. It is easy to see that if ⟨T_{n} : n ∈ ω⟩ is a fusion sequence then there is a standard tree T_{∞} with stem σ such that T_{∞} ≤ T_{n} for all n < ω.

We will need the following lemmas. The first is a standard forcing exercise:

Lemma 1. Let τ̇ be a P-term for an ordinal less than ℵ_{ω+1} and T be a standard tree. Then there is a standard tree T′ ⊂ T with the same stem as T and an n such that for all σ ∈ T′ of length n, if we let T′_{σ} = {σ′ : σ′ ⊃ σ or σ′ ⊂ σ}, then for some β < ℵ_{ω+1}, T′_{σ} ⊩ τ̇ = β.

From this we can see:

Lemma 2. Suppose that G ⊂ P is generic. Then:

(1) V [G] ⊧ cf(ℵ_{ω+1}^{V}) ≥ ω_{1}.

(2) If S ⊂ ℵ_{ω+1}∩cof(ω) is a stationary set in V, then S is stationary in V [G].

⊢ Let ⟨α̇_{n} : n ∈ ω⟩ be a term for an ω-sequence of ordinals and let T be an arbitrary standard tree. Repeatedly applying Lemma 1, we can build a standard tree T′ ⊂ T with the same stem as T with the property that for all infinite branches b through T′, there is a sequence of natural numbers ⟨m_{n} : n ∈ ω⟩ and ordinals ⟨β_{n} : n ∈ ω⟩ such that if T′_{b↾m_{n}} = {σ ∈ T′ : b ↾ m_{n} ⊂ σ or σ ⊂ b ↾ m_{n}} then

T′_{b↾m_{n}} ⊩ α̇_{n} = β̌_{n}

This property is absolute between V and the generic extension determined by P, so if f = ⋃ ⋂ G we know that there is such a sequence for f in V [G].

Fix now any collection of terms {α̇_{n} : n ∈ ω}. Define T′ as in the last paragraph and let A = {β : there is a σ ∈ T′ and some n, T′_{σ} ⊩ α̇_{n} = β̌}. Then |A| = ℵ_{ω} and T′ ⊩ {α̇_{n} : n ∈ ω} ⊂ A. Thus we have established claim 1 of the lemma.

In particular, we have shown that if α̇ is a term for an ordinal less than ℵ_{ω+1} and T is a condition then there is a T′ ⊂ T with the same stem as T and a γ < ℵ_{ω+1} with the property that T′ ⊩ α̇ < γ.

Let Ċ be an arbitrary P-term for a closed unbounded subset of ℵ_{ω+1}, and S be a stationary subset of ℵ_{ω+1} ∩ cof(ω). Let N ≺ ⟨H(λ), ∈, <_{λ} , Ċ, P⟩ for some large λ be such that |N| = ℵ_{ω} and γ^{∗} = N ∩ ℵ_{ω+1} ∈ S, where <_{λ} is a well-ordering of H_{λ}. Choose a sequence ⟨γ_{n} : n ∈ ω⟩ converging to γ^{∗}. Let α̇_{n} ∈ N be a P-term for an element of Ċ above γ_{n}.

Using this one can inductively build a fusion sequence of trees ⟨T_{n} : n ∈ ω⟩ such that:

• T_{n} ∈ N

• T_{n} ⊩ α̇_{n} = δ_{n}.

There is a T^{∗} ∈ P such that for all n, T^{∗} ≤ T_{n}. Clearly T^{∗} ⊩ sup{α̇_{n}} = γ^{∗} and hence T^{∗} ⊩ Ċ ∩ S ≠ ∅. ⊣

We do not know whether forcing with P can ever preserve ℵ_{ω+1}^{V}. Results of Shelah imply that if □_{ℵ_{ω}}^{∗} holds in V then P will collapse ℵ_{ω+1}^{V}. More information about this problem may be found in [1].

In the same spirit as the previous lemma, but with a different proof we have:

Lemma 3. Forcing with P preserves stationary subsets of ℵ_{1}.

There are many places to see the proof of this, for example we cite Namba’s original articles [9, 8], Shelah’s book [10] or [5].

We will also need some standard partial orderings for forcing square. There are several variations on these; basic properties are discussed in [2] among other places. In this paper we will force with partial orderings that add square by initial segments; this partial ordering for adding a square sequence was introduced by Jensen.

Definition 4. Let κ be an uncountable cardinal. Let ℙ(κ) be the partial ordering consisting of all sequences ⟨C_{α} : α ≤ δ⟩ for some δ ∈ κ^{+}, that have the following properties:

(1) The set C_{α} is closed and unbounded in α.

(2) The order type of C_{α} is less than or equal κ.

(3) If β is a limit point of C_{α}, then C_{β} = C_{α} ∩ β.

The ordering of ℙ(κ) is extension.

Proofs of the following facts can be found in [2]:

• The poset ℙ(κ) is countably closed. (In particular forcing with ℙ(κ) preserves stationary subsets of cof(ω) ∩ λ for any regular cardinal λ.)

• The poset ℙ(κ) has the property that player II wins the game G_{κ^{+}}^{II} as defined in [4].

• For all δ ∈ κ^{+} the collection of conditions p ∈ ℙ(κ) which are of length at least δ is a dense set.

• Forcing with ℙ(κ) adds a □_{κ} sequence, without collapsing car-

dinals, or adding any κ-sequences. In our situation we will be adding square successively to each ω_{n}, n ≥

1. With this in mind we define a sequence of partial orderings Sq(n) for n ∈ ω + 1\{0} by induction on n.

(1) Let Sq(1) = ℙ(ω_{1}).

(2) Sq(n + 1) = Sq(n) ∗ ℙ(ω_{n+1}).

Finally we define

(3) Sq(ω) to be the inverse limit of the sequence ⟨Sq(n) : n ∈ ω⟩. It is then standard to show that the iteration Sq(ω) is ω-closed, and

for each n, Sq(ω) ∼ Sq(n) ∗ ℝ, where ℝ is ω_{n+1}-strategically closed. In particular, forcing with Sq(ω) preserves cardinals and cofinalities, and all stationary subets of λ ∩ cof(ω) for any uncountable regular λ. Moreover Sq(ω) adds □_{ω_{n}} sequences for n ≥ 1 and a generic object H ⊂ Sq(ω) corresponds exactly to a sequence ⟨H_{n} : n ∈ ω⟩, with the property that H_{n+1} ⊂ ℙ(ω_{n+1}) is generic over V [⟨H_{k} : 1 ≤ k ≤ n⟩] for each n.

3. The Forcing

In this section we prove:

Theorem 5. Suppose that κ is a supercompact cardinal. Then there is a forcing extension in which:

(1) κ = ℵ_{2}

(2) For all n ∈ ω\{0}, □_{ω_{n}} holds.

(3) Every stationary subset of ℵ_{ω+1} ∩ cof(ω) reflects to an α ∈ cof(ω_{1}).

⊢ We follow the outline of the proof of the consistency of Martin’s Maximum. (Or simply use the Martin’s Maximum partial ordering directly, with slightly more argument.) By the main result of [6], we fix a function f : κ → κ such that for every ordinal γ there is a γ^{+}- supercompact embedding j : V → M with critical point κ, such that j(f)(κ) = γ.

A typical component of our forcing is the partial ordering Sq(ω) ∗ P (as defined in the previous section.) Since Sq(ω) is ω-closed and P preserves stationary subsets of ω_{1} we see that this two-step iteration preserves stationary subsets of ω_{1}.

We define a semi-proper iteration ℙ = ⟨(ℙ_{α}, ℚ_{α}) : α < κ⟩ of length κ with revised countable support so that:

(1) ℚ_{α} = 1 unless α is inaccessible and for all generic G_{α} ⊂ ℝ_{α}, α

is the ω_{2} of V [G_{α}]. In this case

(2) If f(α) = 0 then if

V [G_{α}] ⊧ Sq(ω) ∗ P is semiproper

we let ℚ_{α} = Sq(ω) ∗ P ∗ Col(ω_{1}, ℵ_{ω+1}^{V [G_{α}]}), otherwise ℚ_{α} = 1.

(3) If f(α) > 0 then we let ℚ_{α} = Col(ω_{1}, f(α)).

Exactly as in the proof of the consistency of Martin’s Maximum that appeared in [7], we see that the revised countable support limit at stage κ, ℝ_{κ}, is κ chain condition, makes κ into ω_{2} and is semi-proper. (In particular, in V [G_{κ}], κ^{+ω+1} = ℵ_{ω+1}.) Moreover, if G_{κ} is generic over ℝ_{κ} then by Lemma 3 from [7]:

V [G_{κ}] ⊧ for all partial orderings ℙ, ℙ is semi-proper iff ℙ preserves stationary subsets of ω_{1}

Let G_{κ} ⊂ ℝ_{κ} be generic and let H ⊂ Sq(ω)^{V [G_{κ}]} be generic. We claim that V [G_{κ}, H] satisfies the conclusion of the theorem. Since Sq(ω) adds a square sequence to each ℵ_{n} and preserves ℵ_{ω+1} the only part of the conclusion left to verify is that every stationary subset of ℵ_{ω+1} reflects.

Claim 6. Fix a κ^{+ω+1}-supercompact embedding j : V → M such that j(f)(κ) = 0. Then:

(1) There are generic G′, H′ such that the embedding j can be extended to a ĵ: V [G_{κ}, H] → M[G′, H′].

(2) If S ⊂ κ^{+ω+1} ∩ cof(ω) is stationary in V [G_{κ}, H], then S is stationary in V [G′, H′].

⊢ Standard large cardinal technology shows that if we take any V - generic G′ ⊂ j(ℝ) extending G_{κ}, then j extends to a generic embedding:

j_{0} : V [G_{κ}] → M[G′].

Since in V [G_{κ}] semi-properness is equivalent to preserving stationary subsets of ω_{1}, the partial ordering Sq(ω) ∗ P is semi-proper. Since j is κ^{ω+1}-supercompact, this is absolute between V [G_{κ}] and M[G_{κ}]. Hence, in M[G_{κ}], the partial ordering ℚ_{κ} is taken to be Sq(ω) ∗ P ∗ Col(ω_{1}, κ^{+ω+1}). In particular G′ = G_{κ}∗H∗P ∗C∗G^{∗}, where H∗P ∗C ⊂ Sq(ω) ∗ P ∗ Col(ω_{1}, κ^{+ω+1}) is generic over V [G_{κ}].

We must build a master condition m ∈ j(Sq(ω))^{M[G_{κ}]} with the property that for all q ∈ H, m ≤ j(q). Then for any M[G′]-generic H′ with m ∈ H′ there is an elementary

ĵ: V [G_{κ}, H] → M[G′, H′].

For each n ∈ ω\{0} let γ_{n} = sup(j“κ^{+n}). In M[G′] each γ_{n} has cofinality ω. Choose a cofinal ω-sequence C_{n} in each γ_{n}. If H ∼ ⟨H_{n} : n ∈ ω⟩ where H_{n} is the generic square sequence through κ^{+n}, then we let m_{1} be the canonical term for ⋃ j“H_{1} ∪ {(γ_{1}, C_{1})}. It is then easy to verify by induction on n that:

• If (m_{1}, m_{2}, … m_{n}) ∈ H′_{1} ∗ H′_{2} ∗ … H′_{n} and H′_{1} ∗ H′_{2} ∗ … H′_{n} are generic for Sq(n) then j extends to an elementary embedding

j_{n} : V [G_{κ} ∗ H_{1} ∗ H_{2} ∗ … H_{n}] → M[G′ ∗ H′_{1} ∗ H′_{2} ∗ … H′_{n}].

• m_{n+1} =_{def} j_{n}“H_{n+1}∪{(γ_{n+1}, C_{n+1})} is a condition in the partial ordering for adding □_{j(κ^{+n})} that is stronger than each element of j“H_{n+1}.

(In the jargon, “we take the union of the image of the generic object at κ_{n} and put the sequence C_{n} on top”.)

If we let m be the sequence of M[G′] terms for ⟨m_{n} : n ∈ ω⟩, then we see that m is the desired master condition.

We now show the second part of Claim 6. Let S ⊂ κ^{+ω+1} ∩cof(ω) be stationary. Then by Lemma 2, S is stationary in V [G_{κ} ∗ H ∗ P]. Since S ⊂ cof(ω), S remains stationary after one forces with Col(ω_{1}, κ^{+ω+1}) and hence is stationary in V [G ∗ H ∗ P ∗ C]. Since κ^{+ω+1} is of cofinality ω_{1} in this model, if D ⊂ κ^{+ω+1} is any closed unbounded set of order type ω_{1}, preserving the stationarity of S is equivalent to preserving the stationary subset T of ω_{1} determined by D ∩ S. The forcing that produces G^{∗} over V [G ∗ H ∗ P ∗ C] is semi-proper and hence preserves the stationarity of T. Finally Sq(ω) is ω-closed over V [G′] = V [G_{κ} ∗ H ∗ P ∗ C ∗ G^{∗}] and hence preserves the stationarity of T. This proves the claim. ⊣

Claim 7. Let γ = sup j“κ^{+ω+1} and let S ⊂ κ^{+ω+1}∩cof(ω) be stationary in V [G, H]. Then j“S is stationary in γ in the model V [G′, H′].

⊢ Since γ has cofinality ω_{1} in the model and the forcing for producing G′ over the model V [G_{κ} ∗ H ∗ P] is semi-proper it suffices to show that j“S is stationary in γ in the model V [G_{κ} ∗ H ∗ P]. We work in the model V [G_{κ} ∗ H].

Let Ċ be a term for a closed unbounded subset of γ lying in V [G_{κ} ∗ H]^{P}. Consider Ċ′ = {α ∈ κ^{+ω+1} : j(α) ∈ C}. We now argue that V [G_{κ} ∗ H]^{P} ⊧ C′ ∩ cof(ω)^{V [G_{κ},H]} is closed unbounded relative to (κ^{+ω+1})^{V} ∩ cof(ω)^{V [G_{κ},H]}. This suffices, since forcing with P preserves stationary subsets of κ^{+ω+1} ∩ cof(ω). In particular, there is a δ ∈ C′ ∩ S. Then j(δ) ∈ j(S) ∩ j“λ.

The argument that V [G_{κ} ∗ H]^{P} ⊧ C′ ∩ cof(ω)^{V [G_{κ},H]} is closed unbounded relative to (κ^{+ω+1})^{V} ∩ cof(ω)^{V [G_{κ},H]} follows the argument for Lemma 2. That C′ ∩ cof(ω)^{V [G_{κ},H]} is relatively closed is clear, since j is continuous at points of cofinality ω in V [G_{κ} ∗ H].

We show that C′ ∩ cof(ω)^{V [G_{κ},H]} is unbounded. For each ordinal α ∈ κ^{+ω+1}, let β̇(α) be the P term for the least ordinal β in κ^{+ω+1} such that there is a δ ∈ Ċ with j(α) ≤ δ ≤ β.

Let N ≺ ⟨H(λ), ∈, <_{λ}, Ċ, P⟩ for some large λ be such that |N| = ℵ_{ω} and γ^{∗} = N ∩ ℵ_{ω+1} ∈ cof(ω). Choose a sequence ⟨γ_{n} : n ∈ ω⟩ converging to γ^{∗}. As in Lemma 2, we can build a fusion sequence of trees ⟨T_{n} : n ∈ ω⟩ such that:

• T_{n} ∈ N

• T_{n} ⊩ β̇(γ_{n}) = δ_{n}.

• There is a T^{∗} ∈ P such that for all n, T^{∗} ≤ T_{n}. Then T^{∗} ⊩ sup{β̇(γ_{n})} = γ^{∗}. Again, since j is continuous at points

of cofinality ω, j(γ)^{∗} = sup j“γ^{∗}. In particular, T^{∗} ⊩ j(γ^{∗}) ∈ Ċ. Since N was arbitrary, this shows the claim ⊣

The claim suffices to prove the theorem, since j“S ⊂ j(S), and thus M[G′, H′] ⊧ j(S) reflects to γ. By elementarity, we see that V [G_{κ}, H] ⊧ ∃γ(S reflects to γ).

This finishes the claim and the proof of Theorem 5. ⊣

4. A remark about squares above a supercompact cardinal

In this section we answer a question of Džamonja by showing the following:

Theorem 8. Let κ be a supercompact cardinal and λ ≥ κ a regular cardinal. Then there is no sequence of sets ⟨S_{α} : α < λ^{+}⟩ such that:

(1) S_{α} ⊂ α is unbounded in α

(2) o.t. S_{α} ≤ λ

(3) If cf(α) > ω then S_{α} is stationary in α

(4) If β ∈ S_{α} is a limit point of S_{α}, then S_{β} = S_{α} ∩ β.

We note that this result slightly generalizes a result of Solovay that there can be no square sequence above a supercompact cardinal. ⊢ (Theorem 8) Suppose that there is such a sequence, ⟨S_{α} : α < λ^{+}⟩. We follow the outline of Solovay’s argument with a slight additional twist.

Let j : V ⟶ M be an elementary embedding witnessing that κ is λ^{+}-supercompact; that is to say j has critical point κ, j(κ) > λ^{+} and ^{λ^{+}}M ⊆ M. We set γ = sup j“λ^{+}. Since j“λ^{+} ∈ M it is easy to see that γ has cofinality λ^{+} both in V and in the inner model M; since λ^{+} < j(κ) < j(λ^{+}) and j(λ^{+}) is regular in M, it follows that γ < j(λ^{+}).

Let ⟨T_{β} : β < j(λ^{+})⟩ = j(⟨S_{α} : α < λ^{+}⟩), and let U = T_{γ}. By elementarity, in M the set U is a stationary subset of γ.

Let U^{∗} be the set of those μ < γ such that μ ∈ U, and both U ∩ μ and j“λ^{+} ∩ μ are unbounded in μ. It is clear that in M the set U^{∗} is stationary in γ. We note that if μ ∈ U^{∗} then μ is a point in T_{γ} which is a limit point of T_{γ}, so by elementarity and the coherence property (4) we have U ∩ μ = T_{μ}.

Claim 9. U^{∗} ∩ j“λ^{+} is unbounded in γ.

⊢ We work in M. Given α < γ, let β ∈ U^{∗} be least with β > α. We claim that β has cofinality ω. Suppose for a contradiction that β has uncountable cofinality. Then since U ∩ β = T_{β}, it follows by elementarity that U ∩ β is stationary in β. Since U and j“λ^{+} are both unbounded in β, we may find β^{∗} ∈ U^{∗} with α < β^{∗} < β. This contradicts the minimal choice of β.

The elementary embedding j is continuous at points of cofinality less than κ, and since j“λ^{+} is unbounded in β it follows that β ∈ j“λ^{+}. So U^{∗} ∩ j“λ^{+} is unbounded in γ, as claimed. ⊣

Let X be the unbounded subset of λ^{+} consisting of those η such that j(η) ∈ U^{∗}. Find η ∈ X such that the order type of X ∩ η is greater than λ. Since j(η) ∈ U^{∗}, we see that U ∩ j(η) = T_{j(η)} = j(S_{η}).

For every ζ ∈ X∩η, j(ζ) ∈ U∩j(η) and so by elementarity ζ ∈ S_{η}. So X∩η ⊆ S_{η} and thus the order type of S_{η} is greater than λ, contradicting our assumptions about the sequence ⟨S_{α} : α < λ^{+}⟩. ⊣

Remark: Martin Zeman has pointed out that if κ is subcompact then this suffices to prove Theorem 8 in the case when λ = κ. The cardinal κ is subcompact if for every A ⊆ H_{κ^{+}} there exist δ < κ, a ⊆ H_{δ^{+}} and an elementary π : (H_{δ^{+}}, ∈, a) → (H_{κ^{+}}, ∈, A) such that the critical point of π is δ and π(δ) = κ.

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Department of Mathematical Sciences, CMU, Pittsburgh PA 15213, USA

Department of Mathematics, University of California at Irvine, Irvine CA 92697, USA

Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem 91904, Israel