AN EQUICONSISTENCY RESULT ON PARTIAL SQUARES

JOHN KRUEGER AND ERNEST SCHIMMERLING

Abstract. We prove that the following two statements are equiconsistent: there exists a greatly Mahlo cardinal; there exists a regular uncountable cardinal κ such that no stationary subset of κ^{+} ∩ cof(κ) carries a partial square.

A famous theorem in set theory is the result that the failure of the square principle □_{κ}, for a regular uncountable cardinal κ, is equiconsistent with a Mahlo cardinal. Solovay proved that if λ > κ is a Mahlo cardinal, then in any generic extension by the Lévy collapse Coll(κ, <λ), λ = κ^{+} and ¬□_{κ}. On the other hand, Jensen [6] proved that ¬□_{κ} implies that κ^{+} is Mahlo in L.

Partial square sequences were introduced by Shelah as a weakening of the square principle. Let ν < κ^{+} be regular, and let A ⊆ κ^{+} ∩ cof(ν). We say that A carries a partial square if there exists a sequence ⟨c_{α} : α ∈ A⟩ satisfying: (a) c_{α} is a club subset of α; (b) ot(c_{α}) = ν; (c) if γ is a limit point of c_{α} and c_{β}, then c_{α}∩γ = c_{β} ∩γ.

A significant difference between the square principle and partial squares is that, while □_{κ} is independent of ZFC, the existence of partial squares is provable in ZFC. For example, Shelah [12] proved that if κ is a regular uncountable cardinal, then κ^{+} ∩ cof(< κ) splits into κ many pairwise disjoint subsets each of which carries a partial square.

Another difference is that, unlike the square principle □_{κ}, partial squares on κ^{+} ∩ cof(κ) are consistent with κ being supercompact. For example, suppose κ is indestructibly supercompact. Then the forcing poset for adding a partial square sequence on the set κ^{+} ∩ cof(κ) with initial segments is κ-directed closed and thus preserves the supercompactness of κ. Also if V = L[E] is an extender model, then for any regular uncountable cardinal κ, κ^{+} ∩ cof(κ) carries a partial square. On the other hand, if κ is κ^{+}-supercompact (or even subcompact), then □_{κ} fails ([14], [3]).

Magidor [8] constructed a model of set theory which satisfies a strong form of stationary set reflection, using a weakly compact cardinal. In this model there is no stationary subset of ω_{2} ∩ cof(ω_{1}) which carries a partial square. In [7] we define a forcing iteration which destroys the stationarity of any subset of κ^{+} ∩ cof(κ) which carries a partial square, using a weakly compact cardinal. In this paper we show that the same forcing iteration works assuming only a greatly Mahlo cardinal.

We also obtain the lower bound, by showing that if no stationary subset of κ^{+} ∩ cof(κ) carries a partial square, then κ^{+} is greatly Mahlo in L. Thus we prove the following equiconsistency result.

Theorem 1. The statement that there exists a regular uncountable cardinal κ such that no stationary subset of κ^{+} ∩ cof(κ) carries a partial square is equiconsistent with a greatly Mahlo cardinal.

Date: December 2010.

2010 Mathematical Subject Classification: 03E35, 03E45.

Key words and phrases: partial square sequence, greatly Mahlo cardinal.

We now outline the contents of the paper. In Section 1 we review the basic facts about canonical functions and the Mahlo hierarchy which will be used in the paper. In Section 2 we prove that for a regular uncountable cardinal κ, if κ^{+} is not greatly Mahlo in L, then there exists a stationary subset of κ^{+} ∩ cof(κ) which carries a partial square. The remainder of the paper shows how to force a model in which there are no partial squares using a greatly Mahlo cardinal. In Section 3 we review some facts about elementary substructures and the Lévy collapse. In Section 4 we define the idea of a partial square killing forcing iteration, and prove an absoluteness result. Section 5 describes some of the basic properties of this kind of iteration. Section 6 shows that after Lévy collapsing a greatly Mahlo cardinal, a partial square killing forcing iteration is distributive. In Section 7 we put the pieces together to prove the consistency result. Section 8 describes a related equiconsistency result.

Our notation is standard unless noted otherwise. We assume that the reader has some familiarity with L, but not necessarily with fine structure. We also assume that the reader understands the basics of forcing, iterated forcing, proper forcing, and the Lévy collapse, and is familiar with stationary subsets of P_{λ}(X) = {a ⊆ X : |a| < λ}, where λ is regular and uncountable ([1], [5], [13]).

1. Canonical Functions and the Mahlo Hierarchy

We work out the details about canonical functions which will be used in the paper, and define the Mahlo hierarchy. Most of this material is folklore; also see [2] and [4].

Let λ be a regular uncountable cardinal. For functions f, g : λ → λ, we write f =^{∗} g if there is a club C ⊆ λ such that f(α) = g(α) for all α in C. Clearly =^{∗} is an equivalence relation.

Definition 1.1. Let λ be a regular uncountable cardinal, and let ν < λ^{+}. A function f : λ → λ is said to be a canonical function on λ of rank ν if there exists a surjective function g : λ → ν and a club C ⊆ λ such that

∀α ∈ C (f(α) = ot(g[α])).

If f_{1} is a canonical function on λ of rank ν and f_{1} =^{∗} f_{2}, then f_{2} is a canonical function on λ of rank ν. Also note that if g, h : λ → ν are surjective functions, then there are club many α such that g[α] = h[α]. It follows that if f_{1} and f_{2} are both canonical functions of rank ν, then f_{1} =^{∗} f_{2}. Hence the set of all canonical functions of rank ν is an equivalence class modulo =^{∗}.

Notation 1.2. Let λ be a regular uncountable cardinal and let ν < λ^{+}. Then f_{ν}^{λ} denotes the class of all canonical functions on λ of rank ν.

Clearly f_{ν}^{λ} is non-empty for all ν < λ^{+}. Note that f ∈ f_{0}^{λ} iff f(α) = 0 for club many α.

Lemma 1.3. Let λ be a regular uncountable cardinal, let ν_{0} < ν_{1} < λ^{+}, and let g : λ → ν_{1} be a surjection. Fix f_{ν_{0}} ∈ f_{ν_{0}}^{λ}. Then there is a club C ⊆ λ such that

∀α ∈ C (f_{ν_{0}}(α) = ot(g[α] ∩ ν_{0})).

Proof. If ν_{0} = 0, then the statement is clear. So assume ν_{0} > 0. Define a surjective function h : λ → ν_{0} by

h(α) = \{ \begin{matrix}g(α) & if g(α) ∈ ν_{0} \\ 0 & otherwise. \end{matrix}

Fix γ < λ such that g(γ) = 0. Fix a club C such that

∀α ∈ C (α > γ ∧ f_{ν_{0}}(α) = ot(h[α])).

Then for all α in C, h[α] = g[α]∩ν_{0}, and hence f_{ν_{0}}(α) = ot(h[α]) = ot(g[α]∩ν_{0}). □

In particular, suppose ν_{0} < ν_{1} < λ^{+} and g : λ → ν_{1} is a surjection. If we define f : λ → λ by letting f(α) = ot(g[α] ∩ ν_{0}) for all α, then f ∈ f_{ν_{0}}^{λ}.

It is not hard to show using the last lemma that if f_{γ} and f_{γ+1} are in f_{γ}^{λ} and f_{γ+1}^{λ} respectively, then for club many α, f_{γ+1}(α) = f_{γ}(α) + 1.

If δ < λ^{+} is a limit ordinal and g : λ → δ is a surjection, then there is a club of α such that for all β < α, there is γ < α such that g(β) < g(γ). It follows that if f_{δ} ∈ f_{δ}^{λ}, then f_{δ}(α) is a limit ordinal for club many α.

Given functions f, g : λ → λ, we write f <^{∗} g to mean that there exists a club C ⊆ λ such that f(α) < g(α) for all α in C. Define f ≤^{∗} g similarly. If f_{1} =^{∗} f_{2} and g_{1} =^{∗} g_{2}, then f_{1} <^{∗} g_{1} (f_{1} ≤^{∗} g_{1}) implies f_{2} <^{∗} g_{2} (f_{2} ≤^{∗} g_{2}).

Proposition 1.4. Let λ be a regular uncountable cardinal, and let ν < λ^{+}. Let f_{ν} be in f_{ν}^{λ}. Then f_{β} <^{∗} f_{ν} for all β < ν and f_{β} ∈ f_{β}^{λ}.

Proof. Fix a surjective function g : λ → ν and a club E such that for all α in E, f_{ν}(α) = ot(g[α]). Let β < ν and f_{β} ∈ f_{β}^{λ}. Fix ζ < λ such that g(ζ) = β. By Lemma 1.3, fix a club C ⊆ λ such that for all α in C, α > ζ and f_{β}(α) = ot(g[α]∩β). Then for all α in C ∩ E, f_{β}(α) = ot(g[α] ∩ β) < ot(g[α] ∩ β) + 1 = ot((g[α] ∩ β) ∪ {β}) ≤ ot(g[α]) = f_{ν}(α). □

The next proposition is another basic result about canonical functions; we omit the proof since we will not use this fact.

Proposition 1.5. Let λ be a regular uncountable cardinal, and let ν < λ^{+}. Let f_{β} be in f_{β}^{λ} for all β ≤ ν. If f : λ → λ and f_{β} <^{∗} f for all β < ν, then f_{ν} ≤^{∗} f.

It is straightforward to prove by induction using the proposition that for all ν < λ, if f_{ν} ∈ f_{ν}^{λ}, then f_{ν}(α) = ν for club many α. Also if f_{λ} ∈ f_{λ}^{λ}, then f(α) = α for club many α.

Next we relate the canonical functions on different cardinals.

Lemma 1.6. Let λ be a regular uncountable cardinal, let ν < λ^{+}, and let g : λ → ν be a surjection. For β ≤ ν fix f_{β} ∈ f_{β}^{λ}. For each α < λ, define a function h_{α} on α by letting h_{α}(i) = f_{g(i)}(α) for all i < α. Then there are club many α in λ such that h_{α} is a surjection of α onto f_{ν}(α).

Proof. Fix a club C ⊆ λ such that for all α in C, α is a limit ordinal and f_{ν}(α) = ot(g[α]). By Lemma 1.3, for each β < ν we can fix a club C_{β} ⊆ λ such that

∀α ∈ C_{β} (f_{β}(α) = ot(g[α] ∩ β)).

Let D = C ∩ △{C_{g(i)} : i < λ}.

Let α be in D. We prove that h_{α} is a surjection of α onto f_{ν}(α). First we show that h_{α}(i) ∈ f_{ν}(α) for all i < α. If i < α, then α ∈ C_{g(i)}, and hence

h_{α}(i) = f_{g(i)}(α) = ot(g[α] ∩ g(i)) < ot(g[α]) = f_{ν}(α).

Let π : g[α] → ot(g[α]) be the transitive collapsing map of g[α]. Then for all i < α, π(g(i)) = ot(g[α] ∩ g(i)). To show that h_{α} is a surjection, consider γ in f_{ν}(α) = ot(g[α]). Then π^{−1}(γ) ∈ g[α]. Fix i < α such that π^{−1}(γ) = g(i); then π(g(i)) = γ. As α is in C_{g(i)},

h_{α}(i) = f_{g(i)}(α) = ot(g[α] ∩ g(i)) = π(g(i)) = γ.

Thus h_{α} is a surjection. □

We will prove that a stronger version of the next proposition holds in L, in Lemma 2.3 of the next section.

Proposition 1.7. Let λ be a regular uncountable cardinal. Let ν < λ^{+}, and let f_{ν} be in f_{ν}^{λ}. Then there is a club C ⊆ λ such that for all α in C, if α is regular and uncountable, then f_{ν}(α) < α^{+} and f_{ν} ↾ α is in f_{f_{ν}(α)}^{α}.

Now we define the Mahlo hierarchy. The definition is by recursion.

Definition 1.8. Let λ be a regular uncountable cardinal. Assume that for all regular uncountable cardinals α < λ and ν ≤ α^{+}, we have defined what it means for α to be ν-Mahlo. Let β ≤ λ^{+}. We say that λ is β-Mahlo if:

(1) (β = 0) λ is strongly inaccessible;

(2) (β = γ + 1) the set {α ∈ λ : α is f_{γ}^{λ}(α)-Mahlo} is stationary in λ, for some (any) f_{γ}^{λ} in f_{γ}^{λ};

(3) (β is a limit ordinal) λ is γ-Mahlo for all γ < β.

Definition 1.9. Let λ be a regular uncountable cardinal. Then λ is greatly Mahlo if λ is λ^{+}-Mahlo.

Suppose λ is strongly inaccessible but not greatly Mahlo. Then the least ν such that λ is not ν-Mahlo must be a successor ordinal.

It can be easily proven by induction that if λ is ν-Mahlo, then λ is β-Mahlo for all β < ν.

2. No Partial Squares Implies a Greatly Mahlo Cardinal

We prove that for a regular uncountable cardinal κ, if no stationary subset of κ^{+} ∩ cof(κ) carries a partial square, then κ^{+} is greatly Mahlo in L.

Theorem 2.1. Let κ be a regular uncountable cardinal, and let λ = κ^{+}. Assume that λ is not greatly Mahlo in L. Then there is a stationary subset of κ^{+} ∩ cof(κ) which carries a partial square.

This theorem can be thought of as an extension of Jensen’s theorem that if κ^{+} is not a Mahlo cardinal in L, then □_{κ} holds. We will use the following result.

Theorem 2.2 (Jensen [6]). Let κ be a regular uncountable cardinal, and let λ = κ^{+}. Let W ⊆ λ be a set satisfying that for every limit ordinal η < λ with κ ≤ η < λ, the set

W ∩ [κ · η, κ · (η + 1))

codes a well-ordering of κ of order type η. Let

X = {α ∈ λ : κ · α = α ∧ L[W ∩ α] ⊧ |α| ≤ κ}.

Then there exists a sequence ⟨C_{α} : α ∈ X ∩ lim(X)⟩ satisfying:

(1) C_{α} is in L[W ∩ α];

(2) C_{α} is a club subset of α;

(3) ot(C_{α}) ≤ κ;

(4) C_{α} ⊆ X;

(5) if γ is a limit point of C_{α}, then C_{α} ∩ γ = C_{γ}.

Note that the existence of such a set W in V is an easy consequence of λ being the successor of κ. Also note that for all α < λ, if κ · α = α and

L ⊧ α is singular,

then

L[W ∩ α] ⊧ |α| ≤ κ.

This is true because L[W ∩ α] ⊧ α ≤ κ^{+} by the choice of W; so if α is singular in L[W ∩ α], then α is less than κ^{+} in L[W ∩ α].

Let us begin the proof of Theorem 2.1. Let κ be a regular uncountable cardinal, and let λ = κ^{+}. Assume that λ is not greatly Mahlo in L.

The proof splits into two separate cases. Let

Y = {α ∈ λ ∩ cof(κ) : L ⊧ α is singular}.

We consider first the case that Y is a stationary subset of λ in V.

Let W ⊆ λ be a set as described in the assumptions of Theorem 2.2, and let

X = {α ∈ λ : κ · α = α ∧ L[W ∩ α] ⊧ |α| ≤ κ}.

Let D be the club set of α < λ such that κ · α = α. Then Y ∩ D ⊆ X, as noted above. By Theorem 2.2, fix a sequence ⟨C_{α} : α ∈ X ∩ lim(X)⟩ satisfying:

(1) C_{α} is a club subset of α of order type at most κ;

(2) C_{α} ⊆ X;

(3) if γ is a limit point of C_{α}, then C_{α} ∩ γ = C_{γ}. Consider the restriction of this sequence to Y ∩ D:

⟨C_{α} : α ∈ Y ∩ D⟩.

Then for all α in Y ∩ D, since cf(α) = κ, C_{α} is a club subset of α of order type κ. If α and β are in Y ∩ D and γ is a limit point of C_{α} and C_{β}, then

C_{α} ∩ γ = C_{γ} = C_{β} ∩ γ.

Thus the sequence ⟨C_{α} : α ∈ Y ∩ D⟩ is a partial square sequence on a stationary subset of κ^{+} ∩ cof(κ), and we are done.

From now on we will assume that the set

{α ∈ λ ∩ cof(κ) : L ⊧ α is singular}

is non-stationary in V. Clearly then λ is a strongly inaccessible cardinal in L, so the set of limit cardinals of L below λ is club in λ. Fix a club set C_{0} ⊆ λ in V such that

∀α ∈ C_{0} ∩ cof(κ) (L ⊧ α is strongly inaccessible).

Let us fix canonical functions in L. For each ordinal α ≤ λ which is regular and uncountable in L, and each ordinal δ < (α^{+})^{L}, let g_{δ}^{α} be the L-least surjection of α onto δ. Define f_{δ}^{α} : α → α by

f_{δ}^{α}(γ) = ot(g_{δ}^{α}[γ]).

Then L models that f_{δ}^{α} is a canonical function on α of rank δ. Note that for each γ in α, f_{δ}^{α}(γ) = ot(g_{δ}^{α}[γ]) < (γ^{+})^{L}.

Lemma 2.3. Let β < (λ^{+})^{L}. Then there is a club E ⊆ λ in L such that for all α in E, if α is regular and uncountable in L, then

f_{β}^{λ} ↾ α = f_{f_{β}^{λ}(α)}^{α}.

Proof. Let N be the structure L_{(λ^{+})^{L}}. For each α < λ, let

N_{α} = Sk^{N}(α ∪ {β}).

Let E be the club set of α < λ such that N_{α} ∩ λ = α.

Suppose α is in E, and α is regular and uncountable in L. By the Condensation Lemma, let π : N_{α} → L_{ρ} be the transitive collapsing map, for some ordinal ρ. Since N_{α} ∩ λ = α, π(λ) = α.

We claim that π(β) = f_{β}^{λ}(α). Since N ⊧ |β| ≤ λ, by the L-minimality of g_{β}^{λ}, this function is in N_{α}. It follows by elementarity that

N_{α} ∩ β = g_{β}^{λ}[N_{α} ∩ λ] = g_{β}^{λ}[α].

Hence

π(β) = {π(ξ) : ξ ∈ N_{α} ∩ β} = {π(g_{β}^{λ}(i)) : i < α}.

Since π is an isomorphism, this ordinal is the order type of the set {g_{β}^{λ}(i) : i < α}, which is equal to ot(g_{β}^{λ}[α]) = f_{β}^{λ}(α).

By elementarity, π(g_{β}^{λ}) is the L-least surjection of π(λ) = α onto π(β) = f_{β}^{λ}(α). Therefore π(g_{β}^{λ}) = g_{f_{β}^{λ}(α)}^{α}. It follows that π(f_{β}^{λ}) = f_{f_{β}^{λ}(α)}^{α}. But f_{β}^{λ} ∩ N_{α} = f_{β}^{λ} ↾ α,

and π is the identity on this set. So π(f_{β}^{λ}) = f_{β}^{λ} ↾ α. □

Lemma 2.4. Let β < (λ^{+})^{L}. Let k : λ → λ be a function. Suppose A ⊆ λ is a set which is stationary in V such that

∀α ∈ A (k(α) < f_{β}^{λ}(α)).

Then there is ζ < β and a set B ⊆ A which is stationary in V such that

∀α ∈ B (k(α) = f_{ζ}^{λ}(α)).

Proof. For each α < λ, let h_{α} : α → On be the function

h_{α}(i) = f_{g_{β}^{λ}(i)}^{λ}(α).

By Lemma 1.6, there is a club E ⊆ λ in L such that for all α in E, h_{α} is a surjection of α onto f_{β}^{λ}(α). Define a regressive function m : A ∩ E → λ as follows. Let α be in A ∩ E. Then k(α) < f_{β}^{λ}(α). Since h_{α} is a surjection of α onto f_{β}^{λ}(α), let m(α) be the least ordinal less than α such that h_{α}(m(α)) = k(α). By Fodor’s Lemma, fix i < λ and a set B ⊆ A ∩ C which is stationary in V such that for all α in B, m(α) = i. Let ζ = g_{β}^{λ}(i). Then for all α in B,

k(α) = h_{α}(m(α)) = h_{α}(i) = f_{g_{β}^{λ}(i)}^{λ}(α) = f_{ζ}^{λ}(α).

□

Since λ is not greatly Mahlo in L, let δ < (λ^{+})^{L} be the least ordinal such that λ is not δ + 1-Mahlo in L. Then the set

{α ∈ λ : L ⊧ α is f_{δ}^{λ}(α)-Mahlo}

is non-stationary in L. Let C_{1} be a club subset of λ in L such that

∀α ∈ C_{1} (L ⊧ α is not f_{δ}^{λ}(α)-Mahlo).

Let C = C_{0} ∩ C_{1}. Then for all α in C ∩ cof(κ), α is strongly inaccessible in L, but not f_{δ}^{λ}(α)-Mahlo in L.

For each α in C ∩ cof(κ), let δ(α) < (α^{+})^{L} be the least ordinal such that α not δ(α) + 1-Mahlo in L. Then clearly

∀α ∈ C ∩ cof(κ) (δ(α) < f_{δ}^{λ}(α)).

By Lemma 2.4, there is a set A ⊆ C ∩cof(κ) which is stationary in V and an ordinal ζ < δ such that

∀α ∈ A (δ(α) = f_{ζ}^{λ}(α)).

For each β < δ, define

A_{β} = {α ∈ C ∩ cof(κ) : δ(α) = f_{β}^{λ}(α)}.

We just noted that for some ζ < δ, A_{ζ} is stationary in V. Let β be the least ordinal less than δ such that A_{β} is stationary. By Lemma 2.3, fix a club set E_{β} ⊆ λ in L such that for all α in E_{β}, if α is regular and uncountable in L, then f_{β}^{λ} ↾ α = f_{f_{β}^{λ}(α)}^{α}.

Lemma 2.5. There exists a sequence ⟨d_{α} : α ∈ A_{β} ∩ E_{β}⟩ satisfying:

(1) d_{α} is a club subset of α;

(2) ∀γ ∈ d_{α} (L ⊧ γ is not f_{β}^{λ}(γ)-Mahlo);

(3) if γ is a limit point of both d_{α_{1}} and d_{α_{2}}, then d_{α_{1}} ∩ γ = d_{α_{2}} ∩ γ.

Proof. For each limit ordinal α < λ, let b_{α} be the L-least closed and unbounded subset of α satisfying:

L ⊧ ∀γ ∈ b_{α} (γ is not f_{β}^{λ}(γ)-Mahlo),

if such a set exists. Otherwise let b_{α} be the empty set. Also let φ_{α} be the least ordinal larger than α and f_{β}^{λ}(α) satisfying:

(1) L_{φ_{α}} ⊧ ZF C^{−};

(2) L_{φ_{α}} ⊧ |f_{β}^{λ}(α)| ≤ α;
(3) b_{α} ∈ L_{φ_{α}}.

Note that φ_{α} < (α^{+})^{L}.

Fix α in A_{β} ∩ E_{β}, and we will define d_{α}. Since δ(α) = f_{β}^{λ}(α), α is f_{β}^{λ}(α)-Mahlo in L, but not f_{β}^{λ}(α) + 1-Mahlo in L. Therefore the set

{γ ∈ α : L ⊧ γ is f_{f_{β}^{λ}(α)}^{α}(γ)-Mahlo}

is non-stationary in L. But f_{f_{β}^{λ}(α)}^{α} = f_{β}^{λ} ↾ α. So the set

{γ ∈ α : L ⊧ γ is f_{β}^{λ}(γ)-Mahlo}

is non-stationary in L. It follows that b_{α} is a club subset of α which is disjoint from this set.

Recall that g_{f_{β}^{λ}(α)}^{α} is the L-least surjection of α onto f_{β}^{λ}(α). By (2) in the

definition of φ_{α}, it follows that g_{f_{β}^{λ}(α)}^{α} is in L_{φ_{α}}. Therefore by (1), f_{f_{β}^{λ}(α)}^{α} is in

L_{φ_{α}}, since this function is defined in an absolute way in ZF C^{−} from g_{f_{β}^{λ}(α)}^{α}. But

f_{f_{β}^{λ}(α)}^{α} = f_{β}^{λ} ↾ α. So f_{β}^{λ} ↾ α is in L_{φ_{α}}.

Since α is strongly inaccessible in L, L_{φ_{α}} correctly computes the L-degree of Mahloness of cardinals below α. Therefore L_{φ_{α}} models that b_{α} is the L-least club subset of α such that

∀γ ∈ b_{α} (γ is not f_{β}^{λ}(γ)-Mahlo).

For each γ < α, let N_{γ} be the Skolem hull

Sk^{L_{φα}} (γ ∪ {f_{β}^{λ}(α)}).

Since α is definable in L_{φ_{α}} as the largest cardinal, α is in N_{γ}. Also since f_{β}^{λ} ↾ α and b_{α} are definable in L_{φ_{α}} from f_{β}^{λ}(α), they are in N_{γ}.

Now define

d_{α} = {γ ∈ α : N_{γ} ∩ α = γ}.

Clearly d_{α} is a club subset of α in L.

Consider γ in d_{α}. Since b_{α} ∈ N_{γ}, N_{γ} ∩ α = γ ∈ b_{α} by elementarity. It follows that γ is not f_{β}^{λ}(γ)-Mahlo in L. As the cardinal successor function below α is definable in L_{φ_{α}}, γ is a strong limit cardinal in L.

Since N_{γ} ≺ L_{φ_{α}}, by the Condensation Lemma the transitive collapse of N_{γ} is equal to L_{ψ} for some ordinal ψ. Let

π_{γ} : N_{γ} → L_{ψ}

be the transitive collapsing map of N_{γ}. As N_{γ} ∩ α = γ,

π_{γ}(α) = γ.

We claim that

π_{γ}(f_{β}^{λ}(α)) = f_{β}^{λ}(γ).

Since f_{β}^{λ} ↾ α = f_{f_{β}^{λ}(α)}^{α}, it suffices to show that

π_{γ}(f_{β}^{λ}(α)) = f_{f_{β}^{λ}(α)}^{α}(γ).

As g_{f_{β}^{λ}(α)}^{α} is in N_{γ} by elementarity, and g_{f_{β}^{λ}(α)}^{α} : α → f_{β}^{λ}(α) is a surjection,

N_{γ} ∩ f_{β}^{λ}(α) = g_{f_{β}^{λ}(α)}^{α}[N_{γ} ∩ α] = g_{f_{β}^{λ}(α)}^{α}[γ].

Hence

π_{γ}(f_{β}^{λ}(α)) = {π_{γ}(ξ) : ξ ∈ N_{γ} ∩ f_{β}^{λ}(α)} = {π_{γ}(g_{f_{β}^{λ}(α)}^{α}(i)) : i < γ}.

But π_{γ} is order-preserving, so this ordinal is the order type of the set g_{f_{β}^{λ}(α)}^{α}[γ],

which equals f_{f_{β}^{λ}(α)}^{α}(γ).

Now N_{γ} ∩ (f_{β}^{λ} ↾ α) = f_{β}^{λ} ↾ γ, and π_{γ} is the identity on this set. So

π_{γ}(f_{β}^{λ} ↾ α) = f_{β}^{λ} ↾ γ.

By elementarity, it follows that L_{ψ} models that π_{γ}(b_{α}) is the L-least closed and unbounded subset of γ such that for all ξ in π_{γ}(b_{α}), ξ is not f_{β}^{λ}(ξ)-Mahlo. Since γ is a strong limit cardinal, L_{ψ} correctly computes the L-degree of Mahloness of cardinals below γ. It follows that

π_{γ}(b_{α}) = b_{γ}.

To summarize, we have proven that π_{γ}(α) = γ, π_{γ}(f_{β}^{λ}(α)) = f_{β}^{λ}(γ), and π_{γ}(b_{α}) = b_{γ}. Also, ψ is the least ordinal larger than γ and f_{β}^{λ}(γ) satisfying:

(1) L_{ψ} ⊧ ZF C^{−};

(2) L_{ψ} ⊧ |f_{β}^{λ}(γ)| ≤ γ;
(3) b_{γ} ∈ L_{ψ}.

By elementarity, clearly L_{ψ} satisfies these properties. If χ < ψ also satisfies these properties, then π^{−1}(L_{χ}) = L_{π^{−1}(χ)} would contradict the minimality of φ_{α}. It follows that ψ = φ_{γ}.

Since N_{γ} ≺ L_{φ_{α}}, for all ξ < γ, N_{ξ} = Sk^{N_{γ}} (ξ ∪ {f_{β}^{λ}(α)}). Since π_{γ} : N_{γ} → L_{φ_{γ}} is an isomorphism, it follows that

π_{γ}[N_{ξ}] = Sk^{L_{φγ}} (ξ ∪ {f_{β}^{λ}(γ)});

We now claim that for all ξ < γ,

ξ ∈ d_{α} ⟺ Sk^{L_{φγ}} (ξ ∪ {f_{β}^{λ}(γ)}) ∩ γ = ξ;

in other words,

N_{ξ} ∩ α = ξ ⟺ π_{γ}[N_{ξ}] ∩ γ = ξ.

Suppose N_{ξ} ∩ α = ξ. Since ξ ⊆ N_{ξ} and π_{γ} ↾ γ is the identity, ξ ⊆ π_{γ}[N_{ξ}] ∩ γ. Let ρ be in π_{γ}[N_{ξ}] ∩ γ. Then ρ = π_{γ}(ρ′) for some ρ′ ∈ N_{ξ}. Then π_{γ}(ρ′) = ρ < γ = π_{γ}(α), so ρ′ < α. Therefore ρ′ ∈ N_{ξ} ∩ α = ξ. So ρ′ ∈ ξ. But then ρ = π_{γ}(ρ′) = ρ′, so ρ ∈ ξ. Conversely, assume π_{γ}[N_{ξ}] ∩ γ = ξ. Clearly ξ ⊆ N_{ξ} ∩ α. Let ρ be in N_{ξ} ∩ α. Then π_{γ}(ρ) ∈ π_{γ}(α) = γ, so π_{γ}(ρ) ∈ π_{γ}[N_{ξ}] ∩ γ = ξ = π_{γ}(ξ). So ρ ∈ ξ.

We have proven that for all α in A_{β} ∩ E_{β}, for any γ in d_{α},

d_{α} ∩ γ = {ξ ∈ γ : Sk^{L_{φγ}} (ξ ∪ {f_{β}^{λ}(γ)}) ∩ γ = ξ}.

But the set on the right side is independent of α. It follows that if α_{1} and α_{2} are in A_{β} ∩E_{β}, then for any γ which is a limit point of d_{α_{1}} and d_{α_{2}}, d_{α_{1}} ∩γ = d_{α_{2}} ∩γ. □

Suppose that there exists a club set F ⊆ λ such that for each α in the stationary set A_{β} ∩ E_{β} ∩ lim(F),

ot(d_{α} ∩ F) = κ.

Then letting c_{α} = d_{α} ∩ F for all α in A_{β} ∩ E_{β} ∩ lim(F), clearly the sequence

⟨c_{α} : α ∈ A_{β} ∩ E_{β} ∩ lim(F)⟩

is a partial square sequence on a stationary subset of λ ∩ cof(κ). So it suffices to show that there exists such a set F.

Suppose for a contradiction that for every club set F ⊆ λ, there is an ordinal α in A_{β} ∩ E_{β} ∩ lim(F) such that d_{α} ∩ F has order type different from κ. Let

B = {γ ∈ λ ∩ cof(κ) : ∃α ∈ A_{β} ∩ E_{β} (γ ∈ d_{α})}.

We claim that B is stationary in λ.

To prove that B is stationary, let F ⊆ λ be a club. Then by assumption, there is some α in A_{β} ∩ E_{β} ∩ lim(F) such that d_{α} ∩ F has order type different from κ. Now d_{α} and F ∩ α are both club subsets of α, and α has cofinality κ. So d_{α} ∩ F is club in α, and thus has order type at least κ. Therefore the order type of d_{α} ∩ F is greater than κ. Let γ be the κ-th element of d_{α} ∩ F. Then γ has cofinality κ and is in d_{α}, and α is in A_{β} ∩ E_{β}. So γ is in B, and also γ is in F.

We claim that for all γ in B ∩ C, δ(γ) < f_{β}^{λ}(γ). Consider γ in B ∩ C. Fix α in A_{β} ∩ E_{β} such that γ ∈ d_{α}. Then by the choice of d_{α}, γ is not f_{β}^{λ}(γ)-Mahlo in L. But γ is δ(γ)-Mahlo in L. So δ(γ) < f_{β}^{λ}(γ).

Now we will get a contradiction. By Lemma 2.4, there is a set B′ ⊆ B ∩ C which is stationary in V and an ordinal ζ < β such that

∀γ ∈ B′ (δ(γ) = f_{ζ}^{λ}(γ)).

So B′ ⊆ A_{ζ}, and therefore A_{ζ} is stationary. But ζ < β, and β is the minimal ordinal such that A_{β} is stationary.

3. Elementary Substructures and the Lévy Collapse

We now turn towards proving the other direction of the equiconsistency result. We will start with a model in which κ is a regular uncountable cardinal and λ > κ is greatly Mahlo, and then produce a model by forcing in which λ = κ^{+} and there is no stationary subset of κ^{+} ∩ cof(κ) which carries a partial square. The collapse of λ to become κ^{+} will be achieved by using the Lévy collapse Coll(κ, <λ). After forcing with the Lévy collapse, we will iterate forcing to destroy the stationarity of any subset of κ^{+} ∩ cof(κ) which carries a partial square.

In this section we provide some preliminary results which will be used in the forcing proof, which concern elementary substructures and their interaction with the Lévy collapse.

For a set N and a regular cardinal λ, we let λ_{N} denote N ∩ λ. As the notation suggests, we are interested in the case that λ_{N} is an ordinal less than λ.

Notation 3.1. Let λ be a strongly inaccessible cardinal. Define S_{λ} as the set of N in P_{λ}(H(λ^{+})) satisfying:

(1) N ≺ H(λ^{+});

(2) λ_{N} is strongly inaccessible;

(3) |N| = λ_{N};

(4) N^{<λ_{N}} ⊆ N.

Notation 3.2. Let λ be a strongly inaccessible cardinal, and let ν < λ^{+}. Define S_{ν}^{λ} as the set of N in S_{λ} such that there exists a canonical function f_{ν} on λ of rank ν in N such that λ_{N} is f_{ν}(λ_{N})-Mahlo.

Lemma 3.3. Let λ be a strongly inaccessible cardinal, and let ν < λ^{+}. Suppose λ is ν + 1-Mahlo. Then S_{ν}^{λ} is stationary in P_{λ}(H(λ^{+})).

Proof. Let F : H(λ^{+})^{<ω} → H(λ^{+}) be a function. Build a sequence ⟨N_{i} : i < λ⟩ by induction satisfying:

(1) N_{i} ≺ (H(λ^{+}), ∈, ν, F);

(2) |N_{i}| < λ;
(3) P (N_{i}) ⊆ N_{i+1};
(4) if δ < λ is a limit ordinal, then N_{δ} = ⋃{N_{i} : i < δ}.

Let C be the club set of α < λ such that N_{α} ∩ λ = α = |N_{α}|.

By elementarity, fix a surjective function g_{ν} : λ → ν in N_{0}. Let f_{ν} : λ → λ be the function f_{ν}(α) = ot(g_{ν}[α]). Then f_{ν} is in N_{0} by elementarity, and f_{ν} is a canonical function on λ of rank ν. Since λ is ν + 1-Mahlo, fix α in C such that α is f_{ν}(α)-Mahlo. Then N_{α} is in S_{ν}^{λ} and is closed under F. □

The next definition is standard, although it is usually considered in the case when λ is strongly inaccessible. We will also use this idea when λ is a successor cardinal; see Lemma 3.7 below for the context.

Definition 3.4. Let λ be a regular uncountable cardinal. A set N is a λ-model if: (1) (N, ∈) is a model of ZF C^{−};

(2) N is transitive;

(3) |N| = λ;

(4) λ ∈ N;

(5) N^{<λ} ⊆ N.

Lemma 3.5. Let λ be strongly inaccessible, and suppose N is in S_{λ}. Then \overline{N}, the transitive collapse of N, is a λ_{N}-model.

Proof. Let π : N → \overline{N} be the transitive collapsing map. Since N ≺ H(λ^{+}), N is a model of ZF C^{−}. So \overline{N} is a model of ZF C^{−}, as π is an isomorphism. Obviously \overline{N} is transitive, by the definition of the transitive collapse. Also |\overline{N}| = |N| = λ_{N}. As N ∩ λ = λ_{N}, π(λ) = λ_{N} is in \overline{N}. Finally, since N^{<λ_{N}} ⊆ N and N is isomorphic to \overline{N}, \overline{N}^{<λ_{N}} ⊆ \overline{N}. □

Lemma 3.6. Let λ be strongly inaccessible, and let ν < λ^{+}. Let N be in S_{λ}. Suppose f_{ν} ∈ N is a canonical function on λ of rank ν. Let π : N → \overline{N} be the transitive collapsing map. Then π(ν) = f_{ν}(λ_{N}).

Proof. By elementarity, fix a surjection g_{ν} : λ → ν in N and a club C ⊆ λ in N such that for all α in C, f_{ν}(α) = ot(g_{ν}[α]). Since C ∈ N, λ_{N} ∈ C, so f_{ν}(λ_{N}) = ot(g_{ν}[λ_{N}]). As g_{ν} is in N, clearly N ∩ν = g_{ν}[λ_{N}] by elementarity. So π(ν) = {π(α) : α ∈ N ∩ ν} = {π(g_{ν}(i)) : i ∈ λ_{N}} = π[g_{ν}[λ_{N}]]. Hence π is an order preserving bijection between π(ν) and g_{ν}[λ_{N}], and therefore π(ν) = ot(g_{ν}[λ_{N}]) = f_{ν}(λ_{N}). □

Now we turn to the Lévy collapse. Suppose κ is a regular uncountable cardinal, and λ > κ is strongly inaccessible. The Lévy collapse Coll(κ, < λ) is the forcing poset consisting of conditions p which satisfy:

(1) p : κ × λ → λ is a partial function;

(2) |dom(p)| < κ;

(3) p(α, β) < β for all ⟨α, β⟩ in dom(p). Let q ≤ p if q extends p as a function.

We assume that the reader is familiar with the basic theory of the Lévy collapse; see [5] or [8] . For example, Coll(κ, <λ) is κ-closed and λ-c.c., and collapses λ to become κ^{+}. If κ < \overline{λ} < λ and \overline{λ} is strongly inaccessible, then Coll(κ, <λ) factors as

Coll(κ, <\overline{λ}) × Coll(κ, [\overline{λ}, λ)),

where Coll(κ, [\overline{λ}, λ)) is the suborder of Coll(κ, < λ) consisting of conditions p such that dom(p) ⊆ κ × [\overline{λ}, λ).

Note that if N is a λ-model, then the Lévy collapse Coll(κ, < λ) is a member of N.

Lemma 3.7. Let κ be a regular uncountable cardinal, and assume λ > κ is strongly inaccessible. Let N be a λ-model. Suppose G is a generic filter on Coll(κ, < λ). Then in V [G], N[G] is a λ-model. Moreover, N[G] ∩ V = N.

Proof. By basic facts of forcing, since N is a transitive model of ZF C^{−}, the same is true of N[G]. Also N and N[G] have the same cardinality, which is λ, since λ is preserved in V [G]. As λ ∈ N and N ⊆ N[G], λ ∈ N[G]. It remains to show that N[G]^{<λ} ⊆ N[G].

Since λ = κ^{+} in V [G], it suffices to show that N[G]^{κ} ⊆ N[G] in V [G]. Let f : κ → N[G] be a function. Define g : κ → N in V [G] by choosing for each i < κ a name g(i) in N such that g(i)^{G} = f(i). Since G ∈ N[G], it suffices to show that g is in N[G].

Fix a name ġ satisfying that ġ^{G} = g and Coll(κ, <λ) forces that ġ is a function from κ into names in N. For each i < κ, let A_{i} be a maximal antichain contained in the dense set of conditions which decide ġ(i). Since Coll(κ, <λ) is λ-c.c., |A_{i}| < λ. So A_{i} is in N, because N^{<λ} ⊆ N. Therefore the sequence ⟨A_{i} : i < κ⟩ is in N.

Now define a name ḣ in N by letting ⟨p, ȧ⟩ be in ḣ iff for some i < κ, p ∈ A_{i}, and ȧ is the canonical name for a pair ⟨i, x̌⟩, where p ⊩ ġ(i) = x̌. It is easy to check that ḣ^{G} = g, so g is in N[G] as desired.

The statement that N[G] ∩ V = N follows by a standard argument from the fact that Coll(κ, < λ) is λ-c.c. If ȧ is in N and ȧ^{G} ∈ V, then there is a maximal antichain in N consisting of conditions which either force that ȧ is not in V, or otherwise decide the value of ȧ. Then |A| < λ, so A ⊆ N. Therefore every possible value for ȧ is in N by elementarity. In particular, ȧ^{G} ∈ N. □

Lemma 3.8. Let κ be a regular uncountable cardinal, and let λ > κ be strongly inaccessible. Assume that N is in S_{λ} and κ < λ_{N}. Let π : N → \overline{N} be the transitive collapsing map of N. Suppose G is a generic filter on Coll(κ, <λ). Let σ : N[G] → \overline{N[G]} be the transitive collapsing map of N[G]. Then

σ ↾ N = π

and

\overline{N[G]} = \overline{N}[G ↾ λ_{N}].

In particular, \overline{N[G]} is in V [G ↾ λ_{N}] and is a λ_{N}-model.

The first two assertions were proved in Lemma 6.4 of [7], and we will not repeat the proof here. For the last statement, \overline{N} is a λ_{N}-model by Lemma 3.5, so \overline{N}[G ↾ λ_{N}] = \overline{N[G]} is a λ_{N}-model by Lemma 3.7.

4. Absoluteness of Partial Square Killing

In this section we introduce the type of forcing iteration we will use, which we call a partial square killing forcing iteration. We need to give a general definition of this kind of iteration, rather than just constructing one iteration for the consistency proof. The reason is that we need an absoluteness result which states that if a λmodel believes that ℙ is a partial square killing forcing iteration, then ℙ really is such an iteration.

Let λ be a regular uncountable cardinal. Define X_{λ} by

x ∈ X_{λ} ⟺ x is a closed, bounded, non-empty subset of λ.

Define a relation ≤_{λ} on X_{λ} by

q ≤_{λ} p ⟺ q ∩ (max(p) + 1) = p,

that is, q end-extends p.

For an ordinal ν, define X_{λ,ν} by

p ∈ X_{λ,ν} ⟺ p : ν → X_{λ} is a partial function of size less than λ.

Define a relation ≤_{λ,ν} on X_{λ,ν} by

q ≤_{λ,ν} p ⟺ dom(p) ⊆ dom(q) ∧ ∀α ∈ dom(p) (q(α) ≤_{λ} p(α)).

We also fix formulas in LST which describe X_{λ}, ≤_{λ}, X_{λ,ν}, and ≤_{λ,ν}. Let φ_{0}(x, λ) be the formula of LST expressing that x is a nonempty, closed, bounded subset of λ. Let ψ_{0}(x, y, λ) be the formula of LST which asserts that φ_{0}(x, λ) and φ_{0}(y, λ) hold, and y is an end-extension of x. Similarly fix formulas φ_{1}(x, λ, ν) and ψ_{1}(x, y, λ, ν) of LST to describe X_{λ,ν} and ≤_{λ,ν}. It is easy to check that all these expressions are Δ_{0}, except the part of φ_{1} which asserts that |x| < λ, which is Σ_{1}.

Lemma 4.1. Let λ be a regular uncountable cardinal, and let N be a λ-model. (1) X_{λ} ⊆ N;

(2) for all x, x ∈ X_{λ} iff N ⊧ φ_{0}(x, λ);
(3) for all x, y ∈ X_{λ}, y ≤_{λ} x iff N ⊧ ψ_{0}(x, y, λ).

Proof. (1) follows from the fact that N^{<λ} ⊆ N. (2) and (3) hold because φ_{0} and ψ_{0} are Δ_{0}. □

Lemma 4.2. Let λ be a regular uncountable cardinal, and let N be a λ-model. Let ν be an ordinal in N.

(1) X_{λ,ν} ⊆ N;

(2) for all x, x ∈ X_{λ,ν} iff N ⊧ φ_{1}(x, λ, ν);
(3) for all x, y ∈ X_{λ,ν}, y ≤_{λ,ν} x iff N ⊧ ψ_{1}(x, y, λ, ν).

Proof. (1) follows from the fact that N^{<λ} ⊆ N. (2) If N ⊧ φ_{1}(x, λ, ν), then since φ_{1} is Σ_{1}, x is in X_{λ,ν}. If x is in X_{λ,ν}, then x ∈ N by (1). Since N^{<λ} ⊆ N, N ⊧ |x| < λ. The rest of φ_{1} is Δ_{0}, so N ⊧ φ_{1}(x, λ, ν). (3) follows from (1) and (2) and the absoluteness of end-extension. □

If ℚ is a suborder of X_{λ,ν} and ζ < ν, let

ℚ ↾ ζ = {p ↾ ζ : p ∈ ℚ},

ordered by ≤_{λ,ζ}. It is easy to see that if N is a λ-model, ν ∈ N, and ℚ is a suborder of X_{λ,ν} in N, then for all ζ ∈ N ∩ ν, N computes ℚ ↾ ζ correctly.

Lemma 4.3. Let λ be a regular uncountable cardinal, and let N be a λ-model. Let x be a set in N, and let ℙ be a forcing poset in N. Then N models that Ṫ is a nice ℙ-name for a subset of x iff Ṫ really is a nice ℙ-name for a subset of x.

Proof. The set Ṫ being a nice ℙ-name for a subset of x can be expressed as follows: (1) for all a in Ṫ, there is p ∈ ℙ and z ∈ x such that a = ⟨p, ž⟩;

(2) for all z in x and p, q in ℙ, if ⟨p, ž⟩ and ⟨q, ž⟩ are in Ṫ, then p and q are

incompatible. The statement “y = ž” is absolute for transitive models of ZF C^{−}, and the formula above is Δ_{0} in this statement. □

Suppose ℙ is a forcing poset, ȧ_{1}, …, ȧ_{n} are ℙ-names, and φ(x_{1}, …, x_{n}) is a formula which is absolute for transitive models of ZF C^{−}. Let p ∈ ℙ. Then the statement “p ⊩ φ[ȧ_{1}, …, ȧ_{n}]” is absolute between transitive models of ZF C^{−}.

Lemma 4.4. Suppose κ is a regular uncountable cardinal, λ = κ^{+}, and N is a λ-model. Let ℙ be a forcing poset, and let Ṫ be a nice ℙ-name for a subset of λ ∩ cof(κ). Suppose

N ⊧ ( p ⊩ Ṫ carries a partial square ).

Then p forces that Ṫ carries a partial square.

Proof. Fix a sequence ⟨ċ_{α} : α < λ⟩ in N such that N models that p forces: (1) ċ_{α} is non-empty iff α ∈ Ṫ;

(2) if α ∈ Ṫ, then ċ_{α} is a club subset of α of order type κ;

(3) if γ is a limit point of ċ_{α} and ċ_{β}, then ċ_{α} ∩ γ = ċ_{β} ∩ γ. Statements (1), (2), and (3) are absolute for transitive models of ZF C^{−}. So p really does force (1), (2), and (3). □

In the last lemma we are not assuming any cardinal preservation by the forcing poset ℙ. So in general it might not be clear what is meant by saying that p forces that Ṫ carries a partial square. For clarity, what we mean is exactly what is described in the proof.

If ℙ is a suborder of X_{λ,ν} and β < ν, let

ℙ_{β} = ℙ ∩ X_{λ,β},

considered as a suborder of X_{λ,β}. Note that if N is a λ-model and ℙ ∈ N, then for all β ∈ N ∩ ν, N computes ℙ_{β} correctly.

Now we are ready to introduce the idea of a partial square killing forcing iteration.

Definition 4.5. Suppose κ is a regular uncountable cardinal, λ = κ^{+}, and ν is an ordinal. We say that ℙ is a partial square killing forcing iteration on λ of length ν if there exists a sequence ⟨Ṫ_{i} : i < ν⟩ such that:

(1) ℙ is a suborder of X_{λ,ν};

(2) the empty function is in ℙ;

(3) for all β < ν, Ṫ_{β} is a nice ℙ_{β}-name for a subset of λ ∩ cof(κ);

(4) for all β < ν, ℙ_{β} forces that Ṫ_{β} carries a partial square sequence;

(5) for all p, p is in ℙ iff (p is in X_{λ,ν} and for all α in dom(p), p ↾ α ∈ ℙ and p ↾ α ⊩_{ℙ_{α}} p(α) ∩ Ṫ_{α} = ∅).

Fix a formula θ(ℙ, λ, ν) in LST which asserts that ℙ is a partial square killing forcing iteration on λ of length ν. The formula θ(ℙ, λ, ν) is the conjunction of the statements (1)–(5) above, except that we replace any mention of X_{λ,ν} and ≤_{λ,ν} with the formulas φ_{1} and ψ_{1}.

Proposition 4.6. Suppose κ is a regular uncountable cardinal, λ = κ^{+}, and ν is an ordinal. Assume that N is a λ-model and ν ∈ N. If

N ⊧ θ(ℙ, λ, ν),

then ℙ is a partial square killing forcing iteration on λ of length ν.

Proof. Fix in N a sequence ⟨Ṫ_{i} : i < ν⟩ such that N models (the modified versions using φ_{1} and ψ_{1} of) (1)–(5) of Definition 4.5. Lemma 4.2(2,3) implies that (1) holds for ℙ. (2) is immediate. For (3) and (4), let β < ν. Then N computes ℙ_{β} correctly. Also note that since N^{κ} ⊆ N, the set λ ∩ cof(κ) is computed correctly in N. Lemmas 4.3 and 4.4 now imply (3) and (4). (5) follows easily from Lemma 4.2 and the comments preceding Lemma 4.4. □

Note that if ν < λ^{+}, then

H(λ^{+}) ⊧ θ(ℙ, λ, ν)

iff ℙ is a partial square killing forcing iteration on λ of length ν, because the properties described in Definition 4.5 are absolute between H(λ^{+}) and V.

5. Properties of Partial Square Killing

We will now discuss the basic properties of partial square killing forcing iterations. All of these properties are straightforward to prove, except for distributivity, which is handled in the next section using a large cardinal assumption. The proofs in this section will be brief or omitted, since they were already dealt with rigorously in [7]. In any case, the reader should have little difficulty filling in the details, if interested.

All of the properties developed in this section hold for any forcing iteration which kills subsets of λ ∩ cof(κ), that is, for any ℙ satisfying Definition 4.5(1,2,3,5).

When we say that a forcing poset ℚ is λ-distributive, we mean <λ-distributive, that is, any family of fewer than λ many dense open subsets of ℚ has dense intersection. A set H ⊆ ℚ is said to be N-generic for ℚ if whenever D ∈ N is a dense subset of ℚ, then H ∩ D ∩ N ≠ ∅.

Let κ be a regular uncountable cardinal, and let λ = κ^{+}. Suppose T is a subset of λ∩cof(κ), not necessarily stationary. We let ℙ(T) denote the forcing poset whose conditions are closed, bounded, non-empty subsets of λ which are disjoint from T, ordered by end-extension. In other words,

∀p (p ∈ ℙ(T) ⟺ (p ∈ X_{λ} ∧ p ∩ T = ∅))

and

∀p, q ∈ ℙ(T) (q ≤_{ℙ(T)} p ⟺ q ≤_{λ} p).

It is easy to show that if p and q are compatible in ℙ(T), then either q ≤_{ℙ(T)} p or p ≤_{ℙ(T)} q. So every family of pairwise compatible conditions in ℙ(T) is a chain. Also note that any chain B ⊆ ℙ(T) of size less than κ has a lower bound. For let α = sup{max(p) : p ∈ B}, and let q = ⋃ B ∪ {α}. Then either α belongs to some p in B, which implies that q = p, or α has cofinality less than κ, and hence is not in T. In either case, q is a condition and q ≤_{ℙ(T)} p for all p in B. In particular, ℙ(T) is κ-closed. In general, however, ℙ(T) might collapse λ.

For the rest of the section, fix a regular uncountable cardinal κ and an ordinal ν, and let λ = κ^{+}. Also fix ℙ which satisfies Definition 4.5(1,2,3,5), witnessed by a sequence of names ⟨Ṫ_{i} : i < ν⟩.

Recall that for all β < ν,

ℙ ↾ β = {p ↾ β : p ∈ ℙ}

and

ℙ_{β} = ℙ ∩ X_{λ,β}.

The next two results are easy.

Lemma 5.1. For all β < ν, ℙ_{β} = ℙ ↾ β.

Proof. Definition 4.5(5) easily implies that ℙ is closed under restrictions. So ℙ ↾ β ⊆ ℙ ∩ X_{λ,β} = ℙ_{β}. Conversely, if p is in ℙ ∩ X_{λ,β}, then p = p ↾ β ∈ ℙ ↾ β. □

Lemma 5.2. For all β < ν, ℙ_{β} is a partial square killing forcing iteration on λ of length β.

Proof. It is straightforward to verify Definition 4.5 for ℙ_{β}. (1)–(4) are trivial. (5) follows easily using the fact that (ℙ_{β})_{α} = ℙ_{α}. □

The next lemma describes the relationship between ℙ and its initial segments. The proof is straightforward, and we omit it. The least trivial statement is (2); this follows from Definition 4.5(5).

Lemma 5.3. Let β < ν.

(1) ℙ_{β} ⊆ ℙ;

(2) if q is in ℙ and s ≤ q ↾ β in ℙ_{β}, then letting t = s ∪ q ↾ [β, α), t is in ℙ and t ≤ s, q in ℙ;

(3) the inclusion map ℙ_{β} → ℙ is a complete embedding;

(4) if p and q are in ℙ and q ≤ p in ℙ, then q ↾ β ≤ p ↾ β in ℙ_{β};

(5) if u and t are in ℙ_{β}, then u ≤ t in ℙ_{β} iff u ≤ t in ℙ.

Lemma 5.4. Let p be in X_{λ,ν} but not in ℙ. Then there is some α ∈ dom(p) such that p ↾ α is in ℙ_{α}, but p ↾ α does not force that p(α) is disjoint from Ṫ_{α}.

Proof. Let α be the least ordinal such that Definition 4.5(5) fails for p. By Definition 4.5(5), p ↾ α is in ℙ. □

Lemma 5.5. Let p and q be incompatible conditions in ℙ. Then there is some β in dom(p) ∩ dom(q) such that neither of p(β) nor q(β) is an initial segment of the other.

Proof. If the conclusion of the lemma fails, then it is easy to construct a lower bound of p and q. □

We make a comment about notation. Sometimes when we consider a function p in X_{λ,ν} and an ordinal β < ν, it will be convenient to write “p(β)” without knowing whether or not β is in the domain of p. In the case that it is not, p(β) will denote the empty set.

Lemma 5.6. The forcing poset ℙ is κ-closed. In fact, suppose B is a directed subset of ℙ of size less than κ. Define q in X_{λ,ν} by letting

dom(q) = ⋃{dom(p) : p ∈ B},

and for all β in dom(q),

q(β) = ⋃{p(β) : p ∈ B} ∪ {sup{max(p(β)) : p ∈ B}}.

Then q is in ℙ and q ≤ p for all p in B.

Proof. If {p(β) : p ∈ B} has a largest set s(β), then q(β) = s(β). Otherwise the largest ordinal of q(β) has cofinality less than κ, and therefore is forced to be not in Ṫ_{β}. The lemma follows easily from these observations and Lemma 5.4. □

Lemma 5.7. Suppose 2^{κ} = λ. If ν < λ^{+}, then |ℙ| < λ^{+}. If ν ≥ λ^{+}, then ℙ is λ^{+}-c.c.

Proof. A straightforward computation shows that if 2^{κ} = λ, then |X_{λ}| = λ, and |X_{λ,ν}| ≤ λ for all ν < λ^{+}. Therefore |ℙ| ≤ λ if ν < λ^{+}. In general, a straightforward argument using the Δ-System Lemma and Lemma 5.5 shows that ℙ is λ^{+}-c.c. □

The next lemma describes the purpose of a partial square killing forcing iteration, namely, that it successively destroys the stationarity of sets which carry a partial square.

Lemma 5.8. Let β < ν and suppose that ℙ_{β} is λ-distributive. Then ℙ_{β+1} is isomorphic to a dense subset of ℙ_{β} ∗ ℙ(Ṫ_{β}).

Note that we assume in the lemma that ℙ_{β} is λ-distributive, which will only be true under some special circumstances; see the next section. The isomorphism is given by p ↦ p ↾ β ∗ p(β).

Now we consider a factorization of ℙ. Let β < ν. Suppose G_{β} is a generic filter on ℙ_{β}. In V [G_{β}], define a forcing poset ℙ_{β,ν} as follows. The underlying set of ℙ_{β,ν} is {p ↾ [β, ν) : p ∈ ℙ}. Let q ≤ p in ℙ_{β,ν} if there exists some s ∈ G_{β} such that s ∪ q ≤ s ∪ p in ℙ. Then ℙ is isomorphic to a dense subset of ℙ_{β} ∗ ℙ_{β,ν}, by the map which sends p to p ↾ β ∗ p ↾ [β, ν).

Lemma 5.9. Let β < ν. Then ℙ_{β} forces that ℙ_{β,ν} is κ-closed.

See Lemma 6.16 of [7] for a proof.

In the next section we will show that under some strong assumptions, any partial square killing forcing iteration on λ of length ν < λ^{+} is λ-distributive. Let us observe that this implies that any partial square killing forcing iteration of length λ^{+} is λ-distributive.

Lemma 5.10. Let ℙ be a partial square killing forcing iteration on λ of length λ^{+}, and suppose that for all ν < λ^{+}, ℙ_{ν} is λ-distributive. Then ℙ is λ-distributive.

Proof. Let ḟ be a nice ℙ-name for a function from κ into the ordinals. Then since ℙ is λ^{+}-c.c. and ℙ = ⋃{ℙ_{i} : i < λ^{+}}, ḟ is a ℙ_{ν}-name for some ν < λ^{+}. By assumption, ℙ_{ν} is λ-distributive, and therefore ḟ is forced to be in the ground model. □

6. Distributivity

We prove that assuming that λ is a greatly Mahlo cardinal and κ < λ is regular and uncountable, then any partial square killing forcing iteration on λ of length λ^{+} in a generic extension by the Lévy collapse Coll(κ, <λ) is λ-distributive. The proof is similar to our proof in [7] that a certain forcing iteration in a generic extension by the Lévy collapse is λ-distributive, assuming that λ is weakly compact.

We will use the following two results.

Theorem 6.1 (Magidor [8]). Let \overline{λ} < λ be strongly inaccessible cardinals, and let κ < \overline{λ} be regular. Let G be a generic filter on the Lévy collapse Coll(κ, < λ). In V [G ↾ \overline{λ}], let ℙ be a κ-closed forcing poset of size less than λ. Then in V [G], there exists a V [G ↾ \overline{λ}]-generic filter H on ℙ such that

V [G] = V [G ↾ \overline{λ}][H][K],

where K is a V [G ↾ λ][H]-generic filter on some κ-closed forcing poset.

Note that if p ∈ ℙ, then the suborder ℙ/p = {q ∈ ℙ : q ≤ p} satisfies the same assumptions as does ℙ, so we may choose H to contain p.

Proposition 6.2. Let μ be an ordinal of uncountable cofinality, and let ⟨d_{γ} : γ < μ⟩ be a sequence such that d_{γ} ⊆ γ for all γ < μ. Let ℙ be a proper forcing poset. Then ℙ forces that if c ⊆ μ is a thread of the sequence ⟨d_{γ} : γ < μ⟩, meaning that c is club and for all γ ∈ lim(c), c ∩ γ = d_{γ}, then c is in the ground model.

Proof. Suppose for a contradiction that p forces that ċ is a thread which is not in V. Fix some large enough regular cardinal θ, and let N be a countable elementary substructure of H(θ) which contains as elements ℙ, p, ċ, and ⟨d_{γ} : γ < μ⟩. Since p does not decide ċ, there is ξ in N ∩ μ and conditions s, t ≤ p in N such that s ⊩ ξ ∈ ċ and t ⊩ ξ ∉ ċ.

As ℙ is proper, fix N-generic conditions \overline{s} ≤ s and \overline{t} ≤ t. Let γ = sup(N ∩ μ). Then γ < μ, since μ has uncountable cofinality. Now \overline{s} and \overline{t} force that sup(N[Ġ] ∩ μ) = sup(N ∩ μ) = γ is a limit point of ċ, and hence that ċ ∩ γ = d_{γ}. So \overline{s} and \overline{t} agree about ċ ∩ γ, which contradicts the choice of ξ. □

Theorem 6.3. Let κ be a regular uncountable cardinal, and let λ > κ be strongly inaccessible. Let ν < λ^{+}, and suppose that λ is ν + 1-Mahlo. Let G be a generic filter on the Lévy collapse Coll(κ, < λ). Then in V [G], any partial square killing forcing iteration on λ of length ν is λ-distributive.

Proof. We prove the statement by induction on λ. So let λ > κ be strongly inaccessible, and assume that the statement of the theorem holds for any strongly inaccessible cardinal \overline{λ} with κ < \overline{λ} < λ. Let G be a generic filter on Coll(κ, <λ). We prove the following statement by induction on ν: for all ν < λ^{+}, if λ is ν + 1-Mahlo in V, then in V [G] any partial square killing forcing iteration on λ of length ν is λ-distributive.

Let ν < λ^{+} be given, and assume that the statement holds for all ν_{0} < ν. Suppose that λ is ν + 1-Mahlo in V. In V [G], let ℙ be a partial square killing forcing iteration on λ of length ν, witnessed by a sequence of names ⟨Ṫ_{i} : i < ν⟩.

We prove that ℙ is λ-distributive. Fix a family D of dense open subsets of ℙ with size less than λ, and fix a condition p ∈ ℙ. We will find a condition q ≤ p in ⋂ D.

Since λ is ν + 1-Mahlo in V, for all β < ν, λ is β + 1-Mahlo in V. But ℙ_{β} is a partial square killing forcing iteration on λ of length β, so the induction hypothesis implies that ℙ_{β} is λ-distributive. Hence all of the proper initial segments of ℙ are λ-distributive.

Fix a sequence of names

⟨ċ_{α}^{β} : β < ν, α < λ⟩

in V [G] such that for all β < ν and α < λ,

ℙ_{β} ⊩ ċ_{α}^{β} ≠ ∅ ⟺ α ∈ Ṫ_{β},

and

ℙ_{β} ⊩ ⟨ċ_{α}^{β} : α ∈ Ṫ_{β}⟩ is a partial square sequence.

Fix a Coll(κ, <λ)-name ẋ in H(λ^{+})^{V} such that

ẋ^{G} = ⟨ ℙ, p, D, ⟨Ṫ_{β} : β < ν⟩, ⟨ċ_{α}^{β} : β < ν, α < λ⟩ ⟩.

This is possible, as ν < λ^{+} and H(λ^{+})^{V [G]} = H(λ^{+})^{V} [G].

Recall that in V, S_{ν}^{λ} is the collection of all N satisfying:

(1) N ≺ H(λ^{+});

(2) N ∩ λ = λ_{N} is strongly inaccessible;

(3) |N| = λ_{N};

(4) N^{<λ_{N}} ⊆ N;

(5) there is a canonical function f_{ν} on λ of rank ν in N such that λ_{N} is f_{ν}(λ_{N})-

Mahlo. Since λ is ν + 1-Mahlo in V, by Lemma 3.3 the set S_{ν}^{λ} is a stationary subset of P_{λ}(H(λ^{+})) in V. Fix N in S_{ν}^{λ} such that

N ≺ (H(λ^{+})^{V} , ∈, κ, ν, ẋ).

Fix a canonical function f_{ν} on λ of rank ν in N such that λ_{N} is f_{ν}(λ_{N})-Mahlo.

Let

π : N → \overline{N}

be the transitive collapsing map of N in V. By Lemma 3.6,

π(ν) = f_{ν}(λ_{N}).

Let

σ : N[G] → \overline{N[G]}

be the transitive collapsing map of N[G] in V [G]. By Lemma 3.8,

σ ↾ N = π

and

\overline{N[G]} = \overline{N}[G ↾ λ_{N}].

In particular, \overline{N[G]} is in V [G ↾ λ_{N}], and is a λ_{N}-model. Note that

σ(λ) = λ_{N}.

Since ẋ is in N, the sets

ℙ, p, D, ⟨Ṫ_{β} : β < ν⟩, ⟨ċ_{α}^{β} : β < ν, α < λ⟩

are in N[G]. As D has size less than λ, D ⊆ N[G].

Let θ(a, b, c) be the formula of LST described in Section 4 which asserts that a is a partial square killing forcing iteration on b of length c. Since N[G] ≺ H(λ^{+})^{V [G]},

N[G] ⊧ θ(ℙ, λ, ν).

As σ is an isomorphism,

\overline{N}[G ↾ λ_{N}] ⊧ θ(σ(ℙ), λ_{N}, f_{ν}(λ_{N})).

Since \overline{N}[G ↾ λ_{N}] is a λ_{N}-model, it follows from Lemma 4.6 that σ(ℙ) is a partial square killing forcing iteration on λ_{N} of length f_{ν}(λ_{N}) in V [G ↾ λ_{N}].

Now λ_{N} is f_{ν}(λ_{N})-Mahlo in V. So for all ζ < f_{ν}(λ_{N}), λ_{N} is ζ + 1-Mahlo in V. By the induction hypothesis applied to λ_{N},

∀ζ < f_{ν}(λ_{N}) (σ(ℙ)_{ζ} is λ_{N}-distributive in V [G ↾ λ_{N}]).

Since σ(ℙ) is κ-closed, by Magidor’s Theorem there exists a V [G ↾ λ_{N}]-generic filter \overline{H} on σ(ℙ) containing σ(p) such that

V [G] = V [G ↾ λ_{N}][\overline{H}][K],

where K is a V [G ↾ λ_{N}][\overline{H}]-generic filter on some κ-closed forcing poset in V [G ↾ λ_{N}][\overline{H}].

Note that

σ ↾ (ℙ ∩ N[G])

is an isomorphism between the partial orderings ℙ ∩ N[G] and σ(ℙ). Define

H = σ^{−1}(\overline{H}).

Then H is a filter on ℙ ∩ N[G], and H is N[G]-generic for ℙ. For if D is a dense subset of ℙ in N[G], then σ(D) is a dense subset of σ(ℙ) in V [G ↾ λ_{N}]. As \overline{H} is V [G ↾ λ_{N}]-generic for σ(ℙ), fix s in \overline{H} ∩ σ(D). Then σ^{−1}(s) is in D ∩ H ∩ N[G].

Note that if u ∈ H, then dom(u) ⊆ N[G], since dom(u) has size less than λ. It follows easily from this fact and the N[G]-genericity of H that

⋃{dom(u) : u ∈ H} = N[G] ∩ ν = N ∩ ν.

Similarly, if u ∈ H and β ∈ dom(u), then u(β) is a subset of N[G] ∩ λ = λ_{N}.

Define a partial function q : ν → X with domain N[G] ∩ ν as follows. Given β in N[G] ∩ ν, by the compatibility of conditions in H and the N[G]-genericity of H, the set

⋃{u(β) : u ∈ H}

is a cofinal subset λ_{N} which is closed below λ_{N}. Define

q(β) = ⋃{u(β) : u ∈ H} ∪ {λ_{N}}.

Then q(β) is in X_{λ}. Since |dom(q)| = |N ∩ ν| ≤ |N| < λ, q is in X_{λ,ν}.

We will be done if we can show that q is in ℙ. For then by the definition of q, q ≤ u for all u in H. Also p ∈ H, since σ(p) ∈ \overline{H}, and therefore q ≤ p. To show q ∈ ⋂ D, let D ∈ D be given. Then D is in N[G], so by the N[G]-genericity of H, fix u in D ∩H. Then q ≤ u, so q is in D, since D is open. This completes the proof.

Suppose for a contradiction that q is not a condition in ℙ. Since q is in X_{λ,ν}, by Lemma 5.4 it follows that there is some β in dom(q) such that q ↾ β is in ℙ_{β}, but q ↾ β does not force that q(β) is disjoint from Ṫ_{β}.

Define

H_{β} = {u ↾ β : u ∈ H}.

Since H is upwards closed in ℙ ∩ N[G], H_{β} ⊆ H, and therefore

H_{β} = H ∩ ℙ_{β}.

Also it follows from the N[G]-genericity of H and an easy argument using Lemma 5.3(2) that H_{β} is N[G]-generic for ℙ_{β}.

Note that q ↾ β ≤ u in ℙ_{β} for all u in H_{β}. It follows that q ↾ β forces that every proper initial segment of q(β) is disjoint from Ṫ_{β}. Since q ↾ β does not force that q(β) is disjoint from Ṫ_{β}, there is a condition below q ↾ β in ℙ_{β} which forces that λ_{N} is in Ṫ_{β}.

Let

\overline{H}_{σ(β)} = σ[H_{β}].

It is not hard to show that

\overline{H}_{σ(β)} = \{s ↾ σ(β) : s ∈ \overline{H}\} = \overline{H} ∩ σ(ℙ)_{σ(β)}.

It follows by Lemma 5.3(3) that \overline{H}_{σ(β)} is a V [G ↾ λ_{N}]-generic filter on σ(ℙ)_{σ(β)}. Let

\overline{H}_{σ(β),σ(ν)} = \{s ↾ [σ(β), σ(ν)) : s ∈ \overline{H}\}.

Then \overline{H}_{σ(β),σ(ν)} is a V [G ↾ λ_{N}][\overline{H}_{σ(β)}]-generic filter on σ(ℙ)_{σ(β),σ(ν)}, and

V [G] = V [G ↾ λ_{N}][\overline{H}_{σ(β)}][\overline{H}_{σ(β),σ(ν)}][K].

By Lemma 5.9, it follows that V [G] is a generic extension of V [G ↾ λ_{N}][\overline{H}_{σ(β)}] by a κ-closed forcing poset.

Recall that

ℙ_{β} ⊩ ⟨ċ_{α}^{β} : α ∈ Ṫ_{β}⟩

is a partial square sequence. Fix a sequence of ℙ_{β}-names

⟨ḋ_{γ} : γ < λ⟩

in N[G] satisfying that for all γ < λ,

ℙ_{β} ⊩ ḋ_{γ} = \{ \begin{matrix}ċ_{α}^{β} ∩ γ & if γ ∈ lim(ċ_{α}^{β}) ∩ α \\ ∅ & otherwise. \end{matrix}

Note that by coherence, the definition of ḋ_{γ} in the first case is independent of α.

For each γ < λ, ℙ_{β} forces that ḋ_{γ} has order type less than κ. But ℙ_{β} is κ-closed, so ℙ_{β} forces that ḋ_{γ} is in the ground model V [G]. By the N[G]-genericity of H_{β}, for each γ < λ_{N} we can fix d_{γ} in N[G] such that

∃u ∈ H_{β} (u ⊩_{ℙ_{β}} ḋ_{γ} = ď_{γ}).

This defines a sequence

⟨d_{γ} : γ < λ_{N}⟩

in the model V [G].

Recall that there is a condition below q ↾ β in ℙ_{β} which forces that λ_{N} is in Ṫ_{β}, and hence that ċ_{λ_{N}}^{β} is a club subset of λ_{N} of order type κ. By the induction hypothesis, ℙ_{β} is λ-distributive. So we can find t ≤ q ↾ β in ℙ_{β} and a club set c ⊆ λ_{N} with order type κ such that

t ⊩_{ℙ_{β}} ċ_{λ_{N}}^{β} = č.

Then

∀γ ∈ lim(c) ∩ λ_{N} (t ⊩_{ℙ_{β}} č ∩ γ = ḋ_{γ}).

But t is a lower bound of H ↾ β. So

∀γ ∈ lim(c) ∩ λ_{N} (t ⊩_{ℙ_{β}} č ∩ γ = ď_{γ}).

Therefore

∀γ ∈ lim(c) ∩ λ_{N} (c ∩ γ = d_{γ}).

In other words, c is a thread of the sequence ⟨d_{γ} : γ < λ_{N}⟩.

We claim that the sequence ⟨d_{γ} : γ < λ_{N}⟩ is in the model V [G ↾ λ_{N}][\overline{H}_{σ(β)}]. The sequence of names ⟨ḋ_{γ} : γ < λ⟩ is in N[G], and clearly

σ(⟨ḋ_{γ} : γ < λ⟩) = ⟨σ(ḋ_{γ}) : γ < λ_{N}⟩.

For all γ < λ_{N}, there is some u in H_{β} such that

u ⊩_{ℙ_{β}} ḋ_{γ} = ď_{γ}.

Then

N[G] ⊧ u ⊩_{ℙ_{β}} ḋ_{γ} = ď_{γ}.

It follows that

\overline{N}[G ↾ λ_{N}] ⊧ σ(u) ⊩_{σ(ℙ_{β})} σ(ḋ_{γ}) = σ(ď_{γ}).

Therefore in V [G ↾ λ_{N}],

σ(u) ⊩_{σ(ℙ_{β})} σ(ḋ_{γ}) = σ(ď_{γ}).

Also note that σ(u) ∈ \overline{H}_{σ(β)}. Since d_{γ} is a bounded subset of λ_{N}, σ(d_{γ}) = d_{γ}. It follows that the sequence ⟨d_{γ} : γ < λ_{N}⟩ is the \overline{H}_{σ(β)}-interpretation of ⟨σ(ḋ_{γ}) : γ < λ_{N}⟩, considered as a σ(ℙ_{β})-name. Thus this sequence is in V [G ↾ λ_{N}][\overline{H}_{σ(β)}], as claimed.

In the model V [G ↾ λ_{N}], λ_{N} is equal to κ^{+}. For all ζ < f_{ν}(λ_{N}), σ(ℙ)_{ζ} is λ_{N}- distributive by the induction hypothesis. But β < ν, so σ(β) < σ(ν) = f_{ν}(λ_{N}). It follows that σ(ℙ_{β}) = σ(ℙ)_{σ(β)} is λ_{N}-distributive in V [G ↾ λ_{N}]. So λ_{N} is equal to κ^{+} in the model V [G ↾ λ_{N}][\overline{H}_{σ(β)}].

In the model V [G ↾ λ_{N}][\overline{H}_{σ(β)}], we have a sequence ⟨d_{γ} : γ < λ_{N}⟩, where each d_{γ} is a subset of γ. In the model V [G], c is a thread of this sequence. But V [G] is a generic extension of V [G ↾ λ_{N}][\overline{H}_{σ(β)}] by a κ-closed forcing poset. By Proposition 6.2, it follows that c is in V [G ↾ λ_{N}][\overline{H}_{σ(β)}]. But c has order type κ and is cofinal in λ_{N}. This contradicts that λ_{N} is equal to κ^{+} in the model V [G ↾ λ_{N}][\overline{H}_{σ(β)}]. This contradiction shows that q must be a condition in ℙ as desired, and the proof is complete. □

7. No Partial Squares from a Greatly Mahlo Cardinal

We are now ready to construct a model with no partial square sequences. Let V be a model of set theory satisfying the following assumptions:

(1) κ is a regular uncountable cardinal;

(2) λ is a greatly Mahlo cardinal larger than κ;

(3) 2^{λ} = λ^{+}. We begin by Lévy collapsing λ to become κ^{+}. So let G be a generic filter on the Lévy collapse Coll(κ, < λ). Then in V [G], 2^{κ} = λ = κ^{+} and 2^{λ} = λ^{+}.

Working in V [G], we define by recursion sequences

⟨ℙ_{i} : i ≤ λ^{+}⟩

and

⟨Ṫ_{j}^{i} : i, j < λ^{+}⟩.

For bookkeeping purposes, fix a function f : λ^{+} → λ^{+} × λ^{+} satisfying that f(α) = ⟨i, j⟩ implies i ≤ α. We also define a sequence

⟨Ṫ_{α} : α < λ^{+}⟩

by letting Ṫ_{α} = Ṫ_{j}^{i}, where f(α) = ⟨i, j⟩.

We will maintain the following recursion hypotheses: for all β ≤ λ^{+},

(1) ℙ_{β} is a partial square killing forcing iteration on λ of length β, witnessed by the sequence of names ⟨Ṫ_{α} : α < β⟩;

(2) for all ζ < β, ℙ_{ζ} = ℙ_{β} ↾ ζ;

(3) if β < λ^{+}, then ⟨Ṫ_{i}^{β} : i < λ^{+}⟩ is a list of all nice ℙ_{β}-names for subsets of λ ∩ cof(κ) which are forced by ℙ_{β} to carry a partial square.

Let us summarize the basic properties which ℙ_{β} will satisfy if the above hypotheses hold:

(a) ℙ_{β} is κ-closed (by Lemma 5.6);

(b) |ℙ_{β}| < λ^{+} if β < λ^{+} (by Lemma 5.7); (c) ℙ_{β} is λ^{+}-c.c. (by Lemma 5.7);

(d) ℙ_{β} is λ-distributive (by Theorem 6.3 and Lemma 5.10);

(e) if β = γ + 1, then ℙ_{β} is forcing equivalent to ℙ_{γ} ∗ ℙ(Ṫ_{γ}) (by Lemma 5.8).

Let ℙ_{0} be the trivial forcing consisting of just the empty function.

Suppose ν < λ^{+} and ℙ_{β} is defined for all β ≤ ν. Also assume ⟨Ṫ_{i}^{β} : i < λ^{+}⟩ is defined for all β < ν. Choose a sequence ⟨Ṫ_{i}^{ν} : i < λ^{+}⟩ satisfying (3) above. This is possible, as 2^{λ} = λ^{+} and (b) imply that there are only λ^{+} many nice ℙ_{ν}-names for a subset of λ.

Let f(ν) = ⟨i, j⟩. Then i ≤ ν, so Ṫ_{j}^{i} is defined. Let Ṫ_{ν} = Ṫ_{j}^{i}. Then Ṫ_{ν} is a nice ℙ_{i}-name for a subset of λ ∩ cof(κ) which is forced to carry a partial square. The property of carrying a partial square is easily seen to be upwards absolute. So Ṫ_{ν} is a nice ℙ_{ν}-name for a subset of λ ∩ cof(κ) which carries a partial square.

Now define ℙ_{ν+1} as follows. The underlying set of ℙ_{ν+1} consists of all functions p in X_{λ,ν+1} satisfying that p ↾ ν ∈ ℙ_{ν}, and if ν ∈ dom(p), then p ↾ ν ⊩_{ℙ_{ν}} p(ν)∩Ṫ_{ν} = ∅. The ordering on ℙ_{ν+1} is by ≤_{λ,ν+1}.

Let ν ≤ λ^{+} be a limit ordinal, and suppose that ℙ_{β} is defined for all β < ν. Define ℙ_{ν} as the suborder of X_{λ,ν} consisting of functions p ∈ X_{λ,ν} such that p ↾ β ∈ ℙ_{β} for all β < ν.

Now we verify the recursion hypotheses. (3) is clear. Let us prove (2). This is trivial for ℙ_{0}. Consider ℙ_{ν+1}. Clearly ℙ_{ν} ⊆ ℙ_{ν+1}, and therefore ℙ_{ν+1} ↾ ν = ℙ_{ν}. Then for all β < ν, ℙ_{ν+1} ↾ β = ℙ_{ν} ↾ β = ℙ_{β}, by the recursion hypotheses. Similarly for a limit ordinal ν and β < ν, ℙ_{β} ⊆ ℙ_{ν} follows easily from the definition of ℙ_{ν} and the recursion hypotheses. This implies ℙ_{ν} ↾ β = ℙ_{β}.

It remains to prove (1), by verifying the properties of Definition 4.5. (1)–(4) are immediate from the definition and the recursion hypotheses. The verification of (5) is tedious, but completely trivial. So we will omit it.

This completes the construction. Let ℙ = ℙ_{λ^{+}}.

Let H be a V [G]-generic filter on ℙ. Then in V [G][H], λ = κ^{+}. Suppose T ⊆ λ ∩ cof(κ) carries a partial square in V [G][H]. Then in V [G], there is a nice ℙ-name Ṫ for a subset of λ ∩ cof(κ) such that Ṫ^{G} = T and ℙ forces that Ṫ carries a partial square.

Since ℙ is λ^{+}-c.c. and ℙ = ⋃{ℙ_{i} : i < λ^{+}}, we can find α < λ^{+} such that Ṫ is a nice ℙ_{α}-name for a subset of λ ∩ cof(κ), and moreover ℙ_{α} forces that Ṫ carries a partial square. So for some i < λ^{+}, Ṫ = Ṫ_{i}^{α}. Now choose ν ≥ α such that f(ν) = ⟨α, i⟩. Then Ṫ = Ṫ_{i}^{α} = Ṫ_{ν}. But ℙ_{ν+1} is forcing equivalent to ℙ_{ν} ∗ ℙ(Ṫ_{ν}). So in V [G ∩ ℙ_{ν+1}], T is non-stationary. Thus T is non-stationary in V [G].

We have shown that in V [G], no stationary subset of κ^{+} ∩cof(κ) carries a partial square, which completes the proof.

8. A Related Equiconsistency Result

Let λ be a regular uncountable cardinal. Recall the principle □(λ), which asserts the existence of a sequence

⟨c_{α} : α < λ, α limit⟩

satisfying:

(1) c_{α} is a club subset of α;

(2) if γ is a limit point of c_{α}, then c_{α} ∩ γ = c_{γ};

(3) there does not exist a club set C ⊆ λ such that for all γ in lim(C), C∩γ = c_{γ}. As with partial squares, we can relativize this idea to a stationary set.

Definition 8.1. Let λ be a regular uncountable cardinal, and let A ⊆ λ. The principle □(λ, A) asserts the existence of a sequence

⟨c_{α} : α ∈ A⟩

satisfying:

(1) c_{α} is a club subset of α;

(2) if γ is a limit point of c_{α} and c_{β}, then c_{α} ∩ γ = c_{β} ∩ γ;

(3) there does not exist a club set C ⊆ λ satisfying that whenever γ is in lim(C), there exists α ∈ A such that γ ∈ lim(c_{α}) and C ∩ γ = c_{α} ∩ γ.

Suppose ⟨c_{α} : α ∈ A⟩ is a sequence satisfying (1) and (2). Define ⟨d_{γ} : γ < λ⟩ by

d_{γ} = \{ \begin{matrix}c_{α} ∩ γ & if γ ∈ lim(c_{α}) \\ ∅ & otherwise. \end{matrix}

Let us call ⟨d_{γ} : γ < λ⟩ the derived sequence of ⟨c_{α} : α ∈ A⟩. Then property (3) is equivalent to the assertion that the sequence ⟨d_{γ} : γ < λ⟩ does not have a thread, that is, there does not exist a club C ⊆ λ such that for all γ in lim(C), C ∩ γ = d_{γ}.

Theorem 8.2. The statement that there exists a regular uncountable cardinal κ such that ¬□(κ^{+}, A) holds for every stationary set A ⊆ κ^{+}∩cof(κ) is equiconsistent with a weakly compact cardinal.

Jensen showed that the failure of □(κ^{+}) implies that κ^{+} is weakly compact in L ([11]), and this proves the lower bound. Veličković [15] showed that if λ is weakly compact and κ < λ is regular and uncountable, then Coll(κ, < λ) forces ¬□(κ^{+}) (also see [11]). A variation of the forcing construction given above will show how to obtain ¬□(κ^{+}, A) for any stationary set A ⊆ κ^{+} ∩ cof(κ).

We will use a characterization of weakly compact cardinals which was proven by the first author in [7].

Theorem 8.3. Let λ be a regular uncountable cardinal. Then λ is weakly compact iff there are stationarily many N in P_{λ}(H(λ^{+})) satisfying:

(1) N ≺ H(λ^{+});

(2) N ∩ λ = λ_{N} is strongly inaccessible;

(3) N^{<λ_{N}} ⊆ N;

(4) \overline{N} ≺_{1} H(λ+_{N}).

The forcing construction we use to prove Theorem 8.2 is nearly identical to that given in the previous sections, so we will only point out the differences.

The ground model V satisfies that λ is weakly compact. We define a forcing iteration in V [G], where G is a generic filter on Coll(κ, <λ). Given ℙ_{β}, enumerate all nice ℙ_{β}-names for a subset of λ ∩ cof(κ) as ⟨Ṫ_{α}^{β} : α < λ^{+}⟩. Let f(β) = ⟨i, j⟩, where f is a bookkeeping function. If ℙ_{β} forces □(κ^{+}, Ṫ_{j}^{i}), then let Ṫ_{β} = Ṫ_{j}^{i}. Otherwise let Ṫ_{β} be a name for the empty set. The rest of the definition of the forcing iteration is the same as before. One now shows easily that the forcing iteration satisfies Definition 4.5(1,2,3,5), and thus all the properties discussed in Section 5.

The proof that the iteration is λ-distributive is by induction. So assume ℙ_{β} is λ-distributive for all β < ν, and we show that ℙ_{ν} is λ-distributive. Again we choose an elementary substructure N, but this time we assume that it satisfies the properties listed in Definition 8.3. The definition of q is as before, and it will suffice to show q is a condition. Suppose for a contradiction it is not, and let β in dom(q) be a counterexample.

Fix a name ⟨ċ_{β} : β ∈ Ṫ_{β}⟩ in N which ℙ_{β} forces satisfies Definition 8.1(1,2,3). Let ⟨ḋ_{γ} : γ < λ⟩ be a name for the derived sequence of ⟨ċ_{β} : β ∈ Ṫ_{β}⟩. Then ℙ_{β} forces that the sequence ⟨ḋ_{γ} : γ < λ⟩ does not have a thread. This last statement can be shown to be Π_{1}. Since σ is an isomorphism and \overline{N}[G ↾ λ_{N}] ≺_{1} H(λ+_{N})^{V [G↾λ_{N}]}, in the model V [G ↾ λ_{N}], σ(ℙ_{β}) forces that σ(⟨ḋ_{γ} : γ < λ⟩) does not have a thread.

By the induction hypothesis, ℙ_{β} is λ-distributive. But the property of being λ-distributive is Π_{1}. So again the fact that \overline{N}[G ↾ λ_{N}] ≺_{1} H(λ+_{N})^{V [G↾λ_{N}]} implies that σ(ℙ_{β}) is λ_{N}-distributive in V [G ↾ λ_{N}]. Therefore λ_{N} is equal to κ^{+} in V [G ↾ λ_{N}][\overline{H}_{σ(β)}].

As in the proof of Theorem 6.3, we define a sequence ⟨d_{γ} : γ < λ_{N}⟩ which turns out to be the \overline{H}_{σ(β)}-interpretation of σ(⟨ḋ_{γ} : γ < λ⟩), and find a club c ⊆ λ_{N} in V [G] which threads this sequence. An application of Proposition 6.2 shows that the thread c is in V [G ↾ λ_{N}][\overline{H}_{σ(β)}]. This contradicts that σ(ℙ_{β}) forces in V [G ↾ λ_{N}] that σ(⟨ḋ_{γ} : γ < λ⟩) does not have a thread.

Remark. The results of this paper are related in some ways to Mitchell’s construction of a model in which there is no stationary subset of κ^{+}∩cof(κ) in the approachability ideal I[κ^{+}], using a greatly Mahlo cardinal ([10]). In this model (and also in the model constructed in [9]), there is no stationary subset of κ^{+} ∩ cof(κ) which carries a partial square. However, Mitchell’s argument works only when κ = μ^{+} for some regular cardinal μ. Our forcing construction, on the other hand, assumes only that κ is regular and uncountable. Another difference is that GCH holds in our model, whereas in Mitchell’s model, 2^{μ} = μ^{++}.

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Department of Mathematics, University of North Texas

E-mail address: jkrueger@unt.edu

Department of Mathematical Sciences, Carnegie Mellon University

E-mail address: eschimme@andrew.cmu.edu