A GENERAL MITCHELL STYLE ITERATION

JOHN KRUEGER

Abstract. We work out the details of a schema for a mixed support forcing iteration, which generalizes the Mitchell model [7] with no Aronszajn trees on ω_{2}.

The main purpose of this paper is to present the details of a schema for a mixed support forcing iteration. This schema will provide the technical framework for a variety of consistency results which we establish in [6] and [4].

In [5] we constructed a model in which the properties of being “internally club” and “internally approachable” are distinct, for structures of size the successor of a regular cardinal. Our construction was reminiscent in some ways of Mitchell’s classic model with no Aronszajn trees on ω_{2} ([7]). We used a forcing iteration whose factors are two-step iterations of Cohen forcing followed by a kind of collapse forcing, where the support on the Cohen part and on the collapsing part are of different sizes. However the kind of collapse forcing we used is not even strategically closed.

In Section 2 we describe an iterable property which is satisfied by both the forcing used to construct Mitchell’s model and the forcing we used in [5]. It is this property which we will assume to hold for the factor forcings in our mixed support iteration schema. We work out the details of this iteration in Section 3.

In subsequent papers we use the iterated forcing schema presented below as a tool for establishing consistency results. In [6] we construct a model with a disjoint stationary sequence on the successor of an arbitrary regular cardinal, and also distinguish several variations of the property of being internally approachable for sets of a variety of sizes. In [4] we use the forcing iteration schema to study the approachability ideal at the second successor of a singular cardinal.

1. Preliminaries

We review some background material, notation, and conventions used in the paper. We assume the reader has had some previous exposure to proper forcing and iterated forcing. We refer the reader to [2] for basic facts concerning iterated forcing. In the rest of the section, μ and κ always refer to regular cardinals.

A forcing poset is a pair ⟨ℙ, ≤⟩ such that ≤ is a binary relation on ℙ which is reflexive and transitive. Due to our treatment of iterated forcing, we do not require anti-symmetry as part of the definition of a forcing poset. In the case that p ≤ q and q ≤ p, we say that p and q are equivalent.

A forcing poset ℙ is μ-closed if whenever ⟨p_{i} : i < ξ⟩ is a descending sequence of conditions in ℙ for some ξ < μ, then there is q in ℙ such that q ≤ p_{i} for all i < ξ. The poset ℙ satisfies the κ-covering property if it forces that whenever x is a set of ordinals in the extension with size less than κ, then there is a set a in the ground model with size less than κ in the ground model such that x ⊆ a.

Date: April 2008.

For a forcing poset ℙ and an ordinal α, we define a two-player game G(ℙ, α) as follows. Player I and Player II take turns to define a descending sequence ⟨p_{i} : 1 ≤ i < ζ⟩ of conditions in ℙ, with Player I playing p_{j} for odd ordinals j and Player II playing p_{i} for even ordinals i. The game continues as long as possible, until either p_{i} is defined for all i < α, in which case Player II wins, or otherwise until Player II has no possible move, in which case Player I wins. Clearly Player I wins iff there is a limit ordinal δ < α such that ⟨p_{i} : 1 ≤ i < δ⟩ has no lower bound. We say that ℙ is α-strategically closed if Player II has a winning strategy in the game G(ℙ, α).

The poset ℙ is < κ-distributive if any family of fewer than κ many dense open subsets of ℙ has an intersection which is also dense open. This property is equivalent to ℙ not adding any new sequences of ordinals with order type less than κ. If ℙ is κ-closed then it is κ-strategically closed, and if ℙ is κ-strategically closed then it is < κ-distributive. We say that ℙ is (< κ, ∞, μ)-distributive if for all ξ < κ, ℙ forces that whenever g : ξ → On is a function in the extension, then there is a function h : ξ → V in the ground model such that for all i < ξ, |h(i)|^{V} ≤ μ and g(i) ∈ h(i).

We use Add(μ) to denote the forcing poset which adds a Cohen subset to μ. Conditions in Add(μ) are partial functions f : μ → 2 with domain of size less than μ, and q ≤ p iff q extends p as a function. If μ^{<μ} = μ then Add(μ) has size μ.

Consider a forcing iteration ⟨ℙ_{i}, ℚ̇_{j} : i ≤ α, j < α⟩. We refer to the ℚ̇_{j}’s as the factors of the iteration. In our notation, a condition in ℙ_{α} is a partial function p : α → V whose domain is a subset of α, and the support of p is exactly the domain of p. We do not use the convention that p(i) always denotes the maximal element of ℚ̇_{i} when i is not in the support of p. We also assume that whenever p is a condition in ℙ_{α}, then for all i in the domain of p, ℙ_{i} forces that p(i) is in ℚ̇_{i}.

A useful fact about names in forcing is the so-called maximal principle, which asserts that whenever a condition p in ℙ forces ∃xφ, then there is a ℙ-name ȧ such that p forces φ(ȧ). We apply this fact in the paper in the following situation. Suppose ⟨ℙ_{i}, ℚ̇_{j} : i ≤ α, j < α⟩ is a forcing iteration, p and q are conditions in ℙ_{α}, and q ≤ p. Then there is a condition q′ in ℙ_{α} which is equivalent to q such that for all i in dom(p), ℙ_{i} forces that q′(i) ≤ p(i). Let the domain of q′ be equal to dom(q). If i is in dom(q) \ dom(p), let q′(i) = q(i). Suppose i is in dom(p). Since q ↾ i forces q(i) ≤ p(i), ℙ_{i} forces the statement: ∃x((x ≤ p(i)) & (q ↾ i ∈ Ġ_{i} → x = q(i)). By the maximal principle, let q′(i) be a ℙ_{i}-name ȧ such that ℙ_{i} forces ((ȧ ≤ p(i)) & (q ↾ i ∈ Ġ_{i} → ȧ = q(i)). Then q′ is a condition equivalent to q and satisfies the desired property.

We use Even and Odd to denote the classes of even and odd ordinals respectively. If X is a set, we say that a cardinal θ is much larger than X if θ is larger than |P(P(P(tr(X))))|, where tr(X) is the transitive closure of X.

If μ^{<μ} = μ, then the following statement, known as the Δ-System Lemma, holds for μ: if {a_{i} : i < μ^{+}} is a family of sets each of which has size less than μ, then there is a set a and an unbounded set X ⊆ μ^{+} such that for i < j in X, a_{i} ∩ a_{j} = a.

Let P_{κ}(X) denote the collection of subsets of X which have size less than κ. Usually when this notation is used, κ is a subset of X. A set C ⊆ P_{κ}(X) is club in P_{κ}(X) if it is closed under unions of ⊆-increasing chains of length less than κ, and is cofinal in the sense that any member of P_{κ}(X) is a subset of some member of C. A set S ⊆ P_{κ}(X) is stationary in P_{κ}(X) if it has non-empty intersection with each club set.

We will be interested in elementary substructures of models which are an expansion of ⟨H(λ), ∈⟩ in a countable language, where λ ≥ ω_{2} is a regular cardinal. Suppose N ≺ H(λ) and N ∩ κ ∈ κ. Then any set x in N with size less than κ is a subset of N. For |x| ∈ N by elementarity, and by the fact that N ∩ κ ∈ κ, |x| ⊆ N. Choose f : |x| → x a bijection in N. Then f[ |x| ] = x ⊆ N.

Let N be a set and ξ a limit ordinal. We say that N is internally approachable of length ξ, and is in IA(ξ), if there is a ⊆-increasing and continuous sequence of sets ⟨N_{i} : i < ξ⟩ with union equal to N such that for all j < ξ, ⟨N_{i} : i < j⟩ is in N. We say that N is internally approachable if it is internally approachable of length some limit ordinal.

An internally approachable chain is a ⊆-increasing and continuous sequence of sets ⟨N_{i} : i < ξ⟩, where ξ is a limit ordinal, such that for all j < ξ, ⟨N_{i} : i ≤ j⟩ ∈ N_{j+1}. Note that the union of an internally approachable chain of length ξ is in IA(ξ). Conversely, suppose κ < λ are regular uncountable cardinals, and A is an expansion of H(λ) in a countable language. Assume that N ≺ A, N ∩κ ∈ κ, κ ∈ N, and N is internally approachable. Then there exists an internally approachable chain ⟨N_{i} : i < ξ⟩ with union N, for some limit ordinal ξ, such that for all i < ξ, N_{i} ≺ A and N_{i} ∩ κ ∈ κ. If in addition N is in IA(μ) for a regular cardinal μ, then there is such an internally approachable chain which has length μ. See Section 2 of [3] for more information about internally approachable sets.

Suppose N ≺ ⟨H(λ), ∈, ℙ⟩. A condition q in ℙ is N-generic for ℙ if for any dense open subset D of ℙ in N, q forces that Ġ ∩ N ∩ D is non-empty. This implies that q forces N[Ġ] ∩ V = N, and in particular, N[Ġ] ∩ On = N ∩ On (in fact, these properties are equivalent for q). For suppose G is a generic filter containing q, and let b be in N[G] ∩ V. Fix a ℙ-name ċ in N such that ċ^{G} = b. Let D be the dense open set of conditions which either force ċ is not in V, or decide the value of ċ. By elementarity, D is in N. Fix s in G ∩ N ∩ D. Since b is in V, s decides the value of ċ. Then b is in N, since it is definable from s and ċ.

We use the following generalization of proper forcing. Let ℙ be a forcing poset, and let μ < κ be regular cardinals. We say that ℙ is <κ-proper for IA(μ) if for all sufficiently large regular cardinals θ > κ which are much larger than ℙ, there is a club C ⊆ P_{κ}(H(θ)) such that for any N in C ∩ IA(μ), for all p in N ∩ ℙ there is q ≤ p which is N-generic.

2. Factors of the Iteration

In [8] Mitchell gives a modern description of the construction from [7] of a model with no Aronszajn trees on ω_{2}, as an iteration of adding Cohen reals and ω_{1}-closed collapses up to a weakly compact cardinal, with finite support on the Cohen forcings and countable support on the collapses. In [5] we used a similar kind of mixed support iteration to construct a model in which the notions of internally club and internally approachable are distinct. We iterated alternately adding Cohen subsets to a regular cardinal μ and then collapsing μ^{++} to have size μ^{+}, with supports of size less than μ on the Cohen forcings and supports with size μ on the collapses. However, the kind of collapses we used are not even strategically closed.

Thus we sought an iterable property which is satisfied by the forcing posets used in both constructions. Let μ^{<μ} = μ be a cardinal, and let κ > μ be regular. We consider two-step iterations Add(μ) ∗ ℚ̇, equipped with two partial orderings. The first partial ordering ≤ is the usual ordering on a two-step iteration. The second ordering ≤^{∗} is defined by letting q ∗ ṫ ≤^{∗} p ∗ ṡ if q ∗ ṫ ≤ p ∗ ṡ and q = p. It turns out that the desired property is:

The poset ⟨Add(μ) ∗ ℚ̇ , ≤^{∗}⟩ is κ-strategically closed.

As we see next, if Add(μ) forces that ℚ̇ is κ-strategically closed, then the above property holds. However, we will show at the end of the section that this property does not imply that Add(μ) forces ℚ̇ is κ-strategically closed.

Proposition 2.1. If Add(μ) forces ℚ̇ is κ-strategically closed, then ⟨Add(μ)∗ℚ̇, ≤^{∗} ⟩ is κ-strategically closed. In particular, if Add(μ) forces that ℚ̇ is κ-closed then ⟨Add(μ) ∗ ℚ̇, ≤^{∗}⟩ is κ-strategically closed.

Proof. Let σ̇ be an Add(μ)-name for a strategy for Player II in the game G(ℚ̇, κ). Define a strategy τ for Player II in the game G(⟨Add(μ)∗ℚ̇, ≤^{∗}⟩, κ) as follows. Let Player I begin the game with a condition p_{1} ∗q̇_{1}. Suppose a play ⟨p_{j} ∗q̇_{j} : 1 ≤ j < i⟩ is defined where i < κ is even. By the definition of ≤^{∗}, p_{j} = p_{1} for all 1 ≤ j < i. Assume as an induction hypothesis that p_{1} forces ⟨q̇_{j} : 1 ≤ j < i⟩ is a play of the game G(ℚ̇, κ) according to σ̇. Let τ instruct Player II to play a condition p ∗ q̇_{i} such that p forces q̇_{i} is Player II’s response to ⟨q̇_{j} : 1 ≤ j < i⟩ according to σ̇. Thus Player II is able to pass through every limit stage less than κ and wins the game. □

The next proposition describes the forcing poset we used in [5].

Proposition 2.2. Let λ ≥ κ be a regular cardinal. Let Add(μ) ∗ ℚ̇ be the twostep iterated forcing defined as follows. If G is generic for Add(μ), then in V [G], ℚ̇^{G} is the forcing poset whose conditions are increasing and continuous sequences ⟨a_{i} : i ≤ γ⟩ of sets such that γ < κ and a_{i} is in P_{κ}(H(λ)^{V}) ∩ V for i ≤ γ, ordered by extension of sequences. Then ⟨Add(μ) ∗ ℚ̇, ≤^{∗}⟩ is κ-strategically closed.

Proof. Fix a regular cardinal θ which is much larger than Add(μ) ∗ ℚ̇, and let A denote the structure ⟨H(θ), ∈, Add(μ) ∗ ℚ̇, λ⟩. We describe a strategy σ for Player II in the game G(⟨Add(μ) ∗ ℚ̇, ≤^{∗}⟩, κ) by describing his or her moves on a play ⟨p ∗ q̇_{i} : 1 ≤ i < κ⟩. Simultaneously we define an increasing and continuous sequence ⟨N_{i} : i < κ⟩ of elementary substructures of A with size less than κ. To begin with fix N_{0} ≺ A with size less than κ such that N_{0} ∩ κ ∈ κ and μ ⊆ N_{0}. Let p ∗ q̇_{1} be Player I’s first move.

Suppose that i < κ and we have defined ⟨p ∗ q̇_{j} : 1 ≤ j ≤ i⟩ and ⟨N_{j} : j < i⟩. If i is a limit ordinal let N_{i} = ⋃{N_{j} : j < i}. Otherwise let N_{i} be an elementary substructure of A with size less than κ such that N_{i−1} ⊆ N_{i}, N_{i} ∩ κ ∈ κ, and p ∗ q̇_{i} is in N_{i}. If i is even, let p ∗ q̇_{i+1} be Player I’s next move.

Suppose i is odd. Let σ instruct Player II to play a condition p ∗ q̇_{i+1}, where p forces that q̇_{i+1} is a sequence with domain equal to dom(q̇_{i}) + 1, such that q̇_{i+1} ↾ dom(q̇_{i}) = q̇_{i} and q̇_{i+1}(dom(q̇_{i})) = N_{i} ∩ H(λ)^{V}. Clearly N_{i} ∩ H(λ)^{V} is in P_{κ}(H(λ)^{V}) ∩ V. So to show that this definition makes sense, we need to check p forces that the maximal element of q̇_{i} is a subset of N_{i} ∩ H(λ)^{V}. Let G be generic for Add(μ). Then N_{i}[G] ∩ V = N_{i}, since Add(μ) ⊆ N_{i}. Now q_{i} = q̇_{i}^{G} is a member of N_{i}[G]. Let a be the last element on the sequence q_{i}, which is in N_{i}[G] by elementarity. Since |a| < κ, a ⊆ N_{i}[G] ∩ H(λ)^{V} = N_{i} ∩ H(λ)^{V}.

Now suppose δ < κ is a limit ordinal and we have defined ⟨p ∗ q̇_{j} : 1 ≤ j < δ⟩ and ⟨N_{j} : j < δ⟩. Let N_{δ} = ⋃{N_{i} : i < δ}. Let q̇_{δ}^{∗} be a name for ⋃{q̇_{i} : i < δ}, and let γ̇ be a name for the length of this sequence. Then by construction, for all odd i < δ, p forces that N_{i} ∩ H(λ)^{V} is the maximal element of the sequence q̇_{i+1}. Therefore p forces that the union of the sequence q̇_{δ}^{∗} is equal to ⋃{N_{i} ∩ H(λ)^{V} : i ∈ δ ∩ Odd} = N_{δ} ∩ H(λ)^{V}, which is in P_{κ}(H(λ)^{V}) ∩ V. So let σ instruct Player II to play the condition p ∗ q̇_{δ} such that p forces q̇_{δ} ↾ γ̇ = q̇_{δ}^{∗} and q̇_{δ}(γ̇) = N_{δ} ∩ H(λ)^{V}. Thus Player II is able to pass through all limit stages less than κ, and wins the game. □

We give an example to show that ⟨Add(μ) ∗ ℚ̇, ≤^{∗}⟩ being κ-strategically closed does not imply that Add(μ) forces that ℚ̇ is κ-strategically closed.

Let μ = ω and κ = ω_{1}, and let Add(ω) ∗ ℚ̇ be the forcing poset described in Proposition 2.2 letting λ = ω_{2}. We show that Add(ω) forces that ℚ̇ is not ω_{1}-strategically closed.

Let G be a generic filter for Add(ω) over V. We use the fact due to Abraham and Shelah [1] that in V [G], the set P_{ω_{1}}(H(ω_{2})^{V})\V is stationary in P_{ω_{1}}(H(ω_{2})^{V}). Suppose for a contradiction that there is a strategy σ for the game G(ℚ, ω_{1}) in V [G]. Working in V [G], let θ be a regular cardinal much larger than ω_{2} and let A be the structure ⟨H(θ), ∈, ℚ, σ⟩. Fix a countable N ≺ A such that N ∩ H(ω_{2})^{V} is not in V. Fix an enumeration ⟨x_{n} : n < ω⟩ of N ∩ H(ω_{2})^{V}.

We consider a partial run of the game ⟨p_{j} : 1 ≤ j ≤ ω⟩ defined as follows. We will arrange that p_{j} is in N for all j < ω. Let Player I begin the game by playing a condition p_{1} in N such that x_{0} is in the maximal element of p_{1}. Suppose ⟨p_{j} : 1 ≤ j ≤ i⟩ is defined where 1 ≤ i < ω. If i = 2n is even, let Player I choose some p_{i+1} ≤ p_{i} in ℚ ∩ N such that the maximal element of p_{i+1} contains x_{n} as an element. If i is odd, let Player II play p_{i+1} according to σ. Since σ is in N, p_{i+1} is in N. This defines ⟨p_{i} : i < ω⟩. Let p_{ω} be Player II’s response to this run of the game according to σ.

Let ⋃{p_{n} : n < ω} = ⟨a_{i} : i < γ⟩ and let a = ⋃{a_{i} : i < γ}. For each n < ω, p_{n} is in N. Since p_{n} is countable, p_{n} ⊆ N. So for i < γ, a_{i} is in N, and hence a_{i} ⊆ N since a_{i} is countable. Therefore a ⊆ N ∩ H(ω_{2})^{V}. On the other hand, if z is in N ∩ H(ω_{2})^{V}, then z = x_{n} for some n, and so for some i < γ, z is in a_{i}. Hence a = N ∩ H(ω_{2})^{V}. But then p_{ω}(γ) is equal to N ∩ H(ω_{2})^{V}, since p_{ω} is a continuous sequence. By the definition of ℚ, N ∩ H(ω_{2})^{V} is in V, which is a contradiction.

Here is another way of viewing this argument, pointed out by the referee. The poset Add(ω) adds stationarily many new countable subsets of H(ω_{2})^{V}. The poset ℚ̇ then adds a club set which is disjoint from this stationary set, thus destroying its stationarity. If ℚ̇ is ω_{1}-strategically closed, then it is proper. But proper forcing posets cannot destroy the stationarity of stationary collections of countable sets.

3. Forcing Iteration with Mixed Support

We describe a general schema for a mixed support forcing iteration. Fix a regular cardinal μ such that μ^{<μ} = μ and let κ be a regular cardinal greater than μ such that for all ζ < κ, ζ^{<μ} < κ.

Let us consider a forcing iteration

⟨ℙ_{i}, ℚ̇_{j} : i ≤ α, j < α⟩,

with two partial orderings ≤ and ≤^{∗}. The iteration will alternate between forcing with Add(μ) at even stages, and other posets at odd stages. The iteration will be μ-closed, and therefore the poset Add(μ) is the same in both the ground model and in any intermediate extension. Specifically, suppose the iteration satisfies the following description.

(A) If i < α is even, ℙ_{i} forces ℚ̇_{i} = Add(μ).

(B) If i < α then ℙ_{i} forces ℚ̇_{i} is μ-closed.

(C) If j < α is odd, then ℙ_{j−1} forces that ⟨ ℚ̇_{j−1} ∗ℚ̇_{j}, ≤^{∗} ⟩ = ⟨Add(μ)∗ℚ̇_{j}, ≤^{∗}⟩ is κ-strategically closed.

(D) If i ≤ α is a limit ordinal, ℙ_{i} consists of all partial functions p : i → V such that p ↾ j ∈ ℙ_{j} for j < i, |dom(p) ∩ Even| < μ, and |dom(p) ∩ Odd| < κ.

(E) For i ≤ α and p and q in ℙ_{i}, q ≤ p in ℙ_{i} iff for all γ in the domain of p, γ is in the domain of q and q ↾ γ ⊩ q(γ) ≤ p(γ).

(F) For i ≤ α and p and q in ℙ_{i}, q ≤^{∗} p in ℙ_{i} iff q ≤ p, dom(p) ∩ Even = dom(q) ∩ Even, and for all γ in dom(p) ∩ Even, q ↾ γ forces q(γ) = p(γ).

We will prove that such an iteration preserves all cardinals and cofinalities less than or equal to κ. Hence by recursion, conditions (A), (B), and (C) make sense. Also, since such an iteration is μ-closed, the cardinal arithmetic assumptions we made in the ground model, that μ^{<μ} = μ and ζ^{<μ} < κ for all ζ < κ, will remain true in the extension.

For β ≤ α, let ℙ′_{β} be the subset of ℙ_{β} consisting of conditions p such that for all even i in the domain of p, there is x in Add(μ) such that p(i) is the canonical ℙ_{i}-name for x. Note that if β < γ ≤ α, then ℙ′_{β} ⊆ ℙ′_{γ}.

We make a list of the properties which we prove ℙ_{α} satisfies.

(I) ⟨ℙ_{α}, ≤⟩ is μ-closed.

(II) ⟨ℙ_{α}, ≤^{∗}⟩ is κ-strategically closed.

(III) ℙ′_{α} is a μ-closed dense subset of ℙ_{α}.

(IV) ℙ_{α} is <κ-proper for IA(μ).

(V) If p is in ℙ′_{α} and p ⊩ β̇ is an ordinal, then there is q ≤^{∗} p and a set x in V with size less than or equal to μ such that q ⊩ β̇ ∈ x̌.

(VI) ℙ_{α} is (<κ, ∞, μ)-distributive.

(VII) ℙ_{α} preserves all cardinals and cofinalities less than or equal to κ.

(VIII) For every regular cardinal λ less than or equal to κ, ℙ_{α} preserves the stationarity of all stationary subsets of λ.

(IX) ℙ_{α} forces that whenever X ⊆ V and for all A in ([V]^{<κ})^{V}, X ∩ A ∈ V, then X is in V.

The most difficult properties to verify are (IV), (V), and (IX), so we prove these last. Properties (I), (II), and (III) have fairly standard proofs.

Lemma 3.1. The posets ⟨ℙ_{α}, ≤⟩ and ⟨ℙ′_{α}, ≤⟩ are μ-closed.

Proof. Suppose ⟨p_{i} : i < ξ⟩ is a descending sequence in ℙ_{α}, where ξ < μ is a limit ordinal. We define a lower bound q in ℙ_{α} with domain equal to X = ⋃{dom(p_{i}) : i < ξ}. Suppose q ↾ γ is defined and is below p_{i} ↾ γ for all i < ξ. If γ is in X, then q ↾ γ forces that ⟨p_{i}(γ) : i < ξ, γ ∈ dom(p_{i})⟩ is a descending sequence in ℚ̇_{γ}. By property (B), choose a ℙ_{γ}-name q(γ) for a lower bound of this sequence. If each p_{i} is in ℙ′_{α}, then also make sure that for all even γ in X, q(γ) is the canonical ℙ_{γ}-name for the union of {p_{i}(γ) : γ ∈ dom(p_{i})}. □

Lemma 3.2. The poset ⟨ℙ_{α}, ≤^{∗}⟩ is κ-strategically closed.

Proof. By property (C), for each odd γ + 1 < α, let σ̇_{γ} be a ℙ_{γ}-name for a strategy for Player II in the game G(⟨Add(μ)∗ℚ̇_{γ+1}, ≤^{∗}⟩, κ). Define a strategy σ for Player II in the game G(⟨ℙ_{α}, ≤^{∗}⟩, κ) as follows. Suppose a partial play ⟨p_{i} : 1 ≤ i < ζ⟩ of the game is given where ζ is even. Let X = ⋃{dom(p_{i}) : 1 ≤ i < ζ}. For each odd γ + 1 in X, let i_{γ} be the least ordinal i < ζ such that γ + 1 ∈ dom(p_{i}). Assume inductively that for each odd γ + 1 in X, i_{γ} is odd, and for all i_{γ} ≤ j < ζ, p_{j} ↾ γ forces that ⟨p_{k}(γ) ∗ p_{k}(γ + 1) : i_{γ} ≤ k ≤ j⟩ enumerates a partial play in the game G(⟨Add(μ)∗ℚ̇_{γ+1}, ≤^{∗}⟩, κ) according to σ̇_{γ}, where we interpret p_{k}(γ) to be 1_{Add(μ)} if γ is not in dom(p_{k}). Now let σ instruct Player II to play the condition p_{ζ} with domain equal to X such that p_{ζ} ↾ Even = p_{1} ↾ Even and for all odd γ + 1 in X, p_{ζ} ↾ γ forces p_{ζ}(γ)∗p_{ζ}(γ+1) is Player II’s response to ⟨p_{k}(γ)∗p_{k}(γ+1) : i_{γ} ≤ k < ζ⟩ according to σ̇_{γ}. □

Lemma 3.3. For β ≤ α, ℙ′_{β} is a dense subset of ℙ_{β}.

Proof. We prove the statement by induction on β. Note that for all β ≤ α, ℙ_{β} satisfies (A) – (F), so by Lemma 3.1, ⟨ℙ_{β}, ≤⟩ and ⟨ℙ′_{β}, ≤⟩ are μ-closed. Suppose β ≤ α and for all γ < β, ℙ′_{γ} is dense in ℙ_{γ}. Let p in ℙ_{β} be given.

First assume β is a successor ordinal γ + 1. If γ is not in the domain of p, then p is in ℙ_{γ}, so by the induction hypothesis choose q ≤ p in ℙ′_{γ}. Then q ≤ p in ℙ_{β} and q is in ℙ′_{β}. Otherwise first find p′ ≤ p ↾ γ in ℙ′_{γ} such that for some x in Add(μ), p′ forces p(γ) = x̌. This is possible by the induction hypothesis and the fact that since ℙ_{γ} is μ-closed, ℙ_{γ} forces Add(μ) is the same poset in both the ground model and the extension. Then p′ x̌ ≤ p and p′ x̌ is in ℙ′_{β}.

Assume now that β is a limit ordinal. If cf(β) ≥ μ, then by property (D) there is ξ < β such that dom(p) ∩ Even ⊆ ξ. By the induction hypothesis, find q ≤ p ↾ ξ in ℙ′_{ξ}. Then q (p ↾ [ξ, β)) is a refinement of p in ℙ′_{β}.

Now suppose cf(β) < μ. Fix an increasing and continuous sequence ⟨ξ_{i} : i < cf(β)⟩ cofinal in β with ξ_{0} = 0, and let ξ_{cf(β)} = β. Define by induction a descending sequence ⟨p_{i} : i ≤ cf(β)⟩, with p_{0} = p, so that p_{i} ↾ ξ_{i} is in ℙ′_{ξ_{i}}. Given p_{i}, apply the induction hypothesis and choose q ≤ p_{i} ↾ ξ_{i+1} in ℙ′_{ξ_{i+1}}, and let p_{i+1} = q (p ↾ [ξ_{i+1}, β)). Suppose δ ≤ cf(β) is a limit ordinal and p_{i} is defined for all i < δ. For each i < δ, p_{i} ↾ ξ_{i} is in ℙ′_{ξ_{i}}, and ℙ′_{ξ_{i}} is a subset of ℙ′_{ξ_{δ}}. Since ℙ′_{ξ_{δ}} is μ-closed, let q be a lower bound of ⟨p_{i} ↾ ξ_{i} : i < δ⟩ in ℙ′_{ξ_{δ}}. We claim that q is below p_{i} ↾ ξ_{δ} for i < δ. So let i < δ. Since ξ_{δ} is a limit ordinal, it suffices to show that q ≤ p_{i} ↾ ζ for all ζ < ξ_{δ}. Given ζ < ξ_{δ}, choose i < j < δ large enough so that ζ ≤ ξ_{j}. Then q ≤ p_{j} ↾ ξ_{j} ≤ p_{j} ↾ ζ ≤ p_{i} ↾ ζ. Now let p_{δ} = q (p ↾ [ξ_{δ}, β)). This completes the construction. The condition p_{cf(β)} is a refinement of p in ℙ′_{β}. □

This completes the proof of properties (I), (II), and (III).

Since the proofs of properties (IV) and (V) are complicated, let us first show that property (V) implies properties (VI), (VII), and (VIII). So assume that whenever p is in ℙ′_{α} and p forces β̇ is an ordinal, there is q ≤^{∗} p and a set x with size less than or equal to μ such that q forces β̇ is in x̌. Note that this implies the following property:

(V′) If p is in ℙ′_{α}, q ≤^{∗} p, and q ⊩ β̇ is an ordinal, then there is r ≤^{∗} q and a set x with size less than or equal to μ such that r ⊩ β̇ ∈ x̌.

The point is, if p is in ℙ′_{α} and q ≤^{∗} p, then there is q′ in ℙ′_{α} which is equivalent to q. Namely, let q′ ↾ Odd = q ↾ Odd and q′ ↾ Even = p ↾ Even. Now apply (V) to q′ and β̇.

Fix for the remainder of the section a winning strategy σ for Player II in the game G(⟨ℙ_{α}, ≤^{∗}⟩, κ).

Lemma 3.4. ⟨ℙ_{α}, ≤⟩ is (<κ, ∞, μ)-distributive.

Proof. Let ξ < κ be a limit ordinal, and suppose p is a condition in ℙ′_{α} which forces that ḟ : ξ → On is a function. We define a partial play ⟨p_{i} : 1 ≤ i ≤ ξ⟩ in the game G(⟨ℙ_{α}, ≤^{∗}⟩, κ). Let Player I play p_{1} = p. Suppose ⟨p_{i} : 1 ≤ i < γ⟩ is defined. If γ is even, let p_{γ} be Player II’s response according to σ. Suppose γ = 2 · ζ + 1 is odd. Applying (V′), let Player I choose p_{γ} ≤^{∗} p_{γ−1} such that for a set x_{ζ} with size less than or equal to μ, p_{γ} forces ḟ(ζ) is in x̌_{ζ}. This completes the construction. Let q = p_{ξ}, and define h : ξ → V by h(i) = x_{i}. Then q forces ḟ(i) ∈ h(i) for all i < ξ. □

We note that (VIII) implies (VII). For suppose in some generic extension V [G], there is an ordinal ξ ≤ κ which changed its cofinality. Assume ξ is the least such ordinal. In V [G], cf(ξ) = cf(cf^{V} (ξ)). So by the minimality of ξ, cf^{V} (ξ) = ξ, and ξ is regular in V. Suppose cf(ξ) = β in V [G]. Then β is regular in both V and V [G]. Since β < ξ, the set S = ξ ∩ cof(β) is stationary in ξ in V. But in V [G] there is a club in ξ with order type β, and the limits points of this club have cofinality less than β. Thus S becomes non-stationary in V [G], contradicting (VIII). Now if any cardinals are collapsed, the first collapsed cardinal would have to be a successor cardinal, which would change its cofinality. So indeed (VIII) implies (VII).

Proposition 3.5. For all regular λ ≤ κ and any stationary set S ⊆ λ, ℙ_{α} forces S is stationary.

Proof. If λ < μ then ℙ_{α} does not even add any subsets to λ. If λ = μ, ℙ_{α} preserves stationary subsets of λ since it is μ-closed. So assume μ < λ ≤ κ.

Suppose p is in ℙ′_{α} and p forces Ċ is club in λ. We find q ≤ p which forces S ∩ Ċ is non-empty. We define a sequence ⟨p_{i}, β_{i}, x_{i} : 1 ≤ i < λ⟩, where the sequence of p_{i}’s constitutes a partial play of the game G(⟨ℙ_{α}, ≤^{∗}⟩, κ) according to σ. Let p_{1} = p, β_{1} = 0, and x_{1} = ∅.

Suppose ⟨p_{i}, β_{i}, x_{i} : 1 ≤ i < γ⟩ is defined for some γ < λ, where ⟨β_{i} : 1 ≤ i < γ⟩ is an increasing and continuous sequence of ordinals less than λ. If γ is even, let p_{γ} be Player II’s response according to σ to the sequence of conditions defined thus far. Then let β_{γ} = sup({β_{i} + 1 : 1 ≤ i < γ}) and x_{γ} = ∅.

Suppose γ is odd. Applying (V′), choose p_{γ} ≤^{∗} p_{γ−1} and a set x_{γ} with size less than or equal to μ such that p_{γ} forces min(Ċ \ β_{γ−1}) ∈ x̌_{γ}. Now choose an ordinal β_{γ} in λ larger than both β_{γ−1} and sup(x_{γ}).

By construction the sequence ⟨β_{i} : 1 ≤ i < λ⟩ is strictly increasing and continuous, and so clearly is cofinal in λ. Since S is stationary, fix a limit ordinal δ < λ such that β_{δ} is in S. For every odd γ < δ, p_{γ} forces Ċ ∩ [β_{γ−1}, β_{γ}) is non-empty. Therefore p_{δ} forces β_{δ} is a limit point of Ċ, and hence is in Ċ ∩ S. □

We have proven properties (VI), (VII), and (VIII), assuming (V). It remains to show (IV), (V), and (IX).

The following basic lemma is helpful in the remaining proofs.

Lemma 3.6. Suppose p and q are conditions in ℙ_{α} such that for all γ in dom(p) ∩ dom(q), either p ↾ γ or q ↾ γ forces p(γ) and q(γ) are compatible in ℚ̇_{γ}. Then p and q are compatible.

Proof. Define u below p and q by inductively defining u ↾ γ for γ < α. Assume u ↾ γ is defined and is a condition below p ↾ γ and q ↾ γ. If γ is not in dom(p) ∪ dom(q), then γ is not in the domain of u. If γ is in dom(p) \ dom(q), let u(γ) = p(γ), and if γ is in dom(q) \ dom(p), let u(γ) = q(γ). Suppose γ is in dom(p) ∩ dom(q). By assumption and the induction hypothesis, we can fix a ℙ_{γ}-name u(γ) such that u ↾ γ forces u(γ) ≤ p(γ), q(γ). □

Proposition 3.7. Let θ be a regular cardinal much larger than ℙ_{α}, and let A denote the structure ⟨H(θ), ∈, <_{θ}, ℙ_{α}, σ⟩, where <_{θ} is a well-ordering of H(θ). Suppose N is an elementary substructure of A in P_{κ}(H(θ)), N ∩ κ ∈ κ, and N is internally approachable. Let p be in N ∩ ℙ′_{α}. Then:

(1) There is q ≤^{∗} p such that for any dense subset D of ⟨ℙ_{α}, ≤^{∗}⟩ in N, there is u in N ∩ D such that q ≤^{∗} u.

(2) If, in addition, N is in IA(μ), then there is q ≤^{∗} p which is N-generic for the forcing ⟨ℙ_{α}, ≤⟩.

In particular, ⟨ℙ_{α}, ≤⟩ is <κ-proper for IA(μ).

Conclusion (1) in the proposition asserts the existence of a strongly N-generic condition for the poset ℙ_{α} with the weak ordering ≤^{∗}. Conclusion (2) asserts the existence of an N-generic condition for the poset ℙ_{α} with the usual ordering ≤. The proof of (1) is an easy variation of the standard technique for constructing generic conditions for closed forcings over internally approachable models.

Proof. (1) Let p be in N ∩ ℙ′_{α}. Fix an internally approachable chain ⟨N_{i} : i < ξ⟩, where ξ < κ is a limit ordinal, with union N, such that p is in N_{0}, and for i < ξ, N_{i} is an elementary substructure of A with N_{i} ∩ κ ∈ κ. Write N = N_{ξ}. Define by induction a sequence ⟨p_{i} : 1 ≤ i ≤ ξ⟩ which is partial play in the game G(⟨ℙ_{α}, ≤^{∗} ⟩, κ), such that for all γ ≤ ξ, ⟨p_{i} : 1 ≤ i < γ⟩ is definable in A from ⟨N_{i} : i < γ⟩ and p. Let p_{1} = p.

Suppose ⟨p_{i} : 1 ≤ i < γ⟩ is defined as required. If γ is even, let p_{γ} be Player II’s response to this sequence according to σ. By the induction hypothesis, clearly ⟨p_{i} : 1 ≤ i < γ + 1⟩ is definable in A from ⟨N_{i} : i < γ + 1⟩ and p as required. Assume γ is odd. Since ⟨ℙ_{α}, ≤^{∗}⟩ is κ-strategically closed, it is < κ-distributive.

But |N_{γ−1}| < κ. So let p_{γ} be the <_{θ}-least ≤^{∗}-refinement of p_{γ−1} which is in the intersection of all dense open subsets of ⟨ℙ_{α}, ≤^{∗}⟩ in N_{γ−1}. Clearly ⟨p_{i} : 1 ≤ i < γ+1⟩ is definable in A from ⟨N_{i} : i < γ + 1⟩ and p. Now if ⟨p_{i} : 1 ≤ i < δ⟩ is defined for a limit ordinal δ ≤ ξ, the definition given above witnesses that this sequence is definable in A from ⟨N_{i} : i < δ⟩ and p.

This completes the construction of ⟨p_{i} : 1 ≤ i ≤ ξ⟩. Every proper initial segment of the sequence ⟨p_{i} : 1 ≤ i < ξ⟩ is definable in A from p and a proper initial segment of ⟨N_{i} : i < ξ⟩, and so is in N. In particular, p_{i} is in N for all i < ξ. Now let q = p_{ξ}. By construction, for every dense open subset D of ⟨ℙ_{α}, ≤^{∗}⟩ in N, there is i < ξ such that p_{i} is in N ∩ D, and q ≤^{∗} p_{i}. This implies the conclusion of (1).

(2) Now suppose in addition that N is in IA(μ). Then we can take ξ = μ in the proof of (1). It follows that N^{<μ} ⊆ N. For if y is a subset of N with size less than μ, there is i < μ such that y ⊆ N_{i}. But by assumption on κ, |[N_{i}]^{<μ}| < κ. Since N ∩ κ ∈ κ, this implies [N_{i}]^{<μ} ⊆ N. So y is in N.

Recall that p is a condition in N ∩ ℙ′_{α}. By (1), we can fix q ≤^{∗} p such that for any dense subset D of ⟨ℙ_{α}, ≤^{∗}⟩ in N, there is u in N ∩D such that q ≤^{∗} u. Suppose for a contradiction that q is not N-generic for ⟨ℙ_{α}, ≤⟩. Then there is a dense open subset E of ⟨ℙ_{α}, ≤⟩ in N such that q does not force N ∩ E ∩ Ġ is non-empty. Fix r ≤ q in ℙ′_{α} which forces that N ∩ E ∩ Ġ is empty. It follow that for any y in N ∩ E, r and y are incompatible. Let g : α → Add(μ) be the partial function such that dom(g) = dom(r) ∩ Even ∩ N and for all i in dom(g), r(i) is the canonical ℙ_{i}-name for g(i). By property (D), g has size less than μ. For all i in the domain of g, i is in N, and so ℙ_{i} is in N. Since Add(μ) ⊆ N, it follows that g(i) is in N. Since N^{<μ} ⊆ N, g is in N.

Let D be the set of u in ℙ_{α} such that either u is ≤^{∗}-incompatible with p, or u ≤^{∗} p and there is y ≤ u in E ∩ ℙ′_{α} such that:

(i) dom(g) ⊆ dom(y), and for all i in dom(g), y(i) is the canonical ℙ_{i}-name for a condition in Add(μ) which extends g(i),

(ii) y ↾ Odd = u ↾ Odd. Note that by elementarity D is in N.

We claim that D is dense in ⟨ℙ_{α}, ≤^{∗}⟩. So let s in ℙ_{α} be given. If s is ≤^{∗}- incompatible with p, then s is in D. Otherwise fix t ≤^{∗} p, s.

Define w by letting w ↾ Odd = t ↾ Odd, dom(w)∩Even = dom(g), and for all i in dom(g), w(i) is the canonical ℙ_{i}-name for g(i). We claim w is a condition below t. Since p ∈ N and |dom(p)| < κ, dom(p) ⊆ N. But r ≤ p and dom(r) ∩ Even ∩ N = dom(g). So dom(p)∩Even ⊆ dom(g). Since t ≤^{∗} p, dom(t)∩Even = dom(p)∩Even. Hence dom(t) ⊆ dom(w). Suppose w ↾ γ is a condition below t ↾ γ where γ is in the domain of w, and we show that w ↾ γ + 1 is a condition below t ↾ γ + 1. If γ is odd then w(γ) = t(γ). Suppose γ is even. If γ is not in the domain of t then there is nothing to check. Otherwise γ is in dom(p). Since r ≤ p, r ↾ γ forces r(γ) ≤ p(γ). But this implies g(γ) is below the condition in Add(μ) named by p(γ). Since t ↾ γ forces p(γ) = t(γ), w ↾ γ forces w(γ) ≤ t(γ).

Now extend w to y which is in E ∩ ℙ′_{α}. Without loss of generality we can assume that for all odd γ in dom(t), ℙ_{γ} forces y(γ) ≤ t(γ); for if y did not already have this property, there is a condition equivalent to y which does. Now define u so that u ↾ Even = t ↾ Even and u ↾ Odd = y ↾ Odd. Then clearly y ≤ u ≤^{∗} t. So u ≤^{∗} s, and y witnesses that u is in D; namely, (ii) is by definition, and (i) follows from the fact that y is in ℙ′_{α} and y ≤ w.

Since D is in N and is dense in ⟨ℙ_{α}, ≤^{∗}⟩, by (1) there is u in N ∩ D such that q ≤^{∗} u. Since q ≤^{∗} u, p, u is ≤^{∗}-compatible with p. So by the definition of D and elementarity, there is y ≤ u in E ∩ ℙ′_{α} ∩ N satisfying (i) and (ii) above. Since y is in N ∩ E, by assumption r and y are incompatible. We will get a contradiction by showing that r and y are in fact compatible.

If r and y are incompatible, then by Lemma 3.6 let γ be the least ordinal in dom(r) ∩ dom(y) such that r ↾ γ does not force r(γ) and y(γ) are compatible. Note that dom(y) ⊆ N since |dom(y)| < κ. So γ is in N. If γ is even then γ is in dom(r) ∩ Even ∩ N = dom(g). Then by (i), ℙ_{γ} forces y(γ) is below r(γ), which is impossible. So γ is odd. But then by (ii), u(γ) = y(γ). Since r ≤ q ≤ u, r ↾ γ forces r(γ) ≤ u(γ) = y(γ), which is again impossible. So r and y are compatible. □

Thus we have proven property (IV).

Corollary 3.8. The poset ℙ_{α} has the κ-covering property. In fact, if p is in ℙ′_{α} and p forces ȧ is a set of ordinals with size less than κ, then there is q ≤^{∗} p and a set x with size less than κ such that q ⊩ ȧ ⊆ x̌.

Proof. Suppose p is in ℙ′_{α} and p forces ȧ is a set of ordinals with size less than κ. Let θ be a regular cardinal much larger than ℙ_{α} and ȧ, and let A denote the structure ⟨H(θ), ∈, <_{θ}, ℙ_{α}, σ⟩, where <_{θ} is a well-ordering of H(θ). Fix an elementary substructure N of A such that |N| < κ, N ∩ κ ∈ κ, p and ȧ are in N, and N is in IA(μ). Applying Proposition 3.7(2), let q ≤^{∗} p be N-generic. Let x = N ∩ On. If G is a generic filter for ℙ_{α} which contains q, then N[G] ∩ On = N ∩ On = x. In particular, N[G] ∩ κ = N ∩ κ ∈ κ. Since a = ȧ^{G} has size less than κ and N[G] ≺ H(θ)^{V [G]}, a ⊆ N[G] ∩ On = x. So q forces ȧ ⊆ x̌. □

Proposition 3.9. If p is in ℙ′_{α} and p forces β̇ is an ordinal, then there is q ≤^{∗} p and a set x with size less than or equal to μ such that q ⊩ β̇ ∈ x̌.

Proof. If κ = μ^{+} then the statement follows from Corollary 3.8, letting ȧ = {β̇}. So assume κ > μ^{+}.

Suppose for a contradiction that there is no q ≤^{∗} p and x as in the conclusion of the proposition. We define a sequence ⟨p_{i}, q_{i}, β_{i} : 1 ≤ i < μ^{+}⟩, where the p_{i}’s constitute a partial play in G(⟨ℙ_{α}, ≤^{∗}⟩, κ) according to σ, and the q_{i}’s are in ℙ′_{α}. Let p_{1} = p, β_{1} = 0, and q_{i} = 1_{ℙ_{α}}. Suppose γ < μ^{+} is even and for all 1 ≤ i < γ, p_{i}, β_{i}, and q_{i} are defined. Let p_{γ} be Player II’s response to the play ⟨p_{i} : 1 ≤ i < γ⟩ according to σ, and let β_{γ} = 0 and q_{γ} = 1_{ℙ_{α}}.

Suppose 2 ≤ γ < μ^{+} is even and p_{i}, β_{i}, and q_{i} are defined for all 1 ≤ i ≤ γ. Since p_{γ} ≤^{∗} p, by assumption p_{γ} does not force that β̇ is in the set {β_{i} : 1 ≤ i ≤ γ}. So we can fix q_{γ+1} ≤ p_{γ} in ℙ′_{α} and an ordinal β_{γ+1} not in the set {β_{i} : 1 ≤ i ≤ γ} such that q_{γ+1} forces β̇ = β_{γ+1}. Moreover, choose q_{γ+1} to have the property that for all i in dom(p_{γ}) ∩ Odd, ℙ_{i} forces q_{γ+1}(i) ≤ p_{γ}(i). Now define p_{γ+1} by letting p_{γ+1} ↾ Even = p_{γ} ↾ Even and p_{γ+1} ↾ Odd = q_{γ+1} ↾ Odd. Clearly p_{γ+1} ≤^{∗} p_{γ}.

Since κ > μ^{+}, let q be Player II’s response to the play ⟨p_{i} : 1 ≤ i < μ^{+}⟩ according to σ. Note that {q_{i} : i ∈ μ^{+} ∩ Odd} is an antichain. For if i < j < μ^{+} are odd, then β_{i} ≠ β_{j}, q_{i} ⊩ β̇ = β̌_{i}, and q_{j} ⊩ β̇ = β̌_{j}. Using the fact that μ^{<μ} = μ, apply the Δ-System Lemma to the family {dom(q_{i}) ∩ Even : i ∈ μ^{+} ∩ Odd} to find an unbounded set X ⊆ μ^{+} ∩ Odd and a set a such that for i < j in X, dom(q_{i}) ∩ dom(q_{j}) ∩ Even = a. Since a has size less than μ, Add(μ) has size μ, and each q_{i} is in ℙ′_{α}, there are distinct 1 < i < j in X such that q_{i} ↾ a = q_{j} ↾ a.

We get a contradiction by showing that q_{i} and q_{j} are compatible. Otherwise, by Lemma 3.6 let γ be least such that γ is in dom(q_{i}) ∩ dom(q_{j}) and q_{j} ↾ γ does not force q_{i}(γ) and q_{j}(γ) are compatible. Clearly γ is not even, since otherwise it is in a and therefore q_{i}(γ) = q_{j}(γ). So γ is odd. By construction, q_{i}(γ) = p_{i}(γ). But q_{j} ≤ p_{j−1} ≤ p_{i}, so q_{j} ↾ γ forces q_{j}(γ) ≤ p_{i}(γ) = q_{i}(γ), which is not possible. So indeed q_{i} and q_{j} are compatible, giving us a contradiction. □

This completes the proof of property (V). It remains only to prove property (IX). We use the following lemma.

Lemma 3.10. Let p and q be conditions in ℙ′_{α} and t a condition in ℙ_{α} such that p, q ≤ t. Then there are p′ ≤ p and q′ ≤ q in ℙ′_{α} such that p′ ↾ Odd = q′ ↾ Odd and for all odd i in dom(t), ℙ_{i} forces p′(i) = q′(i) is below t(i).

Proof. Without loss of generality assume p satisfies the property that whenever i is an odd ordinal in dom(t), ℙ_{i} forces p(i) ≤ t(i), since otherwise we can replace p with an equivalent condition which does satisfy this property. We also assume the same for q. Now define p′ and q′ as follows. First choose p′(0) ≤ p(0) and q′(0) ≤ q(0) in Add(μ) so that p′(0) and q′(0) are incompatible. Suppose β > 0 is an even ordinal and p′ ↾ β and q′ ↾ β are defined. Let β be in dom(p′) iff β is in dom(p), in which case let p′(β) = p(β), and similarly with q′. Suppose γ is odd and p′ ↾ γ and q′ ↾ γ are defined. If γ is not in dom(p) ∪ dom(q) then γ is not in the domain of either p′ or q′. If γ is in dom(p) \ dom(q) then let p′(γ) = q′(γ) = p(γ), and if γ is in dom(q) \ dom(p) then let p′(γ) = q′(γ) = q(γ). Suppose γ is in dom(p) ∩ dom(q). Let ẋ_{γ} be a ℙ_{γ}-name such that ℙ_{γ} forces “ẋ_{γ} = p(γ) if p′ ↾ γ is in Ġ_{γ}, and ẋ_{γ} = q(γ) otherwise”. Then ẋ_{γ} is well-defined because p′ ↾ γ and q′ ↾ γ are incompatible, due to the choice of p′(0) and q′(0). Let p′(γ) = q′(γ) = ẋ_{γ}. □

Proposition 3.11. The poset ℙ_{α} forces that whenever X ⊆ V and for all A in ([V]^{<κ})^{V}, X ∩ A ∈ V, then X is in V.

Proof. Suppose for a contradiction that a condition p in ℙ′_{α} forces Ẋ ⊆ V is a set such that for all A in ([V]^{<κ})^{V}, Ẋ ∩ A ∈ V, but Ẋ is not in V.

Let θ be a regular cardinal much larger than ℙ_{α} and Ẋ, and let A denote the structure ⟨H(θ), ∈, <_{θ}, ℙ_{α}, σ⟩, where <_{θ} is a well-ordering of H(θ). Fix an elementary substructure N of A such that |N| < κ, N ∩ κ ∈ κ, p and Ẋ are in N, and N is in IA(μ). Then N^{<μ} ⊆ N. By Proposition 3.7 (1), fix q ≤^{∗} p such that for any dense subset D of ⟨ℙ_{α}, ≤^{∗}⟩ in N, there is u in N ∩ D such that q ≤^{∗} u.

Since N is in ([V]^{<κ})^{V}, fix a condition r ≤ q in ℙ′_{α} which decides Ẋ ∩ N. Let g : α → Add(μ) be the partial function such that dom(g) = dom(r) ∩ Even ∩ N and for all i in dom(g), r(i) is the canonical ℙ_{i}-name for g(i). Since N^{<μ} ⊆ N, g is in N.

Let D be the set of u in ℙ_{α} such that either u is ≤^{∗}-incompatible with p, or u ≤^{∗} p and there are y_{0}, y_{1} ≤ u in ℙ′_{α} such that:

(1) for k = 0, 1, dom(g) ⊆ dom(y_{k}) and for all i in dom(g), y_{k}(i) is the canonical ℙ_{i}-name for a condition in Add(μ) which extends g(i),

(2) y_{0} ↾ Odd = y_{1} ↾ Odd = u ↾ Odd,

(3) there is a such that y_{0} ⊩ ǎ ∈ Ẋ and y_{1} ⊩ ǎ ∉ Ẋ. By elementarity D is in N.

We claim that D is dense in ⟨ℙ_{α}, ≤^{∗}⟩. Let s in ℙ_{α} be given. If s is ≤^{∗}- incompatible with p then s is in D. Otherwise fix t ≤^{∗} p, s. Define w by letting w ↾ Odd = t ↾ Odd, dom(w) ∩ Even = dom(g), and for all i in dom(g), w(i) is the canonical ℙ_{i}-name for g(i). Exactly as in the proof of Proposition 3.7 (2), w is a condition below t. Since w ≤ p, w forces Ẋ is not in V. In particular, there is a such that w does not decide whether or not a is in Ẋ. So there are w_{0}, w_{1} ≤ w in ℙ′_{α} such that w_{0} ⊩ ǎ ∈ Ẋ and w_{1} ⊩ ǎ ∉ Ẋ. Now apply Lemma 3.10 to obtain y_{0} ≤ w_{0} and y_{1} ≤ w_{1} in ℙ′_{α} such that y_{0} ↾ Odd = y_{1} ↾ Odd and for all i in dom(t) ∩ Odd, ℙ_{i} forces y_{0}(i) = y_{1}(i) is below t(i). Define u by letting u ↾ Even = t ↾ Even and u ↾ Odd = y_{0} ↾ Odd = y_{1} ↾ Odd. Clearly then y_{0}, y_{1} ≤ u ≤^{∗} t ≤^{∗} s and (1), (2), and (3) are satisfied.

By the choice of q, there is u in D ∩ N such that q ≤^{∗} u. Since q ≤^{∗} u, p, u and p are ≤^{∗}-compatible. By elementarity and the definition of D we can fix y_{0} and y_{1} below u in N satisfying (1), (2), and (3). Fix a in N as described in (3). Both y_{0} and y_{1} are compatible with r, as follows from (1), (2), and Lemma 3.6, exactly as in the end of the proof of Proposition 3.7 (2). But r decides Ẋ ∩ N, so it decides whether or not a is in Ẋ. So r cannot be compatible with both y_{0} and y_{1}. □

4. Mitchell’s Model

For the convenience of the reader, we give a sketch of Mitchell’s construction of a model with no Aronszajn trees on a successor cardinal^{1}, using the framework described in Section 3. None of the results of this section are due to the author. We give several original applications of the iteration schema of Section 3 in [6].

Let μ be an infinite cardinal such that μ^{<μ} = μ, and suppose κ > μ is a regular cardinal such that for all ζ < κ, ζ^{<μ} < κ. Suppose that λ > κ is a strongly inaccessible cardinal.

We define by induction a forcing iteration ⟨ℙ_{i}, ℚ̇_{j} : i ≤ λ, j < λ⟩ which satisfies (A) – (F) of Section 3. The only thing we need to specify is the value of ℚ̇_{i} when i is odd and check that (B) and (C) hold. Suppose ℙ_{i} is defined for some i < λ. If i is even, then by (A) we let ℚ̇_{i} be a ℙ_{i}-name for Add(μ). If i is odd, let ℚ̇_{i} be a ℙ_{i}-name for Coll(κ, κ^{+}). Then (B) is clear, and (C) is satisfied by Proposition 2.1. By Section 3, the iteration satisfies properties (I) – (IX). By (VII), ℙ_{λ} preserves all cardinals and cofinalities less than or equal to κ.

Since λ is strongly inaccessible, for all i < λ, ℙ_{i} has size less than λ. It follows by a standard argument that ℙ_{λ} is λ-c.c. Since we iterate collapses, every cardinal in the interval (κ, λ) is collapsed to have size κ. So λ is the successor of κ after forcing with ℙ_{λ}. Also ℙ_{λ} forces 2^{μ} = λ.

We show that if λ is a weakly compact cardinal, then ℙ_{λ} forces that there are no Aronszajn trees on λ = κ^{+}. Suppose for a contradiction there is p in ℙ_{λ} which forces that Ṫ is an Aronszajn tree on λ. We may assume that the nodes of Ṫ are exactly the ordinals below λ. Therefore without loss of generality we can assume Ṫ ⊆ V_{λ}. Let C be the club of α less than λ such that p forces (Ṫ ∩ V_{α})Ġ_{α} = Ṫ ↾ α.

Consider the structure ⟨V_{λ}, ∈, ℙ_{λ}, ⊩, p, Ṫ ⟩. This structure models the Π_{1}^{1} statement: for all A, if A is a ℙ_{λ}-name, then for all q ≤ p in ℙ_{λ}, q forces A is not a cofinal branch in Ṫ. Since λ is weakly compact, there is an inaccessible cardinal α < λ in C such that ⟨V_{α}, ∈, ℙ_{α}, ⊩, p, Ṫ ∩ V_{α}⟩ models the same statement.

Write ℙ_{λ} = ℙ_{α} ∗ ℙ̇_{α,λ}. Then ℙ_{α} forces that ℙ̇_{α,λ} is equivalent to a forcing iteration satisfying properties (A) – (F) of Section 3. Let G_{α} ∗ H be a generic filter for ℙ_{α} ∗ ℙ̇_{α,λ} which contains p. Since α is inaccessible, α is the successor of κ in V [G_{α}]. By assumption on α, in V [G_{α}] the tree S = (Ṫ ∩ V_{α})^{G_{α}} has no cofinal branch. In V [G_{α} ∗ H], T = Ṫ^{G_{α}∗H} is a tree on λ and T ↾ α = S.

1As noted in [7], the consistency of no Aronszajn trees on ω_{2} is due in part to J. Silver.

Let c be a node on level α of T in V [G_{α} ∗ H], and let b = {x ∈ T : x <_{T} c}. Then b is a cofinal branch through S, and therefore b is not in V [G_{α}]. Suppose A is in (V [G_{α}]^{<κ})^{V [G_{α}]}. Then A ∩ b is bounded in S, so there is z in b such that x <_{T} z for all x in A ∩ b. But then A ∩ b is exactly the set A ∩ {x ∈ S : x <_{T} z}, which is in V [G_{α}]. By Property (IX) of Section 3 applied to ℙ̇_{α,λ}, b is in V [G_{α}], which is a contradiction.

References

[1] U. Abraham and S. Shelah. Forcing closed unbounded sets. J. Symbolic Logic, 48(3):643–657, 1983.

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

[3] M. Foreman and M. Magidor. Large cardinals and definable counterexamples to the continuum hypothesis. Ann. Pure Appl. Logic, 76(1):47–97, 1995.

[4] M. Gitik and J. Krueger. Approachability at the second successor of a singular cardinal. Submitted.

[5] J. Krueger. Internally club and approachable for larger structures. Submitted.

[6] J. Krueger. Some applications of mixed support iterations. Submitted.

[7] W. Mitchell. Aronszajn trees and the independence of the transfer property. Ann. Math. Logic, 5:21–46, 1972/73.

[8] W. Mitchell. On the Hamkins approximation property. Ann. Pure Appl. Logic, 144(1-3):126– 129, 2006.

Department of Mathematics, University of California, Berkeley

   E-mail address: jkrueger@math.berkeley.edu
   URL: http://www.math.berkeley.edu/∼jkrueger