INTERNALLY CLUB AND APPROACHABLE FOR LARGER STRUCTURES

JOHN KRUEGER

Abstract. We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of P_{κ}(X), where κ ⊆ X. Suppose μ < κ, μ^{<μ} = μ, and κ is supercompact. Then there is a generic extension in which κ = μ^{++}, and for all regular λ ≥ μ^{++}, there are stationarily many N in [H(λ)]^{μ^{+}} which are internally club but not internally approachable.

Suppose μ is an infinite cardinal. A set N is internally approachable with length μ^{+} if N is the union of an increasing and continuous sequence ⟨N_{i} : i < μ^{+}⟩ of sets with size μ such that for all α < μ^{+}, ⟨N_{i} : i < α⟩ is in N. A related idea is that of an internally club set. A set N with size μ^{+} is internally club if N ∩ [N]^{μ} contains a club subset of [N]^{μ}. In other words, N is the union of an increasing and continuous sequence ⟨N_{i} : i < μ^{+}⟩ of sets with size μ such that each N_{i} is in N.

Foreman and Todorčević [3] asked whether the properties of being internally approachable and internally club are equivalent. In [5] we proved that under PFA, for all regular λ ≥ ω_{2} there are stationarily many structures N ≺ H(λ) with size ℵ_{1} such that N is internally club but not internally approachable. In this paper we generalize this result to larger structures.

Theorem 1. Suppose μ < κ, μ^{<μ} = μ, and κ is supercompact. Then there is a μ-closed, μ^{+}-proper forcing poset which collapses κ to become μ^{++}, and forces that for all regular λ ≥ μ^{++}, there are stationarily many N in [H(λ)]^{μ^{+}} which are internally club but not internally approachable.

In the model we construct to prove Theorem 1, we have that 2^{μ} = μ^{++}. In fact, if 2^{μ} = μ^{+}, then any elementary substructure N ≺ H(λ) with size μ^{+} and which contains μ^{+} is internally club iff it is internally approachable; this is shown at the end of the paper.

In Section 1 we review notation and some background material. Section 2 generalizes the idea of a fat subset of a regular cardinal κ to a fat subset of P_{κ}(X), where κ ⊆ X. Section 3 presents the basic forcing poset we use in our consistency result, and in Section 4 we describe how to iterate this poset with a mixed support forcing iteration. In Section 5 we prove Theorem 1.

1. Preliminaries

If κ is regular and κ ⊆ X, we say C ⊆ P_{κ}(X) is club if it is closed under unions of increasing sequences of length less than κ, and is cofinal. A set S ⊆ P_{κ}(X) is stationary if it has non-empty intersection with every club. We will use the fact that if C ⊆ P_{κ}(X) is club, and A ⊆ C is a directed set with size less than κ, then ⋃ A ∈ C (see Lemma 8.25 of [4] for a proof). By directed we mean that if a and b are in A, then there is c in A such that a ∪ b ⊆ c.

Date: November 2006.

If N is a set, ℙ is a forcing poset, and G is a filter on ℙ, then N[G] denotes the set {ȧ^{G} : ȧ ∈ N ∩ V^{ℙ}}. A filter G on ℙ is N-generic if for every dense set D ⊆ ℙ in N, N ∩ D ∩ G is non-empty. A condition q in ℙ is N-generic if q forces Ġ is N-generic, where Ġ is a name for the generic filter. Suppose λ is regular with ℙ ∈ H(λ), and N ≺ ⟨H(λ), ∈, ℙ⟩. Then for any condition q in ℙ, the following are equivalent: (1) q is N-generic, (2) for every dense set D ⊆ ℙ in N, N ∩ D is predense below q, (3) q forces N[Ġ] ∩ On = N ∩ On, and (4) q forces N[Ġ] ∩ V = N. Note that if q is N-generic, then for any set X, q forces N[Ġ] ∩ X̌ = N ∩ X̌.

Suppose ℙ is a forcing poset and λ is regular with ℙ ∈ H(λ). If G is generic for ℙ over V, then H(λ)^{V [G]} = H(λ)^{V} [G]. Suppose N ≺ ⟨H(θ), ∈, ℙ⟩ in V. If G is generic for ℙ over V, then N[G] ≺ H(θ)^{V [G]}.

Let ℙ be a forcing poset and μ a regular cardinal with μ^{<μ} = μ. Then ℙ is μ^{+}-proper if for any regular cardinal θ > 2^{|ℙ|} with ℙ in H(θ), if N is an elementary substructure of ⟨H(θ), ∈, ℙ⟩, N has size μ, and N^{<μ} ⊆ N, then for all p in N ∩ ℙ, there is q ≤ p which is N-generic. Any μ^{+}-proper forcing poset preserves μ^{+}. Note that if ℙ is μ^{+}-c.c. then any condition in ℙ is N-generic, since every maximal antichain of ℙ in N is actually a subset of N.

If μ is a regular cardinal and ℙ is a forcing poset, we say ℙ is μ-distributive if for any collection D of less than or equal to μ many dense open subsets of ℙ, ⋂ D is dense open. This property is equivalent to ℙ not adding any new sequences of ordinals with order type less than or equal to μ. If κ is a cardinal we say ℙ is < κ-distributive if ℙ is μ-distributive for all regular μ < κ.

Let ℙ be a forcing poset and μ a regular cardinal. We say ℙ is μ-closed if whenever ⟨p_{i} : i < ξ⟩ is a descending sequence of conditions in ℙ with ξ < μ, there is q in ℙ such that q ≤ p_{i} for all i < ξ. If A ⊆ ℙ, a greatest lower bound of A, or glb of A, is a condition q such that q ≤ p for all p in A, and whenever r ≤ p for all p in A, r ≤ q. We say ℙ is μ-glb closed if whenever ⟨p_{i} : i < ξ⟩ is a descending sequence of conditions in ℙ with ξ < μ, there exists a greatest lower bound for the set {p_{i} : i < ξ}.

2. Generalized Fat Sets

Let κ be a regular uncountable cardinal. Recall that a set A ⊆ κ is fat if for any club set C ⊆ κ and ξ < κ, A ∩ C contains a closed subset with order type at least ξ.

Fact 2.1 (Abraham and Shelah [1]). Suppose κ is strongly inaccessible or κ = μ^{+} where μ^{<μ} = μ. Then the following are equivalent for a set A ⊆ κ:

(1) A is fat.

(2) There is a < κ-distributive forcing poset ℙ which forces that A contains a club set.

Suppose κ is a regular uncountable cardinal and κ ⊆ X. We generalize the idea of fatness to subsets of P_{κ}(X) with the following definition.

Definition 2.2. Suppose κ is a regular uncountable cardinal and κ ⊆ X. A set A ⊆ P_{κ}(X) is fat if for all regular θ ≥ κ with X ⊆ H(θ), for any club C ⊆ P_{κ}(H(θ)) and ξ < κ, there is an increasing and continuous sequence ⟨N_{i} : i < ξ⟩ such that for all i < ξ, N_{i} ∈ C, N_{i} ∩ X ∈ A, and N_{i} ∈ N_{i+1} when i + 1 < ξ.

Lemma 2.3. Suppose κ = μ^{+}. Then A ⊆ P_{κ}(X) is fat iff for all regular θ ≥ κ with X ⊆ H(θ), for any club C ⊆ P_{κ}(H(θ)), and for any regular cardinal λ ≤ μ, there is an increasing and continuous sequence ⟨N_{i} : i ≤ λ⟩ such that for i ≤ λ, N_{i} ∈ C, N_{i} ∩ X ∈ A, and N_{i} ∈ N_{i+1} when i < λ.

Proof. Suppose A satisfies the second property. Then clearly A is stationary in P_{κ}(X). Fix θ ≥ κ regular with X ⊆ H(θ). We prove by induction on ξ < μ^{+} that for any club set C ⊆ P_{κ}(H(θ)), there is an increasing and continuous sequence ⟨N_{i} : i < ξ⟩ such that for all i < ξ, N_{i} ∈ C, N_{i} ∩ X ∈ A, and N_{i} ∈ N_{i+1} when i + 1 < ξ. The successor step of the induction follows from the fact that A is stationary.

Suppose δ < μ^{+} is a limit ordinal and the claim holds for all δ′ < δ. Let ⟨δ_{i} : i < cf(δ)⟩ be increasing and cofinal in δ. Note that cf(δ) ≤ μ. Let

A = ⟨H(θ), ∈, <, X, A, δ, ⟨δ_{i} : i < cf(δ)⟩⟩,

where < is a well-ordering of H(θ). Fix an increasing and continuous sequence ⟨N_{i} : i ≤ cf(δ)⟩ of sets such that for i ≤ cf(δ), N_{i} ∈ C, N_{i} ≺ A, μ ⊆ N_{i}, N_{i} ∩ X ∈ A, and N_{i} ∈ N_{i+1} when i < cf(δ).

Fix i < cf(δ). By the induction hypothesis, let ⟨M_{j}^{i} : j ≤ δ_{i}⟩ be the <-least increasing and continuous sequence with length δ_{i} + 1 such that μ ∪ {N_{i}} ⊆ M_{0}^{i}, and for j ≤ δ_{i}, M_{j}^{i} ∈ C, N_{i} ≺ M_{j}^{i}, M_{j}^{i} ∩ X ∈ A, and M_{j}^{i} ∈ M_{j+1}^{i} when j < δ_{i}. By elementarity, this sequence is in N_{i+1}. Then the set

{N_{i} : i ≤ cf(δ)} ∪ {M_{j}^{i} : i < cf(δ), j ≤ δ_{i}},

well-ordered by ∈, is increasing and continuous with order type at least δ, and for all N in this set, N ∈ C and N ∩ X ∈ A. □

We will now show that our definition of fatness generalizes the classical notion. Indeed, let A be a fat subset of a regular cardinal κ. We show A is a fat subset of P_{κ}(X), where X = κ, according to Definition 2.2. So let θ ≥ κ be regular, and let C ⊆ P_{κ}(H(θ)) be club. Fix ξ < κ. Define by induction an increasing and continuous sequence ⟨M_{i} : i < κ⟩ such that for i < κ, M_{i} ∩ κ ∈ κ, M_{i} ∈ C, and M_{i} ∈ M_{i+1}. Then ⟨M_{i} ∩ κ : i < κ⟩ is a club subset of κ. Since A is fat, there is a closed set a ⊆ κ with order type at least ξ such that {M_{i} ∩ κ : i ∈ a} ⊆ A. Then ⟨M_{i} : i ∈ a⟩ is as required.

Suppose on the other hand that A ⊆ κ is fat as a subset of P_{κ}(κ) by Definition 2.2, and we show A is fat as a subset of κ. Let C ⊆ κ be club and fix ξ < κ. Let ⟨N_{i} : i ≤ ξ⟩ be an increasing and continuous sequence of sets in P_{κ}(H(κ)) such that for i ≤ ξ, N_{i} ≺ ⟨H(κ), ∈, C⟩, N_{i} ∩ κ ∈ κ, N_{i} ∩ κ ∈ A, and N_{i} ∈ N_{i+1} when i < ξ. Then {N_{i} ∩ κ : i ≤ ξ} is a closed set contained in A ∩ C.

The next theorem generalizes Fact 2.1.

Theorem 2.4. Suppose κ is strongly inaccessible or κ = μ^{+} where μ^{<μ} = μ. Let X be a set containing κ. Then the following are equivalent for a set A ⊆ P_{κ}(X):

(1) A is fat.

(2) There is a < κ-distributive forcing poset which forces there is an increasing and continuous sequence ⟨a_{i} : i < κ⟩ which is cofinal in P_{κ}(X) such that a_{i} ∈ A for i < κ.

Proof. Suppose A ⊆ P_{κ}(X) and ℙ is a < κ-distributive forcing poset which forces that ⟨ȧ_{i} : i < κ⟩ is increasing, continuous, and cofinal in P_{κ}(X) such that ȧ_{i} ∈ A for i < κ. We prove that A is fat. So let θ ≥ κ be regular with X ⊆ H(θ). Suppose C ⊆ P_{κ}(H(θ)) is club. Let G be generic for ℙ over V, and let a_{i} = ȧ_{i}^{G} for i < κ. Since ℙ is < κ-distributive, in V [G] the set C is still a club subset of P_{κ}(H(θ)^{V}).

We work in V [G]. Since X = ⋃{a_{i} : i < κ} and |a_{i}| < κ for all i < κ, X has size κ in the extension. So let ⟨x_{i} : i < κ⟩ enumerate X. We define by induction an increasing and continuous sequence ⟨N_{i} : i < κ⟩ such that for all i < κ, N_{i} ∈ N_{i+1} and N_{i} ∈ C. Choose N_{0} in C arbitrarily. At limits take unions. Suppose N_{i} is defined. Then N_{i} is in H(θ)^{V}, so choose N_{i+1} in C such that N_{i} ∪ {N_{i}} ∪ {x_{i}} ⊆ N_{i+1}. This completes the definition. Now ⟨a_{i} : i < κ⟩ and ⟨N_{i} ∩ X : i < κ⟩ are both club in P_{κ}(X). So there is a club D ⊆ κ such that for all i ∈ D, a_{i} = N_{i} ∩ X. Then ⟨N_{i} : i ∈ D⟩ is an increasing and continuous sequence such that for all i ∈ D, N_{i} ∈ C, N_{i} ∈ N_{i+1}, and N_{i} ∩ X ∈ A. But every initial segment of this sequence is in V since ℙ is < κ-distributive. So A is fat.

In the other direction, suppose A ⊆ P_{κ}(X) is fat. Define a forcing poset ℙ(A) as follows. A condition in ℙ(A) is an increasing and continuous sequence ⟨a_{i} : i ≤ γ⟩, where γ < κ, such that a_{i} ∈ A for all i ≤ γ. The ordering is by extension of sequences. We claim that ℙ(A) is < κ-distributive and ℙ(A) forces that the union of the generic filter is an increasing and continuous sequence cofinal in P_{κ}(X) with order type κ whose elements are in A.

Suppose ⟨D_{i} : i < ξ⟩ is a sequence of dense open subsets of ℙ(A), where ξ < κ is a cardinal. Let p be in ℙ(A), and we find q ≤ p which is in ⋂{D_{i} : i < ξ}. Fix a regular cardinal θ ≫ κ with X ∈ H(θ), and let

A = ⟨H(θ), ∈, X, A, ℙ(A), p, ⟨D_{i} : i < ξ⟩⟩.

Since A is fat we can find an increasing and continuous sequence ⟨N_{i} : i ≤ ξ⟩ such that for all i ≤ ξ, N_{i} ≺ A, N_{i} ∩ κ ∈ κ, ξ ⊆ N_{i}, N_{i} ∩ X ∈ A, and when i < ξ, N_{i} ∈ N_{i+1}.

We define by induction a descending sequence of conditions ⟨p_{i} : i ≤ ξ⟩ in ℙ(A). Our induction hypothesis is that p_{i} is in N_{i+1} and the maximum element of p_{i} is N_{i} ∩ X. Let p_{0} = p (N_{0} ∩ X). Then p_{0} is a condition, because p ∈ N_{0} and thus all the elements of p are subsets of N_{0} ∩ X. Suppose i < ξ, and for all j ≤ i, p_{j} is defined, p_{j} is a member of N_{j+1}, and the maximum element of p_{j} is N_{j} ∩ X. Since ξ ⊆ N_{i+1}, D_{i} is in N_{i+1}. Fix p_{i}^{∗} ≤ p_{i} in D_{i} ∩ N_{i+1}. Since p_{i}^{∗} has size less than κ and N_{i+1} ∩ κ ∈ κ, p_{i}^{∗} ⊆ N_{i+1}, and so every element of p_{i}^{∗} is a subset of N_{i+1} as well. Therefore if we let p_{i+1} = p_{i}^{∗} (N_{i+1} ∩ X), p_{i+1} is a condition in N_{i+2} ∩ D_{i} below p_{i}.

Suppose δ ≤ ξ is a limit ordinal and p_{i} ∈ N_{i+1} is defined for all i < δ. Let

p_{δ} = ⋃{p_{i} : i < δ} (N_{δ} ∩ X),

which is a condition since N_{δ} ∩ X ∈ A and N_{δ} = ⋃{N_{i} : i < δ}. We need to show that p_{δ} is in N_{δ+1} when δ < ξ. The sequence ⟨p_{i} : i < δ⟩ is in N_{δ}^{<ξ}. Since κ is either strongly inaccessible or κ = μ^{+} where μ^{<μ} = μ, N_{δ}^{<ξ} has size less than κ. But N_{δ}^{<ξ} ∈ N_{δ+1}. Since N_{δ+1} ∩ κ ∈ κ, N_{δ}^{<ξ} ⊆ N_{δ+1}. So the sequence ⟨p_{i} : i < δ⟩ is in N_{δ+1}. Clearly then p_{δ} is in N_{δ+1} as well.

This completes the construction of ⟨p_{i} : i ≤ ξ⟩. The condition p_{ξ} is below p and is in ⋂{D_{i} : i < ξ}. So ℙ(A) is < κ-distributive.

For each α < κ let D_{α} be the set of conditions in ℙ(A) with length at least α. Clearly D_{0} is dense open, and if D_{i} is dense open, D_{i+1} is dense open as well.

Assume δ < κ is a limit ordinal and D_{i} is dense open for all i < δ. Since ℙ(A) is < κ-distributive, ⋂{D_{i} : i < δ} is dense open. But if p is in this intersection, p has length at least δ. So ℙ(A) forces the union of the generic filter has length κ. By an easy density argument, ℙ(A) forces the union of the generic filter is cofinal in P_{κ}(X). □

Since we will use the forcing poset from the last theorem in our consistency proof, we describe it explicitly in the following definition.

Definition 2.5. Suppose κ is regular, κ ⊆ X, and A ⊆ P_{κ}(X) is fat. Let ℙ(A) be the forcing poset consisting of increasing and continuous sequences ⟨a_{i} : i ≤ γ⟩, where γ < κ and a_{i} ∈ A for i ≤ γ, ordered by extension of sequences.

The forcing poset ℙ(A) is < κ-distributive and adds an increasing, continuous, and cofinal sequence ⟨a_{i} : i < κ⟩ through P_{κ}(X) such that a_{i} ∈ A for i < κ. In particular, ℙ(A) collapses the size of X to be κ.

If κ = ω_{1} and ω_{1} ⊆ X, one can show using Lemma 2.3 that any stationary set A ⊆ P_{ω_{1}}(X) is fat. Thus ℙ(A) is ω-distributive for any stationary set A ⊆ P_{ω_{1}}(X).

3. The Basic Forcing Poset

We now describe the forcing poset which we will use in our consistency proof. Suppose μ^{<μ} = μ and μ^{+} ⊆ X. The basic forcing poset we will use is Add(μ) ∗

ℙ(Ṡ), where Add(μ) adds a Cohen subset to μ, Add(μ) forces Ṡ = [X]^{μ} ∩ V, and ℙ(Ṡ) is the forcing poset from Definition 2.5. Thus we need to know that Add(μ) forces Ṡ is fat. If μ^{<μ} = μ then Add(μ) is μ^{+}-c.c., so this follows from the next proposition.

Proposition 3.1. Suppose κ is regular and κ ⊆ X. Let ℙ be a κ-c.c. forcing poset. Then ℙ forces P_{κ}(X) ∩ V is fat.

Proof. Let G be generic for ℙ over V. Working in V [G], fix θ ≥ κ regular with X ⊆ H(θ), and let C ⊆ P_{κ}(H(θ)) be club. Fix χ ≫ θ regular such that H(χ) contains C and ℙ as members. Recall that H(χ)^{V [G]} = H(χ)^{V} [G]. Let Ċ be a name for C in H(χ)^{V}.

Now back in V, define by induction an increasing and continuous sequence ⟨N_{i} : i < κ⟩ of elementary substructures of ⟨H(χ), ∈, X, Ċ, ℙ⟩ such that for all i < κ, |N_{i}| < κ, N_{i} ∩ κ ∈ κ, and N_{i} ∈ N_{i+1}. Then in V [G], for all i < κ, N_{i}[G] ≺ ⟨H(χ)^{V [G]}, ∈, C⟩. By elementarity, N_{i}[G] ∩ C is a directed subset of C with size less than κ whose union is equal to N_{i}[G] ∩ H(θ). So N_{i}[G] ∩ H(θ) is in C. Since N_{i} ∈ N_{i+1}, N_{i}[G] ∈ N_{i+1}[G], and therefore N_{i}[G] ∩ H(θ) ∈ N_{i+1}[G] ∩ H(θ). But ℙ is κ-c.c., so N_{i}[G] ∩ V = N ∩ V. For if x ∈ N_{i}[G] ∩ V, there is a name ẋ for x in N_{i}. The maximal antichain of conditions deciding ẋ is in N_{i}, and has size less than κ, so is a subset of N_{i}. But then x is in N_{i}. In particular, N_{i}[G] ∩ X = N_{i} ∩ X, which is in P_{κ}(X) ∩ V. □

The forcing poset Add(μ) is μ-glb closed. Indeed, if ⟨p_{i} : i < ξ⟩ is decreasing in Add(μ) where ξ < μ, then ⋃{p_{i} : i < ξ} is the greatest lower bound. Note that any two step forcing iteration of μ-glb closed forcing posets is μ-glb closed.

Lemma 3.2. Suppose μ^{<μ} = μ, μ^{+} ⊆ X, and Ṡ is an Add(μ)-name for [X]^{μ} ∩ V. Then Add(μ) forces that ℙ(Ṡ) is μ-glb closed. Hence Add(μ)∗ℙ(Ṡ) is μ-glb closed.

Proof. Let G be generic for Add(μ). In V [G], suppose ⟨p_{i} : i < ξ⟩ is a descending sequence of conditions in ℙ(S) where ξ < μ is a limit ordinal. For each i write p_{i} = ⟨a_{j} : j ≤ γ_{i}⟩. Let γ = sup({γ_{i} : i < ξ}) and a = ⋃{a_{i} : i < γ}. Let

q = ⋃{p_{i} : i < ξ} ∪ {⟨γ, a⟩}.

Then q is a condition in ℙ(S) iff a is in V. But since Add(μ) is μ-closed, the sequence ⟨a_{γ_{i}} : i < ξ⟩ is in V, and hence its union a is in V. Clearly any condition which extends each p_{i} must extend q, so q is the greatest lower bound of the sequence. □

4. Iterating the Basic Forcing Poset

We now describe a mixed support iteration of the forcing poset introduced in the last section.

Fix a cardinal μ such that μ^{<μ} = μ. We consider a forcing iteration

⟨ℙ_{i}, ℚ̇_{j} : i ≤ α, j < α⟩,

satisfying the following recursive definition:

(1) If i < α is even, ℙ_{i} forces ℚ̇_{i} = Add(μ), and ℙ_{i} forces Ẋ_{i} is a set containing μ^{+}.

(2) If i = j + 1 < α is odd, ℙ_{i} forces Ṡ_{i} = [Ẋ_{j}]^{μ} ∩ V [Ġ_{j}], where Ġ_{j} is a name for the generic filter for ℙ_{j}, and ℚ̇_{i} = ℙ(Ṡ_{i}) is the poset from Definition 2.5.

(3) If i ≤ α is a limit ordinal, ℙ_{i} is the poset consisting of partial functions p : i → V such that p ↾ j ∈ ℙ_{j} for j < i, |dom(p) ∩ {j < i : j even}| < μ, and |dom(p) ∩ {j < i : j odd}| ≤ μ.

We assume the following recursion hypotheses for all β < α, which guarantee that the definition above makes sense.

(4) ℙ_{β} is μ-glb closed and μ^{+}-proper, and so preserves cardinals and cofinalities less than or equal to μ^{+}.

(5) Let ℙ_{β}^{∗} be the set of p in ℙ_{β} such that for all even j in dom(p), there is x in Add(μ) such that p(j) = x̌. Then ℙ_{β}^{∗} is dense in ℙ_{β}.

(6) If ⟨p_{i} : i < ξ⟩ is a descending sequence of conditions in ℙ_{β}^{∗} with ξ < μ, then the greatest lower bound of this sequence is in ℙ_{β}^{∗}.

We prove that properties (4), (5), and (6) above also hold for ℙ_{α}.

Case 1: α = β + 1 is a successor ordinal.

We show that ℙ_{α} is μ-glb closed. This will follow from the fact that a twostep iteration of μ-glb closed forcing posets is μ-glb closed. If β is even, then ℙ_{α} = ℙ_{β} ∗ Add(μ). Since ℙ_{β} is μ-glb closed by recursion, clearly ℙ_{α} is μ-glb closed as well. Suppose β = γ + 1 is odd. Then ℙ_{α} = ℙ_{γ} ∗ Add(μ) ∗ ℙ(Ṡ_{β}). By recursion, ℙ_{γ} is μ-glb closed, and by Lemma 3.2, ℙ_{γ} forces that Add(μ)∗ℙ(Ṡ_{β}) is μ-glb closed. So ℙ_{α} is μ-glb closed. We prove in Proposition 4.2 below that ℙ_{α} is μ^{+}-proper.

Now we prove that ℙ_{α}^{∗} is dense in ℙ_{α}. Consider a condition p in ℙ_{α}. If β is not in the domain of p or if β is odd, fix q ≤ p ↾ β in ℙ_{β}^{∗}. Then q ≤ p is in ℙ_{α}^{∗} if β is not in dom(p), and q p(β) ≤ p is in ℙ_{α}^{∗} otherwise. Assume β is in dom(p) and β is even. Since ℙ_{β} is μ-closed, it forces that p(β) is an element of Add(μ) in the ground model. So choose r ≤ p ↾ β in ℙ_{β}^{∗} and x in Add(μ) such that r forces p(β) = x̌. Then r x̌ is as required.

Suppose ⟨p_{i} : i < ξ⟩ is a descending sequence of conditions in ℙ_{α}^{∗} with ξ < μ. We show that the greatest lower bound of this sequence is in ℙ_{α}^{∗}. Now ⟨p_{i} ↾ β : i < ξ⟩ is a descending sequence in ℙ_{β}^{∗}. By induction the greatest lower bound q of this sequence is in ℙ_{β}^{∗}. If β is not in dom(p_{i}) for all i < ξ, then q is the greatest lower bound of ⟨p_{i} : i < ξ⟩ in ℙ_{α}^{∗}. Otherwise let γ < ξ be the least ordinal such that β is in dom(p_{γ}). If β is odd, let u̇ be a ℙ_{β}-name for the greatest lower bound of {p_{i}(β) : γ ≤ i < ξ}. Then q u̇ is as required. If β is even, then fix for each γ ≤ i < ξ a condition x_{i} in Add(μ) such that p_{i}(β) = x̌_{i}. Let x = ⋃{x_{i} : γ ≤ i < ξ}. Then q x̌ is as required.

Case 2: α is a limit ordinal.

We show that ℙ_{α} is μ-glb closed. Suppose ⟨p_{i} : i < ξ⟩ is a descending sequence of conditions in ℙ_{α}, with ξ < μ. For each i < α, ℙ_{i} forces ℚ̇_{i} is μ-glb closed. Define q with support equal to ⋃{dom(p_{i}) : i < ξ}, so that for each β in this support, q ↾ β forces q(β) is the greatest lower bound of ⟨p_{i}(β) : γ_{β} ≤ i < ξ⟩, where γ_{β} is the least i < ξ with β in dom(p_{i}). Clearly then q is the greatest lower bound of {p_{i} : i < ξ} in ℙ_{α}. Suppose moreover that p_{i} ∈ ℙ_{α}^{∗} for all i < ξ. Then q can be chosen to be in ℙ_{α}^{∗} as well. Namely, for each even β in dom(q), and for γ_{β} ≤ i < ξ, choose x_{i}^{β} in Add(μ) such that p_{i}(β) = x̌_{i}^{β}. Then let q(β) be a name for ⋃{x_{i}^{β} : γ_{β} ≤ i < ξ}.

Now we show ℙ_{α}^{∗} is dense in ℙ_{α}. First assume cf(α) ≥ μ, and let p be in ℙ_{α}. Then there is ξ < α such that dom(p) ∩ {i < α : i even} ⊆ ξ. By induction we can choose q ≤ p ↾ ξ in ℙ_{ξ}^{∗}. Then q p ↾ [ξ, α) is in ℙ_{α}^{∗} and is below p.

Suppose cf(α) < μ and let p be in ℙ_{α}. Fix an increasing and continuous sequence ⟨ξ_{i} : i < cf(α)⟩ cofinal in α with ξ_{0} = 0, and let ξ_{cf(α)} = α. We define by induction a descending sequence ⟨p_{i} : i ≤ cf(α)⟩ so that p_{i} ↾ ξ_{i} is in ℙ_{ξ_{i}}^{∗}. Let p_{0} = p. Given p_{i}, apply the recursion hypotheses to choose q ≤ p_{i} ↾ ξ_{i+1} in ℙ_{ξ_{i+1}}^{∗}, and let p_{i+1} = q p ↾ [ξ_{i+1}, α). Suppose δ ≤ cf(α) is a limit ordinal and p_{i} is defined for all i < δ. Let q be the greatest lower bound of the sequence ⟨p_{i} ↾ ξ_{i} : i < δ⟩. Since each p_{i} ↾ ξ_{i} is in ℙ_{ξ_{i}}^{∗} ⊆ ℙ_{ξ_{δ}}^{∗}, q is in ℙ_{ξ_{δ}}^{∗}. Now define p_{δ} = q p ↾ [ξ_{δ}, α). This completes the definition. The condition p_{cf(α)} is below p and is in ℙ_{α}^{∗}.

Now we prove that ℙ_{α} is μ^{+}-proper. The proof is the same whether α is a successor or a limit ordinal.

We will use the following basic observation.

Lemma 4.1. 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.

Proposition 4.2. The poset ℙ_{α} is μ^{+}-proper.

Proof. Fix a regular cardinal θ > 2^{|ℙ_{α}|} such that ℙ_{α} is in H(θ). Let N ≺ ⟨H(θ), ∈ , ℙ_{α}⟩ be a set with size μ with N^{<μ} ⊆ N. We would like to show that for every p in N ∩ ℙ_{α}, there is q ≤ p which is N-generic. In Proposition 4.5 we need q to satisfy a slightly stronger property, which we describe in the following claim.

Claim 4.3. For all p in N ∩ ℙ_{α}, there is q ≤ p which satisfies the property that for all r ≤ q, and for any dense set D ⊆ ℙ_{α} in N, there is q′ in D ∩ N compatible with r such that for all odd γ in dom(q′), γ ∈ dom(r) and r ↾ γ forces r(γ) ≤ q′(γ).

Let ⟨⟨D_{i}, f_{i}⟩ : i < μ⟩ be an enumeration of all pairs ⟨D, f⟩ in N such that D ⊆ ℙ_{α} is dense and f : {β < α : β even} → Add(μ) is a partial function with |dom(f)| < μ.

We define by induction a descending sequence ⟨p_{i} : i < μ⟩ of conditions in N ∩ℙ_{α}^{∗} and a sequence ⟨q_{i} : i < μ⟩ of conditions in N ∩ ℙ_{α}^{∗} such that:

(1) for i < μ, dom(p_{i}) ∩ {β < α : β even} = dom(p_{0}) ∩ {β < α : β even},

(2) for i < μ, for all even β in dom(p_{i}), p_{i}(β) = p_{0}(β). Fix p_{0} ≤ p in N ∩ ℙ_{α}^{∗}. If δ < μ is a limit ordinal and p_{i} is defined for all i < δ, let p_{δ} be the greatest lower bound of {p_{i} : i < δ}. Since N^{<μ} ⊆ N, ⟨p_{i} : i < δ⟩ is in N, and therefore p_{δ} is in N ∩ ℙ_{α}^{∗}.

Suppose p_{i} is defined for a fixed i < μ. Consider the pair ⟨D_{i}, f_{i}⟩. If there is q in N ∩ D_{i} below p_{i} such that dom(f_{i}) ⊆ dom(q), and for all β in dom(f_{i}), q(β) is a name for f_{i}(β), then choose q_{i} as such a q. Otherwise just pick q_{i} ≤ p_{i} in N ∩ D_{i}. Now define p_{i+1} with support equal to

(dom(p_{i}) ∩ {β < α : β even}) ∪ (dom(q_{i}) ∩ {γ < α : γ odd })

so that p_{i+1}(β) = p_{i}(β) for even β, and p_{i+1}(γ) = q_{i}(γ) for odd γ.

We define a lower bound q for ⟨p_{i} : i < μ⟩, and prove that q satisfies the requirements of Claim 4.3. Clearly then q is N-generic. The domain of q is ⋃{dom(p_{i}) : i < μ}. In particular, dom(q) ∩ {β < α : β even} = dom(p_{0}) ∩ {β < α : β even}, which has size less than μ. For even β in dom(q), let q(β) = p_{0}(β).

Suppose γ = β + 1 is an odd ordinal in dom(q). Let i_{γ} < μ be the least i such that γ is in dom(p_{i}). For i_{γ} ≤ i < μ, fix a name σ̇_{i}^{γ} so that ℙ_{γ} forces p_{i}(γ) has domain σ̇_{i}^{γ} +1. Let σ̇_{γ} be a ℙ_{γ}-name for sup({σ̇_{i}^{γ} +1 : i_{γ} ≤ i < μ}). Then ℙ_{γ} forces that the union of the conditions in {p_{i}(γ) : i_{γ} ≤ i < μ} is a sequence of length σ̇_{γ}. Let ⟨ȧ_{i}^{γ} : i < σ̇_{γ}⟩ be a sequence of names such that

ℙ_{γ} ⊩ ⋃{p_{i}(γ) : i_{γ} ≤ i < μ} = ⟨ȧ_{i}^{γ} : i < σ̇_{γ}⟩.

Let ȧγ ; σ̇_{γ} be a name for ⋃{ȧ_{i}^{γ} : i < σ̇_{γ}}. Finally, let q(γ) be a name for the sequence ⟨ȧ_{i}^{γ} : i ≤ σ̇_{γ}⟩.

We prove by induction that for all γ ≤ α, q ↾ γ is a condition in ℙ_{γ} and is below p_{i} ↾ γ for all i < μ. Limit stages are clear. Suppose q ↾ γ satisfies this property. If γ is even or if γ is not in dom(q), then clearly q ↾ γ + 1 is as required. Suppose γ = β + 1 is odd and is in dom(q). Then q ↾ γ + 1 is a condition below p_{i} ↾ γ + 1 for all i < μ, provided that q ↾ γ forces that ȧγ ; σ̇_{γ} is in Ṡ_{γ} = [Ẋ_{β}]^{μ} ∩ V [Ġ_{β}].

Let G_{β} ∗H be generic for ℙ_{γ} = ℙ_{β} ∗Add(μ). Since γ is in dom(q), γ is in dom(p_{i}) for some i < μ. Since μ ⊆ N, dom(p_{i}) ⊆ N. Therefore γ, and hence β, is in N. So ℙ_{β} is in N. But Add(μ) is μ^{+}-c.c. in V [G_{β}], so N[G_{β} ∗ H] ∩ V [G_{β}] = N[G_{β}]. In particular, N[G_{β} ∗ H] ∩ X_{β} = N[G_{β}] ∩ X_{β}, which is in [X_{β}]^{μ} ∩ V [G_{β}]. So it suffices to show that a_{σ_{γ}}^{γ} = N[G_{β} ∗ H] ∩ X_{β}.

If i_{γ} ≤ i < μ, then the condition p_{i}(γ) = ⟨a_{j}^{γ} : j ≤ σ_{i}^{γ}⟩ is a member, and hence a subset, of N[G_{β} ∗ H]. Therefore each a_{j}^{γ} is a subset of N[G_{β} ∗ H]. Hence ⋃{a_{j}^{γ} : j < σ_{γ}} ⊆ N[G_{β} ∗ H] ∩ X_{β}. On the other hand, fix x in N[G_{β} ∗ H] ∩ X_{β}. Fix a ℙ_{γ}-name ẋ for x in N. Then there is a dense subset of ℙ_{α} in N of conditions s such that ℙ_{γ} forces ẋ is in some element of the sequence s(γ). Hence for some i < μ, q_{i} is in this dense set. Since q_{i}(γ) = p_{i+1}(γ), ℙ_{γ} forces ẋ appears in some element of p_{i+1}(γ). Therefore x appears in some element of ⟨a_{j}^{γ} : j ≤ σ_{i+1}^{γ}⟩. So x is in a_{σ_{γ}}^{γ}. Thus a_{σ_{γ}}^{γ} = ⋃{a_{i}^{γ} : i < σ_{γ}} = N[G_{β} ∗ H] ∩ X_{β}.

We prove now that q satisfies the property described in Claim 4.3. Let r ≤ q and suppose D is a dense subset of ℙ_{α} in N. Fix s ≤ r in ℙ_{α}^{∗} ∩ D. Let f : α → Add(μ) be the partial function with dom(f) = N ∩ dom(s) ∩ {β < α : β even} such that for all β in dom(f), s(β) is a name for f(β). Since N^{<μ} ⊆ N, f is in N. Fix i < μ such that D_{i} = D and f_{i} = f.

Now H(θ) models that there is u ≤ p_{i} in D_{i} such that dom(f_{i}) ⊆ dom(u), and for all β in dom(f_{i}), u(β) is a name for f_{i}(β), as witnessed by u = s. By elementarity, the same is true in N. Hence by construction, q_{i} also satisfies this property. If γ is odd and is in dom(q_{i}), then γ is in dom(p_{i+1}) and p_{i+1}(γ) = q_{i}(γ). Therefore for all odd γ in dom(q_{i}), γ is in dom(r) and r ↾ γ forces r(γ) ≤ q_{i}(γ).

We show that q_{i} and r are compatible, which finishes the proof. We apply Lemma 4.1 to show q_{i} and s are compatible. Suppose γ is in dom(q_{i}) ∩ dom(s). Since dom(q_{i}) ⊆ N, γ is in N ∩ dom(s). So if γ is even, then γ is in dom(f_{i}). Then q_{i}(γ) and s(γ) are both names for f_{i}(γ) and thus are equal. If γ is odd, then q_{i}(γ) = p_{i+1}(γ), and s ↾ γ forces s(γ) ≤ p_{i+1}(γ). □

This completes the recursion.

The next proposition describes a special property of ℙ_{α} which we will use in the consistency proof of the next section. First we need a technical lemma.

Lemma 4.4. Let p′ and q′ be conditions in ℙ_{α}^{∗}. Then there are p ≤ p′ and q ≤ q′ in ℙ_{α}^{∗} such that dom(p) ∩ {γ < α : γ odd} = dom(q) ∩ {γ < α : γ odd}, and for all odd γ in this set, p(γ) = q(γ).

Proof. First choose p(0) ≤ p′(0) and q(0) ≤ q′(0) in Add(μ) which 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 p(β) = p′(β), and similarly with q. Suppose γ is odd and p ↾ γ and q ↾ γ are defined. If γ is in dom(p′) \ dom(q′) then let p(γ) = q(γ) = p′(γ), and similarly if γ is in dom(q′) \ dom(p′). 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. Let p(γ) = q(γ) = ẋ_{γ}. □

Proposition 4.5. The poset ℙ_{α} forces that whenever f : μ^{+} → V is a function in the extension such that for all i < μ^{+}, f ↾ i is in V, then f is in V.

Proof. Suppose for a contradiction that p forces ḟ : μ^{+} → V is a function which is not in V, but for all i < μ^{+}, ḟ ↾ i is in V.

Fix a regular cardinal θ ≫ μ^{+} with ℙ_{α} ∈ H(θ). Let N be an elementary substructure of ⟨H(θ), ∈, ℙ_{α}, p, ḟ⟩ with size μ and N^{<μ} ⊆ N. By Claim 4.3, fix q ≤ p such that for all r ≤ q and for any dense set D ⊆ ℙ_{α} in N, there is q′ in D ∩ N compatible with r such that for all odd γ in dom(q′), γ ∈ dom(r) and r ↾ γ forces r(γ) ≤ q′(γ).

Let r ≤ q be in ℙ_{α}^{∗} such that r decides ḟ ↾ N ∩ μ^{+}. Let g : α → Add(μ) be the partial function with domain equal to N ∩ dom(r) ∩ {β < α : β even} such that for all β in dom(g), r(β) is a name for g(β). Since N^{<μ} ⊆ N, g is in N.

Define D as the set of s_{0} ≤ p in ℙ_{α}^{∗} for which there exists s_{1} in ℙ_{α}^{∗} such that:

(1) dom(g) ⊆ dom(s_{0}),

(2) there is i < μ^{+} and distinct a_{0} and a_{1} such that s_{0} ⊩ ḟ(i) = a_{0} and s_{1} ⊩ ḟ(i) = a_{1},

(3) dom(s_{0}) ∩ {γ < α : γ odd} = dom(s_{1}) ∩ {γ < α : γ odd},

(4) for all odd γ in dom(s_{0}), s_{0}(γ) = s_{1}(γ),

(5) dom(g) ⊆ dom(s_{1}), and for all β in dom(g), if g(β) is compatible with the condition named by s_{0}(β), then s_{1}(β) is the name for a condition extending g(β). By elementarity, D is in N.

We claim that D is dense below p. So let s ≤ p. Extend s to s′ in ℙ_{α}^{∗} so that dom(g) ⊆ dom(s′). Now define s′′ ≤ s′ with the same domain as s′ as follows. For β ∈ dom(s′) \ dom(g), let s′′(β) = s′(β). Suppose β is in dom(g). If s′(β) names a condition in add(μ) compatible with g(β), let s′′(β) be a name for a condition which extends g(β) and s′(β). Otherwise let s′′(β) = s′(β).

Since ḟ is not in V, there is i < μ^{+} such that s′′ does not decide ḟ(i). Fix s′_{0}, s′_{1} ≤ s′′ in ℙ_{α}^{∗} and distinct a_{0} and a_{1} so that s′_{0} ⊩ ḟ(i) = a_{0} and s′_{1} ⊩ ḟ(i) = a_{1}. Now apply Lemma 4.4 to obtain s_{0} ≤ s′_{0} and s_{1} ≤ s′_{1} in ℙ_{α}^{∗} satisfying (3) and (4). We check that (5) holds. If β is in dom(g) and s_{0}(β) names a condition compatible with g(β), then clearly s′(β) names a condition compatible with g(β). So s′′(β) is a name for a condition refining g(β). Since s_{1} ≤ s′′, s_{1}(β) is a name for a condition refining g(β).

By the genericity property of q, we can fix s_{0} ∈ D ∩ N which is compatible with r, and such that for all odd γ in dom(s_{0}), γ is in dom(r) and r ↾ γ forces that r(γ) ≤ s_{0}(γ). Fix s_{1}, i, a_{0}, and a_{1} in N as described in the definition of D. Since r decides ḟ(i) and r and s_{0} are compatible, r forces ḟ(i) = a_{0}. So r and s_{1} are incompatible. We will get a contradiction by showing r and s_{1} are compatible.

We apply Lemma 4.1. Suppose β is in dom(r) ∩ dom(s_{1}) and β is even. Then β is in N, so β must be in dom(g). Since r and s_{0} are compatible, r(β) and s_{0}(β) are compatible. By (5), s_{1}(β) is the name for a condition extending g(β). Suppose γ is in dom(r) ∩ dom(s_{1}) and γ is odd. Then γ is in dom(s_{0}). So γ is in dom(r), and r ↾ γ forces r(γ) ≤ s_{0}(γ). But s_{0}(γ) = s_{1}(γ). □

5. The Consistency Result

Suppose μ < κ are cardinals, μ^{<μ} = μ, and κ is supercompact. We define a forcing iteration of the form given in the last section which collapses κ to become μ^{++}, and forces that for all regular λ ≥ μ^{++}, there are stationarily many N in [H(λ)]^{μ^{+}} such that N is internally club but not internally approachable.

Fix a Laver function f : κ → V_{κ}. So for all x and λ, there is an elementary embedding j : V → M with critical point κ such that M^{λ} ⊆ M and j(f)(κ) = x.

We define by recursion a forcing iteration

⟨ℙ_{i}, ℚ̇_{j} : i ≤ κ, j < κ⟩.

Suppose ℙ_{i} is defined for a fixed i < κ. If i is an even ordinal, let ℚ̇_{i} be a ℙ_{i}-name for Add(μ). Suppose i = j + 1 is odd. If f(j) is a ℙ_{j}-name for a set which contains μ^{+}, let Ẋ_{j} = f(j), and otherwise let Ẋ_{j} be a ℙ_{j}-name for μ^{+}. Let Ṡ_{i} be a ℙ_{i}-name for [Ẋ_{j}]^{μ} ∩ V [Ġ_{j}], and let ℚ̇_{i} be a ℙ_{i}-name for the poset ℙ(Ṡ_{i}) from Definition 2.5. Suppose δ ≤ κ is a limit ordinal and ℙ_{i} is defined for all i < δ. Then let ℙ_{δ} be the poset consisting of all partial functions p : δ → V such that p ↾ i ∈ ℙ_{i} for all i < δ, |dom(p) ∩ {i < δ : i even }| < μ, and |dom(p) ∩ {i < δ : i odd }| ≤ μ.

Since f is a Laver function, there are stationarily many α < κ such that f(α) is a ℙ_{α}-name and ℙ_{α} forces f(α) = (μ^{++})^{V [Ġ_{α}]}. Indeed, let ẋ be a ℙ_{κ}-name for (μ^{++})^{V [Ġ_{κ}]}. Choose j : V → M with critical point κ such that j(f)(κ) = ẋ and M is sufficiently closed that it models ℙ_{κ} = j(ℙ_{κ}) ↾ κ forces ẋ = (μ^{++})^{M[Ġ_{κ}]}. If C is club in κ, then κ ∈ j(C). Hence by elementarity, there is α < κ in C such that f(α) is as desired. But then ℙ_{α+2} forces |(μ^{++})^{V [Ġ_{α}]}| = μ^{+}. So ℙ_{κ} collapses all cardinals in the interval (μ^{+}, κ).

Since |ℙ_{i}| < κ for all i < κ, there are club many δ < κ such that |ℙ_{i}| < δ for all i < δ. Suppose μ^{+} < δ ≤ κ is inaccessible and satisfies this property. Then ℙ_{δ} is the direct limit of ⟨ℙ_{i} : i < δ⟩, where each ℙ_{i} has size less than δ, and there are stationarily many α < δ such that ℙ_{α} is the direct limit of ⟨ℙ_{i} : i < α⟩. By a standard Δ-system argument, ℙ_{δ} is δ-c.c. (see Theorem 2.2 of [2]). In particular, ℙ_{κ} is κ-c.c. and forces that κ = μ^{++}.

Let G_{κ} be generic for ℙ_{κ}. In V [G_{κ}] let λ ≥ μ^{++} be regular. In V let θ = (2^{λ})^{+}. Let j : V → M be an elementary embedding with critical point κ such that M^{θ} ⊆ M and j(f)(κ) is a ℙ_{κ}-name for H(λ)^{V [Ġ_{κ}]}. Then by choice of j,

j(ℙ_{κ}) = ℙ_{κ} ∗ Add(μ) ∗ ℙ(Ṡ) ∗ ℙ_{tail}

where

ℙ_{κ+1} ⊩ Ṡ = Ṡ_{κ+1} = [H(λ)^{V [Ġ_{κ}]}]^{μ} ∩ M[Ġ_{κ}],

and ℙ_{tail} is forced to be an iteration of the form given in the previous section. Let H ∗ K ∗ G_{tail} be generic over V [G_{κ}] for Add(μ) ∗ ℙ(Ṡ) ∗ ℙ_{tail}. Extend j in V [G_{κ} ∗ H ∗ K ∗ G_{tail}] to

j : V [G_{κ}] → M[G_{κ} ∗ H ∗ K ∗ G_{tail}].

Then j(G_{κ}) = G_{κ} ∗ H ∗ K ∗ G_{tail}. Since ℙ_{κ} is κ-c.c., M[G_{κ}]^{θ} ∩ V [G_{κ}] ⊆ M[G_{κ}]. In particular, H(λ)^{V [G_{κ}]} = H(λ)^{M[G_{κ}]}.

Working in V [G_{κ}], let C ⊆ [H(λ)]^{μ^{+}} be club. We prove there is a set in C which is internally club but not internally approachable. By elementarity, it suffices to prove the same statement about j(C) in M[j(G_{κ})]. We will prove that in M[j(G_{κ})], the set j“H(λ)^{V [G_{κ}]} is in j(C) and is internally club but not internally approachable.

Let N^{∗} = j“H(λ)^{V [G_{κ}]}. First we prove that N^{∗} is in M[j(G_{κ})]. The set j“H(λ)^{V} is in M by the closure of M. But H(λ)^{V [G_{κ}]} = H(λ)^{V} [G_{κ}]. So every element of N^{∗} is of the form j(ȧ^{G_{κ}}) = j(ȧ)^{j(G_{κ})}, where ȧ is in H(λ)^{V}. So N^{∗} = (j“H(λ)^{V})[j(G_{κ})], which is in M[j(G_{κ})]. Also note that in M[j(G_{κ})], |N^{∗}| = |H(λ)^{V [G_{κ}]}| = μ^{+}.

We claim that N^{∗} is in j(C). Since j(C) is closed under unions of directed subsets with size less than j(μ^{++}), it suffices to show that N^{∗} ∩ j(C) is directed and ⋃(N^{∗} ∩ j(C)) = N^{∗}. Suppose j(a) and j(b) are in N^{∗} ∩ j(C). By elementarity, a and b are in C. Fix c in C such that a ∪ b ⊆ c. Then j(a) and j(b) are contained in j(c) and j(c) ∈ N^{∗} ∩ j(C). Hence N^{∗} ∩ j(C) is directed.

We show that ⋃(N^{∗} ∩ j(C)) = N^{∗}. Let j(x) be in N^{∗}. Then x is in H(λ)^{V [G_{κ}]}, so there is a in C such that x ∈ a. Then j(x) ∈ j(a) ∈ N^{∗} ∩j(C). So N^{∗} ⊆ ⋃(N^{∗} ∩ j(C)). On the other hand, suppose y is in ⋃(N^{∗} ∩ j(C)). Fix j(a) ∈ N^{∗} ∩ j(C) so that y ∈ j(a). Then a is in C. In V [G_{κ}], a has size less than μ^{++} = κ, and κ is the critical point of j. So j(a) = j“a, and clearly j“a ⊆ N^{∗}. Thus y is in N^{∗}. Therefore N^{∗} = ⋃(N^{∗} ∩ j(C)) and N^{∗} is in j(C).

Now we show that N^{∗} is internally club but not internally approachable. Let N = H(λ)^{V [G_{κ}]}. Since N is transitive and isomorphic to N^{∗} by the map j ↾ N, N is the transitive collapse of N^{∗} and j^{−1} ↾ N^{∗} = π is the transitive collapse map.

Recall that H ∗ K is generic for Add(μ) ∗ ℙ(Ṡ) over M[G_{κ}], and S = Ṡ^{H} = [N]^{μ} ∩ M[G_{κ}]. Write ⋃ K = ⟨a_{i} : i < μ^{+}⟩. Then N = ⋃{a_{i} : i < μ^{+}}. For all i < μ^{+}, a_{i} is a subset of N = H(λ)^{M[G_{κ}]} which is in M[G_{κ}], and a_{i} has size μ, which is less than λ. Therefore a_{i} is in H(λ)^{M[G_{κ}]} = N. Hence N is internally club. But then ⟨j(a_{i}) : i < μ^{+}⟩ = ⟨j“a_{i} : i < μ^{+}⟩ witnesses that N^{∗} is internally club.

Suppose for a contradiction that N^{∗} is internally approachable in M[j(G_{κ})], as witnessed by a sequence ⟨N_{i}^{∗} : i < μ^{+}⟩. Note that N is then also internally approachable. Indeed, for all i < μ^{+}, let N_{i} = π(N_{i}^{∗}) = π“(N_{i}^{∗}). Clearly ⟨N_{i} : i < μ^{+}⟩ is increasing and continuous and its union is equal to N. For each α < μ^{+}, choose f_{α} in N such that j(f_{α}) = ⟨N_{i}^{∗} : i < α⟩. Then for i < α, j(N_{i}) = N_{i}^{∗} = j(f_{α})(i) = j(f_{α})(j(i)) = j(f_{α}(i)). So N_{i} = f_{α}(i). Therefore ⟨N_{i} : i < α⟩ = f_{α}, which is in N. Hence ⟨N_{i} : i < μ^{+}⟩ witnesses that N is internally approachable.

Let f = ⟨N_{i} : i < μ^{+}⟩. Then for all i < μ^{+}, f ↾ i is in N. Since N ⊆ M[G_{κ}], for all i < μ^{+}, f ↾ i is in M[G_{κ}]. By Proposition 4.5, f = ⟨N_{i} : i < μ^{+}⟩ is in M[G_{κ}]. But this implies N has size μ^{+} in M[G_{κ}], which is false. This completes the proof.

We note that if GCH holds in V, then in V [G], 2^{μ} = μ^{++} and 2^{α} = α^{+} for all infinite cardinals α different from μ. This violation of GCH is necessary, by the following argument:

Suppose 2^{μ} = μ^{+} and λ ≥ μ^{++} is regular. Let N ≺ H(λ) be a model with size μ^{+} such that μ^{+} ⊆ N, and N satisfies the μ^{+}-covering property, that is, every subset of N with size less than μ^{+} is a subset of a member of N with size less than μ^{+}. Then N^{μ} ⊆ N. For if a ⊆ N has size μ, then a is covered by a set b in N with size μ. Since 2^{μ} = μ^{+}, we can enumerate the power set of b by a sequence ⟨x_{i} : i < μ^{+}⟩ in N. But μ^{+} ⊆ N, so x_{i} ∈ N for all i < μ^{+}. In particular, a is in N. Now fix an increasing and continuous sequence ⟨N_{i} : i < μ^{+}⟩ of sets with size μ whose union is N. Since N^{μ} ⊆ N, each N_{i} is in N, and thus every initial segment of this sequence is in N. So N is internally approachable. But if N is internally club, then N satisfies the μ^{+}-covering property.

Remarks. Mixed support iterations similar to that presented in Section 4 appear in Chapter 8 of [8], where an analogue of Proposition 4.2 is proved for iterations of posets of the form ℙ ∗ ℚ̇, where ℙ is ω_{1}-closed, ℙ satisfies a strengthening of ω_{2}-c.c., and ℚ̇ is forced to be ω_{2}-closed. The proof of our consistency result is related to Mitchell’s construction in [6] of a model with no Aronszajn trees on ω_{2}. See [7] for a recent discussion concerning the special property described in Proposition 4.5.

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, pages 1–59. Cambridge Univ. Press, 1983.

[3] M. Foreman and S. Todorčević. A new Löwenheim-Skolem theorem. Trans. Amer. Math. Soc., 357:1693–1715, 2005.

[4] T. Jech. Set Theory. Springer, 2003. Third edition.

[5] J. Krueger. Internally club and approachable. Adv. Math., 213(2):734–740, 2007.

[6] W. Mitchell. Aronszajn trees and the independence of the transfer property. Ann. Math. Logic, pages 21–46, 1972.

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

[8] S. Shelah. Proper and Improper Forcing. Springer, 1998.

Department of Mathematics, University of California, Berkeley

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