INTERNALLY CLUB AND APPROACHABLE

JOHN KRUEGER

Abstract. We prove that PFA implies for all regular λ ≥ ω_{2}, there are stationarily many N ⊆ H(λ) with size ℵ_{1} which are internally club but not internally approachable.

A set N is internally approachable with length and cardinality ℵ_{1} if there is an increasing and continuous sequence ⟨N_{i} : i < ω_{1}⟩ of countable sets with union N such that for all α < ω_{1}, ⟨N_{i} : i < α⟩ is in N. N is internally club if there is an increasing and continuous sequence ⟨N_{i} : i < ω_{1}⟩ of countable sets with union N such that for all α < ω_{1}, N_{α} is in N.

Foreman and Todorčević [2] asked whether these two notions are equivalent for a club of N in [H(λ)]^{ℵ_{1}}, where λ ≥ ω_{2} is regular. Clearly any N ≺ H(λ) which is internally approachable is internally club. We prove the converse fails in general. Namely, if PFA holds, then for all λ ≥ ω_{2} regular, there are stationarily many sets N in [H(λ)]^{ℵ_{1}} which are internally club but not internally approachable.

1. Preliminaries

Suppose κ is regular and X is a set with size at least κ. A set C ⊆ P_{κ}(X) is club if it is closed under unions of ⊆-increasing sequences of length less than κ and cofinal. A set is stationary if it has non-empty intersection with every club. If C ⊆ P_{κ}(X) is club, then there is a function F : [X]^{<ω} → P_{κ}(X) such that for any a in P_{κ}(X), if for all x_{0}, …, x_{n} in a, F (x_{0}, …, x_{n}) ⊆ a, then a is in C. If κ is regular and X has size κ, a set S ⊆ P_{κ}(X) is stationary iff there is an increasing and continuous sequence ⟨a_{i} : i < κ⟩ of sets with size less than κ with union X and a stationary set B ⊆ κ such that {a_{i} : i ∈ B} ⊆ S (or equivalently, for any such sequence there is a stationary set B with {a_{i} : i ∈ B} ⊆ S).

If λ is a regular cardinal, H(λ) is the collection of sets whose transitive closure has size less than λ. Any set in H(λ) can be coded by a bounded subset of λ, so |H(λ)| = λ^{<λ}. Suppose λ ≥ ω_{1} is regular, ℙ is a forcing poset in H(λ), and G is generic for ℙ over V. Then H(λ)[G] = H(λ)^{V [G]}. If ℙ is in N and N ≺ ⟨H(λ), ∈⟩, then N[G] ≺ ⟨H(λ)^{V [G]}, ∈⟩.

If N is a set and G is a filter on ℙ, we say G is N-generic if for any dense set D ⊆ ℙ in N, N ∩ G ∩ D is non-empty. A condition p is N-generic if p forces Ġ is N-generic. Suppose λ ≥ ω_{2} is regular, N ≺ H(λ), and ℙ is in N. Then the following are equivalent: (1) p is N-generic; (2) p ⊩ N[Ġ] ∩ On = N ∩ On; (3) p ⊩ N[Ġ] ∩ V = N ∩ V; and (4) p ⊩ N[Ġ] ∩ H(λ)^{V} = N. A sufficient condition for p being N-generic, known as strong N-genericity, is that for any dense set D ⊆ ℙ in N, there is u in D ∩ N with p ≤ u. If ℙ is ω_{1}-c.c. then any condition in ℙ is N-generic. A condition p ∗ q̇ in a two-step iteration ℙ ∗ ℚ̇ is N-generic iff p is N-generic for ℙ and p forces q̇ is N[Ġ]-generic for ℚ̇.

Date: October 2006.

This work was partially supported by FWF project number P16790-N04.

A forcing poset ℙ is proper if for any set X which contains ω_{1}, ℙ preserves the stationarity of stationary subsets of P_{ω_{1}}(X). Equivalently, ℙ is proper iff for all sufficiently large regular λ ≥ ω_{2}, there is a club of N ≺ H(λ) in P_{ω_{1}}(H(λ)) such that for any p in N ∩ ℙ, there is q ≤ p which is N-generic. Any ω_{1}-c.c. or ω_{1}-closed forcing poset is proper.

A forcing poset ℙ has the countable covering property if whenever p forces ẋ ⊆ On is countable, there is a countable set a and q ≤ p which forces ẋ ⊆ a. Any proper forcing poset satisfies the countable covering property.

We let Add(ω) denote the forcing poset for adding one Cohen real. Conditions in Add(ω) are finite partial functions p : ω → 2, ordered by extension of functions.

If G is a filter on a poset ℙ in V and ȧ is a ℙ-name for a subset of V, then ȧ^{G} is defined as the collection of x for which there is p in G which forces x ∈ ȧ.

Suppose Γ is a class of forcing posets and α ≤ ω_{1} is a cardinal. Then MA_{α}(Γ) is the following statement. Suppose ℙ is in Γ, ⟨Ȧ_{i} : i < α⟩ is a sequence of names for stationary subsets of ω_{1}, and D is a collection of ℵ_{1} many dense subsets of ℙ. Then there is a filter G on ℙ which intersects each dense set in D and for i < α, Ȧ_{i}^{G} is stationary in ω_{1}. If Γ is the class of proper forcing posets, then PFA is the statement MA_{0}(Γ), and PFA^{+} is the statement MA_{1}(Γ).

Fact 1.1 (H. Woodin [5]). Assume α ≤ ω_{1} is a cardinal and MA_{α}(Γ) holds for a class of separative forcing posets Γ. Suppose ℙ is in Γ and ⟨Ȧ_{i} : i < α⟩ is a sequence of ℙ-names for stationary subsets of ω_{1}. Then for any regular θ ≥ ω_{2} with ℙ in H(θ), there is a stationary collection of N in [H(θ)]^{ℵ_{1}} for which there exists a filter G ⊆ ℙ which is N-generic and for all i < α, Ȧ_{i}^{G} is stationary in ω_{1}.

2. Distinguishing Internally Club and Approachable

We prove the following theorem.

Theorem 2.1. PFA implies that for all regular λ ≥ ω_{2}, there are stationarily many sets N in [H(λ)]^{ℵ_{1}} which are internally club but not internally approachable.

The structure of the proof will be as follows. We define a proper forcing poset ℙ and apply Fact 1.1 to obtain stationarily many N in [H(θ)]^{ℵ_{1}} for which there exists an N-generic filter G. We then prove that whenever N is such a model, N ∩ H(λ) is internally club but not internally approachable.

In the proof we will use a forcing poset ℙ(S) defined as follows. Suppose S is a stationary subset of P_{ω_{1}}(Y), where Y is a set containing ω_{2} as a subset. A condition in ℙ(S) is a countable increasing and continuous sequence ⟨a_{i} : i ≤ γ⟩, with each a_{i} in S. The ordering is by extension of sequences.

Lemma 2.2. If S ⊆ P_{ω_{1}}(Y) is stationary then ℙ(S) is ω-distributive.

Proof. Suppose p forces ḟ : ω → On. Let θ ≫ ω_{2} be a regular cardinal such that ḟ and Y are in H(θ). Since S is stationary, we can fix a countable elementary substructure N of the model

⟨H(θ), ∈, S, ℙ(S), p, ḟ⟩

such that N ∩ Y is in S. Let ⟨D_{n} : n < ω⟩ be an enumeration of all the dense subsets of ℙ(S) in N. Inductively define a decreasing sequence ⟨p_{n} : n < ω⟩ of elements of N ∩ℙ such that p_{0} = p and p_{n+1} is a refinement of p_{n} in D_{n} ∩N. Write ⋃{p_{n} : n < ω} = ⟨b_{i} : i < γ⟩. Clearly ⋃{b_{i} : i < γ} = N ∩ Y. Since N ∩ Y is in S, the sequence ⟨b_{i} : i < γ⟩ ∪ {⟨γ, N ∩ Y ⟩} is a condition below p which decides ḟ(n) for all n < ω. □

The poset ℙ(S) adds an increasing and continuous sequence of sets in S with order type ω_{1} and union Y. In particular, ℙ(S) collapses the size of Y to ℵ_{1}.

Let ⟨T, <_{T} ⟩ denote the tree of all increasing countable sequences of ordinals less than ω_{2}, ordered by the initial segment relation. Under PFA, 2^{ω} = 2^{ω_{1}} = ℵ_{2}, so T has height ω_{1}, size ℵ_{2}, and has ℵ_{2} many cofinal branches.

Fix λ ≥ ω_{2} regular. Let Ẋ be an Add(ω)-name for P_{ω_{1}}(H(λ)^{V}) ∩ V. Since Add(ω) is proper, it forces Ẋ is stationary in P_{ω_{1}}(H(λ)^{V}). Let ℙ(Ẋ) be an Add(ω)- name for the forcing poset discussed above which adds an increasing and continuous sequence through Ẋ with order type ω_{1}.

Proposition 2.3. The forcing poset Add(ω) ∗ ℙ(Ẋ) is proper.

Proof. Fix θ ≫ λ regular with Add(ω) ∗ ℙ(Ẋ) in H(θ). Let N ≺ H(θ) be a countable set containing Add(ω) ∗ ℙ(Ẋ) as a member. Let p ∗ q̇ be a condition in N. Since Add(ω) is ω_{1}-c.c., p is N-generic for Add(ω). We will find ṙ such that p forces ṙ ≤ q̇ and ṙ is N[Ġ]-generic for ℙ(Ẋ). Then p∗ṙ ≤ p∗q̇ and p∗ṙ is N-generic for Add(ω) ∗ ℙ(Ẋ).

Let ⟨Ḋ_{n} : n < ω⟩ enumerate all Add(ω)-names in N for dense open subsets of ℙ(Ẋ). Inductively define a sequence ⟨q̇_{n} : n < ω⟩ as follows. Let q̇_{0} = q̇. Given q̇_{n}, let q̇_{n+1} be an Add(ω)-name in N such that p forces q̇_{n+1} ≤ q̇_{n} and q̇_{n+1} is in Ḋ_{n}. This completes the definition.

For each α < N ∩ ω_{1}, there is n such that p forces Ḋ_{n} is the dense open set of conditions in ℙ(Ẋ) whose domain has length at least α. On the other hand, p forces N[Ġ] ∩ ω_{1} = N ∩ ω_{1}, so for all n, p forces dom(q̇_{n}) < N ∩ ω_{1}. So there is an Add(ω)-name ⟨ẋ_{i} : i < N ∩ ω_{1}⟩ for an increasing and continuous sequence which p forces is equal to ⋃{q̇_{n} : n < ω}.

We claim that p forces ⋃{ẋ_{i} : i < N ∩ω_{1}} = N ∩H(λ)^{V}. For if a is in N ∩H(λ)^{V}, then there is n such that p forces Ḋ_{n} is the dense open set of conditions of the form ⟨y_{i} : i ≤ γ⟩, where a is in y_{i} for some i ≤ γ. Since p forces q̇_{n+1} is in Ḋ_{n}, p forces a is in ⋃{ẋ_{i} : i < N ∩ ω_{1}}. On the other hand, if G is generic containing p and a is in ⋃{x_{i} : i < N ∩ ω_{1}}, then there is n such that q_{n} is of the form ⟨x_{i} : i ≤ ζ⟩, where a is in x_{i} for some i ≤ ζ. Since q_{n} is in N[G], a is in N[G] ∩ H(λ)^{V} = N ∩ H(λ)^{V}.

Let ṙ be a name such that p forces ṙ = ⟨y_{i} : i ≤ N ∩ ω_{1}⟩, where y_{i} = ẋ_{i} for i < N ∩ ω_{1} and y_{N∩ω_{1}} = N ∩ H(λ)^{V}. Then p forces ṙ is a condition in ℙ(Ẋ) below q̇ which is N[Ġ]-generic. □

After forcing with Add(ω) ∗ ℙ(Ẋ), ℵ_{2} is collapsed, so T becomes a tree with height and size ℵ_{1}. The next proposition is similar to the Key Lemma of [3]. It implies in particular that Add(ω) ∗ ℙ(Ẋ) does not add any new cofinal branches in T.

Proposition 2.4. The poset Add(ω) ∗ ℙ(Ẋ) forces that whenever ḃ : ω_{1} → On is a function whose initial segments are all in V, ḃ is in V.

Proof. Suppose for a contradiction that p ∗ q̇ forces ḃ : ω_{1} → On is a new sequence whose initial segments are in the ground model. For each i < ω_{1} let X_{i} denote the set of conditions in Add(ω) ∗ ℙ(Ẋ) below p ∗ q̇ which decide for some s in V that ḃ ↾ i = s. Then X_{i} is dense open below p∗q̇. Note that if i < j < ω_{1}, then X_{j} ⊆ X_{i}.

Claim 2.5. There is p_{0} ∗ q̇_{0} ≤ p ∗ q̇ and a sequence ⟨Ė_{i} : i < ω_{1}⟩ of Add(ω)-names for dense open subsets of ℙ(Ẋ) such that whenever p_{0} forces q̇_{1} ≤ q̇_{0} and q̇_{1} ∈ Ė_{i}, then p_{0} ∗ q̇_{1} is in X_{i}.

Proof. Let G ∗ H be generic for Add(ω) ∗ ℙ(Ẋ) containing p ∗ q̇. Then for each i < ω_{1} there is p_{i} ∗ q̇_{i} in X_{i} ∩ (G ∗ H). Since |Add(ω)| = ℵ_{0}, there is p^{∗} such that for cofinally many i < ω_{1}, p_{i} = p^{∗}. But X_{j} ⊆ X_{i} for i < j < ω_{1}. So for all i < ω_{1}, there is u̇_{i} such that p^{∗} ∗ u̇_{i} ∈ X_{i} ∩ (G ∗ H).

In V fix p_{0}∗q̇_{0} ≤ p∗q̇, a condition p^{∗} in Add(ω), and a sequence of Add(ω)∗ℙ(Ẋ)- names ⟨u̇_{i} : i < ω_{1}⟩ such that p_{0} ∗ q̇_{0} forces p^{∗} ∗ u̇_{i} is in X_{i} ∩ (Ġ ∗ Ḣ) for i < ω_{1}. By separativity, p_{0} ≤ p^{∗}. For i < ω_{1}, X_{i} is open, so p_{0} ∗ q̇_{0} forces p_{0} ∗ u̇_{i} is in X_{i} ∩ (Ġ ∗ Ḣ).

Fix i < ω_{1}. Let Ė_{i} be an Add(ω)-name for the dense open set of conditions in ℙ(Ẋ) which decide the value of u̇_{i}. Suppose p_{0} forces q̇_{1} ≤ q̇_{0} and q̇_{1} ∈ Ė_{i}. Then there is an Add(ω)-name v̇_{i} such that p_{0} ⊩ q̇_{1} ⊩ u̇_{i} = v̇_{i} and v̇_{i} ∈ H. By separativity, p_{0} ⊩ q̇_{1} ≤ v̇_{i}. So p_{0} ∗ q̇_{1} ≤ p_{0} ∗ v̇_{i}. Since X_{i} is open and p_{0} ∗ q̇_{1} forces that p_{0} ∗ v̇_{i} is in X_{i}, p_{0} ∗ q̇_{1} forces itself to be in X_{i}. Hence p_{0} ∗ q̇_{1} is in fact in X_{i}. □

Fix θ ≫ λ regular and let N be a countable elementary substructure of ⟨H(θ), ∈⟩ containing as members the objects λ, Add(ω) ∗ ℙ(Ẋ), p_{0} ∗ q̇_{0}, ḃ, ⟨X_{i} : i < ω_{1}⟩, and ⟨Ė_{i} : i < ω_{1}⟩. Let ⟨Ḋ_{n} : n < ω⟩ enumerate all Add(ω)-names in N for dense open subsets of ℙ(Ẋ). Let ⟨γ_{n} : n < ω⟩ be increasing and cofinal in N ∩ ω_{1}.

We define an indexed family {⟨p_{s}, q̇_{s}, ξ_{s}, t_{s}, α_{s}⟩ : s ∈ ^{<ω}2} of elements of N by induction on the length of s in ^{<ω}2. Let p_{⟨⟩} = p_{0}, q̇_{⟨⟩} = q̇_{0}, and ξ_{⟨⟩} = 0. Suppose p_{s}, q̇_{s}, and ξ_{s} are defined. Let t_{s} be the maximal sequence in V such that p_{0} ∗ q̇_{s} forces t_{s} is an initial segment of ḃ, and let α_{s} = dom(t_{s}). Then there are distinct ordinals ξ_{s 0} and ξ_{s 1} in N and for i < 2 a condition p_{s i} ∗ q̇′_{s i} ≤ p_{0} ∗ q̇_{s} which forces ḃ(α_{s}) = ξ_{s i}. Without loss of generality we can assume p_{0} forces q̇′_{s i} ≤ q̇_{s}, otherwise replace it with a name which satisfies this property. Let q̇_{s i} be a name which p_{0} forces is below q̇′_{s i} and in the dense open set

Ḋ_{|s|} ∩ ⋂{Ė_{i} : i ≤ max(γ_{|s|}, α_{s})}.

Then by Claim 2.5, p_{0} ∗ q̇_{s i} is in

⋂{X_{i} : i ≤ max(γ_{|s|}, α_{s})}.

Since p_{0} ∗ q̇_{s i} is compatible with p_{s i} ∗ q̇′_{s i}, p_{0} ∗ q̇_{s i} forces ḃ(α_{s}) = ξ_{s i}. This completes the definition.

If ẋ : ω → 2 is an Add(ω)-name for a real, let qẋ be a name such that p_{0} forces qẋ = ⟨y_{i} : i ≤ N ∩ ω_{1}⟩, where ⟨y_{i} : i < N ∩ ω_{1}⟩ is the union ⋃{q̇_{ẋ↾n} : n < ω} and y_{N∩ω_{1}} = N ∩ H(λ)^{V}. As in the proof of Proposition 2.3, p_{0} forces that qẋ is a condition in ℙ(Ẋ).

Fix an Add(ω)-name for a real ḣ : ω → 2 which is not in V. For each n, p_{0} forces q̇_{ḣ↾(n+1)} is in ⋂{Ė_{i} : i < γ_{n}}, so p_{0} forces qḣ is in ⋂{Ė_{i} : i < N ∩ ω_{1}}. Hence p_{0} ∗ qḣ is in ⋂{X_{i} : i < N ∩ ω_{1}} = X_{N∩ω_{1}}. Fix t such that p_{0} ∗ qḣ forces ḃ ↾ N ∩ ω_{1} = t. Define z : ω → 2 inductively as follows. Given z ↾ n, let z(n) = i for i < 2 iff t(α_{z↾n}) = ξ_{(z↾n) i}. Then p_{0} forces z = ḣ and ḣ is in V, which is a contradiction. □

Let ⟨ȧ_{i} : i < ω_{1}⟩ be an Add(ω) ∗ ℙ(Ẋ)-name for an increasing and continuous sequence of countable sets from Ẋ with union H(λ)^{V} given by a generic for ℙ(Ẋ). Let ⟨ξ̇_{i} : i < ω_{1}⟩ be a name for an increasing and continuous sequence cofinal in ω_{2}^{V} with ξ̇_{0} = 0.

Let G ∗ H be generic for Add(ω) ∗ ℙ(Ẋ) over V. In V [G ∗ H] define a relation R_{T} on T as follows. Let sR_{T} t iff there is i < ω_{1} with t \ ξ_{i} non-empty such that s = t ∩ ξ_{i}. Clearly ⟨T, R_{T} ⟩ is a tree.

Lemma 2.6. The tree ⟨T, R_{T} ⟩ has height and size ℵ_{1} and has no cofinal branches.

Proof. To show T has height ω_{1}, let α < ω_{1}. Fix a ⊆ ω_{2}^{V} in V [G∗H] countable such that {i < ω_{1} : a ∩ [ξ_{i}, ξ_{i+1}) ≠ ∅} has order type at least α + 1. Since Add(ω) ∗ ℙ(Ẋ) has the countable covering property, there is a countable set b in V such that a ⊆ b. Let s be the increasing enumeration of b. Then s is in T and {s′ ∈ T : s′R_{T} s} has order type at least α.

If b is a cofinal branch in ⟨T, R_{T} ⟩, then the set {i < ω_{1} : ran(b) ∩ [ξ_{i}, ξ_{i+1}) ≠ ∅} is cofinal in ω_{1}. So b is cofinal in ω_{2}^{V} and hence is not in V. It is easy to check that every initial segment of b is in T, which contradicts Proposition 2.4. □

In V [G ∗ H] we define ℚ as the forcing poset from [1] which adds a specializing function for the tree ⟨T, R_{T} ⟩. A condition in ℚ is a finite partial function q : T → ω such that whenever u and v are in dom(q) and uR_{T} v, then q(u) ≠ q(v). The ordering of ℚ is by extension of functions. The poset ℚ is ω_{1}-c.c. and forces that there is a total function h : T → ω such that whenever uR_{T} v, h(u) ≠ h(v).

Let ℙ be the proper forcing poset Add(ω) ∗ ℙ(Ẋ) ∗ ℚ̇. Let Ṙ_{T} be an Add(ω) ∗ ℙ(Ẋ)-name for the tree relation defined above, and let ḣ be a ℙ-name for the specializing function for ⟨T, Ṙ_{T} ⟩ given by a generic for ℚ.

Now we are ready to prove Theorem 2.1. Assume PFA. Fix a function F : H(λ)^{<ω} → [H(λ)]^{ℵ_{1}}. We will find a set M ⊆ H(λ) with size ℵ_{1} closed under F which is internally club but not internally approachable.

Fix θ ≫ λ regular. Let A denote the structure

⟨H(θ), ∈, F, ⟨T, <_{T} ⟩, ℙ, ⟨ȧ_{i} : i < ω_{1}⟩, ⟨ξ̇_{i} : i < ω_{1}⟩, Ṙ_{T} , ḣ⟩.

By Fact 1.1 we can choose a set N ≺ A in [H(θ)]^{ℵ_{1}} containing ω_{1} and a filter G ∗ H ∗ I = K ⊆ ℙ which is N-generic for ℙ. It follows that G ∗ H is N-generic for Add(ω) ∗ ℙ(Ẋ). Let M = N ∩ H(λ) and let β = M ∩ ω_{2}.

Suppose x_{0}, …, x_{n} are in M. Since N ≺ A, F (x_{0}, …, x_{n}) ∈ N. But ω_{1} ⊆ N, so F (x_{0}, …, x_{n}) ⊆ N ∩ H(λ) = M. So M is closed under F.

For each i < ω_{1}, let a_{i} = ȧ_{i}^{G∗H} and ξ_{i} = ξ̇_{i}^{G∗H}. Since ȧ_{i} is forced to be in V, a_{i} is the unique set such that there is p in G ∗ H which forces ȧ_{i} = a_{i}, and similarly with ξ_{i}.

Lemma 2.7. The sequence ⟨ξ_{i} : i < ω_{1}⟩ is an increasing and continuous sequence of ordinals cofinal in β with ξ_{0} = 0.

Proof. For each i < ω_{1} there is a dense set in N of conditions which decide the value of ξ̇_{i}. So by N-genericity and the elementarity of N, each ξ_{i} is in N ∩ ω_{2}. For i < j < ω_{1}, since ℙ forces ξ̇_{i} < ξ̇_{j} and conditions in G ∗ H are compatible, ξ_{i} < ξ_{j}. Clearly ξ_{0} = 0. If α < N ∩ ω_{2}, then there is a dense set in N consisting of conditions which decide for some i < ω_{1} that ξ̇_{i} > α. So by N-genericity, the sequence is cofinal in β.

Suppose δ < ω_{1} is limit. Fix p in G ∗ H which forces ξ̇_{δ} = ξ_{δ}. Then for each γ < ξ_{δ}, there is a dense set of conditions below p in N which decide for some i < δ that γ < ξ̇_{i} < ξ_{δ}. By N-genericity there is i < δ with γ < ξ_{i} < ξ_{δ}. So the sequence is continuous. □

Lemma 2.8. The sequence ⟨a_{i} : i < ω_{1}⟩ is an increasing and continuous sequence of countable sets with union M.

Proof. The proof is similar to Lemma 2.7. □

Lemma 2.9. The set M is internally club.

Proof. It suffices to show that each a_{i} is in M. But ȧ_{i} is in N and ℙ forces ȧ_{i} is in H(λ)^{V}. So there is p in (G ∗ H) ∩ N and x in N ∩ H(λ) = M such that p forces ȧ_{i} = x. Then x = a_{i}. □

Let R_{T} = Ṙ_{T}^{G∗H} and h = ḣ^{I}.

Lemma 2.10. For u and v in T ∩N, uR_{T} v iff there is i < ω_{1} with v \ξ_{i} non-empty such that u = v ∩ ξ_{i}. The function h is defined on every u in T ∩ N, and uR_{T} v implies h(u) ≠ h(v).

Proof. Use N-genericity as in the previous lemmas. □

Now suppose for a contradiction that M is internally approachable. Fix an increasing and continuous sequence ⟨M_{i} : i < ω_{1}⟩ of countable sets with union M such that for all i < ω_{1}, ⟨M_{j} : j < i⟩ is in M. Define b : ω_{1} → ω_{2} by letting b(i) = sup(M_{i} ∩ ω_{2}) for i < ω_{1}. Then b is cofinal in β, and every initial segment of b is in N ∩ T. For each i < ω_{1} let b_{i} be the maximal initial segment of b such that ran(b_{i}) ⊆ ξ_{i}. Since b is cofinal in β, {b_{i} : i < ω_{1}} has size ℵ_{1} and is linearly ordered by R_{T}. But then h is injective on this set, contradicting the fact that the range of h is countable. This completes the proof.

Acknowledgements. Our proof of Theorem 2.1 bears some resemblance to the earlier proof in [4] that PFA implies the failure of AP_{ω_{1}}.

References

[1] J. Baumgartner, J. Malitz, and W. Reinhardt. Embedding trees in the rationals. Proceedings of the National Academy of Sciences, 67(4):1748–1753, 1970.

[2] M. Foreman and S. Todorčević. A new Löwenheim-Skolem theorem. Transactions of the American Mathematical Society, 357:1693–1715, 2005.

[3] J. Hamkins. Gap forcing. Israel Journal of Mathematics, 125:237–252, 2001.

[4] B. König and Y. Yoshinobu. Fragments of Martin’s Maximum in generic extensions. Mathematical Logic Quarterly, 50(3):297–302, 2004.

[5] W.H. Woodin. The Axiom of Determinacy, Forcing Axioms, and the Non-Stationary Ideal. Walter de Gruyter, 1999.

Department of Mathematics, University of California, Berkeley

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