INTERNAL APPROACHABILITY AND REFLECTION

JOHN KRUEGER

Abstract. We prove that the Weak Reflection Principle does not imply that every stationary set reflects to an internally approachable set. We show that several variants of internal approachability, namely internally unbounded, internally stationary, and internally club, are not provably equivalent.

Let λ ≥ ω_{2} be a regular cardinal, and consider an elementary substructure N ≺ H(λ) with size ℵ_{1}. Then N is internally approachable if N is the union of an increasing and continuous chain ⟨a_{i} : i < ω_{1}⟩ of countable sets such that for all β < ω_{1}, ⟨a_{i} : i < β⟩ is in N.

The Weak Reflection Principle or WRP is the statement that for all regular λ ≥ ω_{2}, for every stationary set S ⊆ P_{ω_{1}}(H(λ)) there is a set N ⊆ H(λ) with size ℵ_{1} which contains ℵ_{1} such that S ∩ P_{ω_{1}}(N) is stationary in P_{ω_{1}}(N) (that is, S reflects to N). This principle follows from Martin’s Maximum and captures some of its strength; for example, WRP implies Chang’s Conjecture, the presaturation of the non-stationary ideal on ω_{1}, and the Singular Cardinal Hypothesis (see [3] and [6]).

An apparent strengthening of WRP is the statement that for all regular λ ≥ ω_{2}, every stationary subset of P_{ω_{1}}(H(λ)) reflects to an internally approachable set with size ℵ_{1}. In practice it tends to be easier to draw strong consequences from this principle than from WRP. Foreman and Todorčević [4] asked whether the two reflection principles are equivalent. We answer this question in the negative.

Theorem 0.1. Suppose κ is supercompact. Then there is a forcing poset which forces κ = ω_{2}, WRP holds, and for all regular λ ≥ ω_{2} there is a stationary subset of P_{ω_{1}}(H(λ)) which does not reflect to any internally approachable set with size ℵ_{1}.

Foreman and Todorčević [4] described several variations of the notion of internal approachability, and asked whether these properties are equivalent. Let N ≺ H(λ) be a set with size ℵ_{1}. Then N is internally club if N ∩P_{ω_{1}}(N) contains a club subset of P_{ω_{1}}(N). N is internally stationary if N ∩ P_{ω_{1}}(N) is stationary in P_{ω_{1}}(N). N is internally unbounded if N ∩ P_{ω_{1}}(N) is cofinal in P_{ω_{1}}(N). We prove that these properties are not equivalent.

Theorem 0.2. MM implies that for all regular λ ≥ ω_{2}, there is a stationary set of N in P_{ω_{2}}(H(λ)) which are internally unbounded but not internally stationary.

Theorem 0.3. PFA_{2} implies that for all regular λ ≥ ω_{2}, there is a stationary set of N in P_{ω_{2}}(H(λ)) which are internally stationary but not internally club.

I would like to thank James Cummings, Matt Foreman, and Assaf Sharon for discussing the material in this paper with me. I note that Theorem 0.2 was proved independently by Assaf Sharon with essentially the same argument.

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

1. Preliminaries

In this section we review the background material necessary for understanding the paper. Some basic familiarity with iterated forcing, proper forcing, and supercompact cardinals is required.

If κ is a regular uncountable cardinal and X is a set containing κ, P_{κ}(X) denotes the set {a ⊆ X : |a| < κ}. A set C ⊆ P_{κ}(X) is club if C is closed under unions of increasing sequences of length less than κ and C is cofinal in P_{κ}(X). A set S ⊆ P_{κ}(X) is stationary if it has non-empty intersection with each club. If C is a club then there is a function F : X^{<ω} → P_{κ}(X) such that for all a in P_{κ}(X), if F (x_{0}, …, x_{n}) ⊆ a for any x_{0}, …, x_{n} in a, then a is in C. In the case κ = ω_{1}, if C ⊆ P_{ω_{1}}(X) is club then there is H : X^{<ω} → X such that every a in P_{κ}(X) which is closed under H is in C.

For a regular cardinal λ, H(λ) denotes the collection of sets whose transitive closure has size less than λ. The set H(λ) is determined in an absolute way by the bounded subsets of λ. So if W is an outer model of V with the same bounded subsets of λ, then H(λ)^{V} = H(λ)^{W}.

If N ≺ H(λ) for some regular λ ≥ ω_{2}, κ + 1 ⊆ N, x ∈ N, and |x| ≤ κ, then x ⊆ N. For by elementarity there is a surjection f : κ → x in N, hence x = f“κ ⊆ N.

A sequence of sets ⟨N_{i} : i < γ⟩ is an internally approachable sequence if it is increasing and continuous and ⟨N_{j} : j ≤ i⟩ ∈ N_{i+1} for i < γ.

Suppose ℙ is a forcing poset and μ is a cardinal. We say that ℙ is < μ-closed if any descending sequence of conditions in ℙ with length less than μ has a lower bound. We say ℙ is < μ-distributive if any family of fewer than μ many dense open subsets of ℙ has dense open intersection. This is equivalent to the statement that forcing with ℙ does not add any new sets of ordinals with order type less than μ.

If α is an ordinal, we say that ℙ is α-strategically closed if Player II has a winning strategy in the following game. Player I begins the game with a condition p_{0}. After Player I has played a condition p_{i} where i < α, Player II responds with q_{i} ≤ p_{i}; then Player I responds with p_{i+1} ≤ q_{i}. At a limit stage δ < α, Player II attempts to play a condition q_{δ} which is a lower bound of {q_{i} : i < δ}. Player II wins iff he is able to play at all stages less than α.

For a cardinal μ, < μ-closed implies μ-strategically closed, and μ-strategically closed implies < μ-distributive.

A forcing poset ℙ is proper if for any set X which contains ω_{1}, if S is a stationary subset of P_{ω_{1}}(X) then ℙ forces that S is stationary in P_{ω_{1}}(X). A forcing poset ℙ satisfies the countable covering property if ℙ forces that whenever x ⊆ V is countable, there is y in V which is countable in V such that x ⊆ y. Note that any proper forcing poset satisfies the countable covering property, since if ℙ adds a countable set which cannot be covered by a countable set in the ground model, then for some X in V, P_{ω_{1}}^{V}(X) is not stationary in P_{ω_{1}}(X).

Lemma 1.1. Let κ ≥ ω_{1} be a regular cardinal and ℙ a proper forcing poset. Then for any stationary set T ⊆ κ ∩ cof(ω), ℙ forces T is stationary.

Proof. Let S = {a ∈ P_{ω_{1}}(κ) : sup(a) ∈ T }. Since T is stationary in κ, S is stationary in P_{ω_{1}}(κ). For if H : κ^{<ω} → κ is a function then there is a club set of α < κ closed under H. Choose such an α in T, fix x ⊆ α cofinal with order type ω, and let a = cl_{H}(x). Then a is in S and closed under H. If G is generic for ℙ, S remains stationary in P_{ω_{1}}(κ) in V [G]. Let C ⊆ κ be club in V [G]. Then there is a in S closed under the map β ↦ min(C \ β). But then sup(a) ∈ C ∩ T. □

Any ω_{1}-c.c. or countably closed forcing poset is proper. A countable support iteration of proper forcing posets is proper.

If X is an uncountable set, Coll(ω_{1}, X) is a forcing poset for collapsing the size of X to be ℵ_{1}. A condition p is a countable partial function p : ω_{1} → X, and the ordering is by extension of functions. This poset is countably closed hence proper. We write Add(ω) for the forcing to add a Cohen real. A condition is a finite partial function p : ω → 2, and the ordering is by extension of functions. This poset has size ℵ_{0}, hence is ω_{1}-c.c. and so proper.

The Proper Forcing Axiom or PFA is the statement that for any proper forcing poset ℙ and any family D of ℵ_{1} many dense subsets of ℙ, there is a filter G on ℙ which intersects each dense set in D. Martin’s Maximum or MM is the same statement with “proper forcing poset” replaced by “forcing poset which preserves stationary subsets of ω_{1}”.

We review several basic facts used to extend elementary embeddings after forcing. Assume M ⊆ N are inner models, λ is a regular cardinal in N, and M^{<λ} ∩N ⊆ M.

(1) If ℙ is a forcing poset in M which is λ-c.c. in N and G is generic for ℙ over N, then M[G]^{<λ} ∩ N[G] ⊆ M[G].

(2) If ℙ is a forcing poset in M and G ∈ N is generic for ℙ over M, then M[G]^{<λ} ∩ N ⊆ M[G].

Suppose j : M → N is an elementary embedding between inner models. Assume ℙ is a forcing poset, G is generic for ℙ over M, and H is generic for j(ℙ) over N. Then j“G ⊆ H iff there is an extension of j to an elementary embedding j : M[G] → N[H] such that j(G) = H. The extended map satisfies j(ẋ^{G}) = j(ẋ)^{H}.

2. Forcing Axioms

Suppose Γ is a class of forcing posets and α ≤ ω_{1} is a cardinal. Then MA_{α}(Γ) is the following forcing axiom. Suppose ℙ is in Γ, D is a collection of ℵ_{1} many dense subsets of ℙ, and {Ȧ_{i} : i < α} is a collection of ℙ-names for stationary subsets of ω_{1}. Then there is a filter G on ℙ which intersects each dense set in D, and moreover for each i < α, the interpretation Ȧ_{i}^{G} = {β < ω_{1} : ∃p ∈ G p ⊩ β ∈ Ȧ_{i}} is stationary in ω_{1}.

The forcing axiom MA_{1}(Γ) is usually referred to as MA^{+}(Γ) in the literature. In general MA_{α}(Γ) does not imply MA_{α^{+}}(Γ). For more information see [7].

If G is a filter on a poset ℙ and N is a set, then G is N-generic if for any dense set D ⊆ ℙ in N, G ∩ D ∩ N is non-empty.

We will use the following result of Woodin [8].

Proposition 2.1. Assume MA_{α}(Γ) holds for a class Γ of separative forcing posets. Let ℙ be in Γ and suppose {Ȧ_{i} : i < α} is a family of ℙ-names for stationary subsets of ω_{1}. Let θ ≥ ω_{2} be regular such that ℙ is in H(θ).

Then there is a stationary set of N in P_{ω_{2}}(H(θ)) for which there exists an Ngeneric filter G on ℙ such that each Ȧ_{i}^{G} is stationary in ω_{1}.

Proof. Note that every ℙ in Γ preserves ω_{1}.

We claim that for any function H : H(θ)^{<ω} → H(θ) there is a ℙ-name ġ : ω_{1} → H(θ)^{V} such that ℙ forces M = ġ“ω_{1} is closed under H, M contains ω_{1}, and Ġ is M-generic. Let G be generic for ℙ, and we construct g in V [G] by induction.

Suppose g ↾ (ω · α) is defined. For each dense set D ⊆ ℙ in g“(ω · α), choose p_{D} in G ∩ D. Now extend g to have domain ω · (α + 1) by adding a bijection of ω · (α + 1) \ ω · α onto the H-closure of the set

g“(ω · α) ∪ α ∪ {p_{D} : D ∈ g“(ω · α), D ⊆ ℙ dense}.

This verifies the claim.

Let F : H(θ)^{<ω} → P_{ω_{2}}(H(θ)) be given, and we find N as required which is closed under F. Let H : H(θ)^{<ω} → H(θ) be a Skolem function for the structure ⟨H(θ), ∈ , F ⟩. Fix ġ for H as in the claim above. Let D be the collection of dense subsets of ℙ which are definable in the structure A = ⟨H(θ), ∈, ℙ, ġ, H⟩ using countable ordinals as parameters. Apply MA_{α}(Γ) and choose a filter G on ℙ which intersects each dense set in D and such that each Ȧ_{i}^{G} is stationary in ω_{1}.

Now let h = ġ^{G}; in other words, h : ω_{1} → H(θ) and h(α) = x iff there is p in G which forces ġ(α) = x. Let N = h“ω_{1}. We claim that N is closed under F and G is N-generic.

Suppose D ⊆ ℙ is dense and is in N. Fix α such that D = h(α) and p in G which forces ġ(α) = D. Then there is a dense set of conditions q which either force ġ(α) is not dense in ℙ, or there is β such that q forces h(β) = ġ(β) ∈ Ġ∩ġ(α). This dense set is definable in A using α as a parameter. So choose q ∈ G in this dense set. Since p and q are compatible, there is β such that h(β) is in D ∩ N. Since ℙ is separative and q forces h(β) ∈ Ġ, q ≤ h(β) and h(β) is in G. The proof that N is closed under H and contains the countable ordinals is similar.

Since N is closed under H, N ≺ ⟨H(θ), ∈, F ⟩. So for each x_{0}, …, x_{n} in N, F (x_{0}, …, x_{n}) ∈ N. But N contains ω_{1}, so F (x_{0}, …, x_{n}) ⊆ N. □

Suppose ℙ is a forcing poset which satisfies the countable covering property, and ω_{2} ≤ λ < θ are regular cardinals with ℙ ∈ H(θ). Assume ℙ collapses H(λ)^{V} to have size ℵ_{1}, and ⟨ȧ_{i} : i < ω_{1}⟩ is a ℙ-name for an increasing and continuous sequence of countable sets with union H(λ)^{V}. Fix N ≺ ⟨H(θ), ∈, ℙ⟩ with size ℵ_{1} which contains the countable ordinals and has ⟨ȧ_{i} : i < ω_{1}⟩ as an element.

Lemma 2.2. Suppose G is a filter on ℙ which is N-generic. For each i let a_{i} = ȧ_{i}^{G} = {x ∈ H(λ) : ∃p ∈ G p ⊩ x ∈ ȧ_{i}}. Then ⟨a_{i} : i < ω_{1}⟩ is an increasing and continuous sequence of countable sets with union N ∩ H(λ).

Proof. Let i < ω_{1}. Since ℙ satisfies the countable covering property and ȧ_{i} is in N, there is a dense set in N of conditions which decide for some countable set x ⊆ H(λ) that ȧ_{i} ⊆ x. So there is p in G ∩ N and a countable set x ∈ N such that p forces ȧ_{i} ⊆ x. Clearly then a_{i} ⊆ x. But x ⊆ N since x is countable. So each a_{i} is a countable subset of N ∩ H(λ).

Clearly a_{i} ⊆ a_{j} for i < j. Consider a set z in N ∩ H(λ). Let i_{z} be a name for the least index such that z is in ȧ_{i_{z}}. Since ℙ forces ⟨a_{i} : i < ω_{1}⟩ is continuous, i_{z} is forced to be a non-limit ordinal. Fix p in N ∩ G and a non-limit ordinal i such that p forces i_{z} = i. Then z is in a_{i} but not in a_{j} for j < i. It follows that ⟨a_{i} : i < ω_{1}⟩ is an increasing and continuous sequence with union N ∩ H(λ). □

3. Distinguishing Variants of Internal Approachability

Suppose N is a set with size ℵ_{1}. Then N is internally stationary if N ∩ P_{ω_{1}}(N) is stationary in P_{ω_{1}}(N). N is internally club if N ∩ P_{ω_{1}}(N) contains a club subset of P_{ω_{1}}(N). This is equivalent to the existence of an increasing and continuous sequence ⟨a_{i} : i < ω_{1}⟩ of countable sets with union N such that a_{i} ∈ a_{i+1} for all i < ω_{1}.

Lemma 3.1. Suppose N is a set with size ℵ_{1}. Then the following are equivalent. (1) N is internally stationary.

(2) There exists an increasing and continuous sequence ⟨a_{i} : i < ω_{1}⟩ of countable sets with union N and a stationary set A ⊆ ω_{1} such that {a_{i} : i ∈ A} ⊆ N.

(3) For every increasing and continuous sequence ⟨a_{i} : i < ω_{1}⟩ of countable sets with union N, there exists a stationary set A ⊆ ω_{1} such that {a_{i} : i ∈ A} ⊆ N.

Proof. Since N has size ℵ_{1}, N can be written as the union of an increasing and continuous sequence ⟨a_{i} : i < ω_{1}⟩ of countable sets. For any set S ⊆ P_{ω_{1}}(N), S is stationary in P_{ω_{1}}(N) iff {i < ω_{1} : a_{i} ∈ S} is stationary in ω_{1}. If ⟨b_{i} : i < ω_{1}⟩ is another increasing and continuous sequence of countable sets with union N, then there is a club set C ⊆ ω_{1} such that a_{i} = b_{i} for i in C. The lemma follows easily from these facts. □

Lemma 3.2. Suppose N is a set with size ℵ_{1}. Then the following are equivalent. (1) N fails to be internally club.

(2) There is an increasing and continuous sequence ⟨a_{i} : i < ω_{1}⟩ of countable sets with union N and a stationary set B ⊆ ω_{1} such that {a_{i} : i ∈ B} ∩ N = ∅.

(3) For every increasing and continuous sequence ⟨a_{i} : i < ω_{1}⟩ of countable sets with union N there is a stationary set B ⊆ ω_{1} such that {a_{i} : i ∈ B} ∩ N = ∅.

The proof is similar to the proof of Lemma 3.1. The next lemma follows immediately.

Lemma 3.3. Suppose N is a set with size ℵ_{1}. If ℙ is a forcing poset which preserves stationary subsets of ω_{1}, then ℙ preserves each of the following properties of N:

(1) N is internally stationary.

(2) N is not internally club.

We will use the following variant of a theorem of Abraham and Shelah [1].

Theorem 3.4. Suppose ℙ is ω_{1}-c.c. and adds a new real. Then for all regular λ ≥ ω_{2}, ℙ forces P_{ω_{1}}(H(λ)^{V}) \ V is stationary in P_{ω_{1}}(H(λ)^{V}).

Proposition 3.5. Let ω_{2} ≤ λ ≤ θ be regular cardinals, and let ℙ denote the forcing poset Add(ω)∗Coll(ω_{1}, H(θ)^{V}). Then ℙ forces that H(λ)^{V} is internally stationary but not internally club.

Proof. Write N = H(λ). Let G ∗ H be generic for ℙ. By Theorem 3.4, S = P_{ω_{1}}(N) \ V is stationary in P_{ω_{1}}(N) in V [G]. Since Coll(ω_{1}, N) is proper, S is stationary in P_{ω_{1}}(N) in V [G ∗ H]. Now work in V [G ∗ H]. Since |N| = ω_{1}, fix an increasing and continuous chain ⟨a_{i} : i < ω_{1}⟩ of countable sets with union N. As S is stationary in P_{ω_{1}}(N), there is B ⊆ ω_{1} stationary such that {a_{i} : i ∈ B} ⊆ S. But S is disjoint from V and hence disjoint from N, since N ⊆ V. So N is not internally club by Lemma 3.2.

On the other hand, P_{ω_{1}}(N) ∩ V is stationary in P_{ω_{1}}(N) since ℙ is proper. But P_{ω_{1}}(N) ∩ V = P_{ω_{1}}(N) ∩ N since N = H(λ)^{V}. So N is internally stationary. □

Theorem 3.6. The forcing axiom PFA_{2} implies that for every regular λ ≥ ω_{2}, there is a stationary set of N in P_{ω_{2}}(H(λ)) such that N is internally stationary but not internally club.

Proof. Let ℙ = Add(ω) ∗ Coll(ω_{1}, H(λ)^{V}). By Proposition 3.5, Lemma 3.1, and Lemma 3.2, there are ℙ-names Ȧ, Ḃ, and ⟨ȧ_{i} : i < ω_{1}⟩ such that ℙ forces the following statements:

(1) Ȧ and Ḃ are stationary subsets of ω_{1},

(2) ⟨ȧ_{i} : i < ω_{1}⟩ is an increasing and continuous sequence of countable sets with union H(λ)^{V},

(3) {ȧ_{i} : i ∈ Ȧ} ⊆ H(λ)^{V},

(4) {ȧ_{i} : i ∈ Ḃ} ∩ H(λ)^{V} = ∅.

Let F : H(λ)^{<ω} → P_{ω_{2}}(H(λ)) be given. Fix θ ≫ λ regular with ℙ ∈ H(θ), and let A denote the structure ⟨H(θ), ∈, ℙ, Ȧ, Ḃ, F ⟩. By Proposition 2.1 choose N ≺ A with size ℵ_{1} containing the countable ordinals and a filter G on ℙ which is N-generic such that A = Ȧ^{G} and B = Ḃ^{G} are stationary subsets of ω_{1}. Let a_{i} = ȧ_{i}^{G} for each i < ω_{1}. Since ℙ is proper, it satisfies the countable covering property; so Lemma 2.2 implies that ⟨a_{i} : i < ω_{1}⟩ is an increasing and continuous sequence of countable sets with union M = N ∩ H(λ).

Since ω_{1} ⊆ M and N ≺ A, M is closed under F. By Lemmas 3.1 and 3.2 it suffices to show that {a_{i} : i ∈ A} ⊆ M and {a_{j} : j ∈ B} ∩ M = ∅. Let i be in A and fix p in G which forces i ∈ Ȧ. Then p forces ȧ_{i} ∈ H(λ)^{V}. By elementarity there is a dense set in N of conditions which either force i is not in Ȧ or decide for some x that ȧ_{i} = x. Since G is N-generic, there is q in G∩N and x in N such that q forces x = ȧ_{i}. Then x is in H(λ) ∩ N = M. Clearly x = a_{i}. Hence {a_{i} : i ∈ A} ⊆ M.

Suppose j is in B and fix r in G which forces j ∈ Ḃ. For each y in N ∩ H(λ)^{V} = M, there is a dense set in N of conditions which either force j is not in Ḃ or decide for some z that z is in ȧ_{j} △ y. By N-genericity, a_{j} is not equal to y. So a_{j} is not in M. □

Let N be a set of size ℵ_{1}. Then N is internally unbounded if N ∩ P_{ω_{1}}(N) is cofinal in P_{ω_{1}}(N). We prove that internally unbounded does not imply internally stationary, using the forcing poset Add(ω) ∗ ℙ(Ṡ) described in [5].

Suppose λ ≥ ω_{2} is regular and T is a stationary subset of P_{ω_{1}}(H(λ)). Define ℙ(T) as the poset consisting of countable increasing and continuous sequences ⟨b_{i} : i ≤ γ⟩ contained in T, ordered by extension of sequences. The poset ℙ(T) adds an increasing and continuous chain with order type ω_{1} contained in T, collapsing H(λ) to have size ω_{1}.

For proofs of the next two propositions see [5].

Proposition 3.7. Let λ ≥ ω_{2} be regular and let T be a stationary subset of P_{ω_{1}}(H(λ)). Then ℙ(T) is < ω_{1}-distributive.

Proposition 3.8. Let λ ≥ ω_{2} be regular and let Ṡ be an Add(ω)-name for the set P_{ω_{1}}(H(λ)^{V}) \ V. Then Add(ω) ∗ ℙ(Ṡ) preserves stationary subsets of ω_{1}.

Theorem 3.9. The forcing axiom MM implies that for all regular λ ≥ ω_{2}, there is a stationary set of N in P_{ω_{2}}(H(λ)) which are internally unbounded but not internally stationary.

Proof. Let ℙ = Add(ω) ∗ ℙ(Ṡ), where Ṡ is an Add(ω)-name for P_{ω_{1}}(H(λ)^{V}) \ V. Note that by Proposition 3.7, ℙ satisfies the countable covering property (but it is clearly not proper).

Let F : H(λ)^{<ω} → P_{ω_{2}}(H(λ)) be given. Fix θ ≫ λ regular such that ℙ ∈ H(θ) and let A = ⟨H(θ), ∈, ℙ, F ⟩. By Proposition 2.1 choose N ≺ A with size ℵ_{1} containing the countable ordinals and a filter G on ℙ which is N-generic. Let M = N ∩ H(λ). Note that since ω_{1} ⊆ M, M is closed under F.

Fix in N a ℙ-name ⟨ȧ_{i} : i < ω_{1}⟩ for an increasing and continuous sequence disjoint from V given by a generic for ℙ(Ṡ). Let a_{i} = ȧ_{i}^{G} for i < ω_{1}. By Lemma 2.2, ⟨a_{i} : i < ω_{1}⟩ is an increasing and continuous sequence with union M.

To show that M is not internally stationary it suffices to show that {a_{i} : i < ω_{1}} ∩ M = ∅. Fix i < ω_{1}. Then ℙ forces ȧ_{i} is not in H(λ)^{V}. So for each y in M there is a dense set in N of conditions which decide for some z that z is in ȧ_{i} △ y. By N-genericity it follows that a_{i} is not in M.

It remains to prove that M is internally unbounded. Let x be in P_{ω_{1}}(M). Fix j < ω_{1} with x ⊆ a_{j}. By the countable covering property and N-genericity of G, there is y in M countable and p in G which forces ȧ_{j} ⊆ y. Clearly then a_{j} ⊆ y. Hence x ⊆ y ∈ M. □

4. Approachability and Forcing Posets

In this section we present some background material for the consistency result of Section 5. We describe Shelah’s approachability ideal I[λ] and analyze some forcing posets. We will use I[λ] as a tool for preserving the stationarity of certain stationary sets after forcing.

Suppose λ is a regular uncountable cardinal. For a sequence a⃗ = ⟨a_{i} : i < λ⟩ of bounded subsets of λ, let S(a⃗) denote the set of limit ordinals β < λ for which there exists an unbounded set c_{β} ⊆ β with order type cf(β) such that each initial segment of c_{β} is enumerated in the sequence ⟨a_{i} : i < β⟩.

Define I[λ] as the collection of sets S such that there is a⃗ and a club C ⊆ λ with S ∩ C ⊆ S(a⃗). Then I[λ] is a normal ideal on λ which extends the non-stationary ideal.

Oftentimes I[λ] will contain a maximal stationary set. For example, if λ^{<λ} = λ and a⃗ enumerates all bounded subsets of λ, then S(a⃗) is stationary and for any set A in I[λ] there is a club C such that A ∩ C ⊆ S(a⃗). Suppose there is a maximal set S in I[λ]. Then I[λ] = NS_{λ} ↾ (λ \ S). In this case we refer to S as the set of approachable ordinals.

Suppose λ is a Mahlo cardinal and A ⊆ λ is a stationary set of regular cardinals. Define a⃗ = ⟨a_{i} : i < λ⟩ by letting a_{i} = i for i < λ. Then A ⊆ S(a⃗), as witnessed by letting c_{α} = α for α in A. Hence A is in I[λ].

See [2] for more information about I[λ].

Now we discuss some forcing posets we will use in Section 5.

Let κ > ω_{1} be a regular cardinal and suppose A ⊆ κ. We define a forcing poset ℙ(A) which adds a stationary subset of κ ∩ cof(ω) which does not reflect at any ordinal in A. A condition p in ℙ(A) is a bounded subset of κ ∩ cof(ω) such that for all α in A, p ∩ α is not stationary in α. The ordering is by end-extension.

Clearly ℙ(A) is countably closed and hence proper, and has size κ^{<κ}. So by the next lemma, if κ^{<κ} = κ then ℙ(A) preserves all cardinals and cofinalities.

Lemma 4.1. The forcing poset ℙ(A) is κ-strategically closed.

Proof. We describe a strategy for Player II. When Player I plays a condition p_{i}, Player II responds with q_{i} = p_{i} ∪ {sup(p_{i}) + ω + ω}. Also define γ_{i} = sup(p_{i}) + ω. If ξ < κ is a limit ordinal, let x = ⋃{q_{i} : i < ξ}, and define q_{ξ} = x ∪ {sup(x) + ω}. Define γ_{ξ} = sup(x). It is easy to prove by induction that for each limit ordinal ξ < κ, ⋃{q_{i} : i < ξ} is disjoint from {γ_{i} : i < ξ}, and the latter is a club subset of sup(⋃{q_{i} : i < ξ}) = γ_{ξ}. Also q_{ξ} does not reflect at any ordinal in A which is below γ_{ξ}, since by the induction hypothesis q_{i} is a condition for i < ξ. So q_{ξ} is a condition, and the game continues. □

Let Ṡ be a name for ⋃ Ġ.

Lemma 4.2. The poset ℙ(A) forces that Ṡ is stationary.

Proof. Suppose p forces Ċ ⊆ κ is club. Define by induction a descending sequence ⟨p_{n} : n < ω⟩. Let p_{0} = p. Given p_{n}, let p_{n+1} ≤ p_{n} be a condition such that for some α_{n}, max(p_{n+1}) > α_{n} and p_{n+1} forces that α_{n} = min(Ċ \ max(p_{n})). Let x = ⋃{p_{n} : n < ω} and let q = x∪{sup(x)}. Since cf(sup(x)) = ω, q is a condition. Also sup(x) = ⋃{α_{n} : n < ω}, so q forces sup(x) ∈ Ċ ∩ Ṡ. □

Proposition 4.3. If B ⊆ κ is a stationary set in I[κ], then ℙ(A) preserves the stationarity of B. In particular, ℙ(A) preserves the stationarity of any stationary set of regular cardinals in κ.

Proof. Suppose p forces Ċ ⊆ κ is a club. Fix θ ≫ κ and let A denote the structure

⟨H(θ), ∈, <_{θ}, ℙ(A), p, Ċ, ⟨a_{i} : i < κ⟩⟩,

where <_{θ} is a well-ordering of H(θ) and ⟨a_{i} : i < κ⟩ witnesses that B is in I[κ].

Since B is stationary, there is N ≺ A such that N ∩ κ = β is in B. Note that for all i < β, a_{i} is in N. Let x ⊆ β be cofinal with order type cf(β) such that every initial segment of x appears in ⟨a_{i} : i < β⟩.

Define by induction a sequence ⟨p_{i}, q_{i} : i < cf(β)⟩ which is a run in the game in which Player II uses his winning strategy, and each initial segment of the sequence is definable in A from an initial segment of x. Let p_{0} = p, and let q_{0} be Player II’s response. Suppose ⟨p_{j}, q_{j} : j < i⟩ is defined. If i is limit then let p_{i} = q_{i} be Player II’s play according to his strategy. If i is a successor ordinal, let p_{i} be the <_{θ}-least refinement of q_{i−1} such that p_{i} \ q_{i−1} has non-empty intersection with x and p_{i} forces that Ċ ∩ (max(p_{i}) \ max(q_{i−1})) is non-empty. Let q_{i} be Player II’s response.

Since only an initial segment of x is needed at each stage in defining the sequence of conditions, each initial segment of the sequence is in N. Let z = ⋃{q_{i} : i < cf(β)}. Since each q_{i} is in N, z ⊆ β. On the other hand, z ∩ x has size cf(β); since o.t.(x) = cf(β) this implies z ∩ x is unbounded in β. Let r be a lower bound of the sequence of conditions. Then r forces sup(z) is a limit point of Ċ, so r forces β ∈ Ċ ∩ B. □

Now we consider a forcing poset for adding a club subset to a given stationary set. Suppose T ⊆ ω_{2} ∩ cof(ω_{1}) is stationary. Define a forcing poset ℙ_{T} which attempts to add a club subset to T ∪ cof(ω) as follows. A condition is a closed bounded subset of T ∪ cof(ω), and the ordering is by end-extension.

Note that ℙ_{T} is countably closed and hence proper. If 2^{ω_{1}} = ω_{2} then ℙ_{T} has size ω_{2}.

Proposition 4.4. Suppose A is a stationary subset of T which is in I[ω_{2}]. Then ℙ_{T} is < ω_{2}-distributive and preserves the stationarity of A.

For example, if 2^{ω_{1}} = ω_{2}, S is the maximal set in I[ω_{2}], and S ∩ T is stationary, then ℙ_{T} preserves all cardinals and preserves stationary subsets of S ∩ T. If T does not contain a stationary set which is in I[ω_{2}], it is possible that ℙ_{T} will collapse ω_{2}; this was proved in [5].

Proof. Let ⟨D_{i} : i < ω_{1}⟩ be a sequence of dense open subsets of ℙ_{T}. Suppose p forces Ċ ⊆ ω_{2} is club. We find r ≤ p which is in ⋂{D_{i} : i < ω_{1}} and an ordinal β such that r forces β ∈ Ċ ∩ A.

Fix θ ≫ ω_{2} and let A denote the structure

⟨H(θ), ∈, <_{θ}, T, ℙ_{T} , p, Ċ, ⟨D_{i} : i < ω_{1}⟩, ⟨a_{i} : i < ω_{2}⟩⟩,

where <_{θ} is a well-ordering of H(θ) and ⟨a_{i} : i < ω_{2}⟩ witnesses that A is in I[ω_{2}].

Since A is stationary, there is N ≺ A with size ℵ_{1} such that N ∩ ω_{2} = β is in A. Note that {a_{i} : i < β} and {D_{i} : i < ω_{1}} are both subsets of N. Fix x ⊆ β unbounded with order type cf(β) such that every initial segment of x is enumerated in ⟨a_{i} : i < β⟩.

Define by induction a descending sequence ⟨p_{i} : i < ω_{1}⟩ of conditions in ℙ_{T} whose initial segments are definable in A from initial segments of x. Let p_{0} = p. Suppose ⟨p_{i} : i < δ⟩ is defined where δ < ω_{1} is limit. Let y = ⋃{p_{i} : i < ω_{1}}. Then sup(y) has cofinality ω, so p_{δ} = y ∪ {sup(y)} is in ℙ_{T}. Now assume p_{i} is defined for a fixed i. Let p_{i+1} be the <_{θ}-least refinement of p_{i} in D_{i} such that p_{i+1} \ p_{i} has non-empty intersection with x and p_{i+1} forces max(p_{i+1}) \ max(p_{i}) contains an element of Ċ.

This completes the definition. Each initial segment of the sequence is in N because it is definable in A from an initial segment of x. Let z = ⋃{p_{i} : i < ω_{1}}. Then z ⊆ β. Since z ∩ x has size ℵ_{1}, sup(z) = β. Therefore r = z ∪ {β} is a condition since β is in T. Also r is in ⋂{D_{i} : i < ω_{1}} and r forces β ∈ Ċ ∩ A. □

We finish this section by proving two technical results we will need in the next section.

Proposition 4.5. Suppose that κ is a Mahlo cardinal and A ⊆ κ is a stationary set of strongly inaccessible cardinals. Let ⟨ℙ_{i}, ℚ̇_{j} : i ≤ κ, j < κ⟩ be a countable support iteration of proper forcing posets. Assume that for each β in A the following statements hold:

(1) for all γ < β, |ℙ_{γ}| < β,

(2) ℙ_{β} forces β = ω_{2} and 2^{ω} ≤ ω_{2},

(3) ℚ̇_{β} is a name for Coll(ω_{1}, ω_{2}). Then ℙ_{κ} forces that κ = ω_{2}, A ⊆ ω_{2} ∩ cof(ω_{1}), and A is in I[ω_{2}].

Proof. Let G_{κ} be generic for ℙ_{κ}. Clearly κ = ω_{2} in V [G_{κ}] by (2) and the fact that ℙ_{κ} is κ-c.c. Also A ⊆ ω_{2} ∩ cof(ω_{1}) because ℙ_{κ} is proper and hence can only change the cofinality of ordinals in A to ω_{1}.

In V [G_{κ}] construct a sequence ⟨a_{i} : i < ω_{2}⟩ by induction on ordinals in A∪lim(A) such that for each β in A, ⟨a_{i} : i < β⟩ enumerates [β]^{ω} ∩ V [G_{β}]. Suppose α < β are successive ordinals in A and ⟨a_{i} : i < α⟩ is defined. Since ℙ_{β} forces 2^{ω} ≤ ω_{2} = β, [β]^{ω} ∩ V [G_{β}] has size |β| = ω_{1} in V [G_{κ}]. So we can extend the sequence to ⟨a_{i} : i < β⟩ as required.

If β is a limit point of A, then ⟨a_{i} : i < β⟩ is given by induction since the defined sequences are end-extensions of one another. If β is not in A then there are no requirements on this sequence. Suppose β is in A and let x be in [β]^{ω} ∩ V [G_{β}]. Since β = ω_{2} in V [G_{β}], there is α < β such that sup(x) < α. As β is strongly inaccessible in V, there is γ in A with α < γ < β such that x is in V [G_{γ}] ∩ [γ]^{ω}. Then by induction x is in the sequence ⟨a_{i} : i < γ⟩. So ⟨a_{i} : i < β⟩ enumerates [β]^{ω} ∩ V [G_{β}].

For each α in A fix c_{α} in V [G_{α+1}] which is cofinal in α with order type ω_{1}. Since Coll(ω_{1}, ω_{2}) = ℚ_{α} is countably closed, every initial segment of c_{α} is in V [G_{α}], and hence is enumerated in ⟨a_{i} : i < α⟩. So A ⊆ S(a⃗) and A is in I[ω_{2}]. □

Proposition 4.6. Suppose N is a set with size ℵ_{1}.

(1) N is internally stationary iff its transitive collapse is internally stationary.

(2) N is internally club iff its transitive collapse is internally club.

Proof. (1) Suppose N is internally stationary, and let π : N → M be its transitive collapse. Fix an increasing and continuous sequence ⟨N_{i} : i < ω_{1}⟩ of countable sets with union N and a stationary set A ⊆ ω_{1} such that {N_{i} : i ∈ A} ⊆ N. For each i < ω_{1} let M_{i} = π“N_{i}. Then M is the union of the increasing and continuous sequence {M_{i} : i < ω_{1}} of countable sets. If i is in A, then N_{i} ∈ N so M_{i} = π“N_{i} = π(N_{i}) ∈ M by the definition of the transitive collapse map. The converse and (2) are similar. □

5. Internal Approachability and Reflection

Now we are ready to prove the following theorem.

Theorem 5.1. Assume κ is a supercompact cardinal. Then there is a forcing poset which forces that κ = ω_{2} and for all regular λ ≥ ω_{2}:

(1) for every family {U_{i} : i < ω_{1}} of stationary subsets of P_{ω_{1}}(H(λ)), there is an internally stationary set M with size ℵ_{1} such that each U_{i} reflects to M,

(2) there is a stationary set S^{∗} ⊆ P_{ω_{1}}(H(λ)) such that S^{∗} does not reflect to any internally approachable set N with size ℵ_{1}.

Let V denote the ground model, and assume that in V the GCH holds and κ is supercompact. Let B denote the set of measurable cardinals less than κ and let A be the set of non-measurable strongly inaccessible cardinals less than κ. Note that for all α in B ∪ {κ}, A ∩ α is stationary in α.

First we define an Easton support iteration

⟨ℙ_{i}^{S}, ℚ̇_{i}^{S} : i ≤ κ⟩

by recursion. Afterwards we will define another iteration in a generic extension by ℙ_{κ}^{S} ∗ ℚ̇_{κ}^{S}. Suppose ℙ_{α}^{S} is defined for some α ≤ κ. If α is not strongly inaccessible then let ℚ̇_{α}^{S} denote the trivial forcing. Suppose α is strongly inaccessible. Then let ℚ̇_{α}^{S} be a name for the forcing poset ℙ(α \ B) from Section 4. In other words, ℚ̇_{α}^{S} is a name for the α-strategically closed forcing poset for adding a stationary set S_{α} ⊆ α ∩ cof(ω) such that whenever β < α and S_{α} ∩ β is stationary in β, then β is in B.

This completes the definition. Standard arguments show that ℙ_{κ}^{S} preserves all cardinals, cofinalities, the function α ↦ 2^{α}, and for any Mahlo cardinal α ≤ κ, ℙ_{α} is α-c.c.

Let G∗G_{S} be generic for ℙ_{κ}^{S} ∗ℚ_{κ}^{S} over V. Write S = ⋃ G_{S}, which is a stationary subset of κ ∩ cof(ω) such that for all β < κ, if S ∩ β is stationary in β then β is in B. Write W = V [G ∗ G_{S}].

Lemma 5.2. In W, κ is a supercompact cardinal.

Proof. Consider a regular cardinal λ > κ in W. In V fix an elementary embedding j : V → M induced by a normal ultrafilter on P_{κ}(λ). Then crit(j) = κ, j(κ) > λ, and M is closed under λ-sequences. Also λ^{+} < j(κ), and |j(κ)| ≤ κ^{|P_{κ}(λ)|} = κ^{λ} = λ^{+}, so |j(κ)| = λ^{+}. Similarly, |j(κ^{+})| = λ^{+}. Note that κ ∈ j(B) since κ is measurable in M.

Write j(ℙ_{κ}^{S}) = ℙ_{κ}^{S} ∗ ℚ̇_{κ}^{S} ∗ ℙ_{tail}^{S}. The model M[G ∗ G_{S}] is closed under λ-sequences in W. So in W, ℙ_{tail}^{S} is λ^{+}-strategically closed. There are |j(κ)| = λ^{+} many maximal antichains of ℙ_{tail}^{S} in M[G ∗ G_{S}]. So we can construct in W a filter K on ℙ_{tail}^{S} which is generic over M[G ∗ G_{S}]. Since j“G = G ⊆ G ∗ G_{S} ∗ K, j extends to j : V [G] → M[G ∗ G_{S} ∗ K].

We claim that S is a condition in j(ℚ_{κ}^{S}). This is true since κ is in j(B), so it does not matter that S is stationary in κ. The model M′ = M[G∗G_{S} ∗K] is closed under λ sequences. So j(ℚ_{κ}^{S}) is λ^{+}-strategically closed and has λ^{+} many maximal antichains in M′. Construct a filter h on j(ℚ_{κ}^{S}) which contains S and is generic over M′. Then M′[h] is closed under λ-sequences. Since j“G_{S} = G_{S} ⊆ h, we can extend j to j : W → M′[h], which witnesses that κ is λ-supercompact in W. □

Lemma 5.3. In W, if j : W → N is an elementary embedding induced by a normal ultrafilter on P_{κ}(λ) for some λ ≥ κ, then κ is in j(B).

Proof. In N, j(S) ∩ j(κ) = S is stationary in κ. By elementarity, any ordinal to which j(S) reflects is in j(B). □

Lemma 5.4. In W, for all β in B ∪ {κ}, A ∩ β is stationary in β.

Proof. Write ℙ_{κ}^{S} = ℙ_{β}^{S} ∗ ℚ̇_{β}^{S} ∗ ℙ_{tail}^{S}. In V, A ∩ β is stationary in β, and it remains stationary after forcing with the β-c.c. poset ℙ_{β}^{S}. Since A ∩ β consists of regular cardinals, by Proposition 4.3, A ∩ β remains stationary after forcing with ℚ_{β}^{S}. The poset ℙ_{tail}^{S} does not add subsets to β, so A ∩ β is stationary in β in W. □

This completes our analysis of the model W.

In W fix a Laver function l : κ → V_{κ}^{W}. So for every x and every λ ≥ κ, there is an elementary embedding j : W → M with critical point κ such that j(κ) > λ, M is closed under λ-sequences, and j(l)(κ) = x.

In W we define by recursion a countable support iteration

⟨ℙ_{i}, ℚ̇_{j} : i ≤ κ, j < κ⟩

of proper forcing posets. Suppose ℙ_{α} is defined. Let ℚ̇_{α} be a name for Coll(ω_{1}, ω_{2}), unless each of the following conditions are satisfied:

(1) α is in B,

(2) for all β < α, |ℙ_{β}| < α,

(3) l(α) is a pair ⟨ḟ, λ̇⟩ of ℙ_{α}-names and ℙ_{α} forces that α = ω_{2}, ḟ : ω_{2} → H(ω_{2}) is a bijection, and λ̇ ≥ ω_{2} is a regular cardinal. Assume these conditions are satisfied.

Note that ℙ_{α} is α-c.c. Let G_{α} be generic for ℙ_{α} over W, and work in W [G_{α}]. Let f = ḟ^{G} and λ = λ̇^{G}. Since f is a bijection, if N ≺ ⟨H(ω_{2}), ∈, f⟩ then f“(N ∩ ω_{2}) = N. So for each β < ω_{2}, there is at most one such elementary substructure N with N ∩ ω_{2} = β. Let F denote the set of β < ω_{2} for which there exists a unique N(β) ≺ ⟨H(ω_{2}), ∈, f⟩ such that N(β) ∩ ω_{2} = β. Then F contains a club.

Define T ⊆ ω_{2} by letting β be in T iff (a) β is in F, (b) cf(β) = ω_{1}, and (c) if β is in B then N(β) is not internally club.

Lemma 5.5. In W [G_{α}], the set A ∩ T is stationary and is in I[ω_{2}].

Proof. In W let X ⊆ α be club such that for all β in X, if γ < β then |ℙ_{γ}| < β. If β is in A ∩ X, then ℙ_{β} forces β = ω_{2} and ℚ_{β} = Coll(ω_{1}, β). By Proposition 4.5, A ∩ X is in I[ω_{2}] and A ∩ X ⊆ cof(ω_{1}) ∩ ω_{2}. Since A and B are disjoint, A ∩ X ∩ F ⊆ T. □

Let ℙ_{T} be the proper forcing poset from Section 4 for adding a club subset to T ∪ cof(ω). By Lemma 5.5 and Proposition 4.4, ℙ_{T} is < ω_{2}-distributive. Now let ℚ̇_{α} be a name for ℙ_{T} ∗ Add(ω) ∗ Coll(ω_{1}, N), where N is a name for the set H(λ) as computed after forcing with ℙ_{α} ∗ ℙ_{T}.

This completes the definition of ℙ_{κ}. Let G_{κ} be generic for ℙ_{κ} over W. Standard arguments show that in W [G_{κ}], κ = ω_{2} and 2^{ω} = 2^{ω_{1}} = ω_{2}.

Working in W [G_{κ}] fix a bijection f : ω_{2} → H(ω_{2}). Let F be the set of β < κ for which there exists a unique N(β) ≺ H(ω_{2}) closed under f. Then F contains a club. Define T by letting β < ω_{2} be in T iff (a) β ∈ F, (b) cf(β) = ω_{1}, and (c) if β is in B then N(β) is not internally club. Let G_{T} be generic for ℙ_{T} over W [G_{κ}]. Our final model is W′ = W [G_{κ} ∗ G_{T}].

By Lemma 5.5 and Proposition 4.4, ℙ_{T} is < ω_{2}-distributive. Since H(ω_{2}) is determined by P(ω_{1}), H(ω_{2}) is the same in W [G_{κ}] and W′. Also f is a bijection in W′, and F satisfies the same definition in W′ as it does in W [G_{κ}].

Let E denote the club set ⋃ G_{T}. Then E ∩ cof(ω_{1}) ⊆ T ⊆ F.

Lemma 5.6. In W′, A ⊆ ω_{2} ∩ cof(ω_{1}) is stationary, S ⊆ ω_{2} ∩ cof(ω) is stationary, and for all β < ω_{2}, if S ∩ β is stationary in β then β is in B.

Proof. The poset ℙ_{κ} is κ-c.c., so A and S are stationary in W [G_{κ}]. Since ℙ_{T} is proper, ℙ_{T} preserves the stationarity of S by Lemma 1.1. By Lemma 5.5, in W [G_{κ}], A ∩ T is stationary and is in I[ω_{2}]. So Proposition 4.4 implies ℙ_{T} preserves the stationarity of A. The poset ℙ_{κ} ∗ ℙ_{T} is proper, so it can only change the cofinality of ordinals less than κ to ω_{1}. Suppose S ∩ β is stationary in β for some β < ω_{2}. If β is not in B, then S ∩ β is not stationary in β in W and hence also not in W′. So β is in B. □

Lemma 5.7. If β is in E ∩ B and cf(β) = ω_{1}, then N(β) is not internally club.

Proof. By the definition of T, N(β) is not internally club in W [G_{κ}]. Since ℙ_{T} is proper, N(β) is not internally club in W′ by Lemma 3.3. □

We now verify the main claims of Theorem 5.1.

Proposition 5.8. In W′, for all regular λ ≥ ω_{2}, there is a stationary set S^{∗} ⊆ P_{ω_{1}}(H(λ)) such that S^{∗} does not reflect to any internally approachable set with size ℵ_{1}.

Proof. Define S^{∗} as the collection of a in P_{ω_{1}}(H(λ)) such that a ≺ H(λ), a ∩ H(ω_{2}) ≺ ⟨H(ω_{2}), ∈, f⟩, and sup(a ∩ ω_{2}) ∈ S ∩ E.

To show S^{∗} is stationary, consider a function H : H(λ)^{<ω} → H(λ). Then there are club many β < ω_{2} such that for all α_{0} < … < α_{n} < β, cl_{H}({α_{0}, …, α_{n}})∩ω_{2} ⊆ β. Choose such an ordinal β which is in S ∩ E. Let x ⊆ β be cofinal with order type ω and let a = cl_{H}(x). Then sup(a ∩ ω_{2}) = β. This argument shows there are stationarily many a in P_{ω_{1}}(H(λ)) with sup(a ∩ ω_{2}) ∈ S ∩ E. The other two conditions in the definition of S^{∗} are true for club many sets a. So S^{∗} is stationary.

Suppose for a contradiction that S^{∗} ∩ P_{ω_{1}}(N) is stationary in P_{ω_{1}}(N) for an internally approachable set N with size ℵ_{1}. Fix an increasing and continuous chain ⟨N_{i} : i < ω_{1}⟩ of countable sets with union N such that for all i < ω_{1}, ⟨N_{j} : j < i⟩ ∈ N. Then there is a stationary set A ⊆ ω_{1} such that {N_{i} : i ∈ A} ⊆ S^{∗}. For each i in A, sup(N_{i} ∩ ω_{2}) ∈ S ∩ E. Let β = N ∩ ω_{2}. Clearly β is in E ∩ cof(ω_{1}).

For each i in A, N_{i}∩H(ω_{2}) ≺ ⟨H(ω_{2}), ∈, f⟩, hence N ∩H(ω_{2}) ≺ ⟨H(ω_{2}), ∈, f⟩. So N ∩ H(ω_{2}) = N(β). But H(ω_{2}) is a definable class of H(λ), so it is easy to see that N ∩H(ω_{2}) = N(β) is internally approachable by the sequence ⟨N_{i}∩H(ω_{2}) : i < ω_{1}⟩. Lemma 5.7 implies that β is not in B.

Since β is not in B, S does not reflect to β. Choose a club c ⊆ β disjoint from S. Since S^{∗} is stationary in P_{ω_{1}}(N), there is a in S^{∗} closed under the map α ↦ min(c \ α). But then sup(a ∩ ω_{2}) ∈ c ∩ S, which is a contradiction. □

We complete the proof by verifying the following claim in W′: For all regular λ ≥ ω_{2}, for every family {U_{i} : i < ω_{1}} of stationary subsets of P_{ω_{1}}(H(λ)), there is an internally stationary set M with size ℵ_{1} such that U_{i} reflects to M for all i < ω_{1}.

Let λ ≥ ω_{2} be regular and assume {U_{i} : i < ω_{1}} is a family of stationary subsets of P_{ω_{1}}(H(λ)). Then λ is regular in W [G_{κ}]. Fix ℙ_{κ}-names ḟ and λ̇ for f and λ. Back in W let j : W → M be an elementary embedding with critical point κ such that j(κ) > λ, M is closed under λ-sequences, and j(l)(κ) = ⟨ḟ, λ̇⟩.

Note that M[G_{κ} ∗ G_{T}] is closed under λ-sequences in W′.

By the choice of j, we have

j(ℙ_{κ}) = ℙ_{κ} ∗ ℙ_{T} ∗ Add(ω) ∗ Coll(ω_{1}, N) ∗ ℙ_{tail}

where N is a name for H(λ) as computed after forcing with ℙ_{κ} ∗ ℙ_{T}, and ℙ_{tail} is a proper forcing iteration collapsing j(κ) to become ω_{2}.

We would like to extend j to have domain W′ in some outer model of W′. Let K be generic for Add(ω) ∗ Coll(ω_{1}, N) ∗ ℙ_{tail} over W′. Since j“G_{κ} = G_{κ} ⊆ G_{κ} ∗ G_{T} ∗ K, we can extend j to j : W [G_{κ}] → M[G_{κ} ∗ G_{T} ∗ K] in W′[K]. Write M′ = M[G_{κ} ∗ G_{T} ∗ K].

To further extend j to have domain W′, we construct a master condition for j(ℙ_{T}).

Lemma 5.9. The set s = E ∪ {κ} is a condition in j(ℙ_{T}).

Proof. Since M[G_{κ} ∗ G_{T}] is closed under λ-sequences, it contains the set s, so M′ does as well. As E ∩ cof(ω_{1}) ⊆ T, j“(E ∩ cof(ω_{1})) = E ∩ cof(ω_{1}) ⊆ j(T). So it suffices to prove that κ is in j(T).

In M′, κ ∈ j(B) by Lemma 5.3 and cf(κ) = ω_{1}. Also κ is in j(F), so N(κ) exists. By the definition of T, κ is in j(T) iff N(κ) is not internally club in M′.

At stage κ, j(ℙ_{κ}) forces with ℙ_{T} ∗ Add(ω) ∗ Coll(ω_{1}, N). By the closure of M[G_{κ} ∗G_{T}], N is equal to H(λ)^{W′}. Now here is the reason we forced with Add(ω): by Proposition 3.5 and Lemma 3.3, H(ω_{2})^{W′} = H(ω_{2})^{W [G_{κ}]} is not internally club in M′. Now the transitive collapse of j“(H(ω_{2})^{W [G_{κ}]}) is equal to H(ω_{2})^{W [G_{κ}]}, so j“(H(ω_{2})^{W [G_{κ}]}) is not internally club by Proposition 4.6.

We claim j“(H(ω_{2})^{W [G_{κ}]}) is in M′ and is equal to N(κ), which completes the proof. Recall that N(κ) = j(f)“(N(κ) ∩ j(κ)) = j(f)“κ. For each ξ < κ, j(f)(ξ) = j(f)(j(ξ)) = j(f(ξ)). So N(κ) = j(f)“κ = j“(f“κ)) = j“(H(ω_{2})^{W [G_{κ}]}). □

Choose h generic for j(ℙ_{T}) over W′[K] containing the condition s. Now extend j in W′[K ∗ h] to j : W′ → M′[h].

Recall that N = H(λ)^{W′}.

Lemma 5.10. The set j“N is in M′[h].

Proof. The model M′[h] is not closed under λ-sequences in W′[K ∗ h]. However j“(H(λ)^{W}) is in M. Fix a surjection g : [λ]^{<λ} → N in W′. Every set in [λ]^{<λ} is the interpretation of a ℙ_{κ} ∗ ℙ_{T} -name in H(λ)^{W}. Also for any ℙ_{κ} ∗ ℙ_{T} -name ẋ, j(ẋ^{G_{κ}∗G_{T}}) = j(ẋ)^{G_{κ}∗G_{T} ∗K∗h}. So

j“N = {j(g)(y^{G_{κ}∗G_{T} ∗K∗h}) : y ∈ j“(H(λ)^{W})}

and this set is definable in M′[h]. □

By Proposition 3.5 and Lemma 3.3, N is internally stationary in M′[h]. So by Proposition 4.6, j“N is internally stationary in M′[h].

Proposition 5.11. In M′[h], for all i < ω_{1}, j(U_{i}) reflects to j“N.

Proof. Fix i < ω_{1}. By the closure of M[G_{κ} ∗ G_{T}], U_{i} is in M[G_{κ} ∗ G_{T}] and is stationary in P_{ω_{1}}(N). Since K ∗ h is a generic filter for a proper forcing poset over M[G_{κ} ∗ G_{T}], U_{i} is stationary in P_{ω_{1}}(N) in M′[h]. Now j ↾ N is the inverse of the transitive collapse of j“N, therefore is in M′[h] by Lemma 5.10. For each a in U_{i}, a ⊆ N and j(a) = j“a. So j“U_{i} is definable in M′[h] from j ↾ N and U_{i}, hence is in M′[h]. It is easy to check that since U_{i} is stationary in P_{ω_{1}}(N) in M′[h], j“U_{i} is stationary in P_{ω_{1}}(j“N) in M′[h]. But j“U_{i} ⊆ j(U_{i}) ∩ P_{ω_{1}}(j“N), therefore j(U_{i}) reflects to j“N. □

Since j({U_{i} : i < ω_{1}}) = {j(U_{i}) : i < ω_{1}}, by elementarity there is an internally stationary set with size ℵ_{1} to which each U_{i} reflects.

This completes the proof.

We were able to verify stationary reflection in the preceding argument because the iteration ℙ_{κ}∗ℙ_{T} is proper, and so preserves the stationarity of a given stationary set after forcing with a tail of j(ℙ_{κ} ∗ ℙ_{T}). On the other hand, the properness of the iteration guaranteed that the set to which the given stationary set reflects is internally stationary. This raises the following question.

Question 5.12. Does WRP imply that every stationary set S ⊆ P_{ω_{1}}(H(λ)) reflects to an internally stationary set N with size ℵ_{1}?

References

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Department of Mathematics, University of California, Berkeley, CA 94720 USA

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