DENSE NON-REFLECTION FOR STATIONARY COLLECTIONS OF COUNTABLE SETS

DAVID ASPERÓ, JOHN KRUEGER, AND YASUO YOSHINOBU

Abstract. We present several forcing posets for adding a non-reflecting stationary subset of P_{ω_{1}}(λ), where λ ≥ ω_{2}. We prove that PFA is consistent with dense non-reflection in P_{ω_{1}}(λ), which means that every stationary subset of P_{ω_{1}}(λ) contains a stationary subset which does not reflect to any set of size ℵ_{1}. If λ is singular with countable cofinality, then dense non-reflection in P_{ω_{1}}(λ) follows from the existence of squares.

Introduction

A classical consequence of Jensen’s principle □_{λ} is that reflection of stationary subsets of λ^{+} fails densely often; in other words, every stationary subset of λ^{+} contains a non-reflecting stationary set. In this paper we focus on the extrapolation of the above conclusion to the context of stationarity in P_{ω_{1}}(λ): Given a cardinal λ ≥ ω_{2}, we say that dense non-reflection in P_{ω_{1}}(λ) holds if for every stationary S ⊆ P_{ω_{1}}(λ) there is a stationary T ⊆ S which does not reflect to any set of size ℵ_{1}; that is, such that T ∩ P_{ω_{1}}(N) is non-stationary for every set N of size ℵ_{1}.^{1}

One of the purposes of this paper is to contribute to the general project of separating forcing axioms from combinatorial statements, and more specifically to build a model of PFA in which dense non-reflection holds in P_{ω_{1}}(λ) for many instances of λ.

A well–known fact is that PFA implies the failure of □_{λ} for all cardinals λ ≥ ω_{1} ([15]). Yet another fact is that if PFA holds, then it is possible to force in such a way that both PFA is preserved and for every regular cardinal κ ≥ ω_{2} there is a non-reflecting stationary subset of κ. This can be seen by combining the forcing construction in [4] in a class forcing as in Section 5 of this paper. On the other hand, the stronger forcing axiom MM implies, for every regular cardinal κ ≥ ω_{2}, that every stationary subset of κ∩cf(ω) reflects ([8]). And, concerning stationarity in P_{ω_{1}}(λ), MM implies that every stationary subset of P_{ω_{1}}(λ), for every λ ≥ ω_{2}, reflects to a set of size ℵ_{1} ([8]).^{2}

In view of the web of implications given so far, it is natural to ask whether or not there is any connection between PFA and the statement that every stationary subset of P_{ω_{1}}(λ) reflects to a set of size ℵ_{1}. In this paper we show, in analogy to what happens in the context of stationary sets of ordinals, that this is not the case.

Date: October 2008.

2000 Mathematics Subject Classification. Primary 03E35; Secondary 03E05.

Key words and phrases. Dense Non-Reflection, Proper Forcing Axiom.

The second author was partially supported by JSPS Fellowship PE08038. The third author was partially supported by Grant-in-Aid for Scientific Research (C) 20540114, JSPS.

1Dense non-reflection in P_{ω_{1}}(λ) is basically a global failure of weak reflection in P_{ω_{1}}(λ).

2All implications mentioned so far are known to admit stronger, more general, formulations (for example weak forms of □_{λ} suffice to yield that reflection of stationary subsets of λ^{+} fails densely often, etc.). We do not bother to state the stronger statements here, as we just want to display the general situation we were initially interested in.

Another purpose of this paper is to show that, when λ is a singular cardinal of countable cofinality, dense non-reflection in P_{ω_{1}}(λ) follows from fairly general pcf–theoretic assumptions.

The rest of the paper is structured as follows. In Sections 1 and 2 we deal with dense non-reflection in P_{ω_{1}}(λ) for λ ≥ ω_{2} a regular cardinal. In Section 1 we build, for any such λ with λ^{<λ} = λ and any set 𝕏 of size at least λ, a λ–strategically closed poset ℙ(𝕏) preserving cofinalities and adding a non-reflecting stationary subset of P_{ω_{1}}(𝕏) having stationary intersection with every stationary subset of P_{ω_{1}}(𝕏) in the ground model. We also show that a suitable product ℚ(𝕏) of copies of ℙ(𝕏) with < λ–support has the same niceness properties and forces dense non-reflection in P_{ω_{1}}(𝕏). In Section 2 we show that the forcing ℚ(𝕏) from Section 1 preserves PFA. This shows that PFA is consistent with dense non-reflection.

In Sections 3 and 4 we focus on dense non-reflection in P_{ω_{1}}(λ) in the case that λ is a singular cardinal of countable cofinality. In Section 3 we describe general situations implying dense non-reflection in P_{ω_{1}}(λ) for this choice of λ. These situations are phrased in the context of pcf theory. We prove a general result (Theorem 3.2) implying that if λ is a singular cardinal of countable cofinality such that 2^{λ} = λ^{+} and □_{λ}^{∗} holds, then dense non-reflection holds in P_{ω_{1}}(λ) (Corollary 3.11). By a result of Magidor, it is possible to force over any model of PFA in such a way that PFA is preserved, GCH holds above ω, and □_{κ,ℵ_{2}} holds for all κ ≥ ω_{2}. Hence, in Magidor’s model dense non-reflection in P_{ω_{1}}(λ) holds for all singular cardinals λ of countable cofinality (Corollary 3.13). We finish this section by describing a situation which does not assume 2^{λ} = λ^{+} but which nevertheless implies the conclusion that dense non-reflection holds in P_{ω_{1}}(λ) (Corollary 3.19); this time we assume that there is some δ < λ for which the set 𝔞 of regular cardinals between δ and λ is such that pcf(𝔞) is countable and such that □_{κ}^{∗} holds, where κ ≥ λ is such that 2^{λ} = κ^{+}.^{3}

Given a singular cardinal λ of countable cofinality, in Section 4 we introduce a (λ + 1)–strategically closed forcing ℙ for adding a non-reflecting stationary subset of P_{ω_{1}}(λ) having stationary intersection with every stationary subset of P_{ω_{1}}(λ) in the ground model. A suitable product of copies of ℙ with < λ^{+}–support is also (λ + 1)–strategically closed and, if 2^{λ} = λ^{+}, then it has the λ^{++}–c.c. and forces dense non-reflection in P_{ω_{1}}(λ).

Finally, in Section 5 we present two forcing constructions preserving PFA, while at the same time forcing dense non-reflection in P_{ω_{1}}(λ) for all cardinals λ ≥ ω_{2} which are either regular or singular of countable cofinality. In both constructions we start by assuming GCH above ω (which can always be forced preserving PFA). In the first construction we build a reverse Easton iteration in which we force at all relevant stages λ with a forcing as in Section 1 or in Section 4 for getting dense non-reflection in P_{ω_{1}}(λ). The desired forcing is the direct limit of this iteration. In the second construction we start with Magidor’s model of PFA, GCH above ω and □_{κ,ℵ_{2}} for all κ ≥ ω_{2}, and build a reverse Easton iteration in which we keep forcing instances of dense non-reflection in P_{ω_{1}}(λ) only for regular λ.

We should remark that our methods do not seem to work for the case when λ is a singular cardinal of uncountable cofinality. For such a given λ and given any γ < λ with γ^{<γ} = γ, it is possible to force dense non-reflection in P_{ω_{1}}(λ) by a γ– strategically closed forcing. This can be achieved by a special case of the forcing in Section 1. However, this forcing will blow up the power set of all cardinals between γ and λ to at least λ. Hence, iterating this type of forcings in length Ord certainly kills the Power Set Axiom. In fact, we do not know whether dense non-reflection is consistent in P_{ω_{1}}(λ) for any singular strong limit cardinal of uncountable cofinality.

3The existence of such a κ follows already from the existence of a set 𝔞 as above.

The paper should be understandable to a reader familiar with forcing, iterated forcing, proper forcing, and generalized stationarity (see [2] and [3]). If X is a set of size at least ℵ_{1}, a set S ⊆ P_{ω_{1}}(X) = {a ⊆ X : |a| < ω_{1}} is stationary if for any function F : [X]^{<ω} → X, there is a set b in S which is closed under F. A forcing poset ℙ is α-strategically closed, where α is an ordinal, if Player II has a winning strategy in the following game. Players I and II take turns to build a descending sequence of conditions in ℙ, ⟨p_{i} : 1 ≤ i < α⟩, where Player I plays p_{i} for odd i, and Player II plays p_{j} for even j. If at each even stage less than α, Player II is able to play some condition, then Player II wins. The Proper Forcing Axiom, or PFA, is the statement that for any proper forcing poset ℙ, if {D_{i} : i < ω_{1}} is a family of dense subsets of ℙ, then there is a filter G on ℙ which meets D_{i} for all i < ω_{1}.

1. Adding Non-Reflecting Stationary Sets

Let λ ≥ ω_{2} be a regular cardinal. Let 𝕏 be a set of size at least λ. We define a forcing poset ℙ(𝕏) which adds a non-reflecting stationary subset of P_{ω_{1}}(𝕏). A condition in ℙ(𝕏) is a set X such that:

• X ⊆ P_{ω_{1}}(𝕏),

• |X| < λ,
• for any set N in [𝕏]^{ℵ_{1}}, P_{ω_{1}}(N) ∩ X is non-stationary in P_{ω_{1}}(N).

We let Y ≤ X if:

• X ⊆ Y,

• for all y in Y \ X, y is not a subset of ⋃ X. We will prove the following main properties of ℙ(𝕏).

(1) ℙ(𝕏) is λ-strategically closed.

(2) If λ^{<λ} = λ, then ℙ(𝕏) is λ^{+}-c.c., and therefore preserves all cardinals and cofinalities.

(3) The union of a generic filter for ℙ(𝕏) is a stationary subset of P_{ω_{1}}(𝕏) which does not reflect to any set of size ℵ_{1}.

(4) For any stationary set S ⊆ P_{ω_{1}}(𝕏) in the ground model, the union of a

generic filter has stationary intersection with S. After establishing these facts, we will show that a suitable product forcing of ℙ(𝕏) will force dense non-reflection in P_{ω_{1}}(𝕏).

Proposition 1.1. Let ⟨X_{i} : i < δ⟩ be a descending sequence of conditions in ℙ(𝕏), where δ is a limit ordinal less than λ, such that for every limit ordinal ν < δ, X_{ν} = ⋃_{i<ν} X_{i}. Then ⋃_{i<δ} X_{i} is a condition in ℙ(𝕏) which is below X_{i} for all i < δ.

Proof. Let Y = ⋃_{i<δ} X_{i}. Clearly Y is a subset of P_{ω_{1}}(𝕏) of size less than λ, and for all i < δ, X_{i} ⊆ Y. Moreover, if y ∈ Y \ X_{i}, then there is i < j < δ such that y is in X_{j} \ X_{i}, and therefore y is not a subset of ⋃ X_{i}. So if the statement of the proposition fails, then there exists a set N in [𝕏]^{ℵ_{1}} such that P_{ω_{1}}(N) ∩ Y is stationary in P_{ω_{1}}(N).

Fix N in [𝕏]^{ℵ_{1}} and assume for a contradiction P_{ω_{1}}(N)∩Y is stationary in P_{ω_{1}}(N). Let ⟨a_{i} : i < ω_{1}⟩ be an increasing and continuous sequence of countable sets with union equal to N. Then there is a stationary set A ⊆ ω_{1} such that {a_{i} : i ∈ A} is a subset of Y.

Claim 1.2. The cofinality of δ is ω_{1}.

Proof. If cf(δ) > ω_{1}, then there is γ < δ such that {a_{i} : i ∈ A} ⊆ X_{γ}. Thus P_{ω_{1}}(N)∩X_{γ} is stationary, which contradicts that X_{γ} is a condition. Suppose cf(δ) = ω. Fix a cofinal function f : ω → δ. Then P_{ω_{1}}(N) ∩ Y = ⋃_{n<ω}(P_{ω_{1}}(N) ∩ X_{f(n)}). Since the club filter on P_{ω_{1}}(N) is countably complete, there is n < ω such that P_{ω_{1}}(N) ∩ X_{f(n)} is stationary. This contradicts that X_{f(n)} is a condition. □

Let ⟨β_{i} : i < ω_{1}⟩ be an increasing and continuous sequence cofinal in δ. Then X_{δ} = ⋃{X_{β_{i}} : i < ω_{1}}. Define g : A → ω_{1} by letting g(i) be the least ordinal such that a_{i} is in X_{β_{g(i)}}. Note that g(i) is always a successor ordinal, since the map j ↦ β_{j} is normal and X_{ν} = ⋃_{j<ν} X_{j} for every limit ordinal ν < δ.

Claim 1.3. There exists a club C ⊆ ω_{1} such that for all i < j in C, if i is in A then i < g(i) < j.

Proof. Let C_{1} be the club set of limit ordinals α in ω_{1} such that for all i in α ∩ A, g(i) < α. Suppose for a contradiction there does not exist a club set C ⊆ C_{1} such that for all i in C ∩ A, i < g(i). Since g(i) = i is impossible when i is a limit ordinal, there is a stationary set A′ ⊆ A such that for all i in A′, g(i) < i. By Fodor’s Lemma, there is a stationary set A′′ ⊆ A′ and γ < ω_{1} such that for all i in A′′, g(i) = γ. Then for all i in A′′, a_{i} ∈ X_{β_{γ}}. So {a_{i} : i ∈ A′′} is a subset of X_{β_{γ}}, and therefore P_{ω_{1}}(N)∩X_{β_{γ}} is stationary, contradicting that X_{β_{γ}} is a condition. □

Fix a club C as in Claim 1.3. Since A is stationary in ω_{1}, it has non-empty intersection with the limit points of A∩C. So we can choose a closed set x contained in A ∩ C with order type ω + 1. Let ν = max(x). Since ⟨a_{i} : i < ω_{1}⟩ is increasing and continuous, a_{ν} = ⋃{a_{i} : i ∈ x ∩ ν}. Consider i in x ∩ ν. Then g(i) < ν, so a_{i} ∈ X_{β_{g(i)}} ⊆ X_{β_{ν}}. It follows that a_{i} ⊆ ⋃ X_{β_{ν}}. Since this is true for all i in x ∩ ν, a_{ν} ⊆ ⋃ X_{β_{ν}}. On the other hand, ν < g(ν), so a_{ν} is in X_{β_{g(ν)}} \X_{β_{ν}}. By the definition of the ordering on ℙ(𝕏), a_{ν} is not a subset of ⋃ X_{β_{ν}}, which is a contradiction. □

Corollary 1.4. The forcing poset ℙ(𝕏) is λ-strategically closed.

Proof. By Proposition 1.1, the following strategy works: Player II plays anything at successor stages, and plays the union of the previous plays at limit stages. □

Corollary 1.5. The forcing poset ℙ(𝕏) is ω_{1}-closed. In fact, if ⟨X_{n} : n < ω⟩ is a descending sequence of conditions, then ⋃_{n<ω} X_{n} is a condition which is below X_{n} for all n < ω.

Proof. For a descending sequence of order type ω, the hypotheses of Proposition 1.1 hold trivially. □

Lemma 1.6. Suppose X is in ℙ(𝕏) and E is a subset of 𝕏 of size less than λ such that ⋃ X ⊆ E. Then there is Y ≤ X such that ⋃ Y = E.

Proof. Let {E_{i} : i ∈ I} be a partition of E \ ⋃ X into countable sets, where |I| < λ. Let Y = X ∪ {E_{i} : i ∈ I}. If N is in [𝕏]^{ℵ_{1}} and P_{ω_{1}}(N) ∩ Y is stationary, then by the countable completeness of the club filter on P_{ω_{1}}(N), either P_{ω_{1}}(N) ∩ X is stationary or P_{ω_{1}}(N) ∩ {E_{i} : i ∈ I} is stationary. The former is impossible since X is a condition, and the latter is impossible since otherwise there are distinct i and j in I such that E_{i} ⊆ E_{j}. So Y is a condition, and clearly ⋃ Y = E. It is easy to see that Y ≤ X. □

Corollary 1.7. Let Ṫ be a ℙ(𝕏)-name for the union of the generic filter. Then ℙ(𝕏) forces that P_{ω_{1}}(N) ∩ Ṫ is non-stationary in P_{ω_{1}}(N) for every set N in [𝕏]^{ℵ_{1}}.

Proof. Suppose X forces Ṅ is in [𝕏]^{ℵ_{1}}. Since ℙ(𝕏) is λ-strategically closed, there is Y ≤ X and N in [𝕏]^{ℵ_{1}} such that Y forces Ṅ = Ň. By Lemma 1.6 fix Z ≤ Y such that ⋃ Z = ⋃ Y ∪ N. Then Z forces P_{ω_{1}}(Ṅ) ∩ Ṫ = P_{ω_{1}}(N) ∩ Z. For any condition which is compatible with Z does not contain any subsets of ⋃ Z which are not already in Z. Since P_{ω_{1}}(N) ∩ Z is non-stationary, Z forces P_{ω_{1}}(Ṅ) ∩ Ṫ is non-stationary. □

Lemma 1.8. If λ^{<λ} = λ, then ℙ(𝕏) is λ^{+}-c.c.

Proof. Let ⟨X_{i} : i < λ^{+}⟩ be a sequence of conditions in ℙ(𝕏). Then ⟨⋃ X_{i} : i < λ^{+}⟩ is a sequence of sets of size less than λ. Since λ^{<λ} = λ, by the Δ-System Lemma there is an unbounded set A ⊆ λ^{+} and a set a ⊆ 𝕏 such that for all i < j in A, ⋃ X_{i} ∩⋃ X_{j} = a. Now P(a) has size at most λ, so there are at most λ^{<λ} = λ many possibilities for P(a)∩X_{i}, for i in A. Fix i < j in A such that P(a)∩X_{i} = P(a)∩X_{j}.

We claim that X_{i} and X_{j} are compatible. Clearly X_{i}∪X_{j} is a condition. Suppose for a contradiction X_{i}∪X_{j} is not a common refinement of X_{i} and X_{j}. Then without loss of generality, there is y in (X_{i} ∪ X_{j}) \ X_{i} such that y is a subset of ⋃ X_{i}. Since y is not in X_{i}, y is in X_{j}. So y ⊆ ⋃ X_{i} ∩ ⋃ X_{j} = a. Therefore y is in P(a) ∩ X_{j}. But P(a) ∩ X_{j} = P(a) ∩ X_{i}. So y is in X_{i}, which is a contradiction. □

Proposition 1.9. The forcing poset ℙ(𝕏) forces that Ṫ = ⋃ Ġ is stationary in P_{ω_{1}}(𝕏). In fact, for every stationary set S ⊆ P_{ω_{1}}(𝕏), ℙ(𝕏) forces that Ṫ ∩ Š is stationary.

Proof. Let S be a stationary subset of P_{ω_{1}}(𝕏). Let X be a condition in ℙ(𝕏) and suppose X forces Ḟ : [𝕏]^{<ω} → 𝕏 is a function. Fix a regular cardinal θ such that 𝕏, ℙ(𝕏), and Ḟ are in H(θ). Since S is stationary, we can choose a countable set N such that:

• N ≺ H(θ),

• {𝕏, ℙ(𝕏), X, Ḟ } ⊆ N,
• N ∩ 𝕏 ∈ S.

Define a descending sequence ⟨X_{n} : n < ω⟩ of conditions in N ∩ ℙ(𝕏) such that X_{0} = X and for any set D in N which is dense open in ℙ(𝕏), there is n < ω such that X_{n} ∈ D. This is possible since N is countable.

By Corollary 1.5, the set ⋃_{n<ω} X_{n} is a condition in ℙ(𝕏). In particular, ⋃_{n<ω} X_{n} does not reflect to any set of size ℵ_{1}. Let Y = ⋃_{n<ω} X_{n} ∪ {N ∩ 𝕏}. Then Y does not reflect to any set of size ℵ_{1}. We claim Y ≤ X_{n} for all n < ω. Clearly X_{n} ⊆ Y. Suppose y is in Y \ X_{n}. If y is in ⋃_{n<ω} X_{n}, then there is m > n such that y ∈ X_{m} \ X_{n}. Then y is not a subset of ⋃ X_{n}, since X_{m} ≤ X_{n}. Otherwise y = N ∩ 𝕏. Since X_{n} is in N, by elementarity (N ∩ 𝕏) \ ⋃ X_{n} is non-empty. Therefore y = N ∩ 𝕏 is not a subset of ⋃ X_{n}.

Clearly Y forces that N ∩𝕏 is in Ṫ ∩Š. So it suffices to show that Y forces N ∩𝕏 is closed under Ḟ. Let a_{1}, …, a_{n} be in N∩𝕏. Let D be the dense open set of conditions in ℙ(𝕏) which decide the value of Ḟ (a_{1}, …, a_{n}). By elementarity, D is in N. Fix n such that X_{n} is in D, and let b ∈ 𝕏 be such that X_{n} forces Ḟ (a_{1}, …, a_{n}) = b. By elementarity, b is in N. Since Y ≤ X_{n}, Y forces Ḟ (a_{1}, …, a_{n}) = b is in N ∩ 𝕏. □

We now consider a product forcing of the poset ℙ(𝕏) with <λ-support. Let κ be the cardinality of P_{ω_{1}}(𝕏). Define ℚ(𝕏) as the set of partial functions p : κ^{+} → ℙ(𝕏) with domain of size less than λ, and let q ≤ p if dom(p) ⊆ dom(q) and for all i in dom(p), q(i) ≤ p(i) in ℙ(𝕏).

Proposition 1.10. The forcing poset ℚ(𝕏) is ω_{1}-closed and λ-strategically closed.

This proposition follows by an easy argument using the corresponding properties of ℙ(𝕏). So by the next proposition, if λ^{<λ} = λ then ℚ(𝕏) preserves all cardinals and cofinalities.

Proposition 1.11. If λ^{<λ} = λ, then ℚ(𝕏) is λ^{+}-c.c.

Proof. Let ⟨p_{i} : i < λ^{+}⟩ be a sequence of conditions in ℚ(𝕏). Consider i < λ^{+}. Let E_{i} = ⋃{⋃(p_{i}(α)) : α ∈ dom(p_{i})}. Then E_{i} is a subset of 𝕏 of size less than λ, and for each α in dom(p_{i}), ⋃(p_{i}(α)) ⊆ E_{i}. Let q_{i} ≤ p_{i} be a condition such that dom(q_{i}) = dom(p_{i}) and for all α in dom(q_{i}), ⋃(q_{i}(α)) = E_{i}. This is possible by Lemma 1.6.

It suffices to find i < j such that q_{i} and q_{j} are compatible. Since λ^{<λ} = λ, apply the Δ-System Lemma to find an unbounded set A ⊆ λ^{+} and a set a ⊆ κ^{+} such that for all i < j in A, dom(q_{i}) ∩ dom(q_{j}) = a. Applying the Δ-System Lemma again to the collection {E_{i} : i ∈ A}, find an unbounded set B ⊆ A and a set b ⊆ 𝕏 such that for all i < j in B, E_{i} ∩ E_{j} = b.

The set P(b) has size at most λ. So there are at most λ^{<λ} = λ many possibilities for a sequence ⟨P(b) ∩ q_{i}(α) : α ∈ a⟩, for i in B. Fix i < j in B which have the same such sequence.

To show q_{i} and q_{j} are compatible, it suffices to show that if α is in dom(q_{i}) ∩ dom(q_{j}) = a, then q_{i}(α) and q_{j}(α) are compatible in ℙ(𝕏). If not, then without loss of generality there is α in a and a set x in (q_{i}(α)∪q_{j}(α))\q_{i}(α) such that x is a subset of ⋃(q_{i}(α)) = E_{i}. Then x is in q_{j}(α), so x is a subset of ⋃(q_{j}(α)) = E_{j}. Hence x is a subset of E_{i}∩E_{j} = b. Therefore x is in P(b)∩q_{j}(α). But P(b)∩q_{j}(α) = P(b)∩q_{i}(α). So x is in q_{i}(α), which is a contradiction. □

Let α be an ordinal less than κ^{+}. For p in ℚ(𝕏), let p_{α} = p ↾ α and p^{α} = p ↾ [α, κ^{+}). Define ℚ_{α}(𝕏) = {p_{α} : p ∈ ℚ(𝕏)} and ℚ^{α}(𝕏) = {p^{α} : p ∈ ℚ(𝕏)}, with the obvious orderings. Then ℚ(𝕏) is isomorphic to ℚ_{α}(𝕏) × ℚ^{α}(𝕏) by the map p ↦ ⟨p_{α}, p^{α}⟩. Note that ℚ(𝕏) = ⋃_{α<κ^{+}} ℚ_{α}(𝕏).

The following easy lemma is similar to Proposition 1.10.

Lemma 1.12. For all α < κ^{+}, ℚ_{α}(𝕏) and ℚ^{α}(𝕏) are both ω_{1}-closed and λstrategically closed.

Lemma 1.13. Assume λ^{<λ} = λ. Let Ṡ be a ℚ(𝕏)-name for a subset of P_{ω_{1}}(𝕏). Then there is α < κ^{+} such that ℚ(𝕏) forces Ṡ is in V^{ℚ_{α}(𝕏)}.

Proof. Since ℚ(𝕏) does not add any new countable subsets of 𝕏, ℚ(𝕏) forces that Ṡ is a subset of the ground model. Let Ḟ be a ℚ(𝕏)-name for a surjection of κ onto Ṡ. For each i < κ let A_{i} be a maximal antichain of ℚ(𝕏) which is contained in the dense open set of conditions which decide the value of Ḟ (i). Since ℚ(𝕏) is λ^{+}-c.c., each A_{i} has size at most λ. As λ ≤ κ, ⋃_{i<κ} A_{i} has size at most κ. Since ℚ(𝕏) = ⋃_{α<κ^{+}} ℚ_{α}(𝕏), there is α < κ^{+} such that ⋃_{i<κ} A_{i} is a subset of ℚ_{α}(𝕏).

Let G be a generic filter for ℚ(𝕏) over V. Let S = Ṡ^{G} and F = Ḟ^{G}. Then S is equal to the set of a in P_{ω_{1}}(𝕏) such that there is i < κ and a condition X in A_{i} ∩ G such that X forces Ḟ (i) = a. Let G_{α} = {p_{α} : p ∈ G}. Then G_{α} is a generic filter for ℚ_{α}(𝕏) over V. By the choice of α, p_{α} = p for all p in ⋃_{i<κ} A_{i}. So A_{i} ∩ G = A_{i} ∩ G_{α} for all i < κ. Therefore S is equal to the set of a in P_{ω_{1}}(𝕏) such that there is i < κ and a condition X in A_{i} ∩ G_{α} which forces Ḟ (i) = a. So S is in V [G_{α}]. □

Theorem 1.14. Assume λ^{<λ} = λ. Then ℚ(𝕏) preserves all cardinals and cofinalities, and forces that every stationary subset of P_{ω_{1}}(𝕏) contains a stationary subset which does not reflect to any set of size ℵ_{1}.

Proof. Let G be a generic filter for ℚ(𝕏) over V. Let S be a stationary subset of P_{ω_{1}}(𝕏) in V [G]. By the last lemma, there is α < κ^{+} such that S is in V [G_{α}], where G_{α} = {p_{α} : p ∈ G}.

Let H = {p(α) : p ∈ G}. Then H is a generic filter for ℙ(𝕏) over V [G_{α}]. Let T = ⋃ H. By Corollary 1.7 and Proposition 1.9, T does not reflect to any set of size ℵ_{1} and T ∩ S is stationary. Clearly then T ∩ S does not reflect to any set of size ℵ_{1}, and since ℚ(𝕏) does not add any subsets of 𝕏 of size ℵ_{1}, this remains true in V [G]. Since ℚ^{α+1}(𝕏) is ω_{1}-closed in V and ℚ_{α+1}(𝕏) does not add any new countable subsets of V, ℚ^{α+1}(𝕏) is ω_{1}-closed in V [G_{α} ∗ H]. Therefore ℚ^{α+1}(𝕏) is proper in V [G_{α} ∗ H]. It follows that T ∩ S remains stationary in V [G]. □

2. Dense Non-Reflection and PFA

Let λ ≥ ω_{2} be a regular cardinal such that λ^{<λ} = λ and 𝕏 a set of size at least λ. Let κ be the cardinality of P_{ω_{1}}(𝕏). Using the notation of Section 1, let ℙ(𝕏) denote the forcing poset which adds a non-reflecting stationary subset of P_{ω_{1}}(𝕏) with conditions of size less than λ, and let ℚ(𝕏) denote the <λ-support product of κ^{+} many copies of ℙ(𝕏). We will abbreviate ℚ(𝕏) as ℚ in what follows. Recall that ℚ is λ-strategically closed and λ^{+}-c.c., and so preserves all cardinals and cofinalities.

The goal of this section is to prove:

Theorem 2.1. Assume PFA. Then ℚ forces PFA and that every stationary subset of P_{ω_{1}}(𝕏) contains a stationary subset which does not reflect to any set of size ℵ_{1}.

By Theorem 1.14, ℚ forces that every stationary subset of P_{ω_{1}}(𝕏) contains a stationary subset which does not reflect to any set of size ℵ_{1}. So it suffices to show that ℚ forces PFA.

If A and B are subsets of ℚ, we say that A and B are cofinally interleaved if for all p in A there is q in B such that q ≤ p, and for all q in B there is r in A such that r ≤ q. When we write q(i) for a condition q in ℚ, this will denote the empty set if i is not in dom(q), although strictly speaking q(i) is undefined.

Let ℝ̇ be a ℚ-name for a proper forcing poset. We will prove that ℚ forces that PFA holds with respect to ℝ̇. Here is a rough outline of the proof. Suppose q^{∗} is a condition in ℚ which forces {Ḋ_{i} : i < ω_{1}} is a family of dense open subsets of ℝ̇. We will define a ℚ ∗ ℝ̇-name 𝕊̇ for a forcing poset such that the iteration ℚ ∗ ℝ̇ ∗ 𝕊̇ is proper. An application of PFA to this iteration will enable us to obtain a condition in ℚ below q^{∗} which forces there is a filter on ℝ which intersects Ḋ_{i} for all i < ω_{1}.

To define 𝕊̇, consider a generic filter G∗H for ℚ∗ℝ̇ over V. Working in V [G∗H], define 𝕊 as the set of sequences s = ⟨s^{i} : i ≤ γ⟩ satisfying:

• γ < ω_{1},

• s^{i} is in G for all i ≤ γ,

• s^{j} ≤ s^{i} for i ≤ j ≤ γ,

• for every limit ordinal ν ≤ γ, dom(s^{ν}) = ⋃_{i<ν} dom(s^{i}), and for every ξ in dom(s^{ν}), s^{ν}(ξ) = ⋃_{i<ν} s^{i}(ξ).

• for every limit ordinal ν ≤ γ, there is a countable set A ⊆ ℚ in V such that

{s^{i} : i < ν} and A are cofinally interleaved. The ordering on 𝕊 is by extension of sequences.

Note that for every q in G, there is a dense set of conditions t = ⟨t^{i} : i ≤ δ⟩ in 𝕊 such that t^{i} ≤ q for some i ≤ δ. Indeed, let s = ⟨s^{i} : i ≤ γ⟩ in 𝕊 be given. Since s^{γ} and q are both in G, we can choose s^{γ+1} in G which is below s^{γ} and q. Then t = ⟨s^{i} : i ≤ γ + 1⟩ is as desired. Also note that for every ordinal ξ < ω_{1}, there is a dense set of conditions t = ⟨t^{i} : i ≤ δ⟩ in 𝕊 such that δ ≥ ξ. For example, given a condition s = ⟨s^{i} : i ≤ γ⟩ in 𝕊, where γ < ξ, extend s to t = ⟨s^{i} : i ≤ ξ⟩ by letting s^{i} = s^{γ} for γ ≤ i ≤ ξ.

Let 𝕊̇ be a ℚ ∗ ℝ̇-name for the forcing poset 𝕊 described above.

Proposition 2.2. The iteration ℚ ∗ ℝ̇ ∗ 𝕊̇ is proper.

Proof. Let θ be a regular cardinal larger than 2^{|ℚ∗ℝ̇∗𝕊̇|} such that ℚ ∗ ℝ̇ ∗ 𝕊̇ is in H(θ). Let N be a countable elementary substructure of H(θ) with ℚ ∗ ℝ̇ ∗ 𝕊̇ in N. Consider a condition q ∗ ṙ ∗ ṡ in N. We will find a condition q′ ∗ ṙ′ ∗ ṡ′ below q ∗ ṙ ∗ ṡ which is N-generic. First, since ℚ forces that ℝ̇ is proper, choose a ℚ-name ṙ′ for a condition in ℝ̇ which is below ṙ and is N[Ġ]-generic.

To define q′, choose a descending sequence ⟨q_{n} : n < ω⟩ of conditions in N ∩ ℚ such that q_{0} = q and for every D in N which is a dense open subset of ℚ, there is n < ω such that q_{n} is in D. This is possible since N is countable. Define q′ so that the domain of q′ is equal to ⋃_{n<ω} dom(q_{n}), and for α in the domain of q′, q′(α) = ⋃_{n<ω} q_{n}(α). By Corollary 1.5, for all α in dom(q′), q′(α) is a condition in ℙ(𝕏) which is below q_{n}(α) for all n < ω. So clearly q′ is a condition in ℚ and q′ ≤ q_{n} for all n < ω.

In order to define ṡ′, consider a generic filter G ∗ H for ℚ ∗ ℝ̇ over V which contains q′ ∗ ṙ′. Let M = N[G ∗ H]. Then M is an elementary substructure of H(θ) in V [G ∗ H]. Since q′ ∗ ṙ′ is N-generic, M ∩ V = N. Let s = ṡ^{G∗H}.

Choose a descending sequence ⟨s_{n} : n < ω⟩ of conditions in M ∩ 𝕊 such that s_{0} = s and for any D in M which is dense open in 𝕊, there is n < ω such that s_{n} is in D. This is possible since M is countable. For each n let s_{n} = ⟨s^{i} : i ≤ γ_{n}⟩. Let γ = sup{γ_{n} : n < ω}. Note that for each n, s_{n} is countable, and therefore s_{n} is a subset of M. So s^{i} is in M for all i < γ.

Claim 2.3. The set {s^{i} : i < γ} is cofinally interleaved with {q_{n} : n < ω}.

Proof. Consider q_{n}. Since q′ is in G and q′ ≤ q_{n}, q_{n} is in G. Let D be the dense open set of conditions ⟨t_{i} : i ≤ δ⟩ in 𝕊 such that t_{i} ≤ q_{n} for some i ≤ δ. Then D is in M by elementarity. So there is m such that s_{m} = ⟨s^{i} : i ≤ γ_{m}⟩ is in D. Then for some i ≤ γ_{m}, s^{i} ≤ q_{n}.

On the other hand, consider s^{i}. Since s^{i} is in M ∩ ℚ and M ∩ V = N, s^{i} is in N ∩ℚ. Let E be the dense open set of conditions in ℚ which are either incompatible with s^{i} or below s^{i}. By elementarity, E is in N. Fix n such that q_{n} is in E. Since q′ is in G and q′ ≤ q_{n}, q_{n} is in G. Also s^{i} is in G. So q_{n} and s^{i} are compatible. Therefore q_{n} ≤ s^{i}. □

Let us recall the choice of q′. The domain of q′ is equal to ⋃_{n<ω} dom(q_{n}), which by Claim 2.3 is equal to ⋃_{i<γ} dom(s^{i}). For each ξ in the domain of q′, q′(ξ) = ⋃_{n<ω} q_{n}(ξ), which by Claim 2.3 is equal to ⋃_{i<γ} s^{i}(ξ). So clearly q′ ≤ s^{i} for all i < γ. Also q′ is in G. Let s^{γ} = q′ and define s′ = ⟨s^{i} : i ≤ γ⟩. Since {q_{n} : n < ω} is in V, s′ is in 𝕊 and s′ ≤ s_{n} for all n < ω.

Let ṡ′ be a ℚ ∗ ℝ̇-name for s′. Then q′ ∗ ṙ′ ∗ ṡ′ is a condition in ℚ ∗ ℝ̇ ∗ 𝕊̇ which is below q ∗ ṙ ∗ ṡ and is N-generic. □

It follows that ℚ ∗ ℝ̇ ∗ 𝕊̇ preserves ω_{1}.

Fix a sequence ⟨q̇^{i} : i < ω_{1}⟩ of ℚ ∗ ℝ̇ ∗ 𝕊̇-names such that ℚ ∗ ℝ̇ ∗ 𝕊̇ forces: • q̇^{i} is in Ġ for all i < ω_{1},

• for every q in Ġ, there is i < ω_{1} such that q̇^{i} ≤ q,

• q̇^{j} ≤ q̇^{i} for i ≤ j < ω_{1},

• for every limit ordinal ν < ω_{1}, dom(q̇^{ν}) = ⋃_{i<ν} dom(q̇^{i}), and for all ξ in dom(q̇^{ν}), q̇^{ν}(ξ) = ⋃_{i<ν} q̇^{i}(ξ).

• for every limit ordinal ν < ω_{1}, there is a countable set A ⊆ ℚ in V such

that {q̇^{i} : i < ν} and A are cofinally interleaved. We will use the following standard fact (for a proof see Theorem 2.53 of [17]).

Proposition 2.4. Assume PFA and let ℙ be a proper forcing poset. Let θ be a regular cardinal such that ℙ is in H(θ). Then there are stationarily many N in [H(θ)]^{ℵ_{1}} such that ω_{1} ⊆ N, N ≺ H(θ), ℙ ∈ N, and there exists a filter on ℙ which is N-generic.

Suppose q^{∗} is in ℚ, {Ḋ_{i} : i < ω_{1}} is a family of ℚ-names, and q^{∗} forces Ḋ_{i} is a dense open subset of ℝ̇ for all i < ω_{1}.

Fix a regular cardinal θ such that ℚ ∗ ℝ̇ ∗ 𝕊̇, ⟨q̇^{i} : i < ω_{1}⟩, and {Ḋ_{i} : i < ω_{1}} are in H(θ). Applying Proposition 2.4, fix N, K, and K_{1} satisfying:

• N is in [H(θ)]^{ℵ_{1}} and ω_{1} ⊆ N,

• N ≺ H(θ),

• ℚ ∗ ℝ̇ ∗ 𝕊̇, ⟨q̇^{i} : i < ω_{1}⟩, q^{∗}, and {Ḋ_{i} : i < ω_{1}} are in N,

• K is a filter on ℚ ∗ ℝ̇ ∗ 𝕊̇ which is N-generic,

• K_{1} is the set of q for which there is q ∗ ṙ ∗ ṡ in N ∩ K,

• q^{∗} is in K_{1}. The last statement can be arranged since (ℚ/q^{∗}) ∗ ℝ̇ ∗ 𝕊̇ is proper.

Using the fact that K is a filter and is N-generic, it is easy to show that K_{1} is a filter on N ∩ ℚ.

For each i < ω_{1} there is a dense set of conditions in ℚ ∗ ℝ̇ ∗ 𝕊̇ which decide the value of q̇^{i}. By elementarity, this dense set is in N. Since K is N-generic, let q^{i} be the unique condition in ℚ such that there is a condition in N ∩ K which forces q̇^{i} = q^{i}. By elementarity, q^{i} is in N.

Lemma 2.5. The sequence ⟨q^{i} : i < ω_{1}⟩ satisfies:

(1) q^{i} is in K_{1} for all i < ω_{1},
(2) for all q in N ∩ K_{1}, there is i < ω_{1} such that q^{i} ≤ q,
(3) q^{j} ≤ q^{i} for all i ≤ j < ω_{1},

(4) for every limit ordinal ν < ω_{1}, dom(q^{ν}) = ⋃_{i<ν} dom(q^{i}), and for all ξ in

dom(q^{ν}), q^{ν}(ξ) = ⋃_{i<ν} q^{i}(ξ).

Proof. (1) Let D be the dense open set of conditions in ℚ which are either incompatible with q^{i} or below q^{i}. Let E be the dense open set of conditions in ℚ ∗ ℝ̇ ∗ 𝕊̇ of the form q ∗ ṙ ∗ ṡ such that q is in D. By elementarity, D and E are in N. Since K is N-generic, fix q ∗ ṙ ∗ ṡ in N ∩ K ∩ E. Then q is in K_{1} ∩ D.

We claim that q is compatible with q^{i}, and hence q ≤ q^{i} since q is in D. Fix a condition q_{0} ∗ ṙ_{0} ∗ ṡ_{0} in N ∩ K which forces q̇^{i} = q^{i}. Then q_{0} is in K_{1}. Since K_{1} is a filter, let q_{1} be in K_{1} which is below q and q_{0}. Then q_{1} ∗ ṡ_{0} ∗ ṙ_{0} forces that q and q^{i} are both in Ġ, and hence are compatible. So indeed q and q^{i} are compatible.

Now q is in K_{1}, q ≤ q^{i}, and q^{i} is in N ∩ ℚ. Since K_{1} is a filter on N ∩ ℚ, q^{i} is in K_{1}.

(2) Let q be in K_{1}. Fix q ∗ ṙ ∗ ṡ in N ∩ K. Since q ∗ ṙ ∗ ṡ forces that q is in Ġ, it forces that there is i < ω_{1} such that q̇^{i} is below q. Let E be the dense open set of conditions in ℚ ∗ ℝ̇ ∗ 𝕊̇ which are either incompatible with q ∗ ṙ ∗ ṡ, or force for some i < ω_{1} that q̇^{i} ≤ q, and in addition decide the value of q̇^{i}. By elementarity, E is in N. Since K is N-generic, fix q_{0} ∗ ṙ_{0} ∗ ṡ_{0} in N ∩ K ∩ E. Fix i < ω_{1} such that q_{0} ∗ ṙ_{0} ∗ ṡ_{0} forces q̇^{i} ≤ q and q̇^{i} = q^{i}. Then q_{0} ∗ ṙ_{0} ∗ ṡ_{0} forces q^{i} ≤ q. So in fact q^{i} ≤ q.

(3) Choose conditions q_{0} ∗ ṙ_{0} ∗ ṡ_{0} and q_{1} ∗ ṙ_{1} ∗ ṡ_{1} in N ∩ K which decide q̇^{i} and q̇^{j} respectively as q^{i} and q^{j}. Since K is a filter, let q ∗ ṙ ∗ ṡ be a refinement of these two conditions. Since ℚ ∗ ℝ̇ ∗ 𝕊̇ forces q̇^{j} ≤ q̇^{i}, q ∗ ṙ ∗ ṡ forces q^{j} ≤ q^{i}. So indeed q^{j} ≤ q^{i}.

(4) Consider a limit ordinal ν < ω_{1}. Then ℚ∗ℝ̇∗𝕊̇ forces dom(q̇^{ν}) = ⋃_{i<ν} dom(q̇^{i}). Since q^{ν} ≤ q^{i} for all i < ν, ⋃_{i<ν} dom(q^{i}) ⊆ dom(q^{ν}). Let E be the dense open set of conditions in ℚ ∗ ℝ̇ ∗ 𝕊̇ which decide q̇^{ν} and also decide for some countable set A ⊆ ℚ in V that {q̇^{i} : i < ν} and A are cofinally interleaved. By elementarity, E is in N. Since K is N-generic, fix q ∗ ṙ ∗ ṡ in N ∩ K ∩ E. Let A be a countable subset of ℚ which q ∗ ṙ ∗ ṡ forces is cofinally interleaved with {q̇^{i} : i < ν}. Then A is in N, and q ∗ ṙ ∗ ṡ forces that dom(q̇^{ν}) = ⋃{dom(s) : s ∈ A}. Note that since {q̇^{i} : i < ν} is forced to be a subset of Ġ, q ∗ ṙ ∗ ṡ forces A ⊆ Ġ.

Consider ξ in dom(q^{ν}). Let E_{1} be the dense open set of conditions which are either incompatible with q ∗ ṙ ∗ ṡ, or are below it and decide for some s in A that ξ is in dom(s). Since N ∩K ∩E_{1} is non-empty, clearly there is s in A such that ξ is in dom(s). Let E_{2} be the dense open set of conditions which are either incompatible with q ∗ ṙ ∗ ṡ, or are below it and decide for some i < ν that q̇^{i} ≤ s, and moreover decide q̇^{i}. Then there is i < ν such that q^{i} ≤ s, and therefore ξ is in dom(q^{i}). This proves the first part of (4). The proof of the second part of (4) is almost the same argument. □

Define q′ in ℚ as follows. The domain of q′ is equal to ⋃_{i<ω_{1}} dom(q^{i}). For each ξ in dom(q′), let q′(ξ) = ⋃_{i<ω_{1}} q^{i}(ξ). By Proposition 1.1 and Lemma 2.5, for all ξ in dom(q′), q′(ξ) is in ℙ(𝕏) and q′(ξ) ≤ q^{i}(ξ) for all i < ω_{1}. Therefore q′ is in ℚ and q′ ≤ q^{i} for all i < ω_{1}. By Lemma 2.5(2), q′ ≤ q for every q in N ∩ K_{1}. In particular, q′ ≤ q^{∗}.

Let G be a generic filter for ℚ over V which contains q′. Let ℝ = ℝ̇^{G}, and for i < ω_{1} let D_{i} = Ḋ_{i}^{G}. In V [G], define H as the set of conditions r in ℝ such that there is q ∗ ṙ ∗ ṡ in N ∩ K such that r = ṙ^{G}.

We claim that H generates a filter on ℝ. Consider r_{1} and r_{2} in H. Then there are q_{i} ∗ ṙ_{i} ∗ ṡ_{i} in N ∩ K such that r_{i} = ṙ_{i}^{G} for i < 2. Let E be the dense open set of conditions in ℚ ∗ ℝ̇ ∗ 𝕊̇ which are either below q_{i} ∗ ṙ_{i} ∗ ṡ_{i} for i < 2, or incompatible with at least one of them. By elementarity, E is in N. Since K is N-generic, fix q ∗ ṙ ∗ ṡ in N ∩ K ∩ E. Clearly q ∗ ṙ ∗ ṡ is below q_{i} ∗ ṙ_{i} ∗ ṡ_{i} for i < 2. Let r = ṙ^{G}. Then r is in H. Now q is in N ∩ K_{1}, so q′ ≤ q. Therefore q is in G. But q forces ṙ ≤ ṙ_{1}, ṙ_{2}. So r ≤ r_{1}, r_{2}.

To complete the proof, we show that H ∩ D_{i} is non-empty for all i < ω_{1}. Let E be the set of conditions q ∗ ṙ ∗ ṡ in ℚ ∗ ℝ̇ ∗ 𝕊̇ such that either q is incompatible with q^{∗}, or q forces ṙ is in Ḋ_{i}. Then E is dense open, because q^{∗} forces Ḋ_{i} is dense open in ℝ̇. By elementarity, E is in N. So let q ∗ ṙ ∗ ṡ be in N ∩ K ∩ E. Then q is in K_{1} and q forces ṙ is in Ḋ_{i}. Since q′ ≤ q, q is in G. Therefore r = ṙ^{G} is in H ∩ D_{i}. This completes the proof of Theorem 2.1

3. Dense Non-Reflection at a Singular Cardinal

We describe some natural situations in which dense non-reflection holds in P_{ω_{1}}(λ), where λ is a singular cardinal with countable cofinality, and show that such dense non-reflection is consistent with PFA.

First we review some basic pcf theoretic definitions. Let 𝔞 be a set of regular cardinals with |𝔞| < min(𝔞). We write ∏ 𝔞 for the set of functions f with domain 𝔞 such that f(κ) ∈ κ for all κ in 𝔞. Suppose D is a filter on 𝔞. For functions f and g in ∏ 𝔞, let f ≤_{D} g if the set {κ ∈ 𝔞 : f(κ) ≤ g(κ)} is in D, and similarly with f <_{D} g. For a regular cardinal μ, a sequence of functions f⃗ = ⟨f_{i} : i < μ⟩ in ∏ 𝔞 is a scale in ∏ 𝔞/D if f_{i} ≤_{D} f_{j} for i < j < μ, and for any function h in ∏ 𝔞, h ≤_{D} f_{i} for some i < μ. The cobounded filter on 𝔞 is the filter generated by the complements of the proper initial segments of 𝔞.

In Theorem 1.5 of Chapter 2 of [14], Shelah proved:

Theorem 3.1. Let λ be a singular cardinal. Then there is a set 𝔞 of regular cardinals cofinal in λ with order type cf(λ) and cf(λ) < min(𝔞) such that there exists a scale f⃗ = ⟨f_{i} : i < λ^{+}⟩ in ∏ 𝔞/D, where D is the cobounded filter on 𝔞.

A scale f⃗ = ⟨f_{i} : i < μ⟩ in ∏ 𝔞/D is said to be an ω_{1}-better scale if for every limit ordinal α < μ with cofinality ω_{1}, there is a club set c ⊆ α with order type ω_{1} such that for any β in c, there is a set A in D such that for all γ in c ∩ β and κ in A, f_{γ}(κ) < f_{β}(κ). (See [6] for more information on better scales.)

Let F be a filter on a set X and κ a cardinal. We say that F is κ-generated if there is a family {A_{i} : i < ξ} of fewer than κ many sets in F such that for any set A ⊆ X, A is in F iff A_{i} ⊆ A for some i < ξ. We say that F is countably generated if F is ω_{1}-generated. For any infinite cardinal λ, the club filter on P_{ω_{1}}(λ) is (2^{λ})^{+}- generated, since for every club C ⊆ P_{ω_{1}}(λ) there is a function F : [λ]^{<ω} → λ such that C contains the club set of a in P_{ω_{1}}(λ) which are closed under F.

The main theorem of this section is:

Theorem 3.2. Let λ be a singular cardinal with countable cofinality. Let 𝔞 be a countable set of regular uncountable cardinals cofinal in λ, and μ a regular cardinal larger than λ. Suppose that:

(1) The club filter on P_{ω_{1}}(λ) is μ^{+}-generated,

(2) There is a countably generated filter D on 𝔞 and an ω_{1}-better scale ⟨f_{i} : i < μ⟩ in ∏ 𝔞/D.

Then every stationary subset of P_{ω_{1}}(λ) contains a stationary subset which does not reflect to any set of size ℵ_{1}.

Proof. For any countable set b ⊆ λ, let χ_{b} denote the function in ∏ 𝔞 defined by letting χ_{b}(κ) = sup(b ∩ κ) for all κ in 𝔞. Note that for any function h in ∏ 𝔞, there are club many b in P_{ω_{1}}(λ) such that h <_{D} χ_{b}.

Fix a stationary set S ⊆ P_{ω_{1}}(λ). We will find a stationary set T ⊆ S which does not reflect to any set of size ℵ_{1}. Fix a family {C_{i} : i < μ} of club subsets of P_{ω_{1}}(λ) such that for every club C ⊆ P_{ω_{1}}(λ), there is i < μ such that C_{i} ⊆ C. Clearly any set T ⊆ P_{ω_{1}}(λ) is stationary if it has non-empty intersection with C_{i} for all i < μ.

To construct T, we define by induction a sequence ⟨b_{i} : i < μ⟩ and a normal function G : μ → μ such that for all i < μ:

(I) b_{i} is in S ∩ C_{i}, (II) f_{G(i)} <_{D} χ_{b_{i}} <_{D} f_{G(i+1)}.

Suppose i < μ and ⟨b_{j} : j < i⟩ and G ↾ i are defined as required. If i = 0 let G(i) = 0. If i is a limit ordinal, let G(i) = sup{G(j) : j < i}. If i = j + 1, choose G(i) larger than G(j) so that χ_{b_{j}} <_{D} f_{G(i)}. This is possible since f⃗ is a scale. Now applying the stationarity of S, choose b_{i} in S ∩ C_{i} satisfying f_{G(i)} <_{D} χ_{b_{i}}. This completes the construction.

Let T = {b_{i} : i < μ}. By property (I), T is a stationary subset of S. Suppose for a contradiction that N is a subset of λ of size ℵ_{1} and P_{ω_{1}}(N) ∩ T is stationary in P_{ω_{1}}(N).

Let β be the least ordinal such that P_{ω_{1}}(N)∩{b_{i} : i < β} is stationary in P_{ω_{1}}(N). Choose an increasing and continuous sequence ⟨a_{i} : i < ω_{1}⟩ of countable sets with union equal to N. Then {a_{i} : i < ω_{1}} ∩ {b_{i} : i < β} is stationary in P_{ω_{1}}(N). So there is a stationary set A ⊆ ω_{1} such that {a_{i} : i ∈ A} ⊆ {b_{i} : i < β}.

Claim 3.3. The cofinality of β is equal to ω_{1}.

Proof. If cf(β) > ω_{1}, then there is γ < β such that {a_{i} : i ∈ A} ⊆ {b_{i} : i < γ}, which contradicts the minimality of β. Suppose that cf(β) = ω, and fix a cofinal function f : ω → β. Then {b_{i} : i < β} ∩ P_{ω_{1}}(N) is equal to ⋃_{n<ω}({b_{i} : i < f(n)} ∩ P_{ω_{1}}(N)). Since the club filter on P_{ω_{1}}(N) is countably complete, there is n < ω such that {b_{i} : i < f(n)} ∩ P_{ω_{1}}(N) is stationary in P_{ω_{1}}(N). Again this contradicts the minimality of β. □

Fix a collection of sets {X_{n} : n < ω} which generates D. Since G : μ → μ is a normal function, the cofinality of G(β) is equal to ω_{1}. Applying the fact that f⃗ is an ω_{1}-better scale, let c ⊆ G(β) be a club subset of G(β) with order type ω_{1} satisfying: for all α in c, there is n < ω such that for all γ in c ∩ α and κ in X_{n}, f_{γ}(κ) < f_{α}(κ). By intersecting c with the club G[β], we may assume without loss of generality that c is contained in the range of G.

By the normality of G, G^{−1}[c] is a club subset of β with order type ω_{1}. Enumerate G^{−1}[c] in increasing order as ⟨β_{i} : i < ω_{1}⟩. Then the function i ↦ β_{i} is a normal cofinal function from ω_{1} to β.

For i < ω_{1} let h_{i} = f_{G(β_{i})}. Then by the choice of c,

∀i < ω_{1} ∃n < ω ∀j < i ∀κ ∈ X_{n} h_{j}(κ) < h_{i}(κ).

Since A is stationary in ω_{1}, fix a stationary set A_{1} ⊆ A and n_{1} < ω such that:

∀i ∈ A_{1} ∀j < i ∀κ ∈ X_{n_{1}} h_{j}(κ) < h_{i}(κ).

Then in particular:

Statement 3.4. For all j < i in A_{1} and for all κ in X_{n_{1}}, h_{j}(κ) < h_{i}(κ).

Since {a_{i} : i ∈ A_{1}} is a subset of {b_{j} : j < β}, we can define g : A_{1} → β by letting g(i) be an ordinal less than β such that a_{i} = b_{g(i)}.

Claim 3.5. There is a club set C ⊆ ω_{1} such that for all i < j in C, if i is in A_{1} then β_{i} ≤ g(i) < β_{j}.

Proof. Let C_{1} be the set of α in ω_{1} such that for all i in α ∩ A_{1}, g(i) < β_{α}. It follows easily from the fact the map α ↦ β_{α} is increasing that C_{1} is a club set.

Assume for a contradiction that there does not exist a club C ⊆ C_{1} consisting of i such that, if i is in A_{1}, then β_{i} ≤ g(i). Then there is a stationary set A′ ⊆ A_{1} such that g(i) < β_{i} for all i in A′. If ν is any limit ordinal in A′, then g(ν) < β_{ν} = sup_{i<ν} β_{i}, so there is i_{ν} < ν such that g(ν) < β_{i_{ν}}. By Fodor’s Lemma, there is a stationary set A′′ ⊆ A′ and γ < ω_{1} such that for all ν in A′′, g(ν) < β_{γ}. But a_{ν} = b_{g(ν)}. So the set {a_{ν} : ν ∈ A′′} is a stationary subset of P_{ω_{1}}(N) contained in {b_{i} : i < β_{γ}}, contradicting the minimality of β. □

Fix a club C as in Claim 3.5. We now thin out A_{1} ∩ C to A_{2}. For each i in A_{1} ∩C, let i^{∗} be the least ordinal in A_{1} ∩C above i. Given i in A_{1} ∩C, we know that β_{i} ≤ g(i) < β_{i^{∗}}. Therefore g(i) + 1 ≤ β_{i^{∗}}. So by property (II) in the construction of T, we have:

χ_{a_{i}} = χ_{b_{g(i)}} <_{D} f_{G(g(i)+1)} ≤_{D} f_{G(β_{i^{∗}})} = h_{i^{∗}},

and therefore

χ_{a_{i}} <_{D} h_{i^{∗}}.

Choose m(i) < ω so that for all κ in X_{m(i)}, χ_{a_{i}}(κ) < h_{i^{∗}}(κ). Fix a stationary set A_{2} ⊆ A_{1} ∩ C and n_{2} < ω such that for all i in A_{2}, m(i) = n_{2}. Then we have:

Statement 3.6. For all i in A_{2} and κ in X_{n_{2}}, χ_{a_{i}}(κ) < h_{i^{∗}}(κ).

Let X = X_{n_{1}} ∩ X_{n_{2}}, which is in D. Since A_{2} is a stationary subset of ω_{1}, let x be a closed set of ordinals in A_{2} with order type ω + 1, and let ν = max(x). Since the sequence ⟨a_{i} : i < ω_{1}⟩ is increasing and continuous, a_{ν} = ⋃{a_{i} : i ∈ x ∩ ν}. Therefore for all κ in X,

sup(a_{ν} ∩ κ) = sup{sup(a_{i} ∩ κ) : i ∈ x ∩ ν}.

In other words:

Statement 3.7. For all κ in X, χ_{a_{ν}} (κ) = sup{χ_{a_{i}}(κ) : i ∈ x ∩ ν}.

Consider i in x ∩ ν. Then for all κ in X,

χ_{a_{i}}(κ) < h_{i^{∗}}(κ) < h_{ν}(κ).

The first inequality is by Statement 3.6, and the second inequality is by Statement 3.4, noting that i^{∗} < ν. So for all i in x ∩ ν and κ in X,

χ_{a_{i}}(κ) < h_{ν}(κ).

Since this inequality holds for all i in x ∩ ν, by Statement 3.7 we get that for all κ in X,

χ_{a_{ν}} (κ) ≤ h_{ν}(κ).

Now a_{ν} = b_{g(ν)}, so for all κ in X,

χ_{b_{g(ν)}}(κ) ≤ h_{ν}(κ).

By the property of C, β_{ν} ≤ g(ν), and by definition, h_{ν} = f_{G(β_{ν})}. Hence h_{ν} ≤_{D} f_{G(g(ν))}. Therefore

χ_{b_{g(ν)}} ≤_{D} f_{G(g(ν))}.

On the other hand, by property (II) in the construction of T,

f_{G(g(ν))} <_{D} χ_{b_{g(ν)}}.

These last two inequalities give a contradiction. □

In the rest of the section, we consider situations in which the hypotheses of Theorem 3.2 hold.

Proposition 3.8. Let λ be a singular cardinal and μ a regular cardinal larger than λ. Let 𝔞 be a set of regular cardinals cofinal in λ with |𝔞| < min(𝔞). Suppose there exists a filter D on 𝔞 and a scale of length μ in ∏ 𝔞/D. Assume there exist sequences

⟨c_{α} : α ∈ μ ∩ cof(ω_{1})⟩, ⟨C_{β} : β ∈ μ ∩ cof(ω)⟩

such that:

(1) for all α ∈ μ ∩ cof(ω_{1}), c_{α} is a club subset of α with order type ω_{1},

(2) for all β ∈ μ ∩ cof(ω), C_{β} is a family of less than μ many closed countable subsets of β,

(3) for all α ∈ μ ∩ cof(ω_{1}), if β is a limit point of c_{α}, then c_{α} ∩ β is in C_{β}. Then there is an ω_{1}-better scale in ∏ 𝔞/D of length μ.

A similar statement was shown to be true in the proof of Theorem 4.1 of [6], using weak square □_{λ}^{∗} in the particular case when μ = λ^{+}. Our argument is based on their proof.

Proof. Let f⃗ = ⟨f_{i} : i < μ⟩ be a scale in ∏ 𝔞/D. We define by induction a cofinal subsequence g⃗ = ⟨g_{i} : i < μ⟩ of f⃗. Suppose ⟨g_{i} : i < β⟩ is defined for some β < μ. If β is not a limit ordinal of cofinality ω, then choose g_{β} from the set {f_{i} : β ≤ i < μ} so that g_{i} ≤_{D} g_{β} for all i < β.

Suppose β is a limit ordinal of cofinality ω. For each c in C_{β}, define g_{c} in ∏ 𝔞 by letting, for κ in 𝔞, g_{c}(κ) = sup{g_{i}(κ) : i ∈ c}. Note that g_{c}(κ) ∈ κ, since κ is regular and uncountable and c is countable. Since |C_{β}| < μ, choose g_{β} in the set {f_{i} : β ≤ i < κ^{+}} so that g_{c} ≤_{D} g_{β} for all c in C_{β}, and g_{i} ≤_{D} g_{β} for all i < β. This completes the construction of g⃗.

To show g⃗ is an ω_{1}-better scale in ∏ 𝔞/D, consider α in μ ∩ cof(ω_{1}). Let d be the club set of limit points of c_{α}. Consider β in d. Since β is a limit point of c_{α}, c_{α} ∩ β is in C_{β}. So by construction, g_{(c_{α}∩β)} ≤_{D} g_{β}. Fix a set A in D such that for all κ in A, g_{(c_{α}∩β)}(κ) ≤ g_{β}(κ). Then for all γ in d ∩ β and κ in A, γ is in c_{α} ∩ β, therefore g_{γ}(κ) ≤ g_{(c_{α}∩β)}(κ) ≤ g_{β}(κ). □

When μ = κ^{+} is a successor cardinal, then the existence of the sequences described in the last proposition follows from weak square □_{κ}^{∗}.

Definition 3.9. Let κ be an uncountable cardinal and ν ≤ κ^{+}. A sequence ⟨C_{α} : α < κ^{+}⟩ is a □_{κ,ν}-sequence if for all limit ordinals α < κ^{+}:

(1) 1 ≤ |C_{α}| ≤ ν,

(2) for all c in C_{α}, c is a club subset of α, and if cf(α) < κ then o.t.(c) < κ,

(3) for all c in C_{α}, if β is a limit point of c, then c ∩ β is in C_{β}. We say that □_{κ,ν} holds if there exists a □_{κ,ν}-sequence.

We refer to a □_{κ,κ}-sequence as a weak square sequence, or a □_{κ}^{∗}-sequence. We say that □_{κ}^{∗} holds if there exists a □_{κ}^{∗}-sequence. If □_{κ}^{∗} holds, then there is a □_{κ}^{∗}- sequence ⟨C_{α} : α < κ^{+}⟩ such that for each limit ordinal α < κ^{+}, there is a club c in C_{α} with order type equal to cf(α); for a proof of this fact, see page 176 of [7]. Both □_{κ} (that is, □_{κ,1}) and □_{κ}^{∗} are due to Jensen [10]. The definition of □_{κ,ν}, where 1 < ν < κ, is due to Schimmerling [13].

Lemma 3.10. Let κ be an uncountable cardinal and suppose □_{κ}^{∗} holds. Then there exist sequences

⟨c_{α} : α ∈ κ^{+} ∩ cof(ω_{1})⟩, ⟨C_{β} : β ∈ κ^{+} ∩ cof(ω)⟩

such that:

(1) for all α ∈ κ^{+} ∩ cof(ω_{1}), c_{α} is a club subset of α with order type ω_{1},

(2) for all β ∈ κ^{+}∩cof(ω), C_{β} is a family of less than κ^{+} many closed countable subsets of β,

(3) for all α ∈ κ^{+} ∩ cof(ω_{1}), if β is a limit point of c_{α}, then c_{α} ∩ β is in C_{β}.

Proof. Fix a □_{κ}^{∗}-sequence ⟨D_{α} : α < κ^{+}⟩ with the property that for every limit ordinal α < κ^{+}, there is a club in D_{α} with order type cf(α). Define ⟨c_{α} : α ∈ κ^{+} ∩ cof(ω_{1})⟩ by choosing for each α in κ^{+} ∩ cof(ω_{1}) a club set c_{α} in D_{α} with order type ω_{1}. For each β in κ^{+} ∩ cof(ω), define C_{β} as the collection of countable sets in D_{β}. Then properties (1) and (2) are immediate, and (3) follows from Definition 3.9(3). □

Corollary 3.11. Let λ be a singular cardinal with cofinality ω. Assume 2^{λ} = λ^{+} and □_{λ}^{∗} holds. Then every stationary subset of P_{ω_{1}}(λ) contains a stationary subset which does not reflect to any set of size ℵ_{1}.

Proof. It suffices to verify the hypotheses of Theorem 3.2 in the case μ = λ^{+} = 2^{λ}. We know that the club filter on P_{ω_{1}}(λ) is (2^{λ})^{+}-generated. By Theorem 3.1, fix a set 𝔞 of regular uncountable cardinals cofinal in λ with order type ω and a scale f⃗ = ⟨f_{i} : i < λ^{+}⟩ in ∏ 𝔞/D, where D is the cobounded filter on 𝔞. Since the order type of 𝔞 is ω, the cobounded filter on 𝔞 is countably generated. By □_{λ}^{∗}, Lemma 3.10, and Proposition 3.8, there is an ω_{1}-better scale of length λ^{+} in ∏ 𝔞/D. So all the hypotheses of Theorem 3.2 are true. □

We remark that it was pointed out by Toshimichi Usuba that assuming the weaker statement ADS_{λ} in place of □_{λ}^{∗} in Corollary 3.11, one can prove that every stationary subset of P_{ω_{1}}(λ) contains a stationary subset which does not reflect to any set of size ℵ_{1} which contains ℵ_{1} ([16]). (See Section 4 of [6] for the definition of ADS_{λ}.)

We can now prove the consistency of PFA with dense non-reflection in P_{ω_{1}}(λ), where λ is a singular cardinal with cofinality ω. This will follow immediately from Corollary 3.11 and a theorem of Magidor [12].

Theorem 3.12 (Magidor). Suppose PFA is consistent. Then PFA is consistent with the statement that 2^{α} = α^{+} for all α ≥ ω_{1} and □_{κ,ℵ_{2}} holds for all κ ≥ ω_{2}.

Corollary 3.13. Suppose PFA is consistent. Then PFA is consistent with the statement that for every singular cardinal λ with cofinality ω, every stationary subset of P_{ω_{1}}(λ) contains a stationary subset which does not reflect to any set of size ℵ_{1}.

We end this section by describing circumstances in which Theorem 3.2 applies, without assuming 2^{λ} = λ^{+}. First we need to review some facts from pcf theory without proof.

Let 𝔞 be a set of regular cardinals with |𝔞|^{+} < min(𝔞). For any cardinal κ, J_{<κ}(𝔞) denotes the ideal consisting of sets b ⊆ 𝔞 such that for any ultrafilter D on 𝔞 with b ∈ D, the cofinality of (∏ 𝔞, ≤_{D}) is less than κ. Clearly J_{<κ_{1}}(𝔞) ⊆ J_{<κ_{2}}(𝔞) for cardinals κ_{1} < κ_{2}. If κ is a limit cardinal, then J_{<κ}(𝔞) is the union of the ideals J_{<μ}(𝔞), where μ < κ is a cardinal; see page 211 of [5].

Recall that pcf(𝔞) denotes the set of regular cardinals κ for which there exists a filter D on 𝔞 and a scale of length κ in ∏ 𝔞/D. The set pcf(𝔞) always has a maximum element; see 1.8 of [5].

Here are the facts from pcf theory we will use:

Fact 3.14. Let κ be a regular cardinal.

(1) κ is in pcf(𝔞) iff J_{<κ^{+}}(𝔞) \ J_{<κ}(𝔞) is non-empty.

(2) If b is in J_{<κ^{+}}(𝔞) \ J_{<κ}(𝔞), then there is a scale of length κ in ∏ 𝔞/D, where D is the filter on 𝔞 generated by the set b together with the dual filter of J_{<κ}(𝔞).

Fact 3.15. If 𝔞 is an interval of regular cardinals, then so is pcf(𝔞).

Fact 3.16. There exists a sequence ⟨b_{κ} : κ ∈ pcf(𝔞)⟩ of subsets of 𝔞 such that for each κ in pcf(𝔞), J_{<κ^{+}}(𝔞) is the ideal on 𝔞 generated by J_{<κ}(𝔞) ∪ {b_{κ}}.

Fact 3.17. Let λ be a singular cardinal with cofinality ω. Assume there is ω_{1} ≤ δ < λ such that there are only countably many regular cardinals between δ and λ. Let 𝔞 = (δ, λ) ∩ Reg. Then max(pcf(𝔞)) is equal to the cofinality of the partial ordering ([λ]^{ℵ_{0}}, ⊆).

Fact 3.14 is proven as Lemma 1.4 and Corollary 4.4 of [5]. Facts 3.15 and 3.16 are proven as 2.2 and 7.9 of [5] respectively. Fact 3.17 is proven as Theorem 5.11 of [1].

The pcf conjecture is the statement that for any set of regular cardinals 𝔞 with |𝔞| < min(𝔞), pcf(𝔞) has size |𝔞|. It is not known whether the pcf conjecture is a theorem of ZFC.

Lemma 3.18. Let 𝔞 be a countable set of regular cardinals with ω_{1} < min(𝔞) such that pcf(𝔞) is countable. Then for every κ in pcf(𝔞), there is a countably generated filter D on 𝔞 and a scale of length κ in ∏ 𝔞/D.

Proof. Note that for all κ in pcf(𝔞), J_{<κ^{+}}(𝔞) is countably generated. This can be proven by an easy induction using the fact that pcf(𝔞) is countable, together with Fact 3.14(1) and Fact 3.16. Let κ be in pcf(𝔞). Let D_{κ} be the dual filter of J_{<κ^{+}}(𝔞). Clearly D_{κ} is countably generated. By Fact 3.14(1), fix a set b in J_{<κ^{+}}(𝔞) \ J_{<κ}(𝔞). Then by Fact 3.14(2), there is a scale of length κ in ∏ 𝔞/D, where D is the countably generated filter on 𝔞 generated by the set b together with D_{κ}. □

Corollary 3.19. Let λ be a singular strong limit cardinal of cofinality ω. Suppose there is ω_{1} ≤ δ < λ such that there are only countably many regular cardinals between δ and λ. Let 𝔞 = (δ, λ) ∩ Reg. Assume pcf(𝔞) is countable. Then 2^{λ} = κ^{+} for some cardinal κ ≥ λ. Assume that □_{κ}^{∗} holds. Then every stationary subset of P_{ω_{1}}(λ) has a stationary subset which does not reflect to any set of size ℵ_{1}.

Proof. Since λ is a strong limit cardinal of cofinality ω, 2^{λ} = λ^{ω} and λ^{ω} = cf([λ]^{ℵ_{0}})· 2^{ω} = cf([λ]^{ℵ_{0}}). Combining these facts with Fact 3.17, we get that 2^{λ} = max(pcf(𝔞)). Now 𝔞 is an interval of regular cardinals, so pcf(𝔞) is as well by Fact 3.15. As pcf(𝔞) is countable, 2^{λ} = max(pcf(𝔞)) is less than λ^{+ω_{1}}. Since there are no regular limit cardinals between λ and λ^{+ω_{1}}, 2^{λ} = κ^{+} for some κ ≥ λ. By Lemma 3.18, there is a countably generated filter D on 𝔞 and a scale of length 2^{λ} in ∏ 𝔞/D. So assuming □_{κ}^{∗} holds, it follows from Lemma 3.10 and Proposition 3.8 that there is an ω_{1}-better scale of length 2^{λ} in ∏ 𝔞/D. □

4. Adding Non-Reflecting Sets for a Singular Cardinal

Let λ be a singular cardinal with cofinality ω. By Section 1, we can add a nonreflecting stationary subset of P_{ω_{1}}(λ), using for example conditions of size less than ω_{2}. But this poset will add many subsets of ω_{2}, and so λ will not be a strong limit in the extension. In this section we present a (λ + 1)-strategically closed forcing poset for adding a non-reflecting stationary subset of P_{ω_{1}}(λ). Using a suitable product of this forcing, we obtain dense non-reflection in P_{ω_{1}}(λ) with a forcing poset which does not add subsets to λ.

Fix a set 𝔞 of regular cardinals larger than ω_{1} with order type ω which is cofinal in λ. For functions f and g in ∏ 𝔞, we write g ≤^{∗} f to indicate g ≤_{D} f, where D is the cobounded filter on 𝔞, and similarly for g <^{∗} f. Recall that for a countable set b ⊆ λ, χ_{b} is the function in ∏ 𝔞 defined by letting χ_{b}(κ) = sup(b ∩ κ) for all κ in 𝔞.

Define ℙ as the set of pairs ⟨X, f⟩ satisfying:

• X ⊆ P_{ω_{1}}(λ),

• |X| ≤ λ,

• f is in ∏ 𝔞,

• for all a in X, χ_{a} <^{∗} f,

• for any set N in [λ]^{ℵ_{1}}, P_{ω_{1}}(N) ∩ X is non-stationary in P_{ω_{1}}(N). We let ⟨Y, g⟩ ≤ ⟨X, f⟩ if:

• X ⊆ Y,

• f ≤^{∗} g,

• for all b in Y \ X, f ≤^{∗} χ_{b}. Note that ℙ has size 2^{λ}, and so is (2^{λ})^{+}-c.c.

Proposition 4.1. The forcing poset ℙ is (λ + 1)-strategically closed.

Proof. Since λ is singular, by an easy argument it suffices to show that ℙ is νstrategically closed for all ν < λ. So let ν < λ be given. We describe a strategy for Player II. At successor stages, Player II just repeats Player I’s last play. Suppose δ < ν is a limit ordinal and a sequence of plays ⟨⟨X_{i}, f_{i}⟩ : 0 < i < δ⟩ has been determined. Define X_{δ} = ⋃_{0<i<δ} X_{i}. Define f_{δ} by letting f_{δ}(κ) = 0 if κ ∈ 𝔞 ∩ ν, and f_{δ}(κ) = (sup_{0<i<δ} f_{i}(κ)) + 1 if κ is in 𝔞 \ ν.

We claim that ⟨X_{δ}, f_{δ}⟩ is a condition in ℙ which is below ⟨X_{i}, f_{i}⟩ for all 0 < i < δ. Thus the game continues through all stages less than ν. We will prove that for any N in [λ]^{ℵ_{1}}, P_{ω_{1}}(N) ∩ X_{δ} is non-stationary in P_{ω_{1}}(N). The remaining properties are easy to check.

Suppose for a contradiction N is in [λ]^{ℵ_{1}} and P_{ω_{1}}(N)∩X_{δ} is stationary in P_{ω_{1}}(N). Fix an increasing and continuous sequence ⟨a_{i} : i < ω_{1}⟩ of countable sets with union equal to N. Then there is a stationary set A ⊆ ω_{1} such that {a_{i} : i ∈ A} ⊆ X_{δ}.

Claim 4.2. The cofinality of δ is ω_{1}.

The proof is similar to the proof of Claim 3.3.

Fix an increasing and continuous sequence ⟨β_{i} : i < ω_{1}⟩ cofinal in δ. Define g : A → ω_{1} by letting g(i) be the least ordinal such that a_{i} is in X_{β_{g(i)}}. Note that g(i) is always a successor ordinal, since the map j ↦ β_{j} is normal and X_{ξ} = ⋃_{j<ξ} X_{j} for every limit ordinal ξ < δ.

Claim 4.3. There is a club C ⊆ ω_{1} such that for all i < j in C, if i is in A then i < g(i) < j.

The proof is similar to the proof of Claim 3.5.

Fix a club C as in Claim 4.3. We thin out A ∩ C to A_{1}. Consider i in A ∩ C. Let i^{∗} be the least ordinal in A ∩ C above i. Then β_{i} < β_{g(i)} < β_{i^{∗}}. Since a_{i} is in X_{β_{g(i)}}, by the definition of ℙ we have that χ_{a_{i}} <^{∗} f_{β_{g(i)}} ≤^{∗} f_{β_{i^{∗}}}. So there is κ(i) in 𝔞 such that that for all κ in 𝔞 \ κ(i), χ_{a_{i}}(κ) < f_{β_{i^{∗}}} (κ). Since 𝔞 is countable and A ∩ C is stationary, we can fix a stationary set A_{1} ⊆ A ∩ C and κ_{1} in 𝔞 so that for all i in A_{1}, κ(i) = κ_{1}. Then for all i in A_{1} and κ in 𝔞 \ κ_{1}, χ_{a_{i}}(κ) < f_{β_{i^{∗}}} (κ).

By the stationarity of A_{1}, let x be a closed subset of A_{1} with order type ω+1, and let ν = max(x). As ⟨a_{i} : i < ω_{1}⟩ is increasing and continuous, a_{ν} = ⋃{a_{i} : i ∈ x∩ν}. It follows that for all κ in 𝔞, χ_{a_{ν}} (κ) = sup{χ_{a_{i}}(κ) : i ∈ x ∩ ν}.

Consider κ in 𝔞 which is larger than κ_{1} and ν. Then for all i in x ∩ ν, i^{∗} < ν, so χ_{a_{i}}(κ) < f_{β_{i^{∗}}} (κ) ≤ sup_{0<j<β_{ν}} f_{j}(κ). Since this is true for all i in x ∩ ν, χ_{a_{ν}} (κ) ≤ sup_{0<j<β_{ν}} f_{j}(κ). By the definition of f_{β_{ν}}, sup_{0<j<β_{ν}} f_{j}(κ) < f_{β_{ν}} (κ). So χ_{a_{ν}} <^{∗} f_{β_{ν}}. On the other hand, since g(ν) > ν, a_{ν} is in X_{β_{g(ν)}} \ X_{β_{ν}}. By the definition of the ordering on ℙ we have f_{β_{ν}} ≤^{∗} χ_{a_{ν}}, which is a contradiction. □

Let Ṫ be a ℙ-name for the set ⋃{X : ∃f ⟨X, f⟩ ∈ Ġ}.

We claim that for any stationary set S ⊆ P_{ω_{1}}(λ), ℙ forces Ṫ ∩ Š is stationary. For suppose p is in ℙ and p forces Ḟ : [λ]^{<ω} → λ is a function. Since Ḟ is a name for a subset of the ground model of size λ and ℙ is (λ+1)-strategically closed, there is q ≤ p and F : [λ]^{<ω} → λ such that q forces Ḟ = F̌. Let q = ⟨X, f⟩. Since S is stationary, we can choose a in S such that a is closed under F and f ≤^{∗} χ_{a}. Fix g in ∏ 𝔞 such that f <^{∗} g and χ_{a} <^{∗} g. Then ⟨X ∪ {a}, g⟩ is a condition in ℙ which is below q and forces a is in Ṫ ∩ Š and is closed under Ḟ.

Finally, we show ℙ forces Ṫ does not reflect to any set of size ℵ_{1}. Suppose for a contradiction p is in ℙ and Ṅ is a ℙ-name such that p forces Ṅ is in [λ]^{ℵ_{1}} and P_{ω_{1}}(Ṅ) ∩ Ṫ is stationary. Since ℙ is (λ + 1)-strategically closed, there is q ≤ p and N in [λ]^{ℵ_{1}} such that q forces Ṅ = Ň. Let q = ⟨X, f⟩. Define g in ∏ 𝔞 by letting g(κ) = (max{f(κ), sup(N ∩ κ)}) + 1 for κ in 𝔞. Let r = ⟨X, g⟩. Clearly r is a condition and r ≤ q. We claim that r forces P_{ω_{1}}(N) ∩ Ṫ = P_{ω_{1}}(N) ∩ X, which is a contradiction since P_{ω_{1}}(N)∩X is non-stationary. If not, then there is ⟨Y, h⟩ ≤ ⟨X, g⟩ and y in Y \ X which is in P_{ω_{1}}(N). By the definition of the ordering on ℙ, g ≤^{∗} χ_{y}. But for all κ in 𝔞, since y ⊆ N we have χ_{y}(κ) = sup(y ∩ κ) ≤ sup(N ∩ κ) < g(κ). So χ_{y} <^{∗} g, which is a contradiction.

As in Section 1, we can obtain a model satisfying dense non-reflection in P_{ω_{1}}(λ) using a product forcing of the above poset. Assume 2^{λ} = λ^{+}. Define ℚ as the product forcing consisting of partial functions p : λ^{++} → ℙ with domain of size less than λ^{+}, ordered in the usual way. Since ℙ is (λ + 1)-strategically closed, it is easy to show that ℚ is (λ+1)-strategically closed by using Player II’s strategy separately on each coordinate. Since 2^{λ} = λ^{+}, ℙ has size λ^{+}, and a straightforward Δ-system argument shows that ℚ is λ^{++}-c.c. For all α < λ^{++}, ℚ can be factored as ℚ_{α} × ℚ^{α} as in Section 1. The fact that ℚ is λ^{++}-c.c. implies that any stationary subset of P_{ω_{1}}(λ) in V^{ℚ} appears in V^{ℚ_{α}} for some α < λ^{++}. Thus the same argument as given in Section 1 shows that ℚ forces dense non-reflection in P_{ω_{1}}(λ).

5. Global Dense Non-Reflection

In this final section we prove the following theorem.

Theorem 5.1. Suppose PFA holds. Then there is a class-sized ω_{2}–strategically closed partial order P preserving ZFC + PFA and forcing GCH above ω and dense non-reflection in P_{ω_{1}}(λ) for every cardinal λ ≥ ω_{2} such that λ is either regular or of countable cofinality. Furthermore, P preserves cofinalities if GCH above ω holds in the ground model.

We will sketch two proofs of Theorem 5.1. Before starting it will be convenient to fix some pieces of notation. Suppose P is a partial order and Ẋ is a P–name for a subset of some ordinal α. We will say that Ẋ is a nice P–name for a subset of α in case it consists of pairs of the form ⟨p, ξ̌⟩, with p ∈ P and ξ̌ the canonical name for an ordinal ξ ∈ α.

When dealing with set–forcing, the following slightly nonstandard notion of two– step iteration will simplify the parts of the proof of Theorem 5.1 in which we need to compute cardinalities of partial orders: Suppose P is a poset. If Q̇_{0} is a P–name for a poset, then it is clear that, for some ordinal α, the two–step iteration P ∗ Q̇_{0} (in the standard sense) is isomorphic to one of the form P ∗ Q̇ in which Q̇ is forced to consist of subsets of α. And furthermore, it is clear that this second iteration has a dense suborder consisting of pairs ⟨p, q̇⟩ such that q̇ is a nice P–name for a subset of α. When Q̇ is a P–name for a collection of subsets of a minimal ordinal α, we will define the two–step iteration P ∗ Q̇ as the suborder of the corresponding two–step iteration I, taken in the standard sense, consisting precisely of the pairs ⟨p, q̇⟩ ∈ I such that q̇ is a nice P–name for a subset of α. The above remark shows that we do not lose any generality by doing so. Hence, we may and will identify every two–step iteration in the standard sense with an iteration (in the new sense) of the form P ∗ Q̇ for which there is a minimal α such that Q̇ is forced to consist of subsets of α.

The proof of Theorem 5.1 will involve reverse Easton iterations. By such an iteration we mean any forcing iteration in which direct limits are taken at all regular cardinals and inverse limits are taken everywhere else. In other words, a reverse Easton iteration is a forcing iteration of the form ⟨P_{ξ} : ξ ∈ Ω⟩, allowing Ω to be either an ordinal or Ord, based on a sequence ⟨Q̇_{ξ} : ξ ∈ Ω⟩ of names such that each Q̇_{ξ} is a P_{ξ}–name for a poset, and such that sup(supp(p) ∩ \overline{ξ}) < \overline{ξ} whenever \overline{ξ} ∈ Ω is a regular cardinal, ξ_{0} ∈ Ω, and p is a condition in P_{ξ_{0}}.^{4} Every class forcing P preserves all of the ZFC axioms except possibly for the Power Set Axiom and the Axiom scheme of Replacement. In the case that P is the direct limit of a reverse Easton iteration as above with the additional property that for every λ there is some ξ such that the tail forcing P/G is forced to be λ–distributive in V [G] for each P_{ξ}–generic filter G, then P preserves these remaining axioms as well (see Section 2.2 of [9]).

The following easy general fact will be useful.

Lemma 5.2. Let α be a regular cardinal, let Ω be Ord or a member of it, let ξ_{0} ∈ Ω, and let ⟨P_{ξ} : ξ ≤ Ω⟩ be a forcing iteration, based on a sequence ⟨Q̇_{ξ} : ξ ∈ Ω⟩ of names such that, for every ξ ∈ [ξ_{0}, Ω), Q̇_{ξ} is < α–strategically closed in V^{P_{ξ}}. Suppose that P_{ξ_{0}} has the α–chain condition. Suppose in addition that {supp(q)\ξ_{0} : q ∈ P_{Ω}} is closed under unions of ⊆–increasing sequences of length less than α.

Then, P_{Ω}/G is < α–strategically closed in V [G] for every P_{ξ_{0}}–generic filter G over V.

Let us assume PFA. By first forcing with an ω_{2}–directed closed forcing P^{0} if necessary we may assume that GCH holds above ω. P^{0} is the direct limit of a reverse Easton iteration as above in which, at each step ξ, we force with trivial forcing (that is, with {∅}) unless ξ is an infinite V –cardinal above ω_{1}, in which case Q̇_{ξ} is, in V^{P_{ξ}}, the forcing for adding a subset of ξ by initial segments.

If ξ ≥ ω_{2} is a V –cardinal and G is P_{ξ}–generic over V, then P_{ξ+1}/G forces |P (γ)^{V [G]}| ≤ ξ for every γ < ξ. And, if in addition ξ is such that |P_{ξ^{+}}| ≤ ξ (which will happen whenever ξ is a strong limit), then P^{0}/G′ is easily seen to be ξ^{+}–directed closed in V [G′] for every P_{ξ^{+}}–generic G′. It follows that P^{0} preserves ZFC and forces GCH above ω.

By a theorem from [11], any ω_{2}-directed closed forcing poset preserves PFA. Clearly P^{0} is ω_{2}–directed closed. If Ṙ is a P^{0}–name for a proper poset and Ḋ_{i} (for i < ω_{1}) are names for dense subsets of Ṙ, then there is some ξ large enough so that all of these names are in fact P_{ξ}–names, such that each Ḋ_{i} is a P_{ξ}–name for a dense subset of Ṙ and such that, in addition, Ṙ is a P_{ξ}–name for a proper poset. This is true because properness is a local condition (in other words, a poset ℙ is proper if and only if ℙ is proper in H(θ) for any large enough θ with ℙ ∈ H(θ)), and for every θ there is a ξ_{0} such that forcing with P^{0}/Ġ_{ξ_{0}} over V^{P_{ξ_{0}}} leaves H(θ) unchanged. But P_{ξ} is an ω_{2}–directed closed poset and PFA holds in V, which implies that in V^{P_{ξ}}, and therefore in V^{P^{0}}, there is a filter of Ṙ intersecting each Ḋ_{i}.

It may be worth mentioning that the use of reverse Easton supports is not relevant for the task of preserving ZFC and PFA. In fact, the direct limit \overline{P^{0}} of the iteration obtained if we use full supports in the above definition of P^{0} would do as well.^{5} The main use of reverse Easton supports is in ensuring, in a GCH–context, that the relevant posets have the relevant chain condition. This is exemplified in the constructions we are going to consider next.

4Where supp(p), the support of p, is the set of all ξ < ξ_{0} such that p ↾ ξ does not force (in P_{ξ})

that p(ξ) is the weakest condition in Q̇_{ξ}.

We are left with the task of showing, under the assumption that GCH holds above ω and that PFA holds, that there is a class forcing P as in the conclusion of Theorem 5.1 which moreover preserves cofinalities. We will give two constructions of such a P. In what follows, for a regular cardinal ξ, we let ℚ(ξ) denote the product forcing from Section 1 for forcing dense non-reflection in P_{ω_{1}}(ξ) using conditions of size less than ξ.

First construction. Let R be the class of all regular cardinals ξ ≥ ω_{2}. For the first construction we build a reverse Easton iteration ⟨P_{ξ} : ξ ∈ Ord⟩, based on a sequence ⟨Q̇_{ξ} : ξ ∈ Ord⟩ of names, such that each Q̇_{ξ} is forced to be trivial forcing unless it is true in V^{P_{ξ}} that ξ is a regular cardinal above ω_{1} and that 2^{ζ} = ζ^{+} for all ζ with ω_{1} ≤ ζ ≤ ξ.

In that case, if it is not the case in V^{P_{ξ}} that ξ is the successor of a singular cardinal λ of countable cofinality, then Q̇_{ξ} is forced to be ℚ(ξ).

In the other case, if it holds in V^{P_{ξ}} that ξ = λ^{+} and λ is a singular cardinal of countable cofinality, then Q̇_{ξ} is forced to be the two–step iteration 𝔞(λ)∗(ℚ(ξ)×ℚ), where 𝔞(λ) is the atomic forcing that picks an ω–sequence 𝔞 cofinal in λ consisting of regular cardinals in (ω_{1}, λ),^{6} and where ℚ is as in Section 4 for λ and 𝔞.^{7}

Lemma 5.3. For each regular cardinal ξ ≥ ω_{2}, P_{ξ} has size at most ξ.

Proof. This can be proved by induction on ξ using GCH above ω. The result when ξ is inaccessible is straightforward as in that case P_{ξ} is the direct limit of ⟨P_{ξ′} : ξ′ < ξ⟩.

If ξ = ξ_{0}^{+} for ξ_{0} a singular cardinal and ot(R∩ξ_{0}) = \overline{ξ} ≤ ξ_{0}, then P_{ξ} is (isomorphic to) P_{ξ_{0}}, and |P_{ξ_{0}}| is then the cardinality of a collection of 2^{|\overline{ξ}|} = |\overline{ξ}|^{+}–many sets of the form ^{X}(⋃_{ξ′<ξ_{0}} P_{ξ′}) for some X ⊆ \overline{ξ}. It follows that |P_{ξ}| = |P_{ξ_{0}}| ≤ ξ_{0}^{+} = ξ since ⋃_{ξ′<ξ_{0}} P_{ξ′} has size at most sup(R ∩ ξ_{0}) = ξ_{0} by induction hypothesis.

Finally, if ξ = ξ_{0}^{+} for ξ_{0} regular, then P_{ξ} is isomorphic to P_{ξ_{0}}∗Q̇_{ξ_{0}}, with |P_{ξ_{0}}| ≤ ξ_{0} by induction hypothesis and such that Q̇_{ξ_{0}} is forced to be either the empty set or a subcollection of {^{X}(P_{ξ_{0}}(P_{ω_{1}}(ξ_{0}))) : X ∈ ((ξ_{0}^{ℵ_{0}})^{+})^{< ξ_{0}}} or of the same cardinality as a subcollection of the product of {^{X}(P_{ξ_{0}}(P_{ω_{1}}(ξ_{0}))) : X ∈ ((ξ_{0}^{ℵ_{0}})^{+})^{< ξ_{0}}} and {^{X}(P_{λ^{+}}(P_{ω_{1}}(λ)) × ^{ω}λ) : X ∈ (λ^{++})^{λ}} for some λ < ξ_{0}. But in these two cases, ξ_{0}^{<ξ_{0}} = ξ_{0} holds in V^{P_{ξ_{0}}} . It follows, in either case, that there is a certain set X of size ξ_{0}^{+} such that every nice P_{ξ_{0}}–name for a condition in Q̇_{ξ_{0}} can be coded as a function from P_{ξ_{0}} into X. Hence, by |P_{ξ_{0}}| ≤ ξ_{0} and (ξ_{0}^{+})^{ξ_{0}} = ξ_{0}^{+} we have that |P_{ξ}| = |P_{ξ_{0}} ∗ Q̇_{ξ_{0}}| = ξ_{0}^{+} = ξ by our definition of two–step iteration. □

5But \overline{P^{0}} might collapse more cardinals than P^{0}.

6That is, conditions in 𝔞(λ) are sequences 𝔞 as above, and any two distinct conditions are incompatible.

7We will see that P preserves all regular cardinals as well as GCH above ω, but defining the iteration this way makes it clear at the present point that each Q̇_{ξ} is in fact well–defined and makes it easy to find simple inductive proofs of the relevant facts about the iteration.

Note that, by Lemma 5.2, P is < ω_{2}–strategically closed. In particular, it preserves ω_{1} and ω_{2}. Also, since P_{ξ}, for ξ ∈ R, has size at most ξ and forces that Q̇_{ξ} has the ξ^{+}-chain condition (by Propositions 1.11 and by the remarks at the end of Section 4) and each component Q̇_{ζ} on the tail is forced to be < ζ–strategically closed (by Proposition 1.10 and again by the end of Section 4), we get the following result by Lemma 5.2.

Lemma 5.4. For each infinite cardinal ξ ≥ ω_{1}, P_{ξ^{+}} has the ξ^{+}–chain condition and P factors as P_{ξ^{+}} ∗ Ṗ^{∗}, with Ṗ^{∗} a P_{ξ^{+}}–name for an < ξ^{+}–strategically closed forcing.

By an inductive argument using Lemma 5.4 it follows that P preserves the regularity of all ξ ∈ R.

As to the preservation of ZFC by P, we only need to care about Replacement and the Power Set Axiom. But the preservation of these axioms follows also from Lemma 5.4 and the general facts in Section 2.2 of [9].

Using once more Lemmas 5.3 and 5.4 one can prove that P preserves GCH at uncountable regular cardinals. This implies that every singular cardinal λ remains strong limit after forcing with P and hence (2^{λ})^{V^{P}} = (λ^{cf(λ)})^{V^{P}} = (λ^{cf(λ)})^{V} = λ^{+} again by the relevant distributivity of the tail forcings. Hence, P preserves GCH above ω.

Also, from the results in Section 1 we know that dense non-reflection in P_{ω_{1}}(λ) holds in V^{P_{λ^{+}}} for every regular λ ≥ ω_{2}, which implies the same conclusion in V^{P} for every such λ by Lemma 5.4. And, for λ is a singular cardinal of countable cofinality, we know by Section 4 that dense non-reflection in P_{ω_{1}}(λ) holds in V^{P_{λ^{+}}} . The reason why this is true is that the product Q̇_{λ} is forcing–equivalent in V^{P_{λ}} to the two–step iteration ℚ(λ) ∗ Q̃_{λ}, where Q̃_{λ} is the two–step iteration 𝔞(λ) ∗ ℚ, from the definition of the iteration ⟨P_{ξ} : ξ ∈ Ord⟩, as defined in (V^{P_{ξ}})^{ℚ(λ)}.^{8} Again by Lemma 5.4, we get then that dense non-reflection in P_{ω_{1}}(λ) holds in the final extension.

The preservation of PFA is similar to the proof that P^{0} preserves PFA, using also the proof of the results in Section 2. Given a proper poset ℝ̇ ∈ V^{P} and dense subsets Ḋ_{i} of ℝ̇ for i < ω_{1}, also in V^{P}, we can find a high enough ξ such that the above names are in fact P_{ξ}–names, such that all Ḋ_{i} are, in V^{P_{ξ}}, dense subsets of ℝ̇, and such that ℝ̇ is, in V^{P_{ξ}}, a proper poset. Now we can define a P_{ξ} ∗ ℝ̇–name 𝕊̇ for a poset analogous to the name 𝕊̇ in the proof of Theorem 2.1. The corresponding version of Proposition 2.2 holds for this choice of P_{ξ} ∗ ℝ̇ ∗ 𝕊̇ by essentially the same proof, using the fact that all Q̇_{ζ} are forced to be ω_{1}–closed (by Corollary 1.5 and by the proof of Proposition 4.1). Using this one can show, by essentially the same arguments as in the proof of Theorem 2.1, that in V^{P_{ξ}} there is a filter of ℝ̇ intersecting all Ḋ_{i}, but then of course the same is true in V^{P}.

Second construction. For the second construction we start forcing, over our ground model V of PFA and GCH above ω, with Magidor’s partial order – let us call it 𝕄 – for getting □_{κ,ℵ_{2}} for all κ ≥ ω_{1} while preserving PFA ([12]). 𝕄 is the direct limit of a reverse Easton iteration ⟨\overline{P}_{ξ} : ξ ∈ Ord⟩ on which nothing happens unless ξ is a successor cardinal κ^{+} for κ ≥ ω_{1}. In that case we force at stage ξ of the iteration with the natural forcing for adding a □_{κ,ℵ_{2}}–sequence by initial segments. Arguments as in the first construction show that 𝕄 preserves ZFC, GCH above ω and cofinalities, and that it forces □_{κ,ℵ_{2}} for all cardinals κ ≥ ω_{1}. Magidor [12] proved that 𝕄 preserves PFA.

8This is true by the fact that ℚ(λ) is λ^{+}–distributive in V^{P_{λ}}.

Let V_{1} be the extension of V by 𝕄. Now we build a reverse Easton iteration ⟨P̃_{ξ} : ξ ∈ Ord⟩, based on a sequence ⟨Q̇_{ξ} : ξ ∈ Ord⟩ of names for posets. The definition of this sequence is like the definition of the sequence of names making up the iteration ⟨P_{ξ} : ξ ∈ Ord⟩ in the first construction, except that now we only force with ℚ(ξ) when ξ ≥ ω_{2} is a regular cardinal in V P̃_{ξ}. In other words, we do not take any steps now to force dense non-reflection in P_{ω_{1}}(λ) when λ is a singular cardinal of countable cofinality.

By the same arguments as in the first construction we can prove that the direct limit P of this iteration preserves ZFC, GCH above ω, cofinalities and PFA, and that it forces dense non-reflection in P_{ω_{1}}(λ) for all regular λ ≥ ω_{2}.

Finally, when λ is a singular cardinal of countable cofinality, dense non-reflection holds in P_{ω_{1}}(λ) in V_{1}^{P} by Corollary 3.11 because □_{λ,ℵ_{2}} obviously holds in this extension.

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ICREA and Universitat de Barcelona, Departament de Lògica, Universitat de Barcelona, C. Montalegre 6, Barcelona 08001, Spain

E-mail address: david.aspero@icrea.es

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

E-mail address: jkrueger@math.berkeley.edu

Graduate School of Information Science, Nagoya University, Furocho, Chikusa-ku, Nagoya 464-8601, Japan

E-mail address: yosinobu@math.nagoya-u.ac.jp