SQUARES, SCALES AND STATIONARY REFLECTION

JAMES CUMMINGS, MATTHEW FOREMAN, AND MENACHEM MAGIDOR

Abstract. Since the work of Gödel and Cohen, which showed that Hilbert’s First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by Zermelo-Fraenkel Set Theory with the Axiom of Choice, ZFC), there have been a myriad of independence results in many areas of mathematics.

These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond(♢) and square(□) discovered by Jensen. Simultaneously, attempts have been made to find suitable natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen’s principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of “PCF Theory”, a generalization of cardinal arithmetic that is largely determined inside ZFC.

In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales (a fundamental notion in PCF theory), and on consistency results between relatively strong forms of square and stationary set reflection.

1. Introduction

Since the work of Gödel and Cohen which showed that Hilbert’s First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (Zermelo-Fraenkel Set Theory with

1991 Mathematics Subject Classification. Primary 03E35, 03E55; Secondary 03E05.

Key words and phrases. Forcing, large cardinals, combinatorial principles. The first author was partially supported by grant number 517/94 of the Israel

Science Foundation and DMS 9703945.

The second author was partially supported by DMS 9500484. The third author was partially supported by grant number 517/94 of the Israel

Science Foundation.

2 JAMES CUMMINGS, MATTHEW FOREMAN, AND MENACHEM MAGIDOR

the Axiom of Choice, ZFC), there have been a myriad of independence results in many areas of mathematics.

These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond(♢) and square(□) discovered by Jensen. These two principles are far ranging in their application. Among the first applications were the results of Jensen that the principle, ♢_{ω_{1}} implies the existence of a Souslin tree on ω_{1} and hence the failure of Souslin’s Hypothesis. Gregory [17] showed that the principle □_{ω_{1}}, together with the Continuum Hypothesis implies the existence of a Souslin tree on ω_{2}.

Simultaneously, attempts have been made to find suitable natural strengthenings of ZFC, primarily by Large Cardinal (or Reflection) Axioms. These two directions have tension between them, in that Jensen’s principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. One of the basic and most useful types of reflection property that follows from large cardinals is stationary set reflection, which contradicts square.

These issues are particularly piquant at successors of singular cardinals, where Shelah’s “PCF Theory” has become an important tool. This theory provides a generalization of the usual cardinal arithmetic, with the advantage that many of the properties of the PCF theory are determined in ZFC alone. Fundamental to the PCF theory is the existence of various types of “Scales”. A typical application of the PCF theory is the ZFC theorem [46] that ℵ_{ω+1} is not a Jonsson cardinal.

In this paper we consider interactions of these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection.

The differences in behavior of singular and regular cardinals have been recognized for quite some time. For example [9] showed that the behavior of cardinal exponentiation at regular cardinals was essentially arbitrary, subject to the relatively weak requirements of König’s Theorem [26]. The behavior of cardinal exponentiation at singular cardinals is still not fully understood, but it is known to be significantly more complicated. Work of Silver [47] and Galvin and Hajnal [15] put severe restrictions on the size of P(λ) in terms of the behavior of the power function for smaller cardinals, for λ a singular cardinal of uncountable cofinality. In particular, if the Generalized Continuum Hypothesis (GCH) holds below λ then it holds at λ.

Magidor [32] showed that it is consistent to have the GCH hold below ℵ_{ω}, but fail at ℵ_{ω}. Thus singular cardinals of countable cofinality behave differently. Shelah [46] in a particularly dramatic development, showed the inequality:

ℵ_{ω}^{ω} ≤ ℵ_{ω_{4}} × 2^{ℵ_{0}}

For this he developed the theory known as “PCF theory”, a theory primarily concerned with the behavior of reduced products of cofinal sets of regular cardinals A ⊆ λ for a singular cardinal λ.

The differences between regular and singular cardinals are apparent in the behavior of various square principles as well. For example, collapsing a Mahlo cardinal to be the successor of a regular cardinal μ makes square fail at μ^{+}, whereas it is known that the failure of square at the successor of singular cardinal is a much stronger property, though how strong is still not completely settled [40]. It is a theorem of Shelah [13, 41] that even quite weak forms of square (the approachability property) fail at the successor of singular cardinals above a supercompact cardinal. By suitable collapsing it is possible to make these forms of square fail at ℵ_{ω} as well [13].

The main tools for determining lower bounds for the consistency strength of the failure of square are “Core Models” and “Covering Lemmas”. A “core model” is a particular kind of canonical model of ZFC and a large cardinal axiom [38, 48]. A particularly striking feature of these core models is that the failure of “covering” at a singular cardinal for one of the core models yields an inner model with a large cardinal of the strength of that core model. The core models tend to be models of square, or square-like properties [39]. In particular, the failure of square or of the singular cardinals hypothesis implies the existence of inner models of fairly large cardinals.

In his study of core models for Woodin cardinals Schimmerling was able to establish versions of square (□_{κ,λ} and □_{κ,<λ}) that are slightly weaker than the usual square, but stronger than Jensen’s principle “Weak Square”. The usual square provides an organized and coherent way of building bijections between a cardinal κ and ordinals α < κ^{+}, by coherently cofinalizing each α with a closed unbounded set. Roughly speaking, Schimmerling’s alternative square principles provide more than one closed unbounded set for each α < κ^{+}. These principles have much the same utility as the original square. (For a study of weak square principles and their efficacy, see [13].)

By specifying the number of cofinalizing sets involved one gets a hierarchy of square principles. Jensen showed that this hierarchy is strict (in the sense of implication) at successors of regular cardinals, and this paper uses his techniques in a slightly different way to prove the same result for successors of singular cardinals.

These square principles are all stronger then Jensen’s principle “Weak Square”. Weak square, in Schimmerling’s language, is □_{κ,κ}. This principle was shown by Jensen [25] to be equivalent to the existence of a special Aronszajn tree.

It follows from an easy variation on the classical proof that Schimmerling’s □_{κ,<ω} implies the existence of a non-reflecting stationary set. In this paper we show that variations of square that allow infinitely many (but fewer than κ) cofinalizing sets have strong consequences for the existence of various types of scales and their continuity properties. In particular the existence of “Good Scales” and “Very Good Scales” follow from these square principles. These scale principles, in turn, limit what kind of reflection is possible.

On the other hand, we show it is consistent that □_{ℵ_{ω},ω} holds and still every stationary subset of ℵ_{ω+1} reflects. Further it is consistent that weak square holds and every small collection of stationary subsets of ℵ_{ω+1} consisting of points of a bounded cofinality simultaneously reflects.

The first five sections of the paper give “ZFC-results”, that various combinatorial principles imply various others. The rest of the paper is devoted to relative consistency results, showing limitations on what can be proved in ZFC, as well as providing universes with desirable combinations of combinatorial properties. A rough outline of the paper is as follows:

• In section 2, we give the basic definitions explicitly, and describe known results.

• In section 3, we define the principles of reflection and simultaneous reflection, scales, good and very good scales. We show that □_{κ,λ} (for λ < κ) implies the existence of very good scales, and that these in turn imply that simultaneous reflection fails.

• In section 4, we discuss reflection for stationary collections of countable sets and its relation to various square principles. As an intermediary device of independent interest, we consider a principle due to Shelah, called ADS_{κ}. We show that either weak square or the existence of a very good scale implies ADS_{κ}, and that ADS_{κ} implies the negation of stationary reflection for stationary collections of countable sets.

• In section 5, we discuss “Improved Square” and its consequences, and limitations on the consistency of “Improved Square”.

• In section 6, we begin the development of techniques for forcing various squares. Key to our results is the method of “threading” a square. The concept of an “indestructibly generically supercompact cardinal” is reviewed, as well as standard results for constructing models where they exist.

• In section 7, we show that Jensen’s result distinguishing between different □_{κ,λ}’s for κ regular extends to the case of κ singular.

• In section 8, we show that it is sometimes possible to add □_{κ} using forcing of cardinality less than κ.

• In section 9, we show the consistency of various forms of square (e.g. □_{λ,cof(λ)} for λ singular) above a supercompact. The notion of “Indexed Square” is introduced, and it is shown that it implies a square-like principle S_{λ} which Shelah has shown is equivalent to the upwards two cardinal transfer property (ℵ_{1}, ℵ_{0}) → (λ^{+}, λ).

• In section 10, it is shown that it is consistent to have indexed □_{ℵ_{ω},ω} and still have simultaneous reflection for any finite collection of stationary subsets of ℵ_{ω+1}. The proof goes by adding a generic indexed square sequence, and then generically killing “fragile” stationary sets. Those stationary sets that remain are preserved by appropriate thread forcings that recreate generic supercompactness.

A comment on priority is in order here. After the authors had proved the main result of section 10, it was pointed out that a slightly weaker result, namely the consistency of the two cardinal transfer property with every stationary subset of ℵ_{ω+1} reflecting was claimed in [4]. However the proof in that paper contains an error; Lemma 16 (page 2384) is false, since with the stated definition of Q_{S}^{λ_{i}} the sequence ⟨q_{j} : j < ρ⟩ does not supply the “threading” information needed to apply Lemma 6. Moreover a repair to that proof cannot be done easily: results of [13] show that if V is a model of □_{λ}^{∗} where λ is a cardinal of cofinality ω, then there is no μ < λ which is generically supercompact via countably closed forcing in any generic extension by a countably closed forcing (e.g. Shelah and Ben-David’s ℚ_{S}^{λ_{i}}). Nor do these authors know of a suitable threading forcing for S_{λ} or indexed □_{λ,ω} that provably preserves stationary subsets of λ^{+}; at this point in time we do not know how to finish the proof in [4] along the outlines of their argument.

• In section 11, we investigate square and reflection in the model obtained by adding a Prikry sequence to a measurable cardinal κ. We show that there is a non-reflecting stationary subset of κ^{+}, such that any stationary subset of its complement reflects. Further, there is a very good scale in this model.

• In section 12, we show that it is consistent to have weak square at ℵ_{ω} and simultaneous reflection of less than ℵ_{ω} stationary sets consisting of ordinals of a bounded cofinality. The technique here is to fortify a weak square sequence by a “Diamond-Plus Sequence”, and then thread it. The threading forcing is shown to be stationary set preserving (for this special weak square sequence) and to resurrect generically supercompact cardinals.

• In section 13, it is shown that reflection for stationary collections of countable sets does not imply simultaneous reflection, even for two stationary sets. Similar results have been obtained independently by Larson [29].

Our notation is fairly standard, here we give a brief review. |X| is the cardinality of X, ot(X) is the order type of X, and P(X) is the power set of X. ^{Y} X is the set of all functions from Y to X, and if α is an ordinal then ^{<α}X is the set of all sequences of length less than α from X. If κ is a cardinal then [X]^{κ} is the set of subsets of X with cardinality κ, and [X]^{<κ} and [X]^{≤κ} are defined in the natural way. For κ, λ cardinals κ^{λ} is the cardinal |^{λ}κ|. cf(α) is the cofinality of α. Definition 1.1. Let κ be a cardinal. Then

1. cof(κ) = { α : cf(α) = κ }.

2. cof(< κ) = { α : cf(α) < κ }.

3. cof(≤ κ) = { α : cf(α) ≤ κ }.

For forcing we mostly follow the conventions used in Kunen’s book [28]. p ≤ q means that p is stronger than q, a κ-closed poset is one in which any descending sequence of length less than κ has a lower bound, and a (λ, ∞)-distributive poset is one in which any sequence of dense open sets of length less than λ has nonempty intersection. A κ-directed closed poset is one in which any directed set of size less than κ has a lower bound. We write p ∥ ϕ for “p decides ϕ”, which is to say that either p ⊩ ϕ or p ⊩ ¬ϕ.

Add(κ, λ) is the poset for adding λ Cohen subsets of κ, Coll(κ, λ) is the Lévy collapse to make λ have cardinality κ, and Coll(κ, < λ) is the Lévy collapse to make all cardinals in [κ, λ) have cardinality κ.

2. Square sequences and non-reflecting stationary sets

The combinatorial principle □_{κ} was introduced by Jensen [25] who showed that if V = L then □_{κ} holds for every uncountable cardinal κ. □_{κ} is typically used in building objects of size κ^{+}, and was originally motivated by Jensen’s work on constructing κ^{+}-Souslin trees in L.

Definition 2.1. Let κ be an uncountable cardinal. A □_{κ}-sequence is a sequence ⟨C_{α} : α < κ^{+}, lim(α)⟩ such that for all α < κ^{+}

1. C_{α} is closed and unbounded in α.

2. If cf(α) < κ, then ot(C_{α}) < κ.

3. For all β ∈ lim(C_{α}), C_{β} = C_{α} ∩ β. We say that □_{κ} holds iff there exists a □_{κ}-sequence.

Notice that if C⃗ is a □_{κ}-sequence then ot(C_{α}) ≤ κ for all α, and if in addition κ is singular then ot(C_{α}) < κ for all κ. Some writers adopt a slightly different definition, in which property 2 is replaced by the weaker demand that ot(C_{α}) ≤ κ for all α; the existence of a sequence with this weaker property implies the existence of a □_{κ}-sequence.

A classical application of □_{κ} is to show that there are many nonreflecting stationary subsets of κ^{+}.

Definition 2.2. Let κ be an uncountable regular cardinal. Let S be a stationary subset of κ.

1. S reflects at α iff α < κ, cf(α) > ω and S ∩ α is stationary in α.

2. Refl(S) holds iff every stationary subset of S reflects at some α.

3. S is non-reflecting iff S does not reflect at any α.

Stationary reflection is a phenomenon of considerable interest in combinatorial set theory. For example Gregory [17] showed that if GCH holds and there is a non-reflecting stationary subset of ℵ_{2} ∩cof(ω) then there is an ℵ_{2}-Souslin tree.

The next result is well known but we give the proof for the sake of completeness. It will be a paradigm for many of the results that we prove later.

Theorem 1. Let □_{κ} hold and let S be a stationary subset of κ^{+}. Then Refl(S) fails.

Proof. Let C⃗ be a □_{κ}-sequence, and let F (α) = ot(C_{α}) for each limit α < κ^{+}. By Fodor’s lemma there is a stationary set T ⊆ S on which F is constant. We claim that T does not reflect. To see this fix α < κ^{+} with cf(α) > ω, and observe that if β ∈ lim(C_{α}) then F (β) = ot(C_{β}) = ot(C_{α} ∩ β). It follows that F is 1-1 on lim(C_{α}), hence lim(C_{α}) meets T in at most one point and so T ∩ α is not stationary in α.

It is natural to ask whether the hypothesis of Theorem 1 can be weakened. Jensen [25] defined a principle □_{κ}^{∗} (“weak square”) as follows.

Definition 2.3. Let κ be an infinite cardinal. A □_{κ}^{∗}-sequence is a sequence ⟨C_{α} : α < κ^{+}, lim(α)⟩ such that 1. C_{α} ⊆ P(α), 1 ≤ |C_{α}| ≤ κ, and C_{α} is a set of closed and unbounded

subsets of α.

2. If cf(α) < κ then ∀C ∈ C_{α} ot(C) < κ.

3. ∀C ∈ C_{α} ∀β ∈ lim(C) C ∩ β ∈ C_{β}.

We say that □_{κ}^{∗} holds iff there exists a □_{κ}^{∗}-sequence.

It is easy to see that if κ^{<κ} = κ then there is a □_{κ}^{∗}-sequence, so that □_{κ}^{∗} is of greatest interest when κ is singular. It can be shown [13] that if □_{κ}^{∗} holds then we may choose a □_{κ}^{∗}-sequence with the additional property that each C_{α} contains a set of order type cf(α). We would get an equivalent definition of □_{κ}^{∗} if we weakened clause 2 in the definition to demand only that ∀α ∀C ∈ C_{α} ot(C) ≤ κ.

□_{κ}^{∗} is a powerful construction principle, for example Jensen [25] showed that the existence of a □_{κ}^{∗}-sequence is equivalent to the existence of a special κ^{+}-Aronszajn tree. However it turns out that □_{κ}^{∗} is not strong enough to manufacture non-reflecting stationary sets, in fact we see in Theorem 21 that from a suitable large cardinal assumption we can show the consistency of □_{ℵ_{ω}}^{∗} plus a strong form of reflection for stationary subsets of ℵ_{ω+1}.

Schimmerling [39] introduced a natural hierarchy of principles intermediate in strength between □_{κ} and □_{κ}^{∗}, in connection with his work on the core model for one Woodin cardinal.

Definition 2.4. Let κ be a cardinal. A □_{κ,λ}-sequence is a sequence ⟨C_{α} : α < κ^{+}, lim(α)⟩ such that

1. C_{α} ⊆ P(α), 1 ≤ |C_{α}| ≤ λ, and C_{α} is a set of closed and unbounded subsets of α.

2. If cf(α) < κ then ∀C ∈ C_{α} ot(C) < κ.

3. ∀C ∈ C_{α} ∀β ∈ lim(C) C ∩ β ∈ C_{β}.

A □_{κ,<λ}-sequence is defined similarly, only we demand that 1 ≤ |C_{α}| < λ.

It is clear that □_{κ} = □_{κ,1}, □_{κ}^{∗} = □_{κ,κ}, and the strength of □_{κ,λ} decreases with increasing λ. Jensen [24] showed that for κ regular the strength of □_{κ,λ} is strictly decreasing, and in Theorem 13 we prove the corresponding result for singular κ. Schimmerling (and independently Foreman and Magidor) observed that for sufficiently small values of λ we can manufacture non-reflecting stationary sets using □_{κ,<λ}.

Theorem 2. Let κ^{<λ} = κ and let □_{κ,<λ} hold. If S ⊆ κ^{+} is stationary then there exists a stationary set T ⊆ S such that T does not reflect at any α with cf(α) ≥ λ.

Proof. Let C⃗ be a □_{κ,<λ}-sequence. Let F (α) = { ot(C) : C ∈ C_{α} }. By Fodor’s lemma there is T ⊆ S a stationary set on which F is constant, say with value X. Now if cf(α) ≥ λ and C ∈ C_{α} then for every β ∈ lim(C) ∩ T we have ot(C ∩ β) ∈ X. So lim(C) ∩ T is a set of size less than λ, and hence T is not stationary in α.

In particular, □_{κ,<ω} implies the failure of Refl(S) for every stationary S ⊆ κ^{+}.

If κ^{<λ} = κ then necessarily λ ≤ cf(κ). It is worth noticing that we can still get some information in this situation without any cardinal arithmetic assumptions.

Theorem 3. Let κ be singular. Let □_{κ,<cf(κ)} hold, and let S ⊆ κ^{+} be stationary. Then there exist a stationary T ⊆ S and β < κ such that T fails to reflect at points of cofinality greater than β.

Proof. Let C⃗ be a □_{κ,<cf(κ)}-sequence. Define F : κ^{+} ⟶ κ by setting

F : α ⟼ sup({ ot(C) : C ∈ C_{α} }).

Let T ⊆ S be stationary such that F ↾ T is constant, with constant value β < κ. Then if cf(γ) > β and C ∈ C_{γ} we know ot(C) ≥ cf(γ) > β, so that for all sufficiently large δ ∈ lim(C) we have F (δ) > β and hence δ ∉ T.

Remark 2.5. We will see in Theorem 18 that □_{ℵ_{ω},ω} is consistent with “every stationary subset of ℵ_{ω+1} reflects at all sufficiently large cofinalities”.

An important fact about square principles is that they will necessarily hold unless there are inner models of large cardinals. For example, Solovay showed that □_{ℵ_{1}} holds unless ℵ_{2} is Mahlo in L, and a complementary consistency result by Mitchell [37] shows that if κ is Mahlo it can be collapsed to ℵ_{2} in such a way that □_{ℵ_{1}}^{∗} fails in the generic extension.

For κ singular, it can be shown that □_{κ} or something close to it will hold unless there are inner models for sizeable large cardinals. Here we just quote some of the more important results, and refer the reader to Schimmerling’s survey [40] for more details and a comprehensive list of references.

Fact 2.6 (Jensen). If κ is singular and □_{κ} fails then there is an inner model with a strong cardinal.

Fact 2.7 (Mitchell, Schimmerling,Steel). If κ is a measurable cardinal and α < κ is singular and □_{α,<ω} fails then there is an inner model with a Woodin cardinal.

Fact 2.8 (Mitchell, Schimmerling, Steel, Woodin). If α is a singular strong limit cardinal and □_{α,<ω} fails then PD holds (so in particular for every n < ω there is a model with n Woodin cardinals).

However, sufficiently large cardinals will serve to make even weak forms of square fail at a singular cardinal. For example

Fact 2.9 (Solovay). If κ is supercompact and λ is a cardinal with κ < λ then □_{λ} fails.

Fact 2.10 (Shelah). If κ is supercompact and cf(λ) < κ < λ then □_{λ}^{∗} fails.

In fact Shelah proved that under the hypotheses of Fact 2.10 even more is true, the weaker principle AP_{λ} (see Definition 6.13) fails. For more information about these matters see [13].

There is also a connection between weak squares and forcing axioms, as explored by Todorcevic [49] and Magidor [31].

Fact 2.11 (Magidor). PFA implies that □_{κ,ℵ_{1}} fails for κ ≥ ℵ_{1}, while “PFA +∀κ ≥ ℵ_{2} □_{κ,ℵ_{2}}” is consistent.

It is interesting to notice that Martin’s Maximum has much more impact than PFA on the existence of weak squares. Foreman and Magidor [13] introduce a principle VWS_{κ} (“Very Weak Square”) which is substantially weaker than □_{κ}^{∗} and the principle AP_{κ}, but still strong enough for many applications in algebra and topology.

Fact 2.12 (Magidor). Martin’s Maximum implies that VWS_{κ} fails for κ of cofinality ω.

3. Simultaneous reflection and very good scales

We will say that a family of stationary sets “reflects simultaneously” if there is a point at which each set in the family reflects.

Definition 3.1. Let κ be a regular uncountable cardinal, let S ⊆ κ be stationary. Then Refl(μ, S) holds iff for every sequence ⟨T_{i} : i < μ⟩ of stationary subsets of S there exists α such that T_{i} reflects at α for every i < μ.

One reason for taking an interest in simultaneous reflection is that this phenomenon often occurs in the natural models for reflection properties. For example

Fact 3.2 (Baumgartner [2]). If κ is a weakly compact cardinal and ℙ = Coll(ℵ_{1}, < κ) then the reflection property Refl(ℵ_{1}, ℵ_{2} ∩ cof(ω)) holds in V^{ℙ}.

In general simultaneous reflection is stronger than ordinary reflection. For example the principle Refl(ℵ_{2} ∩ cof(ω)) is equiconsistent [18] with the existence of a Mahlo cardinal, while the principle Refl(2, ℵ_{2} ∩ cof(ω)) is equiconsistent [33] with the existence of a weakly compact cardinal.

We will also be interested in the cofinality of the point at which reflection occurs.

Definition 3.3. Let λ = cf(λ) > ω, and let S ⊆ λ be a stationary subset of λ.

1. Refl(S, ν) holds iff every stationary T ⊆ S reflects at some α < λ with cf(α) = ν.

2. Refl(S, ≥ ν) holds iff every stationary T ⊆ S reflects at some α < λ with cf(α) ≥ ν.

3. Refl(μ, S, ν) holds iff for every μ-sequence ⟨T_{i} : i < μ⟩ of stationary subsets of S there exists α < λ such that cf(α) = ν and all the T_{i} reflect at α.

4. Refl(μ, S, ≥ ν) holds iff for every μ-sequence ⟨T_{i} : i < μ⟩ of stationary subsets of S there exists α < λ such that cf(α) ≥ ν and all the T_{i} reflect at α.

We will prove that □_{κ,λ} for λ < κ puts a bound on the extent of simultaneous reflection in κ^{+}. For this we will use some ideas from Shelah’s PCF theory.

PCF stands for “Possible CoFinalities”. PCF theory is a sophisticated combinatorial theory in which a singular cardinal κ is analysed by looking at reduced products of sets of regular cardinals which are unbounded in κ. No detailed knowledge of PCF theory is required to understand our results, as long as the reader is prepared to take on trust the basic Fact 3.6. For more information about PCF theory see [5] or [46].

We begin with the key concept of a scale.

Definition 3.4. Let κ be a singular cardinal. A scale of length β for κ is a triple (κ⃗, f⃗, I) where

1. κ⃗ = ⟨κ_{i} : i < cf(κ)⟩ is an increasing sequence of regular cardinals such that sup_{i} κ_{i} = κ.

2. I is an ideal on cf(κ), from which we define a relation <_{I} on ^{I}ON by

f <_{I} g ⟺ { i : f(i) ≥ g(i) } ∈ I.

3. f⃗ = ⟨f_{α} : α < β⟩ is a sequence of functions such that (a) f_{α} ∈ ∏_{i<cf(κ)} κ_{i}.

(b) If γ < δ < β then f_{γ} <_{I} f_{δ}.

(c) ∀f ∈ ∏_{i<cf(κ)} κ_{i} ∃α < β f <_{I} f_{α}.

This is more generality than we really need, we will only be concerned with scales of length κ^{+} and the ideal of bounded sets.

Definition 3.5. Let κ be singular.

1. A scale for κ is a pair (κ⃗, f⃗) such that (κ⃗, f⃗, I_{bdd}) is a scale of length κ^{+} for κ, where I_{bdd} = { A ⊆ cf(κ) : sup(A) < cf(κ) }

2. Let f, g ∈ ∏_{i<cf(κ)} κ_{i}. Then (a) f <^{∗} g ⟺ f <_{I_{bdd}} g.

(b) f < g ⟺ ∀i f(i) < g(i).

(c) f ≤ g ⟺ ∀i f(i) ≤ g(i).

The following fact is one of the early triumphs of PCF theory.

Fact 3.6 (Shelah). If κ is singular then there exists a scale for κ.

The idea of a “good point” or “flat point” is a key one in analysing the properties of scales.

Definition 3.7. Let κ be singular and let (κ⃗, f⃗) be a scale for κ. A point α < κ^{+} is good for (κ⃗, f⃗) iff there exist A ⊆ α unbounded in α and i < cf(κ) such that ∀β, γ ∈ A ∀j > i (β < γ ⟹ f_{β}(j) < f_{γ}(j)).

It is easy to see that all points of cofinality less than cf(κ) are good.

Definition 3.8. Let g⃗ = ⟨g_{α} : α < β⟩ be increasing in (∏_{i<cf(κ)} κ_{i}, <_{I}). g ∈ ∏_{i<cf(κ)} κ_{i} is an exact upper bound (eub) for g⃗ iff

1. ∀α < β g_{α} <_{I} g.

2. ∀h ∈ ∏_{i<cf(κ)} κ_{i} (h <_{I} g ⟹ ∃α h <_{I} g_{α}).

If cf(α) > cf(κ) then α is good if and only if there is an eub g for ⟨f_{β} : β < α⟩ with the property that cf(g(i)) = cf(α) for all i. We will be interested in the question of how many points are good, where it is worth noticing that the set of good points is essentially a function of κ⃗.

Fact 3.9 (Shelah). The set of good points in a scale (κ⃗, f⃗) is a stationary set. It is determined modulo the club filter on κ^{+} by the sequence κ⃗.

Definition 3.10. (κ⃗, f⃗) is a good scale for κ iff it is a scale for κ, and modulo the club filter on κ^{+} almost every point of cofinality greater than cf(κ) is good for (κ⃗, f⃗). GS_{κ} holds iff there exists a good scale for κ.

It turns out that quite weak versions of square at κ (for example the principle AP_{κ}) imply that all scales for κ are good. See [13] for more on this subject and on situations in which there are scales which fail to be good. For the purposes of this paper we are also interested in a stronger notion, that of a “very good scale”.

Definition 3.11. (κ⃗, f⃗) is a very good scale for κ iff 1. (κ⃗, f⃗) is a scale for κ.

2. For every point α < κ^{+} such that cf(α) > cf(κ) there exists a closed and unbounded set C ⊆ α and i < cf(κ) such that ∀β, γ ∈ C ∀j > i (β < γ ⟹ f_{β}(j) < f_{γ}(j)).

VGS_{κ} holds iff there exists a very good scale for κ. Theorem 4. Let κ be singular, let λ < κ. Then □_{κ,λ} implies VGS_{κ}.

Proof. Let ⟨C_{α} : α < κ^{+}, lim(α)⟩ witness □_{κ,λ}. Let (κ⃗, f⃗) be any scale for κ, where the existence of such a scale is guaranteed by Fact 3.6. Without loss of generality we may assume that λ < κ_{0}. We will inductively build g⃗ such that f_{α} < g_{α} and (κ⃗, g⃗) is a very good scale.

The successor stage is easy, we just choose g_{α+1} such that g_{α+1} > f_{α+1} and g_{α+1} > g_{α}.

For γ a limit, we choose g_{γ} such that

1. g_{γ} > f_{γ}.

2. ∀α < γ g_{α} <^{∗} g_{γ}.

3. For all i, g_{γ}(i) > sup { sup_{β∈C} g_{β}(i) : C ∈ C_{γ}, |C| < κ_{i} }.

We need to show that this gives a very good scale. Fix γ < κ^{+} a limit ordinal with cf(γ) > cf(κ), and choose C ∈ C_{γ}. Fix i such that |C| < κ_{i}, and let j ≥ i. If α, β ∈ lim(C) then C ∩β ∈ C_{β}, α ∈ C ∩β and |C ∩ β| < κ_{j}, so that g_{α}(j) < g_{β}(j). So the club set lim(C) witnesses that (κ⃗, g⃗) has the defining property of a very good scale at the point γ.

We now prove that VGS_{κ} puts some limit on the possibilities for stationary reflection in κ^{+}. Theorem 5. Let κ be singular, and let VGS_{κ} hold. Then for every stationary T ⊆ κ^{+} the principle Refl(cf(κ), T, ≥ cf(κ)^{+}) fails.

Proof. Let (κ⃗, f⃗) be a very good scale for κ. Assume for a contradiction that Refl(cf(κ), T, ≥ cf(κ)^{+}) holds. For each i < cf(κ), consider the function g_{i} from T to κ_{i} given by g_{i}(α) = f_{α}(i), and let S_{i} ⊆ T be stationary such that ∀α ∈ S_{i} g_{i}(α) = g(i).

Now apply the hypothesis Refl(cf(κ), T, ≥ cf(κ)^{+}) to the sequence of sets ⟨S_{i} : i < cf(κ)⟩, and find some α < κ^{+} such that cf(α) > cf(κ) and S_{i} ∩ α is stationary in α for all i < cf(κ).

Since (κ⃗, f⃗) is a very good scale, we may find C ⊆ α club and i < κ such that if γ, δ ∈ C, γ < δ and j > i then f_{γ}(j) < f_{δ}(j).

Let j = i + 1. Since C is club in α and S_{j} ∩ α is stationary in α we may find two points β < γ < α with β, γ ∈ C ∩ S_{j}. But then f_{β}(j) = g(j) = f_{γ}(j), contradicting the choice of C and i.

Remark 3.12. Actually all we needed here was that an unbounded subsequence of the S_{i} should reflect simultaneously.

Remark 3.13. Schimmerling [7] proved that □_{κ,<κ} has some influence on simultaneous reflection.

Remark 3.14. In the model of Theorem 18 we will have □_{ℵ_{ω},ω} (and so VGS_{ℵ_{ω}}) along with simultaneous reflection for any finite family of stationary subsets of ℵ_{ω+1}. It follows that Theorem 5 is fairly sharp.

Remark 3.15. By Theorem 21, □_{ℵ_{ω}}^{∗} is consistent with a strong form of simultaneous reflection and so by Theorem 5 □_{ℵ_{ω}}^{∗} does not imply VGS_{ℵ_{ω}}.

It is sometimes convenient to consider scales with an additional “continuity” property.

Definition 3.16. Let (κ⃗, f⃗) be a scale for κ, let α < κ^{+}. (κ⃗, f⃗) is continuous at α iff

(∃g g is an eub for f⃗ ↾ α) ⟹ f_{α} is an eub for f⃗ ↾ α.

Theorem 6. If there is a very good scale for κ, there is a very good scale which is continuous at every point of cofinality greater than cf(κ).

Proof. Let (κ⃗, f⃗) be a very good scale for κ. We define a new scale (κ⃗, f⃗^{∗}).

f_{γ}^{∗} = f_{γ} unless

• cf(γ) > ω.

• There exists D club in γ and i < cf(κ) such that ⟨f_{α}(j) : α ∈ D⟩

is strictly increasing for all j ≥ i. In this case choose i_{γ} to be the least i such that a club D as above exists and κ_{i} > cf(γ), choose such D with order type cf(γ), and define

f_{γ}^{∗}(j) = \begin{cases}0 & if j < i_{γ} \\ sup\{f_{α}(j) : α ∈ D\} & if j ≥ i_{γ} \end{cases}

This definition is independent of the club D which is chosen.

We claim that the scale f⃗^{∗} is very good and is continuous at every point of cofinality greater than cf(κ). To see this let γ < κ^{+} be such that cf(γ) > cf(κ). Choose C club in γ such that ⟨f_{α}(j) : α ∈ C⟩ is strictly increasing for j ≥ i_{γ}, and ot(C) = cf(γ).

We claim that for j ≥ i_{γ}

• f_{α}^{∗}(j) ≤ f_{α}(j) for all α ∈ lim(C).

• ⟨f_{α}^{∗}(j) : α ∈ lim(C)⟩ is strictly increasing.

• f_{γ}^{∗}(j) = sup{f_{α}^{∗}(j) : α ∈ lim(C)}.

We prove by induction on α ∈ lim(C) that f_{α}^{∗}(j) ≤ f_{α}(j) and that ⟨f_{β}^{∗}(j) : β ∈ lim(C) ∩ (α + 1)⟩ is strictly increasing for every j ≥ i_{γ}. Fix such a j. Successor case: If α is a successor point in lim(C) then cf(α) = ω, so f_{α}^{∗} = f_{α} and in particular f_{α}(j) = f_{α}^{∗}(j). Let α_{0} be the preceding point, then f_{α_{0}}^{∗}(j) ≤ f_{α_{0}}(j) < f_{α}(j) = f_{α}^{∗}(j), and so the sequence ⟨f_{β}^{∗}(j) : β ∈ lim(C) ∩ (α + 1)⟩ is strictly increasing. Limit case: Let α be a limit point in lim(C). By induction the sequence ⟨f_{β}^{∗}(j) : β ∈ lim(C) ∩ α⟩ is strictly increasing. On an unbounded set of β ∈ lim(C) ∩ α (namely the successors) f_{β}^{∗} = f_{β}, and so f_{β}^{∗}(j) = f_{β}(j) < f_{α}(j). This implies that for all β ∈ lim(C) ∩ α, f_{β}^{∗}(j) < f_{α}(j).

There are now two subcases: Subcase 1: cf(α) = ω, and so f_{α}^{∗} = f_{α}. In this case we are done because f_{α}^{∗}(j) = f_{α}(j) and ⟨f_{β}^{∗}(j) : β ∈ lim(C) ∩ (α + 1)⟩ is strictly increasing. Subcase 2: cf(α) > ω. In this case ⟨f_{δ}(k) : δ ∈ lim(C) ∩ α⟩ is increasing for all k ≥ i_{γ}, and also cf(α) < cf(γ) < κ_{i_{γ}}, so we defined i_{α} ≤ i_{γ} and f_{α}^{∗}(k) = sup{f_{δ}(k) : δ ∈ lim(C) ∩ α} for all k ≥ i_{α}.

In particular we defined f_{α}^{∗}(j) = sup{f_{β}(j) : β ∈ lim(C) ∩ α}. So f_{α}^{∗}(j) ≤ f_{α}(j), and ⟨f_{β}^{∗}(j) : β ∈ lim(C) ∩ (α + 1)⟩ is strictly increasing because f_{β}^{∗} = f_{β} unboundedly often in lim(C) ∩ α.

When we reach γ the argument of Subcase 2 shows that f_{γ}^{∗}(j) = sup{f_{α}^{∗}(j) : α ∈ lim(C)}. This implies continuity because cf(γ) > cf(κ).

4. Reflection in [λ]^{ℵ_{0}} and almost disjoint sets

The following generalised form of stationarity is a key idea in modern set theory. This notion is due to Jech [19].

Definition 4.1. Let X be uncountable. S ⊆ [X]^{ℵ_{0}} is stationary iff for all F : ^{<ω}X ⟶ X there exists A ∈ S such that A ≠ ∅ and F “^{<ω}A ⊆ A.

Remark 4.2. This form of the definition was given by Kueker [27], and is equivalent to Jech’s. There is an extensive treatment of generalised stationarity in [14].

An analogue of Fodor’s lemma is true for this version of stationarity.

Fact 4.3. If S ⊆ [X]^{ℵ_{0}} is stationary and F : S ⟶ X is a function such that F (A) ∈ A for all A ∈ S, then F is constant on a stationary subset of S.

There is a notion of reflection appropriate for this notion of stationarity.

Definition 4.4. Let X be an uncountable transitive set.

1. A stationary set S ⊆ [X]^{ℵ_{0}} reflects to Y ⊆ X iff |Y | ⊆ Y and S ∩ [Y]^{ℵ_{0}} is stationary in [Y]^{ℵ_{0}}.

2. Let S ⊆ [X]^{ℵ_{0}} be stationary, then Refl^{∗}(S) holds iff every stationary T ⊆ S reflects to some Y ∈ [X]^{ℵ_{1}} with cf(ot(Y ∩ ON)) = ℵ_{1}.

Typically X is either a cardinal or a set of the form H_{θ}. The following facts are particularly significant for the purposes of this paper.

Fact 4.5 (Foreman, Magidor and Shelah [14]). If κ is a supercompact cardinal and G is generic over V for Coll(ℵ_{1}, < κ), then in V [G] the reflection principle Refl^{∗}([λ]^{ℵ_{0}}) holds for every λ with λ = cf(λ) ≥ ℵ_{2}.

Fact 4.6 (Foreman, Magidor and Shelah [14]). MM (Martin’s Maximum) implies that Refl^{∗}([λ]^{ℵ_{0}}) holds for every λ with λ = cf(λ) ≥ ℵ_{2}.

We will prove that □_{κ}^{∗} and VGS_{κ} each imply the failure of the reflection principle Refl^{∗}([κ^{+}]^{ℵ_{0}}). As in the last section the proof will involve some intermediate combinatorial principles. The following principle was studied by Shelah [42]. We have dubbed it ADS_{κ} where “ADS” stands for “Almost Disjoint Sets”.

Definition 4.7. ADS_{κ} holds iff there exists ⟨A_{α} : α < κ^{+}⟩ such that 1. A_{α} is unbounded in κ, ot(A_{α}) = cf(κ).

2. For all β < κ^{+} there exists g : β ⟶ κ such that the sequence ⟨A_{α} \ g(α) : α < β⟩ consists of pairwise disjoint sets.

The existence of such a sequence A⃗ is a kind of “incompactness” phenomenon in κ^{+}, of the sort studied in [35]. We digress briefly into the relationship between ADS_{κ} and the principles studied in [35]; recall that a transversal for a set of non-empty sets is a one-one choice function for that set.

Definition 4.8. NPT(κ, λ) holds iff there exists a family of sets B such that • |B| = κ.

• ∀B ∈ B |B| < λ.

• For all C ⊆ B, if |C| < κ then C has a transversal.

• B does not have a transversal.

It is clear that any sequence witnessing that ADS_{κ} is true will also witness that NPT(κ^{+}, cf(κ)^{+}) is true. In general the implication will not be reversible; Magidor and Shelah proved [35] that NPT(ℵ_{ω+1}, ℵ_{1}) always holds, while we will see in Theorem 8 that ADS_{ℵ_{ω}} can fail.

Shelah [42, page 440] proved that

Fact 4.9. If κ is regular, or is a singular cardinal such that □_{κ} holds, then ADS_{κ} holds.

Shelah showed that ADS_{κ} has some influence on the cardinal structure of extensions of the universe.

Fact 4.10 (Shelah [42]). If V ⊆ W are two models of set theory such that

V ⊧ “κ is a cardinal and ADS_{κ} holds”

and κ_{V}^{+} is a cardinal in W, then W ⊧ cf(κ) = cf(|κ|).

See [6] for a generalisation of this result.

Theorem 7. If VGS_{κ} or □_{κ}^{∗} holds then ADS_{κ} holds.

Proof. It will be convenient to introduce a concept intermediate between “very good scale” and “good scale”. Let us say that a scale (κ⃗, f⃗) is better iff it has the following property: for all α < κ^{+} with cf(α) > cf(κ) there is D club in α such that

1. ot(D) = cf(α).

2. For all β ∈ D there is i < cf(κ) such that f_{γ}(j) < f_{β}(j) for all j > i and all γ ∈ D ∩ β.

Clearly a very good scale is better and a better scale is good. We start with the easy remark that points of cofinality less than or equal to cf(κ) automatically enjoy the property in question.

Claim. If (κ⃗, f⃗) is any scale for κ, α < κ^{+} with cf(α) ≤ cf(κ), and D is any unbounded subset of α with order type cf(α), then for every β ∈ D there is i < cf(κ) such that f_{γ}(j) < f_{β}(j) for all j > i and all γ ∈ D ∩ β.

Proof. For all γ ∈ D∩β we may choose i(γ) such that ∀j ≥ i(γ) f_{γ}(j) < f_{β}(j). Let i = sup_{γ} i(γ). Since |D ∩ β| < cf(α) ≤ cf(κ), i < cf(κ).

Claim. If □_{κ}^{∗} holds then there is a better scale.

Proof. Let ⟨C_{α} : α < κ^{+}⟩ witness □_{κ}^{∗}. Let (κ⃗, g⃗) be a scale for κ.

We define a new scale (κ⃗, f⃗) by induction. Successor case: Let f_{α+1} = g_{β} for β so large that f_{α} <^{∗} g_{β}. Limit case: For each C ∈ C_{α} define f_{C} by setting f_{C}(i) = sup_{γ∈C} f_{γ}(i) if ot(C) < κ_{i}, and f_{C}(i) = 0 otherwise. Let f_{α} = g_{β} for β so large that

1. ∀C ∈ C_{α} f_{C} <^{∗} g_{β}.

2. ∀γ < α f_{γ} <^{∗} g_{β}.

It is clear that (κ⃗, f⃗) is a scale. Given α with cf(α) > cf(κ) fix some C ∈ C_{α}. Let C_{1} = lim(C). Now if β ∈ C_{1} then C ∩ β ∈ C_{β}, so f_{C∩β} <^{∗} f_{β}. For sufficiently large j we have ot(C ∩ β) < κ_{j} and so f_{C∩β}(j) = sup_{γ∈C∩β} f_{γ}(j). If we now choose D to be a club subset of C_{1} with order type cf(α), then D will witness that (κ⃗, f⃗) is better.

For the rest of the proof we fix a better scale (κ⃗, f⃗). We may assume without loss of generality that cf(κ) < κ_{0} and ∀α ∀i f_{α}(i) ≥ sup_{j<i} κ_{j}. We claim that ⟨range(f_{α}) : α < κ^{+}⟩ exemplifies ADS_{κ}.

We will establish the following fact for all β < κ^{+}.

(∗)_{β}: There exists G : β ⟶ cf(κ) such that if γ < δ < β and i > max{G(γ), G(δ)} then f_{γ}(i) < f_{δ}(i).

To see that this gives the desired property, fix β and let G witness (∗)_{β}. For α < β let B_{α} be the final segment of range(f_{α}) given by { f_{α}(i) : i ≥ G(α) }, then we claim that the sets B_{α} are mutually disjoint.

Suppose that α_{0} < α_{1} < β and x ∈ B_{α_{0}} ∩ B_{α_{1}}, so that x = f_{α_{0}}(i) = f_{α_{1}}(j) for i ≥ G(α_{0}), j ≥ G(α_{1}). i must equal j because otherwise f_{α_{0}}(i) and f_{α_{1}}(j) would lie in disjoint intervals. Therefore i ≥ max{G(α_{0}), G(α_{1})}, but then f_{α_{0}}(i) < f_{α_{1}}(i). So the B_{α} are disjoint, and we have shown that ⟨range(f_{α}) : α < κ^{+}⟩ witnesses ADS_{κ}.

We will prove (∗)_{β} by induction on β, where the case β = 0 is trivial.

Case 1: β = γ + 1. Let G : γ ⟶ cf(κ) witness (∗)_{γ}. Choose H : β ⟶ cf(κ) such that H(γ) = 0, ∀δ < γ G(δ) ≤ H(δ), and ∀δ < γ ∀i > H(δ) f_{δ}(i) < f_{γ}(i). It is easy to check that H witnesses (∗)_{β}.

Case 2: lim(β).

Since the scale (κ⃗, f⃗) is better, we may fix C club in β such that 1. ot(C) = cf(β).

2. For all γ ∈ C there is i < cf(κ) such that f_{δ}(j) < f_{γ}(j) for all δ ∈ C ∩ γ and all j > i.

We enumerate C in increasing order as ⟨β_{ζ} : ζ < cf(β)⟩, and choose ⟨G_{ζ} : ζ < cf(β)⟩ a sequence of functions such that G_{ζ} witnesses (∗)_{β_{ζ}}. We define H(γ) for β_{ζ} ≤ γ < β_{ζ+1} so that

1. H(γ) ≥ G_{ζ+1}(γ).

2. ∀i > H(γ) ∀η < ζ f_{β_{η}}(i) < f_{β_{ζ}}(i).

3. ∀i > H(γ) f_{β_{ζ}}(i) ≤ f_{γ}(i) < f_{β_{ζ+1}}(i).

Claim. H witnesses (∗)_{β}.

Proof. Let γ < δ < β. There are 2 subcases. Subcase a: ∃η β_{η} ≤ γ < δ < β_{η+1}. In this case we know by clause 1 in the definition of H that H(γ) ≥ G_{η+1}(γ) and H(δ) ≥ G_{η+1}(δ). G_{η+1} witnesses (∗)_{β_{η+1}}, so ∀i > max{H(γ), H(δ)} f_{γ}(i) < f_{δ}(i) and we are done. Subcase b: ∃η ∃θ β_{η} ≤ γ < β_{η+1} ≤ β_{θ} ≤ δ < β_{θ+1}. In this case let i > max{H(γ), H(δ)}, then we see that

f_{γ}(i) < f_{β_{η+1}}(i) ≤ f_{β_{θ}}(i) ≤ f_{δ}(i).

Here the inequalities follow from clause 3 in the definition of H(γ), clause 2 in the definition of H(δ) and clause 3 in the definition of H(δ) respectively.

This concludes the proof of Theorem 7.

Finally we return to the problem of stationary reflection.

Theorem 8. Let κ be singular of cofinality ω. If ADS_{κ} holds then Refl^{∗}([κ^{+}]^{ℵ_{0}}) fails.

Proof. Let ⟨A_{α} : α < κ^{+}⟩ be a sequence witnessing ADS_{κ} and define S = { x ∈ [κ^{+}]^{ℵ_{0}} : A_{sup(x)} ⊆ x }.

Claim. S is stationary in [κ^{+}]^{ℵ_{0}}.

Proof. Let F : [κ^{+}]^{<ℵ_{0}} ⟶ κ^{+}, and let α > κ be such that cf(α) = ω and F “[α]^{<ℵ_{0}} ⊆ α. Now choose x to be a countable cofinal subset of α such that A_{α} ⊆ x and x is closed under F, then sup(x) = α so that x ∈ S.

Applying the hypothesis that Refl^{∗}([κ^{+}]^{ℵ_{0}}) holds, let Y ⊆ κ^{+} be such that ℵ_{1} ⊆ Y, cf(ot(Y)) = |Y | = ℵ_{1}, and S ∩[Y]^{ℵ_{0}} is stationary in [Y]^{ℵ_{0}}. Notice that for each X ∈ [Y]^{ℵ_{0}} we have sup(X) < sup(Y).

By the hypothesis that A⃗ witnesses ADS_{κ}, we fix g : sup(Y) ⟶ κ such that ⟨A_{β} \ g(β) : β < sup(Y)⟩ is a sequence of mutually disjoint sets. For each X ∈ S ∩ [Y]^{ℵ_{0}} let h(X) = min(A_{sup(X)} \ g(sup(X))), and observe that ∀X ∈ S ∩ [Y]^{ℵ_{0}} A_{sup(X)} ⊆ X and so ∀X ∈ S h(X) ∈ X.

By Fodor’s theorem there exists T ⊆ S with T stationary in [Y]^{ℵ_{0}} such that h is constant on T. It is easy to see that if X, X′ ∈ T then sup(X) = sup(X′), because otherwise we will have h(X) ≠ h(X′). Suppose sup(X) = β for all X ∈ T.

We know cf(β) = ω and cf(ot(Y)) = ℵ_{1}, so β < sup(Y). It is easy to see that { X ∈ [Y]^{ℵ_{0}} : sup(X) = β } is non-stationary, contradicting the stationarity of T.

Remark 4.11. We have not used the full power of ADS_{κ}, since we only needed to be able to make any subsequence of size ℵ_{1} disjoint in order to make the proof work.

Remark 4.12. The last result uses ideas of Shelah [43]. See the remarks after Theorem 2.12 in Foreman and Magidor’s paper [12].

5. Improved squares

As we remarked after Definition 2.3 the principle □_{κ}^{∗} implies there is a □_{κ}^{∗}-sequence C⃗ with the additional property that each set C_{α} contains a set of order type cf(α). Here we consider the principle obtained by adding this property to the list of properties demanded of a □_{κ,λ} sequence.

Definition 5.1. A □_{κ,λ}^{imp}-sequence (improved □_{κ,λ}-sequence) is a sequence ⟨C_{α} : α < κ^{+}, lim(α)⟩ such that

1. C_{α} ⊆ P(α), 1 ≤ |C_{α}| ≤ λ, and C_{α} is a set of closed and unbounded subsets of α with order types less than or equal to κ.

2. If cf(α) < κ then ∀C ∈ C_{α} ot(C) < κ.

3. ∃C ∈ C_{α} ot(C) = cf(α).

4. ∀C ∈ C_{α} ∀β ∈ lim(C) C ∩ β ∈ C_{β}.

Remark 5.2. If □_{κ,λ}^{imp} holds then □_{μ,λ}^{imp} holds for every μ with λ ≤ μ < κ.

The proof is immediate. This shows that in general □_{κ} does not imply □_{κ,λ}^{imp} for any λ < κ. For example Mitchell [37] starts with a Mahlo cardinal and builds a model where there is no special ℵ_{2}-Aronszajn tree, which is to say that □_{ℵ_{1},ℵ_{1}} fails; it is possible (see Section 6) to force □_{ℵ_{2}} without adding any ℵ_{2}-sequence of ordinals, and doing this forcing construction over Mitchell’s model will give a model of □_{ℵ_{2}} + ¬□_{ℵ_{2},ℵ_{1}}^{imp}.

Improved squares have slightly more influence on stationary reflection than their unimproved cousins. The next result should be contrasted with Theorem 18.

Theorem 9. Let □_{κ,λ}^{imp} hold. Let λ < cf(μ) ≤ μ ≤ κ and let S ⊆ κ^{+} be stationary. Then there exists a stationary T ⊆ S and a β < μ such that T does not reflect at any point α with β < cf(α) ≤ μ.

Proof. Let ⟨C_{α} : α < κ^{+}, lim(α)⟩ witness □_{κ,λ}^{imp}. Define F : κ^{+} ⟶ μ by setting

F : α ⟼ sup({ ot(C) : C ∈ C_{α}, ot(C) < μ }).

Find T ⊆ S a stationary set such that F ↾ T is constant with value β < μ. Now if β < cf(α) ≤ μ, there is C ∈ C_{α} such that ot(C) = cf(α). For each γ ∈ lim(C) we have C ∩ γ ∈ C_{γ} and ot(C ∩ γ) < μ, so that F (γ) ≥ ot(C ∩ γ) for each such γ. It follows that F (γ) > β for all sufficiently large γ ∈ lim(C), so that T cannot be stationary in α.

Next we ask how small λ can be in an improved square.

Lemma 5.3. Let ⟨C_{α} : α < μ⟩ be a sequence such that C_{α} is club in α and ot(C_{α}) = cf(α). If α < μ is such that cf(α) = ℵ_{n} then for every i < n there is β < α such that cf(β) = ℵ_{i} and

|{ C_{γ} ∩ β : γ ≥ β, β ∈ lim(C_{γ}) }| ≥ n − i.

Proof. Let γ_{0} = α. Choose inductively γ_{1}, …, γ_{n} such that 1. γ_{j} ∈ lim(C_{γ_{i}}) for all i < j.

2. cf(γ_{j}) = ℵ_{n−j}.

3. ot(C_{γ_{i}} ∩ γ_{j}) > ℵ_{n−i−1} for all i < j.

For each j and i < j we have arranged that

ℵ_{n−i−1} < ot(C_{γ_{i}} ∩ γ_{j}) < ot(C_{γ_{i}}) = cf(γ_{i}) = ℵ_{n−i},

so that the sets in the sequence ⟨C_{γ_{i}} ∩ γ_{j} : i < j⟩ are all distinct. So γ_{n−i} is the desired point of cofinality ℵ_{i}.

Corollary 5.4. If n < ω then □_{ℵ_{n},j}^{imp} cannot hold if j ≤ n. For κ ≥ ℵ_{ω}, □_{κ,j}^{imp} is false for every j < ω.

This result is fairly sharp, because □_{ρ,<ω}^{imp} is consistent for every ρ. To see this we recall Jensen’s Global □ principle.

Definition 5.5. A Global □-sequence is a sequence ⟨C_{α} : cf(α) < α⟩ such that

1. C_{α} is club in α, with ot(C_{α}) < α.

2. If β ∈ lim(C_{α}) then cf(β) < β and C_{β} = C_{α} ∩ β.

Jensen [25] showed that if V = L then there is a Global □-sequence.

Theorem 10. If there is a Global □-sequence then □_{ρ,<ω}^{imp} holds for every cardinal ρ.

Proof. Let C⃗ be a Global □-sequence. First we derive a form of “Global improved square”, then we derive □_{ρ,<ω}^{imp}-sequences from it.

Claim. There is a sequence ⟨D_{α} : cf(α) < α⟩ such that for all α 1. 1 ≤ |D_{α}| < ω.

2. ∀D ∈ D_{α} ot(D) < α.

3. For all D ∈ D_{α} and all β ∈ lim(D), cf(β) < β and D ∩ β ∈ D_{β}.

4. There exists D ∈ D_{α} with ot(D) = cf(α).

Proof. For each α with cf(α) < α we will define by induction on α a sequence of sets ⟨D_{α}^{i} : i ≤ i(α)⟩ where i(α) < ω.

We start by defining D_{α}^{0} = C_{α}. Let β =_{def} ot(C_{α}). If β = cf(α) then we set i(α) = 0 and stop. Otherwise we know that cf(β) = cf(α) < β, so that we have already defined ⟨D_{β}^{j} : j ≤ i(β)⟩; in this case we define i(α) = i(β) + 1 and D_{α}^{j+1} = { γ ∈ C_{α} : ot(C_{α} ∩ γ) ∈ D_{β}^{j} } for each j ≤ i(β); in other words we copy the sequence at β from β up to α, using C_{α}.

It is easy to see by induction that ⟨D_{α}^{i} : i ≤ i(α)⟩ is a decreasing sequence of club subsets of C_{α}, and that ot(D_{α}^{i(α)}) = cf(α).

Subclaim. If j ≤ i(α) and γ ∈ lim(D_{α}^{j}) then j ≤ i(γ) and D_{α}^{j} ∩ γ = D_{γ}^{j}.

Proof. We prove this by induction on j, for all α simultaneously. The claim is immediate for j = 0 by the coherence properties of the □- sequence C⃗.

Now let j = i + 1. Let β = ot(C_{α}), then we know since i(α) > 0 that • cf(α) = cf(β) < β.

• i ≤ i(β).

• D_{α}^{j} = { δ ∈ C_{α} : ot(C_{α} ∩ δ) ∈ D_{β}^{i} }.

Since D_{α}^{j} ⊆ C_{α} we know that γ ∈ lim(C_{α}), so that cf(γ) < γ and C_{γ} = C_{α} ∩ γ. Let ρ =_{def} ot(C_{γ}).

Since γ is a limit point of D_{α}^{j}, ρ = ot(C_{α} ∩ γ) ∈ lim(D_{β}^{i}). By induction

• i ≤ i(ρ).

• D_{ρ}^{i} = D_{β}^{i} ∩ ρ. Therefore

D_{α}^{j} ∩ γ = { δ ∈ C_{α} ∩ γ : ot(C_{α} ∩ δ) ∈ D_{β}^{i} ∩ ρ }
= { δ ∈ C_{γ} : ot(C_{γ} ∩ δ) ∈ D_{ρ}^{i} }

Now consider the construction of the D_{γ}^{i}’s. cf(γ) = cf(ρ) < ρ = ot(C_{γ}), and i ≤ i(ρ), so that j ≤ i(γ) and D_{γ}^{j} = { δ ∈ C_{γ} : ot(C_{γ} ∩ δ) ∈ D_{ρ}^{i} }.

We have shown that D_{γ}^{j} = D_{α}^{j}∩γ, concluding the proof of the subclaim.

We can now set D_{α} = { D_{α}^{i} : i ≤ i(α) } to conclude the proof of the claim.

Claim. For every cardinal ρ there exists a □_{ρ,<ω}^{imp}-sequence.

Proof. Observe that for every α ∈ (ρ, ρ^{+}) we have cf(α) ≤ ρ < α, so that the family D_{α} defined in the previous claim exists. For each D ⊆ ON define D^{∗} = { γ : ρ + γ ∈ D }, and let

E_{α} = { D^{∗} : D ∈ D_{ρ+α}, ot(D) ≤ ρ }.

It is routine to check that ⟨E_{α} : α < ρ^{+}⟩ is a □_{ρ,<ω}^{imp}-sequence.

This concludes the proof of Theorem 10.

We can also prove results with a more local flavour.

Theorem 11. Suppose that □_{ℵ_{ω},λ} holds, and that □_{ℵ_{n}} holds for all n with 1 ≤ n < ω. Then □_{ℵ_{ω},λ+ℵ_{0}}^{imp} holds.

Proof. Let ⟨C_{α} : α < ℵ_{ω+1}⟩ be a □_{ℵ_{ω},λ}-sequence, and for each n < ω let ⟨C_{β}^{n} : β < ℵ_{n+1}⟩ be a □_{ℵ_{n}}-sequence. We now define a sequence of approximations to an improved square sequence.

Let C_{α}^{0} = C_{α}. Then for each m < ω let

C_{α}^{m+1} = C_{α}^{m} ∪ D_{α}^{m},

where D_{α}^{m} is

{ { γ ∈ C : ot(C ∩ γ) ∈ C_{δ}^{n} } : C ∈ C_{α}^{m}, n < ω, ot(C) = δ < ℵ_{n+1} }.

Finally let C_{α}^{ω} = ⋃_{n<ω} C_{α}^{n}.

Clearly |C_{α}^{ω}| ≤ λ + ℵ_{0}. We claim that ⟨C_{α}^{ω} : α < ℵ_{ω+1}⟩ is a □_{ℵ_{ω},λ+ℵ_{0}}^{imp}- sequence.

First we check that ∀α ∃C ∈ C_{α}^{ω} ot(C) = cf(α). To see this observe that if C ∈ C_{α}^{m} and ot(C) ≠ cf(α) then ot(C) is not a regular cardinal, so for some n we have ot(C) = δ ∈ (ℵ_{n}, ℵ_{n+1}); then if C^{∗} = { γ ∈ C : ot(C ∩ γ) ∈ C_{δ}^{n} } we have ot(C^{∗}) = ot(C_{δ}^{n}) ≤ ℵ_{n} < ot(C) and C^{∗} ∈ C_{α}^{m+1}. It follows that any element of C_{α}^{ω} whose order type is minimal has order type cf(α).

To finish we prove ∀C ∈ C_{α}^{m} ∀γ ∈ lim(C) C ∩ γ ∈ C_{γ}^{m}, by induction on m. If m = 0 this is immediate. Suppose we have it for m. Let C ∈ D_{α}^{m}. Then C = { γ ∈ D : ot(D ∩ γ) ∈ C_{δ}^{n} }, where D ∈ C_{α}^{m} and ot(D) = δ < ℵ_{n+1}. Fix γ ∈ lim(C) and let ρ = ot(D ∩ γ), then ρ ∈ lim(C_{δ}^{n}) and therefore C_{ρ}^{n} = C_{δ}^{n} ∩ ρ.

Now by induction D ∩ γ ∈ C_{γ}^{m}, and we also see that ot(D ∩ γ) < δ < ℵ_{n+1}.

C ∩ γ = { σ ∈ D : ot(D ∩ σ) ∈ C_{δ}^{n} ∩ ρ }
= { σ ∈ D ∩ γ : ot(D ∩ σ) ∈ C_{ρ}^{n} },

so C ∩ γ ∈ C_{γ}^{m+1} because it is defined from D ∩ γ ∈ C_{γ}^{m}.

Remark 5.6. Similar methods show that if n < ω and □_{ℵ_{m}} holds for m ≤ n, then □_{ℵ_{n},n+1}^{imp} holds.

Burke and Kanamori observed [39] that if κ is λ^{+}-strongly compact then □_{λ,<cf(λ)} fails. A similar result is true for the “improved square” principles □_{λ,μ}^{imp}. Both proofs are based on the following well-known fact, which is due to Solovay and is the key to the proof of Fact 2.9.

Fact 5.7. Let γ be regular and uncountable. Let j : V ⟶ M be an embedding such that crit(j) = κ ≤ γ and let ρ =_{def} sup(j“γ). If C is club in ρ, then { α < γ : j(α) ∈ C } is < κ-club in γ.

Theorem 12. Let κ be λ^{+}-strongly compact, where λ > κ. Then □_{λ,<λ}^{imp} fails.

Proof. Suppose that C⃗ witnesses □_{λ,<λ}^{imp}, and choose C_{α} ∈ C_{α} such that ot(C_{α}) = cf(α) for each limit α < λ^{+}. Let j : V ⟶ M be such that crit(j) = κ, and if ρ =_{def} sup(j“λ^{+}) then ρ < j(λ^{+}) and M ⊧ cf(ρ) < j(κ). Let D⃗ = j(C⃗), D = D_{ρ}, and C = { α < λ^{+} : j(α) ∈ D_{ρ} }.

Then C is < κ-club. Let β be the λ + ω^{th} element of C, so that β ∈ lim(C) and |C ∩ β| = λ. Since cf(β) = ω, j is continuous at β and so j(β) ∈ lim(D).

For each α ∈ C ∩ β we have j(α) ∈ D ∩ j(β), so by elementarity there is σ < λ^{+} such that β ∈ lim(C_{σ}), cf(σ) < κ, and α ∈ C_{σ} ∩ β. There are |C_{β}| possibilities for C_{σ} ∩β and each has cardinality less than κ, so that |C ∩ β| ≤ |C_{β}|.κ < λ. This is a contradiction.

6. Preliminaries on forcing

We make several uses of the following theorem.

Fact 6.1 (Laver [30]). Let κ be supercompact, let α < κ. Then there is an α-directed closed and κ-c.c. poset ℙ such that |ℙ| = κ, κ is supercompact in V^{ℙ}, and

V^{ℙ} ⊧ “κ-directed closed forcing preserves the supercompactness of κ”.

A supercompact cardinal κ whose supercompactness is preserved by any κ-directed closed forcing is said to be “Laver indestructible”.

We collect some information related to forcing square sequences and reflection properties, which we will use repeatedly in the sections that follow.

6.1. Forcing square sequences. Let κ be any cardinal, and let 1 ≤ λ ≤ κ. We shall describe a forcing poset ℙ(κ, λ) which adds a □_{κ,λ}- sequence and preserves all cardinals up to and including κ^{+}. The conditions in ℙ(κ, λ) are just proper initial segments of the desired sequence. More precisely

Definition 6.2. p ∈ ℙ(κ, λ) iff

1. p is a function with dom(p) = { β ≤ α : lim(β) } for some limit ordinal α < κ^{+}.

2. For all β ∈ dom(p), 1 ≤ |p(β)| ≤ λ.

3. For all β ∈ dom(p) and C ∈ p(β), C is club in β and ot(C) ≤ κ.

4. If cf(β) < κ then ot(C) < κ for all C ∈ p(β).

5. For all β ∈ dom(p), C ∈ p(β) and γ ∈ lim(C), C ∩ γ ∈ p(γ).

If p, q ∈ ℙ(κ, λ) then q ≤ p iff dom(p) ⊆ dom(q) and q ↾ dom(p) = p.

Remark 6.3. The idea of forcing square sequences by initial segments is due to Jensen.

To show that ℙ(κ, λ) is a reasonable forcing poset, we use the idea of strategic closure.

Remark 6.4. Strategic closure properties of posets and Boolean algebras were introduced by Jech [20, 21] and have also been studied by Foreman [10] and Gray [16].

Definition 6.5. Let ℙ be a partial ordering, let α be an ordinal. Then the game G_{α}^{II}(ℙ) is played as follows: players I and II take turns to write down the entries of a decreasing sequence ⟨p_{β} : β < α⟩ of conditions in ℙ, with I playing at odd stages and II playing at even stages (including all limit stages). If play reaches a stage where player II cannot move then player I wins, otherwise player II wins.

The poset ℙ is α-strategically closed iff player II has a winning strategy for the game G_{α}^{II}(ℙ). ℙ is < λ-strategically closed iff player II wins G_{α}^{II}(ℙ) for all α < λ.

The following fact is well-known.

Fact 6.6. Let κ be a cardinal. If ℙ is (κ + 1)-strategically closed then forcing with ℙ adds no κ-sequences of ordinals.

Lemma 6.7. The poset ℙ(κ, λ) is (κ + 1)-strategically closed.

Proof. We describe a winning strategy for player II in the appropriate game. The only subtle point is that in case κ is singular we cannot let player II play a club of order type κ at move κ; we describe a strategy that works for κ regular or singular.

Let λ = cf(κ) and let C ⊆ κ be a club set of order type λ. We assume that at stage α the players have produced a sequence of conditions ⟨p_{β} : β < α⟩, where max dom(p_{β}) = γ_{β}.

Case 1: α is an even successor ordinal, say α = 2β + 2. In this case player II plays a condition p_{2β+2} ≤ p_{2β+1} such that γ_{2β+2} = γ_{2β+1} + ω and p(γ_{2β+2}) = {{ γ_{2β+1} + n : n < ω }}

Case 2: α is a limit ordinal. We define γ_{α} = sup_{β<α} γ_{β}.

Subcase 2a: sup(C ∩ α) = α. In this case we define p_{α} by setting p_{α} ↾ γ_{α} = ⋃_{β<α} p_{β} and p_{α}(γ_{α}) = { γ_{β} : β ∈ C ∩ α }.

Subcase 2b: sup(C ∩ α) < α. In this case we define p_{α} by setting p_{α} ↾ γ_{α} = ⋃_{β<α} p_{β} and p_{α}(γ_{α}) = { γ_{β} : sup(C ∩ α) < β < α }.

It is routine to verify that this is a winning strategy for player II.

It follows that forcing with ℙ(κ, λ) adds no κ-sequences of ordinals, and so preserves all cardinals and cofinalities up to and including κ^{+}. It is routine to check that if G is ℙ(κ, λ)-generic then ⋃ G is a □_{κ,λ}- sequence in V [G]; the proof that { p : lh(p) > β } is dense for each β < κ^{+} is similar to the proof of Lemma 6.7.

6.2. Threading a generic square sequence. Let C⃗ = ⟨C_{α} : α < κ^{+}⟩ be a □_{κ,λ}-sequence in some model of set theory V. Let W ⊇ V be a larger model of set theory. We say that C ∈ W threads the sequence C⃗ if C is club in κ^{+} and ∀α ∈ lim(C) C ∩ α ∈ C_{α}. It is clear that no such C can exist in V (or any extension of V in which κ^{+} is regular) because every initial segment of C can have order type at most κ. However it is sometimes possible to add such a club C by reasonably distributive forcing; in particular this will be possible if C⃗ was added by the forcing ℙ(κ, λ) given in Definition 6.2.

Definition 6.8. Let C⃗ = ⟨C_{α} : α < κ^{+}⟩ be a □_{κ,λ}-sequence. Let γ = cf(γ) ≤ κ. Then c ∈ 𝕋_{γ}(C⃗) iff

1. c is a closed bounded subset of κ^{+}.

2. ot(c) < γ.

3. ∀β ∈ lim(c) c ∩ β ∈ C_{β}.

If c, d ∈ 𝕋_{γ}(C⃗) then c ≤ d iff c ∩ (max(d) + 1) = d.

It is clear that 𝕋_{γ}(C⃗) will add C which is club and threads C⃗. In general it is not clear what the order type of C will be, and how much damage the forcing will do.

Lemma 6.9. Let κ be a cardinal, let 1 ≤ λ ≤ κ, γ = cf(γ) ≤ κ. Let ℙ = ℙ(κ, λ), and 𝕋 = 𝕋_{γ}(C⃗)_{V^{ℙ}} where C⃗ is the generic □_{κ,λ}-sequence added by ℙ. Then

1. ℙ ∗ 𝕋 has a γ-closed dense subset.

2. 𝕋 adds a generic club which threads C⃗ and has order type γ. κ_{V}^{+} has cofinality γ in V^{ℙ∗𝕋}.

Proof. Observe that 𝕋 ⊆ V, since forcing with ℙ adds no κ-sequences of ordinals and conditions in 𝕋 are bounded subsets of κ^{+}. We take each claim in turn.

1. We define D to be the set of conditions (p, ċ) ∈ ℙ ∗ 𝕋 such that p decides ċ (say p ⊩ ċ = č for some c ∈ V) and max dom(p) = max(c). We claim that D is dense in ℙ ∗ 𝕋 and that it is γ-closed.

For density, let (q, ḋ) be any condition in ℙ ∗ 𝕋. We refine q to p such that p decides ḋ (say p ⊩ ḋ = ď) and γ =_{def} max dom(p) > max(d). Now the condition (p, d ∪ {γ}) refines (q, ḋ) and lies in D.

For closure, fix a decreasing sequence ⟨(p_{δ}, c_{δ}) : δ < β⟩ of conditions in D for some β < γ. Let α_{δ} = max dom(p_{δ}) = max(c_{δ}) for each δ, and define α = sup_{δ<β} α_{δ}. Now define c = ⋃_{δ<β} c_{δ} ∪ {α}, and let p be such that max dom(p) = α, p ↾ α = ⋃_{δ<β} p_{δ}, p(α) = {c ∩ α}. It is routine to check that (p, c) is a condition and that (p, c) ∈ D.

2. An easy density argument shows that 𝕋 adds a club C which threads C⃗. Since each initial segment of C is determined by some condition, ot(C) ≤ γ. On the other hand ℙ ∗ 𝕋 has a γ-closed dense subset, so that

V^{ℙ∗𝕋} ⊧ “γ is regular and κ_{V}^{+} has cofinality at least γ”

and thus V^{ℙ∗𝕋} ⊧ cf(κ_{V}^{+}) = γ = ot(C).

It follows from this that 𝕋 is (γ, ∞)-distributive in V^{ℙ(κ,λ)}.

6.3. Stationary reflection in ℵ_{ω+1}. Recall that “Refl(μ, S, ν)” is the principle stating that any μ stationary subsets of S reflect simultaneously to a point of cofinality ν. Magidor [33] showed that it is possible to start with ω supercompact cardinals and to make a model in which Refl(ℵ_{n+1}, ℵ_{ω+1} ∩ cof(≤ ℵ_{n}), ℵ_{n+1}) holds for each n < ω. In subsequent sections we will be building some related models, so we summarise the key ideas of Magidor’s proof.

We start by fixing ⟨κ_{n} : n < ω⟩ an increasing ω-sequence of supercompact cardinals, and define an iteration ⟨ℙ_{n}, ℚ̇_{n} : n < ω⟩ where ℚ_{0} = Coll(ω, < κ_{0}) and ℚ_{n} = Coll(κ_{n−1}, < κ_{n})_{V^{ℙn}} for n > 0. We let ℙ_{ω} be the inverse limit of this iteration. The final model is V^{ℙ_{ω}}.

The following fact captures some of the key features of V^{ℙ_{ω}}.

Fact 6.10. Let κ =_{def} sup_{n} κ_{n}, and V_{1} =_{def} V^{ℙ_{ω}}.

1. V_{1} ⊧ “κ_{n} = ℵ_{n+1}, κ = ℵ_{ω}, and κ_{V}^{+} = ℵ_{ω+1}”.

2. Let ℝ ∈ V_{1} be a forcing where V_{1} ⊧ “ℝ is ℵ_{n+1}-directed closed”. Let G be ℝ-generic over V_{1} and let λ = cf(λ) > κ_{n}. Then there is ℝ̄ ∈ V_{1}[G] such that

(a) V_{1}[G] ⊧ “ℝ̄ is ℵ_{n}-closed”.

(b) If H is ℝ̄-generic over V_{1}[G] then in V_{1}[G ∗ H] there is an elementary embedding j : V_{1}[G] ⟶ M ⊆ V_{1}[G ∗ H] such that

• crit(j) = κ_{n} = ℵ_{n+1}^{V_{1}}.
• j ↾ λ ∈ M.
• j(κ_{n}) > λ.
• sup(j“λ) < j(λ).
• M ⊧ |κ_{n}| = |λ| = cf(λ) = ℵ_{n} = ℵ_{n}^{V_{1}}.

In the more picturesque language of [11], ℵ_{n+1} is “indestructibly generically supercompact”.

One of the main points in the proof of Fact 6.10 is

Fact 6.11. Let ρ be regular, let ρ < λ < μ, and let

V^{Coll(ρ,<λ)} ⊧ “ℙ is a ρ-closed poset with |ℙ| < μ”.

Let i be the natural complete embedding of Coll(ρ, < λ) into Coll(ρ, < μ). Then i can be extended to a complete embedding j of Coll(ρ, < λ)∗ℙ into Coll(ρ, < μ) in such a way that the quotient forcing Coll(ρ, < μ)/j“[Coll(ρ, < λ) ∗ ℙ] is ρ-closed.

Let us now try to check that V_{1} has the desired stationary reflection property. For simplicity we begin by considering only one stationary set. Let

V_{1} ⊧ “S is a stationary subset of ℵ_{ω+1} ∩ cof(≤ ℵ_{n})”

for some n < ω. Let λ = κ^{+} and apply Lemma 6.10 with ℝ = 0. This gives a map j : V_{1} ⟶ M ⊆ V ℝ̄ ; 1 where

• j ↾ λ ∈ M.

• crit(j) = κ_{n+1} = ℵ_{n+2}.

• M ⊧ cf(λ) = ℵ_{n+1} = ℵ_{n+1}^{V_{1}}.

• V_{1} ⊧ “ℝ̄ is ℵ_{n+1}-closed”.

• sup(j“λ) < j(λ).

Let μ = sup j“λ. It is enough to prove that V ℝ̄ ; 1 ⊧ “S is stationary”; for since j is continuous at points of cofinality less than κ_{n+1} and M ⊆ V_{1}^{ℝ}, it will follow that M ⊧ “j(S) ∩ μ is stationary in μ”. We can then conclude by the elementarity of j that in V_{1} the stationary set S reflects to some point of cofinality ℵ_{n+1}.

Unfortunately it is not true in general [41] that ℵ_{n+1}-closed forcing preserves the stationarity of stationary subsets of ℵ_{ω+1} ∩ cof(≤ ℵ_{n}). We collect some information about preservation of stationarity by sufficiently closed forcing, and then indicate how to finish the argument that Refl(ℵ_{n+1}, ℵ_{ω+1} ∩ cof(≤ ℵ_{n}), ℵ_{n+1}) holds.

Lemma 6.12. Let γ = cf(γ) > ω and let S ⊆ γ be stationary. Then the stationarity of S is preserved by γ-closed forcing.

Proof. Let ℙ be γ-closed and let p ⊩ “Ċ is club in γ”. Define sequences ⟨p_{α} : α < γ⟩ and ⟨β_{α} : α < γ⟩ such that

1. p_{0} ≤ p, and ⟨p_{α} : α < γ⟩ is a decreasing sequence of elements of ℙ.

2. ⟨β_{α} : α < γ⟩ is an increasing and continuous sequence of elements of γ.

3. p_{α} ⊩ β_{α} ∈ Ċ. Since ⟨β_{α} : α < γ⟩ enumerates a club subset of γ which lies in V, there exists α such that β_{α} ∈ S. So p_{α} is a refinement of p which forces Ċ ∩ Š ≠ ∅.

To get an appropriate generalisation of the well-known fact that countably closed forcing preserves the stationarity of a stationary set of cofinality ω ordinals, we introduce yet another variant of □_{κ}. “AP” stands for “APproachability”.

Definition 6.13 (Shelah). ⟨C_{α} : α < κ^{+}⟩ is an AP_{κ}-sequence iff 1. If lim(α) then C_{α} is club in α, ot(C_{α}) = cf(α).

2. For a club set of α < κ^{+}, ∀β < α ∃γ < α C_{α} ∩ β = C_{γ}.

As usual we say that “AP_{κ} holds” iff there is an AP_{κ}-sequence. It is not hard to see that □_{κ}^{∗} ⟹ AP_{κ}. AP_{κ} is known [13] to be substantially weaker than □_{κ}^{∗}, and by Remark 6.15 AP_{κ} says essentially nothing about the existence of non-reflecting stationary sets. On the other hand AP_{κ} does have a strong influence on scales, since for κ singular AP_{κ} implies that every scale for κ is good [13].

Lemma 6.14 (Shelah). Let V ⊧ AP_{κ}. Let S ⊆ κ^{+} ∩ cof(< μ) be stationary, and let ℚ be μ-closed. Then S is stationary in V^{ℚ}.

Proof. Fix C⃗ witnessing AP_{κ}. Suppose for a contradiction that Ḋ names a club and q ⊩ Ḋ ∩ S = ∅. Fix some large regular H_{θ} and a wellordering <_{θ} of H_{θ}, and then let N ≺ (H_{θ}, ∈, <_{θ}) be such that

• κ ⊆ N, δ =_{def} N ∩ κ^{+} ∈ S, |N| = κ.

• q, ℚ, S, Ḋ, C⃗ ∈ N.

• ∀β < δ ∃γ < δ C_{δ} ∩ β = C_{γ}.

Notice that all initial segments of C_{δ} are in N. Enumerate C_{δ} in increasing order as ⟨δ_{i} : i < cf(δ)⟩. Now build ⟨q_{i} : i < cf(δ)⟩ as follows.

• q_{i+1} is the <_{θ}-least condition such that q_{i+1} ≤ q_{i} and there is β greater than δ_{i} such that q_{i+1} ⊩ β ∈ Ḋ.

• For j limit, q_{j} is the <_{θ}-least lower bound for ⟨q_{i} : i < j⟩.

It is clear that ⟨q_{i} : i < j⟩ is definable from the parameters {C_{δ} ∩ δ_{j}, q, Ḋ} which are all in N. So ⟨q_{i} : i < j⟩ ∈ N, as is each condition q_{i}. If we now let q be a lower bound for ⟨q_{i} : i < cf(δ)⟩ then q forces δ ∈ Ḋ ∩ S, contradiction!

The last key point in the argument from [33] is to show that AP_{κ} holds in V^{ℙ_{ω}}. We refer the reader to [33] for this proof; the key points are that for each n ≥ 0

• The cardinal ℵ_{n+1} started off as a supercompact cardinal κ_{n} in V.

• The forcing ℙ_{ω} can be factored as a κ_{n}-c.c. forcing followed by a κ_{n}-closed forcing.

Granted that AP_{ℵ_{ω}} holds, we can argue as outlined above that S reflects to a point of cofinality ℵ_{n+1}.

We still owe a proof that ℵ_{n+1} sets reflect simultaneously; fix an ℵ_{n+1}- sequence ⟨S_{i} : i < ℵ_{n+1}⟩ of stationary subsets of ℵ_{ω+1} ∩ cof(≤ ℵ_{n}), and construct j : V_{1} ⟶ M ⊆ V ℝ̄ ; 1 as before. Since crit(j) = ℵ_{n+2}^{V_{1}} the same argument as before shows that for every i < ℵ_{n+1} = j(ℵ_{n+1}) the set j(S_{i}) ∩ μ is stationary, so by elementarity there is a point of cofinality ℵ_{n+1} at which every set S_{i} reflects.

Remark 6.15. It is also possible to show that in V^{ℙ_{ω}} the principle Refl^{∗}([ℵ_{ω+1}]^{ℵ_{0}}) holds. So AP_{ℵ_{ω}} does not have much power to construct non-reflecting stationary sets either in ℵ_{ω+1} or in [ℵ_{ω+1}]^{ℵ_{0}}.

7. Distinguishing squares

Jensen [24] proved that for λ regular the principles □_{λ,μ} for μ ≤ λ have strictly decreasing strength as μ increases; in this section we prove a similar result for ℵ_{ω}. The case of ℵ_{ω} is considered for simplicity, similar methods will work at other singular cardinals.

Remark 7.1. In the model of Theorem 21 □_{ℵ_{ω}}^{∗} holds but □_{ℵ_{g}o,λ} fails for all λ < ℵ_{ω}. Accordingly we only need to distinguish principles of the form □_{ℵ_{g}o,λ} where λ < ℵ_{ω} in Theorem 13.

Theorem 13. Let κ be supercompact, and suppose 2^{κ^{+ω}} = κ^{+ω+1}. Let μ, ν be two cardinals (one or both can be finite) such that 1 ≤ μ < ν < ℵ_{ω}. Then there is a generic extension in which

1. All cardinals less than or equal to ν are preserved.

2. ℵ_{ω} = κ_{V}^{+ω}.

3. □_{ℵ_{ω},ν} holds.

4. □_{ℵ_{ω},μ} fails.

Proof. Let ρ = max{ν, ℵ_{1}} and let λ = κ^{+ω}. Let ℙ = Coll(ρ, < κ), so that ℙ is ρ-closed and λ = ℵ_{ω}^{V^{ℙ}}.

Working in V^{ℙ} we define a two-step forcing iteration ℚ ∗ ℝ. The model in which the conclusion of the theorem holds will be V^{ℙ∗ℚ}, while ℝ is an auxiliary forcing whose use is explained below.

1. ℚ = ℙ(λ, ν)_{V^{ℙ}}, where ℙ(λ, ν) is the forcing to add a □_{λ,ν}-sequence defined in Definition 6.2.

2. ℝ = 𝕋_{ρ}(C⃗)_{V^{ℙ∗ℚ}} where C⃗ is the □_{λ,ν}-sequence added by ℚ and 𝕋_{ρ}(C⃗)

is the “threading” forcing defined in Definition 6.8. By Lemma 6.9, ℚ ∗ ℝ has a dense ρ-closed subset. Our cardinal arithmetic assumptions imply V^{ℙ} ⊧ |ℚ ∗ ℝ| = λ^{+}. We fix G, H such that G is ℚ-generic over V and H is ℝ-generic over V [G].

Now we fix an elementary embedding j : V ⟶ M such that crit(j) = κ, j(κ) > λ^{+}, ^{λ^{+}}M ⊆ M. j ↾ ℙ is the identity map and gives a complete embedding of ℙ into j(ℙ), with the quotient forcing being Coll(ρ, [κ, j(κ))). By Lemma 6.11 we can extend this to give an embedding of ℙ∗ℚ∗ℝ into j(ℙ) in such a way that the quotient forcing is ρ-closed.

Let I be ℝ-generic over V [G ∗ H] and let J be j(ℙ)/G ∗ H ∗ I-generic over V [G ∗ H ∗ I], then we can build j : V [G] ⟶ M[G ∗ H ∗ I ∗ J] extending j : V ⟶ M. We want to extend this embedding to one with domain V [G ∗ H].

Let γ = ⋃ j“λ^{+}, then γ < j(λ^{+}) and M ⊧ cf(γ) = λ^{+}. Let C be the club set of order type ρ in λ^{+} which is added by I; then C ∉ V [G ∗ H] but every initial segment of C is in V, and so every limit point of C has cofinality less than ρ in V. Now let E = j“C. Clearly E is unbounded in γ, and E is closed because j is continuous at points of cofinality < ρ.

Define D⃗ = ⟨D_{α} : α < γ⟩ = ⋃_{s∈H} j(s). D⃗ is not a condition in j(ℚ) because it lacks a top element. However if α ∈ lim(E) then α = j(β) for some β ∈ lim(C), so that E ∩ α = j“(C ∩ β) = j(C ∩ β) ∈ j(C_{β}) = D_{α}.

It follows that if Q = D⃗ ∪ {(γ, {E})} then Q is a condition in j(ℚ) and what is more Q ≤ j(q) for every q ∈ H.

Now force below the “master condition” Q to get K which is j(ℚ)- generic. Since j“H ⊆ K we may extend j to get an elementary embedding j : V [G ∗ H] ⟶ M[G ∗ H ∗ I ∗ J ∗ K], which we will use to show that there is no □_{λ,μ}-sequence in V [G ∗ H]. Towards a contradiction, suppose that E⃗ = ⟨E_{α} : α < λ^{+}⟩ is such a sequence.

Notice that all the models which we discuss agree in their computation of the set of < ρ-sequences of ordinals, in particular cofinalities and cardinal arithmetic below ρ will look the same in all these models. λ_{V}^{+} is the ℵ_{ω+1} of V [G ∗ H], and is an ordinal of cardinality and cofinality ρ in V [G ∗ H ∗ I].

Claim. In V [G ∗ H ∗ I ∗ J] there exists F club in λ^{+} such that ∀α ∈ lim(F) F ∩ α ∈ E_{α} (that is to say F threads E⃗ in the sense of Section 6.2).

Proof. Let F⃗ = j(E⃗), and observe that because ν ≤ ρ < κ = crit(j) we have F_{j(α)} = j(E_{α}) = j“E_{α}. Choose some A ∈ F_{γ}, and observe that A ∈ M[G ∗ H ∗ I ∗ J] because K is generic over M[G ∗ H ∗ I ∗ J] for (j(λ^{+}), ∞)-distributive forcing; from now on we will work in the model V [G ∗ H ∗ I ∗ J]. j“λ^{+} is an ω-club subset of γ, so that lim(A) ∩ j“λ^{+} is unbounded in γ. Define F_{0} = { α < λ^{+} : j(α) ∈ lim(A) }, so that F_{0} is unbounded in λ^{+}.

If α ∈ F_{0} then j(α) ∈ lim(A), so that A ∩ j(α) ∈ j“E_{α}. For each α ∈ F_{0}, let X_{α} ∈ E_{α} be such that j(X_{α}) = A ∩ j(α). By elementarity, if α < β are two points in F_{0} then α ∈ lim(X_{β}) and X_{α} = X_{β} ∩ α. If F = ⋃_{α∈F_{0}} X_{α} then it is routine to check that F is club and ∀α ∈ lim(F) F ∩ α ∈ E_{α}.

We denote j(ℙ)/G ∗ H ∗ I by 𝕊, so that I ∗ J is ℝ ∗ 𝕊-generic over V [G ∗ H] and 𝕊 is ρ-closed.

Claim. F ∈ V [G ∗ H ∗ I].

Proof. Suppose not. Then F = Ḟ^{J} for some 𝕊-name Ḟ such that

⊩_{𝕊}^{V [G∗H∗I]} “Ḟ is not in the ground model”.

We work in V [G ∗ H ∗ I].

If μ is infinite, let σ be the least cardinal such that 2^{σ} > μ. If μ is finite let σ = ω. In either case:

1. σ is an infinite cardinal, with σ < ρ.

2. ∀τ < σ 2^{τ} < ρ.

3. 2^{σ} > μ.

We will build ⟨p_{t} : t ∈ ^{≤σ}2⟩ a tree of conditions in 𝕊 and a continuous increasing sequence ⟨α_{i} : i ≤ σ⟩ of ordinals less than λ^{+} such that

1. If u extends t then p_{u} ≤ p_{t}.

2. If i < σ then for every t ∈ ^{i}2 the conditions p_{t^{⌢}0} and p_{t^{⌢}1} force incompatible information about Ḟ ∩[α_{i}, α_{i+1}), and they both force that Ḟ ∩ [α_{i}, α_{i+1}) ≠ ∅.

The key points are that 𝕊 is ρ-closed and V [G ∗ H ∗ I] ⊧ cf(λ^{+}) = ρ. If τ ≤ σ is a limit ordinal then we set α_{τ} = sup_{i<τ} α_{i}, and for each t ∈ ^{τ}2 we choose p_{t} to be a lower bound for the decreasing τsequence ⟨p_{t↾i} : i < τ⟩. Suppose now that τ < σ and we have defined { p_{t} : t ∈ ^{τ}2 } and α_{τ}; we choose p_{t^{⌢}i}, γ_{t}, δ_{t}^{i} so that

1. α_{τ} < γ_{t} < δ_{t}^{i} < λ^{+}.

2. p_{t^{⌢}i} ≤ p_{t}.

3. p_{t^{⌢}0} and p_{t^{⌢}1} decide the statement “γ_{t} ∈ Ḟ ” in different ways.

4. p_{t^{⌢}i} ⊩ δ_{t}^{i} ∈ Ḟ. Now we let α_{i+1} = sup_{t∈^{τ} 2,i<2} δ_{t}^{i} + 1.

We have constructed a family { p_{t} : t ∈ ^{σ}2 } of 2^{σ} conditions in 𝕊. For each t ∈ ^{σ}2, p_{t} forces that α_{σ} ∈ lim(Ḟ), so p_{t} forces that Ḟ ∩ α_{σ} ∈ E_{α_{σ}}; we extend p_{t} to p_{t}^{∗} which decides the value of Ḟ ∩ α_{σ}, say p_{t}^{∗} ⊩ Ḟ ∩ α_{σ} = C_{t}. By construction the function which takes t to C_{t} is injective, but this is impossible because 2^{σ} > μ ≥ |E_{α_{σ}}|.

Claim. F ∈ V [G ∗ H].

Proof. The proof is quite similar to that of the last claim; the main difference is that ℝ is not ρ-closed, so we work with ℚ ∗ ℝ instead. Since F ∈ V [G ∗ H ∗ I], we may fix some ℚ ∗ ℝ-name ḟ ∈ V [G] such that ḟ^{H∗I} = F and

⊩_{ℚ∗ℝ}^{V [G]} ḟ^{G_{ℚ}∗G_{ℝ}} ∉ V [G][G_{ℝ}].

Subclaim. For all (q, r) ∈ ℚ ∗ ℝ there exist q̄ ≤ q, r_{0}, r_{1} and γ such that

1. (q̄, r_{0}), (q̄, r_{1}) ≤ (q, r)

2. The conditions (q̄, r_{0}), and (q̄, r_{1}) decide the statement γ̌ ∈ ḟ in different ways.

Proof. Suppose that the subclaim fails. Working in V [G], define a ℚname ḟ_{0} such that for all q̄ ≤ q and all γ,

q̄ ⊩_{ℚ}^{V [G]} γ̌ ∈ ḟ_{0} ⟺ ∃r̄ (q̄, r̄) ≤ (q̄, r), (q̄, r̄) ⊩_{ℚ∗ℝ}^{V [G]} γ̌ ∈ ḟ.

It is now routine to check that (q, r) ⊩ ḟ = ḟ_{0}, contradicting the choice of ḟ.

We now distinguish between the cases where μ is finite and μ is infinite.

Case 1. μ is infinite: Let σ be as in the proof of the last claim. Working in V [G] we will define ⟨r_{t} : t ∈ ^{<σ}2⟩, ⟨q_{i} : i < σ⟩ and ⟨α_{i} : i < σ⟩ such that

• (q_{⟨⟩}, r_{⟨⟩}) ⊩ ∀α ∈ lim(ḟ) ḟ ∩ α ∈ E_{α}.

• For each t, (q_{lh(t)}, r_{t}) ∈ ℚ∗ℝ and max dom(q_{lh(t)}) = max r_{t} = α_{lh(t)}.

• If u extends t then (q_{lh(u)}, r_{u}) ≤ (q_{lh(t)}, r_{t}).

• If i < σ then for every t ∈ ^{i}2 the conditions (q_{lh(t)+1}, r_{t^{⌢}0}) and (q_{lh(t)+1}, r_{t^{⌢}1}) force incompatible information about ḟ ∩ [α_{i}, α_{i+1}), and both force that ḟ ∩ [α_{i}, α_{i+1}) ≠ ∅.

Successor stages: Suppose τ < σ and we have defined the sequences ⟨r_{t} : t ∈ ^{≤τ}2⟩, ⟨q_{i} : i ≤ τ⟩ and ⟨α_{i} : i ≤ τ⟩. We observe that 2^{τ} < λ. Enumerate ^{τ}2 as ⟨s_{j} : j < 2^{τ}⟩. We will build conditions q_{j}^{τ} ∈ ℚ and r_{s_{j}^{⌢}0}^{∗}, r_{s_{j}^{⌢}1}^{∗} ∈ ℝ such that

1. ⟨q_{j}^{τ} : j < 2^{τ}⟩ is decreasing and has a lower bound (this is possible by the strategic closure of ℚ).

2. (q_{j}^{τ}, r_{s_{j}^{⌢}0}^{∗}) and (q_{j}^{τ}, r_{s_{j}^{⌢}1}^{∗}) force incompatible information about ḟ ∩

[α_{τ}, max dom(q_{j})), and also force that ḟ ∩ [α_{τ}, max dom(q_{j})) ≠ ∅. We then let q_{τ+1} be a lower bound for ⟨q_{j}^{τ} : j < 2^{τ}⟩, let α_{τ+1} =

max dom(q_{τ+1}), and r_{s_{j}^{⌢}k} = r_{s_{j}^{⌢}k}^{∗} ∪ {α_{τ+1}}.

Limit stages: Suppose that τ < σ is limit and we have defined the sequences ⟨r_{t} : t ∈ ^{<τ}2⟩, ⟨q_{i} : i < τ⟩ and ⟨α_{i} : i < τ⟩. Let α_{τ} = sup_{i} α_{i}. For each sequence t ∈ ^{τ}2 define r_{t} = ⋃_{i<τ} r_{t↾i} ∪ {α_{τ}}, and let F_{τ} = { r_{t} ∩ α_{τ} : t ∈ ^{τ}2 }. |F_{τ}| = 2^{τ} < μ by the minimal choice of σ, so that we may set q_{τ} = ⋃_{i<τ} q_{i} ∪ {(α_{τ}, F_{τ})}.

After σ steps we choose B ⊆ ^{σ}2 such that |B| = μ^{+}, and let α_{σ} = sup_{i<σ} α_{i}. For each t ∈ B let r_{t} = ⋃_{i<σ} r_{t↾i} ∪ {α_{σ}}, and define q_{σ} = ⋃_{i<σ} q_{i} ∪ {(α_{σ}, { r_{t} ∩ α_{σ} : t ∈ B })}. q_{σ} is a legitimate condition because |B| = μ^{+} ≤ ν, and what is more (q_{σ}, r_{t}) ∈ ℚ ∗ ℝ for all t ∈ B.

We may now extend q_{σ} to a condition q^{∗} which decides the set E_{α_{σ}}, say as some family E of club subsets of α_{σ} where |E| ≤ μ. This is a contradiction, because for every t ∈ B we may extend (q^{∗}, r_{t}) to decide ḟ ∩ α_{σ}, and distinct values of t must give distinct elements of E.

Case 2. μ is finite: In this case we fix (q, r) such that (q, r) ⊩ ∀α ∈ lim(ḟ) ḟ ∩ α ∈ E_{α}. We then fix q̄ ≤ q and r_{0}, … r_{μ} and an ordinal α such that

1. (q̄, r_{i}) ≤ (q, r).

2. The conditions (q̄, r_{i}) force incompatible information about ḟ∩ α.

We now construct q̄_{j}^{i}, r_{j}^{i}, α_{j}^{i} such that

1. q̄ ≥ q̄_{0}^{0} ≥ q̄_{1}^{0} ≥ … ≥ q̄_{μ}^{0} ≥ q̄_{0}^{1} ≥ … q̄_{μ}^{1} ≥ q̄_{0}^{2} ≥ …

2. α < α_{0}^{0} < α_{1}^{0} < … α_{μ}^{0} < α_{0}^{1} < … < α_{μ}^{1} < α_{0}^{2} < …

3. For each i ≤ μ, (q̄, r_{i}) ≥ (q̄_{i}^{0}, r_{i}^{0}) ≥ (q̄_{i}^{1}, r_{i}^{1}) ≥ (q̄_{i}^{2}, r_{i}^{2}) ≥ …

4. For each i ≤ μ and j < ω, (q̄_{j}^{i}, r_{j}^{i}) ⊩ α_{j}^{i} ∈ ḟ.

We then define α^{∗} = sup_{i,j} α_{i,j}, r_{j}^{∗} = ⋃_{i} r_{j}^{i} ∪ {α^{∗}}, and q^{∗} = ⋃_{i,j} q_{j}^{i} ∪ {r_{j}^{∗} ∩ α^{∗} : j ≤ μ}. q^{∗} ∈ ℚ and (q^{∗}, r_{j}) ∈ ℚ ∗ ℝ for all j ≤ μ, and so we may derive a contradiction exactly as in the case where μ is infinite.

This concludes the proof that F ∈ V [G ∗ H].

We have now arrived at a contradiction. Since λ^{+} is regular in V [G ∗ H], ot(F) = λ^{+}. Therefore we may find α ∈ lim(F) such that ot(F ∩ α) > λ. But F ∩ α ∈ E_{α} and E_{α} is a set of club sets which all have order type less than or equal to λ.

It follows that the principle □_{λ,μ} fails in V [G ∗ H]. This concludes the proof of Theorem 13.

8. Adding square by small forcing

In this section we prove that it is sometimes possible to do small forcing over a model in which □_{κ} (or even in some cases □_{κ}^{∗}) fails and obtain a model in which □_{κ} holds.

Theorem 14. Let κ be supercompact. Then there is a generic extension W such that

1. κ = ℵ_{2}^{W}.

2. □_{ℵ_{ω}}^{∗} fails in W.

3. If H is Coll(ω, ω_{1})-generic over W then □_{ℵ_{ω}} holds in W [H].

Proof. By unpublished work of Baumgartner [50], we may force to add a sequence ⟨C_{α} : α ∈ S⟩ such that

1. κ^{+ω+1} ∩ cof(≥ κ) ⊆ S ⊆ κ^{+}.

2. For all α ∈ S (a) ot(C_{α}) ≤ κ^{+ω}.

(b) For all β ∈ lim(C_{α}), β ∈ S and C_{β} = C_{α} ∩ β. preserving all cardinals up to κ^{+ω+1} and the supercompactness of κ. See [50] or [1] for details.

If we then force with Coll(ω_{1}, < κ), we obtain a model W in which □_{ℵ_{ω}}^{∗} fails. Let H be Coll(ω, ω_{1})-generic over W and observe that if α ∈ ℵ_{ω+1} \ S then W [H] ⊧ cf(α) = ω. We may therefore define a □_{ℵ_{ω}}-sequence in W [H] by setting D_{α} = C_{α} if α ∈ S, and letting D_{α} be any set of order type ω cofinal in α when α ∉ S.

Remark 8.1. A similar issue arises in [36]. There the problem is that forcing over a model which has no ℵ_{ω+ω+1}-Aronszajn tree with the poset Coll(ℵ_{0}, ℵ_{ω}) may a priori give a model which has an ℵ_{ω+1}-Aronszajn tree.

A similar argument shows that for λ regular we can create a situation in which □_{λ} fails, and forcing with Coll(ω, ω_{1}) makes □_{λ} hold. However, in general it may not be possible to force □_{λ} with mild forcing if we demand that □_{λ}^{∗} should fail in the ground model.

Theorem 15. Let 1 ≤ n < ω and let λ = ℵ_{n}. Let ℙ be λ-c.c. and suppose that ⊩_{ℙ} “□_{λ} holds”. Then □_{λ}^{∗} holds in V.

Proof. By results of Shelah [45] if T = { α < λ^{+} : cf(α) < λ } then there exists ⟨D_{α} : α ∈ T ⟩ such that

1. D_{α} is a nonempty family of size at most λ of club subsets of α with order type less than λ.

2. If D ∈ D_{α} and β ∈ lim(D) then D ∩ β ∈ D_{β}.

Let ⟨Ċ_{α} : α < λ^{+}⟩ name a □_{λ}-sequence. If α < λ^{+} has cofinality λ then Ċ_{α} names a club subset of α with order type λ, and so since ℙ is λ-c.c. it is easy to see that

C_{α}^{∗} = { γ < α : ⊩_{ℙ} γ ∈ Ċ_{α} }

is a club subset of α with order type λ.

If γ ∈ lim(C_{α}^{∗}) then ⊩_{ℙ} γ ∈ lim(Ċ_{α}), and so ⊩_{ℙ} Ċ_{γ} = Ċ_{α} ∩ γ and

C_{α}^{∗} ∩ γ = { δ < γ : ⊩_{ℙ} δ ∈ Ċ_{γ} }.

Let

S = { γ < λ^{+} : ∃α γ ∈ lim(C_{α}^{∗}) },

and for γ ∈ S let

C_{γ}^{∗} = { δ < γ : ⊩_{ℙ} δ ∈ Ċ_{γ} }.

Now let

E_{α} = \begin{cases}\{C_{α}^{∗}\} & if cf(α) = λ \\ D_{α} & if cf(α) < λ, α ∉ S \\ D_{α} ∪ \{C_{α}^{∗}\} & if cf(α) < λ, α ∈ S \end{cases}

It is routine to verify that E⃗ is a □_{λ}^{∗}-sequence.

9. Square above a supercompact cardinal

As we mentioned in Section 5, the existence of a supercompact cardinal puts some constraints on the possibilities for the existence of square sequences. To be more precise if κ is supercompact and λ ≥ κ then

• □_{λ,<cf(λ)} fails.

• If cf(λ) < κ then □_{κ}^{∗} fails. Here we will show that if κ is supercompact and κ ≤ cf(λ) < λ then

□_{λ,cf(λ)} can hold.

Theorem 16. Let λ be a singular cardinal. Then there exists a forcing poset ℙ such that

1. ℙ is cf(λ)-directed closed.

2. ℙ is < λ-strategically closed.

3. ⊩_{ℙ} “□_{λ,cf(λ)} holds”

Proof. Let cf(λ) = μ and fix ⟨λ_{i} : i < μ⟩ an increasing sequence of regular cardinals such that μ < λ_{0} and sup_{i} λ_{i} = λ.

Conditions in ℙ have the form

p = ⟨C_{α,i} : lim(α), α ≤ γ, i(α) ≤ i < μ⟩,

where

1. γ is a limit ordinal less than λ^{+}. We refer to γ as the length of p and write γ = lh(p).

2. i(α) < μ for each limit α ≤ γ.

3. If i(α) ≤ i < μ then C_{α,i} is club in α and ot(C_{α,i}) < λ_{i}.

4. If i(α) ≤ i < j < μ then C_{α,i} ⊆ C_{α,j}.

5. If i(β) ≤ i < μ and α ∈ lim(C_{β,i}) then i(α) ≤ i and C_{α,i} = C_{β,i}∩α.

6. If α and β are limit ordinals with α < β ≤ γ then α ∈ lim(C_{β,i}) for all sufficiently large i < μ.

If

p = ⟨C_{α,i}^{p} : α ≤ γ_{p}, i_{p}(α) ≤ i < μ⟩
q = ⟨C_{α,i}^{q} : α ≤ γ_{q}, i_{q}(α) ≤ i < μ⟩

then q ≤ p iff 1. γ_{p} ≤ γ_{q}.

2. For all limit α ≤ γ_{p}, i_{p}(α) = i_{q}(α) and C_{α,i}^{p} = C_{α,i}^{q} for all i with i_{p}(α) ≤ i < μ.

Claim. ℙ is μ-directed closed.

Proof. The ordering on ℙ is treelike, that is to say that if p ≤ q, r then q is comparable with r. So the directed subsets of ℙ are exactly the linearly ordered ones and it suffices to show that ℙ is μ-closed.

Let ρ < μ and let ⟨p_{σ} : σ < ρ⟩ be a strictly decreasing sequence of conditions with γ(σ) =_{def} lh(p_{σ}). Let γ =_{def} sup_{σ} γ(σ).

By the definition of the ordering on ℙ, the union of the p_{σ} for σ < ρ is a matrix of sets ⟨C_{α,i} : α < γ, i(α) ≤ i < μ⟩. We need to define i(γ) and ⟨C_{γ,i} : i(γ) ≤ i < μ⟩ in such a way that ⟨C_{α,i} : α ≤ γ, i(α) ≤ i < μ⟩ is a condition in ℙ.

For each pair of ordinals (σ, τ) with σ < τ < ρ, let f(σ, τ) be the least i < μ such that γ(σ) ∈ lim(C_{γ(τ),i}). Such an i must exist by clause 6 in the definition of ℙ.

Observe that if f(σ, τ) ≤ i < μ then γ(σ) ∈ lim(C_{γ(τ),i}) and so C_{γ(σ),i} = C_{γ(τ),i} ∩ γ(σ). We define i(γ) =_{def} sup_{σ,τ} f(σ, τ), where i(γ) < μ because ρ < μ = cf(μ). We let C_{γ,i} =_{def} ⋃_{σ<ρ} C_{γ(σ),i} for all i ≥ i(γ).

It is now routine to check that ⟨C_{α,i} : α ≤ γ, i(α) ≤ i < μ⟩ is a condition. The key points are that

• ρ < μ < λ_{0} so that ot(C_{γ,i}) < λ_{i}.

• The choice of i(γ) guarantees that C_{γ,i} ∩ γ(σ) = C_{γ(σ),i} for all σ < ρ and i ≥ i(γ).

Claim. ℙ is < λ-strategically closed.

Proof. For a fixed θ < λ, we describe a winning strategy for player II in G_{θ+1}^{II}(ℙ). We begin by fixing j < μ such that θ < λ_{j}. We will denote the condition played at move ρ in the game by

p_{ρ} = ⟨C_{α,i} : α ≤ γ(ρ), i(α) ≤ i < μ⟩.

Player II will play in such a way that

• i(γ(2α)) = j for all α.

• If α < β and i ≥ j then C_{γ(2α),i} = C_{γ(2β),i} ∩ γ(2α).

Successor stages: Suppose that player I plays p_{2α+1}. Player II responds with p_{2α+2} where

1. γ(2α + 2) = γ(2α + 1) + ω, i(γ(2α + 2)) = j.

2. If i ≥ j and γ(2α) ∈ lim(C_{γ(2α+1),i}) then C_{γ(2α+2),i} = C_{γ(2α+1),i} ∪ { γ(2α + 1) + n : n < ω }. Notice that this case will apply for all sufficiently large i < μ.

3. If γ(2α) ∉ lim(C_{γ(2α+1),i}) then C_{γ(2α+2),i} = C_{γ(2α),i} ∪ {γ(2α)} ∪ { γ(2α + 1) + n : n < ω }.

We have arranged matters so that C_{γ(2α+2),i} increases with i, always end-extends C_{γ(2α),i} and end-extends C_{γ(2α+1),i} for all large i. Using this it is routine to check that p(2α + 2) is a condition. Limit stages: Let δ ≤ θ be a limit stage in the game. Let γ(δ) = sup_{α<δ} γ(α), i(γ(δ)) = j, and C_{γ(δ),i} = ⋃_{2α<δ} C_{γ(2α),i} for i ≥ j. Player II played at the even stages 2α preceding δ in such a way that C_{γ(δ),i} ∩ γ(2α) = C_{γ(2α),i}. Using this it is routine to check that p_{δ} is a condition.

Since ℙ is < λ-strategically closed, forcing with ℙ adds no < λsequence of ordinals. Since λ is singular, forcing with ℙ adds no λsequences of ordinals, and so preserves all cardinals and cofinalities up to and including λ^{+}.

Essentially the same argument as for strategic closure shows that for all γ < λ^{+} the set { p ∈ ℙ : lh(p) > γ } is dense in ℙ. So the generic object is a □_{λ,cf(λ)}-sequence in V^{ℙ}.

The following is an immediate consequence of Theorem 16.

Theorem 17. Let κ be a Laver indestructible supercompact cardinal and let κ ≤ cf(λ) < λ. Then there is a forcing extension in which κ is still supercompact, cardinals and cofinalities up to λ^{+} are preserved, and □_{λ,cf(λ)} holds.

Remark 9.1. In the model of Theorem 17 • □_{λ,<λ}^{imp} fails as a result of Theorem 12.

• If S = λ^{+} ∩ cof(< κ) then the supercompactness of κ implies that any family of size less than κ of stationary subsets of S reflects simultaneously. Theorem 5 implies that there are cf(λ) subsets of S which do not reflect simultaneously.

We observe that in Theorem 16 we have actually forced a principle which is (formally) stronger than □_{λ,cf(λ)}.

Definition 9.2. A □_{λ,cf(λ)}^{ind}-sequence (indexed □_{λ,cf(λ)} sequence) is a matrix of sets ⟨C_{α,i} : α < λ^{+}, i(α) ≤ i < cf(λ)⟩ such that for some increasing sequence ⟨λ_{i} : i < cf(λ)⟩ of regular cardinals with limit λ

1. i(α) < cf(λ) for all α < λ^{+}.

2. ot(C_{α,i}) < λ_{i} for all α.

3. C_{α,i} is club in α.

4. If i(α) ≤ i < j < cf(λ) then C_{α,i} ⊆ C_{α,j}.

5. If i(β) ≤ i < cf(λ) and α ∈ lim(C_{β,i}) then i(α) ≤ i and C_{α,i} = C_{β,i} ∩ α.

6. If α and β are limit ordinals with α < β < λ^{+} then α ∈ lim(C_{β,i}) for all sufficiently large i < μ.

Remark 9.3. Shelah [44] introduces a combinatorial principle S_{λ} and shows that S_{λ} is equivalent to the model-theoretic transfer principle (ℵ_{1}, ℵ_{0}) ⟶ (λ^{+}, λ). If ⟨C_{α,i} : α < λ^{+}, i(α) ≤ i < cf(λ)⟩ is a □_{λ,cf(λ)}^{ind}- sequence, and we set S_{α,i} = { γ < α : ωγ ∈ lim(C_{ωα,i}) } for i ≥ i(α) and S_{α,i} = ∅ otherwise, then ⟨S_{α,i} : α < λ^{+}, i < cf(λ)⟩ is a S_{λ}-sequence.

Remark 9.4. Cummings and Schimmerling [7] have studied a number of variations on the concept of an indexed square sequence.

For use in the next section we develop a theory of “threading” appropriate for generic □_{λ,cf(λ)}^{ind}-sequences.

We fix a singular cardinal λ, and an increasing sequence ⟨λ_{i} : i < μ⟩ of regular cardinals such that μ = cf(λ) and sup_{i} λ_{i} = λ. Let ℙ be the forcing to add a generic □_{λ,μ}^{ind}-sequence which was defined in the proof of Theorem 16. Let C⃗ = ⟨C_{α,i} : α < λ^{+}, i(α) ≤ i < cf(λ)⟩ be the generic □_{λ,μ}^{ind}-sequence added by ℙ. Working in V^{ℙ} we define a family of forcing posets.

Definition 9.5. Let j < μ. c ∈ ℚ_{j}(C⃗) iff 1. c is a closed bounded subset of λ^{+}.

2. If γ = max(c) then i(γ) ≤ j and c = C_{γ,j} ∪ {γ}.

ℚ_{j}(C⃗) is ordered by end-extension.

It is worthwhile to notice that each condition in ℚ_{j}(C⃗) lies in V. The following result is analogous to Lemma 6.9.

Lemma 9.6. ℙ ∗ ℚ_{j}(C⃗) has a λ_{j}-directed closed dense subset.

Proof. It will suffice to produce a dense λ_{j}-closed subset on which the ordering is treelike. Let D be the set of conditions of the form (p, č) where lh(p) = max(c).

If p = ⟨C_{α,i} : α ≤ γ, i(α) ≤ i < μ⟩ and (p, č) ∈ D then γ = max(c), i(γ) ≤ j and c = C_{γ,j} ∪ {γ}.

Claim. D is dense in ℙ ∗ ℚ_{j}(C⃗).

Proof. Let (p, ċ) ∈ ℙ∗ℚ_{j}(C⃗). Extending p if necessary, we may assume that p decides ċ (say p ⊩ ċ = č) and also that lh(p) > max(c). Let p = ⟨C_{α,i} : α ≤ γ, i(α) ≤ i < μ⟩. Let δ = max(c), so that c = C_{δ,j} ∪{δ}.

We will define a condition q = ⟨C_{α,i} : α ≤ γ + ω, i(α) ≤ i < μ⟩ of length γ + ω extending p, with i(γ + ω) = j.

Let k be minimal such that k > j and δ ∈ lim(C_{γ,k}). We define

C_{γ+ω,i} = C_{δ,i} ∪ {δ} ∪ {γ + n : n < ω}

for j ≤ i < k, and

C_{γ+ω,i} = C_{γ,i} ∪ {γ + n : n < ω}

for k ≤ i < μ.

Now let d = C_{γ+ω,j} ∪ {γ + ω}. (q, ď) ∈ D and (q, ď) ≤ (p, ċ) so that D is dense.

Claim. D is λ_{j}-closed.

Proof. Let ρ < λ_{j} and let ⟨(p_{σ}, c_{σ}) : σ < ρ⟩ be a strictly decreasing sequence of conditions in D, where lh(p_{σ}) = γ(σ). Let γ = sup_{σ} γ(σ), and let the union of the sequence ⟨p_{σ} : σ < ρ⟩ be the matrix of sets ⟨C_{α,i} : α < γ, i(α) ≤ i < μ⟩. Let c = ⋃_{σ} c_{σ}.

For each σ < ρ, γ(σ) ∈ lim(c) and c ∩ γ(σ) = C_{γ(σ),j}. It follows that if σ < τ < ρ and i ≥ j then γ(σ) ∈ lim(C_{γ(τ),i}), and so C_{γ(τ),i} ∩ γ(σ) = C_{γ(σ),i}. We now define p = ⟨C_{α,i} : α ≤ γ, i(α) ≤ i < μ⟩ by setting i(γ) = j and C_{γ,i} = ⋃_{σ<ρ} C_{γ(σ),i} for all i ≥ j; in particular, C_{γ,j} = c.

Finally we let d = c ∪ {γ}. (p, d) ∈ D and (p, d) ≤ (p_{σ}, c_{σ}) for all σ < ρ, so that D is λ_{j}-closed.

This concludes the proof that ℙ ∗ ℚ_{j}(C⃗) has a λ_{j}-closed subset.

We conclude this section by proving that if i < j then in V^{ℙ} there is natural projection from ℚ_{i} to ℚ_{j}.

Definition 9.7. In V^{ℙ} define a map π_{ij} : ℚ_{i}(C⃗) ⟶ ℚ_{j}(C⃗) by π_{ij}(c) = C_{max(c),j} ∪ {max(c)}.

Lemma 9.8. In V^{ℙ} 1. π_{ij} is a projection from ℚ_{i} to ℚ_{j}.

2. If i < j < k then π_{ik} = π_{jk} ∘ π_{ij}.

3. If (p, c) ∈ D then p determines π_{ij}(c) for all j ≥ i.

Proof. We prove that π_{ij} is a projection, the remaining claims are immediate. Let p be a condition which forces that d ∈ ℚ_{j}, c ∈ ℚ_{i} and d ≤ π_{ij}(c). Let max(c) = δ, max(d) = γ. We may as well assume that lh(p) = γ. If p = ⟨C_{α,k} : α ≤ γ, i(α) ≤ k < μ⟩ then we have c = C_{δ,i} ∪ {δ}, d = C_{γ,j} ∪ {γ}, and δ ∈ lim(C_{γ,j}).

Arguing exactly as in the proof of Lemma 9.6 we may build q ≤ p of height γ + ω such that i(γ + ω) = i, C_{γ+ω,i} ∩ δ = c and C_{γ+ω,j} ∩ γ = d. If c^{∗} = C_{γ+ω,i} ∪ {γ + ω} then q forces that c^{∗} ∈ ℚ_{i}, c^{∗} ≤ c and π_{ij}(c^{∗}) ≤ d.

10. Square and reflection

Theorem 2 shows that if □_{κ,<ω} holds then every stationary subset of κ^{+} has a non-reflecting stationary subset. Here we show it is consistent with □_{ℵ_{ω},ω}^{ind} that every finite family of stationary subsets of ℵ_{ω+1} should reflect simultaneously. This result is close to optimal, because it follows from Theorem 5 that if □_{ℵ_{ω},ω} holds then any stationary subset of ℵ_{ω+1} has ω stationary subsets which do not reflect simultaneously.

Remark 10.1. We have presented the case of ℵ_{ω} here but the idea of the proof works in more general situations.

Theorem 18. Let ⟨κ_{n} : n < ω⟩ be an increasing sequence of supercompact cardinals. Let κ = sup_{n} κ_{n}. Then there is a generic extension in which

1. κ = ℵ_{ω}, and κ^{+} is preserved.

2. □_{ℵ_{ω},ω}^{ind} holds.

3. For every finite set f of stationary subsets of ℵ_{ω} there exists N < ω such that if N ≤ n < ω then there exists α of cofinality ℵ_{n} such that all sets in f reflect at α.

Proof. The idea of the proof is to collapse the supercompact cardinals as in section 6.3, force □_{ℵ_{ω},ω}^{ind} as in Theorem 16, and finally to do an iteration destroying certain “bad” stationary sets. Those stationary subsets of ℵ_{ω+1} which survive the final stage of the construction will be shown to reflect by an argument in the same spirit as that of 6.3.

Let

V_{0} ⊧ “⟨κ_{n} : n < ω⟩ is an increasing sequence of supercompact cardinals”.

Let V = V_{0}^{ℝ} where ℝ is the iterated Lévy collapse described in section 6.3. In V we have κ_{n} = ℵ_{n+1}, κ = ℵ_{ω}, κ_{V_{0}}^{+} = ℵ_{ω+1}.

Until further notice we will consider V as the ground model. The fact that ℵ_{n+1}^{V} was supercompact in V_{0} will not be important until we come to prove the reflection properties of the final model.

Working in V we will define an iteration 𝕊∗ℙ, and our final model will be V^{𝕊∗ℙ}. Let 𝕊 be the forcing to add a □_{ℵ_{ω},ω}^{ind}-sequence from Theorem 16, where we set λ_{n} = ℵ_{n} for each n < ω. Let 𝕋_{n} = ℚ_{n}(C⃗)_{V^{ℙ}}, where C⃗ is the generic □_{ℵ_{ω},ω}^{ind}-sequence added by ℙ and ℚ_{n}(C⃗)_{V^{ℙ}} is the threading forcing from Definition 9.5.

Working in V^{𝕊} we will define posets ⟨ℙ_{α} : α ≤ ℵ_{ω+2}⟩ by induction on α, maintaining a certain inductive hypothesis which will imply that 𝕊 ∗ ℙ_{α} adds no ℵ_{ω}-sequences of ordinals. We will also be inductively defining Ṡ_{α} ∈ V^{𝕊∗ℙ_{α}} such that ⊩ “Ṡ_{α} is a subset of ℵ_{ω+1}”.

Conditions in ℙ_{α} are functions p such that 1. dom(p) ⊆ α, | dom(p)| ≤ ℵ_{ω}.

2. If β ∈ dom(p) then (a) p(β) is a closed bounded subset of ℵ_{ω+1}.

(b) p ↾ β ⊩_{ℙ_{β}} p(β) ∩ Ṡ_{β} = ∅. If p, q ∈ ℙ_{α} then p ≤ q iff

1. dom(q) ⊆ dom(p).

2. For all β ∈ dom(q), p(β) ∩ (max q(β) + 1) = q(β).

Remark 10.2. Since ℙ_{α} adds no ℵ_{ω}-sequences, ℙ_{α+1} is equivalent to the iteration ℙ_{α} ∗ CU(ℵ_{ω+1} \ Ṡ_{α}) where CU(ℵ_{ω+1} \ Ṡ_{α}) is the standard forcing to shoot a closed unbounded subset through ℵ_{ω+1} \ S_{α}.

Remark 10.3. Similar iterations appear in [33, Theorem 1] and [23, Theorem 3].

Remark 10.4. It is easy to see that ℙ_{ℵ_{ω+2}} will be ℵ_{ω+2}-c.c. in V^{𝕊}.

Definition 10.5. Let α ≤ ℵ_{ω+2} and suppose that

V [𝕊 ∗ ℙ_{α}] ⊧ S ⊆ ℵ_{ω+1}.

S is fragile iff

V [𝕊 ∗ ℙ_{α} ∗ 𝕋_{n}] ⊧ “S is non-stationary”

for all n < ω.

We will choose the sequence of names Ṡ_{α} so that each one names a fragile set, in such a way that

V^{𝕊∗ℙ_{ℵ_{ω+2}}} ⊧ “every fragile set is non-stationary”.

This is possible because

V^{𝕊} ⊧ “2^{ℵ_{ω}} = ℵ_{ω+1} and ℙ_{ℵ_{ω+2}} is ℵ_{ω+2}-c.c.”

As the construction proceeds we will fix names Ċ_{α}^{n} such that

⊩_{𝕊∗ℙ_{α}∗𝕋_{n}} Ċ_{α}^{n} ∩ Ṡ_{α} = ∅.

By an argument like that for Lemma 9.6 It is easy to see that the set of conditions in 𝕊 ∗ ℙ_{α} ∗ 𝕋_{n} which are of the form (s, p̌, f̌) and have lh(s) = max(f) is dense. We abuse notation by using “𝕊 ∗ ℙ_{α} ∗ 𝕋_{n}” to refer to this dense set of conditions.

Definition 10.6. Let A be a set with |A| ≤ ℵ_{ω}. P⃗ is a good partition of A iff P⃗ = ⟨P_{n} : n < ω⟩ where the P_{n} form a partition of A and |P_{n}| < ℵ_{n} for all n.

If P⃗ is a good partition then “P⃗ ↾ k = 0” means that P_{i} = ∅ for all i < k.

If P⃗ is a good partition of A ⊆ α and β < α, then P⃗ ↾↾ β will denote the good partition Q⃗ of A ∩ β given by Q_{j} = P_{j} ∩ β.

Definition 10.7. Let q = (s, p, f) ∈ 𝕊 ∗ ℙ_{α} ∗ 𝕋_{n}. q is correct for P⃗ iff 1. P⃗ is a good partition of supp(p) and P⃗ ↾ n = 0.

2. For all k ≥ n and all β ∈ P_{k}, (s, p ↾ β, π_{nk}(f)) ⊩ max p(β) ∈ Ċ_{β}^{k}.

Definition 10.8. Let q = (s, p, f) and q′ = (s′, p′, f′) be two conditions in 𝕊∗ℙ_{α}∗𝕋_{n} Let P⃗, P⃗′ be good partitions of supp(p) and supp(p′) respectively such that P⃗ ↾ n = 0 and P⃗′ ↾ n = 0.

Then (q′, P′) ≤ (q, P) iff

1. q′ ≤ q in 𝕊 ∗ ℙ_{α} ∗ 𝕋_{n}.

2. P′_{k} ∩ supp(p) = P_{k} for all k ≥ n.

Lemma 10.9. Let ⟨(q_{i}, P⃗^{i}) : i < μ⟩ be such that 1. q_{i} = (s_{i}, p_{i}, f_{i}) ∈ 𝕊 ∗ ℙ_{α} ∗ 𝕋_{n}.

2. q_{i} is correct for P⃗^{i}.

3. μ < ℵ_{n}.

4. If i < j < μ then (q_{j}, P⃗^{j}) ≤ (q_{i}, P⃗^{i}).

Then there exists (q, P⃗) such that q is correct for P⃗ and (q, P⃗) ≤ (q_{i}, P⃗^{i}) for all i < μ.

Proof. We are assuming that lh(s_{i}) = max(f_{i}) for all i < μ, so by Lemma 9.6 there is (s, f) ∈ 𝕊 ∗ 𝕋_{n} such that lh(s) = max(f) and (s, f) ≤ (s_{i}, f_{i}) for all i < μ.

We will define a condition p ∈ ℙ_{α} with supp(p) = ⋃_{i} supp(p_{i}). We define P⃗ by setting P_{j} = ⋃_{i<μ} P_{j}^{i} for all j < ω, where P⃗ is good because μ < ℵ_{n} and P⃗ ↾ n = 0. To define p we will determine p ↾ β by induction on β. Limit stages in the construction of p are not problematic because the intended support of p is of cardinality ℵ_{ω}.

Suppose that β ∈ ⋃_{i} supp(p_{i}) and we have defined p ↾ β so that (s, p ↾ β, f) ≤ (s_{i}, p_{i} ↾ β, f_{i}) for all i < μ. There is a fixed N < ω such that β ∈ P_{N}^{i} for all large i < μ. Since q_{i} is correct for P⃗^{i},

(s_{i}, p_{i} ↾ β, π_{nN}(f_{i})) ⊩ max p_{i}(β) ∈ C_{β}^{N}

and so

(s, p ↾ β, π_{nN}(f)) ⊩ max p_{i}(β) ∈ C_{β}^{N}.

Let δ_{β} = sup_{i<μ} p_{i}(β) and p(β) = ⋃_{i<μ} p_{i}(β) ∪ {δ_{β}}. Since Ċ_{β}^{N} is forced to be club,

(s, p ↾ β, π_{nN}(f)) ⊩ δ_{β} ∈ C_{β}^{N}.

Claim. (s, p ↾ β + 1) ∈ 𝕊 ∗ ℙ_{β+1}.

Proof. It suffices to check that (s, p ↾ β) ⊩ p(β) ∩ Ṡ_{β} = ∅. Since (s_{i}, p_{i} ↾ β) ⊩ p_{i}(β) ∩ Ṡ_{β} = ∅ and (s, p ↾ β) ≤ (s_{i}, p_{i} ↾ β), we need only to check (s, p ↾ β) ⊩ δ_{β} ∉ Ṡ_{β}. Suppose for a contradiction that this is not the case and choose (t, q) ≤ (s, p ↾ β) such that (t, q) ⊩ δ_{β} ∈ Ṡ_{β}. Then (t, q, π_{nN}(f)) ≤ (s, p ↾ β, π_{nN}(f)), so (t, q, π_{nN}(f)) ⊩ δ_{β} ∈ C_{β}^{N} contradicting the choice of Ċ_{β}^{N} as a name for a club disjoint from Ṡ_{β}.

This construction produces a condition q = (s, p, f) which is correct for P⃗ and has the property that (q, P⃗) ≤ (q_{i}, P⃗^{i}) for all i < μ.

The next lemma is the key to the analysis of 𝕊 ∗ ℙ_{α}. It will follow from this lemma that 𝕊 ∗ ℙ_{α} ∗ 𝕋_{n} adds no < ℵ_{n}-sequences of ordinals, which in turn implies that 𝕊 ∗ ℙ_{α} adds no ℵ_{ω}-sequences of ordinals.

Lemma 10.10. Let q = (s, p, f) ∈ 𝕊 ∗ ℙ_{α} ∗ 𝕋_{n}. Let k ≥ n. Let P⃗ be a good partition of supp(p) such that P⃗ ↾ k = 0. Then there exists q^{∗} = (s^{∗}, p^{∗}, f^{∗}) ∈ 𝕊 ∗ ℙ_{α} ∗ 𝕋_{n} and P⃗^{∗} such that

1. (q^{∗}, P⃗^{∗}) ≤ (q, P⃗).

2. q^{∗} is correct for P⃗^{∗}.

3. P⃗^{∗} ↾ k = 0.

Proof. We prove the Lemma by induction on α for all n and k simultaneously. There are various cases.

Case 1: cf(α) ≥ ℵ_{ω+1}. | supp(p)| ≤ ℵ_{ω} so there exists β < α such that supp(p) ⊆ β. Apply the induction hypothesis to (s, p ↾ β, f).

Case 2: α is a successor, say α = β + 1. If β ∉ supp(p) then apply the induction hypothesis to (s, p ↾ β, f). If β ∈ supp(p) then β ∈ P_{m} for some m ≥ k. Ċ_{β}^{m} is a 𝕊 ∗ ℙ_{β} ∗ 𝕋_{m}-name for a club subset of ℵ_{ω+1} so we may find (s′, p′, f′) ∈ 𝕊 ∗ ℙ_{β} ∗ 𝕋_{n} and γ > max p(β) such that (s′, p′, f′) ≤ (s, p ↾ β, f) and (s′, p′, π_{nm}(f′)) ⊩ γ̌ ∈ Ċ_{β}^{m}. Arguing as in the proof of Lemma 10.9, (s′, p′) ⊩ γ̌ ∉ Ṡ_{β}.

Now choose Q⃗ a good partition of supp(p′) such that

• Q⃗ ↾ k = 0.

• Q_{j} ∩ supp(p) = P_{j}.

• Q_{m} ∩ supp(p) = P_{m} \ {β}.

By the induction hypothesis we may find (s′′, p′′, f′′) ∈ 𝕊 ∗ ℙ_{β} ∗ 𝕋_{n} and R⃗ such that

1. (s′′, p′′, f′′) is correct for R⃗.

2. R⃗ ↾ k = 0.

3. ((s′′, p′′, f′′), R⃗) ≤ ((s′, p′, f′), Q⃗).

Now let s^{∗} = s′′, f^{∗} = f′′ and let p^{∗} ∈ ℙ_{β+1} be such that p^{∗} ↾ β = p′′, p^{∗}(β) = p(β) ∪ {γ}. Define a good partition P⃗^{∗} of supp(p^{∗}) by setting P_{j}^{∗} = R_{j} for j ≠ m, P_{m}^{∗} = R_{m} ∪ {β}. Then q^{∗} = (s^{∗}, p^{∗}, f^{∗}) and R⃗^{∗} are as required.

Case 3: cf(α) = ℵ_{N} where N < ω.

Subcase 3a: N < k. Fix a sequence ⟨α_{i} : i < ℵ_{N}⟩ which is continuous, increasing and cofinal in α. Let α_{ℵ_{N}} = α. We will define q_{i} = (s_{i}, p_{i}, f_{i}) and P⃗^{i} for i ≤ ℵ_{N} so that

1. q_{i} ∈ 𝕊 ∗ ℙ_{α_{i}} ∗ 𝕋_{k}.

2. q_{i} is correct for P⃗^{i}.

3. If i < j then

((s_{j}, p_{j} ↾ α_{i}, f_{j}), P⃗^{j} ↾↾ α_{i}) ≤ ((s_{i}, p_{i}, f_{i}), P⃗^{i}).

4. ((s_{i}, p_{i}, f_{i}), P⃗^{i}) ≤ ((s, p ↾ α_{i}, π_{nk}(f)), P⃗ ↾↾ α_{i}).

For i = 0 we choose q_{0} and P⃗^{0} by applying the induction hypothesis to the condition (s, p ↾ α_{0}, π_{nk}(f)) and the partition P⃗ ↾↾ α_{0}.

Suppose that we have defined q_{i} and P⃗^{i}. Consider the condition (s_{i}, p_{i}^{⌢}p ↾ [α_{i}, α_{i+1}), f_{i}) and the partition Q⃗ of supp(p ↾ [α_{i}, α_{i+1})) given by Q_{m} = P_{m}^{i} ∪ (P_{m} ∩ [α_{i}, α_{i+1})). Applying the induction hypothesis gives q_{i+1} and P⃗^{i+1} as desired.

Suppose now that i is limit. Choose (s_{i}, f_{i}) to be a lower bound for ⟨(s_{j}, f_{j}) : j < i⟩ as in Lemma 9.6. We now define p_{i} with support ⋃_{j<i} supp(p_{j}). If β ∈ ⋃_{j<i} supp(p_{j}) then let γ_{j}(β) = max p_{j}(β) for all j such that β ∈ supp(p_{j}), and let γ(β) = sup_{β∈supp(p_{j})} γ_{j}(β). Define p_{i}(β) = ⋃_{j<i} p_{j}(β) ∪ {γ(β)}. An inductive proof similar to that of 10.9 shows that q_{i} = (s_{i}, p_{i}, f_{i}) is a condition in 𝕊 ∗ ℙ_{α_{i}} ∗ 𝕋_{k}, and that if we define P⃗^{i} by P_{m}^{i} = ⋃_{j<i} P_{m}^{j} then q_{i} is correct for P⃗^{i}.

Now let s^{∗} = s_{ℵ_{n}}, p^{∗} = p_{ℵ_{n}}, P⃗^{∗} = P⃗^{ℵ_{n}}, and choose f^{∗} such that π_{nk}(f^{∗}) ≤ f_{ℵ_{n}}. Then q^{∗} = (s^{∗}, p^{∗}, f^{∗}) and P⃗^{∗} are as required.

Subcase 3b: k ≤ N. Let k̄ = N + 1 and define a new partition Q⃗ of supp(p) by Q⃗ ↾ k̄ = 0, Qk̄ = ⋃_{m≤k̄} Pk̄, Q_{m} = P_{m} for m > k̄.

By the construction of Subcase 3a we may find (q′, R⃗) ≤ (q, Q) where q′ = (s′, p′, f′) is correct for R⃗. Observe that since P⃗ is a good partition, | ⋃_{m<k̄} P_{m}| < ℵ_{N} and so ⋃_{m<k̄} P_{m} is bounded in α, say ⋃_{m<k̄} P_{m} ⊆ β.

Define a good partition S⃗ of supp(p^{∗} ↾ β) by setting

S_{i} = \begin{cases}∅ & i < k \\ P_{i} ∩ β & k ≤ i < k̄ \\ (Rk̄ \ ⋃_{m<k̄} P_{m}) ∩ β & i = k̄ \\ R_{i} ∩ β & i > k̄ \end{cases}

By induction we find q′′ = (s′′, p′′, f′′) and T⃗ such that (q′′, T⃗) ≤ ((s′, p′ ↾ β, f′), S⃗) and q′′ is correct for T⃗. Finally we let s^{∗} = s′′, f^{∗} = f′′, p^{∗} = p′′ ⌢ p′ ↾ [β, α) and define P⃗^{∗} by setting P_{i}^{∗} = T_{i} ∪ (S_{i} ∩ [β, α)).

Now we use results of Foreman [10] about games played on posets and Boolean algebras.

Definition 10.11. Let ℙ be a partial ordering, let μ be regular. The game G_{μ}^{I}(ℙ) is played as follows: player I and player II take turns to write down p_{α} ∈ ℙ for α < μ, where ∀α < β p_{β} ≤ p_{α}. Player I plays at all even α, including limit α. Player II loses if there exists a limit λ < μ such that ⟨p_{α} : α < λ⟩ has no lower bound, and wins otherwise. If 𝔹 is a Boolean algebra then G_{μ}^{I} is G_{μ}^{I}(𝔹 − {0}).

Lemma 10.12. Player II has a winning strategy for the game G_{ℵ_{n}}^{I}(𝕊∗ ℙ_{ω+2} ∗ 𝕋_{n}).

Proof. Whenever II plays a condition q_{i} = (s_{i}, p_{i}, f_{i}), they will also record a partition P⃗^{i} of supp(p_{i}) such that q_{i} is correct for P⃗^{i}. Lemmas 10.10 and 10.9 ensure that this is possible, and gives a winning strategy in the game.

We are finally ready to prove that Refl(ℵ_{ω+1}) holds in V^{𝕊∗ℙ}. Recall that V_{0} was the original model in which the cardinals κ_{i} were supercompact, and that κ_{i} became ℵ_{i+1}. Let V_{1} = V, V_{2} = V_{1}^{𝕊}, V_{3} = V_{2}^{ℙ_{ω+2}}.

Suppose that V_{3} ⊧ “S is a stationary subset of ℵ_{ω+1}”. Since S is not fragile we may find n so large that

• S ∩ cof(< ℵ_{n}) is stationary.

• S is stationary in V_{3}^{𝕋_{n+1}}. Without loss of generality we replace S by S ∩ cof(< ℵ_{n}).

Now let ℚ = Coll(ℵ_{n+1}, ℵ_{n+2})_{V_{1}}. It is easy to see that II still wins G_{ℵ_{n+1}}^{I}(𝕊 ∗ ℙ_{ω+2} ∗ 𝕋_{n+1}) in V_{ℚ}^{1}, and that V_{ℚ}^{1} ⊧ |𝕊 ∗ ℙ_{ω+2} ∗ 𝕋_{n+1}| = ℵ_{n+1}.

We now appeal to a theorem of Foreman [10].

Fact 10.13. If μ = μ^{<μ}, |ℙ| = μ, and II has a winning strategy for G_{μ}^{I}(ℙ) then the Boolean algebra ro(ℙ) has a μ-directed closed dense subset.

So if V_{4} = V_{1}^{ℚ×𝕊∗ℙ_{ω+2}∗𝕋_{n+1}} then we can view V_{4} as an extension of V_{1} by an ℵ_{n+1}-directed closed poset ℝ.

At this point we will use the “indestructible generic supercompactness” property described in Lemma 6.10. This gives us a generic supercompact embedding j : V_{4} ⟶ M ⊆ V ℝ̄ ; 4, where

• crit(j) = κ_{n}.

• j(κ_{n}) > κ^{+}.

• sup(j“κ^{+}) < j(κ^{+}).

• j ↾ κ^{+} ∈ M.

• M ⊧ cf(κ^{+}) = ℵ_{n}.

• V_{4} ⊧ “ℝ̄ is ℵ_{n}-closed”.

Now V ℝ̄ ; 4 is an extension of V_{1} by ℵ_{n}-closed forcing, and V_{1} is a model of □_{ℵ_{ω},ω} and so a fortiori is a model of AP_{ℵ_{ω}}. By Lemma 6.14, it follows that S is stationary in V ℝ̄ ; 4. If μ = sup j“κ^{+} then

M ⊧ “j(S) ∩ μ is stationary in μ, cf(μ) = ℵ_{n}”.

By elementarity,

V_{4} ⊧ “there is α < κ^{+} such that S ∩ α is stationary, cf(α) = ℵ_{n}”.

V_{4} = V_{3}^{𝕋_{n}∗ℚ} ⊇ V_{3}, so in V_{3} we see that S reflects to some ordinal of cofinality ℵ_{n}.

This concludes the proof of the theorem for one stationary set. Essentially the same proof will show that any finite family of stationary sets reflects simultaneously; however, as we would expect in the light of Theorems 4 and 5, the proof breaks down for an ω-sequence of stationary sets because we may not be able to choose a single value of n which works for every set in the sequence.

11. Prikry forcing

In this section we investigate the model obtained by doing Prikry forcing at κ where κ is κ^{+}-supercompact. This continues the work of Magidor and Ben-David [34].

Throughout this section κ is a measurable cardinal, U is a normal measure on κ, and ℙ is the Prikry forcing defined from U. Magidor and Ben-David [34] showed that if U is the normal measure generated by a κ^{+}-supercompact embedding then V^{ℙ} ⊧ □_{κ}^{∗} + ¬□_{κ}.

Theorem 19. In V let S_{0} =_{def} { α < κ^{+} : cf(α) < κ } and let S_{1} =_{def} { α < κ^{+} : cf(α) = κ }. Then in V^{ℙ}

1. S_{1} is a non-reflecting stationary set of cofinality ω ordinals.

2. If κ is κ^{+}-supercompact, Refl(< ω, S_{0}) holds.

3. Refl(ω, S_{0}) fails.

Proof. We take each claim in turn.

1. S_{0} and S_{1} are stationary in V and ℙ is κ^{+}-c.c. so that S_{0} and S_{1} are stationary in V^{ℙ}. cf(κ) = ω in V^{ℙ} so that S_{1} is a set of ordinals with cofinality ω in V^{ℙ}. Finally if cf(α) > ω in V^{ℙ} then ω < cf(α) < κ in V, so that if C ∈ V is a club set in α with ot(C) = cf(α) then C ∩ S_{1} = ∅.

2. Suppose that Refl(n, S_{0}) fails for some n < ω, and let

⊩_{ℙ} “⟨Ṫ_{i} : i < n⟩ are a counterexample to Refl(n, S_{0})”.

Let G be a ℙ-generic filter and for each i let T_{i} ∈ V [G] be the interpretation of Ṫ_{i}. If j < ω let G_{j} be the set of conditions (s, A) ∈ G such that lh(s) = j. Working in V [G] we define

T_{i}^{j} = { α ∈ S_{0} : ∃p ∈ G_{j} p ⊩ α̌ ∈ T_{i} }.

It is easy to see that ⟨T_{i}^{j} : j < ω⟩ is an increasing sequence of sets, whose union is T_{i}.

We choose j so large that T_{i}^{j} is stationary in V [G] for each i < n. Let s be the sequence consisting of the first j elements of the generic ω sequence added by G; every element of G_{j} has form (s, A) for some A ∈ U. Working in V, define

U_{i} = { α ∈ S_{0} : ∃A ∈ U (s, A) ⊩ α̌ ∈ Ṫ_{i} }.

Clearly T_{i}^{j} ⊆ U_{i}, and so U_{i} must be stationary in V.

κ is κ^{+}-supercompact so there exists α < κ^{+} such that ω < cf(α) < κ and U_{i} ∩ α is stationary in V for all i < n. Let D be club in α with ot(D) = cf(α), and for each β ∈ U_{i} ∩ D choose A_{β,i} ∈ U such that (s, A_{β,i}) ⊩ β ∈ Ṫ_{i}. Let A =_{def} ⋂_{β,i} A_{β}, where A ∈ U because |D| < κ.

Now (s, A) ⊩ U_{i} ∩ D ⊆ Ṫ_{i} ∩ α. U_{i} ∩ D is still a stationary set in V^{ℙ} because cf(α) < κ and ℙ does not affect the universe below κ. So

(s, A) ⊩ “Ṫ_{i} reflects at α for all i < n”,

which is a contradiction. So Refl(< ω, S_{0}) holds.

3. By a well known result of Solovay, in V we may write S_{0} = ⋃_{i<κ} T_{i} where the T_{i} are disjoint and stationary. In V^{ℙ} consider the sequence of stationary sets ⟨T_{κ_{n}} : n < ω⟩ where ⟨κ_{n} : n < ω⟩ is the cofinal ω-sequence in κ added by ℙ.

Suppose that in V^{ℙ} there is γ with cf(γ) > ω such that T_{κ_{n}} ∩ γ is stationary for each n. In V we have ω < cf(γ) < κ. Working in V define B =_{def} { i < κ : T_{i} ∩ γ is stationary }.

Now on the one hand κ_{n} ∈ B for all n, so that B is cofinal in κ. On the other hand |B| ≤ cf(γ) because the T_{i} are disjoint, and since cf(γ) < κ this is a contradiction.

Theorem 20. There is a very good scale at κ in V^{ℙ}.

Proof. Let ⟨κ_{n} : n < ω⟩ be the generic ω-sequence added by ℙ. Jech proved [22] that if we choose f_{ζ} with [f_{ζ}]_{U} = ζ for each ζ < κ^{+}, and then define f_{ζ}^{∗} ∈ ∏_{n} κ_{n}^{+} by f_{ζ}^{∗} : n ⟼ f_{ζ}(κ_{n}), then V^{ℙ} ⊧ “⟨f_{ζ}^{∗} : ζ < κ^{+}⟩ is a scale”.

We will show that this scale is very good. Fix ζ < κ^{+} such that V^{ℙ} ⊧ cf(ζ) = λ > ω. Then λ < κ and V ⊧ cf(ζ) = λ, so we choose D ∈ V such that D is club in ζ and ot(D) = λ. We will use D to witness that the scale is very good.

Let A =_{def} { α < κ : ⟨f_{γ}(α) : γ ∈ D⟩ is strictly increasing }. κ ∈ j(A), because j(D) = j“D and so

⟨j(f)_{γ}(κ) : γ ∈ j(D)⟩ = ⟨j(f_{γ})(κ) : γ ∈ D⟩ = ⟨γ : γ ∈ D⟩.

Therefore A ∈ U, and hence for all sufficiently large n we have κ_{n} ∈ A. This is precisely what is demanded in the definition of very good scale.

Remark 11.1. Cummings and Schimmerling show in [7] that Prikry forcing at κ always adds a □_{κ,ω}-sequence.

12. Square and reflection revisited

In this section we prove the following theorem.

Theorem 21. Let ⟨κ_{n} : n < ω⟩ be an increasing sequence of supercompact cardinals. Let κ = sup_{n} κ_{n}, and assume that GCH holds above κ.

Then there is a forcing ℙ such that in V^{ℙ}

1. κ = ℵ_{ω} and GCH holds.

2. □_{ℵ_{ω}}^{∗} holds.

3. Refl(ℵ_{n}, ℵ_{ω+1} ∩ cof(< ℵ_{n}), ℵ_{n}) holds for 1 ≤ n < ω.

We start by giving a rough outline of the proof.

1. We iterate Laver’s indestructibility forcing [30] to make each κ_{n} be indestructible under κ_{n}-directed closed forcing, and then force a certain combinatorial principle ♢_{κ^{+}}^{+} using κ^{+}-directed closed forcing. Let the resulting model be V′.

2. Working in V′ we iterate Coll(κ_{n}, < κ_{n+1}) as in Section 6, so that κ_{n} becomes ℵ_{n+1} and κ becomes ℵ_{ω}. We argue that ♢_{ℵ_{ω+1}}^{+} now holds, and we know that each κ_{n} is “generically indestructibly supercompact” in the sense of Lemma 6.10. Let the resulting model be V_{1}.

3. Working in V_{1} we force □_{ℵ_{ω}}^{∗} using some forcing 𝕊 whose definition involves the ♢_{ℵ_{ω+1}}^{+}-sequence; the aim is to add a □_{ℵ_{ω}}^{∗}-sequence with some strong closure properties. Let V_{2} = V_{1}^{𝕊}.

4. Now we need to prove that stationary reflection holds in V_{2}. Given a sequence ⟨S_{k} : k < ℵ_{n}⟩ ∈ V_{2} as in the statement of the theorem, we force over V_{2} with 𝕋_{n+1} which is the forcing to add a “threading” of the generic □_{ℵ_{ω}}^{∗}-sequence in order type ℵ_{n+1}.

The strong closure properties of the generic □_{ℵ_{ω}}^{∗}-sequence guarantee that 𝕋_{n+1} preserves the stationarity of the sets S_{k}. 𝕊 ∗ 𝕋_{n+1} has an ℵ_{n+1}-directed closed dense subset, the S_{k} are still stationary in V_{2}^{𝕋_{n+1}} = V_{1}^{𝕊∗𝕋_{n+1}}, and this enables us to appeal to the indestructible generic supercompactness of κ_{n} and show that in V_{2}^{𝕋_{n+1}} there is a single β of cofinality ℵ_{n} at which every set S_{k} reflects. Stationarity is downwards absolute, so the S_{k} reflect simultaneously in V_{2}.

12.1. Strong diamond. The principle ♢_{λ^{+}}^{+} was introduced by Jensen [8]. He showed that it holds in L and gave a forcing construction to add a ♢_{λ^{+}}^{+}-sequence; we give a different forcing construction here, which is better adapted to our purposes.

Definition 12.1. A ♢_{λ^{+}}^{+}-sequence is a sequence ⟨S_{α} : α < λ^{+}⟩ such that

1. S_{α} ⊆ P(α), |S_{α}| ≤ λ.

2. For every X ⊆ λ^{+} there is C ⊆ λ^{+} closed and unbounded, such that if α ∈ lim(C) then X ∩ α ∈ S_{α} and C ∩ α ∈ S_{α}.

We describe a forcing construction which will add such a sequence.

Theorem 22. Let 2^{λ} = λ^{+} and 2^{λ^{+}} = λ^{++}. Then there exists a λ^{+}- directed-closed and λ^{++}-c.c. forcing ℙ such that ♢_{λ^{+}}^{+} holds in V^{ℙ}.

Proof. The idea of the proof will be to add a sequence S⃗ of the right general form, and then to add club sets which will eventually witness that S⃗ is a ♢_{λ^{+}}^{+}-sequence.

Definition 12.2. p ∈ ℚ_{0} if 1. p is a function with dom(p) ∈ λ^{+}, dom(p) a successor ordinal.

2. For each α ∈ dom(p), p(α) ⊆ P(α).

3. |p(α)| ≤ λ for all α ∈ dom(p). If p, q ∈ ℚ_{0} then p ≤ q iff dom(q) ≤ dom(p) and p ↾ dom(q) = q.

It is easy to see that ℚ_{0} is λ^{+}-directed-closed and λ^{++}-c.c. In fact ℚ_{0} is just a version of the forcing to add one Cohen subset of λ^{+}. Forcing with ℚ_{0} adds a sequence of sets which has the right form to be a ♢_{λ^{+}}^{+}-sequence.

We will now build a forcing iteration ⟨ℙ_{α}, ℚ̇_{α} : α < λ^{++}⟩ of length λ^{++}, with supports of size ≤ λ. Let ℚ_{0} be the forcing we just defined, and let S⃗ be the λ^{+}-sequence of sets added by ℚ_{0}. At stages α ≥ 1 we will choose a ℙ_{α}-name Ẋ_{α} for a subset of λ^{+}; we will see that ℙ_{λ^{++}} is a λ^{++}-c.c. forcing poset of size λ^{++}, and so will be able to choose ⟨Ẋ_{α} : α < λ^{++}⟩ in such a way that every subset of λ^{+} in V^{ℙ_{λ^{++}}} is denoted by some Ẋ_{α}. ℚ_{α} will then be the forcing defined in V^{ℙ_{α}} whose conditions are closed bounded sets c ⊆ λ^{+} such that

∀β (β ∈ lim(c) ⟹ c ∩ β, X_{α} ∩ β ∈ S_{β}),

and whose ordering is end-extension.

A priori it is not clear that ℚ_{α} will be well-behaved, for example it seems that it may add a generic club set of order type ω. We will see that this does not happen; the key point in the proof will be that ℙ_{α} adds no λ-sequences of ordinals to V, so that each condition in ℚ_{α} and each initial segment of X_{α} actually lies in V.

To analyse the properties of the iteration ℙ_{α}, we will study a certain subset of the conditions.

Definition 12.3. Let 1 ≤ α < λ^{++}, and let ℙ_{α} be stage α of the forcing iteration described above. Let p ∈ ℙ_{α}. Then p is flat iff there exists β < λ^{+} such that

1. dom(p(0)) = β + 1.

2. For all α ∈ dom(p) with α > 0, (a) p ↾ α decides Ẋ_{α} ∩ β.

(b) p ↾ α ⊩ p(α) = Č_{α}, where C_{α} ∈ V is a closed set with

maximum element β. ℙ_{α}^{∗} is the subset of ℙ_{α} consisting of flat conditions. If p ∈ ℙ_{α}^{∗},

the height of p is the unique β with the properties above (that is dom(p(0)) − 1)

If p is a flat condition, we will not distinguish between p(α) (which is technically a ℙ_{α}-name) and the set in V to which p ↾ α forces it to be equal. Notice that if p ∈ ℙ_{α}^{∗} then p ↾ β ∈ ℙ_{β}^{∗} for all β < α.

Lemma 12.4. Let 1 ≤ α < λ^{++}. Then

1. ℙ_{α}^{∗} is λ^{+}-directed-closed. Moreover, if ⟨p_{γ} : γ < μ⟩ is a directed set of flat conditions for some μ < λ^{+}, and the height of p_{γ} is σ_{γ}, then there is a greatest lower bound (in ℙ_{α}) p which is given by (a) supp(p) = ⋃ supp(p_{γ}).

(b) dom(p(0)) = σ + 1, where σ = sup_{γ} σ_{γ}.

(c) p(0) ↾ σ = ⋃ p_{γ}(0).

(d) For β > 0, p(β) = ⋃ { p_{γ}(β) : β ∈ dom(p_{γ}) } ∪ {σ}.

(e) p(0)(σ) = { p(β) ∩ σ : β ∈ dom(p) } ∪ { x_{β} : β ∈ dom(p) }, where x_{β} is the subset of σ such that p ↾ β ⊩ Ẋ_{β} ∩ σ = x_{β}.

p ∈ ℙ_{α}^{∗} and p has height σ.

2. ℙ_{α}^{∗} is dense in ℙ_{α}.

Proof. The first claim is easy to verify. We prove the second claim by induction on α. Suppose ℙ∗ ; ᾱ is dense in ℙᾱ for all ᾱ < α. We show ℙ_{α}^{∗} is dense in ℙ_{α}.

Case 1: α = 1, ℙ_{1}^{∗} = ℙ_{1} ≃ ℚ_{0} and there is nothing to prove.

Case 2: α = β + 1. Fix a condition p ∈ ℙ_{β+1}^{∗}.

By induction, we know that ℙ_{β}^{∗} is λ^{+}-closed and dense in ℙ_{β}. In particular ℙ_{β} adds no bounded subsets of λ^{+}, and so we may choose p_{0} ≤ p ↾ β such that p_{0} decides the value of p(β), say that p_{0} forces p(β) to equal c where max(c) = γ. As ℙ_{β}^{∗} is dense, we may choose p_{0} ∈ ℙ_{β}^{∗} and by extending if necessary may also assume that the height of p_{0} is greater than γ.

Now we argue in a similar vein to build a decreasing ω-sequence of conditions p_{0} > p_{1} > p_{2} > … and an increasing ω-sequence of ordinals ρ_{0} < ρ_{1} < …, where p_{n} is a condition in ℙ_{β}^{∗} of height ρ_{n}, and p_{n+1} decides Ẋ_{β} ∩ ρ_{n}; say p_{n+1} ⊩ Ẋ_{β} ∩ ρ_{n} = x_{n}. By the first claim of the Lemma, we may form a greatest lower bound for p⃗ which will be a condition q ∈ ℙ_{β}^{∗} of height ρ = sup ρ_{n}.

Now we define q^{+} as follows. dom(q^{+}) = β + 1, and q^{+}(γ) = q(γ) for 0 < γ < β. q^{+}(β) = c ∪ {ρ_{0}, ρ_{1}, …, ρ}. q^{+}(0) ↾ ρ = q(0) ↾ ρ, and q^{+}(0)(ρ) = q(0)(ρ) ∪ {q^{+}(β), ⋃_{n} x_{n}}.

It is now routine to check that q^{+} ∈ ℙ_{β+1}, q^{+} is flat of height ρ, and q^{+} refines p.

Case 3: α is limit with cf(α) ≥ λ^{+}. Fix p ∈ ℙ_{α}, then the support of p is bounded by some β < α. By induction we may find q ≤ p ↾ β with q ∈ ℙ_{β}^{∗}; if q^{+} ∈ ℙ_{α} is defined by q^{+} ↾ β = q and q^{+} ↾ [β, α) = 0 then q^{+} ∈ ℙ_{α}^{∗} and q^{+} ≤ p, as required.

Case 4: α is limit and cf(α) ≤ λ. Choose a sequence ⟨α_{i} : i < cf(α)⟩ which is increasing, continuous and cofinal in α. Fix p ∈ ℙ_{α}.

We will define a decreasing sequence of conditions ⟨p_{i} : i ≤ cf(α)⟩ such that p_{0} ≤ p, p_{i} ↾ α_{i} ∈ ℙ_{α_{i}}^{∗} for each i, and p_{cf(α)} ∈ ℙ_{α}^{∗}. We let σ_{i} denote the height of p_{i} ↾ α_{i}. i = 0. Let q_{0} ≤ p ↾ α_{0}, q_{0} ∈ ℙ_{α_{0}}^{∗}. Let p_{0} ↾ α_{0} = q_{0}, p_{0} ↾ [α_{0}, α) = p ↾ [α_{0}, α)

i = j + 1. Let q_{i} ≤ p_{j} ↾ α_{i}, q_{i} ∈ ℙ_{α_{i}}^{∗}. Now let p_{i} ↾ α_{i} = q_{i}, and p_{i} ↾ [α_{i}, α) = p ↾ [α_{i}, α).

i is limit. For each j < i consider the sequence ⟨p_{k} ↾ α_{j} : j ≤ k < i⟩. This is a decreasing sequence from ℙ_{α_{j}}^{∗} so by the first claim of the Lemma we can form a greatest lower bound r_{j}, where r_{j} ∈ ℙ_{α_{j}}^{∗} and r_{j} has height σ = sup_{k<i} σ_{k}.

It is easy to see that if m < n then r_{m} ↾ (0, α_{m}) = r_{n} ↾ (0, α_{m}), r_{m}(0) ↾ σ = r_{n}(0) ↾ σ, and r_{m}(0)(σ) ⊆ r_{n}(0)(σ). We define q_{i} such that dom(q_{i}) = α_{i}, q_{i} ↾ (0, α_{j}) = r_{j} ↾ (0, α_{j}) for all j, q_{i}(0) ↾ σ is the common value of r_{j}(0) ↾ σ, and q_{i}(0)(σ) = ⋃_{j<i} r_{j}(0)(σ).

It is routine to check that q_{i} ∈ ℙ_{α_{i}}^{∗} and q_{i} has height σ, also that q_{i} ≤ p_{j} ↾ α_{i} for each j < i. Now let p_{i} ↾ α_{i} = q_{i}, p_{i} ↾ [α_{i}, α) = p ↾ [α_{i}, α).

The construction for the limit step also works for i = cf(α), and produces p_{cf(α)} which is in ℙ_{α}^{∗} and refines p.

Using Lemma 12.4, it is easy to see that ℙ =_{def} ℙ_{λ^{++}}^{∗} is λ^{+}-directed-closed however the sets Ẋ_{α} are chosen. ℙ is λ^{++}-c.c. by a standard Δ-system argument. Therefore we may choose the names Ẋ_{α} in such a way that every subset of λ^{+} in V^{ℙ} is denoted by some Ẋ_{α}.

Now if X ∈ V^{ℙ} is any subset of λ^{+}, and X = X_{α}, then the club set C_{α} added at stage α in the iteration will have the property that ∀β ∈ lim(C) X ∩β, C ∩β ∈ S_{β}. This is to say that S⃗ is a ♢_{λ^{+}}^{+}-sequence in V^{ℙ}.

Remark 12.5. Jensen also defined ♢_{κ}^{+} for κ inaccessible, but that definition demands |S_{α}| ≤ α. In general (say for κ a Laver indestructible supercompact) it will not be possible to force ♢_{κ}^{+} with κ-directed closed forcing, because ♢_{κ}^{+} implies the existence of a κ-Kurepa tree.

We now prove that forcing of size λ^{+} which preserves the regularity of λ^{+} will also preserve ♢_{λ^{+}}^{+}. For technical reasons we need some precise information about the definability of a ♢_{λ^{+}}^{+}-sequence in the extension from data in the ground model and the generic filter.

Lemma 12.6. Let ℙ be such that |ℙ| = λ^{+} and ⊩_{ℙ} “λ^{+} is regular”. Let ⟨q_{α} : α < λ^{+}⟩ be such that ℙ ⊆ { q_{α} : α < λ^{+} }.

Let f be a bijection between λ^{+} and λ^{+} × λ^{+} × 3, and let g be a bijection between λ^{+} and λ^{+} × λ^{+} × λ^{+} × 3. Let ⟨S_{α} : α < λ^{+}⟩ be a ♢_{λ^{+}}^{+}-sequence.

Then 1. There exists ⟨Ṫ_{α} : α < λ^{+}⟩ such that if G is ℙ-generic and T_{α} =

Ṫ_{α}^{G} then

• ⟨T_{α} : α < λ^{+}⟩ is a ♢_{λ^{+}}^{+} sequence in V [G].

• T_{α} is uniformly Δ_{0}-definable from { γ < α : q_{γ} ∈ G }, S_{α},

and f.

2. There exists ⟨U̇_{α} : α < λ^{+}⟩ such that if G is ℙ-generic and U_{α} = U̇_{α}^{G} then

• For all α < λ^{+}, |U_{α}| ≤ λ.

• For all A ∈ V [G], if A ⊆ λ^{+} × λ^{+} there is C ∈ V [G] such that C is club in λ^{+} and ∀α ∈ lim(C) A ∩ α × α, C ∩ α ∈ U_{α}.

• U_{α} is uniformly Δ_{0} in { γ < α : q_{γ} ∈ G }, S_{α}, g.

Proof. We describe the construction of Ṫ_{α}, the construction for U̇_{α} will be exactly parallel (and so we omit it). Let E be the club set of α < λ^{+} such that f ↾ α is a bijection between α and α × α × 3.

For α ∉ E we let Ṫ_{α} name ∅. For α ∈ E and x, y ∈ S_{α} we define

B_{0}(x) = { β < α : ∃γ < α q_{γ} ∈ G, (γ, β, 1) ∈ f“x },

and

B_{1}(x, y) = { δ ∈ y ∩ E : ∀β < δ ∃γ < δ ∃i < 2 q_{γ} ∈ G, (γ, β, i) ∈ f“x }.

We now define Ṫ_{α} to be a name for

{ B_{0}(x) : x ∈ S_{α} } ∪ { B_{1}(x, y) : x, y ∈ S_{α} }.

Notice that this is uniformly Δ_{0} in the parameters { γ < α : q_{γ} ∈ G }, S_{α}, f.

We now check that we have defined a name for a ♢_{λ^{+}}^{+}-sequence. Let G be ℙ-generic, and for each α let T_{α} = Ṫ_{α}^{G}. Let A ∈ V [G] be a subset of λ^{+}, and let A = τ̇^{G} for some ℙ-name τ̇.

Define Y ⊆ λ^{+} × λ^{+} × 3 as follows; (γ, β, i) ∈ Y iff q_{γ} ∈ ℙ and one of the following holds.

• i = 0 and q_{γ} ⊩ β̌ ∉ τ̇.

• i = 1 and q_{γ} ⊩ β̌ ∈ τ̇.

• i = 2 and q_{γ} ∦ β̌ ∈ τ̇. Let X = { α : f(α) ∈ Y }.

Since S⃗ is a ♢_{λ^{+}}^{+}-sequence we may find C ⊆ λ^{+} club in λ^{+} such that ∀α ∈ lim(C) X ∩ α, C ∩ α ∈ S_{α}. Working in V [G], we now define

D = { α : ∀β < α ∃γ < α q_{γ} ∈ G, q_{γ} ∥ β̌ ∈ τ̇ }.

Since V [G] ⊧ “λ^{+} is regular”, D is a club subset of λ^{+}. Let F = C ∩ D ∩ E, then F is a club subset of λ^{+}.

Claim. ∀α ∈ lim(F) A ∩ α, F ∩ α ∈ T_{α}.

Proof. Fix α ∈ lim(F). As α ∈ lim(C), X ∩ α ∈ S_{α} and C ∩ α ∈ S_{α}.

As α ∈ E, f“X ∩ α = Y ∩ α × α × 3. That is to say, for all γ, β < α if q_{γ} ∈ ℙ then

• q_{γ} ⊩ β̌ ∈ τ̇ ⟺ (γ, β, 1) ∈ f“X ∩ α.

• q_{γ} ∥ β̌ ∈ τ̇ ⟺ ∃i < 2 (γ, β, i) ∈ f“X ∩ α.

Now let β < α.

β ∈ A ⟺ ∃p ∈ G p ⊩ β ∈ τ̇
⟺ ∃γ < α q_{γ} ∈ G, q_{γ} ⊩ β̌ ∈ τ̇ (as α ∈ D)
⟺ ∃γ < α q_{γ} ∈ G, (γ, β, 1) ∈ f“X ∩ α
⟺ β ∈ B_{0}(X ∩ α)

Hence A ∩ α = B_{0}(X ∩ α) ∈ T_{α}.

Similarly if δ < α then

δ ∈ D ⟺ ∀β < δ ∃γ < δ q_{γ} ∈ G, q_{γ} ∥ β̌ ∈ τ̇
⟺ ∀β < δ ∃γ < δ q_{γ} ∈ G, ∃i < 2 (γ, β, i) ∈ f“X ∩ α

Hence F ∩ α = B_{1}(X ∩ α, C ∩ α) ∈ T_{α}. The claim is proved.

This concludes the proof of Lemma 12.6.

12.2. The models V′ and V_{1}. Recall that we are beginning with a model V in which GCH holds above κ, and ⟨κ_{n} : n < ω⟩ is an increasing sequence of supercompact cardinals. We start by iterating the forcing described in Fact 6.1 to make each κ_{n} be Laver indestructible, and then force ♢_{κ^{+}}^{+} using the forcing iteration of the last section. We obtain a model V′ in which ♢_{κ^{+}}^{+} holds, and in which it is still true that each κ_{n} is supercompact and that GCH holds above κ.

Now we force over V′ with an iteration of the sort described in Fact 6.10. The iteration has the form ⟨ℙ_{n}, ℚ̇_{n} : n < ω⟩ where ℚ_{0} = Coll(ω, < κ_{0}) and ℚ_{n} = Coll(κ_{n−1}, < κ_{n})_{(V′)^{ℙn}} for n > 0. We let ℙ_{ω} be the inverse limit of this iteration, and then let V_{1} = (V′)^{ℙ_{ω}}.

We refer the reader back to Fact 6.10 for a detailed account of the properties of V_{1}. Recall in particular that in V_{1} we have κ_{n} = ℵ_{n+1}, κ = ℵ_{ω}, κ^{+} = ℵ_{ω+1}. Since |ℙ_{ω}| = ℵ_{ω+1} and ℙ_{ω} preserves the regularity of κ^{+}, Lemma 12.6 tells us that ♢_{ℵ_{ω+1}}^{+} holds in V_{1}.

12.3. The model V_{2}. Working in V_{1}, we define a forcing 𝕊 to add a □_{ℵ_{ω}}^{∗}-sequence. For technical reasons (see Lemma 12.15) we will need to use a forcing more complex than the one defined in Definition 6.2. Conditions will be functions where dom(p) = α + 1 for some limit ordinal α < ℵ_{ω+1} (we call α the height of p and write α = ht(p)) and we define the set of conditions of height α by induction on α. We will fix some large regular cardinal λ and a well ordering <_{λ} of H_{λ}; given X ⊆ H_{λ} we denote by Sk_{λ}(X) the Skolem hull of X in the structure (H_{λ}, ∈, <_{λ}). Let S⃗ be the <_{λ}-least ♢_{ℵ_{ω+1}}^{+}-sequence.

Remark 12.7. Since ℵ_{ω} is strong limit, if ℵ_{ω} ⊆ X ⊆ H_{λ} then every bounded subset of ℵ_{ω} is in Sk_{λ}(X).

Remark 12.8. In what follows we will consider Sk_{λ}(A) for various sets A defined in a generic extension V_{1}^{𝕊} of V_{1}. Every A for which we do this will actually lie in V_{1} and will be a subset of H_{λ}^{V_{1}}.

Definition 12.9. p ∈ 𝕊 iff

1. p = ⟨C_{β} : β ≤ α, lim(β)⟩ where α < ℵ_{ω+1}, |C_{β}| ≤ ℵ_{ω}, and C_{β} is a nonempty set of club subsets of β each with order type less than ℵ_{ω}.

2. For β ≤ α, if C ∈ C_{β} and γ ∈ lim(C) then C ∩ γ ∈ C_{γ}.

3. For all β < α, p ↾ β + 1 ∈ 𝕊.

4. There is a set A_{α}^{p} ⊆ H_{λ} such that ℵ_{ω} ∪ α ⊆ A_{α}^{p}, |A_{α}^{p}| = ℵ_{ω} and C_{α} is equal to the set of C such that

(a) C is club in α, ot(C) < ℵ_{ω}.

(b) C ∈ Sk_{λ}(A_{α}^{p} ∪ {p ↾ α} ∪ {S⃗ ↾ α + 1}).

(c) ∀γ ∈ lim(C) C ∩ γ ∈ C_{γ}. If p, q ∈ 𝕊 then p ≤ q iff ht(q) ≤ ht(p) and p ↾ ht(q) + 1 = q.

Lemma 12.10. Let p = ⟨C_{β} : β ≤ α⟩ ∈ 𝕊. If C ∈ C_{β} then every closed and unbounded subset of C is also in C_{β}. In particular, C_{β} contains a set of order type cf(β).

Proof. Fix ⟨A_{β}^{p} : β ≤ α⟩ witnessing clause 4 above for β ≤ α. Let D be a closed and unbounded subset of C. Since ot(C) < ℵ_{ω} and ℵ_{ω} ⊆ A_{β}^{p} we see by Remark 12.7 that { ot(C ∩ γ) : γ ∈ D } ∈ Sk_{λ}(A_{β}^{p}), so that D ∈ Sk_{λ}(A_{β}^{p} ∪ {p ↾ β} ∪ {S⃗ ↾ β + 1}). We need to check that ∀γ ∈ lim(D) D ∩ γ ∈ C_{γ}, which we will do by induction.

If γ ∈ lim(D) then γ ∈ lim(C), so C ∩ γ ∈ C_{γ}. D ∩ γ is a closed unbounded subset of C ∩ γ, so by induction D ∩ γ ∈ C_{γ} and we are done.

It will be important to know that 𝕊 adds no new ℵ_{ω}-sequences of ordinals. Since ℵ_{ω} is singular the next lemma will suffice for this.

Lemma 12.11. If n < ω then 𝕊 is (ℵ_{n} + 1)-strategically closed.

Proof. We describe a strategy for player II in the game G_{ℵ_{n}+1}^{II}(𝕊). The strategy is similar to that for the forcing ℙ(κ, λ) from Definition 6.2. Let the move made at stage i be p_{i} = ⟨C_{β} : β ≤ α_{i}⟩.

At limit stages i: let α_{i} = sup_{j<i} α_{j}, and let C = { α_{j} : j < i }. Let A = ℵ_{ω} ∪ α_{i} ∪ {C} and define C_{α_{i}} to be the set of all those D such that

1. D is club in α_{i} with order type less than ℵ_{ω}.

2. ∀β ∈ lim(D) D ∩ β ∈ C_{β}.

3. D ∈ Sk_{λ}(A ∪ {⟨C_{β} : β < α_{i}⟩} ∪ {S ↾ α_{i} + 1}). II then plays p_{i+2} = ⟨C_{β} : β ≤ α_{i}⟩.

The key point here is that by induction { α_{j} : j < i } ∈ C_{α_{i}} for every limit i ≤ ℵ_{n}, so that at each limit stage the strategy gives a nonempty set C_{α_{i}} and player II makes a legal move. At successor stages i+2, i even: let α_{i+2} = α_{i+1}+ω, C = α_{i+2}\α_{i+1}. Let A = ℵ_{ω} ∪ α_{i+2} ∪ {C} and define C_{α_{i}} to be the set of all those D such that

1. D is club in α_{i} with order type less than ℵ_{ω}.

2. ∀β ∈ lim(D) D ∩ β ∈ C_{β}.

3. D ∈ Sk_{λ}(A ∪ {p_{i+1}} ∪ {S ↾ α_{i+2} + 1}). II then plays p_{i} = ⟨C_{β} : β ≤ α_{i+2}⟩. In this case it is clear that C ∈ C_{α_{i}} since C has no limit points.

Now let q⃗ = ⟨q_{α} : α < ℵ_{ω+1}⟩ be the <_{λ}-least sequence such that 1. q⃗ enumerates the set of all functions q such that dom(q) is a

successor ordinal less than ℵ_{ω+1} and range(q) ⊆ H_{ℵ_{ω+1}}.

2. For all α, dom(q_{α}) ≤ α. Notice that 𝕊 ⊆ { q_{α} : α < ℵ_{ω+1} }.

Let g be the <_{λ}-minimal bijection between ℵ_{ω+1} and ℵ_{ω+1} × ℵ_{ω+1} × ℵ_{ω+1} × 3. Applying Lemma 12.6 we fix ⟨U̇_{α} : α < ℵ_{ω+1}⟩ such that if G is 𝕊-generic over V_{1} and U_{α} = U̇_{α}^{G} then

• For all α < ℵ_{ω+1}, |U_{α}| ≤ ℵ_{ω}.

• For all A ∈ V_{1}[G], if A ⊆ ℵ_{ω+1} × ℵ_{ω+1} there is C ∈ V_{1}[G] such that C is club in ℵ_{ω+1} and ∀α ∈ lim(C) A ∩ α × α, C ∩ α ∈ U_{α}.

• U_{α} is uniformly Δ_{0} in { γ < α : q_{γ} ∈ G }, S_{α}, g.

Lemma 12.12. Let α < ℵ_{ω+1}.

1. If G is generic and U_{β} = U̇_{β}^{G} for all β, then U_{α} ∈ V_{1} and U⃗ ↾ α ∈ V_{1}.

2. If G is 𝕊-generic over V_{1}, then { γ < α : q_{γ} ∈ G } ∈ V_{1}.

3. If p ∈ 𝕊 is a condition of height α and G is some 𝕊-generic filter with p ∈ G then

{ γ < α : q_{γ} ∈ G } = { γ < α : q_{γ} is an initial segment of p ↾ α }

and so { γ < α : q_{γ} ∈ G } ∈ Sk_{λ}({p ↾ α}).

4. If p ∈ 𝕊 is a condition of height α and G is some 𝕊-generic filter with p ∈ G then U_{α} ∈ Sk_{λ}({p ↾ α, S_{α}}).

Proof. The first two claims are immediate from the fact that 𝕊 adds no ℵ_{ω}-sequences of ordinals. For the third claim, observe that if γ < α then dom(q_{γ}) ≤ γ < α, so that q_{γ} ∈ G exactly when q_{γ} is an initial segment of p ↾ α. q⃗ ∈ Sk_{λ}(∅) and { γ < α : q_{γ} ∈ G } is definable in H_{λ} from p ↾ α and q⃗, so { γ < α : q_{γ} ∈ G } ∈ Sk_{λ}({p ↾ α}).

For the last claim, observe that U_{α} is definable (in fact Δ_{0}-definable) from the parameters { γ < α : q_{γ} ∈ G }, S_{α}, and g. By the choice of g, g ∈ Sk_{λ}(∅). We saw in the third claim that { γ < α : q_{γ} ∈ G } ∈ Sk_{λ}({p ↾ α}), so U_{α} ∈ Sk_{λ}({p ↾ α, S_{α}}).

Lemma 12.13. Let G be 𝕊-generic and let ⟨C_{α} : α < ℵ_{ω+1}⟩ be the □_{ℵ_{ω}}^{∗}- sequence added by G. Let α < ℵ_{ω+1}. Let C ∈ V_{1} be such that

1. C is club in α, with order type less than ℵ_{ω}.

2. ∀β ∈ lim(C) C ∩ β ∈ C_{β}.

3. C ∈ Sk_{λ}(ℵ_{ω} ∪ U_{α} ∪ C_{α}). Then C ∈ C_{α}.

Proof. Let p = ⟨C_{β} : β ≤ α⟩. From the definition of 𝕊 we may fix a set A_{α}^{p} ⊇ ℵ_{ω} such that C_{α} is equal to the set of D such that

1. D is club in α, ot(D) < ℵ_{ω}.

2. D ∈ Sk_{λ}(A_{α}^{p} ∪ {p ↾ α} ∪ {S⃗ ↾ α + 1}).

3. ∀γ ∈ lim(D) D ∩ γ ∈ C_{γ}.

Since |U_{α}| ≤ ℵ_{ω}, we see by Remark 12.12 that U_{α} ⊆ Sk_{λ}(ℵ_{ω} ∪ {p ↾ α} ∪ {S_{α}}). Thus C ∈ Sk_{λ}(A_{α}^{p} ∪ {p ↾ α} ∪ {S⃗ ↾ α + 1}), and it follows that C ∈ C_{α}.

Let V_{2} = V_{1}^{𝕊}. Working in V_{2}, we define a forcing 𝕋_{n} for each n by setting 𝕋_{n} =_{def} 𝕋_{ℵ_{n}}(C⃗)_{V_{2}} where 𝕋_{ℵ_{n}}(C⃗) is the thread forcing defined in Definition 6.8. Of course 𝕋_{n} ∉ V_{1}, but the strategic closure of 𝕊 implies that 𝕋_{n} ⊆ V_{1}.

In V_{1} let x⃗ = ⟨x_{α} : α < ℵ_{ω+1}⟩ be the <_{λ}-minimal sequence such that • x⃗ enumerates all bounded subsets of ℵ_{ω+1}.

• For all α < ℵ_{ω+1}, x_{α} ⊆ α. Notice that 𝕋_{n} ⊆ { x_{α} : α < ℵ_{ω+1} }.

The following lemma is analogous to Lemmas 6.9 and 9.6.

Lemma 12.14. 𝕊 ∗ 𝕋_{n} has an ℵ_{n}-directed closed subset.

Proof. We let D be the set of conditions (p, č) such that ht(p) = max(c). Arguing as in the proof of Lemma 6.9 we may show that D is dense and the ordering on D is treelike, so to finish the proof it suffices to show that D is ℵ_{n}-directed closed.

Let γ < ℵ_{n} and let ⟨(p_{i}, c_{i}) : i < γ⟩ be a decreasing sequence of conditions in D, where ht(p_{i}) = α_{i} and p_{i} = ⟨C_{δ} : δ ≤ α_{i}⟩. We define α = sup_{i<γ} α_{i} and C = ⋃_{i<γ} c_{i}, from which it follows easily that C is club in α and ∀β ∈ lim(C) C ∩ β ∈ C_{β}.

Let A = ℵ_{ω} ∪ α ∪ {C} and define C_{α} to be the set of all those D such that

1. D is club in α_{i} with order type less than ℵ_{ω}.

2. ∀β ∈ lim(D) D ∩ β ∈ C_{β}.

3. D ∈ Sk_{λ}(A ∪ {⟨C_{β} : β < α_{i}⟩} ∪ {S ↾ α_{i} + 1}). Notice that C ∈ C_{α}.

Then (⟨C_{i} : i ≤ α⟩, C ∪ {α}) is a condition in D which is a lower bound for ⟨(p_{i}, c_{i}) : i < γ⟩.

The following lemma is the key to the proof of Theorem 21.

Lemma 12.15. Let 1 ≤ n < ω. If

V_{2} ⊧ “S is a stationary subset of ℵ_{ω+1} ∩ cof(< ℵ_{n})”

then S is stationary in V_{2}^{𝕋_{n}}.

Proof. Let us fix G which is 𝕊-generic over V_{1}, and let C⃗ be the weak square sequence added by G. Let V_{2} = V_{1}[G], and work in V_{2}. Suppose that S is a counterexample to the statement of the lemma, and fix p_{0}, Ḋ such that

p_{0} ⊩_{𝕋_{n}}^{V_{2}} Ḋ is club in ℵ_{ω+1} and Ḋ ∩ Š = ∅.

For each α < ℵ_{ω+1} let

D_{α} = { q ∈ 𝕋_{n} : max(q) ≥ α, ∃β > α q ⊩ β̌ ∈ Ḋ }.

D_{α} is dense below p_{0} in 𝕋_{n}. Now we define Z ⊆ ℵ_{ω+1} × ℵ_{ω+1} by Z = { (β, γ) : x_{γ} ∈ D_{β} }.

By the construction of U⃗, we may fix C club in ℵ_{ω+1} such that for all δ ∈ lim(C)

• C ∩ δ ∈ U_{δ}.

• Z ∩ δ × δ ∈ U_{δ}.

It follows that if δ ∈ lim(C) then ⟨D_{β} ∩ { x_{γ} : γ < δ } : β < δ⟩ ∈ Sk_{λ}(U_{δ}).

For each δ < ℵ_{ω+1} let

N_{δ} = Sk_{λ}(ℵ_{ω} ∪ δ ∪ { U_{α} : α < δ } ∪ { C_{α} : α < δ }),

and observe that N_{δ} ∩ ℵ_{ω+1} ∈ ℵ_{ω+1} and that ⟨N_{δ} ∩ ℵ_{ω+1} : δ < ℵ_{ω+1}⟩ is continuous. Also ⟨N_{δ} ∩ ℵ_{ω+1} : δ < ℵ_{ω+1}⟩ ∈ V_{2}.

Since S is stationary in V_{2} we may find ζ such that

1. p_{0} ∈ N_{ζ}.

2. ζ ∈ lim(C) ∩ S.

3. N_{ζ} ∩ ℵ_{ω+1} = ζ.

4. N_{ζ} ∩ 𝕋_{n} = { x_{α} : α < ζ, x_{α} ∈ 𝕋_{n} }.

5. For all β < ζ, D_{β} ∩ N_{ζ} is dense below p_{0} in 𝕋_{n} ∩ N_{ζ}.

As ζ ∈ S we know that cf(ζ) < ℵ_{n}, say cf(ζ) = ℵ_{m} for some m < n. Using Lemma 12.13, we choose D ∈ C_{ζ} such that ot(D) = ℵ_{m}. Let D be enumerated in increasing order as ⟨α_{i} : i < ℵ_{m}⟩. We will build ⟨p_{i} : i ≤ ℵ_{m}⟩ in such a way that

1. For i ≤ ℵ_{m}, p_{i} ∈ 𝕋_{n}.

2. For i < ℵ_{m}, p_{i} ∈ N_{ζ} and p⃗ ↾ i ∈ N_{ζ}. Let γ_{j} = max(p_{j}). We describe the construction of p⃗. Case 1: Suppose that i < ℵ_{m} and we have defined ⟨p_{j} : j ≤ i⟩. Choose β_{i} to be the least ordinal greater than γ_{i} such that both C and D have nonempty intersection with the interval (γ_{i}, β_{i}).

D_{β_{i}} ∩N_{ζ} is dense in 𝕋_{n} ∩N_{ζ} and N_{ζ} ∩𝕋_{n} = { x_{α} : α < ζ, x_{α} ∈ 𝕋_{n} }. We choose p_{i+1} to be x_{γ}, where γ < ζ is chosen minimal such that x_{γ} ≤ p_{i} and x_{γ} ∈ D_{β_{i}}. Case 2: i < ℵ_{m} is limit and we have defined ⟨p_{j} : j < i⟩. Let γ_{i} = sup_{j<i} γ_{j}, c_{i} = ⋃_{j} p_{j}, p_{i} = c_{i} ∪ {γ_{i}}. We need to show that p_{i} ∈ 𝕋_{n} ∩ N_{ζ} and that ⟨p_{j} : j < i⟩ ∈ N_{ζ}.

Notice that by construction γ_{i} ∈ lim(C)∩lim(D). Since γ_{i} ∈ lim(D), D ∩ γ_{i} ∈ C_{γ_{i}}. Since γ_{i} ∈ lim(C), C ∩ γ_{i} ∈ U_{γ_{i}} and

⟨D_{β} ∩ ⟨x_{δ} : δ < γ_{i}⟩ : β < γ_{i}⟩ ∈ Sk_{λ}(ℵ_{ω} ∪ U_{γ_{i}}).

Now ⟨p_{j} : j < i⟩ is definable in H_{λ} from the parameters D∩γ_{i}, C ∩γ_{i}, ⟨D_{β} ∩ ⟨x_{δ} : δ < γ_{i}⟩ : β < γ_{i}⟩. Therefore ⟨p_{j} : j < i⟩ ∈ Sk_{λ}(ℵ_{ω} ∪ U_{γ_{i}} ∪ C_{γ_{i}}). Since c_{i} and p_{i} are definable from ⟨p_{j} : j < i⟩, we see that

c_{i}, p_{i} ∈ Sk_{λ}(ℵ_{ω} ∪ U_{γ_{i}} ∪ C_{γ_{i}})

By the definition of N_{ζ}, we see that p_{i} and p⃗ ↾ i are in N_{ζ}.

By construction c_{i} is a club subset of γ_{i} such that ∀δ ∈ lim(c_{i}) c_{i}∩δ ∈ C_{δ}, so by Lemma 12.13 c_{i} ∈ C_{γ_{i}}. Since i < ℵ_{m} < ℵ_{n} and each p_{j} is in 𝕋_{n}, ot(c_{i}) < ℵ_{n} and so p_{i} ∈ 𝕋_{n}. Case 3: We have defined ⟨p_{j} : j < ℵ_{m}⟩. By the construction, ζ = sup_{j<ℵ_{m}} γ_{j}. Let c_{ℵ_{m}} = ⋃_{j<ℵ_{m}} p_{j}, p_{ℵ_{m}} = c_{ℵ_{m}} ∪ {ζ}.

We chose D to be in C_{ζ}. Since ζ ∈ lim(C), C ∩ ζ ∈ U_{ζ} and

⟨D_{β} ∩ ⟨x_{δ} : δ < ζ⟩ : β < ζ⟩ ∈ Sk_{λ}(ℵ_{ω} ∪ U_{ζ}).

Now ⟨p_{j} : j < ℵ_{m}⟩ is definable in H_{λ} from the parameters D, C ∩ ζ, ⟨D_{β} ∩ ⟨x_{δ} : δ < ζ⟩ : β < ζ⟩. Therefore ⟨p_{j} : j < ℵ_{m}⟩ ∈ Sk_{λ}(ℵ_{ω}∪U_{ζ}∪C_{ζ}). Since c_{ℵ_{m}} and p_{ℵ_{m}} are definable from ⟨p_{j} : j < ℵ_{m}⟩,

c_{ℵ_{m}}, p_{ℵ_{m}} ∈ Sk_{λ}(ℵ_{ω} ∪ U_{ζ} ∪ C_{ζ}).

By construction c_{ℵ_{m}} is a club subset of ζ such that

∀δ ∈ lim(c_{ℵ_{m}}) c_{ℵ_{m}} ∩ δ ∈ C_{δ},

so by Lemma 12.13 c_{ℵ_{m}} ∈ C_{ζ}. Since ℵ_{m} < ℵ_{n} and each p_{j} is in 𝕋_{n}, ot(c_{ℵ_{m}}) < ℵ_{n} and so p_{ℵ_{m}} ∈ 𝕋_{n}.

The construction of p gives us that p ⊩ ζ ∈ lim(Ḋ), and since p_{0} forces Ḋ to be club p ⊩ ζ ∈ Ḋ. This contradicts the choice of Ḋ as a name for a club set disjoint from S.

Therefore forcing with 𝕋_{n} over V_{2} preserves the stationarity of S.

12.4. Reflection holds in V_{2}. In V_{2}, let ⟨S_{k} : k < ℵ_{n}⟩ be a sequence of stationary subsets of the stationary set ℵ_{ω+1} ∩cof(< ℵ_{n}). We finish the proof of Theorem 21 by proving that there exists β such that cf(β) = ℵ_{n} and all of the S_{k} reflect at β.

We saw in the last section that 𝕊 ∗ 𝕋_{n+1} has an ℵ_{n+1}-directed closed subset, and that forcing over V_{2} with 𝕋_{n+1} preserves the stationarity of each S_{k}. Notice that 𝕋_{n+1} has added a club set of order type ℵ_{n+1} to ℵ_{ω+1}, and that ℵ_{ω+1} has become an ordinal of cofinality ℵ_{n+1} in (V_{2})^{𝕋_{n}}.

Let C⃗ be 𝕊-generic over V_{1} and let c be 𝕋_{n+1}-generic over V_{2} = V_{1}[C⃗]. By 6.10 there is a generic extension of V_{2}[c] by ℵ_{n}-closed forcing, V_{3} say, in which there exists j : V_{2}[c] ⟶ M ⊆ V_{3} such that

• crit(j) = ℵ_{n+1}^{V_{1}}.

• j ↾ ℵ_{ω+1}^{V_{1}} ∈ M.

• j(ℵ_{n+1}^{V_{1}}) > ℵ_{ω+1}^{V_{1}}.

• sup(j“ℵ_{ω+1}^{V_{1}}) < j(ℵ_{ω+1}^{V_{1}}).

• M ⊧ cf(ℵ_{ω+1}^{V_{1}}) = ℵ_{n} = ℵ_{n}^{V_{1}}.

We claim that each S_{k} is stationary in V_{3}. To see this observe that since cf(ℵ_{ω+1}^{V_{1}}) = ℵ_{n+1} in V_{2}[c], we can fix f : ℵ_{n+1} ⟶ ℵ_{ω+1}^{V_{1}} continuous increasing and cofinal. Let T_{k} = { α : f(α) ∈ S_{k} }, then T_{k} is a stationary subset of ℵ_{n+1} ∩ cof(< ℵ_{n}) in V_{2}[c]. It will be enough to prove that T_{k} is stationary in V_{3}.

Now V_{2}[c] is a model of GCH and so □_{ℵ_{n}}^{∗} holds in V_{2}[c]. A fortiori AP_{ℵ_{n}} holds in V_{2}[c] and so we can appeal to Lemma 6.14.

Since V_{3} is an extension of V_{2}[c] by ℵ_{n}-closed forcing, each of the sets T_{k} (and hence each of the sets S_{k}) is stationary in V_{3}. Now we can use a familiar line of argument to finish the proof. Let μ = sup j′′ℵ_{ω+1}, then in V_{3} each of the sets j“S_{k} is stationary in μ (j has critical point ℵ_{n+1} so is continuous at points of smaller cofinality). μ has cofinality ℵ_{n} in V_{3}. Let ⟨U_{k} : k < ℵ_{n}⟩ = j(⟨S_{k} : k < ℵ_{n}⟩), then M ⊧ “for all k < ℵ_{n}, U_{k} ∩ μ is stationary”. By the elementarity of j, there is β such that in V_{2} the cofinality of β is ℵ_{n} and each S_{k} ∩ β is stationary.

This concludes the proof of Theorem 21.

Corollary 12.16. By Theorem 5, VGS_{ℵ_{ω}} fails in the model of Theorem 21. So □_{ℵ_{ω}}^{∗} does not imply VGS_{ℵ_{ω}}.

Remark 12.17. It is natural to ask whether Theorem 21 can be improved to give the consistency of Refl^{∗}([κ^{+}]^{ℵ_{0}}) with □^{∗} (say by making the 𝕋_{n} forcings proper). This is impossible because by Theorem 7 □_{κ}^{∗} ⟹ ADS_{κ}, while by Theorem 8 Refl^{∗}([κ^{+}]^{ℵ_{0}}) ⟹ ¬ADS_{κ}.

13. More on reflection

The model of Theorem 21 has a very strong form of simultaneous reflection for stationary subsets of ℵ_{ω+1}, but by Remark 12.17 the principle Refl^{∗}([ℵ_{ω+1}]^{ℵ_{0}}) fails there. It is easy to see that Refl^{∗}([κ^{+}]^{ℵ_{0}}) implies Refl(T), where T = { α < κ^{+} : cf(κ) = ω }. In this section we show that Refl^{∗} does not imply any form of simultaneous reflection. We use the forcing axiom MM^{+} from [14].

Theorem 23. Let MM^{+} hold. Let κ be regular and uncountable. Then there is a forcing extension in which

• Refl(2, κ) fails.

• Refl^{∗}([λ]^{ℵ_{0}}) holds for every λ.

Proof. Let V ⊧ MM^{+}. Define a forcing ℙ to add a pair of stationary subsets of κ which do not reflect simultaneously. A condition in ℙ is a pair p = (p_{0}, p_{1}) where

• dom(p_{0}) = dom(p_{1}) < κ (we abuse notation by referring to the common domain as “dom(p)”).

• range(p_{i}) ⊆ 2 for i = 0, 1.

• ∀α p_{i}(α) = 1 ⟹ cf(α) = ω for i = 0, 1.

• If cf(β) > ω and β ≤ dom(p) then there exist c club in β and

i < 2 such that ∀α ∈ c p_{i}(α) = 0. The ordering on ℙ is extension.

It is easy to check that ℙ is countably closed, and is also < κstrategically closed. ℙ adds a pair of sets X_{0}, X_{1} which do not reflect simultaneously; a straightforward density argument shows that each of the sets X_{0}, X_{1} and Y =_{def} κ − (X_{1} ∪ X_{2}) is stationary in V^{ℙ}.

We now claim that Refl^{∗}([λ]^{ℵ_{0}}) holds in V^{ℙ} for every uncountable λ. Let λ be uncountable and suppose

⊩_{ℙ} “S is a stationary subset of [λ]^{ℵ_{0}}”.

Let ℚ = Coll(ℵ_{1}, max{κ, λ}). If H : ℵ_{1} ⟼ max{κ, λ} is the generic surjection added by ℚ then we can define “collapsed” versions of the sets X_{i}, Y and S in V^{ℙ∗ℚ}. To be more precise let

s = { α < ℵ_{1} : H“α ∩ λ ∈ S },
x_{i} = { α < ℵ_{1} : sup(H“α ∩ κ) ∈ X_{i} },

and

y = { α < ℵ_{1} : sup(H“α ∩ κ) ∈ Y }.

Since ℚ is countably closed each of the sets X_{i}, Y, S is stationary in V^{ℙ∗ℚ}, and so easily each of the sets x_{i}, y, s is also stationary.

Let z be the first set in the list x_{0}, x_{1}, y such that s ∩ z is stationary, and define ℝ to be the standard forcing [3] to add a closed unbounded subset of z; conditions in ℝ are closed bounded subsets of z, ordered by extension. We make the remark that (by the closure of ℙ and of ℚ) conditions in the forcings ℚ and ℝ are actually sets in V; we will identify ℙ ∗ ℚ ∗ ℝ with the dense subset of conditions (p, q, r) such that q and r are in V, and in which dom(p) = sup(range(q) ∩ κ) (the latter assumption will be technically convenient later).

Claim. Forcing with ℙ ∗ ℚ ∗ ℝ preserves stationary subsets of ℵ_{1}.

Proof. Let T ⊆ ℵ_{1} be stationary. Let (p, q, r) be a condition in ℙ∗ℚ∗ℝ and let Ċ be a name for a club set in ℵ_{1}. We may assume that p is strong enough to determine the choice of z, and for the sake of definiteness we assume that p ⊩ z = x_{0} (the other cases will be similar). As usual we fix some large regular θ and some well ordering <_{θ} of H_{θ}, and then choose N ≺ (H_{θ}, ∈, <_{θ}) such that (p, q, r), Ċ, ℙ ∗ ℚ ∗ ℝ ∈ N, |N| = ω and δ =_{def} N ∩ ℵ_{1} ∈ T.

Now we build a decreasing chain of conditions ⟨(p_{n}, q_{n}, r_{n}) : n < ω⟩ from ℙ ∗ ℚ ∗ ℝ ∩ N so as to meet every dense subset of ℙ ∗ ℚ ∗ ℝ which lies in N. Let p_{ω} =_{def} ⋃_{n} p_{n} and q_{ω} =_{def} ⋃_{n} q_{n}, then (p_{ω}, q_{ω}) ∈ ℙ ∗ ℚ; p_{ω} is a pair of functions with common domain α =_{def} sup(N ∩ κ), and q_{ω} is a function with dom(q_{ω}) = δ and range(q_{ω}) = N ∩ max{κ, λ}. r_{ω} = ⋃_{n} r_{n} is a closed and unbounded subset of δ.

In particular q_{ω} forces that sup(H“δ ∩ κ) = α. This means that if we extend p_{ω} to a condition p^{∗} to force α ∈ X_{0}, then (p^{∗}, q_{ω}) ⊩ δ ∈ x_{0}; since p ⊩ x_{0} = z this implies that (p^{∗}, q_{ω}, r_{ω} ∪ {δ}) is a condition in ℙ∗ℚ∗ℝ. As usual the condition (p^{∗}, q_{ω}, r_{ω} ∪{δ}) forces that δ ∈ Ċ ∩T, so we have shown that the stationarity of T is preserved.

Claim. s is a stationary subset of ℵ_{1} in V^{ℙ∗ℚ∗ℝ}.

Proof. It is a standard fact that shooting a club set through a stationary subset A ⊂ ℵ_{1} preserves the stationarity of stationary subsets of A. We chose z so that s∩z was stationary in V^{ℙ∗ℚ}, hence s∩z (and so a fortiori s itself) is stationary in V^{ℙ∗ℚ∗ℝ}.

We are now in a position to apply MM^{+} to the stationary-preserving forcing ℙ∗ℚ∗ℝ and the name s. We define some dense sets in ℙ∗ℚ∗ℝ (here α < ℵ_{1}).

D_{0} = { (p, q, r) : “p decides whether z is x_{0}, x_{1} or y”. }
D_{1}^{α} = { (p, q, r) : α < sup(r) }
D_{2}^{α} = { (p, q, r) : α ⊆ dom(q), ∃β ∈ dom(q) q(β) ∈ λ − sup(q“α ∩ λ) }
D_{3}^{α} = { (p, q, r) : α ⊆ dom(q), ∃β ∈ dom(q) q(β) ∈ κ − sup(q“α ∩ κ) }

Applying MM^{+} to ℙ ∗ ℚ ∗ ℝ and {D_{0}} ∪ { D_{i}^{α} : α < ℵ_{1}, 1 ≤ i ≤ 3 } we get a filter F on ℙ ∗ ℚ ∗ ℝ which meets all these dense sets and which realises s as a stationary set (we will denote the realisation of s by s^{∗}).

Let P = ⋃ { p : (p, q, r) ∈ F } and define Q, R similarly.

Claim. P ∈ ℙ and P ⊩_{ℙ} “S ∩ [range(Q) ∩ λ]^{ℵ_{0}} is stationary”

Proof. Since dom(p) = sup(range(q)∩κ) for all the conditions (p, q, r) ∈ F, dom(P) = sup(range(Q) ∩ κ). Let α denote dom(P), then we met enough dense sets to guarantee that cf(α) = ℵ_{1}. We also met enough sets to ensure that | range(Q) ∩ λ| = ℵ_{1}. From P we can read off X_{0}^{∗}, X_{1}^{∗} and Y^{∗} which are subsets of α.

We need to check that P is a condition. The point here is that R is a closed and unbounded subset of ℵ_{1}, and so { sup(Q“α ∩ κ) : α ∈ R } is a closed unbounded subset of one of the sets X_{0}^{∗}, X_{1}^{∗} or Y^{∗}. This guarantees that at least one of the sets X_{i}^{∗} is not stationary in α, and hence that P ∈ ℙ.

Since s^{∗} is the realisation of s, for each β ∈ s^{∗} we have P ⊩ Q“β∩λ ∈ S. Let T = { Q“β ∩ λ : β ∈ s^{∗} }, then clearly

V ⊧ “T is stationary in [range(Q) ∩ λ]^{ℵ_{0}}”.

Since ℙ is countably closed it is proper, so that

V^{ℙ} ⊧ “T is stationary in [range(Q) ∩ λ]^{ℵ_{0}}”.

Finally P ⊩ T ⊆ S ∩ [range(Q) ∩ λ]^{ℵ_{0}}, so that the claim is proved.

This concludes the proof that Refl^{∗}([λ]^{ℵ_{0}}) holds in V^{ℙ}.

Remark 13.1. Larson [29] has independently used quite similar methods to study the relationship between various reflection principles for ordinals and countable sets of ordinals. In particular Theorem 5.5 of [29] and Theorem 23 are proved in the same way, though neither one literally subsumes the other.

14. A picture and some problems

The following picture indicates how some of the principles considered in this paper are related to each other, in the case when κ is a strong limit cardinal of cofinality ω. For the benefit of readers of [13] we have indicated the relationship of the Not So Very Weak Square principle to the other principles.

There are many interesting questions still open. For example

1. How much of □_{κ} is needed in the various classical applications of square?

2. Does VGS_{κ} imply □_{κ}^{∗}?

3. Does □_{κ,<κ} imply VGS_{κ}?

SQUARES 67

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Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA 15213-3890, USA

  E-mail address: jcumming@andrew.cmu.edu

Mathematics Department, University of California, Irvine, Irvine CA 92717, USA

  E-mail address: mforeman@math.uci.edu

Mathematics Institute, Hebrew University, Givat Ram, 91904 Jerusalem, ISRAEL

  E-mail address: menachem@math.huji.ac.il