a very weak square principle

A VERY WEAK SQUARE PRINCIPLE

Matthew Foreman and Menachem Magidor

First Author: University of California, Irvine,Second Author: Hebrew University of Jerusalem,

Research supported by The Israel Science Foundation administered by TheIsrael Academy of Sciences and Humanities. Grant number 517/94

§1 Very Weak Square

In this paper we explicate a very weak version of the principle � discovered byJensen who proved it holds in the constructible universe L. This principle is strongenough to include many of the known applications of �, but weak enough that itis consistent with the existence of very large cardinals. In this section we showthat this principle is equivalent to a common combinatorial device, which we call aJensen matrix. In the second section we show that our principle is consistent with asupercompact cardinal. In the third section of this paper we show that this principleis exactly equivalent to the statement that every torsion free Abelian group has afiltration into σ-balanced subgroups. In the fourth section of this paper we showthat this priciple fails if you assume the Chang’s Conjecture: (ℵω+1,ℵω) � (ℵ1,ℵ0).In the fifth section of the paper we review the proofs that the various weak squareswe consider are strictly decreasing in strength. Section 6 was added in an ad hocmanner after the rest of the paper was written, because the subject matter ofTheorem 6.1 fit well with the rest of the paper. It deals with a principle dubbed“Not So Very Weak Square”, which appears close to Very Weak Square but turnsout not to be equivalent.

The now-classical principle �κ is the following statement:

There is a sequence 〈Cα : α ∈ κ+ and α a limit ordinal〉 such that:(1) Cα ⊂ α and Cα is closed unbounded in α.(2) If the cofinality of α is less than κ then the order type of Cα is less than κ(3) If β is a limit point of Cα, then Cα ∩ β = Cβ .

� is useful for proving many combinatorial results, yielding a general method forpassing singular limit cardinals in inductive constructions. There is a strictly weakerprinciple than � that is useful in many applications:

A weak square sequence is a sequence 〈Cα : α ∈ κ+ and α is a limit ordinal〉 suchthat:

(1) Cα ⊂ P (α) and |Cα| ≤ κ(2) every C ∈ Cα is closed and unbounded in α and if α has cofinality less than

κ, then C has order type less than κ.

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1

2 MATTHEW FOREMAN AND MENACHEM MAGIDOR

(3) If C ∈ Cα and β is a limit point of C, then C ∩ β ∈ Cβ .

Jensen proved [J] that weak square is equivalent to the existence of a specialAronszajn tree on κ+. Weak square really is weaker than square (see [B-D-M]).

We now remark that if weak square holds then we can assume without loss ofgenerality that for each α there is a C ∈ Cα such that order type of C is exactly thecofinality of α. To see this let 〈Dα : α ∈ κ+〉 be an arbitrary weak square sequence.For each ordinal δ < κ, choose a closed unbounded set Eδ ⊂ δ of ordertype thecofinality of δ. Note that for each β < κ, |{Eδ ∩ β : δ < κ}| ≤ κ. We now modifyour weak square sequence to have the additional property that for each α there isa C ∈ Cα with o.t.C = cof(α). Given an α < κ+ and an element D ∈ Dα of ordertype β, each Eδ∩β with Eδ∩β unbounded in β, gives us a closed unbounded subsetof D. The collection of all of these subsets of D has cardinality κ. We define Cα

to be the collection of all subsets of α obtained this way as D varies through all ofthe elements of Dα. Then Cα has cardinality at most κ and the resulting sequenceclearly has the coherence properties required of a weak square sequence. Finally, ifD ∈ Dα has ordertype δ, then Eδ induces a subset of D of ordertype the cofinalityof α, so we see that our new sequence has the additional property desired.

Suppose now that 2κ = κ+. Enumerate [κ+]<κ = 〈xα : α ∈ κ+〉. An ordinal αis approachable with respect to this enumeration iff there is an unbounded subsetof α, C, with order type the cofinality of α such that for all β < α there is aγ < α,C ∩ β = xγ . Let S∗ = {α : α is approachable }. It is easy to check that if〈yα : α ∈ κ+〉 is another enumeration of [κ+]<κ and T ∗is the collection of ordinalsapproachable with respect to 〈yα : α ∈ κ+〉 then S∗∆T ∗ is non-stationary. Hence S∗

is a well defined stationary set that is independent of the enumeration 〈xα : α ∈ κ+〉modulo closed unbounded sets. We say that κ+ has the approachability property iffS∗ is closed and unbounded. (These properties were first studied by Shelah [SH].)Clearly this can be rephrased in the following way, which makes sense without theGCH and is in a form that the approachability property is obviously a consequenceof weak square:

There is a sequence 〈Cα : α < κ+〉 such that for a closed unbounded set ofα ∈ κ+:

(1) Cα is closed and unbounded in α(2) The ordertype of Cα is the cofinality of α.(3) For all β < α there is a γ < α,Cα ∩ β = Cγ .

(The requirement in (1) that Cα be closed as well as unbounded is harmless:There is a closed unbounded set of β < κ+ such that for all α < β there is anα′ < β with xα′ the closure of xα. Hence if we have a sequence satisfying (1)-(3)save for the fact that the Cα’s are not closed, then the new sequence obtained byclosing the Cα’s is a witness to the approachability property.)

We note that if the G.C.H. holds then the existence of a sequence satisfying item(2) in the definition of the approachability property is equivalent to the existenceof a sequence satisfying the following statement:

(2’) If the cofinality of α is less than κ, then the order type of Cα is lessthan κ.

A VERY WEAK SQUARE PRINCIPLE 3

We note the connection of the approachability property with Shelah’s ideal I[κ+].The approachability principle is weaker than weak square, but is still inconsistentwith a supercompact cardinal. We review a proof of this in section 5.

In this paper we introduce a principle we call “Very Weak Square”. Let κ be asingular cardinal and suppose that 2κ = κ+. Let [κ+]<κ = 〈xα : α ∈ κ+〉. Thenthe Very Weak Square Principle holds iff for a closed unbounded set of α < κ+

there is an unbounded Cα ⊂ α such that for all bounded countable x ⊂ Cα, thereis a γ < α with x = xγ .

The following definition is equivalent to this property in the case the GCH holdsand clearly follows from �κ (and even the approachability property, hence fromweak square) if one assumes that κ is a strong limit cardinal.

Definition. Let κ be a singular cardinal. A sequence 〈Cα : α ∈ κ+〉 is a VeryWeak Square Sequence iff for a closed unbounded set of α ∈ κ+

(1) Cα ⊂ α, Cα is unbounded in α,(2) for all bounded x ∈ [Cα]<ω1 , there is a β ∈ α such that x = Cβ .

We note that passing to an unbounded subset of Cα preserves the last condi-tion, so we may assume without loss of generality that each Cα has ordertype thecofinality of α.

Since we do not require any coherence of the Cα’s, the existence of a Very WeakSquare sequence is really a property of the countable sets appearing on the sequence.

Perhaps it ought also be remarked that if κ is a strong singular limit cardinal ofuncountable cofinality, then Very Weak Square holds at κ+. To see this let λ = κ+

and enumerate λ<ω1 as 〈xα : α ∈ λ〉. Then for a closed unbounded set D, if γ ∈ Dhas uncountable cofinality then 〈xα : α ∈ γ〉 is an enumeration of γ<ω1 . Now definea new sequence 〈yα : α ∈ λ〉, by setting yα+1 = xα for all α and for α a limitchoosing yα cofinal in α of ordertype the cofinality of α.

Definition. A Jensen matrix on λ is a matrix 〈Anα : n ∈ ω, α ∈ λ〉 such that:

(1) Anα ⊂ λ and for n < m,An

α ⊂ Amα .

(2) For α < β and all m, there is an n with Amα ⊂ An

β .(3) For each α, set Bα =

⋃[An

α]<ω1 . Then |Bα| < λ and the sequence 〈Bα :α ∈ λ〉 is increasing and continuous and [λ]<ω1 =

⋃Bα.

(In fact condition 2) is easily seen to follow from conditions 1) and 3). For ifnot, then for each n, we can choose an γn ∈ Am

α \Anβ . But then the sequence

{γn : n ∈ ω} ⊂ Amα , but for no k is {γn : n ∈ ω} ⊂ Ak

β . )

Jensen proved that � together with GCH implies the existence of a Jensenmatrix. We note that there is a variation on Jensen matrices: If λ = κ+ and κ issingular, then define a weak Jensen matrix on λ to be:

(1) Anα ⊂ κ and for n < m, An

α ⊂ Amα .

(2) For α < β and all m, there is an n with Amα ⊂ An

β .(3) For each α, set Bα =

⋃[An

α]ℵ0 . Then |Bα| < λ and the sequence 〈Bα : α ∈λ〉 is increasing and continuous and [κ]<ω1 =

⋃Bα.


It is easy to see that to show the existence of a Jensen matrix, or a weak Jensenmatrix it suffices to define the An

α’s on a closed unbounded set of α. We also notethat for singular κ of cofinality ω and λ = κ+ |Bα| < λ implies that each |An

α| < κ.

A similar notion was defined by Hajnal, Juhasz and Weiss ([HJW]), who showeddirectly that it followed from square. Their principle is equivalent to the existenceof a Jensen Matrix.

Theorem 1. Assume that κ is a singular strong limit cardinal and λ = κ+ ≥ κω.Then the following are equivalent:

(1) There is a Very Weak Square sequence on λ(2) There is a Jensen matrix on λ(3) There is a weak Jensen matrix on λ

Proof. We note that this theorem is non-trivial only in the case where κ has cofi-nality ω. For A a structure with definable Skolem functions we define SkA(X) tobe the Skolem hull of X in A.

1) ⇒ 2) We show that the existence of a Very Weak Square sequence impliesthat there is a Jensen matrix on λ.

Let A0 = 〈H(κ+2),∈,∆0, 〈xα : α ∈ λ〉, 〈Cα : α ∈ λ〉 . . . 〉, be a structure in acountable language, where ∆0 is a well ordering of H(κ+2), and 〈Cα : α ∈ λ〉 isthe Very Weak Square sequence and 〈xα : α < λ〉 is an enumeration of [λ]<ω1 . LetA1 = 〈H(κ+ω+2),∈,∆1, 〈xα : α ∈ λ〉, 〈Cα : α ∈ λ〉,A0 . . . 〉 be a similar structureon H(κ+ω+2).

Let 〈κn : n ∈ ω〉 be an increasing cofinal sequence in κ. We define the Anα for the

closed unbounded set of α where SkA1(α) ∩ λ = α and where Cα has the definingproperty for a Very Weak Square sequence. If α has countable cofinality, choose〈αn : n ∈ ω〉 ⊂ Cα increasing and cofinal in α. Let Bn

α = SkA0(κn∪{α0, α1 . . . αn}).If the cofinality of α is uncountable, we let Bn

α = SkA0(κn∪Cα). Let Anα = Bn

α∩λ.Clearly

⋃Bn

α = SkA0(α). We now verify that this is a Jensen matrix.

The first clause in the definition of a Jensen matrix holds since the sequence〈κn : n ∈ ω〉 is increasing. To see the second clause, fix m,α < β. Since

⋃Bn

β =SkA(β), there is an n > m such that α ∈ Bn

β and the order type of Cα < κn. Butthen Bm

α ⊂ Bnβ and hence Am

α ⊂ Anβ .

To see the third clause, we claim that⋃

[Anα]<ω1 = SkA0(α) ∩ [λ]<ω1 , hence the

Bα’s are continuous and increasing. Since⋃{SkA0(α) : α < λ} ⊃ [λ]<ω1 , this claim

suffices.

The claim is equivalent to the statement that⋃

[Anα]<ω1 = 〈xβ : β < α〉. If

y ∈ [Anα]<ω1 then there is a bounded countable set of ordinals Z in Cα such that

y ∈ [SkA0(κn ∪ Z)]<ω1 (If α is of countable cofinality then Z is finite.) Fromthe defining property of the Very Weak Square sequence, we see that there is aγ < α,Z = xγ . Hence SkA0(κn ∪ Z) ∈ SkA1(α). Since κ<ω1

n < κ, we see that[SkA0(κn ∪ Z)]<ω1 ⊂ SkA1(α) and hence that for some β < α, y = xβ .

To see the other inclusion, we note that⋃

Anα = α. If β < α, then for some

n, β ∈ Anα. But then xβ ∈ SkA0(κn∪Cα), so xβ ⊂ SkA0(κn∪Cα). Hence xβ ⊂ An

α.


2) ⇒ 3) This is immediate.

3) ⇒ 1)

Let A0 = 〈H(κ+2),∈,∆0, 〈xα : α ∈ λ〉, 〈Anα : α ∈ λ, n ∈ ω〉 . . . 〉, be a structure

in a countable language, where 〈Anα : α ∈ λ, n ∈ ω〉 is the weak Jensen Matrix and

〈xα : α < λ〉 is an enumeration of [λ]<ω1 .. Let A1 = 〈H(κ+ω+2),∈,∆1, 〈xα : α ∈λ〉, 〈An

α : α ∈ λ, n ∈ ω〉,A0, . . . 〉.

Let D = {α : SkA1(α) ∩ λ = α and Bα = 〈xβ : β < α〉 ∩ [κ]<ω1}. Then D isa closed unbounded set in λ. For α ∈ D, since SkA1(α) |= |κω| = |λω| there areunboundedly many β < α, xβ ⊂ κ.

As remarked after the definition, Very Weak Square is really a property of anyenumeration of the < ω1-sequences from λ. In view of this, for α ∈ λ, we setCα+1 = xα. For α of cofinality ω, we let Cα be any cofinal sequence of order typeω.

We must define Cα for α ∈ D of uncountable cofinality. Fix such an α. Since⋃[An

α]<ω1 = {xβ : β < α} ∩ [κ]<ω1 and the latter set is cofinal in {xβ : β < α}there is an n such that [An

α]<ω1 is cofinal in the sequence 〈xβ : β < α〉, andhence SkA0([An

α]<ω1) is cofinal in α. Let Cα = SkA0([Anα]<ω1) ∩ λ. For each

x ∈ [Cα]<ω1 we must find a β < α, x = xβ . Since each element of x is definablein A0 from parameters in [An

α]<ω1 , there is a countable subset Z of [Anα]<ω1 such

that x ⊂ SkA0(Z). Since Z is countable there is a countable subset W ⊂ Anα such

that Z ⊂ P (SkA0(W )). Since there is a γ < α,W = xγ , we see that SkA0(W ) ∈SkA1(α). Since 2ℵ0 < κ, we see that Z ∈ SkA1(α) and hence SkA0(Z) ∈ SkA1(α).Again, by 2ℵ0 < κ, we see that x ∈ SkA1(α) and thus there is a β < α, x = xβ . �

Hajnal, Juhasz and Shelah showed in [HJS], assuming the consistency of a su-percompact cardinal, that it is consistent for Very Weak Square to fail at ℵω.Their model is an example of the failure of the weaker property where one justasks that the crucial property holds at α ∈ ℵω+1 of cofinality ω1. This raisesthe question of whether it is a theorem of ZFC + GCH that there is a sequence〈Cα : α ∈ ℵω+1〉 such that for almost all α of cofinality ω2, Cα is unbounded in αand [Cα]<ω1 ⊂ 〈Cβ : β < α〉. At this time this question is open, but Shelah hasshown that this property implies the stronger approachability property that thereis such a sequence where for almost all α of cofinality ω2 and all β < α there is aγ < α,Cα∩β = Cγ . In fact, for ℵω+1, Very Weak Square implies the approachabil-ity property. By the results of the next section this is not true for larger cardinalssuch as the ω + 1st successor of a supercompact cardinal.

In section 4, we show that the Chang’s Conjecture:

(ℵω+1,ℵω) � (ω1, ω)

implies the failure of Very Weak Square at ℵω+1. This gives another proof of theconsistency of the failure of Very Weak Square, since this Chang’s Conjecture wasshown to be consistent by Levinski, Magidor and Shelah [LMS].

§2 The consistency Proof


In this section we prove that it is consistent with a supercompact cardinal thatVery Weak Square holds at the successor of every singular cardinal. Similar argu-ments show in a routine way that Very Weak Square is consistent with huge andother very large cardinals.

Definition 2.1. Let κ be a singular strong limit cardinal of cofinality ω. Supposethat |κω| = κ+. Let 〈xα : α ∈ κ+〉 be an enumeration of [κ+]<ω1 . Let P(κ+) bethe collection of sequences 〈Cα : α ∈ β +1, and α a limit ordinal 〉 for some β ∈ κ+

satisfying:(1) Cα is unbounded in α and has order type less than κ(2) If there is a bounded countable subset of Cα not appearing as xβ for some

β < α then for all limit points γ of Cα, Cγ = Cα ∩ γ.Order P(κ+) by end- extension.

The purpose of P(κ+) is to create Very Weak Square on κ+ in some very min-imal way. We first show that P(κ+) is <-κ-strategically closed. Since P(κ+) hascardinality κ+ this implies that P(κ+) preserves cardinals.

Lemma 2.2. P(κ+) is <-κ- strategically closed.

Proof. (We use the standard proof that the partial ordering usually used for forcingsquare is strategically closed.) Recall the game for <-κ strategic closure. Fix acardinal µ < κ and a condition p ∈ P(κ+). Players I and II take turns playing adescending sequence 〈pα : α ∈ β〉 of elements of P(κ+) with player II playing firstat limit stages. Player II wins the µ-game if at every stage β ≤ µ, there is a qbelow each element in the sequence 〈pα : α ∈ β〉. So fix such a µ and we describe awinning strategy for II in this game: At successor stages II choses any element ofthe partially ordered set which is less than, but not equal to I’s most recent move.At a limit stage β, if {pα : α < β} is the sequence of conditions played so far, letν be the supremum of the lengths of the pα’s. In view of II’s plays ν is a limitordinal. Put

Cν = {δ : for some α ∈ β the length of pα is δ}.

Now at stage β, II plays⋃{pα : α < β} ∪ {(ν, Cν)}.

�

Lemma 2.3. If p ∈ P(κ+) and δ ∈ κ+ then there is an extension of p to a q thathas length greater than δ.

Proof. We prove this by induction on δ. Clearly we may assume that δ > κ. Assumethat the lemma is true for all p and all δ′ < δ. Let µ be the the cofinality of δ. LetD ⊂ δ be an unbounded set of order type µ. Define a play of the game below p oflength µ by letting II play according to his winning strategy and at stage 2α + 1,player I plays an element q2α+1 of length longer than the αth element of D. (We canassume that such elements of p ∈ P(κ+) exist by our induction hypothesis.) SinceII has played according to his winning strategy there is a q below each element ofthe play. This q clearly must have length at least δ.


Lemma 2.4. Let κ be a singular strong limit cardinal of cofinality ω and supposethat 2κ = κ+. If G ⊂ P(κ+) is generic over V , then V [G] |=“Very Weak Square atκ+”.

Proof. First note that the previous lemmas imply that P(κ+) adds no new countablesequences to V . Suppose that G = 〈Cα : α ∈ κ+〉. Define a Very Weak Squaresequence by setting

Dα ={

Cα, if α is a limit ordinalxα−1, otherwise.

Let λ be a regular cardinal much bigger then κ and let N ≺ 〈H(λ) ∈,∆, 〈Dα :α ∈ κ+〉〉 be an elementary substructure of cardinality κ such that α = N∩κ+ ∈ κ+.Then if there is a bounded countable subset y of Dα not enumerated in 〈xβ : β ∈ α〉we must have that for all γ a limit point of Dα, Dγ = Dα ∩ γ. Since y is bounded,we may assume that y is a countable subset of some Dγ for γ < α. Since κ is astrong limit ordinal and Dγ has cardinality less than κ, P (Dγ) ⊂ N . Hence y isenumerated before α, contradiction. �

Theorem 2.5. Suppose that the GCH holds and κ is a supercompact cardinal.Then there is a class forcing extension of V that preserves cardinals and cofinalitiesin which Very Weak Square holds at all singular cardinals and where κ remainssupercompact.

Proof. Let κ be a supercompact cardinal and assume the G.C.H. We now define abackwards Easton forcing iteration that adds Very Weak Square at all successorsof singular cardinals of countable cofinality.

To specify the iteration P = lim−→〈Pα : α ∈ OR〉 we must specify the supportand the factors of the iteration Qα which determine Pα+1 = Pα ∗Qα. We will usestandard Easton supports.

For each singular cardinal α of cofinality ω we fix an enumeration x of [α+]<ω1 .This determines a well defined partial ordering P(α+).

If α is a singular cardinal of cofinality ω, working in V Pα we define Qα = P(α+).In all other cases we define Qα to be the trivial partial ordering {1}.

Standard arguments show that if µ is the successor of a regular cardinal thenP ≈ Pl ∗Pr where Pl has cardinality less than µ and Pr is µ++- strategically closed.Hence P preserves ZFC and all cardinals and cofinalities. Further it is clear fromlemma 2.4 and the fact that we preserve cardinals and cofinalities that if we forcewith P then resulting model is a model of Very Weak Square. What we have leftto show is that in the resulting model κ is still supercompact.

Let µ > κ be the successor of a singular cardinal γ of countable cofinality. Thenwe can factor P ≈ Pµ ∗R, where R is 22µ

-strategically closed. Hence, if we can showthat in Pµ, κ is µ- supercompact, we know that κ remains µ-supercompact in V P.Thus it suffices to show this latter statement for all successors of singulars µ. Sofix such a µ and let j : V → M be a 2λ+

-supercompact embedding, where λ is acardinal much bigger than µ. In M , we can factor


j(Pµ) = j(P)µ ∗ R,

where R is 22µ

- strategically closed. Since M is closed under λ-sequences we seethat j(P)µ = Pµ. Let G ⊂ Pµ be generic. We work in M [G] to build a strongmaster condition m ∈ R; namely a condition m such that for all p ∈ G, j(p) ≥ m.Standard large cardinal theory then implies that if we take H ⊂ R to be V [G]-generic with m ∈ H, then j can be extended to a j : V [G] → M [H].

Since each p ∈ Pµ has Easton support, j(p) has an Easton support and thesupport of j(p) has empty intersection with the interval (κ, j(κ)). There are onlyµ conditions in Pµ and the ideal of Easton supports above µ is < −µ+complete.Hence S = {α > µ : for some p ∈ G, j(p)α 6= 1} is an Easton support. Thus itsuffices to show that for all α < j(µ) there is a condition rα ∈ j(P)α such that forall p ∈ G, rα ≤ j(p)α, for then we can take mα = rα for α ∈ S.

Note that if α ∈ S then α is a singular cardinal of cofinality ω in the sense of M .We temporarily define an ordinal γ between a singular cardinal and its successor tobe good if there is a cofinal subset D ⊂ γ such that all bounded countable subsetsof D occur in the (fixed) enumeration x of [α+]<ω1 before γ. We claim that for allsingular α between j(κ) and j(µ) of cofinality ω,

γ = sup{j(f)(α) : there is an Easton support X ⊂ µ, f : X → µ, andfor all β ∈ X, f(β) ∈ β+}

is a good ordinal.

Let D = {j(f)(α) : there is an Easton support X ⊂ µ, f : X → µ, and for allβ, f(β) ∈ β+}. Let x = 〈xδ : δ < α+〉 be the enumeration of [α+]<ω1 we chose forα. (Note that this enumeration, which lies in M , is determined by the choice ofenumerations in V ; in fact it is the αth term in the j image of the sequence of choicesin V .) We now show that if y ⊂ D is countable, then there is a δ < γ, y = xδ.Suppose y = {βn : n ∈ ω}. Then there are {fn : n ∈ ω} such that βn = j(fn)(α).Let Xn be the Easton support which forms the domain of fn. For each ordinalα′ ∈

⋂Xn, the set yα′ = {fn(α′) : n ∈ ω} is equal to an xδ′ on the chosen

enumeration for α′ for some δ′ < (α′)+. Define g(α′) = δ′.

By the elementarity of j, if we set δ = j(g)(α) then xδ = y. Clearly, δ < γ whichestablishes the claim.

We now define the condition rα. A simple density argument shows that the γwe defined above is also sup{j(f)(α) : there is a condition p ∈ G, and for all α ∈supp(p), f(α) is the length of pα}. If we take

⋃{j(p)(α) : p ∈ G} we get a sequence

of length γ with the properties required to be a condition, except that it has limitlength (it has no “top”). Let C ⊂ D be cofinal in γ of length the cofinality of γ.Then every countable subset of C appears in the enumeration of [α+]<ω1 before γ.Define

rα =⋃{j(p)(α) : p ∈ G} ∪ {(γ, C)}.


Then, since γ is a good ordinal, this is a condition in P(α+)M . We have shownthe existence of a strong master condition m and hence the existence of the genericelementary embedding.

We now use the 22µ

-strategic closure to show that V [G] has a supercompactultrafilter on Pκ(µ). Let 〈Aα : α ∈ 2µ〉 and 〈fα : α ∈ 2µ〉 be enumerationsof the subsets of Pκ(µ) and the regressive functions defined on Pκ(µ). In V [G],every condition in R below m forces that j extends to a j as above. For eachcondition r ∈ R and all α there is a condition s ≤ r so that s decides the statementj“µ ∈ j(Aα) and such that for some ordinal δ, s j(fα)(j“µ) = δ.

Fix a winning strategy for player II in the game of length 22µ

. Define a decreas-ing sequence of conditions:

m = s0 ≥ r0 ≥ s1 ≥ r1 ≥ . . . rα ≥ sα+1 ≥ rα+1 . . .

so that for all α, sα+1 decides the statement j“µ ∈ j(Aα) and such that for someordinal δ, sα+1 j(fα)(j“µ) = δ, and for all α, rα is II’s response to the previousconditions according to his winning strategy. (We use the existence of such astrategy to define such a sequence.)

Then in V [G], U = {A ⊂ Pκ(µ) : for some α, sα j“µ ∈ A} is easily checked tobe a normal ultrafilter on Pκ(µ). �

§3 An application to Abelian groups

In this section we show that a certain property of Abelian groups is equivalentto Very Weak Square. For applications and motivation for this property we referthe reader to [Fu-M].

Recall that if H is a subgroup of G then H is pure iff G/H is torsion free. If Gis a group of uncountable cardinality the the sequence 〈Gα : α < δ〉 is a filtrationof G iff

(1) |Gα| < |G| for all α < δ.(2) For α < β,Gα ⊂ Gβ .(3) If α is a limit ordinal then Gα =

⋃β{Gβ : β < α} and

(4) G =⋃

α{Gα : α < δ}

Definition 3.1. Let G and H be Abelian groups with H a pure subgroup of G.Then H is a balanced subgroup of G provided that for all g ∈ G and all countablesubsets {hn : n ∈ ω} of H there is an h ∈ H such that for all n and all primenumbers p and all k ∈ ω, pk|g−hm iff pk|h−hm. A subgroup H ⊂ G is σ-balancedif H =

⋃{Hn : n ∈ ω}, where {Hn} is an increasing sequence of balanced subgroups

of G.

Theorem 3.2. Let κ be a singular strong limit cardinal of cofinality ω and supposethat κω = κ+. Then the following are equivalent:

(1) Every torsion free Abelian group G of cardinality κ+ has a filtration 〈Gα :α ∈ κ+〉 such that each Gα is σ-balanced.

(2) Very Weak Square on κ


Proof.

Assume that every Abelian group of cardinality κ+ has a filtration as described.We now define a group G of cardinality κ+ such that a filtration of G into σ-balancedgroups yields a Jensen matrix on κ+.

Let pn be the nth prime number in N.

Let {aα : α ∈ κ+} ∪ {bη : η ∈ [κ+]ω} be a collection of distinct objects.

We will define our group as the subgroup G of the Q-vector space

Σα∈κ+Qaα ⊕ Ση∈[κ+]ω Qbη

generated by:

{aα : α ∈ κ+} ∪ {{1/pn(bη − aη(n)), bη} : η ∈ [κ+]ω}

Claim 3.3. Let {βn : n ∈ ω} ∈ [κ+]ω and suppose that h ∈ G is such that for alln, pn|(h − aβn

). If h = Σi≤kriaαi⊕ Σj≤lsjbηj

then for all n, βn ∈⋃{range(ηj) :

j ≤ l} ∪ {αi : i ≤ k}.

Proof. Suppose not. Choose an βm such that βm /∈⋃{range(ηj) : j ≤ l} ∪ {αi :

i ≤ k} . Suppose now that h− aβm = pg for some g, where p = pm. Write:

h− aβm= A + B + C

where A = Σpnxax, B = Σpmybηyand C = Σ(pts,j/pj)(bηs

− aηs(j)), all of thesums being finite and without repetitions.

Since aβmappears on the left hand side of this equality with coefficient −1, it

must appear in a term in C with non-zero coefficient. Hence there is a bη occuringin C and an j ∈ ω with βm = η(j). For each such η, let Sη be the collection ofall of the coefficients of bη occuring in C . For each s, every coefficient in Sηs

is ofthe form pts,j/pj . Each of these coefficients have numerator divisible by p, unlessj = m, and this only occurs once in each Sη. Since the sum of the coefficientsof aβm

in C is equal to −1 modulo p, the sum of all of the coefficients of aβm

occuring in all of the Sη’s must be an integer equivalent to −1 modulo p. Hencethere must be some η with βm = η(m). Among these η’s there must be one suchthat the numerator of the sum of Sη is not divisible by p. Since all of the non-zerocoefficients of bη occuring in B are divisible by p, bη must occur with non-zerocoefficient in h, contradicting the choice of βm. �

We say that a subgroup H ⊂ G is almost ω-closed iff whenever 〈βn : n ∈ ω〉 ∈[κ+]ω is such that for all n, aβn

∈ H, then there is an h ∈ H,h = Σi≤kriaαi⊕

Σj≤lsjbηj such that for all n, βn ∈⋃{range(ηj) : j ≤ l} ∪ {αi : i ≤ k}.

Claim 3.4. If H ⊂ G is balanced then H is almost ω-closed.

Proof. Let η = 〈βn : n ∈ ω〉 ∈ [κ+]ω be such that {aβ(n) : n ∈ ω} ⊂ H. Note thatevery prime number pn divides bη − aβn

. Since H is balanced there is an h ∈ H


such that for all n, pn|h − aβn . By the Claim 3.3, we see that h has the desiredproperty. �

Let A0 = 〈H(κ+2),∈,∆, 〈xα : α ∈ λ〉, 〈aα, bη : α ∈ κ+, η ∈ [κ+]ω〉 . . . 〉, bea structure in a countable language, where ∆ is a well ordering of H(κ+2), and〈xα : α ∈ λ〉 is an enumeration of the ω-sequences from κ+.

Let A1 = 〈H(κ+ω+2),∈,∆1, 〈xα : α ∈ λ〉, 〈aα, bη : α ∈ κ+, η ∈ [κ+]ω〉,A0 . . . 〉 bea similar structure on H(κ+ω+2).

Let 〈Gα : α ∈ κ+〉 be a filtration of G. By standard arguments we can assumethat on a closed unbounded set C ⊂ κ+, Gα = G ∩ SkA1(α) = G ∩ SkA1(Gα) andalso α = SkA1(α) ∩ κ+. Further we may assume that for a closed unbounded setof α for all β < α, P (xβ) is enumerated before α and that all finite variations of xβ

using elements of α appear in the enumeration before α.

For each α ∈ C, let 〈Hnα : n ∈ ω〉 be an increasing sequence of balanced subsets

of G with union Gα.

Since each Hnα is balanced, it is almost ω-closed. Since |Hn

α | ≤ κ, we know that|Hn

α ∩ {aβ : β ∈ κ+}| < κ. Define

Anα = SkA0(Hn

α ∩ {aα : α ∈ κ+}) ∩ κ+

.

Clearly for n < m, Anα ⊂ Am

α .

Claim 3.5.⋃

[Anα]ω = {xβ : β < α}.

Proof. Let ζ ∈ [Anα]ω. Then there is a countable sequence of ordinals η such that

〈aη(n) : n ∈ ω〉 ⊂ Hnα ∩ {aδ : δ ∈ κ+} and ζ ⊂ SkA0(η). Since Hn

α is almostω-closed, there is an h ∈ Hn

α ⊂ Gα , such that η ⊂ SkA0({h}). Thus η ∈ SkA1(Gα)and so SkA0(η) ∈ SkA1(Gα). From this we see that [SkA0(η)]ω ⊂ SkA1(Gα). Henceζ ∈ SkA1(Gα). Since SkA1(Gα) ∩ κ+ = α there is a β < α, ζ = xβ .

To see the other inclusion, let β < α. Then for some n, aβ ∈ Hnα l. Hence,

xβ ∈ SkA0(Hnα ∩ {aδ : δ ∈ κ+}) and thus xβ ⊂ An

α. �

We have shown clauses 1.) and 3.) in the definition of a Jensen Matrix. As weremarked after the definition of a Jensen Matrix the second clause in the definitionis a consequence of these two properties.

This finishes the proof that condition 1 of the theorem implies condition 2 of thetheorem.

To see that condition 2 in the theorem implies condition 1, fix 〈κn : n ∈ ω〉be increasing and cofinal in κ. Let G = 〈gα : α ∈ κ+〉 be an arbitrary tor-sion free Abelian group of cardinality κ+. Similar to our earlier arguments we letA0 = 〈H(κ+2),∈,∆, 〈xα : α ∈ λ〉, 〈Cα : α ∈ κ+〉, G . . . 〉 and A1 = 〈H(κω+2),∈,∆1,A0, 〈xα : α ∈ λ〉, 〈Cα : α ∈ κ+〉, G . . . 〉, where 〈Cα : α ∈ κ+〉 is a Weak Squaresequence.

Then for all α ∈ κ+ such that SkA1(α) ∩ κ+ = α, let Hα = 〈gβ : β < α〉.Then the sequence 〈Hα : α ∈ κ+〉 is a continuous increasing sequence of subgroups


of G of cardinality κ. Since SkA1(α) ∩ κ+ = α = SkA0(α) ∩ κ+ we see that[Hα]ω ∩ SkA1(α) = [Hα]ω ∩ SkA0(α). We must show that each Hα is σ-balanced.

Claim 3.5 There is an increasing sequence of elementary substructures Bn of A0

that are closed under ω-sequences and are such that⋃

Bn ∩G = SkA0(α) ∩G.

We distinguish two cases:

Case 1 α has cofinality ω.

Let 〈αn : n ∈ ω〉 be an increasing cofinal sequence in α. Let Xn be the elementarysubstructure of A0 generated by κn ∪ {α0 . . . αn−1}. Then each Xn ∈ SkA1(α).Hence, the smallest elementary substructure, Bn of A0 including Xn as a subsetand closed under ω-sequences is an element of SkA1(α). Since Bn has cardinalityless than κ, Bn ⊂ SkA1(α). Hence, Bn ∩ G is a subset of SkA1(α), and thus ofSkA0(α). Since

⋃Xn = SkA0(α),

⋃Bn ∩G = SkA0(α) ∩G.

Case 2 Otherwise.

Let Xn be the elementary substructure of A0, generated by κn ∪ Cα. Let Bn

be the smallest elementary substructure of A0 including Xn as a subset and closedunder ω-sequences. Then for every element g of Bn ∩G there is a countable subsetZ of Cα such that g is in the smallest countable closed elementary substructure ofA0 including Z ∪ κn as a substructure. By the defining property of a Very WeakSquare sequence, Z ∈ SkA0(α). Hence, g ∈ SkA1(α), and thus in SkA0(α). Since⋃

κn = κ,⋃

Bn ∩G = SkA0(α) ∩G.

Now let Hn = Bn ∩Hα. We claim that Hn is balanced. Let 〈hn : n ∈ ω〉 be acountable subset of Hn. Let g ∈ G. Then 〈hl : l ∈ ω〉 and {(p, k, l) : p is a primeand pk|g − hl} are both in Bn. Since Bn ≺ A0, there is an h ∈ Hn such that forall primes p and all natural numbers k, pk|h− hl iff pk|g − hl. �

§4 Chang’s Conjecture implies the failure of Very Weak Square

In this section we show that the Chang’s Conjecture (ℵω+1,ℵω) � (ℵ1,ℵ0)implies the failure of the Very Weak Square Principle at ℵω. Since this Chang’sConjecture was shown consistent in [L-M-S], this shows the consistency of the failureof Very Weak Square. Another model where Very Weak Square fails is producedin the final section of [H-J-S]. Our tool for proving the main result of this sectionis Shelah’s theory of reduced products. While we give complete proofs of ourassertions here, they are strongly motivated by Shelah’s work. For references tothis see Shelah’s papers or [B-M]. Also relevant to the results in this section is thefirst section of the paper [M-S].

Theorem 4.1. Suppose that (ℵω+1,ℵω) � (ℵ1,ℵ0). Then the Very Weak Squarefails.

Recall that a sequence 〈fα : α ∈ ℵω+1〉 ⊂ Π{ℵn : n ∈ ω} is called a scale iff forevery g ∈ Π{ℵn : n ∈ ω}, there is an α, g(n) < fα(n) for all but finitely many n.A sequence of elements 〈fβ : β < α〉 is said to have (strong) least upper bound hprovided that h ∈ Π{ℵn : n ∈ ω} and has the property that for all β < α, and allbut finitely many n, fβ(n) < h(n) and if g ∈ Π{ℵn : n ∈ ω} is such that for allbut finitely many n, g(n) < h(n) then there is a β < α such that for all but finitely


many n, g(n) < fβ(n).

In this context we note that it is natural to consider equivalence classes of func-tions modulo the Frechet filter, i.e. two functions in the product are equivalent iffthey are equal for all but finitely many n. If a least upper bound for the sequence〈fβ : β < α〉 exists then its equivalence class is unique. Thus we will talk of theleast upper bound.

A scale 〈fα : α ∈ ℵω+1〉 is continuous iff whenever 〈fβ : β ∈ α〉 has a leastupper bound then fα is the least upper bound. Note that it is an easy consequenceof the G.C.H. that continuous scales exist, and a theorem in Shelah’s PCF theory([B-M]) that ZFC proves that there is a set b ⊂ ω such that there is a continuousscale of length ℵω+1 in Π{ℵn : n ∈ b}. To simplify notation we will work withΠ{ℵn : n ∈ ω}, the proof works in the other case with no serious mutations. Anordinal α ∈ ℵω+1 is good iff cof(α) > ω and there is an unbounded subset X of α,and a cofinite set A ⊂ ω such that γ < β ∈ X and n ∈ A implies fγ(n) < fβ(n).

Lemma 4.2. Let 〈fα : α ∈ ℵω+1〉 be a continuous scale. Then for all α, if α isgood then fα is the least upper bound of 〈fβ : β < α〉 and for a cofinal set of n ∈ ω,the cofinality of fα(n) is equal to the cofinality of α.

Proof. If α is good there is an unbounded set X ⊂ α and a cofinal set A ⊂ ωsuch that for all n ∈ A, {fβ(n) : β ∈ X} is strictly increasing. For n ∈ A, defineh(n) = sup{fβ(n) : β ∈ X}. We show that h is a least upper bound for {fβ :β < α} and hence fα = h for almost all n ∈ ω. This clearly suffices since for alln ∈ A, cof(h(n)) = cof(X) = cof(α).

Let g ∈ Π{ℵn : n ∈ ω} be such that for all but finitely many n, g(n) < h(n).Then for each such n there is a βn ∈ X such that for all β > βn, g(n) < fβ(n).Since α has uncountable cofinality, there is a β ∈ X such that for all n, β > βn.Then for all but finitely many n, g(n) < fβ(n). �

Fix a continuous scale 〈fα : α ∈ ℵω+1〉. To prove that the Chang’s Conjectureimplies the failure of Very Weak Square we establish the following two claims:

Claim 4.3. Suppose that (ℵω+1,ℵω) � (ℵ1,ℵ0). Then there is a stationary set ofordinals α ∈ ℵω+1 of cofinality ω1 that are not good.

Claim 4.4. Suppose that Very Weak Square holds. Then there is a closed un-bounded set C ⊂ ℵω+1 such that every ordinal in C of cofinality ω1 is good. Further,if the approachability property holds then there is a closed unbounded set C suchthat every ordinal in C of uncountable cofinality is good.

.

Proof of 4.3. Let C ⊂ ℵω+1 be a closed unbounded set. Let λ be a regular cardinalmuch bigger then ℵω+1, and let N ≺ 〈H(λ),∈,∆, 〈fα : α ∈ ℵω+1〉, C〉 be anelementary substructure such that N ∩ ℵω+1 has cardinality ω1 and N ∩ ℵω iscountable. (Note that this implies that the order type of N ∩ ℵω+1 is ω1.) Theexistence of such a substructure is a consequence of Chang’s Conjecture. Let γ =


sup(N ∩ ℵω+1). Then γ ∈ C and γ has cofinality ω1. It suffices to show that γ isnot a good ordinal.

Note that for each β ∈ γ ∩ N, fβ ∈ N . Since fβ is a countable collection ofordered pairs, fβ ⊂ N . Hence Π{N ∩ ℵn : n ∈ ω} is cofinal in 〈fβ : β ∈ γ〉. Notethat for each n ∈ ω, N ∩ ℵn is countable.

Suppose now that γ is good. Then fγ is the least upper bound for {fβ : β ∈ N∩γ}and for all but finitely many n, the cofinality of fγ(n) is constantly ω1. There isa cofinal set X ⊂ N ∩ γ and a fixed K such that for all n > K, β ∈ X, fβ(n) <fγ(n). Note that for all β ∈ X, fβ(n) ∈ N ∩ ℵn, which is a countable set. Sincefγ(n) has uncountable cofinality, for each n > K, there is a δn such that for allβ ∈ X, fβ(n) < δn < fγ(n). Define g(n) = δn. Then g is a counterexample to fγ

being the least upper bound for {fβ : β ∈ X}. �

We now prove claim 4.4:

Proof. Recall that a set N is internally approachable of lenght γ iff there is asequence 〈Nα : α ∈ γ〉 such that for all β < γ, 〈Nα : α ∈ β〉 ∈ N and N =

⋃{Nα :

α ∈ γ}. (See [Fo-M] for information about internally approachable sets.)

We split the proof of claim 4.4 into two parts. In the first part we show that ifA is a structure on some H(λ) and α is a typical approachable ordinal of cofinalityµ > ω then there is an internally approachable N ≺ A of length and cardinalityµ with N ∩ ℵω+1 = α. In the second part we show that if N is an internallyapproachable elementary substructure of 〈H(λ),∈, 〈fα : α ∈ ℵω+1〉〉 of length andcardinality µ > ω, then sup(N ∩ ℵω+1) is a good ordinal.These two parts togetherprove the claim, since Very Weak Square implies that there is a closed unboundedset C such that any limit point of C of cofinality ω1 is approachable. Further theapproachability property implies this true for a closed unbounded set relative toordinals of uncountable cofinality.

Let λ be a regular cardinal much bigger than ℵω+1 and suppose that 〈Cα :α ∈ ℵω+1〉 is a sequence of sets such that Cα ⊂ α. Let A = 〈H(λ),∈, 〈Cα : α ∈ℵω+1〉,∆, ...〉 be a structure on H(λ). Let 〈Bα : α ∈ ℵω+1〉 be a continuous sequenceof elementary substructures of A with the property that Bα+1 = SkA(〈Bβ : β ≤α〉∪Bα. Note that for a closed unbounded set of α, Bα ⊃ SkA(α), and Bα∩ℵω+1 =α. Now suppose that α is an ordinal where Cα is unbounded in α, has ordertypethe cofinality of α, and every initial segment of Cα occurs in 〈Cβ : β < α〉 (i.e. αis approachable with respect to the sequence 〈Cα〉.) Since every initial segment ofthe closure of Cα in α occurs in Bα we assume that Cα is closed. Suppose thatCα = 〈αi : i ∈ cof(α)〉. Define Cα = SkA({〈(αi,Bαi) : i < j〉 : j < cof(α)}). Thenclearly the cardinality of Cα is the cofinality of α.

Define Cj = SkA({〈(αi,Bαi) : i < j′〉 : j′ < j}). If we see that for all j <cof(α), 〈Ci : i < j〉 ∈ Cα we will have shown that Cα is internally approachable oflength the cofinality of α and taking N = Cα we will have finished the first half ofthe proof of 4.4.

For each j < cof(α), there is some β ∈ Cα such that 〈(αi,Bαi) : i < j〉 ∈ Bβ .

Since β ∈ Cα,Bβ ∈ Cα. Further Cj = SkA({〈(αi,Bαi) : i < j′〉 : j′ < j}) =SkBβ ({〈(αi,Bαi

) : i < j′〉 : j′ < j}), since Bβ ≺ A. Hence, Cj is in Cα and


uniformly definable in the parameters {〈(αi,Bαi) : i < j〉,Bβ}. Further, each Cj′

is uniformly definable from j′ in Cα using the parameters {〈(αi,Bαi) : i < j〉,Bβ}.

Hence 〈Cj′ : j′ < j〉 ∈ Cα.

To finish the proof of 4.4 we must show that if N ≺ 〈H(λ),∈, 〈fα : α < ℵω+1〉〉is internally approachable and |N | = cof(N ∩ ℵω+1) = µ is uncountable, thensup(N ∩ ℵω+1) is a good ordinal. Suppose that 〈Nα : α < µ〉 witnesses the factthat N is internally approachable. An easy argument (see [Fo-M]) shows that wecan assume that |Nα| < µ. (This is clear in the internally approachable structureswe constructed for the first part of the proof.) By passing to a subsequence we canassume that 〈Nβ : β ≤ α〉 ∈ Nα+1. Further, for a closed unbounded subset C ofµ,Nα ≺ N .

Define χα(n) = sup(Nα ∩ ℵn). For α < β elements of C, for all n ∈ ω, χα(n) <χβ(n). Enumerate C in increasing order as 〈αi : i ∈ µ〉. Since χαi

∈ Nαi+1 , there isa βi ∈ Nαi+1 such that for all but finitely many n ∈ ω, χαi

(n) < fβi(n). Note that

βi > sup(Nαi∩ℵω+1) since χαi

dominates all fβ , β < sup(Nαi∩ℵω+1). Then for all

i ∈ µ, n ∈ ω, fβi(n) < χαi+1(n). There is a cofinal set Y ⊂ µ and some K such thatfor all i < j with i, j ∈ Y, n > K, we have χαi(n) < fβi(n) < χαi+1(n) < χαj (n).Let X = {βi : i ∈ Y }. Then for all β < γ with β, γ ∈ X and all n > K, fβ(n) <fγ(n). Hence α is good. �

§5 The table

In this section we consider the properties �, weak– �, the approachability prop-erty and the Very Weak Square property and review the models which witness thatthey are strictly decreasing in strength. Many of the results we use in this sectionare known but scattered through the papers of Shelah. We collect them here inresponse to several requests.

As mentioned earlier in the paper, Ben-David and Magidor, in [B-D-M] showedthat it is consistent to have a model where the weak square property holds at ℵω,but the square property fails.

We now describe a model where the approachability property holds but the weaksquare property fails. Begin with a model containing a supercompact cardinal κ.Standard techniques ([B]) show that it is consistent to have the G.C.H. togetherwith the following square property at the supercompact cardinal κ.

There is a sequence of sets 〈Cα : α ∈ κ+ω+1〉 such that:

(1) For each α ∈ κ+ω+1 of cofinality bigger than or equal to κ, Cα isclosed and unbounded in α and has order type less than κ+ω,

(2) If α ∈ κ+ω+1 has cofinality at least κ and δ is a limit point ofCα, then Cα ∩ δ = Cδ.

Having produced this model, we can use the construction of section 2 to force aVery Weak Square sequence 〈Dβ〉 at κ+ω, while keeping κ supercompact. Note thatthis doesn’t destroy the above mentioned square for ordinals of cofinality bigger orthan equal to κ. The final model is then constructed by collapsing κ to be ℵ2 usingcountably closed conditions.


It is now easy to see that the approachability property holds at κ+ω+1: Forordinals of cofinality at least κ (which is now ℵ2) the old square sequence witnessesthe approachability property. For ordinals β of cofinality ω1 (where the Very WeakSquare property holds), we consider the element of the Very Weak Square sequenceat β, Dβ . Since the final forcing used countably closed conditions, if we let D′

β bea subset of Dβ of order type ω1, then every initial segment of D′

β occurs as someDα for α < β. Hence the approachability property holds at β.

To finish the proof we need to argue that the weak square property fails at κ+ω.

Claim. Let κ be a supercompact cardinal and suppose the G.C.H. holds. Let G ⊂Col(ω1, < κ) be generic. Then the weak square property fails at κ+ω in V [G].

Proof. Suppose to the contrary that 〈Cα : α ∈ κ+ω+1 and α is a limit ordinal 〉 is aweak square sequence in the generic extension V P . So:

(1) Cα ⊂ P (α) and |Cα| ≤ κ+ω

(2) every C ∈ Cα is closed and unbounded in α and has order type less thanκ+ω.

(3) If C ∈ Cα and β is a limit point of C, then C ∩ β ∈ Cβ .By the remarks in the beginning of the paper we may assume that for all α

there is a C ∈ Cα such that the order type of C is exactly the cofinality of α.Further, since κ+ω is a strong limit, we can assume that if C ∈ Cα, then everyclosed unbounded subset of C is in Cα.

Let P = Col(ω1, < κ). Let j : V → M be a κ+ω+1 supercompact embeddingand G ⊂ P be generic. Since P has the κ-chain condition we can find a genericH ⊂ j(P ) so that j can be extended to a j : V [G] → M [H]. Let 〈Cj

α : α ∈ j(κ+ω+1)〉be the j image of 〈Cα : α ∈ κ+ω+1〉.

Let γ be the supremum of j“κ+ω+1. Then by the definition of weak square,M [H] |= there is a Dγ ⊂ γ that is club in γ and Dγ ∈ Cj

γ . Since in M [H]the cofinality of κ+ω+1 is ω1 we can assume that Dγ has order type exactly ω1.Further, since j“κ+ω+1 is ω-closed, we can assume that Dγ ⊂ j“κ+ω+1.

Let δ′ ∈ j“κ+ω+1. Then Dδ′ = Dγ ∩ δ′ ∈ Cjδ′ . Since Dδ′ is a countable subset

of the range of j, there is an x ∈ V such that j(x) = Dδ′ . If j(δ) = δ′, then byelementarity, x ∈ Cδ.

Thus, by forcing over V [G] with the forcing j(P )/G to get H, you have added a“thread” through the weak square sequence of order type ω1 that is cofinal throughκ+ω+1, namely there is a set D ∈ V [H] cofinal in κ+ω+1 of order type ω1 such thatfor all limit points δ of D,D ∩ δ ∈ Cδ.

Using the fact that the forcing j(P )/G used to get H over V [G] we now derivea contradiction. Let Q = j(P )/G. Then Q is countably closed. Let D be a Q termfor the “thread”.

We first claim that for all q ∈ Q there is a δ < κ+ω+1 such that |{s : for somer < q, r “s = D ∩ δ”| = κ+ω.

Otherwise let q be a counterexample. For each δ < κ+ω+1, let Tδ = {s : forsome r < q, r “s = D ∩ δ”. Then for δ < δ′, |Tδ| ≤ |Tδ′ |. Hence, if q is a


counterexample, there is an n for all δ < κ+ω+1, |Tδ| < κ+n.

For each δ of cofinality κ+n there is a β < δ for all s, t ∈ Tδ, if s 6= t thens ∩ β 6= t ∩ β. This induces a regressive function on ordinals of cofinality κ+n,which must be constant on an unbounded set. From this we conclude that there isa β < κ+ω+1 such that for an unbounded set U of δ < κ+ω+1 for all s, t ∈ Tδ, ifs 6= t then s ∩ β 6= t ∩ β.

Let r ≤ q be such that for some s, r D ∩ β = s. Then for each δ ∈ Uthere is only only possibility in Tδ compatible with r. Hence D ∈ V , which is acontradiction.

We now build a tree of conditions in Q indexed by finite sequences of elementsof κ+ω . We go by induction on the length of the sequences to produce {qσ :σ ∈ (κ+ω)n} and ordinals δn so that for all σ of length n, qσ determines D ∩ δn.Suppose we have done this for n. Fix σ of length n and choose a collection ofκ+ω of conditions {rτ : σ ⊂ τ, τ ∈ (κ+ω)n+1} below qσ such that for some δσ andall τ 6= τ ′, rτ and rτ ′ force incompatible information about D ∩ δσ. Let δn+1 bethe supremum of {δσ : σ ∈ (κ+ω)n}. For each τ ∈ (κ+ω)n+1 extend rτ to a qτ

deciding D ∩ δn+1. We may assume, by enlarging δn+1, and extending qτ ω times,if necessary that each qτ forces D ∩ δn+1 to be unbounded in δn+1.

Let δω = supnδn. For each f ∈ (κ+ω)ω, let qf =⋃

qf�n. Then for all such f , qf

determines D ∩ δω and forces that D ∩ δω is unbounded in δω.

This yields a contradiction, since if qf D ∩ δω = s then s ∈ Cδω. Further if

f 6= g ∈ (κ+ω)ω and qf D ∩ δω = s and qg D ∩ δω = t then s 6= t. Hence theremust be (κ+ω)ω many distinct elements of Cδω

, which is a contradiction.

�

We now discharge our final debt, by showing that the approachability propertyfails at the ωth successor of a supercompact. By the results of section 2, where itis proved that the Very Weak Square is consistent at a supercompact, this showsthat Very Weak Square does not imply the approachability property.

We do note that it is a theorem of Shelah that the Very Weak Square is equivalentto the approachability property at ℵω.

Claim. Suppose that κ is supercompact and that the G.C.H. holds. Then the ap-proachability property fails at κ+ω+1.

Proof. Let λ = κ+ω+1. Suppose that 〈Cα : α ∈ λ〉 is a witness to the approachabil-ity property ( i.e. for a closed unbounded set of α, Cα is club in α, has order typeless than κ+ω, and for a club set of α and all β < α there is a γ < α,Cα ∩β = Cγ .)Let 〈xη : η < λ〉 be a one to one enumeration of [λ]<ω1 . Define a function F : λ → λby setting F (δ) to be the least γ such that for all β < δ, [Cβ ]<ω1 ⊂ 〈xη : η < γ〉.

Let j : V → M be a κ+ω+1 supercompact embedding. Let 〈Cjα : α ∈ j(λ)〉

denote j(〈Cα : α ∈ λ〉) and 〈xjη : η < j(λ)〉 denote j(〈xη : η < λ〉).

Let γ be the supremum of j“κ+ω+1. Consider Cjγ . Since Cj

γ is closed andunbounded in γ, there is a δ < λ having countable cofinality such that Cγ ∩ j(δ)


has limit order type bigger than κ+ω.

By the approachability property there is a β0 < γ such that Cjγ ∩ j(δ) = Cj

β0.

Since β0 < γ, there is a δ′ < λ, β0 < j(δ′). However, this implies that j(F )(j(δ′)) <γ, and hence there is a β < γ such that [Cj

γ ∩ j(δ)]<ω1 ⊂ 〈xjη : η < β〉. On the other

hand, if x is a countable subset of j“δ, then there is a countable y ⊂ δ, x = j(y).Hence every countable subset of j“δ is of the form xj(η) for some η < λ.

This is a contradiction since there are λ many countable subsets of Cjγ ∩ j“δ.

They all have to be enumerated in the sequence 〈xjη : η < β〉. Each must be

enumerated as some xjj(η). But there are less than λ such xj

j(η). �

§6 Not So Very Weak Square

In this section we revisit Very Weak Square and discuss the principle we havedubbed “Not So Very Weak Square”.

Not So Very Weak Square has the same definition as Very Weak Square exceptthat we ask the Cα’s be closed in their supremum. Formally, a sequence 〈Cα : α <κ+〉 is a “Not So Very Weak Square Sequence” provided that:

(1) Cα ⊂ α, Cα is closed and unbounded in α,(2) For all bounded x ∈ [Cα]<ω1 , there is a β < α such that x = Cβ .

.

This apparently innocuous additional requirement makes no difference for α ofcofinality less than or equal ω1, since one can always close such a Cα. Nor does thisdistinction come up for the approachability property for the same reason: Given alist of the bounded subsets of Cα one can “compute”, in a very concrete way, a listof the bounded subsets of the closure of Cα in α. In virtue of Shelah’s theorem thatVery Weak Square is equivalent to the approachability property at ℵω, it is clearthat the Not So Very Weak Square and the Very Weak Square are equivalent atℵω. The theorem in this section shows that the two properties are not equivalent ingeneral. We show that if λ is the limit of an increasing sequence of supercompactcardinals 〈κn : n ∈ ω〉, then the Not So Very Weak Square fails at λ. In view ofthe results of section 2 (or rather the canonical extension of these results to get theconsistency of the Very Weak Square with countably many supercompacts), thisshows that the two properties are not equivalent.

Theorem 6.1. Suppose 〈κn : n ∈ ω〉 is an increasing sequence of supercompactcardinals and λ is the supremum of 〈κn : n ∈ ω〉. Then the Not So Very WeakSquare property fails at λ.

We begin the proof with a standard lemma:

Lemma 6.2. Let 〈κn : n ∈ ω〉 be an increasing sequence of supercompact cardinalswith supremum λ. Let 〈Sα : α ≤ β < κn〉 be a sequence of stationary subsets of λ+

consisting of points of cofinality less then κn. Then there is a γ < λ+ such that forall α < β, Sα is stationary in γ.

Proof of 6.2. Let j : V → M be a κn-supercompact embedding. Let γ′ =sup j“λ.Since j“λ is < κn-closed in γ′, j“Sα is stationary in γ′ for all α < β. Hence in M,


the statement “there is a γ′ for all α < β, Sα is stationary in γ′” holds. Thus, usingthe elementarity of j and the fact that j(β) = β, we get that the statement holdsin V .

�

Proof of 6.1.

Consider the reduced product Π〈κn : n ∈ ω〉/{finite sets}. Since the singularcardinals hypothesis holds above supercompact cardinals, and in particular at λ wecan find a scale 〈fα : α ∈ λ+〉. Without loss of generality we may assume that ifthe sequence 〈fβ : β < α〉 has a least upper bound (in the sense of section 4) thenfα is this least upper bound.

Let j : V → M be the κ0-supercompact embedding. Let 〈Cα : α < κ+〉 be a NotSo Very Weak Square Sequence. Denote j(〈fα : α ∈ λ+〉) by 〈f j

α : α < j(λ+)〉, andlet 〈Cj

α : α < j(λ+)〉 = j(〈Cα : α < λ+〉). Let γ =sup j“λ+ and γn =sup j“κn.Since j is a supercompact embedding, j“Vλ+ is closed under ω-sequences. Thus itis easy to check that the function g(n) = γn is a least upper bound for 〈f j

α : α < γ〉and hence that f j

γ = g modulo finite sets.

Given an α < j(λ+) of uncountable cofinality define hα(n) to be the infemumover all closed unbounded sets C ⊂ α of the supremum {f j

β(n) : β ∈ C}, i.e.h(n) =infC supβ∈C f j

β(n). Then, since the cofinality of α is uncountable there is aclosed unbounded Dα ⊂ α such that for all n, h(n) =supβ∈Dα

f jβ(n). Further, we

can assume that for all α, Dα ⊂ Cjα and the order type of Dα is the cofinality of α.

Clearly for all β < α, hα(n) ≥ f jβ(n) for all but finitely many n. Hence we see

that for a cofinite set of n, hγ(n) ≥ f jγ(n), since f j

γ is a least upper bound. Werestrict our attention to that cofinite set of n where f j

γ(n) = γn and hγ(n) ≥ f jγ(n).

Since f jγ(n) = γn, for all δ ∈ κn, there is a stationary set Fδ ⊂ γ ∩Dγ such that

for all β ∈ Fδ, fjβ(n) > j(δ). Then 〈Fδ : δ < κn〉 is a sequence of stationary subsets

of Dγ .

We can refine the Fδ’s for cofinally many δ’s to assume that for some m and acofinal set En ⊂ κn of δ all the elements of Fδ have cofinality less then or equal κm

(and Fδ is still stationary.) Since Dγ is closed and unbounded in γ and the ordertype of Dγ is λ+, we can apply Lemma 6.2 to find a ηn ∈ Dγ such that for all δ inthe cofinal set En, Fδ is stationary in ηn. Hence hηn(n) ≥ sup (j“En) = γn.

Let A = 〈H(χ),∈,∆, 〈fα : α < λ+〉, 〈Cα : α < λ+〉〉, for some regular χ muchgreater then λ+. There is a closed unbounded set U of β < λ+ such that SkA(β)∩λ+ = β. Since γ ∈ j(U), γ = Skj(A)(γ) ∩ j(λ+).

Since ηn ∈ Dγ ⊂ Cjγ , the sequence 〈ηn : n ∈ ω〉 is a subsequence of Cj

γ . Hence,by the defining property of the Not So Very Weak Square the sequence 〈ηn : n ∈ ω〉appears as a Cj

µ for some µ < γ. Since j(A) has a relation symbol denoting thevery weak square sequence, 〈ηn : n ∈ ω〉 ∈ Skj(A)(γ). Thus the function defined ash(n) = fηn

(n) lies in Skj(A)(γ). But then there is a δ < λ+, f jj(δ)(n) > h(n) for all

but finitely many n.


Hence for large n, h(n) ≥ γn, and γn > f jj(δ)(n) > h(n), a contradiction. �

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