canonical structure in the universe of set theory: part one

Annals of Pure and Applied Logic 129 (2004) 211–243

www.elsevier.com/locate/apal

Canonical structure in the universe of set theory:part one

James Cummings∗;1 , Matthew Foreman2;3 , Menachem Magidor3

Department of Mathematical Sciences, CMU, Pittsburgh, PA 15213-3890, USA

Received 1 March 2003; accepted 11 March 2004

Communicated by T. Jech

Abstract

We start by studying the relationship between two invariants isolated by Shelah, the sets ofgood and approachable points. As part of our study of these invariants, we prove a form of“singular cardinal compactness” for Jensen’s square principle. We then study the relationshipbetween internally approachable and tight structures, which parallels to a certain extent therelationship between good and approachable points. In particular we characterise the tight struc-tures in terms of PCF theory and use our characterisation to prove some covering results fortight structures, along with some results on tightness and stationary re6ection. Finally, we provesome absoluteness theorems in PCF theory, deduce a covering theorem, and apply that theoremto the study of precipitous ideals.c© 2004 Elsevier B.V. All rights reserved.

MSC: primary 03E35; 03E55; secondary 03E05

Keywords: PCF theory; Good ordinal; Approachable ordinal; The ideal I [�]; Internally approachablestructure; Tight structure; Square sequence; Covering properties; Precipitous ideal; Mutual stationarity;Stationary re6ection

1. Introduction

It is a distinguishing feature of modern set theory that many of the most interestingquestions are not decided by ZFC, the theory in which we profess to work; to put

∗ Corresponding author. Fax: +1-412-268-6380.E-mail address: jcumming@andrew.cmu.edu (J. Cummings).

1 The Erst author was partially supported by NSF Grants DMS-9703945 and DMS-0070549.2 The second author was partially supported by DMS-9803126 and DMS-0101155.3 The second and third authors were partially supported by the US–Israel Binational Science Foundation.

212 J. Cummings et al. / Annals of Pure and Applied Logic 129 (2004) 211–243

it another way, ZFC admits a large variety of models. A natural response to this isto identify invariants which may take diKerent values in diKerent models, and whichcodify a large amount of information about a model.

Of particular interest are invariants which are canonical, in the sense that the Axiomof Choice is needed to show that they exist, but once shown to exist they are indepen-dent of the choices made. For example the uncountable regular cardinals are canonicalin this sense.

Shelah discovered a large class of canonical invariants, the study of which he labeledPCF theory. These invariants include two which are central in this paper; Shelah [24,26](under some mild cardinal arithmetic assumptions on the singular cardinal �) deEnedtwo stationary subsets of �+, the sets of good and approachable points. The deEnitionsof these sets appear to depend on certain arbitrary choices, but (modulo the club Elter)are in fact independent of these choices. Other canonical structures we study in thispaper include the stationary sets of tight and internally approachable structures, andthe collection of good points on a scale.

It is known that every approachable point is good and that weak forms of square,for example Jensen’s weak-square principle ∗

�, imply that every point is approachable.Foreman and Magidor [16] showed that their principle “Very weak square”, whichcaptures some of the approachability implied by ∗

�, implies many of the interestingconsequences of ∗

�. Cummings [4] showed that the assertion that every point is goodseverely constrains all �+-preserving extensions of V .

Our Erst motivation for the work in this paper is the problem of the relationshipbetween the sets of good and approachable points. This problem is trivial when weaksquares exist, but non-trivial in general. For example it is consistent relative to largecardinals that not every point of coEnality ℵ1 in ℵ!+1 is good. We have speculatedthat perhaps the sets of good and approachable points coincide, and in Section 3 weprove that under some strong structural hypotheses this is the case.

The concept of an approachable ordinal is closely linked to that of an internallyapproachable (IA) structure. To be more precise, the set of approachable ordinals ofcoEnality � can be characterised [16] as the set of ordinals which have the formsup(N ∩ �+) for some internally approachable N of length and cardinality �.

Foreman and Magidor [17] isolated the concept of tight structure in their workon mutual stationarity and the non-saturation of the non-stationary ideal on P��, andtightness turns out to be closely related to the issues of goodness and approachability. Inparticular internally approachable structures are tight, and if N is tight then sup(N ∩ �+)is good. Our second motivation for the work in this paper is the analogy

Tight structuresIA structures

=Good ordinals

Approachable ordinals:

Here is an outline of the paper. Section 2 contains some background material.

• In Section 3 we prove a technical result about square-like sequences using the ma-chinery of PCF theory. We use this to show that under some structural hypothesesall good points in ℵ!+1 of coEnality greater than ℵ1 are approachable, and also toshow a kind of “singular cardinal compactness” for square sequences. For example

J. Cummings et al. / Annals of Pure and Applied Logic 129 (2004) 211–243 213

we show that if CH holds and ℵn holds for all n¡!, then there is a sequence〈C�: �∈ ℵ!+1 ∩ cof (ℵ2)〉 where C� is a club subset of � with order type ℵ2 and theC� cohere at common limit points of uncountable coEnality.

• In Section 4 we study the important property of uniformity for a structure, andshow that suOciently uniform structures can be reconstructed from their characteristicfunctions.

• In Section 5 we characterise tight structures in terms of PCF theory. We also showthat the properties of uniformity and tightness can sometimes be propagated from aset of regular cardinals K to the set pcf (K).

• In Section 6 we explore the relationship between tightness, covering properties andinternal approachability. We prove theorems showing that under some circumstancestightness and internal approachability are equivalent. We also record a remarks onthe connection between tightness and stationary re6ection.

• In Section 7 we prove some absoluteness results in PCF theory. We deduce a cov-ering theorem, and use it to show that if I is an ideal on ℵ1 such that forcing withPℵ1=I is suOciently mild then I is precipitous.

This paper contains only ZFC results. In the sequel [9] we prove a series of comple-mentary consistency results. We would like to thank John Krueger for comments andcorrections on an earlier draft of this paper.

2. Preliminaries

After a brief review of notation, we discuss the “canonical” concepts which arecentral in this paper: internally approachable and tight structures, and approachableand good ordinals.

We write ON for the class of ordinals, LIM for the limit ordinals and SUCC forthe successor ordinals. We write CARD for the class of inEnite cardinals, REG forthe regular cardinals and SING for the singular cardinals. We denote by cof (�) theclass of ordinals of coEnality �, and if A is a set of ordinals we write cf (A) for thecoEnality of A considered as an ordered set. An interval of regular cardinals is a setof the form REG ∩ [�; �) for cardinals � and �. We denote by �M the interpretation ofa term � in a model M , and by �M the relativisation of a formula � to M . An algebraon a set X is a structure for some countable Erst-order language which has X as itsunderlying set.

2.1. Internally approachable and tight structures

When � is an uncountable regular cardinal, we will denote by H� the transitive setof those X such that the transitive closure of X has size less than �. We denote by¡� some Exed well-ordering of H�. By convention when �¡! we will assume that¡� is the restriction of ¡! to H�.

We will frequently be interested in structures of the form A = (H�; ∈ ;¡�); theadvantage of building in a well-ordering is that if X ⊆H� and SkA(X ) is the set ofelements of H� deEnable in A with parameters from X , then SkA(X ) ≺ A and SkA(X )is the smallest substructure of A containing X .

The deEnition of internally approachable structure appears in Foreman et al.’s paper[18] on Martin’s Maximum. These structures are ubiquitous in modern set theory; seeLemma 2.3 and the remarks which follow it for some motivation.

De�nition 2.1. Let � be regular and let A be some algebra expanding (H�; ∈ ;¡�).Let N ≺ A. N is internally approachable (IA) if and only if there exist a limit ordinal" and a sequence 〈N#: #¡"〉 such that

(1) N =⋃

#¡" N#.(2) For all $¡", 〈N#: #¡$〉 ∈N .

In this case we will say that N is IA of length " and that 〈N#: #¡"〉 is an approachingsequence for N . We note that if we can always take an approaching sequence to becontinuous in the sense that if $ is a limit ordinal, then N$ =

⋃#¡$ N#.

The length " of the approaching sequence is not uniquely determined, but it is easyto see that cf (") is uniquely determined; by a mild abuse of language we refer tothis coEnality as the co3nality of N . In their paper on deEnable counterexamples tothe continuum hypothesis Foreman and Magidor [15] give a detailed discussion of thelengths and coEnalities of approaching sequences and the sizes of the structures thatappear in them.

All countable N are IA, and if N is approached by a sequence of length " then"⊆N . We are often concerned with IA structures N which have length and cardinality� for some regular uncountable cardinal �. In this case we can cast the deEnition in aslightly diKerent form, using the following easy lemma.

Lemma 2.2. Let � be regular and uncountable, and let N be an IA substructure ofA with length and cardinality �. Then there is a continuous approaching sequence〈M#: #¡�〉 for M , such that M$ is an elementary substructure of A with cardinalityless than � and 〈M#: #6$〉 ∈M$+1 for all $¡�.

We will refer to increasing and continuous sequences 〈M#: #¡$〉 of elementarysubmodels of A such that 〈M#: #6�〉 ∈M�+1 for � + 1¡$ as “continuous internallyapproaching chains of submodels”.

The following lemma encapsulates some of the key properties of IA structures. Forsimplicity we only consider IA substructures with approaching sequences consisting ofcontinuous internally approaching chains of submodels.

Lemma 2.3. Let � be a regular cardinal. Let 〈M#: #¡�〉 be a continuous increas-ing chain of elementary submodels of A such that 〈M#: #6�〉 ∈M�+1 for �¡�. LetM =

⋃#¡� M# and let �= |M |. Then

(1) �⊆M , so in particular �6�.(2) For all ordinals �∈M with cf (�)¿�

(a) cf (M ∩ �) = �.(b) There is a closed unbounded set C in sup(M ∩ �) with C ⊆M .

(3) For every set X ⊆M with |X |¡�, there is Y ∈M with |Y |¡� and X ⊆Y .

(4) Let K ∈M be a set of regular cardinals such that �¡min(K) and |K |¡�. Thenevery function in

∏�∈K M ∩ � is dominated pointwise by a function in M ∩ ∏

K .(5) Let P∈M be a �-closed and (�;∞)-distributive forcing poset. Then there is a

decreasing sequence 〈pi: i¡�〉 of conditions in M ∩P which meets every denseopen subset of P lying in M .

Proof. We sketch the proof.

(1) If $¡� then 〈M#: #¡$〉 in M , and so by elementarity the length $ of this sequenceis in M . We note that as a consequence M$ ∈M also.

(2) Since cf (�)¿�= |M |, M ∩ � is bounded in �. Since �∈M and each M# is inM , the sequence 〈sup(M# ∩ �): #¡�〉 is easily seen to be a continuous increasingsequence which is coEnal in sup(M ∩ �) and consists of ordinals in M .

(3) Since |X |¡�, there exists #¡� such that X ⊆M#.(4) Let f be a function in

∏�∈K M ∩ �. We note that since |K |¡�⊆M and K ∈M ,

we have K ⊆M . Since |K |¡�, we may End #¡� such that f is in∏

�∈K M# ∩ �.We deEne g with domain K by g(�) = sup(M# ∩ �). Since |M#|6�¡min(K) wesee that g∈ ∏

K , and since K and M# both lie in M we also have g∈M . Clearlyf(�)¡g(�) for all �∈K .

(5) By induction we choose pi to be the ¡�-least condition in P which is a lowerbound for 〈pj: j¡i〉 and lies in all the dense open subsets of P which are in Mi.Since 〈Mj: j¡i〉 lies in M it is easy to see that 〈pj: j¡i〉 lies in M , and hencepi lies in M .

Property 5 is not especially relevant in this paper, but it is highly signiEcant inthe context of [18,15]. Properties 2–4 will all be of interest to us in what follows. InSection 4 we will make a detailed study of structures which have uniform coEnality asin Property 2. Property 3 is a kind of internal covering which we discuss at length inSection 6. Property 4 we call tightness and we axiomatise it in the following deEnition:actually we axiomatise something a little more general.

De�nition 2.4. Let K be a set of regular cardinals, let �= cf (�)¿ sup(K), and letA = (H�; ∈ ;¡�). Let M ≺ A.

Then M is tight for K if and only if

(1) K ∈M .(2) For all g∈ ∏

� ∈ M ∩ K (M ∩ �) there exists h∈M ∩ ∏K such that g(�)¡h(�) for

all �∈M ∩K .

We note that if |K | ⊆M then K ⊆M , and in this case tightness has the simpler formappearing in Property 4 above: when K ⊆M , M is tight for K exactly when M ∩ ∏

Kis coEnal in

∏�∈K M ∩ �. It is natural to phrase this deEnition in terms of a standard

idea, the characteristic function of a structure.

De�nition 2.5. Let K be a set of regular cardinals and let M be a set. The characteristicfunction of M (on K) is the function !K

M with domain K given by !KM : � �→ sup(M ∩ �).

We will usually drop the superscript K and write !M when the set K is clear fromthe context. Typically we will be in a situation where |M |¡min(K) and so !M ∈ ∏

K .If a structure M is such that K ⊆M , then tightness of M amounts to saying that everyfunction in

∏K which is pointwise dominated by !M is pointwise dominated by some

function in M ∩ ∏K .

There are several reasons why it seems worthwhile to isolate the property of tight-ness. One reason is that there are many arguments in PCF theory which employ IAstructures, and on closer inspection these arguments are typically just using the tight-ness guaranteed by Lemma 2.3. Another reason is that tight structures arise naturallyin Foreman and Magidor’s theory of mutual stationarity. We only give a cursory de-scription of the theory of mutual stationarity, for more information see [17] and thesequel to this paper [9].

De�nition 2.6. Let K be a set of regular uncountable cardinals. Let 〈S�: �∈K〉 be suchthat S� ⊆ � for all �∈K .

(1) If N is a set, then N meets 〈S�: �∈K〉 if and only if sup(N ∩K) ∈ S� for all�∈N ∩K (equivalently !N � N ∩K ∈ ∏

�∈N ∩ K S�).(2) 〈S�: �∈K〉 is mutually stationary if and only if for every algebra A on sup(K)

there exists N ≺ A such that N meets 〈S�: �∈K〉.It is easy to see that if 〈S�: �∈K〉 is mutually stationary then S� is stationary for

each �. Foreman and Magidor showed [17] that the converse is false in general, butis true if S� ⊆ �∩ cof (!) for all �.

Mutual stationarity can be seen as intermediate between the classical concept ofstationarity for subsets of a regular uncountable cardinal, and the very general conceptof stationarity introduced in [18]. We recall that if S ⊆P(X ) then S is a stationarysubset of P(X ) if and only if for every algebra A on X there is B∈ S such thatB≺ A. This is easily seen to be equivalent to demanding that for every F : ¡!X →Xthere is a non-empty B∈ S which is closed under F . When we need to distinguishbetween diKerent 6avours of stationarity we will refer to this last concept as generalstationarity.

The sequence 〈S�: �∈K〉 is mutually stationary if and only if the set of subsetsof sup(K) which meet 〈S�: �∈K〉 is a stationary subset of P(sup(K)). By standardfacts [18, Lemma 0] about general stationarity, if X is any set with sup(K) ⊆X then〈S�: �∈K〉 is mutually stationary if and only if the set of subsets of X which meet〈S�: �∈K〉 is a stationary subset of P(X ). Subsequently we will often let X =H� for� some regular cardinal greater than sup(K).

Two of the most useful facts about stationary subsets of a regular uncountablecardinal � are Fodor’s lemma [13] and Solovay’s splitting theorem [27]. It is opento what extent these results may be generalised to arbitrary mutually stationary se-quences; Foreman and Magidor [17] identiEed a class of mutually stationary sequences,the tightly stationary sequences, for which versions of Fodor’s lemma andSolovay’s splitting theorem are available. As one might expect, a tightly station-ary sequence is a sequence whose mutual stationarity is witnessed by tightstructures.

De�nition 2.7. Let K be a set of regular cardinals and let 〈S�: �∈K〉 be such thatS� ⊆ � for all �∈K . Let �= sup(K)+. The sequence 〈S�: �∈K〉 is tightly stationaryif and only if for every algebra A on H� there is N ≺ A such that N is tight for Kand N meets 〈S�: �∈K〉.

See Foreman and Magidor’s paper [17] for the statements and proofs of Fodor’slemma and Solovay’s theorem in the context of tight stationarity.

2.2. Approachable and good points

There is a close connection between internally approachable structures and the normalideal I [�] deEned by Shelah [24–26].

De�nition 2.8. Let � be a regular uncountable cardinal. S ∈ I [�] if and only if thereexists a club subset E of � and a sequence 〈a#: #¡�〉 of bounded subsets of �, suchthat for all "∈E ∩ S there is A⊆ " unbounded in " such that ot(A) = cf (")¡" and forevery $¡" there is �¡" such that A∩ $ = a�.

We discuss some alternative characterisations of I [�] in Section 7. The followingresult appears in [16] as part of the proof of Claim 4.4 in that paper, and gives onedirection of the connection between I [�] and approachable structures.

Lemma 2.9. Let � be regular and uncountable. Let �¿� and let A be an algebraexpanding (H�; ∈ ;¡�). Let S ∈ I [�]. Then there is a club subset F of � such thatfor every "∈F ∩ S there is M ≺ A such that M is IA of length and cardinality cf (")and sup(M ∩ �) = ".

To get a reasonable converse we Ex a regular cardinal � less than � and assumethat �¡� = �. We let I [�; �] be the restriction of I [�] to coEnality �, that is the idealof those X ⊆ � such that X ∩ cof (�) ∈ I [�]. We enumerate [�]¡� as 〈a#: #¡�〉, and letS be the set of "∈ �∩ cof (�) such that there is A⊆ " unbounded in " with ot(A) = �and every proper initial segment of A enumerated as a� for some �¡". If we choose adiKerent enumeration 〈b#: #¡�〉 of [�]¡� then {a#: #¡$} = {b#: #¡$} for a club setof $¡�, so modulo the club Elter S is independent of the choice of the enumeration〈a#: #¡�〉.

It is not diOcult to see that S generates I [�; �] modulo the club Elter on �, or toput it another way S is (modulo club sets) the largest subset of �∩ cof (�) which is inI [�]. We will refer to S as “the set of approachable points of coEnality � in �” withthe understanding that this set is well-deEned modulo the club Elter. In this situationthere is an easy converse to Lemma 2.9.

Lemma 2.10. Let �, � and � be regular with �¡�¡� and �¡� = �. Let 〈a#: #¡�〉be an enumeration of [�]¡� in order type �, and let S be de3ned as above. LetN ≺ (H�; ∈ ;¡�; {〈a#: #¡�〉}) be IA of length and cardinality �. Then sup(N ∩ �) ∈ S.

Proof. Let N be approached by 〈Ni: i¡�〉, and let �i = sup(Ni ∩ �). Then every properinitial segment of 〈�i: i¡�〉 lies in N , so is enumerated as a# for some #∈N ∩ �. Itfollows that sup(N ∩ �) ∈ S.

Summarising, if �¡� = � then (modulo club sets) S is the set of ordinals of theform sup(N ∩ �) where N is IA of length and cardinality �. If C is any club subsetof � we can add C to the structure in the proof of Lemma 2.10, and End N such thatsup(N ∩ �) ∈C ∩ S; it follows that S is stationary. It is interesting to note that thereis an attractive characterisation of S in terms of forcing: results of Shelah [24] implythat every �+-closed forcing poset preserves the stationarity of subsets of S, but thereis a �+-closed poset which destroys the stationarity of �∩ cof (�)\S.

The set S is an example of the sort of “canonical invariant” discussed in the intro-duction. We will compare the set of approachable points with the set of good points,but before we can deEne the set of good points we need some PCF-theoretic prelim-inaries. For more information about PCF theory we refer the reader to Shelah’s book[26], or the survey papers [3,1].

Given a set X and an ideal I on X , we refer to the sets in I as I -small sets. Wesay that a set Y ⊆X is I -large if X \Y ∈ I and is I -positive if Y �∈ I . Given ordinalvalued functions f and g with domain X , we say that g dominates f modulo I (andwrite f ¡I g) if f(x)¡g(x) for an I -large set of values of x; the relation ¡I is astrict partial ordering. Similarly we say g dominates f pointwise (and write f¡g)if f(x)¡g(x) for all x. Given an ordinal valued function f a scale of length $ in(∏

x f(x);¡I ) is an increasing and coEnal sequence of length $ in (∏

x f(x);¡I ).For Y a subset of X , a sequence of functions in XON is pointwise increasing on Y

if it is strictly increasing on every coordinate in Y . We note that a sequence which ispointwise increasing on an I -large set is ¡I -increasing, but that in general the converseis not true. Two sequences 〈f#: #¡�〉 and 〈g#: #¡�〉 which are increasing with respectto ¡I are co3nally interleaved (modulo I) if for all #¡� there is $¡� such thatf# ¡I g$ and g# ¡I f$.

The function g is a an exact upper bound (eub) for the ¡I -increasing sequence〈f#: #¡$〉 if and only if f# ¡I g for all #¡$, and for every h with h ¡I g there exists#¡$ with h ¡I f#. This is equivalent to 〈f#: #¡$〉 being a scale in (

∏x g(x);¡I )

(with the caveat that the functions f# need only be dominated by g modulo I , ratherthan literally being members of

∏x g(x)). It is easy to see that an exact upper bound,

if one exists, is well-deEned modulo I .A point $ is good for a ¡I -increasing sequence f (of length at least $) if and

only if cf ($)¿|X | and there exists an exact upper bound h for 〈f#: #¡$〉 with theproperty that cf (h(x)) = cf ($) for all x. The following lemma gives some useful equiv-alent characterisations of goodness: the implication from (3) to (1) gives an importantconstruction principle for exact upper bounds.

Lemma 2.11. The following are equivalent for 〈f#: #¡$〉 a ¡I -increasing sequencewith cf ($)¿|X |.(1) $ is good.

(2) There is a sequence of functions of length cf ($) which is pointwise increasingand is co3nally interleaved with 〈f#: #¡$〉 modulo I.

(3) There is a sequence of functions of length cf ($) which is pointwise increasing onan I -large subset of X and is co3nally interleaved with 〈f#: #¡$〉 modulo I.

Proof. For (1) implies (2), let h be an eub such that cf (h(x)) = cf ($) for all x andEx for each x a sequence 〈#x

i : i¡cf ($)〉 increasing and coEnal in h(x); now deEnegi(x) = #x

i and check that 〈gi: i¡cf ($)〉 is pointwise increasing and coEnally inter-leaved with 〈f#: #¡$〉. The implication from (2) to (3) is immediate. For (3) im-plies (1) Ex 〈gi: i¡cf ($)〉 pointwise increasing on an I -large set and coEnally in-terleaved with 〈f#: #¡$〉; check that the pointwise supremum of 〈gi: i¡cf ($)〉 isan exact upper bound for 〈f#: #¡$〉 which has coEnality cf ($) on an I -large set,and then alter it to get an exact upper bound which has coEnality cf ($) onall x.

As an immediate corollary of Lemma 2.11, if $ is a good point then there is C clubin $ such that every point of C with coEnality greater than |X | is good. So the setof ungood points of coEnality greater than |X | is quite thin, in the sense that if it isstationary then its stationarity can only re6ect at points of itself.

Example 2.12. If 〈f#: #¡�〉 and 〈f′#: #¡�〉 are two ¡I -increasing sequences with the

same exact upper bound g then it is easy to see that they are coEnally interleaved. Itfollows that if � is a regular uncountable cardinal there is a club set of $¡� suchthat 〈f#: #¡$〉 and 〈f′

#: #¡$〉 are coEnally interleaved. Therefore, the sets of goodpoints for 〈f#: #¡�〉 and 〈f′

#: #¡�〉 are equivalent modulo the club Eller and so givean example of “canonical” structure.

The following result by Shelah is central in PCF theory.

Fact 2.13. If � is a singular cardinal then there is a set K ⊆ � of regular cardinalssuch that ot(K) = cf (�) and there is a scale of length �+ in

∏K under the eventual

domination ordering.

As we just pointed out, the set of good points in such a scale is essentially inde-pendent of the choice of the scale so has some claim to be considered a canonicalinvariant. Of course there is still a dependence on K but for small values of � we canalso make a canonical choice for K . The case of most interest to us is � = ℵ!, and inthis case work of Shelah shows that modulo Enite sets there is a largest K ⊆ {ℵn: n¡!}such that

∏K has a scale of length ℵ!+1 in the eventual domination ordering. In this

situation we refer to the set of good points in such a scale as the good points in ℵ!+1.The next lemma makes the connection between scales, good points and IA structures.

It should be compared with Lemma 2.10.

Lemma 2.14. Let |X |¡�¡�¡� with �, � and � regular. Let f = 〈f#: #¡�〉 be a¡I -increasing sequence in XON=I , and suppose there is an exact upper bound g for

f such that cf (g(x))¿� for all x∈X . Let N ≺ (H�; ∈ ;¡�; {f; g}) be an IA structureof length and cardinality �. Then sup(N ∩ �) is a good point for f.

Proof. We Ex an internally approaching chain 〈Ni: i¡�〉 such that X ∈N0, X ⊆N0,|Ni|¡� for all i and the union of the chain is N . We let gi(x) = sup(Ni ∩ g(x)) fori¡�, and claim that 〈gi: i¡�〉 will serve as a witness that sup(N ∩ �) is good. SinceX ⊆N0 and cf (g(x))¿� for all x∈X , we see that gi¡g; moreover if i¡j then Ni ∈Nj,so gi ∈Nj and hence gi¡gj.

Since gi ∈N and gi is dominated by the exact upper bound g, it follows by ele-mentarily that gi ¡I f# for some #∈N ∩ �. On the other hand if #¡ sup(N ∩ �) thenthere exist i¡� and $∈Ni ∩ � with #¡$, so that range(f$) ⊆Ni and f$¡gi.

If C is club in � we may add a predicate for C to A and produce a good pointof coEnality � lying in C. It follows that the set of good points of coEnality � isstationary.

We can now prove that, as we mentioned in the introduction, approachable pointsare good.

Corollary 2.15. Let � be the successor of a singular cardinal �, let K ⊆ � be anunbounded set of regular cardinals with ot(K) = cf (�)¡min(K). Let f be a scale oflength � in

∏K under eventual domination. If S ∈ I [�] then almost all points of S

with co3nality greater than cf (�) are good for f.

Proof. Immediate from Lemmas 2.9 and 2.14.

Recalling that the set of approachable points (when it can be deEned) is the maximalset in I [�], we see that modulo the club Elter every approachable point of coEnalitygreater than cf (�) is good.

3. Goodness, approachability and compactness for squares

One theme of this paper is the relationship between the concepts of goodness andapproachability. As we showed in Corollary 2.15, approachable points are good. In thissection we show that under certain circumstances the implication from approachabilityto goodness can be reversed. It is notable that the result presented here shows thatscales can be used to derive squarelike principles at � from squares below �.

Since approachable points are good, the problem “which good points are approach-able” becomes trivial if almost every point is approachable. We digress brie6y to reviewwhat is known about the extent of I [�].

It is known that if Jensen’s weak square principle ∗� holds then �+ ∈ I [�+], so

that large cardinals will be required to make models in which I [�+] is non-trivial.If �¡� = � then ∗

� holds, so GCH trivialises I [�] for � the successor of a regularcardinal. Shelah has shown that if � is regular then �+ ∩ cof (¡�) is in I [�+], and thatif � is singular then for all regular �¡� there is a stationary subset of �+ ∩ cof (�)lying in I [�+].

If � is supercompact and cf (�)¡�¡�, then �+ ∩ cof (¡�) �∈ I [�+]. By doing somesuitable Levy collapses (see [24] or [20] for details) this can be used to producea model in which ℵ!+1 ∩ cof (ℵ1) �∈ I [ℵ!+1]. As for successors of regular cardinals,Mitchell’s model [22] in which there is no ℵ2-Aronszajn tree also has the propertythat ℵ2 ∩ cof (ℵ1) �∈ I [ℵ2]. In recent work [23] Mitchell has produced a model in whichI [ℵ2] is generated modulo the club Elter by the set ℵ2 ∩ cof (!).

We now narrow our focus to the cardinal ℵ!+1, where the points of coEnality ℵ1

seem to play a special role. It is known (see [16] or [9]) to be consistent that stationarilymany points of coEnality ℵ1 are not good, and it is open whether ℵ!+1 ∩ cof (�= ℵ1) isalways in I [ℵ!+1]. As we see shortly all points of coEnality greater than 2ℵ0 are good.In particular under CH all points of coEnality greater than ℵ1 are good.

Before we prove the main result of this section we need some technical preliminaries.We start with the concept of a continuous sequence. Let f be a ¡I -increasing sequence.The sequence f is continuous at $ if and only either there is no exact upper boundfor 〈f#: #¡$〉 or f$ is such a bound. f is continuous if and only if it is continuousat every limit $. Given an arbitrary ¡I -increasing sequence f of limit length �, wemay replace f$ for $¡� limit by an exact upper bound for 〈f#: #¡$〉 whenever sucha bound exists, and get a continuous sequence which is coEnally interleaved with theoriginal one.

If $ is a good point for f and 〈gi: i¡cf ($)〉 is increasing on an I -large set andcoEnally interleaved with 〈f#: #¡$〉 then let g be the pointwise supremum of thesequence 〈gi: i¡cf ($)〉. As we saw in Lemma 2.11 g is an exact upper bound for〈f#: #¡$〉 and so by continuity f$ is also an exact upper bound: since exact upperbounds are unique modulo I , the functions f$ and g must agree on an I -large set.

Next we need an alternative characterisation of good points in scales of a specialkind. When X is an ordered set with no last element, and I is the ideal of boundedsubsets of X , we usually write ¡∗ for ¡I and = ∗ for ¡I . We refer to ¡∗ as theeventual domination ordering. This is the context of the “good scales” and “very goodscales” studied in our paper [8] on scales, squares and re6ection.

In a scale under eventual domination the deEnition of good point can be simpliEed.To be more precise the following statements are equivalent:

• The ordinal $ is good for f.• The coEnality of $ is greater than |X |, and for every unbounded A⊆ $ there exists

an unbounded B⊆A and x∈X such that 〈f#: #∈B〉 is pointwise increasing on{y: x¡y}.

The proof of the forward implication uses an “interleaving” argument of a type whichis ubiquitous in PCF theory, so we give it in detail.

Since $ is good, we may Ex x0 ∈X and 〈g#: #¡cf ($)〉 which is pointwise increasingon {y: x0¡y} and is coEnally interleaved with 〈f#: #¡$〉. Thinning out the sequenceof g# we may also assume that for every # there is �# ∈A with g# ¡∗ f�# ¡∗

g#+1. Since cf ($)¿|X | we may End x¿x0 and an unbounded set B0 ⊆ cf ($) such thatg#(y) ¡ f�#(y)¡g#+1(y) for all #∈B0 and y¿x. Let B= {�#: #∈B0}, and observethat if # and #′ are in B0 with #¡#′ and y¿x then f�#(y)¡g#+1(y)6g#′(y)¡f�′

We will also need Shelah’s important Trichotomy theorem [26].

Fact 3.1. Let I be an ideal on a set X of cardinality �, and let � be regular with�+¡�. Let 〈f#: #¡�〉 be a ¡I -increasing sequence. Then one of the following mustoccur:

(1) There is an eub g for 〈f#: #¡�〉 such that cf (g(x))¿� for all x.(2) There exist an ultra3lter U on X disjoint from I and sets 〈Sx: x∈X 〉 with

|Sx|6�, such that for all #¡� there is g∈ ∏x∈X Sx and $¡� such that f#¡Ug

¡Uf$.(3) There exists a function h such that if D# = {x:f#(x)¡h(x)} then the sequence

〈D#: #¡�〉 is not eventually constant modulo I .

If 2�¡� then it is not hard to see that the second and third cases are impossible, sothat there must exist an eub g for 〈f#: #¡�〉 such that cf (g(x))¿� for all �.

For the rest of this section we Ex some inEnite A⊆! such that there is a scale oflength ℵ!+1 in

∏n ∈ A ℵn under eventual domination. We also Ex 〈f#: #¡ℵ!+1〉 which

is such a scale. Altering the scale at limits if necessary, we may assume that it has thefollowing strengthened form of the continuity property: if 0¡k¡! and # is a goodpoint of coEnality ℵk , then f# is an exact upper bound for 〈f5: 5¡#〉 (this is justcontinuity) and in addition cf (f#(n)) = ℵk for all n¿k.

We already know that there are stationarily many good points for this scale inany uncountable coEnality but with some cardinal arithmetic assumptions we can saymore.

If 2ℵ0¡ℵk and $∈ ℵ!+1 is a point of coEnality ℵk then it follows from Trichotomythat there is an eub g for 〈f#: #¡$〉 with cf (g(n))¿ℵ0 for all n∈A. A little analysis(see [21] or [5] for the details) shows that cf (g(n)) = ℵk for coEnitely many n, so that$ is a good point: it follows from our assumptions on the scale that f$ is an eub for〈f#: #¡$〉 and cf (f$(n)) = ℵk for all n∈A with n¿k.

The structural hypothesis which we need for our main result is a weakening ofsquare which only refers to ordinals below ℵ! with a Exed coEnality.

De�nition 3.2. Let k be a natural number with k¿1. A (ℵ!; cof (!k))-sequence is asequence 〈C#: #∈ ℵ! ∩ cof (!k)〉 such that

(1) For all #; C# is club in # and ot(C#) =!k .(2) For all #; $ and �, if �∈ lim(C#) ∩ lim(C$) then C# ∩ �=C$ ∩ �.

The next lemma is similar in spirit to the results on “improving squares” in ourpaper on scales, squares and re6ection [8].

Lemma 3.3. If ℵn holds for all n with k6n¡! then there is a (ℵ!; cof (!k))-sequence.

Proof. Fix 〈Dn# : #¡ℵn+1〉 witnessing ℵn , where we assume without loss of generality

that Dn# ⊆#\(ℵn + 1) for #∈ ℵn+1\(ℵn + 1). We deEne C# inductively.

Base case: If #∈ ℵk+1 ∩ cof (!k), let C# =Dk# .

Successor case: Let #∈ (ℵn+1\(ℵn+1))∩cof (!k) for n¿k and let #∗ = ot(Dn# ). Since

cf (#) =!k and k �= n, we see that #∗¡ℵn. Since Dn# is club in #, cf (#∗) = cf (#) =!k .

By induction C#∗ has already been deEned, and we deEne C# by copying C#∗ into Dn# .

To be more precise we set C# = {�∈Dn# : ot(Dn

# ∩ �) ∈C#∗}.We now check that this deEnition succeeds. Let �∈ lim(C�) ∩ lim(C"). By our as-

sumption on the sets Dn# , either � and " are both less than ℵk+1 or they both lie in

ℵn+1\(ℵn + 1) for some n¿k.Case 1: �; "¡ℵk+1. In this case C� =Dk

� and C" =Dk" , and it follows that C� ∩ �=Dk

�=C" ∩ � by the deEning property of a ℵk -sequence.

Case 2: ℵn¡�; "¡ℵn+1 for n¿k. In this case �∈ lim(Dn� ) ∩ lim(Dn

" ) and so Dn� ∩ �

=Dn" ∩ �. Moreover if 5= ot(Dn

� ∩ �) then by the deEnition of C� we have that5∈ lim(C�∗), and similarly 5∈ lim(C"∗). By induction C�∗ ∩ 5=C"∗ ∩ 5, and so bydeEnition C� ∩ �=C" ∩ �.

The main theorem of this section shows that starting with a (ℵ!; cof (!k))-sequence,we may lift it via PCF theory to a square-like sequence deEned on the set of goodpoints of coEnality !k in some scale of length ℵ!+1. To be more precise we deEnethe following square-like principle, which is obtained by allowing the club sets to bedeEned only at points of some set S and weakening the coherence requirement so thatit only applies at common limit points of uncountable coEnality.

De�nition 3.4. Let S⊆ℵ!+1 ∩ cof (!k) for some k¿1. A −!ℵ!

(S)-sequence is a se-quence 〈E#: #∈ S〉 such that

(1) For every #∈ S, E# is club in # and ot(E#) =!k .(2) For all �; "∈ S and every � of uncountable coEnality which is a common limit

point of E� and E"; E� ∩ �=E" ∩ �.

Theorem 3.5. Let k be an integer with k¿2, and assume that there is a (ℵ!;cof (!k))-sequence. If G is the set of good points of co3nality !k for the scale〈f#: #¡ℵ!+1〉, then there exists a −!

ℵ!(G)-sequence.

Proof. Dropping Enitely many points from the set A if necessary, we may assumethat all points in A are greater than k. Fix a (ℵ!; cof (!k))-sequence 〈C#: #∈ ℵ! ∩cof (!k)〉. By our assumptions on the scale and the set A, for all �∈G the function f�

is an exact upper bound for 〈f#: #¡�〉 and cf (f�(n)) = ℵk for all n∈A.Given �∈G, we deEne functions g�

i for i¡!k by setting g�i (n) equal to the ith

member of Cf�(n). By construction the sequence of functions 〈g�i : i¡!k〉 is pointwise

increasing and is coEnally interleaved with 〈f#: #¡�〉.For almost every �¡� of uncountable coEnality, there is j¡!k such that 〈g�

i : i¡j〉is coEnally interleaved with 〈f#: #¡�〉; the sequence 〈g�

i : i¡j〉 witnesses that � isgood. Fix such � and j. By the argument of Theorem 2.11 and the uniqueness ofexact upper bounds f� =∗ supi¡j g�

i , and since Cf�(n) is closed we also see thatsupi¡j g�

i (n) = g�j(n). We conclude that f� =∗ g�

For every �∈G, we now deEne D� = {�¡�: ∃jf� =∗ g�j}. The set D� need not be

club in �, and so we let E� be the closure of D� in �. We will show that 〈E�: �∈G〉is a −!

ℵ!(G)-sequence.

If � is an accumulation point of D� with uncountable coEnality, then there is aunique k such that the functions {g�

j : j¡k} are coEnally interleaved with {f#: #¡�}.It follows that � is good, and so by continuity we have f� =∗ g�

k and thus �∈D�.It is easy to see that ot(D�) =!k and we just showed that D� is closed under

supreme of uncountable coEnality. Now let �; " be members of G. We claim that if� is a common accumulation point of D� and D" with uncountable coEnality, thenD� ∩ �=D" ∩ �.

Since � is in D�, there is j with cf (�) = cf (j) such that f�(n) = g�j(n) for all large n,

so that f�(n) ∈ lim(Cf�(n)) for all large n. Similarly there is k such that f�(n) = g"k(n)

and f�(n) ∈ lim(Cf"(n)) for all large n.It follows that Cf�(n) ∩f�(n) =Cf"(n) ∩f�(n) for all large n, and hence that j = k

and that g�i = g"

i for i¡j. If 5∈D� ∩ � then f5 =∗ g�i for some i¡j, and so f5 =∗ g"

iand 5∈D" ∩ �; similarly D" ∩ �⊆D� ∩ �, so D� ∩ �=D" ∩ �.

It is now routine to verify that 〈E�: �∈ S〉 is a −!ℵ!

(G)-sequence.

Theorem 3.5 has the following striking corollary.

Corollary 3.6. Let CH hold and let ℵn hold for all n¡!. Then for every integerm¿2 there exists a −!

ℵ!(cof!m)-sequence.

Remark 3.7. With a little work we can put Corollary 3.6 in a more pleasing way. Ex-tending our notation, let −!

ℵn(cof (ℵk)) be the statement that there exists 〈C#: #∈ ℵn+1 ∩

cof (ℵk)〉 with C# club in #, and the C# cohering at common limit points of uncountablecoEnality. Let CH hold, let k¿2; then if −!

ℵn(cof (ℵk)) holds for all suOciently large

Enite n, it holds for n=!. The proof is an easy variation on the one given above.Theorem 3.5 also supplies a partial answer to the problem which motivates this

paper, the relationship between goodness and approachability. It is easy to see that ifthere is a −!

ℵ!(S)-sequence then S ∈ I [ℵ!+1], and so Theorem 3.5 has the following

corollary.

Corollary 3.8. If ℵn holds for all 3nite n then in ℵ!+1 all good points of co3nalitygreater than ℵ1 are approachable.

We can also deduce some results about stationary re6ection.

Corollary 3.9. Let CH hold and let ℵn hold for all n¡!. For all integers m andn such that 0¡m¡n there is a stationary subset of ℵ!+1 ∩ cof (ℵm) which does notre:ect at any point of ℵ!+1 ∩ cof (ℵn).

Proof. Let 〈C#: #∈ ℵ!+1 ∩ cof (ℵn)〉 be a −!ℵ!

(cof !n)-sequence. Let S be the set ofpoints of coEnality ℵm which occur as limit points of some C#, and given $∈ S letD$ be the unique set such that D$ =C# ∩ $ when $∈ lim(C#).

For any club subset D of ℵ!+1, we can End #∈ lim(D) with cf (#) = ℵn. Then D∩C#

is club in # and so we can End $∈ lim(C# ∩D) with cf ($) = ℵm. Clearly $∈D∩ Sso that S is stationary.

Applying Fodor’s lemma we may End T⊆S stationary and � such that ot(D$) = �for all $∈T . For each # of coEnality ℵn the club set lim(C#) meets T exactly once,so that the stationarity of T does not re6ect at any point of coEnality ℵn.

In our paper [8] it is shown that it is consistent that ℵn holds for 0¡n¡! butℵ! fails. This is achieved by arranging that every stationary subset of ℵ!+1 ∩ cof (!)

re6ects at some point of ℵ!+1 ∩ cof (ℵ1). In the sequel [9] to this paper we show that itis consistent that the least � for which � fails should be the Erst inaccessible cardinal.

We can also combine the −!ℵ!

(cof!m)-sequences of Corollary 3.6 for diKerent valuesof m and get a weakening of the principle ℵ!;! [7].

Corollary 3.10. Let CH hold and let ℵn hold for all n¡!. Then there exists〈C�: �¡ℵ!+1〉 such that for every limit ordinal �

(1) C� is a countable family of subsets of �, each with order type less than ℵ!.(2) For every C ∈ C� and every $∈ lim(C) with uncountable co3nality, C ∩ $∈ C$.

Similar arguments can be used with some other square like principles. The following“Strong Non-Re6ection” principle was introduced by Dzamonja and Shelah [11], andhas been used by Cummings et al. [6,10] in the investigation of stationary re6ection.

De�nition 3.11. Let � and � be regular and uncountable with �¡�. Then SNR(�; �)holds if and only if there is f: � → � such that for all #∈ �∩ cof (�) there is C clubin # such that f�C is strictly increasing.

This principle is true in L for all � which are not weakly compact and all �¡�. Itimplies that every stationary subset of � has a stationary subset which re6ects at nopoint of coEnality �.

Dzamonja and Shelah [12] have studied the cardinal u(�) which is deEned to bethe least �¿� such that SNR(�; �) fails. In particular they have shown by an elaborateforcing argument that u(�) can be the successor of a singular cardinal. The followingtheorem, which is proved in the same way as Theorem 3.5, seems to have some bearingon problems of this type.

Theorem 3.12 (Cummings [5]). Suppose that k¡! and that SNR(ℵn;ℵk) holds forall large n¡!. Then there is f: ℵ!+1 → ℵk such that for all good #∈ ℵ!+1 ∩ cof (ℵk)there is C club in # such that f�C ∩ {�:!¡cf (�)¡ℵk} is strictly increasing.

One rather unsatisfying feature of some results in this section is the appearance ofthe Continuum Hypothesis among the hypotheses. CH is only being used to derive “allpoints of ℵ!+1 ∩ cof (¿ℵ1) are good”, and it is quite possible that this statement isactually true in ZFC. For some discussion of the diOculties associated with attemptingto show that this statement is consistently false see [4].

4. Uniformity

In this section we develop the idea of uniformity for a structure. The main point isthat suOciently uniform structures can be reconstructed from a small amountof data.

We recall from Lemma 2.3 that if M is an IA structure of size � with a continuousapproaching sequence of length � then cf (M ∩ �) = � for all ordinals �∈M such thatcf (�)¿�. This is the prototype for the uniformity properties we will consider.

De�nition 4.1. Let M≺(H�; ∈ ;¡�), let K be a set of regular cardinals and let � bea cardinal. Then M is �-uniform on K if and only if cf (M ∩ �) = � for all �∈K ∩M ,and is weakly �-uniform on K if and only if cf (M ∩ �)¿� for all �∈K ∩M .

This property arises naturally in the study of mutual stationarity. A structure whichmeets a sequence of stationary sets all consisting of ordinals of coEnality � will auto-matically be �-uniform on the relevant set of regular cardinals.

A particularly interesting case for our purposes will occur when K is an interval ofcardinals and M contains a large enough initial segment of the ordinals.

De�nition 4.2. Let M≺(H�; ∈;¡�) and let �; � and � be cardinals with �6�¡�6�.The structure M is �-uniform (resp. weakly �-uniform) between � and � if and onlyif

(1) �⊆M .(2) The structure M is �-uniform (resp. weakly �-uniform) on the interval of regular

cardinals {�∈REG: �¡�¡�}.

If M is �-uniform between � and � we say that M is �-uniform past �, and similarlyfor weak uniformity.

This kind of uniformity arises naturally in the study of IA structures. If M is an IAstructure with an approaching chain of length � and �= |M |⊆M , then M is �-uniformpast �.

Let M≺(H�; ∈;¡�). If � is any limit ordinal in M , then cf (�) is in M and thereexists in M an increasing coEnal map f from cf (�) to �. Restricting f to cf (�) ∩Mwe get a coEnal map from cf (�) ∩M to �∩M , so that cf (�) ∩M and �∩M havethe same coEnality. In particular if M is �-uniform on K then cf (�∩M) = � for allordinals �∈M such that cf (�) ∈K , and similarly for weak uniformity. If M is �-weaklyuniform between � and � and 9∈M is an ordinal of coEnality less than or equal to� it follows that M ∩ 9 is unbounded in 9. If cf (9) is strictly between � and �, thencf (M ∩ 9)¿�.

Before discussing the implications of weak uniformity, we recall a well-known vari-ation on the closed unbounded Elter. Given a regular cardinal � and a limit ordinal 9with �6cf (9), we say that a set A⊆9 is a ¡�-club subset of 9 if A is unbounded in9, and sup(x) ∈A for every x⊆A with |x|¡�. If � is uncountable then the collectionof ¡�-club sets generates a cf (9)-complete Elter, which is in fact the restriction ofthe club Elter to the set of points of coEnality less than �.

Lemma 4.3. Let M be �-weakly uniform between � and �. Then

(1) For every bounded subset x of M ∩ � with |x|¡�; sup(x) ∈M .(2) For every regular � with �¡�6 sup(M ∩ �)

(a) cf (M ∩ �)¿�.(b) There exists E⊆M ∩ � which is ¡�-club in sup(M ∩ �).

(Note that in item (2), we do not require that �∈M .)

Proof. For claim (1), we may as well assume that x has limit order type. Let $ =sup(x) and �= min(M\$), where clearly both $ and � are limit ordinals less than �and M ∩ �=M ∩ $. Suppose for a contradiction that �¿$. Then cf (�)¿�, becauseotherwise M ∩ � would be unbounded in �. Since �∈M and �¡cf (�)¡� we havecf (M ∩ �)¿�, but this is impossible because cf (M ∩ $) = cf ($)6|x|¡� andM ∩ $ =M ∩ �.

For claim (2), we Ex a regular � with �¡�6 sup(M ∩ �). Suppose for a contradic-tion that cf (M ∩ �)¡� and Ex x⊆M ∩ � coEnal in M ∩ � with |x|¡�. Since �6�¡�,M ∩ � is bounded in �. Since �6 sup(M ∩ �), x is a bounded subset of M ∩ � andso by the Erst claim sup(x) = sup(M ∩ �) ∈M . This is a contradiction since M ∩ � isbounded in �. If we now Ex any coEnal set D⊆M ∩ � and let E be the closure of Dunder suprema of size less than �, then E⊆M ∩ � and E is a ¡�-club set as required.

It is notable that claim (2) of Lemma 4.3 applies to regular cardinals which do notlie in M . Before we can exploit Lemma 4.3 we need some more or less standard factsabout rebuilding structures.

The idea of Lemmas 4.4 and 4.5 Erst appears in the proof by Solovay that theSingular Cardinals Hypothesis holds above a strongly compact cardinal. Lemma 4.5will End an immediate application in Theorem 4.6 where we show that suOcientlyuniform structures are determined by their characteristic functions: the more generalLemma 4.4 will be useful later in the covering results of Section 7.

Lemma 4.4. Let � be a cardinal, let K be an interval of regular cardinals withmin(K) = �+ and let � be a regular cardinal with sup(K)¡�. Let M0 and M1 betwo elementary substructures of (H�; ∈;¡�) such that

• �⊆M0 ∩M1.• For every �∈K , M0 ∩M1 ∩ � is co3nal in M0 ∩ �.

Then M0 ∩ sup(K)⊆M1 ∩ sup(K).

Proof. Clearly M0 ∩ �=M1 ∩ �= �. We show by induction on �∈K that M0 ∩ �⊆M1 ∩ �. There is nothing to do when � is a limit cardinal, so let �= 5+ where M0 ∩ 5⊆M1 ∩ 5 by induction. If #∈M0 ∩ � there is $∈M0 ∩M1 ∩ � with #¡$, and we Exf the ¡�-least bijection from $ to |$|. Since $∈M0 ∩M1 and f is deEned from $we have f∈M0 ∩M1, and since $¡5+ we have |$|65. So f(#) ∈M0 ∩ 5, and sinceM0 ∩ 5⊆M1 ∩ 5 we conclude that f(#) ∈M1 ∩ 5 and so #∈M1. Thus M0 ∩ �⊆M1 ∩ �and the induction goes through.

We note that we are not assuming that K⊆M here.

Lemma 4.5. If we strengthen the hypotheses of Lemma 4.4 by adding the demandthat M0 ∩M1 ∩ � is also co3nal in M1 ∩ � for all �∈K , then we may strengthen theconclusion to M0 ∩ sup(K) =M1 ∩ sup(K).

Proof. Immediate from Lemma 4.4.

Theorem 4.6. Let � be an uncountable cardinal. Let K be an interval of regularcardinals with min(K) = �+ and let � be a regular cardinal with sup(K)¡�. LetM≺(H�; ∈;¡�) be �-weakly uniform between � and sup{�+: �∈K}. Then M ∩ sup(K) is determined by !K

M and sup(M ∩K).

Proof. Suppose that M and N are both substructures of (H�; ∈;¡�) which are �-weakly uniform between � and sup{�+: �∈K}, with !K

M = !KN and sup(N ∩K) =

sup(M ∩K). Then N ∩K =M ∩K . By Lemma 4.3, for every �∈K ∩M each of thesets M ∩ � and N ∩ � contains a set which is ¡�-club in sup(M ∩ �) = sup(N ∩ �). Theintersection of two ¡�-club sets is ¡�-club, so M ∩N ∩ � is unbounded in M ∩ � andN ∩ �. By Lemma 4.5 M ∩ sup(K) =N ∩ sup(K).

We will only be using Theorem 4.6 in the special case when K⊆M . It will beuseful later to know that the process of reconstructing M ∩ sup(K) from !K

M is simplydeEnable.

Lemma 4.7. Let K; �; �; � be as in the last lemma and suppose that M;N≺(H�; ∈;¡�) with !K

M ∈N . Then M ∩ sup(K) ∈N .

Proof. Since K = dom(!KM ) we see that K ∈N , and thus sup(K) ∈N and �∈N . We

may also End �∗ ∈N such that M is �∗-weakly uniform.Let 〈g$: $¡ sup(K)〉 be such that g$ is the ¡�-least bijection from $ to |$|, and

deEne partial functions g and h from sup(K) × sup(K) to sup(K) by

g($; #) = g$(#); h($; �) = g−1$ (�):

Since g and h are deEned from parameters in N , they are members of N . The argumentin the proof of Lemma 4.5 shows that M ∩ sup(K) can be computed as follows: foreach C = 〈C�: �∈K〉 such that C� is ¡�∗-club in !M (�) for each �, compute theclosure X (C) of �∪ (

⋃� C�) under g and h, and then take the intersection of all the

sets X (C). It follows that M ∩ sup(K) ∈N .

5. Tight structures and PCF

Recall from deEnition 2.4 that if K is a set of regular cardinals and M is a structurewith K ∈M and K⊆M , then M is tight for K if and only if M ∩ ∏

K is coEnal in∏� ∈ K M ∩ �. For the rest of this section we will make the following

Blanket assumption. We are given a set K of regular cardinals and a structureM≺(H�; ∈;¡�) such that K ∈M , K⊆M and |M |¡min(K).

We note that since |M |¡min(K), we have !KM ∈ ∏

K . Also since if K⊆M we have|K |¡min(K) so that K is a “progressive” set of regular cardinals, and the methods ofPCF theory can be applied to K .

The main idea will be to analyse tightness for M in terms of the PCF-theoreticproperties of K . The key new points in the analysis (which are closely related) willbe that under some reasonable circumstances we can discern whether a structure M istight by inspecting M ∩ON , and can reconstruct a tight structure M from a Enite setof ordinal parameters.

We summarise the main interest in tight structures by the following points:

(1) Tight structures are canonically determined by a Enite number of canonical ordinalparameters, i.e. good ordinals. (Theorem 5.3) In particular, the stationary set oftight structures is canonically well-ordered. This is clearly not possible for arbitrarystructures as they outnumber the possible collections of ordinal parameters.

(2) A tight elementary substructure of an H� is determined by the its “cardinal struc-ture”. This is implicit in the statement and explicit in the proof of Theorem 7.3.

(3) Internally approachable structures share the two previous points (which is whythey were used in the original PCF theory). However, given a structure N≺(H�;∈;¡�), to determine whether N is tight (in an absolute way from knowledgeof the regular cardinals) one must only know !N�ℵ!+1 (i.e. N ’s trace on theordinals), but to determine whether N ∩H (ℵ!+1) is internally approachable seemsto require considerably more information and conceivably might not be an absolutequestion.

We summarise the last point by the slogan that being tight is an “exterior” question,but being internally approachable is an “internal” question.

The results of this section generalise a result by Foreman and Magidor [17, Section7.3] which analyses uniform structures which are tight for K = {ℵn: n¡!} under theassumption that max pcf (K) = ℵ!+1. They are also related to work by Shelah analysingthe characteristic functions of certain IA structures in terms of PCF.

For our purposes the most important results are Theorems 5.2 and 5.3, which give adetailed analysis of tightness for M under the assumptions that pcf (K)⊆M and M isuniform on K . Theorem 5.5 shows that we can drop the assumption that pcf (K)⊆Mand demand only weak uniformity for M on K ; the price we pay is that we need extratechnical assumptions and we get a somewhat weaker conclusion. Theorem 5.6 showsthat we can sometimes propagate tightness from K to pcf (K).

As a warmup for the style of argument which we will be using in this section, weshow that if M is tight for K then some uniformity properties can be propagated fromK to pcf (K).

Theorem 5.1. Let M be tight for K and let � be a cardinal with �¿|K |. If M is�-weakly uniform on K , then M is �-weakly uniform on pcf (K). If M is �-uniformon K , then M is �-uniform on pcf (K).

Proof. Suppose that M is �-weakly uniform on K . Given �∈M ∩ pcf (K), let usEx U ∈M an ultraElter on K and 〈f#: #¡�〉 ∈M increasing and coEnal in

∏K=U .

Suppose for a contradiction that cf (M ∩ �)¡�. We Ex A⊆M ∩ � coEnal with ordertype cf (M ∩ �), and note that if #∈A then f#(�) ∈M ∩ � for all �∈K . If we deEneg(�) = sup# ∈ A f#(�) for all �∈K , then g(�)¡ sup(M ∩ �) because cf (M ∩ �)¿�¿|A|.Since M is tight there is h∈M ∩ ∏

K with g¡h, and by elementarity there is #∈M ∩ �with h¡Uf#. So g¡Uf#, but this is impossible because by construction f#6g.

Now suppose that M is �-uniform on K , and use this to End a pointwise increasingsequence 〈gi: i¡�〉 of functions whose ranges are contained in M and whose point-wise supremum is !M . As above let �∈M ∩ pcf (K) and Ex U ∈M an ultraElter onK and 〈f#: #¡�〉 ∈M increasing and coEnal in

∏K=U . It is easy to see by tight-

ness and elementarity that 〈f#: #∈M ∩ �〉 is coEnally interleaved with 〈gi: i¡�〉, socf (M ∩ �) = �.

We recall that we have a set K of regular cardinals such that |K |¡min(K). This isa situation in which Shelah’s PCF theory gives an analysis of pcf (K) in terms of thePCF generators, and we give a brief review of this analysis.

As usual we let J¡� be the ideal of sets A⊆K such that every ultraElter U containingA gives an ultrapower

∏K=U of coEnality less than �. The sequence of ideals J¡� is

increasing with � and is continuous at limit cardinals.By standard results from PCF theory we Ex a sequence of sets 〈B�: �∈ pcf (K)〉 with

B�⊆K for each �, such that B� generates J¡�+ over J¡�: that is to say that for A⊆Kwe have A∈ J¡�+ ⇔ A\B� ∈ J¡�. An easy induction argument shows that J¡� is theideal of sets which are covered by a Enite union of sets in {B�: �¡�}.

It is known that pcf (K) has a maximum element, and accordingly we will chooseBmax pcf (K) =K . By standard facts from PCF theory we may Ex a matrix of functions〈f�

# : #¡�, �∈ pcf (K)〉 such that 〈f�# : #¡�〉 is a continuous scale in

∏B�=J¡� for

every �∈ pcf (K).We Ex � some suOciently large regular cardinal and let A be the structure (H�; ∈;

¡�; {K}; 〈B�〉; 〈f�# 〉). We will analyse tightness of M for K in terms of PCF theory,

assuming that M≺A; pcf (K) ⊆M and M is uniform on K .

Theorem 5.2. Let � be regular and uncountable, and suppose that |K |¡�¡min(K).Suppose that M≺A where M is �-uniform on K and pcf (K) ⊆M . Then the followingare equivalent:

(1) M is tight for K .(2) For every �∈ pcf (K), sup(M ∩ �) is a good point of co3nality � for 〈f�

# : #¡�〉and f�

sup(M ∩ �) =J¡ � !B�M .

Proof. For the forward direction, assume M is tight. Since cf (M ∩ �) = � for all �∈K ,we may End a pointwise increasing sequence 〈gi: i¡�〉 in

∏K whose pointwise supre-

mum is !KM .

Fix �∈ pcf (K). We claim that 〈gi�B�: i¡�〉 is coEnally interleaved with 〈f�# : #∈

M ∩ �〉 modulo J¡�. For all #∈M ∩ �; f�# ∈ M and so f�

# is pointwise dominated by

!B�M . Since |K |¡�, there exists i¡� such that f�

# is dominated pointwise by gi�B�. Con-versely if i¡� then it follows from the tightness of M that gi is dominated pointwiseby some function in M , and so by elementarity gi¡J¡ �f

�# for some #∈M ∩ �.

It follows that cf (M ∩ �) = � and sup(M ∩ �) is a good point of coEnality �. ByLemma 2.11 f�

sup(M ∩ �) agrees modulo J¡� with the pointwise supremum of the gi�B�,

that is to say f�sup(M ∩ �) =J¡� !

B�M .

For the converse direction, assume that for every �∈ pcf (K), sup(M ∩ �) is a goodpoint of coEnality � for 〈f�

# : #¡�〉 and f�sup(M ∩ �) =J¡� !

B�M . Let h be a function in

which is pointwise dominated by !KM . For every � in pcf (K) we have h¡J¡ �f

�sup(M ∩ �).

Since sup(M ∩ �) is a good point it follows that f�sup(M ∩ �) is an exact upper bound

for 〈f#: #∈M ∩ �〉. Hence for every �∈ pcf (K) there is #∈M ∩ � with h¡J¡ �f�# .

We will inductively build a decreasing sequence �0¿�1¿�2¿�j of members ofpcf (K) together with ordinals #i ∈M ∩�i. We will also build a decreasing sequence ofsets D0 ⊇ D1 ⊇ D2 : : : such that Di ∈ J¡�i , halting when we reach a stagewith Di = ∅.

We let �0 = max pcf (K), and recall that B�0 =K . We choose #0 ∈M ∩ �0 such thath¡J¡ �0

f�0#0

. Let D0 = {�: h(�)¿f�0#0

(�)}, so that D0 ∈ J¡�0 .Suppose that we have deEned �i; #i and Di for i6j. We stop if Dj = ∅. Otherwise

we choose �j+1 to be the unique member of pcf (K) with Dj ∈ J¡�+j+1

and Dj =∈ J¡�j+1 ,

and choose #j+1 ∈M ∩ �j+1 such that h¡J¡ �j+1f�j+1

#j+1 . Now we let

Dj+1 = {�∈Dj: � =∈B�j+1 or h(�)¿f�j+1#j+1 (�)}:

Since B�j+1 generates J¡�+j+1

and h¡J¡ �j+1f�j+1

#j+1 , we see that Dj+1 ∈ J¡�j+1 .Since the descending sequence of �i can only have Enite length, the construction

must terminate. Let j be the last stage, so Dj = ∅. For each �∈K , let i be minimalwith � =∈Di. By deEnition we must have �∈B�i and h(�)¡f�i

#i (�). We conclude thath is pointwise dominated by the pointwise supremum of {f�i

#i : i6j}, and since thisfunction lies in M we have proved that M is tight.

The following result will play an important role when we analyse the relationshipbetween tightness and internal approachability in Section 6. It is a generalisation ofShelah’s analysis of internally approachable structures used to bound powers of singularcardinals. It is important for the purposes of that section to note that each of the setsB�i\Ei is in M , and each of the indices sup(M ∩ �i) is computed in a uniform wayfrom M .

Theorem 5.3. Let � be regular and uncountable, and suppose that |K |¡�¡min(K).Suppose that M is �-uniform on K and pcf (K)⊆M .If M is tight for K then there exist �0; : : : ; �j ∈ pcf (K) and E0; : : : ; Ej with Ei ∈M ∩

J¡�i such that K =⋃

i(B�i\Ei) and !M is the pointwise supremum of the functionsf�i

sup(N ∩ �i)�(B�i\Ei) for i6j.

Proof. By Theorem 5.2, f�sup(M ∩ �) =J¡� !

B�M for all �∈ pcf (K).

We argue in a way which parallels the last part of the proof of Theorem 5.2.As there we build a decreasing sequence �0¿�1¿ · · · of elements of pcf (K), andalso sets Di with Di ∈ J¡�i which this time are not necessarily decreasing. We alsobuild sets Ei ∈M ∩ J¡�i such that Di⊆Ei, and choose Di+1 as a subset of Ei ratherthan Di.

Let �0 = max pcf (K) and D0 = {�: !M (�) �= f�0sup(M ∩ �0)(�)}, so that D0 ∈ J¡�0 . Sup-

pose now that we have constructed the ordinals �0; �1; : : : ; �j, together with the setsD0; : : : ; Dj and E0; : : : ; Ej−1.

If Dj = ∅ we set Ej = ∅ and stop the construction. If Dj �= ∅ we choose �j+1 as theunique member of pcf (K) such that Dj ∈ J¡�+

j+1and Dj =∈ J¡�j+1 . We choose Ej to be

some Enite union of sets in {B�: �6�j+1} such that Dj⊆Ej.We note that Ej ∈ J¡�+

j+1∩M , and that since Dj =∈ J¡�j+1 we must have B�j+1⊆Ej.

Dj+1 = {�∈Ej: � =∈B�j+1 or !M (�) �= f�j+1

sup(M ∩ �j+1)(�)}:

Since Ej\B�j+1 is covered by a Enite union of sets from {B�: �¡�j+1}, and the func-

tions !M and f�j+1

sup(M ∩ �j+1)(�) agree on a J¡�j+1 -large subset of B�j+1 , we see thatDj+1 ∈ J¡�j+1 .

Since the sequence of �i is decreasing we eventually reach a stage j with Dj = ∅,and so we halt the construction after setting Ej = ∅. To Enish we need to check that!M is the pointwise supremum of the functions f�i

sup(M ∩ �i)�(B�i\Ei) for i6j.

Let �∈B�i\Ei for some i6j. Since Di⊆Ei, �∈B�i\Di. By the construction of Di

we know that {�∈B�i :f�isup(M ∩ �i)

(�) �= !M (�)} is a subset of Di, so f�isup(M ∩ �i)

(�) =!M (�).

Given �∈K , let i be minimal such that � =∈Ei. If i = 0 then �∈B�0 because wechose B�0 =K . If i¿0 then �∈Ei−1, and since � =∈Di we see that �∈B�i . It followsthat every element of K appears in B�i\Ei for some i.

Remark 5.4. For an application which we will make of Theorems 5.2 and 5.3 inSection 7, we note that we only needed the scales {f�

# : #¡�} to be continuous atgood points of coEnality �.

Now we give an alternative characterisation of tightness for M in the style of The-orem 5.2, dropping the assumption that pcf (K) ⊆M and weakening the uniformityassumption to weak uniformity. As we remarked earlier, this analysis owes a debt toShelah’s analysis of the characteristic functions of IA structures. For technical reasonswe will not be using continuous scales, but rather scales with a technical propertycalled !-club minimality.

Let us be given an index set X , an ideal I on X and a ¡I -increasing sequence f. Ifcf ($)¿|X |, we deEne a function f∗

$ by letting f∗$ (�) be the least ordinal of the form

sup# ∈ E f#(�) for E an !-club subset of $. We note that if E is some !-club subset

of $ with f∗$ (�) = sup#∈E f#(�), then f∗

$ (�) = sup#∈F f#(�) for any F ⊆E with F an!-club subset of $.

Since the intersection of |X |-many !-club subsets of $ is !-club, we may Ex a sin-gle !-club subset F of $ such that f∗

$ (�) = sup#∈F f#(�) for all �∈F ; it follows that

f∗$ is an upper bound modulo I for 〈f#: #¡$〉. We say that the sequence f is !-clubminimal at $ if f∗

$ =f$. Returning to the context of this section, it is routine to checkthat if I is an ideal on K with tcf(

∏K=I) = � then we may End a scale 〈f#: #¡�〉

K=I which is !-club minimal at every $¡� such that |K |¡cf ($)¡min(K);by a slight abuse of language we will say that such a scale is !-club minimalin

∏K=I .

We will Ex a matrix of functions 〈f�# : #¡�, �∈ pcf (K)〉 such that 〈f�

# : #¡�〉 isan !-club minimal scale in

∏B�=J¡� for every �∈ pcf (K). Let A be the structure

(H�; ∈ ;¡�; {K}; 〈B�〉; 〈f�# 〉). We will assume that M≺ A.

We are now ready to use our technical assumption of !-club minimality. Supposethat M is |K |+-weakly uniform between 9 and ;, where all cardinals in M∩ pcf (K)are greater than 9 and less than ;. Let �∈M∩ pcf (K) and let �= sup(M ∩ �). ByLemma 4.3 there is E ⊆M ∩ � which is !-club in �. Since |M |¡min(K) we see thatcf (�)¡min(K), and since � lies in an interval where M is weakly uniform we alsosee that cf (�)¿|K |. So by the assumption of !-club minimality, we may End F ⊆Esuch that F is !-club in � and f�

� (�) = sup#∈F f�# (�) for all �∈K . Since F ⊆E ⊆M ,

K ⊆M and �∈M it follows easily that f��6!B�

M .We can now give a characterisation of tightness.

Theorem 5.5. Let M be |K |+-weakly uniform between 9 and ;, where all cardinalsin M∩ pcf (K) are greater than 9 and less than ;. Then the following conditions onM are equivalent:

(1) M is tight for K .(2) For every �∈M∩ pcf (K), if �= sup(M∩�) then

(a) f��6!B�

M .(b) There is A⊆B� such that A∈M∩ J¡� and {�∈B�:f�

� (�)¡!M (�)} ⊆A.(c) f�

� is an exact upper bound for 〈f�# : #¡�〉 modulo J¡� in the following

strengthened sense: for all f¡f�� there exist $∈M ∩ � and B∈M ∩ J¡� such

that {�∈B�:f�$ (�)6f(�)} ⊆B.

Proof. (1) implies (2). We have already seen that the uniformity hypothesis on Mimplies that f�

�6!B�M . Since M is tight we may End a function g∈M∩∏

K such thatfor all �∈K , f�

� (�)¡!M (�) implies that f�� (�)¡g(�). By !-club minimality we know

that f�� is the pointwise supremum of {f�

# : #∈F} for some F ⊆M which is !-clubin �, and we may End #∈F so large that f�

# dominates g modulo J¡�. Now sincef�# 6f�

� we see that for �∈K

f�� (�) ¡ !M (�) ⇒ f�

� (�) ¡ g(�) ⇒ f�#(�) ¡ g(�);

so that if we set A= {�∈B�:f�# (�)¡g(�)} then A is as required. Now let f¡f�

� .Since f�

�6!M and M is tight, we may End h∈M such that f¡h and then End $∈Fsuch that f�

$ dominates h modulo J¡�. Now

f�$(�) 6 f(�) ⇒ f�

$(�) 6 h(�)

and if we set B= {�∈B�:f�$ (�)6h(�)} then B is as required.

(2) implies (1). Let f∈ ∏K with f¡!K

M . The construction is very similar to thatfor Theorem 5.2. We construct a decreasing sequence of ordinals �0¿�1¿ · · · with�i ∈M∩ pcf (K), together with ordinals #i ∈M∩ �i and sets Bi ∈M∩ J�i , in such a waythat if �∈Bi−1\Bi then f(�)¡f�i

#i (�).Let �0 = max pcf (K) and �0 = sup(M∩�0), and observe that by (b) the functions

f�0�0

and !M agree outside a set in M∩ J¡�0 . By (c) we can End a set B0 ∈M∩ J¡�0

such that f(�)¡f�0�0

(�) for � =∈B0.If at stage i we have Bi �= ∅, then choose �i+1 minimal with Bi =∈ J¡�i+1 , noting

that �i+1 ∈M because Bi ∈M . Now we apply (b) and (c) to End Bi+1 ∈ J¡�i+1 and#i+1 ∈M∩�i+1 such that f(�)¡f�i+1

#i+1 (�) for �∈Bi\Bi+1.As usual, the construction must terminate with Bi = ∅. Then f is pointwise dominated

by the supremum of the functions f�j#j for j¡i, and this function lies in M .

Under certain circumstances we can do an analysis of tightness as in Theorem 5.5under a weaker uniformity hypothesis. It is known that if K is an interval of regularcardinals then pcf (K) is also an interval of cardinals. If K is an interval, M is tightfor K and M is |K |+-weakly uniform in some interval containing K then it followsfrom Theorem 5.1 that M is weakly uniform in an interval containing pcf (K), and sothe analysis of Theorem 5.5 applies.

It follows from Theorem 5.1 that if M is tight some information about coEnalitiescan be propagated from K to pcf (K). The next result shows that under slightly strongerassumptions the same is true for the property of tightness itself. The idea of takingpointwise suprema of functions from many scales comes from Shelah’s proof thatpcf (pcf (K)) = pcf (K).

Theorem 5.6. Let pcf (K) ⊆M and let M be |pcf (K)|+-weakly uniform on K . If Mis tight for K then M is tight for pcf (K).

Proof. Let L= pcf (K), and let F ∈ ∏�∈L M ∩ �. We deEne f∈ ∏

f : � �→ sup�∈L

f�F(�)(�) + 1:

Since L⊆M , we see that f�F(�) ∈M and f�

F(�)(�) ∈M ∩ � for all �∈L and �∈K .Since cf (M∩�)¿|L| it follows that f(�)¡ sup(M∩ �) for all �∈K .

Since M is tight for K , there is g∈M∩∏K which dominates f pointwise. Since

〈f�# : #¡�, �∈ pcf (K)〉 ∈M and the sequence 〈f�

# : #¡�〉 is a scale for each �, it fol-lows by elementarity that there is G ∈M∩∏

L such that g¡J¡�f�G(�) for all �∈L.

We now see that for each �∈L we have

f�F(�) ¡ f ¡ g ¡J¡� f�

G(�);

so that G dominates F pointwise and we have shown that M is tight for L.

Theorem 5.6 allows us a practical alternative to Theorem 5.5:

Corollary 5.7. Let M be |K |+-weakly uniform (resp. �-uniform with |K |¡�¡min(K)) between p and ;, where all cardinals in M∩ pcf (K) are greater than 9and less than ;. Let M ′ be the Skolem hull of M∪ pcf (K) in (H�;∈;¡�). Then thefollowing conditions on M are equivalent.

(1) M is tight for K .(2) M ′ is tight for pcf (K) and |K |+-weakly uniform (resp. �-uniform) on pcf (K).(3) For every �∈ pcf (K), sup(M ′∩ �) is a good point of co3nality at least �+ (resp.

of co3nality �) for 〈f�# : #¡�〉 and f�

sup(M ′∩�) = J¡�!B�M ′ .

Proof. By Fact 6.2 we know that |pcf (K)|¡|K |+4. Thus M ′ ⊂ skH�(M∪|K |+3). Stan-dard arguments show that for all �∈K\|K |+3, sup(M∩ �) = sup(skH�(M∪|K |+3)∩�).Suppose now that M is tight for K .

By the results in the last paragraph, M ′ is tight for K and by Theorem 5.6, M ′

is tight for pcf (K). Moreover, by Theorem 5.1, M ′ is |K |+-weakly uniform (resp. �-uniform) on pcf (K). Hence 1 implies 2. Again by the previous paragraph it is clearthat 2 implies 1.

By Theorem 5.2, 2 and 3 are equivalent for M ′. .

6. Tightness, approachability and re1ection

In this section we prove a general covering theorem for tight structures, and use itto show that certain tight uniform structures are IA. We also discuss the connectionbetween tightness and stationary re6ection.

We start with some general discussion of covering properties of structures. Supposethat �, � and � are regular cardinals with �¡�¡�. We consider substructures N≺(H�;∈;¡�; {�; �}) such that |N | = � and �⊆N .

Given a set Z with �⊆Z , we let P�Z be the set of x⊆Z such that |x|¡� andx∩ � ∈ �. In increasing order of strength we may consider the following properties ofN :Internally co3nal in P��: N∩P�(N∩�) is coEnal (in the inclusion ordering) in

P�(N∩�)Internally stationary in P��: N∩P�(N∩�) is stationary in P�(N∩�).Internally club in P��: N∩P�(N∩�) contains a club in P�(N∩�).Internally approachable in P�H�: N∩H� is IA of length and cardinality �.The following easy lemma shows that some cardinal arithmetic assumptions simplify

the picture.

Lemma 6.1. Let �, � and � be regular cardinals with �¡�¡� and let N≺(H�;∈;¡�; {�; �}) be such that |N | = � and �⊆N .If �¡� = � and N is internally co3nal in P��, then P�(N∩ �) ⊆N , from which it

follows that N is IA in P�H�.

For the rest of this section we will be studying structures N which are tight forsome interval K of regular cardinals. One reason for this is that we wish to apply theresults of Section 4 on recovering structures from their characteristic functions, andthese results require an interval of regular cardinals. Another reason is that we can usethe following result of Shelah.

Fact 6.2 (Shelah). Let K be an interval of regular cardinals with |K |¡min(K). Then

• pcf (K) is an interval of regular cardinals, which has a largest element.• |pcf (K)|6|K |+3.• pcf (pcf (K)) = pcf (K).

Remark 6.3. If K is an inEnite interval of regular cardinals with |K |¡min(K), thenit follows from Fact 6.2 that by deleting a suitable Enite initial segment k of K wecan obtain an interval L with |pcf (L)|¡min(L).

Remark 6.4. It is a viable conjecture that |pcf (A)| = |A| for all A with |A|¡min(A).

Before stating and proving the very general Theorem 6.5, we digress brie6y toconsider what is perhaps the most interesting special case. Let K = {ℵn: 0¡n¡!},and suppose for simplicity that pcf (K) is countable, say pcf (K) = {ℵ#+1: #6$} forsome countable $.

Let m¿0 and let � be a large regular cardinal. Let N≺H� be such that ℵm = |N | ⊆N ,and let N be ℵm-uniform and tight on {ℵn:m¡n¡!}. By Theorems 5.1 and 5.6 N isℵm-uniform and tight on {ℵ#+1:m6#6$}.

Under these circumstances, it will follow from Theorem 6.5 that if m¿1 then Nis internally stationary in Pℵm(ℵ$+1). In conjunction with some extra cardinal arith-metic assumptions as in Lemma 6.1, it will follow that N ∩Hℵ$+1 is IA of length andcardinality ℵm. This is an instance of one of the motivating theses of this paper, thattightness plus uniformity is very close to internal approachability.

Theorem 6.5. Let � be an uncountable regular cardinal. Let L be an interval ofregular cardinals such that min(L) = �++, |L|¡�, and L= pcf (L). Let � be a largeregular cardinal and let N≺(H�;∈;¡� {L}) be such that |N | = �+ and N is �+-uniform between �+ and max(L)+ and N is tight for L. Then N is internally stationaryin P�+(max(L)).

Proof. Fix some algebra A on N∩ max(L). We build an increasing sequence〈Mi: i¡�〉 such that for all i

(1) Mi≺N , with �⊆Mi and |Mi| = �.(2) Mi∩ max(L)≺A.

(3) Mi∩�+¡Mi+1∩�+.(4) !L

M iis pointwise dominated by some function in Mi+1∩

The last demand on Mi is the crucial one, and it is possible to satisfy it because N istight and �+-uniform. We let M =

⋃i Mi and L∗ = {�+}∪L, and note that pcf (L∗) =L∗.

As usual we will let 〈B�: �∈L∗〉 be the ¡�-minimal sequence of generators for L∗,〈f�

# : �∈L∗; #¡�〉 the ¡�-least matrix of functions such that 〈f�# : #¡�〉 is a continuous

scale in∏

B�=J¡�.By construction X L∗

Miis pointwise dominated by !L∗

Mi+1, so that M is �-uniform

between �+ and max(L)+. Since |L∗|¡� the construction also guarantees that M istight for L∗.

By Theorem 5.3 the function !L∗M is the pointwise supremum of Enitely many func-

tions of the form f�sup(M∩�) � (B�\E�), where E� ∈M . By Lemma 4.3 N∩ max(L) is

�-closed in sup(N∩ max(L)), so in particular sup(M∩�) ∈N for all �.It follows that !L∗

M ∈N . By Lemma 4.7 we see that M∩ max(L) ∈N , and by construc-tion M∩ max(L)≺A and M ∈P�+(N ∩ �). This shows that N ∩P�+(N ∩ �) is stationary.

As we mentioned before proving Theorem 6.5, we can use the theorem to show thatsometimes tight plus uniform equals IA. The following corollary gives the simplestinteresting case.

Corollary 6.6. Let 2ℵ1 = ℵ2 and 2ℵ! = ℵ!+1. Let � be a large regular cardinal andlet N≺(H�;∈;¡�) be ℵ2-uniform between ℵ2 and ℵ!, and tight for {ℵn: 2¡n¡!}.Then N∩Hℵ!+1 is IA of length and cardinality ℵ2.

For structures of size ℵ1 we have less than satisfactory results which we illustratewith the following example:

Example 6.7. Suppose that ℵ! is a strong limit and 2ℵ! = ℵ!+1. Let N≺H (ℵ!+2) havecardinality ℵ1 and uniform coEnality !1. Suppose that sup(N∩ℵ!+1) is an approachableordinal � and !N = ∗f�. Then N is internally approachable.

To see this, let M be the internally approachable structure that has cardinality !1

with !M = ∗f�. Then for some !l, the Skolem hull of N∪!l and the Skolem hull ofM∪!l have the same characteristic function and hence the two Skolem hulls haveequal intersection in ℵ!+1. In particular, the Skolem hull of N∪ℵl is closed under!-sequences below ℵ!+1. One can sees inductively that if Skolem hull of N∪ℵl−i isclosed under !-sequences then so is the Skolem hull of N∩ℵl−(i+1). In particular, N isclosed under !-sequences below ℵ!+1 and is thus internally approachable below ℵ!+1.

Foreman and Magidor [15] showed that there is a close connection between internalapproachability, uniformity and stationary re6ection for sets of structures. In particularthey showed that

• Let � be supercompact, let �, and ? be regular uncountable cardinals less than �with �¡?. Let G be generic for Col(?;¡�), then in V [G] the following stationary

re6ection principle holds:Every stationary set of IA substructures of (H�;∈;¡�) of length and cardinality �re6ects to some substructure of cardinality ?.

• Let � ¿ !3 and let Sij be the set of N≺(H�;∈;¡�) such that |N | = ℵ1, ℵ1 ⊆N ,cf (N∩ℵ2) = ℵi and cf (N∩ℵ3) = ℵj.Then only S11 can have the property that every stationary subset is re6ecting.

The following result indicates that tightness is also relevant to problems about sta-tionary re6ection.

Theorem 6.8. Let K be a countable set of regular cardinals and let S be a stationaryset of elementary substructures of (H�;∈;¡�), such that every element of S is tightfor K . Let M be such that S re:ects to M . Then M≺(H�;∈;¡�) and M is tightfor K .

Proof. Let t be a Skolem term and let a be a Enite sequence of parameters from M .There is N ⊆M such that N ∈ S and all the members of a come from N . Since N isclosed under t and N ⊆M , t(a) ∈M as required.

Note that since K is countable, K ⊆N for all N ∈ S. Let f ∈ ∏�∈K K∩M , and End

N ∈ S such that N ⊆M and f⊆N . Since N is tight for K there is g∈N∩∏K which

dominates f, and g∈M since N ⊆M .

Foreman and Todorcevic [19] have deEned a notion of tightness for countable struc-tures, and have used this to investigate stationary re6ection in [H�]ℵ0 .

7. PCF absoluteness, covering and precipitous ideals

In this section we prove a version of the covering lemma and apply it to a problemabout precipitous ideals. Our covering lemma states roughly that if V and W are innermodels with V ⊆W , and the PCF structures of V and W are similar enough then everyset of ordinals in W is covered by a set of the same size lying in V . The key ideaof the proof is that the characteristic functions of certain IA structures in W can (asin Section 5) be described in terms of the PCF structure of W , and so (by the PCFresemblance hypothesis) these characteristic functions will be elements of V .

We start with a result which says that suOciently similar universes have similar PCFstructures.

Theorem 7.1. Let V be an inner model of W , and in V let K be a set of regularcardinals with |K |¡min(K). Assume that for all �∈ [min(K);max(pcf (K))V ], if � isregular in V then � is regular in W . Assume also that for every f∈ (

∏K)W there

is g∈ (∏

K)V with f¡g. Then

(1) JV¡� = JW

¡� ∩V for all cardinals �¿min(K).(2) Any sequence of PCF generators for K in V is still a sequence of PCF generators

for K in W .(3) pcf (K)V = pcf (K)W .

Proof. We start by proving that JV¡� = JW

¡� ∩V for all �∈ [min(K);max(pcf (K))V ],by induction on �. This is clear for � = min(K). For � singular we have J¡� = J¡�+

?¡� J¡?, so it remains to show that if � is regular and JV¡� = JW

¡� ∩V thenJV¡�+ = JW

¡�+ ∩V .Suppose Erst that A∈ JV

¡�+ . If A∈ JV¡� then we are done, otherwise we Ex a scale

f of length � in (∏

A=J¡�)V . By the induction hypothesis and our assumptions on Vand W , f is a scale in (

∏A=J¡�)W and so A∈ JW

¡�+ .Now suppose that A∈ JW

¡�+ ∩V . Let D∈V be any V -ultraElter on K with A∈D,and in W extend D to RD a W -ultraElter. By our assumptions on V and W the map : [f]D �→ [f] RD is a coEnal order-preserving function from (

∏A=D)V to (

∏A= RD)W .

Since A∈ JW¡�+ the W -coEnality of (

∏A= RD)W is at most �, and the existence of

and our assumptions on V and W imply that the V -coEnality of (∏

A=D)V is at most�. Therefore A∈ JV

¡�+ .Next we show that PCF generators agree between V and W . We note that we can

identify the maximum of pcf (K) as the least � such that K ∈ J¡�+ , so that V and Wagree on the value of the maximum element in pcf (K). We call this common value�max. In V we Ex a sequence 〈BV

� : min(K)6�6�max〉 such that BV� generates JV

¡�+

over JV¡�, and similarly we Ex in W a sequence 〈BW

� : min(K)6�6�max〉 such that BW�

generates JW¡�+ over JW

¡�. We choose these generators so that BV� = ∅ for � �∈ pcf (K)V ,

and similarly BW� = ∅ for � �∈ pcf (K)W . We also choose BV

�max=BW

�max=K .

We will show by induction that BW� and BV

� are equal modulo JW¡� for all �. Since

BV� is in JW

¡�+ and BW� is a generator, we see that BV

� is contained in BW� modulo

JW¡� for all �. Let � be least such that BW

� and BV� are unequal modulo JW

¡�. We nowadjust the BW

� , replacing BW� by BV

� for �¡�; this is legitimate by the choice of �. LetC =BW

� \BV� so that C �∈ JW

¡� but C ∈ JW¡�+ .

Now let � be least such that C is covered by some set BV�0

∪ · · ·BV�n

∪BV� with

�0¡ · · ·¡�n¡�; such a � exists because BV� =K . Suppose for a contradiction that

�6�; then �¡� since C is disjoint from BV� . Now BV

? =BW? for ?¡�, and it follows

that C ∈ JW¡�, which is a contradiction: so �¿�. Note that BV

� �= ∅ so that �∈ pcf (K)V .Let D =BV

�0∪ · · ·BV

�n∪BV

� . Working in V End 〈g#: #¡�〉 such that g# ∈ (∏

D)V and〈g# � BV

� : #¡�〉 is increasing and coEnal in (∏

BV� )V modulo JV

¡� � BV� . Working in

W End 〈h$: $¡�〉 such that h$ ∈ (∏

D)V and 〈h$ � C : $¡�〉 is increasing and coEnalin (

∏C)W modulo JW

¡� � C.Since �¿� we may End h$ which dominates g# on C modulo JW

¡� for unboundedlymany #¡�, and may then End g# such that

(1) h$ dominates g# modulo JW¡� � C.

(2) g# dominates h$ modulo JV¡� � BV

We End "1; : : : ; "n such that "1¡ · · · "n¡� and {�∈C: h$(�)6g#(�)} ⊆BW"1

∪ · · · ∪BW"n

. Note that BW"i

. We also End D1; : : : ; Dp such that D1¡ · · · Dp¡�and {�∈BV

� : h$(�)¿g#(�)} ⊆BVD1

∪ · · · ∪BVDp .

We claim that C is covered by the union of the BV�i

, BV"i

and BVDi . To see this observe

that if �∈C then at least one of the following must hold:

• �∈BV�i

for some i.• �∈C ∩BV

� and h$(�)6g#(�), in which case �∈BV"i

for some i.• �∈C ∩BV

� and h$(�)¿g#(�), in which case �∈BVDi for some i.

This contradicts the minimal choice of �. It follows that as we claimed the sets BV�

and BW� agree modulo JW

� for all �. In particular the BV� will serve as a sequence of

generators in W .It remains to be seen that pcf (K)V = pcf (K)W . This is immediate by the following

computation: given any sequence B? such that B? generates J¡?+ over J¡?, pcf (K) isthe set of ? such that B? is not covered by a Enite union of B� for �¡?.

A small technical diOculty is caused by the fact that the property of being thecharacteristic function of some structure is not obviously downwards absolute. We willresolve this diOculty using Lemma 4.4 and the following result.

Lemma 7.2. Let V and W be inner models of set theory with V ⊆W . In V let Kbe an interval of regular cardinals with |K |¡� and min(K) = �+ for some regularuncountable �. Assume that in W the cardinal � is still regular and K is still aninterval of regular cardinals. Let f∈V be such that for all # with �¡#¡ sup(K),f(#;−) is a bijection between # and |#|.

Let � be some su>ciently large regular cardinal of W , and in W let N ≺ (H�; ∈;¡�; {K}; f) be an IA structure of length and cardinality �. If !K

N ∈V then there isB∈V such that N ∩ sup(K) ⊆B⊆ sup(K) and |B| = �.

Proof. Since N is IA of length �, we may Ex 〈C� : �∈K〉 lying in W with C� ⊆N ∩ �and C� club in sup(N ∩ �) of order type �. Since !K

N ∈V and coEnalities agree,we may End in V a sequence 〈D� : �∈K〉 with D� club in sup(N ∩ �) of ordertype �.

Now let M0 =N ∩ sup(K) and note that M0 ≺ (sup(K); f). Let M1 be the hullin (sup(K); f) of �∪ ⋃

� D�, and note that M1 ∈V and |M1| = �. The hypothesesof Lemma 4.4 are satisEed because for each �∈K the set C� ∩D� is contained inM0 ∩M1 ∩ � and is coEnal in M0 ∩ �. It follows that M0 ⊆M1 so we may set B=M1.

Theorem 7.3. Let V and W be inner models of set theory with V ⊆W . In V let � beregular and let �¿� be a cardinal. Let K =REGV ∩ [�; �] and suppose that V andW agree on regular cardinals in the interval [�; max(pcf (K)V )], K is progressive and(∏

K)V is co3nal in (∏

K)W .Let A⊆ �. Then there is B∈V such that A⊆B and |B|W6max{�; |A|W}.

Proof. We will prove this by induction on sup(A). We way assume that sup(A)¿�,since otherwise we may set B= � + 1. For the rest of this proof cardinalities and

coEnalities should be understood as computed in W , though we will tacitly use theagreement between cardinalities of V and W at several points.Case 0: sup(A) is not a cardinal. Let f∈V be a bijection between sup(A) and

|sup(A)|, and let A∗ be the image of A under f. By induction we may End B∗ ⊆ |sup(A)|such that A∗ ⊆B∗ and |B∗|6max{�; |A|}, and we may then let B be the inverse imageof B∗ under f.Case 1: sup(A) is regular. In this case |A| = sup(A) and we let B= sup(A).Case 2: sup(A) is singular. If |A| = sup(A) then we may set B= sup(A), so we

now assume that |A|¡sup(A). By our hypotheses on K we may End � regular withmax{�; |A|; |K |+4}¡�¡sup(A). We set K0 = (�+; sup(A)) ∩REG.

Let L= pcf (K0) where by Fact 6.2 |L|¡� and L= pcf (L). We Ex in V a sequenceB= 〈B?: ?∈L〉 of PCF generators for K0, and a matrix of functions f = 〈f?

i : ?∈L; i¡?〉such that 〈f?

i : i¡?〉 is a continuous scale in∏

B?=J¡?.By Theorem 7.1, B is still a sequence of PCF generators in W . It is clear that

〈f?i : i¡?〉 is still a scale in W , and we claim that additionally this scale is still con-

tinuous at good points of coEnality �.We use Fact 3.1, the Trichotomy Theorem of Shelah. Let ?∈L, and in W let 9¡?

be a point of coEnality � which is good for the scale 〈f?i : i¡?〉. We begin by argu-

ing that in V there must exist an eub for 〈f?i : i¡9〉 whose values have coEnality at

least �.If this is not the case, then, in V we must be either in Case 2 or 3 from Fact 3.1.

It is easy to see that the properties of being in Case 2 or 3 are upwards absolutefrom V to W , in fact the witnesses from V will work in W as long as we extend theultraElter in V for Case 2 to an ultraElter in W . Since we have an eub for 〈f?

i : i¡9〉of uniform coEnality � in W , it follows from Fact 3.1 that in W we are not in Case2 or 3, so that in V there is an eub for 〈f?

i : i¡9〉 whose values have coEnality atleast �.

By continuity in V , f?9 is an eub whose values have coEnality at least � almost

everywhere in V . By the hypothesis on the resemblance between V and W , f?9 retains

these properties in W . It follows that the scale 〈f?i : i¡?〉 is continuous at 9.

We also Ex in V a function f from sup(K0)2 to sup(K0) coding some informationabout cardinalities, as in Lemma 7.2.

We now build N ≺ (H�; ∈;¡�; {K0}; f) where A⊆N and N is IA of length andcardinality �. By theorem 5.3 !K0

N is the pointwise supremum of Enitely many functionsof the form f?

i � (B?\E?) where E? is the union of a Enite subset of {B5: 5 ¡ ?}.It follows that !K0

N ∈V . The hypotheses of Lemma 7.2 are satisEed so we may EndC in V such that |C| = �¡sup(A) and A⊆C. Now let g∈V be a bijection betweenC and �, and let A∗ be the image of A under g. Now we may use the inductionhypothesis to cover A∗ by a suitable B∗, and then pull back along g to get a suitableB covering A.

Remark 7.4. Theorem 7.3 is actually an equivalence. The covering statement in theconclusion implies that V and W agree on regular cardinals between � and �, and alsoimplies that (

∏K)V is coEnal in (

∏K)W .

In order to be able to state the next result in a compact way, we make the followingad hoc deEnition.

De�nition 7.5. Let V and W be inner models of set theory with V ⊆W . In V , let �be regular and let �¿� be a cardinal. We say that W weakly resembles V on [�; �)if and only if the hypotheses of Theorem 7.3 are satisEed.

For background on the theory of precipitous ideals we refer the reader to Foreman’ssurvey paper [14]. The following result belongs to a genre of theorems in which we aregiven an ideal I on a cardinal � and some information about preservation of cardinalswhen forcing with P�=I , and we conclude that I must be precipitous. This line ofinquiry was begun by Baumgartner and Taylor [2], who proved for example that underGCH a countably closed ideal on ℵ1 whose quotient algebra preserves ℵ2 is necessarilyprecipitous.

Theorem 7.6. Let I be a countably complete ideal on ℵ1. Let �= 2ℵ1 , and suppose itis forced by Pℵ1=I that V [G] weakly resembles V on [ℵ2; �). Then I is precipitous.

Proof. Let G be a generic ultraElter on (Pℵ1)V and let M =Ult(V;G). We observethat by our hypotheses ℵV

2 = ℵV [G]1 and recall the standard fact that ℵV

2 is an initialsegment of the well-founded part of M . It must be the case that ℵV

2 = ℵM1 , for if not

then in M there is a surjection from ! onto some larger M -ordinal, contradicting thefact that ℵV

2 = ℵV [G]1 .

Suppose for a contradiction that M is not well-founded, and in V [G] choose asequence 〈fi: i∈!〉 such that fi ∈V , fi: ℵV

1 →ON and fi+1¡Gfi for all i. SincePℵ1=I has the �+-c:c: we may End C ∈V such that |C|6� and range (fi) ⊆C forall i.

By Theorem 7.3 we may End B⊆C such that B∈V , |B|V = ℵV2 and range(fi) ⊆B

for all i. Working in V , we write B= ∪j¡ℵ2 Bj where the Bj are increasing and|Bj| = ℵ1. In V [G] this gives a representation of B as an increasing union of countablesets. Since

⋃i range(fi) is countable in V [G] we can Ex a j such that range(fi) ⊆Bj

for all i. Now let � be the order type of Bj, let h : Bj � � be order preserving andlet f∗

i = h ◦ fi. Then 〈f∗i : i∈!〉 is a G-decreasing sequence of functions from ℵ1 to

�, and so to get a contradiction we need only show that jG(�) is well-founded for all�¡ℵV

2 .Suppose for a contradiction that jG(�) is ill-founded, and choose a bijection F ∈M

between ℵM1 and jG(�).

Since ℵM1 = ℵV [G]

1 , we may End "¡ℵV [G]1 such that in V [G] the set F”" contains an

inEnite decreasing sequence of M -ordinals.Working in V [G], let T be the tree of all Enite sequences (#0; : : : ; #i) from " such

that F(#i)¿F(#i+1) for all i. The tree T is in the well-founded part of M . Clearly Tis well-founded in M , and is not well-founded in V [G].

The tree T is countable in M , and so in M there is a rank function 9:T → ℵM1 .

Since ℵM1 is well-founded, this contradicts the existence of a branch of T in V [G]. We

conclude that M is well-founded and so that I is a precipitous ideal.

References

[1] U. Abraham, M. Magidor, Cardinal arithmetic, Handbook of Set Theory, to appear.[2] J.E. Baumgartner, A.D. Taylor, Saturation properties of ideals in generic extensions. II, Trans. Amer.

Math. Soc. 271 (2) (1982) 587–609.[3] M. Burke, M. Magidor, Shelah’s pcf theory and its applications, Ann. Pure Appl. Logic 50 (1990)

207–254.[4] J. Cummings, Collapsing successors of singulars, Proc. Amer. Math. Soc. 125 (9) (1997) 2703–2709.[5] J. Cummings, Compactness and incompactness in set theory, in: Proc. Logic Colloq. 2001, to appear.[6] J. Cummings, M. DSzamonja, S. Shelah, A consistency result on weak re6ection, Fund. Math. 148 (1)

(1995) 91–100.[7] J. Cummings, M. Foreman, M. Magidor, Squares, scales and stationary re6ection, J. Math. Logic 1 (1)

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Appl. Logic, accepted.[10] J. Cummings, S. Shelah, Some independence results on re6ection, J. London Math. Soc. 59 (1) (1999)

37–49.[11] M. DSzamonja, S. Shelah, Saturated Elters at successors of singular, weak re6ection and yet another

weak club principle, Ann. Pure Appl. Logic 79 (3) (1996) 289–316.[12] M. Dzamonja, S. Shelah, Weak re6ection at the successor of a singular cardinal, J. London Math. Soc.

67 (2003) 1–15.[13] G. Fodor, Eine Bemerkung zur Theorie der regressiven Funktionen, Acta Universitatis Szegediensis,

Acta Sci. Math. 17 (1956) 139–142.[14] M. Foreman, Ideals and generic embeddings, Handbood of Set Theory, to appear.[15] M. Foreman, M. Magidor, Large cardinals and deEnable counterexamples to the continuum hypothesis,

Ann. Pure Appl. Logic 76 (1995) 47–97.[16] M. Foreman, M. Magidor, A very weak square principle, J. Symbolic Logic 62 (1) (1997) 175–196.[17] M. Foreman, M. Magidor, Mutually stationary sequences and the non-saturation of the non-stationary

ideal on P��, Acta Math. 186 (2) (2001) 271–300.[18] M. Foreman, M. Magidor, S. Shelah, Martin’s Maximum, saturated ideals and non-regular ultraElters I,

Ann. Math. 127 (1988) 1–47.[19] M. Foreman, S. Todorcevic, A new Lowerheim–Skolem theorem, to appear.[20] A. Hajnal, I. JuhTasz, S. Shelah, Splitting strongly almost disjoint families, Trans. Amer. Math. Soc. 295

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(1972) 21–46.[23] W.J. Mitchell, I [!2] can be the non-stationary ideal on conEnality !1, in preparation.[24] S. Shelah, On successors of singular cardinals, in: M. BoKa, D. van Dalen, K. McAloon (Eds.), Logic

Colloquium ’78, North-Holland, Amsterdam, 1979, pp. 357–380.[25] S. Shelah, Re6ecting stationary sets and successors of singular cardinals, Arch. Math. Logic 31 (1)

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canonical structure in the universe of set theory: part one

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