CANONICAL STRUCTURE IN THE UNIVERSE OF SET THEORY: PART ONE

JAMES CUMMINGS, MATTHEW FOREMAN, AND MENACHEM MAGIDOR

Abstract. We start by studying the relationship between two invariants isolated by Shelah, the sets of good 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 relationship between internally approachable and tight structures, which parallels to a certain extent the relationship between good and approachable points. In particular we characterize the tight structures in terms of PCF theory and use our characterisation to prove some covering results for tight structures, along with some results on tightness and stationary reflection. Finally we prove some absoluteness theorems in PCF theory, deduce a covering theorem, and apply that theorem to the study of precipitous ideals.

1. Introduction

It is a distinguishing feature of modern set theory that many of the most interesting questions are not decided by ZFC, the theory in which we profess to work; to put it another way, ZFC admits a large variety of models. A natural response to this is to identify invariants which may take different values in different models, and which codify a large amount of information about a model.

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

Shelah discovered a large class of canonical invariants, the study of which he labeled PCF theory. These invariants include two which are central in this paper; Shelah [24, 26] (under some mild cardinal arithmetic assumptions on the singular cardinal μ) defined two stationary subsets of μ^{+}, the sets of good and approachable points. The definitions of these sets appear to depend on certain arbitrary choices, but (modulo the club filter) are in fact independent of these choices. Other canonical structures we study in this paper include the stationary sets of tight and internally approachable structures, and the 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 □_{μ}^{∗}, imply that every point is approachable. Foreman and Magidor [16] showed that their principle “Very weak square”, which captures some of the approachability implied by □_{μ}^{∗}, implies many of the interesting consequences of □_{μ}^{∗}. Cummings [5] showed that the assertion that every point is good severely constrains all μ^{+}-preserving extensions of V.

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

Key words and phrases. PCF theory, good ordinal, approachable ordinal, the ideal I[λ], internally approachable structure, tight structure, square sequence, covering properties, precipitous ideal, mutual stationarity, stationary reflection.

The first author was partially supported by NSF Grants DMS-9703945 and DMS-0070549.

The second author was partially supported by DMS-9803126 and DMS-0101155. The second and third authors were partially supported by the US-Israel Binational Science Foundation.

Our first motivation for the work in this paper is the problem of the relationship between the sets of good and approachable points. This problem is trivial when weak squares exist, but non-trivial in general. For example it is consistent relative to large cardinals that not every point of cofinality ℵ_{1} in ℵ_{ω+1} is good. We have speculated that perhaps the sets of good and approachable points coincide, and in Section 3 we prove that under some strong structural hypotheses this is the case.

The concept of an approachable ordinal is closely linked to that of an internally approachable (IA) structure. To be more precise, the set of approachable ordinals of cofinality η can be characterized [16] as the set of ordinals which have the form sup(N ∩ μ^{+}) for some internally approachable N of length and cardinality η.

Foreman and Magidor [17] isolated the concept of tight structure in their work on mutual stationarity and the non-saturation of the non-stationary ideal on P_{κ}λ, and tightness turns out to be closely related to the issues of goodness and approachability. In particular internally approachable structures are tight, and if N is tight then sup(N ∩ μ^{+}) is good. Our second motivation for the work in this paper is the analogy

\frac{Tight structures}{IA structures} = \frac{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 machinery of PCF theory. We use this to show that under some structural hypotheses all good points in ℵ_{ω+1} of cofinality greater than ℵ_{1} are approachable, and also to show a kind of “singular cardinal compactness” for square sequences. For example we show that if CH holds and □_{ℵ_{n}} holds for all n < ω, then there is a sequence ⟨C_{γ} : γ ∈ ℵ_{ω+1} ∩ cof(ℵ_{2})⟩ where C_{γ} is a club subset of γ with order type ℵ_{2} and the C_{γ} cohere at common limit points of uncountable cofinality.

• In section 4 we study the important property of uniformity for a structure, and show that sufficiently uniform structures can be reconstructed from their characteristic functions.

• In section 5 we characterize tight structures in terms of PCF theory. We also show that the properties of uniformity and tightness can sometimes be propagated from a set of regular cardinals K to the set pcf(K).

• In section 6 we explore the relationship between tightness, covering properties and internal approachability. We prove theorems showing that under some circumstances tightness and internal approachability are equivalent. We also record a remarks on the connection between tightness and stationary reflection.

• In section 7 we prove some absoluteness results in PCF theory. We deduce a covering theorem, and use it to show that if I is an ideal on ℵ_{1} such that forcing with Pℵ_{1}/I is sufficiently mild then I is precipitous.

This paper contains only ZFC results. In the sequel [7] we prove a series of complementary consistency results. We would like to thank John Krueger for comments and corrections on an earlier draft of this paper.

2. Preliminaries

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

We write ON for the class of ordinals, LIM for the limit ordinals and SUCC for the successor ordinals. We write CARD for the class of infinite cardinals, REG for the regular cardinals and SING for the singular cardinals. We denote by cof(κ) the class of ordinals of cofinality κ, and if A is a set of ordinals we write cf(A) for the cofinality of A considered as an ordered set. An interval of regular cardinals is a set of the form REG∩[κ, λ) for cardinals κ and λ. We denote by τ^{M} the interpretation of a term τ in a model M, and by ϕ^{M} the relativisation of a formula ϕ to M. An algebra on a set X is a structure for some countable first-order language which has X as its underlying set.

2.1. Internally approachable and tight structures. When θ is an uncountable regular cardinal, we will denote by H_{θ} the transitive set of those X such that the transitive closure of X has size less than θ. We denote by <_{θ} some fixed wellordering of H_{θ}. By convention when θ < χ we will assume that <_{θ} is the restriction of <_{χ} to H_{θ}.

We will frequently be interested in structures of the form A = (H_{θ}, ∈, <_{θ}); the advantage of building in a well-ordering is that if X ⊆ H_{θ} and Sk^{A}(X) is the set of elements of H_{θ} definable in A with parameters from X, then Sk^{A}(X) ≺ A and Sk^{A}(X) is the smallest substructure of A containing X.

The definition of internally approachable structure appears in Foreman, Magidor and Shelah’s paper [18] on Martin’s Maximum. These structures are ubiquitous in modern set theory; see Lemma 2.3 and the remarks which follow it for some motivation.

Definition 2.1. Let θ be regular and let A be some algebra expanding (H_{θ}, ∈, <_{θ}). Let N ≺ A. N is internally approachable (IA) if and only if there exist a limit ordinal δ and a sequence ⟨N_{α} : α < δ⟩ such that

(1) N = ⋃_{α<δ} N_{α}.
(2) For all β < δ, ⟨N_{α} : α < β⟩ ∈ N.

In this case we will say that N is IA of length δ and that ⟨N_{α} : α < δ⟩ is an approaching sequence for N. We note that if we can always take an approaching sequence to be continuous in the sense that if β is a limit ordinal, then N_{β} = ⋃_{α<β} N_{α}.

The length δ of the approaching sequence is not uniquely determined, but it is easy to see that cf(δ) is uniquely determined; by a mild abuse of language we refer to this cofinality as the cofinality of N. In their paper on definable counterexamples to the continuum hypothesis Foreman and Magidor [15] give a detailed discussion of the lengths and cofinalities of approaching sequences and the sizes of the structures that appear in them.

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

Lemma 2.2. Let κ be regular and uncountable, and let N be an IA substructure of A with length and cardinality κ. Then there is a continuous approaching sequence ⟨M_{α} : α < κ⟩ for M, such that M_{β} is an elementary substructure of A with cardinality less than κ and ⟨M_{α} : α ≤ β⟩ ∈ M_{β+1} for all β < κ.

We will refer to increasing and continuous sequences ⟨M_{α} : α < β⟩ of elementary submodels of A such that ⟨M_{α} : α ≤ γ⟩ ∈ M_{γ+1} for γ + 1 < β as “continuous internally approaching chains of submodels”.

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

Lemma 2.3. Let κ be a regular cardinal. Let ⟨M_{α} : α < κ⟩ be a continuous increasing chain of elementary submodels of A such that ⟨M_{α} : α ≤ γ⟩ ∈ M_{γ+1} for γ < κ. Let M = ⋃_{α<κ} M_{α} and let λ = |M|. Then

(1) κ ⊆ M, so in particular κ ≤ λ.

(2) For all ordinals γ ∈ M with cf(γ) > λ (a) cf(M ∩ γ) = κ.

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

(3) For every set X ⊆ M with |X| < κ, there is Y ∈ M with |Y | < λ and X ⊆ Y.

(4) Let K ∈ M be a set of regular cardinals such that λ < min(K) and |K| < κ. Then every function in ∏_{γ∈K} M ∩ γ is dominated pointwise by a function in M ∩ ∏ K.

(5) Let ℙ ∈ M be a κ-closed and (λ, ∞)-distributive forcing poset. Then there is a decreasing sequence ⟨p_{i} : i < κ⟩ of conditions in M ∩ ℙ which meets every dense open subset of ℙ lying in M.

Proof. We sketch the proof.

(1) If β < κ then ⟨M_{α} : α < β⟩ in M, and so by elementarity the length β of this sequence is in M. We note that as a consequence M_{β} ∈ M also.

(2) Since cf(γ) > λ = |M|, M ∩γ is bounded in γ. Since γ ∈ M and each M_{α} is in M, the sequence ⟨sup(M_{α} ∩ γ) : α < κ⟩ is easily seen to be a continuous increasing sequence which is cofinal in sup(M ∩ γ) and consists of ordinals in M.

(3) Since |X| < κ, there exists α < κ such that X ⊆ M_{α}.

(4) Let f be a function in ∏_{κ∈K} M ∩ κ. We note that since |K| < κ ⊆ M and K ∈ M, we have K ⊆ M. Since |K| < κ, we may find α < κ such that f is in ∏_{κ∈K} M_{α} ∩ κ. We define g with domain K by g(κ) = sup(M_{α} ∩ κ). Since |M_{α}| ≤ λ < min(K) we see that g ∈ ∏ K, and since K and M_{α} both lie in M we also have g ∈ M. Clearly f(κ) < g(κ) for all κ ∈ K.

(5) By induction we choose p_{i} to be the <_{θ}-least condition in ℙ which is a lower bound for ⟨p_{j} : j < i⟩ and lies in all the dense open subsets of ℙ which are in M_{i}. Since ⟨M_{j} : j < i⟩ lies in M it is easy to see that ⟨p_{j} : j < i⟩ lies in M, and hence p_{i} lies in M.

□

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

Definition 2.4. Let K be a set of regular cardinals, let θ = cf(θ) > sup(K), and let A = (H_{θ}, ∈, <_{θ}). Let M ≺ A.

Then M is tight for K if and only if

(1) K ∈ M.

(2) For all g ∈ ∏_{κ∈M∩K}(M ∩κ) there exists h ∈ M ∩∏ K such that g(κ) < h(κ)

for all κ ∈ M ∩ K.

We note that if |K| ⊆ M then K ⊆ M, and in this case tightness has the simpler form appearing in Property 4 above: when K ⊆ M, M is tight for K exactly when M ∩ ∏ K is cofinal in ∏_{κ∈K} M ∩ κ. It is natural to phrase this definition in terms of a standard idea, the characteristic function of a structure.

Definition 2.5. Let K be a set of regular cardinals and let M be a set. The characteristic function of M (on K) is the function χ_{M}^{K} with domain K given by χ_{M}^{K} : κ ⟼ sup(M ∩ κ).

We will usually drop the superscript K and write χ_{M} when the set K is clear from the context. Typically we will be in a situation where |M| < min(K) and so χ_{M} ∈ ∏ K. If a structure M is such that K ⊆ M, then tightness of M amounts to saying that every function in ∏ K which is pointwise dominated by χ_{M} is pointwise dominated by some function in M ∩ ∏ K.

There are several reasons why it seems worthwhile to isolate the property of tightness. One reason is that there are many arguments in PCF theory which employ IA structures, and on closer inspection these arguments are typically just using the tightness guaranteed by Lemma 2.3. Another reason is that tight structures arise naturally in Foreman and Magidor’s theory of mutual stationarity. We only give a cursory description of the theory of mutual stationarity, for more information see [17] and the sequel to this paper [7].

Definition 2.6. Let K be a set of regular uncountable cardinals. Let ⟨S_{κ} : κ ∈ K⟩ be such that S_{κ} ⊆ κ for all κ ∈ K.

(1) If N is a set, then N meets ⟨S_{κ} : κ ∈ K⟩ if and only if sup(N ∩ κ) ∈ S_{κ} for all κ ∈ N ∩ K (equivalently χ_{N} ↾ N ∩ K ∈ ∏_{κ∈N∩K} S_{κ}).

(2) ⟨S_{κ} : κ ∈ K⟩ is mutually stationary if and only if for every algebra A on sup(K) there exists N ≺ A such that N meets ⟨S_{κ} : κ ∈ K⟩.

It is easy to see that if ⟨S_{κ} : κ ∈ K⟩ is mutually stationary then S_{κ} is stationary for each κ. Foreman and Magidor showed [17] that the converse is false in general, but is true if S_{κ} ⊆ κ ∩ cof(ω) for all κ.

Mutual stationarity can be seen as intermediate between the classical concept of stationarity for subsets of a regular uncountable cardinal, and the very general concept of stationarity introduced in [18]. We recall that if S ⊆ P(X) then S is a stationary subset of P(X) if and only if for every algebra A on X there is B ∈ S such that B ≺ A. This is easily seen to be equivalent to demanding that for every F : ^{<ω}X ⟶ X there is a non-empty B ∈ S which is closed under F. When we need to distinguish between different flavors of stationarity we will refer to this last concept as general stationarity.

The sequence ⟨S_{κ} : κ ∈ K⟩ is mutually stationary if and only if the set of subsets of sup(K) which meet ⟨S_{κ} : κ ∈ K⟩ is a stationary subset of P(sup(K)). By standard facts [18, Lemma 0] about general stationarity, if X is any set with sup(K) ⊆ X then ⟨S_{κ} : κ ∈ K⟩ is mutually stationary if and only if the set of subsets of X which meet ⟨S_{κ} : κ ∈ K⟩ is a stationary subset of P(X). Subsequently we will often let X = H_{θ} for θ some regular cardinal greater than sup(K).

Two of the most useful facts about stationary subsets of a regular uncountable cardinal κ are Fodor’s lemma [13] and Solovay’s splitting theorem [27]. It is open to what extent these results may be generalised to arbitrary mutually stationary sequences; Foreman and Magidor [17] identified a class of mutually stationary sequences, the tightly stationary sequences, for which versions of Fodor’s lemma and Solovay’s splitting theorem are available. As one might expect, a tightly stationary sequence is a sequence whose mutual stationarity is witnessed by tight structures.

Definition 2.7. Let K be a set of regular cardinals and let ⟨S_{κ} : κ ∈ K⟩ be such that S_{κ} ⊆ κ for all κ ∈ K. Let θ = sup(K)^{+}. The sequence ⟨S_{κ} : κ ∈ K⟩ is tightly stationary if and only if for every algebra A on H_{θ} there is N ≺ A such that N is tight for K and N meets ⟨S_{κ} : κ ∈ K⟩.

See Foreman and Magidor’s paper [17] for the statements and proofs of Fodor’s lemma 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 normal ideal I[λ] defined by Shelah [24, 25, 26].

Definition 2.8. Let λ be a regular uncountable cardinal. S ∈ I[λ] if and only if there exists a club subset E of λ and a sequence ⟨a_{α} : α < λ⟩ of bounded subsets of λ, such that for all δ ∈ E ∩ S there is A ⊆ δ unbounded in δ such that ot(A) = cf(δ) < δ and for every β < δ there is γ < δ such that A ∩ β = a_{γ}.

We discuss some alternative characterisations of I[λ] in Section 7. The following result appears in [16] as part of the proof of Claim 4.4 in that paper, and gives one direction of the connection between I[λ] and approachable structures.

Lemma 2.9. Let λ be regular and uncountable. Let θ > λ and let A be an algebra expanding (H_{θ}, ∈, <_{θ}). Let S ∈ I[λ]. Then there is a club subset F of λ such that for every δ ∈ F ∩ S there is M ≺ A such that M is IA of length and cardinality cf(δ) and sup(M ∩ λ) = δ.

To get a reasonable converse we fix a regular cardinal κ less than λ and assume that λ^{<κ} = λ. We let I[λ, κ] be the restriction of I[λ] to cofinality κ, that is the ideal of those X ⊆ λ such that X ∩ cof(κ) ∈ I[λ]. We enumerate [λ]^{<κ} as ⟨a_{α} : α < λ⟩, and let S be the set of δ ∈ λ ∩ cof(κ) such that there is A ⊆ δ unbounded in δ with ot(A) = κ and every proper initial segment of A enumerated as a_{γ} for some γ < δ. If we choose a different enumeration ⟨b_{α} : α < λ⟩ of [λ]^{<κ} then {a_{α} : α < β} = {b_{α} : α < β} for a club set of β < λ, so modulo the club filter S is independent of the choice of the enumeration ⟨a_{α} : α < λ⟩.

It is not difficult to see that S generates I[λ, κ] modulo the club filter on λ, or to put it another way S is (modulo club sets) the largest subset of λ ∩ cof(κ) which is in I[λ]. We will refer to S as “the set of approachable points of cofinality κ in λ” with the understanding that this set is well-defined modulo the club filter. In this situation there is an easy converse to Lemma 2.9.

Lemma 2.10. Let κ, λ and θ be regular with κ < λ < θ and λ^{<κ} = λ. Let ⟨a_{α} : α < λ⟩ be an enumeration of [λ]^{<κ} in order type λ, and let S be defined as above. Let N ≺ (H_{θ}, ∈, <_{θ}, {⟨a_{α} : α < λ⟩}) be IA of length and cardinality κ. Then sup(N ∩ λ) ∈ S.

Proof. Let N be approached by ⟨N_{i} : i < κ⟩, and let γ_{i} = sup(N_{i} ∩ λ). Then every proper initial segment of ⟨γ_{i} : i < κ⟩ lies in N, so is enumerated as a_{α} for some α ∈ N ∩ λ. It follows that sup(N ∩ λ) ∈ S. □

Summarising, if λ^{<κ} = λ then (modulo club sets) S is the set of ordinals of the form sup(N ∩ λ) where N is IA of length and cardinality κ. If C is any club subset of λ we can add C to the structure in the proof of Lemma 2.10, and find N such that sup(N ∩λ) ∈ C ∩S; it follows that S is stationary. It is interesting to note that there is an attractive characterisation of S in terms of forcing: results of Shelah [24] imply that every κ^{+}-closed forcing poset preserves the stationarity of subsets of S, but there is a κ^{+}-closed poset which destroys the stationarity of λ ∩ cof(κ) \ S.

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

Given a set X and an ideal I on X, we refer to the sets in I as I-small sets. We say that a set Y ⊆ X is I-large if X \Y ∈ I and is I-positive if Y ∉ I. Given ordinal valued functions f and g with domain X, we say that g dominates f modulo I (and write f <_{I} g) if f(x) < g(x) for an I-large set of values of x; the relation <_{I} is a strict 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 β in (∏_{x} f(x), <_{I}) is an increasing and cofinal sequence of length β in (∏_{x} f(x), <_{I}).

For Y a subset of X, a sequence of functions in ^{X}ON is pointwise increasing on Y if it is strictly increasing on every coordinate in Y. We note that a sequence which is pointwise increasing on an I-large set is <_{I}-increasing, but that in general the converse is not true. Two sequences ⟨f_{α} : α < γ⟩ and ⟨g_{α} : α < γ⟩ which are increasing with respect to <_{I} are cofinally interleaved (modulo I) if for all α < γ there is β < γ such that f_{α} <_{I} g_{β} and g_{α} <_{I} f_{β}.

The function g is a an exact upper bound (eub) for the <_{I}-increasing sequence ⟨f_{α} : α < β⟩ if and only if f_{α} <_{I} g for all α < β, and for every h with h <_{I} g there exists α < β with h <_{I} g_{α}. This is equivalent to ⟨f_{α} : α < β⟩ being a scale in (∏_{x} g(x), <_{I}) (with the caveat that the functions f_{α} need only be dominated by g modulo I, rather than literally being members of ∏_{x} g(x)). It is easy to see that an exact upper bound, if one exists, is well-defined modulo I.

A point β is good for a <_{I}-increasing sequence f⃗ (of length at least β) if and only if cf(β) > |X| and there exists an exact upper bound h for ⟨f_{α} : α < β⟩ with the property that cf(h(x)) = cf(β) for all x. The following lemma gives some useful equivalent characterisations of goodness: the implication from 3) to 1) gives an important construction principle for exact upper bounds.

Lemma 2.11. The following are equivalent for ⟨f_{α} : α < β⟩ a <_{I}-increasing sequence with cf(β) > |X|.

(1) β is good.

(2) There is a sequence of functions of length cf(β) which is pointwise increasing and is cofinally interleaved with ⟨f_{α} : α < β⟩ modulo I.

(3) There is a sequence of functions of length cf(β) which is pointwise increasing on an I-large subset of X and is cofinally interleaved with ⟨f_{α} : α < β⟩ modulo I.

Proof. For 1) implies 2), let h be an eub such that cf(h(x)) = cf(β) for all x and fix for each x a sequence ⟨α_{i}^{x} : i < cf(β)⟩ increasing and cofinal in h(x); now define g_{i}(x) = α_{i}^{x} and check that ⟨g_{i} : i < cf(β)⟩ is pointwise increasing and cofinally interleaved with ⟨f_{α} : α < β⟩. The implication from 2) to 3) is immediate. For 3) implies 1) fix ⟨g_{i} : i < cf(β)⟩ pointwise increasing on an I-large set and cofinally interleaved with ⟨f_{α} : α < β⟩; check that the pointwise supremum of ⟨g_{i} : i < cf(β)⟩ is an exact upper bound for ⟨f_{α} : α < β⟩ which has cofinality cf(β) on an I-large set, and then alter it to get an exact upper bound which has cofinality cf(β) on all x. □

As an immediate corollary of Lemma 2.11, if β is a good point then there is C club in β such that every point of C with cofinality greater than |X| is good. So the set of ungood points of cofinality greater than |X| is quite thin, in the sense that if it is stationary then its stationarity can only reflect at points of itself.

Example 2.12. If ⟨f_{α} : α < λ⟩ and ⟨f′_{α} : α < λ⟩ are two <_{I}-increasing sequences with the same exact upper bound g then it is easy to see that they are cofinally interleaved. It follows that if λ is a regular uncountable cardinal there is a club set of β < λ such that ⟨f_{α} : α < β⟩ and ⟨f′_{α} : α < β⟩ are cofinally interleaved. Therefore the sets of good points for ⟨f_{α} : α < λ⟩ and ⟨f′_{α} : α < λ⟩ are equivalent modulo the club filter and so give an example of “canonical” structure.

The following result by Shelah is central in PCF theory.

Fact 2.13. If μ is a singular cardinal then there is a set K ⊆ μ of regular cardinals such that ot(K) = cf(μ) and there is a scale of length μ^{+} in ∏ K under the eventual domination ordering.

As we just pointed out, the set of good points in such a scale is essentially independent of the choice of the scale so has some claim to be considered a canonical invariant. Of course there is still a dependence on K but for small values of μ we can also make a canonical choice for K. The case of most interest to us is μ = ℵ_{ω}, and in this case work of Shelah shows that modulo finite sets there is a largest K ⊆ {ℵ_{n} : n < ω} such that ∏ K has a scale of length ℵ_{ω+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_{ω+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| < κ < λ < θ with κ, λ and θ regular. Let f⃗ = ⟨f_{α} : α < λ⟩ be a <_{I}-increasing sequence in ^{X}ON/I, and suppose there is an exact upper bound g for f⃗ such that cf(g(x)) > κ for all x ∈ X. Let N ≺ (H_{θ}, ∈, <_{θ}, {f⃗, g}) be an IA structure of length and cardinality κ. Then sup(N ∩ λ) is a good point for f⃗.

Proof. We fix an internally approaching chain ⟨N_{i} : i < κ⟩ such that X ∈ N_{0}, X ⊆ N_{0}, |N_{i}| < κ for all i and the union of the chain is is N. We let g_{i}(x) = sup(N_{i}∩g(x)) for i < κ, and claim that ⟨g_{i} : i < κ⟩ will serve as a witness that sup(N ∩λ) is good.

Since X ⊆ N_{0} and cf(g(x)) > κ for all x ∈ X, we see that g_{i} < g; moreover if i < j then N_{i} ∈ N_{j}, so g_{i} ∈ N_{j} and hence g_{i} < g_{j}.

Since g_{i} ∈ N and g_{i} is dominated by the exact upper bound g, it follows by elementarity that g_{i} <_{I} f_{α} for some α ∈ N ∩λ. On the other hand if α < sup(N ∩λ) then there exist i < κ and β ∈ N_{i} ∩ λ with α < β, so that range(f_{β}) ⊆ N_{i} and f_{β} < g_{i}. □

If C is club in λ we may add a predicate for C to A and produce a good point of cofinality κ lying in C. It follows that the set of good points of cofinality κ is stationary.

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

Corollary 2.15. Let λ be the successor of a singular cardinal μ, let K ⊆ μ be an unbounded set of regular cardinals with ot(K) = cf(μ) < min(K). Let f⃗ be a scale of length λ in ∏ K under eventual domination. If S ∈ I[λ] then almost all points of S with cofinality greater than cf(μ) are good for f⃗.

Proof. Immediate from Lemmas 2.9 and 2.14 □

Recalling that the set of approachable points (when it can be defined) is the maximal set in I[λ], we see that modulo the club filter every approachable point of cofinality greater than cf(μ) is good.

3. Goodness, approachability and compactness for squares

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

Since approachable points are good, the problem “which good points are approachable” becomes trivial if almost every point is approachable. We digress briefly to review what is known about the extent of I[λ].

It it is known that if Jensen’s weak square principle □_{μ}^{∗} holds then μ^{+} ∈ I[μ^{+}], so that large cardinals will be required to make models in which I[μ^{+}] is non-trivial. If μ^{<μ} = μ then □_{μ}^{∗} holds, so GCH trivialises I[λ] for λ the successor of a regular cardinal. Shelah has shown that if λ is regular then λ^{+} ∩ cof(< λ) is in I[λ^{+}], and that if λ is singular then for all regular κ < λ there is a stationary subset of λ^{+} ∩ cof(κ) lying in I[λ^{+}].

If κ is supercompact and cf(λ) < κ < λ, then λ^{+} ∩ cof(< κ) ∉ I[λ^{+}]. By doing some suitable Levy collapses (see [24] or [20] for details) this can be used to produce a model in which ℵ_{ω+1} ∩ cof(ℵ_{1}) ∉ I[ℵ_{ω+1}]. As for successors of regular cardinals, Mitchell’s model [23] in which there is no ℵ_{2}-Aronszajn tree also has the property that ℵ_{2} ∩ cof(ℵ_{1}) ∉ I[ℵ_{2}]. In recent work [22] Mitchell has produced a model in which I[ℵ_{2}] is generated modulo the club filter by the set ℵ_{2} ∩ cof(ω).

We now narrow our focus to the cardinal ℵ_{ω+1}, where the points of cofinality ℵ_{1} seem to play a special role. It is known (see [16] or [7]) to be consistent that stationarily many points of cofinality ℵ_{1} are not good, and it is open whether ℵ_{ω+1} ∩ cof(≠ ℵ_{1}) is always in I[ℵ_{ω+1}]. As we see shortly all points of cofinality greater than 2^{ℵ_{0}} are good. In particular under CH all points of cofinality greater than ℵ_{1} 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 f⃗ be a <_{I}-increasing sequence. The sequence f⃗ is continuous at β if and only either there is no exact upper bound for ⟨f_{α} : α < β⟩ or f_{β} is such a bound. f⃗ is continuous if and only if it is continuous at every limit β. Given an arbitrary <_{I}-increasing sequence f⃗ of limit length λ, we may replace f_{β} for β < λ limit by an exact upper bound for ⟨f_{α} : α < β⟩ whenever such a bound exists, and get a continuous sequence which is cofinally interleaved with the original one.

If β is a good point for f⃗ and ⟨g_{i} : i < cf(β)⟩ is increasing on an I-large set and cofinally interleaved with ⟨f_{α} : α < β⟩ then let g be the pointwise supremum of the sequence ⟨g_{i} : i < cf(β)⟩. As we saw in Lemma 2.11 g is an exact upper bound for ⟨f_{α} : α < β⟩ and so by continuity f_{β} is also an exact upper bound: since exact upper bounds are unique modulo I, the functions f_{β} and g must agree on an I-large set.

Next we need an alternative characterisation of good points in scales of a special kind. When X is an ordered set with no last element, and I is the ideal of bounded subsets of X, we usually write <^{∗} for <_{I} and =^{∗} for <_{I}. We refer to <^{∗} as the eventual domination ordering. This is the context of the “good scales” and “very good scales” studied in our paper [8] on scales, squares and reflection.

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

• The ordinal β is good for f⃗.

• The cofinality of β is greater than |X|, and for every unbounded A ⊆ β there exists an unbounded B ⊆ A and x ∈ X such that ⟨f_{α} : α ∈ B⟩ is pointwise increasing on {y : x < y}.

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

Since β is good, we may fix x_{0} ∈ X and ⟨g_{α} : α < cf(β)⟩ which is pointwise increasing on {y : x_{0} < y} and is cofinally interleaved with ⟨f_{α} : α < β⟩. Thinning out the sequence of g_{α} we may also assume that for every α there is γ_{α} ∈ A with g_{α} <^{∗} f_{γ_{α}} <^{∗} g_{α+1}. Since cf(β) > |X| we may find x ≥ x_{0} and an unbounded set B_{0} ⊆ cf(β) such that g_{α}(y) < f_{γ_{α}}(y) < g_{α+1}(y) for all α ∈ B_{0} and y ≥ x. Let B = {γ_{α} : α ∈ B_{0}}, and observe that if α and α′ are in B_{0} with α < α′ and y ≥ x then f_{γ_{α}}(y) < g_{α+1}(y) ≤ g_{α′}(y) < f_{γ′_{α}}(y).

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

Fact 3.1. Let I be an ideal on a set X of cardinality κ, and let λ be regular with κ^{+} < λ. Let ⟨f_{α} : α < λ⟩ be a <_{I}-increasing sequence. Then one of the following must occur:

(1) There is an eub g for ⟨f_{α} : α < λ⟩ such that cf(g(x)) > κ for all x.

(2) There exist an ultrafilter U on X disjoint from I and sets ⟨S_{x} : x ∈ X⟩ with |S_{x}| ≤ κ, such that for all α < λ there is g ∈ ∏_{x∈X} S_{x} and β < λ such that f_{α} <_{U} g <_{U} f_{β}.

(3) There exists a function h such that if D_{α} = {x : f_{α}(x) < h(x)} then the sequence ⟨D_{α} : α < λ⟩ is not eventually constant modulo I.

If 2^{κ} < λ then it is not hard to see that the second and third cases are impossible, so that there must exist an eub g for ⟨f_{α} : α < λ⟩ such that cf(g(x)) > κ for all κ.

For the rest of this section we fix some infinite A ⊆ ω such that there is a scale of length ℵ_{ω+1} in ∏_{n∈A} ℵ_{n} under eventual domination. We also fix ⟨f_{α} : α < ℵ_{ω+1}⟩ which is such a scale. Altering the scale at limits if necessary, we may assume that it has the following strengthened form of the continuity property: if 0 < k < ω and α is a good point of cofinality ℵ_{k}, then f_{α} is an exact upper bound for ⟨f_{ζ} : ζ < α⟩ (this is just continuity) and in addition cf(f_{α}(n)) = ℵ_{k} for all n > k.

We already know that there are stationarily many good points for this scale in any uncountable cofinality but with some cardinal arithmetic assumptions we can say more.

If 2^{ℵ_{0}} < ℵ_{k} and β ∈ ℵ_{ω+1} is a point of cofinality ℵ_{k} then it follows from Trichotomy that there is an eub g for ⟨f_{α} : α < β⟩ with cf(g(n)) > ℵ_{0} for all n ∈ A. A little analysis (see [21] or [4] for the details) shows that cf(g(n)) = ℵ_{k} for cofinitely many n, so that β is a good point: it follows from our assumptions on the scale that f_{β} is an eub for ⟨f_{α} : α < β⟩ and cf(f_{β}(n)) = ℵ_{k} for all n ∈ A with n > k.

The structural hypothesis which we need for our main result is a weakening of square which only refers to ordinals below ℵ_{ω} with a fixed cofinality.

Definition 3.2. Let k be a natural number with k ≥ 1. A □(ℵ_{ω}, cof(ω_{k}))-sequence is a sequence ⟨C_{α} : α ∈ ℵ_{ω} ∩ cof(ω_{k})⟩ such that

(1) For all α, C_{α} is club in α and ot(C_{α}) = ω_{k}.

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

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

Lemma 3.3. If □_{ℵ_{n}} holds for all n with k ≤ n < ω then there is a □(ℵ_{ω}, cof(ω_{k}))- sequence.

Proof. Fix ⟨D_{α}^{n} : α < ℵ_{n+1}⟩ witnessing □_{ℵ_{n}}, where we assume without loss of generality that D_{α}^{n} ⊆ α \ (ℵ_{n} + 1) for α ∈ ℵ_{n+1} \ (ℵ_{n} + 1). We define C_{α} inductively.

Base case: If α ∈ ℵ_{k+1} ∩ cof(ω_{k}), let C_{α} = D_{α}^{k}.

Successor case: Let α ∈ (ℵ_{n+1} \(ℵ_{n} +1))∩cof(ω_{k}) for n > k and let α^{∗} = ot(D_{α}^{n}). Since cf(α) = ω_{k} and k ≠ n, we see that α^{∗} < ℵ_{n}. Since D_{α}^{n} is club in α, cf(α^{∗}) = cf(α) = ω_{k}. By induction C_{α^{∗}} has already been defined, and we define C_{α} by copying C_{α^{∗}} into D_{α}^{n}. To be more precise we set C_{α} = {γ ∈ D_{α}^{n} : ot(D_{α}^{n} ∩ γ) ∈ C_{α^{∗}}}.

We now check that this definition succeeds. Let η ∈ lim(C_{γ}) ∩ lim(C_{δ}). By our assumption on the sets D_{α}^{n}, either γ and δ are both less than ℵ_{k+1} or they both lie in ℵ_{n+1} \ (ℵ_{n} + 1) for some n > k.

Case 1: γ, δ < ℵ_{k+1}. In this case C_{γ} = D_{γ}^{k} and C_{δ} = D_{δ}^{k}, and it follows that C_{γ} ∩ η = D_{η}^{k} = C_{δ} ∩ η by the defining property of a □_{ℵ_{k}}-sequence.

Case 2: ℵ_{n} < γ, δ < ℵ_{n+1} for n > k. In this case η ∈ lim(D_{γ}^{n}) ∩ lim(D_{δ}^{n}) and so D_{γ}^{n} ∩ η = D_{δ}^{n} ∩ η. Moreover if ζ = ot(D_{γ}^{n} ∩ η) then by the definition of C_{γ} we have that ζ ∈ lim(C_{γ^{∗}}), and similarly ζ ∈ lim(C_{δ^{∗}}). By induction C_{γ^{∗}} ∩ ζ = C_{δ^{∗}} ∩ ζ, and so by definition C_{γ} ∩ η = C_{δ} ∩ η. □

The main theorem of this section shows that starting with a □(ℵ_{ω}, cof(ω_{k}))- sequence, we may lift it via PCF theory to a square-like sequence defined on the set of good points of cofinality ω_{k} in some scale of length ℵ_{ω+1}. To be more precise we define the following square-like principle, which is obtained by allowing the club sets to be defined only at points of some set S and weakening the coherence requirement so that it only applies at common limit points of uncountable cofinality.

Definition 3.4. Let S ⊆ ℵ_{ω+1} ∩ cof(ω_{k}) for some k > 1. A □_{ℵ_{ω}}^{−ω}(S)-sequence is a sequence ⟨E_{α} : α ∈ S⟩ such that

(1) For every α ∈ S, E_{α} is club in α and ot(E_{α}) = ω_{k}.

(2) For all γ, δ ∈ S and every η of uncountable cofinality which is a common limit point of E_{γ} and E_{δ}, E_{γ} ∩ η = E_{δ} ∩ η.

Theorem 3.5. Let k be an integer with k ≥ 2, and assume that there is a □(ℵ_{ω}, cof(ω_{k}))-sequence. If G is the set of good points of cofinality ω_{k} for the scale ⟨f_{α} : α < ℵ_{ω+1}⟩, then there exists a □_{ℵ_{ω}}^{−ω}(G)-sequence.

Proof. Dropping finitely many points from the set A if necessary, we may assume that all points in A are greater than k. Fix a □(ℵ_{ω}, cof(ω_{k}))-sequence ⟨C_{α} : α ∈ ℵ_{ω} ∩ cof(ω_{k})⟩. By our assumptions on the scale and the set A, for all γ ∈ G the function f_{γ} is an exact upper bound for ⟨f_{α} : α < γ⟩ and cf(f_{γ}(n)) = ℵ_{k} for all n ∈ A.

Given γ ∈ G, we define functions g_{i}^{γ} for i < ω_{k} by setting g_{i}^{γ}(n) equal to the i^{th} member of C_{f_{γ}(n)}. By construction the sequence of functions ⟨g_{i}^{γ} : i < ω_{k}⟩ is pointwise increasing and is cofinally interleaved with ⟨f_{α} : α < γ⟩.

For almost every η < γ of uncountable cofinality, there is j < ω_{k} such that ⟨g_{i}^{γ} : i < j⟩ is cofinally interleaved with ⟨f_{α} : α < η⟩; the sequence ⟨g_{i}^{γ} : i < j⟩ witnesses that η is good. Fix such η and j. By the argument of Theorem 2.11 and the uniqueness of exact upper bounds f_{η} =^{∗} sup_{i<j} g_{i}^{γ}, and since C_{f_{γ}(n)} is closed we also see that sup_{i<j} g_{i}^{γ}(n) = g_{j}^{γ}(n). We conclude that f_{η} =^{∗} g_{j}^{γ}.

For every γ ∈ G, we now define D_{γ} = { η < γ : ∃j f_{η} =^{∗} g_{j}^{γ} }. The set D_{γ} need not be club in γ, and so we let E_{γ} be the closure of D_{γ} in γ. We will show that ⟨E_{γ} : γ ∈ G⟩ is a □_{ℵ_{ω}}^{−ω}(G)-sequence.

If η is an accumulation point of D_{γ} with uncountable cofinality, then there is a unique k such that the functions {g_{j}^{γ} : j < k} are cofinally interleaved with {f_{α} : α < η}. It follows that η is good, and so by continuity we have f_{η} =^{∗} g_{k}^{γ} and thus η ∈ D_{γ}.

It is easy to see that ot(D_{γ}) = ω_{k} and we just showed that D_{γ} is closed under suprema of uncountable cofinality. Now let γ, δ be members of G. We claim that if η is a common accumulation point of D_{γ} and D_{δ} with uncountable cofinality, then D_{γ} ∩ η = D_{δ} ∩ η.

Since η is in D_{γ}, there is j with cf(η) = cf(j) such that f_{η}(n) = g_{j}^{γ}(n) for all large n, so that f_{η}(n) ∈ lim(C_{f_{γ}(n)}) for all large n. Similarly there is k such that f_{η}(n) = g_{k}^{δ}(n) and f_{η}(n) ∈ lim(C_{f_{δ}(n)}) for all large n.

It follows that C_{f_{γ}(n)} ∩ f_{η}(n) = C_{f_{δ}(n)} ∩ f_{η}(n) for all large n, and hence that j = k and that g_{i}^{γ} = g_{i}^{δ} for i < j. If ζ ∈ D_{γ} ∩ η then f_{ζ} =^{∗} g_{i}^{γ} for some i < j, and so f_{ζ} =^{∗} g_{i}^{δ} and ζ ∈ D_{δ} ∩ η; similarly D_{δ} ∩ η ⊆ D_{γ} ∩ η, so D_{γ} ∩ η = D_{δ} ∩ η.

It is now routine to verify that ⟨E_{γ} : γ ∈ S⟩ is a □_{ℵ_{ω}}^{−ω}(G)-sequence. □

Theorem 3.5 has the following striking corollary.

Corollary 3.6. Let CH hold and let □_{ℵ_{n}} hold for all n < ω. Then for every integer m ≥ 2 there exists a □_{ℵ_{ω}}^{−ω}(cof ω_{m})-sequence.

Remark 3.7. With a little work we can put Corollary 3.6 in a more pleasing way. Extending our notation, let □_{ℵ_{n}}^{−ω}(cof(ℵ_{k})) be the statement that there exists ⟨C_{α} : α ∈ ℵ_{n+1} ∩ cof(ℵ_{k})⟩ with C_{α} club in α, and the C_{α} cohering at common limit points of uncountable cofinality. Let CH hold, let k ≥ 2; then if □_{ℵ_{n}}^{−ω}(cof(ℵ_{k})) holds for all sufficiently large finite n, it holds for n = ω. 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 if there is a □_{ℵ_{ω}}^{−ω}(S)-sequence then S ∈ I[ℵ_{ω+1}], and so Theorem 3.5 has the following corollary.

Corollary 3.8. If □_{ℵ_{n}} holds for all finite n then in ℵ_{ω+1} all good points of cofinality greater than ℵ_{1} are approachable.

We can also deduce some results about stationary reflection.

Corollary 3.9. Let CH hold and let □_{ℵ_{n}} hold for all n < ω. For all integers m and n such that 0 < m < n there is a stationary subset of ℵ_{ω+1} ∩ cof(ℵ_{m}) which does not reflect at any point of ℵ_{ω+1} ∩ cof(ℵ_{n}).

Proof. Let ⟨C_{α} : α ∈ ℵ_{ω+1} ∩ cof(ℵ_{n})⟩ be a □_{ℵ_{ω}}^{−ω}(cof ω_{n})-sequence. Let S be the set of points of cofinality ℵ_{m} which occur as limit points of some C_{α}, and given β ∈ S let D_{β} be the unique set such that D_{β} = C_{α} ∩ β when β ∈ lim(C_{α}).

For any club subset D of ℵ_{ω+1}, we can find α ∈ lim(D) with cf(α) = ℵ_{n}. Then D ∩ C_{α} is club in α and so we can find β ∈ lim(C_{α} ∩ D) with cf(β) = ℵ_{m}. Clearly β ∈ D ∩ S so that S is stationary.

Applying Fodor’s lemma we may find T ⊆ S stationary and γ such that ot(D_{β}) = γ for all β ∈ T. For each α of cofinality ℵ_{n} the club set lim(C_{α}) meets T exactly once, so that the stationarity of T does not reflect at any point of cofinality ℵ_{n}. □

In our paper [9] it is shown that it is consistent that □_{ℵ_{n}} holds for 0 < n < ω but □_{ℵ_{ω}} fails. This is achieved by arranging that every stationary subset of ℵ_{ω+1} ∩ cof(ω) reflects at some point of ℵ_{ω+1} ∩ cof(ℵ_{1}). In the sequel [7] to this paper we show that it is consistent that the least λ for which □_{λ} fails should be the first inaccessible cardinal.

We can also combine the □_{ℵ_{ω}}^{−ω}(cof ω_{m})-sequences of Corollary 3.6 for different values of m and get a weakening of the principle □_{ℵ_{ω},ω} [8].

Corollary 3.10. Let CH hold and let □_{ℵ_{n}} hold for all n < ω. Then there exists ⟨C_{γ} : γ < ℵ_{ω+1}⟩ such that for every limit ordinal γ

(1) C_{γ} is a countable family of subsets of γ, each with order type less than ℵ_{ω}.

(2) For every C ∈ C_{γ} and every β ∈ lim(C) with uncountable cofinality, C ∩β ∈ C_{β}.

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

Definition 3.11. Let λ and κ be regular and uncountable with λ < κ. Then SNR(κ, λ) holds if and only if there is f : κ ⟶ λ such that for all α ∈ κ ∩ cof(λ) there is C club in α such that f ↾ C is strictly increasing.

This principle is true in L for all κ which are not weakly compact and all λ < κ. It implies that every stationary subset of κ has a stationary subset which reflects at no point of cofinality λ.

Dzamonja and Shelah [12] have studied the cardinal u(λ) which is defined to be the least κ > λ such that SNR(κ, λ) fails. In particular they have shown by an elaborate forcing argument that u(λ) can be the successor of a singular cardinal. The following theorem, which is proved in the same way as Theorem 3.5, seems to have some bearing on problems of this type.

Theorem 3.12 (Cummings [4]). Suppose that k < ω and that SNR(ℵ_{n}, ℵ_{k}) holds for all large n < ω. Then there is f : ℵ_{ω+1} ⟶ ℵ_{k} such that for all good α ∈ ℵ_{ω+1} ∩ cof(ℵ_{k}) there is C club in α such that f ↾ C ∩ {γ : ω < cf(γ) < ℵ_{k}} is strictly increasing.

One rather unsatisfying feature of some results in this section is the appearance of the Continuum Hypothesis among the hypotheses. CH is only being used to derive “all points of ℵ_{ω+1} ∩ cof(> ℵ_{1}) are good”, and it is quite possible that this statement is actually true in ZFC. For some discussion of the difficulties associated with attempting to show that this statement is consistently false see [5].

4. Uniformity

In this section we develop the idea of uniformity for a structure. The main point is that sufficiently uniform structures can be reconstructed from a small amount of data.

We recall from Lemma 2.3 that if M is an IA structure of size λ with a continuous approaching sequence of length κ then cf(M ∩ γ) = κ for all ordinals γ ∈ M such that cf(γ) > λ. This is the prototype for the uniformity properties we will consider.

Definition 4.1. Let M ≺ (H_{θ}, ∈, <_{θ}), let K be a set of regular cardinals and let κ be a cardinal. Then M is κ-uniform on K if and only if cf(M ∩ η) = κ for all η ∈ K ∩ M, and is weakly κ-uniform on K if and only if cf(M ∩ η) ≥ κ for all η ∈ K ∩ M.

This property arises naturally in the study of mutual stationarity. A structure which meets a sequence of stationary sets all consisting of ordinals of cofinality κ will automatically be κ-uniform on the relevant set of regular cardinals.

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

Definition 4.2. Let M ≺ (H_{θ}, ∈, <_{θ}) and let κ, λ and μ be cardinals with κ ≤ λ < μ ≤ θ. The structure M is κ-uniform (resp. weakly κ-uniform) between λ and μ if and only if

(1) λ ⊆ M.

(2) The structure M is κ-uniform (resp. weakly κ-uniform) on the interval of

regular cardinals {η ∈ REG : λ < η < μ}. If M is κ-uniform between λ and θ we say that M is κ-uniform past λ, and similarly for weak uniformity.

This kind of uniformity arises naturally in the study of IA structures. If M is an IA structure with an approaching chain of length κ and λ = |M| ⊆ M, then M is κ-uniform past λ.

Let M ≺ (H_{θ}, ∈, <_{θ}). If γ is any limit ordinal in M, then cf(γ) is in M and there exists in M an increasing cofinal map f from cf(γ) to γ. Restricting f to cf(γ) ∩ M we get a cofinal map from cf(γ) ∩ M to γ ∩ M, so that cf(γ) ∩ M and γ ∩ M have the same cofinality. In particular if M is κ-uniform on K then cf(γ ∩ M) = κ for all ordinals γ ∈ M such that cf(γ) ∈ K, and similarly for weak uniformity. If M is κ-weakly uniform between λ and μ and ρ ∈ M is an ordinal of cofinality less than or equal to λ it follows that M ∩ ρ is unbounded in ρ. If cf(ρ) is strictly between λ and μ, then cf(M ∩ ρ) ≥ κ.

Before discussing the implications of weak uniformity, we recall a well-known variation on the closed unbounded filter. Given a regular cardinal κ and a limit ordinal ρ with κ ≤ cf(ρ), we say that a set A ⊆ ρ is a < κ-club subset of ρ if A is unbounded in ρ, and sup(x) ∈ A for every x ⊆ A with |x| < κ. If κ is uncountable then the collection of < κ-club sets generates a cf(ρ)-complete filter, which is in fact the restriction of the club filter to the set of points of cofinality less than κ.

Lemma 4.3. Let M be κ-weakly uniform between λ and μ. Then (1) For every bounded subset x of M ∩ μ with |x| < κ, sup(x) ∈ M.

(2) For every regular η with λ < η ≤ sup(M ∩ μ)

(a) cf(M ∩ η) ≥ κ.

(b) There exists E ⊆ M ∩ η which is < κ-club in sup(M ∩ η). (Note that in item 2, we do not require that η ∈ M.)

Proof. For claim 1, we may as well assume that x has limit order type. Let β = sup(x) and γ = min(M \ β), where clearly both β and γ are limit ordinals less than μ and M ∩ γ = M ∩ β. Suppose for a contradiction that γ > β. Then cf(γ) > λ, because otherwise M ∩γ would be unbounded in γ. Since γ ∈ M and λ < cf(γ) < μ we have cf(M ∩ γ) ≥ κ, but this is impossible because cf(M ∩ β) = cf(β) ≤ |x| < κ and M ∩ β = M ∩ γ.

For claim 2, we fix a regular η with λ < η ≤ sup(M ∩ μ). Suppose for a contradiction that cf(M ∩ η) < κ and fix x ⊆ M ∩ η cofinal in M ∩ η with |x| < κ. Since κ ≤ λ < η, M ∩ η is bounded in η. Since η ≤ sup(M ∩ μ), x is a bounded subset of M ∩ μ and so by the first claim sup(x) = sup(M ∩ η) ∈ M. This is a contradiction since M ∩ η is bounded in η. If we now fix any cofinal set D ⊆ M ∩ η and let E be the closure of D under suprema of size less than κ, then E ⊆ M ∩ η and E is a < κ-club set as required. □

It is notable that claim 2) of Lemma 4.3 applies to regular cardinals which do not lie in M. Before we can exploit Lemma 4.3 we need some more or less standard facts about rebuilding structures.

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

Lemma 4.4. Let λ be a cardinal, let K be an interval of regular cardinals with min(K) = λ^{+} and let θ be a regular cardinal with sup(K) < θ. Let M_{0} and M_{1} be two elementary substructures of (H_{θ}, ∈, <_{θ}) such that

• λ ⊆ M_{0} ∩ M_{1}.
• For every η ∈ K, M_{0} ∩ M_{1} ∩ η is cofinal in M_{0} ∩ η.

Then M_{0} ∩ sup(K) ⊆ M_{1} ∩ sup(K).

Proof. Clearly M_{0} ∩ λ = M_{1} ∩ λ = λ. We show by induction on η ∈ K that M_{0} ∩ η ⊆ M_{1} ∩ η. There is nothing to do when η is a limit cardinal, so let η = ζ^{+} where M_{0} ∩ ζ ⊆ M_{1} ∩ ζ by induction. If α ∈ M_{0} ∩ η there is β ∈ M_{0} ∩ M_{1} ∩ η with α < β, and we fix f the <_{θ}-least bijection from β to |β|. Since β ∈ M_{0} ∩ M_{1} and f is defined from β we have f ∈ M_{0} ∩ M_{1}, and since β < ζ^{+} we have |β| ≤ ζ. So f(α) ∈ M_{0} ∩ ζ, and since M_{0} ∩ ζ ⊆ M_{1} ∩ ζ we conclude that f(α) ∈ M_{1} ∩ ζ and so α ∈ M_{1}. Thus M_{0} ∩ η ⊆ M_{1} ∩ η 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 demand that M_{0} ∩ M_{1} ∩ η is also cofinal in M_{1} ∩ η for all η ∈ K, then we may strengthen the conclusion to M_{0} ∩ sup(K) = M_{1} ∩ sup(K).

Proof. Immediate from Lemma 4.4. □

Theorem 4.6. Let κ be an uncountable cardinal. Let K be an interval of regular cardinals with min(K) = λ^{+} and let θ be a regular cardinal with sup(K) < θ. Let M ≺ (H_{θ}, ∈, <_{θ}) be κ-weakly uniform between λ and sup{η^{+} : η ∈ K}. Then M ∩ sup(K) is determined by χ_{M}^{K} and sup(M ∩ K).

Proof. Suppose that M and N are both substructures of (H_{θ}, ∈, <_{θ}) which are κweakly uniform between λ and sup{η^{+} : η ∈ K}, with χ_{M}^{K} = χ_{N}^{K} and sup(N ∩K) = sup(M ∩K). Then N ∩K = M ∩K. By Lemma 4.3, for every η ∈ K ∩M each of the sets M ∩ η and N ∩ η contains a set which is < κ-club in sup(M ∩ η) = sup(N ∩ η). The intersection of two < κ-club sets is < κ-club, so M ∩ N ∩ η is unbounded in M ∩ η and N ∩ η. 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 be useful later to know that the process of reconstructing M ∩ sup(K) from χ_{M}^{K} is simply definable.

Lemma 4.7. Let K, κ, λ, θ be as in the last lemma and suppose that M, N ≺ (H(θ), ∈, <_{θ}) with χ_{M}^{K} ∈ N. Then M ∩ sup(K) ∈ N.

Proof. Since K = dom(χ_{M}^{K}) we see that K ∈ N, and thus sup(K) ∈ N and λ ∈ N. We may also find κ^{∗} ∈ N such that M is κ^{∗}-weakly uniform.

Let ⟨g_{β} : β < sup(K)⟩ be such that g_{β} is the <_{θ}-least bijection from β to |β|, and define partial functions g and h from sup(K) × sup(K) to sup(K) by

g(β, α) = g_{β}(α), h(β, γ) = g_{β}^{−1}(γ).

Since g and h are defined from parameters in N, they are members of N. The argument in the proof of Lemma 4.5 shows that M ∩ sup(K) can be computed as follows: for each C⃗ = ⟨C_{η} : η ∈ K⟩ such that C_{η} is < κ^{∗}-club in χ_{M}(η) for each η, compute the closure X(C⃗) of λ ∪ (⋃_{η} C_{η}) under g and h, and then take the intersection of all the sets X(C⃗). It follows that M ∩ sup(K) ∈ N. □

5. Tight structures and PCF

Recall from definition 2.4 that if K is a set of regular cardinals and M is a structure with K ∈ M and K ⊆ M, then M is tight for K if and only if M ∩ ∏ K is cofinal in ∏_{κ∈K} M ∩ κ. For the rest of this section we will make the following

Blanket assumption: We are given a set K of regular cardinals and a structure M ≺ (H_{θ}, ∈, <_{θ}) such that K ∈ M, K ⊆ M and |M| < min(K).

We note that since |M| < min(K), we have χ_{M}^{K} ∈ ∏ K. Also since K ⊆ M we have |K| < min(K) so that K is a “progressive” set of regular cardinals, and the methods of PCF theory can be applied to K.

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

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

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

(2) A tight elementary substructure of an H(θ) is determined by the its “cardinal structure”. 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 why they were used in the original PCF theory.) However, given a structure N ≺ (H(θ), ∈, <_{θ}), to determine whether N is tight (in an absolute way from knowledge of the regular cardinals) one must only know χ_{N} ↾ ℵ_{ω+1} (i.e. N’s trace on the ordinals), but to determine whether N ∩ H(ℵ_{ω+1}) is internally approachable seems to require considerably more information and conceivably might not be an absolute question.

We summarize 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, Section 7.3] which analyses uniform structures which are tight for K = {ℵ_{n} : n < ω} under the assumption that max pcf(K) = ℵ_{ω+1}. They are also related to work by Shelah analysing the 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 a detailed analysis of tightness for M under the assumptions that pcf(K) ⊆ M and M is uniform on K. Theorem 5.5 shows that we can drop the assumption that pcf(K) ⊆ M and demand only weak uniformity for M on K; the price we pay is that we need extra technical assumptions and we get a somewhat weaker conclusion. Theorem 5.6 shows that 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, we show that if M is tight for K then some uniformity properties can be propagated from K to pcf(K).

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

Proof. Suppose that M is η-weakly uniform on K. Given λ ∈ M ∩pcf(K), let us fix U ∈ M an ultrafilter on K and ⟨f_{α} : α < λ⟩ ∈ M increasing and cofinal in ∏ K/U. Suppose for a contradiction that cf(M ∩ λ) < η. We fix A ⊆ M ∩ λ cofinal with order type cf(M ∩ λ), and note that if α ∈ A then f_{α}(κ) ∈ M ∩ κ for all κ ∈ K. If we define g(κ) = sup_{α∈A} f_{α}(κ) for all κ ∈ K, then g(κ) < sup(M ∩ κ) because cf(M ∩ κ) ≥ η > |A|. Since M is tight there is h ∈ M ∩ ∏ K with g < h, and by elementarity there is α ∈ M ∩ λ with h <_{U} f_{α}. So g <_{U} f_{α}, but this is impossible because by construction f_{α} ≤ g.

Now suppose that M is η-uniform on K, and use this to find a pointwise increasing sequence ⟨g_{i} : i < η⟩ of functions whose ranges are contained in M and whose pointwise supremum is χ_{M}. As above let λ ∈ M ∩ pcf(K) and fix U ∈ M an ultrafilter on K and ⟨f_{α} : α < λ⟩ ∈ M increasing and cofinal in ∏ K/U. It is easy to see by tightness and elementarity that ⟨f_{α} : α ∈ M ∩ λ⟩ is cofinally interleaved with ⟨g_{i} : i < η⟩, so cf(M ∩ λ) = η. □

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

As usual we let J_{<λ} be the ideal of sets A ⊆ K such that every ultrafilter U containing A gives an ultrapower ∏ K/U of cofinality less than λ. The sequence of ideals J_{<λ} is increasing with λ and is continuous at limit cardinals.

By standard results from PCF theory we fix a sequence of sets ⟨B_{λ} : λ ∈ pcf(K)⟩ with B_{λ} ⊆ K for each λ, such that B_{λ} generates J_{<λ^{+}} over J_{<λ}: that is to say that for A ⊆ K we have A ∈ J_{<λ^{+}} ⟺ A \ B_{λ} ∈ J_{<λ}. An easy induction argument shows that J_{<λ} is the ideal of sets which are covered by a finite union of sets in {B_{μ} : μ < λ}.

It is known that pcf(K) has a maximum element, and accordingly we will choose B_{max pcf(K)} = K. By standard facts from PCF theory we may fix a matrix of functions ⟨f_{α}^{λ} : α < λ, λ ∈ pcf(K)⟩ such that ⟨f_{α}^{λ} : α < λ⟩ is a continuous scale in ∏ B_{λ}/J_{<λ} for every λ ∈ pcf(K).

We fix θ some sufficiently large regular cardinal and let A be the structure (H_{θ}, ∈, <_{θ}, {K}, ⟨B_{λ}⟩, ⟨f_{α}^{λ}⟩). 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 η be regular and uncountable, and suppose that |K| < η < min(K). Suppose that M ≺ A where M is η-uniform on K and pcf(K) ⊆ M. Then the following are equivalent:

(1) M is tight for K.

(2) For every λ ∈ pcf(K), sup(M ∩ λ) is a good point of cofinality η for ⟨f_{α}^{λ} : α < λ⟩ and f_{sup(M∩λ)}^{λ} =_{J_{<λ}} χ_{M}^{B_{λ}}.

Proof. For the forward direction, assume M is tight. Since cf(M ∩ κ) = η for all κ ∈ K, we may find a pointwise increasing sequence ⟨g_{i} : i < η⟩ in ∏ K whose pointwise supremum is χ_{M}^{K}.

Fix λ ∈ pcf(K). We claim that ⟨g_{i} ↾ B_{λ} : i < η⟩ is cofinally interleaved with ⟨f_{α}^{λ} : α ∈ M ∩ λ⟩ modulo J_{<λ}. For all α ∈ M ∩ λ, f_{α}^{λ} ∈ M and so f_{α}^{λ} is pointwise dominated by χ_{M}^{B_{λ}}. Since |K| < η, there exists i < η such that f_{α}^{λ} is dominated pointwise by g_{i} ↾ B_{λ}. Conversely if i < η then it follows from the tightness of M that g_{i} is dominated pointwise by some function in M, and so by elementarity g_{i} <_{J_{<λ}} f_{α}^{λ} for some α ∈ M ∩ λ.

It follows that cf(M ∩ λ) = η and sup(M ∩ λ) is a good point of cofinality η. By Lemma 2.11 f_{sup(M∩λ)}^{λ} agrees modulo J_{<λ} with the pointwise supremum of the g_{i} ↾ B_{λ}, that is to say f_{sup(M∩λ)}^{λ} =_{J_{<λ}} χ_{M}^{B_{λ}}.

For the converse direction, assume that for every λ ∈ pcf(K), sup(M ∩ λ) is a good point of cofinality η for ⟨f_{α}^{λ} : α < λ⟩ and f_{sup(M∩λ)}^{λ} =_{J_{<λ}} χ_{M}^{B_{λ}}. Let h be a function in ∏ K which is pointwise dominated by χ_{M}^{K}. For every λ in pcf(K) we have h <_{J_{<λ}} f_{sup(M∩λ)}^{λ}. Since sup(M ∩ λ) is a good point it follows that f_{sup(M∩λ)}^{λ} is an exact upper bound for ⟨f_{α} : α ∈ M ∩ λ⟩. Hence for every λ ∈ pcf(K) there is α ∈ M ∩ λ with h <_{J_{<λ}} f_{α}^{λ}.

We will inductively build a decreasing sequence λ_{0} > λ_{1} > λ_{2} > λ_{j} of members of pcf(K) together with ordinals α_{i} ∈ M ∩ λ_{i}. We will also build a decreasing sequence of sets D_{0} ⊇ D_{1} ⊇ D_{2} … such that D_{i} ∈ J_{<λ_{i}}, halting when we reach a stage with D_{i} = ∅.

We let λ_{0} = max pcf(K), and recall that B_{λ_{0}} = K. We choose α_{0} ∈ M ∩ λ_{0} such that h <_{J_{<λ_{0}}} f_{α_{0}}^{λ_{0}}. Let D_{0} = {κ : h(κ) ≥ f_{α_{0}}^{λ_{0}}(κ)}, so that D_{0} ∈ J_{<λ_{0}}.

Suppose that we have defined λ_{i}, α_{i} and D_{i} for i ≤ j. We stop if D_{j} = ∅. Otherwise we choose λ_{j+1} to be the unique member of pcf(K) with D_{j} ∈ J_{<λ_{j+1}^{+}}

and D_{j} ∉ J_{<λ_{j+1}}, and choose α_{j+1} ∈ M ∩ λ_{j+1} such that h <_{J_{<λ_{j+1}}} f_{α_{j+1}}^{λ_{j+1}}. Now we let

D_{j+1} = {κ ∈ D_{j} : κ ∉ B_{λ_{j+1}} or h(κ) ≥ f_{α_{j+1}}^{λ_{j+1}}(κ)}.

Since B_{λ_{j+1}} generates J_{<λ_{j+1}^{+}} and h <_{J_{<λ_{j+1}}} f_{α_{j+1}}^{λ_{j+1}}, we see that D_{j+1} ∈ J_{<λ_{j+1}}.

Since the descending sequence of λ_{i} can only have finite length, the construction must terminate. Let j be the last stage, so D_{j} = ∅. For each κ ∈ K, let i be minimal with κ ∉ D_{i}. By definition we must have κ ∈ B_{λ_{i}} and h(κ) < f_{α_{i}}^{λ_{i}}(κ). We conclude that h is pointwise dominated by the pointwise supremum of {f_{α_{i}}^{λ_{i}} : i ≤ j}, and since this function lies in M we have proved that M is tight. □

The following result will play an important role when we analyse the relationship between tightness and internal approachability in Section 6. It is a generalization of Shelah’s analysis of internally approachable structures used to bound powers of singular cardinals. It is important for the purposes of that section to note that each of the sets B_{λ_{i}} \ E_{i} is in M, and each of the indices sup(M ∩ λ_{i}) is computed in a uniform way from M.

Theorem 5.3. Let η be regular and uncountable, and suppose that |K| < η < min(K). Suppose that M is η-uniform on K and pcf(K) ⊆ M.

If M is tight for K then there exist λ_{0}, … λ_{j} ∈ pcf(K) and E_{0}, … E_{j} with E_{i} ∈ M ∩ J_{<λ_{i}} such that K = ⋃_{i}(B_{λ_{i}} \ E_{i}) and χ_{M} is the pointwise supremum of the functions f_{sup(N∩λ_{i})}^{λ_{i}} ↾ (B_{λ_{i}} \ E_{i}) for i ≤ j.

Proof. By Theorem 5.2, f_{sup(M∩λ)}^{λ} =_{J_{<λ}} χ_{M}^{B_{λ}} for all λ ∈ 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 λ_{0} > λ_{1} > … of elements of pcf(K), and also sets D_{i} with D_{i} ∈ J_{<λ_{i}} which this time are not necessarily decreasing. We also build sets E_{i} ∈ M ∩ J_{<λ_{i}} such that D_{i} ⊆ E_{i}, and choose D_{i+1} as a subset of E_{i} rather than D_{i}.

Let λ_{0} = max pcf(K) and D_{0} = {κ : χ_{M}(κ) ≠ f_{sup(M∩λ_{0})}^{λ_{0}}(κ)}, so that D_{0} ∈ J_{<λ_{0}}. Suppose now that we have constructed the ordinals λ_{0}, λ_{1} … λ_{j}, together with the sets D_{0}, … D_{j} and E_{0}, … E_{j−1}.

If D_{j} = ∅ we set E_{j} = ∅ and stop the construction. If D_{j} ≠ ∅ we choose λ_{j+1} as the unique member of pcf(K) such that D_{j} ∈ J_{<λ_{j+1}^{+}} and D_{j} ∉ J_{<λ_{j+1}}. We choose E_{j} to be some finite union of sets in {B_{λ} : λ ≤ λ_{j+1}} such that D_{j} ⊆ E_{j}.

We note that E_{j} ∈ J_{<λ_{j+1}^{+}} ∩ M, and that since D_{j} ∉ J_{<λ_{j+1}} we must have B_{λ_{j+1}} ⊆ E_{j}. Let

D_{j+1} = {κ ∈ E_{j} : κ ∉ B_{λ_{j+1}} or χ_{M}(κ) ≠ f_{sup(M∩λ_{j+1})}^{λ_{j+1}}(κ)}.

Since E_{j} \ B_{λ_{j+1}} is covered by a finite union of sets from {B_{λ} : λ < λ_{j+1}}, and the functions χ_{M} and f_{sup(M∩λ_{j+1})}^{λ_{j+1}}(κ) agree on a J_{<λ_{j+1}}-large subset of B_{λ_{j+1}}, we see that D_{j+1} ∈ J_{<λ_{j+1}}.

Since the sequence of λ_{i} is decreasing we eventually reach a stage j with D_{j} = ∅, and so we halt the construction after setting E_{j} = ∅. To finish we need to check that χ_{M} is the pointwise supremum of the functions f_{sup(M∩λ_{i})}^{λ_{i}} ↾ (B_{λ_{i}} \ E_{i}) for i ≤ j.

Let κ ∈ B_{λ_{i}} \ E_{i} for some i ≤ j. Since D_{i} ⊆ E_{i}, κ ∈ B_{λ_{i}} \ D_{i} . By the construction of D_{i} we know that {κ ∈ B_{λ_{i}} : f_{sup(M∩λ_{i})}^{λ_{i}}(κ) ≠ χ_{M}(κ)} is a subset of D_{i}, so f_{sup(M∩λ_{i})}^{λ_{i}}(κ) = χ_{M}(κ).

Given κ ∈ K, let i be minimal such that κ ∉ E_{i}. If i = 0 then κ ∈ B_{λ_{0}} because we chose B_{λ_{0}} = K. If i > 0 then κ ∈ E_{i−1}, and since κ ∉ D_{i} we see that κ ∈ B_{λ_{i}}. It follows that every element of K appears in B_{λ_{i}} \ E_{i} for some i. □

Remark 5.4. For an application which we will make of Theorems 5.2 and 5.3 in Section 7, we note that we only needed the scales {f_{α}^{λ} : α < λ} to be continuous at good points of cofinality η.

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

Let us be given an index set X, an ideal I on X and a <_{I}-increasing sequence f⃗. If cf(β) > |X|, we define a function f_{β}^{∗} by letting f_{β}^{∗}(κ) be the least ordinal of the form sup_{α∈E} f_{α}(κ) for E an ω-club subset of β. We note that if E is some ω-club subset of β with f_{β}^{∗}(κ) = sup_{α∈E} f_{α}(κ), then f_{β}^{∗}(κ) = sup_{α∈F} f_{α}(κ) for any F ⊆ E with F an ω-club subset of β.

Since the intersection of |X|-many ω-club subsets of β is ω-club, we may fix a single ω-club subset F of β such that f_{β}^{∗}(κ) = sup_{α∈F} f_{α}(κ) for all κ ∈ F; it follows that f_{β}^{∗} is an upper bound modulo I for ⟨f_{α} : α < β⟩. We say that the sequence f⃗ is ω-club minimal at β if f_{β}^{∗} = f_{β}. Returning to the context of this section, it is routine to check that if I is an ideal on K with tcf(∏ K/I) = λ then we may find a scale ⟨f_{α} : α < λ⟩ in ∏ K/I which is ω-club minimal at every β < λ such that |K| < cf(β) < min(K); by a slight abuse of language we will say that such a scale is ω-club minimal in ∏ K/I.

We will fix a matrix of functions ⟨f_{α}^{λ} : α < λ, λ ∈ pcf(K)⟩ such that ⟨f_{α}^{λ} : α < λ⟩ is a an ω-club minimal scale in ∏ B_{λ}/J_{<λ} for every λ ∈ pcf(K). Let A be the structure (H_{θ}, ∈, <_{θ}, {K}, ⟨B_{λ}⟩, ⟨f_{α}^{λ}⟩). We will assume that M ≺ A.

We are now ready to use our technical assumption of ω-club minimality. Suppose that M is |K|^{+}-weakly uniform between ρ and σ, where all cardinals in M ∩pcf(K) are greater than ρ and less than σ. Let λ ∈ M ∩ pcf(K) and let γ = sup(M ∩ λ). By Lemma 4.3 there is E ⊆ M ∩ λ which is ω-club in γ. Since |M| < min(K) we see that cf(γ) < min(K), and since λ lies in an interval where M is weakly uniform we also see that cf(γ) > |K|. So by the assumption of ω-club minimality, we may find F ⊆ E such that F is ω-club in γ and f_{γ}^{λ}(κ) = sup_{α∈F} f_{α}^{λ}(κ) for all κ ∈ K. Since F ⊆ E ⊆ M, K ⊆ M and λ ∈ M it follows easily that f_{γ}^{λ} ≤ χ_{M}^{B_{λ}}.

We can now give a characterisation of tightness.

Theorem 5.5. Let M be |K|^{+}-weakly uniform between ρ and σ, where all cardinals in M ∩ pcf(K) are greater than ρ and less than σ. Then the following conditions on M are equivalent.

(1) M is tight for K.

(2) For every λ ∈ M ∩ pcf(K), if γ = sup(M ∩ λ) then (a) f_{γ}^{λ} ≤ χ_{M}^{B_{λ}}.

(b) There is A ⊆ B_{λ} such that A ∈ M ∩ J_{<λ} and {κ ∈ B_{λ} : f_{γ}^{λ}(κ) < χ_{M}(κ)} ⊆ A.

(c) f_{γ}^{λ} is an exact upper bound for ⟨f_{α}^{λ} : α < γ⟩ modulo J_{<λ} in the following strengthened sense: for all f < f_{γ}^{λ} there exist β ∈ M ∩ λ and B ∈ M ∩ J_{<λ} such that {κ ∈ B_{λ} : f_{β}^{λ}(κ) ≤ f(κ)} ⊆ B.

Proof. 1) implies 2). We have already seen that the uniformity hypothesis on M implies that f_{γ}^{λ} ≤ χ_{M}^{B_{λ}}. Since M is tight we may find a function g ∈ M ∩ ∏ K such that for all κ ∈ K, f_{γ}^{λ}(κ) < χ_{M}(κ) implies that f_{γ}^{λ}(κ) < g(κ). By ω-club minimality we know that f_{γ}^{λ} is the pointwise supremum of {f_{α}^{λ} : α ∈ F } for some F ⊆ M which is ω-club in γ, and we may find α ∈ F so large that f_{α}^{λ} dominates g modulo J_{<λ}. Now since f_{α}^{λ} ≤ f_{γ}^{λ} we see that for κ ∈ K

f_{γ}^{λ}(κ) < χ_{M}(κ) ⟹ f_{γ}^{λ}(κ) < g(κ) ⟹ f_{α}^{λ}(κ) < g(κ),

so that if we set A = {κ ∈ B_{λ} : f_{α}^{λ}(κ) < g(κ)} then A is as required. Now let f < f_{γ}^{λ}. Since f_{γ}^{λ} ≤ χ_{M} and M is tight, we may find h ∈ M such that f < h and then find β ∈ F such that f_{β}^{λ} dominates h modulo J_{<λ}. Now

f_{β}^{λ}(κ) ≤ f(κ) ⟹ f_{β}^{λ}(κ) ≤ h(κ),

and if we set B = {κ ∈ B_{λ} : f_{β}^{λ}(κ) ≤ h(κ)} then B is as required.

2) implies 1). Let f ∈ ∏ K with f < χ_{M}^{K}. The construction is very similar to that for Theorem 5.2. We construct a decreasing sequence of ordinals λ_{0} > λ_{1} > … with λ_{i} ∈ M ∩ pcf(K), together with ordinals α_{i} ∈ M ∩ λ_{i} and sets B_{i} ∈ M ∩ J_{λ_{i}}, in such a way that if κ ∈ B_{i−1} \ B_{i} then f(κ) < f_{α_{i}}^{λ_{i}}(κ).

Let λ_{0} = max pcf(K) and γ_{0} = sup(M ∩λ_{0}), and observe that by b) the functions f_{γ_{0}}^{λ_{0}} and χ_{M} agree outside a set in M ∩J_{<λ_{0}}. By c) we can find a set B_{0} ∈ M ∩J_{<λ_{0}} such that f(κ) < f_{γ_{0}}^{λ_{0}}(κ) for κ ∉ B_{0}.

If at stage i we have B_{i} ≠ ∅, then choose λ_{i+1} minimal with B_{i} ∉ J_{<λ_{i+1}}, noting that λ_{i+1} ∈ M because B_{i} ∈ M. Now we apply b) and c) to find B_{i+1} ∈ J_{<λ_{i+1}} and α_{i+1} ∈ M ∩ λ_{i+1} such that f(κ) < f_{α_{i+1}}^{λ_{i+1}}(κ) for κ ∈ B_{i} \ B_{i+1}.

As usual, the construction must terminate with B_{i} = ∅. Then f is pointwise dominated by the supremum of the functions f_{α_{j}}^{λ_{j}} for j < i, and this function lies in M. □

Under certain circumstances we can do an analysis of tightness as in Theorem 5.5 under a weaker uniformity hypothesis. It is known that if K is an interval of regular cardinals then pcf(K) is also an interval of cardinals. If K is an interval, M is tight for K and M is |K|^{+}-weakly uniform in some interval containing K then it follows from Theorem 5.1 that M is weakly uniform in an interval containing pcf(K), and so the analysis of Theorem 5.5 applies.

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

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

Proof. Let L = pcf(K), and let F ∈ ∏_{λ∈L} M ∩ λ. We define f ∈ ∏ K by

f : κ ↦ sup_{λ∈L} f_{F (λ)}^{λ}(κ) + 1.

Since L ⊆ M, we see that f_{F (λ)}^{λ} ∈ M and f_{F (λ)}^{λ}(κ) ∈ M ∩ κ for all λ ∈ L and κ ∈ K. Since cf(M ∩ κ) > |L| it follows that f(κ) < sup(M ∩ κ) for all κ ∈ K.

Since M is tight for K, there is g ∈ M ∩∏ K which dominates f pointwise. Since ⟨f_{α}^{λ} : α < λ, λ ∈ pcf(K)⟩ ∈ M and the sequence ⟨f_{α}^{λ} : α < λ⟩ is a scale for each λ, it follows by elementarity that there is G ∈ M ∩ ∏ L such that g <_{J_{<λ}} f_{G(λ)}^{λ} for all λ ∈ L. We now see that for each λ ∈ L we have

f_{F (λ)}^{λ} < f < g <_{J_{<λ}} f_{G(λ)}^{λ},

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. η-uniform with |K| < η < min(K)) between ρ and σ, where all cardinals in M ∩ pcf(K) are greater than ρ and less than σ. Let M′ be the Skolem hull of M ∪ pcf(K) in (H(θ), ϵ, <_{θ}). Then the following conditions on M are equivalent.

(1) M is tight for K.

(2) M′ is tight for pcf(K) and |K|^{+}-weakly uniform (resp. η-uniform) on pcf(K).

(3) For every λ ∈ pcf(K), sup(M′ ∩ λ) is a good point of cofinality at least κ^{+} (resp. of cofinality η) for ⟨f_{α}^{λ} : α < λ⟩ and f_{sup(M′∩λ)}^{λ} =_{J_{<λ}} χ_{M′}^{B_{λ}}.

Proof. By Fact 6.2 we know that | pcf(K)| < |K|^{+4}. Thus M′ ⊂ sk^{H(θ)}(M ∪|K|^{+3}). Standard arguments show that for all κ ∈ K\|K|^{+3}, sup(M ∩ κ) = sup(sk^{H(θ)}(M ∪ |K|^{+3}) ∩ κ). 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. η-uniform) on pcf(K). Hence 1 implies 2. Again by the previous paragraph it is clear that 2 implies 1

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

6. Tightness, approachability and reflection

In this section we prove a general covering theorem for tight structures, and use it to show that certain tight uniform structures are IA. We also discuss the connection between tightness and stationary reflection.

We start with some general discussion of covering properties of structures. Suppose that κ, λ and θ are regular cardinals with κ < λ < θ. We consider substructures N ≺ (H_{θ}, ∈, <_{θ}, {κ, λ}) such that |N| = κ and κ ⊆ N.

Given a set Z with κ ⊆ Z, we let P_{κ}Z be the set of x ⊆ Z such that |x| < κ and x ∩ κ ∈ κ. In increasing order of strength we may consider the following properties of N:

Internally cofinal in P_{κ}λ: N ∩ P_{κ}(N ∩ λ) is cofinal (in the inclusion ordering) in P_{κ}(N ∩ λ)

Internally stationary in P_{κ}λ: N ∩ P_{κ}(N ∩ λ) is stationary in P_{κ}(N ∩ λ).

Internally club in P_{κ}λ: N ∩ P_{κ}(N ∩ λ) contains a club in P_{κ}(N ∩ λ).

Internally approachable in P_{κ}H_{λ}: N ∩ H_{λ} is IA of length and cardinality κ.

The following easy lemma shows that some cardinal arithmetic assumptions simplify the picture.

Lemma 6.1. Let κ, λ and θ be regular cardinals with κ < λ < θ and let N ≺ (H_{θ}, ∈, <_{θ}, {κ, λ}) be such that |N| = κ and κ ⊆ N.

If κ^{<κ} = κ and N is internally cofinal in P_{κ}λ, then P_{κ}(N ∩ λ) ⊆ N, from which it follows that N is IA in P_{κ}H_{λ}.

For the rest of this section we will be studying structures N which are tight for some interval K of regular cardinals. One reason for this is that we wish to apply the results of Section 4 on recovering structures from their characteristic functions, and these results require an interval of regular cardinals. Another reason is that we can use the 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)| ≤ |K|^{+3}.

• pcf(pcf(K)) = pcf(K).

Remark 6.3. If K is an infinite interval of regular cardinals with |K| < min(K), then it follows from Fact 6.2 that by deleting a suitable finite initial segment k of K we can 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 briefly to consider what is perhaps the most interesting special case. Let K = {ℵ_{n} : 0 < n < ω}, and suppose for simplicity that pcf(K) is countable, say pcf(K) = {ℵ_{α+1} : α ≤ β} for some countable β.

Let m > 0 and let θ be a large regular cardinal. Let N ≺ H_{θ} be such that ℵ_{m} = |N| ⊆ N, and let N be ℵ_{m}-uniform and tight on {ℵ_{n} : m < n < ω}. By Theorems 5.1 and 5.6 N is ℵ_{m}-uniform and tight on {ℵ_{α+1} : m ≤ α ≤ β}.

Under these circumstances, it will follow from Theorem 6.5 that if m > 1 then N is internally stationary in P_{ℵ_{m}}(ℵ_{β+1}). In conjunction with some extra cardinal arithmetic assumptions as in Lemma 6.1, it will follow that N ∩ H_{ℵ_{β+1}} is IA of length and cardinality ℵ_{m}. This is an instance of one of the motivating theses of this paper, that tightness plus uniformity is very close to internal approachability.

Theorem 6.5. Let η be an uncountable regular cardinal. Let L be an interval of regular cardinals such that min(L) = η^{++}, |L| < η, and L = pcf(L). Let θ be a large regular cardinal and let N ≺ (H_{θ}, ∈, <_{θ}, {L}) be such that |N| = η^{+} and N is η^{+}-uniform between η^{+} and max(L)^{+} and N is tight for L. Then N is internally stationary in P_{η^{+}}(max(L)).

Proof. Fix some algebra A on N ∩ max(L). We build an increasing sequence ⟨M_{i} : i < η⟩ such that for all i

(1) M_{i} ≺ N, with η ⊆ M_{i} and |M_{i}| = η.

(2) M_{i} ∩ max(L) ≺ A.

(3) M_{i} ∩ η^{+} < M_{i+1} ∩ η^{+}.

(4) χ_{M_{i}}^{L} is pointwise dominated by some function in M_{i+1} ∩ ∏ L.

The last demand on M_{i} is the crucial one, and it is possible to satisfy it because N is tight and η^{+}-uniform. We let M = ⋃_{i} M_{i} and L^{∗} = {η^{+}} ∪ L, and note that pcf(L^{∗}) = L^{∗}. As usual we will let ⟨B_{λ} : λ ∈ L^{∗}⟩ be the <_{θ}-minimal sequence of generators for L^{∗}, ⟨f_{α}^{λ} : λ ∈ L^{∗}, α < λ⟩ the <_{θ}-least matrix of functions such that ⟨f_{α}^{λ} : α < λ⟩ is a continuous scale in ∏ B_{λ}/J_{<λ}.

By construction χ_{M_{i}}^{L^{∗}} is pointwise dominated by χ_{M_{i+1}}^{L^{∗}}, so that M is η-uniform between η^{+} and max(L)^{+}. Since |L^{∗}| < η the construction also guarantees that M is tight for L^{∗}.

By Theorem 5.3 the function χ_{M}^{L^{∗}} is the pointwise supremum of finitely many functions of the form f_{sup(M∩λ)}^{λ} ↾ (B_{λ} \ E_{λ}), where E_{λ} ∈ M. By Lemma 4.3 N ∩ max(L) is η-closed in sup(N ∩ max(L)), so in particular sup(M ∩ λ) ∈ N for all λ.

It follows that χ_{M}^{L^{∗}} ∈ N. By Lemma 4.7 we see that M ∩ max(L) ∈ N, and by construction M ∩max(L) ≺ A and M ∈ P_{η^{+}}(N ∩λ). This shows that N ∩P_{η^{+}}(N ∩λ) is stationary. □

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

Corollary 6.6. Let 2^{ℵ_{1}} = ℵ_{2} and 2^{ℵ_{ω}} = ℵ_{ω+1}. Let θ be a large regular cardinal and let N ≺ (H_{θ}, ∈, <_{θ}) be ℵ_{2}-uniform between ℵ_{2} and ℵ_{ω}, and tight for {ℵ_{n} : 2 < n < ω}. Then N ∩ H_{ℵ_{ω+1}} is IA of length and cardinality ℵ_{2}.

For structures of size ℵ_{1} we have less than satisfactory results which we illustrate with the following example:

Example 6.7. Suppose that ℵ_{ω} is a strong limit and 2^{ℵ_{ω}} = ℵ_{ω+1}. Let N ≺ H(ℵ_{ω+2}) have cardinality ℵ_{1} and uniform cofinality ω_{1}. Suppose that sup(N ∩ℵ_{ω+1}) is an approachable ordinal γ and χ_{N} =^{∗} f_{γ}. Then N is internally approachable.

To see this, let M be the internally approachable structure that has cardinality ω_{1} with χ_{M} =^{∗} f_{γ}. Then for some ω_{l}, the Skolem hull of N ∪ω_{l} and the Skolem hull of M ∪ ω_{l} have the same characteristic function and hence the two Skolem hulls have equal intersection in ℵ_{ω+1}. In particular, the Skolem hull of N ∪ ℵ_{l} is closed under ω-sequences below ℵ_{ω+1}. One can sees inductively that if Skolem hull of N ∪ ℵ_{l−i} is closed under ω-sequence then so is the Skolem hull of N ∩ ℵ_{l−(i+1)}. In particular, N is closed under ω-sequences below ℵ_{ω+1} and is thus internally approachable below ℵ_{ω+1}.

Foreman and Magidor [15] showed that there is a close connection between internal approachability, uniformity and stationary reflection for sets of structures. In particular they showed that

• Let κ be supercompact, let μ and ν be regular uncountable cardinals less than κ with μ < ν. Let G be generic for Col(ν, < κ), then in V [G] the following stationary reflection principle holds: Every stationary set of IA substructures of (H_{θ}, ∈, <_{θ}) of length and cardinality μ reflects to some substructure of cardinality ν.

• Let θ ≥ ω_{3} and let S_{ij} be the set of N ≺ (H_{θ}, ∈, <_{θ}) such that |N| = ℵ_{1},

ℵ_{1} ⊆ N, cf(N ∩ ℵ_{2}) = ℵ_{i} and cf(N ∩ ℵ_{3}) = ℵ_{j}.

Then only S_{11} can have the property that every stationary subset is reflecting.

The following result indicates that tightness is also relevant to problems about stationary reflection.

Theorem 6.8. Let K be a countable set of regular cardinals and let S be a stationary set of elementary substructures of (H_{θ}, ∈, <_{θ}), such that every element of S is tight for K. Let M be such that S reflects to M. Then M ≺ (H_{θ}, ∈, <_{θ}) and M is tight for K.

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

Note that since K is countable, K ⊆ N for all N ∈ S. Let f ∈ ∏_{κ∈K} K ∩ M, and find 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 defined a notion of tightness for countable structures, and have used this to investigate stationary reflection in [H_{λ}]^{ℵ_{0}}.

7. PCF absoluteness, covering and precipitous ideals

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

We start with a result which says that sufficiently similar universes have similar PCF structures.

Theorem 7.1. Let V be an inner model of W, and in V let K be a set of regular cardinals with |K| < min(K). Assume that for all μ ∈ [min(K), max(pcf(K))^{V}], if μ is regular in V then μ is regular in W. Assume also that for every f ∈ (∏ K)^{W} there is g ∈ (∏ K)^{V} with f < g. Then

(1) J_{<μ}^{V} = J_{<μ}^{W} ∩ V for all cardinals μ ≥ 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 J_{<μ}^{V} = J_{<μ}^{W}∩V for all μ ∈ [min(K), max(pcf(K))^{V}], by induction on μ. This is clear for μ = min(K). For λ singular we have J_{<λ} = J_{<λ^{+}} = ⋃_{ν<λ} J_{<ν}, so it remains to show that if μ is regular and J_{<μ}^{V} = J_{<μ}^{W} ∩ V then J_{<μ^{+}}^{V} = J_{<μ^{+}}^{W} ∩ V.

Suppose first that A ∈ J_{<μ^{+}}^{V}. If A ∈ J_{<μ}^{V} then we are done, otherwise we fix a scale f⃗ of length μ in (∏ A/J_{<μ})^{V}. By the induction hypothesis and our assumptions on V and W, f⃗ is a scale in (∏ A/J_{<μ})^{W} and so A ∈ J_{<μ^{+}}^{W}.

Now suppose that A ∈ J_{<μ^{+}}^{W} ∩ V. Let D ∈ V be any V -ultrafilter on K with A ∈ D, and in W extend D to D̄ a W -ultrafilter. By our assumptions on V and W the map ψ : [f]_{D} ↦ [f]D̄ is a cofinal order-preserving function from (∏ A/D)^{V} to (∏ A/D̄)^{W}. Since A ∈ J_{<μ^{+}}^{W} the W -cofinality of (∏ A/D̄)^{W} is at most μ, and the existence of ψ and our assumptions on V and W imply that the V -cofinality of (∏ A/D)^{V} is at most μ. Therefore A ∈ J_{<μ^{+}}^{V}.

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 λ such that K ∈ J_{<λ^{+}}, so that V and W agree on the value of the maximum element in pcf(K). We call this common value λ_{max}. In V we fix a sequence ⟨B_{μ}^{V} : min(K) ≤ μ ≤ λ_{max}⟩ such that B_{μ}^{V} generates J_{<μ^{+}}^{V} over J_{<μ}^{V}, and similarly we fix in W a sequence ⟨B_{μ}^{W} : min(K) ≤ μ ≤ λ_{max}⟩ such that B_{μ}^{W} generates J_{<μ^{+}}^{W} over J_{<μ}^{W}. We choose these generators so that B_{μ}^{V} = ∅ for μ ∉ pcf(K)^{V}, and similarly B_{μ}^{W} = ∅ for μ ∉ pcf(K)^{W}. We also choose B_{λ_{max}}^{V} = B_{λ_{max}}^{W} = K.

We will show by induction that B_{μ}^{W} and B_{μ}^{V} are equal modulo J_{<μ}^{W} for all μ. Since B_{μ}^{V} is in J_{<μ^{+}}^{W} and B_{μ}^{W} is a generator, we see that B_{μ}^{V} is contained in B_{μ}^{W} modulo J_{<μ}^{W} for all μ. Let λ be least such that B_{λ}^{W} and B_{λ}^{V} are unequal modulo J_{<λ}^{W}. We now adjust the B_{μ}^{W}, replacing B_{μ}^{W} by B_{μ}^{V} for μ < λ; this is legitimate by the choice of λ. Let C = B_{λ}^{W} \ B_{λ}^{V} so that C ∉ J_{<λ}^{W} but C ∈ J_{<λ^{+}}^{W}.

Now let μ be least such that C is covered by some set B_{μ_{0}}^{V} ∪ … B_{μ_{n}}^{V} ∪ B_{μ}^{V} with μ_{0} < … < μ_{n} < μ; such a μ exists because B_{λ}^{V} = K. Suppose for a contradiction that μ ≤ λ; then μ < λ since C is disjoint from B_{λ}^{V}. Now B_{ν}^{V} = B_{ν}^{W} for ν < λ, and it follows that C ∈ J_{<λ}^{W}, which is a contradiction: so μ > λ. Note that B_{μ}^{V} ≠ ∅ so that μ ∈ pcf(K)^{V}.

Let D = B_{μ_{0}}^{V} ∪ … B_{μ_{n}}^{V} ∪ B_{μ}^{V}. Working in V find ⟨g_{α} : α < μ⟩ such that g_{α} ∈ (∏ D)^{V} and ⟨g_{α} ↾ B_{μ}^{V} : α < μ⟩ is increasing and cofinal in (∏ B_{μ}^{V})^{V} modulo J_{<μ}^{V} ↾ B_{μ}^{V}. Working in W find ⟨h_{β} : β < λ⟩ such that h_{β} ∈ (∏ D)^{V} and ⟨h_{β} ↾ C : β < λ⟩ is increasing and cofinal in (∏ C)^{W} modulo J_{<λ}^{W} ↾ C.

Since μ > λ we may find h_{β} which dominates g_{α} on C modulo J_{<λ}^{W} for unboundedly many α < μ, and may then find g_{α} such that

(1) h_{β} dominates g_{α} modulo J_{<λ}^{W} ↾ C.

(2) g_{α} dominates h_{β} modulo J_{<μ}^{V} ↾ B_{μ}^{V}.

We find δ_{1}, … δ_{n} such that δ_{1} < … δ_{n} < λ and {κ ∈ C : h_{β}(κ) ≤ g_{α}(κ)} ⊆ B_{δ_{1}}^{W} ∪…∪B_{δ_{n}}^{W}. Note that B_{δ_{i}}^{W} = B_{δ_{i}}^{V}. We also find ϵ_{1}, … ϵ_{p} such that ϵ_{1} < … ϵ_{p} < μ and {κ ∈ B_{μ}^{V} : h_{β}(κ) ≥ g_{α}(κ)} ⊆ B_{ϵ_{1}}^{V} ∪ … ∪ B_{ϵ_{p}}^{V}.

We claim that C is covered by the union of the B_{μ_{i}}^{V}, B_{δ_{i}}^{V} and B_{ϵ_{i}}^{V}. To see this observe that if κ ∈ C then at least one of the following must hold:

• κ ∈ B_{μ_{i}}^{V} for some i.

• κ ∈ C ∩ B_{μ}^{V} and h_{β}(κ) ≤ g_{α}(κ), in which case κ ∈ B_{δ_{i}}^{V} for some i.

• κ ∈ C ∩ B_{μ}^{V} and h_{β}(κ) ≥ g_{α}(κ), in which case κ ∈ B_{ϵ_{i}}^{V} for some i. This contradicts the minimal choice of μ. It follows that as we claimed the sets

B_{μ}^{V} and B_{μ}^{W} agree modulo J_{μ}^{W} for all μ. In particular the B_{μ}^{V} 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_{ν} such that B_{ν} generates J_{<ν^{+}} over J_{<ν}, pcf(K) is the set of ν such that B_{ν} is not covered by a finite union of B_{μ} for μ < ν. □

A small technical difficulty is caused by the fact that the property of being the characteristic function of some structure is not obviously downwards absolute. We will resolve this difficulty 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 K be an interval of regular cardinals with |K| < μ and min(K) = μ^{+} for some regular uncountable μ. Assume that in W the cardinal μ is still regular and K is still an interval of regular cardinals. Let f ∈ V be such that for all α with μ < α < sup(K), f(α, −) is a bijection between α and |α|.

Let θ be some sufficiently large regular cardinal of W, and in W let N ≺ (H_{θ}, ∈ , <_{θ}, {K}, f) be an IA structure of length and cardinality μ. If χ_{N}^{K} ∈ V then there is B ∈ V such that N ∩ sup(K) ⊆ B ⊆ sup(K) and |B| = μ.

Proof. Since N is IA of length μ, we may fix ⟨C_{κ} : κ ∈ K⟩ lying in W with C_{κ} ⊆ N ∩ κ and C_{κ} club in sup(N ∩ κ) of order type μ. Since χ_{N}^{K} ∈ V and cofinalities agree, we may find in V a sequence ⟨D_{κ} : κ ∈ K⟩ with D_{κ} club in sup(N ∩ κ) of order type μ.

Now let M_{0} = N ∩ sup(K) and note that M_{0} ≺ (sup(K), f). Let M_{1} be the hull in (sup(K), f) of μ ∪ ⋃_{κ} D_{κ}, and note that M_{1} ∈ V and |M_{1}| = μ. The hypotheses of Lemma 4.4 are satisfied because for each κ ∈ K the set C_{κ} ∩ D_{κ} is contained in M_{0} ∩ M_{1} ∩ κ and is cofinal in M_{0} ∩ κ. It follows that M_{0} ⊆ M_{1} so we may set B = M_{1}. □

Theorem 7.3. Let V and W be inner models of set theory with V ⊆ W. In V let κ be regular and let λ > κ be a cardinal. Let K = REG^{V} ∩ [κ, λ] and suppose that V and W agree on regular cardinals in the interval [κ, max(pcf(K)^{V})], K is progressive and (∏ K)^{V} is cofinal in (∏ K)^{W}.

Let A ⊆ λ. Then there is B ∈ V such that A ⊆ B and |B|^{W} ≤ max{κ, |A|^{W} }.

Proof. We will prove this by induction on sup(A). We way assume that sup(A) > κ, since otherwise we may set B = κ + 1. For the rest of this proof cardinalities and cofinalities should be understood as computed in W, though we will tacitly use the agreement 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 find B^{∗} ⊆ | sup(A)| such that A^{∗} ⊆ B^{∗} and |B^{∗}| ≤ max{κ, |A|}, and we may then let B be the inverse image of 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 find μ regular with max{κ, |A|, |K|^{+4}} < μ < sup(A). We set K_{0} = (μ^{+}, sup(A)) ∩ REG.

Let L = pcf(K_{0}) where by Fact 6.2 |L| < μ and L = pcf(L). We fix in V a sequence B⃗ = ⟨B_{ν} : ν ∈ L⟩ of PCF generators for K_{0}, and a matrix of functions f⃗ = ⟨f_{i}^{ν} : ν ∈ L, i < ν⟩ such that ⟨f_{i}^{ν} : i < ν⟩ is a continuous scale in ∏ B_{ν}/J_{<ν}.

By Theorem 7.1, B⃗ is still a sequence of PCF generators in W. It is clear that ⟨f_{i}^{ν} : i < ν⟩ is still a scale in W, and we claim that additionally this scale is still continuous at good points of cofinality μ.

We use Fact 3.1, the Trichotomy Theorem of Shelah. Let ν ∈ L, and in W let ρ < ν be a point of cofinality μ which is good for the scale ⟨f_{i}^{ν} : i < ν⟩. We begin by arguing that in V there must exist an eub for ⟨f_{i}^{ν} : i < ρ⟩ whose values have cofinality at least κ.

If this is not the case then in V we must be either in Case 2 or Case 3 from Fact 3.1. It is easy to see that the properties of being in Case 2 or Case 3 are upwards absolute from V to W, in fact the witnesses from V will work in W as long as we extend the ultrafilter in V for Case 2 to an ultrafilter in W. Since we have an eub for ⟨f_{i}^{ν} : i < ρ⟩ of uniform cofinality μ in W, it follows from Fact 3.1 that in W we are not in Case 2 or Case 3, so that in V there is an eub for ⟨f_{i}^{ν} : i < ρ⟩ whose values have cofinality at least κ.

By continuity in V, f_{ρ}^{ν} is an eub whose values have cofinality at least κ almost everywhere in V. By the hypothesis on the resemblance between V and W, f_{ρ}^{ν} retains these properties in W. It follows that the scale ⟨f_{i}^{ν} : i < ν⟩ is continuous at ρ.

We also fix in V a function f from sup(K_{0})^{2} to sup(K_{0}) coding some information about cardinalities, as in Lemma 7.2.

We now build N ≺ (H_{θ}, ∈, <_{θ}, {K_{0}}, f) where A ⊆ N and N is IA of length and cardinality μ. By theorem 5.3 χ_{N}^{K_{0}} is the pointwise supremum of finitely many functions of the form f_{i}^{ν} ↾ (B_{ν} \ E_{ν}) where E_{ν} is the union of a finite subset of {B_{ζ} : ζ < ν}.

It follows that χ_{N}^{K_{0}} ∈ V. The hypotheses of Lemma 7.2 are satisfied so we may find C in V such that |C| = μ < sup(A) and A ⊆ C. Now let g ∈ V be a bijection between C and μ, and let A^{∗} be the image of A under g. Now we may use the induction hypothesis to cover A^{∗} by a suitable B^{∗}, and then pull back along g to get a suitable B covering A. □

Remark 7.4. Theorem 7.3 is actually an equivalence. The covering statement in the conclusion implies that V and W agree on regular cardinals between κ and λ, and also implies that (∏ K)^{V} is cofinal in (∏ K)^{W}.

In order to be able to state the next result in a compact way, we make the following ad hoc definition.

Definition 7.5. Let V and W be inner models of set theory with V ⊆ W. In V, let κ be regular and let λ > κ be a cardinal. We say that W weakly resembles V on [κ, λ) if and only if the hypotheses of Theorem 7.3 are satisfied.

For background on the theory of precipitous ideals we refer the reader to Foreman’s survey paper [14]. The following result belongs to a genre of theorems in which we are given an ideal I on a cardinal κ and some information about preservation of cardinals when forcing with Pκ/I, and we conclude that I must be precipitous. This line of inquiry was begun by Baumgartner and Taylor[2], who proved for example that under GCH a countably closed ideal on ℵ_{1} whose quotient algebra preserves ℵ_{2} is necessarily precipitous.

Theorem 7.6. Let I be a countably complete ideal on ℵ_{1}. Let λ = 2^{ℵ_{1}}, and suppose it is forced by Pℵ_{1}/I that V [G] weakly resembles V on [ℵ_{2}, λ). Then I is precipitous.

Proof. Let G be a generic ultrafilter on (Pℵ_{1})^{V} and let M = Ult(V, G). We observe that by our hypotheses ℵ_{2}^{V} = ℵ_{1}^{V [G]} and recall the standard fact that ℵ_{2}^{V} is an initial segment of the well-founded part of M. It must be the case that ℵ_{2}^{V} = ℵ_{1}^{M}, for if not then in M there is a surjection from ω onto some larger M-ordinal, contradicting the fact that ℵ_{2}^{V} = ℵ_{1}^{V [G]}.

Suppose for a contradiction that M is not well-founded, and in V [G] choose a sequence ⟨f_{i} : i ∈ ω⟩ such that f_{i} ∈ V, f_{i} : ℵ_{1}^{V} ⟶ ON and f_{i+1} <_{G} f_{i} for all i. Since Pℵ_{1}/I has the λ^{+}-c.c. we may find C ∈ V such that |C| ≤ λ and range(f_{i}) ⊆ C for all i.

By Theorem 7.3 we may find B ⊆ C such that B ∈ V, |B|^{V} = ℵ_{2}^{V} and range(f_{i}) ⊆ B for all i. Working in V, we write B = ⋃_{j<ℵ_{2}} B_{j} where the B_{j} are increasing and |B_{j}| = ℵ_{1}. In V [G] this gives a representation of B as an increasing union of countable sets. Since ⋃_{i} range(f_{i}) is countable in V [G] we can fix a j such that range(f_{i}) ⊆ B_{j} for all i. Now let γ be the order type of B_{j}, let h : B_{j} ≃ γ be order preserving and let f_{i}^{∗} = h ∘ f_{i}. Then ⟨f_{i}^{∗} : i ∈ ω⟩ is a G-decreasing sequence of functions from ℵ_{1} to γ, and so to get a contradiction we need only show that j_{G}(γ) is well-founded for all γ < ℵ_{2}^{V}.

Suppose for a contradiction that j_{G}(γ) is ill-founded, and choose a bijection F ∈ M between ℵ_{1}^{M} and j_{G}(γ).

Since ℵ_{1}^{M} = ℵ_{1}^{V [G]}, we may find δ < ℵ_{1}^{V [G]} such that in V [G] the set F “δ contains an infinite decreasing sequence of M-ordinals.

Working inV [G], let T be the tree of all finite sequences (α_{0}, … α_{i}) from δ such that F (α_{i}) > F (α_{i+1}) for all i. The tree T is in the well-founded part of M. Clearly T is 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 ρ : T ⟶ ℵ_{1}^{M}. Since ℵ_{1}^{M} 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. □

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