PCF Theory and the Tukey Spectrum

In this paper, we investigate the relationship between the Tukey order and PCF theory, as applied to sets of regular cardinals. We show that it is consistent that for all sets $A$ of regular cardinals that the Tukey spectrum of $A$, denoted $\operatorname{spec}(A)$, is equal to the set of possible cofinalities of $A$, denoted $\operatorname{pcf}(A)$; this is to be read in light of the $\mathsf{ZFC}$ fact that $\operatorname{pcf}(A)\subseteq\operatorname{spec}(A)$ holds for all $A$. We also prove results about when regular limit cardinals must be in the Tukey spectrum or must be out of the Tukey spectrum of some $A$, and we show the relevance of these for forcings which might separate $\operatorname{spec}(A)$ from $\operatorname{pcf}(A)$. Finally, we show that the strong part of the Tukey spectrum can be used in place of PCF-theoretic scales to lift the existence of Jonsson algebras from below a singular to hold at its successor. We close with a list of questions.


Introduction
The Tukey order has become a very useful tool for comparing directed, partially ordered sets.The order is sufficiently coarse that is allows us to compare many different partial orders, yet it is fine enough to preserve a number of order-theoretic properties of interest (such as calibre properties; see Proposition 2.4 below).The Tukey order works by comparing partial orders in terms of what happens "eventually", or more precisely, in terms of what happens cofinally.
The Tukey order arose in the study of Moore-Smith convergence in topology ( [21] and [35]), with [2] and [5] following shortly after.Schmidt ( [25]) and Isbell ([15] and [16]) later studied cofinal types among the class of directed posets.Later, Todorčević ([32]) showed that it is consistent that there are only five cofinal types of directed sets of size ≤ ℵ 1 .Todorčević also showed that under the CH, there are 2 c -many such cofinal types, and in [34] he extended this result to all transitive relations on ω 1 .
Cofinal structure has been studied by set theorists coming from a different angle.Especially important for us is Shelah's theory of possible cofinalities, or PCF theory for short.The main objects of study in PCF theory are reduced products of sets of regular cardinals, modulo an ideal.Shelah has developed the theory in a series of papers which culminated in the book [29].PCF theory has had dramatic implications for our understanding of cardinal arithmetic (see [1]), as well as plenty of applications both inside and outside of set theory, such as [17].See [28] for a discussion of further applications.
These two ways of studying cofinal structure are related (and we will discuss this more later): given a set A of regular cardinals, we consider the Tukey spectrum of A, which consists of all regular cardinals which are Tukey below ( A, <) (i.e., A with the pointwise domination ordering).We denote this by spec(A).It follows quickly from the definitions (which we give later) that for any set A of regular cardinals, pcf(A) ⊆ spec(A).In this paper, we are concerned with the following general question: Question 1.1.Does ZFC prove that for any set A of regular cardinals, pcf(A) = spec(A)?
To our knowledge, the only result, so far, which addresses this question is due to Gartside and Mamatelashvili ( [11]) who have a proof showing that if A is any progressive set of regular cardinals, then pcf(A) = spec(A) ("progressive" is a common assumption when doing PCF theory).However, there is a gap in their proof.We address this gap later, observing that their argument rather shows that if A is progressive, then spec(A) ⊆ pcf(A) ∪ lim(pcf(A)). 1 Additional assumptions on pcf(A) then guarantee equality.However, the status of Question 1.1 when A is not progressive is far from clear.
In this work, we prove various results related to Question 1.1.After a review of the basics of the Tukey order and PCF theory in Section 2, we turn in Section 3 to the question of how much bigger spec(A) can be than pcf(A).We review the theorem from [11] and address the gap in their proof.Then we turn to showing that Question 1.1 has a consistent positive answer.We also discuss circumstances under which, for all A, spec(A) is no worse than pcf(A) ∪ lim(A).In Section 4 we address the role that small large cardinals (Mahlo and weakly compact) have in excluding a regular limit κ from spec(A) (where A ⊆ κ) or for ensuring that κ ∈ spec(A).The upshot of these results is that they may reduce the options for showing that Question 1.1 has a consistent negative answer which is witnessed by a forcing separating spec(A) and pcf(A) (if such exists).In the last main section, Section 5, we show that a subset of the Tukey spectrum (what we call the "strong 1 Since spec(A) consists, by definition, of regular cardinals, if A is progressive and spec(A) = pcf(A), then pcf(A) has a regular limit point.
part" of the Tukey spectrum) is sufficiently strong to be able to "lift" the existence of Jónsson algebras; this generalizes Shelah's celebrated result [27] that scales in PCF theory can lift the existence of Jónsson algebras.

Acknowledgements
We would like to thank Will Brian, James Cummings, Todd Eisworth, and Paul Gartside for many helpful conversations about the Tukey order and PCF theory and for suggesting ways of extending this line of research.

A quick overview of Tukey and PCF
In this section we review the basics of the Tukey order and PCF theory which are relevant for this paper.
Remark 2.1.Throughout the paper, all posets are assumed to be directed.
First we recall the definition of the Tukey order (see [12] for a detailed development of these ideas).Definition 2.2.A poset Q is said to be a Tukey quotient of P if there exists a function ϕ : P −→ Q which preserves cofinal sets.We denote this by P ≥ T Q. P ≥ T Q is equivalent to the existence of a map ψ : Q −→ P which preserves unbounded sets.Definition 2.3.Suppose that κ ≥ λ ≥ µ are cardinals.We say that a poset P has calibre (κ, λ, µ) if for all κ-sized P ⊆ P there is a λ-sized R ⊆ P so that every µ-sized B ⊆ R is bounded in P .
We say simply that P has calibre κ if P has calibre (κ, κ, κ).
Note that the definition of P having calibre κ simplifies to the following: every κ-sized P ⊆ P has a κ-sized subset which is bounded in P .
The following item connects the ideas of ≥ T and calibre.
Proposition 2.4.Suppose that κ is regular, that P has calibre (κ, λ, µ), The next item is particularly relevant for us.
Proposition 2.5.For a regular cardinal κ, a poset P fails to have calibre κ iff P ≥ T κ.
Now we define the Tukey spectrum of a poset.
Definition 2.6.The Tukey spectrum of a poset P is denoted by spec(P ) and defined to be spec(P ) := {κ : P ≥ T κ ∧ κ is regular}.When A is a set of regular cardinals, we let spec(A) abbreviate spec ( A, <).
Remark 2.7.spec(P ) consists of all regular κ so that P does not have calibre κ.
It is helpful to get a better handle on what κ ∈ spec(A) means in the specific case that A is a set of regular cardinals.Indeed, κ ∈ spec(A) iff there exists a set F of κ-many functions in A so that every F 0 ∈ [F] κ is unbounded in ( A, <), i.e., A with the pointwise domination ordering.That is to say, there is at least one coordinate a ∈ A so that {f (a) : f ∈ F 0 } is unbounded in the regular cardinal a.We give a name to these coordinates in the next definition.
We let ub(F) denote the set of unbounded coordinates of F.
We address the question of how many coordinates are in ub(F) in Section 5.
The following lemma is a standard part of Tukey-ology.Lemma 2.9.spec(P × Q) = spec(P ) ∪ spec(Q).Hence if A and B are sets of regular cardinals, spec(A ∪ B) = spec(A) ∪ spec(B).This captures all of the basics of the Tukey order that we need.We now review some of the central results in PCF theory, beginning with the central definition (see [1] for a clear and detailed exposition of these and related ideas).Definition 2.10.Let A be a set of regular cardinals.

pcf(A) := cf
A/D : D is an ultrafilter on A .
The following are routine facts about the pcf function.
Fact 2.11.Suppose that A and B are sets of regular cardinals.
(2) If A ⊆ B, then pcf(A) ⊆ pcf(B) (since any ultrafilter on A can be extended to one on B). (3) pcf(A∪B) = pcf(A)∪pcf(B) (since any ultrafilter on A∪B contains either A or B).
The next lemma follows almost immediately from the definitions: Lemma 2.12.For any set A of regular cardinals, pcf(A) ⊆ spec(A).
A very useful assumption when doing PCF theory is the following: We next define certain ideals which are naturally associated to the cardinals in pcf(A).A being progressive plays an important role in the development of the ideas that follow.Definition 2.14.Let A be a set of regular cardinals and λ a cardinal (singular or regular).Define the ideal J <λ [A] to consist of all B ⊆ A so that for any ultrafilter D on A with B ∈ D, cf( A/D) < λ.
If the set A is clear from context, we write J <λ instead of J <λ [A].A crucial fact about the ideal J <λ is the following: Proposition 2.15.Suppose that A is progressive.Then for any cardinal λ, A/J <λ is < λ-directed (that is, any set of fewer than λ-many functions in A has an upper bound mod J <λ ).
A major theorem in PCF theory is the existence of generators, which we define now.Definition 2.16.Let A be a set of regular cardinals and λ ∈ pcf(A).A generator for λ is a set Thus a generating set B λ is a maximal set in J <λ + , modulo J <λ ; they are unique modulo J <λ .The following is often proven using universal cofinal sequences: Proposition 2.17.Suppose that A is progressive.Then for any λ ∈ pcf(A), there is a generator for λ.

Generators give an instance of compactness:
Proposition 2.18.Suppose that A is progressive and B ⊆ A. Let B λ : λ ∈ pcf(A) be a sequence of generators.Then there exists a finite decreasing sequence Another application of generators and related ideas is the following fact: Proposition 2.19.Suppose that A is progressive.Then pcf(A) has a maximum element, and moreover, The last collection of background facts to review concerns weak compactness.
Definition 2.20.Let α be an inaccessible cardinal.We say that a transitive set M is an α-model if M |= ZFC − , M has size α, M is closed under < αsequences, and α ∈ M .Now we recall the following characterization of weak compactness (see [14]): Fact 2.21.An inaccessible cardinal κ is weakly compact if and only if for any κ-model M , there exist a κ-model N and an elementary embedding j : M → N with crit(j) = κ.
In the context of the previous fact, note that j : M → N gives rise to an M -normal ultrafilter

How bad can Spec be?
In this section, we study various conditions which guarantee that either (a) spec(A) is no worse than pcf(A) together with (regular) limit points of pcf(A) or (b) spec(A) is no worse than pcf(A) together with regular limit points of A. These results, when coupled with anti large cardinal hypotheses, give a consistent positive answer to Question 1.1.
We begin by looking at the theorem from [11] which we mentioned in the introduction.Addressing a small gap in their argument, what their proof shows is that for any progressive A, spec(A) can at worst add regular limits of pcf(A).Additional assumptions on pcf(A) then give the equality of pcf(A) and spec(A).We first have some notation.Notation 3.1.Given functions g 0 , . . ., g n in some product A of regular cardinals, we let max(g 0 , . . ., g n ) be the function h ∈ A defined by h(a) := max {g 0 (a), . . ., g n (a)} .Theorem 3.2.(Almost entirely [11]) Suppose that A is a progressive set of regular cardinals.Then spec(A) ⊆ pcf(A) ∪ lim(pcf(A)). 2n particular, if pcf(A) is closed under regular limits, or if pcf(A) has no regular limits (for example, if pcf(A) is itself progressive), then pcf(A) = spec(A).
Proof.We begin with some set-up.Since A is progressive, we may apply Proposition 2.17 to fix a sequence B λ : λ ∈ pcf(A) of generators for pcf(A).For each λ ∈ pcf(A), B λ is progressive, being a subset of the progressive set A. Thus λ = maxpcf(B λ ) = cf ( B λ , <), using B λ ∈ J <λ + \ J <λ for the first equality and applying Proposition 2.19 for the second equality.Therefore, for each λ ∈ pcf(A), we may choose a sequence ). Fix a cardinal κ ∈ spec(A), suppose that κ is not a limit point of pcf(A), and we will show that κ ∈ pcf(A).Because κ is not a limit point of pcf(A), we know that | pcf(A) ∩ κ| < κ; this will permit us to apply a pigeonhole argument at a crucial step later in the proof.
Since κ ∈ spec(A), let f γ : γ < κ enumerate a set F of κ-many functions in A so that every κ-sized subset of F is unbounded in ( A, <).To show that κ ∈ pcf(A), it suffices to show that J <κ = J <κ + .Since F is bounded in A/J <κ + (because this reduced product is < κ + -directed, by Proposition 2.15), it in turn suffices to show that F is unbounded in A/J <κ .
We work by contradiction and suppose that F is bounded in A/J <κ .Our goal is to define a set X of fewer than κ-many functions in A so that every f ∈ F is pointwise below some g ∈ X .Supposing we can define such an X , we obtain a contradiction as follows: since κ is regular, there is a single g ∈ X which is pointwise above κ-many elements of F, and this contradicts the fact that F witnesses that κ ∈ spec(A).
We begin the construction of X as follows.Since F is bounded in A/J <κ , let g be a bound.For each n ∈ ω, each sequence λ = λ 0 > • • • > λ n in pcf(A) ∩ κ, and each sequence α = α 0 , . . ., α n with α i ∈ λ i for all i ≤ n, we let h( α, λ) be the function max(f λ 0 α 0 , . . ., f λn αn , g); see Notation 3.1.We let X be the set of such h.Since by assumption pcf(A) ∩ κ is bounded below κ, we see that X consists of fewer than κ-many functions.Now we verify that X has the desired property.Recalling that f α : α < κ enumerates F, fix some α < κ.Since f α < J<κ g, we know that is in J <κ .Thus max pcf(z α ) < κ, and so by Proposition 2.18, we may find a sequence , this holds pointwise on B λ i ).Then we see that . On the other hand, if a ∈ z α , there is some i ≤ n so that a ∈ B λ i , and hence f α (a) < f λ i α i (a).Since the function max(f λ 0 α 0 , . . ., f λn αn , g) is in X , this completes the main part of the proof.For the "in particular" part of the theorem, observe that if pcf(A) is progressive, then there are no regular limit κ in lim(pcf(A)).Otherwise, In the rest of this section, we work to isolate a condition (Theorem 3.5) which is consistent with ZFC and which implies that spec(A) ⊆ pcf(A) ∪ lim(pcf(A)) for all A. In fact, it implies the stronger result that spec(A) ⊆ pcf(A) ∪ lim(A) for all A. We have some preliminary lemmas.The first of these illustrates a "dropping" phenomenon in the Tukey spectrum; we will use this lemma as part of a later inductive argument.Lemma 3.3.Suppose that A is a set of regular cardinals.Let κ ∈ spec(A)∩ sup(A).Then κ ∈ spec(A ∩ (κ + 1)).
Proof.Set µ := sup(A); note that we are making no assumption about whether or not µ ∈ A. Since κ ∈ spec(A), let F be a set of κ-many functions in A so that for all F 0 ∈ [F] κ , F 0 is unbounded in ( A, <).Let Note that since κ < µ = sup(A), A \ (κ + 1) is a non-empty set of regular cardinals.Also, note that F >κ is bounded in (A \ (κ + 1), <), since F >κ consists of at most κ-many functions, and we can take a sup of κ-many elements on each coordinate in A \ (κ + 1).We next argue that F ≤κ has size κ and that every κ-sized subset is unbounded in ( A ∩ (κ + 1), <); this will show the desired result.First suppose for a contradiction that F ≤κ has size < κ.Then there is a single f ∈ A ∩ (κ + 1) so that for κ-many g ∈ F, g (κ + 1) = f .Let F 0 be this set of g ∈ F. But then F 0 is bounded in the entire product ( A, <), using the observation from the previous paragraph to bound the elements of F 0 on coordinates in A above κ.A nearly identical argument shows that F ≤κ must satisfy that every κ-sized subset is unbounded in ( A ∩ (κ + 1), <).
The next lemma will also be used in the proof of Theorem 3.5: Lemma 3.4.Suppose that κ is a regular limit cardinal.Let A ⊆ κ be a nonstationary set of regular cardinals which is unbounded in κ, and let D be an ultrafilter on A which extends the tail filter on A. Then cf( A/D) ≥ κ + .Proof.To begin, let µ ν : ν < κ be the increasing enumeration of A, and let ζ i : i < κ enumerate a club C ⊆ κ with C ∩ A = ∅.By relabeling if necessary, we take ζ 0 = 0.
Suppose for a contradiction that cf( A/D) ≤ κ; then the cofinality must equal exactly κ since D extends the tail filter on A. Let f = f α : α < κ be a sequence of functions in A which is increasing and cofinal modulo D. We obtain our contradiction by showing that f is bounded in A, modulo the tail filter on A, and hence modulo D.
We finish the proof by showing that g bounds each of the f ξ on a tail.To this end, fix α < κ, and let i < κ so that α ≤ ζ i .We claim that for all where the last inequality follows since α ≤ ζ i ≤ ζ j .
Theorem 3.5.Let V be a model of ZFC in which 2 µ = µ + for all limit cardinals µ and in which there are no Mahlo cardinals.Then V satisfies that for any set A of regular cardinals, spec(A) ⊆ pcf(A) ∪ lim(A).Consequently, if 2 µ = µ + for all singular µ and if there are no regular limit cardinals, then for all A, spec(A) = pcf(A).
Proof.Fix such a V (for example, work in L up to the first λ which is Mahlo in L, if such a λ exists, and otherwise work in all of L).Let A be a set of regular cardinals, and we will prove the result by induction on the order type of A.
We first dispense with the case when max(A) exists.Let λ := max(A).Then, applying Lemma 2.9 and Fact 2.11, as well as our inductive assumption, we get Now we suppose that A does not have a max.Let µ := sup(A), and fix κ ∈ spec(A).We have a few cases on κ.
First suppose that κ < µ.Then by Lemma 3.3, κ ∈ spec(A ∩ (κ + 1)).Since the order type of Ā := A ∩ (κ + 1) is less than the order type of A, we apply the induction hypothesis to conclude that Now suppose that κ ≥ µ.If κ = µ (and in particular, µ is a regular limit cardinal), then because A is unbounded in µ, κ = µ ∈ lim(A).
The final case is that κ > µ (here µ may be either regular or singular).Since µ is a limit cardinal, our cardinal arithmetic assumption implies that A has size µ µ = µ + .Hence no cardinal greater than µ + is in spec(A) or in pcf(A).
To finish the proof in this final case, we will argue that µ + ∈ pcf(A).Towards this end, let D be an ultrafilter on A which extends the tail filter on A. Then cf( A/D) ≥ µ = sup(A).If µ is singular, then the regular cardinal cf( A/D) is greater than µ.On the other hand, if µ is regular, then µ is not a Mahlo cardinal by assumption.This in turn, with the help of Lemma 3.4, implies that cf( A/D) > µ.Thus in either case on µ, cf( A/D) > µ.This cofinality must be exactly µ + , however, since A has size µ + .
For the "consequently" part of the theorem, recall that spec(A) consists, by definition, of regular cardinals.Thus if there are no regular limit cardinals, lim(A) contains no regular cardinals, and this implies that spec(A) ⊆ pcf(A).By Lemma 2.12, we conclude that pcf(A) = spec(A).
Thus Question 1.1 has a consistent positive answer.

Small Large Cardinals and the Tukey Spectrum
In this section, we prove some results showing the relationship between certain small large cardinals (Mahlo and weakly compact) and the Tukey spectrum.The first of this gives a sufficient condition for including a regular limit cardinal in the Tukey spectrum.After this, we prove Theorem 4.2 which gives a sufficient condition for excluding a cardinal from spec(A).After the proof of Theorem 4.2, we comment on applications.Proposition 4.1.Suppose that κ is a Mahlo cardinal and that A ⊆ κ is any stationary set of regular cardinals.Then κ ∈ spec(A).
Proof.We show that functions which are constant on a tail witness the result.For each α < κ, let f α be the function in A which takes value 0 on all a ∈ A with a ≤ α, and which takes value α on all a ∈ A with α < a.We claim that the sequence f α : α < κ enumerates a set which witnesses that κ ∈ spec(A).
Towards this end, let X ∈ [κ] κ , and we will show that f α : α ∈ X is unbounded in ( A, <).Since A is stationary, we may find some a ∈ A ∩ lim(X).Then for all α ∈ X ∩ a, f α (a) = α.Since a is a limit point of X, we have that {f α (a) : α ∈ X ∩ a} is cofinal in a.This completes the proof.
The next result shows that we can use weak compactness to exclude a regular limit κ from spec(A), for certain A. Theorem 4.2.Suppose that κ is weakly compact and that A ⊆ κ is a non-stationary set of regular cardinals which is unbounded in κ.Then κ / ∈ spec(A).
Therefore, if κ is weakly compact and A ⊆ κ is an unbounded set of regular cardinals, κ ∈ spec(A) iff A is stationary.
Proof.Let C ⊆ κ be a club with C ∩A = ∅.Enumerate C in increasing order as ζ i : i < κ , where we assume ζ 0 = 0. Also, enumerate A in increasing order as µ i : i < κ .As in the proof of Lemma 3.4, we have that for each i < κ, µ Next, let f α : α < κ be an enumeration of a set F of κ-many functions in A, and we will show that F does not witness that κ ∈ spec(A).Fix a κ-model M (see Definition 2.20) which contains A, C, and f α : α < κ as elements.By the weak compactness of κ, let U be an M -normal ultrafilter on P(κ) ∩ M .
Our strategy is to use U to successively freeze out longer and longer initial segments of many functions on the sequence f α : α < κ .We will then bound their tails using the non-stationarity of A.
For each i < κ, the product is a member of M , and hence a subset of M .Since κ is strongly inaccessible, this product has size < κ.Applying the fact that U is a κ-complete ultrafilter on P(κ) ∩ M , we may find a function Next, define an increasing sequence β j : j < κ below κ so that for all j < κ, This also uses the completeness of U to see that for each j < κ, i<j Z i is in U and has size κ.
As a result of this freezing out, we have the following: for a fixed i < κ, and all j > i, i.e., all values of all of the f β j on the column µ ν ∈ A. By applying the argument in the previous paragraph, we conclude that R(ν) in fact equals is finite, and hence has size smaller than µ ν (which is, after all, an infinite cardinal).On the other hand, if i is infinite, then But by definition of R(ν), this means that for all ν < κ, f β j (µ ν ) : j < κ is bounded in µ ν .Thus f β j : j < κ enumerates a set of κ-many functions from F which is bounded in ( A, <).Since F was arbitrary, this shows that κ / ∈ spec(A).
For the final statement of the theorem, note that if A ⊆ κ is nonstationary, then since κ is Mahlo, Proposition 4.1 shows that κ / ∈ spec(A).
Note that the converse of the above theorem may fail, since a regular limit cardinal which is not weakly compact may also fail to be in spec(A): Corollary 4.3.Suppose that κ is weakly compact and that A ⊆ κ is a nonstationary set of regular cardinals unbounded in κ.Let P := Add(ω, κ), the poset to add κ-many Cohen subsets of ω.Then P forces that κ / ∈ spec(A).
Proof.Let ḟα : α < κ be a sequence of P-names for elements of A. We will find an X ∈ [κ] κ in V so that P forces that ḟα : α ∈ X is bounded.Indeed, using the c.c.c. of P, for each α < κ, we may find a function Namely, let ϕ α (a) be above the sup of the countably-many γ ∈ a so that γ is forced to be the value of ḟα (a) by some condition in P.
By the Theorem 4.2, let X ∈ [κ] κ so that ϕ α : α ∈ X is bounded in the product ( A, <), say with h as a bound.Then P forces that for each α ∈ X, ḟα is pointwise below h.
We conclude this section with a discussion of a promising suggestion of James Cummings about separating pcf(A) and spec(A).Given that pcf(A) ⊆ spec(A) always holds, we'd like to create a forcing extension in which, for some A, there is a cardinal κ ∈ spec(A) \ pcf(A).
The strategy is to start with a cardinal κ which is at least Mahlo.Then let A = {µ + : µ < κ} and attempt to force the existence of a set F of κ-many functions in A which witnesses that κ ∈ spec(A).This strategy appears promising due to the next observation.Lemma 4.4.Suppose that κ is strongly inaccessible and that A ⊆ κ is a nonstationary set of regular cardinals unbounded in κ.Then κ / ∈ pcf(A).
Proof.If D is an ultrafilter on A that concentrates on a bounded subset of A, then the strong inaccessibility of κ implies that cf( A/D) < κ.
On the other hand, if D extends the tail filter on κ, then by Lemma 3.4, cf( A/D) > κ.
While the above strategy is natural, problems remain.First, natural Easton-style forcings to add a witness to κ ∈ spec(A) seem either to fail to add such a witness, or seem to change the Mahlo κ into a weakly, nonstrongly inaccessible cardinal, i.e., they increase the continuum function below κ to take values at or above κ.Thus the crucial assumption of Lemma 4.4 fails.Or phrased differently, ultrafilters which concentrate on bounded subsets may give rise to reduced products with very high cofinality.
Moreover, Theorem 4.2 provides another obstacle: if a forcing P places κ inside spec(A) \ pcf(A) for some A which is non-stationary and unbounded in κ, then one of two things needs to happen.Either κ starts off as nonweakly compact (and this assumption plays a role in the argument) or P must ensure that κ loses its weak compactness.

The Strong Part of the Tukey Spectrum
In this section we introduce the idea of the strong part of the Tukey spectrum, and then we will show how this idea can be used in place of scales to lift the property of not being a Jónsson cardinal.Recall the notation ub(F) from Definition 2.8.
First we make a simple observation about having infinitely-many unbounded coordinates.
Lemma 5.1.Let A be a set of regular cardinals without a max, and let κ ∈ spec(A) with κ ≥ sup(A).Let F be any witness that κ ∈ spec(A).Then for all F 0 ∈ [F] κ , ub(F 0 ) is infinite.
Proof.Suppose otherwise, with F 0 as a counterexample.Then since ub(F 0 ) is finite and has a max below κ, ub(F 0 ) has size below κ.Let F 1 ∈ [F 0 ] κ so that the function f ∈ F 1 → f ub(F 0 ) is constant, say with value f .Then we can bound all of F 1 in the entire product A using f on the coordinates in ub(F 0 ) ⊇ ub(F 1 ).
Of course, ub(F) can very well be finite, or even a singleton, for instance, if κ is a member of A.
We want to isolate cases in which there are plenty of unbounded coordinates.This leads to the next definition.Definition 5.2.Suppose that A is a set of regular cardinals.The strong part of the Tukey spectrum of A, denoted spec * (A), consists of all regular λ satisfying the following: there is a set F ⊆ A of size λ, so that for every Thus λ ∈ spec * (A) iff there is a witness F to λ ∈ spec(A) with the additional property that every λ-sized subset has unboundedly-many unbounded coordinates.
Observe that if A is a set of regular cardinals without a max, then spec * (A) ∩ sup(A) = ∅.Indeed, if λ < sup(A) and F ⊆ A has size λ, then we can bound F on all coordinates in A \ λ + .
Under cardinal arithmetic assumptions, it is easy to see that every λ ∈ spec(A) \ sup(A) is in spec * (A): Lemma 5.3.Suppose that A is a set of regular cardinals with no max and that sup(A) is a strong limit cardinal (regular or singular).Then spec(A) \ sup(A) ⊆ spec * (A).
Proof.Fix λ ∈ spec(A) at least as large as sup(A).It suffices to show that if F is any witness that λ ∈ spec(A), then ub(F) is unbounded.Fix δ ∈ A. Then the product (A ∩ (δ + 1)) has size below sup(A).Thus there is F 0 ∈ [F] λ so that the function taking f ∈ F 0 to f (A ∩ (δ + 1)) is constant on F 0 .Since F witnesses that λ ∈ spec(A), we must have that ub(F 0 ) is non-empty.Let δ * be the least element of ub(F 0 ), and note that δ * > δ, since we froze out the values of the functions in F 0 on A ∩ (δ + 1).next item is almost exactly Lemma 5.6 from [9]; we have added a parameter to the statement, which does not change the proof.In the statement of the lemma, < χ denotes a wellorder of H(χ).

Shelah ([27]
) has proven the following remarkable theorem: Theorem 5.9.(Shelah) Suppose that µ is singular and that ( µ, f ) is a scale (modulo the ideal of bounded sets) of length µ + .Additionally, suppose that each µ i carries a Jónsson algebra (i.e., is not a Jónsson cardinal).Then µ + carries a Jónsson algebra.
Here we show that it suffices to assume that µ + is in the strong part of the Tukey spectrum of A, provided the order type of A is not too high.Theorem 5.10.Suppose that A is a set of regular cardinals with ot(A) < µ := sup(A) so that every a ∈ A carries a Jónsson algebra, and suppose that µ + ∈ spec * (A).Then µ + carries a Jónsson algebra.
Proof.We will show that µ + carries a Jónsson algebra by showing that (2) of Lemma 5.8 is false.
Applying the elementarity of M , we may find a set F ⊆ A of functions witnessing µ + ∈ spec * (A) with F ∈ M .Using F, we will show that there are unboundedly-many a ∈ A so that |M ∩ a| = a.The upshot of this is that for each such a, since a is not a Jónsson cardinal, a ⊆ M .Since there are unboundedly-many such a, we conclude that sup(A) = µ ⊆ M .And finally, since |M ∩ µ + | = µ + , we can conclude that µ + ⊆ M .
To show the existence of unboundedly-many such a, let Now let a ∈ ub(F M ).Then for all f ∈ F M , f (a) ∈ M , since f and a are each members of M .Since {f (a) : f ∈ F M } is unbounded in a (by definition of a being an unbounded coordinate) and a subset of M , we conclude that M ∩ a has size a.This completes the proof.
We close this section by providing a bound on the strong part of the Tukey spectrum.First note that it follows almost immediately from the definitions that sup(spec(A)) ≤ cf( A, <).Now let J bd denote the ideal of bounded sets on A. Note that A/J bd does have a cofinality, but it needn't have a true cofinality (i.e., a linearly-ordered, cofinal subset).
Proposition 5.11.Let A be a set of regular cardinals, and let λ ∈ spec(A).
Then either λ ≤ cf( A/J bd ) or λ ∈ spec( Ā) for some proper initial segment Ā of A.
Thus F 0 := {h ξ : ξ ∈ Y } is bounded on every coordinate in A \ ā.Since F 0 ∈ [F] λ , F 0 is unbounded in ( A, <).From this, one can argue that {f ξ (A ∩ ā) : ξ ∈ Y } has size λ and witnesses that λ ∈ spec(A ∩ ā).For the "in particular" part of the proposition, note that if λ > cf( A/J bd ), then the previous argument shows that there is an F 0 ∈ [F] λ so that ub(F 0 ) is bounded in A.

Questions
Here we record a few questions which we find interesting.The first question restates Question 1.1, the main one driving this line of research: Question 6.1.Does ZFC prove that for all sets A of regular cardinals, pcf(A) = spec(A)?
A restricted version of Question 6.1, to be read in light of Theorem 3.2, is this: Question 6.2.Does ZFC prove that for all progressive sets A of regular cardinals, pcf(A) = spec(A)?
One can also ask about the relationship between spec * (A) and pcf(A), as in the next two questions: Question 6.3.Does ZFC prove that spec * (A) ⊆ pcf(A)?Question 6.4.Does ZFC prove that if A is a set of regular cardinals without a max, then spec(A) \ sup(A) ⊆ spec * (A)?
The following question should be read in light of Theorem 3.5: Question 6.5.Does ZFC prove that for all sets A of regular cardinals, spec(A) ⊆ pcf(A) ∪ lim(pcf(A))?
The next question connects to Theorem 4.2 and Corollary 4.3: Question 6.6.Is a weakly compact necessary to get a model in which κ is a regular limit, A ⊆ κ is unbounded and non-stationary, and κ / ∈ spec(A)?
Theorem 5.10 showed that the strong part of the Tukey spectrum can be used in place of PCF-theoretic scales to lift the property of not being a Jónsson cardinal.Where else, if at all, can spec * (A) be used in this way?In particular, we ask whether the strong part of the Tukey spectrum is enough to generalize a classic result of Todorcevic ( [33]; see the treatments in [3] and [9]).Question 6.7.Suppose that A is a set of regular cardinals cofinal in a singular µ so that every κ ∈ A fails to satisfy κ → [κ] 2 κ .Suppose that µ + ∈ spec * (A).Does this imply that µ + fails to satisfy µ + → [µ + ] 2 µ + ?