THE PSEUDOPOWER DICHOTOMY

Abstract We investigate pseudopowers of singular cardinals and deduce some consequences for covering numbers at singular cardinals of uncountable cofinality.

1. Introduction 1.1.Overview.We use pcf theory to establish some equalities between various pseudopowers at a singular cardinal, and use these to derive some ZFC conclusions in cardinal arithmetic.Our proofs rest on a principle we call the Pseudopower Dichotomy.This generalization of Fact 1.9 on page 324 of [12]) tells us that an arbitrary singular cardinal falls into one of two classes: either it is a strong limit in a very weak sense, or it is not.The majority of the paper is concerned with obtaining results in pcf theory that will allow us to derive meaningful conclusions in both situations.
Our main application of the dichotomy is a ZFC result about the behavior of covering numbers at singular cardinals of uncountable cofinality.The result is quite general, but a typical special case (in which the parameters are chosen for amusement), tells us, for example, that if µ is a singular cardinal of cofinality ℵ 6 , then (1.1.1)cov(µ, µ, ℵ 9 , ℵ 2 ) = cov(µ, µ, ℵ 7 , ℵ 2 ) + cov(µ, µ, ℵ 9 , ℵ 6 ). 1   We will also use recent work of Gitik [5] on the Shelah Weak Hypothesis to provide complementary consistency results, showing us that in many such equations both terms on the right-hand side are required.This equation (1.1.1)is unlikely to mean much to someone unfamiliar with Shelah's Cardinal Arithmetic [12], so our goal for the remainder of this introductory section is lead a reader with a basic knowledge of pcf theory to the point where the preceding paragraphs are understandable.

Basic pcf theory.
So what do we consider to be "a basic knowledge of pcf theory"?Certainly the material covered in Abraham and Magidor's chapter [1] in the Handbook of Set Theory [4] is more than enough.We use their work as a starting point, and any notation that we neglect to define comes from their presentation.The book [8] is an comprehensive account of the same material, and both Kojman's unpublished [9] and the classic paper of Burke and Magidor [2] provide ample coverage of the background material as well.The author's paper [3] also addresses of some of the topics under consideration here.Most of the pcf material we need has to do with properties of the ideals J <λ [A] and associated pcf generators.More precisely, we will need to use some results of Shelah showing that well-structured sets of generators can be found in certain circumstances.
1.3.Covering numbers.Our equation (1.1.1)expresses an equality between covering numbers at a singular cardinal.These covering numbers are cardinal characteristics introduced by Shelah [12] in his analysis of cardinal arithmetic, and they arise naturally when one considers structures of the form ([µ] <κ , ⊆).Indeed, these covering numbers are a refinement of the idea of cofinality in such structures.Definition 1.3.1.Suppose µ, κ, θ, and σ are cardinals satisfying A σ-cover of [µ] <θ in [µ] <κ is a family P ⊆ [µ] <κ with the property that every member of [µ] <θ can be covered by a union of fewer than σ elements drawn from P, that is The covering number cov(µ, κ, θ, σ) is defined to be the least cardinality of a σ-cover of [µ] <θ in [µ] <κ .
We assume (1.3.1) in order to avoid uninteresting cases.Since it is clear that (1.3.3)cov(µ, κ, θ, 2) = cov(µ, κ, θ, ℵ 0 ), we may as well assume that all four parameters are infinite cardinals.Note as well that cov(µ, κ + , κ + , 2) is just the cofinality of the structure ([µ] κ , ⊆), so covering numbers refine this familiar notion.Section 5 of Chapter II in [12] contains a comprehensive list of basic properties satisfied by covering numbers.Our work in this paper focuses on covering numbers in which the first two arguments are the same, so we take µ equal to κ.Such covering numbers describe the way in which [µ] <θ sits inside of [µ] <µ .This is interesting only in the case where µ is singular: if µ is regular, then the initial segments of µ will automatically cover [µ] <µ so the covering number is just µ.Furthermore, we assume σ ≤ cf(µ) < θ to avoid similar trivialities.Shelah has shown that general covering numbers can be computed from those where the first two components are the same, so our restriction is done without loss of generality.It is the behavior at singular cardinals that is important.1.4.Pseudopowers.One of Shelah's many surprising discoveries in cardinal arithmetic is that covering numbers can often be computed using pcf theory, and we exploit this link to obtain results like (1.1.1).The connection occurs through pseudopowers of singular cardinals.Pseudopowers are best thought of as pcf-theoretic versions of cardinal exponentiation.Computing a pseudopower at a singular cardinal µ involves examining the cardinals that can be represented in specific ways as the cofinality of reduced products of sets of regular cardinals cofinal in µ.The following definition and subsequent discussion fix our vocabulary.Definition 1.4.1.Suppose µ < λ are cardinals with µ singular and λ regular.A representation of λ at µ is a pair (A, J) where • λ is the true cofinality of the reduced product A/J, that is, there is a sequence We abbreviate this by writing λ = tcf( A/J).
• λ = max pcf(A).A cardinal λ is representable at µ if such a representation exists.More generally, given cardinals σ < θ ≤ µ with σ regular, we say λ is Γ(θ, σ)-representable at µ if there is a representation (A, J) of λ at µ with |A| < θ and σ-complete ideal J. 2  We say that |A| is the size of the representation, and the completeness of the representation is the completeness of the ideal J, that is, the least cardinal τ such that τ is not closed under unions of size τ .
When speaking about Γ(θ, σ)-representability at a cardinal µ, we will always assume σ ≤ cf(µ) < θ (as otherwise things degenerate), and that σ and θ are regular.This is reminiscent of the assumption (1.3.1), and it will guarantee that the associated pseudopowers obey some useful rules.
Moving on, we come to the actual definition of the pseudopower operation: There are a few notational variants used in the literature, all due to Shelah.For example, (1.4.5) pp θ (µ) := pp Γ(θ + ,ℵ0) (µ) (so in this case we are computing the supremum of all cardinals that can be represented at µ using a set of size at most θ), and (which computes the supremum of the set of cardinals that can be represented at µ using a σ-complete ideal on a set of size cf(µ)).Finally, the pseudopower pp(µ) of µ is defined by A little discussion may help the reader digest the preceding definitions.First, we note the obvious monotonicity property: if σ ≤ σ ′ < θ ′ ≤ θ are regular cardinals and µ is singular with cofinality in the interval [σ ′ , θ ′ ), then (1.4.8)PP Γ(θ ′ ,σ ′ ) (µ) ⊆ PP Γ(θ,σ) (µ). 2 To highlight a point of potential confusion, note that σ may be larger than ℵ 1 , so we are not using "σ-complete" as a synonym for "countably complete".
Given a singular cardinal µ, a well-known theorem of Shelah on the existence of scales at successors of singular cardinals provides us with a cofinal subset A of µ ∩ Reg such that • |A| = cf(µ), and The bounded ideal is trivially cf(µ)-complete, so this means that µ + is Γ(cf(µ))representable, and then an appeal to (1.4.8) shows us that in fact µ + is Γ(θ, σ)representable at µ for all relevant σ and θ.Thus, pp Γ(θ,σ) (µ) is always at least µ + .Lying a little deeper is a result of Shelah that PP Γ(θ,σ) (µ) consists of an interval of regular cardinals, that is This is known as the No Holes Conclusion (see 2.3 in Chapter II of [12]).The interval of regular cardinals PP Γ(θ,σ) (µ) enjoys some nice closure properties: if A is a subset of PP Γ(θ,σ) (µ) of cardinality less than θ, then that is, the interval of regular cardinals PP Γ(θ,σ) (µ) is closed under computing σcomplete pcf on sets of cardinality less than θ. 3 This is a critical property for us, and it is usually expressed in terms of an inverse monotonicity property of Γ(θ, σ)-pseudopowers: Proposition 1.4.3 (Inverse Monotonicity).Suppose σ ≤ cf(µ) < θ with σ and θ regular.If η < µ satisfies • σ ≤ cf(η) < θ, and and therefore The preceding result can be found as ⊗ 1 in Section II.2 of [12].To see why this implies the closure property expressed in (1.4.10), suppose A ⊆ PP Γ(θ,σ) (µ) satisfies |A| < θ and J is σ-complete ideal on A for which A/J has true cofinality equal to some λ.Let η be the least cardinal such that A ∩ η / ∈ J. Then η is singular with σ ≤ cf(η) < θ, and λ is Γ(θ, σ)-representable at η by way of the pair (A ∩ µ ′ , J ↾ A ∩ µ ′ ).Inverse Monotonicity implies that λ also Γ(θ, σ)-representable at µ, and hence it is a member of PP Γ(θ,σ) (µ) by definition.
The final ingredient of the calculus of pseudopowers that we need is denoted Continuity, and can be found as ⊗ 2 in Section II.2 of [12]: Proposition 1.4.4 (Continuity).Assume σ ≤ cf(µ) < θ with σ and θ regular, and let λ be a regular cardinal greater than µ.If λ is Γ(θ, σ)-representable at η for an of A/J equal to λ. See, e.g., the chapter [1].
1.5.The cov vs. pp Theorem.With the above material in hand, we can state the connection between covering numbers and pseudopowers.This theorem is due to Shelah (Theorem 5.4 of Chapter II of [12]); our paper [3] gives another proof of the result, in addition to providing much more background about pseudopowers and their computation.
The proof of (1.5.2) is the hard part of the result, and the question of whether the uncountability of σ is necessary to prove the equality is a major open question in pcf theory known as the "cov vs. pp problem".Shelah has shown that in many cases this condition can be dropped, and that pcf theory in the neighborhood of a counterexample µ must be very badly behaved.1.6.Overview revisited.Given the above discussions, we hope the reader is now in a better position to understand the summary given in Subsection 1.1.For example, to obtain our sample equation (1.1.1),we prove that if a cardinal λ is Γ(ℵ 9 , ℵ 2 )-representable at µ of cofinality ℵ 6 , then either λ is Γ(ℵ 7 , ℵ 2 )-representable at µ, or it is Γ(ℵ 9 , ℵ 6 )-representable at µ.In other words, if λ can be represented at a singular cardinal µ of cofinality ℵ 6 using an ℵ 2 -complete ideal on a set of cardinality at most ℵ 8 , then it can either be represented using an ℵ 2 -complete ideal on a set of cardinality ℵ 6 (the minimum possible size of a representation at µ), or it can be represented on a set of cardinality at most ℵ 8 using an ℵ 6 -complete ideal (the maximum possible completeness of a representation at µ).We then apply the cov vs. pp Theorem to draw conclusions about the corresponding covering numbers (see Corollary 7.1.2and the subsequent discussion). 4seudopower Dichotomy: in the "strong limit" option of the dichotomy, we will actually get a representation of λ using a cf(µ)-complete ideal on a set of size cf(µ), and it is the other option of the dichotomy that is responsible for the complexity of (1.1.1).

1.7.
Structure of the paper.The remainder of the paper is structured as follows: • In Section 2 we chain together some theorems of Shelah (scattered throughout several papers) concerning the existence of well-organized sets of generators in order to formulate a precise and user-friendly result (Corollary ??) that we then use to prove a generalization of the main result of [11], tailored to σ-complete pcf.
• Section 3 builds on this work and analyzes pseudopowers at singular cardinals that are eventually Γ(θ, σ)-closed.We are able to show that cardinals Γ(θ, σ)-representable at such a µ can be represented in a well-organized way.
• In Section 5, we use the previous sections to obtain results in ZFC about equalities between various types of pseudopowers at a singular cardinal µ.
In particular, we show and, for σ < cf(µ), • Section 6 uses recent work of Gitik [5] to provide complementary independence results related to the formulas derived in Section 5.
• In Section 7 we map out consequences of these results for covering numbers, arriving at the formula (1.1.1)and its relatives, and also discussing its consequences.We conclude with questions raised by this work.

2.
Reducing the size of a representations 2.1.On generators.In this section, we prove a generalization of Theorem 1.1 of [11] that extends Shelah's result to σ-complete pcf.We prove the theorem by manipulating a suitably nice collection of pcf generators rather than by working directly with characteristic functions of models as in [11].
The basic definitions follow, and we refer the reader to [1] and [2] for more detailed discussion of these matters.Definition 2.1.1.Let A be a progressive set of regular cardinals. ( (2) A generating sequence for pcf(A) is a sequence B λ : λ ∈ pcf(A) where B λ is a generator λ in A.
(3) More generally, if C is a progressive subset of pcf(A) and Λ is a subset of pcf(C), we say that In (3), our assumptions imply pcf(C) ⊆ pcf(A) so Λ is a subset of pcf(A) as well.We will usually assume that C includes A, and in this situation we know that if B λ is a generator for λ in C, then B λ ∩ A will be a generator for λ in A. Where important for clarity, we may write B λ [C] to emphasize that the corresponding set is a generator for λ in C.
It is a fundamental result of pcf theory that for a progressive A, any λ in pcf(A) has a corresponding generator in A, and thus we can always find a generating sequence B λ [A] : λ ∈ pcf(A) for pcf(A) in A. Obtaining transitive generating sequences is a more complicated issue.Shelah shows in Claim 6.7 of [14] that there is a transitive generating sequence for A (not pcf(A)!) in A whenever A is progressive.Thus, if pcf(A) happens to be progressive, then a transitive generating sequence for pcf(A) in pcf(A) will exist.Abraham and Magidor prove something a little more general in Section 6 of [1].A corollary of their presentation is that if κ is a regular cardinal with |A| < κ < min(A) and Λ is a subset of pcf(A) of cardinality at most κ, then we can find a transitive generating sequence for Λ in A.
Our proof requires more than this, and we need some results of Shelah appearing in Claims 6.7A and B of [14].These are quite technical, so in the interest of readability we summarize our requirements in a "black box" result: • C ξ is a subset of pcf σ-com (A ξ ) of cardinality less than σ, and Part (1) of the conclusion gives more than what Abraham and Magidor obtain in [1], as they produce a transitive generating sequence for N ∩ pcf(A) in A, and not in the (possibly larger) set N ∩ pcf(A).Said another way, the argument in [1] provides sets B λ for λ ∈ N ∩pcf(A) that are subsets of A and such that B λ generates the ideal J ≤λ [A] over J <λ [A].The stronger version we need provides generators that function in the progressive set N ∩ pcf(A), rather than just in A, with the corresponding . This extra power will be important for us, and it follows from parts (1) and ( 2) of Claim 6.7B of [14].
Part (2) of the conclusion says that even though the generators we produce may not be in N , they do interact well with N in the sense that pcf compactness arguments work as long as we are trying to cover sets in N .This follows from part (3) of Claim 6.7B of [14], and it is arranged by showing that the generators produced have suitable "internal reflections" inside of N .

2.2.
Reducing the size of a representation.We turn now to the main result of this section: a relative of Theorem 1.1 from Chapter VIII of [12].
Then can find a subset This is a template for what we might call a reduction-in-size theorem.The point is that we are replacing the set A by set C whose size is under our control, and doing it in such a way that λ is still captured by pcf σ-com (C).
Proof.Looking at our assumptions, it is clear that we may assume that η is at most the cardinality of A. The assumptions also imply that η is at least σ, as otherwise pcf σ-com (A) would simply be the union of the various pcf σ-com (A ξ ) for ξ < η, and this would violate (2.2.2).Thus, we can assume σ ≤ η ≤ |A|.
We may also assume that |A| + is strictly less than min(A), and so (by setting κ = |A| + ) we can find a model N as in Theorem 2.1.3containing A and the sequence A ξ : ξ < η .Note that since κ + 1 ⊆ N , we know that A is a subset of N and each individual element A ξ of our sequence will be in N as well.Since |A ξ | < κ for each ξ, it follows that A ξ will be a subset of N ∩ pcf(A) too.
Let B λ : λ ∈ N ∩ pcf(A) be the transitive sequence from Theorem 2.1.3.Conclusion (2) of this theorem tells us there is a sequence We set Then |C| ≤ η • σ = η, and C is definable from the sequence C ξ : ξ < η in N .
We now show that λ is in pcf σ-com (C), which will finish the proof.To do this, we apply conclusion (2) of Theorem 2.1.3again, this time to the single set C ∈ N .We obtain a set D ∈ N of cardinality less than σ such that (2.2.7) This is the crucial point where we need for our generators to work in N ∩ pcf(A) rather than just in the potentially smaller set A, as we are applying a compactness argument to cover a subset of N ∩ pcf(A).We know that max pcf(C) is at most λ because (2.2.9) max pcf(pcf(A)) = max pcf(A).
Thus, if λ fails to be in pcf σ-com (C), it must be the case that (2.2.10) that is, all members of D must be less than λ.
We now have an untenable situation.Since λ is in pcf σ-com (A), the σ-complete ideal on A generated by J <λ [A] must be a proper ideal.But for ǫ ∈ D, we know Theorem 3.1.1(Shelah [11]).Suppose ℵ 0 < cf(µ) ≤ θ < µ, and for every sufficiently large η < µ, Then PP θ (µ) = PP Γ(cf(µ)) (µ).In fact, any λ ∈ PP θ (µ) can be represented as the true cofinality of The above is part of Corollary 1.6 on page 321 of [12], and we will shortly derive it from our own work in this section.For now, we wish to highlight the assumption (3.1.1):it expresses that µ is (in a weak sense) a type of strong limit cardinal, and we will be working with such assumptions a lot in this section.The conclusion of the above theorem can be described informally in terms of upgrading the representation of λ: we are able to move from an arbitrary representation of λ based on a set of cardinality at most θ to one based on a set of cardinality cf(µ), with the added bonus that the ideal used in the representation is as simple as possible.The results we prove in this section will have a similar flavor.
We start with our main hypothesis, a definition that is natural given the preceding discussion.Definition 3.1.2.Let σ and θ be regular cardinals, and suppose µ is singular with σ ≤ cf(µ) < θ < µ.
Shelah has looked at such concepts in a more general setting.In particular, the third section of [13] briefly examines the idea of "pcf inaccessibility".The following lemma shows that our definition and his approach are essentially the same.Lemma 3.1.3.Suppose σ < θ are regular cardinals, and µ is a singular cardinal satisfying σ ≤ cf(µ) < θ < µ.Then the following conditions are equivalent: (1) µ is eventually Γ(θ, σ)-closed.
(2) There is an η < µ such that if A is a set of regular cardinals from the interval (η, µ) bounded below µ, then Proof.Assume (1), and choose η < µ such that µ is Γ(θ, σ)-closed beyond η.Given a set A satisfying the assumptions of (2), suppose λ is in pcf σ-com (A) \ µ.By passing to the generator B λ [A], we may assume λ is max pcf(A), and so if J is the σ-complete ideal on A generated by Now let ξ be least ordinal with A∩ξ / ∈ J. Our assumptions imply that ξ is a singular cardinal of cofinality at most |A| < θ, and λ is Γ(θ, σ)-representable at ξ. Since η < ξ < µ, we have a contradiction.The proof that (2) implies ( 1) is even easier: if µ is not eventually Γ(θ, σ)-closed, then the associated representations show that (2) must fail.Before we give the proof, note that this result is a natural counterpart to Theorem 3.1.1,with the only difference being the inclusion of the parameter σ.This theorem asks for a weaker closure condition than (3.1.1)in the situation where σ is uncountable, but it also has a weaker conclusion: it says only that if µ is eventually Γ(θ, σ)-closed, then any cardinal representable at µ via a σ-complete ideal on a set of cardinality less than θ is in fact representable at µ via a σ-complete ideal on a set of cardinality cf(µ), the minimum possible, but the proof does not let us obtain a representation using the bounded ideal.
Proof.Suppose A and J witness that the cardinal λ is Γ(θ, σ)-representable at µ. Since we may remove an initial segment of A if necessary, we may assume that µ is Γ(θ, σ)-closed beyond min(A), and by restricting to a suitable generator if necessary, we may assume that λ = max pcf(A) as well.Given µ α : α < κ increasing and cofinal in µ, if we define (3.2.2) then the hypotheses of Theorem 2.2.1 are satisfied by A α : α < cf(µ) and λ, and we obtain a set of cardinality less than θ such that We can do better than this, though, and the next lemma lies is the heart of our results.It shows that with assumptions similar to those used in Theorem 3.2.1,we are able to obtain representations of cardinals that are "well-organized".This will then allow us to show that the corresponding ideals in the representation satisfy stronger completeness conditions.Lemma 3.2.2.Assume µ is eventually Γ(θ, σ)-closed, where σ and θ are regular, and σ ≤ cf(µ) < θ.Suppose λ is Γ(θ, τ )-representable at µ for some regular cardinal τ in the interval [σ, cf(µ)].Then we can find a cardinal σ * < σ and a set (3.2.6) C = {λ α ς : α < cf(µ) and ς < σ * } of regular cardinals less than µ such that • pcf σ-com {λ β ς : β < α and ς < σ * } ⊆ µ for each α < cf(µ), and • if X is an unbounded subset of cf(µ) then (3.2.7) λ = max pcf({λ α ς : α ∈ X and ς < σ * }) and (3.2.8) λ ∈ pcf τ -com ({λ α ς : α ∈ X and ς < σ * }).The above is essentially a generalization of Shelah's result Theorem 3.1.1.We will discuss this after the proof, and even show how his result follows easily from the lemma.
Proof.Suppose A and J are a Γ(θ, σ)-representation of λ at µ. Just as in the proof of Theorem 3.2.1,we may assume µ is Γ(θ, σ)-closed beyond min(A) and that λ is max pcf(A).Let µ α : α < cf(µ) be an increasing sequence cofinal in µ, and let A α be A ∩ µ α .
We implement the argument of Theorem 2.2.1 and make use of transitive generators.To do this, we assume (without loss of generality) that |A| < κ = cf(κ) < min(A), and let N and b be as inTheorem 2.1.3with N containing all objects under discussion in the preceding paragraph.By properties of b, in the model N there is a sequence of sets C α for α < cf(µ) such that • C α is a subset of pcf σ-com (A α ) of cardinality less than σ, and Since the sequence A α : α < cf(µ) is increasing, it follows that whenever X is an unbounded subset of cf(µ) we have Since each C α has cardinality less than σ < cf(µ), we can (by passing to an unbounded subset of cf(µ)) assume each C α is of some fixed cardinality σ * < σ, say (3.2.12) C α = {λ α ς : ς < σ * }, and we are done by letting C be the union of the sets C α .Note that we do not claim that the sets C α are disjoint, and they may very well overlap.It is helpful to visualize C as an array of regular cardinals with cf(µ) rows and σ * columns.In this interpretation, row α corresponds to C α , and we get a corresponding column for each fixed ς < σ * .
The pcf structure of C transfers to the index set Λ = cf(µ) × σ * in the natural way, and we may define an ideal J on Λ by With this point of view, we see that for X ⊆ cf(µ), (3.2.14) {(α, ς) : α ∈ X and ς < σ * } ∈ J ⇐⇒ X is bounded in cf(µ).
Note as well that the τ -complete ideal generated by J is a proper ideal because of (3.2.11).
It is helpful to look back and compare our situation with the conclusion of Theorem 3.1.1.We have not managed to represent λ as the true cofinality of a product of cardinals modulo the bounded ideal, but we have come close!What goes wrong is that the rows C α in our array are not singletons, and instead all we know is that they all have a fixed cardinality σ * less than σ.What Shelah does in [11] is note that if this cardinality happens to be finite, then we can improve the situation and get the ideal to consist of just the bounded sets: Corollary 3.2.3(Theorem 3.1.1,due to Shelah [11]).Suppose ℵ 0 < cf(µ) ≤ θ < µ and µ is eventually Γ(θ + , ℵ 0 )-closed, that is, for all sufficiently large ν < µ, Then any member of PP θ (µ) has a representation of the form (C, J bd [C]) where C is unbounded in µ ∩ Reg of order-type cf(µ) and J bd [C] is the ideal of bounded subsets of C.
Proof.Suppose λ = tcf A/J where A is cofinal in µ of cardinality at most θ and J is an ideal on A extending the bounded ideal.We apply Lemma 3.2.2 to Γ(θ + , ℵ 0 ) and obtain n < ω and C = λ α i : i ≤ n such that • sup pcf{λ β i : β < α and i ≤ n} is less than µ for each α < cf(µ), and • λ = max pcf({λ α i : α ∈ X and i ≤ n}) for any unbounded X ⊆ cf(µ).We now claim there is an i ≤ n and an unbounded X ⊆ cf(µ) such that (3.2.16) λ = max pcf{λ α i : α ∈ Y } for every unbounded Y ⊆ X.This is enough, as letting D = {λ α i : i ∈ Y }, it follows easily that the ideal J <λ [D] consists of the bounded ideal J bd [D].
We establish this by contradiction: if there are no such i and X, then for every i ≤ n and every unbounded X ⊆ cf(µ) there is an unbounded Y ⊆ X such that (3.2.17) max pcf{λ α i : α ∈ Y } < λ.
Working by induction, we find a single unbounded X ⊆ cf(µ) such that as any ultrafilter on this set must contain one of the columns, and we have a contradiction.
We note that Shelah obtains even nicer representations in Chapter VIII of Cardinal Arithmetic, but the above observation is at the heart of his argument.
Before presenting a proof of this theorem, we note that condition (3.3.2) holds if υ is less than τ +ω , so at the very least, the theorem allows us to find representations that are "more complete", while simultaneously ensuring that the size of the set involved is at most cf(µ).In order to do this, we need to make the stronger assumption that σ is strictly less than cf(µ), and this is what allows us to achieve greater completeness in our representation.
Suppose by way of contradiction this is not the case.Then there is a least cardinal ρ < υ such that for some sequence of sets D i : i < ρ and unbounded X ⊆ cf(µ), we have for each i < ρ, and • {λ α ς : α ∈ X and ς < σ * } ⊆ i<ρ D i .Clearly τ must be less than or equal to ρ, and by shrinking C if necessary we may assume there are such sets D i for i < ρ with Our assumptions tell us that cov(ρ, ρ, σ, 2) is less than cf(µ), so there is a family P such that • |P| < cf(µ), and Again, note that Theorem 3.2.1 can only tell us that PP Γ(θ,τ ) (µ) = PP Γ(τ ) (µ) under the same assumptions as Theorem 3.3.1.The improvement we obtain here is that we find a representation using an υ-complete ideal on a set of size cf(µ) rather than simply a τ -complete one.
We close this section with the following corollary, which gives us a little information about the assumptions we make relating σ, τ , and υ.Corollary 3.3.2.Suppose µ is eventually Γ(σ)-closed, where σ < cf(µ).If τ < υ are regular cardinals in the interval [σ, cf(µ)] and then there is a singular cardinal ρ of cofinality less than σ such that If we work with cardinals whose cofinality is ℵ n for some n, then the above situation cannot happen because there is no place for such a cardinal ρ.Thus, the following is immediate.

The Pseudopower Dichotomy
4.1.Introducing the dichotomy.In the preceding section we analyzed the behavior of pp Γ(θ,τ ) (µ) where µ is a singular cardinal that is eventually Γ(θ, σ)-closed and σ ≤ τ ≤ cf(µ).The analysis showed that if a cardinal is Γ(θ, τ )-representable at µ, then it is has a σ-complete representation using the minimum possible size, cf(µ), and the completeness of the representation can be increased if σ is strictly less than τ .In this section, we will analyze what happens when µ is not eventually Γ(θ, σ)-closed.We start with the following result which formulates a fundamental dichotomy about singular cardinals.This is not a new idea (in fact, the result we present is a relative of Fact 1.9 in [11]), and Shelah has made and used similar observations in other places.However, such dichotomies are extraordinarily useful for proving theorems in ZFC, as they allow one to analyze situations by breaking into cases where each option carries non-trivial information.We will refer to this result as the Pseudopower Dichotomy.
Lemma 4.1.1(Pseudopower Dichotomy).Suppose µ is a singular cardinal, and let σ < θ be regular cardinals with σ ≤ cf(µ) < θ.Then exactly one of the following statements hold: Option 2: µ is a limit of eventually Γ(θ, σ)-closed cardinals η for which Proof.It is clear that the two options are mutually exclusive as µ + is always Γ(θ, σ)representable at µ. Suppose that Option 1 fails for a cardinal µ.By definition, this means that for each ξ < µ there is a singular cardinal η greater than ξ such that By Inverse Monotonicity (Proposition 1.4.3),this implies Note as well that if η is the LEAST such cardinal above ξ, then η is Γ(θ, σ)-closed beyond ξ.Thus, µ is a limit of eventually Γ(θ, σ)-closed cardinals η for which (4.1.3)is true, and for which Let X be the collection of such cardinals η < µ, and for each η in X , we define ( If η is in X , then Y η is an non-empty interval of regular cardinals with minimum µ + .Furthermore, if η < ν in X then by Inverse Monotonicity we have (4.1.8) Thus, the sequence Y η : η ∈ X must eventually stabilize, say with value Y.
If λ ∈ Y, then by an application of the continuity property of Proposition 1.4.4 we know λ ∈ PP Γ(θ,σ) (µ), and so for all sufficiently large η ∈ X , we have Combining this with (4.1.6)establishes that Option 2 of the Pseudopower Dichotomy holds.
As a corollary, we have the following result which we will use later in the paper.
Corollary 4.1.2.Suppose µ is singular, while σ < θ are regular cardinals with then µ is a limit of singular cardinals η such that • η is eventually Γ(θ, σ)-closed, and Proof.Our assumption implies that Option 2 of the Pseudopower Dichotomy holds by way of Theorem 3.2.1, and now the result follows immediately.

4.2.
More on improving representations.The proof of the Pseudopower Dichotomy provides a template for its applications.For example, suppose that a cardinal λ is Γ(θ, σ)-representable at µ and look at the Pseudopower Dichotomy.If Option 1 holds, then our work in the preceding section applies and we can get nice representations of λ at µ. If, on the other hand, Option 2 holds sway, then we know instead that λ has nice representations for many cardinals below µ, and we will be able to combine these representations (using the Continuity Property of pseudopowers from Proposition 1.4.4) to produce a nice representation of λ at µ itself.The following definition will help us take advantage of these ideas.
We define X Γ(θ,σ) (µ) to be the collection of cardinals η satisfying • η is eventually Γ(θ, σ)closed, and This set might very well be empty, but note that the Pseudopower Dichotomy can be reformulated as the statement "either X Γ(θ,σ) (µ) is unbounded in µ or it is not".If X Γ(θ,σ) (µ) is unbounded, then it will exert an influence over representations at µ through continuity.For example, we have the following observation which yields the same conclusion as Theorem 3.2.1.
The next result builds on this idea, and shows how assumptions on the structure of X Γ(θ,σ) (µ) let us improve representations of cardinals in PP Γ(θ,σ) (µ).
Theorem 5.1.1.Suppose µ is singular, and cf(θ) ≤ θ < µ.Then We make a couple of comments before presenting the proof.First, note that the above theorem shows us that if λ has a representation at µ using a set of size θ, then λ can be represented at µ using either a set of cardinality cf(µ) (the minimum possible size) or a cf(µ)-complete ideal (the maximum possible).These options are not mutually exclusive (for example, both hold simultaneously for µ + ), but the power of the theorem is in the statement that at least of these two things must occur.
Our second comment is to note that because PP(µ) and PP Γ(θ,cf(µ)) (µ) are both intervals of regular cardinals, the equation (5.1.2) implies that PP θ (µ) must in fact be equal to one of these two sets.
Proof of Theorem 5.1.1.We may assume that µ has uncountable cofinality, as otherwise (5.1.2) holds automatically.We apply the pseudopower dichotomy to Γ(θ + , ℵ 0 ).If µ is eventually Γ(θ + , ℵ 0 )-closed, then Corollary 3.2.3gives us more than we need because any λ in PP θ (µ) can be represented using the bounded ideal on a set of cardinality cf(µ) cofinal in µ ∩ Reg.It follows that in this situation, all three of the sets from (5.1.2) are equal, and which is more than we require.Thus, we may assume that Option 2 of the Pseudopower Dichotomy is in force and the corresponding set X Γ(θ + ,ℵ0) is unbounded in µ.Abbreviating this set as "X ", we split into two cases.
For each n, let η n be the least such cardinal.The sequence η n : n < ω is nonincreasing, hence eventually constant, say with value η.This particular η will be eventually Γ(θ n , σ)-closed for each n < ω, and hence This result will reappear in the final section of our paper, when we formulate some natural open questions.
6. Independence Results 6.1.Gitik's theorem.This short section will focus on obtaining independence results complementary to the theorems we established in the previous section.Specifically, we will examine the formula established in Corollary 5.2.2.We rely almost completely on recently published work of Moti Gitik [5].Since we do not have the expertise to discuss his proof in detail, our approach will be to quote his results liberally.We start with his main theorem: Theorem 6.1.1 (Theorem 1.3 of [5]).Assume GCH.Let η be an ordinal and δ be a regular cardinal.Let κ α : α < η be an increasing sequence of strong cardinals, and let λ be a cardinal greater than the supremum of {κ α : α < η}.Then there is a cardinal preserving extension in which, for every α < η, (1) cf(κ α ) = δ, and (2) pp(κ α ) ≥ λ.
The final section of his paper (Section 8) is the part most relevant for us, as he examines the cardinal arithmetic structure of his model in some detail.We will follow his notation, and point out what need.6.2.Computations.We assume that δ and η are regular cardinals with ℵ 2 ≤ δ and δ + < η, and we work in the generic extension V [G] from Theorem 6.1.1.In V [G], the cardinals κ α for α < η will all have cofinality δ, and each satisfies pp(κ α ) ≥ λ.We need a little more: as Gitik notes prior to his Proposition 8.11, in fact we have the stronger result that (6.2.1) pp Γ(δ) (κ α ) ≥ λ as all of the ideals involved in the computation are δ-complete.
In V [G], if α < η is a limit ordinal, then κα is singular with cofinality less than η.Our choices of δ and η guarantee that there are limit ordinals α < η with cf(α) greater than δ, and others of cofinality less than δ but greater than ω 1 .The corresponding κα for these two sorts of α are the places of interest to us, and we analyze each situation separately.Again, relying on Gitik's work we have: Proposition 6.2.1.Let α be a limit ordinal less than η, and let V [G] be as in Theorem 6.1.1.Then the model V [G] satisfies: (1) If cf(α) < δ then (6.2.3) pp(κ α ) < pp Γ(δ + ,cf(α)) (κ α ). ( Proof.For (1), suppose α < η is a limit ordinal of cofinality less than δ.We know that {κ β : β < α} is cofinal in κα , and each κ β is singular of cofinality δ with (6.2. as required.For (2), suppose α < η is a limit ordinal of cofinality greater than δ.Again, the set {κ β : β < α} is unbounded in κα , and so an application of continuity tells us (6.2.10) pp Γ(δ) (κ α ) ≥ λ.One the one hand, if we define µ = κω1 , θ = δ + , and σ = ℵ 0 , then by equation (6.2.3), we have and so for this choice of parameters we have On the other hand, if we let µ = κδ + , θ = δ + , and σ = δ, then from equation (6.2.4) we conclude (6.3.3)pp and so (remembering that µ is of cofinality θ) we have Taken together, (6.3.2) and (6.3.4) show us that both summands in equation (6.1.1)are important if we want a theorem that holds in ZFC.We have not pushed the analysis of the cardinal arithmetic in Gitik's model beyond what was presented above; it may be that there are similar examples for many other values of θ and σ.
7. On covering numbers 7.1.Back to the beginning.We now look at what Theorem 5.2.1 and its relatives tell us about covering numbers at singular cardinals.Once again, we assume that σ and θ are infinite regular cardinals, while µ is a singular cardinal satisfying Recall that the covering number cov(µ, µ, θ, σ) is defined to be the minimum cardinality of a subset P of [µ] <µ that σ-covers [µ] <θ , that is, such that for every X ∈ [µ] <θ there is a subset Y of P of cardinality less than σ with (7.1.2)X ⊆ Y.
and we finally arrive at (1.If we let P be M ∩ [µ] <µ , then EXACTLY one of the following three things MUST occur: (1) Some X ∈ [µ] ℵ6 cannot be covered by a union of ℵ 1 sets from P.
(2) Every X ∈ [µ] ℵ8 can be covered by a union of ℵ 1 sets from P.
(3) Every X ∈ [µ] ℵ6 is covered by a union of ℵ 1 sets from P, but some Y ∈ [µ] ℵ8 cannot be covered by a union of ℵ 5 sets from P.
More generally, we can consider this view of our conclusion (7.1.5).Given a model M as above, if every member of [µ] cf(µ) can be covered by a union of fewer than σ sets from M ∩ [µ] <µ , and every member of [µ] <θ can be covered by a union of at most σ sets from M ∩ [µ] <µ , then in fact every member of [µ] <θ can be covered by fewer than σ sets from M ∩ [µ] <µ .7.3.The role of pseudopowers.Looking back at the discussion ending the previous section, the conclusions speaks about elementary submodels and the combinatorics of [µ] <µ without any reference at all to the pseudopowers that were used in the proofs.The fact that we are forced to use pseudopowers and then rely on the cov vs. pp theorem means that any conclusions on covering numbers are currently limited to situations where the last argument σ is uncountable.Thus, eliminating the reliance on pseudopowers might unlock a more general theorem: Question 1. Can results like Theorem 7.1.1be proved directly without the use of pseudopowers?
It is possible that an argument like Shelah's original proof of the cov vs. pp theorem (pp.87-93 of [12]) may work under these circumstances.7.4.Gitik-Shelah revisited.Look back at Theorem 5.3.1.This tells us that for a singular cardinal µ, if we fix a regular cardinal σ ≤ cf(µ) then as θ ranges over the interval (cf(µ), µ) ∩ Reg there are only finitely many distinct pseudopowers pp Γ(θ,σ) (µ) achieved.If we instead fix θ ∈ (cf(µ), µ) ∩ Reg and let σ range over the regular cardinals in [ℵ 0 , cf(µ)], then we achieve only finitely many distinct values for different reasons: as σ increases, the corresponding sequence of pseudopowers is non-increasing.Thus, it is natural to ask if there is something deeper going on.Question 2. Suppose µ is a singular cardinal.Note that this is true for µ satisfying cf(µ) < ℵ ω , so the first interesting case occurs when µ is singular of cofinality ℵ ω+1 .For a more specific question that may shed light, suppose µ is singular of cofinality ℵ ω+1 , and we have a cardinal λ that is Γ(ℵ n )-representable at µ for all n < ω.Is λ also Γ(ℵ ω+1 )-representable at µ? 7.5.A conjecture.Finally, we ask if the behavior we observed for singular cardinals with cofinality less than ℵ ω holds in general.Question 3. Suppose µ is singular, and σ < θ are regular cardinals with σ ≤ cf(µ) < θ < µ.

Definition 2 . 1 . 2 .
Let A be a progressive set of regular cardinals, and suppose b is a generating sequence B λ [C] : λ ∈ Λ for Λ in C, where Λ ⊆ pcf(A) and C is a progressive subset of pcf(A) containing A. The sequence b is transitive if

Theorem 2 . 1 . 3 (
Black Box).Let A be a progressive set of regular cardinals, and suppose κ is a regular cardinal satisfying |A| < κ < min(A).Then for any sufficiently large regular χ and x ∈ H(χ), there is an elementary submodel N of H(χ) of cardinality κ containing A and x and a sequence b = B λ : λ ∈ N ∩ pcf(A) such that (1) b is a transitive generating sequence for N ∩ pcf(A) in N ∩ pcf(A), and(2) given a sequence Ā = A ξ : ξ < η of sets and a regular cardinal σ such that (a) Ā ∈ N (b) η and σ are less than κ, and

( 3 .
2.4) λ ∈ pcf σ-com (C).Now let I be the σ-complete ideal generated by J <λ [C].We claim that C and I witness that λ is Γ(θ)-representable at µ. Certainly C is a subset of µ ∩ Reg because of (3.2.3) and Lemma 3.1.3.We know I is a proper ideal on C because λ is in pcf σ-com (C), and (3.2.5) λ = tcf C/I because λ = max pcf(C) and I extends J <λ [C].To finish, we need to show that C is unbounded in µ and I includes all initial segments of C, but this follows from Lemma 3.1.3as well: any subset of C bounded below µ must be in I because λ is greater than µ.
a progressive set of regular cardinals, and suppose σ is a regular cardinal.Suppose and since |D| < σ it follows that A cannot be contained in the union of the sets B ǫ for ǫ in D, that is, In this section we work with cardinals that are strong limits in a sense measured by pseudopowers.Our aim is to generalize one of the main results of Chapter VIII of Cardinal Arithmetic, where Shelah addresses basic questions about improving representations of cardinals.Simply recalling a small part of what Shelah establishes provides us with a good starting point.
But now we have a contradiction, as we have shown that α∈X C α can be covered by |Y | < ρ sets from J <λ α∈X C α ⊆ i∈Y D i .* [C].

5 .
Applications of the Pseudopower Dichotomy 5.1.Computing PP θ (µ).In this section, we put together pieces from our preceding work to obtain theorems in ZFC based on the Pseudopower Dichotomy.Our first result echoes Theorem 3.1.1,as tells us that PP θ (µ) can be computed from pseudopowers that involve the cofinality of µ in two different ways.PP Γ(θ + ,cf(µ)) ⊆ PP θ (µ).
1.1) from the introduction.What important for this choice of parameters is (7.1.11)σ ≤ cf(µ) < σ +ω , 7.2.What does it mean?The above formula (7.1.10)hides a trichotomy about the ways in which certain elementary submodels interact with subsets of µ.To see why, let χ be a sufficiently large regular cardinal, and suppose M is an elementary submodel of H(χ) containing µ and such that (7.2.1) |M | + 1 ⊆ M.