aa r X i v : . [ m a t h . L O ] J a n UPGRADING PSEUDOPOWER REPRESENTATIONS
TODD EISWORTH
Abstract.
We investigate pseudopowers of singular cardinals, and show thatdeduce some consequences for cardinal arithmetic. For example, we show thatin
ZFC thatcov( µ, µ, θ, σ ) = cov( µ, µ, (cf µ ) + , σ ) + cov( µ, µ, θ, σ + )whenever ℵ ≤ σ = cf( σ ) < cf( µ ) < θ < µ , and use recent work of Gitik toshow that both summands in the equation are required. Background and Definitions
This paper uses pcf theory to prove some theorems about pseudopowers of sin-gular cardinals, and then uses these results to draw conclusions about coveringnumbers and conventional cardinal arithmetic. For example, we are able to show(1.1) cov( µ, µ, ℵ , ℵ ) = cov( µ, µ, ℵ , ℵ ) + cov( µ, µ, ℵ , ℵ )whenever µ is a singular cardinal of cofinality ℵ . This introductory section isintended to provide enough information so that readers who are not intimatelyfamiliar with Shelah’s
Cardinal Arithmetic [8] can understand what the previoussentences mean.We do need to assume that the reader has at least a basic familiarity with pcftheory. Certainly the material covered in Abraham and Magidor’s chapter [1] inthe Handbook of Set Theory [3] is more than enough, and any undefined notationcomes from their exposition. The book [5] and unpublished [6] are also excellentresources for background material, while the author’s paper [2] contains a detailedtreatment of some of the topics we consider here.
Covering Numbers
Covering numbers are cardinal characteristics introduced by Shelah [8] in hisanalysis of cardinal arithmetic at singular cardinals, and they arise naturally whenone considers structures of the form ([ µ ] <κ , ⊆ ). Indeed, covering numbers are justa refinement of the idea of cofinality in such structures. Definition 1.1.
Suppose µ , κ , θ , and σ are cardinals satisfying(1.2) 2 ≤ σ < θ ≤ κ ≤ µ. A σ -cover of [ µ ] <θ in [ µ ] <κ is a family P ⊆ [ µ ] <κ with the property that everymember of [ µ ] <θ can be covered by a union of fewer than σ elements drawn from P , Mathematics Subject Classification.
Key words and phrases. pcf theory, pseudopowers, covering numbers, cardinal arithmetic,singular cardinals. Of course, the choice of cardinals in (1.1) has been made for amusement, but this equation isa good example of the sort of relationships between covering numbers that we uncover. that is(1.3) ( ∀ X ∈ [ µ ] <θ )( ∃ Y ⊆ P ) h | Y | < σ and X ⊆ [ Y i . The covering number cov( µ, κ, θ, σ ) is defined to be the least cardinality of a σ -coverof [ µ ] <θ in [ µ ] <κ .We assume (1.2) in order to avoid some uninteresting cases. Since it is clear that(1.4) cov( µ, κ, θ,
2) = cov( µ, κ, θ, ℵ ) , we may as well assume that all four parameters are infinite cardinals. Note as wellthat cov( µ, κ + , κ + ,
2) is just the cofinality of the structure ([ µ ] κ , ⊆ ), so we can seeimmediately that covering numbers are a refinement of this familiar notion, andthat they have relevance for computations in classical cardinal arithmetic. Section5 of Chapter II in [8] contains a comprehensive list of basic properties satisfied bycovering numbers.Our work in this paper focuses on the situation where µ = κ , so we are analyzinghow [ 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 thecovering number is just µ . Furthermore, we assume σ ≤ cf( µ ) < θ to avoid similartrivialities. Shelah has shown that general covering numbers can be computedfrom those where the first two components are the same, so our restriction is donewithout loss of generality. It is the behavior at singular cardinals that is important. Pseudopowers
One of Shelah’s many surprising discoveries in cardinal arithmetic is that cov-ering numbers can often be computed using pcf theory, and we exploit this linkto obtain results like 1.1. The connection occurs through pseudopowers of singularcardinals. Pseudopowers should be imagined as pcf-theoretic versions of cardinalexponentiation, and their computation at a singular cardinal µ involves examiningthe cardinals that arise as the cofinality of certain reduced products of sets of reg-ular cardinals cofinal in µ . The following definition and subsequent discussion fixour vocabulary. Definition 1.2.
Suppose µ < λ are cardinals with µ singular and λ regular. A representation of λ at µ is a pair ( A, J ) where • A is a progressive (that is, satisfying | A | < min( A )) cofinal subset of µ ∩ Reg • J is an ideal on A extending the bounded ideal J bd [ A ] • λ is the true cofinality of the reduced product Q A/J , that is, there is asequence h f α : α < λ i in Q A such that(1.5) α < β < λ = ⇒ f α < J f β , and(1.6) ( ∀ g ∈ Y A )( ∃ α < λ ) [ g < J f α ] . We abbreviate this by writing λ = tcf( Q 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 PGRADING PSEUDOPOWER REPRESENTATIONS 3 J . We say that | A | is the size of the representation, and the completeness of therepresentation is the completeness of the ideal J , that is, the least cardinal τ suchthat τ is not closed under unions of size τ .When speaking about Γ( θ, σ )-representability at a cardinal µ , we will alwaysassume σ ≤ cf( µ ) < θ (as otherwise things degenerate) and that σ and θ areregular. This is reminiscent of (1.2), and the assumption will guarantee that theassociated pseudopowers obey some useful rules.Moving on, we come to the actual definition of the pseudopower operation: Definition 1.3.
Suppose µ is singular, and σ ≤ cf( µ ) < θ ≤ µ with σ and θ regular. We define(1.7) PP Γ( θ,σ ) ( µ ) := { λ : λ is Γ( θ, σ )-representable at µ } , and the Γ( θ, σ )-pseudopower of µ , pp Γ( θ,σ ) ( µ ), is defined by(1.8) pp Γ( θ,σ ) ( µ ) := sup PP Γ( θ,σ ) ( µ ) . There are a few notational variants used in the literature, all due to Shelah. Forexample,(1.9) pp θ ( µ ) := pp Γ( θ + , ℵ ) ( µ )(so we are looking at the supremum of cardinals that can be represented at µ usinga set of size at most θ ), and(1.10) pp Γ( σ ) ( µ ) = pp Γ((cf µ ) + ,σ ) ( µ ) , the supremum of the set of cardinals that can be represented at µ using a σ -completeideal on a set of size cf( µ ). Finally, the pseudopower pp( µ ) of µ is defined by(1.11) pp( µ ) := pp cf( µ ) ( µ ) . 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 cardinalsand µ is singular with cofinality in the interval [ σ ′ , θ ′ ), then(1.12) PP Γ( θ ′ ,σ ′ ) ( µ ) ⊆ PP Γ( θ,σ ) ( µ ) . Now given a singular cardinal µ , a well-known theorem of Shelah on the existenceof scales at successors of singular cardinals provides us with a cofinal subset A of µ ∩ Reg such that • | A | = cf( µ ), and • tcf( Q A/J bd [ A ]) = µ + , where J bd [ A ] is the bounded ideal on A .The bounded ideal is trivially cf( µ )-complete, so this means that µ + is in PP Γ(cf( µ )) ( µ ),and then an appeal to (1.12) shows us that µ + is Γ( θ, σ )-representable at µ for allrelevant 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(1.13) λ ∈ PP Γ( θ,σ ) ( µ ) = ⇒ [ µ + , λ ] ∩ Reg ⊆ PP Γ( θ,σ ) ( µ ) . This is known as the
No Holes Conclusion (see 2.3 in Chapter II of [8]). To highlight a point of potential confusion, note that σ may be larger than ℵ , so we are notusing “ σ -complete” as a synonym for “countably complete”. TODD EISWORTH
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(1.14) pcf σ -com ( A ) ⊆ PP Γ( θ,σ ) ( µ ) , that is, the interval of regular cardinals PP Γ( θ,σ ) ( µ ) is closed under computing σ -complete pcf on sets of cardinality less than θ . This is a critical property for us,and it is usually expressed in terms of an inverse monotonicity property of theΓ( θ, σ ) pseudopowers:
Proposition 1.4 ( Inverse Monotonicity) . Suppose σ ≤ cf( µ ) < θ with σ and θ regular. If η < µ satisfies • σ ≤ cf( η ) < θ , and • µ ≤ pp Γ( θ,σ ) ( η ),then(1.15) PP Γ( θ,σ ) ( µ ) ⊆ PP Γ( θ,σ ) ( η ) , and therefore(1.16) pp Γ( θ,σ ) ( µ ) ≤ pp Γ( θ,σ ) ( η ) . The preceding result can be found as ⊗ in Section II.2 of [8]. To see whythis implies the closure property expressed in (1.14), suppose A ⊆ PP Γ( θ,σ ) ( µ )satisfies | A | < θ and J is σ -complete ideal on A for which Q A/J has true cofinalityequal to some λ . Let η be the least cardinal such that A ∩ η / ∈ J . Then η issingular with σ ≤ cf( η ) < θ , and λ is Γ( θ, σ )-representable at η by way of thepair ( A ∩ µ ′ , J ↾ A ∩ µ ′ ). Using Inverse Monotonicity , it follows that λ also Γ( θ, σ )-representable at µ , and hence 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 ⊗ in Section II.2 of [8]: Proposition 1.5 ( Continuity) . Assume σ ≤ cf( µ ) < θ with σ and θ regular, andlet λ be a regular cardinal greater than µ . If λ is Γ( θ, σ )-representable at η for anunbounded set of singular cardinals η < µ (satisfying σ ≤ cf( η ) < θ < η ), then λ isalso Γ( θ, σ )-representable at µ . In other words, if(1.17) µ = sup { η < µ : σ ≤ η < θ < η and λ ∈ PP Γ( θ,σ ) ( η ) } , then(1.18) λ ∈ PP Γ( θ,σ ) ( µ ) . The proof is not difficult: one can paste together suitable representations of λ at cardinals η < µ to obtain a representation at µ itself. The Plan
With the above material in hand, we can state to connection between coveringnumbers and pseudopowers. This theorem is due to Shelah (Theorem 5.4 of Chap-ter II of [8]); our paper [2] gives another proof of the result, in addition to providingmuch more background about pseudopowers and their relationship to pcf theory. A cardinal λ is in pcf σ -com ( A ) if there is a σ -complete ideal J on A with the true cofinalityof Q A/J equal to λ . See, e.g., the chapter [1]. PGRADING PSEUDOPOWER REPRESENTATIONS 5
Theorem 1 ( The cov vs. pp Theorem) . Suppose σ < θ are infinite regular cardinals,and µ is singular with σ ≤ cf( µ ) < θ . Then (1.19) pp Γ( θ,σ ) ( µ ) ≤ cov( µ, µ, θ, σ ) . If σ > ℵ (so µ has uncountable cofinality) then (1.20) pp Γ( θ,σ ) ( µ ) = cov( µ, µ, θ, σ ) . We hope the reader is now in a better position to understand the summarygiven in the first paragraph: in this paper, we will use results in the calculusof pseudopowers (in particular, those discussed above) to establish some equalitiesbetween pseudopowers at a singular cardinal µ , and then use the cov vs. pp Theorem to draw some conclusions about the corresponding covering numbers.In more detail, the paper is structured as follows: • In Section 2 we lay out some results about pcf generators taken from workof Shelah and use them to prove a relative of the main result of [7] tailoredfor σ -complete pcf. • Section 3 builds on this work, and uses it to analyze pseudowers at singu-lar cardinals that are eventually Γ( θ, σ ) -closed . We are able to show thatcardinals λ that are Γ( θ, σ )-representable at such a µ can be represented ina well-organized way. • Section 4 defines what we call the
Pseudopower Dichotomy , and then ana-lyzes PP Γ( θ,σ ) ( µ ) when the singular cardinal µ is NOT eventually Γ( θ, σ )-closed. • In Section 5, we prove some theorems in
ZFC about equalities betweenvarious types of pseudopowers at a singular cardinal µ without makingassumptions about cardinal arithmetic below µ . In particular, we show(1.21) pp θ ( µ ) = pp( µ ) + pp Γ( θ + , cf( µ )) ( µ ) , and, for σ < cf( µ ),(1.22) pp Γ( θ,σ ) ( µ ) = pp Γ( σ ) ( µ ) + pp Γ( θ,σ + ) ( µ ) . • Section 6 uses recent work of Gitik [4] to provide complementary indepen-dence results related to the formulas derived in Section 5. • Finally, in Section 7 we map out consequences of these results for coveringnumbers, arriving at the formula (1.1) and its relatives, and also discussingits consequences.2.
Reflecting Generating Arrays
This section serves two purposes. Our primary goal is to prove a theorem gener-alizing Theorem 1.1 of [7] to pcf σ -com ( A ) for a progressive set of regular cardinals A . Our argument needs the existence of a suitably nice generating sequence forpcf( A ), but we require more than can be found in the Abraham-Magidor chapterfrom Handbook of Set Theory and need to rely on a more complicated theorem ofShelah from the last section of [9]. We realize that much of pcf theory has onlybeen published in an unpolished form, so this section has a secondary goal of intro-ducing helpful terminology and providing some discussion around Shelah’s result.We begin by reviewing definitions and basic facts concerning generating sequences TODD EISWORTH for pcf( A ), and again refer the reader to [1] if a more detailed discussion of thismaterial is desired. Definition 2.1.
Let A be a progressive set of regular cardinals.(1) If λ ∈ pcf( A ), then a subset B of A is a generator for λ ∈ A if(2.1) J ≤ λ [ A ] = J <λ [ A ] + B. (2) A generating sequence for pcf( A ) is a sequence h B λ : λ ∈ pcf( A ) i where B λ is a generator λ in A .(3) More generally, if A ⊆ C ⊆ pcf( A ), | C | < min( A ), and Λ is a subset ofpcf( A ), we say that h B λ : λ ∈ Λ i is a generating sequence for Λ in C if B λ is a generator for λ in C for all λ ∈ Λ, that is(2.2) J ≤ λ [ C ] = J <λ [ C ] + B λ for all λ ∈ Λ.Note that in (3), since C is progressive we know(2.3) pcf( C ) ⊆ pcf( A )and pcf( C ) will have a generating sequence, so the definition makes sense. Note aswell that if B λ is a generator for λ in C , then B λ ∩ A is a generator for λ in A . Whereimportant for clarity, we may write B λ [ C ] to emphasize that a set is a generatorfor λ in C . Definition 2.2.
Let A be a progressive set of regular cardinal, and suppose ¯ b is a generating sequence h B λ [ C ] : λ ∈ Λ i for Λ in C , where Λ ⊆ pcf( A ) and A ⊆ C ⊆ pcf( A ) with | C | < min( A ).(1) The sequence ¯ b is transitive if(2.4) θ ∈ B λ [ C ] ∩ Λ = ⇒ B θ [ C ] ⊆ B λ [ C ] . (2) The sequence ¯ b is closed if(2.5) λ ∈ Λ = ⇒ pcf( B λ [ C ]) ∩ C = B λ [ C ] . One of the nicest aspects of pcf theory is that generators actually exist for every λ ∈ pcf( A ) when A is a progressive set of regular cardinals, that is, generating se-quences for pcf( A ) in A exist. Obtaining closed and transitive generating sequencesis a little more delicate, though. Abraham and Magidor’s treatment of this materialshows that, given a progressive set of regular cardinals A and a suitable model N containing A , one can obtain a transitive generating sequence for N ∩ pcf( A ) in A .Claim 6.7 in Shelah’s [9] contains a stronger result that shows, among other things,that if | pcf( A ) | < min( A ), one can find a closed and transitive generating sequencefor pcf( A ) in pcf( A ). In other words, if pcf( A ) is progressive, then we can obtainextraordinarily well-behaved generating sequences. Our plan is to state a theorem,and then prove it under the additional assumption that pcf( A ) is progressive forthe set A in question. After we do this, we will work to show how the additionalassumption can be removed using more powerful results of Shelah. The exact assumptions are that N is κ -presentable where | A | < κ = cf κ < min( A ) PGRADING PSEUDOPOWER REPRESENTATIONS 7
Theorem 2.
Suppose A = S ξ<η A ξ is a progressive set of regular cardinals, andsuppose σ is a regular cardinal. Suppose (2.6) λ = max pcf( A ) , and (2.7) λ ∈ pcf σ -com ( A ) \ [ ξ<η pcf σ -com ( A ξ ) . Then can find a subset C of S ξ<η pcf σ -com ( A ξ ) such that (2.8) | C | ≤ η, and (2.9) λ ∈ pcf σ -com ( C ) . One should think of the above theorem as providing a template for what wemight call “reduction in size results”. The point is that we are replacing the set A with a set C whose size is under our control, and doing it so that λ is still capturedby pcf σ -com ( C ). As mentioned previously we first present a proof of this theoremusing the following strong assumption: Assumption:
The set pcf( A ) is progressive. Proof of Theorem 2 under the assumption.
Restricting to the case where pcf( A ) isprogressive lets us conclude pcf(pcf( A )) = pcf( A ), and therefore we have corre-sponding ideals J <τ [pcf( A )] and generators B τ [pcf( A )] for τ ∈ pcf( A ) such that(2.10) J ≤ τ [pcf( A )] = J <τ [pcf( A )] + B τ [pcf A ] . Furthermore, by Theorem 6.7 of [9], we may assume that the generating sequenceis transitive. The reader should keep in mind that the point here is that thegenerators are subsets of pcf( A ) that function for the full set pcf( A ), instead ofbeing constrained as subsets of A .Now recall we assume A is a progressive set of regular cardinals expressed as S ξ<η A ξ , the cardinal λ is max pcf( A ), and(2.11) λ ∈ pcf σ -com ( A ) \ [ ξ<η pcf σ -com ( A ξ ) . Note that if η were less than σ , then pcf σ -com ( A ) would simply be the union of thevarious pcf σ -com ( A ξ ) for ξ < η , and this violates (2.11). Thus, our assumptionsimply σ ≤ η , and we may also assume η ≤ | A | .For each ξ < η , there is a subset C ξ of pcf σ -com ( A ξ ) of cardinality less than σ such that(2.12) A ξ ⊆ [ τ ∈ C ξ B τ . This is Claim 6.7F of [9], and is a version of “pcf compactness” (see Theorem 4.11of [1]) tailored for σ -complete pcf. This is one of the many ways generators areused in pcf theory.Let C be the union of the C ξ for ξ < η . Clearly | C | ≤ σ · η = η , so we need onlyprove that λ is in pcf σ -com ( C ). We do not know that C is a subset of A , but we cannow use the fact that our transitive generating sequence functions for the entire set TODD EISWORTH pcf( A ) to apply Claim 6.7F of [9] again, and obtain a subset D of pcf σ -com ( C ) ofcardinality less than σ with(2.13) C ⊆ [ τ ∈ D B τ . This is the point where we need the generators to work in pcf( A ), as C need notbe a subset of A .We know that max pcf( C ) is at most λ because(2.14) max pcf(pcf( A )) = max pcf( A ) . Thus, if λ fails to be in pcf σ -com ( C ), it must be the case that(2.15) D ⊆ pcf σ -com ( A ) ∩ λ, 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 -completeideal on A generated by J <λ [ A ] must be a proper ideal. But for ǫ ∈ D , we know B ǫ ∩ A is in the ideal J <λ [ A ], and since | D | < σ it follows that A cannot be containedin the union of the sets B ǫ for ǫ in D , that is,(2.16) A * [ ǫ ∈ D B ǫ . On the other hand, though, for each ζ ∈ A , there are a τ ∈ C and ǫ ∈ D such that(2.17) ζ ∈ B τ , and(2.18) τ ∈ B ǫ . Here is where the transitivity of our generating sequence is used, because the con-junction of these two statements tells us(2.19) ζ ∈ B ǫ and therefore(2.20) A ⊆ [ ǫ ∈ D B ǫ , which contradicts (2.16). Thus, λ must be in pcf σ -com ( C ), and we are done. (cid:3) Now in general, it is still unknown if we can just eliminate the assumption thatpcf( A ) is progressive. The issue is that we may have | pcf( A ) | > | A | , and in sucha situation the techniques for building a nice generating sequence for all of pcf( A )simply don’t work. To push the proof of Theorem 2 through in ZFC , we need tomove from Claim 6.7 to Claim 6.7A in [9].The idea is that we should work to obtain a closed transitive generating sequencefor a certain progressive subset of pcf( A ), and argue that this set is rich enough tocontain all the generators required for the proof of Theorem 2 to go through. Wewill introduce some terminology here to make the discussion more understandable.Let us assume for now that A is a progressive set of regular cardinals, and that | A | < κ = cf( κ ) < min( A ). Let χ be a sufficiently large regular cardinal, and fix asequence ¯ N = h N α : α < κ i of elementary submodels of H ( χ ) such that • A ∈ N , • | N α | < κ PGRADING PSEUDOPOWER REPRESENTATIONS 9 • N α ∩ κ is an initial segment of κ , and • h N β : β ≤ α i ∈ N α +1 .and let(2.21) N := [ α<κ N α . This is a typical situation, so to save space we may in the future simply say that N (or ¯ N ) is as usual .Consider now the set N ∩ pcf( A ). This is of cardinality at most κ , and so isprogressive given our assumptions. Shelah’s Claim 6.7A in [9] provides us with aclosed transitive generating sequence for N ∩ pcf( A ) in N ∩ pcf( A ), which is almostenough for the argument to go through.Why almost? This is actually an important point that lies behind the complexityof the results we need to use, and so we will pause for a little explanation. Supposewe have a generating sequence for N ∩ pcf( A ) in N ∩ pcf( A ), and think about howwe used generators in the proof of Theorem 2. You will see they came up twicein arguments using compactness. First, we found a set C that indexed a coveringof A by the generators, and this part goes through fine under the assumptionswe just made. However, our proof that λ is in pcf σ -com ( C ) required us to use thecompactness property of generators once more in a proof by contradiction, and hereis where we run into difficulty. The issue is that the set C is built using objectsthat are not in N ( N does not have the generating sequence available), and so wehave no way of guaranteeing that C is in N . This means that there is no reason tobelieve that C can be covered by the generators at our disposal: we have generatorscorresponding to elements of N ∩ pcf( A ), but we have no reason to believe thatthis collection of generators will contain enough to cover the set C . This is subtle,but it is an important ingredient in Shelah’s work and critical for the argument.It also explains the complexity of the of the full version of Shelah’s Claim 6.7A: tomake the argument work, we will need a way of ensuring that the externally definedgenerators for N ∩ pcf( A ) have corresponding internal reflections that will give ussome control over the set C we build. This is the point of the next few definitions. Definition 2.3.
Let A , N , and h N α : α < κ i be as usual. A generating array for A over N is a doubly-indexed family(2.22) [¯ b ] = h B αλ : α < κ and λ ∈ N α ∩ pcf( A ) i such that(1) for each α < κ , the family ¯ b α = h B αλ : λ ∈ N α ∩ pcf( A ) i is a generatingsequence for N α ∩ pcf( A ) in N α ∩ pcf( A ) that is an element of N α +1 , and(2) for a fixed λ ∈ N ∩ pcf( A ), the sequence h B αλ : α < κ ∧ λ ∈ N α ∩ pcf( A ) i is increasing.We say that [¯ b ] is transitive (respectively closed) if each ¯ b α is transitive (respectivelyclosed).For α < κ , we should think of N α ∩ pcf( A ) as an approximation to the set N ∩ pcf( A ) that is an element of N α +1 , and the generators ¯ b α are an approximationin N α +1 to the final set of generators we are trying to build. The definition demandsthat these approximations increase with α < κ . Note as well that if λ is in N ∩ pcf( A ), then λ is in N α for all sufficiently large α < κ , and so B αλ will be defined for all sufficiently large α < κ . With this point of view, the following definition isa natural next step: Definition 2.4.
Let A , N , and ¯ N be as usual, and suppose [¯ b ] is a generatingarray for A over N . We define the limit of the array [¯ b ] to be the sequence(2.23) ¯ b = h B λ : λ ∈ N ∩ pcf( A ) i where each λ ∈ N ∩ pcf( A ), we define(2.24) B λ = [ α<κ ∧ λ ∈ N α B αλ . The limit of a generating array inherits nice properties from its components, andwe capture this in the next proposition.
Proposition 2.5.
Let [¯ b ] be a generating array for A over N , with correspondinglimit ¯ b .(1) ¯ b is a generating sequence for N ∩ pcf( A ) in N ∩ pcf( A ).(2) ¯ b is closed if each ¯ b α is closed.(3) ¯ b is transitive if each ¯ b α is transitive. Proof.
These all have easy proofs relying on the localization property of pcf (see,for example, Theorem 6.6 of [1]): if C is a progressive subset of pcf( A ) (possiblyof cardinality greater than that of A ) and λ ∈ pcf( C ), then λ ∈ pcf( C ) for somesubset C of C with | C | ≤ | A | . For example, suppose that λ is in N ∩ pcf( A ).Since | A | < κ and κ is regular, by the localization property we have(2.25) pcf( B λ ) = [ α<κ pcf( B αλ ) , and therefore max pcf( B λ ) is at most λ . Since N ∩ pcf( A ) is progressive, thereexists a generator B for λ in this set, so to finish the proof of (1), it suffices toprove(2.26) max pcf( B \ B λ ) < λ. If this fails, then by localization there is a subset C of B \ B λ such that | C | ≤ | A | and λ = max pcf( C ). Choose α < κ such that λ ∈ N α and C ⊆ N α ∩ pcf( A ). Since C and B λ are disjoint, it follows that(2.27) λ ∈ pcf( N α ∩ pcf( A )) \ B αλ , and this is a contradiction as B αλ is a generator for λ in N α ∩ pcf( A ). The proofs ofthe other two statements in the proposition are similar, and left to the reader. (cid:3) The next definition addresses the “internal reflection” requirement that we dis-cussed prior to Definition 2.3. The property tells us that the generating array [¯ b ]is large enough to cover sets that are members of N . Definition 2.6.
Let [¯ b ] be a generating array for A over N . We say that [¯ b ] hasthe internal covering property if given C ⊆ N α ∩ pcf( A ) in N α +1 and a regularcardinal σ ∈ N α +1 , there is a set D such that • D ∈ N α +1 , • D ⊆ N α +1 ∩ pcf σ -com ( C ), • | D | < σ , and PGRADING PSEUDOPOWER REPRESENTATIONS 11 • C ⊆ S λ ∈ D B α +1 λ .We are now ready to tackle Shelah’s result Claim 6.7A from [9], which we refor-mulate using our newly minted vocabulary. Theorem 3 (Claim 6.7A of [9], reformulated) . Let A be a progressive set of regularcardinals, and suppose κ is a regular cardinal such that | A | < κ < min( A ) . Givena sufficiently large regular cardinal χ and some x ∈ H ( χ ) , there are objects N , ¯ N = h N α : α < κ i , and [¯ b ] such that • A , N , and ¯ N are as usual, with x ∈ N , and • [¯ b ] is a closed and transitive generating array for A over N with the internalcovering property.Proof. The proof is just a sketch showing how we have reformulated Shelah’sClaim 6.7A. His result starts with a sequence of models of length κ ++ (which heassumes, without loss of generality, is less than min( A )). He then uses an I [ κ ++ ]-argument to pull out a “thin enough” sequence of models of length κ , and it is thissequence that is the N in our statement. Translating the rest of the conclusion ofShelah’s claim into our language of generating arrays is straightforward; we havesimply given a name to the object he constructs. (cid:3) The following corollary is an attempt to capture the power of the above resultin a way that removes the language of generating arrays completely. This corollaryis sufficient for our purposes, and for other similar arguments.
Corollary 2.7.
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 largeregular 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 = h B λ : λ ∈ N ∩ pcf( A ) i such that • ¯ b is a closed transitive generating sequence for N ∩ pcf( A ) in N ∩ pcf( A ),and • given a sequence ¯ A = h A ξ : ξ < η i of sets and a regular cardinal σ suchthat – ¯ A ∈ N – η and σ are less than κ , and – A ξ is a subset of N ∩ pcf( A ) for each ξ < η ,there is a family ¯ C = h C ξ : ξ < η i such that • ¯ C ∈ N , • C ξ is a subset of pcf σ -com ( A ξ ) of cardinality less than σ , and • A ξ ⊆ S λ ∈ C ξ B λ Proof.
We apply Theorem 3 to obtain a model N , sequence h N α : α < κ i , andgenerating array [¯ b ] as there, and let ¯ b be the limit of [¯ b ]. We know that ¯ b is aclosed transitive generating sequence for N ∩ pcf( A ) in N ∩ pcf( A ), so we need onlycheck the remaining property. Suppose ¯ A = h A ξ : ξ < η i is a sequence as in thestatement of the corollary. For those referring to [9], the various cardinals have a different notation there, for example,his σ is our κ , and his κ is our κ ++ . But any reader wrestling with [9] will have no troublereconciling his statement with ours. Choose α < κ such that ¯ A ∈ N α . Note that each A ξ is a subset of N α ∩ pcf( A )and an element of N α +1 , so by the internal covering property of [¯ b ], there is a set C ξ such that • C ξ ∈ N α +1 , • C ξ ⊆ N α +1 ∩ pcf σ -com ( A ξ ), • | C ξ | < σ , and • A ξ ⊆ S λ ∈ C ξ B α +1 λ .The above construction can be carried out in N α +2 using the sequences ¯ A and h B α +1 λ : λ ∈ N α ∩ pcf( A ) i , and therefore we can find the needed ¯ C in N . (cid:3) We now give a proof of Theorem 2 in
ZFC without the additional assumptionthat pcf( A ) is progressive, basing the argument on Corollary 2.7. Proof of Theorem 2.
Given A and the sequence h A ξ : ξ < η i , by the above corollarythere is a model N containing both of these objects that admits a correspondinggenerating sequence ¯ b . Note that since κ + 1 ⊆ N , we know that A is a subsetof 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 is a subset of N ∩ pcf( A ) too.By properties of ¯ b , there is a sequence h C ξ : ξ < η i in N such that each C ξ is asubset of pcf σ -com ( A ξ ) of cardinality less than σ and(2.28) A ξ ⊆ [ λ ∈ C ξ B λ . Now let(2.29) C = [ ξ<η C ξ . Clearly | C | ≤ η · σ = η , and C is definable from the sequence h C ξ : ξ < η i in N .We now show that λ is in pcf σ -com ( C ), which will finish the proof. To do this, weuse the internal reflection property of ¯ b once more, this time applied to the singleset C ∈ N . We obtain a set D ∈ N of cardinality less than σ such that(2.30) D ⊆ pcf σ -com ( C ) ⊆ pcf σ -com ( A )and(2.31) C ⊆ [ λ ∈ D B λ . Once we have D , the rest of the proof is exactly the same. (cid:3) Eventually Γ( θ, σ ) -closed cardinals In this section we will work with cardinals that are strong limits in a sensemeasured by pseudopowers. Our aim is to generalize one of the main results ofChapter VIII of
Cardinal Arithmetic , where Shelah addresses basic questions aboutimproving representations of cardinals. Simply recalling a small part of what Shelahestablishes provides us with a good starting point.
PGRADING PSEUDOPOWER REPRESENTATIONS 13
Theorem 4 (Shelah [7]) . Suppose ℵ < cf( µ ) ≤ θ < µ , and for every sufficiently η < µ , (3.1) cf( η ) ≤ θ = ⇒ pp θ ( η ) < µ. Then PP θ ( µ ) = PP Γ(cf( µ )) ( µ ) . In fact, any λ ∈ PP θ ( µ ) can be represented as thetrue cofinality of Q A/J bd [ A ] where A is a cofinal subset of µ ∩ Reg of cardinal-ity cf( µ ) . The above is part of Corollary 1.6 on page 321 of [8], and we will shortly deriveit from our own work in this section. For now, we wish to highlight the assump-tion (3.1), as it expresses that µ is (in a weak sense) a type of strong limit cardinaland note that we will be working with such assumptions a lot in this section. Noteas well that the conclusion of the above theorem can be described in general termsas “upgrading” the representation of λ to one with a very simple form: we areable to move from an arbitrary representation of λ based on a set of cardinality atmost θ to one based on a set of cardinality cf( µ ), with the added bonus that theideal used in the representation is as simple as possible. We will be hunting similarupgrades in the presence of assumptions related to (3.1).Moving ahead, with (3.1) and the above theorem as motivation, the followingdefinition is natural. Definition 3.1.
Let σ and θ be regular cardinals, and suppose µ is singular with σ ≤ cf( µ ) < θ < µ .(1) We say that µ is eventually Γ( θ, σ ) -closed if for all sufficiently large ν < µ ,if ν is singular with σ ≤ cf( ν ) < θ then pp Γ( θ,σ ) ( ν ) < µ .(2) We say that µ is eventually Γ( θ, σ )-closed beyond η if η < µ and the aboveholds for all ν between η and µ .The following elementary result captures the combinatorial essence of the pre-ceding concept. The proof is left as an exercise, but should cause no difficulty to areader who has made it to this point. Proposition 3.2.
The following conditions are equivalent:(1) µ is Γ( θ, σ )-closed beyond η .(2) If A is a bounded subset of ( η, µ ) ∩ Reg with | A | < θ , then(3.2) sup pcf σ -com ( A ) < µ Turning to the main topic of this section, we begin with an application of ourTheorem 2 to the question of representation.
Theorem 5.
Suppose σ ≤ cf( µ ) < θ < µ with σ and θ regular. If µ is eventually Γ( θ, σ ) -closed, then (3.3) PP Γ( θ,σ ) ( µ ) = PP Γ( σ ) ( µ ) . Before we give the proof, note that this result is a natural counterpart to The-orem 4, with the only difference being the inclusion of the parameter σ . Thistheorem asks for a weaker closure condition than (3.1) in the situation where σ isuncountable, but it also has a weaker conclusion: it says only that if µ is eventuallyΓ( θ, σ )-closed, then any cardinal representable at µ via a σ -complete ideal on a setof 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 obtaina 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 isΓ( 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 h µ α : α < κ i increasing andcofinal in µ , if we define(3.4) A α := A ∩ µ α , then the hypotheses of Theorem 2 are satisfied by h A α : α < cf( µ ) i and λ , and weobtain a set(3.5) C ⊆ [ α< cf( µ ) pcf σ -com ( A α )of cardinality less than θ such that(3.6) λ ∈ pcf σ -com ( C ) . (cid:3) We can do better than this, though. The next lemma lies at the heart of many ofour results. It shows that with marginally stronger assumptions than we use in The-orem 5, we are able to obtain “well-organized” representations of cardinals, whichwe use in which the corresponding ideals satisfy stronger completeness conditions.
Lemma 3.3.
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(3.7) C = { λ ας : α < cf( µ ) and ς < σ ∗ } of regular cardinals less than µ such that • sup pcf σ -com { λ βς : β < α and ς < σ ∗ } < µ for each α < cf( µ ), and • if X is an unbounded subset of cf( µ ) then(3.8) λ = max pcf( { λ ας : α ∈ X and ς < σ ∗ } )and(3.9) λ ∈ pcf τ -com ( { λ ας : α ∈ X and ς < σ ∗ } ) . The above lemma can be viewed as an relative of Shelah’s result Theorem 4. Wewill discuss this after the proof, and even show how Theorem 4 follows easily fromthe lemma.
Proof.
Suppose A and J are a Γ( θ, σ )-representation of λ at µ . Just as in the proofof Theorem 5, we may assume µ is Γ( θ, σ )-closed beyond min( A ) and that λ ismax pcf( A ). Let h µ α : α < cf( µ ) i be an increasing sequence cofinal in µ , and let A α be A ∩ µ α .We implement the argument of Theorem 2 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 in Corollary 2.7 with N containing all objects under discussionin the preceding paragraph. By properties of ¯ b , in the model N there is a sequenceof 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 PGRADING PSEUDOPOWER REPRESENTATIONS 15 • A α ⊆ S λ ∈ C α B λ .Since the sequence h A α : α < cf( µ ) i is increasing, it follows that whenever X is anunbounded subset of cf( µ ) we have(3.10) A ⊆ [ α ∈ X [ λ ∈ C α B λ , and so the transitivity arguments from the proof of Theorem 2 tells us(3.11) λ = max pcf [ α ∈ X C α ! and(3.12) λ ∈ pcf τ -com [ α ∈ X C α ! . Since each C α has cardinality less than σ < cf( µ ), we can (by passing to anunbounded subset of cf( µ )) assume each C α is of some fixed cardinality σ ∗ < σ ,say(3.13) C α = { λ ας : ς < σ ∗ } , and we are done by letting C be the union of the sets C α . (cid:3) Note that we do not claim that the sets C α are disjoint, and they may very welloverlap. It is helpful to visualize C as an array of regular cardinals with cf( µ ) rowsand σ ∗ columns. In this interpretation, row α corresponds to C α , and we get acorresponding column for each fixed ς < σ ∗ .The pcf structure of C transfers to the index set Λ = cf( µ ) × σ ∗ in the naturalway, and we may define an ideal J on Λ by(3.14) X ∈ J ⇐⇒ max pcf (cid:0) { λ ας : ( α, ς ) ∈ X } (cid:1) < λ ∗ . With this point of view, we see that for X ⊆ cf( µ ),(3.15) { ( 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 becauseof (3.12).It is helpful to look back and compare our situation with the conclusion of The-orem 4. We have not managed to represent λ as the true cofinality of a product ofcardinals modulo the bounded ideal, but we have come close! What goes wrong isthat the rows C α in our array are not singletons, and instead all we know is thatthey all have a fixed cardinality σ ∗ less than σ . What Shelah does in [7] is notethat if this cardinality happens to be finite, then we can improve the situation andget the ideal to consist of just the bounded sets: Corollary 3.4 (Theorem 4, due to Shelah [7]) . Suppose ℵ < cf( µ ) ≤ θ < µ and µ is eventually Γ( θ + , ℵ )-closed, that is, for all sufficiently large ν < µ ,(3.16) cf( ν ) ≤ θ = ⇒ pp θ ( 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 boundedsubsets of C . Proof.
Suppose λ = tcf Q 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 3.3 to Γ( θ + , ℵ )and obtain n < ω and C = h λ αi : i ≤ n i such that • sup pcf { λ βi : β < α and i ≤ n } is less than µ for each α < cf( µ ), and • λ = max pcf( { λ αi : 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(3.17) λ = max pcf { λ αi : α ∈ Y } for every unbounded Y ⊆ X . This is enough, as letting D = { λ αi : i ∈ Y } , it followseasily 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(3.18) max pcf { λ αi : α ∈ Y } < λ. Working by induction, we find a single unbounded X ⊆ cf( µ ) such that(3.19) ( ∀ i ≤ n ) (max pcf { λ αi : α ∈ X } ) < λ. But then(3.20) max pcf { λ αi : α ∈ X and i ≤ n } < λ, as any ultrafilter on this set must contain one of the columns, and we have acontradiction. (cid:3) We note that Shelah obtains even nicer representations in Chapter VIII of
Car-dinal Arithmetic , but the above observation is at the heart of his argument. Movingon to the more general situation, we now apply Lemma 3.3 to obtain the followingimprovement of Theorem 5.
Theorem 6.
Suppose µ is eventually Γ( θ, σ ) -closed for some regular σ and θ with (3.21) σ < cf( µ ) < θ < µ, and τ < υ are regular cardinals in the interval [ σ, cf( µ )] . If (3.22) τ ≤ ρ < υ = ⇒ cov( ρ, ρ, σ, < cf( µ ) then (3.23) PP Γ( θ,τ ) ( µ ) = PP Γ( υ ) ( µ ) . Before presenting a proof of this theorem, we note that (3.22) will hold if υ isless than τ + ω , so at a minimum the theorem allows us to find representations thatare “more complete”, as well as making sure that the size of the set involved is atmost cf( µ ). In contrast with Theorem 5, note we assume that σ is strictly less thanthe cofinality of µ . Proof.
Suppose λ is Γ( θ, τ )-representable at µ , and apply Lemma 3.3 to obtain acardinal σ ∗ < σ and a set C of regular cardinals λ ας (for α < cf( µ ) and ς < σ ∗ ) asthere. In particular, if X is any unbounded subset of cf( µ ), then(3.24) λ ∈ pcf τ -com { λ ας : α ∈ X and ς < σ ∗ } . It suffices to prove that the υ -complete ideal on C generated by J <λ ∗ [ C ] is proper.Suppose by way of contradiction this is not the case. Then there is a leastcardinal ρ < υ such that for some sequence of sets h D i : i < ρ i and unbounded X ⊆ cf( µ ), we have PGRADING PSEUDOPOWER REPRESENTATIONS 17 • D i ∈ J <λ ∗ [ C ] for each i < ρ , and • { λ ας : α ∈ X and ς < σ ∗ } ⊆ S i<ρ D i .Clearly τ must be less than or equal to ρ , and by shrinking C if necessary we mayassume there are such sets D i for i < ρ with(3.25) C ⊆ [ i<ρ D i . Our assumptions tell us that cov( ρ, ρ, σ,
2) is less than cf( µ ), so there is a family P such that • P ⊆ [ ρ ] <ρ , • |P| < cf( µ ), and • ( ∀ U ∈ [ ρ ] <σ )( ∃ V ∈ P )[ U ⊆ V ].For each α < cf( µ ), there is a set Y α ∈ [ ρ ] <σ such that(3.26) C α = { λ ας : ς < σ ∗ } ⊆ [ i ∈ Y α D i , Since |P| < cf( µ ), there is a single set Y ∈ P and an unbounded subset X of cf( µ )such that(3.27) [ α ∈ X C α ⊆ [ i ∈ Y D i . But now we have a contradiction, as we have shown that S α ∈ X C α can be coveredby | Y | < ρ sets from J <λ ∗ [ C ]. (cid:3) Again, note that Theorem 5 tells us only that PP Γ( θ,τ ) ( µ ) = PP Γ( τ ) ( µ ) underthe same assumptions as Theorem 6, and that the improvement we obtain here isthat we find a representation using an υ -complete ideal on a set of size cf( µ ) ratherthan simply a τ -complete one.We close this section with the following corollary, which gives us a little infor-mation about the assumptions we make relating σ , τ , and υ . Corollary 3.5.
Suppose µ is eventually Γ( σ )-closed, where σ < cf( µ ). If τ < υ are regular cardinals in the interval [ σ, cf( µ )] and(3.28) PP Γ( υ ) ( µ ) $ PP Γ( τ ) ( µ ) , then there is a singular cardinal ρ of cofinality less than σ such that(3.29) τ < ρ < υ ≤ cf( µ ) < cov( ρ, ρ, σ, , hence(3.30) τ < ρ < υ ≤ cf( µ ) < cf([ ρ ] <σ , ⊆ ) ≤ ρ <σ . If we work with cardinals whose cofinality is ℵ n for some n , then the abovesituation cannot happen because there is no place for such a cardinal ρ . Thus, thefollowing is immediate. Corollary 3.6.
Suppose µ is singular with ℵ < cf( µ ) < ℵ ω . If µ is eventuallyΓ( θ, σ )-closed for some σ < cf( µ ), then(3.31) PP Γ( θ,σ ) ( µ ) = PP Γ(cf( µ )) ( µ ) . The Psuedopower Dichotomy
In the preceding section we analyzed the behavior of pp Γ( θ,τ ) ( µ ) where µ is asingular cardinal that is eventually Γ( θ, σ )-closed and σ ≤ τ ≤ cf( µ ). The analysisshowed that if a cardinal is Γ( θ, τ )-representable at µ , then it is has a σ -completerepresentation using the minimum possible size, cf( µ ), and the completeness of therepresentation can be increased if σ < τ . In this section, we will analyze whathappens when µ is not eventually Γ( θ, σ )-closed. We start with the following resultwhich formulates a fundamental dichotomy about singular cardinals. This is not anew idea (in fact, the result we present is a relative of Fact 1.9 in [7]), and Shelahhas made and used similar observations in other places. However, such dichotomiesare extraordinarily useful for proving theorems in ZFC , as they allow one to analyzesituations by breaking into cases where each option carries non-trivial information.We will refer to this result as the Pseudopower Dichotomy . Lemma 4.1 ( 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 followstatements hold: Option 1: µ is eventually Γ( θ, σ )-closed. Option 2: µ is a limit of eventually Γ( θ, σ )-closed cardinals η for which(4.1) σ ≤ cf( η ) < θ, and(4.2) PP Γ( θ,σ ) ( µ ) ⊆ PP Γ( θ,σ ) ( η ) . Proof.
It is clear that the two options are mutually exclusive. Suppose that Option 1fails for µ . By definition, this means that for each ξ < µ there is a singular cardinal η greater than ξ such that(4.3) σ ≤ cf( η ) < θ, and(4.4) µ ≤ pp Γ( θ,σ ) ( η ) . By Inverse Monotonicity (Proposition 1.4), this implies(4.5) PP Γ( θ,σ ) ( µ ) ⊆ PP Γ( θ,σ ) ( η ) . Note as well that if η is the LEAST such cardinal above ξ , then η is Γ( θ, σ )-closedbeyond ξ , and therefore Option 2 holds. (cid:3) It is easy to see how the preceding dichotomy might be useful. For example, if λ is Γ( θ, σ )-representable at µ and Option 1 holds, then our work in the precedingsection applies. If, on the other hand, Option 2 is in play, then we know that λ has nice representations for many cardinals below µ , and we will be able tocombine these representations (using the Continuity Property of pseudopowers fromProposition 1.5) to obtain representations of λ at µ itself. The following definitionwill help us to make this idea clear. Definition 4.2.
Suppose µ is singular, and σ < θ are regular with σ ≤ cf( µ ) < θ .We define X Γ( θ,σ ) ( µ ) to be the collection of cardinals η satisfying • σ ≤ cf( η ) < θ < η < µ , • η is eventually Γ( θ, σ )closed, and PGRADING PSEUDOPOWER REPRESENTATIONS 19 • PP Γ( θ,σ ) ( µ ) ⊆ PP Γ( θ,σ ) ( η ).This set might very well be empty, but note that the Pseudopower Dichotomy canbe 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 exerts an influence over representations at µ .For example, we have the following observation which yields the same conclusionas Theorem 5. Proposition 4.3.
Suppose { η ∈ X Γ( θ,σ ) ( µ ) : cf( η ) ≤ cf( µ ) } is unbounded in µ .Then(4.6) PP Γ( θ,σ ) ( µ ) = PP Γ( σ ) ( µ ) . Proof.
Suppose λ is Γ( θ, σ )-representable at µ . Since Option 2 holds, we knowthat λ is also Γ( θ, σ )-representable at each η ∈ X . Given such an η , we canapply Theorem 5 to conclude λ is also Γ((cf η ) + , σ ))-representable at η . By theassumption of the lemma, we have(4.7) µ = sup { η < µ : σ ≤ cf( η ) < (cf( µ )) + and λ ∈ PP Γ((cf µ ) + ,σ ) ( η ) } , and hence(4.8) λ ∈ PP Γ( σ ) ( µ )by an application of Continuity (Proposition 1.5). (cid:3) The next result builds on this idea, and shows how assumptions on the structureof X Γ( θ,σ ) ( µ ) let us improve representations of cardinals in PP Γ( θ,σ ) ( µ ). Theorem 7.
Suppose µ is singular, and σ , τ , χ , and θ are regular cardinals satis-fying (4.9) σ < τ ≤ cf( µ ) < χ ≤ θ < µ. If (4.10) µ = sup { η ∈ X Γ( θ,σ ) ( µ ) : sup { cov( ρ, ρ, σ, + : σ ≤ ρ < τ } ≤ cf( η ) < χ } , then (4.11) PP Γ( θ,σ ) ( µ ) = PP Γ( χ,τ ) ( µ ) , hence (4.12) pp Γ( θ,σ ) ( µ ) = pp Γ( χ,τ ) ( µ ) Proof.
Clearly PP Γ( χ,τ ) ( µ ) is contained in PP Γ( θ,σ ) ( µ ), so we will show the reverseinclusion holds. To do this, suppose η satisfies the following conditions: • λ is Γ( θ, σ )-representable at η , • η is eventually Γ( θ, σ )-closed, and • σ ≤ ρ < τ = ⇒ cov( ρ, ρ, σ, < τ ≤ cf( η ).An application of Theorem 6 tells us that λ is Γ((cf η ) + , τ )-representable at η , andsince cf( η ) < χ we know(4.13) λ ∈ PP Γ( χ,τ ) η. We have assumed that the set of such η is unbounded in µ , so Continuity (Proposi-tion 1.5) tells us that λ is Γ( χ, τ )-representable at µ as well. (cid:3) Applications of the Pseudopower Dichotomy
In this section, we put together pieces from our preceding work to obtain theo-rems in
ZFC based on the Pseudopower Dichotomy. Our first result echoes Theo-rem 4, as it speaks about PP θ ( µ ).Clearly, we know(5.1) PP( µ ) ∪ PP Γ( θ + , cf( µ )) ⊆ PP θ ( µ )because of the definitions involved (recall PP( µ ) is defined as PP cf( µ ) ( µ )). The firsttheorem in this section shows that the two sides of (5.1) are actually equal. Theorem 8.
Suppose µ is singular, and cf( θ ) ≤ θ < µ . Then (5.2) PP θ ( µ ) = PP( µ ) ∪ PP Γ( θ, cf( µ )) ( µ ) . hence (5.3) pp θ ( µ ) = pp( µ ) + pp Γ( θ + , cf( µ )) ( µ ) . We make a couple of comments before presenting the proof. First, note that theabove 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 minimumpossible size) or a cf( µ )-complete ideal (the maximum possible). These optionsare not mutually exclusive (for example, both hold simultaneously for µ + ), but thepower 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 bothintervals of regular cardinals, the equation (5.2) implies that PP θ ( µ ) must in factbe equal to one of these two sets. Proof of Theorem 8.
We may assume that µ has uncountable cofinality, as other-wise (5.2) holds automatically. We apply the Pseudopower Dichotomy to Γ( θ + , ℵ ).If µ is eventually Γ( θ + , ℵ )-closed, then Corollary 3.4 gives us more than we need,as any λ in PP θ ( µ ) can be represented using the bounded ideal on a set of cardi-nality cf( µ ) cofinal in µ ∩ Reg . It follows that in this situation, all three of the setsfrom (5.2) are equal, and(5.4) PP Γ( θ + , ℵ ) ( µ ) = PP Γ(cf( µ )) ( µ ) ⊆ PP( µ ) = PP Γ( θ + , cf( µ )) ( µ ) , which is more than we require.Thus, we may assume that Option 2 of the Pseudopower Dichotomy is in forceand the set X Γ( θ + , ℵ ) ) is unbounded in µ . Abbreviating this set as “ X ”, we splitinto two cases: Case a : { η ∈ X : cf( η ) ≤ cf( µ ) } is unbounded in µ In this situation, Proposition 4.3 tells us(5.5) PP Γ( θ + , ℵ ) ( µ ) = PP( µ ) , and (5.2) is immediate. Case b : { η ∈ X : cf( η ) > cf( µ ) } is unbounded in µ .In this situation, we apply Theorem 7 with χ = θ + , υ = cf( µ ), and σ and τ bothequal to ℵ to conclude(5.6) PP θ ( µ ) = PP Γ( θ + , cf( µ )) ( µ ) , PGRADING PSEUDOPOWER REPRESENTATIONS 21 which establishes (5.2). Note that Theorem 7 does apply here: since σ = ℵ , weknow cov( ρ, ρ, σ,
2) = ρ for any ρ < cf( µ ).Clearly at least one of these cases must happen, and therefore (5.2) holds underOption 2 of the Pseudopower Dichotomy , finishing the proof. (cid:3)
The preceding theorem and proof can be summarized as follows:(1) If µ is eventually Γ( θ + , ℵ )-closed, then all three sets in (5.2) are equal, and(5.7) PP Γ( θ + , ℵ ) ( µ ) = PP( µ ) = PP Γ( θ + , cf( µ )) ( µ ) = PP Γ(cf( µ )) ( µ ) . (2) If µ is not eventually Γ( θ + , ℵ )-closed, then X Γ( θ + , ℵ ) ( µ ) is unbounded in µ ,and we can break into the following subcases:(a) If the set of η ∈ X Γ( θ + , ℵ ) ( µ ) with cf( η ) ≤ cf( µ ) is unbounded in µ ,then(5.8) PP θ ( µ ) = PP( µ ) . (b) If the set of η ∈ X Γ( θ + , ℵ ) ( µ ) with cf( η ) > cf( µ ) is unbounded in µ ,then(5.9) PP θ ( µ ) = PP Γ( θ + , cf( µ )) ( µ ) . Again, (1) and (2) form a true dichotomy, but nothing prevents 2(a) and 2(b) fromholding simultaneously. The following corollary reformulates the above discussionin less technical language:
Corollary 5.1. If µ is singular and λ is representable at µ (for some θ < µ ), thenat least one of the following is true:(1) either λ can be represented at µ using a set of cardinality cf( µ ),(2) or λ can be represented at µ using a cf( µ )-complete ideal.The next theorem examines the more general situation where σ may be un-countable. The conclusion is weaker than that of Theorem 8, but the fact that σ isuncountable will allow us to use the cov vs. pp Theorem to transfer the result intoa statement about covering numbers. Theorem 9.
Suppose µ is singular, and let σ and θ be regular cardinals with σ < cf( µ ) < θ < µ . Then (5.10) PP Γ( θ,σ ) ( µ ) = PP Γ( σ ) ( µ ) ∪ PP Γ( θ,τ ) ( µ ) for any regular τ ∈ ( σ, cf( µ )] such that (5.11) σ ≤ ρ < τ = ⇒ cov( ρ, ρ, σ, < cf( µ ) . In particular, for such τ we have (5.12) pp Γ( θ,σ ) ( µ ) = pp Γ( σ ) ( µ ) + pp Γ( θ,τ ) ( µ ) . Proof.
The proof mirrors that of Theorem 8. If µ is eventually Γ( θ, σ )-closed,then all three sets mentioned in (5.10) are equal, and the result follows. Thus, wemay assume that Option 2 of the Pseudopower Dichotomy is in force, and the set X = X Γ( θ,σ ) ( µ ) from Definition 4.2 is unbounded in µ . We break into cases just asin Theorem 8, that is, we conser Case (a):
The set of η ∈ X with cf( η ) ≤ cf( µ ) is unbounded in µ .and Case (b):
The set of η ∈ X with cf( µ ) < cf( η ) is unbounded in µ .In Case (a), Proposition 4.3 tells usPP Γ( θ,σ ) ( µ ) = PP Γ( σ ) ( µ ) , while Theorem 7 handles Case (b) and tells us that in this situation, we will have(5.13) PP Γ( θ,σ ) ( µ ) = PP Γ( θ,τ ) ( µ ) . (cid:3) We conclude this section with a pair of corollaries that apply the above theoremin special situations. In particular, note that condition (5.11) holds if σ ≤ τ < σ + ω ,so we are able to conclude the following: Corollary 5.2.
Suppose µ is singular, and σ and θ are regular cardinals such that σ + n ≤ cf( µ ) < θ < µ. Then(5.14) PP Γ( θ,σ ) ( µ ) = PP Γ( σ ) ( µ ) ∪ PP Γ( θ,σ + n ) ( µ ) , and thus(5.15) pp Γ( θ,σ ) ( µ ) = pp Γ( σ ) ( µ ) + pp Γ( θ,σ + n ) ( µ ) . In particular, if σ < cf( µ ), then(5.16) pp Γ( θ,σ ) ( µ ) = pp Γ( σ ) ( µ ) + pp Γ( θ,σ + ) ( µ ) . This is similar in spirit to results we saw earlier: if we know a cardinal is Γ( θ, σ )-representable at µ with σ < cf( µ ), then either it is representable using a σ -completeideal on a set of cardinality cf( µ ) (so the size is as small as possible), or it isrepresentable on a set of cardinality less than θ , but using a σ + -complete ideal(so we are able to find a representation with greater completeness). The equation(5.16) will be important in the next section, in which we use work of Gitik to showthat both terms appearing on the right-hand side of the equality are needed.6. Independence Results
This short section focus on obtaining independence results complementary tothe theorems we established in the previous section. Our focus will be the equation(6.1) pp Γ( θ,σ ) ( µ ) = pp Γ( σ ) ( µ ) + pp θ,σ + ( µ )established in Corollary 5.2. This section relies almost completely on recently pub-lished work of Moti Gitik [4]. Since we do not have the expertise to discuss hisproof in detail, our approach will be to quote his results liberally. We start withhis main theorem: Theorem 10 (Theorem 1.3 of [4]) . Assume GCH. Let η be an ordinal and δ bea regular cardinal. Let h κ α : α < η i be an increasing sequence of strong cardinals,and let λ be a cardinal greater than the supremum of { κ α : α < η } . Then there isa cardinal preserving extension in which, for every α < η , (1) cf( κ α ) = δ , and (2) pp( κ α ) ≥ λ . PGRADING PSEUDOPOWER REPRESENTATIONS 23
The final section of his paper (Section 8) is the part most relevant for us, as heexamines the cardinal arithmetic structure of his model in some detail. We willfollow his notation, and point out what need.We assume that δ and η are regular cardinals such that ℵ ≤ δ and δ + < η , andwe work in the generic extension V [ G ] from Theorem 10. 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 littlemore: as Gitik notes prior to his Proposition 8.11, in fact we have the strongerresult that(6.2) pp Γ( δ ) ( κ α ) ≥ λ as all of the ideals involved in the computation are δ -complete.For each α < η , we let(6.3) ¯ κ α = sup { κ β : β < α } . In V [ G ], if α < η is a limit ordinal, then ¯ κ α is singular with cofinality less than η . Our choices of δ and η guarantee that there are limit ordinals α < η withcf( α ) greater than δ , and others of cofinality less than δ but greater than ω . Thecorresponding ¯ κ α for these two sorts of α are the places of interest to us, and weanalyze each situation separately. Again, relying on Gitik’s work we have: Proposition 6.1.
Let α be a limit ordinal less than η , and let V [ G ] be as inTheorem 10. Then the model V [ G ] satisfies:(1) If cf( α ) < δ then(6.4) pp(¯ κ α ) < pp Γ( δ + , cf( α )) (¯ κ α ) . (2) If δ < cf( α ) then(6.5) pp Γ( δ + ) (¯ κ α ) < pp Γ( δ ) (¯ κ α ) . Proof.
For (1), suppose α < η is a limit ordinal of cofinality less than δ . We knowthat { κ β : β < α } is cofinal in ¯ κ α , and each κ β is singular of cofinality δ with(6.6) pp Γ( δ ) ( κ β ) ≥ λ, hence(6.7) pp Γ( δ + , cf( α )) ( κ β ) ≥ λ. An application of Continuity tells us(6.8) pp Γ( δ + , cf( α )) (¯ κ α ) ≥ λ as well. On the other hand, Gitik’s Proposition 8.6 tells us(6.9) pp(¯ κ α ) = κ α , and so(6.10) pp(¯ κ α ) = κ α < λ ≤ pp Γ( δ + , cf( α )) (¯ κ α )as required.For (2), suppose α < η is a limit ordinal of cofinality greater than δ . Again, theset { κ β : β < α } is unbounded in ¯ κ α , and so an application of continuity tells us(6.11) pp Γ( δ ) (¯ κ α ) ≥ λ. On the other hand, Gitik’s Proposition 8.11 tells us(6.12) pp Γ( δ + ) (¯ κ α ) ≤ κ α , and (6.5) follows immediately. (cid:3) Let us now return to the equation (6.1), and work with Gitik’s extension V [ G ]in the situation where δ is at least ω and η is greater than δ + (these restrictionsare simply to make sure we have enough room to manipulate parameters of interestto us).One the one hand, if we define µ = ¯ κ ω , θ = δ + , and σ = ℵ , then by equa-tion (6.4), we have(6.13) pp Γ( σ ) ( µ ) ≤ pp( µ ) = pp(¯ κ α ) < pp Γ( δ + , cf( α )) (¯ κ α ) = pp Γ( θ,σ + ) ( µ ) . and so for this choice of parameters we have(6.14) pp Γ( θ,σ ) ( µ ) = pp Γ( θ,σ + ) ( µ ) > pp Γ( σ ) ( µ ) . On the other hand, if we let µ = ¯ κ δ + , θ = δ + , and σ = δ , then from equation (6.5)we conclude(6.15) pp Γ( σ + ) ( µ ) < pp Γ( σ ) ( µ ) , and so (remembering that µ is of cofinality θ ) we have(6.16) pp Γ( θ,σ ) ( µ ) = pp Γ( σ ) ( µ ) > pp Γ( θ,σ + ) ( µ ) . Taken together, (6.14) and (6.16) show us that both summands in equation (6.1)are important if we want a theorem that holds in
ZFC . We have not pushed theanalysis of the cardinal arithmetic in Gitik’s model beyond what was presentedabove; it may be that there are similar examples for many other values of θ and σ .7. Consequences for covering numbers
In this final section, we look at what Theorem 9 tells us about covering numbersat singular cardinals. Once gain, work in the situation where σ and θ are infiniteregular cardinals, and µ is a singular cardinal satisfying(7.1) σ ≤ cf( µ ) < θ < µ. Recall that the covering number cov( µ, µ, θ, σ ) is defined to be the minimumcardinality 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(7.2) X ⊆ [ Y . Recall as well that the cov vs. pp Theorem of Shelah (Theorem 1 mentioned in ourintroduction) tells us that if σ is uncountable, then(7.3) cov( µ, µ, θ, σ ) = pp Γ( θ,σ ) ( µ ) . The following theorem translates Theorem 9 into the language of covering num-bers.
Theorem 11.
Suppose µ is a singular cardinal, and let σ and θ be regular cardinalssuch that (7.4) ℵ < σ ≤ cf( µ ) < θ. Then (7.5) cov( µ, µ, θ, σ ) = cov( µ, µ, (cf µ ) + , σ ) + cov( µ, µ, θ, σ + ) . PGRADING PSEUDOPOWER REPRESENTATIONS 25
Proof.
The theorem is trivial if σ is the cofinality of µ , so we assume cf( µ ) is strictlyless than σ . By (5.16), we know(7.6) pp Γ( θ,σ ) ( µ ) = pp Γ( σ ) ( µ ) + pp Γ( θ,σ + ) ( µ ) . Because σ is uncountable, we invoke Theorem 1 and convert the pseudopowers intocovering numbers, yielding(7.7) cov( µ, µ, θ, σ ) = cov( µ, µ, (cf µ ) + , σ ) + cov( µ, µ, θ, σ + )as required. (cid:3) Similarly, based on Corollary 5.2, we have the following:
Corollary 7.1.
Suppose µ is a singular cardinal such that ℵ < cf( µ ) < ℵ ω . Forany regular cardinals σ and θ with(7.8) ℵ < σ ≤ cf( µ ) < θ < µ, we have(7.9) cov( µ, µ, θ, σ ) = cov( µ, µ, (cf µ ) + , σ ) + cov( µ, µ, θ, cf( µ )) . It is a little difficult to see exactly what this means simply based on the defi-nitions. Note that if µ is singular of cofinality ℵ , then Corollary 5.2 implies theequation deployed for shock effect in the first part of the paper:(7.10) cov( µ, µ, ℵ , ℵ ) = cov( µ, µ, ℵ , ℵ ) + cov( µ, µ, ℵ , ℵ ) . There are a few ways to think about what this means. One way is to picture itas expressing a trichotomy about interactions between elementary submodels andsubsets of µ . To see why, let χ be a sufficiently large regular cardinal, and suppose M is an elementary submodel of H ( χ ) such that containing µ and such that(7.11) | M | + 1 ⊆ M. If we let P be M ∩ [ µ ] <µ , then EXACTLY one of the following three things MUSToccur:(1) Some X ∈ [ µ ] ℵ cannot be covered by a union of ℵ sets from P .(2) Every X ∈ [ µ ] ℵ can be covered by a union of ℵ sets from P .(3) Every X ∈ [ µ ] ℵ is covered by a union of ℵ sets from P , but some Y ∈ [ µ ] ℵ cannot be covered by a union of ℵ sets from P .The above is quite easy: if we let κ be the cardinality of M , then (7.10) tells usthat exactly one of the following must be true:(1) ′ κ < cov( µ, µ, ℵ , ℵ ), or(2) ′ cov( µ, µ, ℵ , ℵ ) ≤ κ , or(3) ′ cov( µ, µ, ℵ , ℵ ) ≤ κ < cov( µ, µ, ℵ , ℵ )The rest follows easily.Reorganizing this a little, and considering more general values of the parameters,given such a model M , 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 aunion of at most σ sets from M ∩ [ µ ] <µ , then in fact every member of [ µ ] <θ can becovered by fewer than σ sets from M ∩ [ µ ] <µ . It is unclear if such considerations haveapplications, other than simply imparting surprise by being true, but we intend tocontinue this line of investigation in future work. References [1] Uri Abraham and Menachem Magidor. Cardinal arithmetic. In
Handbook of set theory. Vols.1, 2, 3 , pages 1149–1227. Springer, Dordrecht, 2010.[2] Todd Eisworth. Xxxx.
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Introduction to cardinal arithmetic . Birkh¨auser AdvancedTexts: Basler Lehrb¨ucher. [Birkh¨auser Advanced Texts: Basel Textbooks]. Birkh¨auser Verlag,Basel, 1999.[6] Menachem Kojman. The A,B,C of pcf: a companion to pcf theory, part I. unpublished(https://arxiv.org/abs/math/9512201).[7] Saharon Shelah. Advanced: cofinalities of small reduced products. In
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