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CARDINAL CHARACTERISTICS AND THE

arXiv:math/9209203v1 [math.LO] 8 Sep 1992CARDINAL CHARACTERISTICS AND THEPRODUCT OF COUNTABLY MANYINFINITE CYCLIC GROUPSAndreas BlassAbstract. We study, from a set-theoretic point of view, those subgroups of the in-finite direct product Zℵ0 for which all homomorphisms to Z annihilate all but finitelymany of the standard unit vectors.

Specifically, we relate the smallest possible sizeof such a subgroup to several of the standard cardinal characteristics of the con-tinuum. We also study some related properties and cardinals, both group-theoreticand set-theoretic.

One of the set-theoretic properties and the associated cardinal arecombinatorially natural, independently of any connection with algebra.IntroductionLet Π = Zℵ0 be the direct product of a countable infinity of copies of the infinitecyclic group Z. Specker [19] proved that Π as well as many of its subgroups G havethe following property, in which en is the element of Π whose nth component is1 and whose other components are all zero (the nth standard unit vector). If his a homomorphism from Π (or G) to Z, then h(en) = 0 for all but finitely manyn.

The subgroups G for which Specker established this property all have, like Πitself, the cardinality of the continuum, c = 2ℵ0, and the question naturally ariseswhether any smaller subgroups G of Π also have this property. Eda [7] showed thatthis question is undecidable on the basis of the usual (Zermelo-Fraenkel) axioms ofset theory (ZFC).

Specifically, he proved that the answer is negative in models ofMartin’s Axiom but positive in models obtained by adjoining many random reals.In fact, his proofs give somewhat more precise information about the minimumcardinality κ of a subgroup of Π satisfying Specker’s theorem.In terms of thecardinal characteristics of the continuum introduced in [6] and described in Section1 below, Eda’s proofs establish that p ≤κ ≤d.One purpose of this paper is to improve these estimates; we shall show that theadditivity of Lebesgue measure add(L) is another lower bound for κ and that thebounding number b is an upper bound. (Definitions of these cardinals are recalledin Section 1.) The new upper bound subsumes Eda’s, since b ≤d; the new lowerbound is incomparable with Eda’s, since neither of p and add(L) is provably largerthan the other.These proofs suggest some additional group-theoretic questions concerning ho-momorphisms into free groups.

We say that an abelian group G “binds” a subgroupH if every homomorphism of G into a free group maps H to a group of finite rank.1991 Mathematics Subject Classification. 20K25, 03E05, 03E75.Partially supported by NSF grant DMS-9204276.Tt bAMS T X

2ANDREAS BLASSWe consider questions about the smallest cardinalities in which various sorts ofbinding (a group binding an infinite rank subgroup, a group binding itself, etc. )can occur non-trivially.

A rather surprising result is that one of these cardinalitiesis ≥add(L) but ≤the additivity of Baire category, add(B). These questions areconnected to the Specker phenomenon described above and also to questions con-sidered by Eklof and Shelah [9] about groups that satisfy G ∼= G ⊕F with F freeof finite rank.Finally, we consider some purely set-theoretic questions arising out of these prob-lems.

These questions seem quite natural from a combinatorial point of view, butdo not seem to have been previously considered.This paper is organized as follows.Section 1 presents the definitions of thecardinal characteristics of the continuum that we will need later and some relevantknown theorems about them. Section 2 is devoted to the Specker phenomenon andthe proof that it occurs in cardinality b, i.e., the upper bound mentioned above.The lower bound is established in Section 4 as a consequence of a stronger resultabout binding.Section 3 begins with a general discussion of binding,describesits connection with the Specker phenomenon and with the cardinal p, and endswith the connection with the Eklof-Shelah work mentioned above.

Finally, Section4 contains the purely set-theoretic notions of “predicting” and “evading,” theirconnection with cardinal characteristics and applications to the group-theoretictopics of the previous sections.AcknowledgementI thank John Irwin for many helpful conversations about abelian groups and forhis tireless efforts to keep my attention focused on Π.Terminology and NotationBy “group” we mean abelian group, and we write the group operation as addi-tion. Z is the group of integers.

For any set I, ZI is the group of functions I →Zwith addition of corresponding values as the operation; it is the direct product of|I| copies of Z, where |I| means the cardinality of I. Z(I) is the subgroup of ZIconsisting of functions whose values are 0 at all but finitely many elements of I; itis the direct sum of |I| copies of Z.

When I is the set ω of natural numbers, wewrite Π and Σ for Zω and Z(ω), respectively. If x and y are in ZI and at least oneof them is in Z(I), then the sum Pi∈I x(i)y(i) is finite, and we denote it by theinner product notation ⟨x, y⟩.A group G is torsionless if for every non-zero x ∈G there is a homomorphismh : G →Z with h(x) ̸= 0.

This is equivalent to requiring that G be embeddablein ZI for some set I. (For one direction of the equivalence, use the projectionhomomorphisms ZI →Z that evaluate functions in ZI at a specific element of I.For the other direction, take I to be the set of all homomorphisms G →Z andembed G in ZI by g 7→(i 7→i(g)).) In any group G, the elements that are mappedto 0 by all homomorphisms G →Z form a subgroup N, and the quotient G/N isthe largest torsionless quotient of G.We use the standard (among set-theorists) notation ω for the set of naturalnumbers.

When discussing functions on ω or subsets of ω, we often use an asterisk ∗to indicate that finitely many exceptions are allowed. For example, if A and B aresubsets of ω then A ⊆∗B means that A⧹B is finiteSimilarly if f and g are

CARDINAL CHARACTERISTICS AND Π3functions from ω into an ordered set, then f ≤∗g means that f(n) ≤g(n) for allbut finitely many n.1. Cardinal Characteristics of the ContinuumFor information about cardinal characteristics of the continuum, including thefacts stated without proof below, we refer to Vaughan [20] and the earlier workscited there.

We shall have occasion to refer to nine of the standard cardinal char-acteristics of the continuum, three related to Lebesgue measure, three related toBaire category, and three of a more combinatorial nature.The characteristics related to Lebesgue measure are(1) add(L), the additivity of measure, i.e., the smallest number of measure zerosets whose union is not of measure zero,(2) cov(L), the covering number for measure, i.e., the smallest number of mea-sure zero sets that can cover the real line, and(3) unif(L), the uniformity number for measure, i.e., the smallest cardinalityof a set not of measure zero.To avoid possible confusion, we emphasize that “not of measure zero” is not syn-onymous with “of non-zero measure,” because the former includes the possibilityof not being measurable.The analogous characteristics for Baire category are(4) add(B), the smallest number of first category sets whose union is of secondcategory,(5) cov(B), the smallest number of first category sets that can cover the realline, and(6) unif(B), the smallest cardinality of a set of second category.These definitions referred to measure and category in the real line, but thecardinals defined here would be the same if we used either of the following twospaces instead.2ω is the set of infinite sequences of zeros and ones, with theproduct topology obtained from the discrete topology on {0, 1} and the productmeasure obtained from the measure on {0, 1} that gives each of the two elementsmeasure 1/2. Similarly ωω is the set of infinite sequences of non-negative integers,with the product topology obtained from the discrete topology on ω and with theproduct measure obtained from the measure on ω that gives each point n measure2−(n+1).To define the three remaining characteristics that we will need, we recall thenotations ⊆∗and ≤∗introduced above, and we also introduce the following termi-nology.

A family F of sets is said to have the strong finite intersection property ifthe intersection of every finite subfamily of F is infinite. In terms of these conceptswe define(7) p is the smallest cardinality of a family F of subsets of ω such that F has thestrong finite intersection property but there is no infinite A ⊆ω satisfyingA ⊆∗F for all F ∈F,(8) d is the smallest cardinality of a family D of functions from ω to ω suchthat every function from ω to ω is ≤∗some function in D, and(9) b is the smallest cardinality of a family B of functions from ω to ω such thatno function from ω to ω is ≥∗all the functions in B.

(In other words, forevery g : ω →ω there is f ∈B such that g(n) < f(n) for infinitely many n )

4ANDREAS BLASSClearly, p would be unchanged if ω in its definition were replaced by any count-ably infinite set. Similarly, d and b would be unchanged if their definitions referredto functions A →ω instead of ω →ω, where A is countably infinite.

Also, d wouldbe unchanged if the relation of “almost everywhere majorization,” ≤∗, were re-placed with “everywhere majorization,” for we can adjoin to D, without increasingits cardinality, all those functions that differ only finitely from functions in D.For orientation, we point out that all nine of these cardinals lie between ℵ1and the cardinality c of the continuum, inclusive. In particular, if the continuumhypothesis holds, then all are equal to ℵ1 = c. It follows from results of Martin andSolovay [14] that under Martin’s axiom all these cardinals equal c.We shall need some information about the relationships between these nine car-dinals.

Most of this information is summarized in the following diagram, in whichan arrow from one cardinal to another indicates that the former is provably lessthan or equal to the latter; that is →means ≤. (Except for the part involving p,this is part of what is called Cicho´n’s diagram; see [3, 10, 20].

The use of arrowsinstead of inequality signs is due to the custom of working not with the cardinalsκ themselves but with the hypotheses κ = c; inequalities between the cardinalsobviously yield implications between these hypotheses, and in practice the conversealso holds, i.e., the known proofs of implications between these hypotheses alsoestablish the inequalities. )cov(L)−→unif(B)xxb−→dxxadd(L)−→add(B)−→cov(B)−→unif(L)xpA few of the inequalities are obvious from the definitions, namely add ≤covand add ≤unif for both measure and category, and b ≤d.The inequalitiescov(B) ≤d and b ≤unif(B) follow easily from the observation that, for eachg ∈ωω, the set of all f ≤∗g is of first category in ωω.Among the non-trivial inequalities, cov(L) ≤unif(B) and cov(B) ≤unif(L)are due to Rothberger [18], add(L) ≤add(B) is due to Bartoszy´nski [1] andindependently to Raisonnier and Stern [17], p ≤add(B) is due to Martin andSolovay [14], and add(B) ≤b is due to Miller [16].

In fact, Miller proved a strongerresult, which we shall need later and therefore display here for emphasis:(10)add(B) = min{cov(B), b}.The diagram above is complete in the sense that every provable (in ZFC) in-equality between our nine cardinals is given by a chain of arrows in the diagram.For inequalities not involving p, the proofs are summarized in [3]. The consistencyof p < add(L) is proved in [13], and the consistency of p > cov(L) is stated in [11],attributed to Miller (unpublished).To close this section, we retract our earlier statement that p, d, and b are morecombinatorial than the measure and category characteristics.

Not that the formeraren’t combinatorial but it turns out that the latter have combinatorial descriptions

CARDINAL CHARACTERISTICS AND Π5as well. We shall need these descriptions, due to Bartoszy´nski [1,2], for add(L) andcov(B); for related results, see also [15,16, 17].For the first of these descriptions, we use Bartoszy´nski’s notion of a slalom,namely a function s with domain ω, such that s(n) is a set of cardinality (n+1)2 foreach n. We say that a function f : ω →ω goes through the slalom s if f(n) ∈s(n)for all but finitely many n. With this terminology, we have that add(L) is thesmallest possible cardinality of a family F of functions ω →ω such that, for eachslalom s, some f ∈F does not go through s.We say that two functions f and g on ω are infinitely often equal if f(n) = g(n)for infinitely many n. Then cov(B) is the smallest possible cardinality of a familyF of functions ω →ω such that no single function g : ω →ω is infinitely oftenequal to each of the functions in F.2.

The Specker PhenomenonWe say that a subgroup G of Π = Zω exhibits the Specker phenomenon ifit contains a sequence (an)n∈ω of linearly independent elements such that everyhomomorphism G →Z maps an to 0 for all but finitely many n. (The concept,but not the name, is from [7].) When we need to be more specific, we shall saythat G exhibits the Specker phenomenon witnessed by (an)n∈ω.

Thus, Specker’stheorem [19, Satz III] asserts that Π exhibits the Specker phenomenon witnessedby the sequence of standard unit vectors (en).Definition. The smallest cardinal of any subgroup of Π exhibiting the Speckerphenomenon is denoted by se.The symbol se stands for Specker and Eda.

Eda studied this cardinal in [7],proving that se = c follows from Martin’s axiom but is false in a model obtainedby adding many random reals.In fact, his argument for the random real caseshows that se ≤d. (It is well-known that d < c in the random real model.

)His proof that Martin’s axiom implies se = c also establishes somewhat more,though indirectly. The application of Martin’s axiom in the proof uses a partiallyordered set which does not merely (as Martin’s axiom requires) satisfy the countableantichain condition but in fact is σ-centered.

Bell [4] showed that Martin’s axiomfor σ-centered orderings is equivalent to p = c. In conjunction with Eda’s argument,this proves that p = c implies se = c, and a trivial variation of the argument showsthat p ≤se. (We shall give a simpler proof of this, not using Bell’s theorem butfollowing the lines of Eda’s proof, in Section 3.) Thus, we can summarize Eda’sresults (in the light of Bell’s theorem) as p ≤se ≤d.In this section, we shall improve the upper bound on se from d to b.

Later, inSection 4, we shall give a new lower bound, namely add(L).Before proceeding, it seems appropriate to make some remarks about the def-inition of the Specker phenomenon. On the one hand, the definition might seemtoo restrictive — why are only subgroups of Π considered?

On the other hand, onemight be even more restrictive and bring the definition closer to Specker’s result byrequiring that the witnesses an be the standard unit vectors en. We wish to pointout that, once some trivialities are removed, such variations in the definition do notaffect se.

As an example of a triviality to be removed, we note that, if G were notrequired to be a subgroup of Π, then every divisible group of infinite rank wouldexhibit the Specker phenomenon simply because it has no non-zero homomorphismsto Z

6ANDREAS BLASSQuite generally, if G is not torsionless, then its homomorphisms to Z are “thesame” as those of its largest torsionless quotient G/N, so, when dealing with homo-morphisms to Z (e.g., when discussing the Specker phenomenon), it is reasonable todivide out the irrelevant N and work with the torsionless G/N. Thus, when relax-ing the restriction, in the definition of Specker phenomenon, that G be a subgroupof Π, we should still require that it be torsionless.Theorem 1.

Either of the following two modifications of the definition of se doesnot change its value. (a) Instead of requiring G to be a subgroup of Π, require only that G be torsionless.

(b) Require the witness sequence to be the sequence (en) of all standard unit vectors.Proof. (a) The assumption that G is torsionless is equivalent to requiring that Gbe embeddable in ZI for some index set I.

Suppose now that G ⊆ZI exhibits theSpecker phenomenon, witnessed by (an). For each non-zero linear combination c of(any finitely many) an’s, select one coordinate i ∈I such that the ith component ofc is not zero.

Let J be the set of i’s so selected; note that J is a countable subset of I.Let p : ZI →ZJ be the canonical projection map. By our choice of J, the elementsp(an) are, like the an, linearly independent.

For any homomorphism h : p(G) →Z,we know that hp : G →Z annihilates almost all an, so h annihilates almost allp(an). Thus, p(G) exhibits the Specker phenomenon witnessed by (p(an)).

Since Jis countable, p(G) is isomorphic to a subgroup of Π, so we have an instance of theSpecker phenomenon as originally defined, involving a group p(G) no larger thanthe given G. Thus, se defined with the relaxed condition on G in (a) is no smallerthan se as originally defined. That it is no larger is trivial, so (a) is proved.

(b) Consider any G ⊆Π exhibiting the Specker phenomenon witnessed by (an).The first part of the proof of Specker’s [19, Satz I] can be used to show that, givenany countably many elements of Π, like the an, we have an endomorphism of Πmapping all these elements into the subgroup Σ of elements having only finitelymany non-zero components. (See also Chase [5, Corollary 3.3].) Thus, we mayassume without loss of generality, replacing G by its image under a suitable endo-morphism of Π, that our witnesses an all lie in Σ.

Let us write Sn for the (finite)set of coordinates where an has non-zero components. Each coordinate i is in onlyfinitely many Sn’s, because the projection of G ⊆Π to the ith coordinate is a ho-momorphism to Z, to which the definition of the Specker phenomenon is applicable.This observation allows us to find an infinite subsequence of (an) for which the sup-ports Sn are pairwise disjoint.

Indeed, we can put a0 into our subsequence, put outthose finitely many other an whose Sn meets S0, put in the first ak different fromall these, put out the finitely many other an whose Sn meets Sk, etc. Of course,the resulting subsequence, like any infinite subsequence of (an), still witnesses theSpecker phenomenon for G, so we may assume, without loss of generality, that (an)is such that the supports Sn are pairwise disjoint.

By applying suitable automor-phisms of Π, composed from automorphisms of all the finite rank free groups ZSnseparately, we can arrange that each an is an integer multiple of a standard unitvector en. Replacing G by its smallest supergroup pure in Π (which does not in-crease the cardinality), we can arrange that the Specker phenomenon is witnessedby a sequence of (some of the) standard unit vectors, say the en for n in a certaininfinite subset T of ω.

We can arrange further that G is a subgroup of ZT , justby replacing it with its image under the projection Π = Zω →ZT . Finally, usinga bijection between T and ω we obtain a subgroup of Π of no greater cardinality

CARDINAL CHARACTERISTICS AND Π7than the original G, for which the Specker property is witnessed by the sequenceof all the standard unit vectors en. This proves (b).□The preceding theorem shows that the definition of se is robust under fairly widevariations in the sort of group and the sort of witnessing sequence considered.

Nowwe turn to our first estimate of se; the proof uses ideas from Eda [8].Theorem 2. Π has a subgroup of cardinality b that exhibits the Specker phenom-enon (witnessed by the sequence (en) of standard unit vectors).

Thus se ≤b.Proof. Let B be a family of b functions ω →ω as in the definition (9) of b, i.e., forany g : ω →ω there is f ∈B such that g(n) < f(n) for infinitely many n. We canarrange that the functions in B are monotone non-decreasing and nowhere zero;just replace each f ∈B with the larger function f ′(n) = 1 + max{f(k) | k ≤n}.Henceforth, we assume that this replacement has been made, so 0 < f(m) ≤f(n)whenever m ≤n and f ∈B.Lemma.

For every infinite A ⊆ω and every function g : A →ω, there is f ∈Bsuch that g(n) < f(n) for infinitely many n ∈A.Proof of lemma. Extend the given g to a function on all of ω by defining its value atany k /∈A to be the (already defined) value of g at the next larger element of A. Byour choice of B, it contains an f that is greater than (the extended) g at infinitelymany k. If infinitely many of these k’s are in A, we are done.

Otherwise, for each ofthese k’s that is not in A, let n be the next larger element of A, and observe that,by definition of g(k) and monotonicity of f, we have g(n) = g(k) < f(k) ≤f(n).So f is greater than g at infinitely many n ∈A.□For each f ∈B, define a function x : ω →ω by the recursion x(0) = 1 and(11)x(n + 1) = 2 · f(n + 1) · x(n) ·nXi=0x(i). (The motivation for this strange definition will become clear in the course of theproof.) We note for future reference that x(n) is never zero, that x(n) →∞asn →∞, and that, because of the factor x(n) in the definition, x(n) divides x(m)whenever n ≤m.As B has cardinality b, we have obtained b functions x in this manner.

LetG be the smallest pure subgroup of Π containing all these x’s and containing the(countably many) elements of Σ (the functions in Π with finite support). ThenG has cardinality b, and we complete the proof by showing that G exhibits theSpecker phenomenon witnessed by (en).We proceed by contradiction, so suppose h : G →ω is a homomorphism and theset A = {n | h(en) ̸= 0} is infinite.

Define a function g by g(n) = maxk≤n |h(en)|,and apply the lemma to obtain an f ∈B such that M = {n ∈A | f(n) > g(n)}is infinite.Let x be the function defined, using this f, by (11).Thus x ∈G.Since x(n) →∞as n →∞and since M is infinite, we can fix a non-zero n ∈Mwith x(n) > 2|h(x)|.We shall prove that h(en) = 0; this will be the desiredcontradiction, because n ∈M ⊆A.We split x ∈G as the sumx =Xx(i)ei + y =Xx(i)ei + x(n)z,

8ANDREAS BLASSwhere y is the element of Π that agrees with x from coordinate n on but is zeroat the earlier coordinates. Since all components x(m) for m ≥n are divisible byx(n), we can write y as x(n)z for some z ∈Π.

Furthermore, as G contains x andPi

So does the right side, because(13)Xi

So we have(14)h(x) =Xi

Binding SubgroupsDefinition. A group G binds a subgroup H if every homomorphism from G to afree group maps H into a group of finite rank.This definition has some trivial cases, e.g., if H has finite rank or if G admits nonon-zero homomorphisms to free groups.

As in the discussion preceding Theorem 1,elements of G whose image under every homomorphism to Z is 0 are irrelevantdistractions in connection with binding, so it is reasonable to divide G by thesubgroup of these elements, i.e., to replace G by its largest torsionless quotient.Thus, when we discuss binding, we shall always assume that G is torsionless andtherefore embeddable in ZI for some set I. We shall also assume, to avoid triviality,that H has infinite rank.Notice that, if G binds H, then every (torsionless) supergroup of G binds every(infinite rank) subgroup of H as well

CARDINAL CHARACTERISTICS AND Π9Definition. If a torsionless group G binds some subgroup of infinite rank, then wesay G is binding.

If G has infinite rank and binds itself, then we call it self-binding.Clearly, every self-binding group is binding.Proposition 3. A torsionless group G binds a subgroup H if and only if everyhomomorphism G →Σ maps H to a group of finite rank.

G binds itself if and onlyif it does not admit a homomorphism onto Σ, if and only if it does not have Σ asa direct summand.Proof. For the non-trivial half of the first statement, suppose f : G →Z(I) maps Hto a group of infinite rank.

We must achieve the same situation with a countableset in place of I. As f(H) has infinite rank, we can choose a countable infinityof linearly independent elements in it and then choose, for each non-zero linearcombination x of these elements, one element i ∈I with x(i) ̸= 0.

Let J consistof the countably many i so chosen, and compose f with the canonical projectionZ(I) →Z(J). The resulting homomorphism G →Z(J) ∼= Σ is as desired.For the non-trivial half of the second statement, suppose G does not bind itself,and choose, by what we just proved, f : G →Σ with f(G) of infinite rank.

Butany subgroup of Σ of infinite rank is isomorphic to Σ, so we can compose f withan isomorphism f(G) →Σ to obtain a surjection as claimed. The last part of theproposition, about direct summands, follows because any homomorphism onto afree group, such as Σ, splits.□We shall need to consider groups G with Σ ⊆G ⊆Π.

We call such a groupΣ-binding if it binds Σ, and we call it weakly Σ-binding if every homomorphismG →Σ that extends to an endomorphism of Π maps Σ to a group of finite rank. Itis clear, by the first part of Proposition 3, that Σ-binding implies weakly Σ-binding.Although Σ-binding is a special case of the general notion of binding, the followingtheorem shows that it is not very special.Theorem 4.

If a torsionless group G binds a subgroup of infinite rank, then G hasa homomorphic image G′ ⊆Π such that G′ + Σ binds Σ.Proof. Let G ⊆ZI bind a subgroup H of infinite rank.

As G binds all subgroups ofH, we may assume that H is countable. Choosing for each non-zero element x ∈Hone i ∈I such that x(i) ̸= 0, letting J ⊆I be the set of these countably manychosen i’s, and replacing G and H by their projections in ZJ, we arrange that, upto isomorphism, G ⊆Π, G still binds H, and H still has infinite rank.The next step in the proof resembles an argument in Specker’s proof of [19, SatzI] and even more closely resembles Theorem 3.2 of [5].Lemma.

If H is any subgroup of Π of countably infinite rank, then there existsan endomorphism f of Π and there exist positive integers (dn)n∈ω such that f(H)includes the subgroup Ln dnZ of Σ.Proof of lemma. We define the endomorphism f as the composition of an infinitesequence of endomorphisms f = · · ·◦f2◦f1◦f0; the composition will be well-definedbecause fk will not change the nth component of x for any x ∈Π and any n < k.Thus the nth component of f(x) is the nth component of fn ◦· · · ◦f2 ◦f1 ◦f0(x).To define f0, choose a non-zero element a ∈H such that the greatest commondivisor d of all its components a(n) is as small as possible.

Notice that d is thelargest integer by which a is divisible in ΠThis provides a description of d that

10ANDREAS BLASSis invariant under automorphisms of Π. Also, d is an integral linear combinationof finitely many a(n)’s, say Pn

Let g be the automorphism of Π thatleaves the first r components of its argument unchanged but transforms the restaccording tog(x)(m) = x(m) −a(m)dXn

(The components of g(a) have greatest common divisor d,thanks to the automorphism-invariance of d noted above. )Let f0 be the composite of g and the automorphism just described.

So f0(a) =d · e0. Also, let d0 = d.Consider any element b ∈f0(H).

As f0 is an automorphism of Π, the greatestcommon divisor of the components of b is the same as for f −10 (b) ∈H, hence is atleast d. In particular, |b(0)| is at least d, unless it is zero. It follows that b(0) isdivisible by d, for otherwise we could write b(0) = qd + r with 0 < r < d, and thenb′ = b −qde0 ∈f0(H) would have 0th component r, a contradiction because theconclusion of the previous sentence applies to b′ as well as to b.Thus, every element of f0(H) is expressible as d0e0 + x, where x ∈f0(H) hasx(0) = 0.

Therefore, f0(H) = ⟨d0e0⟩⊕H1, where H1 is a subgroup of Π1 = Zω⧹{0}.Now proceed with H1 just as we did with H in the preceding paragraphs, obtain-ing an automorphism f1 of Π1 (which we extend to Π by letting it act trivially on the0 coordinate) sending H1 to ⟨d1e1⟩⊕H2, where H2 is a subgroup of Π2 = Zω⧹{0,1}and where d1 is the smallest g.c.d. of all the components of any element of H1.Continuing this process inductively, we obtain fn’s whose composite sends H toa subgroup containing all the dnen.

(Notice that, although each fn is an automor-phism of Π, we can only claim that the composite f is an endomorphism.) Thiscompletes the proof of the lemma.□Returning to the proof of the theorem, we apply the lemma to the H thatwas bound by our (modified) G ⊆Π, and we set G′ = f(G), where f is givenby the lemma.

Then G′ is a homomorphic image of our modified G, hence alsoof our original G. It binds f(H), so its supergroup G′ + Σ binds the subgroupS = Ln dnZ of f(H). But S is a subgroup of Σ such that the quotient Σ/S isa torsion group, because S contains a multiple of each of the generators en of Σ.It follows immediately that any homomorphism that is defined on Σ and maps Sinto a group of finite rank must also map Σ into a group of (the same) finite rank.Thus, binding S implies binding Σ, and the proof is therefore complete.□Corollary 5.

Each of the following cardinals is less than or equal to the next:(1) ℵ1(2) the smallest cardinality of a weakly Σ-binding group(3) the smallest cardinality of a binding group(4) the smallest cardinality of a self-binding group(5) the cardinality c of the continuum.Proof. (1)≤(2): We must show that, if Σ ⊆G ⊆Π and G is countable, then thereis an endomorphism of Π mapping G into Σ and mapping Σ onto a group of infinite

CARDINAL CHARACTERISTICS AND Π11rank.For this, it more than suffices to get an automorphism of Π mapping Ginto Σ. Furthermore, it suffices to do this under the additional assumption that Gis a pure subgroup of Π, because G can be enlarged to a pure subgroup withoutincreasing its cardinality.

But now what we need is given by [5], Corollary 3.3. (2)≤(3): By Theorem 4, the cardinal in (3) is also the smallest cardinality of aΣ-binding group.

Since Σ-binding implies weakly Σ-binding, the desired inequalityfollows. (3)≤(4): This is immediate, as every self-binding group is binding.

(4)≤(5): Π has cardinality c and binds itself. (If it had Σ as a direct summand,then its dual, which is isomorphic to Σ by [19, Satz III], [12, Cor.

94.6], wouldhave a summand isomorphic to the dual of Σ, i.e., to Π. This is absurd as Σ iscountable and Π is not.

The same conclusion also follows instantly from the moredetailed information in [12, pp.159–160] about homomorphisms from products tosums. )□The next two results relate binding to the Specker phenomenon discussed inSection 2.Theorem 6.

A group that exhibits the Specker phenomenon is binding.Proof. Suppose G exhibits the Specker phenomenon witnessed by (an), and let Hbe the subgroup of G generated by these witnesses an.

Suppose also, toward acontradiction, that G does not bind H. So let f : G →Σ map H to a groupof infinite rank. We inductively construct an element x ∈Π and an increasingsequence of natural numbers km as follows.At stage m of the construction, we have already defined ki for i < m, and wehave already defined some finite initial segment of x.

As the group f(H) generatedby the f(an)’s has infinite rank, we can choose km so that f(akm) has a non-zerocomponent in some position, say the ith, such that x(i) is not yet defined. Thenwe can easily extend x (or, rather, the finite part of x already defined) so that(a) x(j) becomes defined for all j such that f(akm)(j) ̸= 0 and so that (b) the innerproduct ⟨f(akm), x⟩is not zero.

(The inner product will not depend on future stepsin the definition of x, because of (a). Making it non-zero is easy, by appropriatelychoosing the value of x(i).

)Notice that all the km are distinct, because the mth stage is the first one wherex is defined at all coordinates where f(akm) has a non-zero component, so we canrecover m from km.Now the homomorphism z 7→⟨z, x⟩maps Σ and hence f(G) into Z and mapsnone of the f(akm)’s to zero. So its composite with f violates the choice of the anas witnessing the Specker phenomenon.□This theorem immediately implies that se is at least as big as the cardinal labeled(3) in the Corollary 5.

In fact, we can do a bit better, replacing (3) with (4).Theorem 7. The smallest cardinality of a self-binding group is ≤se.Thus, se could be inserted into Corollary 5 as item (4.5).

Theorems 6 and 7cannot be combined to assert that a group exhibiting the Specker phenomenonbinds itself; a counterexample is given by Σ ⊕Π regarded as a subgroup of Π in theobvious way (sequences in which only finitely many odd-numbered components arenon zero)

12ANDREAS BLASSProof of Theorem 7. By Theorem 1, let G1 be a group of cardinality se such thatΣ ⊆G1 ⊆Π and G1 exhibits the Specker phenomenon witnessed by the sequenceof standard unit vectors en.

By Theorem 6, G1 is binding, and by Corollary 5 thereis a weakly Σ-binding group G2 of cardinality at most se. Let G3 = G1 +G2.

ThenΣ ⊆G3 ⊆Π, G3 exhibits the Specker phenomenon witnessed by the en’s, G3 isweakly Σ-binding, and |G3| = se.The next step of the proof is based on an idea from [8]. Fix two disjoint increasingsequences of prime numbers, say (pn) and (qn).

For each n, obtain from the Chineseremainder theorem an integer dn that is divisible by all pk for k ≤n but is congruentto −1 modulo all qk for k ≤n. Extend G3 to a pure subgroup G of Π that is closedunder (component-by-component) multiplication by the sequence d = (dn).

Thiscan be done with |G| = se, because we are simply closing G3 under countably manypartial functions (multiplication by d and division by each positive integer). Thefollowing lemma is due to Eda [8].Lemma.

A homomorphism f : G →Z is completely determined if the values f(en)are known.Proof. It suffices to show that, if f(en) = 0 for all n then f(x) = 0 for all x ∈G.Fix any such f and x.

Also, temporarily fix a positive integer n. Decompose thecomponentwise product dx as the sum of Pi

So f(dx) is divisible bypn. Now un-fix n. f(dx) is divisible by arbitrarily large primes pn, hence is zero.We can repeat the same argument with the sequence d + 1 = (dn + 1) in place ofd and with qn in place of pn (since qk divides dn + 1 when k ≤n).

We obtainf((d + 1)x) = 0. But then, as x = (d + 1)x −dx, it follows that f(x) = 0, asdesired.□Combining the lemma with the Specker phenomenon witnessed by the en’s, wefind that every homomorphism f : G →Z has the form f(x) = Pi

It follows, by applyingthis observation to each component, that every homomorphism G →Σ extends toa homomorphism Π →Π. We use this to complete the proof of the theorem byshowing that G binds itself.Let f : G →Σ.

As f extends to a homomorphism Π →Π and as G is weaklyΣ-binding, f(Σ) must have finite rank. Therefore, for all but finitely many n ∈ω,the nth component fn : G →Z of f is zero on Σ.

But, by the lemma, this impliesthat these (all but finitely many) fn are zero on G. So f(G) has finite rank, asrequired.□The following corollary extends Corollary 5 to incorporate Theorems 2 and 7.Corollary 8. Each of the following cardinals is less than or equal to the next:(1) ℵ1(2) the smallest cardinality of a weakly Σ-binding group(3) the smallest cardinality of a binding group(4) the smallest cardinality of a self-binding group(5) se(6) b(7) c□

CARDINAL CHARACTERISTICS AND Π13We are now in a position to fulfill our promise, from Section 2, to give anotherproof, not using Bell’s theorem, of the inequality p ≤se.In fact, we obtain astronger result, showing that p ≤all the cardinals except ℵ1 mentioned in Corol-lary 8.Theorem 9. A group of cardinality < p cannot be weakly Σ-binding.Proof.

Suppose Σ ⊆G ⊆Π and |G| < p. Recall from Section 1 that if C is acountable set, if F is a family of fewer than p subsets of C, and if every finitesubfamily of F has infinite intersection, then there is an infinite subset A of C suchthat A ⊆∗F for all F ∈F. We apply this with C = Σ⧹{0} and with the following|G| sets Zx as the family F. For each x ∈G, letZx = {c ∈C | ⟨c, x⟩= 0}.To check that the family F = {Zx | x ∈G} has the strong finite intersec-tion property, let any finitely many of its elements, say Zx1, Zx2, .

. .

, Zxn, begiven; we must show that their intersection is infinite. For each k ∈ω, let yk bethe n-component integer vector (x1(k), x2(k), .

. ., xn(k)).These infinitely manyvectors yk, lying in the finite-dimensional vector space Qn, must be linearly de-pendent; fix a non-zero r-tuple (c(0), c(1), .

. ., c(r)) of rational numbers such thatPrk=0 c(k)xi(k) = 0 for all i = 1, 2, .

. ., n. Clearing denominators, we can arrangethat the c(k) are integers.

Extending the definition of c(k) to k > r by making thesec(k) = 0, we obtain an element c ∈Σ⧹{0} with ⟨c, xi⟩= 0 for all i = 1, 2, . .

., n.This shows that the intersection of the Zxi is non-empty; in fact, it is infinitebecause we can multiply c by any non-zero integer.Having checked the strong finite intersection property, we use the fact that |F| ≤|G| < p to obtain an infinite set A = {a0, a1, . .

., an, . .

.} ⊆Σ⧹{0} such that A ⊆∗Zx for all x ∈G.

Now we define a homomorphism f : Π →Π by f(x)(n) = ⟨an, x⟩.For each x ∈G, we have, for all but finitely many n ∈ω, that an ∈Zx, whichmeans that f(x)(n) = 0. Thus, f(x) ∈Σ.

We have shown that (the restriction of)f maps G into Σ and extends to an endomorphism (namely the unrestricted f) ofΠ. We shall show that f(Σ) has infinite rank, thereby completing the proof that Gis not weakly Σ-binding.Suppose, toward a contradiction, that f(Σ) had finite rank.

Then there wouldbe an m ∈ω such that, for all n ≥m and all k ∈ω, f(ek)(n) = 0. This equationmeans 0 = ⟨an, ek⟩= an(k).

This would make all the an for n > m equal to zero,contradicting the fact that the an are all in C which doesn’t contain zero.□This theorem allows us to insert p as item (1.5) in Corollaries 5 and 8.We close this section by connecting the notion of binding to a question studiedby Eklof and Shelah in [9]. One of their results (not the main one) is that, under theassumption of Martin’s axiom, if G is a group of cardinality < c and if G ∼= G⊕Zmfor some positive integer m, then G ∼= G ⊕Σ (and therefore G ∼= G ⊕Zm forevery positive integer m).

In fact, their application of Martin’s axiom involved a σ-centered partially ordered set, so, by Bell’s theorem, the hypotheses that Martin’saxiom holds and |G| < c can be weakened to the hypothesis that |G| < p. Here isan alternate approach, not using Bell’s theorem, to (a stronger form of) this result.Assume G ∼= G ⊕Zm for some positive integer m. Instead of assuming |G| < p,we assume only that |G| is smaller than the smallest cardinality of any self-bindinggroup(This hypothesis would follow if |G| < p by virtue of Corollary 8 and

14ANDREAS BLASSTheorem 9.) Let N be, as before, the subgroup of G consisting of elements mappedto 0 by all homomorphisms G →Z.

The analogous subgroup of G ⊕Zm is clearlyN ⊕0, as Zm is torsionless. So the assumed isomorphism from G onto G ⊕Zmmust send N to N ⊕0 and must therefore induce an isomorphism from the largesttorsionless quotient G/N onto (G/N) ⊕Zm.By comparing the ranks of thesegroups, we conclude that G/N has infinite rank.

(This is the only use we need tomake of the assumption that G ∼= G⊕Zm.) As |G/N| ≤|G| < the smallest cardinalof any self-binding group, Proposition 3 tells us that G/N admits a surjection to Σ,and therefore so does G, and therefore G ∼= A⊕Σ for some A.

But since Σ ∼= Σ⊕Σ,it follows that G ∼= G ⊕Σ, as desired.4. Predicting and EvadingThis section is devoted to combinatorial concepts, prediction and evasion, moti-vated by some of the group-theoretic concepts of the preceding section, particularlythe weakest of these, weak Σ-binding.Definition.

For any set S, an S-valued predictor is a pair π = (Dπ, (πn)n∈Dπ)where Dπ is an infinite subset of ω and πn is, for each n ∈Dπ, a function Sn →S.We say that the predictor π predicts the function x : ω →S if, for all but finitelymany n ∈Dπ, we have x(n) = πn(x ↾n). Otherwise, we say that x evades π.Here x ↾n means the n-tuple (x(0), x(1), .

. ., x(n −1)).

Intuitively, we regardpredictors π as follows. The values of an unknown function x : ω →S are beingrevealed, one at a time, in order, and we are to guess x(n) just before it is to berevealed, i.e., just after we have seen x ↾n.

The predictor π provides a strategy formaking these guesses when n ∈Dπ, namely, if we have seen the n-tuple t so far,then we are to guess πn(t). The functions predicted by π are just those for whichall but finitely many of the guesses provided by this strategy are correct.Definition.

A Z-valued predictor π is called linear if, for each n ∈Dπ, the functionπn : Zn →Z is a linear function with rational coefficients. A predictor is forgetfulif, whenever m < n are consecutive elements of Dπ, the function πn depends onlyon the last n −m −1 components of its argument n-tuple.In terms of the intuitive picture of predictors, forgetfulness means that, whenguessing x(n), the strategy considers only the values of x revealed since its lastguess, namely x(m + 1) through x(n −1).Definition.

e, the evasion number, is the smallest possible cardinality of a familyE of functions ω →ω such that every ω-valued predictor is evaded by some x ∈E.el (resp. ef, resp.

efl) is the smallest possible cardinality of a family E of functionsω →Z such that every linear (resp.forgetful, resp.forgetful linear) Z-valuedpredictor is evaded by some x ∈E.Note that e would be unchanged if its definition were phrased in terms of func-tions ω →D and D-valued predictors, where D is any countably infinite set. Inparticular, D could be Z, and so it is clear that e is greater than or equal to bothel and ef, which are in turn greater than or equal to efl.The general notion of predictor and the associated cardinal e seem quite natural,but it is the more specialized notions involving linearity and the associated cardinalsthat connect directly with the group-theoretic concepts of the preceding section, asthe following theorem shows

CARDINAL CHARACTERISTICS AND Π15Theorem 10. The following three cardinals are equal.

(1) The smallest cardinality of a weakly Σ-binding group(2) el(3) eflProof. (1)≥(2): Let G be a weakly Σ-binding group.

In particular, G ⊆Π, so G isa family of functions ω →Z. We shall show that every linear predictor π is evadedby some element of G, so el ≤|G|, as required.Suppose, toward a contradiction, that π is linear and predicts every element ofG.

Let Dπ = {d0, d1, . .

. }.

Thus, for each x ∈G, all but finitely many k ∈ωhave x(dk) = πdk(x ↾dk). As πdk is linear with rational coefficients, we can cleardenominators to rewrite this equation as a linear relation with integer coefficients(15k)dkXi=0ckix(i) = 0.Note that the coefficients cki here depend only on π, not on x.

Note also thatckdk ̸= 0, bacause the rational linear relation from which we got (15k) really involvedx(dk).We define an endomorphism C of Π by settingC(x)(k) =dkXi=0ckix(i)for all x ∈Π. Regarding the elements of Π as infinite column vectors, this en-domorphism is given by left multiplication by the infinite matrix C = (cni).

Ifx ∈G, then (15k) holds, and therefore C(x)(k) = 0, for all but finitely many k, i.e.,C(x) ∈Σ. So C is an endomorphism of Π mapping G into the free group Σ.

As Gis weakly Σ-binding, C must map Σ into a group of finite rank. But C(Σ) includesthe elements C(en), the columns of the matrix C; among these are the columnsindexed by the elements dk of Dπ.

The kth of these columns has a non-zero entryckdk in the kth row and zero entries in all earlier rows. Thus, these columns bythemselves form a lower triangular matrix with non-zero diagonal entries.

Theyare therefore linearly independent, contrary to the fact that they lie in a group offinite rank. (2)≥(3): trivial.

(3)≥(1): Suppose E is a family of functions ω →Z such that every forgetfullinear predictor is evaded by some element of E. So E is a subset of Π; let G be thesubgroup that it and Σ generate. G has the same cardinality as E, so to completethe proof it suffices to show that G is weakly Σ-binding.Suppose it were not.Fix an endomorphism f of Π mapping G into Σ andmapping Σ onto a group of infinite rank.

Each component of f, mapping Π to Z,has, by Specker’s theorem [19 Satz III], the form f(x)(n) = Pi cnix(i), where foreach fixed n only finitely many cni are non-zero. Thus, in the matrix C = (cni),each row has only finitely many non-zero entries.

So does each column, for the ithcolumn is f(ei) ∈f(G) ⊆Σ. (Recall that G was defined so as to contain all theei.

)We inductively choose infinitely many rows, the rows indexed by i0, i1, etc., asfollowsChoose iso that the i th row of C isn’t zero(This is possible as f is

16ANDREAS BLASSnot identically zero.) For the induction step, suppose we have already chosen ik,an index of a non-zero row in C. Let dk be the number of the column in whichthe last non-zero entry of row ik occurs.

Then choose ik+1 so that row ik+1 of C isnon-zero, but its entries in columns 0 through dk are all zero. The first half of thisconstraint is satisfied by infinitely many rows, for otherwise f(Π) would have finiterank, whereas we are assuming that even f(Σ) has infinite rank.

The second halfof the constraint is satisfied by all but finitely many rows, because each column ofC has only finitely many non-zero entries. So the choice of ik+1 is possible.Define a forgetful linear predictor π by Dπ = {dk | k ∈ω} (where dk is, as inthe preceding paragraph, the largest d for which cikd ̸= 0) andπdk(t) = −1cikdkXr

This means that, for infinitely many k,x(dk) ̸= −1cikdkXr

Theorems 9 and 10.□By Theorem 10, any lower bound for el = efl is also a lower bound for all thecardinals except ℵ1 listed in Corollary 8, in particular for se. Apart from p, wehave one more such bound, given by the following theorem.Theorem 12. add(L) ≤el.Proof.

We use the combinatorial description of add(L), at the end of Section 1,as the smallest number of of functions that do not all go through a single slalom.We observe that, although that description is phrased in terms of functions whosevalues are elements of ω and slaloms whose values are subsets of ω, it would makeno difference if ω were replaced in both places with some other countably infiniteset A. (The domains of the slaloms and of the functions under consideration arestill ω; only the ranges are modified.

Thus, the requirement in the definition ofslalom that |s(n)| = (n + 1)2 still makes sense. )We begin by describing a convenient A for this proof.

Partition ω into finiteblocks of consecutive numbers, say I0 = [0, a1), I1 = [a1, a2), etc., in such a waythat In has more than (n + 1)2 elements. (For example, one could take an = 2n3.

)Let A be the set of all functions with domain equal to one of the In’s and withvalues in Z. As A is countably infinite, the remarks in the preceding paragraphapply to it.

The proof will be complete if we find el functions ω →A that do notall go through any single slalom s (whose values are subsets of A)

CARDINAL CHARACTERISTICS AND Π17Let E be a family of el functions ω →Z as in the definition of el, i.e., everylinear predictor is evaded by some f ∈E. For each f ∈E, define f ′ : ω →A byf ′(n) = f ↾In.

Suppose s were a slalom that all these f ′’s go through; we shallcomplete the proof by deducing a contradiction.Temporarily fix some n ∈ω. Let T be the set of functions In →Z that areelements of s(n).

(s(n) may also have elements with domain Im for m ̸= n, but theseare irrelevant to our purpose as well as to the assumption that all f ′ go through s.)T consists of at most |s(n)| = (n + 1)2 elements of the rational vector space ofrational-valued functions on In. This vector space has dimension |In| > (n + 1)2,so there is a non-zero linear functional φn on this vector space annihilating all theelements of T. It has the form φn(t) = Pi∈In cnit(i), for some rational coefficientscni, not all zero.

Let dn be the largest i for which cni ̸= 0. We record for future usethe fact that, if f ′(n) ∈s(n) then φn(f ↾In) = 0.

This is because f ↾In = f ′(n)belongs to s(n) and maps In into Z and therefore belongs to T.Now unfix n, but keep the notations φn, cni, and dn from the preceding para-graph. (The subscripts n, superfluous before, now prevent ambiguity.) Define apredictor π by setting Dπ = {dn | n ∈ω} and, for each dn ∈Dπ, settingπdn(t) = −1cndnXi

If some f : ω →Z and some n ∈ω satisfyφn(f ↾In) = 0, then trivial algebraic manipulation of this equation gives f(dn) =πdn(f ↾dn).For each f ∈E, the assumption that f ′ goes through s means that for all butfinitely many n we have f ′(n) ∈s(n), and so φn(f ↾In) = 0, and so f(dn) =πdn(f ↾dn). But that means that π predicts every f ∈E, contrary to the choice ofE.□Upper bounds for the evasion cardinals e and el are less interesting than lowerbounds, since they do not imply corresponding bounds for the more natural group-theoretic cardinals in Corollary 8.

Nevertheless, we record for the sake of complete-ness the upper bounds which we have been able to obtain.Theorem 13. All three of unif(L), unif(B), and d are upper bounds for e. Fur-thermore, min{e, b} ≤add(B).Proof.

e ≤unif: We handle the measure and category cases together. It is easyto verify that, for any predictor π, the set of functions x ∈2ω (i.e., functions on ωwith only 0 and 1 as values) that are predicted by π is a set of first category andmeasure zero in 2ω.

The desired inequalities follow immediately.e ≤d: Let D be a family of d functions from ω × ω (the set of pairs of naturalnumbers) to ω such that every function ω × ω →ω is majorized (everywhere) by afunction from D. (See the remarks in Section 1 after the definition of d.) To eachfunction g ∈D, we associate a function x : ω →ω by the recursion(16)x(n) = g(n, 1 + max{x(p) | p < n}).We shall show that every ω-valued predictor π is evaded by one of these d functionsx

18ANDREAS BLASSLet π = (Dπ, (πn)) be given, and define a function f : ω × ω →ω byf(n, k) = max{πn(t) | t ∈ωn and all values of t are < k},if n ∈Dπarbitrary,otherwise.Choose g ∈D such that g(n, k) > f(n, k) for all n and k, and let x be the functionobtained from g by (16). We shall show that this x evades π.Consider any n ∈Dπ.

Let k = 1 + max{x(p) | p < n} and note that x ↾n,being in ωn and having all its values < k, is one of the t’s involved in the definitionof f(n, k). So f(n, k) ≥πn(x ↾n).

On the other hand, by the definition (16) ofx and the choice of g, we also have x(n) ≥g(n, k) > f(n, k). Comparing theseinequalities, we find that x(n) ̸= πn(x ↾n).

As n was an arbitrary element of Dπ,this shows that x evades π. (It actually shows more: not only are infinitely manyof π’s predictions wrong, as required for evasion, all of them are wrong.

)min{e, b} ≤add(B): Since add(B) = min{cov(B), b}, it suffices to prove thatmin{e, b} ≤cov(B). For this we use the combinatorial description of cov(B) atthe end of Section 1.

We must show that, if F is a family of functions ω →ωof cardinality smaller than min{e, b}, then there exists g : ω →ω such that everyx ∈F is infinitely often equal to g.Let such an F be given. As its cardinality is smaller than b, fix d : ω →ωeventually majorizing (i.e., ≥∗) every x ∈F.

For any natural number k we willwrite max(d, k) for the function whose value at any n is max{d(n), k}. For anyx ∈F, we can find k so large that x is majorized everywhere by max(d, k).

(dmajorizes all but finitely many values of x, so we just choose k to majorize thosefinitely many values. )Partition ω into finite blocks of consecutive numbers, say I0 = [0, a1), I1 =[a1, a2), etc., in such a way that the cardinality of Ik is(17)|Ik| =Yj

Note that none of these blocks are empty.Let A be the set of functions into ω whose domains are blocks in this partition.For each x ∈F, define an associated function x′ : ω →A by x′(k) = x ↾Ik.As |F| < e, all these functions x′ are predicted by a single A-valued predictorπ = (Dπ, (πn)).We use π to define the desired g (infinitely often equal to each x ∈F) as follows.If k /∈Dπ, define g ↾Ik arbitrarily. If k ∈Dπ, then proceed as follows.

List thefunctions I0 ∪· · · ∪Ik−1 →ω that are majorized (on their domain) by max(d, k) ast0, t1, . .

. tr−1; the number of such functions, here called r, is the cardinality of Ik,by (17).

For each j < r, obtain t′j from tj analogously to the way x′ was definedfrom x, i.e., t′j : {0, 1, . .

., k −1} →A and t′j(i) = tj ↾Ii. Consider the set X ofthose πk(t′j) (for our fixed k and arbitrary j < k) that are functions Ik →ω.

Thisis a set of at most r functions on a set Ik of size r. So there is a function Ik →ωthat agrees at least once with each of the functions in X. Let g ↾Ik be such afunction.To show that g has the desired property, consider any x ∈F; we must showthat g and x are infinitely often equalTemporarily fix some k ∈Dso large

CARDINAL CHARACTERISTICS AND Π19that max(d, k) majorizes x everywhere and so large that π correctly predicts x′(k),i.e., x′(k) = πk(x′ ↾k). Then, in the definition of g ↾Ik, one of the tj’s underconsideration was x ↾I0 ∪· · · ∪Ik−1 and the corresponding t′j was x′ ↾k.

So, as πcorrectly predicted x′(k), we have thatx ↾Ik = x′(k) = πk(x′ ↾k) = πk(t′j) ∈Xis one of the functions with which we made g ↾Ik agree at least once. This provesthat g and x agree at least once in Ik.

Now un-fix k. The preceding argumentapplies to infinitely many k’s, namely all sufficiently large elements of Dπ. So gand x are infinitely often equal.□Corollary 14. el ≤add(B)Proof.

Combine Theorem 13 with the facts that el ≤e (as we remarked rightafter defining these cardinals) and el ≤b, which follows from Theorem 10 andCorollary 8.□Remark. Referring to the intuitive interpretation of predictors, which was describedjust after their definition, we point out that there is a natural extension of theconcept of predictor that matches even better the intuition of a guessing strategy.Instead of having a fixed set Dπ of numbers n for which the strategy tries to predictthe value x(n) of an unknown function, given the list x ↾n of prior values, we couldlet the decision whether to attempt a prediction of x(n) for a particular n dependalso on x ↾n.

Of course, we should either require that, for each x, the strategyattempts infinitely many predictions or else declare that any x for which this failsis deemed to have evaded π. The cardinal e+ associated to this concept, namelythe smallest number of functions ω →ω needed to evade all ω-valued predictors ofthis generalized sort, is clearly ≥e.

All the upper bounds we have given for e applyalso to e+, with only minor modifications of the proofs.We also remark that, if we were to weaken the definition of “predicts” by re-quiring only that infinitely many (rather than all but finitely many) predictions arecorrect, then the new e would be ≥the old one, but it would still be ≤d and itsminimum with b would still be ≤add(B), by the same proofs as for the original e.5. QuestionsThe results presented in this paper raise a multitude of questions, for we have in-troduced numerous cardinals and many possible connections between them remainundecided.

Here are some rather strong conjectures, whose proof would greatlysimplify our picture of these cardinals. (1) e−= se.

(Notice that this would imply equality of cardinals (2) through(5) in Corollary 8. )(2) e ≤b.

(3) se = add(B).A more specific problem, whose solution might throw considerable light on thegeneral situation, is to compute the value of se in the model obtained by an ℵ2-stage, countable support iteration of Mathias forcing over a model of the generalizedcontinuum hypothesis. This model has b and both unif’s equal to ℵ2 while bothcov’s are ℵ1.

Therefore e = ℵ1 by Theorem 13, but our results do not determinese

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