On inverse gamma-systems and the number of L_{infty,lambda}-equivalent, non-isomorphic mode
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a r X i v :m a t h /9807181v 1 [m a t h .L O ] 15 J u l 1998On inverse γ-systems and the number of L ∞λ-equivalent,non-isomorphic models for λsingularSaharon Shelah∗Pauli V¨a is¨a nen†February 8,2008AbstractSuppose λis a singular cardinal of uncountable cofinality κ.For a model M of cardinalityλ,let No(M )denote the number of isomorphism types of models N of cardinality λwhich are L ∞λ-equivalent to M .In [She85]Shelah considered inverse κ-systems A of abelian groups and their certain kind of quotient limits Gr(A )/Fact(A ).In particular Shelah proved in [She85,Fact 3.10]that for every cardinal µthere exists an inverse κ-system A such that A consists of abelian groups having cardinality at most µκand card Gr(A )/Fact(A ) =µ.Later in [She86,Theorem 3.3]Shelah showed a strict connection between inverse κ-systems and possible values of No (under the assumption that θκ<λfor every θ<λ):if A is an inverse κ-system of abelian groups having cardinality <λ,then there is a model M such that card(M )=λand No(M )=card Gr(A )/Fact(A ) .The following was an immediate consequence (when θκ<λfor every θ<λ):for every nonzero µ<λor µ=λκthere is a model M µof cardinality λwith No(M µ)=µ.In this paper we show:for every nonzero µ≤λκthere is an inverse κ-system A of abelian groups having cardinality <λsuch that card Gr(A )/Fact(A )=µ(under the assumptions 2κ<λand θ<κ<λfor all θ<λwhen µ>λ),with the obvious new consequence concerning the possible value of No.Specifically,the case No(M )=λis possible when θκ<λfor every θ<λ.11IntroductionSuppose λis a cardinal.For a model M we let card(M )denote the cardinality of the universe of M .When M and N are models of the same vocabulary and they satisfy the same sentences of the infinitary language L ∞λ,we write M ≡∞λN .For any model M of cardinality λwe define No(M )to be the cardinality of the setN /∼=|card(N )=λand N ≡∞λM,where N /∼=is the equivalence class of N under the isomorphism relation.Our principal purposeis to study the possible values of No(M )for models M of singular cardinality with uncountable cofinality.When M is countable,No(M )=1by [Sco65].This result extends to structures of cardinality λwhen λis a singular cardinal of countable cofinality [Cha68].If V=L,λis an uncountable regular cardinal which is not weakly compact,and M is a model of cardinalityλ,then No(M)has either the value1or2λ.Forλ=ℵ1this result was first proved in[Pal77a].Later in[She81]Shelah extended this result to all other regular non-weakly compact cardinals.The possibility No(M)=ℵ0is consistent with ZFC+GCH in case λ=ℵ1,as remarked in[She81].The values No(M)∈ω {0,1}are proved to be consistent with ZFC+GCH in the forthcoming paper of the authors[SV97](number646in Shelah’s publications).The case M has cardinality of a weakly compact cardinal is dealt with in[She82]by Shelah. The result is that forκweakly compact there is for every1≤µ≤κa model Mµsuch that No(Mµ)=µ.There is in preparation by the authors a paper where the question forκweakly compact is revisited.The case M is of singular cardinalityλwith uncountable cofinalityκwasfirst treated in[She85], where the relations of M have infinitely many ter in[She86]Shelah improved the result by showing that ifθκ<λfor everyθ<λand0<µ<λthen No(M)=µis possible for a model M having cardinalityλand relations offinitely many places only.The main idea in those papers was to transform the problem of possible values of No(M)into a question concerning possible cardinalities of“quotient limit”Gr(A)/Fact(A)of an inverse system A of groups[She86,Theorem3.3]:Theorem1(λcardinal withλ>cf(λ)=κ>ℵ0)Ifθκ<λfor everyθ<λand A is an inverse κ-system of abelian groups having cardinality<λ,then there is a model M of cardinalityλ(with relations havingfinitely many places only)such that No(M)=card Gr(A)/Fact(A) . Actually the groups in[She86,Theorem3.3]are not limited to be abelian.However,abelian groups suffice for the present purposes.The recent paperfills a gap left open since the paper[She86].We present a uniform way to construct inverseκ-system of abelian groups having a quotient limit of desired cardinality.The most important new case is that the cardinality of a quotient limit can beλfor some inverse system(in other cases,where the result below can be applied,the Singular Cardinal Hypothesis fails).The result of this paper is:Theorem2(λcardinal withλ>cf(λ)=κ>ℵ0)For every nonzeroµ≤λthere is an inverse κ-system A= G i,h i,j|i<j<κ of abelian groups satisfying that card(G i)<λfor every i<κand card Gr(A)/Fact(A) =µ.The same conclusion holds also for the valuesλ<µ≤λκunder the assumption that2κ<λandθ<κ<λfor everyθ<λ.So the general method used here tofind new possibilities for the values of No(M)is the same as in[She86].As an immediate consequence of the last theorem we get:Theorem3Supposeλis a singular cardinal of uncountable cofinalityκ.For each nonzeroµ≤λκthere is a model M(with relations havingfinitely many places only)satisfying card(M)=λand No(M)=µ,provided thatθκ<λfor everyθ<λ.We give all necessary definitions concerning inverseκ-systems A of abelian groups and their special kind of quotient limits Gr(A)/Fact(A)in the next section.2PreliminariesDefinition2.1Supposeγis a limit ordinal and for every i<j<γ,G i is a group and h i,j is a homomorphism from G j into G i.The family A= G i,h i,j|i<j<γ is called an inverse γ-system when the equation h i,j◦h j,k=h i,k holds for every i<j<k<γ.As in[She85]we assume that all the groups G i,i<γ,are additive abelian groups.To simplify our notation we make an agreement that the letters i,j,k,and l always denote ordinals smaller thanγ.Hence“for all i<j”means“for all ordinals i and j with i<j<γ”and so on.The main objects of our study are the following two sets:Gr(A)= a i,j|i<j<γ |a i,j∈G i and for all k>j,a i,k=a i,j+h i,j(a j,k) ; Fact(A)= a i,j|i<j<γ |for some¯y∈ k<γG k,a i,j=¯y i−h i,j(¯y j) .We consider Gr(A)and Fact(A)as additive abelian groups where the group operation+and the unit element0are pointwise defined.The factor group Gr(A)/Fact(A)is well-defined since Fact(A)⊆Gr(A)by the requirements h i,j◦h j,k=h i,k for all i<j<k.For any inverse γ-system A,the group Gr(A)/Fact(A)is called the quotient limit of A.Definition2.2We letγ⋆γbe the set{(i,j)∈γ×γ|i<j}.For every subset I ofγ⋆γwe defineI1st={i<γ|(i,j)∈I for some j<γ}and for each i∈I1st,I[i]={j<γ|(i,j)∈I}.We also say thatI is cobounded ifγ I1st andγ I[i],for all i∈I1st,are bounded subsets ofγ;I is coherent if I1st is unbounded inγand for every i∈I1st,I[i]=I1st (i+1);I is eventually coherent if it is unbounded and for every i∈I1st,I1st I[i]is a boundedsubset ofγ.Remark.Suppose I is an eventually coherent subset ofγ⋆γand S is a subset of I1st.If card(S)<cf(γ),then I1st ( i∈S I[i])is a bounded subset ofγ.If S is unbounded inγ,then I∩(S×S)is an eventually coherent subset of I.In[She86,Claim1.12]Shelah proved(note the remark given after the following lemma)that if two sequences a and b from Gr(A)agree on a coherent set of indices,then a≡b mod Fact(A). The following slight improvement of this condition has an essential role in the proof of Theorem2. Lemma2.3Suppose A is an inverseγ-system,and a,b∈Gr(A).Then a≡b mod Fact(A) holds if there is an eventually coherent subset I ofγ⋆γsuch that a i,j=b i,j for all(i,j)∈I.Proof.We shall need an eventually coherent subset J of I having the property that J[i]|i∈J1st is a decreasing chain of end segments of J1st.Let S be an unbounded subset of I having the order type cf(γ).Define a subset J of I by J1st=S and for all j∈S,J[j]=S∩ i∈S∩(j+1) I[i] (i∗+1) ,where i∗is the supremum of the bounded subset I1st I[i]ofγ.The set J is well-defined since I is eventually coherent and card S∩(j+1) <cf(γ)for all j<γ.Now J is also eventually coherent,and furthermore,for all i∈J1st,J[i]=S min(J[i])and for all j∈J1st i, min(J[i])≤min(J[j]).Define for every i<γ,i′to be min J1st (i+1) and i′′=min(J[i′]).Then the following are satisfied for all i<j:i<i′<i′′,j<j′<j′′,i′≤j′,i′′≤j′′,and also i′<j′′;j′′∈I1st and(i′,i′′),(j′,j′′),(i′,j′′)∈I.)we haveSince a and b are in Gr(A TRa i,j′′=a i,j+h i,j(a j,j′′),b i,j′′=b i,j+h i,j(b j,j′′).Therefore the following equations hold:a i,j−b i,j=(a i,j′′−b i,j′′)− h i,j(a j,j′′)−h i,j(b j,j′′)(A)=(a i,j′′−b i,j′′)−h i,j(a j,j′′−b j,j′′).Because of i<i′<j′′we also have thata i,j′′=a i,i′+h i,i′(a i′,j′′),b i,j′′=b i,i′+h i,i′(b i′,j′′).Since(i′,j′′)∈I,a i′,j′′=b i′,j′′holds.Hence we get(B)a i,j′′−b i,j′′=a i,i′−b i,i′.Moreover,i<i′<i′′yieldsa i,i′′=a i,i′+h i,i′(a i′,i′′),b i,i′′=b i,i′+h i,i′(b i′,i′′).Now(i′,i′′)∈I implies that a i′,i′′=b i′,i′′,and consequentlya i,i′−b i,i′=a i,i′′−b i,i′′.This equation together with(A)and(B)implies that for all i<ja i,j−b i,j=(a i,i′′−b i,i′′)−h i,j(a j,j′′−b j,j′′).So the sequence¯y= a i,i′′−b i,i′′|i<γ ∈ i<γG i exemplifies that a−b∈Fact(A),and we have a≡b mod Fact(A). 2.3Remark.In[She86,Claim1.12]the groups of an inverse system A need not to be abelian groups.Hence instead of the factor group Gr(A)/Fact(A)a partition Gr(A)/≈A with a special kind of equivalence relation≈A were considered there.However,it is straightforward to prove, by means of the preceding proof,also the more general case of Lemma2.3where“equivalent modulo Fact(A)”is replaced by≈A.In the next section we shall need a notion of a tree,so we shortly describe our notation.Definition2.4Suppose T= T,¡ is a tree of heightγ.For every i<γ,T i is the i th level of the tree.When i<j<γandη∈T j,thenη↾i denotes the unique elementν∈T i for whichν¡ηholds.For each i<γandν∈T i,T j[ν]is the set{η∈T j|ν¡η}.The set of allγ-branches of T,i.e.,the set{t∈ i<γT i|for all i<j,t(i)¡t(j)},is denoted by Brγ(T).3The inverseγ-system of free R-modulesIn this section we define special kind of inverseγ-systems A TRand prove a result concerningcardinalities of their quotient limit Gr(A TR )/Fact(A TR)(Conclusion3.12).A direct consequenceof the result will be Theorem2.Definition3.1Supposeγis a limit ordinal,R is a ring,and T is a tree of heightγ.We define an inverseγ-system A TR= G i,h i,j|i<j<γ by the following stipulations:a)for each i<γ,G i is the R-module freely generated by{xν,l|ν∈T i and i<l<γ};b)for every i<j<γ,h i,j is the homomorphism from G j into G i determined by the valuesh i,j(xη,l)=xη↾i,l−xη↾i,j,for allη∈T j and l>j.(It is easy to check that the equationsh i,k=h i,j◦h j,k are satisfied for all i<j<k).We consider Gr(A TR ),Fact(A TR),and Gr(A TR)/Fact(A TR)as R-modules where the operations+,·,and the unit element0for addition are pointwise defined.For each t∈Brγ(T),we define t to be the sequence x t(i),j|i<j<γ .Directly by thedefinitions of G i and h i,j,t belongs to Gr(A TR )for every t∈Brγ(T).We let t t∈Brγ(T)Rbethe submodule of Gr(A TR)generated by the elements t,t∈Brγ(T).When Brγ(T)is emptyt t∈Brγ(T)Ris the trivial submodule{0}.Remark.Each G i is nonempty when T has heightγ.Hence i<γG i is nonempty,and alsoFact(A TR)= ¯y i−h i,j(¯y j)|i<j<γ |¯y∈ i<γG iis nonempty.So Gr(A TR )⊇Fact(A TR)is nonempty for every ring R and tree T of heightγ.Observe also that the inverseγ-system A TRis the same as used in[She85,Claim3.8]when R is the trivial ring{0,1}and T consists ofµmany disjointγ-branches.So the proof given in this section offers an alternative proof for[She85,Claim3.8],and even more information,namely that card Gr(A T R)/Fact(A T R) must be exactlyµnot only≥µ.Definition3.2Suppose a∈Gr(A TR)and i<j<γ.By the definition of G i and the requirement a i,j∈G i,we define a i,jν,l forν∈T i and l>i,to be the coefficients from R(with onlyfinitely many of them nonzero)which satisfy the equationa i,j= ν∈T i l>i a i,jν,l·xν,l.Thefinite set{(ν,l)∈T i× γ (i+1) |a i,jν,l=0}is called the support of a i,j,and it is denoted by supp(a i,j).Suppose S is a subset ofγ,e∈G i,and eν,l∈R for everyν∈T i and l>i are elements such thate= ν∈T i l>i eν,l·xν,l.Then we write e↾S for the following element of G i:ν∈T il∈S (i+1)eν,l·xν,l.The following simple lemma has an important corollary.Lemma3.3a)The restriction h i,j(e)↾j equals0for every i<j and e∈G j.b)For every a∈Fact(A TR) {0},there are i<j<γsuch that a i,j↾j=0.Proof.a)Straightforwardly by the definitions of G j and h i,j.b)By the definition of Fact(A TR),let¯y∈ i<γG i be such that for all i<j,a i,j=¯y i−h i,j(¯y j). In addition to that let y iν,l∈R,for i<γ,ν∈T i and l>i,be such that¯y i= ν∈T i l>i y iν,l·xν,l.Since a=0there must be i<γwith¯y i=0.Define j to be min{l>i|y iν,l=0for someν∈T i}+1.Then¯y i↾j is nonzero and because h i,j(¯y j)↾j=0,we have a i,j↾j=¯y i↾j−h i,j(¯y j)↾j=¯y i↾j=0.Corollary3.4The elements t,t∈Brγ(T),are independent over Fact(A TR),i.e.,t t∈Brγ(T)R ∩Fact(A TR)={0}.Hence A TR satisfies card Gr(A T R)/Fact(A T R) ≥card t t∈Brγ(T)R .Proof.Directly by the definition of t,t i,j=x t(i),j and hence t i,j↾j=0,for all t∈Brγ(T)and i<j.So for any nonzero a= 1≤m≤n d m·t m,where n<ω,d m∈R {0},and t m∈Brγ(T), the restrictions a i,j↾j are equal to0for all i<j.So by the preceding lemma a can not be in Fact(A TR). Next we derive equations of weighty significance.Lemma3.5Suppose b∈Gr(A TR)and i<j<k<γ.Then the following equations are satisfied for allν∈T i:b i,kν,l=b i,jν,l when i<l<j;(A)b i,kν,j=b i,jν,j− η∈T j[ν]l>j b j,kη,l;(B)b i,kν,l=b i,jν,l+ η∈T j[ν]b j,kη,l when l>j.(C)Proof.By dividing the sum into groups we get thatb i,j= ν∈T i l>i b i,jν,l·xν,l= ν∈T i i<l<j b i,jν,l·xν,l+b i,jν,j·xν,j+ l>j b i,jν,l·xν,l .Similarly the following equation is satisfied,b i,k= ν∈T i i<l<j b i,kν,l·xν,l+b i,kν,j·xν,j+ l>j b i,kν,l·xν,l .From the definition of h i,j we may infer thath i,j(b j,k)= (η∈T j,l>j)b j,kη,l·h i,j(xη,l)= (η∈T j,l>j)b j,kη,l·(xη↾i,l−xη↾i,j)= (η∈T j,l>j)b j,kη,l·xη↾i,l− (η∈T j,l>j)b j,kη,l·xη↾i,j= ν∈T i l>j η∈T j[ν]b j,kη,l ·xν,l− (η∈T j[ν],l>j)b j,kη,l ·xν,j .So the equations(A),(B),and(C)for all i<j<k follow by comparing the coefficients of each generator xν,l in the equation b i,k=b i,j+h i,j(b j,k). 3.5).Lemma3.6Suppose a∈Gr(A TRa)For all i<j<k,a i,j↾j=a i,k↾j.b)(cf(γ)>ℵ0)For every i<γ,the union i<j<γsupp(a i,j↾j)is offinite cardinality(wheresupp(a i,j↾j)=supp(a i,j)∩(T i×j)of course).)satisfying the following conditions:c)(cf(γ)>ℵ0)There is b∈Gr(A TR),a≡b mod Fact(A TRI={(i,j)∈γ⋆γ|b i,j↾j=0}is cobounded(in fact I1st=γ),andfor every(i,j)∈I,b i,j=b i,j↾{j}.Proof.a)The claim holds directly by Lemma3.5(A).b)Suppose the union is infinite.Since cf(γ)>ℵ0there is some k<γfor which already j<k supp(a i,j↾j)is infinite.By(a),supp(a i,j↾j)⊆supp(a i,k)for each j<k.Consequently j<k supp(a i,j↾j)⊆supp(a i,k)contrary to thefiniteness of supp(a i,k).c)By(a)and(b)there must be for every i<γa bound i∗∈γ (i+1)such that for every)byj≥i∗,a i,i∗↾i∗=a i,j↾j.Define an element c∈Fact(A TRc i,j=a i,i∗↾i∗−h i,j(a j,j∗↾j∗),)and for every i<γand j≥i∗for all i<j.Let b be a−c.Then a≡b mod Fact(A TRb i,j=a i,j−c i,j=a i,j↾(γ j)+a i,j↾j−a i,i∗↾i∗+h i,j(a j,j∗↾j∗)=a i,j↾(γ j)+h i,j(a j,j∗↾j∗).It follows from Lemma 3.3(a)that b i,j ↾j =0for all i <γand j ≥i ∗,and thus I is cobounded.Now suppose,contrary to the last claim in (c),that b i,j ν,l =0for some i <γ,j ≥i ∗,ν∈T i ,andl >j .Let k be max {i ∗,j ∗,l +1}.Then both b i,k ↾k and b j,k ↾k are 0.By Lemma 3.5(C)the following equation holds:η∈T j [ν]b j,k η,l =b i,k ν,l −b i,jν,l .Since b i,j ν,l =0and l <k implies b i,kν,l =0the sumη∈T j [ν]b j,kη,l must be nonzero.So there isη∈T j [ν]with b j,k η,l =0.This contradicts the facts l <k and b j,k ↾k equals 0.3.6Lemma 3.7Suppose b ∈Gr(A T R )and I is a subset of {(i,j )∈γ⋆γ|bi,j↾{j }=b i,j }.Then for all (i,j )∈I ,ν∈T i ,and k ∈I [i ]∩I [j ],b i,j ν,j = η∈T j [ν]b j,k η,k =b i,kν,k .Proof.Since (i,k )and (j,k )are in I ,both b i,k ν,j and b j,kη,l are equal to 0for all η∈T j when l =k .Hence Lemma 3.5(B)can be reduced to the form b i,jν,j = η∈T j [ν]b j,k η,k .Now (i,j )∈I guarantees that b i,j ν,k =0.Thus the reduced form together with Lemma 3.5(C)(applied for l =k )yieldb i,j ν,j =b i,k ν,k .Lemma 3.8Suppose b is an element of Gr(A T R ).a)If b is not in Fact(A T R )and I is an eventually coherent subset of γ⋆γsuch that bi,j=b i,j ↾{j }for all (i,j )∈I ,then there is an eventually coherent subset J of I with b i,j =b i,j ↾{j }=0whenever (i,j )∈J .b)(cf(γ)>ℵ0)If J is an eventually coherent subset of γ⋆γsuch that b i,j =b i,j ↾{j }=0for all (i,j )∈J ,then there are a bound n ∗<ωand an eventually coherent subset K of J such that card(supp(b i,j ))<n ∗for all (i,j )∈K .Proof.a)Since b ≡0mod Fact(A T R )it follows by Lemma 2.3that there is no subset of{(i,j )∈I |b i,j=0}which would be eventually coherent.Hence there is an unbounded subset S of I 1st such that for each i ∈S there is j i ∈I [i ]with b i,j i nonzero.Fix any i ∈S .Sinceb i,j i =b i,j i ↾{j i }=0,let νi be an element of T i with b i,j iνi ,j i =0.By Lemma 3.7,b i,k νi ,k =b i,j i νi ,j i =0for all k ∈I [i ]∩I [j i ].Because I was eventually coherent,we have shown that J =I ∩(S ×S )is an eventually coherent set as wanted in the claim.b)First of all we claim that for each i ∈J 1st the unionj ∈J [i ]supp(b i,j )is of finite cardinality.Observe that for every (i,j )∈J ,supp(b i,j )=supp(b i,j )∩(T i ×{j }).Assume,contrary to this subclaim,that i ∈J 1st , j m |m <ω is an increasing sequence ofordinals in J [i ],and {νm |m <ω}is a set of distinct elements from T i such that b i,j mνm ,j m nonzero for every m <ω.Since J is eventually coherent and γis of uncountable cofinality let k <γbe the minimal element in J [i ]∩m<ωJ [j m ].Now for each m <ω,the pairs (i,j m ),(i,k ),and(j m ,k )are in J ,and by Lemma 3.7,the equation b i,j mνm ,j m =b i,k νm ,k =0holds.So the infinite set {(νm ,k )|m <ω}is a subset of supp(b i,k ),a contradiction.It follows from the subclaim that for each i∈J1st,thefinite ordinaln i=card j∈J[i]supp(b i,j) +1satisfies card(supp(b i,j))<n i for all j∈J[i].Since J1st is uncountable,there are n∗<ωand an unbounded subset S of J1st such that n i=n∗for all i∈S.So n∗and the set K=J∩(S×S) meet the requirements of the claim. 3.8 Lemma3.9(cf(γ)>card(R))Suppose b is in Gr(A TR)and I is an eventually coherent subset of{(i,j)∈γ⋆γ|b i,j↾{j}=b i,j=0}.Then there are d∈R,t∈Brγ(T),and an eventuallycoherent subset J of I for which b i,jt(i),j=d=0whenever(i,j)∈J.Proof.We define by induction onα<cf(γ)the following objects:an increasing sequence iα|α<cf(γ) of ordinals in I1st with limitγ;an increasing sequence να|α<cf(γ) ∈ α<cf(γ)T iα;subsets Kαof I[iα]such that I1st Kαare bounded inγ;elements dα∈R {0}such that for every k∈Kα,b iα,kνα,k=dα.This suffices since card(R)<cf(γ)implies that there are d∈R and H⊆cf(γ)unbounded in cf(γ)such that dα=d for everyα∈H.Moreover,the claim is satisfied by t∈Brγ(T)andJ⊆I defined as follows.For every i<γ,t(i)=νβi ↾i,whereβi=min{α<cf(γ)|iα≥i},and J= α∈H {iα}×(S∩Kα) ,where S is{iα|α∈H}.Let γα|α<cf(γ) be an increasing sequence with limitγ.Defineiα=min (I1st∩ β<αKβ) γαandj=min I1st∩I[iα]∩ β<αI[iβ] ,where both β<αKβand β<αI[iβ]are equal toγwhenα=0.This pair(iα,j)is well-defined since I is eventually coherent,α<cf(γ),and whenα>0,I1st Kβis bounded for eachβ<αby the induction hypothesis.Ifα=0,then(i0,j)∈I guarantees that b i0,j↾{j}=b i0,j=0.Hence we canfindν0∈T iwithb i0,jν0,j=0.Whenα>0we define elementsηβ∈T iα[νβ]for eachβ<αas follows.Fixβ<α.Sinceiα∈Kβwe get by the induction hypothesis that b iβ,iανβ,iα=dβ=0.Furthermore(iβ,iα)∈I (because Kβ⊆I[iβ]),(iβ,j)∈I,and(iα,j)∈I together with Lemma3.7yieldη∈T iα[νβ]b iα,jη,j=b iβ,iανβ,iα=0.Therefore we canfindηβ∈T iα[νβ]for which b iα,jηβ,j=0.Ifα>0is a successor ordinal defineναto beηα−1.Whenαis a limit ordinal,thefiniteness of the support supp(b iα,j)ensures that there areνα∈T iαand an unbounded subset H ofαsuch thatηβ′=ναfor allβ′∈H.By the induction hypothesisνβ¡νβ′for allβ<β′<α.Hence νβ¡νβ′¡ηβ′=ναholds for everyβ<αandβ′=min(H β).Let dαbe b iα,jνα,j .By Lemma3.7,every k∈I[iα]∩I[j]satisfies that b iα,kνα,k=b iα,jνα,j=dα.Hence iα,να,and dαtogether with the set Kα=I[iα]∩I[j]meet the requirements given at the beginning of the proof. 3.9Corollary3.10(cf(γ)>ℵ0)If Brγ(T)is empty,then Gr(A TR )=Fact(A TR).Proof.Suppose a∈Gr(A TR ) Fact(A TR).By Lemma3.6(c)together with Lemma3.8(a)thereis b∈Gr(A TR )such that a≡b mod Fact(A TR)and the set{(i,j)∈γ⋆γ|b i,j↾{j}=b i,j=0}is eventually coherent.By Lemma3.9there is aγ-branch through the tree T,i.e.,Brγ(T)=∅. Observe that the assumption card(R)<cf(γ)is not needed,as can be seen from the proof of Lemma3.9. Lemma3.11(cf(γ)>max{ℵ0,card(R)})The elements t,t∈Brγ(T),generate Gr(A TR)mod-ulo Fact(A TR).Proof.We show that for every a∈Gr(A TR )with a∈Fact(A TR)we canfind n<ω,d1,...,d n∈R {0}and t1,...,t n∈Brγ(T)satisfyinga≡ 1≤m≤n d m·t m mod Fact(A T R).(A)Suppose a∈Gr(A TR ) Fact(A TR).By Lemma3.6(c)and Lemma3.8(a)let b be an element ofGr(A TR )and I1an eventually coherent subset ofγ⋆γsuch that a≡b mod Fact(A TR)and foreach(i,j)∈I1,b i,j=b i,j↾{j}=0.Furthermore,we may assume by Lemma3.8(b)that n∗<ωis a bound for which card(supp(b i,j))<n∗hold for all(i,j)∈I1.By Lemma3.9there are d1∈R,t1∈Brγ(T),and an eventually coherent set J1⊆I1havingthe property that b i,jt1(i),j =d1=0whenever(i,j)∈J1.Since d1·t1∈Gr(A TR),the sequencec=b−d1·t1is in Gr(A TR ).If c is in Fact(A TR),then b≡d1·t1mod Fact(A TR),and becauseof a≡b mod Fact(A TR),also(A)holds for n=1.Suppose1≤n<ωand objects d m∈R {0},t m∈Brγ(T),and J m⊆J1for m≤n are already defined.Assume also that these objects satisfy the following conditions:1)J m′⊇J m for all1≤m′≤m≤n;2)for all1≤m′<m≤n and i∈(J m)1st,t m′(i)=t m(i);3)for every1≤m≤n and(i,j)∈J m,b i,jt m(i),j=d m=0;4)c=b− 1≤m≤n d m·t m∈Fact(A T R).Clearly c i,j=c i,j↾{j}and card(supp(c i,j))≤card(supp(b i,j))<n∗for all(i,j)∈J n.Again by Lemma3.8(a),there is an eventually coherent set I n+1⊆J n such that for each(i,j)∈I n+1, c i,j=0.Moreover,by Lemma3.9,there are d n+1∈R,t n+1∈Brγ(T),and an eventuallycoherent set J n+1⊆{(i,j)∈I n+1|c i,jt n+1(i),j=d n+1=0}.The properties(2),(3)and(4)above imply that c i,jt m(i),j =b i,jt m(i),j−d m=0for every m≤nand(i,j)∈J m.On the other hand,c i,jt n+1(i),jis nonzero for each(i,j)∈J n+1.Thus t n+1(i)cannot be in{t m(i)|1≤m≤n}if i∈(J n+1)1st.So for all(i,j)∈J n+1,x tn+1(i),j ∈{x tm(i),j|1≤m≤n},and consequently b i,jt n+1(i),j =c i,jt n+1(i),j.Thus also J n+1,t n+1,and d n+1satisfy theproperties(1),(2),and(3)(but not necessarily(4)).We claim that there must be n<n∗such thatb− 1≤m≤n d m·t m∈Fact(A T R).(B)Assume,contrary to this subclaim,that the process introduced above has been carried out n∗many times and objects J m,t m,d m for i≤m≤n∗are defined.In addition to that suppose they satisfy the conditions(1),(2),and(3).Define i=min (J n∗)1st and j=min(J n∗[i]).Then for every m≤n∗,(i,j)∈J m yields b i,jt m(i),j=d m=0.This contradicts the condition card(supp(b i,j))<n∗,since the set{(t m(i),j)|m≤n∗}⊆supp(b i,j)is of cardinality n∗. Now suppose n<ωis afinite ordinal satisfying(B).Then b≡ 1≤m≤n d m·t m mod Fact(A T R),and because a≡b mod Fact(A TR)also(A)is satisfied. 3.11Conclusion3.12For any ordinalγof uncountable cofinality,ring R with card(R)<cf(γ), and tree T of heightγ,the inverseγ-system A TR= G i,h i,j|i<j<γ has the properties thatcard(G i)=max{card(γ),card(T i),card(R)}for all i<γ,andcard Gr(A T R)/Fact(A T R) =card t t∈Brγ(T)R .Proof of Theorem2.Remember thatλandκwere cardinals withℵ0<κ=cf(λ)<λ.We wanted to study possible cardinalitiesµof the quotient limit Gr(A)/Fact(A),where A is an inverseκ-system consisting of abelian groups having cardinality<λ.Now Conclusion3.12gives a complete solution to this problem because ofλ>cf(λ)=κ=cf(κ)>ℵly,in order to meet the requirements card(G i)<λfor all i<κ,it is needed only to ensure that R and the i th level of T are small enough.On the other hand,a suitable choice of R and T yields any desired value forµ=card Gr(A T R)/Fact(A T R) .We briefly describe methods to choose suitable R and T for every nonzeroµ≤λκ.For any R,card Gr(A T R)/Fact(A T R) equals1when Brκ(T)is empty.Soµ=1is possible since obviously there exists a tree of heightκwithoutκ-branches and having levels of cardinality<λwhenλsingular of cofinalityκ.Also all thefinite valuesµ>1are possible by taking T with only oneκ-branch and R with card(R)=µ.Furthermore the case of infiniteµ<λis satisfied by any R with card(R)<min{κ,µ}and T with exactlyµmanyκ-branches.The valueµ=λis possible for any R with card(R)<κbecause a suitable tree can be constructed,for example,as follows.Let λi|i<κ be an increasing sequence of ordinals<λwith limitλ.Then the treeT={t↾α|α<κ,t∈ i<κλi,and t(i)is nonzero only forfinitely many i<κ}, ordered by inclusion,satisfies card(Brκ(T))=λand card(T i)=λi<λfor each i<κ.Also the cardinalitiesµof the quotient limit,whenλ<µ≤λκ,are possible for any ring of cardinality<κ.Existence of a suitable tree is proved for example in[She89,Fact10]under the assumption that2κ<λandθ<κ<λfor everyθ<λ(other sources for a proof are given in [She94,Analytical Guide§10]). 2 References[Cha68] C.C.Chang.Some remarks on the model theory of infinitary languages.In J.Barwise, editor,The syntax and semantics of infinitary languages,Lecture Notes in Math.,72, pages36–63.Springer-Verlag,Berlin,1968.theories,II.Algebra i Logika,16(4):443–[Pal77a] E.A.Palyutin.Number of models in L∞,ω1456,1977.English translation in[Pal77b].[Pal77b]E.A.Palyutin.Number of models in L∞,ωtheories,II.Algebra and Logic,16(4):299–1309,1977.[Sco65]Dana Scott.Logic with denumerably long formulas andfinite strings of quantifiers.In J.W.Addison,Leon Henkin,and Alfred Tarski,editors,Theory of Models(Proc.1963Internat.Sympos.Berkeley),pages329–34.North-Holland,Amsterdam,1965. [She81]Saharon Shelah.On the number of nonisomorphic models of cardinalityλL∞,λ-equivalent to afixed model.Notre Dame J.Formal Logic,22(1):5–10,1981.[She82]Saharon Shelah.On the number of nonisomorphic models in L∞,λwhenκis weakly compact.Notre Dame J.Formal Logic,23(1):21–26,1982.[She85]Saharon Shelah.On the possible number no(M)=the number of nonisomorphic models L∞,λ-equivalent to M of powerλ,forλsingular.Notre Dame J.Formal Logic, 26(1):36–50,1985.[She86]Saharon Shelah.On the no(M)for M of singular power.In Around classification theory of models,Lecture Notes in Math.,1182,pages120–134.Springer-Verlag,Berlin,1986. 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