Saharon Shelah and Pauli Vaisanen- On inverse gamma-systems and the number of L-infinity-lambda...

8/3/2019 Saharon Shelah and Pauli Vaisanen- On inverse gamma-systems and the number of L-infinity-lambda equivalent, n

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On inverse -systems and the number ofL-equivalent,

non-isomorphic models for singular

Saharon Shelah Pauli Vaisanen

October 6, 2003

Abstract

Suppose is a singular cardinal of uncountable cofinality . For a model M of cardinality

, let No(M) denote the number of isomorphism types of models N of cardinality whichare L-equivalent to M. In [She85] Shelah considered inverse -systems A of abeliangroups and their certain kind of quotient limits Gr(A)/Fact(A). In particular Shelah provedin [She85, Fact 3.10] that for every cardinal there exists an inverse -system A such thatA consists of abelian groups having cardinality at most and card

Gr(A)/Fact(A)

= .

Later in [She86, Theorem 3.3] Shelah showed a strict connection between inverse -systemsand possible values of No (under the assumption that < for every < ): ifA is aninverse -system of abelian groups having cardinality < , then there is a model M suchthat card(M) = and No(M) = card

Gr(A)/Fact(A)

. The following was an immediate

consequence (when < for every < ): for every nonzero < or = there is amodel M of cardinality with No(M) = . In this paper we show: for every nonzero there is an inverse -system A of abelian groups having cardinality < such thatcard

Gr(A)/Fact(A)

= (under the assumptions 2 < and ), with the obvious new consequence concerning the possible value of No. Specifically,the case No(M) = is possible when < for every < .1

1 Introduction

Suppose is a cardinal. For a model M we let card(M) denote the cardinality of the universeof M. When M and N are models of the same vocabulary and they satisfy the same sentencesof the infinitary language L, we write M N. For any model M of cardinality wedefine No(M) to be the cardinality of the set

N/= | card(N) = and N M

,

whereN/= is the equivalence class ofN under the isomorphism relation. Our principal purposeis to study the possible values of No(M) for models M of singular cardinality with uncountablecofinality.

When M is countable, No(M) = 1 by [Sco65]. This result extends to structures of cardinality when is a singular cardinal of countable cofinality [Cha68].

Thanks to GIF for its support of this research and also to University of Helsinki for funding a visit of the

first author to Helsinki in August 1996. Pub. No. 644.The second author wishes to thank Tapani Hyttinen under whose supervision he did his share of the paper.11991 Mathematics Subject Classification: primary 03C55; secondary 03C75. Key words: number of models,

infinitary logic, inverse -system.

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If V = L, is an uncountable regular cardinal which is not weakly compact, and M is amodel of cardinality , then No(M) has either the value 1 or 2. For = 1 this result wasfirst proved in [Pal77a]. Later in [She81] Shelah extended this result to all other regular non-weakly compact cardinals. The possibility No(M) = 0 is consistent with ZFC + GCH in case = 1, as remarked in [She81]. The values No(M) {0, 1} are proved to be consistent

with ZFC + GCH in the forthcoming paper of the authors [SV97] (number 646 in Shelahspublications).

The case M has cardinality of a weakly compact cardinal is dealt with in [She82] by Shelah.The result is that for weakly compact there is for every 1 a model M such thatNo(M) = . There is in preparation by the authors a paper where the question for weaklycompact is revisited.

The case M is of singular cardinality with uncountable cofinality was first treated in [She85],where the relations of M have infinitely many places. Later in [She86] Shelah improved theresult by showing that if < for every < and 0 < < then No(M) = is possiblefor a model M having cardinality and relations of finitely many places only. The main idea

in those papers was to transform the problem of possible values of No(M) into a questionconcerning possible cardinalities of quotient limit Gr(A)/Fact(A) of an inverse system A ofgroups [She86, Theorem 3.3]:

Theorem 1 ( cardinal with > cf() = > 0)If < for every < andA is an inverse

-system of abelian groups having cardinality < , then there is a model M of cardinality (with relations having finitely many places only) such that No(M) = card

Gr(A)/Fact(A)

.

Actually the groups in [She86, Theorem 3.3] are not limited to be abelian. However, abeliangroups suffice for the present purposes.

The recent paper fills a gap left open since the paper [She86]. We present a uniform way to

construct inverse -system of abelian groups having a quotient limit of desired cardinality. Themost important new case is that the cardinality of a quotient limit can be for some inversesystem (in other cases, where the result below can be applied, the Singular Cardinal Hypothesisfails). The result of this paper is:

Theorem 2 ( cardinal with > cf() = > 0)For every nonzero there is an inverse-system A = Gi, hi,j | i < j < of abelian groups satisfying that card(Gi) < for everyi < andcard

Gr(A)/Fact(A)

= . The same conclusion holds also for the values <

under the assumption that 2 < and


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2 Preliminaries

Definition 2.1 Suppose is a limit ordinal and for every i < j < , Gi is a group and hi,j isa homomorphism from Gj into Gi. The family A = Gi, hi,j | i < j < is called an inverse-system when the equation hi,j hj,k = hi,k holds for every i < j < k < . As in [She85] we

assume that all the groups Gi, i < , are additive abelian groups.To simplify our notation we make an agreement that the letters i, j, k, and l always denoteordinals smaller than . Hence for all i < j means for all ordinals i and j with i < j < and so on.

The main objects of our study are the following two sets:

Gr(A) =

ai,j | i < j < | ai,j Gi and for allk > j, ai,k = ai,j + hi,j (a

j,k)

;

Fact(A) =

ai,j | i < j < |for some y k


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where i is the supremum of the bounded subset I1st I[i] of . The set J is well-defined sinceI is eventually coherent and card

S (j + 1)

< cf() for all j < . 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 < , 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.

Since a and b are in Gr(ATR) we have

ai,j = ai,j + hi,j (a

j,j),

bi,j = bi,j + hi,j(b

j,j).

Therefore the following equations hold:

ai,j

b

i,j

= (a

i,j

b

i,j

)

hi,j (a

j,j

) hi,j (b

j,j

)

(A)= (ai,j

bi,j

) hi,j (aj,j bj,j

).

Because of i < i < j we also have that

ai,j = ai,i

+ hi,i(a

i,j),

bi,j = bi,i

+ hi,i(bi,j).

Since (i, j) I, ai,j = bi

,j holds. Hence we get

(B) ai,j

bi,j

= ai,i

bi,i

.

Moreover, i < i < i yieldsa

i,i

=a

i,i

+ hi,i(a

i,i

),b

i,i = bi,i

+ hi,i(bi,i).

Now (i, i) I implies that ai,i = bi

,i , and consequently

ai,i bi,i

= ai,i

bi,i

.

This equation together with (A) and (B) implies that for all i < j

ai,j bi,j = (ai,i

bi,i

) hi,j (aj,j bj,j

).

So the sequence y = ai,i

bi,i

| i <

i


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3 The inverse -system of free R-modules

In this section we define special kind of inverse -systems ATR and prove a result concerningcardinalities of their quotient limit Gr(AT

R)/Fact(AT

R) (Conclusion 3.12). A direct consequence

of the result will be Theorem 2.

Definition 3.1 Suppose is a limit ordinal, R is a ring, and T is a tree of height . We definean inverse -system ATR = Gi, hi,j | i < j < by the following stipulations:

a) for each i < , Gi is the R-module freely generated by {x,l | Ti andi < l < };

b) for every i < j < , hi,j is the homomorphism from Gj into Gi determined by the valueshi,j (x,l) = xi,l xi,j, for all Tj and l > j. (It is easy to check that the equationshi,k = hi,j hj,k are satisfied for all i < j < k).

We consider Gr(ATR

), Fact(ATR

), andGr(ATR

)/Fact(ATR

) as R-modules where the operations +, ,

and the unit element 0 for addition are pointwise defined.For each t Br(T), we define t to be the sequence xt(i),j | i < j < . Directly by the

definitions of Gi and hi,j , t belongs to Gr(ATR

) for every t Br(T). We let ttBr(T)R be

the submodule of Gr(ATR

) generated by the elements t, t Br(T). When Br(T) is empty

ttBr(T)R is the trivial submodule {0}.

Remark. Each Gi is nonempty when T has height . Hence

ii ai,j

,l x,l.

The finite set {(, l) Ti

(i + 1)

| ai,j,l = 0} is called the support ofai,j, and it is denoted

by supp(ai,j ).

Suppose S is a subset of , e Gi, and e,l R for every Ti and l > i are elements suchthat

e =

Til>i e,l x,l.

Then we write eS for the following element of Gi:

Ti

lS(i+1)e,l x,l.

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The following simple lemma has an important corollary.

Lemma 3.3

a) The restriction hi,j (e)j equals 0 for every i < j and e Gj.

b) For everya Fact(ATR) {0}, there are i < j < such thatai,jj = 0.

Proof. a) Straightforwardly by the definitions of Gj and hi,j .

b) By the definition of Fact(ATR), let y

i i, be such that

yi =

Til>i

yi,l x,l.

Since a = 0 there must be i < with yi = 0. Define j to be min{l > i | yi,l = 0 for some

Ti} + 1. Then yij is nonzero and because hi,j (y

j )j = 0, we have ai,jj = yij hi,j (yj )j =

yij = 0.

Corollary 3.4 The elements t, t Br(T), are independent over Fact(ATR), i.e.,

ttBr(T)R Fact(A

T

R) = {0}.

Hence ATR satisfies card

Gr(ATR)/Fact(ATR)

card

ttBr(T)R

.

Proof. Directly by the definition of t, ti,j = xt(i),j and hence ti,jj = 0, for all t Br(T) and

i < j. So for any nonzero a =

1mn dm tm, where n < , dm R {0}, and tm Br(T),

the restrictions ai,jj are equal to 0 for all i < j. So by the preceding lemma a can not be in

Fact(AT

R).

Next we derive equations of weighty significance.

Lemma 3.5 Suppose b Gr(ATR) andi < j < k < . Then the following equations are satisfiedfor all Ti:

bi,k,l = bi,j,l when i < l < j;(A)

bi,k,j = bi,j,j

Tj []

l>jbj,k,l ;(B)

bi,k,l = bi,j,l +

Tj []b

j,k,l when l > j.(C)

Proof. By dividing the sum into groups we get that

bi,j =

Ti

l>ibi,j,l x,l

=Ti

i


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From the definition of hi,j we may infer that

hi,j (bj,k) =

(Tj , l>j)

bj,k,l hi,j (x,l)

=

(Tj , l>j)bj,k,l (xi,l xi,j)

=(Tj , l>j)

bj,k,l xi,l (Tj , l>j)bj,k,l xi,j

=

Ti

l>j

Tj []

bj,k,l

x,l

(Tj [], l>j)bj,k,l

x,j

.

So the equations (A), (B), and (C) for all i < j < k follow by comparing the coefficients of eachgenerator x,l in the equation b

i,k = bi,j + hi,j (bj,k). 3.5

Lemma 3.6 Suppose a Gr(ATR).

a) For all i < j < k, ai,jj = ai,kj.

b) (cf() > 0) For every i < , the union

i 0 there is some k < for which alreadyj


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Lemma 3.7 Suppose b Gr(ATR) and I is a subset of {(i, j) | bi,j{j} = bi,j}. Then

for all(i, j) I, Ti, and k I[i] I[j],

bi,j,j =

Tj []bj,k,k = bi,k,k.

Proof. Since (i, k) and (j,k) are in I, both bi,k,j and bj,k,l are equal to 0 for all Tj when l = k.Hence Lemma 3.5(B) can be reduced to the form bi,j,j =

Tj []

bj,k,k. Now (i, j) I guarantees

that bi,j,k = 0. Thus the reduced form together with Lemma 3.5(C) (applied for l = k) yield

bi,j,j = bi,k,k.

Lemma 3.8 Suppose b is an element of Gr(ATR).

a) If b is not in Fact(ATR) and I is an eventually coherent subset of such that bi,j =

bi,j{j} for all (i, j) I, then there is an eventually coherent subset J of I with bi,j =b

i,j{j} = 0 whenever (i, j) J.

b) (cf() > 0) If J is an eventually coherent subset of such that bi,j = bi,j{j} = 0

for all(i, j) J, then there are a bound n < and an eventually coherent subset K of Jsuch that card(supp(bi,j )) < n for all(i, j) K.

Proof. a) Since b 0 mod Fact(ATR) it follows by Lemma 2.3 that there is no subset of{(i, j) I | bi,j = 0} which would be eventually coherent. Hence there is an unbounded subsetS of I1st such that for each i S there is ji I[i] with b

i,ji nonzero. Fix any i S. Since

bi,ji = bi,ji{ji} = 0, let i be an element of Ti with b

i,jii,ji

= 0. By Lemma 3.7, bi,ki,k = bi,jii,ji

= 0for all k I[i] I[ji]. 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 J1st the union

jJ[i] supp(bi,j ) is of finite cardinality.Observe that for every (i, j) J, supp(bi,j ) = supp(bi,j ) (Ti {j}).

Assume, contrary to this subclaim, that i J1st, jm | m < is an increasing sequence ofordinals in J[i], and {m | m < } is a set of distinct elements from Ti such that b

i,jmm,jm

nonzerofor every m < . Since J is eventually coherent and is of uncountable cofinality let k < bethe minimal element in J[i]

m card(R)) Suppose b is in Gr(ATR) and I is an eventually coherent subsetof {(i, j) | bi,j{j} = bi,j = 0}. Then there are d R, t Br(T), and an eventually

coherent subset J of I for which bi,jt(i),j

= d = 0 whenever (i, j) J.

Proof. We define by induction on < cf() the following objects:

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an increasing sequence i | < cf() of ordinals in I1st with limit ;

an increasing sequence | < cf()


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Lemma 3.11 (cf() > max{0, card(R)}) The elements t, t Br(T), generate Gr(ATR) mod-

ulo Fact(ATR).

Proof. We show that for every a Gr(ATR

) with a Fact(ATR

) we can find n < , d1, . . . , dn R {0} and t1, . . . , tn Br(T) satisfying

(A) a

1mndm tm mod Fact(A

T

R).

Suppose a Gr(ATR

) Fact(ATR). By Lemma 3.6(c) and Lemma 3.8(a) let b be an element ofGr(ATR) and I1 an eventually coherent subset of such that a b mod Fact(A

TR) and for

each (i, j) I1, bi,j = bi,j{j} = 0. Furthermore, we may assume by Lemma 3.8(b) that n <

is a bound for which card(supp(bi,j )) < n hold for all (i, j) I1.

By Lemma 3.9 there are d1 R, t1 Br(T), and an eventually coherent set J1 I1 having

the property that bi,jt1(i),j

= d1 = 0 whenever (i, j) J1. Since d1 t1 Gr(ATR

), the sequence

c = b d1 t1 is in Gr(ATR

). Ifc is in Fact(ATR), then b d1 t1 mod Fact(ATR

), and becauseofa b mod Fact(ATR), also (A) holds for n = 1.

Suppose 1 n < and objects dm R{0}, tm Br(T), and Jm J1 for m n are alreadydefined. Assume also that these objects satisfy the following conditions:

1) Jm Jm for all 1 m m n;

2) for all 1 m < m n and i (Jm)1st, tm(i) = tm(i);

3) for every 1 m n and (i, j) Jm, bi,jtm(i),j

= dm = 0;

4) c = b

1mn dm tm Fact(ATR).

Clearly ci,j = ci,j{j} and card(supp(ci,j)) card(supp(bi,j )) < n for all (i, j) Jn. Again byLemma 3.8(a), there is an eventually coherent set In+1 Jn such that for each (i, j) In+1,c

i,j = 0. Moreover, by Lemma 3.9, there are dn+1 R, tn+1 Br(T), and an eventually

coherent set Jn+1 {(i, j) In+1 | ci,jtn+1(i),j

= dn+1 = 0}.

The properties (2), (3) and (4) above imply that ci,jtm(i),j

= bi,jtm(i),j

dm = 0 for every m n

and (i, j) Jm. On the other hand, ci,jtn+1(i),j

is nonzero for each (i, j) Jn+1. Thus tn+1(i) can

not be in {tm(i) | 1 m n} if i (Jn+1)1st. So for all (i, j) Jn+1, xtn+1(i),j {xtm(i),j |

1 m n}, and consequently bi,jtn+1(i),j

= ci,jtn+1(i),j

. Thus also Jn+1, tn+1, and dn+1 satisfy the

properties (1), (2), and (3) (but not necessarily (4)).

We claim that there must be n < n such that

(B) b

1mn

dm tm Fact(AT

R).

Assume, contrary to this subclaim, that the process introduced above has been carried out n

many times and objects Jm, tm, dm for i m n are defined. In addition to that supposethey satisfy the conditions (1), (2), and (3). Define i = min

(Jn)

1st and j = min(Jn [i]).Then for every m n, (i, j) Jm yields b

i,jtm(i),j

= dm = 0. This contradicts the condition

card(supp(bi,j )) < n, since the set {(tm(i), j) | m n} supp(bi,j ) is of cardinality n.

Now suppose n < is a finite ordinal satisfying (B). Then b

1mn dm tm mod Fact(ATR),

and becausea

b

mod Fact(AT

R) also (A) is satisfied.

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Conclusion 3.12 For any ordinal of uncountable cofinality, ring R with card(R) < cf(),and tree T of height , the inverse -system ATR = Gi, hi,j | i < j < has the properties that

card(Gi) = max{card(), card(Ti), card(R)}

for alli < , and

card

Gr(ATR)/Fact(AT

R)

= card

ttBr(T)R

.

Proof of Theorem 2. Remember that and were cardinals with 0 < = cf() < . Wewanted to study possible cardinalities of the quotient limit Gr(A)/Fact(A), where A is aninverse -system consisting of abelian groups having cardinality < . Now Conclusion 3.12 givesa complete solution to this problem because of > cf() = = cf() > 0. Namely, in order tomeet the requirements card(Gi) < for all i < , 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 desiredvalue for = card

Gr(ATR)/Fact(A

TR

)

. We briefly describe methods to choose suitable R andT for every nonzero .

For any R, card

Gr(ATR)/Fact(A

TR)

equals 1 when Br(T) is empty. So = 1 is possible sinceobviously there exists a tree of height without -branches and having levels of cardinality < when singular of cofinality . Also all the finite values > 1 are possible by taking T withonly one -branch and R with card(R) = .

Furthermore the case of infinite < is satisfied by any R with card(R) < min{, } and Twith exactly many -branches. The value = is possible for any R with card(R) < because a suitable tree can be constructed, for example, as follows. Let i | i < be anincreasing sequence of ordinals < with limit . Then the tree

T = {t | < , t i


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[She81] Saharon Shelah. On the number of nonisomorphic models of cardinality L,-equivalent to a fixed model. Notre Dame J. Formal Logic, 22(1):510, 1981.

[She82] Saharon Shelah. On the number of nonisomorphic models in L, when is weaklycompact. Notre Dame J. Formal Logic, 23(1):2126, 1982.

[She85] Saharon Shelah. On the possible number no(M) = the number of nonisomorphicmodels L,-equivalent to M of power , for singular. Notre Dame J. Formal Logic,26(1):3650, 1985.

[She86] Saharon Shelah. On the no(M) for M of singular power. In Around classification theoryof models, Lecture Notes in Math., 1182, pages 120134. Springer-Verlag, Berlin, 1986.

[She89] Saharon Shelah. The number of pairwise non-elementarily-embeddable models. J.Symbolic Logic, 54(4):14311455, 1989.

[She94] Saharon Shelah. Cardinal arithmetic, volume 29 of Oxford Logic Guides. The Claren-don Press Oxford University Press, New York, 1994. Oxford Science Publications.

[SV97] Saharon Shelah and Pauli Vaisanen. On the number of L,1-equivalent, non-isomorphic models. Reports of the Department of Mathematics 147, University ofHelsinki, 1997.

Saharon Shelah:Institute of MathematicsThe Hebrew UniversityJerusalem. Israel

Rutgers University

Hill Ctr-BuschNew Brunswick. New Jersey [email protected]

Pauli Vaisanen:Department of MathematicsP.O. Box 400014 University of [email protected]

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