saharon shelah and pauli vaisanen- the number of l-infinity-kappa equivalent non-isomorphic models...

8/3/2019 Saharon Shelah and Pauli Vaisanen- The number of L-infinity-kappa equivalent non-isomorphic models for kappa w

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The number ofL-equivalent non-isomorphic

models for weakly compact

Saharon Shelah Pauli Vaisanen

October 6, 2003

Abstract

For a cardinal and a model M of cardinality let No(M) denotethe number of non-isomorphic models of cardinality which are L,-equivalent to M. We prove that for a weakly compact cardinal, thequestion of the possible values of No(M) for models M of cardinality is equivalent to the question of the possible numbers of equivalenceclasses of equivalence relations which are 11-definable over V. By [SV]it is possible to have a generic extension, where the possible numbersof equivalence classes of 11-equivalence relations are in a prearrangedset. Together these results settle the problem of the possible values ofNo(M) for models of weakly compact cardinality. 1

1 Introduction

Suppose is a cardinal and M is a model of cardinality . Let No(M)denote the number of non-isomorphic models of cardinality which areelementary equivalent to M over the infinitary language L. We studythe possible values of No(M) for different models M.

When M is countable, No(M) = 1 by [Sco65]. This result extends to all

structures of singular cardinality provided that is of countable cofinalityResearch supported by the United States-Israel Binational Science Foundation. Pub-

lication 718.The second author wishes to thank Tapani Hyttinen under whose supervision he did

his share of the paper.12000 Mathematics Subject Classification: primary 03C55; secondary 03C75. Key

words: number of models, infinitary logic.

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[Cha68]. The case M is of singular cardinality with uncountable cofinality

was first treated in [She85] and later on in [She86]. In these papers Shelahshowed that if > 0,

< for every < , and 0 < < or = , thenNo(M) = for some model M of cardinality . In [SV00] of the authorsthe singular case is revisited, and particularly, it is established, under thesame assumptions as above, that the values with < are possiblefor No(M) with M of cardinality .

IfV = L, is an uncountable regular cardinal which is not weakly compact,and M is a model of cardinality , then No(M) {1, 2} [She81]. For = 1 this result was first proved in [Pal77a]. The values No(M) {0, 1}for a model of cardinality 1 are consistent with ZFC + GCH as noted in

[She81]. All the nonzero finite values of No(M) for models of cardinality 1are proved to be consistent with ZFC + GCH in [SV01].

When is a weakly compact cardinal and is a nonzero cardinal thereis a model M of cardinality with No(M) = [She82]. In the present paperwe show that for weakly compact, the possible values of No(M) for modelsof cardinality depend only on the possible numbers of equivalence classesof equivalence relations ,P having the following form (11-definable overV): for some first order sentence in the vocabulary {, R0, R1, R2, R3}and a subset P of V, it is the case that for all s, t

2,

s t iff for some r 2, V, ,P,s,t ,r |= ,

where P, r, s, and t are the interpretations of the symbols R0, R1, R2, andR3 respectively.

Theorem 1 When is a weakly compact cardinal, the following two con-ditions are equivalent for all cardinals :

(A) there is an equivalence relation on 2, which is 11-definable over Vand it has equivalence classes;

(B) No(M) = for some model M of cardinality .

In [SV] we proved that for every nonzero cardinal {, +, 2} there

is a 11-equivalence relation (as defined above) with equivalence classes.Moreover, it is possible to have a generic extension where the possible num-bers of equivalence classes of any second order definable equivalence relationsare completely controlled [SV, Theorem 1]. It follows that, the question ofpossible value of No(M) is completely solved, when M is of weakly compactcardinality. More formally, the conclusion is the following.

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Theorem 2 Suppose that the following conditions are satisfied:

is a weakly compact cardinal and 2 = + holds;

remains a weakly compact cardinal in the standard Cohen forcingadding a new subset of ;

> + is a cardinal with = ;

is a set of cardinals below ;

contains all nonzero cardinals ;

for every with > +

and < , the inequality

+

holds; is closed under unions and products of cardinals.

Then there is a forcing extension, where there are no new sets of cardinality< , all cardinals and cofinalities are preserved, remains weakly compact,2 = , and for all cardinals ,

there exists a model M of cardinality with No(M) = if and onlyif is in .

When is a weakly compact cardinal, it is possible to have, using theupward Easton forcing, a generic extension where is still a weakly compactcardinal, and remains weakly compact in the Cohen forcing adding asubset of (by an unpublished result of Silver). The forcing needed inthe conclusion is the standard way to add, for every , a Kurepa treeof height having branches of length through it. As noted in [SV,Fact 5.1], this forcing is locally Cohen, and therefore, remains a weaklycompact cardinal in the composite forcing of the upward Easton forcing andthe addition of new Kurepa trees.

Note also, that the closure properties mentioned in the conclusion are nec-essary by the fact that the possible numbers of equivalence classes of 11-

equivalence relations are always closed under unions of length andproducts of length < [SV, Lemma 3.4].

There are several parts in the paper. In Section 2 we start with a defini-tion of a Ehrenfeucht-Frasse-game EF;(M,N) generalizing the elemen-tary equivalence between two models over an infinitary language L. We

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note that ifM is a model of cardinality , there is a 11-equivalence relation

having No(M) equivalence classes.

The remaining sections are dedicated to the other half of the proof of thetheorem, namely to the proof that the existence of a 11-equivalence rela-tion with equivalence classes implies the existence of a model M withcard(M) = and No(M) = (Lemma 7.8 at the end of Section 7).

First in Section 3 we introduce a coding tree which is a skeleton for thebasic functions defined in Section 4. The basic functions form partial iso-morphisms between models. We want those partial isomorphisms to havea strong extension property ((1.2) below). The construction of the partialisomorphisms and the models are similar to [She82]. The role of the coding

tree is to provide a control over which basic functions are extended. Thebasic functions are defined by induction along the branches of the codingtree. For example, a -branch through the coding tree yields an isomorphismbetween two models ((1.3) below).

In Section 5 we define a special family of functions, which is used to buildmodels in the last section. Roughly speaking the family is the closure ofthe basic functions under the composition. However, in order to managethe functions in the family, we cannot allow arbitrary compositions. For anexplanation why we are forced to consider compositions, see the beginningof Section 5. This part might feel quite technical. The reader may skip all

the lemmas of this section in the first reading, and return to those when theyare referred from the last section. The following intuition might help thereader: The basic functions are build from simple ordinal addition preservingshifts of ordinals by putting a shift on top of another. A basic functionconsisting of finitely many shifts maps an interval [0, ) into finitely manyintervals [0, 1), . . . , [2n, 2n+1) only. Moreover, if the basic functionconsist of infinitely many shifts, then it is a permutation of an interval ofthe form [0, ). Similar type of properties are preserved under compositionsof basic functions as proved in the last three lemmas of Section 5. Thesetype of properties are mainly applied in the proof of (1.4) below.

In Section 6 we briefly sketch a proof that the extension properties of the

basic functions are preserved under compositions.

The content of Section 7 is as follows. Assuming that is a strongly inac-cessible cardinal and defines a 11-equivalence relation ,P on

2 with aparameter P V we construct models Mt for t

2 satisfying that

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(1.1) the models are of cardinality and they have a common vocabulary

consisting of relation symbols each of having arity < ;

(1.2) all the models are pairwise L-equivalent, and even more, they arepairwise M;-equivalent for any previously fixed regular cardinal < (Definition 2.1);

(1.3) for all s, t 2, the models Ms and Mt are isomorphic if, and onlyif, s and t are equivalent with respect to ,P.

Furthermore, when is a weakly compact cardinal the models satisfy theadditional property that

(1.4) if a model N has vocabulary , N is of cardinality , and N is L-equivalent to one (all) of the models Mt, t

2, then N is isomorphicto Ms for some s

2.

This is the main difference between the strongly inaccessible non-weaklycompact case and the weakly compact case: the 11-indescribability propertyof a weakly compact cardinal ensures that the isomorphism type of anymodel N with domain is determined by the isomorphism types of thebounded parts N, < , alone (Lemma 7.7).

2 Preliminaries

Definition 2.1 Suppose is a cardinal and is a infinite regular car-dinal. Let M and N be models of a common relational vocabulary. TheEhrenfeucht-Frasse-game EF;(M,N) is defined as follows. The game hastwo players, player I and player II. A play of the game continues for at most rounds. On round i < player I first chooses Xi {M,N } and Ai Xiof cardinality < . Then player II replies with a partial isomorphism pi suchthat

dom(pi) M, ran(pi) N,j


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Let M and N be models of a common relational vocabulary and be a

cardinal. The game EF;(M,N) is the usual Ehrenfeucht-Frasse-game oflength which characterizes the existence of a nonempty family of partialisomorphism with the fewer than at the time back-and-forth property. IfM andN satisfy the same sentences of the infinitary language L, we writeM N. By the Karps theorem [Kar65], player II has a winning strategyin EF;(M,N) if, and only if, M N. So the game EF;(M,N), foran infinite regular cardinal < , is a generalized version of the fewer than at the time back-and-forth property. There are so-called infinitely deeplanguages M; with the property that M ; N if, and only if, M andN satisfy the same sentences of M; [Hyt90, Kar84, Oik97].

For a model N we let card(N) denote the cardinality of the universe of N.For any model M of cardinality and a regular cardinal 0 < , wedefine No(M) to be the cardinality of the set

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

,

where N/= is the equivalence class of N under the isomorphism relation.

Next we recall from [SV, Definition 3.1] the definition of an equivalencerelation on 2, which is second order definable over the set H() (all setshereditarily of cardinality < ). In this paper we need only equivalencerelations which are definable using one second order existential quantifier.Hence the definition is presented in a restricted form.

Definition 2.2 Suppose is a regular cardinal. We say that defines a11-equivalence relation ,P on

2 with a parameter P H(), providedthat

is a first order sentence in a vocabulary consisting of , one unaryrelation symbol R0, and binary relation symbols R1, R2, and R3;

the following definition yields an equivalence relation on 2: for alls, t 2

s ,P t iff for some r

2, V, ,P,s,t ,r |= holds,where P, s, t, and r are the interpretations of the symbols R0, R1, R2,and R3 respectively.

By [SV, Lemma 3.4] the possible numbers of equivalence classes of a 11-relation over H() contains all nonzero cardinals in { | + or = 2}.

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In the next lemma we point out that the possible numbers of classes contains

the cardinal No(M) for every model M of cardinality .

Lemma 2.3 Suppose is an uncountable regular cardinal, M is a modelof cardinality , and is the cardinal No(M). Then there is an equivalencerelation ,P on2, which is 11-definable overH() and it has equivalenceclasses.

Proof. Without loss of generality M has domain and is infinite. Wemay also assume that the vocabulary of M consist of one n-place relationP with n < (Lemma 7.8).

Since there exists a definable bijection from into 3, we may define, for

every r 2, that r codes a triple r1, r2, r3 so that for every i {1, 2, 3},{ri | r 2} = 2 holds. Similarly, let Rs denote the n-place relation on coded by s 2, and let hr denote the binary relation on coded byr 2.

Since for all models N, the game EF;(M,N) is determined, M Nholds iff player I has a winning strategy in EF;(M,N). Moreover, there

is a definable bijection from into []


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[She82]. However, this time we want the functions to have more properties.

Hence the definition of the family is more complicated.

To make our models strongly equivalent we shall guarantee that for everypair Ms and Mt, s, t

2, a certain subfamily of all the functions forms awinning strategy for player II in the game EF;(Ms, Mt) (Definition 3.5).So we shall also need a stronger extension property for the functions thanwas needed in [She82].

We want that models Ms and Mt, s, t 2, are isomorphic if, and only if,the indices s and t are equivalent with respect to some previously fixed 11-equivalence relation (Lemma 7.4). Hence we code the equivalence relationinto a special tree (Definition 3.2 and Definition 3.3). This tree is a steering

apparatus in the construction of the family of functions (Definition 4.3 andDefinition 5.2).

Throughout the paper denotes a sentence which defines a 11-equivalencerelation ,P on 2 with a parameter P V (Definition 2.2).

Without loss of generality we may assume that for every s,t,r 2, thesame element r witnessing that s ,P t holds also witnesses that t ,P sholds, i.e.,

(3.1) V, ,P,s,t ,r |= iffV, ,P,t ,s,r |= .

Because ,P is a relation on

2 and it has a 11-definition, the most natural

object to consider is the tree of triples such that the first two elementsare initial segments of potentially equivalent functions in 2 and the thirdelement is a potential witnessing function for the equivalence. For technicalreasons we also have fourth member in the nodes. It is convenient to useonly a restricted part of 2 to get more room for coding different initialsegments.

Remark. The reader may skip the next definition, and safely think duringthe first reading of the paper, that for every , the defined set Fun(, 2)equals the collection 2 of all functions from into 2.

Definition 3.1 Suppose (remember =). We denote byFun(, 2)the family of functions from into 2 satisfying that

() = 0 if


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Remark. This restriction is harmless: We want to show that it is possible

to have a model M of cardinality such that No(M) equals the number ofequivalence classes w.r.t. ,P. Because is assumed to be a strongly inac-cessible cardinal, we may assume that ,P satisfies the following property:for all s, t 2,

(3.2) if s() = t() for every < then s ,P t.

The restriction turned out to be useful in Definition 4.3, and most impor-tantly, in Lemma 5.7(e) (we want that for a successor u T1,2 with c

u in-creasing, the information end(pu) is determined by a single point ran(c

u)alone, for the unexplained notation see the definitions 3.6, 4.3 and 5.2).

Definition 3.2 LetT[0] be {, , , } and for every nonzero < definethat T[] consists of all tuples ,,,C such that

,, Fun(, 2);

= ;

C is a closed subset of .

Then define that

T =


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Remember that defines an equivalence relation. So in the next definition

we fix the part T1 of T whose nodes are initial segments of equivalentfunctions.

Definition 3.3 Suppose s, t, r are in Fun(, 2). If V, ,P,s,t ,r |= does not hold, define Cs,t,r to be {0}, and otherwise, define

Cs,t,r = {0}

< | V, , P V, s, t, r V, ,P,s,t ,r |=

.

Let T1 be the set of all u T such that for some s,t,r Fun(, 2) the following conditions are satisfied:

V, ,P,s,t ,r |= ;

fun1(u) s, fun2(u) t, and fun3(u) r;

ord(u) Cs,t,r and cst(u) = Cs,t,r ord(u).

Note that for all nonzero Cs,t,r, = . Notice also that if r witnessesthat s ,P t holds, then the following elements form a -branch in the treeT1:

s, t, r, Cs,t,r | Cs,t,r

.

In the definition below, we fix a direction for the first two functions in the

nodes of T. This is needed to keep track of how the partial isomorphismsare extended between models.

Definition 3.4 For each < define a lexicographic order as follows: for all elements , Fun(, 2),

iff = and() < (),for = min{


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equivalent) and we do not restrict ourselves to the initial segments of equiva-

lent functions. Remember that is a fixed regular cardinal below , and isthe length of the Ehrenfeucht-Frasse-game played between the forthcomingmodels.

Definition 3.5 Define Suc+ to be the set

{+ 1 | is a successor ordinal}.

Choose a surjective function from {0}Suc+ onto {, , , }T[] suchthat

if () = u then either = 0 and u = , , , , or else ord(u) < ;

for every u {, , , } T[], the set { Suc+ | () = u} isunbounded in .

We define T2 , for fixed regular < , to be the smallest subset ofT satisfyingthe following conditions.

(1) , , , is in T2 .

(2) If u T2 then u1 T2 .

(3) T2

contains every u T[

] having the properties:

(i) If sup cst(u) < ord(u) then ord(u) Suc+ and for the maxi-mal element = supcst(u) (which is in cst(u)), the equation

ord(u)

= u holds (this element is in T1 T2);

(ii) ifsup cst(u) = ord(u), thencst(u)Suc+ is nonempty, cf(ord(u)) sup+(ran(fj)). Note that sup

+(dom(fj1)) . Thensup+(dom(fj)) < sup

+(ran(fj)) since dom(f0) = dom(f) and ran(fi) =dom(fi+1) for every i < j . However, from (c) it follows that sup

+(ran(fj)) sup+(dom(fk)) < sup

+(ran(fk)) for all k {j + 1, . . . , lh(f) 1}, and sosup+(ran(f)) = sup+(ran(flh(f)1)) > , a contradiction.

(e) By applying (b) to fn1, for n = lh(f) 1, we get that sup+(dom(fn))

sup+(ran(fn)) = . By (d), sup+(dom(fn)) . Hence sup

+(dom(fn)) =. In the same way it can be shown sup+(dom(fi)) = for all i n.

Suppose dom(f) = . By Lemma 5.5, ind(fi) is a limit point, say ui T1,2 ,

and sup+(ran(fi)) = = dom(pui) = ran(pui) for every i n. However,when n > 1, f0 pu0 together with dom(f0) = dom(f) = = dom(pu)imply that f0 = pu0 and ran(f0) = = dom(f1). A similar reasoning showsdom(fi) = ran(fi) = for every i n. 5.6

The next lemma is the main tool in the proof of the most important lemmaof the paper, namely Lemma 7.6(b) (in particular, the last three items).

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Lemma 5.7

(a) Suppose f , g F are such that dom(f) = dom(g) = {}, f() = g(),and fi is increasing for every i < lh(f). Then lh(f) lh(g) and fork = lh(g) lh(f), both fi = gk+i and beg(fi) = beg(gk+i) hold for everyi < lh(f). Moreover, if beg(g) = beg(f) or for every j < lh(g), gj isincreasing, then lh(f) = lh(g).

(b) Suppose q F1, < dom(q), and that both q{} and q{} areincreasing. If there is no d E with {, } dom(d) then card() sup

+(dom(f)) = +1 contrary to the assumption dom(f) =dom(g) = {}. Thus gm is increasing. Since ran(fn) = ran(gm), fn = gmby Fact 4.2(a). By Lemma 5.6(a), beg(fn) = beg(gm). When n > 0,

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ran(fn1) = dom(fn) = dom(gm) = ran(gm1). Hence we can repeat the

same argument and we get that fni = gmi and beg(fni) = beg(gmi)for every i min{m, n}. However m n since otherwise dom(g) ={f n and beg(g) = beg(f) then gmn1() = and end(gmn1) =beg(gmn) = beg(f0) = beg(f) = beg(g) = beg(g0) contrary to the mini-mality of g.

If m > n and gj is increasing, for every j < lh(g), then ran(gmn1) =dom(gmn) = dom(f0) = {} and dom(g) = {g0

1 . . . gmn11()} =

{}, a contradiction.

(b) Let u ind(q) be the smallest element with dom(pu), and v ind(q)be the smallest element with dom(pv). Then u and v are successors,u v, dom(cu), dom(cv), and q{, } cu cv pu c

v.Assume cu = cv. Then u v. Since cu is increasing dom(pu) = where = ref(cu). Because dom(cv) dom(cu) we have dom(cv) = . Socard()


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too. Similarly, the latter claim, concerning end(f), is a consequence of the

facts that for n = lh(u) 1, f() ran(gu,) = ran(gu,n ) n+1n andfun2(un) is a constant function on the interval n+1 n.

(f) Abbreviate sup+(dom(f)) by , lh(f) 1 by n, and for every i n,ind(fi) by ui. If sup

+(ran(f)) there is nothing to prove. So assumesup+(ran(f)) > . There must be the smallest index j n satisfyingsup+(dom(fj)) < sup+(ran(fj)), uj is a successor, and cujdom(fj),abbreviated by d, is increasing. Let be mindom(d). Then for all dom(fj)dom(d), fj() < fj(), by the definition ofcuj . Besides, dom(fj) together with Fact 4.2(c) ensure that for all dom(d), fj() fj() =d() d() = < . So the claim holds in case j = n.

Suppose n > j. From (d) it follows that for every i {j + 1, . . . , n}, uiis a successor, cui is increasing, and dom(fi) f


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namely otherwise, we reach a contradiction in the following manner. Assume

hj{j} is decreasing. There are two subcases:

(i) Assume that sup+(dom(hj)) > sup+(ran(hj)) or sup

+(dom(hj1)) >sup+(ran(hj1)).

(ii) By Lemma 5.5 the following holds:

sup+(dom(hj)) = sup+(ran(hj));

sup+(dom(hj1)) = sup+(ran(hj1)).

So suppose sup+(dom(hj)) < sup+(ran(hj)) and sup

+(dom(hj1)) , a contradiction.

(ii) The function hj1{j1} is increasing, since otherwise,

sup+(dom(hj1)) j1 + 1 hj1(

j1) + 1 =

j + 1 = sup

+(ran(hj1)).

Let be ref(hj1{j1}). If j1 , then j1 >

j1 and h(j1) =

j1. Moreover, ref(hj1{j1}) > and sup+(dom(hj1)) = j1 +

1 > +1 > j + 1 = sup+(ran(hj1)), a contradiction. On the other

hand, if j1 < , then it follows from the assumption j > hj(j) that

hj1{j1} =

hj{

j}1

. By Lemma 5.6(a), hj = hj11 and beg(hj1) =

end(hj) contrary to the minimality of h.

Hence hj{j} is increasing, and by (d), ind(hi), abbreviated by vi, is suc-

cessor, cvi is increasing, and i dom(cvi) for every i { j , . . . , m}. We

show by induction on i { j , . . . , m} that i + i where + is the ordinal

addition.

Since j = < , j , and is cardinal, we have j +

j . Suppose

i < m and i + i. If hi(i) = i = i+1 then i+1 + = i +

i l for infinitely many l J, a contradiction. Hence ind(fj) isa limit. By (b), there is an end segment K of J such that for every l Jand k { j , . . . , n 1}, dlk is increasing. For every l K, f(l) = l holds,

and thus the compositions (dl0)1

. . . (dlj1)1

and dln1 . . . dlj are

equal. Since end(fj1) = beg(fj) and the sequences (dlj1)

1, . . . , (dl0)

1

and dlj , . . . , dln1 are in F (in both of the sequences all the functions are

increasing), it follows from Lemma 5.7(a), that these sequences are equal.

Particularly, dlj = (dlj1)1 for every l K. From (a) it would follow that

fj1 = fj and beg(fj1) = end(fj), contrary to the minimality of f. 5.8

6 The back and forth properties of the family

To make sure that certain submodels are already strongly equivalent, weneed the following closed unbounded set.

Definition 6.1 We define D to be the following closed and unboundedsubset of :

( + 1) |

V, , V, X V, Y V

V, , , X , Y

,

where is the function from Definition 3.5 and

X =

pu, beg(pu), end(pu)

| u T1,2

;

Y =

v, pv, beg(pv), end(pv)

| v T1,2

.

Note that for all D, = .

The following notation is used for all compositions of basic functions be-ginning from and ending to . This is needed to define the closure ofrelations in each model M as described in the beginning of Section 5 (andformally presented in Definition 7.1).

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Definition 6.2 For all D {} and , Fun(, 2) define:

F[, ] =

f F | ind(fi) T1,2 [< ] for alli < lh(f),

beg(f) , andend(f)

;

F1[, ] =

f | f F[, ] andlh(f) = 1

.

In the lemma below we present the promised back and forth properties ofthe basic functions. Naturally the properties extends to the compositions ofbasic functions too. The lemma is applied in the proof of Lemma 7.5.

Lemma 6.3 Suppose D {}.

(a) For every u T1,2 , if u is a limit point or a successor in T2 , then

pu V implies u T1,2 [< ]. For all successors u T

1, if pu is in Vthen there is v T1,2 [< ] such that pv = pu, beg(pv) = beg(pu), andend(pv) = end(pu).

(b) For everyv T1,2 [< ] and < , there is (Suc+)(+1) such

that for every , Fun(, 2) with fun1(v) and fun2(v) , wecan find u

, T1,22

[] satisfying that fun1(u

,) = , fun2(u,) =

, and u, is a successor of v.

(c) If , Fun(, 2) and q F1[, ] is such that for v = ind(q) both

fun1(v) and fun2(v) hold, then there is u T1,2 [< ] satisfyingthat ind(q) u (implying q pu), pu F1[, ], and dom(pu) ran(pu).

(d) Suppose , Fun(, 2), q F1[, ], and < . There is f F[, ]such that q f and dom(f) ran(f).

(e) Suppose , Fun(, 2), f F[, ], and < . There is g F[, ]with g f and dom(g) ran(g).

Proof. The properties (a)(c) are straightforward consequences of the defi-nition of the functions pu. We sketch proofs of the rest of the properties.

(d) Here we need the small detail that we used id(u) in Definition 4.3.Denote ind(q) by v. If both fun1(v) and fun2(v) hold, then theclaim follows from (c).

Let , Fun(, 2) be such that fun1(v) and fun2(v)

. Fixelements u0, u1, u2 from T2 so that

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u0 is the smallest in -order with fun1(u0) , fun2(u

0) , and

dom(pu0);


1) , v u1

and ran(pu0) dom(pu1);


2) ,ran(pu1ran(pu0)) dom(pu2).

Define f to be gw,W, where w = ui | 0 i 2. Then f is in F[, ].

Define 1 to be min{ + 1 | dom(q) and () = fun1(v)()}, and 2to be min{ + 1 | dom(q) and () = fun2(v)()}. Since beg(q)

and end(q) , we have that 1 sup+

(dom(q)) and 2 sup+

(ran(q)).So 1 =

1 and 2 = 2 ensure that f0

1dom(q) is identity andf2ran(q) is identity. Therefore q f.

(e) Since dom(f) ran(f) is bounded in it follows from Lemma 5.6(d)that dom(fi) ran(fi) are bounded in for all i < lh(f). Hence for everyi < lh(f), fi V, and by (a) we may assume, ind(fi) V. The claimfollows from (d) by induction on i < lh(f). 6.3

7 The strongly equivalent non-isomorphic models

Recall that is a fixed strongly inaccessible cardinal and is a fixed regularcardinal below .

For ordinals < and subsets A of , [A] is the set of all -sequencesof ordinals in A. For every < < define that (sup+(a) is defined inDefinition 5.1)

[]B

=a [] | sup+(a) < and for all i < j < ,ai = aj

.

Denote the union


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if there is f F[0, ] with dom(f) = a, then f(a) Ra.

Suppose D{}. Define to be the vocabulary {Ra | a []


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Lemma 7.4 For all s, t Fun(, 2),

(a) s ,P t implies Ms = Mt (,P is given in Definition 2.2), and

(b) if Ms = Mt then s ,P t.

Proof. (a) Suppose s ,P t, and let r : 2 be such that V,

,P,s,t ,r |= . For every C = Cs,t,r D define u to be the tuples, t, r, Cs,t,r (Cs,t,r is given in Definition 3.3). Directly by Defini-tion 3.3, for all < C, u, u are in T

1 and u u. Hence pu pu for < C, and moreover, for the function h =

C pu both of the equa-

tions dom(h) = and ran(h) = hold. Consequently h is an isomorphism

from Ms onto Mt.

(b) Suppose s = t and for fixed < , s() = t(). Let h be an iso-morphism from Ms onto Mt, and let S

be the set { ( + 1) |h[] = is a cardinal of cofinality }. Since h is an isomorphism ands = t for all S, it follows from Fact 7.3(b) that for every S

there is a sequence f F[s, t] such that f = h. For all < S,f = h h = f. Since S is stationary in , there are n < and astationary subset S of S such that the equation lh(f) = n holds for every S.

Consider some Sand i < n. Abbreviate ind(fi ) by ui . By Lemma 5.6(e),

dom(fi ) = ran(fi ) = . Moreover, by Lemma 5.8(c), ui ui and fi fifor all S . By Fact 4.4(f), ui is in T

1. So fi = puiand dom(fi ) =

= ord(ui ) =. Define for each i < n,

si =S

fun1(ui );

ri =S

fun3(ui );

and let sn beS fun2(u

n1). Then s = s0 and t = sn.

We claim that s ,P t. By the transitivity of ,P it is enough to show that

for every i < n, ri witnesses si ,P si+1. Contrary to this subclaim assumethat for some i < n,

V, , P , si, si+1, ri |= .

Then there is S for which

V , , P V, si, si+1, ri V, , P , si, si+1, ri.

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However si = fun1(ui ), si+1 = fun2(u

i ), and ri = fun3(u

i ), and so

V, , P V, fun1(ui ), fun2(u

i ), fun3(u

i ) |= ,

contrary to the fact that ui is in T1. 7.4

In the following two lemmas we assume existence of a regular cardinal in D.Such does not necessarily exists, if is an arbitrary strongly inaccessiblecardinal. However, these lemmas are only preliminaries for the main lemma,Lemma 7.7, where we assume to be a weakly compact cardinal. Note, when = in Lemma 7.5(a) below, it suffices that is a strongly inaccessiblecardinal.

Lemma 7.5 Suppose D is a regular cardinal or = , and that , are functions from into 2.

(a) M ; M.

(b) For every < , the model M satisfies the L-sentence

xi | i < a[]B

Ra

xi | i <

(c) For alla []


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be the smallest ordinal which is strictly greater than any ordinal in ji Aj( < since is regular, i < , and card(Aj) < for every j i). ByLemma 6.3(d), there is ui in T1,2 [< ] satisfying that v u

i, fun1(ui) ,

fun2(ui) , and dom(pui) ran(pui). Sincej and for every i < , bi < card(bi+1). This ispossible since = cf(), D implies is an uncountable limit cardinal,and card(A) = implies that A is unbounded in . By Lemma 7.5(b),there is a []

Bsuch that M |= Ra(b). Since b A also M

A satisfies

Ra(b). By Lemma 7.5(d), there is < such that M |= Ra( b).

By Lemma 7.5(c), there should be A with M A |= Ra( b).

However, then by Fact 7.3(b), there should be f F satisfying dom(f) ={} b, f() = = , and f(b) = b contrary to Lemma 5.8(d).

(b) Our proof has the following structure:

When A the claim follows from (a) and Fact 7.2(b).

The case that A is not a subset of + for any A is shown to beimpossible.

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Lastly we prove that when A + for some A, there are

Fun(, 2) and g F such that dom(g) = , ran(g) = A, beg(g) = ,and end(g) 0. So g is an isomorphism between M and M

0A.

Suppose there is an -sequence b such that b0 > and for all l < , bl Aand bl+1 bl+. By the equivalence M

0A M0 and Lemma 7.5(b),

there is a []B

such that M0A |= Ra(b). Hence there should be f F

with dom(f) = a and f(a) = b contrary to Lemma 5.7(f).

Suppose A and A + . As above, there are f F and < with dom(f) = {}, f() = , beg(f) = 0(+ 1), and end(f) = 0( + 1).Since < = f(), there is the smallest index k < lh(f) such thatf


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Then M0 = M0 M, and by the isomorphism M =

M0 A, we have M0 M0 A and for all Fun(, 2), M0 A =M. This contradicts Lemma 7.6(b). 7.7

Lemma 7.8 Suppose is a weakly compact cardinal, < is a regularcardinal, and there is a 11-equivalence relation on

2 having differentequivalent classes. Then there exists a model M such that the vocabularyof M consists of one relation symbol of finite arity, card(M) = , andNo(M) = .

Proof. By the preceding lemmas the model M0 defined as in Definition 7.1satisfies the claim, except that the vocabulary ofM is overly large. However,

by [She85, Claim 1.3(1)], the inaccessibility of ensures that there is amodel N of cardinality with relations of finite arity satisfying No(N) =No(M0) (the proof is a simple coding). Furthermore, by [She85, Claim1.4(2)], relations can be coded by one relation so that the other propertiesare preserved. Actually, the claims cited concern the case = 0, but thereis no problem to preserve No(M0) in the cases 0 < < , too.

References

[Cha68] C. C. Chang. Some remarks on the model theory of infinitary lan-

guages. In J. Barwise, editor, The syntax and semantics of infini-tary languages, Lecture Notes in Math., 72, pages 3663. Springer-Verlag, Berlin, 1968.

[Hyt90] Tapani Hyttinen. Model theory for infinite quantifier languages.Fund. Math., 134(2):125142, 1990.

[Kar65] Carol R. Karp. Finite-quantifier equivalence. In Theory of Models(Proc. 1963 Internat. Sympos. Berkeley), pages 407412. North-Holland, Amsterdam, 1965.

[Kar84] Maaret Karttunen. Model theory for infinitely deep languages.

Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, (50):96, 1984.

[Oik97] Juha Oikkonen. Undefinability of-well-orderings in L. J. Sym-bolic Logic, 62(3):9991020, 1997.

[Pal77a] E. A. Palyutin. Number of models in L,1 theories, II. Algebra iLogika, 16(4):443456, 1977. English translation in [Pal77b].

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[Pal77b] E. A. Palyutin. Number of models in L,1 theories, II. Algebra

and Logic, 16(4):299309, 1977.

[Sco65] Dana Scott. Logic with denumerably long formulas and finitestrings of quantifiers. In J. W. Addison, Leon Henkin, and AlfredTarski, editors, Theory of Models (Proc. 1963 Internat. Sympos.Berkeley), pages 32934. North-Holland, Amsterdam, 1965.

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

[She82] Saharon Shelah. On the number of nonisomorphic models in L,

when is weakly compact. Notre Dame J. Formal Logic, 23(1):2126, 1982.

[She85] Saharon Shelah. On the possible number no(M) = the numberof nonisomorphic models 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 Aroundclassification theory of models, Lecture Notes in Math., 1182, pages120134. Springer-Verlag, Berlin, 1986.

[SV] Saharon Shelah and Pauli Vaisanen. On equivalence relations 11-

definable over H(). Fund. Math. To appear. No. 719 in the listof Shelahs publications.

[SV00] Saharon Shelah and Pauli Vaisanen. On inverse -systems and thenumber of L-equivalent, non-isomorphic models for -singular.J. Symbolic Logic, 65(1):272284, 2000. No. 644 in the list of She-lahs publications.

[SV01] Saharon Shelah and Pauli Vaisanen. On the number of L1-equivalent non-isomorphic models. Trans. Amer. Math. Soc.,353:17811817, 2001. No. 646 in the list of Shelahs publications.

Saharon Shelah:Institute of MathematicsThe Hebrew UniversityJerusalem, Israel

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Department of Mathematics

Rutgers UniversityNew BrunswickNJ, [email protected]

Pauli Vaisanen:Department of MathematicsUniversity of [email protected]

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saharon shelah and pauli vaisanen- the number of l-infinity-kappa equivalent non-isomorphic models...

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