steven givant and saharon shelah- universal theories categorical in power and kappa-generated models

8/3/2019 Steven Givant and Saharon Shelah- Universal Theories Categorical in Power and kappa-Generated Models

1/22

404

re

vision:1994-01-21

modified:1994-01-29

UNIVERSAL THEORIES CATEGORICAL

IN POWER AND -GENERATED MODELS

Steven Givant and Saharon Shelah

Abstract. We investigate a notion called uniqueness in power that is akin tocategoricity in power , but is based on the cardinality of the generating sets ofmodels instead of on the cardinality of their universes. The notion is quite usefulfor formulating categoricity-like questions regarding powers below the cardinality of

a theory. We prove, for (uncountable) universal theories T, that ifT is -unique forone uncountable , then it is -unique for every uncountable ; in particular, it iscategorical in powers greater than the cardinality ofT.

It is well known that the notion of categoricity in power exhibits certain irregu-larities in small cardinals, even when applied to such simple theories as universalHorn theories. For example, a countable universal Horn theory categorical in oneuncountable power is necessarily categorical in all uncountable powers, by Morleystheorem, but it need not be countably categorical.

Tarski suggested that, for universal Horn theories T, this irregularity might beovercome by replacing the notion of categoricity in power by that of freeness inpower. T is free in power , or -free, if it has a model of power and if all suchmodels are free, in the general algebraic sense of the word, over the class of allmodels of T. T is a free theory if each of its models is free over the class of allmodels of T. It is trivial to check that, for > |T|, categoricity and freeness inpower are the same thing. For |T| they are not the same thing. For example,the (equationally axiomatizable) theory of vector spaces over the rationals is anexample of a free theory, categorical in every uncountable power, that is -free, butnot -categorical. Tarski formulated the following problem: Is a universal Horn

theory that is free in one infinite power necessarily free in all infinite powers? Is ita free theory?Baldwin, Lachlan, and McKenzie, in Baldwin-Lachlan [1973], and Palyutin, in

Abakumov-Palyutin-Shishmarev-Taitslin [1973], proved that a countable -categoricaluniversal Horn theory is 1-categorical, and hence categorical in all infinite powers.Thus, it is free in all infinite powers. Givant [1979] showed that a countable -free,but not -categorical, universal Horn theory is also 1-categorical, and in fact it isa free theory. Further, he proved that a universal Horn theory, of any cardinality, that is -free, but not -categorical, is necessarily a free theory.

Shelahs research was partially supported by the National Science Foundation and the United

States-Israel Binational Science Foundation. This article is item number 404 in Shelahs bibliog-raphy. The authors would like to thank Garvin Melles for reading a preliminary draft of the paperand making several very helpful suggestions.

Typeset by AMS-TEX

1


2/22


3/22

404

re

vision:1994-01-21

modified:1994-01-29

UNIVERSAL THEORIES CATEGORICAL IN POWER AND -GENERATED MODELS 3

some of the notions and results of stability theory that are developed in Shelah[1990] (see also Shelah [1978]). We will assume that the reader is acquainted withthe elements of model theory and with such basic notions from stability theory

as superstability, a-saturatedness, strong type, regular type, and Morley sequence.We begin by reviewing some notation and terminology, and then proving a fewelementary lemmas.

The letters m and n shall denote finite cardinals, and and infinite cardinals.The cardinality of a set U is denoted by |U|. The set-theoretic difference of A andB is denoted by A B. If is a function, and a = a0, . . . , an1 is a sequence ofelements in the domain of , then (a) denotes the sequence (a0), . . . , (an1).We denote the restriction of to a subset X of its domain by X, and a similarnotation is employed for the restriction of a relation. A sequence X : < ofsets is increasing if X X for < < , and continuous if X =


4/22

404

re

vision:1994-01-21

modified:1994-01-29

4 STEVEN GIVANT AND SAHARON SHELAH

same elementary formulas.Fix a linear ordering V = V , < , and set W = V =

V. Let bethe (proper) initial segment relation on W, and, for each , let P be the set

of elements in W with domain , i.e., P =

V. There is a natural lexicographicordering, < , on W induced by the ordering of V: f < g iff either f g or elsethere is a natural number n in the domain of both f and g such that fn = gnand f(n) < g(n) in V. Take h to be the binary function on W such that, for anyf, g in W, h(f, g) is the greatest common initial segment of f and g. We shall callthe structure W= W ,


5/22

404

re

vision:1994-01-21

modified:1994-01-29


n such that the equation f(x0,...,xn1) = holds in T.

For every infinite ordering U= U , and (complete) Ehrenfeucht-Mostowskiset of formulas of L compatible with T, there is a model M = EM(U, )

of T

called a (standard) Ehrenfeucht-Mostowski model of T

such that M

isgenerated by U (in particular, U M), and U is a set of -indiscernibles in M

with respect to the ordering ofU, i.e., if(x0,...,xn1) , and if a nU satisfies

ai < aj for i < j < n, then M |= [a].

Shelah [1990], Chapter VII, Theorem 3.6, establishes the existence of certain gen-eralizations of Ehrenfeucht-Mostowski models. Suppose that T is, e.g., nonsuper-stable, and that the sequence n(x, y) : n < of formulas witnesses this non-superstability (see, e.g., Shelah [1990], Chapter II, Theorems 3.9 and 3.14). Thenthere is a generalized Ehrenfeucht-Mostowski set from which we can construct,for every tree U, a generalized Ehrenfeucht-Mostowski model M = EM(U, ) ofT. In other words, given a tree U, there a model M of T generated by a sequenceau : u U of -indiscernibles with respect to the atomic formulas ofU; in par-ticular, if w and v in nU are atomically equivalent in U, then aw0 , . . . , awn1 and

av0 , . . . , avn1 are elementarily equivalent in M. Moreover, for w in PM

n and v

in PM

, we have that M |= n[aw, av] iff w v. In general, the sequences au may

be of length greater than 1. However, to simplify notation we shall act as if theyall have length 1, and in fact, we shall identify au with u. Thus, we shall assumethat M is generated by U and that two sequences from U which are atomicallyequivalent in U are elementarily equivalent in M.

Here are some well-known facts about Ehrenfeucht-Mostowski models. Let U,Vbe infinite orderings or trees, and M = EM(U, ) ,N = EM(V, ). (In the caseof trees, we must assume that M and N exist.)

Fact 1. If is any embedding of V into U, then the canonical extension of toN is an elementary embedding ofN into M.

In particular,

Fact 2. Any automorphism ofU extends to an automorphism ofM.

Fact 3. IfV U, thenSgM

(V) M , andSgM

(V) is isomorphic to N via acanonical isomorphism that is the identity on V.

Fact 4. If U is a linear order, then U is irredundant inM. If U is a tree, then

for any element f and subset X of U, if f is not an initial segment of any elementof X, then f is not generated by X inM.

We shall always denote the reduct ofM to L by M, in symbols, M = ML.Facts 1, 2, and 4 transfer automatically from M to M. However, M is usually notgenerated by U. We shall therefore formulate versions of the above facts that applyto SgM(U), and, more generally, to a collection of models that lie between SgM(U)

and SgM

(U) = M.

Definition. Suppose M = EM(U, ) , and K L L.

(i) For eachf K, letRf be the relation corresponding to fM , i.e., Rf[a0, . . . , an]

iff fM[a0, . . . , an1] = an. SetMK = M , Rf fK.(ii) For each V U set

K V = V {fM

[a] : f K, a from V} .


6/22

404

re

vision:1994-01-21

modified:1994-01-29


Thus, the elements ofKV are just the elements ofV, together with the elementsofM that can be obtained from sequences of V by single applications of f in K.

Lemma 1. LetU,V be dense orderings or dense trees, M = EM(U, ) ,N =EM(V, ), and K L L. (Thus, in the case of dense trees, we postulate thatgeneralized Ehrenfeucht-Mostowski models for exist.)

(i) U is a set of indiscernibles in SgM(K U) for the atomic formulas of (with respect to the atomic formulas of U); in particular, if a and b aretwo atomically equivalent sequences inU, then they satisfy the same atomic

formulas inSgM(K U).

(ii) U is a set of indiscernibles inSgM(K U) for some complete Ehrenfeucht-Mostowski set compatible with T, i.e., if a and b are two atomically equiva-lent sequences inU, then they are elementarily equivalent inSgM(K U).

(iii) If embeds V into U, then the canonical extension of to SgN(K V)

elementarily embeds SgN(K V) into SgM(K U).

In particular:

(iv) Every automorphism ofUextends canonically to an automorphism ofSgM(KU).

(v) IfV U, thenSgM(K V) SgM(K U).

The lemma continues to hold if we replace M and N everywhere by MK and NK respectively.

Proof. Since satisfaction of atomic formulas is preserved under submodels, part (i)

is trivial. Part (i) directly implies (iv), and part (v) implies (iii). Thus, we onlyhave to prove (ii) and (v). We begin with (v).First, assume that U is -homogeneous. We easily check that the Tarski-Vaught

criterion holds. Here are the details. Suppose a is an n-termed sequence fromSg(K V) satisfying x(x, y) in Sg(K U); say, b is an element ofSg(K U)such that b a satisfies (x, y). Let c and d be sequences of elements from V andU that generate a and b respectively. Thus, there are f0, . . . , f k1 and g0, . . . , gl1in K, and a term (z, w) and terms 0(u, v), . . . , n1(u, v) in L such that (addingdummy variables to simplify notation)

ai = if0[c], . . . , f k1[c], c for i < n, and b = g0[d], . . . , gl1[d], d .

Hence, the sequence

g0[d ], . . . , gl1[d ], d

, 0

f0[c], . . . , f k1[c], c

, . . . , n1

f0[c], . . . , f k1[c], c

satisfies in Sg(KU). Since V is dense, there is a sequence e from V such that cdis atomically equivalent to ce in U. By -homogeneity there is an automorphism ofUtaking cd to ce. In view of (iv), this automorphism extends to an automorphismofSg(K U). Thus,

g0[e], . . . , gl1[e], e

, 0

f0[c], . . . , f k1[c], c

, . . . , n1

f0[c], . . . , f k1[c], c

satisfies in Sg(K U). Since [g0[e], . . . , gl1[e], e] is in Sg(K V), this showsthat the Tarski-Vaught criterion is satisfied. Hence, Sg(K V) Sg(K U).


7/22

404

re

vision:1994-01-21

modified:1994-01-29


Now let U be an arbitrary dense ordering or dense tree. Take W to be an -homogeneous extension ofU, and set P = EM(W, ). Thus, M P, and henceM P, by Fact 3. By the case just treated we have

SgM(K U) = SgP(K U) SgP(K W)

andSgM(K V) = SgN(K V) SgN(K W) .

Hence, SgM(K V) SgM(K U), as was to be shown. This proves (v).For (ii), suppose a, b nU are atomically equivalent in U. Let W be an -

homogeneous elementary extension ofU that includes the elements of a and b, andset P = EM(W, ). By Fact 3 we may assume that M P, and hence thatM P. Since W is -homogeneous, there is an automorphism ofW taking a to

b. By Fact 2, this extends to an automorphism ofP

. Now maps W one-oneonto itself and preserves all the operations of P. In particular, an appropriaterestriction is an automorphism of SgP(K W). Thus, a and b are elementarily

equivalent in SgP(K W). By part (v) they remain elementarily equivalent in

SgP(K U) = SgM(K U).The above proof obviously remains valid if we replace (implicit and explicit oc-

currences of) M and N everywhere by MK and NK respectively.

Lemma 2. Suppose U has power , M = EM(U, ), and K L has power at

most . ThenSgM(K U) is strictly -generated.

Proof. Since |U| = |K|, we have |K U| = . Thus, Sg(K U) is -generated.Suppose now, for contradiction, that Sg(K U) is -generated for some (finite orinfinite) < . A standard argument then gives us a set V U, of power when , and finite when < , such that K V generates Sg(K U). In particular,K V generates U in M. But then V generates U in M, which is impossible, byFact 4. Indeed, either U is irredundant in M or else a cardinality argument givesus an f in U that is not an initial segment of any element of V.

Main Hypothesis. Throughout the remainder of the paper we fix an uncountable

cardinal and assume that T is a -unique universal theory in a language L of

arbitrary cardinality.

As before, L will be an arbitrary expansion ofL with built-in Skolem functions,T a Skolem theory in L extending T, and a standard or generalized Ehrenfeucht-Mostowski set compatible with T. Again, recall that ifM = EM(U, ), then weset M = ML. Our first goal is to show that Tthe theory of the infinite modelsof Tis complete.

Lemma 3. Let U be a dense ordering of power , and M = EM(U, ). Then for every n and every countable set H L L, there is countable set K, withH K L L, such that

SgM(K U) n M .

The same is true ifU is a dense tree, provided thatM exists.

Proof. The proof is by induction on n, for all H at once. The case n = 0 is trivial:take K = H. Assume, now, that the lemma holds for a given n 0 and for all


8/22

404

re

vision:1994-01-21

modified:1994-01-29


countable sets H L L. Suppose, for contradiction, that H0 is a countablesubset of L L such that,

(1) For every countable K with H0 K L L we have

SgM(K U) n+1 M .

We construct an increasing sequence H : < 1 of countable subsets of L L

such that, settingA = Sg

M(H U) ,

we have

(2) A n M and A n+1 A+1 for all < 1 .

The set H0 is given. Suppose H has been constructed. Since A n+1 M, by(1), there is a n-formula (x0, . . . , xp1, y0, . . . , yq1) in L, appropriate terms0, . . . , q1 in L and function symbols f0, . . . , f r1 in H , and a sequence a = a0, . . . , as1 from U such that (adding dummy variables to simplify notation)

(3) f0[a], . . . , f r1[a], a0, . . . , as1 satisfies x0 . . . xp1(x0, . . . , xp1, 0, . . . , q1)in M, but not in A .

(Here we are using the fact that H U generates A .) Thus, there are Skolemfunctions g0, . . . , gp1 in L

such that

(4) g0[a], . . . , gp1[a], f0[a], . . . , f r1[a], a0, . . . , as1 satisfies the n-formula(x0, . . . , xp1, 0, . . . , q1) in M .

By the induction hypothesis there is a countable H+1 with

H {g0, . . . , gp1} H+1 L L

and such that, setting A+1 = Sg(H+1 U), we have A+1 n M. However,we put witnesses in A+1 to insure that A n+1 A+1 . At limit stages takeH =


9/22

404

re

vision:1994-01-21

modified:1994-01-29


satisfies (x0, . . . , xp1, 0, . . . , q1) in M, and hence also in B, by (5). Suppose,for contradiction, that Sg(H X) n+1 B. Then

(7) f0[b], . . . , f r1[b], b0, . . . , bs1 satisfies x0 . . . xp1(x0, . . . , xp1, 0, . . . , q1)in Sg(H X) .

Now we have Sg(HX) A , by Lemma 1(v). Therefore, (7) holds with Sg(HX) replaced by A. Since a and b are elementarily equivalent in the expansion

SgMH (H U) ofA , by Lemma 1(ii), we conclude that f0[a], . . . , f r1[a], a0, . . . , as1

satisfiesx0 . . . xp1(x0, . . . , xp1, 0, . . . , q1)

in A. But this contradicts (3); so the proof of (6) is completed.We now work towards a contradiction of (6) by producing a < 1 and a

countable, dense set X U such that

(8) Sg(H X) B .

Since B = SgM(G U) and SgM(U) are both strictly -generated, by Lemma 2,they are isomorphic, by -uniqueness. Thus, there is a set V B and a denseordering on V such that V = V , and U are isomorphic, V generates B,and V is a set of indiscernibles in B with respect to atomic formulas (under theordering ).

We define increasing sequences, Xn : n and Yn : n , of countable,dense subsets ofU and V respectively, and an increasing sequence n : n ofcountable ordinals, such that

(9) SgB(Hn Xn) SgB(Yn) Sg

B(Hn+1 Xn+1) .

Indeed, set 0 = 0 and take X0 to be an arbitrary countable, dense subset of U.Since H0 is countable, so is H0 X0 . Therefore, there is a countable Y0 V thatgenerates H0 X0 in B. By throwing in extra elements, we may assume that Y0 isdense.

Now suppose that n , Xn and Yn have been defined. Since Yn is countable andB =

n . Since Hn+1 U generates An+1 , there is acountable subset Xn+1 U such that Hn+1Xn+1 generates Yn in B. Of course we

may choose Xn+1 so that it is dense and includes Xn . As before, we now choosea countable, dense Yn+1 V that generates Xn+1 in B and includes Yn . Thiscompletes the construction.

Set = sup{n : n } , X =

nXn , Y =

nYn .

Then < 1 and H =

n Hn , by definition of the sequence H : < 1 .Also, X , and Y , are countable, dense submodels ofUand V respectively.By (5), (9), and Lemma 1(v) we have

SgM(HX) = SgB(HX) =

n


10/22


11/22

404

re

vision:1994-01-21

modified:1994-01-29

UNIVERSAL THEORIES CATEGORICAL IN POWER AND -GENERATED MODELS11

elementarily equivalent in Nand therefore also in Nand in SgN(KV) by Fact3 and Lemma 1(ii). Hence,

N |= [a] iff N |= [a]iff P |= [a] by (6),

iff M |= [(a)] by (8),

iff SgM(K U) |= [(a)] by (2),

iff SgP(K W) |= [a] by (9),

iff SgN(K V) |= [a] by (7),

iff SgN(K V) |= [a] .

Theorem 5. T is complete.

Proof. Let B1 and B2 be two infinite models of T. For i = 1, 2, let Ti be thetheory ofBi , let L

be an expansion of the language of Ti with built-in Skolemfunctions, Ti a Skolem theory in L

extending Ti , and i a standard Ehrenfeucht-Mostowski set in L compatible with Ti . Let U be a dense ordering of power ,and set Mi = EM(U, i) and Mi = M

i L. By Lemma 4 there is a countable

Ki L L such that

(1) SgMi(Ki U) Mi .

Now SgM1(K1 U) and SgM2(K2 U) are strictly -generated models of T, by

Lemma 2. Therefore, they are isomorphic, by the -uniqueness of T. From thisand (1) we conclude that M1 and M2 are elementarily equivalent. But M

i , and

hence also Mi , is a model of the theory ofBi . In consequence, B1 and B2 areelementarily equivalent, as was to be shown.

Our next goal is to prove:

Theorem 6. T is superstable.

Proof. Set U = ( + 1) Q and U= U , < , where < is the lexicographic orderingon U. We define a substructure V of the full tree over U by specifying its universe.It is the smallest set V satisfying the following conditions: all finite levels of the fulltree are included in V, i.e.,

n

nU V; all eventually constant functions fromU are in V; for each limit ordinal < 1 we choose a strictly increasing functionf in

1 such that sup{f(n) : n } = , and we put into V a copy of f fromV called g and determined by: g(n) = f(n) , 0 for each n.

(1) For every countable set W U, the set V W is also countable.

Indeed, the finite levels, nW, and the set of eventually constant functions of Ware all countable. Moreover, the domain of W, i.e., { : , q W for some q Q}, is countable, so the set ofg in W, with < 1, is also countable. This proves(1).

Let be a standard Ehrenfeucht-Mostowski set compatible with T (our Skolemextension of T), and set M = EM(U, ). Assume, for contradiction, that T


12/22

404

re

vision:1994-01-21

modified:1994-01-29


is not superstable, and let n(x, y) : n < be a sequence of formulas wit-nessing this nonsuperstability. Then, as mentioned in the preliminaries, there isan Ehrenfeucht-Mostowski set compatible with T such that the generalized

Ehrenfeucht-Mostowski model N

= EM(V, ) exists, and, for u from PVn and

v from PV , we have N |= n[u, v] iff u v. Set M = ML and N = NL.

By Lemma 4, there are countable sets J, K L L such that

(2) Sg(J U) M and Sg(K V) N .

By Lemma 2, Sg(J U) and Sg(K V) are both strictly -generated. Hence,-uniqueness implies that they are isomorphic. Let be such an isomorphism.

We now define two strictly increasing, continuous sequences, X : < 1 andY : < 1, of countable, dense subsets of U and V respectively, each Y beingdownward closed, i.e., closed under initial segments. (i) Let X0 = {} Q and

take Y0 to be any countable, dense, downward closed subset of V. Suppose, now,that X and Y have been defined. Since both of these sets are countable, byassumption, so are J X and K Y. For any a in J U there is obviously a finitesubset Ca V such that K Ca generates (a) (in Sg(K V)). (ii) For each a inJ X, put the elements of Ca into Y+1. Since J X is countable, this adds onlycountably many elements to Y+1. (iii) Similarly, for every b in K Y, choose afinite set Db U such that J Db generates

1(b) (in Sg(J U)), and put eachelement of Db into X+1. (iv) For each , q in X, put all of {} Q intoX+1. (v) Put all of

( Q) V into Y+1. The latter set remains countable,by (1). (vi) If necessary, add countably many more elements to X+1 and Y+1 to

insure that these sets are dense, that X X+1 and Y Y+1, and that Y+1 isdownward closed. (vii) For limit ordinals , set X =


13/22

404

re

vision:1994-01-21

modified:1994-01-29


(6) (1 Q) Y

0 such that

(1 Q) Y0 (1 Q) .

Continuing in this fashion, we obtain a strictly increasing sequence n : n < ofcountable ordinals such that

(1 Q) Yn (n+1 Q) .

Set = sup{n : n < }. Then

(1 Q) Y

.Let h1 and h2 be the extensions of gn to n + 1 determined by

h1(n) = g(n) = f(n) , 0 and h2(n) = f(n) , 1 .

Notice that h1 = g (n + 1). Clearly, h1 and h2 are in Y, by conditions (v) and(vii). Because is in E, and f(n) > , we can apply the definition ofE to concludethat h1 and h2 are not in Y, and hence are not initial segments of any elements inY. Therefore, they are not in B, by Fact 4. They realize the same type over B,by tree indiscernibility. Moreover, N |= n+1[h1, g] and N |= n+1[h2, g], sincen+1 codes the initial segment relation between elements in P

Vn+1 and P

V . Thus,

both n+1(h1, x) and n+1(h2, x) are in tp(g, B). This completes the proof of(7).

We now work towards a contradiction to (7).

(8) For every in E and every a in Sg(J U) A, there is an in E such thattp(a, A) does not split over A .

To see that (8) contradicts (7), take any in E, and set a = 1(g

). Since gis in Sg(K V) B, we get that a is in Sg(J U) A, by (4). Hence, there is an

in E such that tp(a, A) does not split over A, by (8). But then, applying to a, tp(a, A), and A, we get that tp(g, B) does not split over B, by (4). Thiscontradicts (7).


14/22

404

re

vision:1994-01-21

modified:1994-01-29


To prove (8), fix a in E and an a in Sg(J U) A. Then there is a sequenceu = u0, . . . , ur1 from U, function symbols f0, . . . f n1 in J, and a term fromL, such that (adding dummy variables)

(9) a =

f0[u], . . . , f n1[u], u

.

Each ui has the form ui = i , qi for a unique i and qi in Q. The set

i =

: {} Q X and i

is nonempty, since it contains , by condition (i). We shall denote the smallestelement of this set by i. Notice that ui X iff

i = i, by condition (iv). Now,

by definition, {i} Q X for each i < r. Since is in E, there is, for each

i < r, a in E such that {i} Q X . Take to be the maximum of these. Then is in E and {i} Q X for each i < r.

To verify that tp(a, A) does not split over A, let (x, y) be a formula ofL, andlet b and c be sequences of equal length from A, such that (x, b) and (x, c) arein tp(a, A), i.e.,

(10) M |= [a, b] [a, c] .

We must prove that

(11) tp(b, A) = tp(c, A) .

Since b and c come from A, there is a finite sequence v from X that generates band c with the help of some elements ofJ. Using the ordinals i , we shall constructa sequence u = u0, . . . , u

r1 in X such that, for each i, we have u

i = ui iff ui is

in X, and

(12) ui < uk iff ui < uk and u

i = u

k iff ui = uk ,

ui < vj iff ui < vj and ui = vj iff ui = vj .

for all appropriate j, k.

The construction of u

is not difficult: we shall set ui =

i , q

i , where q

i is

chosen from Q. Since {i} Q X, this assures that ui will be in X. If i is

in i, then i = i. In this case, take q

i = qi, so that u

i = ui. Suppose, now, that

vj = j , rj . If ui < vj , then we have either i < j , or else i = j and qi < rj .In the first case, observe that j i , since vj is in X (here we are using againcondition (iv)). Therefore, i j . If

i < j , then we may choose q

i however we

wish. Ifi = j , then we must choose qi < rj . This is possible since {

i} Q X.

In the second case, when i = j , we have i in X, so ui = ui. In both cases we

obtain ui < vj . The other cases in (12) are treated similarly. Notice, however, thatwe must choose qi so that (12) holds for all appropriate j and k at once. To dothis, we use the density of

Q.

From (12) we conclude that uv and uv are atomically equivalent in U. BecauseU is a set of order indiscernibles in M, it follows that uv and uv are elementarilyequivalent in M. Recalling (9), set a =

f0[u

], . . . , f n1[u], u

, and notice that

a is in A, since u is from X.


15/22

404

re

vision:1994-01-21

modified:1994-01-29


Since u generates a in the same way that u generates a, and since v generatesb and c, we conclude that ab c and ab c are elementarily equivalent in M. Inparticular, from (10) we get that

M |= [a, b] [a, c] .

Thus, (a, y) is in tp(b, A), while (a, y) is in tp(c, A). This proves (11), and

hence (8).

As is well-known, the fact that T is superstable (and hence stable) implies thatorder indiscernibles are totally indiscernible. Thus, ifM = EM(U, ), then anytwo one-one sequences in nU satisfy the same formulas in M (but not necessarily inM). It follows that Lemma 1 (with K = ) goes through without any referencesto order. Thus, ifM and N are as in Lemma 1, then any injection of V into

U extends canonically to an elementary embedding ofSgN

(V) into SgM

(U), andthis extension is an isomorphism iff is onto. In particular, the isomorphismtype of a subalgebra (of some very large model) generated by an infinite set ofindiscernibles with respect to is uniquely determined by the cardinality of theset of indiscernibles. To give a precise definition of such algebras, we fix a verylarge dense ordering Z (as large as we will need for any argument in this paper),set C = EM(Z, ), and let Z : an infinite cardinal |Z| be an increasingsequence of subsets of Z such that |Z| = .

Definition 7. LetF = SgC(Z). The set Z is referred to as a -basis ofF

The algebras F , for , have many properties in common with free algebras.For example, ifU and V are -bases ofF and F respectively, and if maps V one-one into U then the canonical extension of maps F elementarily into F . This isjust a reformulation of what was said above. For another example, notice that F isstrictly-generated, by Lemma 2. It follows that any infinite -basis of F must havecardinality ; otherwise we would have F = F for some = , forcing the strictly-generated model F to be -generated and visa-versa.

For the case when T is superstable, we readily extend Lemma 1(v) to covera:

Lemma 8. Let U,V be dense orderings, M

= EM(U, ), and K L

L. IfV U, thenSgM(K V) a Sg

M(K U) .

The lemma continues to hold if we replace M by MK .

Proof. We begin with the case when U is 1-homogeneous. Let p be a strong typeofM based on a finite subset E of Sg(K V), and suppose that pE is realized inSg(K U) by b. Let v = v0, . . . , vm1 be a finite sequence in V that generates Ein M, with the help of finitely many f K, and let u = u0, . . . , un1 be a finitesequence from Uthat generates b. Because of the density ofV, we can certainly finda sequence u = u

0

, . . . , un1

in V such that vu and vu are atomically equivalentin U. Using again the density ofV, is not difficult to construct an -sequence w inV such that vwu and vwu are atomically equivalent and the range, X, of vw isdense. Assume, for the moment, that this done. Since U is 1-homogeneous, thereis an automorphism ofU taking vwu to vwu. Extend it to an automorphism


16/22

404

re

vision:1994-01-21

modified:1994-01-29


ofM. Then also induces an automorphism ofSg(K U), and it is the identitymapping on Sg(KX). Because X is dense, Sg(KX) is an elementary submodelofSg(K V), and therefore p is stationary over this submodel. Since fixes an

elementary submodel over which p is stationary, we see that must map p

E toitself. Thus, (b) realizes p E in Sg(K U). But Sg(K V) is an elementarysubstructure ofSg(K U), by Lemma 1(v), and all the formulas of p E are overSg(K V). Thus, (b) must realize pE in Sg(K V).

Now suppose that U is arbitrary. Let Wbe an 1-homogeneous extension ofU,and set N = EM(W, ). Then by the previous case,

SgM(K V) = SgN(K V) a SgN(K W)

andSgM(K U) = SgN(K U) a Sg

N(K W) .

Therefore, SgM(K V) a SgM(K U).

Returning to w, a simple example should suffice to indicate how it is to beconstructed. Suppose, for example, that

vi < u0 < u1 < u2 < u3 < vj ,

and that no other elements of the sequences v and u are in the interval between viand vj . Of course,

vi < u0 < u

1 < u

2 < u

3 < vj .

If u0 u0, or else if u3 u3 , then include in the range of w a dense set betweenvi and u

0 , or else between u

3 and vj , respectively. Now consider the case when

u0 < u0 < u

3 < u3. If u

1 u1 , or else if u

2 u2 then include in the range of

w a dense set between u0 and u1 , or else between u

2 and u

3 , respectively. There

remains the case when u1 < u1 < u

2 < u2 . In this case, include in the range of w

a dense set between u1 and u2.

The next lemma is the analogue of Lemma 3 for the notion a.

Lemma 9. LetU be a dense ordering of power , andM = EM(U, ). Then forevery countable H L L there is a countable K with H K L L such that

SgM(K U) a M .

Proof. The proof is quite similar to the proof of Lemma 3. Suppose, for contradic-tion, that H is a counterexample to the assertion of the lemma. Analogously to theproof of Lemma 3, we construct an increasing, continuous sequence H : < 1 of countable subsets of L L such that

(1) Sg(H U) Sg(H+1 U) M , but Sg(H U) a Sg(H+1 U) .

Indeed, by Lemma 4 we can take H0 to be a countable extension of H such thatSg(H0U) M. Suppose, now, that H has been defined, and that Sg(H U) M. By the assumption on H, Sg(H U) is not a-saturated in M. Thus, there isa strong type p

based on a finite subset E of Sg(H U), such that p

E isrealized in Msay by abut p

E is not realized in Sg(H U). Since a is


17/22

404

re

vision:1994-01-21

modified:1994-01-29


generated by U in M, there is a finite set F L L such that F U generatesaxi in M. Take H+1 to be a countable extension of H F in L

L such thatSg(H+1U) M. Then Sg(HU) Sg(H+1U) . By construction, a realizes

p

E in Sg(H+1 U), so Sg(H U) a Sg(H+1 U). This completes theverification of (1).

Set G =


18/22

404

re

vision:1994-01-21

modified:1994-01-29


countable subsets ofU and V respectively, and an increasing sequence n : n < of countable ordinals, such that

Sg(Hn Xn) Sg(Yn) Sg(Hn+1 Xn+1)

for every n < . Set

= sup{n : n < } , X =

n

Xn , and Y =

n


19/22

404

re

vision:1994-01-21

modified:1994-01-29


Lemma 12. There is a cardinal such that every model of T of power > is

a-saturated.

Proof. Let be the Hanf number for omitting types in languages of cardinalityat most |T|. Suppose, for contradiction, that there is a model A of T of power> max{, } that is not a-saturated. Then there is a strong type p = p E (overA) based on a finite subset E of A such that p is omitted by A. Since there are atmost |T| many inequivalent formulas in p, by Shelah [1990], Chapter III, Lemma2.2(2), we may assume that p is a (possibly incomplete) typeas opposed to astrong typeover some subset B of A of cardinality at most |T|.

Let L be a language with built-in Skolem functions that extends L and thatincludes constants for the elements of B. Let T be a Skolem theory in L thatextends the theory of A , b bB. By Morleys Omitting Types Theorem and thechoice of, there is an Ehrenfeucht-Mostowski model N ofT over a dense ordering

U of indiscernibles of cardinality such that N omits the type p (see Morley[1965], Theorem 3.1, or Chang-Keisler [1973], Exercise 7.2.4). Let V be a dense

submodel ofU of power , and set M = SgN

(V). Then M is the correspondingEhrenfeucht-Mostowski model over V, and M N, by Fact 3. Obviously, M,and hence also M, omits p.

Let H L L be a finite set containing all the individual constant symbols forelements of E. By Lemma 4, there is a countable K with H K L L suchthat Sg(K V) M. Thus, E is a subset of the universe ofSg(K V). Since p,as a strong type, is based on a subset of the model Sg(K V), every formula ofp is equivalent to a formula with parameters from Sg(K V), by Shelah ibid.,

Lemma 2.15(1). Let p be the set of these formulas . Then p is a strong typebased on E that is equivalent to p. Because M omits p, it also omits p. Therefore,the elementary substructure Sg(K V) must omit p. This shows that Sg(K V)is not a-saturated. But that is impossible: since Sg(K V) is strictly -generated(by Lemma 2), it is isomorphic to F , and we have seen that F is a-saturated.

Lemma 13. F is a-prime over any -basis, for . Consequently, F is alsoa-prime over.

Proof. Let U be a -basis ofF . Then U generates any given sequence a from F ,so tp(a, U) is clearly atomic. In particular, F is a-atomic over U. Moreover, F

is also a-saturated, by Lemma 11, and there are obviously no Morley sequences inF over U. By Shelahs second characterization theorem for a-prime models, F isa-prime over U (see Shelah [1990], Chapter IV, Definition 4.3 and Theorem 4.18).

Now an a-prime model over a countable set is also a-prime over (see ibid.).Hence, F is also a-prime over .

We now work inside of some monster model. Let p0, . . . , pn1 be pairwise or-thogonal regular types, all based, say, on a finite set E. By realizing the strong typeof E over in Pra(), and then passing to automorphic copies of p0, . . . , pn1 ,we may assume that E is a subset of the universe ofPra(). For each i < n, let Iibe an infinite Morley sequence built from pi

E.

Lemma 14. Pra(E

i


20/22

404

re

vision:1994-01-21

modified:1994-01-29


over . Hence, these two models are isomorphic, by the uniqueness of a-primemodels. Since F is strictly -generated, by Lemma 2, so is Pra(E

i


21/22

404

re

vision:1994-01-21

modified:1994-01-29


Now let ( |T|) be so large that all the models of T of power > are a-saturated (see Lemma 12). Then T is +-categorical. Indeed, if M and N aremodels of power +, then they both are a-saturated, by choice of , and have p-

dimension +

, by unidimensionality. Hence, they must be isomorphic, by Shelah[1990], Chapter V, Theorem 2.10. This shows that T is categorical in some power> |T|. By the results of Shelah [1974], it is categorical in every power > |T|.

Theorem 16. T is -unique for every uncountable .

Proof. If T is definitionally equivalent to a countable theory, and is categorical inevery infinite power, i.e., if T is a totally categorical theory, then the theorem isobvious. Suppose, now, that it is not totally categorical. We shall use Theorems13, including the proof of Theorem 2, from Laskowski [1988]. According to these,T must have a minimal prime model M0. Moreover, there is a type p based onM0 with the following properties:

(1) If N is any elementary extension ofM0, then any two maximal Morley sequences inN built from pM0 must have the same cardinality. This is called the p-dimensionofN.

(2) If N and N are elementary extensions ofM0, with the same p-dimension, thenthere is an isomorphism ofN onto N that fixes M0 .

(3) If M0 N N, then pN is realized in N.

To simplify notation, we shall assume that p is a 1-type.

Since T is a universal-existential theory categorical in power, it is model com-plete, by Lindstroms theorem (see, e.g., Chang-Keisler [1973], Theorem 3.1.12).Fix > , and let M be a strictly -generated model ofT, say X is a generating

set of power . Our goal is to prove that M has p-dimension . Let W be a count-ably infinite subset of X. Then Sg(W) M, by model completeness. Withoutloss of generality, we may suppose that M0 is an elementary submodel ofSg(W).We now define a strictly increasing, continuous sequence, Y : < , of subsetsof X of size < , such that

(4) M0 Sg(Y) Sg(Y+1) M for < .

Indeed, take Y0 = W. IfY has been defined, then Sg(Y) = M, since M is strictly-generated and Y has power < . Hence, there is an element u in X that is notgenerated by Y. We set Y+1 = Y {u}. Property (4) follows by our choice ofY0 and u, for each , and by model completeness.

For each < , the type p Sg(Y) is realized in Sg(Y+1), by (3) and (4).Therefore, using our strictly increasing chain Sg(Y) : < of elementary sub-structures ofM, we can build from pM0 a Morley sequence in M of length at least. Thus, M has p-dimension at least .

Suppose now that I is a maximal Morley sequence in M built from pM0. ThenSg(M0 I) = M. For otherwise we would have Sg(M0 I) M, by modelcompleteness. Hence, we could realize pSg(M0 I) in M, by (3), and thus extendI to a larger Morley sequence in M built from pM0, contradicting its maximality.Since X also generates M, and has cardinality > , there is a subset J of I ofcardinality such that M0 J generates X, and hence also M. In particular,M0J generates I. But then J = I, since any element in IJ would be independent


22/22

04

re

vision:1994-01-21

modified:1994-01-29


over M0J, and hence could not be generated by this set. We conclude that |I| .In other words, M has p-dimension at most , and hence exactly .

We have shown:

(6) Any strictly -generated model of T extending M0 has p-dimension .

Now let M and N be any two strictly -generated models of T. By passing toisomorphic copies, we may assume that M0 is an elementary substructure of each.Hence, by (2) and (6), M and N are isomorphic over M0. This shows that T is-unique.

By the preceding theorem, the noncountably generated models of T are, up toisomorphisms, precisely the structures F, for > .

Bibliography

[1973] Baldwin, J.T. and Lachlan, A.H., On universal Horn classes categorical in some infinitepower, Algebra Universals 3 (1973), 98111.

[1973] Abakumov, A.I., Palyutin, E.A., Shishmarev, Yu. E., and Taitslin, M.A., Categoricalquasivarieties, Algebra and Logic 11 (1973), 121.

[1973] Chang, C.C. and Keisler, H.J., Model theory, Studies in Logic and the Foundations ofMathematics 73, North-Holland Publishing Company, Amsterdam, 1973, xii + 550 pp.

[1979] Givant, S.R., Universal Horn classes categorical or free in power, Annals of MathematicalLogic 15 (1979), 153.

[1988] Laskowski, M.C., Uncountable theories that are categorical in a higher power, Journal ofSymbolic Logic 53 (1988), 512530.

[1965] Morley, M., Omitting classes of elements, in The theory of models, Proceedings of the

1963 International Symposium at Berkeley (J.W. Addison et al., eds.), Studies in Logic,North-Holland Publishing Company, Amsterdam, 1965, pp. 265274.[1974] Shelah, S., Categoricity of uncountable theories, in Proceedings of the Tarski Symposium

(L. Henkin et al., eds.), Proceedings of Symposia in Pure Mathematics XXV, AmericanMathematical Society, Providence, R.I., 1974, pp. 187203.

[1978] , Classification Theory and the number of non-isomorphic models, Studies in Logicand the Foundations of Mathematics 92, North-Holland Publishing Company, Amster-dam, 1978, xvi + 544 pp.

[1987] , Existence of manyL,-equivalent non-isomorphic models of power . Proceed-

ings of (Trento) Model Theory Conference, June, 1986, Annals of Pure and Applied Logic34 (1987), 291310.

[1987a] , On the number of strongly-saturated models of power ., Annals of Pure andApplied Logic 36 (1987), 279287.

[1990] , Classification Theory and the number of non-isomorphic models, Studies in Logicand the Foundations of Mathematics 92, second edition, North-Holland Publishing Com-pany, Amsterdam, 1990, xxxiv + 705 pp.

Mills College

Hebrew University, Rutgers University, and MSRI

steven givant and saharon shelah- universal theories categorical in power and kappa-generated models

Documents