saharon shelah and alexander usvyatsov- model theoretic stability and categoricity for complete...

8/3/2019 Saharon Shelah and Alexander Usvyatsov- Model Theoretic Stability and Categoricity for Complete Metric Spaces

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MODEL THEORETIC STABILITY AND CATEGORICITY

FOR COMPLETE METRIC SPACES

SH837

Saharon ShelahThe Hebrew University of Jerusalem

Einstein Institute of MathematicsEdmond J. Safra Campus, Givat Ram

Jerusalem 91904, Israel

Department of MathematicsHill Center-Busch Campus

Rutgers, The State University of New Jersey110 Frelinghuysen Road

Piscataway, NJ 08854-8019 USA

Alexander UsvyatsovUCLADepartment of Mathematics

P.O. Box 951555Los Angeles, CA 90095-1555 USA

Centro de Matematica e Aplicac

oes FundamentaisAv. Prof. Gama Pinto, 21649-003 Lisboa, Portugal

We would like to thank Alice Leonhardt for the beautiful typing.The first author would like to thank the Israel Science Foundation for partial support of thisresearch (Grant No. 242/03).We thank the anonymous referee for very helpful comments and suggestions which greatly im-proved the exposition, especially in the last section of the paper.Latest Revision - 09/May/18

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Abstract. We deal with systematic development of stability for the context ofapproximate elementary submodels of a monster metric space, which is not far, butstill very different from first order model theory. In particular we prove the analogueof Morleys theorem for classes of complete metric spaces.


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MODEL THEORETIC STABILITY AND CATEGORICITY 3

1 Introduction and Preliminaries

We work in the context of a compact homogeneous modelC

which is also acomplete metric space with a definable metric d(x, y) and all the predicates andfunction symbols respect the metric. Such a monster model will be called a mon-ster metric space (a momspace), Definition 2.17. We investigate the class K ofalmost elementary complete submodels ofC, Definition 2.19.

This paper is devoted to categoricty of such classes K in uncountable cardinalities(generalizing Morleys theorem to this context). Since we believe that isometry istoo strong as a notion of isomorphism for classes of metric structures, we try toweaken the assumptions and work with -embeddings instead (Definitions 6.2, 6.4).

Several suggestions for a framework suitable for model theoretic treatment ofclasses arising in functional analysis and dynamics have been made in the last 40years by Chang, Keisler, Stern, Henson, and more recently by Ben-Yaacov. Allthese attempts were concerned with allowing a certain amount of compactness(e.g. capturing ultra-products of Banach spaces introduced by Krivine), withouthaving to deal with non-standard elements. In this paper the authors choose to workin the most general context which still allows compactness, therefore generalizingall the above frameworks. The main tools and techniques used here are borrowedfrom homogeneous model theory.

Homogeneous model theory was introduced and first studied by Keisler, devel-oped further by the first author, Grossberg, Lessmann, Hyttinen and others. Itinvestigates classes of elementary (sometimes somewhat saturated) submodels of

a big homogeneous model (monster), see precise definitions later.In [Sh 54], the first author classified such monsters with respect to the amount

of compactness they admit. Monsters which are compact in a language closed undernegation are called monsters of kind II. Later Hrushovski suggested the nameRobinson Theories for such classes, see [Hrxz]. Monsters which are compactin a language not necessarily closed under negations are called monsters of kindIII. Recently Ben-Yaacov has studied this context in great detail. He called suchmonsters compact abstract theories, in short CATS, see [BY03].

A simple generalization (replacing equality by definable metric) allows us tospeak of a monster model of a class of metric spaces. We call such monsters mon-

ster metric spaces, see Definition 2.17 below. Several results are proven in thisgeneral context, but some require compactness, Definition 2.15(2). Therefore, ourmain theorem (Theorem 8.2) holds in the metric analogue of monsters of kindIII, metric cats in Ben-Yaacovs terminology.

Independently of our work (and simultaneously), Ben-Yaacov investigated cat-egoricity for metric cats under the additional assumption of the topology on thespace of types being Hausdorff, which is the metric analogue of monsters of kind


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II (Robinson theories). In this context one can reconstruct most of classical stabil-ity theory (e.g., independence based on non-dividing), see [BY05]. These methodsfail in the more general context we were working at. Therefore, techniques de-

veloped and used here are very different, and rely heavily on non-splitting andEhrenfeucht-Mostowski constructions. These tools do not make any significant useof compactness, and we believe that this assumption can be eliminated by modify-ing our methods slightly. At this point we decided not to make the effort, but wetry to mention where exactly compactness is used.

In a recent work [BeUs0y], Ben-Yaacov and the second author introduce a frame-work of continuous first order logic, closely related to [ChKe66] and show that oncemodified slightly, most model-theoretic approaches to classes of metric spaces (suchas Hensons logic, see [HeIo02], Hausdorff metric cats, see [BY05]) are equivalentto continuous logic. Although if continuous model theory had been discovered ear-lier, this paper might have looked differently, we would still like to point out thatequivalences shown in [BeUs0y] do not include monster metric spaces, not evencompact ones. The assumption of the topology on the type space being Hausdorffis absolutely crucial in [BeUs0y] and [BY05]; it provides us with the ability to ap-proximate negations, which makes continuous logic very similar to classical firstorder logic (of course, this has many advantages). Lacking Hausdorff assumption,one has to use more general methods in order to reobtain basic properties. This iswhy monsters of kind III (general cats) have been studied and understood muchless than first order or Robinson theories, even in the discrete (non-continuous,classical first order) context.

Some work has been done, though: the first author proved the analogue of

Morleys theorem for existentially closed models in [Sh 54], classes of existentiallyclosed models were investigated further by Pillay in [Pi00], general cats were studiedby Ben-Yaacov in [BY03], [BY03a] and other papers. Our work continues thisinvestigation in the more general metric context (classical model theory can beviewed as a particular case with discrete metric).

Several words should be said also about the difference between the discrete andthe metric context. For example, why could we have not simply modified the proofin [Sh 54] slightly and obtain Theorem 8.2? The answer is that our categoricityassumption is significantly weaker, as we assume categoricity only for completestructures. For instance, the class of infinite-dimensional Hilbert spaces is categor-

ical in all densities, but not so is the class of inner-product spaces, not necessarilycomplete. Starting with a weaker assumption we aim for a weaker conclusion; butconsequences of our assumption are not as powerful as what one gets in [Sh 54](e.g. 0-stability is lost, we only have a topological version), which complicates lifesignificantly.

This work was originally carried out as a Ph.D. thesis of the second authorunder the supervision of the first one. The paper is an expanded version of the


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thesis which was written in Hebrew and submitted to the Hebrew University.

The paper is organized as follows:We introduce our context in 2. In particular, we define the class of models

which is being studied (almost elementary submodels ofC, M 1 C). This notiongeneralizes Hensons approximate elementary submodels (see [HeIo02]). It has thefollowing important property: if M 1 C, then its completion (metric closure) Nalso satisfies N 1 C. We will be interested mostly in complete (as metric spaces)almost elementary submodels ofC.

The next section, 3, is devoted to different kinds of approximations to formulaeand types. The importance of considering these approximations, i.e. topological

neighborhoods of partial types, becomes clear later, when stability, isolation, andother central notions are discussed.In 4 we generalize the notion of stability in a cardinal to our topological

context. We say that C is 0+ -stable if for no > 0 can we find an -net of +

types over a set of cardinality , i.e. the space of types over a set of cardinality has (in a sense) density . This is a generalization of-stability for Banach spacesstudied by Iovino in [Io99]. It is equivalent to the definitions given by Ben-Yaacovfor Hausdorff cats in [BY05], and in the context of continuous theories it coincideswith the definitions given in [BeUs0y].

We prove equivalence of several similar definitions for 0+ -stability, connect0+ 0-stability to non-splitting of types, classical stability in homogeneous modeltheory, existence of average types, saturation of a (closure of a) union of (D, 1)-homogeneous models. Notions of isolation are developed and density of strictlyisolated types is proved under the assumption of 0+ 0-stability.

In 5 we develop the theory of Ehrenfeucht-Mostowski models in our context. Aswe lack forking calculus, some basic facts (e.g. existence of (D, 1)-homogeneousmodels in all uncountable cardinalities) require a different approach, which is pro-vided by the EM-models techniques. Also in the proof of the main theorem (8) wetake advantage of the representation of the homogeneous model as an EM-modelin order to find inside it a converging sequence which is close to a subsequenceof a given uncountable sequence.

6 is devoted to notions of-embeddings and -isomorphisms. We try to weakenour assumptions as much as we can, and choose to work with the following no-tion of weakly uncountably categorical classes: for every > 0 there exists > 0 in which every two models and -isomorphic to each other. This prop-erty seems a priori weaker than uncountable categoricity in some > 0, and eventhan there exists > 0 such that for every > 0 any two models of density are -isomorphic. But the main theorem (8.2) states the following: assume C is


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weakly uncountably categorical. Then every model of density > 0 is (D, )-homogeneous (in particular, unique up to an isometry). So all the above notionsturn out to be equivalent.

In 7 we prepare the ground for the proof of 8.2, showing that any wu-categorical(weakly uncountably categorical) momspace is uni-dimensional (in the sense of [Sh3]), i.e., any (D, 1)-homogeneous model of density character is, in fact, (D, )-homogeneous.

The last section, 8, contains the proof of the main theorem, Theorem 8.2. Weassume that C is weakly uncountably categorical, but has a non-(D, )-homogeneousmodel in some density character > 0. By 7 this model is not (D, 1)-homogeneous.Applying the analysis done in 4 and 5 (and some infinitary combinatorics), weshow that it is possible to construct non-(D, 1)-homogeneous models of arbitrarilylarge density characters. By 4 and 6, this contradicts weak uncountable cate-

goricity.

We recall now basic definitions considering homogeneous model theory (see [Sh3],[Sh 54] and [GrLe02]):

1.1 Definition. 1) Let be a vocabulary, D a set of complete -types over infinitely many variables. A -model M is called a D-model if D(M) D (whereD(M) = the set of complete finite -types over realized in M). M is called

(D, )-homogeneous if D(M) = D and M is -homogeneous.2) We call D as in (1) a finite diagram if for some model M, D = D(M).3) A is a D-set in M if A M and a >A tp(a, , M) D. For A a D-set,p SmD(A, M) if there are N, a such that M N, a N realizes p and A a is aD-set in N.

1.2 Remark. Let D be a finite diagram. M is (D, )-homogeneous iff M is uni-versal for D-models of cardinality (or just -sets when < || + 0) and-homogeneous iffD(M) = D and M is (D, )-saturated (i.e. every D-type over a

subset of cardinality < is realized in M).

Proof. Easy (or see [GrLe02](2.3),(2.4)).

This motivates the following


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1.3 Definition. Let be big enough. A (D, )-homogeneous model C will becalled a D-monster (or a homomogeneous monster for D) . We usually assumeC = .

Recall:

1.4 Definition. 1) A finite diagram D is called -good if there is a (D, )-homogeneousmodel M of cardinality .2) D is good if it is -good for every .

1.5 Remark. So D is good iff it has a monster.

1.6 Convention. In this paper we will fix a good finite diagram D and a D-monster

model C.

1.7 Observation. 1) If (D is good, C a D-monster model) p SD(A), A B thenthere is q, p q SD(B).

Question: Why can we use good D? There are several answers.Basically, Claim 1.8 says that every D which is the finite diagram of a compact

momspace (see Definitions 2.17, 2.21) is good. Claim 1.9 says that even withoutcompactness, categoricity implies stability, which implies D is good by [Sh 3]. The

reader can easily omit these claims in the first reading.

1.8 Claim. Assume

(a) D is a finite diagram, i.e., a set of complete n-types inL(C)

(b) C is (D, 0)-homogeneous

(c) is full forC (see Definition 2.15)

(d) C is -compact.

Then

() () D is good() if (D) < then (D) (|C| + 0)

+ ((D) as in [Sh 54]).

Proof. See [Sh 54], the discussion of monsters of kind III or [BY03], existence of auniversal domain.


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1.9 Claim. 1) Assume that

(a) a countable metric vocabulary and let () = (20)+

(b) = D(c) D is a finite diagram

(d) there is a (D, 1)-homogeneous model M

(e) there is a D-model of cardinality () (probably less is enough) and

(f) K1D = {M : M Kc1} (see Definition 2.19) is categorical in > 0 or is

wu-categorical (see Definition 6.4).

Then D is stable, hence is good.

Proof. By 6.6, 4.10 hence we get stability, D is good follows by [Sh 3].

1.10 Notations.,, infinite cardinals,, infinite ordinals limit ordinals, sequences of ordinals ,, formulaeC the monster model

M, N models (in the monster)A,B,C sets (in the monster) , , positive realsI, J order typesI, J indiscernible sequencesd a metric


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2 Main context - monster metric spaces

In this section we discuss our main context. We start with some notations.2.1 Definition. Let (X, d) be a metric space.1) We extend the metric to n-tuples: for ai : i < n, bi : i < n we defined(a, b) = max{d(ai, bi) : i < n}.2) For a an n-tuple, A a set ofn-tuples, d(a, A) = d(A, a) = inf{d(a, b) : b A}.3) For sets A, B Xn, we define two versions of distances:

d1(A, B) = inf{d(a, b) : a A, b B}

d2(A, B) = sup{{d1(a, B) : a A} {d1(A, b) : b B}}.

4) We denote the density (the density character) of X by Ch(X). So Ch(X) is theminimal cardinality of a dense subset of X.5) We denote the topological (metric) closure of a set A by A or mcl(A).

2.2 Definition. 1) We call a vocabulary metric if it contains predicates Pq1,q2(x, y)for all 0 q1 q2 rationals. We call the collection of these predicates a metric scheme.

2) Given a metric vocabulary , we call a -structure M semi-metric if for some(unique) d:

(i) M is a metric space with the metric d

(ii) d is definable in M by the -metric scheme, i.e., d(a, b) [q1, q2] iff M |=Pq1,q2(a, b) for all rationals q1, q2 and a, b M.

3) We call a semi-metric structure complete if (M, d) is complete as a metric space.

2.3 Remark. Given a semi-metric -structure M, we will usually write M |=d(a, b) q, etc., forgetting the -metric scheme.

Note that in a semi-metric structure for each r1, r2 R, the property r1 d(x, z) r2 is 0-type-definable by a set of atomic formulas.


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2.4 Definition. 1) (M, d) is a metric structure (or model) when:

(a) (M) is a metric vocabulary and (M, d) is a semi-metric structure

(b) PM

arity(P)

M is closed (with respect to d) for every predicate P M(c) FM : arity(F)M N is a continuous function for every function symbol

F M.

2) (C, d) is a (homogeneous) metric monster if: (C, d) is a metric model, C is aD-monster for some finite diagram D, constant here.

2.5 Remark. Each homogeneous monster admits the discrete definable metric, i.e.,d(a, b) = 1 for all a = b C, and it is definable by the equality and inequality. Soeach homogeneous monster is a metric monster with the discrete metric.

2.6 Notations. We will often identify = (x, a) (for a formula, a C) with theset of the realizations in C of(x, a), i.e., (C, a) = C =: {b g(x)C : C |= [b, a]}.So d1(, ) in fact means d1(

C, C), etc.

The following definition is the analogue of abstract elementary classes (see [Sh 88r])in our context.

2.7 Definition. Let (K, K) be an ordered class of = (K) complete metricstructures ( is a metric vocabulary), K closed under -isomorhisms. We call(K, K) an abstract metric class (a.m.c.) if

(1) Mi : i < is a K-increasing sequence (from K, of course), then M =

mcl(

i


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2.9 Definition. In C, , are contradictory if d1(C, C) > 0.

2.10 Remark. Later (see 3.7) we show that in our context this is equivalent toC C = .

2.11 Example. If d is discrete, then each subset of C is closed, so the set of allformulas = L(C) is admissible.

In order to give a non-trivial example, we define

2.12 Definition. For a metric model (M, d) and a formula (x, a) with parametersa M, we say that M |= x(x, a) (there almost exists x such that (x, a)) if forevery > 0 there exist b, a such that M |= (b, a) and d(a, a) .

2.13 Definition. 1) We define positive formulae by induction: each atomic formulais positive, for , positive, , , x, x are positive. So negation andimplication are not allowed.2) Positive existential formulae are defined similarly without allowing x.

2.14 Observation. For (M, d) a metric model and positive (x) L(M), M ={a g(x)M : M |= [a]} is a closed subset of g(x)M (under d), i.e., = thepositive formulae is admissible.

2.15 Definition. 1) For a homogeneous monster C, a set of formulae is calledfull if

(i) for all a C, A C, tp(a,A, C) is determined by the -type tp(a,A,C)

(ii) if , a C and C |= (a), then there exists such that (, ) iscontradictory (see 2.9) and C |= (a).

2) We call C -compact (where is a set of formulae, i.e., L(C)) if each set of-formulae with parameters from C of cardinality < |C| which is finitely satisfiablein C, is realized in C. We omit if constant, and abusing notation write = (C).

2.16 Definition. We say that is full+ if: as in 2.15 but in (ii) the quantifierdepth of is the quantifier depth of .

Now we make the main definition of this section, introducing the context of thispaper.


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2.17 Definition. 1) A metric homogeneous class (K, K) (equivalently: its metrichomogeneous monster C) is called -momspace (monster metric space) if = (C)is a set of formulae containing the metric scheme, such that

(a) L(C) is closed under , and subformulae

(b) is admissible

(c) is full for C.

2) Let C is momspace mean for some and choose such = (C) (well, it isnot necessarily unique but we ignore this).3) The metric d = dC can be defined from C so we can forget to mention d,but still will usually say (C, d), e.g. to distinguish from C when we use the (D, )-homogeneous context.

2.18 Convention. (C, d) is a fixed momspace.

The class of models we are interested in is defined below.

2.19 Definition. 1) K1 = K1C

is the class of M such that:

(a) M C

(b) M 1 C which means:if > 0, C |= [b, a], (y, x) , a g(x)M then for some b g(y)M we

have C |= (x, y)((y, x) d(yx, ba) ).

2) Kc1 is the class of members of K1 which are complete.

2.20 Claim. 1) Assume M C and N = mcl(M). Then

(a) M K1 iff N K1 iff N Kc1.

2) (Kc1, ) is an abstract metric class .

2.21 Definition. A momspace (C, d) is called compact ifC is (C)-compact.

2.22 Convention. We work in a compact -momspace (C, d), M , N denote submod-els which are from K1; though really interested in the closed (complete) ones, i.e.versions of categoricity are defined using complete models. BUT we try to mentionwhen we use compactness.


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2.23 Observation. [C compact] For (x, y) , a C, we have C |= x(x, a) C |= x(x, a) [so we can forget about the new existential quantifier and use theclassical one].

Proof. Assume C |= x(x, a). Then the set {(x, y), d(y, a) 1n : n < } isconsistent in C (by compactness), so C |= x(x, a).

2.24 Examples. 1) Let T be first order, C a big saturated model of T, then C is acompact momspace (with discrete metric), (C) = L (all formulae).2) Let T be a Robinson theory (see [Hrxz]), C its universal domain. Then it is acompact momspace.3) Consider the unit ball of a monster Banach space as in [HeIo02], Chapter 12.

Then (C, ) is a compact momspace (we write the norm here instead of the met-ric) where (C) = positive formulae (more precisely positive bounded formulae,see [HeIo02], Chapter 5), K = Kc1 = {M : M A C}, see [HeIo02](6.2) for thedefinition of A. Compactness is proven in Chapter 9 there. One can easily showthat A=

1, see also 3.9, [HeIo02](6.6).

4) Metric Hausdorff CATs, see [BY03] for a definition of a Hausdorff CAT. A CATis metric if its monster is metric. See [BY05] for more on Hausdorff metric cats.5) The main example we have in mind: (C, d) is a compact momspace for (C) =positive existential formulae, K= Kc1. So in particular it is a metric cat (not nec-essarily Hausdorff).


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3 Approximations to formulas and types

The following notions of -approximations of formulae is of major importance.We give two different definitions and will use both for different purposes. Note that3.1 simply defines topological neighborhoods, while in 3.2 by moving the parameterswe allow the formula to change a little, see also 3.9. Let denote a non-negativerational number.

3.1 Definition. 1) For a formula (x) possibly with parameters, we define [](x) =x((x) d(x, x) ). So [](x, a) = (x)[d(x, x) (x, a)].

2) For a (partial) type p, define p[] = {(

0, we say that c B realizes p[] ifC |= p[](c). We say that p[] is realized in

B if some c B realizes it.2) Similar for p.

3.4 Remark. 1) Note that [](x, a) is equivalent to []1 (x), when we expand C by

individual constants for a and let 1(x) = (x, a).2) [] |= for all .3) For a formula without parameters , [] = .4) For 0, [] |= [], |= .5) For = 0, [] .

3.5 Observation. 1) If = (x, b) (as usual is admissible), then

>0

[].

2) If = (x, a) then

>0

(0,)

[].

3) Similar for .


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Proof. 1) [a |= a |=

{[] : > 0}] is obvious.Suppose a |= [] for all , so for each n there exists an such that C |= [an] and

d(a, an) 1n . So an : n converges to a, therefore a |= , as

C is a closed set.

2),3) Similar.

3.6 Claim. 1) ((x)) (x).2) (p(x)) p.3) Assume d(b, b) 1 2 then

(x, b) |= (x, b).

3.7 Claim. 1) If C is -compact then a pair (of admissible -formulas) =(x, a), = (x, b) is contradictory (see definition 2.9) iff C C = .2) [C not necessarily compact] A pair as in (1) , is contradictory iff for some > 0, [] [] = .

3) If, in (2) are without parameters, we can add iff = for some > 0.4) IfC is compact, (3) holds for formulae with parameters.

Proof. 1) Obviously if (, ) is a contradictory pair then C C = . Now assumed1(, ) = 0. So for each n > 0 there exists an, bn C such that d(an, bn) 0, so the set {(x, a)(x, b) : > 0}is finitely satisfiable (by 3.4(4)) and therefore realized in C, now by 3.5(3) = ,therefore and are not contradictory. The other direction is trivial. 3.7

3.8 Claim. [Let (C, d) be a compact momspace], p a type over a set A.Then a C realizes p[] iff there exists a C realizing p, d(a, a) (so p[] is justthe -neighborhood of p).

Proof. The if direction is obvious. Now suppose a |= p[], so [(a)][] holds for all p, but there is no a |= p such that d(a, a) . Then the set

{d(x, a) } {(x) : p, is a (C)-formula}

is inconsistent (as is full for C), and by compactness we get a contradiction(because in the definition of p[] we essentially close p under conjunctions).


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We connect the relevant notion of submodel defined in 2.19 with the logical notionof approximation:

3.9 Observation. M 1 N (M, N structures, see 2.19 for the definition of 1) iff

for every (y, x) , > 0, if N |= y(y, a), a M, then N |= (b, a) forsome b M.

Note that the assumption above ofM being complete in fact follows from (D, 1)-homogeneous:

3.10 Observation. Let M K1 be (D, 1)-homogeneous, then M Kc1 , i.e., iscomplete.

Proof. Let an : n < converge to a C, an M. So the set {d(x, an) d(a, an) : n N} is realized in M, so an : n < converges in M (to the samelimit, of course). 3.10

3.11 Definition. We call M Kc1 pseudo (D, )-homogeneous if for every A M, |A| < , for every p SmD (A), for every > 0, p

() is realized in M (seeDefinition 3.3).Note that replacing p above by p[], we get (D, )-homogeneous, see 3.13 below.

3.12 Definition. We say that a type p S(A) is almost realized in B (or B almost

realizes p) if for all > 0 there exists b B, b |= p[].

3.13 Claim. [(C, d) compact] Let M Kc1 be such that every 1-type over a subsetof M of cardinality less than is almost realized in M ( infinite). Then M is(D, )-homogeneous.

Proof. Let p S1D(A), A M, |A| < . Choose by induction bn |= p[ 12n ], bn

M, d(bn+1, bn) 1

2n1.

Why is this possible? Let qn = p(x) {d(x, bn) 12n}. As bn |= p

[ 12n ], some

b C realizes qn (see 3.8) and let qn = tpD(b/A {bn}), so qn S1D(A {bn}),

and by the assumption there exists bn+1 M realizing (qn)[ 12n+1

]. So bn+1 |=

p[1

2n+1], d(bn+1, bn)

12n +

12n+1

12n1 , as required.

Now, bn : n < is a Cauchy sequence, let b M be its limit (M Kc1 , so is

complete). Now b |= p[] for all > 0, so b |= p (see 3.5(1)), and we are done.


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3.14 Corollary. Let M Kc1 be non-(D, )-homogeneous. Then there exists A M, |A| < , p S1D(A) and > 0 such that p

[] is omitted in M.

Proof. This is just restating 3.13; but we prefer this form for later use.

3.15 Remark. We will not use the notion of pseudo homogeneity (Definition 3.11),in this paper (as a postoriori all the models will turn out to be (D, )-homogeneous),but it is interesting to point out what non-categorical classes can look like. See 4.23later.

3.16 Claim. If A C, p SD(mcl(A)) and c C realizes p A then it realizes p.

Proof. If (x, a1, . . . , am) p ( of course, parameters not suppressed) thenfor each n < we can choose bn A (for = 1, . . . , m) such that d(a, b

n ) 0, whichcontradicts and being a contradictory pair, see 3.7(4).


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4 Stability in momspaces

We define a topological version of stability. The intuition behind the definitionis that there may be many types, but the density of the space of types is small. Ourdefinition generalizes Iovinos stability for Banach spaces, see [Io99]. For Hausdorffcats and continuous theories, it coincides with definitions given in [BY05], [BeUs0y]respectively. Note that for elementary homogeneous class (i.e. discrete metric), thisdefinition coincides with the usual one (so certainly for an elementary class, i.e.,C saturated). For a non-discrete metric the classical -stability of D (countingtypes, as in [Sh 3]) is stronger than the topological relative we define here, and isequivalent if and only if = 0 . Stability in our sense (i.e., -stable for some )is equivalent to stability for D, but for a specific (e.g. = 0) the notions differ.

4.1 Hypothesis. (C, d) is a momspace for = C.

4.2 Definition. 1) A momspace (C, d) is called 0+ -stable if for all A Cof cardinality there exists B C of cardinality such that each p SD(A) isrealized in mcl(B) = B (i.e., the topological closure of B).

4.3 Claim. [(C, d) compact] For a compact momspace C, the following are equiv-alent:

(A) C is 0+

-stable(B) for each A C, |A| , there exists B, |B| such that for each > 0

and p SD(A), p[] is realized in B (B almost realizes all types over A, see

Definition 3.12)

(C) there are no A, |A| , pi : i < + a sequence of members of SD(A), i.e.,

complete D-types over A and > 0 such that p[]i p

[]j = for alli, j <

+.

Proof. (A) (C).Assume (A). Suppose (C) fails, so we have p

i: i < + over A as there. By

(A) we can find B of cardinality such that each pi is realized in mcl(B) = B,by, say, ai. As {ai : i <

+} form an -net (by the nature of the pis), it is obviousthat density of B is at least +, contradiction (|B| = ).

(C) (B).Let A be given. We construct Bn by induction:B0 = A


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Bn+1 realizes all complete D-types over Bn up to1

n+1 , i.e.

if p SD(Bn) then for some a Bn+1, tp(a, Bn)[ 1n+1 ](C) p[

1n+1 ](C) =

|Bn| =

B = Bn is obviously as required in (B).How is Bn+1 constructed? Let = 1/2(n + 1) and let p

ki : i < be a maximal

-disjoint set of complete types of -tuples over Bn, (I.e. (pki )

[] (pkj )[] = for all

i, j)Bn+1 = Bn {a

ki : a

ki realizes p

ki for some i < , < }.

Now each complete k-type p over Bn is1

n+1-realized in Bn+1, as there is i <

such that (pki )[](C) p[](C) is non-empty, let b be in the intersection. We can

replace b by any b realizing tp(b, Bn) hence without loss of generality d(b, aki ) < .

Also we can find c realizing p such that d(b, c) < . So aki witnesses p is1

n+1 -realizedin Bn+1.

(B) (A).Let A be given. Define B0 = A, Bn+1 almost realizes all types over Bn, Bn

Bn+1, |Bn| = . Let B =

n 1 (possible by 3.8

as Bn almost realizes all types over

i 2, tp(a/A) = tp(b/A).


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Proof. p[] splits over A there exist [](x, a), [](x, b) p[] (where (x, a), (x, b) p) such that a A b and (

[](x, y), [](x, y)) is a contradictory pair [so (, ) is a2-contradictory pair].

4.8 Observation. 1) The type p splits over A iff for some > 0, the type p[] splitsover A.

Proof. p splits over A iff some contradictory pair (, ) witnesses this, now pick

= d1(,)3 and use 4.7.

4.9 Lemma. Let C be 0+ 0-stable. Then for any B C, p SD(B) and

> 0, p[]

does not split over a finite subset of B.

Proof. Suppose p[] splits over every finite subset of its domain. We construct finitesets An for n < and elementary maps F for >2 as follows:

(1) A0 =

(2) An+1 = An{an, bn}, where there are n(x, y), n(x, y) such that n(x, an), n(x, bn)exemplify p[] splits over An, i.e. an An bn, n(x, an), n(x, bn) p

[] and

[]n (C, an)

[]n (C, bn) =

(3) for n2, F : An C is an elementary mapping

(4) for n2, F 0(an) = F 1(bn) and F 0, F 1 extend F.

The construction is straightforward. Now denote for 2, F =

n2} (so it is countable) and choose1 p, p p

SD(A). Obviously, = 2 p[] p

[] = , contradicting 0+ 0-stability by

4.3 by an implication not using compactness.

4.10 Claim. Assume (C, d) is 0+

0-stable. Then

(a) if p SD(B) then p does not split over some countable A B

(b) D is stable (in the sense of [Sh 3]), (D) 1.

1this uses D is good


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Remark. Note that it is close but not as in first order; there may be 0 exceptions.

Proof.Clause (a):

By the previous claim for every > 0 for some finite B A, p does not -splitover B. Let B = {B1/(n+1) : n < }, so B is a countable subset of A and by 4.8and the obvious monotonicity of non--splitting, p does not split over B.

Clause (b): Follows from (a).

4.11 Definition. Given an uncountable indiscernible set I mC and a set A C,define the average type of I over A, Av(A, I) as follows:

Av(A, I) = {(x, a) :(x, y) , g(x) = m, a A, and for infinitely many

c I, (c, a) holds}.

4.12 Fact. IfC is stable, then any indiscernible sequence is an indiscernible set.

Proof. Standard.

We often say I is indiscernible meaning an indiscernible sequence, which is thesame as indiscernible set.

4.13 Claim. LetC be 0+ 0-stable.1) If I mC is indiscernible uncountable, A C a set, then Av(A, I) SmD (A).2) If A, I are as in (1), then

Av(A, I) = {(x, a) :(x, y) , g(x) = m, a A and for all but

countably many c I does (c, a) hold}.

Proof. Using the standard argument, one shows that for a given contradictory pair(, ) = ((x, z), (x, z)) and a M, one of the sets {c I :|= (c, a)}, {c I :|=(c, a)} is finite (otherwise, let = d1(, ), and construct 20 -distant types over


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a countable set, contradictory 0+ 0-stability; in fact, one constructs 20 pairwisedistinct (, )-types, i.e., types mentioning only and ).

Now given a formula (x, a) over M, if{c I : (c, a) holds} is infinite, then for

each (x, z) such that (, ) is contradictory, the set J = {c I : (c, a) holds} isnecessarily finite, so taking the union of J over all such , we obtain a countableset of exceptions

J = J = {c I :|= (c, a)}.

This completes the proof of clause (2). For clause (1), let (x, a) be a formula overA,g(x) = m. If for uncountably many c I, (c, a) holds, then (x, a) Av(A, I),otherwise for some (x, a) such that (, ) is a contradictory pair, (c, a) holdsfor uncountably many c I, so (x, a) Av(A, I). Clearly, only one of the two

options above is possible, so (1) follows.

Discussion: Why countable and not finite in the definition of averages? Even ifthe majority satisfies (x, a), for each there can be finitely many c I such thatC |= (c, a) and (, ) are -contradictory, and this finite number can increasewhen goes to 0.

4.14 Lemma. If M is (D, 1)-homogeneous, p Sm(M) then for some uncount-able I mM, we have p = Av(M, I).

Proof. Let B be as in 4.10, clause (a). Let m = 1 for simplicity. Choose a Mrealizing p B {a : < } by induction on < 1. Now I = a : < 1 isindiscernible over B by [Sh:c, I,2].

If q = Av(M, I) = p still q SD(M) (by 4.13(1)) and we can find (x, b) q, (x, b) p and they are contradictory. So u = { < 1 : C |= [a, b]} is infinitelet v u be of cardinality 0, let v () < 1 and choose a M( [(), 1)

realizing p (B {ai : i < ()} b {a : ((), )}.

Easy contradiction: for a given contradictory pair (, ) all but finitely many ele-ments of I have to make a choice, see also 4.13(1).

4.15 Definition. 1) A momspace (C, d) is called (, )-superstable if given Mi :i < an increasing chain of (D, )-homogeneous models (Mi Kc1 , of course),

mcl(

i |C| + 0.


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Remark. This definition generalizes superstability for C a saturated model of a firstorder theory, and C a homogeneous monster.

The following claim will be mainly of interest for us when = 0:4.16 Claim. Let (C, d) be 0+ 0-stable and compact. Then (C, d) is (+, )-superstable for every 0.

Proof. Let Mn : n < be an increasing sequence of (D, +)-homogeneous models,

and assume p S(A), A

n 0 trying to builda tree p : >2 of -contradictory types) without loss of generalityp has

a unique extension in SD(M), call it q. By 3.14 as (C, d) is compact we canadd that for some > 0, p[] is not realized in M. Without loss of generality

A mcl(A

n


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4.17 Definition. We say that M K1 (< )-omits p(x), a type over A M,when for no [0, )R and b M, does b realize p

[].

4.18 Definition. 1) We say that a formula (x, b) pseudo (, ) isolates a typep(x) if (x, b) |= p[](x) (note that the roles of , are not symmetric and thedifferent notions of approximation!). In other words, if C |= [a, b] then forsome a p(C) we have d(a, a) (so a realizes p, a realizes (x, b)).2) We say that A is a pseudo (< )-support for p(x) or A pseudo (< )-supports pwhen there is a consistent (x, b), b A and positive 1, 2 such that (x, b) |=p[2](x). So (x, b) pseudo ( 2, 1)-isolates p(x).3) We say that A really (< )-omits p(x) if it does not pseudo (< )-support p(x).

4.19 Claim.1) If M K

c

1 (< )-omits p(x), p(x) Sm

(A), A M then M really(< )-omits p(x).2) If p(x) is a type over M and M really (< )-omits p(x), then M(< )-omits p.

Proof. 1) Assume M is a pseudo (< )-support for p(x), i.e., there exist (x, b)over M and 1, 2 > 0 such that

(x, b) |= p[2](x). C |= x(x, b), M Kc1,so for some a M, C |= (a, b) (see 3.9), therefore p[2](a) holds, and wehave p( 0, a M. Then the formula x = a isover M and pseudo ( 2, 1)-isolates p(x) for 2 = 1 = 3 .

4.20 Definition. 1) We say (x, b) strictly (, )-isolates a type p if (x, b) pand (x, b) pseudo (, )-isolates p.2) We say (x, b) is strictly (, )-isolating over A when

(a) b A

(b) if (x, b) p Sg(x)D (A) then (x, b) strictly isolates p.

3) We say that p SmD (A) is strictly (, )-isolated if some strictly (, )-isolatesit.3A) Strictly -isolate means for some > 0, strictly (, )-isolate.4) We say that p SmD(A) is strictly isolated when for every > 0 for some(x, a) p and some > 0 the formula (x, a) strictly (, )-isolates the type p(i.e., p is -strictly isolated for all > 0).


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4.21 Claim. [(C, d) is compact]Assume that

(a) p Sm

D(A) or just p is an m-type closed under conjunctions(b) (x, b) p

(c) > 0 and 0.

Then one of the following occurs

() there is a pair (1(x, y), 2(x, y)) of formulas and a sequence b from A

such that b b, g(b) = g(y) such that (x, b) 1(x, b), (x, b) 2(x, b

) are -contradictory (and both consistent, of course), hence for noa, C |= (x)(d(x, a) /2 (x, b) 1(x, b)) and C |= (x)[d(x, a) /2 (x, b) 2(x, b)]

() for every a such thatC |= [a, b] there is a sequence a realizingp suchthat d(a, a) (so (x, b) strictly (, )-isolates p, see 4.20).

Proof. We can assume that clause () fails and let a exemplify it. So for every a

realizing p we have d(a, a) > .Let q(y) = tp(a, A) so (y, b) q as C |= (a, b), and let r(x, y) =

p(x) q(y) {d(x, y) }. If r(x, y) is consistent, so is r(x, a), and any a

realizing r(x, a) is as required in clause (), contradicting our assumption. So

r(x, y) is inconsistent. As (C, d) is compact and p(x), q(y) are closed under con-junctions, (x, b) p(x), (y, b) q(y) and as we can add dummy vari-ants, there are b A, b b and 1(x, b

) p(x), 2(y, b) q(y) such that

{(x, b) 1(x, b), (x, b) 2(y, b), d(x, y) } is contradictory. So we get clause ().

4.22 Isolation Claim. [(C, d) is 0+ 0-stable and compact]1) If a A and (x, a) is consistent and > 0 then we can find 1(x, a1) and > 0 such that a1 A and (x, a) 1(x, a1) is strictly (, )-isolating over A,see Definition 4.20.

1A) Similarly omitting getting strictly -isolating.2) The set of strictly isolated p SmD (A) is dense, i.e. for every (x, a) with a A,there exists p SmD (A), (x, a) p, p is strictly isolated.

Proof of 4.22. Part (1A) is restating Part (1). Also part (2) follows from part (1)by choosing n(x, an) such that (x, a) 1(x, a1) . . . n(x, an) is

1n+1

-isolating


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over A (iterating 1A) and applying compactness ofC. So we concentrate on provingpart (1) .

We can choose n > 0 for n < such that, e.g. {n : n < } /5

Assume that (x, b), b A is a counterexample. Now we choose (x, a) : n2 by induction on n such that

(a) a A

(b) (x, a) is consistent

(c) (x, a) = (x, b)

(d) if 0, 1 n2 then 0(x, a0), 1(x, a1) are-contradictory

(e) if = 0 n2 then (x, a) |= (x, a)

(f) if = 1 n2 then (x, a) |= (x, a).

By 4.21 there is no problem to carry the definition, i.e., having (x, a), clause() of 4.21 cannot hold (with , a here standing for , b there) as (x, a) is acounterexample. Hence clause () there holds, let us choose (x, a)for = 0, 1.

Now let n = {m : m [n, )}, so clearly

() ifn(1) < n(2) < and n()2 for = 1, 2 and 12 then

2 (x, a2) |=

1 (x, a1)

[Why? By 3.6 using clauses (e) + (f) of we get 2 |=

1 , wheren(1),n(2) = {m : m [n(1), n(2))}. Now use 3.6 again.]

Now let C = {a : >2}.

Hence

() for 2 the set {n (x, an) : n < } is consistent, hence is includedin some p S(C)

() if 2 for = 0, 1 then p[/5]1 (x) p

[/5]2 (x) is inconsistent.

[Why? By (d) and the choice of n], a contradiction to 0+ 0-stability.

4.22

The following is not used at present but clarifies non-categoricity. Recall Defi-nition 3.11 and Claim 3.13. Note that 3.14 says that a non-(D, )-homogeneuosmodel omits some p[]. Here we clarify what happens in the case of pseudo (D, )-homogeneous non-(D, )-homogeneous model.


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4.23 Claim. Assume M Kc1 , > 0 + |C| and M is not (D, )-homogeneous.Then () or ()

() (a) N

1

M, |N| < (b) p SD(N)

(c) ifC is 0+ -stable for some [0 + |C|, ) then p has aunique extension in SD(M)

(d) > 0 and M omits p

() (a) for every B M, |B| < and q SD(B) and > 0, the type q

is realized in M (so M is pseudo -saturated)

(b) N 1 M, |C| + 0 N <

(c) p SD(N)

(d) ifC is 0+ -stable for some [|C| + 0, ) then p has a uniqueextension in SD(M)

(e) > 0, p[] is omitted by M

(f) for every n, for some cn, n, n we have 1/(n + 1) > n > n> 0, cn M realizes p

and d(cn, p(C)) 10 n n.

Proof. Clearly there is A M such that |A| < and p S1D(A) is omitted. Let = |A| + |C| + 0 so < . IfC is 0+ -stable for some [|C| + 0, ), easilywithout loss of generalityp has a unique extension in SD(M) and A = |N|, N

1

M. If for some > 0, p is also omitted by M, the case () holds. So we mayassume () fails, in other words, clause (a) of () holds.

Let n < , = 12 (any > 0 works). We are going to find cn , n , n as

required in clause (f) above. First, we try to choose bn M by induction on n < such that

n (a) bn M

(b) bn realizes p

(c) if n = m + 1 then d(bn, bm) 10 /2n.

For n = 0 there is b0 M realizing p = p by clause (a) of (). So wecan begin.

Point 1: We cannot succeed to choose bn : n < .Why? Suppose we have succeeded. Then bn : n < is a Cauchy sequence andtherefore converges to some b M. We will show that b |= p. If(x, a) p, thenfor each n, C |= (bn, a) so there is an >C, bn C such that


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(a) d(bn, bn) /(n + 1)

(b) d(an, a) /(n + 1)

(c) |= [bn, an].

Now

bn : n < converges to b

an : n < converges to a

hence |= [b, a]. So b |= p, b M, a contradiction.

Point 2:So we are stuck in some n = m+1 so let cn := bm, n = /2n, n =

/2n+2. Ifthe demand in (f) of () fails, then there is bm C realizing p

(x){d(x, bm) 10 n n} hence p(x) {d(x, bm) 10 n n} is consistent henceit is contained in some qn S(N {bm}).

So for every > 0 (we use small enough) there is bn M realizing q(x)

(recall we are assuming ()(a)!). So bn realizes (p) hence p hence

if is small enough, p. Also d(bn, bm) (10 n n) + + < 10

n = 10/2n (because if (a, b) realized d(x, y) then d(a, b) ++).So bn is as required in n(a) (c) above, so we could have continued choosing

the bn. 4.22


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5 Ehrenfeucht-Mostowski models

In this section we adapt the technique of constructing Ehrenfeucht-Mostowskimodels to our context. The reader should have a look at chapter 7 of [Sh:c] for thebasic definitions ( proper, etc.). The basic idea is the following: we start with Cin vocabulary . Adding skolem functions, we obtain vocabulary . Choosing anindiscernible sequence and taking its type (its EM - blueprint) , for each ordertype J we can construct EM(J, ) (like in chapter 7 of [Sh:c]), which will be anelementary submodel ofC expanded to , therefore its restriction to , EM(J, )is an elementary submodel of C, although not necessarily complete. Taking thecompletion, we obtain a model in Kc1. Adding more structure to the language wecan make it (D, )-homogeneous, and more, see below.

Given a vocabulary with skolem functions, and a -diagram of indiscernibles

(EM-blueprint), we denote for each order-type I, the EM-model (the -skolemhull of a sequence ai : i I) by EM(I, ) or EM(I, ) if

is clear from thecontext. We denote by EM0(I, ) the restriction of EM(I, ) to the vocabulary0 .

Let C be a momspace. Let be the vocabulary of C. It is clear that for any (with skolem functions) expanding , a -diagram of indiscernibles (in Cexpanded to ), I an order, we can think of EM(I, ) as an elementary submodelofC, so EM(I, ) L((C)) C. This is not necessarily true for the completion, but

EM(I, ) 1 C by 2.20.

5.1 Claim. Let (C, d) be a momspace, the vocabulary of C, || 0, , with Skolem functions, a -blueprint.0A) For every linear order J, EM(J,

) K1and mcl(EM(J,

)) Kc1.0B) If J1 J2 then EM(J1,

) 1 EM(J2, ); moreover EM(J1,

) EM(J1,

) and mcl(EM(J1, )) 1 mcl(EM(J2,

)).1) There exists expanding, || = 20 and a -diagram such that for each finite orderJ, EM(J,

) is (D, 1)-homogeneous.2) IfC is (1, )-superstable, then

as in (1) works for all orders J, but we haveto take the closure, i.e., mcl(EM(J,

)) which Kc1 is (D, 1)-homogeneous for

all J.3) IfC is 0+ -stable, then in (1) and (2) can be chosen of cardinality 1.

Proof. 0) (A),(B) straight.1) Choose for i < 1, i, i, |i| = 20 with skolem functions expanding increas-ing continuous such that for each i-type p over a finite subset of the skeleton


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of EM(I, i), say a1, . . . , an, there exists a function symbol fp in i+1 such thatfp(a1, . . . , an) realizes p.

More precisely, we do the following: for any consistent set p of formulas of

the form = (x, y1, . . . , yn) i (y1, . . . , yn are the parameters; some of theyis may be dummy variables) such that p is closed under conjunctions and forevery p, x(x, y1, . . . , yn) i, we add a function symbol fp to i+1 suchthat for every p the following formula is in i+1 : x(x, y1, . . . , yn) (fp(y1, . . . , yn), y1, . . . , yn).

Now let =

i 0. Consider mcl(EM(, )), as in 5.1(3) (note that || 1)

and use 5.1(2) + 4.16.

5.3 Discussion: If (C, d) is 0+0-stable, does it have a (D, ) homogeneous modelin every ? By 7.6 this follows from categoricity, which is good enough for our

purposes.


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6 Embeddings, isomorphisms and categoricity

In this section we introduce notions of -embedding and -isomorphism which

are weaker than isometry. This will lead us to the notion of weak uncountablecategoricity that we investigate in 8.

6.1 Convention. Models are from K = K1.

6.2 Definition. For two metric structures in the same vocabulary and 0 wesay

(1) f : M1 M2 is an -embedding if for every -formula , a M1, M1 |=(a) M2 |= [](a)

(2) f : M1 M2 is an -isomorphism if it is an -embedding which is one-to-oneand onto

(3) M1, M2 are +-isomorphic if there exists a -isomorphism f : M1 M2

for all >

6.3 Observation. 1) 0-embedding is a regular notion of (elementary) embedding,0-isomorphisms is regular isomorphism (in particular isometry).2) If there exists a -isomorphism f : M1 M2 for all > , then there exists

a -isomorphism g : M2 M1 for all > (so clause (3) of the definition abovemakes sense).

Proof. Clear.

The following definition is the central one.

6.4 Definition. Let (C, d) be a momspace, 0, a cardinal.1) We say C is +-categorical in if every two complete M1, M2 Kc1 of density are +-isomorphic.2) We say C is categorical in if every two complete M1, M2 Kc1 of density areisomorphic.3) We say that C is possibly categorical (+-categorical) if it is categorical (+-categorical) in some > 0.


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4) We say that C is weakly uncountably categorical (wu-categorical) if the followingholds: for each > 0 there exists a cardinal such that C is +-categorical in .

6.5 Observation. 1) If then +-categoricity implies +-categoricity (for aspecific ).2) Possible 0+-categoricity implies weak uncountable categoricity.3) Categoricity implies all the other notions (for a specific ).

6.6 Theorem. [(C, d) compact]Let K = Kc1(C) be a wu-categorical momspace with countable language. ThenC is0+ 0-stable.

Proof. Let be (K), is expanded with skolem functions, -proper for K. So is countable.

6.7 Subclaim. Under these assumptions, let Ibe a well-ordered set, M0 = EM(I, ),A M0 is countable > 0. Then each -disjoint set P of types from SD(A),

2

-

realized in M0 (so p1, p2 P p[]1 p

[]2 is contradictory and for each p P, p

[ 2 ]

is realized in M0) is countable.

Proof of the Subclaim. If not, let pi : i < 1 be -disjoint types over A, p[ 2 ]i

realized in M0 by bi. Pick b0i EM(I, ), d(b

0i , bi)

100

. Without loss of generalityA = EM(J, ) for J I, |J| 0. As J is well ordered, by the standard argument,

there are uncountably many b0i s satisfying the same type over A, but b0i |= p

[]i and

p[]i , p

[]j are contradictory for i = j, a contradiction. 6.7

Now we prove the theorem. Assuming C is not 0+-stable, we get A C countableand pi : i < 1 -disjoint types over A for some > 0 (remember 4.3). Let besuch that C is +-categorical in for


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6.8 Corollary. Let C and K be as in 6.6, then K has a (D, 1)-homogeneousmodel in (recall this means of density ) for all > 0.

Proof. By 6.6 and 5.2.

6.9 Observation. [(C, d) compact].1) Every two (D, )-homogeneous models at density character and isomorphic.2) No (D, )-homogeneous model is 0+-isomorphic to a non-(D, )-homogeneousmodel.

Proof. 1) Standard and does not require compactness.

2) By Corollary 3.14.


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7 Uni-dimensionality

The following notion was explored in [Sh 3] but not defined there:

7.1 Definition. A (good) finite diagram D is uni-dimensional if for some regular there is no (D, )-homogeneous model of K which is not +-homogeneous incardinality +.

In [Sh 3] it is essentially proven (see [Sh 3, 6]) that:

7.2 Theorem. Assume D is stable. Then the following are equivalent:

(1) D is not uni-dimensional

(2) there is some regular such that there are maximally (D, )-homogeneousmodels of arbitrary large cardinalities

(3) for all large enough regular < , there is a (D, )-homogeneous model Mand ai : i < , bi : i < mutually indiscernible sequences in M suchthat bi : i < is a maximal indiscernible sequence in M, i.e., can not beextended in M

(4) there is a cardinal and a model M of cardinality which is (D, 1)-homogeneous, but not (D, )-homogeneous.

7.3 Remark. In our context we will say C is uni-dimensional or K is uni-dimensional, meaning that D is.

7.4 Reminder. 1) For an indiscernible set I C of cardinality > |C| + 0 and a setA C, we define

Av(I, A) = {(x, a) :a A, infinitely many elements

of I satisfy (x, a)}.

We call this set the average type of I over A. (See 4.10, all but finitely many iswrong.)2) For stable C, for any indiscernible I, |I| > |C| + 0 and set A, Av(I, A) is acomplete type, see 4.13(1).3) Let I = ai : i < , where ai : i is indiscernible. Then a |= Av(I, I).4) Let I be indiscernible, I = ai : i < and let a |= Av(I, I). Then ai : i


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is indiscernible.5) It follows from (3) + (4) that I = ai : i < M is a maximal indiscerniblesequence (set) in M iff Av(I, I) is omitted in M.

6) (x, ai1 , . . . , ain) Av(I, I) for I = ai : i < (-limit ordinal) iff aj |=(x, ai1 , . . . , ain) for some/every j / {i1, . . . , in}.

7.5 Theorem. 1) Let (C, d) be a momspace, (C) countable, 0+-categorical in, > 0. ThenC is uni-dimensional.2) The same is true ifC is wu-categorical.

Proof. 1) If not, choose 0 < 1 > 1) inthe beginning, then by the Erdos-Rado theorem (as in the proof of [Sh:c, VIII,5.3],or using [BY03a](1.2)) we can choose a diagram of indiscernibles (EM-blueprint) in vocabulary such that for any , denoting the skeleton of M0 := EM(, ) byI = ai : i < , we have

1 I is a -indiscernible sequence (set), moreover, it is -indiscernible over

PM

0

2 for each n < , for some i1, . . . , in < 2, a0, . . . , a

n1 has the same

-typeas ai1 , . . . , ain.

Now:

3 let M0 = EM(, ), then M0 C, so mcl(M0) K = Kc1 (as on the one

hand has skolem functions, and on the other hand M0 does not realize-types over that were not realized in M, so M0 is a D-model)

4 M0 |= T

(skolem functions, so M0 M)


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5 PM0 is a -indiscernible set (by 4 and 0 above)

6 PM0 is countable. Why? Each b PM

0 is of the form (ai1 , . . . , ain) for

some -term . But as I is -indiscernible over PM

0 , each such b depends

only on , and there are countably many -terms ( is countable, and sois ).

Denote J = PM

0 , a -indiscernible set. Denote p = Av(J, J). Then

[p][] is omitted in M0.Why? Pick (aij , . . . , a

in

) M0. Let j1, . . . , jn be such that ai1

, . . . , ain

aj1 , . . . , ajn, (aj1, . . . , ajn) M does not realize p[], so for some (x) p, M |=

d1((aj), ) .

Note: (x) = (x, c), c PM

. Call that (x, c) p M |= (d, c) for some/alld PM

, dc = (as PM

= J is an indiscernible set, see 7.4(6)). So M |= c P

such that [d P\c, (d, c) ] & [d1((aj), (x, c)) ]. Therefore, M0 satisfiesthe same formula with (ai), which obviously means that (a

i) does not satisfy

[p][], as required.

We have finished now: let = , so in M0 = EM(, ) we have a countableindiscernible set whose average is omitted (as (p)[] is omitted in M0), so M0 is anon-(D, 1)-homogeneous model in K of density , but by 6.8 we have a (D, 1)-homogenous model of density > 0. So categoricity in fails, moreover, 0

+-categoricity fails, as M0 is at least

2

-distant from any (D, 1)-homogeneous model.2) Repeat the proof of (1), and at the end choose in which, say, ( 2 )-categoricityholds, and get the same contradiction.

7.6 Claim. LetC be a wu-categorical momspace, K = Kc1(C).1) There exists a (D, )-homogeneous model in K for all > 0.2) Each (D, 1)-homogeneous model in K of density is (D, )-homogeneous.3) If K is 0+-categorical in > 0, then each K-model of density is (D, )-homogeneous.

Proof. 1) By 6.8 and uni-dimensionality.2) By the equivalence 7.2.

3) Otherwise by a Lowenheim-Skolem argument we will get a non-(D, )-homogeneousmodel of density , and together with (1) this will lead to a contradiction.


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8 The main theorem

8.1 Hypothesis. C is a compact momspace with countable vocabulary.

In this section we prove the main theorem of the paper. The proof is rather longand technical, but the ideas behind it are quite simple, so let us give an outline ofthe main steps.

We begin with the following situation; a model N, a countable subset B and atype p SD(B) which is really (< )-omitted in N. We would like to constructsuch a model of an arbitrarily large density character . The strategy is as follows:construct by induction on < tuples c such that A = N {c : < } stillreally (< )-omits p.

We have to make sure that A is a model, and that A has cardinality . Thenthe metric closure of A, mcl(A), will be the desired model.Making sure that A is a model is not hard. For this purpose, whenever A is

not in K1, we choose c to realize a formula over A whose realization is missingon A. In order not to create a pseudo-support for p while doing that, we choosec carefully, namely, c realizes a strictly isolated type over A. So that part ofthe proof is very similar to the classical proof of Morleys thoerem. The situationis slightly complicated by technical issues related to strict isolation, and some cal-culations are needed in order to show that p is still really (< )-omitted. This isthe content of Claim 8.3.

Making sure that A has the right density is more difficult. We take care ofthis requirement at stages when A K1. At these stages we need to pickc close enough to elements of A (to make sure it does not provide pseudo(< )-support for p) and yet far from A in terms of the metric in order to makesure that the density character increases. In classical model theory one would atthis point take c to continue an indiscernible sequence in A, more specifically,c |= Av(A, I) with I A. Unfortunately, we do not have true -stability, soexistence of indiscernible sequences is not guaranteed.

What we do in Claim 8.4 is immitating existence of indiscernible sequences.Specifically, we recall that our model M = mcl(A) can be thought of as a sub-model of an Ehrenfeucht-Mostowski structure M. Using this representation, we

can find an indiscernible sequence in M which is close enough to elements ofM.More precisely, we do the following. Since M has uncountable density character,

we can find a : < 1 in M -distant from each other. If we knew thatthis sequence had an indiscernible subsequence, we would choose b = c to satisfyits average over A. Since it is not necessarily possible to do that, we use therepresentations ofM as a subset of the closure of EM(I, ) and choose a sequenceb,n : n < of elements of EM(I, ) converging to a for each . Then using


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some infinitary combinations, we construct in an extension of M an indiscerniblesequence which is similar to the diagonal of the matrix b,n : < 1, n < ,hence converging to a subsequence of a : < 1. Realizing the average type

of that sequence will give us the desired new element b = c.Note that the second step would be simplified significantly if we assumed cat-

egoricity above the continuum. In that case, any sequence would have an indis-cernible subsequence (since C is truly stable in = 0), and one could apply anargument along the lines of the classical ones.

8.2 Theorem. Assume K = Kc1(C) wu-categorical. Then K is -categorical forall > 0, moreover, any model of K of density > 0 is (D, )-homogeneous.

Proof. Suppose not, soC

is 0

+

0-stable, (1, )-superstable, uni-dimensional by6.6, 4.16, 7.5, and there are > 0, N Kc1, ch(N) = > 0, N is not (D, )-homogeneous. By 7.6, N is not (D, 1)-homogeneous.

Then there exists B N, |B| 0, p S1D(B), p is omitted in N, and in fact by3.14, p[2] is omitted in N for some > 0. Therefore (by 4.19) p has no pseudo(< )-support in N, see Definition 4.18.

Let > be a large enough regular cardinal.We choose c by induction on < such that

A = N {c : < } really (< )-omits p(x).

Case (a): If A =: N {c : < } is not in K1. Then first choose (x, y) , a A such that (x, a) witnesses A / K1, and second choose c realize some

strictly isolated q Sg(x)D (A) which contains (x, a).

By 8.3 below, i.e., the next claim, this is possible and is preserved.

Case (b): Not (a), then c / mcl(A) and holds, using 8.4 below.Having carried out the construction, let M = mcl(A).

Let us show that M is a non-(D, 1)-homogeneous model of density character. Since the proof is an easy version of some arguments which appear in Claim 8.4below, we decided to give all the details.

First we note:

() there is a club E of such that E A K1.

In order to prove (), assume towards a contradiction that there is a stationary setS such that S M / K1. Hence for every S there is a formula


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(x, y) and a A such that (x, a) is not realzied in A but (c, a)holds.

Since there are only countably many formulae, without loss of generality (x, y) =

(x, z) for all S. Now define f : S by f() = min{ : a A+A}.Clearly, f is regressive, so for some S S stationary and < we have a Afor all S. Since |A| < , it must be the case that a = a for some < S, so the formula (x, a) is taken care of twice in our construction,which is of course impossible ((x, a) is realized by c A).

So we have shown (). It is now easy to derive:

()1 A K1, hence M Kc1

()2 char(M) = (this is because by () there is a club E of on which case(b) holds)

()3 M is not (D, 1)-homogeneous. In fact, it cannot be ( 2 )+-isomorphic to a(D, 1)-homogeneous model, because p(x) is (< )-omitted by it (by and4.19(2)).

But for each > 0 there is M Kc1, Ch(M) = , M is (D, 1)-homogeneous (by5.1(2)). So for arbitrarily large enough , there are two models of density character, which are not ( 2 )

+-isomorphic (see ()3), a contradiction to wu-categoricity.

In order to complete the proof of the main theorem, we only need to show that theconstruction above (both Case (a) and Case (b)) is possible, which is done in the

following two claims.

8.3 Claim. [(C, d) is 0+ 0-stable, compact]Let p S1D(B), B countable, A B, A is not a pseudo (< )-support for p, see

Definition 4.18. Let(x, a) be a consistent formula overA. Then there exists b Csuch thatC |= (b, a) and A b is not a pseudo (< )-support for p. In fact, it isenough to choose b such that tp(b, A) is strictly isolated andC |= (b, a).

Proof. By 4.22(2) for some b (C, a), tp(b, A) is strictly isolated. So assume

toward contradiction

()1 A b is a pseudo (< )-support for p(x).

Hence (by Definition 4.18) there are (1), (2), 1(x, b, c)

()2 1 (x, b, c1) |= p

[(2)](x) and c1 A and 1(x, b, c1) is consistent.


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As tp(b, A) is strictly isolated (see Definition 4.20), there are (3) > 0 and (y, c2)such that

()3 (i) (y, c2) tp(b, A) so c2 A

(ii) (y, c2) pseudo ((1), (3))-isolates tp(b, A), i.e. (y, c2] |=

tp(b, A)[(1)].

Let

()4 2(x, c1, c2) = (y)[(y, c2) 1(x, y, c1)].

Clearly

()5 2(x, c1, c2) is consistent.

Choose (4) such that

()6 0 < (4) < (3) and (4) < (1).

We shall now show that 2(x, c1, c2) pseudo ( (2), (4))-isolates p (and is overA), i.e.

2 (x, c1, c2) |= p

[(2)](x).

This will give a contradiction to the assumption that A is not a pseudo (< )-support for p. So assume

()7 C |=

2

[a, c1, c2].

By the definition of 2 , there are a

, c1, c2 such that

()8 (i) C |= 2[a, c1, c2]

(ii) d(a, a) (4) (1)

(iii) d(c1, c1) (4) (1)

(iv) d(c2, c2) (4) (3).

By the choice of 2, i.e. ()4, for some b

()9 (i) C |= [b, c2]

(ii) C |= 1[a, b, c1].

By the choice of (y, c2), i.e., ()3(ii) (note that as 0 < (4) (3), we have(b, c2)) there is b

C such that

()10 (i) b realizes tp(b, A)

(ii) d(b, b) (1).


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So d(a, a) (4) (1), d(b, b) (1) and d(c1, c1) (4) (1), therefore by()9(ii) we get

()11 C |=

1 [a,b

, c1].

So by ()2, replacing b with b which has the same type over A

()12 a realizes p[(2)]

so we have finished proving , hence getting the desired contradiction. 8.3

8.4 Claim. [(C, d) is 0+ 0-stable,uni-dimensional]1) Assume

(a) M = mcl(M) C

(b) Ch(M) > 0

(c) B M is countable

(d) p SmD(B)

(e) M really (< )-omits p(x), see Definition 4.18.

Then for some b C\M, also M {b} really (< )-omits p.2) Assume clauses (a)-(e) and b C, b / M, then M b really (< )-omits p(x)when: for every c >M and > 0 there is b realizing tp(b, B c) such that

d(b, g(b)M) < .

3) In clause (2) it is enough to assume that for every c >M and > 0 thereexist b, b such that d(b, b) < , d(b, M) < and b realizes tp(b, B c).

Remark. Assuming just |C| + 0 + |B| < Ch(M) suffices. Assuming Ch(M) >|B| + 20 + |C|, we can waive the assumption on 0

+ 0-stability.

Proof. 1) Let be as in 5.1(3), so () of cardinality 1. Recall

() for every uncountable (large enough if we do not assume countable) linear

order I, MI = mcl(EM(I, )) is (D, |I|)-homogeneous.

[It is (D, 1)-homogeneous and use uni-dimensionality.] Choose I = for regular, > Ch(M), and let M = M .

So without loss of generality M M. As Ch(M) > 0, we can by 8.7 belowfind > 0 and a M for < 1 such that < < 1 d(a, a) > .Let {a1+n : n < } M include B. Now for each < 1 + and n <


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we can find b,n EM(I, ) such that d(a, b,n) < 1/(n + 1). Let b,n =,n(at,n,0 , . . . , at,n,k(,n)1).

Let (for < 1)

S = {t,n, : ( < ) [1, 1 + ) and n < , < k(, n)} .

Let (for < 1, n < , < k(, n))

,n, = Min{ S {} : t,n, }.

8.5 Subclaim. Under these assumptions, there exists C, a club of 1, such that

() if C and m < then the following set is stationary

W,m = { C :for every n m, ,n = ,n, hence k(, n) = k(, n) and

(,n, = ,n,) (,n, S ,n, S) for < k(, n)}.

Proof. By a standard coding argument, there exist functions fn : 1 1 (for

n < ) such that for < 1, fn() encodes the finite sequence

,m : m n (,m,, m

,m,,

,m,) : m n, < k(, m)

where ,m, < 1 + , m,m, and

,m, are natural numbers, satisfying:

(,m,, m,m,,

,m,) is the minimal (lexicographically) triple (

, m, )such that t,m, = ,m,.

In fact, there exists such coding fn : 1 1 such that on a club Cn, fn is

regressive. Now by Fodors lemma, the following set contains a club (as its 1complement cannot contain a stationary set):

Cn = { Cn : { C

n : fn() = fn()} is stationary}.

We call this club Cn and let C =

n


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8.6 Subclaim. Under these assumptions, there exist () C such that (m min W(),m for all m; now if for all < 1m < such that = min W,m we set an easy contradiction (e.g. byFodors lemma, although it is an overkill here). 8.6

By a similar argument we can find () C such that there exists a sequencen : n < , n C, n < n+1 < () for all n and W(),n = Wn,n for all n, so inparticular m n Wm,m = Wn,m = W(),m.We obtain (by the choice of n, C , W n,m, etc.):

1 bn,m : n m is an indiscernible sequence over [Bm = B {b,i : 0, c M and a formula (x, b, c) such that](x, b, c) |= p[](x).

Choose 1, 2 > 0 such that 1 < 2 < and 1 < 2. By the assumptionthere are b, b such that

() (a) b realizes tp(b, B c)

(b) b M of length g(b)

(c) d(b, b) < 1 < 2.

By clause (a) of () as (x, b, c) |= p[](x) also (x, b, c) |= p[](x).Hence (by (c) and 3.6(3)) (x, b, c) |= p[](x), a contradiction to clause (e)of the assumptions of the Claim, that is, p being really (< )-omitted by M.

3) Similar proof to (2), first using 3.6(3) to show that (x, b, c) |= p[](x)for some . 8.4

8.7 Observation. Let (X, d) be a non-separable metric space. Then there exist

ai : i < 1 X and > 0 such that d(ai, aj)

i, j < 1.

Proof. Choose by induction on i < 1, ai such that ai / {aj : j < i}. Choose

0 < i d(ai, {aj : j < i}), i Q. Without loss of generality i = for all i, andwe are done. 8.2


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REFERENCES.

[BY03] Itay Ben-Yaacov. Positive model theory and compact abstract theories.Journal of Mathematical Logic, 3:85118, 2003.

[BY03a] Itay Ben-Yaacov. Simplicity in compact abstract theories. J. Math. Log.,3:163191, 2003.

[BY05] Itay Ben-Yaacov. Uncountable dense categoricity in cats. J. SymbolicLogic, 70:829860, 2005.

[BeUs0y] Itay Ben-Yaacov and Alexander Usvyatsov. Continuous first order logicand local stability. Transactions of the American Mathematical Society,to appear.

[ChKe66] Chen-Chung Chang and H. Jerome Keisler. Continuous Model Theory,

volume 58 ofAnnals of Mathematics Studies. Princeton University Press,Princeton, NJ, 1966.

[GrLe02] Rami Grossberg and Olivier Lessmann. Shelahs stability spectrum andhomogeneity spectrum in finite diagrams. Archive for MathematicalLogic, 41:131, 2002.

[HeIo02] C. Ward Henson and Jose Iovino. Ultraproducts in analysis. In Analysisand logic (Mons, 1997), volume 262 of London Math. Soc. Lecture NoteSer., pages 1110. Cambridge Univ. Press, Cambridge, 2002.

[Hrxz] Ehud Hrushovski. Simplicity and the Lascar Group. Preprint, 1997.[Io99] Jose Iovino. Stable Banach spaces and Banach space structures. I. Fun-

damentals. In Models, algebras, and proofs (Bogota, 1995), volume 203of Lecture Notes in Pure and Appl. Math., pages 7795. Dekker, NewYork, 1999.

[Pi00] Anand Pillay. Forking in the category of existentially closed structures.In Connections between model theory and algebraic and analytic geome-try, volume 6 of Quad. Mat., pages 2342. Dept. Math., Seconda Univ.Napoli, Caserta, 2000.

[Sh 88r] Saharon Shelah. Abstract elementary classes near 1. Chapter I.0705.4137. 0705.4137.

[Sh 3] Saharon Shelah. Finite diagrams stable in power. Annals of MathematicalLogic, 2:69118, 1970.

[Sh 54] Saharon Shelah. The lazy model-theoreticians guide to stability. Logiqueet Analyse, 18:241308, 1975.


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[Sh:c] Saharon Shelah. Classification theory and the number of nonisomorphicmodels, volume 92 ofStudies in Logic and the Foundations of Mathemat-ics. North-Holland Publishing Co., Amsterdam, xxxiv+705 pp, 1990.

saharon shelah and alexander usvyatsov- model theoretic stability and categoricity for complete...

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