john t. baldwin and saharon shelah- the stability spectrum for classes of atomic models

8/3/2019 John T. Baldwin and Saharon Shelah- The Stability spectrum for classes of atomic models

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The Stability spectrum for classes of atomic

models

John T. Baldwin

University of Illinois at Chicago

Saharon Shelah

Hebrew University of Jerusalem

Rutgers University

April 6, 2010

Abstract

We prove two results on the stability spectrum for L1, . Here Smi (M) de-

notes an appropriate notion (at or mod) of Stone space ofm-types over M. The-orem A. (unstable case) Suppose that for some positive integer m and for every

< (T), there is an M K with |Smi (M)| > |M|(|T|). Then for ev-

ery |T|, there is an M with |Smi (M)| > |M| = . Theorem B. (strictlystable case) Suppose that for every < (T), there is M K such that = |M| and |S

mi (M)| > . Then for any with

0 > , K

is not i-stable in . These results provide a new kind of sufficient condition for

the unstable case and shed some light on the spectrum of strictly stable theories

in this context. The methods avoid the use of compactness in the theory under

study. In the Section 4, we expound the construction of tree indiscernibles for sen-

tences ofL1, . Further we provide some context for a number of variants on the

Ehrenfeucht-Mostowski construction.

1 Context

For many purposes, e.g., the study of categoricity in power, the class of models of a

sentence of L1, can be profitably translated to the study of the class of models ofa first order theory T that omit a collection of first order types over the empty set.

In particular, if is complete (i.e. a Scott sentence) can be taken as the collectionof all non-principal types and the study is of the atomic models ofT. This translationdates from the 60s; it is described in detail in Chapter 6 of [Bal09]. The study of finite

We give special thanks to the Mittag-Leffler Institute where this research was conducted. This is pa-

per 959 in Shelahs bibliography. Baldwin was partially supported by NSF-0500841. Shelah thanks the

Binational Science Foundation for partial support of this research.

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diagrams (see below) is equivalent to studying sentences ofL1,; the study of atomic

models of a first order theory is equivalent to studying complete sentences ofL1,.The stability hierarchy provides a crucial tool for first order model theory. She-lah [She78] and Keisler [Kei76] show the function fT() = sup{|S(M)| : |M| =, M |= T} has essentially only six possible behaviors (four under GCH). In [She70],Shelah establishes a similar result for homogeneous finite diagrams. The homogene-

ity assumption is tantamount to assuming amalgamation over all sets. This is a strong

hypothesis that is avoided in Shelahs further investigation of categoricity in L1,([She83a, She83b]), which is expounded as Part IV of [Bal09]. Important examples,

due to Marcus and Zilber, which do not satisfy the homogeneity hypothesis are also

described in [Bal09]. As we explain below, this investigation begins by identifying the

appropriate notion of type over a set (and thus of-stability). Shelah [She83a, Bal09]showed that -stability implies stability in all powers. And assuming 20 < 21 ,-stability was deduced from 1-categoricity. But further questions concerning the

stability hierarchy for this notion of type for arbitrary sentences of L1, had not beeninvestigated. We do so now. In fact our results hold for arbitrary finite diagrams, the

class of models of first order theory that omit a given set of types over the empty set.

But our results are by no means as complete as in homogeneous case.

There are (at least) two a priori reasonable notions of Stone space for studying

atomic models of a first order theory. (As noted, we could more generally replace

atomic by finite diagram.) Recall that for a first order theory T (with a monstermodel M) A M is an atomic set if each finite sequence from A realizes a principaltype over the empty set. An atomic set is an atomic model if it is also a model of the

theory T.

Definition 1.1 LetK be the class of atomic models of a complete first order theory.

1. LetA be an atomic set; Sat(A) is the collection ofp S(A) such that ifa Mrealizes p, Aa is atomic.

2. Let A be an atomic set; Smod(A) is the collection of p S(A) such that p isrealized in some M Kwith A M.

In [Bal09] we wrote S for the notion called Smod here. The latter notation is moreevocative. We will simultaneously develop the results for both notions of Stone space

and indicate the changes required to deal with the two cases. We will write Si(M)where i can be either at or mod.

We sometimes write |T| for || where is the vocabulary of T. K = KT is theclass of atomic models of T. We write H = H() for the Hanf number for atomicmodels of all theories with |T| = . By [She78] H equals (T), where (T), thewell-ordering number of the class of models of a theory T omitting a family of types,is defined in VII.5 of [She78]. It is also shown there that ifT is countable, H evaluatesas 1 while for uncountable T H = (2|T|)+ . Fix = (|T|).

Remark 1.2 In [She70], Shelahs definition of stability makes a stronger requirement;

it implies by definition the existence of homogeneous models in certain cardinals. We

do not make that assumption here so we are considering a larger class of theories.

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Definition 1.3 1. K is i-stable in (for i = at or mod) if for every m < , and

M K

with |M| = , |S

m

i (M)| = .2. Stability classes. For eitheri = at ormod,

(a) K is i-stable if it is i-stable in some .

(b) K is i-superstable if it is i-stable in all H.

(c) K is strictly i-stable if it is i-stable but noti-superstable.

For any M, Sat(M) contains Smod(M) so at-stability in implies mod-stabilityin . Thus for both notions -stability implies stability in all powers by results of[She83a, She83b], expounded in [Bal09].

We prove Theorem A in Section 2 and Theorem B in Section 3. The proof of

Theorem B uses an application of omitting types in Ehrenfeucht-Mostowski models

generated by trees of the form |M|

(|T|). Then for every |T|, there is an M

with |Smi (M)| > |M| = .

Remark 2.2 (Proof Sketch) Before the formal proof we outline the argument. We

start with a sequence of models M and many distinct types over each of them. Byan argument which is completely uniform in , we construct triples a,i,b,i,d,ifor i < + with the a,i,b,i M and d,i in an elementary extension M

of

M of the same cardinality and so that Md,i is atomic and the distinctness of thetypes of the d,i is explicitly realized by formulas. Then we apply Morleys omitting

types theorem to the M and extract from this sequence a countable sequence of orderindiscernibles with desirable properties. Finally, this set of indiscernibles easily yields

models of all cardinalities with the required properties.

Remark 2.3 The idea of the proof can be seen by ignoring the and proving a slightlyweaker result from one model of size (T).

Notation 2.4 = |M|+2(|T|); = (|T|); = +2(|T|).

Lemma 2.5 There is , proper for linear orders, in a vocabulary extending with|| = ||, with fixed additional unary predicates P, P1 and binary R such that:

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1. For every linear ordering I, NI = EM(I, ) |= T and MI = EM(I, )

P K

. Naturally, J I implies NJ NI andMJ MI.2. The skeleton ofNI is aibici : i I andlg(ci) = m.3. For some first order:

NI |= ((ct,as) (ct,bs)) iffs = |M|

+2(|T|). Fix p,i fori < + , a list of distinct types in S

mi (M). We work throughout in a monster model

M ofT.

Notation 2.6 In the following construction, we choose by induction triples

a,i,b,i,d,i for i < + . We use the following notation for initial segments ofthe sequences.

1. D,i = {d,j : j < i}.

2.X,i = {

a,j,

b,j : j < i}

.

3. q,i is the type ofd,i overX,i.

The following variant on splitting is crucial to carry out the construction. We call it

ex-splitting (for external) because the elements which exemplify splitting are required

to satisfy the same type over a set D which is not in (so external to) the model M and,in particular, is not required to be realized in an atomic set.

Definition 2.7 Let M be a model, X M and D M. We say that p Smi (M)ex-splits over(D, X) if there exista,b M, f M so thatf realizes p X, a D bbut(a, f) and(b, f) realize different types over.

We will apply the next claim to M, X,i, and D,i when carrying out the con-struction in paragraph 2.10. Note that this computation does not depend on |M|.

Claim 2.8 For any model M, the number of types in Smi (M) that do not ex-split overa pair(D, X) with |X| = |D| is at most+2.

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Proof. Let P denote the collection of tp(e/M) with lg(e) = m that do not ex-

split over a pair (D, X). Each type r in P is determined by knowing r X and foreach formula i(x1, . . . xki) for i < |T| the restriction of r to one ki-tuple from eachequivalence class of the equivalence relation Ek on M defined by aEkib ifa and brealize the same ki-type over D. So, since |D| = , there are at most

2 (22|D|

)|T| = (22

)|T| = +2

possible such r. 2.8

As noted, for each M we will be constructing by induction on i < + , setsX,i, D,i of cardinality . We need to choose in advance a type p which does notex-split over any (X,i, D,i) that arises. In order to do that we restrict the source ofD,i; clearly X,i M. That is, we will fix M with M M

, |M

| = and

M

is +

-saturated and choose D,i

M

. (Note then that M

is not in general

atomic.)

The number of types in Smi (M) that do not ex-split over any pair (D, X) with|X| = |D| = is bounded by the number of such sets, |M

| , times the number

of types in Smi (M) that do not ex-split over a particular choice of (D, X), which is+2 by Claim 2.8. That is, the bound is |M|

+2. Since this number is lessthan + , we can fix a type p S

Mi (M) which does not ex-split over any of the

relevant (D, X).

Definition 2.9 For each < (T) , fix M with M M, |M

| = , andM

is

+ saturated. Choose, by induction on i < +, triples e,i = a,i,b,i,d,i where

a) d,i M.

b) a,i,b,i are sequences of the same length from M that realize the same typeoverD,i = {d,j : j < i}.

c) The types over the empty set of(a,i,d,i) and(b,i,d,i) differ.

d) q,i = p X,i = tp(d,i/X,i) so ifj < i, q,j q,i.

e) Mdi is an atomic set for each i. (In the mod-version N,i is an atomic modelcontaining Mdi .)

Construction 2.10 Choose d,i to realize p X,i. By Claim 2.8 and since|Si(M)| > we can choose a,i and b,i to satisfy conditions b) and c). So wehave

tp(d,i,a,j) = tp(d,i,b,j) if and only ifi < j. (1)We want this order condition for a single formula. For each i < + , the types of

(a,i,d,i) and (b,i,d,i) differ. That is, ,i(a,i,d,i) and ,i(b,i,d,i) forsome ,i. By the pigeon-hole principal we may assume the ,i is always the same. (Further, since |T| is not cofinal in (T), we can assume the is the same forall .)

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Now the construction is completed. We expand to a language by adding

predicates P,


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clearly satisfy all the conditions of the requirements in Lemma 2.5 and we complete

the proof. 2.5Proof of Theorem 2.1: To show instability in , let I be a dense linear ordering

with cardinality and choose J I, that realizes more than |I| cuts over I. ThenEM(J, ) realizes more than types in Smi (P(EM(I, )). To see this, considerfor any cut in I realized by an element j J the type:

{(ai,bi, ci,x, ) : i < j} {(ai,bi, ci,x, ) : i j}.

Then aj ,bj , cj realizes the type in EM(J, ) and P(EM(J, ))cj is an atomicset since Q was omitted. For the mod-case, use the interpretation ofR to define N,i.2.1

Question 2.13 Must an atomic class that is unstable in all have the order property?

We say a class of atomic models has the order property if there is a sequence as in

Lemma 2.5.3 but with the set of all the sequences contained in atomic set. Condition

3) only requires each triple to be atomic. In particular we dont know the various cscan appear together in any atomic model.

3 Strictly stable case

As the following examples show, it is easy to have superstable (incomplete) sentences

of L1, that are not superstable for some values below the Hanf number H. Thefollowing theorem has two easily stated corollaries. IfK is not superstable then it is

not stable in every with

> . IfK is superstable then it is stable in some < H.The results here are related to those in [GS86] but the combinatorics here is con-siderably simpler than in [GS86] for two related reasons. First, we construct tree indis-

cernibles indexed by


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Theorem 3.3 Suppose that for every < (T) , there is M K such that =

|M

| andS

m

i (M

) > . Then for any with

0

> ,K

is not stable in .

Proof. Fix for each < (T), M K such that |Smi (M)| > . Fix p,i for

i < + , a list of distinct types in Smi (M). We work throughout in a monster model

M ofT.To prepare for the application of an appropriate version of Morleys omitting types

theorem we construct a sequence of models and certain types. For this, we construct

trees of types that arise from failure of stability. The combinatorics slightly extends

the classical arguments and avoids compactness. Note that this stage of the construc-

tion takes place in the original language. We will apply the following general result

uniformly to each M.

Fact 3.4 Suppose |M| +1 and P is a collection of > = |M| members of

Smi (M). Then there exists a sequence bj :j < with each bj M and a formula

(x,y) = P such that for each j < ,

|{p P: i < j (x,bi) p but (x,bj) p}| > . (2)

Proof. We consider many possibilities for and prove one works. We choose{ : Ti} by induction on i < where each Ti is a subset of i2 and eachb M so that

1. j < i and Ti implies j Tj .

2. if Ti then p = {j(x,bj)(j) :j < i} is included in > members of

P.

3. For limit i,

Ti = { i2 : (j < i) j Tj and p is included in > members of P}

4. ifi = j + 1 then Ti = {0, 1 : Tj}.For the successor step in the induction recall the following crucial observation of

Morley. Suppose there are more than |M| types over M extending a partial type p.Then there exists a formula (x,a) with a M such that both p {(x,a) andp {(x,a)} have more than |M| extensions to complete types over M. (We areextending Morleys analysis to types in Smi (M) but the argument is just counting; thereis a unique type which has more than extensions.)

The interesting point in the induction is the limit stage. We cannot guarantee that

individual paths survive. But at each stage in the induction, we have defined types over

a set of cardinality . So there are at most +1 types over {b :lg() < }. So oneof the paths must have more than extensions to Smi (M).

So T = . Choose T . Let j(x,bj) = j(x,bj)(j) for j < .

Since the path has length = (T), by the pigeonhole principle we may assumethere is a single formula . This completes the construction of the and the bj . Wehave the result by condition 4. 3.4

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Now we apply this fact to construct from the original M given in the hypothesis

of Theorem 3.3 a sequence of models

M

and associated sequences b, and c, for extensions to Smx (M). Let P = {r Sx(M) : q r} so |P| > .

By Fact 3.4, we find b,bj :j < and satisfying displayed statement 2.Let Mn+1 be a submodel of M

with Mn {b, : n+1()} M

n+1 and

with cardinality . Mn+1 M so is an atomic model and each q

extends to an

atomic type over M.For n() and i < first define

pbi = q {(x,b,bj) :j < i} { (x,b,bi)}.

Since < |{r Sx(M) : p r}|, we can find pbi S

mx (Mn ) extending p

bi

such that Pbi = {r Sx(M) :pbi r} has cardinality > . Note that

pbi q {(x,b,bj) :j < i} { (x,b,bi)}.

This completes the n + 1st stage of the construction. So we can construct theMn and {q,i :


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With the construction complete, we expand to a language in two stages.

Form

by adding predicates P, Pn, < , then there is an I with |I| = such that Smat(MI) > , where

MI = EM(I, ) P.

Proof. Note that by displayed statement 3 and Claim 3.7 we have:

1. If, i I, tp(Fn()/Pn(M)) tp(Fn+1(i)/Pn+1(M)).2. If I and i = j,

tp(Fn+1(j)/Pn+1(M)) = tp(Fn+1(i)/Pn+1(M)).Now in any MI = EM(I, ) for any J define p S

mat(PN(MI) =

tp(Fn(), Pn(M)). Now letting p Smat(P(MI)) bei then there is an I with |I| = such thatSmmod(MI) > , whereMI = EM(I, ) P.

Proof. We need to construct an atomic model N containing MIc (from the proofof Claim 3.8). The natural choice is the -Skolem closure of MIc. The reasonthe reduct of this structure to is atomic is that any finite sequence is of the form a,bwhere thea come from Pn(MI) (for a fixed n) and each of the b has the form G(a, c)where G is a -Skolem function. But then the type ofab is the same as the -type ofa sequence ab where a Pn(M) and each b N, is of the form G(a, cn).

3.9

Remark 3.10 We investigate the difference in hypotheses between Theorem 2.1 andTheorem 3.32. We first study Theorem 2.1.

Let = |M|.Case 1. : then is equal to = 2

= +1. The assumptionof the theorem is that |Smi (M)| >

= 2 . This case is not possible since|Smi (M)| 2

2 .Case 2. > . On one hand wehave ; on the other+1 =

.

Thus, max(+1, ). The hypothesis in the theorem says that |Smi (M)| >

, so |Smi (M)| > max(+1, ).This leads to two cases:

Case 2a. +1: then |Smi (M)| > max(+1,

) . So the requirementis at least instability in .

Case 2b. < max(+1,

) =+1 > .This yields instability in . (Under GCH, of course, this case is empty.)

In general, the hypothesis in case 2a) requires more than instability in : if hascofinality less than or equal to the cofinality of, then > , and the number oftypes needs to be (possibly) much greater than .

2This analysis was worked out by the first author and Alexei Kolesnikov.

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Theorem 3.3 asserts that K is unstable in some cardinal then it is unstable in any

with

> so it is analogous to the first order case. Further it asserts that the firststability cardinal for a superstable class is less than H.Thus, in Theorem 2.1 we assume serious instability and get instability everywhere

and in Theorem 3.3 we assume just instability, and get instability for cardinals of

countable cofinality only.

We further analyze case 2a under GCH. The possible values of , given that > +1, become and 2

= + (the first is the case when the cofinality of isgreater than the cofinality of; otherwise, the second alternative holds).

Under the GCH the difference between serious and just instability disappears.

Moreover, we can expect to find M satisfying the hypothesis only for |M| of cofi-nality greater than the cofinality of. So under the GCH, the difference between the

hypotheses in 2.1 and 3.2 disappears, but the conclusion of 3.2 is weaker.

4 Tree Indiscernibility

The main result of this section is the existence of tree indiscernibles as needed in the

previous section. But we take the occasion to discuss the role of various types of

index sets for indiscernible collections and to make explicit the role of expanding the

vocabulary when finding indiscernibles in various contexts.

The theorem reported here is implicit in the literature (e.g. [She78, GS86]) but we

could not find an explicit statement. Theorem VII.3.6 of [She78] finds an indiscernible

tree in the first order case on but we want to omit types as well. The basic plan ofthe proof dates to Morley [Mor65]. We indicate the modifications needed for the more

complicated combinatorics to build models to omit types that are over indiscernible

trees instead of over linear orders.

Many variants of tree indiscernibles are used in various parts of model theory; we

sketch the contexts to point out where the current version lies.

linear order


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the difference in stability spectrum for stable but not superstable theories, we want (3)

the first row, third column. But to count the number of models of superstable theory in-volves (4) trees of height +1. The proof here differs from [She78], where the numberof models of an unsuperstable theory is computed, because in working with L1,, wemust omit types. In VII.3.6 of [She78], Erdos-Rado is applied to show the existence of

a uniform -tree implies the existence of a tree of indiscernibles indexed by


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We rely on a combinatorial lemma that follows from Erdos-Rado. The result is

proved as Theorem 2.6 in the appendix to [She78]. A stronger result (the bound onk(m, n) is smaller) with a shorter proof appears in the appendix of [GS86].

Lemma 4.3 ([She78]) For every n, m < , there is a k = k(n, m) < such that if = k()+ the following is true. For any function f : [n]m , there exists aT n such that

1. Each T has + immediate successors in T.

2. If and are m-tuples from T with atp(/) = atp(/), then

f() = f().

We now prove the theorem on the existence of tree-indiscernibles. In order to be

clear about the definability of the tree in the original vocabulary we extend Notation 4.1

and are quite pedantic about the vocabularies involved.

Notation 4.4 1. includes both and and includes Skolem functions for ,

where the Skolem axioms and relations with crucial -formulas are axiomatizedin a -theory T1.

2. The set of constants C which guarantee the consistency of the order are addedto .

3. i denotes the set of {Pj :j > i} with at mosti free variables.

Tree-indiscernibles are a special case of generalized indiscernibility as defined in

VII.2 of [She78]. Indiscernibles indexed by other types of structure appear for exam-

ple in [LS03, D04, Sco]. The following notion of

modeling property, based on oneintroduced by Scow[Sco] in a slightly different context is helpful for stating the results

here. The point is that although the type of an infinite collection of indiscernibles may

not be realized in any of the input models, each type of a finite subsequence is. Thus

properties of finite character (such as realizing a finite type) follow immediately if the

indiscernibles have the modeling property. We use for isomorphic.

Definition 4.5 A collection of -tree-indiscernibles B = {b : T} has the mod-eling property if it is derived from a sequence (M, X) (where M X = {a : T}, andT T for < H) such that for every finite sequence from T andevery sequence b from B and some there is a sequence a X with havingthe same -type as and such thata andb have the same -type.

Note that in the argument below when the X are refined using Lemma 4.3 a tuplea Xi,n was originally named a X

0+m,n (where m

< can be easilycomputed). But, and realize the same -type.

Remark 4.6 There are at least four approaches to the proof of Morleys omitting types

theorem that differ subtly. In [CK73, Mar02]3 the language is countable and there are

3Compare comments on the proof in [CK73]. The stated result is the existence of large models omitting

types without mentioning indiscernibility.

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separate steps to guarantee indiscernibility and omission of the types (meeting indis-

cernibility type omission requirements in turn for each formula and for each type). Inthe argument here, we use the Skolemization of the models M to deduce the omissionof types from the indiscernibility. This argument strategy is forced because in dealing

with uncountable languages, working with one formula at each step makes the induc-

tion too long. We replace this by working with all formulas with n free variables at stepn. The arguments in [She78, GL02, GS86] employ nonstandard-models of set theory.Finally, the arguments in [Hod87, Kei71], work directly in infinitary logic using Hin-

tikka sets or consistency properties. The arguments of [She78, GS86, Hod87] make the

connection with well-ordering numbers explicit. Tsuboi [Tsu08] shows that a family

of < 20 complete types that is omitted up to 1 can be omitted in arbitrarily largemodels; this argument introduces some new combinatorial ideas.

Recall that = (|T|). Writing rather than and considering M for < (T) = (2|T|)+ is part of the price for dealing with uncountable T.

Note that when applying this theorem in Section 3, the M here are the M+ (asSkolemized) there.

Theorem 4.7 Let T1 be a theory with Skolem functions in a vocabulary . Supposefor < (T), there exists a model M of T1 with |M| such that M omits afamily of-types. contains the vocabulary

andX is a set of elements in Mthat form a tree of type


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Let T


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[Bal09] John T. Baldwin. Categoricity. Number 51 in University Lecture Notes.

American Mathematical Society, 2009. www.math.uic.edu/ jbaldwin.[CK73] C.C. Chang and H.J Keisler. Model theory. North-Holland, 1973. 3rd edition

1990.

[D04] Dzamonja+Shelah. On


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[She74] S. Shelah. Categoricity of uncountable theories. In Proc Tarski Symposium

(Univ. of California, Berkeley, Calif., 1971), Proceedings of Symposia inPure Mathematics XXV, pages 187203. American Mathematical Society,

1974.

[She75] S. Shelah. A two-cardinal theorem. Proc American Math Soc, 48:207213,

1975.

[She76] S. Shelah. A two-cardinal theorem and a combinatorial theorem. Proc Amer-

ican Math Soc, 62:134136, 1976.

[She78] S. Shelah. Classification Theory and the Number of Nonisomorphic Models.

North-Holland, 1978.

[She83a] S. Shelah. Classification theory for nonelementary classes. I. the number of

uncountable models of L1 part A. Israel Journal of Mathematics,46:3:212240, 1983. paper 87a.

[She83b] S. Shelah. Classification theory for nonelementary classes. I. the number of

uncountable models of L1 part B. Israel Journal of Mathematics,46;3:241271, 1983. paper 87b.

[Tsu08] Akito Tsuboi. Models omitting given complete types. Notre Dame Journal

of Formal Logic, 49:393399, 2008.

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john t. baldwin and saharon shelah- the stability spectrum for classes of atomic models

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