U.U.D.M. Project Report 2018:12

Examensarbete i matematik, 15 hpHandledare: Vera Koponen Examinator: Martin HerschendJuni 2018

Department of MathematicsUppsala University

Uncountable categoricity

Henrik Jonasson

Uncountable categoricity

Henrik Jonasson

June 16, 2018

1 IntroductionModel theory is the study of the models of a mathematical theory T , in termsof the properties of T . Model theory also provides a variety of classifications oftheories T , based on how their models behave.

One important such classification is κ-categoricity. A model is said to havecardinality (or synonymously, power) κ if the underlying set has cardinality(power) κ. A theory T is called κ-categorical (or categorical in κ) if T has amodel of cardinality κ, and all models of T of cardinality κ are isomorphic -that is if T determines an essentially unique structure of size κ in which T istrue. Model theory then asks about the consequence of such a property on themodels of T .

The object of this text is to present the proof of one important consequenceof κ-categoricity, namely Morley’s Categoricity Theorem, which states that if Tis a complete theory in a countable vocabulary, and T is categorical in someuncountable power κ, then T is categorical in every uncountable power.

The proof follows the one given in [1], but with a more thorough expositionof the arguments and an attempt to explain all the details, which will hopefullymake the ideas of the proof more accessible.

It is worth mentioning that the proof in [1] is not the original proof presentedby Morley in [4], but an easier one by Baldwin and Lachlan. Morley’s result isan interesting fact in itself, but the original proof also introduced ideas aboutstable theories which were expanded by Shelah to develop a new field in modeltheory, namely stability theory.

There are actually not many theories occurring naturally in mathematicswhich are known to be categorical in some uncountable power, but the followingare some examples of such theories, which by Morley’s theorem are categoricalin all uncountable powers:

• For any countable division ring K, the theory of vector spaces over K.

• The theory of torsion-free divisible Abelian groups.

• The theory of infinite Abelian groups where all elements have order p(where p is a prime).

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• The theory of algebraically closed fields of characteristic 0 or p.

• The complete theory of the natural numbers with the successor function.

2 BackgroundWe will recall some basic definitions in mathematical logic and model theory,and state theorems which will be needed in the proof of Morley’s Theorem. Theproofs belong in an under graduate course and can be found in an introductionon the subject, like [2], or in the basic chapters of a more advanced treatment,like [1].

We will work in first-order logic, which uses the logical symbols ∧, ∨, ¬, →↔, (, ), ∃ and ∀, which are interpreted as usual, and a set of proof-rules, whichcan be said to be those we use in ordinary mathematics.

A vocabulary V is a set of constant symbols, relation symbols, and functionsymbols.

A V-structureM is an underlying set M , called the universe or domain ofM, with an interpretation of every constant symbol c, relation symbol R (ofarity k) and function symbol f (of arity l) as an element cM ∈ M , a relationR ⊂Mk and a function f : M l →M , respectively.

Introducing a set of variables v0, v1, . . . we can construct V-formulas as fol-lows:

• V-terms: Every constant symbol in V and variable vi is a V-term. Ift1, . . . , tk are V-terms and f ∈ V a k-ary function symbol, then f(t1, . . . , tk)is a V-term.

• Atomic V-formulas: If t1 and t2 are V-terms, then t1 ≡ t2 is an atomicformula. If R is a k-ary relation symbol and t1, . . . , tk are terms, thenR(t1, . . . , tk) is an atomic formula.

• V-formulas: Every atomic V-formula is a V-formula. If φ and ψ are for-mulas, and � ∈ {∧,∨,→,↔}, then ¬φ and φ�ψ are formulas. If φ is aformula and v a variable, then (∃v)φ and (∀v)φ are formulas.

We write φ(v1, . . . , vn) if the free variables of the formula φ are amongv1, . . . , vn. By φ(c) we mean the formula obtained by replacing every free oc-currence of x in φ(x) by the constant symbol c. If a V-formula contains no freevariables, it is called a V-sentence.

Now given a V-structure M and a V-sentence φ, we write M |= φ, or saythatM is a model of φ, if φ is true inM. Given a set Γ of sentences, we writethatM |= Γ ifM |= φ for each sentence φ ∈ Γ. IfM |= Γ implies thatM |= φ,we write Γ |= φ. If φ can be deduced from Γ using the proof rules of first-orderlogic, we write Γ ` φ.

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Theorem 1 (Soundness of first order logic). If Γ ` φ, then Γ |= φ.

Theorem 2 (Completeness of first order logic). If Γ |= φ, then Γ ` φ.

A set of V-sentences Γ is said to be inconsistent if Γ ` φ ∧ ¬φ for somesentence φ. Otherwise it is called consistent. Γ is maximal consistent if everyset of sentences Σ such that Γ ⊂ Σ is inconsistent. We call Γ complete if forevery V-formula φ, either Γ ` φ or Γ ` ¬φ.

Theorem 3 (Model existence theorem). A set of sentences Γ has a model ifand only if Γ is consistent.

Theorem 4 (Compactness theorem). A set of sentences Γ has a model if andonly if every finite subset of Γ has a model.

If a set of sentences is consistent, it is called a theory, usually denoted T .The theory Th(M) of a modelM is the set of all sentences φ such thatM |= φ.

Let M be a V-structure and N a W-structure. Suppose V ⊂ W, then wecall N an expansion ofM (orM a reduct of N ) if they have the same universeand the interpretation of every symbol in V is the same in both. Given a subsetA ⊂ M we denote the expansion of M to the vocabulary V ∪ {ca : a ∈ A} asMA (sometimes (M, a)a∈A, or if A is enumerated by ordinals, (M, aα)α<κ). Ifφ(v1, . . . , vn) is a formula and (a1, . . . , an) ∈M , then we byM |= φ(a1, . . . , an)mean thatMM |= φ(ca1 , . . . , can).

The theory Th(MM ) is called the elementary diagram ofM.The following useful fact is called the "Lemma on constants", and can be

found on page 43 in [3].

Lemma 5. Let T be a V-theory, φ(v1, . . . , vn) a V-formula and c1, . . . , cn asequence of constants which do not occur in V. If T |= φ(c1, . . . , cn), thenT |= (∀x1, . . . , xn)φ(x1, . . . , xn).

LetM andN be V-structures. Then they are elementary equivalent, denotedM≡ N , if Th(M) = Th(N ).

They are said to be isomorphic, denoted M ∼= N , if there is a bijectivefunction f : M → N such that for every constant symbol c, relation sym-bol R (of arity k) and function symbol g (of arity l) in V, we have thatf(cM) = cN , (a1, . . . , ak) ∈ RM if and only if (f(a1), . . . , f(ak)) ∈ RN andf(gM(a1, . . . , al)) = gN (f(a1), . . . , f(al)).M is said to be a substructure of N (or N an extension ofM), denotedM⊂

N ifM ⊂ N and for every constant symbol c, relation symbol R (of arity k) andfunction symbol f (of arity l) in V we have that cM = cN , RM = RN ∩Mk andfor all (a1, . . . , al) ∈M l, fM(a1, . . . , al) = fN (a1, . . . , al). (M=N andM = Nis allowed.)M is said to be an elementary substructure of N , denoted M ≺ N , if

M⊂ N and for every V-formula φ(v1, . . . , vn) and every (a1, . . . , an) ∈Mk, wehave thatM |= φ(a1, . . . , an) ⇔ N |= φ(a1, . . . , an). The following criterion isoften useful:

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Proposition 6 (The Tarski-Vaught criterion). Let M and N be V-structuressuch thatM⊂ N . ThenM≺ N if and only if for every V-formula φ(x1, . . . , xn,y) and every (a1, . . . , an) ∈Mn the following holds:

N |= ∃yφ(a1, . . . , an, y)⇒ N |= φ(a1, . . . , an, b) for some b ∈M.

Recall that a partial ordering ≤ on a set M is an binary relation such thatfor all x, y, z ∈M : x ≤ x, (x ≤ y∧y ≤ z)⇒ x ≤ z and (x ≤ y∧y ≤ x)⇒ x = y.A linear ordering is a partial order such that (∀x, y ∈ M)(x ≤ y ∨ y ≤ x). Awell ordering is a linear ordering such that for every subset X ⊂ M there is aleast element with respect to ≤. The well ordering principle, which is equivalentto the axiom of choice, tells us that every set can be well ordered.

An ordinal is a set α such that for every a ∈ α, a ⊂ α and α is well orderedby ’∈’ as a relation. The class of ordinals can be shown to be strictly wellordered by ’∈’ (α 6∈ α), and by α < β we mean that α ∈ β. The successorof an ordinal α is the ordinal α ∪ {α}. An ordinal β which is not a successorordinal, i.e (∀α ∈ β)(α ∪ {α} ∈ β), is called a limit ordinal. It can be shownthat all ordinals are either the empty set, a successor ordinal or a limit ordinal.Furthermore, induction can be carried out over ordinals similarly as over thenatural numbers:

Proposition 7 (Transfinite induction). Let P (α) be a property of ordinals.Assume that for every ordinal β, if P (γ) holds for each γ < β, then P (β) holds.Then P (α) holds for every ordinal α.

A cardinal κ = α is an ordinal such that for each β < α, there is no injectivefunction from α to β. For each set A, there is a unique cardinal such that thereis a bijective function from A to κ. This κ is called the cardinality of A, and itis denoted |A|.

If either κ or λ is an infinite cardinal, then κ+ λ = max{κ, λ}, and κ · λ =max{κ, λ}.

The least cardinal larger than κ is denoted by κ+. The least infinite cardinalis ω, and ω1 = ω+ denotes the least uncountable cardinal.

We define a cardinal to be regular if for every set A ⊂ κ, if |A| < κ, thensup(A) < κ.

Proposition 8. Any infinite successor cardinal κ+ is regular.

The power of a V-structure M is the cardinality of the universe M . Asmentioned before, a theory T is κ-categorical if all models of T of power κare isomorphic. We conclude the background with the following importanttheorems:

Theorem 9 (Upward Löwenheim-Skolem theorem). If a theory T has a modelof infinite power, then T has models of arbitrarily large powers.

If T is taken to be Th(MM ) for a modelM, the theorem yields that everymodel has arbitrarily large elementary extensions.

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Theorem 10 (Downward Löwenheim-Skolem theorem). LetM be a V-structureand A ⊂ M . Then there is a an elementary substructure N ≺ M such thatA ⊂ N and |N | ≤ ω + |V |+ |A|

3 TypesDefinition. We say that a model M realizes a set Σ(v1, . . . , vn) of formulasσ(v1, . . . , vn) if there is an n-tuple a1, . . . , an ofM such thatM |= σ(a1, . . . , an)for each σ ∈ Σ. IfM does not realize Σ, thenM is said to omit Σ.

Definition. A set Σ(v1, . . . , vn) of V-formulas in the variables v1, . . . , vn is saidto be a type if it is maximal consistent.

Note that given any model M and any n-tuple a1, . . . , an ∈ M , the setΣ(v1, . . . , vn) of all formulas σ(v1, . . . , vn) satisfied by a1, . . . , an is a type. Be-cause given any formula σ(v1, . . . , vn), either a1, . . . , an satisfies or does notsatisfy σ, hence σ ∈ Σ or ¬σ ∈ Σ. That is, any Γ such that Σ is a proper subsetof Γ must contain both σ and ¬σ for some formula σ, hence it is inconsistentand Σ is maximal consistent. Furthermore, this is the unique type realized bya1, . . . , an, for if Σ′ is another type realized by a1, . . . , an, then all σ ∈ Σ′ is alsoin Σ, and Σ′ ⊂ Σ. As Σ′ is maximal consistent and Σ is not inconsistent, Σ′

can’t be a proper subset of Σ and Σ′ = Σ.

Definition. A formula σ(v1, . . . , vn) is consistent with a theory T if there is amodelM of T which realizes σ.

Definition. Let Σ(v1, . . . , vn) be a set of formulas in a vocabulary V. A V-theory T is said to locally realize Σ if there is a V-formula φ(v1, . . . , vn) suchthat:

(1). φ is consistent with T .

(2). For all σ ∈ Σ, T |= φ→ σ

That is, in every model of T , an n-tuple which satisfies φ realizes Σ.T is said to locally omit Σ if T does not locally realize Σ, that is, for every

formula φ(v1, . . . , vn) which is consistent with T , there exists a σ ∈ Σ such that(φ ∧ ¬σ) is consistent with T .

To limit the exposition on types, we state the following proposition withouta proof; it can be read in [2] on page 271.

Proposition 11. Let T be a complete theory and let M |= T , then each typeconsistent with T is realized in some elementary extension of M .

The following consequence of Proposition 11 will be of use later. Let T =Th(M). Let SM be the set of all types Σ(x) consistent with Th(M) and indexthem as Σ(x)γ , γ < δ = |SM|. By the proposition there is a N0 such thatM ≺ N0 and Σ(x)0 is realized in N0. By N0 being an elementary extension

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ofM, Th(N0) = Th(M) hence SM = SN0and there is a model N1 such that

N0 ≺ N1 and Σ(x)1 is realized in N1. Inductively, there is a model Nδ suchthat M ≺ Nδ and Nδ realizes Σ(x)γ for all γ < δ, hence realizes every typeconsistent with Th(M).

Theorem 12 (Omitting types theorem). Let T be a consistent theory in acountable vocabulary V and Σ(x1, . . . , xn) be a set of formulas. If T locallyomits Σ, then T has a countable model which omits Σ.

Proof. To simplify notation, let Σ(x) be a set of formulas in one variable. Theproof is completely analogous in the general case Σ(x1, . . . , xn). Suppose Tlocally omits Σ(x). Let C = {co, c1, . . .} be a countable set of constant symbolsnot already in V and define V ′ = V ∪ C, which will be countable. Hence thereare countably many V ′-sentences, and we can enumerate them as φ0, φ1, . . .. Wewill now construct an increasing sequence of theories

T = T0 ⊂ T1 ⊂ . . . ⊂ Tm ⊂ . . .

which satisfy:

(1). Each Tm is a consistent theory of V such that Tm \ T is finite.

(2). Either φm ∈ Tm+1 or ¬φm ∈ Tm+1.

(3). If φm = (∃x)ψ(x) and φm ∈ Tm+1, then ψ(cp) ∈ Tm+1, where cp is thefirst constant not occuring in Tm or φm.

(4). There is a formula σ(x) ∈ Σ(x) such that (¬σ(cm)) ∈ Tm+1.

Given a theory Tm = T ∪ {θ1, . . . , θr} a finite extension of T , we constructTm+1 as follows. Let θ = (θ1 ∧ . . . ∧ θr) and n be such that all (finitely many)constants from C occurring in θ is among c1, . . . , cn. Then replacing each ciin θ by xi yields a V-formula θ(x1, . . . , xn) which is consistent with T , as anymodel of Tm is a model of T which realizes θ. As T locally omits Σ(x), there isa σ(x) ∈ Σ(x) such that (θ ∧¬σ) is consistent with T . By putting the sentence¬σ(cm) in Tm+1, (4) holds. If φm is consistent with Tm ∪ {¬σ(cm)}, put φmin Tm+1. Otherwise, put ¬φm in Tm+1. Then Tm+1 is consistent, and also, (2)is assured. If φm = (∃x)ψ(x) and φm is consistent with Tm ∪ {¬σ(cm)}, putψ(cp) into Tm+1. Then (3) holds. Finally, as Tm+1 is obtained by adding afinite number of sentences consistent with Tm, (1) holds for Tm+1.

Let Tω =⋃n<ω Tn. By (1), Tω is consistent, for if we assume otherwise,

there would be some finite ∆ ⊂ Tω such that ∆ deduces a contradiction, but aswe also have ∆ ⊂ Tm for some m, this would contradict (1). Furthermore, (2)gives that Tω is maximal consistent.

Now let N ′ = (N , b0, b1, . . .) be a countable model of Tω, which exists bythe Model existence theorem and the Downward Löwenheim-Skolem theorem.Let M′ = (M, b0, b1, . . .) be the substructure of N ′ generated by the inter-pretation {b0, b1, . . .} of all constants. Given any V ′-function f(x1, . . . , xn) and

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(a1, . . . , an) ∈ {b0, b1, . . .}n, if f(a1, . . . , an) = y, then the formula (∃y)(f(ba1 , . . ., ban) = y) is true in N ′. As N ′ |= Tω and Tω is maximal consistent, we musthave that (∃y)(f(ba1 , . . . , ban) = y) ∈ Tω, hence (∃y)(f(ba1 , . . . , ban) = y) ∈Tm for some m and by (3), N |= f(ba1 , . . . , ban) = bp for some p. Hencewe must have y = bp and {b0, b1, . . .} is closed under all functions, that isM = {b0, b1, . . .}.

One can then prove thatM′ |= φ⇔ Tω |= φ via induction over the complex-ity of sentences. The base case with atomic formulas is immediate, asM′ ⊂ N ′,and the interpretation of all constants of V ′ are in M . The induction step isstraight forward and completely analogous to the one used in [2] in the proof ofthe model existence theorem.

So we get thatM′ |= Tω. LetM be the reduct ofM′ to the vocabulary V.We have that M |= T , as T ⊂ Tω was a theory of V-sentences. By condition(4), there is for every element m ∈M a formula σ ∈ Σ such that the V ′-sentence¬σ(cm) ∈ Tm+1 ⊂ Tω. So asM′ models the sentences of Tω, there is no elementofM which satisfies Σ(x) inM. Hence T has a countable modelM which omitsΣ(x), as was to be shown.

4 Atomic and saturated modelsDefinition. Consider a complete theory T . A formula φ(x1, . . . , xn) is calledcomplete, or an atom, (in T ) if for every formula ψ(x1, . . . , xn), exactly one of

T |= (∀x1, . . . , xn)(φ(x1, . . . , xn)→ ψ(x1 . . . xn)),T |= (∀x1, . . . , xn)(φ(x1, . . . , xn)→ ¬ψ(x1 . . . xn))

holds.A formula ψ(x1, . . . , xn) is called completable (in T ) if there exists an atom

φ(x1, . . . , xn) such that T |= (∀x1, . . . , xn)(φ(x1, . . . , xn) → ψ(x1 . . . xn)). Oth-erwise, ψ is called incompletable.

A theory T is said to be atomic if every V-formula consistent with T iscompletable in T .

A model M is said to be an atomic model if every n-tuple a1, . . . , an ∈ Msatisfies an atom in Th(M).

Definition. A modelM is κ-saturated if for every subset X ⊂M with |X| < κ,the expansionMX realizes every type Σ(v) of the vocabulary V ∪ {ca : a ∈ X}which is consistent with Th(MX). M is simply called saturated if it is |M |-saturated.

Lemma 13 (Back and forth lemma). Suppose that κ is infinite,M and N areboth κ-saturated models and M ≡ N . Let aξ and bξ (ξ < κ) be sequences ofelements of the underlying sets M and N ofM and N , respectively. Then there

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are two sequences aξ and bξ such that:

range(aξ : ξ < κ) ⊂ range(aξ : ξ < κ)

range(bξ : ξ < κ) ⊂ range(bξ : ξ < κ)

(M, aξ)ξ<κ ≡ (N , bξ)ξ<κ.

Proof. Write every ordinal ξ < κ uniquely as ξ = ω · λ + n where n ∈ ω andλ < κ. (That this can be done is a known set theoretical result, which can beshown by induction over ordinals.) ξ is called even if n is even, and odd if nis odd. We will construct two sequences aξ and bξ such that for each ordinalξ < κ the following hold:

(a). For all η < ξ, if η = ω · λ+ 2n is even, then aη = a(ω·λ+n)

(b). For all η < ξ, if η = ω · λ+ (2n+ 1) is odd, then bη = b(ω·λ+n)

(c). (M, aη)η<ξ ≡ (N , bη)η<ξ.

We construct such a sequence by induction.For the base case ξ = 0, (a), (b) and (c) are trivially satisfied.For the inductive step, assume that we have found aη and bη (η < ξ) such

that (a), (b) and (c) hold for ξ. We then find aξ and bξ as follows. If ξ = ω·λ+2nis even, then let aξ = a(ω·λ+n), which satisfies (a). Now let Σ(v) be the typeof aξ in (M, aη)η<ξ. By induction hypothesis, (M, aη)η<ξ ≡ (N , bη)η<ξ, henceΣ(v) is consistent with Th(N , bη)η<ξ = Th(M, aη)η<ξ. By ξ < κ and N beingκ-saturated, there is an element b in (N , bη)η<ξ which satisfies Σ(v). Let bξ = b;as ξ is even, (b) still holds.

Now consider any "new" sentence γ containing aξ in the vocabulary of theexpansion (M, aη)η≤ξ. This can be written as γ = σ(aξ) for some formula σ inthe original vocabulary of (M, aη)η<ξ. Now as aξ and bξ satisfy the same typeΣ(v), we get that γ = σ(bξ) is true in (N , bη)η≤ξ if and only if γ = σ(aξ) is true in(M, aη)η≤ξ. Hence (M, aη)η≤ξ ≡ (N , bη)η≤ξ and (c) holds. If ξ = ω ·λ+(2n+1)is odd, let bξ = b(ω·λ+n) and find aξ in the same way.

As aξ = a(ω·λ+n) = a(ω·λ+2n) for all ξ < κ, we have that range(aξ : ξ <

κ) ⊂ range(aξ : ξ < κ). Similarly range(bξ : ξ < κ) ⊂ range(bξ : ξ < κ).(M, aξ)ξ<κ ≡ (N , bξ)ξ<κ follows immediately as (c) holds for all ξ < κ, and thelemma is proved

Theorem 14 (Uniqueness of saturated models). Let M and N be elementaryequivalent, saturated models of power κ. ThenM∼= N .

Proof. Let aξ and bξ (ξ < κ) be an enumeration of the underlying sets M andN ofM and N , respectively. By Lemma 13, there are aξ and bξ such that

(M, aξ)ξ<κ ≡ (N , bξ)ξ<κ

and aξ and bξ contains the enumerations aξ and bξ. Hence any a ∈M equals aξfor some ξ. Defining the function g by sending a to bξ yields thatM∼= N .

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5 Elementary chainsAn elementary chain is a chain of modelsM0 ≺ M1 ≺ . . . ≺ Mβ ≺ . . . whereβ < α andMγ ≺ Mβ whenever γ < β < α. They will be used in the proof ofthe following:

Theorem 15. Let M = (M,V, . . .) be a model, where V is a unary relationsuch that ω ≤ |V | < |M |. Then there are two models N = (N,W, . . .) andO = (O,W, . . .) such that N ≺ M,N ≺ O, and |N | = ω, |O| = ω1, where theinterpretation of W is the same in both N and O.

Proof. By the Löwenheim Skolem theorem, we can assume that |M | = |V |+.By Proposition 8, |M | is regular. By the well-ordering principle, there is awell ordering ≤ of M of type |M |. Let a1, . . . , an ∈ M and ψ(x, y, v1, . . . , vn)be an arbitrary formula of the vocabulary of M. Assume that in the model(M,≤) there are arbitrarily large a ∈ M such that for some b ∈ V , M |=ψ(b, a, a1, . . . , an).

That |M | is regular yields that there is a fixed b ∈ V such that there arearbitrarily large a ∈ M such that M |= ψ(b, a, a1, . . . , an), for if we assumetowards a contradiction that there is no such b, there would for each b ∈ V bean ab ∈ M such that for each a ∈ M , M |= ψ(b, a, a1, . . . , an) ⇔ a ≤ ab. LetA = {ab : b ∈ V }. For each c ∈ M , there is by assumption an element ac suchthat c ≤ ac and for some b ∈ V , M |= ψ(b, ac, a1, . . . , an). Hence there is foreach c ∈ M some ab ∈ A such that c ≤ ab. Mapping A to the correspondingordinals of ab in |M | yields a subset A′ ⊂ |M | such that |A′| = |V | < |M |, butsup(A′) = |M |, contradicting that |M | is regular.

Let the relation V (x) be true if x ∈ V , then we have shown that the followingsentence holds in (M,≤):

(1) (∀v1, . . . , vn)[(∀z∃y, x)(z ≤ y ∧ V (x) ∧ ψ(x, y, v1, . . . , vn)

→ (∃x∀z∃y)(z ≤ y ∧ V (x) ∧ ψ(x, y, v1, . . . , vn))].

Let us prove that:

(2) Every countable model (N0,≤0) ≡ (M,≤) has a countable properelementary extension (N1,≤1) such that the interpretation W ofV in N0 and N1 is the same.

The Löwenheim Skolem theorem immediately gives a model (N0,≤0) ≺ (M,≤)of power ω. So if (2) holds, we can construct a ω1-termed elementary chain ofcountable models (Nξ,≤ξ), ξ < ω1 such that (Nξ,≤ξ) ≡ (M,≤) for each ξ, theinterpretation of V in each Nξ is W , and Nξ+1 −Nξ 6= ∅.

Thus in the model (O′,≤′) =⋃ξ<ω1

(Nξ,≤ξ), V is interpreted asW , and theunderlying set O has power ω1 (as each union "adds" a non-zero, countable num-ber of elements to the underlying set, which yields ω1 elements in total). Fur-thermore (O′,≤′) ≡ (M,≤), so the reduct of (N0,≤0) and (O′,≤′) to the orig-inal vocabulary V are the desired models N = (N,W, . . .) and O = (O,W, . . .).

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To prove (2), let N0 = (N0,W, . . .) be a model such that (N0,≤0) ≡ (M,≤).Note that ≤0 need not be a well ordering on N0. Let V ′ be the expansion ofV by adding {≤, c} and {cb : b ∈ N0}. Let T be the union of the elementarydiagram of (N0,≤0) and the set of sentences {cb < c : b ∈ N0}. Let Σ(x) be theset of formulas {V (x)} ∪ {x 6≡ cb : b ∈W}.

We look for a countable model N ′ of T which omits Σ(x), for if (N ′0,≤′0) isthe reduct of N ′ to the vocabulary of (N0,≤0), then we will have that

(N ′0,≤′0) � (N0,≤0)

as N ′ models the elementary diagram of (N0,≤0). Furthermore, by it modelling{cb < c : b ∈ N0} there is an element b ∈ N ′0 −N0 , and by omitting {V (x)}∪{x 6≡ cb : b ∈W}, every element which satisfies V (x) must be interpreted as anelement in the set W of N0, that is the interpretation of V is the same in bothN0 and N1, and (2) will be proved.

To find such a N ′, first note that T is a consistent theory. Because givenany finite subset ∆ of T , only a finite number of the sentences {cb < c} lie in∆, hence there is an interpretation of c in N0 such that c > cb for all cb, and(N0,≤0) models ∆ . The Compactness theorem then yields that T has a model,hence by the Model existence theorem T is a consistent theory.

We then prove that T locally omits Σ, as then the Omitting types theo-rem (Theorem 12) will yield the existence of such a model N ′. Let N ∗0 bethe expanded model (N0,≤, b)b∈N0 and let ψ(c) be a V ′-sentence and ψ(y) thecorresponding formula.

Assume that T ∪ {ψ(c)} is consistent. Given any b ∈ N0, as cb < c ∈ Twe have that T ∪ {ψ(c) ∧ cb < c} is consistent, and there exists a model O∗of T ∪ {ψ(c) ∧ cb < c}. Thus O∗ models the sentence (∃y)(ψ(y) ∧ cb < y),and as T contained the elementary diagram of (N0,≤), the restriction of O∗to the vocabulary of N ∗0 is an elementary extension of N ∗0 , and it follows thatN ∗0 |= (∃y)(ψ(y) ∧ cb < y).

Now assume that ψ(y) is satisfied by arbitrarily large elements of N ∗0 . Everyfinite subset of T ∪ {ψ(c)} is contained in some subset ∆ ∪ {ψ(c)}, where ∆ isthe union of the elementary diagram of N0 and {cb < c : b ∈ B}, where Bis a suitable finite subset of N0. Now as ψ(y) is satisfied by arbitrarily largeelements, there is some y such that cb < y for all b ∈ B and N ∗0 |= ψ(y). Byinterpreting c as this y, we get that N ∗0 |= ∆ ∪ {ψ(c)}, so every finite subset ofT ∪{ψ(c)} is consistent, and by compactness T ∪{ψ(c)} is consistent. We havethus shown that:

(3) ψ(c) is consistent with T ⇔ N ∗0 |= (∀x∃y)(x ≤ y ∧ ψ(y)).

Now let θ(x, c) be any formula consistent with T . That is, there is a model ofT which realizes θ(x, c). But in this model, (∃x)θ(x, c) also holds. Hence thesentence (∃x)θ(x, c) is consistent with T . For any x which satisfies the formulaθ(x, c), x is either in W or not. Hence at least one of the following holds:

(4) (∃x)(θ(x, c) ∧ ¬V (x)) is consistent with T

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or(5) (∃x)(θ(x, c) ∧ V (x)) is consistent with T .

If (4) holds, σ(x) = V (x) is a formula in Σ(x) such that θ(x, c) ∧ ¬σ(x) isconsistent with T , and T locally omits Σ(x). Suppose (5) holds. Then byletting ψ(c) = (∃x)(θ(x, c) ∧ V (x)), (3) yields that:

N ∗0 |= (∀z∃xy)(z ≤ y ∧ (θ(x, y) ∧ V (x))).

By (1) and the fact that (N0,≤0) and (M,≤) are elementary equivalent, theabove yields that:

N ∗0 |= (∃x∀z∃y)(z ≤ y ∧ (θ(x, y) ∧ V (x)).

So for some b ∈W , where cb is the corresponding constant:

N ∗0 |= (∀z∃y)(z ≤ y ∧ (θ(cb, y) ∧ V (cb))

which gives that:

N ∗0 |= (∀z∃y)(z ≤ y ∧ (∃x)(θ(x, y) ∧ x ≡ cb)).

So by (3), the sentence (∃x)(θ(x, c) ∧ x ≡ cb) is consistent with T . But asσ(x) = (x 6≡ cb) ∈ Σ(x), we have found a σ ∈ Σ such that θ(x, c) ∧ ¬σ(x) isconsistent with T . Thus T locally omits Σ(x). We have thus shown the existenceof two models N = (N,W, . . .) and O = (O,W, . . .) such that N ≺M,N ≺ O;|N | = ω, |O| = ω1 and W is interpreted the same in both.

6 Skolem functions and IndiscerniblesGiven a vocabulary V, the Skolem expansion of V is obtained by adding a newfunction symbol Fψ corresponding to each formula ψ of the form ψ(x1, . . . , xn) =(∃x)φ(x, x1, . . . , xn), where Fψ has the same arity as ψ. The Skolem expansionis denoted by V∗ and the Fψ are called the Skolem functions.

Now for each Skolem function Fψ, let y1, . . . , yn be variables not occurringin ψ and construct the following V∗-sentence σψ:

(∀(y1, . . . , yn)(ψ(y1, . . . , yn)→ φ(Fψ(y1, . . . , yn), y1, . . . , yn))

That is, if ψ is true, the Skolem function of ψ gives the element which ψ saysexists. The Skolem theory ΣV of V is the theory with the σψ above as axioms.

Given a V-structureM and an expansionM∗ to the vocabulary V∗,M∗ iscalled the Skolem expansion of M if M∗ |= ΣV . Furthermore, given a theoryT , the Skolem expansion of T is the deductive closure of T ∪ ΣV .

Given a subset X ⊂M of a Skolem expansionM∗, the Skolem hull H(X) isthe smallest set X ⊂ H(X) ⊂M which is closed under all the Skolem functions.The corresponding substructure ofM∗ is denoted H(X).

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A V-theory T is said to have built-in Skolem functions if there already isa term tψ in V of arity n for each formula ψ of the form ψ(x1, . . . , xn) =(∃x)φ(x, x1, . . . , xn) such that:

T ` (∀(y1, . . . , yn)(ψ(y1, . . . , yn)→ φ(tψ(y1, . . . , yn), y1, . . . , yn))

Proposition 16. Every V-structureM has a Skolem expansionM∗

Proof. Let M be a V-structure and well-order M . Given a V-formula of theform ψ(x1, . . . , xn) = (∃x)φ(x, x1, . . . , xn), we let the interpretation Gψ of theSkolem function Fψ in the expansion M∗ = (M, {Fψ : ψ = (∃x)φ}) be asfollows. Given any a1, . . . , an ∈ M , if M |= ψ(a1, . . . , an), let Gψ(a1, . . . , an)be the least element a such that M |= φ(a, a1, . . . , an) (well-defined in a wellordering). If M 6|= ψ(a1, . . . , an) , let Gψ(a1, . . . , an) be arbitrary. It followsimmediately thatM∗ models the axioms of ΣV , henceM∗ is a Skolem expansionofM.

Proposition 17. Let T be a V-theory. Then there exists an expansion V of Vand an extension T of T such that T has built-in Skolem functions. Furthermore,every model of T has a Skolem expansion which models T .

Proof. Given the vocabulary V0 = V, define a sequence of expansions Vn byVn+1 = (Vn)∗. Let V =

⋃n Vn.

Now define a V -theory T by having the set T ∪⋃n ΣVn as axioms. Any

V-formula of the form ψ = (∃x)φ contains a finite number of symbols, henceit is a Vn-formula for some n and there is a corresponding Skolem functionFψ ∈ Vn+1 ⊂ V. Now as T |= ΣVn , Fψ is a built-in Skolem function of ψ in T .

Also, note that by Proposition 16, ifMn is a model of Tn = T ∪⋃k<n ΣVk ,

then there is a Skolem expansion Mn+1 = (Mn)∗ which models Tn+1 = Tn ∪⋃k≤n ΣVk . Hence there is inductively an expansion M of M which models

T .

Note that if Vn in the proof is countable, then there are at most countablymany ψ = (∃x)φ(x) in Vn, and the Skolem expansion (Vn)∗ of Vn is countable.Hence if V is a countable vocabulary, the expansion V above is countable.

Given a V-structure M, we now define a set of indiscernibles in M to bea subset X ⊂ M such that X is strictly and linearly ordered by < (< is notnecessarily in V) and for every pair of increasing sequences x1 < . . . < xnand y1 < . . . < yn in X, (M, x1, . . . , xn) ≡ (M, y1, . . . , yn). That is, a set Xis indiscernible in M if no V-formula can distinguish increasing sequences ofelements in X.

The proof of the following lemma relies on Ramsey’s theorem, but we willnot delve into that theory in this text. The proof can be found in [1], on page148.

Lemma 18. Let V ′ = V ∪ {cn : n ∈ ω} where each cn is a new constantsymbol. Let T be a V-theory with infinite models. Then the following set T ′ of

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V ′-sentences is consistent:

T ′ = T∪{φ(ci1 , . . . , cin)↔ φ(cj1 , . . . , cjn) : φ(v1, . . . , vn) is a V-formula,n ∈ ω and i1 < . . . < in, j1 < . . . < jn} ∪ {¬c1 ≡ c2}

Theorem 19. Let T be a V-theory with infinite models and let 〈X,<〉 be anylinearly ordered set. Then there is a modelM of T such that X ⊂M and X isa set of indiscernibles inM.

Proof. Expand V to V ′ = V ∪ {cx : x ∈ X}. Note that every finite subset of Xcan be embedded in 〈ω,<〉, hence given a finite number x1, . . . , xn of elementsfrom X, they can be said to be the interpretation of some cxi

, cyi ∈ {cn : n ∈ ω}.Define T ′ as follows

T ′ = T∪{φ(cx1 , . . . , cxn)↔ φ(cy1 , . . . , cyn) : φ(v1, . . . , vn) is a V-formula,n ∈ ω and x1 < . . . < xn, y1 < . . . < yn} ∪ {¬cx1 ≡ cx2 : x1, x2 ∈ X}

and note that every finite subset ∆ of T ′ only contains the sentence φ(cx1 , . . . , cxn)↔ φ(cy1 , . . . , cyn) for finitely many formulas φ and finitely many sequencesx1 < . . . < xn, y1 < . . . < yn. Hence ∆ can be interpreted as a subset ofsome T ′′ of the form given in Lemma 18, and ∆ is consistent. So by com-pactness, T ′ is consistent. Now let M′ be a model of T ′, and M the reductof M′ to V, which must be a model of the V-theory T . If we in M′ inter-pret each cx by the element x, T ′ gives that for each formula φ(v1, . . . , vn) inV and sequences x1 < . . . < xn, y1 < . . . < yn, M′ |= φ(x1, . . . , xn) if andonly ifM′ |= φ(y1, . . . , yn). And as φ(v1, . . . , vn) was in V, this holds inM, so(M, x1, . . . , xn) ≡ (M, y1, . . . , yn) and X is a set of indiscernibles inM.

We make the following observation. Let V be a countable vocabulary and Ta V-theory with infinite models. Proposition 17 gives us a countable vocabularyV and a V-theory T with built-in Skolem functions. By Theorem 19, given anylinearly ordered set 〈X,<〉, we can find a model M of T where X is a set ofindiscernibles. Now the Skolem expansion M∗ is essentially the same as M ifwe interpret the Skolem functions as the built-in Skolem functions in T . Hencethe substructure generated by X is the Skolem hull H(X) of X, and by theTarski-Vaught criterion (Proposition 6), it is an elementary substructure ofM.So by setting M = H(X), this is also a model of T in which X is a set ofindiscernibles.

Theorem 20. Let V be a countable vocabulary and T a V-theory with infinitemodels. Then for every infinite cardinal κ, T has a model M of power κ suchthat for every subset N ⊂M , the expanded modelMN realizes at most |N |+ωtypes in the expanded vocabulary V ∪ {cb : b ∈ N}

Proof. As in the observation above, we extend T to a theory T which has built-in Skolem functions in an expanded vocabulary V. Let (X,<) be a well-orderedset of order type κ. Then there is a modelM = H(X) of T in which X is a setof indiscernibles. Note thatM is of power κ, because as V is countable, there

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are at most countably many functions and the closure H(X) of X under thesefunctions thus has at most κ · ω = κ elements.

Let N ⊂ M = H(X) and choose for each b ∈ N a term t(v1, . . . , vn) and(y1, . . . , yn) ∈ Xn such that b = t(y1, . . . , yn). This is called a standard rep-resentation of b. Let Y be the set of all y ∈ X which appear in one of thesestandard representations. As there are finitely many y in each standard repre-sentation, we have |Y | ≤ |N | if |N | ≥ ω, and |Y | ≤ ω if N is countable, hence|Y | ≤ |N |+ ω.

Call two sequences x1, . . . , xn and y1, . . . , yn of elements in X equivalent overY if for all k ≤ n and all z ∈ Y , we have xk, yk 6= z, and xk < z if and only ifyk < z.

Form the expanded vocabulary V ′ = V ∪ {cz : z ∈ Y }. Assume thatx1, . . . , xn and y1, . . . , yn are equivalent over Y . Any V ′-formula φ(v1, . . . , vn)can be written as φ(v1, . . . , vn) = ψ(v1, . . . , vn, cz1 , . . . , czm) with some suit-able V-formula ψ(v1, . . . , vn, w1, . . . , wm), where cz1 , . . . , czm are the constantsappearing in φ which are not in V. As x1, . . . , xn and y1, . . . , yn are equiva-lent over Y , the elements z1, . . . , zm can be "ordered in between" the sequencex1 < . . . < xn in the same way as in y1 < . . . < yn, which gives two newsequences x′1 < . . . < x′n+m and y′1 < . . . < y′n+m of elements in X. Assumethat x1 < . . . < xn < z1 < . . . < zn - if the ordering is otherwise, rear-range the free variables v1, . . . , vn, w1, . . . , wn so their appearance in the for-mula ψ(v1, . . . , vn, w1, . . . , wn) matches the order of x′1 < . . . < x′n+m and y′1 <. . . < y′n+m. As X is a set of indiscernibles, M |= ψ(x1, . . . , xn, z1, . . . , zm) ⇔M |= ψ(y1, . . . , yn, z1, . . . , zm). Hence x1, . . . , xn and y1, . . . , yn satisfy the sameformulas φ(v1, . . . , vn) in the expanded model (M, z)z∈Y .

It follows that for any term t(v1, . . . , vn) of V ′, the two elements t(x1, . . . , xn)and t(y1, . . . , yn) realize the same type in the model (M, z)z∈Y . Note that asb = t(z1, . . . , zn) for some t and z1, . . . , zn ∈ Y , any type in the vocabulary of(M, b)b∈N can be written in the vocabulary of (M, z)z∈Y , so the two elementst(x1, . . . , xn) and t(y1, . . . , yn) also realize the same types in (M, b)b∈N . LetMbe the reduct ofM to V. Then t(x1, . . . , xn) and t(y1, . . . , yn) realizes the sametype in (M, b)b∈N whenever x1, . . . , xn and y1, . . . , yn are equivalent over Y .

Let x′ = "least z ∈ Y such that x < z" if such a z ∈ Y exists, otherwiselet x′ = ∞. It follows immediately that two n-tuples x1, . . . , xn and y1, . . . , ynare equivalent over Y if and only if x′1 = y′1, . . . , x

′n = y′n. Thus there is at most

|Y | + ω non-equivalent n-tuples over Y , as this is the number of all possible{x′1, . . . , x′n} ⊂ Y . Hence for a term t, evaluating t for all possible n-tuplesnot in Y gives a set whose elements realizes at most |Y | + ω different types in(M, z)z∈Y . Call this set At.

Note that the vocabulary of (M, z)z∈Y has cardinality |Y |+ ω, so there areat most |Y |+ ω different terms t. Hence the elements of A =

⋃tAt realizes at

most |Y |+ ω different types in (M, z)z∈Y .Every element of M = H(X) is equal to some term t(x1, . . . , xn) in M,

and if some xi ∈ Y , we can replace xi by a corresponding constant zi to getequality to a term in the model (M, z)z∈Y . Continuing like this, we get that

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our element is equal to a term t′(x′1, . . . , x′l) in (M, z)z∈Y where x′1, . . . , x′l 6∈ Y .

Hence M = A and the model (M, z)z∈Y realizes at most (|Y |+ ω) ≤ (|N |+ ω)different types, hence (M, b)b∈N realizes at most (|N |+ ω) different types.

7 Morley’s Categoricity TheoremTheorem 21 (Morley’s theorem). Suppose V is countable and T is a completeV-theory. If T is categorical in some uncountable power, then T is categoricalin every uncountable power.

The following lemma shows that it will be enough to prove that if T iscategorical in some uncountable power, then every model of T of power ω1 issaturated. This will be done in the end of the section, using Lemmas 24, 25 and29.

Lemma 22. Suppose V is countable and T is a complete V-theory, such thatevery model of T of power ω1 is saturated. Then every uncountable model of Tis saturated, and it follows that T is categorical in every uncountable power.

Proof. If T has a model of some infinite power, then T has models of any infinitepower. Given any infinite power κ, any two modelsM,N of T of the same powerκ are elementary equivalent. This follow from the completeness of T , as for everyV-sentence φ, T |= φ or T |= ¬φ. Hence ifM |= φ, we must have that T |= φ, asT |= ¬φ would give thatM |= T ⇒M |= φ∧¬φ, a contradiction. Now N |= Tyields that N |= φ, and Th(M) ⊂ Th(N ). Analogously, Th(N ) ⊂ Th(M) andTh(M) = Th(N ).

If we show that every uncountable model of T is saturated, the uniquenessof saturated models (Theorem 14) yields that the models are isomorphic, andhence T is κ-categorical.

Given any κ > ω1, assume towards a contradiction that T has a model Mof power κ which is not saturated. Then there is a subset X ⊂ M of power|X| < κ and a set Σ(v) of formulas in the expanded vocabulary VX consistentwith Th(M) such that Σ(v) is not satisfiable inMX , but every finite subset ofΣ is satisfied inMX . Since V is countable, |VX | = |V|+ |X| < κ, hence |Σ| < κand we can choose a subset U ⊂M such that |U | = |Σ|, i.e. there is a bijectionφ : U → Σ.

Let U(u) be the relation which is true if u ∈ U . The function φ associatesa formula φu in the vocabulary VX to each element u ∈ U . Define the relationR(v1, v2) to be true for all pairs (u, x) ∈ M × M such that u ∈ U , x ∈ Xand the constant cx occurs in the formula φu. Thus for every u, there are onlyfinitely many x ∈ M such that R(u, x). Let S(v1, v2) be the relation which istrue for all (u, v) ∈ M ×M such that u ∈ U andMX |= φu(v). ExpandM toM′ = (M, U,R, S).

The idea of the proof is that sentences that hold in a given model will holdin elementary extensions and substructures of that model. It follows by thedefinition of S that the formula

(1) (∀v)(S(u, v)↔ φu(v))

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is satisfied by each u ∈ U in the modelM′X . Now define the sentence

(2) (∀u1, . . . , un)[U(u1) ∧ . . . ∧ U(un)→ (∃v)(S(u1, v) ∧ . . . ∧ S(un, v)].

This holds inM′, as every finite subset of Σ is satisfiable inM′, i.e. (∀u1, . . . , un)[U(u1) ∧ . . . ∧ U(un) → (∃v)(φu1(v) ∧ . . . ∧ φun(v)], which by (1) is equivalentto (2). Also, the sentence

(3) ¬(∃v∀u)(U(u)→ S(u, v))

holds in M′ as Σ is not realized by any v ∈ M , i.e ¬(∃v∀u)(U(u) → φu(v))which by (1) is equivalent to (3).

By Theorem 15, since |U | < |M |, there exists two models

N ′ = (N,U ′, R′, S′) , and O′ = (O,U ′, R′′, S′′)

such that N ′ ≺M′,N ′ ≺ O′, |N | = ω, |O| = ω1 and the interpretation U ′ of Uis the same. U ′ ⊂ N must be countable. Furthermore, if we put X ′ = X ∩N ,it holds for the expanded models that:

(4) N ′X′ ≺M′X′

and(5) N ′X′ ≺ O′X′ .

(As X ′ is a subset of N , and hence of M and O too.)By (4) and (5), the sentences (2) and (3) hold in O′. If for each u ∈ U ′ we

have that the formula (1) is satisfied in O′X′ , the sentences (2) and (3) will beequivalent to (∀u1, . . . , un) [U(u1) ∧ . . . ∧ U(un) → (∃v)(φu1(v) ∧ . . . ∧ φun(v)]and ¬(∃v∀u)(U(u)→ φu(v)) respectively, i.e. there is a set

Σ′(v) = {φa : a ∈ U ′}

which is not satisfied in the model O′X′ , but every finite subset of Σ′(v) is.For each u ∈ U ′ ⊂ N , the set {x ∈ M : R(u, x)} is finite, say |{x ∈ M :

R(u, x)}| = n. The following formula states that "there are exactly n differentelements x such that R(y, x) is true":

ψ(y) =(∃x1, . . . , xn)[(∧i6=j

(xi 6≡ xj) ∧ (R(y, x1) ∧ . . . ∧R(y, xn))]∧

(∀x1, . . . , xn, xn+1)[R(y, x1) ∧ . . . ∧R(y, xn+1)→ (∨i 6=j

xi ≡ xj)].

Now as M′X′ |= ψ(u) and u ∈ U ′ ⊂ N , (4) gives that N ′X′ |= ψ(u). Hence|{x ∈ N : R(u, x)}| = |{x ∈ M : R(u, x)}|. Note that by N ⊂ M we have that{x ∈ N : R(u, x)} ⊂ {x ∈ M : R(u, x)}, hence by the equal cardinality of thesets we get that

{x ∈M : R(u, x)} = {x ∈ N : R′(u, x)} ⊂ X ′.

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This yields that for each u ∈ U ′, all constants cx appearing in the correspondingformula φu lie in the vocabulary V ′X of N ′X′ . So by (4) and (5) we get that eachu ∈ U ′ actually satisfies (1) in O′X′ .

By (4) and (5), O is a model of T of power ω1. By above, there is a subsetX ′ ⊂ O such that |X ′| < ω1 (as X ′ ⊂ N and N is countable), but there is aset of formulas Σ′(v) which is not satisfied in O′X′ . Thus O is not saturated,contradicting the hypothesis of the lemma.

So every uncountable modelM of T must be saturated.

The proof of Morley’s theorem relies heavily on the notion of a stable theory.

Definition. A theory is T is stable in power κ if for every modelM of T andevery set X ⊂M of power κ, the expansionMX realizes no more than κ typesin a single variable v.

Lemma 23. Let T be a theory in V which is stable in power ω. Then T isstable in every infinite power.

Proof. Assume towards a contradiction that there is a κ > ω such that T is notstable in power κ. That is assume T has a model M with a subset X ⊂ Msuch that |X| = κ and the expanded model MX realizes more than κ types,that is at least κ+ types. We can assume without loss of generality thatM haspower κ+, as otherwise the Löwenheim Skolem theorem will give an elementarysubstructure of power κ+. This gives thatMX can’t realize more than κ+ types(as each element inM can realize at most one type), henceMX realizes exactlyκ+ types.

Since the vocabulary V is countable, the vocabulary VX = V ∪ {cx : x ∈ X}has power κ. Thus the set Σ of all formulas φ(v) of VX has power κ. Given asubset U ⊂M of power κ, we can find a one to one function from U to Σ, thatis, an enumeration φu(v) of all formulas in Σ, where u ∈ U .

As in the previous lemma, we define the relations U(u) if u ∈ U , R(u, x) ifu ∈ U and cx occurs in the formula φu, and S(u, v) ifMX |= φu(v).

Furthermore, for each of the κ+ different types realized by MX , chooseone element which satisfies this type. Let V ⊂ M be the subset of power|V | = κ+ containing these elements, and define the relation V (v) if v ∈ V .As an element can’t satisfy two different types, our construction yield that twodistinct elements in V will realize different types in MX . As M and V havethe same power, there is a one to one function G from M into V . Expand Mto the model

M′ = (M, U, V,R, S,G)

As in the previous lemma, in the modelM′X all u ∈ U satisfies the formula

(1) (∀v)(S(uv)↔ φu(v)).

We can write that "two distinct elements of V realize different types" and "Gis a bijection from the underlying set M of the model into V " as the followingsentences, which will hold inM′:

(2) (∀v, w)[(v 6≡ w ∧ V (v) ∧ V (w))→ (∃u)(U(u) ∧ ¬(S(u, v) ∧ S(u,w))]

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(3) (∀v, w)[(v 6≡ w)→ (V (G(v)) ∧G(v) 6≡ G(w))].

Using Theorem 15, that |U | < |M | gives the existence of two models

N ′ = (N,U ′, V ′, R′, S′, G′) , and O′ = (O,U ′, V ′′, R′′, S′′, G′′)

such that N ′ ≺ M′,N ′ ≺ O′, |N | = ω, |O| = ω1 and the interpretation of U ′is the same in both N ′ and O′. The sentences (2) and (3) must now hold in Nand O. (3) gives that G′′ is a bijection between O and V ′′, hence

(4) |V ′′| = |O| = ω1.

Now for all u ∈ U ′, if x ∈ X and R(u, x) holds, then we as in the previouslemma have that x ∈ X ′ = X ∩ N . Thus every formula φa(v) is a formula inVX′ , and (1) is satisfied by each u ∈ U ′ in the models N ′X′ and O′X′ .

This gives that (∃u)(U ′(u)∧¬(S′′(u, v)∧S′′(u,w))⇔ (∃u)(U ′(u)∧¬(φu(v)∧φu(w)) in O′X′ , hence (2) says that any two distinct elements of V ′′ realizedifferent formulas, and they must realize different types in the model O′X′ . Soby (4), O′X′ realizes uncountably many types. But asX ′ ⊂ N , |X ′| is a countableset, which contradicts that T is a ω-stable theory.

Thus T is stable in every infinite power.

Lemma 24. If a theory T is categorical in some uncountable power κ, then Tis stable in power ω.

Proof. Assume towards a contradiction that T is not stable in power ω. Thenthere is a model M and a subset X ⊂ M of power ω such that the simpleexpansion MX realizes more than ω types in a single variable, i.e. at leastω1 types. By the Löwenheim-Skolem theorem, we can assume that |M | = ω1,so MX realizes at most, hence exactly, ω1 types. Using Theorem 20, T has amodel N of power κ with the property that for all Y ⊂ N , the expanded modelNY realizes at most |Y |+ω types. As T is κ-categorical, this property holds forall models of power κ. By the Löwenheim-Skolem theorem, we get that thereis an elementary extension M′ � M of power κ. Then by letting Y = X,M′X realizes at most |X|+ ω = ω types, whileMX relizes ω1 types. This is acontradiction, as every element of M realizes the same type in both MX andM′X

Lemma 25. Suppose T is a theory which has infinite models and is stable inpower ω, then for every regular cardinal κ > ω, T has an κ-saturated model ofevery power β ≥ κ.

Proof. Let M be a model of T of power β ≥ κ, and TM the complete theoryof the expansion MM . By Proposition 11 there is a model NM such thatMM ≺ NM and every type consistent with TM is realized in NM .

By Lemma 23 , T is stable in power β. Hence NM realizes at most |M | =β types. Let A ⊂ N be a set containing for each type realized in NM oneelement which realizes that type. Let A ∪M = B ⊂ N and note that |B| = β.

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The Löwenheim-Skolem theorem gives the existence of an elementary submodelM′M ≺ NM such that B ⊂M ′ and |M ′| = β. By A ⊂M ′, every type consistentwith NM , hence every type consistent with TM, is realized inM′M .

ByM ⊂M ′ we immediately getM⊂M′M . Given any formula φ(x1, . . . , xn)and (a1, . . . , an) ∈ Mn, MM |= φ(a1, . . . , an) ⇔ NM |= φ(a1, . . . , an) byMM ≺ NM . As M ⊂ M ′ and M′M ≺ NM , we also have that M′M |=φ(a1, . . . , an)⇔ NM |= φ(a1, . . . , an), henceMM ≺M′M .

So we have shown the existence of an elementary extension M′M � MM

such that |M ′| = β and every type consistent with TM is realized inM′M . Forreference, call this result (1).

We use (1) κ times to construct an elementary chain Mγ , γ < κ, such thateach Mγ is a model of T of power β; also (1) gives that any type consistentwith the theory of Mγ is realized in Mγ+1. If γ is a limit ordinal, we simplydefineMγ =

⋃δ<γMδ Let N be the union N =

⋃δ<κMδ. As β ≥ κ, N is of

power β.We now show that N is κ-saturated. Let X be any set X ⊂ N of power

|X| < κ. Well-order N and let X ′ be the corresponding ordinals of X. Sinceκ is regular it follows that sup(X ′) < κ, hence there exists a γ < κ suchthat X ⊂ Mγ . Now every type Σ(v) consistent with the complete theory of(Mγ , x)x∈X is realized in (Mγ+1, x)x∈X , and hence realized in (N , x)x∈X . ButasMγ ≺ N , the two models have the same complete theories, hence every typeΣ(v) consistent with the complete theory of (N , x)x∈X is realized in (N , x)x∈X .So N is κ-saturated.

The following two lemmas will be needed to prove Lemma 29.

Lemma 26. Let T be a theory in a countable vocabulary V which is stable inpower ω. Then for every modelM of T and every subset X ⊂M , the completetheory TX of the expanded modelMX is atomic.

Proof. Assume towards a contradiction that TX is not atomic. Then thereis a smallest positive integer n such that there is an incompletable formulaψ(v1, . . . , vn) in n variables, consistent with TX .

We only need to show a contradiction in the simplest case when n = 1, forif we assume that n > 1, there exists an expansion ofMX and a theory whichhas an incomplete formula in one variable, hence the contradiciton will follow.

To get this expansion, note that if n > 1, then the formula ∃vnψ(v1, . . . , vn)has n-1 variables, and thus an atom φ(v1, . . . , vn−1). Add new constant symbolsto get the vocabulary V ′X = VX ∪ {c1, . . . , cn−1}, and define the V ′X -theory:

T ′X = TX ∪ {φ(c1, . . . , cn−1)}

Note that any sentence δ in V ′X containing some of c1, . . . , cn−1 can be writtenas δ = γ(c1, . . . , cn−1) for some formula γ(v1, . . . , vn−1). As φ(v1, . . . , vn−1) isatomic, either TX |= φ(v1, . . . , vn−1)→ γ(v1, . . . , vn−1) or TX |= φ(v1, . . . , vn−1)

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→ ¬γ(v1, . . . , vn−1), hence either δ or ¬δ is a consequence of the sentenceφ(c1, . . . , cn−1), and T ′X is a complete theory.

We now claim that the formula ψ(c1, . . . , cn−1, vn) has no atoms in T ′X . Forif there is an atom θ(c1, . . . , cn−1, vn) of ψ(c1, . . . , cn−1, vn), then:

T ′X |= θ(c1, . . . , cn−1, vn)→ ψ(c1, . . . , cn−1, vn)

Note that {c1, . . . , cn} was not in the original vocabulary VX . By Lemma 5 theabove yields that T ′X |= (∀v1, . . . , vn−1)(θ(v1, . . . , vn) → ψ(v1, . . . , vn)). Henceas TX |= φ(v1, . . . , vn−1)→ ∃vnψ(v1, . . . , vn), we get that:

TX |= φ(v1, . . . , vn−1) ∧ θ(v1, . . . , vn)→ ψ(v1, . . . , vn)

As φ(v1, . . . , vn−1) ∧ θ(v1, . . . , vn) → φ(v1, . . . , vn−1), φ ∧ θ is an atom of TX .But then the above contradicts that ψ(v1, . . . , vn) has no atoms. Now leta1, . . . , an−1 be elements such thatMX |= φ(a1, . . . , an−1), thenM′ = (MX , a1,. . . , an−1) is a model of T ′X , and the formula ψ(c1, . . . , cn−1, v) = ψ(v) has noatoms in the complete theory ofM′.

So assume that n = 1 and that ψ(v) is consistent with and incompletable in TX .ψ(v) is not an atom (as an atom is trivially completable), hence there is someformula γ(v1, . . . , vn) such that TX 6|= ψ → γ and TX 6|= ψ → ¬γ. Thus thesentences ψ0(v) = ψ(v) ∧ γ(v) and ψ1 = ψ(v) ∧ ¬γ(v) are formulas consistentwith TX such that:

TX |= ψ0(v)→ ψ(v), TX |= ψ1(v)→ ψ(v), and TX |= ¬(ψ0(v) ∧ ψ1(v))

Furthermore, they are not atomic in TX ; for if say ψ0(v) was an atom, thenTX |= ψ0 → ψ contradicts that ψ(v) is incompletable.

Thus we can repeat the above argument for ψ0(v) to get formulas ψ00(v)and ψ01(v) with the same properties as above, and so on infinitely. We will getinfinite sequences of zeroes and ones, each which corresponds to a consistent setof non-atomic formulas. As there are 2ω such sequences, we get 2ω sets of suchsentences. Given any two different sequences, let "s" denote all initial elementswhich are identical. If A and B are the corresponding sets of sentences, ψs0 ∈ Aand ψs1 ∈ B. As TX |= ¬(ψs0 ∧ ψs1) we get that the two sets can not realizedby the same element. Hence if we extend each set to a type, we get 2ω differenttypes.

Let Y be the set of all constants x ∈ X which occur in some formula in ourtree of formulas. There is a finite number of sequences of a given length n. Ifthe tier Σn(v) is the set of all ψs(v) in the tree such that s has length n, therecan only be a finite number of constants appearing in the formulas of Σn(v).As

⋃n<ω Σn(v) are all formulas appearing in the tree, we get that Y contains

at most ω elements.Let TY be the complete theory of MY . Then TY has 2ω different types in

V. By Proposition 11, TY has a model NY which realizes all 2ω types, so T isnot ω stable, a contradiction. Thus TX must be atomic.

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Lemma 27. Let T be a theory such that for every model M of T and everysubset X ⊂M , the complete theory of the expanded modelMX is atomic. Thenfor every model M of T and every subset X ⊂M , the complete theory of MX

has an atomic model.

Proof. Given the model MX in the vocabulary V, where ||V|| = |V| + ω = α,we will construct a sequence aβ , β < α · ω of elements of M such that:

(1) For all β < α · ω, aβ realizes an atom in the complete theory of (MX , aγ)γ<β .

(2) For all n < ω and every atom φ(v) in the complete theory of (MX , aγ)γ<α·n,there exists δ < α · (n+ 1) such that aδ realizes φ(v)

By then letting NX be the submodel ofMX with underlying set N = {aβ : β <α · ω}, we will in fact get an elementary submodel ofMX which is atomic.

We first construct the sequence aβ via induction, over m < ω.For the base case m = 0, (1) is trivially satisfied for all β < α ·m and so

is (2) for each n < m. For the induction step, suppose we have a sequence aβ ,β < α · m such that (1) holds for all β < α · m and (2) holds for all n < m.Given the model (MX , aγ)γ<α·m, its vocabulary has cardinality α, hence theset of all atomic formulas φ(v) for the complete theory of (MX , aγ)γ<α·m can beordered as φδ(v), δ < α. We choose for aα·m an element inM which satisfies theatom φ0(v) in (MX , aγ)γ<α·m. By the assumption of the lemma, the completetheory of (MX , aγ)γ<(α·m)+1 is atomic. Hence we can find an atom φ′1(v) in thecomplete theory of (MX , aγ)γ<(α·m)+1 for our formula φ1(v). (Although φ1(v)is an atom for the complete theory of (MX , aγ)γ<α·m, it does not need to beone of (MX , aγ)γ<(α·m)+1.) Choose a(α·m)+1 to be an element satisfying φ′1(v).

Continuing in this manner, we get elements a(α·m)+δ, δ < α such thata(α·m)+δ realizes an atom φ′δ(v) in the complete theory of (MX , aγ)γ<(α·m)+δ,hence (1) holds for all β < α · (m + 1). And given any atom φ(v) = φδ(v) of(MX , aγ)γ<(α·m), that φ′δ(v) is an atom of φδ(v) in (MX , aγ)γ<(α·m)+δ yieldsthat φδ(v) is realized by the element a(α·m)+δ ∈M , hence (2) holds for n = m.By the induction principle, (1) holds for all β < α · ω and (2) for all n < ω.

To show that NX ≺ MX , we use the Tarski-Vaught criterion. Recall thatN = {aβ : β < α · ω}, hence given a formula φ(x1, . . . , xn, v) and any tu-ple (a1, . . . , an) ∈ Nn, thatMX |= ∃vφ(a1, . . . , an, v) implies that the formulaφ(v) = φ(a1, . . . , an, v) is consistent with the complete theory of (MX , aγ)γ<α·ω.

There is an n < ω such that for all constants aβ not in X appearing inφ(v), β < α · n. Hence φ(v) is also consistent with the complete theory of(MX , aβ)β<α·n, which is atomic by hypothesis, and there is an atom φ′(v) ofφ(v) in this theory. By (2), there is an element aδ, δ < α · (n+ 1) which satisfiesφ′(v), hence satisfies φ(v). As aδ lies in N , we get that MX |= φ(b) for someb ∈ N , and the Tarski-Vaught test thus yields that NX ≺MX .

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Finally, we prove by induction that for all β < α · ω, the following hold.

(3) For all δ1, . . . , δn < β, the n-tuple aδ1 , . . . , aδn satisfies an atom inMX .

That is, every n-tuple of elements in N satisfies an atom in the complete theoryof the elementary substructure NX ≺MX , which gives that NX is atomic.

(3) is trivially true for the base case β = 0. Suppose (3) holds for all β < γ.If γ is a limit ordinal, then (3) holds for γ. Let γ = η + 1 be a successorordinal. If aη is not in the n-tuple, (3) holds by the induction hypothesis, sowe only need to consider (n + 1)-tuples aδ1 , . . . , aδn , aη. By (1), aη realizesan atom φ(v) in the complete theory of (MX , aβ)β<η. If cλ1

, . . . , cλnare the

constant symbols not inX appearing in φ(v), then φ(v) can be written as φ(v) =φ(v, cλ1

, . . . , cλn), where φ(v, u1, . . . , um) is a formula in the theory of MX .

Also, given the induction hypothesis, the (n+m)-tuple aδ1 , . . . , aδn , aλ1 , . . . , aλn

satisfies an atom θ(v1, . . . , vn, u1, . . . , um) in MX . By φ(v) being an atom of(MX , aβ)β<η, in that model one of the following holds for an arbitrary formulaψ(v):

(4) (∀v)(φ(v)→ ψ(v)) ,or (∀v)(φ(v)→ ¬ψ(v))

Given any formula ψ(v, v1, . . . , vn, u1, . . . , um) in VX , (∀v)(φ(v, u1, . . . , um) →ψ) is a formula in the variables v1, . . . , vn, u1, . . . , um, hence as θ(v1, . . . , vn, u1,. . . , um) is an atom, it follows that either

MX |= θ(v1, . . . , vn, u1, . . . , um)→ (∀v)(φ(v, u1, . . . , um)→ ψ)

orMX |= θ(v1, . . . , vn, u1, . . . , um)→ ¬(∀v)(φ(v, u1, . . . , um)→ ψ)

Suppose we are in the second case and assume towards a contradiction that(∃v)(φ(v, u1, . . . , um)→ ψ). Let v = b be such an element. By setting (v1, . . . , vn,u1, . . . , un) = (aδ1 , . . . , aδn , aλ1 , . . . , aλn), our choice of b yields that

(MX , aβ)β<η |= φ(b, cλ1 , . . . , cλn)→ ψ(b, cδ1 , . . . , cδn , cλ1 , . . . , cλn).

Now as φ(v, cλ1 , . . . , cλn) is our atom φ(v), (4) gives that we must have

(∀v)(φ(v, cλ1 , . . . , cλn)→ ψ(v, cδ1 , . . . , cδn , cλ1 , . . . , cλn)).

But by the choice of θ, (aδ1 , . . . , aδn , aλ1 , . . . , aλn) satisfies θ, and this contradictsthat MX |= θ(v1, . . . , vn, u1, . . . , um) → ¬(∀v)(φ(v, u1, . . . , um) → ψ). Hencewe can rewrite the second case above as:

MX |= θ(v1, . . . , vn, u1, . . . , um)→ (∀v)(φ(v, u1, . . . , um)→ ¬ψ).

Now Θ = θ(v1, . . . , vn, u1, . . . , um) ∧ φ(v, u1, . . . , um) is an atom in the theoryof MX , as Θ → θ. Note that ψ(v, v1, . . . , vn, u1, . . . , um) = ψ(v, v1, . . . , vn) isallowed, and for any such ψ, we still have that either Θ → ψ or Θ → ¬ψ.Define Θ′(v, v1, . . . , vn) = (∃u1, . . . , um)Θ(v, v1, . . . , vn, u1, . . . , um). Thus forall (v, v1, . . . , vn) such that Θ′(v, v1, . . . , vn) is true, we can pick (u1, . . . , um) =

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(d1, . . . , dm) such that Θ(v, v1, . . . , vn, d1, . . . , dm) is true, hence either Θ′(v, v1,. . . , vn) → ψ(v, v1, . . . , vn) or Θ′(v, v1, . . . , vn) → ¬ψ(v, v1, . . . , vn), so Θ′ is anatom inMX .

Finally, aδ1 , . . . , aδn , aη satisfies Θ′, so (3) holds for all β ≤ γ, and by induc-tion, it holds for all β < α · ω.

The previous two lemmas immediatly yield the following corollary.

Corollary 28. Let T be a theory in a countable vocabulary such that T is stablein power ω. Then for any modelM of T and every subset X ⊂M the completetheory ofMX has an atomic model.

Lemma 29. Suppose T is a complete ω-stable V-theory andM is an uncount-able model of T . Then there is a proper elementary extension N �M such thatevery countable set Γ(v) of formulas of VM which is realized in NM is realizedinMM .

Proof. We first prove the existence of an atomic model N which is a properelementary extension ofM.

We need a formula ψ(v) such that:

(1) The set A = {b ∈M :MM |= ψ(b)} is uncountable(2) For any formula φ(v) of VM exactly one of the sets

B = {b ∈M :MM |= ψ(b) ∧ φ(b)},C = {b ∈M :MM |= ψ(b) ∧ ¬φ(b)}is countable

Suppose that such a formula does not exist. Then for every formula ψ(v) whichis satisfied by ucountably many elements inM, (2) does not hold. Thus thereis a formula φ(v) such that both B and C are uncountable. Let ψ0 = ψ ∧φ andψ1 = ψ ∧ ¬φ. Then the following holds:

MM |= ψ0 → ψ,MM |= ψ1 → ψ,MM |= ¬(ψ0 ∧ ψ1).

Note that both ψ0 and ψ1 are satisfied by uncountably many elements, hencewe can repeat the argument above to get ψ00, ψ01 and so on. As in the proof ofLemma 26, this yields 2ω different sequences, hence 2ω different sets realized bydifferent elements, which can be extended to 2ω different types consistent withTh(MY ) for some countable Y ⊂M . There is furthermore a model NY whichrealizes all these types, but this contradicts the assumption that T is ω-stable.As there exist formulas which are satisfied by uncountable many elements, forexample ’v = v’, there must be a formula ψ(v) satisfying (1) and (2).

Now if c is a new constant symbol, let ∆ be the set of all sentences φ(c)in the vocabulary VM ∪ {c} such that for the formula φ(v), all but countablymany elements which satisfy ψ(v) satisfy φ(v). Note that given any finite subset∆′ ⊂ ∆, the set of all elements which satisfy ψ(v) but not φ(v) for some φ ∈ ∆′

is a union of finitely many countable sets, hence it is a countable set. Thus there

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are uncountably many elements which satisfy ψ(v) and φ(v) for every φ ∈ ∆′,and we can always find an interpretation of c such that the expansion of MM

models ∆′. Hence ∆′ is consistent, and by compactness, so is ∆.Note that Th(MM ) trivially is in ∆, as a sentence γ such thatMM |= γ is

trivially satisfied by all elements as a formula γ(v), hence the set of all b ∈ Msuch that MM |= ψ(b) ∧ ¬γ(b) is empty, and γ(c) = γ is in ∆. So γ ∈ ∆or ¬γ ∈ ∆ for each VM -sentence γ. Now each sentence γ containing the newconstant symbol c can be written as γ = φ(c), where φ is a formula of VM . IfC is countable for φ(v), φ(c) ∈ ∆. If B is countable for φ(v), C is countable for¬φ(v) and ¬φ(c) ∈ ∆. So by exactly one of B or C being countable for everyφ(v), γ ∈ ∆ or ¬γ ∈ ∆.

As ∆ is consistent, the Model existence theorem gives some VM ∪ {c}-structure OM∪{c} which models ∆. As γ ∈ ∆ or ¬γ ∈ ∆, for each VM ∪ {c}-sentence γ, ∆ is the complete theory of OM∪{c}. Note that O |= T , asT ⊂ Th(MM ) ⊂ ∆, so by Corollary 28, there is an atomic model NM∪{c}of ∆.

Let N be the reduct of NM∪{c} to V. By Th(M) ⊂ ∆ we get thatM≺ N .Furthermore, N is a proper extension ofM, as for each a ∈M , all but countablyelements satisfy the formula φ(v) = (¬a ≡ v), hence φ(c) ∈ ∆ and ∆ |= ¬c ≡ a.

Now we show that every countable set Γ(v) of formulas which is realized inthis NM is in fact realized in MM . Suppose b ∈ N satisfies Γ(v). By NM∪{c}being atomic, b satisfies an atomic formula φ(c, v). Now as NM∪{c} models thesentence (∃v)φ(c, v), it is consistent with the complete theory ∆, and must bea consequence of ∆. Also, ∆ |= φ(c, v) → γ(v) for all γ(v) ∈ Γ(v), for givensuch a formula γ(v), that φ(c, v) is an atom implies that either φ(c, v) → γ(v)or φ(c, v)→ ¬γ(v) is a consequence of the complete theory ∆, but as b satisfiesboth φ(c, v) and γ(v), the second implication is impossible.

Furthermore, since every consequence of ∆ can be derived in a finite sequencefrom some finite subset of ∆, and Γ(v) is countable, there is a countable ∆0 ⊂ ∆such that

∆0 |= (∃v)φ(c, v), and ∆0 |= φ(c, v)→ γ(v)

for all γ(v) ∈ Γ(v).By our construction, each δ(v) such that δ(c) ∈ ∆0 is satisfied by all but

countably many of the uncountably many elements which satisfy ψ(v) inMM ,hence that ∆0 is countable implies that only countably many elements in M donot satisfy all such δ(v). So there is a c0 ∈M that satisfy all δ(v), which givesthat (MM , c0) models ∆0. Thus MM models the existence of a d0 ∈ M suchthat φ(c0, d0) holds, hence d0 satisfies every γ(v) ∈ Γ(v), and Γ(v) is realized inMM .

We finally give the proof of Morley’s theorem.

Proof. (Morley’s Theorem) Let T be categorical in the uncountable power κ.By Lemma 22 it suffices to show that every model of T of power ω1 is saturated.Let M be any model of T of power ω1. By Lemma 24, T is ω-stable. Thus

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we can use Lemma 29 κ times, taking unions at limit ordinals, to obtain anelementary extension N �M of power κ such that every countable set Γ(v) offormulas of VM which is realized in NM is realized inMM . By Lemma 25, thatT is ω-stable and κ ≥ ω1 implies that T has a ω1-saturated model of power κ.As T is κ-categorical, N must be isomorphic to this model, hence ω1-saturated.Now, given any subset Y ⊂ M with |Y | < ω1 and any set Γ(v) of formulasconsistent with Th(MY ), and hence consistent with Th(NY ), Γ(v) is realizedin NY . But as |VY | ≤ ω, |Γ(v)| is countable, which implies that Γ(v) is realizedinMY , henceM is ω1-saturated.

References[1] C.C Chang and H.J. Keisler, Model Theory, Studies in Logic and the Foun-

dations of Mathematics, North-Holland Publishing Company, 2nd edition,1973.

[2] S. Hedman, A First Corse in Logic: An Introduction to Model Theory, ProofTheory, Computability, and Complexity, Oxford University press, 1st edi-tion, 2004.

[3] W. Hodges, Model Theory, Cambridge University press, 1st edition, 1993.

[4] M. Morley, Categoricity in power, Transactions of the American Mathemat-ical Society, vol. 114, pp.514-538, 1965.

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