A SUBSTITUTE FOR SATURATION IN ABSTRACT ELEMENTARY CLASSES

MONICA VANDIEREN

Abstract. This paper continues work of Shelah and Villaveces from [ShVi]. Shelah and Villaveces work towards a downward categoricity transfer theorem in abstract elementary classes (AEC) with no maximal models under GCH and a version of the weak diamond.

One of the main conjectures in [ShVi] was the uniqueness of limit models. Here we solve this problem under an additional assumption. Suppose that K is an AEC categorical in some λ above the Hanf number and that for μ < λ, K_{μ}^{am} is closed under unions of length < μ^{+}.

Theorem 0.1 (Uniqueness of Limit Models). Let θ_{1}, θ_{2} be limit ordinals < μ^{+}. If M_{1} is a (μ, θ_{1})-limit model over M and M_{2} is a (μ, θ_{2})- limit model over M, then M_{1} is isomorphic to M_{2}.

In order to prove the uniqueness theorem, we develop the theory of Galois-splitting in AECs. In particular we prove the extension property for Galois-splitting in AECs:

Theorem 0.2 (Extension Property for Splitting). Suppose that ga-tp(a/M) does not μ-split over N and that M is universal over N. For every M′ ∈ K_{μ}^{am} with M ≺_{K} M′, we have that there exists q ∈ ga-S(M′) such that q ⊇ ga-tp(a/M) and q does not μ-split over N.

We generalize Theorem 0.2 and prove the <^{c}-extension property for towers. This is also a partial solution to a problem from [ShVi]. In our proof of Theorem 0.1, we develop the concept of full towers. We show that the union of full towers is full.

1. Introduction

Shelah’s paper, [Sh 702] is based on a series of lectures given at Rutgers University. In the lectures, Shelah elaborates on open problems in model theory which he has attempted but which have not yet been solved. There Shelah refers to the subject of Section 13, “Classification of Non-elementary Classes,” as the major problem of model theory. He points out that one of the main steps in classifying non-elementary classes is the development of stability theory. In first order logic, solutions to Łǒs’ Conjecture produced machinery that advanced the study of stability theory. It is natural, then, to

Date: January 11, 2002. Preliminary Draft This paper is part of the author’s Ph.D. thesis written under the

guidance of Rami Grossberg.

consider a generalization of this conjecture as a test question for a proposed stability theory for AECs:

Conjecture 1.1 ([Sh 702]). If K is an abstract elementary class that is categorical in some λ > Hanf(K), then K is categorical in every μ > Hanf(K).

Despite the existence of over 500 published pages of partial results towards this conjecture, it remains very open. Since the mid-eighties, model theorists have approached Shelah’s conjecture from two different directions. Shelah, M. Makkai and O. Kolman attacked the conjecture with set theoretic assumptions (see [MaSh], [KoSh] and [Sh 472]). On the other hand, Shelah also looked at the conjecture under additional model theoretic assumptions in [Sh 394] and [Sh 600]. More recent work of Shelah and A. Villaveces [ShVi] profits from both model theoretic and set theoretic assumptions, however these assumptions are weaker than the hypotheses made in [MaSh], [KoSh], [Sh 472], [Sh 394], and [Sh 600]. A main feature of their context is that they work in AECs where the amalgamation property is not known to hold. This paper focuses on resolving problems from [ShVi]. Here we recall the context of [ShVi] (Assumptions 1.2.(1) through 1.2.(5)).

Assumption 1.2. We make the following assumptions for the remainder of the paper:

(1) K is an abstract elementary class,

(2) K has no maximal models,

(3) K is categorical in some λ > LS(K),

(4) GCH holds and

(5) Φ_{μ^{+}}(S_{θ}^{μ^{+}}) holds for every cardinal μ < λ and every regular θ with θ < μ^{+}.

Assumption 1.2.(5) is not explicitly made in [ShVi], but it is implicit in Hypothesis 1.3.8 from [ShVi]. We provide a complete proof of the theorem which uses Hypothesis 1.3.8 (see Theorem 4.7) and give an exposition of the strength of Assumption 1.2.5 in Section 4.

In 1985 Rami Grossberg made the following conjecture:

Conjecture 1.3. If K is an AEC categorical above the Hanf number, then every M ∈ K is an amalgamation base.

This conjecture encouraged Shelah to produce a partial solution to the categoricity conjecture under the assumption that every model M ∈ K is an amalgamation base [Sh 394]. This result directs future work towards the categoricity conjecture to solving Conjecture 1.3. The underlying goal of [ShVi] was to make progress towards Conjecture 1.3 under Assumption 1.2. Not knowing that every model is an amalgamation base presents several obstacles in applying known notions and techniques. For instance, there may exist some models over which we cannot even define the most basic notion of a type.

One approach to Conjecture 1.3 is to see if arguments from [KoSh] can be carried out in this more general context. Shelah and Kolman prove Conjecture 1.3 for L_{κ,ω} theories where κ is a measurable cardinal. They first introduce limit models as a substitute for saturated models, and then prove the uniqueness of limit models. A major objective of [ShVi] was to show the uniqueness of limit models:

Conjecture 1.4 (Uniqueness of Limit Models). Suppose Assumption 1.2 holds. For θ_{1}, θ_{2} < μ^{+} < λ, if M_{1} and M_{2} and (μ, θ_{1})-, (μ, θ_{2})-limit models over M, respectively, then M_{1} is isomorphic to M_{2}.

While limit models were used to prove that every model is an amalgamation base in [KoSh], limit models played a behind-the-scenes role in Shelah’s downward solution to the categoricity conjecture in [Sh 394]. Furthermore, there is evidence that the uniqueness of limit models provides a basis for the development of a notion of non-forking for abstract elementary classes. Such a notion could be used in the development of a stability theory for AECs.

In the Fall of 1999, I identified a gap in Shelah and Villaveces’ proof of uniqueness of limit models. As of the Fall of 2001, Shelah and Villaveces could not resolve the problem.

This paper includes an exposition of selected results from [ShVi] and includes a partial solution to the uniqueness of limit models (see Theorem 2.28). In order to prove Theorem 2.28, we prove several facts about splitting and full towers which stand alone.

The author would like to thank John Baldwin for his unlimited willingness to discuss and comment on this paper. Additionally the author would like to thank Olivier Lessmann and Andres Villaveces for their comments on preliminary versions of this paper.

2. Background

Definition 2.1. K is an abstract elementary class (AEC) iff K is a class of models for some vocabulary τ and is equipped with a binary relation, ⪯_{K} satisfying the following:

(1) Closure under isomorphisms.

(2) ⪯_{K} refines the submodel relation.

(3) ⪯_{K} is a partial order on K.

(4) If ⟨M_{i} | i < δ⟩ is a ≺_{K}-increasing and chain of models in K (a) ⋃_{i<δ} M_{i} ∈ K,

(b) for every j < δ, M_{j} ≺_{K} ⋃_{i<δ} M_{i} and

(c) if M_{i} ≺_{K} N for every i < δ, then ⋃_{i<δ} M_{i} ≺_{K} N.

(5) If M_{0}, M_{1} ⪯_{K} N and M_{0} is a submodel of M_{1}, then M_{0} ⪯_{K} M_{1}.

(6) (Downard Löwenheim-Skolem Axiom) There is a Löwenheim-Skolem number of K, denoted LS(K) which is the minimal κ such that for every N ∈ K and every A ⊂ N, there exists M with A ⊆ M ≺_{K} N of cardinality κ + |A|.

Notation 2.2. If λ is a cardinal and K is an abstract elementary class, K_{λ} is the collection of elements of K with cardinality λ.

Definition 2.3. For models M, N in an AEC, K, the mapping f : M → N is an ≺_{K}-embedding iff f is an injective L(K)-homomorphism and f[M] ⪯_{K} N.

Using the axioms of AEC, one can show that Axiom 4 has an alternative formulation (see [Sh 88] or Chapter 13 of [Gr]):

Proposition 2.4 (P.M. Cohn 1965). Let (I, ≤) be a directed set. If ⟨M_{t} | t ∈ I⟩ and {h_{t,s} | t ≤ s ∈ I} are such that

(1) for t ∈ I, M_{t} ∈ K

(2) for t ≤ s ∈ I, h_{t,s} : M_{t} → M_{s} is a ≺_{K}-embedding and

(3) for t_{1} ≤ t_{2} ≤ t_{3} ∈ I, h_{t_{1},t_{3}} = h_{t_{2},t_{3}} ∘ h_{t_{1},t_{2}} and h_{t,t} = id_{M_{t}}, then, whenever s = lim_{t∈I} t, there exist M_{s} ∈ K and ≺_{K}-mappings {h_{t,s} | t ∈ I} such that

h_{t,s} : M_{t} → M_{s} and
for t_{1} ≤ t_{2} ≤ s, h_{t_{1},s} = h_{t_{2},s} ∘ h_{t_{1},t_{2}} and h_{s,s} = id_{M_{s}}.

Definition 2.5. A partially ordered set (I, ≤) is directed iff for every a, b ∈ I, there exists c ∈ I such that a ≤ c and b ≤ c.

We will make use of a related lemma about directed limits:

Proposition 2.6. K^{≺_{K}} := {(N, M) | M, N ∈ K, M ≺_{K} N} is an abstract elementary class with L(K^{≺_{K}}) = L(K) ⋃{P } where P is a unary predicate and ≺_{K}^{≺_{K}} is defined by

(N, M) ≺_{K}^{≺_{K}} (N′, M′) ⇔ (N ≺_{K} N′ and M ≺_{K} M′).

From this proposition and the existence and uniqueness of direct limits for AECs, we have:

Lemma 2.7. Suppose that ⟨M_{t} ≺_{K} N_{t} | t ∈ I⟩ and ⟨f_{t,s} | t ≤ s ∈ I⟩ is a directed system with f_{t,s} : N_{t} → N_{s} and f_{t,s} ↾ M_{t} : M_{t} → M_{s}. If M^{∗} and N^{∗} are the direct limits of the directed sets ⟨M_{t}, f_{t,s} ↾ M_{t} | t ≤ s ∈ I⟩ and ⟨N_{t}, f_{t,s} | t ≤ s ∈ I⟩, respectively, then M^{∗} ≺_{K} N^{∗} and f_{t,sup(I)} ↾ M_{t} → M^{∗}.

We will use Lemma 2.7 as well as the trivial observation (Claim 2.8) in the proof of the Conjecture 1.4.

Claim 2.8. If ⟨N_{t} | t < s⟩ and ⟨f_{r,t} | r < t < s⟩ form a directed system and for every r ≤ t < s we have that N_{t} = N_{r} = N and f_{r,t} ∈ Aut(N). Then the direct limit (N_{s}, ⟨f_{t,s} | t ≤ s⟩) is such that f_{t,s} : N_{t} ≅ N_{s} for every t ≤ s. Moreover we can choose N_{s} = N.

The following gives a characterization of AECs as PC-classes. Theorem 2.10 is often referred to as Shelah’s Presentation Theorem.

Definition 2.9. A class K of structures is called a P C −class if there exists a language L_{1}, a first order theory, T_{1}, in the language, L_{1}, and a collection of types without parameters, Γ, such that L_{1} is an expansion of L(K) and

K = P C(T_{1}, Γ, L) := {M ↾ L : M ⊧ T_{1} and M omits all types from Γ}.

When |T_{1}| + |L_{1}| + |Γ| + ℵ_{0} = μ, we say that K is P C_{μ}.

Theorem 2.10 (Lemma 1.8 of [Sh 88] or [Gr]). If (K, ≺_{K}) is an AEC, then there exists μ ≤ 2^{LS(K)} such that K is P C_{μ}.

In Section 3 we will see that this presentation of AECs as PC-classes allows us to construct Ehrenfuecht-Mostowski models.

Definition 2.11. Let K be an abstract elementary class.

(1) Let μ,κ_{1} and κ_{2} be cardinals with μ ≤ κ_{1}, κ_{2}. We say that M ∈ K_{μ} is a (κ_{1}, κ_{2})-amalgamation base if for every N_{1} ∈ K_{κ_{1}} and N_{2} ∈ K_{κ_{2}} and g_{i} : M → N_{i} for (i = 1, 2), there are ≺_{K}-embeddings f_{i}, (i = 1, 2) and a model N such that the following diagram commutes:

\xymatrix{N_{1} \ar[r]_{f_{1}} & N \\ M \ar[u]^{g_{1}} \ar[r]_{g_{2}} & N_{2} \ar[u]_{f_{2}}}

(2) We say that a model M ∈ K_{μ} is an amalgamation base if M is a (μ, μ)-amalgamation base.

(3) We write K^{am} for the class of amalgamation bases which are in K.

(4) We say K satisfies the amalgamation property iff for every M ∈ K, M is an amalgamation base.

Remark 2.12. We get an equivalent definition of amalgamation base, if we additionally require that g_{i} ↾ M = id_{M} for i = 1, 2, in the definition above. See [Gr] for details.

Amalgamation bases are central in the definition of types. Since we are not working in a fixed logic, we will not define types as collections of formulas. Instead, we will define types as equivalence classes with respect to images under ≺_{K}-mappings:

Definition 2.13. For triples (ā_{l}, M_{l}, N_{l}) where ā_{l} ∈ N_{l} and M_{l} ⪯_{K} N_{l} ∈ K for l = 0, 1, we define a binary relation E as follows: (ā_{0}, M_{0}, N_{0})E(ā_{1}, M_{1}, N_{1}) iff M_{0} = M_{1} and there exists N ∈ K and ≺_{K}-mappings f_{0}, f_{1} such that f_{l} : N_{l} → N and f_{l} ↾ M = id_{M} for l = 0, 1 and f_{0}(ā_{0}) = f_{1}(ā_{1}):

\xymatrix{N_{1} \ar[r]_{f_{1}} & N \\ M \ar[u]^{id} \ar[r]_{id} & N_{2} \ar[u]_{f_{2}}}

Remark 2.14. E is an equivalence relation on the set of triples of the form (ā, M, N) where M ⪯_{K} N, ā ∈ N and M, N ∈ K_{μ}^{am} for fixed μ ≥ LS(K).

In AEC with the amalgamation property, we are often limited to speak of types only over models. Here we are further restricted to deal with types only over models which are amalgamation bases.

Definition 2.15. Let μ ≥ LS(K) be given.

(1) For M, N ∈ K_{μ}^{am} and ā ∈ ^{ω>}N, the Galois-type of ā in N over M, written ga-tp(ā/M, N), is defined to be (ā, M, N)/E.

(2) For M ∈ K_{μ}^{am}, ga-S^{1}(M) := {ga-tp(a/M, N) | M ⪯ N ∈ K_{μ}^{am}, a ∈ N}.

(3) We say p ∈ ga-S(M) is realized in N whenever M ≺_{K} N and there exist a ∈ N and N′ ∈ K_{μ}^{am} such that p = (a, M, N′)/E.

Remark 2.16. We refer to these types as Galois-types to distinguish them from notions of types defined as a collection of formulas.

Proposition 2.17 (see [Gr]). When K = Mod(T) for T a complete first order theory, the above definition of ga-tp(a/M, N) coincides with the classical first order defintion where c and a have the same type over M iff for every first order formula φ(x, b̄) with parameters from M,

⊧ φ(c, b̄) ↔⊧ φ(a, b̄).

Proof. By Robinson’s Consistency Theorem. ⊣

Definition 2.18. We say that K is stable in μ if for every M ∈ K_{μ}^{am}, | ga-S^{1}(M)| = μ.

Fact 2.19 (Fact 2.1.3 of [ShVi]). Since K is categorical in λ, for every μ < λ, we have that K is stable in μ.

Definition 2.20. (1) Let κ be a cardinal. We say N is κ-universal over M iff for every M′ ∈ K_{κ} with M ≺_{K} M′ there exists a ≺_{K}-embedding g : M′ → N such that g ↾ M = id_{M}:

\xymatrix{M′ \ar[dr]_{g} \\ M \ar[u]^{id} \ar[r]_{id} & N}

(2) We say N is universal over M iff N is ∥M∥-universal over M.

Universal extensions exist in many stable contexts [Sh 600]. We provide a reference to [ShVi] where a proof is provided from stronger assumptions including categoricity and GCH (see Lemma 2.21).

Lemma 2.21 (Theorem 1.3.1 from [ShVi]). For every μ with LS(K) < μ < λ, if M ∈ K_{μ}^{am}, then there exists M′ ∈ K_{μ}^{am} such that M′ is universal over M.

Notice that the following proposition asserts that it is unreasonable to prove a stronger existence statement than Lemma 2.21, without having proved the amalgamation property.

Proposition 2.22. If M′ is universal over M, then M is an amalgamation base.

As mentioned in the introduction, limit models were introduced by Kolman and Shelah in [KoSh]. After proving the uniqueness of limit models in their context, Shelah and Kolman derive the Amalgamation Property. The main goal of this paper is to prove the uniqueness of limit models in the context of [ShVi] under an additional assumption.

Definition 2.23. For M′, M ∈ K_{μ} and σ a limit ordinal with σ < μ^{+}, we say that M′ is a (μ, σ)-limit over M iff there exists a ≺_{K}-increasing and continuous sequence of models ⟨M_{i} ∈ K_{μ} | i < σ⟩ such that

(1) M ⪯_{K} M_{0},

(2) M′ = ⋃_{i<σ} M_{i}

(3) for i < σ, M_{i} is an amalgamation base and

(4) M_{i+1} is universal over M_{i}.

Remark 2.24. (1) Notice that in Definition 2.23, for i < σ and i a limit

ordinal, M_{i} is a (μ, i)-limit model.

(2) Notice that Condition (4) implies Condition (3) of Definition 2.23.

Definition 2.25. We say that M′ is a (μ, σ)-limit iff there is some M ∈ K such that M′ is a (μ, σ)-limit over M.

Notation 2.26. (1) For μ a cardinal and σ a limit ordinal with σ < μ^{+},

we write K_{μ}^{σ} for the collection of (μ, σ)-limit models of K.

(2) We define

K_{μ}^{∗} := {M ∈ K | M is a (μ, θ)−limit model for some limit ordinal θ < μ^{+}}.

as the collection of limit models of K.

Limit models also exist in certain abstract elementary classes. By repeated applications of Lemma 2.21, the existence of (μ, ω)-limit models can be proved:

Proposition 2.27 (Theorem 1.3.1 from [ShVi]). Let μ be a cardinal such that μ < λ. For every M ∈ K_{μ}^{am}, there exists M′ ∈ K such that M ≺_{K} M′ and M′ is a (μ, ω)-limit over M.

In order to extend this argument further to yield the existence of (μ, σ)- limits for arbitrary limit ordinals σ < μ^{+}, we need to be able to verify that limit models are in fact amalgamation bases. We will examine this in Section 4.

While the existence of certain limit models is relatively easy to derive from the categoricity assumption, the uniqueness of limit models is more difficult. Here we recall two easy uniqueness facts which state that limit models of the same length are isomorphic:

Proposition 2.28 (Fact 1.3.6 from [ShVi]). Let μ ≥ LS(K) and σ < μ^{+}. If M_{1} and M_{2} are (μ, σ)-limits over M, then there exists an isomorphism g : M_{1} → M_{2} such that g ↾ M = id_{M}. Moreover if M_{1} is a (μ, σ)-limit over M_{0}; N_{1} is a (μ, σ)-limit over N_{0} and g : M_{0} ≅ N_{0}, then there exists a ≺_{K}-mapping, ĝ, extending g such that ĝ : M_{1} ≅ N_{1}.

Proposition 2.29 (Fact 1.3.7 from [ShVi]). Let μ be a cardinal and σ a limit ordinal with σ < μ^{+} ≤ λ. If M is a (μ, σ)-limit model, then M is a (μ, cf(σ))-limit model.

A more challenging uniqueness question, which was a major goal of [ShVi], is:

Conjecture 2.30. Suppose K is categorical in some λ > LS(K). Let μ ≥ LS(K) and θ_{1}, θ_{2} < μ^{+} ≤ λ be given. If M_{1} is a (μ, θ_{1})-limit over M and M_{2} is a (μ, θ_{2})-limit over M, then there exists g : M_{1} ≅ M_{2} with g ↾ M = id_{M}.

A main result of this paper, Corollary 8.1, is a solution to this conjecture under Assumptions 1.2.(1)-(5) plus K_{μ}^{am} is closed under unions of length < μ^{+}.

We will need one more notion of limit model, which will appear implicitly in the proofs of Theorem 6.7 and Theorem 7.8. This notion is a mild extension of the notion of limit models already defined:

Definition 2.31. Let μ be a cardinal < λ, we say that M̌ is a (μ, μ^{+}) limit over M iff there exists a ≺_{K}-increasing and continuous chain of models ⟨M_{i} ∈ K_{μ}^{am} | i < μ^{+}⟩ satisfying

(1) M_{0} = M

(2) ⋃_{i<μ^{+}} M_{i} = M̌ and

(3) for i < μ^{+}, M_{i+1} is universal over M_{i}

Remark 2.32. While it is known that (μ, θ)-limit models are amalgamation bases when θ < μ^{+}, it is open as to whether or not (μ, μ^{+})-limits are amalgamation bases. To avoid confusion between these two concepts of limit models, we will always denote (μ, μ^{+})-limit models with aˇ above the model’s name (ie. M̌)

The existence of (μ, μ^{+})-limit models follows from the fact that (μ, θ)- limit models are amalgamation bases when θ < μ^{+}, see Corollary 4.9. The uniqueness of (μ, μ^{+})-limit models (Proposition 2.33) can be shown using an easy back and forth construction as in the proof of Proposition 2.28.

Proposition 2.33. Suppose M̌_{1} and M̌_{2} are (μ, μ^{+})-limits over M_{1} and M_{2}, respectively. If there exists an isomorphism h : M_{1} ≅ M_{2}, then h can be extended to an isomorphism g : M̌_{1} ≅ M̌_{2}.

(μ, μ^{+})-limit models turn to be useful because they are weakly model homogeneous (in Section 9 we will make this definition more explicit) providing a replacement for monster models:

Proposition 2.34. If M̌ is a (μ, μ^{+})-limit, then for every N ≺_{K} M̌ with N ∈ K_{μ}^{am}, we have that M̌ is universal over N. Moreover, M̌ is a (μ, μ^{+})- limit over N.

3. Ehrenfeucht-Mostowski Models

Since K has no maximal models, K has models of cardinality Hanf(K). Then by Theorem 3.1, we can construct Ehrenfeucht-Mostowski models.

Theorem 3.1 (Claim 0.6 of [Sh 394] or see [Gr]). Assume that K is an AEC that contains a model of cardinality ≥ ℶ_{(2^{LS(K)})^{+}}. Then, there is a Φ, proper for linear orders, such that for linear orders I ⊆ J we have that

(1) EM(I, Φ) ↾ L(K) ≺_{K} EM(J, Φ) ↾ L(K) and

(2) ∥EM(I, Φ) ↾ L(K)∥ = |I| + LS(K).

We describe an index set which appears often in work toward the categoricity conjecture. This index set was used in [KoSh], [Sh 394] and [ShVi].

Notation 3.2. Let α < λ be given. We define

I_{α} := { η ∈ ^{ω}α : {n < ω | η[n] ≠ 0} is finite }

Associate with I_{α} the lexicographical ordering ⋖. If X ⊆ α, we write I_{X} := {η ∈ ^{ω}X : {n < ω | η[n] ≠ 0} is finite}}.

The following proposition is proved in several papers e.g. [ShVi].

Proposition 3.3. If M ≺_{K} EM(I_{λ}, Φ) ↾ L(K) is a model of cardinality μ^{+} with μ^{+} < λ, then there exists a ≺_{K}-mapping f : M → EM(I_{μ^{+}}, Φ) ↾ L(K).

A variant of this universality property is (implicit in Lemma 3.7 of [KoSh]):

Proposition 3.4. Suppose κ is a regular cardinal. If M ≺_{K} EM(I_{κ}, Φ) ↾ L(K) is a model of cardinality < κ and N ≺_{K} EM(I_{λ}, Φ) ↾ L(K) is an extension of M of cardinality ∥M∥, then there exists a ≺_{K}-embedding f : N → EM(I_{κ}, Φ) ↾ L(K) such that f ↾ M = id_{M}.

4. Amalgamation Bases

Since the amalgamation property for abstract elementary classes is inherent in the definition of types, most work towards understanding AECs has been under the assumption that the class K has the amalgamation property. In [ShVi], Shelah and Villaveces begin to tackle the categoricity problem with an approach that does not require the amalgamation property as an assumption. Shelah and Villaveces, however, prove a weak amalgamation property, which they refer to as density of amalgamation bases, summarized here:

Theorem 4.1 (Theorem 1.2.4 from [ShVi]). For every M ∈ K_{<λ}, there exists N ∈ K_{∥M∥}^{am} with M ≺_{K} N.

We can now improve Lemma 2.21 slightly. This improvement is used throughout this paper.

Lemma 4.2. For every μ with LS(K) < μ < λ, if M ∈ K_{μ}^{am}, N ∈ K and ā ∈ ^{μ^{+}>}N are such that M ≺_{K} N, then there exists M ā ∈ K_{μ}^{am} such that M ā is universal over M and M ⋃ ā ⊆ M ā.

Proof. By Axiom 6 of AEC, we can find M′ ≺_{K} N_{λ} of cardinality μ containing M ⋃ ā. Applying Theorem 4.1, there exists an amalgamation base of cardinality μ, say M′′, extending M′. By Lemma 2.21 we can find a universal extension of M′′ of cardinality μ, say M ā.

Notice that M ā is also universal over M. Why? Suppose M^{∗} is an extension of M of cardinality μ. Since M is an amalgamation base we can amalgamate M′′ and M^{∗} over M. WLOG we may assume that the amalgam, M^{∗∗}, is an extension of M′′ of cardinality μ and f^{∗} : M^{∗} → M^{∗∗} with f^{∗} ↾ M = id_{M}.

\xymatrix{M^{∗} \ar[r]_{f^{∗∗}} & M^{∗∗} \\ M \ar[u]^{id} \ar[r]_{id} & M′′ \ar[u]_{id}}

Now, since M ā is universal over M′′, there exists a ≺_{K}-mapping g such that g : M^{∗∗} → M ā with g ↾ M′′ = id_{M′′}. Notice that g ∘ f^{∗} gives us the desired mapping of M^{∗} into M ā. ⊣

While Theorem 4.1 asserts the existence of amalgamation bases, it is unknown (in this context) what characterizes amalgamation bases. Shelah and Villaveces have claimed that every limit model is an amalgamation base (Fact 1.3.10 of [ShVi]), under the hypothesis of ♢_{S_{cf(μ)}^{μ^{+}}} (which does not follow

from GCH). We believe that Assumption 1.2.(5) is needed. We provide a proof that every limit model is an amalgamation base under this additional assumption.

Definition 4.3. Let θ be a regular ordinal < μ^{+}. We denote

S_{θ}^{μ^{+}} := {α < μ^{+} | cf(α) = θ}.

Definition 4.4. For μ a cardinal and S ⊆ μ^{+} a stationary set, Φ_{μ^{+}}(S) is said to hold iff for all F : ^{λ^{+}>}2 → 2 there exists g : λ^{+} → 2 so that for every f : λ^{+} → 2 the set

{δ ∈ S | F (f ↾ δ) = g(δ)} is stationary.

We will be using a consequence of Φ_{μ^{+}}(S), called Θ_{μ^{+}}(S) (see [Gr]).

Definition 4.5. For μ a cardinal S ⊆ μ^{+} a stationary set, Θ_{μ^{+}}(S) is said to hold if and only if for all families of functions

{f_{η} : η ∈ ^{μ^{+}}2 where f_{η} : μ^{+} → μ^{+}}

and for every club C ⊆ μ^{+}, there exist η ≠ ν ∈ ^{μ^{+}}2 and there exists a δ ∈ C ∩ S such that

(1) η ↾ δ = ν ↾ δ,

(2) f_{η} ↾ δ = f_{ν} ↾ δ and

(3) η[δ] ≠ ν[δ].

Fact 4.6. ♢_{μ^{+}}(S_{θ}^{μ^{+}}) ⟹ Φ_{μ^{+}}(S_{θ}^{μ^{+}}) ⟹ Θ_{μ^{+}}(S_{θ}^{μ^{+}}).

Theorem 4.7. If M is a (μ, θ)-limit for some θ with θ < μ^{+} ≤ λ, then M is an amalgamation base.

This Theorem appears in [ShVi] with a one-line proof. J. Baldwin has requested that I supply the complete proof here. Before we begin the proof notice that:

Remark 4.8 (Invariance). By Axiom 1 of AEC, if M is an amalgamation base and f is an ≺_{K}-embedding, then f(M) is an amalgamation base.

Proof of Theorem 4.7. Given μ, suppose that θ is the minimal infinite ordinal < μ^{+} such that there exists a model M which is a (μ, θ)-limit and not an amalgamation base. Notice that by Proposition 2.29, we may assume that cf(θ) = θ.

Now we define by induction on the length of η ∈ ^{μ^{+}>}2 a tree of structures, ⟨M_{η} | η ∈ ^{μ^{+}>}2⟩, satisfying:

(1) for η ⋖ ν ∈ ^{μ^{+}>}2, M_{η} ≺_{K} M_{ν}

(2) for l(η) a limit ordinal with cf(l(η)) ≤ θ, M_{η} = ⋃_{α<l(η)} M_{η↾α}

(3) for η ∈ ^{α}2 with α ∈ S_{θ}^{μ^{+}}, (a) M_{η} is a (μ, θ)-limit model

(b) M_{ηˆ 0}, M_{ηˆ 1} cannot be amalgamated over M_{η}

(c) M_{ηˆ 0} and M_{ηˆ 1} are amalgamation bases of cardinality μ

(4) for η ∈ ^{α}2 with α ∉ S_{θ}^{μ^{+}}, (a) M_{η} is an amalgamation base

(b) M_{ηˆ 0}, M_{ηˆ 1} are universal over M_{η} and

(c) M_{ηˆ 0} and M_{ηˆ 1} are amalgamation bases of cardinality μ (it may

be that M_{ηˆ 0} = M_{ηˆ 1} in this case). This construction is possible: η = ⟨⟩: By Theorem 4.1, we can find M′ ∈ K_{μ}^{am} such that M ≺_{K} M′.

Define M_{⟨⟩} := M′.

l(η) is a limit ordinal: When cf(l(η)) > θ, let M′_{η} := ⋃_{α<l(η)} M_{η↾α}. M′_{η} is not necessarily an amalgamation base, but for the purposes of this construction, continuity at such limits is not important. Thus we can find an extension of M′_{η}, say M_{η}, of cardinality μ where M_{η} is an amalgamation base.

For η with cf(l(η)) ≤ θ, we require continuity. Define M_{η} := ⋃_{α<l(η)} M_{η↾α}.

We need to verify that if l(η) ∉ S_{θ}^{μ^{+}}, then M_{η} is an amalgamation base. In fact, we will show that such a M_{η} will be a (μ, cf(l(η)))-limit model. Let ⟨α_{i} | i < cf(l(η))⟩ be an increasing and continuous sequence of ordinals converging to l(η) such that cf(α_{i}) < θ for every i < cf(l(η)). Condition (4b) guarantees that for i < cf(l(η)), M_{η↾α_{i+1}} is universal over M_{η↾α}. Additionally, condition (2) ensures us that ⟨M_{η↾α_{i}} | i < cf(l(η))⟩ is continuous. This sequence of models witnesses that M_{η} is a (μ, cf(l(η)))-limit model. By our minimal choice of θ, we have that (μ, cf(l(η)))-limit models are amalgamation bases.

ηˆ i where l(η) ∈ S_{θ}^{μ^{+}}: We first notice that M_{η} := ⋃_{α<l(η)} M_{η↾α} is a

(μ, θ)-limit model. Why? Since l(η) ∈ S_{θ}^{μ^{+}} and θ is regular, we can find an increasing and continuous sequence of ordinals, ⟨α_{i} | i < θ⟩ converging to l(η) such that for each i < θ we have that cf(α_{i}) < θ. Condition (4b) of the construction guarantees that for each i < θ, M_{η↾α_{i+1}} is universal over M_{η↾α_{i}}. Thus ⟨M_{η↾α_{i}} | i < θ⟩ witnesses that M_{η} is a (μ, θ)-limit model.

Since M_{η} is a (μ, θ)-limit, we can fix an isomorphism f : M ≅ M_{η}. By Remark 4.8, M_{η} is not an amalgamation base. Thus there exist M_{ηˆ 0} and M_{ηˆ 1} extensions of M_{η} which cannot be amalgamated over M_{η}. WLOG we can choose, M_{ηˆ 0} and M_{ηˆ 1} to be elements of K_{μ}^{am}.

ηˆ i where l(η) ∉ S_{θ}^{μ^{+}}: Since M_{η} is an amalgamation base, we can choose M_{ηˆ 0} and M_{ηˆ 1} to be extensions of M_{η} such that M_{ηˆ l} ∈ K_{μ}^{am} and M_{ηˆ l} is universal over M_{η}, for l = 0, 1.

This completes the construction. For every η ∈ ^{μ^{+}}2, define M_{η} := ⋃_{α<μ^{+}} M_{η↾α}. By categoricity in λ and Proposition 3.3, we can fix a ≺_{K}- mapping g_{η} : M_{η} → EM(I_{μ^{+}}, Φ) ↾ L(K) for each η ∈ ^{μ^{+}}2. Now apply Θ_{μ^{+}}(S_{θ}^{μ^{+}}) to find η, ν ∈ ^{μ^{+}}2 and α ∈ S_{θ}^{μ^{+}} such that

· ρ := η ↾ α = ν ↾ α, · η[α] = 0, ν[α] = 1 and · g_{η} ↾ M_{ρ} = g_{ν} ↾ M_{ρ}.

By Axiom 6 (the Löwenheim-Skolem property) of AEC, there exists N ≺_{K} EM(I_{μ^{+}}, Φ) ↾ L(K) of cardinality μ such that the following diagram commutes:

\xymatrix{M_{ρˆ 1} \ar[r]_{g_{ν}↾M_{ρˆ 1}} & N \\ M_{ρ} \ar[u]^{id} \ar[r]_{id} & M_{ρˆ 0} \ar[u]_{g_{η}↾M_{ρˆ 0}}}

Notice that g_{η} ↾ M_{ρˆ 0}, g_{ν} ↾ M_{ρˆ 1} and N witness that M_{ρˆ 0} and M_{ρˆ 1} can be amalgamated over M_{ρ}. Since l(ρ) = α ∈ S_{θ}^{μ^{+}}, we contradict condition (3b) of the construction.

⊣

Corollary 4.9 (Existence of limit models and (μ, μ^{+})-limit models). For every cardinal μ and limit ordinal θ with θ ≤ μ^{+} ≤ λ, if M is an amalgamation base of cardinality μ, then there exists M′ ∈ K which is a (μ, θ)-limit over M.

Proof. By repeated applications of Lemma 2.21 and Theorem 4.7. ⊣

5. Weak Disjoint Amalgamation

Shelah and Villaveces prove a version of weak disjoint amalgamation in an attempt to prove an extension property for towers. We will be using weak disjoint amalgamation and provide a proof here for completeness.

Theorem 5.1 (Weak Disjoint Amalgamation [ShVi]). Given λ > μ ≥ LS(K) and α, θ_{0} < μ^{+} with θ_{0} regular. If M_{0} is a (μ, θ_{0})-limit and M_{1}, M_{2} ∈ K_{μ} are ≺_{K}-extensions of M_{0}, then for every b̄ ∈ ^{α}( M_{1}\M_{0}), there exist M_{3}, a model, and h, a ≺_{K}-embedding, such that

(1) h : M_{2} → M_{3};

(2) h ↾ M_{0} = id_{M_{0}} and

(3) h(M_{2}) ∩ b̄ = ∅ (equivalently h(M_{2}) ∩ M_{1} = ∅).

Shelah and Villaveces provide a proof of this theorem in [ShVi]. It has been suggested that I elaborate on the proof here:

Proof. Suppose that M_{0}, M_{1}, M_{2} and b̄ ∈ M_{1} form a counter-example. Since M_{0} is a μ amalgamation base, we may assume that there exists M^{∗} ∈ K_{μ} with M_{1}, M_{2} ≺_{K} M^{∗}. Let θ be regular and < μ^{+} such that M_{0} is a (μ, θ)- limit. We define a ≺_{K}-increasing and continuous sequence of models ⟨N_{i} | i < μ^{+}⟩ satisfying:

(1) N_{i} ∈ K_{μ}^{am}

(2) N_{i+1} is universal over N_{i} and

(3) when cf(i) = θ, we additionally define N_{i}^{1}, N_{i}^{2}, N_{i}^{∗} and b̄_{i} ∈ N_{i}^{1} such that there exists an isomorphism f : M^{∗} ≅ N_{i}^{∗} with f(M_{0}) = N_{i}, f(M_{1}) = N_{i}^{1}, f(M_{2}) = N_{i}^{2} and f(b̄) = b̄_{i}.

The construction is possible by Lemma 2.21, Theorem 4.7 and Proposition 2.28.

Let N_{μ^{+}} := ⋃_{i<μ^{+}} N_{i}. Since K is categorical in λ, Proposition 3.3 allows us to find a ≺_{K} g : N_{μ^{+}} → EM(I_{μ}^{+}, Φ) ↾ L(K). So WLOG, we may assume that N_{μ^{+}} ≺_{K} EM(I_{μ}^{+}, Φ) ↾ L(K).

Let E ⊆ μ^{+} be a club such that

δ ∈ E ⇒ N_{δ} ≺_{K} EM(I_{δ}, Φ) ↾ L(K).

For each i ∈ S_{θ}^{μ^{+}}, choose a Skolem-term τ_{i} and a sequence of indices α_{i,0}, …, α_{i,n_{i}−1} such that b̄ = τ_{i}(α_{i,0}, …, α_{i,n_{i}−1}). Let m_{i} < n_{i} be such

k < m_{i} ⇔ α_{i,k} ∈ I_{i}.

Set α_{i,<m_{i}} := ⟨α_{i,k} | 0 ≤ k < m_{i}⟩ and α_{i,≥m_{i}} := ⟨α_{i,k} | m_{i} ≤ k < n_{i}⟩. Let δ_{0} ∈ E ∩ S_{θ}^{μ^{+}}. For every δ_{1}, with δ_{0} < δ_{1} < μ^{+}. Define g_{δ_{1}} to be the ≺_{K}-mapping from

EM(I_{δ_{1}}, Φ) ↾ L(K) to EM(I_{μ^{+}}, Φ) ↾ L(K) induced by the mapping from μ^{+} to μ^{+} defined by

j ↦ \{ \begin{matrix}j & if j < δ_{0} \\ δ_{1} + j & if δ_{0} ≤ j < δ_{1} \end{matrix}

Subclaim 5.2. There exists some δ_{1} < μ^{+} such that g_{δ_{1}}(N_{δ_{0}}^{1}) ∩ b̄_{δ_{0}} = ∅.

Proof. Suppose the claim fails. Then for every δ with δ_{0} < δ < μ^{+}, there exists a Skolem term σ_{δ} and a sequence of indices

β_{δ,0}, …, β_{δ,m_{δ}−1}, β_{δ,m_{δ}}, …, β_{δ,n_{δ}−1}

such that

k < m_{δ} ⇔ β_{δ,k} ∈ I_{δ_{0}}

and b = σ_{δ_{0}}(β_{δ,0}, …, β_{δ,n_{δ}−1}).

Let β_{δ,<m_{δ}} := ⟨β_{δ,k} | 0 ≤ k < m_{δ}⟩ and β_{δ,≥m_{δ}} := ⟨β_{δ,k} | m_{δ} ≤ k < n_{δ}⟩. Notice that

EM(I_{μ^{+}}, Φ) ↾ L(K) ⊧ b = σ_{δ_{0}}(β_{δ,<m_{δ}}; β_{δ,≥m_{δ}}) = τ_{δ_{0}}(α_{δ_{0},<m_{δ_{0}}} ; α_{δ_{0},≥m_{δ_{0}}} ).

By our definition of g_{δ}, we have that

(∗)_{δ} k ≥ m_{δ} ⇔ β_{δ,k} ∈ I_{δ\δ_{1}∪δ_{0}}.

In other words when k ≥ m_{δ}, every term from the sequence β_{δ,k} which is larger than δ_{0} is also larger than δ_{1}. Thus, for k ≥ m_{δ}, β_{δ,k} and α_{δ_{0}≥m_{δ_{0}}} are not intertwined above δ_{0}.

Since all our indices are finite sequences and δ_{0} is a limit ordinal, there exists δ^{∗} < δ_{1} such that α_{δ_{0},<m_{δ_{0}}} , β_{δ,<m_{δ}} ∈ I_{δ^{∗}}. This allows us to find a sequence of indices α^{∗} ∈ I_{δ_{0}} which have the same type over I_{δ^{∗}} (with respect to the lexicographical ordering) as α_{δ_{0},≥m_{δ_{0}}} . So by indiscernibility and (∗)_{δ}

EM(I_{μ^{+}}, Φ) ↾ L(K) ⊧ σ_{δ_{0}}(β_{δ,<m_{δ}}; β_{δ,≥m_{δ}}) = τ_{δ_{0}}(α_{δ_{0},<m_{δ_{0}}} ; α^{∗}).

Notice that σ_{δ_{0}}(β_{δ,<m_{δ}}; β_{δ,≥m_{δ}}) = b. Thus we have found a way to construct b from I_{δ_{0}} (by τ_{δ_{0}}(α_{δ_{0},<m_{δ_{0}}} ; α^{∗})). This contradicts our choice of b ∉ EM(I_{δ_{0}}) ↾ L(K).

⊣

Let g′ be an order preserving mapping and α_{2} an ordinal such that

(1) α_{1} < α_{2} < μ^{+},

(2) g′ : α_{1} → α_{2},

(3) g′ ↾ α_{0} = id_{α_{0}} and

(4) g′(α_{1}) ∩ α_{1} = α_{0}.

Let g^{∗} be the ≺_{K}-mapping taking M_{1}^{∗} to EM(I_{α_{2}}, Φ) ↾ L(K) determined by g′. Notice that by Subclaim 5.2, g^{∗} witnesses that M_{0}^{∗}, M_{1}^{∗}, M_{2}^{∗} and b̄^{∗} can be weakly disjointly amalgamated.

⊣

Let us state an easy corollary of Theorem 5.1 that will simplify future constructions:

Corollary 5.3. Suppose μ, M_{0}, M_{1}, M_{2} and b̄ are as in the statement of Theorem 5.1. If M̌ is universal over M_{1}, then there exists a ≺_{K}-mapping h such that

(1) h : M_{2} → M̌,

(2) h ↾ M_{0} = id_{M_{0}} and

(3) h(M_{2}) ∩ b̄ = ∅ (equivalently h(M_{2}) ∩ M_{1} = ∅).

Proof. By Theorem 5.1, there exists a ≺_{K}-mapping g and a model M_{3} of cardinality μ such that

· g : M_{2} → M_{3}
· g ↾ M_{0} = id_{M_{0}}
· g(M_{2}) ∩ b̄ = ∅ and
· M_{1} ≺_{K} M_{3}.

Since M̌ is universal over M_{1}, we can fix a ≺_{K}-mapping f such that

· f : M_{3} → M̌ and
· f ↾ M_{1} = id_{M_{1}}

Notice that h := g ∘ f is the desired mapping from M_{2} into M̌.

⊣

6. <_{μ,α}^{b}-extension property for K_{μ,α}^{∗}

Shelah introduced towers in [Sh 48] and [Sh 87b] as a tool to build a model of cardinality μ^{+} from models of cardinality μ. Roughly speaking, we will use an increasing and continuous chain (of length σ_{1}) of towers which have length σ_{2} to construct an array of models of height σ_{1} and width σ_{2} in such a way that the union will simultaneously be a (μ, σ_{1})-limit model and a (μ, σ_{2})-limit model. The construction of such a model is sufficient to prove the uniqueness of limit models by Lemma 2.28. First we fix the following notation:

Definition 6.1 (Towers Definition 3.1.1 of [ShVi]). Let μ > LS(K) and α, θ < μ^{+} (1)

K_{μ,α} := \{ (M̄, ā) | \begin{matrix}(M̄, ā) := (⟨M_{γ} | γ < α⟩, ⟨a_{γ} | γ < α⟩); \\ M̄ is ≺_{K} −increasing; \\ for every γ < α, a_{γ} ∈ M_{γ+1}\M_{γ}; \\ for every γ < α, M_{γ} ∈ K_{μ} \end{matrix} \}

(2) K_{μ,α}^{θ} := {(M̄, ā) ∈ K_{μ,α} | for every γ < α, M_{γ} is a (μ, θ)-limit}

(3) K_{μ,α}^{∗} := ⋃_{θ<μ^{+}} K_{μ,α}^{θ}

Fact 6.2 (Fact 3.17 from [ShVi]). Suppose K is categorical in λ. Given λ > μ ≥ LS(K), α < μ^{+} and θ a regular cardinal with θ < μ^{+}, we have that K_{μ,α}^{θ} ≠ ∅.

In order to have some control during the construction of the array of models, an ordering on the towers is defined. Namely, we would like to have continuity at limit stages of the chain. So we would like that the union of the increasing chain of towers, should be a tower. The problem is that the models at the limit stages may not be limit models. This motivates the following ordering on towers:

Definition 6.3 (Definition 3.1.3 of [ShVi]). For (M̄, ā), (N̄, b̄) ∈ K_{μ,α}^{∗} we say that

(1) (M̄, ā) ≤_{μ,α}^{b} (N̄, b̄) if and only if (a) ā = b̄;

(b) for every γ < α, M_{γ} ⪯_{K} N_{γ} and

(c) whenever M_{γ} ≺_{K} N_{γ}, then N_{γ} is universal over M_{γ}.

(2) (M̄, ā) <_{μ,α}^{b} (N̄, b̄) if and only if (M̄, ā) ≤_{μ,α}^{b} (N̄, b̄) and for every γ < α, M_{γ} ≠ N_{γ}.

Remark 6.4. If ⟨(M̄, ā)_{σ} ∈ K_{μ,α}^{∗} | σ < γ⟩ is a <_{μ,α}^{b}-increasing and continuous chain with γ < μ^{+}, then ⋃_{σ<γ}(M̄, ā)_{σ} ∈ K_{μ,α}^{∗}. Why? Notice that for i < α, M_{i,γ} := ⋃_{σ<γ} M_{i,σ} is a limit model, witnessed by ⟨M_{i,σ} | σ < γ⟩.

In order to construct a non-trivial chain of towers, we need to be able to take proper <_{μ,α}^{b}-extensions.

Definition 6.5. We say the <_{μ,α}^{b}-extension property holds iff for every (M̄, ā) ∈ K_{μ,α}^{∗} there exists (M̄′, ā) ∈ K_{μ,α}^{∗} such that (M̄, ā) <_{μ,α}^{b} (M̄′, ā).

Remark 6.6. Shelah and Villaveces claim the <_{μ,α}^{b}-extension property as Fact 3.19(1) in [ShVi]. Their proof does not converge. As of the Fall of 2001, they were unable to produce a proof of this claim.

We will prove the <_{μ,α}^{b}-extension property for a particular class of towers:

Theorem 6.7 (The <_{μ,α}^{b}-extension property for nice towers). For every nice (M̄, ā) ∈ K_{μ,α}^{∗}, there exists a nice tower (M̄′, ā) ∈ K_{μ,α}^{∗} such that (M̄, ā) <_{μ,α}^{b} (M̄′, ā)

Definition 6.8. (⟨M_{i} | i < α⟩, ā) ∈ K_{μ,α}^{∗} is nice provided that for every limit ordinal i < α, we have that ⋃_{j<i} M_{j} is an amalgamation base.

Notice that in the definition of towers, we do not require continuity at limit ordinals i of the sequence of models. This allows for towers in which M_{i} ≠ ⋃_{j<i} M_{j}. Since we only require that M_{i} is an amalgamation base, there are towers which are not necessarily nice.

Remark 6.9. Why isn’t Theorem 6.7 sufficient? What is needed to conclude the uniqueness of limit models?

(1) We would like to ultimately construct a <_{μ,α}^{b}-increasing and continuous chain of towers. The problem occurs at limit stages. Suppose we have constructed a <_{μ,α}^{b}-increasing and continuous sequence of towers, ⟨(M̄^{n}, ā) | n ≤ ω⟩. We may not be able to extend this chain of towers because we are not guaranteed that the limit (M̄^{ω}, ā) is a nice tower. A sufficient resolution would be to show that every tower has a ≤_{μ,α}^{b}-extension which is nice.

(2) Alternatively, if we try to carry out the arguments of [ShVi] towards the uniqueness of limit models using only nice towers, we see that the only obstacle is to prove that reduced towers are continuous.

(3) If we make an additional assumption, that K_{μ}^{am} is closed under unions of length < μ^{+}, then we can adjust the proof of Theorem 6.7 and eliminate the assumption of niceness, thereby getting the full <_{μ,α}^{b}- extension property.

Proof of Theorem 6.7. Let μ be a cardinal and α a limit ordinal such that α < μ^{+} ≤ λ. Let a nice tower (M̄, ā) ∈ K_{μ,α}^{∗} be given. Denote by M_{α} a model in K_{μ}^{am} extending ⋃_{i<α} M_{i}. Fix M̌ to be a (μ, μ^{+})-limit model over M_{α}.

We define by induction on i < α a sequence of models ⟨M′_{i} | i < α⟩ and a sequences of ≺_{K}-mappings, ⟨f′_{j,i} | j < i < α⟩ and ⟨f̌_{j,i} | j < i < α⟩ such that for i ≤ α:

(1) (⟨f′_{j,i}(M′_{j}) | j ≤ i⟩, ā ↾ i) is a <_{μ,i}^{b}-extension of (M̄, ā) ↾ i,

(2) (⟨M′_{j} | j < i⟩, ⟨f′_{j,i} | j ≤ i⟩) forms a directed system,

(3) M′_{i} is universal over M_{i},

(4) M′_{i+1} is universal over f′_{i,i+1}(M′_{i}),

(5) f′_{j,i} ↾ M_{j} = id_{M_{j}},

(6) M′_{i} ≺_{K} M̌,

(7) f_{j,i} can be extended to an automorphism of M̌, f̌_{j,i}, for j ≤ i and (8) (⟨M̌_{j} = M̌ | j < i⟩, ⟨f̌_{k,j} | k ≤ j < i⟩) forms a directed system. Notice that the M′_{i}’s will not necessarily form an extension of the tower

(M̄, ā). Rather, for each i < α, we find some image of ⟨M_{j} | j < i⟩ which will extend the initial segment of length i of (M̄, ā) (see condition (1) of the construction).

The construction is possible:

i = 0: Since M_{0} is an amalgamation base, we can find M′′_{0} ∈ K_{μ}^{∗} (a first approximation of the desired M′_{0}) such that M′′_{0} is universal over M_{0}. By Corollary 5.3 (applied to M_{0}, M_{α}, M′′_{0} and ā), we can find a ≺_{K}-mapping h : M′′_{0} → M̌ such that h ↾ M_{0} = id_{M_{0}} and h(M′′_{0})∩ā = ∅. Set M′_{0} := h(M′′_{0}), f′_{0,0} := id_{M′_{0}} and f̌_{0,0} := idM̌.

i = j + 1: Suppose that we have completed the construction of all k ≤ j. Since M′_{j} and M_{j+1} are both ≺_{K}-substructures of M̌, we can get M′′_{j+1} (a first approximation to the desired M′_{j+1}) such that M′′_{j+1} ∈ K_{μ}^{∗} is universal over M′_{j} and universal over M_{j+1}. How? By the Downward Löwenheim Skolem Axiom (Axiom 6) of AEC and the density of amalgamation bases (Theorem 4.1), we can find an amalgamation base L of cardinality μ such that M′_{j}, M_{j+1} ≺_{K} L. By Lemma 2.21 and Corollary 4.9, there exists M′′_{j+1}, a (μ, ω)-limit over L.

Subclaim 6.10. M′′_{j+1} is universal over M′_{j} and is universal over M_{j+1}.

Proof. It suffices to show that when L_{0} ≺_{K} L_{1} ≺_{K} L are amalgamation bases of cardinality μ, if L is universal over L_{1}, then L is universal over L_{0}. Let L′ be an extension of L_{0} of cardinality μ. Since L_{0} is an amalgamation base, we can find an amalgam L′′ such that the following diagram commmutes:

\xymatrix{L′ \ar[r]_{h} & L′′ \\ L_{0} \ar[u]^{id} \ar[r]_{id} & L_{1} \ar[u]_{id}}

Since L is universal over L_{1}, there exists g : L′′ → L with g ↾ L_{1} = id_{L_{1}}. Notice that g ∘ h : L′ → L with g ∘ h ↾ L_{0} = id_{L_{0}}. ⊣

M′′_{j+1} may serve us well if it does not contain any a_{l} for j + 1 ≤ l < α, but this is not guaranteed. So we need to make an adjustment. Notice that M̌ is universal over M_{j+1}. Thus we can apply Corollary 5.3 to M_{j+1}, M_{α}, M′′_{j+1} and ⟨a_{l} | j + 1 ≤ l < α⟩. This yields a ≺_{K}-mapping h such that

· h : M′′_{j+1} → M̌

· h ↾ M_{j+1} = id_{M_{j+1}} and

· h(M′′_{j+1}) ∩ {a_{l} | j + 1 ≤ l < α} = ∅.

Set M′_{j+1} := h(M′′_{j+1}), f′_{j+1,j+1} = id_{M_{j+1}}, f̌_{j+1,j+1} = idM̌ and f′_{j,j+1} := h ↾ M′_{j}. Since M̌ is a (μ, μ^{+})-limit over both M′_{j} and f′_{j,j+1}(M′_{j}), by Proposition 2.33 we can extend f′_{j,j+1} to an automorphism of M̌, denoted by f̌_{j,j+1}.

To guarantee that we have a directed system, for k < j, define f′_{k,j+1} := f′_{j,j+1} ∘ f′_{k,j} and f̌_{k,j+1} := f̌_{j,j+1} ∘ f̌_{k,j}.

i is a limit ordinal: Suppose that (⟨M′_{j} | j < i⟩, ⟨f′_{k,j} | k ≤ j < i⟩) and (⟨M̌ | j < i⟩, ⟨f̌_{k,j} | k ≤ j < i⟩) have been defined. Since they are both directed systems, we can take direct limits.

Claim 6.11. We can choose the direct limits (M_{i}^{∗}, ⟨f_{j,i}^{∗} | j ≤ i⟩) and (M̌_{i}^{∗}, ⟨f̌_{j,i}^{∗} | j ≤ i⟩) of (⟨M′_{j} | j < i⟩, ⟨f′_{k,j} | k ≤ j < i⟩) and (⟨M̌ | j < i⟩, ⟨f̌_{k,j} | k ≤ j < i⟩) respectively such that

· M_{i}^{∗} ≺_{K} M̌_{i}^{∗}

· f_{j,i}^{∗} is an automorphism for every j ≤ i
· M̌_{i}^{∗} = M̌
· f_{j,i}^{∗} ↾ M_{j} = id_{M_{j}} for every j < i.

Proof. We will realize each of these conditions one at a time. By Lemma 2.7 we may choose direct limits (M_{i}^{∗∗}, ⟨f_{j,i}^{∗∗} | j ≤ i⟩) and (M̌_{i}^{∗∗}, ⟨f̌_{j,i}^{∗∗} | j ≤ i⟩) such that M_{i}^{∗∗} ≺_{K} M̌_{i}^{∗∗}. By Claim 2.8 we have that for every j ≤ i, f̌_{j,i}^{∗∗} is an automorphism and M̌_{i}^{∗∗} = M̌. Notice that this forms a direct limit satisfying the first three properties.

Subclaim 6.12. ⟨f_{j,i}^{∗∗} ↾ M_{j} | j < i⟩ is increasing.

Proof. Let j < k < i be given. By construction

f′_{j,k} ↾ M_{j} = id_{M_{j}}.

An application of f_{k,i}^{∗∗} yields

f_{k,i}^{∗∗} ∘ f′_{j,k} ↾ M_{j} = f_{k,i}^{∗∗} ↾ M_{j}.

By the definition of directed limits, we have

f_{j,i}^{∗∗} ↾ M_{j} = f_{k,i}^{∗∗} ∘ f′_{j,k} ↾ M_{j} = f_{k,i}^{∗∗} ↾ M_{j}.
⊣

By the subclaim, we have that g := ⋃_{j<i} f_{j,i}^{∗∗} ↾ M_{j} is a partial autmorphism of M̌ from ⋃_{j<i} M_{j} onto ⋃_{j<i} f_{j,i}^{∗∗}(M_{j}). Since M̌ is a (μ, μ^{+})- limit model and since ⋃_{j<i} M_{j} is an amalgamation base we can extend g to G ∈ Aut(M̌) by Proposition 2.33. Now consider the direct limit defined by M_{i}^{∗} := G^{−1}(M_{i}^{∗∗}) with ⟨f_{j,i}^{∗} := G^{−1} ∘ f_{j,i}^{∗∗} | j < i⟩ and f_{i,i}^{∗} = id_{M_{i}^{∗}} and the direct limit M̌_{i}^{∗} := M̌ with ⟨f̌_{j,i}^{∗} := G^{−1} ∘ f̌_{j,i}^{∗∗} | j < i⟩ and f̌_{i,i}^{∗} := id_{N_{i}^{∗}}. Notice that f_{j,i}^{∗} ↾ M_{j} = G^{−1} ∘ f_{j,i}^{∗∗} ↾ M_{j} = id_{M_{j}} for j < i.

⊣

By Condition (4) of the construction, notice that M_{i}^{∗} is a (μ, i)-limit model witnessed by ⟨f_{j,i}^{∗}(M′_{j}) | j < i⟩. Hence M_{i}^{∗} is an amalgamation base. Since M_{i}^{∗} and M_{i} both live inside of M̌, we can find M′′_{i} ∈ K_{μ}^{∗} which is universal over M_{i} and universal over M_{i}^{∗}.

By Corollary 5.3 applied to M_{i}, M_{α}, M′′_{i} and ⟨a_{l} | l ≤ i < α⟩ we can find h : M′′_{i} → M̌ such that h ↾ M_{i} = id_{M_{i}} and h(M′′_{i}) ∩ {a_{l} | i ≤ l < α} = ∅.

Set M′_{i} := h(M′′_{i}), f′_{i,i} := id_{M_{i,i}}, f̌_{i,i} := idM̌ and for j < i, f′_{j,i} := h ∘ f_{j,i}^{∗}. We need to verify that for j ≤ i, f′_{j,i}(M′_{j}) ⋂{a_{l} | j ≤ l < α} = ∅. Clearly by our application of weak disjoint amalgamation, we have that for every l with i ≤ l < α and every j ≤ i, a_{l} ∉ M′_{i} ⊇ f′_{j,i}(M′_{j}). Suppose that j < i and l is such that j ≤ l < i. By construction a_{l} ∉ f′_{j,l+1}(M′_{j}) and f′_{l+1,i}(a_{l}) = a_{l}. So f′_{j,i}(M′_{j}) = f′_{l+1,i} ∘ f′_{j,l+1}(M′_{j}) implies that a_{l} ∉ f′_{j,i}(M′_{j}).

Notice that for every j < i, M̌ is a (μ, μ^{+})-limit over both M′_{j} and f′_{j,i}(M′_{j}). Thus by the uniqueness of (μ, μ^{+})-limit models, we can extend f′_{j,i} to an automorphism of M̌, denoted by f̌_{j,i}.

The construction is enough: Let M′_{α} and ⟨f_{i,α} | i ≤ α⟩ be the direct limit of (⟨M′_{i} | i < α⟩, ⟨f_{j,i} | j ≤ i < α⟩). By Subclaim 6.12 we may assume that ⋃_{i<α} M_{i} ≺_{K} M′_{α}. It is routine to verify that (⟨f_{i,α}(M′_{i}) | i < α⟩, ā) is a <_{μ,α}^{b}-extension of (M̄, ā).

⊣

7. <_{μ,α}^{c} Extension Property for K_{μ,α}^{∗}

Unfortunately, it seems that working with the relatively simple K_{μ,α}^{∗} towers is not sufficient to carry out the proof for the uniqueness of limit models. Shelah and Villaveces have idenitified a more elaborate tower. The extension proprerty for these towers is also missing from [ShVi]. We provide a partial solution to this extension property, analagous to the solution for K_{μ,α}^{∗} in the previous section.

While there is no known notion of forking in the general context of AECs, Shelah has generalized the notion of splitting to AECs in [Sh 394]:

Definition 7.1. Let μ be a cardinal with μ < λ. For M ∈ K^{am} and p ∈ ga-S(M), we say that p μ-splits over N iff N ≺_{K} M and there exist N_{1}, N_{2} ∈ K_{μ} and a ≺_{K}-mapping h : N_{1} ≅ N_{2} such that

(1) h(p ↾ N_{1}) ≠ p ↾ N_{2},

(2) N ≺_{K} N_{1}, N_{2} ≺_{K} M and

(3) h ↾ N = id_{N}.

This notion behaves nicely under stability assumptions. Shelah and Villaveces draw a connection between categoricity and superstability-like properties with:

Theorem 7.2 (Theorem 2.2.1 from [ShVi]). Assume that (1) ⟨M_{i} | i ≤ σ⟩ is ≺_{K}-increasing and continuous,

(2) for all i ≤ σ, M_{i} ∈ K_{μ}^{am},

(3) for all i < σ, M_{i+1} is universal over M_{i}

(4) cf(σ) = σ ≤ μ^{+} ≤ λ and

(5) p ∈ ga-S(M_{σ}). Then there exists i < σ such that p does not μ-split over M_{i}.

We derive the following extension property for non-splitting types:

Theorem 7.3 (Extension of non-splitting types). Suppose that M, M̌ ∈ K_{μ} is universal over N and ga-tp(a/M, M̌) does not μ-split over N. Let M̌ ∈ K_{μ} be a (μ, μ^{+})-limit containing ā ⋃ M.

Let M′ be an extension of M of cardinality μ. Then there exists a ≺_{K}- mapping f such that f : M′ → M̌, f ↾ M = id_{M} and ga-tp(a/f(M′)) does not μ-split over N.

Proof. Notice that M̌ is universal over M. So we may assume that M̌ contains M′. Since M is universal over N, there exists a ≺_{K} mapping h′ : M′ → M with h′ ↾ N = id_{N}. Let M′′ ≺_{K} M̌ be an extension of M′ such that h′ can be extended to h′′ : M′′ ≅ M. Further extend h′′ to h so that dom(h) ≺_{K} M̌ and a ∈ rge(h). By invariance, ga-tp(h^{−1}(a)/M′′) does not μ-split over N.

Subclaim 7.4. ga-tp(h^{−1}(a)/M) = ga-tp(a/M).

Proof. Let N_{1} := M′′ and N_{2} = M. Let p := ga-tp(h^{−1}(a)/M′′). Consider the mapping h : N_{1} ≅ N_{2}. Since p does not μ-split over N, h(p ↾ N_{1}) = p ↾ N_{2}. Let us calculate this

h(p ↾ N_{1}) = ga-tp(h(h^{−1}(a))/g(M′′) = ga-tp(a/M).

While,

p ↾ N_{2} = ga-tp(h^{−1}(a)/M).

Thus ga-tp(h^{−1}(a)/M) = ga-tp(a/M) as required. ⊣

From the subclaim, we can find a ≺_{K}-mapping g and a model M^{∗} ≺_{K} M̌ such that g : M′′ → M^{∗}, g ↾ M = id_{M} and g ∘ h^{−1}(a) = a. Notice that ga-tp(a/g(M′′), M̌) does not μ-split over M. ⊣

Definition 7.5.

+ K_{μ,α}^{∗} := \{ (M̄, ā, N̄) | \begin{matrix}(M̄, ā) ∈ K_{μ,α}^{∗}; \\ N̄ = ⟨N_{i} | i + 1 < α⟩; \\ for every i + 1 < α, N_{i} ≺_{K} M_{i},; \\ M_{i} is universal over N_{i} and; \\ ga-tp(a_{i}, M_{i}, M_{i+1}) does not μ-split over N_{i}. \end{matrix} \}

Similar to the case of K_{μ,α}^{∗} we define an ordering,

Definition 7.6. For (M̄, ā, N̄) and (M̄′, ā′, N̄′) ∈ ^{+} K_{μ,α}^{∗}, we say (M̄, ā, N̄) <_{μ,α}^{c} (M̄′, ā′, N̄′) iff

(1) (M̄, ā) <_{μ,α}^{b} (M̄′, ā′)

(2) N̄ = N̄′ and

(3) for every i < α, ga-tp(a_{i}/M′_{i+1}, M′_{i+1}) does not μ-split over N_{i}.

Notation 7.7. We say that (M̄, ā, N̄) is nice iff when i is a limit ordinal ⋃_{j<i} M_{j} is an amalgamation base.

The following theorem is a partial solution to a problem from [ShVi]:

Theorem 7.8 (The <_{μ,α}^{c}-extension property for nice towers). If (M̄, ā, N̄) ∈ + K_{μ,α}^{∗} is nice, then there exists a nice (M̄′, ā, N̄′) ∈ ^{+} K_{μ,α}^{∗} such that (M̄, ā, N̄) <_{μ,α}^{c} (M̄′, ā, N̄′).

Proof. Let μ be a cardinal and α a limit ordinal such that α < μ^{+} ≤ λ. Let (M̄, ā, N̄) ∈ ^{+} K_{μ,α}^{∗} be given. Denote by M_{α} a model in K_{μ}^{am} extending ⋃_{i<α} M_{i}. Fix M̌ to be a (μ, μ^{+})-limit model over M_{α}.

We will define by induction on i < α a sequence of models ⟨M′_{i} | i < α⟩ and a sequences of ≺_{K}-mappings, ⟨f′_{j,i} | j < i < α⟩ and ⟨f̌_{j,i} | j < i < α⟩ such that for i ≤ α:

(1) (⟨f′_{j,i}(M′_{j}) | j ≤ i⟩, ā ↾ i, N̄ ↾ i) is a <_{μ,i}^{c} of (M̄, ā, N̄) ↾ i,

(2) (⟨M_{j} | j < i⟩, ⟨f′_{j,i} | j ≤ i⟩) forms a directed system,

(3) M′_{i} is universal over M_{i},

(4) M′_{i+1} is universal over f′_{i,i+1}(M′_{i}),

(5) f′_{j,i} ↾ M_{j} = id_{M_{j}},

(6) M′_{i} ≺_{K} M̌,

(7) f_{j,i} can be extended to an autmorphism of M̌, f̌_{j,i}, for j ≤ i and (8) (⟨M̌ | j < i⟩, ⟨f̌_{k,j} | k ≤ j < i⟩) forms a directed system. The construction is enough: We can take M′_{α} and ⟨f′_{i,α} | i < α⟩ to be the

direct limit of (⟨M′_{i} | i < α⟩, ⟨f′_{j,i} | j ≤ i < α⟩). Since f′_{j,i} ↾ M_{j} = id_{M_{j}}, for every j ≤ i < α, we may assume that f′_{i,α} ↾ M_{i} = id_{M_{i}} for every i < α. Notice that (⟨f′_{i,α}(M′_{i}) | i < α⟩, ā) is a <_{μ,α}^{c}-extension of (M̄, ā).

The construction is possible:

i = 0: Since M_{0} is an amalgamation base, we can find M′′_{0} ∈ K_{μ}^{∗} (a first approximation of the desired M′_{0}) such that M′′_{0} is universal over M_{0}. By Theorem 7.3, we may assume that ga-tp(a_{0}/M′′_{0}) does not μ-split over N_{0} and M′′_{0} ≺_{K} M̌. Since a_{0} ∉ M_{0} and ga-tp(a_{0}/M_{0}) does not μ-split over N_{0}, we know that a_{0} ∉ M′′_{0}. But, we might have that for some l > 0, a_{l} ∈ M′′_{0}. We use weak disjoint amalgamation to avoid {a_{l} | 0 < l < α}. By the Downward Löweneim-Skolem Axiom for AECs (Axiom 6) we can choose M^{2} ∈ K_{μ} such that M′′0, M_{1} ≺_{K} M^{2} ≺_{K} M̌.

By Corollary 5.3 (applied to M_{1}, M_{α}, M^{2} and ⟨a_{l} | 0 < l < α⟩), we can find a ≺_{K}-mapping f such that

· f : M^{2} → M̌ · f ↾ M_{1} = id_{M_{1}} · f(M^{2}) ∩ {a_{l} | 0 < l < α} = ∅

Define M′_{0} := f(M′′_{0}). Notice that a_{0} ∉ M′_{0} because a_{0} ∉ M′′_{0} and f(a_{0}) = a_{0}. Clearly M′_{0} ∩ {a_{l} | 0 ≤ l < α} = ∅. We need only verify that ga-tp(a_{0}/M′_{0}) does not μ-split over N_{0}. By invariaince, ga-tp(a_{0}/M′′_{0}) does not μ-split over N_{0} implies that ga-tp(f(a_{0})/f(M′′_{0})) does not μ-split over N_{0}. But recall f(a_{0}) = a_{0} and f(M′′_{0}) = M′_{0}. Thus ga-tp(a_{0}/M′_{0}) does not μ-split over N_{0}.

Set f̌_{0,0} := idM̌ and f′_{0,0} := id_{M′_{0}}.

i = j + 1: Suppose that we have completed the construction of all k ≤ j. Since M′_{j}, M_{j+1} ≺_{K} M̌, we can apply the Downward-Löwenheim Axiom for AECs to find M′′′_{j+1} (a first approximation to M′_{j+1}) a model of cardinality μ extending both M′_{j} and M_{j+1}. WLOG we may assume that M′′′_{j+1} is a limit model of cardinality μ and M′′′_{j+1} is universal over M_{j+1} and M′_{j}. By Theorem 7.3, we can find a ≺_{K} mapping f : M′′′_{j+1} → M̌ such that f ↾ M_{j+1} = id_{M_{j+1}} and ga-tp(a_{j+1}/f(M′′′_{j+1})) does not μ-split over N_{j+1}. Set M′′_{j+1} := f(M′′′_{j+1}).

Subclaim 7.9. a_{j+1} ∉ M′′_{j+1}

Proof. Suppose that a_{j+1} ∈ M′′_{j+1}. Since M′_{j+1} is universal over N_{j+1}, there exists a ≺_{K}-mapping, g : M′′_{j+1} → M_{j+1} such that g ↾ N_{j+1} = id_{N_{j+1}}. Since ga-tp(a_{j+1}/M′′_{j+1}) does not μ-split over N_{j+1}, we have that

ga-tp(a_{j+1}/g(M′′_{j+1}) = ga-tp(g(a_{j+1})/g(M′′_{j+1}).

Notice that because g(a_{j+1}) ∈ g(M′′_{j+1}), we have that a_{j+1} ∈ g(M′′_{j+1}). But g(M′′_{j+1}) ≺_{K} M_{j+1}. This contradicts the definition of towers: a_{j+1} ∉ M_{j+1}.

⊣

M′′_{j+1} may serve us well if it does not contain any a_{l} for j + 1 ≤ l < α, but this is not guaranteed. So we need to make an adjustment. Let M^{2} be a model of cardinality μ such that M_{j+2}, M′′_{j+1} ≺_{K} M^{2} ≺_{K} M̌. Notice that M̌ is universal over M_{j+2}. Thus we can apply Corollary 5.3 to M_{j+2}, M_{α}, M^{2} and ⟨a_{l} | j + 2 ≤ l < α⟩. This yields a ≺_{K}-mapping h such that

· h : M^{2} → M̌

· h ↾ M_{j+2} = id_{M_{j+2}} and

· h(M^{2}) ∩ {a_{l} | j + 2 ≤ l < α} = ∅. Set M′_{j+1} := h(M′′_{j+1}). Notice that by invariance, ga-tp(a_{j+1}/M′′_{j+1})

does not μ-split over N_{j+1} implies that ga-tp(h(a_{j+1})/h(M′′_{j+1}) does not μ-split over h(N_{j+1}). Recalling that h ↾ M_{j+2} = id_{M_{j+2}} we have that ga-tp(a_{j+1}/M′′_{j+1}) does not μ-split over N_{j+1}. We need to verify that a_{j+1} ∉ M′_{j+1}. This holds because a_{j+1} ∉ M′′_{j+1} and h(a_{j+1}) = a_{j+1}.

Set f′_{j+1,j+1} = id_{M_{j+1,j+1}} and f̌_{j+1,j+1} = idM̌ and f′_{j,j+1} := h ∘ f ↾ M′_{j}. Since M̌ is a (μ, μ^{+})-limit over both M′_{j} and f′_{j,j+1}(M′_{j}), we can extend f′_{j,j+1} to an automorphism of M̌, denoted by f̌_{j,j+1}.

To guarantee that we have a directed system, for k < j, define f′_{k,j+1} := f′_{j,j+1} ∘ f′_{k,j} and f̌_{k,j+1} := f̌_{j,j+1} ∘ f̌_{k,j}.

i is a limit ordinal: Suppose that (⟨M′_{j} | j < i⟩, ⟨f′_{k,j} | k ≤ j < i⟩) and (⟨M̌ | j < i⟩, ⟨f̌_{k,j} | k ≤ j < i⟩) have been defined. Since they are both directed systems, we can take direct limits. By Claim 6.11, we may assume that (M_{i}^{∗}, ⟨f_{j,i}^{∗} | j < i⟩) and (M̌, ⟨f̌_{j,i}^{∗} | j < i⟩) are the respective direct limits such that M_{i}^{∗} ≺_{K} M̌ and ⋃_{j<i} M_{j} ≺_{K} M_{i}^{∗}. By Condition (4) of the construction, notice that M_{i}^{∗} is a (μ, i)-limit model witnessed by ⟨f_{j,i}^{∗}(M′_{j}) | j < i⟩. Hence M_{i}^{∗} is an amalgamation base. Since M_{i}^{∗} and M_{i} both live inside of M̌, we can find M′′′_{i} ∈ K_{μ}^{∗} which is universal over M_{i} and universal over M_{i}^{∗}.

By Theorem 7.3 we can find a ≺_{K}-mapping f : M′′′_{i} → M̌ such that f ↾ M_{i} = id_{M_{i}} and ga-tp(a_{i}/f(M′′′_{i})) does not μ-split over N_{i}. Set M′′_{i} := f(M′′′_{i}). By a similar argument to Subclaim 7, we can see that a_{i} ∉ M′′_{i}.

M′′_{i} may contain some a_{l} when i ≤ l < α. We need to make an adjustment using weak disjoint amalgamtion. Let M^{2} be a model of cardinality μ such that M′′_{i}, M_{i+1} ≺_{K} M^{2} ≺_{K} M̌. By Corollary 5.3 applied to M_{i}, M_{α}, M^{2} and ⟨a_{l} | i < l < α⟩ we can find h : M′′_{i} → M̌ such that h ↾ M_{i+1} = id_{M_{i+1}} and h(M^{2}) ∩ {a_{l} | i < l < α} = ∅.

Set M′_{i} := h(M′′_{i}). We need to verify that a_{i} ∉ M′_{i} and ga-tp(a_{i}/M′_{i}) does not μ-split over N_{i}. Since a_{i} ∉ M′′_{i} and h(a_{i}) = a_{i}, we have that a_{i} ∉ h(M′′_{i}) = M′_{i}. By invariance of non-splitting, ga-tp(a_{i}/M′′_{i}) not μsplitting over N_{i} implies that ga-tp(h(a_{i})/h(M′′_{i})) does not μ-split over h(N_{i}). Recalling our definition of h and M′_{i}. This yields ga-tp(a_{i}/M′_{i}) does not μ-split over N_{i}.

Set f′_{i,i} := id_{M_{i},i}, f̌_{i,i} := idM̌ and for j < i, f′_{j,i} := h ∘ f ∘ f_{j,i}^{∗}.

Notice that for every j < i, M̌ is a (μ, μ^{+})-limit over both M′_{j} and f′_{j,i}(M′_{j}). Thus by the uniqueness of (μ, μ^{+})-limit models, we can extend f′_{j,i} to an automorphism of M̌, denoted by f̌_{j,i}.

⊣

8. Uniqueness of Limit Models

Recall the running assumptions:

(1) K is an abstract elementary class,

(2) K has no maximal models,

(3) K is categorical in some λ > LS(K),

(4) GCH and Φ_{μ^{+}}(S_{θ}^{μ^{+}}) holds for every cardinal μ < λ and every regular θ with θ < μ^{+}.

Using Theorem 7.8, arguments as in [ShVi] and some new results we can conclude the uniqueness of limit models. This is a partial solution to a conjecture from [ShVi].

Theorem 8.1 (Uniqueness of Limit Models). Suppose K_{μ}^{am} is closed under unions of length < μ^{+}. Let μ be a cardinal θ_{1}, θ_{2} limit ordinals such that θ_{1}, θ_{2} < μ^{+} ≤ λ. If M_{1} and M_{2} are (μ, θ_{1}) and (μ, θ_{2}) limit models over M, respectively, then there exists an isomorphism f : M_{1} ≅ M_{2} such that f ↾ M = id_{M}.

We begin working towards the theorem by introducing a more general notion of towers:

Definition 8.2 (Definition 3.3.1 of [ShVi]). For 𝔘 a set of intervals of ordinals < μ^{+}, let

+ K_{μ,𝔘}^{∗} := \{ (M̄, ā, N̄) | \begin{matrix}M̄ = ⟨M_{i} | i ∈ u for some interval u ∈ 𝔘⟩; \\ M̄ is ≺_{K} increasing, but not \\ necessarily continuous; \\ a_{i} ∈ M_{i+1}\M_{i} when i, i + 1 ∈ ⋃ 𝔘; \\ N̄ = ⟨N_{i} | i ∈ ⋃ 𝔘⟩; \\ M_{i} is universal over N_{i} when i ∈ ⋃ 𝔘 and \\ ga-tp(a_{i}, M_{i}, M_{i+1}) does not μ − split over N_{i} \end{matrix} \}

Notice that these scattered towers are in some sense subtowers of the towers ^{+} K_{μ,α}^{∗}. Hence we can consider the restriction of <_{μ,α}^{c} to this class:

Definition 8.3 (Definition 3.3.2 of [ShVi]). Let 𝔘_{1}, 𝔘_{2} be such that ⋃ 𝔘_{1} ⊆ ⋃ 𝔘_{2}. Let (M̄^{l}, ā^{l}, N̄^{l}) ∈ ^{+} K_{μ,𝔘_{l}}^{∗} for l = 1, 2. (M̄^{1}, ā^{1}, N̄^{1}) ≤^{c} (M̄^{2}, ā^{2}, N̄^{2}) iff for i ∈ ⋃ 𝔘_{1},

(1) M_{i}^{1} ⪯_{K} M_{i}^{2}, a_{i}^{1} = a_{i}^{2} and N_{i}^{1} = N_{i}^{2} and

(2) if M_{i}^{1} ≠ M_{i}^{2}, then M_{i}^{2} is universal over M_{i}^{1}. We say that (M̄^{1}, ā^{1}, N̄^{1}) <^{c} (M̄^{2}, ā^{2}, N̄^{2}) provided that for every i ∈ ⋃ 𝔘_{1}, M_{i}^{1} ≠ M_{i}^{2}.

We first need a generalization of the <_{μ,α}^{c}-extension property:

Definition 8.4. A scattered tower (M̄, ā, N̄) ∈ ^{+} K_{μ,𝔘}^{∗} is said to be nice provided that whenever i ∈ ⋃ 𝔘 is a limit of some sequence from 𝔘, then ⋃_{j∈𝔘, j<i} M_{j} is an amalgamation base.

Theorem 8.5 (<^{c}-Extension Property for Nice Scattered Towers). Let 𝔘^{1}, 𝔘^{2} be such that ⋃ 𝔘^{1} ⊆ ⋃ 𝔘^{2}. Let (M̄^{1}, ā^{1}, N̄^{1}) ∈ ^{+} K_{μ,𝔘^{1}}^{∗} be a nice scattered tower. There exists a nice scattered tower (M̄^{2}, ā^{2}, N̄^{2}) ∈ ^{+} K_{μ,𝔘^{2}}^{∗} such that (M̄^{1}, ā^{1}, N̄^{1}) <^{c} (M̄^{2}, ā^{2}, N̄^{2}).

Proof. WLOG we can rewrite 𝔘^{1} and 𝔘^{2} as collections of disjoint intervals. Moreover, we can write 𝔘^{1} := {u_{i}^{1} | i < α} where sup{u_{i}^{1}} < min{u_{i+1}^{1}}. We then can enumerate 𝔘^{2} conveniently as {u_{i,j}^{2} | i < α, j < | 𝔘^{2} |} such that u_{i,0}^{2} is an end-extension of u_{i}^{1} and for j > 0,

u_{i,j}^{2} := \{ \begin{matrix}u & ; when the number of intervals of 𝔘^{2} between \\ & u_{i}^{1} and u_{i+1}^{1} is ≥ j and u is the j^{th} interval of \\ & 𝔘^{2} between u_{i}^{1} and u_{i+1}^{1} \\ ∅ & ; otherwise. \end{matrix}

Given (M̄^{1}, ā, N̄) ∈ ^{+} K_{μ,𝔘^{1}}^{∗} a nice tower, we will find a <^{c}-extension in + K_{μ,𝔘^{2}}^{∗} by using direct limits inside a (μ, μ^{+})-limit model as we have done in the proofs of Theorem 6.7 and Theorem 7.8. We will leave out some details so as not to be bogged down with notation. The details we hide will be no different than those already described in the proofs of Theorem 6.7 and Theorem 7.8. As before, fix M̌ a (μ, μ^{+})-limit model containing ⋃_{t∈⋃ 𝔘^{1}} M_{t}^{1}. We will demonstrate how to find an extension of (M̄^{1}, ā, N̄) ↾ u_{0}^{1} to {u_{0,j}^{2} | j < | 𝔘^{2} |∥ such that the extension lies inside of M̌ and contains no illegal a_{i}.

By Theorem 7.8, we can find a <^{c}-extension of (M̄^{1}, ā, N̄) ↾ u_{0}^{1}, say (M̄′, ā, N̄) ∈ ^{+} K_{μ,{u_{0}^{1}}}^{∗}, such that M̄′ lies inside of M̌ and avoids the bad a_{i}’s. By Weak Disjoint Amalgamation (Corollary 5.3) and niceness, we can find an extension of M_{min{u_{1}^{1}}}^{1}, denoted by M_{u_{1,0}}^{2} such that

(1) M_{min{u_{1,0}}}^{2} avoids the bad a_{i},

(2) M_{min{u_{1,0}}}^{2} is universal over M_{min{u_{1,0}}}^{1}

(3) M_{min{u_{1,0}}}^{2} is a (μ, | 𝔘^{2} |)-limit over ⋃_{t∈u_{0}^{1}} M_{t}^{1} witnessed by ⟨M_{j}^{∗} | j < | 𝔘^{2} |⟩ where each M_{j}^{∗} is a limit model over ⋃_{t∈u_{0}^{1}} M_{t}^{1} as well,

(4) M_{min{u_{1,0}}}^{2} ≺_{K} M̌ and

(5) M_{min{u_{1,0}}}^{2} contains an image of M̄′ fixing M̄^{1} ↾ u_{0}^{1}.

Set M_{t}^{2} to be the image of M′_{t} for t ∈ u_{0}^{1}. For t ∈ u_{0,j}^{2}, if t is the j^{th}-element of 𝔘^{2} which lies between sup{u_{0}^{1}} and min{u_{1}^{1}}, then set M_{t}^{2} := M_{j+1}^{∗}. For each such t ∈ u_{0,j}^{2}, we can fix a_{t} ∈ M_{t+1}^{2}\M_{t}^{2}. Since M_{t}^{2} is a limit model over ⋃_{s∈u_{0}^{1}} M_{s}^{1}, there exists s ∈ u_{0}^{1} such that ga-tp(a_{t}/M_{t}^{2}) does not μ-split over M_{s}^{1} and M_{t}^{2} is universal over M_{s}^{1}. Thus we can set N_{t}^{2} := M_{s}^{1}. This completes the definition of a <^{c}-extension of (M̄^{1}, ā, N̄) ↾ u_{0}^{1} to {u_{0,j}^{2} | j < | 𝔘^{2} |} inside M̌.

We can now apply an argument as in the successor case of Theorem 7.8 to first find an extension of (M̄^{1}, ā, N̄)↾ {u_{0}^{1}, u_{1}^{1}} to {u_{0,j}^{2} | j < | 𝔘^{2} |} ⋃ u_{1,0}^{2} inside of M̌ avoiding the bad a′_{i}s and containing an image of our previous work. We then can proceed as in the base case above to further extend (M̄^{1}, ā, N̄) ↾ {u_{0}^{1}, u_{1}^{1}} to a tower with index set {u_{i,j}^{1} | i ∈ {0, 1}, j < | 𝔘^{2} |} inside M̌.

This construction is done in a coherent manner so that we can take direct limits in limit stages as we did in the proof of Theorem 7.8.

⊣

The following concept was introduced in [ShVi] for ^{+} K_{μ,α}^{∗} towers.

Definition 8.6. A tower (M̄, ā, N̄) ∈^{+} K_{μ,𝔘}^{∗} is said to be reduced provided that for every (M̄′, ā, N̄) ∈ ^{+} K_{μ,𝔘}^{∗} with (M̄, ā, N̄) ≤^{c} (M̄′, ā, N̄) we have that for every i ∈ ⋃ 𝔘,

M′_{i} ∩ ⋃ ; j∈⋃ 𝔘M_{j} = M_{i}.

Notice that for a tower in ^{+} K_{μ,𝔘}^{∗} to be reduced, it only depends on other towers with the same index set. However, if we take a <^{c}-increasing and continuous chain of reduced towers with increasing index sets, the union will be reduced:

Proposition 8.7. Let ⟨𝔘_{γ} | γ < β⟩ be an increasing and continuous sequence of sets of intervals. If ⟨(M̄, ā, N̄)^{γ} ∈ ^{+} K_{μ,𝔘_{γ}}^{∗} | γ < β⟩ is <^{c}- increasing and continuous sequence of reduced towers, then the union of these towers is reduced.

Proof. Denote by (M̄, ā, N̄)^{β} the limit of the sequence of towers and 𝔘_{β} the limit of the intervals. Suppose that it is not reduced. Let (M̄′, ā, N̄) ∈ + K_{μ,𝔘_{β}}^{∗} witness this. Then there exists an i ∈ ⋃ 𝔘_{β} and an element a such

that a ∈ (M′_{i} ∩ ⋃_{j∈𝔘_{β}} M_{j}^{β})\M_{i}^{β}. There exists γ < β such that i ∈ 𝔘_{γ} and there exists j ∈ 𝔘_{γ} such that a ∈ M_{j}^{γ}. Now consider the tower in ^{+} K_{μ,𝔘_{γ}}^{∗}, (M̄′, ā, N̄) ↾ 𝔘_{γ}. Notice that (M̄′, ā, N̄) ↾ 𝔘_{γ} witnesses that (M̄, ā, N̄)^{γ} is not reduced. ⊣

This is where we need to introduce the additional assumption. It will allow us to take <^{c}-extensions of all scattered towers (not just the nice ones). For the remainder of the paper we assume that

Assumption 8.8. K_{μ}^{am} is closed under unions of length μ^{+}.

The proofs of the following two results on reduced towers (Proposition 8.9 and Theorem 8.10) rely on the <^{c}-extension property. The proofs were outlined in [ShVi] for the particular case of 𝔘 = {[0, α)} when α is a limit ordinal < μ^{+} (see Fact 3.1.13 and Theorem 3.1.15 of their work). However, Shelah and Villaveces were unable to prove the <^{c}-extension property. The proofs of Proposition 8.9 and Theorem 8.10 for scattered towers are the same as for ^{+} K_{μ,α}^{∗}-towers.

Proposition 8.9 (Density of reduced towers). Let (M̄, ā, N̄) ∈^{+} K_{μ,𝔘}^{∗} be given. Then there exists (M̄′, ā, N̄) ∈^{+} K_{μ,𝔘}^{∗} such that

· (M̄, ā, N̄) <^{c} (M̄′, ā, N̄) and
· (M̄′, ā, N̄) is reduced.

Theorem 8.10 (Reduced towers are continuous). If (M̄, ā, N̄) ∈^{+} K_{μ,𝔘}^{∗} is reduced, then M̄ is continuous.

Definition 8.11 (Definition 3.2.1 of [ShVi]). For M a (μ, θ)-limit model, (1) Let

𝔖t(M) := \{ (p, N) | \begin{matrix}N ≺_{K} M; \\ N is a (μ, θ) − limit model; \\ M is universal over N and \\ p ∈ ga-S(M) does not μ − split over N. \end{matrix} \}

and (2) For types (p_{l}, N_{l}) ∈ 𝔖t(M) (l = 1, 2), we say (p_{1}, N_{1}) ∼ (p_{2}, N_{2}) iff

for every M′ ∈ K_{μ}^{am} extending M there is a q ∈ S(M′) extending both p_{1} and p_{2} such that q does not μ-split over N_{1} and q does not μ-split over N_{2}.

Notice that ∼ is an equivalence relation on 𝔖t(M). This gives us another notion of types.

By Fact 2.19, we have

Fact 8.12. For M ∈ K_{μ}^{am}, | 𝔖t(M)/ ∼ | ≤ μ.

We can then consider towers which are saturated with respect to 𝔖t(M):

Definition 8.13. A tower (M̄, ā, N̄) ∈^{+} K_{μ,𝔘}^{∗} is said to be full iff (1) μ divides cf(sup{⋃ 𝔘}) and

(2) if β ∈ ⋃ 𝔘 and (p, N^{∗}) ∈ 𝔖t(M_{β}), then for some i < μ with β + i ∈ ⋃ 𝔘, we have that (ga-tp(a_{β+1}, M_{β+i}, M_{β+i+1}), N_{β+i}) ∼ (p, N^{∗}).

Remark 8.14. Definition 8.13 appears in [ShVi] for the special case when 𝔘 = {[0, α)} for α a limit ordinal < μ^{+} (see Definition 3.2.3 of their paper).

The following theorem is proved in [ShVi] under the particular instance of 𝔘 = {[0, α)} for α a limit ordinal < μ^{+} (Theorem 3.2.4 of their work). We require the more general result, but the proof is similar to Shelah and Villaveces’ argument.

Theorem 8.15. If (M̄, ā, N̄) ∈^{+} K_{μ,𝔘}^{∗} is full and M̄ is continuous, then ⋃_{i∈⋃ 𝔘} M_{i} is a (μ, cf(sup{⋃ 𝔘}))-limit model over M_{0}.

In addition to these generalized results we need the following new theorem which is an analog to the statement that the union of κ(T)-many saturated models is saturated in first order stable theories. We are not implying that fullness is equivalent to saturation, but that the spirit of the results is similar. The following theorem was not stated in [ShVi] and is new:

Theorem 8.16 (Union of Full Towers is Full). Let α be a limit ordinal < μ^{+} and let 𝔘 be set of intervals such that | 𝔘 | < μ^{+}. If ⟨(M̄^{β}, ā, N̄) ∈^{+} K_{μ,𝔘}^{∗} | β < α⟩ is a <^{c}-increasing and continuous chain of full towers for α < μ^{+}, then the union is a full tower.

Proof. Let ⟨(M̄^{β}, ā, N̄) ∈^{+} K_{μ,𝔘}^{∗} | β < α⟩ be a <^{c}-increasing and continuous chain of towers. We need to verify that for i ∈ 𝔘 and (p, N) ∈ 𝔖t(⋃_{β<α} M_{i}^{β}), that there exists j < μ such that i + j ∈ 𝔘 and (p, N) ∼ (ga-tp(a_{i+j}, ⋃_{β<α} M_{i+j}^{β}), N_{i+j}).

By the definition of <^{c}, we have that ⋃_{β<α} M_{i}^{β} is a (μ, α)-limit witnessed by ⟨M_{i}^{β} | β < α⟩. By Theorem 7.2, there exists β < α such that p does not μ-split over M_{i}^{β}. Thus (p ↾ M_{i}^{β+1}, M_{i}^{β}) ∈ 𝔖t(M_{i}^{β+1}). By the assumption of fullness of the β + 1^{st} tower, there exists a j < μ such that

(p ↾ M_{i}^{β+1}, M_{i}^{β}) ∼ (ga-tp(a_{i+j}/M_{i+j}^{β+1}), N_{i+j}).

Recalling the definition of ∼, we know that there exists q ∈ ga-S(⋃_{γ<α} M_{i+j}^{γ}) such that

· p ↾ M_{i}^{β} ⊆ q

· ga-tp(a_{i+j}/M_{i+j}^{β+1}) ⊆ q

· q does not μ-split over M_{i}^{β} and · q does not μ-split over N_{i+j}.

Notice that it suffices to show

Subclaim 8.17. (p, N) ∼ (ga-tp(a_{i+j}/ ⋃_{γ<α} M_{i+j}^{γ}), N_{i+j}).

Proof of Subclaim 8.17. By definition of ∼, we have that

(p ↾ M_{i}^{β+1}, M_{i}^{β}) ∼ (p, N).

Recalling that ga-tp(a_{i+j}/ ⋃_{γ<α} M_{i+j}^{γ}) does not μ-split over N_{i+j}, we see that

(ga-tp(a_{i+j}/M_{i+j}^{β+1}), N_{i+j}) ∼ (ga-tp(a_{i+j}/ ⋃_{γ<α} M_{i+j}^{γ}), N_{i+j}).

Since ∼ is transitive, we have that (p, N) ∼ (ga-tp(a_{i+j}/ ⋃_{γ<α} M_{i+j}^{γ}), N_{i+j}).

⊣
⊣

Now we are ready to prove Theorem 8.1.

Proof of Theorem 8.1. WLOG θ_{1} and θ_{2} are regular. By Proposition 2.27 it suffices to construct a model which is simultaneously a (μ, θ_{1})-limit model and a (μ, θ_{2})-limit model. The idea is to build a (scattered) array of models such that at some point in the array, we will find a model which is a (μ, θ_{1})- limit model witnessed by its height in the array and is a (μ, θ_{2})-limit model witnessed by its horizontal position in the array. We will define a chain of scattered towers of length μ^{+} × θ_{1} while increasing the index set of the towers as we proceed.

We will consider the index set 𝔘^{(α,ζ)} at stage (α, ζ) ∈ μ^{+} × θ_{1} where

𝔘^{(α,ζ)} := ⋃{[βμ, βμθ_{1} + μζ) | β < α}.

Define by induction on (α, ζ) ∈ μ^{+} × θ_{1} the <^{c}-increasing and continuous sequence of scattered towers, ⟨(M̄, ā, N̄)^{(α,ζ)} ∈^{+} K_{μ,𝔘^{(α,ζ)}}^{∗} | (α, ζ) ∈ μ^{+} × θ_{1}⟩, such that

(1) (M̄, ā, N̄)^{(α,ζ)} is reduced,

(2) (M̄, ā, N̄)^{(α+1,0)} := ⋃_{ζ<θ_{1}}(M̄, ā, N̄)^{(α,ζ)} and

(3) in successor stages in new intervals of length μ put in representatives of all 𝔖t-types from the previous stages.

This construction is possible by Theorem 8.5, Theorem 8.9, Theorem 8.10 and Fact 8.12.

Consider the mapping f : μ^{+} → μ^{+} defined by

f(α) := min \{ γ | \begin{matrix}for every β < α, i ∈ ⋃ 𝔘_{β} and \\ for every (p, N) ∈ 𝔖t(M_{i}^{β}) there \\ exists β′ < γ and j < μ such that \\ (ga-tp(a_{i+j}/M_{i+j}^{β′}), N_{i+j}) ∼ (p, N) \end{matrix} \}

By condition (3) of the construction, f can be defined. Then there exists a club C such that

δ ∈ C ⇒ f ↾ δ : δ → δ.

Notice that by the definition of f, this implies

δ ∈ C ⇒ (M̄, ā, N̄)^{(δ,0)} is full.

Pick α ∈ C ∩ S_{θ_{2}}^{μ^{+}}. Subclaim 8.18. We can find ⟨α_{ζ} | ζ < θ_{1}⟩ increasing and continuous such that (M̄, ā, N̄)^{(α_{ζ},0)} ↾ 𝔘^{(α,0)} is full. Proof of Subclaim. By Theorem 8.16. ⊣

Take such a sequence. We will see that

M^{∗} := ⋃_{ζ<θ_{1}} ⋃ ; i∈⋃ 𝔘^{(α,0)} M_{i}^{(α_{ζ},0)}

is a (μ, θ_{1})-limit witnessed by ⟨⋃_{i∈⋃ 𝔘^{(α,0)}} M_{i}^{(α_{ζ},0)} | ζ < θ_{1}⟩. Notice that by Theorem 8.16 M^{∗} is full. Furthermore, we see that M^{∗} is continuous since it is reduced. Now we can apply Theorem 8.15 to conclude that M^{∗} is a (μ, cf(sup{⋃ 𝔘^{(α,0)}}))-limit model. But by our choice of α, we have that cf(sup{⋃ 𝔘^{(α,0)}}) = θ_{2}. Thus M^{∗} is also a (μ, θ_{2})-limit model.

⊣

The above proof implicitly shows the existence of full towers: Corollary 8.19. There exists an interval 𝔘 and a tower (M̄, ā, N̄) ∈ ^{+} K_{μ,𝔘}^{∗} such that (M̄, ā, N̄) is full.

9. Future Work

(1) Removing the extra assumption from Theorem 8.1 by proving either: Conjecture 9.1 (Density of nice towers). For every (M̄, ā) ∈ K_{μ,α}^{∗}, there exists (M̄′, ā) ∈ K_{μ,α}^{∗} such that (M̄, ā) ≤_{μ,α}^{b} (M̄′, ā) and (M̄′, ā) is nice. or

Conjecture 9.2. If (M̄, ā) ∈ K_{μ,α}^{∗} is reduced, then M̄ is continuous.

(2) Deriving the amalgamation property and/or Conjecture 1.1 from Theorem 8.1.

(3) Analyzing the magnitude of extra assumption in Theorem 8.1.

Conjecture 9.3. Extra assumption implies that all amalgamation bases are (μ, κ) amalgamation bases for μ ≤ κ < λ.

Conjecture 9.4. The categoricity model is weakly model homogeneous.

Definition 9.5. N is weakly model homoegeneous iff for every M ≺_{K} N with M ∈ K_{<∥N∥}^{am} we have that N is universal over M.

Corollary 9.6 (to Conjecture 9.4). The categoricity model is Galoissaturated.

References

[BaSh] J. Baldwin and S. Shelah, Abstract Classes with Few Models have ’Homogeneous-Universal’ Models. JSL, 60, 1995.

[Gr] R. Grossberg. A Course in Model Theory. Book in Preparation.

[Gr1] R. Grossberg. Classification Theory for Abstract Elementary Classes. Preprint.

    URL: www.math.cmu.edu/~rami/Bilgi.pdf.

[KoSh] O. Kolman and S. Shelah. Categoricity of Theories in L_{κ,ω} when κ is a measurable cardinal. Part I. Fundamentae Mathematicae, 151, 209–240, 1996.

[MaSh] M. Makkai and S. Shelah. Categoricity of theories L_{κω} with κ a compact cardinal. Annals of Pure and Applied Logic, 47, 41-97, 1990.

[Sha] Saharon Shelah, Classification Theory and the Number of Nonisomorphic Models. North-Holland Publishing Co., Amsterdam, 1978.

[Sh 48] S. Shelah. Categoricity in ℵ_{1} of sentences in L_{ω_{1},ω}(Q). Israel J. Math., 1975. [Sh 87b] S. Shelah. Classification theory for nonelementary classes, I. The number of un-

countable models of ψ ∈ L_{ω_{1},ω}. Part B. Israel J. Math., 1983. [Sh 88] S. Shelah. Classification of nonelementary classes, II. Abstract Elementary Classes.

In Classification Theory (Chicago IL 1985), volume 1292 of Lecture Notes in Mathematics, pages 419–497. Springer, Berlin, 1987. Proceedings of the USA-Israel Conference on Classification Theory, Chicago, December 1985; ed. Baldwin, J.T.

[Sh 394] S. Shelah. Categoricity of abstract classes with amalgamation. Annals of Pure

and Applied Logic, 1999. [Sh 472] S. Shelah. Categoricity of Theories in L_{κ^{∗},ω} when κ^{∗} is a measurable cardinal.

Part II. Fundamenta Mathematica. [Sh 576] S. Shelah. Categoricity of an abstract elementary class in two successive cardinals.

Israel J. of Math. 102 pages. [Sh 600] S. Shelah. Categoricity in abstract elementary classes: going up inductive step.

Preprint. 92 pages. [Sh 702] S. Shelah. On what I do not understand (and have something to say), model

theory. Math Japonica, 51, pages 329–377, 2000.

[ShVi] S. Shelah and A. Villaveces. Categoricity in abstract elementary classes with no maximal models.Annals of Pure and Applied Logic, 1999. E-mail address, Monica VanDieren: monicav@andrew.cmu.edu

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213