MORLEY SEQUENCES IN ABSTRACT ELEMENTARY CLASSES

RAMI GROSSBERG AND MONICA VANDIEREN

Abstract. In this paper, we explore stability results in the new context of tame abstract elementary classes with the amalgamation property. The main result is:

Theorem 0.1. Let K be a tame abstract ementary class satisfying the amalgamation property without maximal models. There exists a cardinal μ_{0}(K) such that for every μ ≥ μ_{0}(K) and every M ∈ K_{>μ}, A, I ⊂ M such that |I| ≥ μ^{+} > |A|, if K is Galois stable in μ, then there exists J ⊂ I of cardinality μ^{+}, Galois indiscernible indiscernible over A. Moreover J can be chosen to be a Morley sequence over A.

This result strengthens Claim 4.16 of [Sh 394] as we do not assume categoricity. This is also an improvement of a result from [GrLe] concerning the existence of indiscernible sequences.

This is used to get a stability spectrum theorem.

Introduction

Already in the fifties model theorists studied non-elementary classes of structures (e.g. Jónsson [Jo1], [Jo2] and Fraïssé [Fr]). In [Sh 88], Shelah introduced the framework of abstract elementary classes and embarked on the ambitious program of developing a classification theory for Abstract Elementary Classes. For a survey of some of the basics, see [Gr2] or Chapter 13 of [Gr1]. While much is known about abstract elementary classes, especially when K is an AEC under the additional assumption that there exists a cardinal λ > Hanf(K) such that K is categorical in λ, little progress has been made towards a full-fledged stability theory. One of the open problems from [Sh 394] (Remark 4.10(1)) is to identify of a good (forking-like) notion of independence for abstract elementary classes. This is open even for classes that have the amalgamation property and are categorical above the Hanf number. In [Sh 394], several weak notions of independence are introduced under the assumption that the class is categorical. Among these notions is the Galois-theoretic notion of non-splitting. Here we study the notion of non-splitting in a more general context than categorical AEC: Tame stable classes. We plan to use Morley sequences for non-splitting as a bootstrap to define a dividing-like concept for these classes.

Date: This draft draft draft is from January 11, 2002.

AMS Subject Classification: Primary: 03C45, 03C52, 03C75. Secondary: 03C05, 03C55 and 03C95.

This paper is part of the second author’s Ph.D. thesis written under the guidance of Rami Grossberg.

We are greateful to John Baldwin for reading a preliminary version, many discussions and suggesting to consider the stability spectrum problem for AECs.

1. Background

Definition 1.1. Let K be a class of structures all in the same similarity type L(K), and let ≺_{K} be a partial order on K. The ordered pair ⟨K, ≺_{K}⟩ is an abstract elementary class, AEC for short iff

A0 (Closure under isomorphism)

(a) For every M ∈ K and every L(K)-structure N if M ≅ N then N ∈ K.

(b) Let N_{1}, N_{2} ∈ K and M_{1}, M_{2} ∈ K such that there exist f_{l} : N_{l} ≅ M_{l} (for l = 1, 2) satisfying f_{1} ⊆ f_{2} then N_{1} ≺_{K} N_{2} implies that M_{1} ≺_{K} M_{2}.

A1 For all M, N ∈ K if M ≺_{K} N then M ⊆ N.

A2 Let M, N, M^{∗} be L(K)-structures. If M ⊆ N, M ≺_{K} M^{∗} and N ≺_{K}

M^{∗} then M ≺_{K} N. A3 (Downward Löwenheim-Skolem) There exists a cardinal

LS(K) ≥ ℵ_{0} + |L(K)| such that for every

M ∈ K and for every A ⊆ |M| there exists N ∈ K such that N ≺_{K}

M, |N| ⊇ A and ∥N∥ ≤ |A| + LS(K).

A4 (Tarski-Vaught Chain)

(a) For every regular cardinal μ and every N ∈ K if {M_{i} ≺_{K} N : i < μ} ⊆ K is ≺_{K}-increasing (i.e. i < j ⟹ M_{i} ≺_{K} M_{j}) then ⋃_{i<μ} M_{i} ∈ K and ⋃_{i<μ} M_{i} ≺_{K} N.

(b) For every regular μ, if {M_{i} : i < μ} ⊆ K is ≺_{K}-increasing then ⋃_{i<μ} M_{i} ∈ K and M_{0} ≺_{K} ⋃_{i<μ} M_{i}.

Notation 1.2. If λ is a cardinal and K is a class of models, K_{λ} is the subclass of models from K of cardinality λ.

Definition 1.3. For models M, N ∈ K, f : M → N is an ≺_{K}-embedding iff f[M] ≺_{K} N.

Definition 1.4. (1) We say that a model M is an amalgamation base if for every N_{i} ≻ M with ∥N_{i}∥ = ∥M∥ (i = 1, 2), there are ≺_{K}- embeddings g_{i}, (i = 1, 2) and a model N such that the following diagram commutes:

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

(2) K^{am} := {M ∈ K | M is an amalgamation base}.

Definition 1.5. Let β > 0 be an ordinal. For triples (ā_{l}, M_{l}, N_{l}) where ā_{l} ∈ ^{β}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 elementary 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 1.6. E is an equivalence relation on the class of triples of the form (ā, M, N) where M ≺_{K} N, ā ∈ N and both M, N ∈ K^{am}. When only M ∈ K^{am}, E may fail to be transitive, but the transitive closure of E could be used instead.

While it is standard to use the E relation to define types in abstract elementary classes, we will discuss and make use of stronger relations between triples in section 3 of this paper.

In order to avoid confusing this new notion of “type” with the conventional one (i.e. set of formulas) we will follow [Gr1] and [Gr2] and introduce it below under the name of Galois type.

Definition 1.7. Let β be a positive ordinal (can be one).

(1) For M, N ∈ K^{am} and ā ∈ ^{β}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) We abbreviate ga-tp(ā/M, N) by ga-tp(ā/M).

(3) For M ∈ K^{am},

ga-S^{β}(M) := {ga-tp(ā/M, N) | M ≺ N ∈ K_{∥M∥}^{am}, ā ∈ ^{β}N}.

We write ga-S(M) for ga-S^{1}(M).

(4) Let p := ga-tp(ā/M′, N) for M ≺_{K} M′ we denote by p ↾ M′ the type ga-tp(ā/M, N). The domain of p is denoted by dom p and it is by definition M′.

(5) Let p = ga-tp(ā/M, N), suppose that M ≺_{K} N′ ≺_{K} N and let b̄ ∈ ^{β}N′ we say that b̄ realizes p iff ga-tp(b̄/M, N′) = p ↾ M.

(6) For types p and q, we write p ≤ q if dom(p) ⊆ dom(q) and there exists ā realizing p in some N extending dom(p) such that (ā, dom(p), N) ∈ q ↾ dom(p).

Definition 1.8. We say that K is β-stable in μ if for every M ∈ K_{μ}^{am}, | ga-S^{β}(M)| = μ. The class K is Galois stable in μ iff K is 1-stable in μ.

Definition 1.9. We say that M ∈ K is Galois saturated if for every N ≺_{K} M of cardinality < ∥M∥, and every p ∈ ga-S(N), we have that M realizes p.

Remark 1.10. When K = Mod(T) for a first-order T, using the compactness theorem one can show (Theorem 2.2.3 of [Gr1]) that for M ∈ K, the model M is Galois saturated iff M is saturated in the first-order sense.

It is interesting to mention

Theorem 1.11 (Shelah [Sh 300]). Let λ > LS(K). Suppose that K has the amalgamation property and N ∈ K_{λ}. The following are equivalent

(1) N is Galois staurated.

(2) N is model-homogenous. I.e. if M ≺_{K} N and M′ ≻ M of cardinality less than λ then there exists a K-embedding over M from M′ into N.

Unfortunately [Sh 300] has an incomplete skeleton of a proof, a complete and correct proof appeared in [Sh 576]. See also [Gr2].

In first order logic, it is natural to consider saturated models for a stable theory. In this context, saturated models are model homogeneous and hence unique. In abstract elementary classes, the existence and even the uniqueness of saturated models is often difficult to derive. To combat this, Shelah introduced a replacement for saturated models, namely, limit-models, which in certain contexts can be shown to exist and be unique.

Definition 1.12. For M′, M ∈ K_{λ}, we say that M′ is universal over M (abbreviated M ≺_{K}^{univ} M′) iff for every N ∈ K_{λ} with M ≺ N, there exists a ≺_{K}-embedding, g, such that g : N → M′ and g ↾ M = id_{M}.

Definition 1.13. For M′, M ∈ K_{μ}, we say that M′ is a (μ, σ)-limit over M iff there exists a ≺_{K}-increasing and continuous sequence of amalgamation bases ⟨M_{i} ∈ K_{μ} | i ≤ σ⟩ such that

(1) M ≺_{K} M_{0},

(2) M′ = M_{σ} and

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

When K = Mod(T) for a first-order and stable T then automatically (by Theorem III.3.12 of [Shc]):

M ∈ K_{μ} is saturated ⟹ M is (μ, σ)-limit for all σ < μ^{+} of cofinality ≥ κ(T).

When T is countable, stable but not superstable then the saturated model of cardinality μ is (μ, ℵ_{1})-limit but not (μ, ℵ_{0})-limit.

Claim 1.14 (Claim 1.14.1 from [Sh 600]). Suppose K is an abstract elementary class with the amalgamation property. If K is Galois stable in μ, then for every M ∈ K_{μ}, there exists M′ ∈ K_{μ} such that M′ is universal over M. Moreover M′ can be chosen to be a (μ, σ)-limit over M for any σ < μ^{+}.

The following lemma holds in the context of abstract elementary classes with no maximal models and provides an alternative proof to the existence of universal extensions.

Lemma 1.15 (Theorem 1.3.1 from [ShVi 635]). Categoricity in some λ > LS(K) implies the existence universal extensions in every μ such that LS(K) < μ < λ. Furthermore, this implies the existence of (μ, σ)-limits for ω ≤ σ < μ^{+}.

While the existence of limit models is useful, it is desirable that these models be unique. The following Lemma provides one level of uniqueness, namely, limit models of the same length are isomorphic:

Lemma 1.16 (Fact 1.3.6 from [ShVi 635]). 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}.

Showing that two (μ, σ)-limit models are isomorphic for different σs is a central (dichotomy) property even for first-order theories: E.g. Let T be a countable (for simplicity), stable, non-superstable first-order theory. Suppose T is stable in μ. By Harnik’s theorem ([Har]) T has a saturated model of cardinality μ, in fact for σ < μ^{+} the theory has (μ, σ)-limit models. Since T is not superstable there exists a union of an ω-sequence of saturated models which is not saturated (see [AlGr]) thus it is not isomorphic to a (μ, ω_{1})-limit model (that was obtained by taking a union of saturated models, see Theorem III.3.12 of [Shc]).

2. Existence of Indiscernibles

Assumption 2.1. For the remainder of this section, we will fix K, an abstract elementary class with the amalgamation property.

Remark 2.2. The focus of this paper are classes with the amalgamation property. Several of the proofs in this section can be adjusted to the context of abstract elementary classes with density of amalgamation bases as in [ShVi 635].

The most obvious attempt to generalize Shelah’s argument from Lemma I.2.5 of [Shc] does not apply since the notion of type cannot be identified with a set of first order formulas. Moreover, the notion of a type over an arbitrary set does not exist in the context of abstract elementary classes.

From [Sh 394]:

Definition 2.3. A type p ∈ S^{β}(N) μ-splits over M ≺_{K} N if and only if ∥M∥ ≤ μ, there exist N_{1}, N_{2} ∈ K_{≤μ} and h, a K-embedding such that M ≺_{K} N_{l} ≺_{K} N for l = 1, 2 and h : N_{1} → N_{2} such that h ↾ M = id_{M} and p ↾ N_{2} ≠ h(p ↾ N_{1}).

Notice that non splitting is monotonic: I.e. If p ∈ ga-S(N) does not split over M (for some M ≺_{K} N) then p does not split over M′ for every M ≺_{K} M′ ≺_{K} N.

Similarly to κ(T) when T is first-order the following is a natural cardinal invariant of K:

Definition 2.4. Let β > 0. We define an invariant κ_{μ}^{β}(K) to be the minimal κ such that for every ⟨M_{i} ∈ K_{μ} | i ≤ κ⟩ which satisfies

(1) κ = cf(κ) < μ^{+},

(2) ⟨M_{i} | i ≤ κ⟩ is ≺_{K}-increasing and continuous and

(3) for every i < κ, M_{i+1} is a (μ, θ)-limit over M_{i} for some θ < μ^{+},

and for every p ∈ ga-S^{β}(M_{κ}), there exists i < κ such that p does not μ-split over M_{i}. If no such κ exists, we say κ_{μ}^{β}(K) = ∞.

The following theorem in conjunction with Lemma 1.15 states that categorical abstract elementary classes with no maximal models satisfy κ_{μ}^{1}(K) ≤ ω, for various μ.

Theorem 2.5 (Theorem 2.2.1 from [ShVi 635]). Suppose K is categorical in λ > LS(K). Let μ < λ. If ⟨M_{i} ∈ K_{μ} | i ≤ κ⟩ and p ∈ S(M_{κ}) satisfy

(1) κ = cf(κ) < μ^{+},

(2) ⟨M_{i} | i ≤ κ⟩ is ≺_{K}-increasing and continuous and

(3) for every i < κ, M_{i+1} is a (μ, θ)-limit over M_{i} for some θ < μ^{+}, then there exists i < κ such that p does not μ-split over M_{i}.

A slight modification of the argument of Claim 3.3 from [Sh 394] can be used to prove:

Theorem 2.6. Let β > 0. Suppose that K is β-stable in μ. For every p ∈ ga-S^{β}(N) there exists M ≺_{K} N of cardinality μ such that p does not μ-split over M. Thus κ_{μ}^{β}(K) ≤ μ.

In Theorem 4.7 below we present an improvement of Theorem 2.6 for tame AECs: In case K is β-stable in μ for some μ above its Hanf number then κ_{μ}^{β}(K) is bounded by the Hanf number. Notice that the bound does not depend on μ.

The following is a new Galois-theoretic notion of indiscernible sequence.

Definition 2.7. (1) ⟨ā_{i} | i < i^{∗}⟩ is a Galois indiscernible sequence over M iff for every i_{1} < ⋯ < i_{n} < i^{∗} and every j_{1} < ⋯ < j_{n} < i^{∗},

ga-tp(ā_{i_{1}} … ā_{i_{n}}/M) = ga-tp(ā_{j_{1}} … ā_{j_{n}}/M).

(2) ⟨ā_{i} | i < i^{∗}⟩ is a Galois indiscernible sequence over A iff for every i_{1} < ⋯ < i_{n} < i^{∗} and every j_{1} < ⋯ < j_{n} < i^{∗}, there exists M_{i}, M_{j}, M^{∗} ∈ K and ≺_{K}-mappings f_{i}, f_{j} such that

(a) A ⊆ M_{i}, M_{j};

(b) f_{l} : M_{l} → M^{∗}, for l = i, j;

(c) f_{i}(ā_{i_{0}}, …, ā_{i_{n}}) = f_{j}(ā_{j_{0}}, …, ā_{j_{n}}) and

(d) and f_{i} ↾ A = f_{j} ↾ A = id_{A}.

Remark 2.8. This is on the surface a weaker notion of indiscernible sequence than is presented in [Sh 394]. However, this definition coincides with the first order definition. Additionally, it is suspected that, under some reasonable assumptions, this definition and the definition in [Sh 394] are equivalent.

The following lemma provides us with sufficient conditions to find an indiscernible sequence.

Lemma 2.9. Let μ ≥ LS(K), κ, λ be ordinals and β a positive ordinal. Suppose that ⟨M_{i} | i < λ⟩ and ⟨ā_{i} | i < λ⟩ satisfy

(1) ⟨M_{i} | i < λ⟩ are ⪯_{K}-increasing;

(2) M_{i} ∈ K_{μ};

(3) M_{i+1} is a (μ, κ)-limit over M_{i};

(4) ā_{i} ∈ ^{β}M_{i+1};

(5) p_{i} := ga-tp(ā_{i}/M_{i}, M_{i+1}) does not μ-split over M_{0} and

(6) for i < j < λ, p_{i} ≤ p_{j}. Then, ⟨ā_{i} | i < λ⟩ is a Galois indiscernible sequence over M_{0}.

Definition 2.10. A sequence ⟨ā_{i}, M_{i} | i < λ⟩ satisfying conditions (1) − (6) of Lemma 2.9 is called a Morley sequence.

Remark 2.11. While the statement of the lemma is similar to Shelah’s Lemma I.2.5 in [Shc], the proof differs, since types are not sets of formulas.

Proof. We prove that for i_{0} < ⋯ < i_{n} < λ and j_{0} < ⋯ < j_{n} < λ, ga-tp(ā_{i_{0}}, …, ā_{i_{n}}/M_{0}, M_{i_{n+1}}) = ga-tp(ā_{j_{0}}, …, ā_{j_{n}}/M_{0}, M_{j_{n+1}}) by induction on n < ω.

n = 0: Let i_{0}, j_{0} < λ be given. Condition 6, gives us

ga-tp(ā_{i_{0}}/M_{0}, M_{i_{0}+1}) = ga-tp(ā_{j_{0}}/M_{0}, M_{j_{0}+1}).

n > 0: Suppose that the claim holds for all increasing sequences ī and j̄ ∈ λ of length n. Let i_{0} < ⋯ < i_{n} < λ and j_{0} < ⋯ < j_{n} < λ be given. Without loss of generality, i_{n} ≤ j_{n}. Define M^{∗} := M_{1}. ¿From condition 3 and uniqueness of (μ, ω)-limits, we can find a ≺_{K}-isomorphism, g : M_{j_{n}} → M_{i_{n}} such that g ↾ M_{0} = id_{M_{0}}. Moreover we can extend g to g : M_{j_{n}+1} → M_{i_{n}+1}. Denote by b̄_{j_{l}} := g(ā_{j_{l}}) for l = 0, …, n. Notice that b_{j_{l}} ∈ M_{i_{n}} for l < n. Since ga-tp(b̄_{j_{0}}, …, b̄_{j_{n}}/M_{0}, M_{i_{n}+1}) = ga-tp(ā_{j_{0}}, …, ā_{j_{n}}/M_{0}, M_{j_{n}+1}) it suffices to prove that ga-tp(b̄_{j_{0}}, …, b̄_{j_{n}}/M_{0}, M_{i_{n}+1}) = ga-tp(ā_{i_{0}}, …, ā_{i_{n}}/M_{0}, M_{i_{n}+1}).

Also notice that the ≺_{K}-mapping preserves some properties of p_{j}. Namely, since p_{j} does not μ-split over M_{0}, g(p_{j} ↾ M_{j_{n}}) = p_{j} ↾ M_{i_{n}}. Thus, ga-tp(b̄_{j_{n}}/M_{i_{n}}, M_{i_{n}+1}) = ga-tp(ā_{j_{n}}/M_{i_{n}}, M_{i_{n}+1}). In particular we have that ga-tp(b̄_{j_{n}}/M_{i_{n}}, M_{i_{n}+1}) does not μ-split over M_{0}.

By the induction hypothesis

ga-tp(b̄_{j_{0}}, …, b̄_{j_{n−1}}/M_{0}, M_{i_{n}}) = ga-tp(ā_{i_{0}}, …, ā_{i_{n−1}}/M_{0}, M_{i_{n}}).

Thus we can find h_{i} : M_{i_{n}+1} → M^{∗} and h_{j} : M_{i_{n}+1} → M^{∗} such that h_{i}(ā_{i_{0}}, …, ā_{i_{n−1}}) = h_{j}(b̄_{j_{0}}, …, b̄_{j_{n−1}}). Let us abbreviate b̄_{j_{0}}, …, b̄_{j_{n−1}} by b̄j̄. Similarly we will write āī for ā_{i_{0}}, …, ā_{i_{n−1}}.

By appealing to condition 5, we derive several equalities that will be useful in the latter portion of the proof. Since p_{j} does not μ-split over M_{0}, we have that p_{j} ↾ h_{j}(M_{i_{n}}) = h_{j}(p_{j} ↾ M_{i_{n}}), rewritten as

(∗) ga-tp(b̄_{j_{n}}/h_{j}(M_{i_{n}}), M_{i_{n}+1}) = ga-tp(h_{j}(b̄_{j_{n}})/h_{j}(M_{i_{n}}), M^{∗}).

Similarly as p_{i} does not μ-split over M_{0}, we get p_{i} ↾ h_{j}(M_{i_{n}}) = h_{j}(p_{i} ↾ M_{i_{n}}) and p_{i} ↾ h_{i}(M_{i_{n}}) = h_{i}(p_{i} ↾ M_{i_{n}}). These equalities translate to

(∗∗)_{j} ga-tp(ā_{i_{n}}/h_{j}(M_{i_{n}}), M_{i_{n}+1}) = ga-tp(h_{j}(ā_{i_{n}})/h_{j}(M_{i_{n}}), M^{∗}) and

(∗∗)_{i} ga-tp(ā_{i_{n}}/h_{i}(M_{i_{n}}), M_{i_{n}+1}) = ga-tp(h_{i}(ā_{i_{n}})/h_{i}(M_{i_{n}}), M^{∗}), respectively. Finally, from condition 6., notice that

(∗ ∗ ∗) ga-tp(ā_{i_{n}}/M_{i_{n}}, M_{i_{n}+1}) = ga-tp(b̄_{j_{n}}/M_{i_{n}}, M_{i_{n}+1}).

Applying h_{j} to (∗ ∗ ∗) yields

(†) ga-tp(h_{j}(b̄_{j_{n}})/h_{j}(M_{i_{n}}), M^{∗}) = ga-tp(h_{j}(ā_{i_{n}})/h_{j}(M_{i_{n}}), M^{∗}).

Since h_{i}(āī) = h_{j}(b̄j̄) ∈ h_{j}(M_{i_{n}}), we can draw from (†) the following:

(1) ga-tp(h_{j}(b̄_{j_{n}})ˆ h_{j}(b̄j̄)/M_{0}, M^{∗}) = ga-tp(h_{j}(ā_{j_{n}})ˆ h_{i}(āī)/M_{0}, M^{∗}).

Equality (∗∗)_{i} allows us to see

(2) ga-tp(ā_{i_{n}}ˆ h_{i}(āī)/M_{0}, M^{∗}) = ga-tp(h_{i}(ā_{i_{n}})ˆ h_{i}(āī)/M_{0}, M^{∗}).

Since ga-tp(h_{j}(ā_{i_{n}})/h_{j}(M_{i_{n}}), M^{∗}) = ga-tp(ā_{i_{n}}/h_{j}(M_{i_{n}}), M_{i_{n}+1}) (equality (∗∗)_{j})) and h_{i}(āī) = h_{j}(b̄j̄) ∈ h_{j}(M_{i_{n}}), we get that

(3) ga-tp(h_{j}(ā_{i_{n}})ˆ h_{i}(āī)/M_{0}, M^{∗}) = ga-tp(ā_{i_{n}}ˆ h_{i}(āī)/M_{0}, M^{∗}).

Combining equalities (1), (2) and (3), we get

(††) ga-tp(h_{i}(āī)ˆ h_{i}(ā_{i_{n}})/M_{0}, M^{∗}) = ga-tp(h_{j}(b̄j̄)ˆ h_{j}(b̄_{j_{n}})/M_{0}, M^{∗}).

Recall that h_{i} ↾ M_{0} = h_{j} ↾ M_{0} = id_{M_{0}}. Thus (††), witnesses that

ga-tp(ā_{i_{0}}, …, ā_{i_{n}}/M_{0}, M_{i_{n}+1}) = ga-tp(b̄_{j_{0}}, …, b̄_{j_{n}}/M_{0}, M_{i_{n}+1}).
⊣

3. Tame Abstract Elementary Classes

By Lindström’s Theorem, one obvious feature of non-elementary abstract elementary classes is the absence of the compactness theorem. A method of combating this is to view types as equivalences classes of triples (Definition 1.7) instead of sets of formulas. While this notion of type has led to several profound results in the study of abstract elementary classes, a stronger equivalence relation (denoted E_{μ}) is eventually utilized in various partial solutions to Shelah’s Categoricity Conjecture (see [Sh 394] and [Sh 576])?.

Shelah identified E_{μ} as an interesting relation in [Sh 394]. Here we recall the defintion.

Definition 3.1. Triples (ā_{1}, M, N_{1}) and (ā_{2}, M, N_{2}) are said to be E_{μ}- related provided that for every M′ ≺_{K} M with M′ ∈ K_{<μ},

(ā_{1}, M′, N_{1})E(ā_{2}, M′, N_{2}).

Notice that in first order logic, the finite character of consistency implies that two types are equal if and only if they are E_{ω}-related.

In Main Claim 9.3 of [Sh 394], Shelah ultimately proves that, under categoricity in some λ > Hanf(K) and under the assumption that K has the amalgamation property, for types over saturated models, E-equivalence is the same as E_{μ} equivalence for some μ < Hanf(K).

We now define a context for abstract elementary classes where consistency has small character.

Definition 3.2. Let χ be a cardinal number. We say the abstract elementary class K with the amalgamation property is χ-tame provided that for types, E-equivalence is the same as the E_{χ} relation. In other words, for M ∈ K_{>Hanf(K)}, p ≠ q ∈ ga-S(M) implies existence of N ≺_{K} M of cardinality χ such that p ↾ N ≠ q ↾ N.

K is tame iff there exists such that K is χ-tame for some χ < Hanf(K)

Remark 3.3. We actually only use that E-equivalence is the same as E_{χ}- equivalence for types over limit models.

Notice that if K is a finite diagram (i.e. we have amalgamation not only all models but also over subsets of models) then it is a tame AEC.

Example 3.4. There are tame AECs with amalgamation which are not finite diagrams. WILL BE ADDED

Example 3.5. For μ_{1} < μ_{2} < ℶ_{ω_{1}}, there is a class which is μ_{2}-tame but not μ_{1}-tame. WILL BE ADDED

4. The order property

Remark 4.1. For the remainder of this section we will assume that K is an abstract elementary class which satisfies the amalgamation property.

The order property, defined next, is an analog of the first order definition of order property using formulas. The order property for non-elementary classes was introduced by Shelah in [Sh 394].

Definition 4.2. K is said to have the κ-order property provided that for every α, there exists ⟨d̄_{i} | i < α⟩ and where d̄_{i} ∈ ^{κ}ℭ such that if i_{0} < j_{0} < α and i_{1} < j_{1} < α,

(∗) then for no f ∈ Aut(ℭ) do we have f(d̄_{i_{0}} ˆ d̄_{j_{0}}) = d̄_{j_{1}} ˆ d̄_{i_{1}}.

Remark 4.3 (Trivial monotonicity). Notice that for κ_{1} < κ_{2} if a class has the κ_{1}-order property then it has the κ_{2}-order property.

Claim 4.4 (Claim 4.6.3 of [Sh 394]). We may replace the phrase every α in Definition 4.2 with every α < ℶ_{(2^{κ+LS(K)})^{+}} and get an equivalent definition.

Theorem 4.5 (Claim 4.8.2 of [Sh 394]). If K has the κ-order property and μ ≥ κ, then for some M ∈ K_{μ} we have that | ga-S^{κ}(M)/E_{κ}| ≥ μ^{+}. Moreover, we can conclude that K is not Galois stable in μ.

Question 4.6. Can we get a version of the stability spectrum theorem for tame stable classes?

Theorem 4.7. Let β > 0. Suppose that K is a κ-tame abstract elementary class. If K is β-stable in μ with ℶ_{(2^{κ+LS(K)})^{+}} ≤ μ, then κ_{χ}^{β}(K) < ℶ_{(2^{κ+LS(K)})^{+}}.

Proof. Let χ := ℶ_{(2^{κ+LS(K)})^{+}}. Suppose that the conclusion of the theorem does not hold. Let ⟨M_{i} ∈ K_{μ} | i ≤ χ⟩ and p ∈ ga-S^{β}(M_{χ}) witness the failure. Namely, the following hold:

(1) ⟨M_{i} | i ≤ χ⟩ is ≺_{K}-increasing and continuous,

(2) for every i < χ, M_{i+1} is a (μ, θ)-limit over M_{i} for some θ < μ^{+} and (3) for every i < μ^{+}, p μ-splits over M_{i}. For every i < χ let f_{i}, N_{i}^{1} and N_{i}^{2} witness that p μ-splits over M_{i}. Namely,

M_{i} ≺_{K} N_{i}^{1}, N_{i}^{2} ≺_{K} M,
f_{i} : N_{i}^{1} ≅ N_{i}^{2} with f_{i} ↾ M_{i} = id_{M_{i}}
and f_{i}(p ↾ N_{i}^{1}) ≠ p ↾ N_{i}^{2}.

By κ-tameness, there exist B_{i} and A_{i} := f_{i}^{−1}(B_{i}) of size < κ such that

f_{i}(p ↾ A_{i}) ≠ p ↾ B_{i}.

By renumbering our chain of models, we may assume that

(4) A_{i}, B_{i} ⊂ M_{i+1}.

Since M_{i+1} is a limit model over M_{i}, we can additionally conclude that (5) c̄_{i} ∈ M_{i+1} realizes p ↾ M_{i}. For each i < μ, let d̄_{i} := A_{i}ˆ B_{i} ˆ c̄_{i}.

Claim 4.8. ⟨d̄_{i} | i < χ⟩ witnesses the κ-order property.

Proof. Suppose for the sake of contradiction that there exist g ∈ Aut(ℭ), i_{0} < j_{0} < χ and i_{1} < j_{1} < χ such that

g(d̄_{i_{0}} ˆ d̄_{j_{0}}) = d̄_{j_{1}} ˆ d̄_{i_{1}}.

Notice that since i_{0} < j_{0} < α we have that c̄_{i_{0}} ∈ M_{j_{0}}. So f_{j_{0}}(c̄_{i_{0}}) = c̄_{i_{0}}. Recall that f_{j_{0}}(A_{j_{0}}) = B_{j_{0}}. Thus, f_{j_{0}} witnesses that

(∗) ga-tp(c̄_{i_{0}}ˆ A_{j_{0}}/∅) = ga-tp(c̄_{i_{0}}ˆ B_{j_{0}}/∅).

Applying g to (∗) we get

(∗∗) ga-tp(c̄_{j_{1}}ˆ A_{i_{1}}/∅) = ga-tp(c̄_{j_{1}}ˆ B_{i_{1}}/∅).

Applying f_{i_{1}} to the RHS of (∗∗), we notice that

(♯) ga-tp(f_{i_{1}}(c̄_{j_{1}})ˆ B_{i_{1}}/∅) = ga-tp(c̄_{j_{1}}ˆ B_{i_{1}}/∅).

Because i_{1} < j_{1}, we have that c̄_{j_{1}} realizes p ↾ M_{i_{1}}. Thus, (♯) implies

(♯♯)f_{i_{1}}(p ↾ A_{i_{1}}) = p ↾ B_{i_{1}},

which contradicts our choice of f_{i_{1}}, A_{i_{1}} and B_{i_{1}}.

⊣

By Claim 4.4 and Theorem 4.5, we have that K is unstable in μ, contradicting our hypothesis.

⊣

5. Morley’s sequences

Hypothesis 5.1. For the rest of the paper we make the following assumption: K is a tame abstract elementary class, has no maximal models and satisfies the amalgamation property.

Theorem 5.2. Suppose μ ≥ ℶ_{(2^{Hanf(K)})^{+}}. Let M ∈ K_{>μ}, A, I ⊂ M be given such that |I| ≥ μ^{+} > |A|. If K is Galois stable in μ, then there exists J ⊂ I of cardinality μ^{+}, Galois indiscernible over A. Moreover J can be chosen to be a Morley sequence over A.

Proof. Fix κ := cf(μ). Let {ā_{i} | i < μ^{+}} ⊆ I be given. Define ⟨M_{i} ∈ K_{μ} | i < μ^{+}⟩ ≺_{K}-increasing and continuous satisfying

(1) A ⊆ |M_{0}|

(2) M_{i+1} is a (μ, κ)-limit over M_{i}

(3) ā_{i} ∈ M_{i+1}

Let p_{i} := ga-tp(ā_{i}/M_{i}, M_{i+1}) for every i < μ^{+}. Define f : S_{κ}^{μ^{+}} → μ^{+} by

f(i) := min{j < μ^{+} | p_{i} does not μ- split over M_{j}}.

By Theorem 4.7, f is regressive. Thus by Fodor’s Lemma, there are a stationary set S ⊆ S_{κ}^{μ^{+}} and j_{0} ∈ I such that for every i ∈ S,

(†) p_{i} does not μ-split over M_{j_{0}}.

By stability and the pigeon-hole principle there exists p^{∗} ∈ ga-S(M_{j_{0}}) and S^{∗} ⊆ S of cardinality μ^{+} such that for every i ∈ S^{∗}, p^{∗} = p_{i} ↾ M_{j_{0}}. Enumerate and rename S^{∗}. Let M^{∗} := M_{1}. Again, by stability we can find J ⊂ S^{∗} of cardinality μ^{+} such that for every i ∈ S^{∗}, p^{∗∗} = p_{i} ↾ M^{∗}. Enumerate and rename J.

Subclaim 5.3. For i < j ∈ J, p_{i} = p_{j} ↾ M_{i}.

Proof. Let 0 < i < j ∈ J be given. Since M_{i+1} and M_{j+1} are (μ, κ)- limits over M_{i}, there exists an isomorphism g : M_{j+1} → M_{i+1} such that g ↾ M_{i} = id_{M_{i}}. Let b̄_{j} := g(ā_{j}). Since the type p_{j} does not μ-split over M_{j_{0}}, g cannot witness the splitting. Therefore, it must be the case that ga-tp(b̄_{j}/M_{i}, M_{i+1}) = p_{i} ↾ M_{i}. Then, it suffices to show that ga-tp(b̄_{j}/M_{i}, M_{i+1}) = p_{i}.

Since p_{i} ↾ M_{0} = p_{j} ↾ M_{0}, we can find ≺_{K}-mappings witnessing the equality. Furthermore since M^{∗} is universal over M_{0}, we can find h_{l} : M_{l+1} → M^{∗} such that h_{l} ↾ M_{0} = id_{M_{0}} for l = i, j and h_{i}(ā_{i}) = h_{j}(b̄_{j}).

We will use (†) to derive several inequalities. Consider the following possible witness to splitting. Let N_{1} := M_{i} and N_{2} := h_{i}(M_{i}). Since p_{i} does not μ-split over M_{0}, we have that p_{i} ↾ N_{2} = h_{i}(p_{i} ↾ N_{1}), rewritten as

(∗) ga-tp(ā_{i}/h_{i}(M_{i}), M_{i+1}) = ga-tp(h_{i}(ā_{i})/h_{i}(M_{i}), M^{∗}).

Similarly we can conclude that

(∗∗) ga-tp(b̄_{j}/h_{j}(M_{i}), M_{i+1}) = ga-tp(h_{j}(b̄_{j})/h_{j}(M_{i}), M^{∗}).

By choice of J, we know that

(∗ ∗ ∗) ga-tp(b̄_{j}/M^{∗}) = ga-tp(ā_{i}/M^{∗}).

Now let us consider another potential witness of splitting. N_{1}^{∗} := h_{i}(M_{i}) and N_{2}^{∗} := h_{j}(M_{i}) with H^{∗} := h_{j} ∘ h_{i}^{−1} : N_{1}^{∗} → N_{2}^{∗}. Since p_{j} ↾ M_{i} does not μ-split over M_{0}, p_{j} ↾ N_{2}^{∗} = H^{∗}(p_{j} ↾ N_{1}^{∗}). Thus by (∗∗) we have

(♯) H^{∗}(p_{j} ↾ N_{1}^{∗}) = ga-tp(h_{j}(b̄_{j})/h_{j}(M_{i}), M^{∗}).

Now let us translate H^{∗}(p_{j} ↾ N_{1}^{∗}). By monotonicity and (∗ ∗ ∗), we have that p_{j} ↾ N_{1}^{∗} = ga-tp(b̄_{j}/h_{i}(M_{i}), M_{i+1}) = ga-tp(ā_{i}/h_{i}(M_{i}), M_{i+1}). We can then conclude by (∗) that p_{j} ↾ N_{1}^{∗} = ga-tp(h_{i}(ā_{i})/h_{i}(M_{i}), M_{i+1}). Applying H^{∗} to this equality yields

(♯♯) H^{∗}(p_{j} ↾ N_{1}^{∗}) = ga-tp(h_{j}(ā_{i})/h_{j}(M_{i}), M^{∗}).

By combining the equalities from (♯) and (♯♯) and applying h_{j}^{−1} we get that

ga-tp(b̄_{j}/M_{i}, M_{i+1}) = ga-tp(ā_{i}/M_{i}, M_{i+1}).
⊣

Notice that by Subclaim 5.3 and our choice of J, ⟨M_{i} | i ∈ J⟩ and ⟨ā_{i} | i ∈ J⟩ satisfy the conditions of Lemma 2.9. Applying Lemma 2.9, we get that ⟨ā_{i} | i ∈ J⟩ is a morley sequence over M_{0}. In particular, since A ⊂ M_{0}, we have that ⟨ā_{i} | i ∈ J⟩ is a Morley sequence over A.

⊣

6. Bounded multiplicity

Definition 6.1. Suppose M ≺_{K} N and let ā, b̄ ∈ |N|.

ā ∼_{M} b̄ iff there are n < ω and there exists a sequence ⟨J_{k} | k ≤ 2n⟩ satisfying the following conditions:

(1) every J_{k} is an infinite sequence, Galois indiscernible over M;

(2) ā ∈ J_{0} and b̄ ∈ J_{2n};

(3) For every k < n the sequence J_{2k}ˆ J_{2k+1} is Galois indiscernible over M;

(4) For every k < n the sequence J_{2k+2}ˆ J_{2k+1} is Galois indiscernible over M.

Theorem 6.2. Let K be as in Hypothesis 5.1 Suppose μ ≥ ℶ_{(2^{Hanf(K)})^{+}}. Let M ∈ K_{>μ}, N ≺_{K} M. If K is stable in μ then

|{p ∈ ga-S(M) : p does not split over N}| ≤ 2^{μ}.

Corollary 6.3 (partial stability spectrum).

7. Dividing

In this section we define dividing and derive some of its properties.

Definition 7.1. Let p ∈ ga-S(M) and N ≺_{K} M. We say that p divides over N iff there are ā ∈ M non-algebraic over N and a Morley sequence, {ā_{n} | n < ω} for the ga-tp(ā/N, M) such that for every collection {f_{n} ∈ Aut_{M}ℭ | n < ω} with f_{n}(ā) = ā_{n} we have

{f_{n}(p) | n < ω} is inconsistent.

Theorem 7.2 (Existence). Suppose that K is stable in μ and κ-tame for some κ < μ. For every p ∈ ga-S(M) with M ∈ K_{≥μ} there exists N ≺_{K} M of cardinality μ such that p does not divide over N.

Proof. Suppose that p and M form a counter-example. WLOG we may assume that M = ℭ. Through the proof of Claim 3.3.1 of [Sh 394], in order to contradict stability in μ, it suffices to find N_{i}, N_{i}^{1}, N_{i}^{2}, h_{i} for i < μ satisfying

(1) ⟨N_{i} ∈ K_{μ} | i ≤ μ⟩ is a ≺_{K}-increasing and continuous sequence of models;

(2) N_{i} ≺_{K} N_{i}^{l} ≺_{K} N_{i+1} for i < μ and l = 1, 2;

(3) for i < μ, h_{i} : N_{i}^{1} ≅ N_{i}^{2} and h_{i} ↾ N_{i} = id_{N_{i}} and

(4) p ↾ N_{i}^{2} ≠ h_{i}(p ↾ N_{i}^{1}).

Suppose that N_{i} has been defined. Since p divides over every substructure of cardinality μ, we may find ā, {ā_{n} | n < ω} and {f_{n} | n < ω} witnessing that p divides over N_{i}. Namely, we have that {f_{n}(p) | n < ω} is inconsistent. Let n < ω be such that f_{0}(p) ≠ f_{n}(p). Then p ≠ f_{0}^{−1} ∘ f_{n}(p). By κtameness, we can find N^{∗} ≺_{K} ℭ of cardinality μ containing N such that p ↾ N^{∗} ≠ (f_{0}^{−1} ∘ f_{n}(p)) ↾ N^{∗}. WLOG f_{0}^{−1} ∘ f_{n} ∈ Aut_{N}N^{∗}.

Let h_{i} := f_{0}^{−1} ∘ f_{n}, N_{i}^{1} := N^{∗} and N_{i}^{2} := N^{∗}. Choose N_{i+1} ≺_{K} ℭ to be an extension of N^{∗} of cardinality μ. ⊣

Next step:

(1) extension?

(2) symmetry?

(3) uniqueness (over very saturated models)?

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