LOCAL ORDER PROPERTY IN NONELEMENTARY CLASSES

RAMI GROSSBERG AND OLIVIER LESSMANN

Abstract. We study a local version of the order property in several frameworks, with an emphasis on frameworks where the compactness theorem fails: (1) Inside a fixed model, (2) for classes of models where the compactness theorem fails and (3) for the first order case. Appropriate localizations of the order property, the independence property, and the strict order property are introduced. We are able to generalize some of the results that were known in the case of local stability for the first order theories, and for stability for nonelementary classes (existence of indiscernibles, existence of averages, stability spectrum, equivalence between order and instability). In the first order case, we also prove the local version of Shelah’s Trichotomy Theorem. Finally, as an application, we give a new characterization of stable types when the ambient first order theory is simple.

1. Introduction

In the first order case, Victor Harnik and Leo Harrington in [HH], while presenting an alternative approach of forking to that of Saharon Shelah [Sh a], started a localized generalization of stability theory extending Saharon Shelah’s Unstable Formula Theorem (Theorem II 2.2 [Sh a]). This work was later continued and extended by Anand Pillay in [P]. About ten years later Zoe Chatzidakis and Ehud Hrushovski in their deep study of the model theory of fields with an automorphism [CH] as well as Ehud Hrushovski and Anand Pillay [HP] discovered natural examples of this phenomenon in algebra and obtained results in local stability for first order simple theories.

In parallel, Rami Grossberg and Saharon Shelah continued their study of stability and the order property in contexts where the compactness theorem fails; inside a model and for nonelementary classes (see for example [Gr1], [Gr2], [GrSh1], [GrSh2], [Sh12], and [Sh300]).

Date: February 12, 1999. The second author would like to express his gratitude to the first author for his constant

guidance during his PhD thesis.

The goal of this paper is to continue the study of local stability both in the first order case and in cases where the compactness theorem fails. When possible, we have tried to merge first order local stability with nonelementary stability theory and obtain results improving existing theorems in two directions. Three frameworks, listed in decreasing order of generality, are examined: (1) Inside a fixed structure; (2) For a nonelementary class of structures; (3) For the first order case. To explain what we mean, consider the property (*) of a model, that the model has no definable order of uncountable length. Studying (*) in (1) means that we study some model M which has property (*). In particular, this allows for the first order theory of M to be unstable. This framework was studied in [Gr1] and [Gr2]. In (2), this means that we assume that all the models of a given nonelementary class have property (*) (which still allows for the first order theory of the models to be unstable). In particular, all the results of (1) can be used. In addition, since (*) becomes a property of the class, some of the tools of infinitary logic can be used. In (3), this means that we assume that all the models of a first order theory have property (*). By compactness, this implies that the theory is stable. All the results of (1) and (2) can be used in (3).

We will focus on local versions of (*), namely local versions of stability and the order property. In addition to this, in (3) we also study local versions of the independence property and the strict order property. By local, we mean inside the set of realizations of a fixed type.

In (1) and (2), since the compactness theorem fails, we cannot use the forking machinery or definability of types, as [HH], [P] and [Sh a] do. Hence, the methods used have a combinatorial and set-theoretic flavor. Also, by (2) we mean the study of models of an infinitary logic, or of the class PC(T_{1}, T, Γ) (see the beginning of Section 3 for a definition). Hence, in addition to the absence of compactness, we have to do without the existence of saturated or even homogeneous models, as such models do not exist in general. Thus, frameworks (1) and (2) are more general than the study of finite diagrams [Sh3], also known as stability inside a homogeneous model, which was recently the focus of some activity, for example by Tapani Hyttinen and Saharon Shelah or Olivier Lessmann. In fact, for a treatment of local stability in the context of finite diagrams including the complete local stability spectrum and the local homogeneity spectrum, see [GrLe].

The basic structure assumption will be the impossibility of coding, via a formula in a given logic, a linear order of a certain length inside the set of realizations of a fixed type p. In the context (1), this means that this impossibility happens in a particular model. For example, a model M may have orders of length ω but not ω_{1} inside the set of realizations of p. We are not concerned with what happens in other models with the same theory (in particular by compactness there will be elementarily equivalent models that encode uncountable linear orders inside p). In the context of (2), we consider when the impossibility of coding a long order happens for all the models of a nonelementary class (for example, we can consider a class of structures which has a predicate p(x) containing a nonarchimedean field; no linear order of size more than the continuum can be embedded inside p). In (3), this will be considered for all the models of a first order theory. In this case, because of the compactness theorem, the structure assumption is that no infinite linear order exist inside the set of realizations of p. In the first order case, there are currently two notions of stable types in use. Our definition is equivalent to the following: p is stable if there do not exist ϕ(x̄, ȳ), ā_{i} ⊧ p for i < ω and b̄_{j} ⊧ p for j < ω such that ⊧ ϕ(ā_{i}, b̄_{j}) if and only if i < j. Another notion of stable type, sometimes called persistently stable (see for example [Bo]) is like the definition above, except that the b̄_{j} are not required to realize p. Hence, a persistently stable type is stable, but the converse may fail. We chose the first definition for two reasons: as a structure assumption it is weaker than the other and still allows us to derive the expected conclusions, and as a nonstructure assumption, the existence of long orders implies the existence of many nonisomorphic models (see Theorem VIII 3.2 in [Sh a]), even in nonelementary cases (see for example [Sh12] and [GrSh1]).

The paper is organized as follows:

In Section 2, we study stability and order for the realizations of a type p inside a fixed model M. In particular, the model M may omit many types. Denote by p(M) the set of realizations of p in M. We prove that the impossibility of coding a linear order of a certain length inside p(M) implies local stability (Theorem 5). By local stability, we mean the usual definitions in terms of the number of types extending the fixed type p. This is used to prove the existence of indiscernibles (Theorem 9), as well as averages (Theorem 12). It may be helpful for the reader to keep in mind that in Section 2, only what happens in M is relevant. Even if p is unstable in the first order sense, for each cardinal λ there will be arbitrarily large models M, where it is impossible to encode linear orders of size λ inside p(M). For results of this nature, see Chapter VII section 4 of [Sh a].

In Section 3, we study these local notions for classes of models that fail to satisfy the compactness theorem. We obtain a characterization of local stability for such a class of models in terms of the failure of the local order property, and a version of the stability spectrum (Theorem 16).

Finally, in Section 4, we particularize our discussion to the first order case. We introduce local version of the independence property and the strict order property. We prove the local version of Shelah’s Trichotomy Theorem: the local order property is equivalent to the disjunction of the local independence property and the local strict order property (Corollary 21). We characterize the local independence property in terms of averages (Theorem 23) and give, as an application, a characterization of stable types in terms of averages when the ambient first order theory is simple (Corollary 26).

Credits have been given throughout the text when particular cases of these results were known, either in the local first order case, or the nonlocal nonelementary case.

We are thank the anonymous referee for a carefull reading and many constructive comments.

2. Local notions inside a fixed model

In this section, we work inside a fixed structure M. Denote by L(M) the set of formulas in the language of M. These formulas may be considered to be in first order logic, or in a fragment of a larger logic, or even some subset of these with some weak closure properties. The basic assumption is that in this given model M, some type-definable set does not contain a large L(M)-definable order. Hence, exactly what L(M) is plays little role here, as we make no assumption on the theory of M or its level of homogeneity. Thus, even if L(M) is the set of first order formulas, these results are new since we make no assumption on the behavior of elementary equivalent models. This is a key difference with the work done by [HH] or [P].

Let p be a fixed set of formulas (maybe with parameters in M) such that p is realized in M. Denote by p(M) the set of elements of M realizing p.

Recall the notion of complete type inside a model. Let A ⊆ M, Δ be a set of L(M)-formulas and c̄ ∈ M. We let

tp_{Δ}(c̄/A, M) = {ϕ(x̄, ā) | ā ∈ A, ϕ(x̄, ȳ) ∈ Δ or ¬ϕ(x̄, ȳ) ∈ Δ, M ⊧ ϕ[c̄, ā]}.

We omit Δ when Δ = L(M).

For A ⊆ M and Δ a set of formulas, we let

S_{Δ,p}(A, M) = {tp_{Δ}(c̄/A, M) | c̄ ∈ M and c̄ realizes p }.

We omit Δ when Δ = L(M).

For a type q and a set A, we denote by q ↾ A the set of formulas in q with parameters in A. For a set of formulas Δ, we denote by q ↾ Δ the set of instances in q of formulas of Δ.

The next two definitions are the main concept of this paper.

Definition 1. For an infinite cardinal λ ≥ |L(M)|, the model M is said to be (λ, p)-stable if |S_{p}(A, M)| ≤ λ for each A ⊆ p(M) of cardinality at most λ.

Note that in the above definition we make demands only on subsets of p(M). In fact, throughout the rest of this paper, we will only deal with types q ∈ S_{p}(A, M) such that A ⊆ p(M).

Definition 2. M has the (λ, p)-order property if there exists a formula ϕ(x̄, ȳ) ∈ L(M) and a set {ā_{i} | i < λ} ⊆ p(M), such that

M ⊧ ϕ[ā_{i}, ā_{j}] if and only if i < j < λ.

The first theorem (Theorem 5) is a local version inside a model of Shelah’s Theorem that the failure of the order property implies stability for complete, first order theories. A generalization of Shelah’s theorem for nonelementary classes and in the local case will appear in the next section (Theorem 5). Theorem 5 will also be used in a key way to prove existence of indiscernibles (Theorem 9) and averages (Theorem 12) in this section. The technical tool needed to prove it is splitting. Recall the definition.

Definition 3. Let q ∈ S_{p}(B, M), with B ⊆ p(M). Let Δ_{1}, Δ_{2} ⊆ L(M). The type q is said to (Δ_{1}, Δ_{2})-split over A, if there exist elements b̄, c̄ ∈ B and a formula ϕ(x̄, ȳ) ∈ Δ_{2} such that tp_{Δ_{1}}(b̄/A, M) = tp_{Δ_{1}}(c̄/A, M) and both ϕ(x̄, b̄) and ¬ϕ(x̄, c̄) belong to q. We simply say splits for (L(M), L(M))- splits.

The next fact is a variation on Exercise I.2.3 from [Sh a].

Proposition 4. Let Δ_{1} and Δ_{2} be sets of L(M) formulas (closed under negation). Let B ⊆ C ⊆ p(M) and let A ⊆ M. Suppose that B realizes all the types in S_{Δ_{2},p}(A) that are realized in C. Let q, r ∈ S_{Δ_{1},p}(C) such that q, r do not (Δ_{1}, Δ_{2})-split over A. If q ↾ B = r ↾ B, then q = r.

The following theorem localizes results from [Sh12] and [Gr1]. The proof appearing in [Sh12] uses generalizations of a theorem of Paul Erdős and Michael Makkai appearing in [EM]. The proof given here is simpler and closer to [Gr1].

Theorem 5. Let μ and λ be cardinals such that μ ≥ |L(M)|, λ^{μ} = λ, and λ ≥ 2^{2^{μ}}. If M does not have the (μ^{+}, p)-order property, then M is (λ, p)- stable.

Proof. Suppose that M is not (λ, p)-stable. Then, there exists A ⊆ p(M) of cardinality λ such that |S_{p}(A, M)| > λ.

For each q ∈ S_{p}(A, M), we have (q ↾ ϕ) ∈ S_{ϕ,p}(A, M). Define
f: S_{p}(A, M) → Π_{ϕ∈L}S_{ϕ,p}(A, M), by f(q) = (q ↾ ϕ)_{ϕ∈L(M)}.

Then, f is a well-defined injection. Observe that

|Π_{ϕ∈L(M)}S_{ϕ,p}(A, M)| ≤ λ^{|L(M)|} ≤ λ^{μ} < λ^{+} ≤ |S_{p}(A, M)|.

By the pigeonhole principle, we can find ϕ ∈ L(M) such that |S_{ϕ,p}(A, M)| > λ.

Fix ϕ(x̄, ȳ) as above and choose {ā_{i} | i < λ^{+}} ⊆ p(M) such that

i ≠ j implies tp_{ϕ}(ā_{i}/A, M) ≠ tp_{ϕ}(ā_{j}/A, M).

Write χ(ȳ, x̄) := ϕ(x̄, ȳ). Define ⟨A_{i} | i < λ⟩ an increasing continuous sequence of subsets of p(M) containing A, each of cardinality at most λ, such that

(*) A_{i+1} realizes every type in S_{p}(B, M), for each B ⊆ A_{i} with |B| ≤ μ.

This is possible: Having constructed A_{i} of cardinality at most λ, there are at most λ^{μ} = λ subsets B of A_{i} of cardinality μ. Further, for each such B, we have |S_{p}(B, M)| ≤ 2^{μ} ≤ λ, so we can add the needed realizations in A_{i+1} from p(M) while keeping |A_{i+1}| ≤ λ.

We now claim that (*) allows us to choose, for every i < λ^{+}, an index j, with i < j < λ^{+}, such that for each l < μ^{+} the type tp_{ϕ}(ā_{j}/A_{l+1}, M) (χ, ϕ)-splits over each B ⊆ A_{l} of cardinality at most μ.

Otherwise, there is i < λ^{+} such that for every index j, with i < j < λ^{+}, there exists l < μ^{+} and B^{j} ⊆ A_{l} of cardinality μ such that tp_{ϕ}(ā_{j}/A_{l+1}, M) does not (χ, ϕ)-split over B^{j}. By the pigeonhole principle (since λ^{+} ≥ μ) we can find S ⊆ λ^{+} of cardinality λ^{+}, an ordinal l < μ^{+}, and B ⊆ A_{l+1} of cardinality μ such that tp_{ϕ}(ā_{j}/A_{l+1}, M) does not (χ, ϕ)- splits over B, for every j ∈ S. By (*) we can choose C ⊆ A_{l+1} of cardinality at most 2^{μ} such that C realizes every type in S_{χ,p}(B, M). Then, since |S_{ϕ,p}(C, M)| ≤ 2^{2^{μ}} < λ^{+}, by the pigeonhole principle, we may assume that tp_{ϕ}(ā_{j}/C, M) is constant for j ∈ S. By Proposition 4, we must have tp_{ϕ}(ā_{j}/A_{l+1}, M) = tp_{ϕ}(ā_{i}/A_{l+1}, M), for i, j ∈ S. This contradicts the choice of ā_{i}s and the fact that A ⊆ A_{l+1}.

Define {c̄_{l}, d̄_{l}, b̄_{l} | l < μ^{+}} ⊆ A_{2l+2} and B_{l} = ⋃{c̄_{k}, d̄_{k}, b̄_{k} | k < l} such that:

(1) B_{l} ⊆ A_{2l} and |B_{l}| ≤ μ;

(2) tp_{χ}(c̄_{l}/B_{l}, M) = tp_{χ}(d̄_{l}/B_{l}, M);

(3) Both ϕ(x̄, c̄_{l}) and ¬ϕ(x̄, d̄_{l}) belong to tp_{ϕ}(ā_{j}/A_{2l}, M);

(4) b̄_{l} ∈ A_{2l+1} realizes both ϕ(x̄, c̄_{l}) and ¬ϕ(x̄, d̄_{l}).

This is possible: Let B_{0} = ∅ and B_{l} = ⋃_{k<l} B_{k} when l is a limit ordinal. Having constructed B_{l} ⊆ A_{2l} of cardinality at most μ, the type tp_{ϕ}(ā_{j}/A_{2l}) (χ, ϕ)-splits over B_{l} and hence there are c̄_{l}, d̄_{l} ∈ A_{2l} with tp_{χ}(c̄_{l}/B_{l}, M) = tp_{χ}(d̄_{l}/B_{l}, M) and ϕ(x̄, c̄_{l}) and ¬ϕ(x̄, d̄_{l}) ∈ tp_{ϕ}(ā_{j}/A_{2l}, M). Then, by construction we can find b̄_{l} ∈ A_{2l+1} realizing tp_{ϕ}(ā_{j}/c̄_{l}d̄_{l}, M) so (4) is automatically satisfied.

Now, the set {b̄_{l} ˆ c̄_{l} ˆ d̄_{l} | l < μ^{+}} ⊆ p(M) and the formula

ψ(x̄_{0}, x̄_{1}, x̄_{2}, ȳ_{0}, ȳ_{1}, ȳ_{2}) := ϕ(x̄_{0}, ȳ_{1}) ↔ ϕ(x̄_{0}, ȳ_{2})

demonstrate that M has the (μ^{+}, p)-order property.

The following definition generalizes the notion of relative saturation.

Definition 6. We say that a set C ⊆ M is relatively (λ, p)-saturated if C realizes every q ∈ S_{p}(B, M) for every B ⊆ C such that |B| < λ.

The following lemma is a version of κ(T) ≤ |T |^{+} for the notion of splitting.

Lemma 7. Let μ be a cardinal such that μ ≥ |L(M)|. Suppose that M does not have the (μ^{+}, p)-order property. Suppose that B ⊆ p(M) is relatively (μ^{+}, p)-saturated. Then for each q ∈ S_{p}(B, M) there is A ⊆ B of cardinality at most μ such that q does not split over A.

Proof. Suppose, for a contradiction, that there exist a relatively (μ^{+}, p)- saturated set B and a type q ∈ S_{p}(B, M), such that q splits over every A ⊆ B of cardinality at most μ.

We will show that M has the (μ^{+}, p)-order property. Construct a sequence of sets ⟨A_{i} | i < μ^{+}⟩ such that:

(1) A_{0} = ∅;

(2) A_{i} = ⋃_{j<i} A_{j}, when i is a limit ordinal;

(3) A_{i} ⊆ B, for each i < μ^{+};

(4) |A_{i}| ≤ μ, for each i < μ^{+};

(5) There are ϕ_{i} ∈ L(M) and ā_{i}, b̄_{i} ∈ A_{i+1}, such that tp(ā_{i}/A_{i}, M) = tp(b̄_{i}/A_{i}, M) and ϕ(x̄, ā_{i}) and ¬ϕ(x̄, b̄_{i}) are in q;

(6) A_{i+1} contains c̄_{i} realizing q ↾ (A_{i} ∪ ā_{i}b̄_{i}).

This is possible: For i = 0 or a limit ordinal, it is obvious. Suppose that A_{i} has been constructed. Since |A_{i}| ≤ μ and A_{i} ⊆ B, q splits over A_{i}. Hence, there exist a formula ϕ_{i} ∈ L(M), and ā_{i}, b̄_{i} ∈ B demonstrating this. Since B is relatively (μ^{+}, p)-saturated, and q ↾ (A_{i} ∪ā_{i}b̄_{i}) ∈ S_{p}(A_{i} ∪ā_{i}b̄_{i}, M), there exists c̄_{i} ∈ B realizing q ↾ (A_{i} ∪ ā_{i}b̄_{i}). Let A_{i+1} = A_{i} ∪ {ā_{i}, b̄_{i}, c̄_{i}}. All the conditions are satisfied.

This is enough: By the pigeonhole principle, since μ ≥ |L(M)|, we may assume that there exists ϕ ∈ L(M) such that ϕ_{i} = ϕ, for each i < μ^{+}. Now consider {c̄_{i} ˆ ā_{i} ˆ b̄_{i} | i < μ^{+}} and the formula

ψ(x̄_{0}, x̄_{1}, x̄_{2}, ȳ_{0}, ȳ_{1}, ȳ_{2}) := ϕ(x̄_{0}, ȳ_{1}) ↔ ϕ(x̄_{0}, ȳ_{2}).

It is easy to see that they demonstrate that M has the (μ^{+}, p)-order property.

The following fact is Lemma I.2.5 of [Sh a].

Fact 8. Let B ⊆ p(M) and let {ā_{i} | i < α} ⊆ p(M) be given. Consider the type q_{i} = tp(ā_{i}/B ∪ {ā_{j} | j < i}, M) ∈ S_{p}(B ∪ {ā_{j} | j < i}, M) and suppose that

(1) If i < j < α then q_{i} ⊆ q_{j};

(2) For each i < α the type q_{i} does not split over B.

Then {ā_{i} | i < α} is indiscernible over B.

The next theorem is a generalization of two theorems. (1) When p is stable for every model of a first order theory, a version of this theorem appears in [P]. (2) When p := {x̄ = x̄}, it appears in [Gr1].

Theorem 9. Let μ and λ be cardinals such that μ ≥ |L(M)|, λ^{μ} = λ, and λ ≥ 2^{2^{μ}}. If M does not have the (μ^{+}, p)-order property, then for every I ⊆ p(M) and every A ⊆ p(M) such that |I| > λ ≥ |A|, there exists J ⊆ I of cardinality λ^{+} indiscernible over A.

Proof. Let I = {ā_{i} | i < λ^{+}}. By the pigeonhole principle, we may assume that ℓ(ā_{i}) = ℓ(ā_{j}), for i, j < λ^{+}.

Define ⟨A_{i} | i < λ^{+}⟩ ⊆ p(M) such that:

(1) A_{0} = A;

(2) A_{i} = ⋃_{j<i} A_{j}, when j is a limit ordinal;

(3) A_{i} ⊆ p(M);

(4) |A_{i}| ≤ λ, for every i < λ^{+};

(5) A_{i+1} contains ā_{i};

(6) A_{i+1} realizes every type in S_{p}(B, M), for each B ⊆ A_{i} of cardinality at most μ.

This is possible: For i = 0 it is clear. If i is a limit ordinal it is easy. Let us concentrate on the successor stage. Assume that A_{i} of cardinality λ has been constructed. By cardinality assumption, there are λ = λ^{μ} subsets B of A_{i} of cardinality μ, and for each such B we have |S_{p}(B, M)| ≤ 2^{μ} ≤ λ. Hence, A_{i+1} satisfying (3)–(6) can be found.

Consider the following stationary subset of λ^{+}

S = {i < λ^{+} | cf(i) ≥ μ^{+}}.

Let r_{i} := tp(ā_{i}/A_{i}, M). Then clearly r_{i} ∈ S_{p}(A_{i}, M). Now, for each i ∈ S, since cf(i) ≥ μ^{+}, the set A_{i} is relatively (μ^{+}, p)-saturated. Hence, by Lemma 7, there exists B_{i} ⊆ A_{i} of cardinality at most μ such that r_{i} does not split over B_{i}. Furthermore, since cf(i) = μ^{+}, there exists j < i such that B_{i} ⊆ A_{j}.

This shows that the function f: S → λ^{+} defined by

f(i) = min{j | B_{i} ⊆ A_{j}},

is regressive. Hence, by Fodor’s lemma (see Theorem 22 of [Je]), there is S′ ⊆ S of cardinality λ^{+} and i_{0} < λ^{+} such that for each i ∈ S′ we have B_{i} ⊆ A_{i_{0}}. Since there are only λ^{μ} = λ subsets of A_{i_{0}} of size μ, we may assume, by the pigeonhole principle, that there exists a set B ⊆ A_{i_{0}} such that B_{i} = B for each i ∈ S′. Now, M does not have the (μ^{+}, p)-order property, and λ^{μ} = λ, so Theorem 5 implies that M is (λ, p)-stable. Hence, |S_{p}(A_{i_{0}}, M)| ≤ λ, and thus by the pigeonhole principle, we may further assume that tp(ā_{i}/A_{i_{0}}, M) = tp(ā_{j}/A_{i_{0}}, M), for every i, j ∈ S′.

By re-enumerating if necessary, we may assume that S′\(i_{0}+1) = λ^{+}. Now let

q_{i} := tp(ā_{i}/A_{i_{0}} ∪ {ā_{j} | j < i}) ∈ S_{p}(Ai_{0} ∪ {ā_{j} | j < i}).

By Proposition 4 we have that q_{i} ⊆ q_{j} if i < j. Thus, all the assumptions of Fact 8 are satisfied, so J = {ā_{i} | i < λ^{+}} is indiscernible over A, since A ⊆ A_{i_{0}}. This finishes the proof.

In the previous theorem, we demanded that A be a subset of p(M). The next remark summarizes what we can do when A ⊆ M is not necessarily contained in p(M). It follows from the previous theorem by considering an expansion of L(M) with constants for elements in A.

Remark 10. Let μ ≥ |L(T)| be a cardinal. Let A ⊆ M be given and suppose that M does not have the (μ^{+}, p)-order property even allowing parameters from A. Let λ^{μ} = λ and λ ≥ 2^{2^{μ}}. Then, for every I ⊆ p(M) of cardinality λ^{+}, there exists J ⊆ I of cardinality λ^{+} indiscernible over A.

Definition 11. Let I be an infinite set of finite sequences. Let A ⊆ M. We define the average of I over A in M as follows

Av(I, A, M) := {ϕ(x̄, ā) | ā ∈ A, ϕ(x̄, ȳ) ∈ L(M),
and M ⊧ ϕ[c̄, ā] for |I| elements c̄ ∈ I}.

We will be interested in conditions guaranteeing that averages are well-defined. It is a known fact (see Lemma III 1.7 (1) of [Sh a]) that if M is a model of a complete, first order, stable theory T, then for every infinite set of indiscernibles I and A ⊆ M, the average Av(I, A, M) is a complete type over A. Also, if |I| > |A| + κ(T), then the average is realized by an element of I (this is essentially Lemma III 3.9 of [Sh a]). A corresponding local result (Theorem 23) in the presence of compactness will be given in Section 4. Inside a fixed model, the situation is more delicate. The next theorem is a localization of Conclusion 1.11 in [Sh300]. Notice the similarity with the assumptions of Theorem 9. However while in Theorem 9 the set J depends on the set A in the following theorem we mange to find a subset J that is independent of A.

Theorem 12. Let μ and λ be cardinals such that μ ≥ |L(M)|, λ^{μ} = λ, and λ ≥ 2^{2^{μ}}. If M does not have the (μ^{+}, p)-order property, then for every I ⊆ p(M) of cardinality λ^{+}, there exists J ⊆ I of cardinality λ^{+} such that for each A ⊆ p(M) the average Av(J, A, M) is a complete type over A. Moreover, if |J| > |A|, then Av(J, A, M) ∈ S_{p}(A, M).

Proof. Let I = {ā_{α} | α < λ^{+}}. We may assume by the pigeonhole principle that there exists n < ω such that ℓ(ā_{α}) = n, for each α < λ^{+}.

We first essentially repeat the proof of Theorem 9 and construct a sequence ⟨A_{α} | α ≤ λ^{+}⟩ such that:

(1) A_{0} = ∅, A_{δ} = ⋃_{α<δ} A_{α} when δ is limit, and A_{α} ⊆ A_{α+1}.

(2) A_{α} ⊆ p(M).

(3) |A_{α}| ≤ λ, for every α < λ^{+}.

(4) A_{α+1} contains ā_{α}.

(5) A_{α+1} realizes all types in S_{p}(A_{α}, M).

This is possible: Since M does not have the (μ^{+}, p)-order property, then M is (λ, p)-stable by Theorem 5. Hence, |S_{p}(A_{α}, M)| ≤ λ inductively, for each α < λ^{+}.

Now (5) implies that

(6) If cf(δ) ≥ μ^{+} then A_{δ} is relatively (μ^{+}, p)-saturated.

As in the proof of Theorem 9, we can find a set S ⊆ {δ < λ^{+} | cf(δ) ≥ μ^{+}} of cardinality λ^{+} and an ordinal α(∗) = min S such that

(7) For each α ∈ S, the type tp(ā_{α}/A_{α}, M) does not split over A_{α(∗)}.

(8) If α, β ∈ S and α < β then tp(ā_{α}/A_{α}, M) ⊆ tp(ā_{β}/A_{β}, M).

We claim that the set J = {ā_{α} | α ∈ S} is as desired. To show this, we will show that

(*) For every c̄ ∈ p(M) and ϕ(x̄, ȳ) ∈ L(M), either

|{α ∈ S : M ⊧ ϕ[ā_{α}, c̄]}| ≤ μ or |{α ∈ S : M ⊧ ¬ϕ[ā_{α}, c̄]}| ≤ μ.

This implies the conclusion of the theorem: For A ⊆ p(M), condition (*) implies that Av(J, A, M) is a consistent set of formulas over A, as each finite subset is realized by all but μ many elements of J. Since averages are always complete, we need only show that Av(J,A,M) is consistent. For the last sentence, notice that all but |A| + |L(M)| + μ elements of J realize Av(J, A, M). Hence, if λ^{+} > |A|, then there exists ā_{α} ∈ J ⊆ p(M) realizing Av(J, A, M) (as λ ≥ μ+|L(M)|). This shows that Av(J, A, M) ∈ S_{p}(A, M).

Let c̄ ∈ p(M) and ϕ(x̄, ȳ) ∈ L(M) be given. Since c̄ ∈ p(M) then tp(c̄/A_{α}, M) ∈ S_{p}(A_{α}, M). Hence, by (5), we can find {c̄_{α} | α ∈ S} ⊆ p(M) satisfying

(9) c̄_{α} ∈ A_{α+2}.

(10) c̄_{α} realizes tp(c̄/A_{α+1}, M).

We will prove (*) by finding a set of ordinals E of cardinality μ such that either {α ∈ S : M ⊧ ϕ[ā_{α}, c̄]} ⊆ E or {α ∈ S : M ⊧ ¬ϕ[ā_{α}, c̄]} ⊆ E.

We construct the set E, as well as a set C ⊆ A_{λ^{+}} with the following properties:

(11) |E| ≤ μ and |C| ≤ μ.

(12) λ^{+} ∈ E.

(13) If α+1 ∈ E then α ∈ E and if δ ∈ E and cf(δ) ≤ μ then sup(E∩δ) = δ.

(14) If δ ∈ E and cf(δ) ≥ μ^{+}, then tp(c̄/A_{δ}, M) does not split over C ∩A_{δ}. Moreover, C ∩ A_{δ} ⊆ A_{sup(E∩δ)}.

This is possible: Construct E_{n} and C_{n} of cardinality at most μ by induction on n < ω. Let E_{0} = {λ^{+}} and C_{0} = ∅. Then, by (6) and Lemma 7 we can find C_{n+1} of cardinality μ such that tp(c̄/A_{δ}, M) does not split over C_{n+1} ∩ A_{δ} for each δ ∈ E_{n} with cf(δ) ≥ μ^{+}. Furthermore, we can add at most μ many ordinals to E_{n+1} to ensure that C_{n+1} ⊆ A_{sup(E_{n+1}∩δ)}. Thus, E = ⋃_{n<ω} E_{n} and C = ⋃_{n<ω} C_{n} are as desired.

This is enough to prove (*). In fact, to show that {α ∈ S : M ⊧ ϕ[ā_{α}, c̄]} ⊆ E or {α ∈ S : M ⊧ ¬ϕ[ā_{α}, c̄]} ⊆ E, it clearly suffices to show

(**) M ⊧ ϕ[ā_{α}, c̄] ↔ ϕ[ā_{β}, c̄], for every α, β ∈ S \ E.

Notice that by construction (11)–(14) the set S \ E is partitioned into at most μ intervals of the form {α ∈ S | sup(E ∩ δ) ≤ α < δ}, where δ ∈ E with cf(δ) ≥ μ^{+}. If such an interval is nonempty, then it must have size at least μ^{+}. We will make use of this and prove (**) in two stages. In the first part, we will show that (**) holds, provided α and β belong to the same interval, and then in the second part, that (**) holds also when α and β belong to different intervals.

Let δ ∈ E be such that cf(δ) ≥ μ^{+}. Denote by δ_{0} = sup(E ∩ δ). Now let α, β ∈ S such that δ_{0} ≤ α < β < δ. Without loss of generality, assume that M ⊧ ϕ[ā_{α}, c̄]. Then ϕ(ā_{α}, ȳ) ∈ tp(c̄/A_{δ}, M). By (14) the type tp(c̄/A_{δ}, M) does not split over C ∩ A_{δ} ⊆ A_{δ_{0}}. But, by (8), we have tp(ā_{α}/A_{δ_{0}}, M) = tp(ā_{β}/A_{δ_{0}}, M). Hence, by nonsplitting ϕ(ā_{β}, ȳ) ∈ tp(c̄/A_{δ}, M) and so M ⊧ ϕ[ā_{β}, c̄].

To prove the second part, we first claim that

(†) M ⊧ ϕ[ā_{α_{1}}, c̄_{β_{1}}] ↔ ϕ[ā_{α_{2}}, c̄_{β_{2}}], for every α_{1} > β_{1} and α_{2} > β_{2} in S.

To see this, let γ = max(α_{1}, α_{2}). Then by (8) and (9) (recall that ordinals in S are limit), we have M ⊧ ϕ[ā_{α_{1}}, c̄_{β_{1}}] ↔ ϕ[ā_{γ}, c̄_{β_{1}}] and also M ⊧ ϕ[ā_{α_{2}}, c̄_{β_{2}}] ↔ ϕ[ā_{γ}, c̄_{β_{2}}]. Now by (10) we have that tp(c̄_{β_{1}}/A_{α(∗)}, M) = tp(c̄_{γ}/A_{α(∗)}, M), and by (9), both c̄_{β_{1}}, c̄_{β_{2}} ∈ A_{γ}. But by (7) the type tp(ā_{γ}/A_{γ}, M) does not split over A_{α(∗)}. Hence, ϕ(x̄, c̄_{β_{1}}) ∈ tp(ā_{γ}/A_{γ}, M) if and only if ϕ(x̄, c̄_{β_{2}}) ∈ tp(ā_{γ}/A_{γ}, M), and therefore, M ⊧ ϕ[ā_{γ}, c̄_{β_{1}}] ↔ ϕ[ā_{γ}, c̄_{β_{2}}]. This proves (†).

Now for the second part, let δ, ξ ∈ E with cf(δ) ≥ μ^{+} and cf(ξ) ≥ μ^{+}. Denote by δ_{0} = sup(E ∩ δ) and ξ_{0} = sup(E ∩ ξ). Assume that δ_{0} < ξ_{0} and let i ∈ S with δ_{0} ≤ i < δ and j ∈ S with ξ_{0} ≤ j < ξ. To show: M ⊧ ϕ[ā_{i}, c̄] ↔ ϕ[ā_{j}, c̄]. Suppose M ⊧ ¬(ϕ[ā_{i}, c̄] ↔ ϕ[ā_{j}, c̄]). We will derive a contradiction by showing that M has the (μ^{+}, p)-order property.

Assume, without loss of generality, that M ⊧ ϕ[ā_{i}, c̄] and M ⊧ ¬ϕ[ā_{j}, c̄]. We distinguish two cases.

Case 1: Suppose M ⊧ ϕ[ā_{j}, c̄_{i}] (recall j > i). Then, by (†), we have that M ⊧ ϕ[ā_{α}, c̄_{β}], for every α, β ∈ S with α > β. On the other hand since M ⊧ ¬ϕ[ā_{j}, c̄], the first part of this argument shows that M ⊧ ¬ϕ[ā_{α}, c̄], for each α ∈ S with ξ_{0} ≤ α < ξ. Hence, by (10), for each β ∈ S with α ≤ β we have that M ⊧ ¬ϕ[ā_{α}, c̄_{β}]. Thus, for α, β ∈ S ∩ [ξ_{0}, ξ), we have

M ⊧ ¬ϕ[ā_{α}, c̄_{β}] if and only if α ≤ β.

This implies easily that M has the (μ^{+}, p)-order property.

Case 2: Suppose M ⊧ ¬ϕ[ā_{j}, c̄_{i}]. Similarly to Case 1, we obtain the (μ^{+}, p)-order property by using the interval S ∩ [δ_{0}, δ) and the fact that M ⊧ ϕ[ā_{i}, c̄].

3. Local order and stability for nonelementary classes

In this short section, we will examine the stability of p with respect to all the models of a given class of models K. This specializes the context of Section 2 in that we will assume that all models M ∈ K fail to have the (λ, p)-order property, for some λ. This allows us to use infinitary model theoretic techniques. Let us fix the concepts. We will work inside the class K = PC(T_{1}, T, Γ). Recall that for T ⊆ T_{1} and Γ a set of T_{1}-types over the empty set, we let

PC(T_{1}, T, Γ) = {M ↾ L(T) : M ⊧ T_{1} and M omits every type in Γ}

We will denote by μ(K) = μ(|T_{1}|, |Γ|), the Hanf-Morley number for K. Recall that μ(λ, κ) is the least cardinal μ with the property that for every PC(T_{1}, T, Γ) with |T_{1}| ≤ λ and |Γ| ≤ κ, if PC(T_{1}, T, Γ) contains a model of cardinality μ, then it contains models of arbitrarily large cardinality. When κ = |Γ| = 0 there is no type to omit and so by the Upward Löwenheim-Skolem Theorem we have μ(K) = ℵ_{0}. For |Γ| ≥ 1, then μ(K) = ℶ_{δ(|T_{1}|,|Γ|)}. Recall that δ(λ, κ) is the least ordinal δ with the property that for every PC(T_{1}, T, Γ) with |T_{1}| ≤ λ and |Γ| ≤ κ, if PC(T_{1}, T, Γ) contains a model with a predicate whose order type is δ, then it contains a model where this predicate is not well-ordered. Much is known about such numbers. Here are some of the known facts. First δ(λ, 0) = ω and δ(λ, κ) is always a limit ordinal. We have monotonicity properties: if λ_{1} ≤ λ_{2} and κ_{1} ≤ κ_{2}, then δ(λ_{1}, κ_{1}) ≤ δ(λ_{2}, κ_{2}). Also, if 1 ≤ κ ≤ λ then δ(λ, κ) = δ(λ, 1). In general δ(λ, κ) ≤ (2^{λ})^{+}. Finally, suppose κ ≤ λ and λ is a strong limit cardinal of cofinality ℵ_{0}, then δ(λ, κ) = λ^{+}. See Lemma VII.5.1 and Theorem VII.5.5 of [Sh a] or [Gr b]. All these results about δ(λ, κ) and μ(λ, κ) and more can be found in [Sh a] Chapter VII.5.

Choosing to carry out the theorems of this section in a PC-class is arbitrary. We could have chosen to study any sufficiently general class of models extending the first order case in which the compactness theorem fails. For example, the class of models of an infinitary sentence ψ ∈ L_{ω_{1}ω} or L_{λ^{+}ω}. All the results of this section hold for such classes and the proofs can usually be used verbatim.

As in the previous section, we will fix p a set of L(T)-formulas (with parameters).

We expand the definitions we made in the first section for the class K.

Definition 13. (1) Let λ be a cardinal. We say that p is stable in λ, if

for every M ∈ K, M is (λ, p)-stable.

(2) We say that p is stable if there exists a cardinal λ such that p is stable in λ.

Definition 14. (1) We say that p has the λ-order property if there exists

M ∈ K such that M has the (λ, p)-order property.

(2) We say that p has the order property if p has the λ-order property for every λ.

Using the tools of [Sh12] (finer results are in [GrSh1] and [GrSh2]) we observe:

Fact 15. The following conditions are equivalent.

(1) p has the order property;

(2) p has the λ-order property for every λ < μ(K);

(3) p has the μ(K)-order property;

(4) There exists a model M ∈ K, a formula ϕ(x̄, ȳ), and an indiscernible sequence {ā_{i} | i < μ(K)} ⊆ p(M), such that

M ⊧ ϕ[ā_{i}, ā_{j}] if and only if i < j < μ(K).

We now prove a version of the stability spectrum and the equivalence between local instability and local order. Nonlocal theorems of this vein appear in [Sh12].

Theorem 16. The following conditions are equivalent.

(1) p is stable;

(2) There exists a cardinal κ(K) < μ(K) + |L(T)|^{+} such that p is stable in every λ ≥ μ(K) satisfying λ^{κ(K)} = λ.

(3) p does not have the order property.

Proof. (2) ⇒ (1) trivially.

(3) ⇒ (2): Since p does not have the order property, by Fact 15 there exists a cardinal κ < μ(K) such that no model of K has the (κ^{+}, p)-order property. Let λ ≥ μ(K). Then, automatically, since κ < μ(K) and μ(K) is either ℵ_{0} or a strong limit, we have λ ≥ 2^{2^{κ}}. Let κ(K) = κ + |L(T)|. Hence, if λ ≥ μ(K) satisfies λ^{κ(K)} = λ, and M ∈ K, then Theorem 5 implies that M is (λ, p)-stable. Thus, p is stable in λ.

(1) ⇒ (3): This is again a standard application of Hanf number techniques. We give just a sketch. Suppose p is stable in λ. Let T^{∗} be an expansion of T_{1} with Skolem functions, such that |T^{∗}| = |T_{1}|. Let κ be smallest such that 2^{κ} > λ. Using the order property and the methods of Morley, we can find M^{∗} ⊧ T^{∗} such that M = M^{∗} ↾ L(T) ∈ K, with ϕ(x̄, ȳ), and {ā_{i} | i < ω} ⊆ p(M) demonstrating the p-order property. Furthermore {ā_{i} | i < ω} ⊆ p(M) is T^{∗}-indiscernible. Hence, by compactness, we can find a model N^{∗} ⊧ T^{∗} and a set {ā_{η} | η ∈ ^{κ≥}2} ⊆ p(N^{∗}) demonstrating the p-order property with respect to the lexicographic order. Furthermore, for every n < ω

tp(ā_{ν_{0}}, …, ā_{ν_{n}}/∅, N^{∗}) = tp(ā_{0}, …, ā_{n}/∅, M^{∗}), for every ν_{0} < ⋯ < ν_{n}.

We may assume that N^{∗} is the Ehrenfeucht-Mostowski closure of {ā_{η} | η ∈ ^{κ≥}2}, since T^{∗} has Skolem functions. Let N = N^{∗} ↾ L(T). Then N ∈ K. Consider A = ⋃_{η∈ κ>2} ā_{η} ⊆ p(N). Then |A| ≤ 2^{<κ} ≤ λ and |S_{p}(A, N)| = 2^{κ} > λ. Thus, N is not (λ, p)-stable, a contradiction.

Remark 17. The first order case can be read from the results above by setting T_{1} = T and Γ = ∅. We have μ(K) = ℵ_{0} and so p is stable if and only if p is stable in every λ such that λ^{|L(T)|} = λ. In the first order case, most authors define stable types using (3) with μ(K) = ℵ_{0}.

4. Local order, independence, and strict order in the first order case

In this section, we will fix a complete, first order theory T and obtain results for the class of models of T. As usual, we work inside the monster model ℭ, a model which is κ̄-saturated, for a cardinal κ̄ larger than any cardinality mentioned in this paper. Hence, all sets will be assumed to be inside ℭ and satisfaction is defined with respect to ℭ. We will write S_{p}(A) for S_{p}(A, ℭ) and Av(I, A) for Av(I, A, ℭ) as is customary. As before, we fix a (nonalgebraic) T -type p. Denote by dom(p) the set of parameters of p.

Given that PC(T_{1}, T, Γ) with T_{1} = T and Γ = ∅, all the results we have obtained so far hold with μ(K) = ℵ_{0}.

We first give local versions of Saharon Shelah’s first order notion of independence and strict order property (see [Sh a]). Several variations of these definitions can be thought of, according to which variables should be relativized to p. Only the one given below allows one to recover Shelah’s Trichotomy Theorem (see Theorem 20 and the Corollary immediately following).

For a statement t and a formula ϕ, we use the following notation: ϕ^{t} = ¬ϕ if the statement t is false and ϕ^{t} = ϕ, if the statement t is true. We will use the same notation when t ∈ {0, 1}, where 0 stands for false and 1 stands for truth.

Definition 18. (1) We say that ϕ(x̄, ȳ) has the p-independence property if for every n < ω there exists {ā_{i} | i < n} ⊆ p(ℭ) such that

p(x̄) ∪ {ϕ(x̄, ā_{i})^{i∈w} | i < n} is consistent, for every w ⊆ n.

We say that p has the independence property if there exists a formula ϕ(x̄, ȳ) with the p-independence property;

(2) A formula ϕ(x̄, ȳ) is said to have the p-strict order property if for every n < ω there exists {ā_{i} | i < n} ⊆ p(ℭ) such that

⊧ ∃x̄(¬ϕ(x̄, ā_{i}) ∧ ϕ(x̄, ā_{j})) if and only if i < j < n.

We say that p has the strict order property if there exists a formula ϕ(x̄, ȳ) with the p-strict order property.

The next proposition is left for the reader.

Proposition 19. If p has the independence property or the strict order property, then p has the order property.

The next two results depend explicitly on the parameters of p. The proof of the next theorem is a minor variant of the proof of Theorem II 4.7 (2) of [Sh a]. One has to make sure to choose parameters that realize p.

Theorem 20. Let ϕ(x̄, ȳ) be a formula with the p-order property. Then, either ϕ(x̄, ȳ) has the p-independence property, or there exist χ(x̄), the conjunction of finitely many formulas of p, an integer n < ω and a sequence η ∈ ^{n}2 such that the formula χ(x̄) ∧ ⋀_{l<n} ϕ(x̄, ȳ_{l})^{η[l]} has the p-strict order property (maybe with parameters from dom(p)).

The next corollary is the local version of Shelah’s Trichotomy Theorem (see Theorem II 4.7 of [Sh a]).

Corollary 21. Assume that p has no parameters. The type p has the order property if and only if p has the independence property or p has the strict order property.

Proof. Suppose p has the order property. Then some formula ϕ has the porder property. Thus, by Theorem 20 p has the independence property or the strict order property (without parameters, since dom(p) = ∅).

The converse is Proposition 19.

The following is an improvement of Theorem II.2.20 of [Sh a]. The proof is immediate, but we include it as we could not find it in the literature, even when p is x = x.

Lemma 22. The following conditions are equivalent

(1) p does not have the independence property;

(2) For every infinite indiscernible sequence I ⊆ p(ℭ) and for every formula ϕ(x̄, ȳ) ∈ L(T) there exists an integer n_{ϕ} < ω such that for every c̄ ∈ p(M) either

|{ā ∈ I : ⊧ ϕ[ā, c̄]}| ≤ n_{ϕ} or |{ā ∈ I : ⊧ ¬ϕ[ā, c̄]}| ≤ n_{ϕ}.

Proof. (1) ⇒ (2) Let ϕ(x̄, ȳ) and I be given. Suppose (2) fails. Then, by compactness, we can find c̄ ∈ p(ℭ) and a sequence {ā_{i} | i < ω} ⊆ p(ℭ) indiscernible over dom(p) such that

|{i < ω : ⊧ ϕ[ā_{i}, c̄]}| = ℵ_{0} and |{i < ω : ⊧ ¬ϕ[ā_{i}, c̄]}| = ℵ_{0}. (*)

We are going to show that ϕ(x̄, ȳ) has the p-independence property. Let n < ω and w ⊆ n. It is enough to show that

(**) p(ȳ) ∪ {ϕ(ā_{i}, ȳ)^{i∈w} | i < n} is consistent.

To see this, construct a strictly increasing sequence of integers ⟨i_{m} | m < n⟩ such that ℭ ⊧ ϕ[ā_{i_{m}}, c̄] if and only if m ∈ w. This is easily done by induction using (*). By indiscernibility of {ā_{i} | i < ω}, (**) holds if and only the set of formulas p(ȳ) ∪ {ϕ(ā_{i_{m}}, ȳ)^{m∈w} | m < n} is consistent, which is the case, since it is realized by c̄.

(2) ⇒ (1) Suppose that ϕ(x̄, ȳ) has the p-independence property and I = {ā_{i} | i < ω} ⊆ p(ℭ) demonstrate this. Then, for each n < ω, and for each w ⊆ n we have

p(x̄) ∪ {ϕ(x̄, ā_{i})^{i∈w} | i < n} is consistent.

Hence, by compactness, we can find an indiscernible sequence J = {b̄_{i} | i < ω} ⊆ p(ℭ) and c̄ ∈ p(ℭ) such that both {i < ω : ⊧ ϕ[c̄, b̄_{i}]} and {i < ω : ⊧ ¬ϕ[c̄, b̄_{i}]} are infinite. Hence both ϕ(c̄, ȳ) and ¬ϕ(c̄, ȳ) belong to Av(J, c̄). Thus Av(J, c̄) is not consistent, which contradicts (2).

We can now answer the question of when averages are well-defined and characterize types without the independence property.

Theorem 23. The following conditions are equivalent:

(1) p does not have the independence property;

(2) For every infinite indiscernible sequence I ⊆ p(ℭ) and every subset A ⊆ p(ℭ) the average Av(I, A) is a complete type. Furthermore, Av(I, A) ∈ S_{p}(A).

Proof. (1) ⇒ (2): Let I, A ⊆ p(ℭ) and I be an infinite indiscernible sequence. By Lemma 22 (1) ⇒ (2), we have that Av(I, A) ∈ S(A). Furthermore, since I ⊆ p(ℭ), we have Av(I, A) ∈ S_{p}(A).

(2) ⇒ (1): We prove the contrapositive. Suppose that p has the independence property. Then, by Lemma 22 (2) ⇒ (1), there exists an infinite indiscernible sequence I ⊆ p(ℭ) and ā ∈ p(ℭ) such that both ϕ(x̄, ā) and ¬ϕ(x̄, ā) belong to Av(I, ā). This contradicts (2).

We now give an easy characterization of stable types in simple theories. The following fact is due to Shelah and appears in [Sh93].

Fact 24. If T is simple then T does not have the strict order property.

We make use of the following observation.

Proposition 25. If the formula ϕ(x̄, ȳ, b̄) with parameter b̄ ∈ ℭ has the p-strict order property, then T has the strict order property.

Proof. Show that T has the strict order property, by showing that ϕ(x̄, ȳ, z̄) has the strict order property.

Corollary 26. Let T be simple. The following conditions are equivalent:

(1) p is stable; (2) For every infinite indiscernible sequence I ⊆ p(ℭ) and for every A ⊆

p(ℭ), we have Av(I, A) ∈ S_{p}(A).

Proof. (1) ⇒ (2): Let p be stable, then p does not have the order property by Theorem 16. Hence p does not have the independence property by Proposition 19. Hence, (2) follows from Theorem 23.

(2) ⇒ (1): Suppose p is not stable. Then p has the order property by Theorem 16. Thus, p has either the independence property or the strict order property (maybe with parameters) by Theorem 20. Since T is simple, by Fact 24, we have that T does not have the strict order property. But, if p has the strict order property with parameters, then T has the strict order property by Proposition 25. Therefore, p has the independence property, and so (2) fails by Lemma 23.

References

[Bo] Elisabeth Bouscaren, An introduction to independence and local modularity, in Algebraic Model Theory edited by B. Hart, A. Lachlan and M. Valeriote, NATO ASI Series, Mathematical and Physical Sciences - Vol 496, Kluwer Academic Publishers 1997.

[CH] Zoe Chatzidakis and Ehud Hrushovski, ACFA, preprint.

[EM] Paul Erdős and Michael Makkai, Some remarks in set theory, part X, Studia Scientarum Math. Hungarica, 1 (1966) pages 157–159.

[Gr1] Rami Grossberg, Indiscernible sequences in a model which fails to have the order property, Journal of Symbolic Logic, 56 (1991) pages 115–122.

[Gr2] Rami Grossberg, On chains of relatively saturated submodels of a model without the order property, Journal of Symbolic Logic 56 (1991) 123-128.

[Gr b] Rami Grossberg, Lecture notes on Classification for nonelementary classes,

in preparation.

[GrLe] Rami Grossberg and Olivier Lessmann, Shelah’s Stability Spectrum and Homogeneity Spectrum, Preprint.

[GrSh1] Rami Grossberg and Saharon Shelah, On the number of nonisomorphic models of an infinitary theory which has the infinitary order property I, Journal of Symbolic Logic, 51 (1986) pages 302–322.

[GrSh2] Rami Grossberg and Saharon Shelah, On Hanf numbers of the infinitary property, Preprint.

[HH] Victor Harnik and Leo Harrington, Fundamentals of forking, Annals of Pure and Applied Logic, 26(1984) pages 245-286.

[HP] Ehud Hrushovski and Anand Pillay, Groups definable in local fields and pseudofinite fields. Israel J. Math., 85 (1994) pages 203–262.

[Je] Thomas Jech, Set Theory, Academic Press, New York, 1978.

[P] Anand Pillay, Forking, normalizations and canonical bases, Annals of Pure and Applied Logic, 32 (1986) pages 61-82.

[Sh3] Saharon Shelah, Finite Diagrams Stable in Power, Annals of Pure and Applied Logic, 2 (1970) pages 69-118,

[Sh12] Saharon Shelah, A Combinatorial Problem, Order for Models and Infinitary Languages, Pacific Journal of Mathematics, 41 (1972) pages 247–261.

[Sh93] Saharon Shelah, Simple Unstable Theories, Annals of Mathematical Logic, 19 (1980) pages 177-203.

[Sh300] Saharon Shelah, Universal Classes, Proc. of the USA-Israel Symp. in Classification Theory (1985), ed. J. Baldwin, Springer-Verlag Lectures Notes 1292 (1987) pages 264–418.

[Sh a] Saharon Shelah, Classification Theory and the Number of Nonisomorphic Models, Rev. Ed., North-Holland, 1990, Amsterdam.

Department of Mathematical Sciences, Carnegie Mellon University,

Pittsburgh, PA 15213