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Title Independence Relations in Theories with the Tree Property

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Author Harrison-Shermoen, Gwyneth Fae

Publication Date 2013

Peer reviewed|Thesis/dissertation

eScholarship.org Powered by the California Digital Library University of California Independence Relations in Theories with the Tree Property

Gwyneth Fae Harrison-Shermoen

A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy

Logic and the Methodology of Science

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor Thomas Scanlon, Chair Professor Leo Harrington Professor John Steel Associate Professor Antonio Montalb´an Professor Martin Olsson

Fall 2013 Independence Relations in Theories with the Tree Property

Abstract

Independence Relations in Theories with the Tree Property by Gwyneth Fae Harrison-Shermoen Doctor of Philosophy in Logic and the Methodology of Science University of California, Berkeley Professor Thomas Scanlon, Chair

This thesis investigates theories with the tree property and, in particular, notions of independence in such theories. We discuss the example of the two-sorted theory of an infinite dimensional vector space over an algebraically closed field and with a bilinear form (which we refer to as T∞), examined by Granger in [9]. Granger notes that there is a well-behaved notion of independence, which he calls Γ-non-forking, in this theory, and that it can be viewed as the limit of the non-forking independence in the theories of its finite dimensional subspaces, which are ω-stable. He defines a notion of an “approximating sequence” of substructures, and shows that Γ-non-forking in a model of T∞ corresponds to “eventual” non-forking in an approximating sequence. We generalize his notion of approximation by substructures, and in the case of a theory whose large models can be approximated in this way, define a lim relation ^| , which is the “limit” of the non-forking independence in the theories of the approximating substructures. We show that if the approximating substructures have simple lim theories, ^| satisfies invariance, monotonicity, base monotonicity, transitivity, normality, lim extension, finite character, and symmetry. Under certain additional assumptions, ^| also satisfies anti-reflexivity and the Independence Theorem over algebraically closed sets. We also consider the two-sorted theory of infinitely many cross-cutting equivalence rela- ∗ tions, Tfeq. We give a proof, explaining in detail the argument of Shelah and Usvyatsov for ∗ Theorem 2.1 in [23], that Tfeq does not have SOP2 (equivalently, TP1). The argument makes use of a theorem of Kim and Kim, from [14], along with several other lemmas involving tree indiscernibility. i

To my parents. ii

Contents

Contents ii

1 Introduction 1 1.1 Notation and conventions ...... 3 1.2 Tree properties, and the SOP hierarchy ...... 4 1.3 Independence ...... 7 1.4 Forking ...... 9 1.5 Tree indiscernibility ...... 12

2 Parameterized Equivalence Relations 22 2.1 The Theory ...... 22 2.2 Demonstration of subtlety ...... 29

3 Limit Independence 36 3.1 Motivation ...... 36 3.2 Deﬁnitions ...... 42 3.3 Properties of the relation ...... 46 3.4 Example ...... 55

Bibliography 69 iii

Acknowledgments

First and foremost, I thank my adviser, Tom Scanlon, for his constant encouragement, his extraordinary generosity with his time and ideas, and his patience, which rivals Penelope’s. I am also deeply grateful to Zo´eChatzidakis for advising me during my time in France, taking such an active interest in my progress, and putting up with my ignorance about ﬁelds. I am grateful to my mathematical siblings and fellow Logic Group students for providing me with a wonderful intellectual and social community during my time at Berkeley, and to Hipline for providing me with a community outside of grad school, and plenty of opportunities to dance and laugh. I thank my family and friends for their unfailing support. Finally, I thank the Institute of International Education and the Franco-American Com- mission for the Fulbright fellowship, UC Berkeley for the Chancellor’s and Dissertation Year Fellowships, the Embassy of France in the USA for the Chateaubriand fellowship, Scan- lon’s NSF grant for the semester and summer of GSR support, and the ASL, the Graduate Division, the Graduate Assembly, and the AMS for travel support. 1

Chapter 1

Introduction

Classiﬁcation theory is a far-reaching project within model theory, originating with the work of Morley in the 1960s and Shelah in the 1970s. Its goal is, roughly, to categorize ﬁrst-order theories based on how much variation there is among their models and, in the other direction, to recover structural information about the models of a theory from facts about the theory. One measure of the “amount of variation among models” is the number of types consistent with the theory.

Definition 1.0.1. A first-order theory, T , is stable if there is some infinite cardinal κ such that for all models M T , subsets A of M with |A| ≤ κ, and n < ω, the number of complete n-types in M with parameters from A is no more than κ, i.e.

M |Sn (A)| ≤ κ.

A rich theory has been developed around the notion of stability, and as a result, stable theories are relatively well understood. There are many mathematical structures with unstable theories, however, and so there has been a concerted effort to generalize various results of stability theory in several different directions (for example, to simple theories, theories without the independence property, and rosy theories). One of the aspects of stable theories that pops up in many unstable theories is the existence of a well-behaved independence relation, similar to linear independence in vector spaces. (In fact, the abstract notion of independence generalizes both linear independence in vector spaces and algebraic independence in fields, as well as many other geometric notions of independence.) Essentially, given sets A, B, and C of elements in whatever structure we are working in, A is “independent” from B over C if B ∪C has no more “information” about A than C does. (In the case of a vector space over a fixed field, “A is independent from B over C” boils down to span(A ∪ C) ∩ span(B ∪ C) = span(C).) Independence relations give us a way to determine what behavior is “generic,” even in theories with many models. In particular, the relation of non-forking independence satisfies, in stable theories, many of the properties that are desirable for an independence relation. In the late 1990s, Byung- han Kim showed ([12], [13]) that non-forking independence satisfies almost all of the same CHAPTER 1. INTRODUCTION 2 properties in simple theories, and, in fact, that a theory is simple if and only if non-forking independence is a symmetric relation. He and Anand Pillay showed in [16] that if a theory has an abstract independence relation satisfying a certain set of axioms, then that theory is simple and the independence relation is non-forking independence. In more recent years, model theorists have discovered various theories that are not simple but still have a nice notion of independence. Since these theories are not simple, the independence relation(s) in question cannot satisfy all the axioms satisfied by non-forking independence in a simple theory, but they may satisfy almost all of them. One example of such a theory is the theory of ω-free pseudo algebraically closed fields, studied by Chatzidakis in [4], [3], and [2]. (Chatzidakis actually identifies several distinct notions of independence, satisfying different subsets of the properties of non-forking independence in a simple theory.) Another example is the two-sorted theory of an infinite dimensional vector space over an algebraically closed field and with a bilinear form, studied by Granger in his thesis [9]. In the case of the infinite dimensional vector spaces with a bilinear form, the independence relation that Granger singles out (which he refers to as “Γ-non-forking”) can be viewed as the limit of the non-forking independence in the finite dimensional subspaces, whose theories are stable. There is a sense in which the finite dimensional subspaces collectively “approximate” the infinite dimensional vector space. In chapter 3, I generalize this notion of approximation by substructures, and define an independence relation that is the limit of the non-forking independence in the theories of the approximating substructures.

Definition 1.0.2. Given a theory T , a model, M, of T , and a directed system, H, of | lim substructures N of M, we define the following notion of independence: A^C B (A is limit independent from B over C) if for each finite subset A0 of A, there is NA0 ∈ H such that N M N | M | A0 ⊆ NA0 and for all NA0 ⊆ N ∈ H, A0âcl (C)∩N acl (B) ∩ N, where ^ is non-forking independence as computed in N , and acl is the model-theoretic notion of algebraic closure. We say that H approximates M if the following conditions hold:

•H covers M.

•H is closed under automorphism.

• Any sentence that is true in M is “eventually true” in H.

• Every formula either “eventually forks” in H or “eventually does not fork” in H.

/| lim • If A^C B, this dependence is witnessed by the “eventual forking” of some formula. • (optional) Algebraically closed sets in M are “eventually algebraically closed” in H.

• (optional) For each N ∈ H, a, b ∈ N, and A ⊂ M, if tpM(a/A) = tpM(b/A), then tpN (a/A ∩ N) = tpN (b/A ∩ N). CHAPTER 1. INTRODUCTION 3

I then show that if the theories of the approximating substructures are simple, “limit independence” satisﬁes many of the properties satisﬁed by non-forking independence in simple theories.

Theorem 1.0.3. Given a κ-saturated, strongly κ-homogeneous model M of T approximated lim by H, where each N ∈ H has a simple theory, ^| has the following properties: automorphism invariance, transitivity, finite character, symmetry, existence, and extension. If H satisfies the optional conditions and in the theories of its elements, non-forking indepen- lim dence satisfies the Independence Theorem over algebraically closed sets, then ^| satisfies anti-reflexivity and the Independence Theorem over algebraically closed sets.

This thesis is structured as follows: in the remainder of this chapter, we present our notational conventions and necessary background from stability and simplicity theory, as well as some definitions and results of a combinatorial nature on “tree indiscernibility.” Chapter ∗ 2 examines the two-sorted theory of infinitely many cross-cutting equivalence relations (Tfeq). ∗ Following Shelah and Usvyatsov [23], we give a proof that Tfeq does not have the tree property of the first kind (making heavy use of the tree indiscernibility material from this chapter). In Chapter 3, we introduce the example of the infinite dimensional bilinear spaces, define notions of “approximation” and “limit independence,” and prove Theorem 1.0.3. We then Γ look at the relationship between Granger’s Γ-non-forking (^| ) and limit independence, and lim Γ see to what extent ^| generalizes ^| .

1.1 Notation and conventions

We assume knowledge of basic model theory, at the level of chapters 1 through 5 of [17]. Given a complete theory T , we work in a large, κ-saturated, strongly κ-homogeneous (for some large regular cardinal κ) model C of T . We consider only models M of T of cardinality less than κ and such that M ≺ C. (This is without loss of generality, as every model of T of cardinality less than κ can be elementarily embedded into C.) We consider only sets A ⊂ C of cardinality less than κ. Hence, we may write acl(A) or ϕ(a) without specifying a model. Usage of lowercase and uppercase Greek and Latin letters varies somewhat with context, but the most common (and default) uses are as follows. Uppercase A, B, C, and D denote sets, while M and N denote models with universes M and N, respectively. L is a first-order (usually countable) language, and T is a first-order theory. R is a relation symbol, E an equivalence relation, and P a unary predicate. I and J are index sets. K is a field (L and F may be, as well). Concerning (lowercase) Greek letters, λ and κ are infinite cardinals, ϕ and ψ (and occasionally θ and χ) are formulae, and η, ν are finite sequences of natural numbers (i.e. initial segments of tree branches). The lowercase letters a, b, c, d are elements (or tuples) of the ambient structure; f, g, and h are functions; i, j, k, l (or `), m, and n are natural numbers; p and q are types. The letters r, s, and t are occasionally types, but are CHAPTER 1. INTRODUCTION 4

more often natural numbers, and u, v, w, x, y, z are variables. In languages with multiple sorts, we may use uppercase letters X,Y, and Z as variables, too. The symbols x, a, and so on, may denote tuples of variables and parameters - in general, we do not distinguish a tuple of length greater than 1 by use of an overline. We shall warn the reader of exceptions as they arise. We denote the length of the tuple a by lg(a). (That is, if a = (a0, . . . , an−1), lg(a) = n.) We abuse notation by writing a ∈ A (where a is a tuple of length n) rather than a ∈ An. We often denote a union by concatenation, e.g. we may write BC for B ∪ C. We use acl(A) to denote the model-theoretic algebraic closure of A, and Kalg to denote the field-theoretic algebraic closure of K. An automorphism over A or A-automorphism is an automorphism of C (or perhaps some smaller model, where specified) fixing A pointwise. B ≡A C means that tp(B/A) = tp(C/A) and hence, by the strong κ-homogeneity of C, that there is a A-automorphism of C sending B to C. If ∆ ⊂ L, B ≡∆ C means that B and C have the same ∆-types. We may use (∃=1x)ϕ(x) to abbreviate ∃x(ϕ(x) ∧ ∀y(ϕ(y) → y = x)), (and similarly with (∃=nx)ϕ(x), for n > 1). We work quite a bit with trees in Section 1.5 and Chapter 2. The set of functions f : ω → ω, ωω, can be viewed as an infinitely branching tree of height ω. Hence, β ∈ ωω is a <ω branch from such a tree, and α ∈ ω is an initial segment of such a branch, that is, β k for some β ∈ ωω, k < ω. We also work with finitely-branching trees ωq for q < ω, especially ω 2. In all cases, we denote the tree order by E: that is, η E ν if there are k ≤ ` ∈ ω and ω ω β ∈ ω (respectively, β ∈ 2) such that η = β k and ν = β `.

1.2 Tree properties, and the SOP hierarchy

In the effort to push the results of stability theory beyond the boundaries of stable theories, Shelah has identified a number of properties that an unstable theory might have (or lack). First of all, a theory is unstable if and only if it has at least one of the independence property ω (that is, there are a formula ϕ(x; y) and sequences hai : i < ωi, hcσ : σ ∈ 2 i such that ϕ(ai; cσ) if and only if σ(i) = 0) or the strict order property [19] (that is, there are a formula ϕ(x, y) and a sequence hbi : i < ωi such that ∀x(ϕ(x; bi) → ϕ(x; bj)) if and only if i < j). In this thesis, we shall focus primarily on unstable theories without the strict order property. In this section, we introduce a number of properties weaker than the strict order property. All definitions are due to Shelah. Definition 1.2.1. A set of formulae is k-inconsistent if every subset of size k is inconsistent. That is, a set of formulae {ψi(x): i ∈ I} (possibly with parameters) is k-inconsistent if for V any i0, . . . , ik−1 ∈ I, ¬∃x( ψij (x)). Definition 1.2.2 ([20], Definition 0.1). 1. A theory T has the tree property if there are <ω a formula ϕ(x; y), k < ω, and sequences aη ∈ C (η ∈ ω) such that: CHAPTER 1. INTRODUCTION 5

<ω • for any η ∈ ω, {ϕ(x, aη_l): l < ω} is k-inconsistent, but ω • for every β ∈ ω, {ϕ(x, aβn): n < ω} is consistent. 2. A theory T is simple if it does not have the tree property.

Remark 1.2.3. The semicolon in ϕ(x; y) above is significant: it separates the object variables (x) from the parameter variables (y). Instances of ϕ are formulae ϕ(x; b), in which the parameter variables have been replaced by parameters. The tree property is a robust dividing line among unstable theories: the class of simple theories is exactly the class of theories in which non-forking independence is symmetric, as we shall see in Section 1.4. Simple theories can also be characterized by having an independence relation (to be defined in Section 1.3) satisfying a certain type amalgamation property (analogous to stationarity of types in stable theories). The theories of the random graph and of pseudo-finite fields are two examples of simple unstable theories. In [19], Shelah proves that if a theory has the tree property, then it has one of two “extreme” tree properties. The tree property specifies that instances along a branch are consistent, while instances at nodes that are siblings are inconsistent. It leaves open the consistency of instances at incomparable, non-sibling nodes. It is therefore natural to consider tree properties that do make a decision on the consistency of these instances.

Deﬁnition 1.2.4. 1. A theory T has the k-tree property of the ﬁrst kind (k−TP1) if <ω there are a formula ϕ(x; y) and tuples {aα : α ∈ ω} such that:

<ω • for α0, . . . , αk−1 ∈ ω pairwise incomparable, {ϕ(x, aαi ) : 1 ≤ i ≤ k} is inconsistent, but ω • for β ∈ ω, {ϕ(x, aβn): n ∈ ω} is consistent.

2. A theory has TP1 if it has 2-TP1.

3. A theory is called NTP1 or subtle if it does not have TP1.

∗ The two-sorted theory of infinitely many cross-cutting equivalence relations, Tfeq (to be discussed extensively in Chapter 2), is NTP1 [23] but not simple. Recently, Chernikov has identified a sufficient condition for a theory to be NTP1, and used it to show that both the two-sorted theory of infinite dimensional vector spaces over algebraically closed fields with a bilinear form (described in Section 3.1) and the theory of ω-free PAC fields of characteristic 0 are NTP1 [5]. This makes use of results from [3], [2], [9], and [6]. Each of these theories is not simple, and so by Theorem 1.2.7 below, has the tree property of the second kind.

Deﬁnition 1.2.5. A theory T has the tree property of the second kind (TP2) if there are a <ω formula ϕ(x; y), tuples {aα : α ∈ ω}, and k < ω such that:

<ω • for any α ∈ ω, {ϕ(x; aα_i): i < ω} is k-inconsistent, but CHAPTER 1. INTRODUCTION 6

<ω • for any n and any α0, . . . , αn−1 ∈ ω, no two of which are siblings, {ϕ(x; aαi ): i < n} is consistent.

A theory without TP2 is called NTP2.

Remark 1.2.6. TP2 is equivalent to the following condition (which is more commonly given as the deﬁnition of TP2): there are a formula ϕ(x; y) and tuples {ai,j : i, j < ω} such that

• for any f : ω → ω, {ϕ(x; ai,f(i)): i < ω} is consistent, but

• for all i < ω, {ϕ(x; ai,j): j < ω} is k-inconsistent for some k. Various theories of valued fields - for example, any theory of an ultraproduct of p-adics - are NTP2 but not simple [6]. Theorem 1.2.7 ([19], Theorem III.7.11). If a theory T has the tree property, then either T has TP1 or T has TP2. Definition 1.2.8 ([22], Definitions 2.1, 2.2, and 2.5). 1. A theory T has the strict order property if some formula ϕ(x, y) (where x and y are tuples of the same length) defines, in some model M T , a partial order with infinite chains. 2. A (complete) theory T has the strong order property (SOP) if there is some sequence n n ϕ = hϕn(x ; y ): n < ωi of formulae such that for every λ, a) lg(xn) = lg(yn) are finite, and xn (respectively yn) is an initial segment of xn+1 (respectively yn+1); n+1 n+1 n n b) T ∪ {ϕn+1(x , y )} ` ϕn(x , y ); n,0 n,m−1 V n,k n,` c) for m ≤ n, ¬(∃x , . . . , x )[ {ϕn(x , x ): k = ` + 1 mod m}] belongs to T ; n n d) there is a model M of T and aα ∈ M (of length y , for n < ω, α < λ) such that n n+1 n n n aα = aα lg(y ) and M ϕn[aβ, aα] for n < ω and α < β < λ.

3. A theory T has the n-stronger order property (SOPn) if there are a formula ϕ(x, y) (where x and y are tuples of the same length), a model M of T , and tuples lg(x) {ak ∈ M : k < ω} such that

•M ϕ(ak, am) for k < m < ω, and V •M ¬∃x0, . . . , xn−1( {ϕ(xl, xk): l, k < n and k = l + 1 mod n}).

Deﬁnition 1.2.9 ([7], Deﬁnition 2.2). 1. T has SOP2 if there are a formula ϕ(x, y) and <ω tuples aη for η ∈ 2 such that

ω • for every ρ ∈ 2, the set {ϕ(x, aρn): n < ω} is consistent, while <ω • if η, ν ∈ 2 are incomparable, {ϕ(x, aη), ϕ(x, aν)} is inconsistent. CHAPTER 1. INTRODUCTION 7

<ω 2. T has SOP1 if there are a formula ϕ(x, y) and tuples aη for η ∈ 2 such that

ω • for ρ ∈ 2 the set {ϕ(x, aρn): n < ω} is consistent, but _ <ω • if ν h0i E η ∈ 2, then {ϕ(x, aη), ϕ(x, aν_h1i)} is inconsistent.

Fact 1.2.10. A theory T has TP1 if and only if it has SOP2. (See, e.g., [14].) The various notions deﬁned in this section are related as follows: the strict order property implies the strong order property (SOP), and in turn,

SOP ⇒ ... ⇒ SOPn+1 ⇒ SOPn ⇒ ... ⇒ SOP3 ⇒ SOP2(⇔ TP1) ⇒ SOP1 ⇒ TP

It is unknown whether the implications SOP3 ⇒ SOP2 and SOP2 ⇒ SOP1 are strict. All of the other implications (i.e. strict order property ⇒ SOP, SOP ⇒ SOPn, SOPn+1 ⇒ SOPn for n ≥ 3, and SOP1 ⇒ TP) are strict. See [22] for more details.

1.3 Independence

As mentioned above, notions of independence provide us with a measure of genericity. There is some disagreement about which properties should be required of a ternary relation in order for it to be called an independence relation. Below we give Adler’s deﬁnition from [1]. In general, we shall use the phrase notion of independence as a catch-all term for a relation satisfying a “reasonable” number of the axioms below.

Deﬁnition 1.3.1. An independence relation is a ternary relation (which we denote by ^| ) on small sets satisfying the following axioms:

0 0 0 0 0 • invariance: If A^| C B and (A ,B ,C ) ≡ (A, B, C), then A ^| C0 B .

0 0 0 0 • monotonicity: If A^| C B, A ⊆ A, and B ⊆ B, then A ^| C B .

• base monotonicity: Suppose D ⊆ C ⊆ B. If A^| DB, then A^| C B.

• transitivity: Suppose D ⊆ C ⊆ B. If B^| C A and C^| DA, then B^| DA.

• normality: A^| C B implies AC^| C B.

0 0 0 0 • extension: If A^| C B and B ⊇ B, then there is A ≡BC A such that A ^| C B .

• finite character: If A0^| C B for all finite A0 ⊆ A, then A^| C B. • local character: For every A there is a cardinal κ(A) such that for any set B there is a subset C ⊆ B of cardinality |C| < κ(A) such that A^| C B. An independence relation is strict if it also satisfies anti-reflexivity:

a^| Ba implies a ∈ acl(B). CHAPTER 1. INTRODUCTION 8

Recalling the slogan from the beginning of this chapter - that A should be independent from B over C just in case B ∪ C has no more “information” about A than C has on its own - many of these axioms should seem intuitive. Remark 1.3.2. Some additional properties one might ask a notion of independence to satisfy are:

0 0 • existence: For any A, B, and C there is A ≡C A such that A ^| C B.

• symmetry: A^| C B if and only if B^| C A.

• strong ﬁnite character: If A^/| C B, then there are ﬁnite tuples a ∈ A, b ∈ B, and c ∈ C, and a formula ϕ(x, y, z) without parameters such that

– ϕ(a, b, c), and 0 /| 0 0 – a ^C b for all a such that ϕ(a , b, c). 0 0 0 0 • stationarity: If a^| C B and B ⊆ B , there is a ≡BC a such that a ^| C B , and if 0 0 d^| C B and d ≡BC a, then d ≡B0C a . (A relation with the property of “stationarity over algebraically closed sets” is one that satisﬁes the property of stationarity when C is algebraically closed.)

• independence theorem over a model: If a ≡M b, a^| M A, b^| M B (for M a model and A, B ⊇ M), and A^| M B, then there is c such that c ≡A a, c ≡B b, and c^| M AB. (One should read this as a weakening of the previous property. Roughly, if ^| satisﬁes stationarity, independent extensions of types are unique. If ^| satisﬁes the independence theorem (over a model), independent extensions (of types over models) can be amalgamated.)

In the next section, we shall see that the syntactic relations of non-forking and non- dividing are notions of independence (and, in certain theories, satisfy all of the properties mentioned here). For now, we give a few concrete examples from [1]. Example 1.3.3. 1. The trivial relation holding of all triples is an independence relation (as deﬁned above), but not a strict independence relation.

2. In the theory of an equivalence relation with infinitely many infinite classes (and no finite classes), the relation given by

A^| C B if and only if A ∩ B ⊆ C

is a strict independence relation.

3. The relation given by

| a A^C B if and only if acl(AC) ∩ acl(BC) = acl(C) CHAPTER 1. INTRODUCTION 9

satisﬁes base monotonicity if and only if whenever A, B, C are algebraically closed sets such that B ⊇ C, B ∩acl(AC) = acl((B ∩A)C). In this case, it is a strict independence relation.

4. In the theory of the random graph, the relation given by

A^| C B if and only if A ∩ B ⊆ C and there is no edge from A \ C to B \ C

satisﬁes all the axioms of a strict independence relation except for local character.

1.4 Forking

In this section, we discuss the concepts of forking and dividing, and some of the properties they satisfy, paying particular attention to the behaviour of forking and dividing in simple theories. We give proofs of several easy facts, as this may provide the reader with a better sense of dividing and forking, which can be diﬃcult to grasp. The reader should assume that these facts have already been proven many times, by many people, even where no speciﬁc reference is given.

Deﬁnition 1.4.1 ([19], Chapter III, Deﬁnitions 1.3-1.4). • A formula ϕ(x; b) divides over A if there are {bi : i < ω} and k < ω such that

– tp(bi/A) = tp(b/A) for all i < ω, and

– {ϕ(x; bi): i < ω} is k-inconsistent. • A type p(x) divides over A if there is a formula ϕ(x; b) such that p(x) ` ϕ(x; b) and ϕ(x; b) divides over A.

• A type p(x) forks over A if there are formulae ϕ0(x; a0), . . . , ϕn−1(x; an−1) such that W 1. p(x) ` i

2. ϕi(x; ai) divides over A for each i.

Deﬁnition 1.4.2. A sequence (ai : i < α) (where for all i, j, lg(ai) = lg(aj)) is indiscernible over A or A-indiscernible if for every n < ω and every i0 < . . . < in−1, j0 < . . . < jn−1,

tp(ai0 , . . . , ain−1 /A) = tp(aj0 , . . . , ajn−1 /A).

Remark 1.4.3 ([19], Lemma III.1.1(3)). In the ﬁrst part of Deﬁnition 1.4.1, we could have taken {bi : i < ω} to be an indiscernible sequence. In that case, the second condition is equivalent to {ϕ(x; bi): i < ω} is inconsistent. CHAPTER 1. INTRODUCTION 10

Both non-dividing and non-forking can be viewed as notions of independence. We write | d | f a^C B for “tp(a/BC) does not divide over C” and a^C B for “tp(a/BC) does not fork over C.” Fact 1.4.4. For a set C, parameters b∈ / C and c ∈ C, formulae ψ(x, y, z), ϕ(x, w) ∈ L, and n < ω, if ψ(x, b, c) divides over C, then the following formula does, as well:

(∃=nx)ϕ(x, w) ∧ ∃x(ϕ(x, w) ∧ ψ(x, b, c)).

Proof. By the deﬁnition of dividing and Remark 1.4.3, we have an inﬁnite, C-indiscernible sequence hbici : i ∈ Ii = hbic : i ∈ Ii, with b0c0 = bc (since c ∈ C, ci = c for all i ∈ I) and such that {ψ(x, bi, c): i ∈ I} is inconsistent. We show that

=n {(∃ x)ϕ(x, y) ∧ ∃x(ϕ(x, y) ∧ ψ(x, bi, c)) : i ∈ I} is inconsistent. If not, let d be a realization. That is, for each i ∈ I,

=n (∃ x)ϕ(x, d) ∧ ∃x(ϕ(x, d) ∧ ψ(x, bi, c))

Suppose {a0, . . . , an−1} is the set deﬁned by ϕ(x, d). Then for each i ∈ I, there is j < n, such that ψ(aj, bi, c). It follows that for at least one j < n, there is an inﬁnite subset I0 ⊆ I such that

0 ψ(aj, bi, c) for all i ∈ I .

However, there is some k < ω such that {ψ(x, bi, c): i ∈ I} is k-inconsistent, so in particular, 0 {ψ(x, bi, c): i ∈ I } is k-inconsistent. Contradiction. | d | d Corollary 1.4.5. If A^C B, then acl(A)^C B. /| d Proof. We prove the contrapositive. If acl(A)^C B, then there is some formula ψ(x, b, c) ∈ tp(acl(A)/BC)

(where ψ(x, y, z) ∈ L, b ∈ B, and c ∈ C) that divides over C. Let a0 ∈ acl(A) realize ψ(x; b, c). Since a0 ∈ acl(A), there are n < ω, ϕ(x, w) ∈ L, and a ∈ A such that

ϕ(x, a) ∧ (∃=nx)ϕ(x, a) ∈ tp(a0/A).

It follows that (∃=nx)ϕ(x, w) ∧ ∃x(ϕ(x, w) ∧ ψ(x, b, c)) ∈ tp(a/BC). /| d By fact 1.4.4 this formula divides over C, so A^C B. CHAPTER 1. INTRODUCTION 11

Fact 1.4.6. Dividing over A and forking over A are preserved under A-automorphism.

Proof. This is clear from the deﬁnitions. W Fact 1.4.7. If ψ(x, d) forks over C, and d0, . . . , dn−1 are C-conjugates of d, then ψ(x, di) i

Proof. This follows from the deﬁnition and Fact 1.4.6.

Fact 1.4.8. If d ∈ acl(B), ϕ(y, b) isolates tp(d/B) (for ϕ(y, z) ∈ L and b ∈ B), and ψ(x, d) forks over C ⊆ B, then ∃y(ϕ(y, b) ∧ ψ(x, y)) forks over C.

Proof. By assumption, there are d0 = d, d1, . . . , dn−1 such that {di : i < n} = ϕ(C, b), and so _ ∃y(ϕ(y, b) ∧ ψ(x, y)) ` ψ(x, di). i

By the fact that for i, j < n, tp(di/B) = tp(dj/B), strong κ-homogeneity of C, and Fact 1.4.7, ∃y(ϕ(y, b) ∧ ψ(x, y)) forks over C.

| f | f Corollary 1.4.9 ([19], Lemma III.6.3(3)). If A^C B, then A^C acl(B). Fact 1.4.10. An inﬁnite, A-indiscernible sequence is acl(A)-indiscernible.

| | | | d | | f Fact 1.4.11. A^C B if and only if A^acl(C)B (for ^ = ^ or ^ = ^ ). Proposition 1.4.12 ([12], Proposition 2.1). If T is simple, ϕ(x, a) divides over A if and only if ϕ(x, a) forks over A.

Proposition 1.4.13 (Properties of forking independence). Let T be a simple theory. The | f independence relation given by A^C B if tp(A/BC) forks over C satisfies the axioms of existence, anti-reflexivity, monotonicity, base monotonicity, finite character, local character, extension, symmetry, and transitivity.

Proof. Anti-reflexivity, monotonicity, base monotonicity, and finite character follow from the definition of forking (and are true in any theory). Existence follows from the equivalence of forking and dividing in a simple theory. Local character is proved in [20], Claim 4.2. Extension is proved in [19], chapter III, Theorem 1.4. Symmetry and transitivity are proved in [12], Theorems 2.5 and 2.6, respectively.

Remark 1.4.14. If T is simple, we shall write ^| for forking (equivalently, dividing) independence. Remark 1.4.15. In a simple theory, Corollary 1.4.5 can be strengthened to the following: if A^| C B, then acl(A)^| C acl(B). This is by the equivalence of forking and dividing and symmetry of ^| . CHAPTER 1. INTRODUCTION 12

The next two theorems give us ways to identify simple theories by looking at the behaviour of either non-forking independence or an abstract independence relation. Theorem 1.4.16 ([13], Theorem 2.4). Let T be arbitrary. The following are all equivalent. 1. T is simple. 2. Forking (Dividing) satisfies local character. 3. Forking (Dividing) satisfies symmetry. 4. Forking (Dividing) satisfies transitivity. Remark 1.4.17. The equivalence of T ’s simplicity and local character of forking was proved by Shelah in [20] (Claim 4.2 and Theorem 4.3). Theorem 1.4.18 (Characterization of Simplicity: [16], Theorem 4.2). Let T be an arbitrary ◦ theory. Then T is simple if and only if T has a notion of independence ^| which satisfies ◦ the Independence Theorem over a model. Moreover, if ^| is such, then for all a, B, C, | ◦ | ◦ a^C B if and only if tp(a/BC) does not fork over C (namely, ^ -non-forking coincides with non-forking). Remark 1.4.19. In [16], the authors take ‘notion of independence’ to mean ternary relation satisfying invariance, monotonicity, base monotonicity, local character, finite character, extension, symmetry, and transitivity. An analogous result for stable theories was proved by Harnik and Harrington in 1984. Rephrasing to match the terminology of the previous theorem, the result states: Theorem 1.4.20 ([10], Theorem 5.8). Given a theory T and a notion of independence ◦ ◦ ^| , if ^| satisfies invariance, monotonicity, base monotonicity, transitivity, extension, local ◦ character, and stationarity over algebraically closed sets, then T is stable and ^| is non- forking independence.

1.5 Tree indiscernibility

In general, it is not easy to prove directly that a theory does not have a tree property (of any kind). The problem becomes somewhat more tractable if we may assume certain facts about the tree. In particular, it is helpful to assume that the tree in question has some level of indiscernibility. There are several kinds of tree indiscernibility, but the idea behind all of them is that given two configurations of nodes on the tree that “look alike,” the parameters decorating the nodes of those two configurations should have the same type. The differences among the various notions of tree indiscernibility come from the different interpretations of what it means for two tree configurations to “look alike.” (For comparison, in the case of an indiscernible sequence, two sets of elements from the sequence “look alike” if they have the same order type.) For an in-depth discussion of “generalized indiscernibles,” see [18]. CHAPTER 1. INTRODUCTION 13

Remark 1.5.1 (Notation). 1. In this section, we will make use of overlines to denote tuples of nodes, and tuples of the tuples decorating those nodes. For example, η denotes, for

some n < ω, a sequence (η0, . . . , ηn−1), and aη denotes (aη0 , . . . , aηn−1 ). Note that each

aηi may itself be a tuple, but we shall follow the same convention we use elsewhere, and not use an overline in this case. For η = (η0, . . . , ηn−1) and i < n, we write ηi ∈ η. 2. For a single node η, we will use |η| to refer to the height of the node. That is, if η is a node in a q-branching tree, |η| = k if η ∈ kq. The following deﬁnition is taken from [14], although an equivalent version of parts 1 through 3 appeared in [7] several years earlier. As the authors of [14] note, the terminology of part 4 of this deﬁnition is due to Scow [18].

<ω Deﬁnition 1.5.2. 1. A tuple η ∈ q is ∩-closed if for each ηi, ηj ∈ η, ηi ∩ ηj ∈ η, too.

<ω 2. Given tuples η, ν ∈ q, η ≈1 ν if: • η and ν are ∩-closed tuples of the same arity, and •∀ i, j < lg(η) and ∀t < q,

– ηi E ηj iﬀ νi E νj, and _ _ – ηi hti E ηj iﬀ νi hti E νj.

We shall read η ≈1 ν as “η is 1-similar to ν.”

<ω <ω 3. We say haη|η ∈ qi is 1-fti (or 1-fully tree indiscernible) if for all η, ν ∈ q, η ≈1 ν implies aη ≡ aν.

<ω <ω 4. We say a sequence haη|η ∈ qi is 1-modeled by a sequence hbη|η ∈ qi if, for any d < ω, finite set ∆(x0, . . . , xd−1) of L-formulae, and ∩-closed tuple <ω <ω η = hη0, . . . , ηd−1i ∈ q, there exists ν ∈ q such that η ≈1 ν and bη ≡∆ aν. Remark 1.5.3. The concept of 1-modeling allows us to “transform” one tree into another tree that is 1-fti (see Kim and Kim’s Theorem 1.5.6, below). Importantly, this transformation preserves the property of being SOP2 (Remark 1.5.7, below), so if we are given an SOP2 tree, we can find a 1-fti SOP2 tree. Before stating the results that will be important in Chapter 2, we give a few examples of the properties defined above. Example 1.5.4. Let η = (h1i, h101i, h1001i). The tuple η is not ∩-closed, because h101i ∩ h1001i = h10i ∈/ η. CHAPTER 1. INTRODUCTION 14

• η3

• η2

◦ •

• η1

•

Example 1.5.5. Let

η = (hi, h0i, h001i, h01i, h111i) ν = (h1i, h101i, h1010i, h10110i, h110i)

Then η ≈1 ν.

ν4 •

ν3 •

• η3 • η5 • ν2 ν5 •

• η4 • • • •

η2 • • ν1

η1 •

<ω Theorem 1.5.6 ([14], Proposition 2.3). Any sequence hαη|η ∈ qi can be 1-modeled (in a <ω suﬃciently saturated model) by some 1-fti sequence hbη|η ∈ qi.

<ω Remark 1.5.7 (See also [15], Remark 5.2, for the ω-branching case). If haη|η ∈ 2i witnesses <ω <ω SOP2 for a formula ψ(x, y) and hbη|η ∈ 2i 1-models haη|η ∈ 2i in some model M, then <ω hbη|η ∈ 2i is also an SOP2 tree for ψ(x, y).

ω Proof. 1. First we show that for any β ∈ 2, {ψ(x, bβm)|m ∈ ω} is consistent. Fix such a β, and consider a finite subset of {ψ(x, bβm)|m ∈ ω}. It is contained in a set of the form {ψ(x, bβm)|m < n} for some n ∈ ω, so it suffices to show that sets of this form are consistent. Choose n ∈ ω, and let η = hβ 0, . . . , β n − 1i. This is an ∩-closed V <ω tuple. Consider ∆(y) := {∃x( ψ(x, yi))}. By hypothesis, we can find a ν ∈ 2 such i

<ω on a branch. Since haη|η ∈ 2i is an SOP2 tree for ψ, {ψ(x, aν )|i < n} is consistent, V V i and thus M ∃x( ψ(x, aνi )). It follows that M ∃x( ψ(x, bηi )), and thus the set i

∆(y) := {∃x(ψ(x, y0) ∧ ψ(x, y1))}.

<ω We can ﬁnd ν ∈ 2 such that ν ≈1 η and bη ≡∆ aν. Since α ∩ γ is the most recent common ancestor of α and γ, it must be that one of α, γ is a descendant of (α ∩ γ)_h0i, while the other is a descendant of (α ∩ γ)_h1i. Without loss of generality, _ _ _ assume (α ∩ γ) h0i E α. It follows that ν2 h0i E ν0, while ν2 h1i E ν1, and thus, that <ω ν0 and ν1 are incomparable. Since haη|η ∈ 2i is an SOP2 tree for ψ, it follows that

{ψ(x, aν0 ), ψ(x, aν1 )} is inconsistent, and thus that M ¬∃x(ψ(x, aν0 ) ∧ ψ(x, aν1 )). Since bη ≡∆ aν, M ¬∃x(ψ(x, bα) ∧ ψ(x, bγ)), and {ψ(x, bα), ψ(x, bγ)} is inconsistent, as desired.

Remark 1.5.8. 1-modeling does not necessarily preserve TP, however. We omit a formal argument, and simply observe that in a tree 1-modeled on a tree with TP, nodes that are siblings might be modeled on nodes that are not siblings in the original tree. Since TP only guarantees inconsistency among instances of the formula at sibling nodes, the new tree might not satisfy the inconsistency requirement.

The above facts tell us that if a theory T has TP1 (and hence, SOP2), then there are <ω a formula ψ(x; y) and a sequence haα|α ∈ 2i which is a 1-fti tree witnessing SOP2 for ψ(x; y). We would also like to be able to narrow down the set of formulae which we must show do not have SOP2 in order to show that no formula has SOP2. To that end, we show that we can “eliminate the disjunction” - that is, if a disjunction of formulae has SOP2, then one of the disjuncts has SOP2.

<ω Lemma 1.5.9. If haα : α ∈ 2i is a 1-fti tree witnessing SOP2 for ψ0 ∨ ψ1, then we can ﬁnd a tree witnessing SOP2 for either ψ0 or ψ1.

Proof. We may assume that both ψ0(x, ahi) and ψ1(x, ahi) are consistent - otherwise <ω haα : α ∈ 2i is already a tree for one or the other. (This uses the fact that the tree is <ω 1-fti: if, say, ψ0(x, ahi) is inconsistent, then so is ψ0(x, aα) for every α ∈ 2. It follows that consistency of the instances of ψ0∨ψ1 along a branch implies consistency of the corresponding instances of ψ1. Inconsistency of incomparable instances of the disjunction automatically implies inconsistency of incomparable instances of ψ1, so the original tree witnesses SOP2 for ψ1.) Consider the leftmost branch (i.e. {hi, h0i, h00i, h000i,...}). Let b realize

{(ψ0 ∨ ψ1)(x; ah0ni): n < ω}, CHAPTER 1. INTRODUCTION 16 where h00i = hi, h01i = h0i, h02i = h00i, etc. Let f be a function from ω to {0, 1} deﬁned as follows: ( 0 if ψ (b; a j ) f(j) = 0 h0 i 1 otherwise The function f must take at least one of {0, 1} as its value for arbitrarily large j. We may assume f(j) = 0 for arbitrarily large j. Now, for any m ∈ ω, consider η := {hi, h0i,... h0mi}. By our assumption, we can ﬁnd j0, . . . , jm ∈ ω with j0 < . . . < jm and such that

{ψ0(x, ah0ji i): i ≤ m}

j0 j1 jm is consistent. Let ν := {h0 i, h0 i,..., h0 i}. Clearly, η ≈1 ν. Since our tree is 1-fti, it follows that {ψ0(x, ah0ii): i ≤ m} is consistent. By compactness, {ψ0(x, ah0ii): i ∈ ω} is consistent. We now deﬁne a function g : <ω2 → <ω2 by induction, as follows:

g(hi) := hi g(α_h0i) := g(α)_h00i g(α_h1i) := g(α)_h01i

This is our pruning function: it picks out the nodes from the original tree that will form our new tree. We illustrate the ﬁrst few steps of the function below. Think of the tree on the left as the tree we are constructing (the candidate for an SOP2 tree for ψ0(x; y)), and the tree on the right as the original tree. The black dots are the nodes in the image of g, and these are the nodes that form our new tree. The circles are nodes not in the image of g. • • • •

◦ ◦ ◦ ◦

• • • • • •

• • ◦ ◦

g • −→ • Notice that we are picking nodes from every other level of the original tree, and each node that appears in the image of g lies above the left child of the previous node appearing in the image. This construction gives us a tree all of whose branches “look like” (are 1-similar to) the leftmost branch of the original tree. We shall show that the new tree witnesses that ψ0(x; y) has SOP2. <ω <ω For α ∈ 2, we let bα := ag(α). We claim that hbα : α ∈ 2i is a tree witnessing SOP2 for ψ0. CHAPTER 1. INTRODUCTION 17

<ω Claim 1.5.10. Given η, ν ∈ 2, η E ν if and only if g(η) E g(ν). (In particular, if η and ν are incomparable, so are g(η) and g(ν).)

Proof of claim. →: Suppose η E ν. We show that g(η) E g(ν) by induction on the distance between η and ν.

• Distance = 0; i.e. η = ν. Then g(η) = g(ν), and so g(η) E g(ν). _ • Distance = n + 1; i.e. ν = α hii for some i ∈ {0, 1} and α D η, with the distance between η and α equal to n. By the induction hypothesis, g(η) E g(α). By the deﬁnition of g, _ _ g(ν) = g(α hii) = g(α) h0ii D g(α) D g(η), as desired.

←: We prove the contrapositive. Suppose η 5 ν. There are two cases. _ • ν / η. Then there is i ∈ {0, 1} such that ν hii E η. By the forward direction of _ _ this claim, g(ν hii) E g(η), hence g(ν) h0ii E g(η). It follows that g(η) 5 g(ν), as desired. • ν and η are incomparable. We may assume without loss of generality that:

_ (ν ∩ η) h0i E ν; _ (ν ∩ η) h1i E η. Again by the forward direction of the claim, we have:

_ _ _ _ g(ν) D g((ν ∩ η) h0i) = g(ν ∩ η) h00i = (g(ν ∩ η) h0i) h0i; _ _ _ _ g(η) D g((ν ∩ η) h1i) = g(ν ∩ η) h01i = (g(ν ∩ η) h0i) h1i.

Hence g(η) and g(ν) are incomparable, and g(η) 5 g(ν), as desired.

_ Remark 1.5.11. In fact, if η / ν, then g(η) h0i E g(ν). _ Proof of remark. If η / ν, then η hii E ν, for i = 0 or i = 1. By Claim 1.5.10 and the deﬁnition of g, _ _ _ _ g(η hii) = g(η) h0ii = (g(η) h0i) hii E ν

<ω Claim 1.5.12. If η, ν ∈ 2 are incomparable, then ψ0(x, bη) ∧ ψ0(x, bν) is inconsistent.

Proof of claim. Recall that bη = ag(η) and bν = ag(ν). By the above, g(η) and g(ν) are incomparable, so ψ0(x, ag(η)) ∧ ψ0(x, ag(ν)) - i.e. ψ0(x, bη) ∧ ψ0(x, bν) - is inconsistent, since (ψ0 ∨ ψ1(x, ag(η))) ∧ (ψ0 ∨ ψ1(x, ag(ν))) is. CHAPTER 1. INTRODUCTION 18

ω Claim 1.5.13. If β ∈ 2, {ψ0(x, bβi): i ∈ ω} is consistent.

Proof of claim. It suﬃces to show that for any n ∈ ω, {ψ0(x, bβi): i ≤ n} is consistent. By _ Remark 1.5.11, if i < j, then g(β i) h0i E g(β j). Let η := (g(β i): i ≤ n). We just observed that η lies on a branch, and so it is ∩-closed. Now let ν := (h0ii : i ≤ n). This is also ∩-closed. Furthermore, for i, j ≤ n,

i j i j i _ j h0 i E h0 i iff h0 i = h0 i or h0 i h0i E h0 i iff i ≤ j _ iff g(β i) = g(β j) or g(β i) h0i E g(β j) iff g(β i) E g(β j)

<ω It follows that η ≈1 ν. Since {ψ0(x, ah0ii): i ≤ n} is consistent and haα : α ∈ 2i is a 1-fti tree, {ψ0(x, ag(βi)): i ≤ n} = {ψ0(x, bβi): i ≤ n} is also consistent, as desired. <ω Claims 1.5.12 and 1.5.13 tell us that hbα : α ∈ 2i is a tree witnessing SOP2 for ψ0, so the proof of the lemma is ﬁnished.

Corollary 1.5.14. If T is ℵ0-categorical and T has SOP2, then there is a complete formula (that is, a formula isolating a complete type) ϕ(x; y) with SOP2.

Proof. Suppose ψ(x; y) has SOP2, and ψ(x; y) is incomplete. By ℵ0-categoricity of T , there are ﬁnitely many complete types in the variables of ψ. Of these, let p0, . . . , pn−1 list the ones that contain ψ. Again by ℵ0-categoricity, each pi is isolated by some formula - say, ψi. Clearly, ψi ` ψ for each i. Further, since any realization of ψ realizes one of p0, . . . , pn−1, W W ψ ≡ ψi. By Lemma 1.5.9 and induction on n, ψi has SOP2 if and only if one of the ψi i

<ω Lemma 1.5.15. Suppose hbη : η ∈ 2i is a 1-fti tree, where the bη are tuples of length n. <ω i i <ω i i If for some η, ν ∈ 2 and some i ≤ n, bη = bν, then for all ρ ∈ 2, bρ = bη. Proof. We begin by considering the case where η and ν are siblings, and have the same ith coordinate as each other and as their parent. <ω i i i <ω i i Claim 1.5.16. If for some ρ ∈ 2, bρ = bρ_h0i = bρ_h1i, then for all ξ ∈ 2, bξ = bρ. Proof of claim. We split the proof into two cases, depending on whether or not ρ is the root of the tree. CHAPTER 1. INTRODUCTION 19

• Case 1: ρ = hi. For all ξ ∈ <ω2, either

hiξ ≈1 hih0i

or hiξ ≈1 hih1i i i i i i i Since both bhi = bh0i and bhi = bh1i, tree-indiscernibility implies that bhi = bξ.

• Case 2: ρ 6= hi. Then either h0i E ρ or h1i E ρ, and hence either

_ hiρ ≈1 ρρ h0i

or _ hiρ ≈1 ρρ h1i. i i i i i i Since both bρ = bρ_h0i and bρ = bρ_h1i, bhi = bρ by tree-indiscernibility. Then, since

_ _ (hi, h0i, h1i) ≈1 (ρ, ρ h0i, ρ h1i),

i i i tree-indiscernibility implies that bh0i = bhi = bh1i, and we are now in case 1 (giving the <ω i i i result that for all ξ ∈ 2, bξ = bhi = bρ).

We split the proof of the lemma into two cases, depending on whether or not η and ν are i i i comparable. In the following, refer to bη = bν as b .

• Case 1. Suppose η and ν lie on the same branch. Without loss of generality, suppose _ 0 _ 0 i i i η h0i E ν. Then for any ν D η h0i, ην ≈1 ην , and so bν0 = bη = b . In particular, i i i i _ bη_h0i = bη_h00i = bη_h01i = b . By Claim 1.5.16, we are done (take ρ to be η h0i). • Case 2. Suppose η and ν are incomparable. Without loss of generality, suppose _ _ 0 _ i i i (η ∩ ν) h0i E η and (η ∩ ν) h1i E ν. Then for all η D (η ∩ ν) h0i, bη0 = bν = b 0 0 _ i i i (since (ν, η, η ∩ ν) ≈1 (ν, η , η ∩ ν)), and for all ν D (η ∩ ν) h1i, bν0 = bη = b (since 0 (η, ν, η ∩ ν) ≈1 (η, ν , η ∩ ν)). In particular,

i i b = b(η∩ν)_h0i i = b(η∩ν)_h00i i = b(η∩ν)_h01i

We apply Claim 1.5.16 again, this time taking ρ to be (η ∩ ν)_h0i. CHAPTER 1. INTRODUCTION 20

<ω Corollary 1.5.17. Suppose hbη : η ∈ 2i is a 1-fti tree, where the bη are tuples of length <ω n. If for some η, ν ∈ 2 c ∈ bη ∩ bν (that is, c is a coordinate in each tuple, though not <ω necessarily the same coordinate), then for all ρ ∈ 2, c ∈ bρ.

i j Proof. Take i 6= j < n such that c = bη = bν. (If i = j, the statement reduces to Lemma 1.5.15.) We split the proof into two cases, depending on whether or not η and ν are comparable.

• Case 1: Suppose η and ν are comparable. Without loss of generality, we may assume _ that η h0i E ν. Then

_ _ (η, ν) ≈1 (η h0i, η h00i) _ ≈1 (η, η h00i)

i j Since bη = bν, we have

i j bη_h0i = bη_h00i i j bη = bη_h00i

i i i i <ω Hence bη = bη_h0i, and by Lemma 1.5.15, c = bη = bρ for all ρ ∈ 2. • Case 2: Suppose η and ν are incomparable. Without loss of generality, we may assume _ _ that (η ∩ ν) h0i E η and (η ∩ ν) h1i E ν. It follows that

_ (η, ν, η ∩ ν) ≈1 ((η ∩ ν) h0i, ν, η ∩ ν).

i j i j i i Since bη = bν, b(η∩ν)_h0i = bν, and we have c = b(η∩ν)_h0i = bη. By Lemma 1.5.15, i <ω c = bρ for all ρ ∈ 2.

<ω Remark 1.5.18. If hbη : η ∈ 2i is a 1-fti tree (where the bη are tuples of length n) and c <ω <ω is an element of the common intersection of hbη : η ∈ 2i (that is, c ∈ bρ for all ρ ∈ 2), <ω then for any η, ν ∈ 2, tp(bη/c) = tp(bν/c).

i <ω Proof. By the proof of Corollary 1.5.17, there is some i < n such that c = bρ, all ρ ∈ 2. <ω i i Since hbη : η ∈ 2i is 1-fti, tp(bη/c) = tp(bη/bη) = tp(bν/bν) = tp(bν/c) for all η, ν ∈ <ω2. We can generalize Corollary 1.5.17 a bit, so that it applies to the intersections of the deﬁnable closures of tuples at distinct nodes on a 1-fti tree. We begin by showing that we can add elements of the deﬁnable closure (in a reasonable way) to a 1-fti tree, and the resulting tree will also be 1-fti. CHAPTER 1. INTRODUCTION 21

<ω Lemma 1.5.19. If hbη : η ∈ 2i is a 1-fti tree (where the bη are tuples of length n), ϕ(x, y) =1 <ω is a formula such that M (∃ x)ϕ(x, bhi), and for each η ∈ 2 cη is the unique realization <ω of ϕ(x, bη), then the tree hbηcη : η ∈ 2i is 1-fti.

=1 Proof. First note that the statement of the lemma makes sense: if M (∃ x)ϕ(x; bhi), then =1 <ω M (∃ x)ϕ(x; bη) for all η ∈ 2, so the cη’s exist. We must show that if η and ν are <ω tuples from 2 such that η ≈1 ν, then bηcη ≡ bνcν. Let η ≈1 ν with m = lg(η) = lg(ν), and let ψ be any formula (to which we add dummy variables, if necessary). Suppose that M ψ(bη, cη). Then ^ M ∃x(ψ(bη, x) ∧ ϕ(xi, bηi )). i

Since cν is the only realization of ϕ(xi, bν ), cν is the only possible realization of ψ(bν, x) ∧ V i i ϕ(xi, bνi )). Since we know this formula has a realization, M ψ(bν, cν). Since ψ was i

<ω Corollary 1.5.20. If hbη : η ∈ 2i is a 1-fti tree and for some η, ν (η 6= ν)

c ∈ dcl(bη) ∩ dcl(bν),

<ω then c ∈ dcl(bρ) for all ρ ∈ 2.

Proof. Suppose ϕ1 witnesses that c ∈ dcl(bη) and ϕ2 witnesses that c ∈ dcl(bν). For each <ω ρ ∈ 2, let cρ be the unique realization of ϕ1(x, bρ), and let dρ be the unique realization <ω of ϕ2(x, bρ). (So cη = c = dν.) By Lemma 1.5.19, the tree hbρcρdρ : ρ ∈ 2i is 1-fti. Since <ω c ∈ bηcηdη ∩ bνcνdν, c ∈ bρcρdρ - and hence, c ∈ dcl(bρ) - for all ρ ∈ 2, by Corollary 1.5.17. <ω (In fact, by the proof of Corollary 1.5.17, c = cρ = dρ for all ρ ∈ 2.) 22

Chapter 2

Parameterized Equivalence Relations

In this chapter, we consider the two-sorted theory of infinitely many cross-cutting parama- terized equivalence relations, which is known to have the tree property, but not the tree property of the first kind. This example was introduced by Shelah in [21]. In [1] and [14], ∗ Adler and Kim and Kim (respectively) discuss possible notions of independence for Tfeq. In ∗ 1 [23], Shelah and Usvyatsov show that Tfeq does not have SOP2. We give a version of this proof, making use of the results on indiscernible trees presented in the first chapter.

2.1 The Theory

We take L to be a language whose signature has two sorts: one, P , for points, and another, E, for equivalence relations, and one three-place relation, R, on P × P × E. Given an E- element r, and two P -elements, a and b, we shall also write r(a, b) for R(a, b, r). We shall use lowercase letters x, y, z to denote variables of sort P , uppercase letters X,Y,Z to denote variables of sort E, lowercase a, b, c, d to denote parameters of sort P , and lowercase r, s, t to denote parameters of sort E. The universal L-theory T0 says that for each r ∈ E, r(·, ·) is an equivalence relation on P :

T0 :={∀X, y(X(y, y)), ∀X, y, z(X(y, z) → X(z, y)), ∀X, y, z, w((X(y, z) ∧ X(z, w)) → X(y, w))}

Fact 2.1.1. The class K of ﬁnite models of T0 has the Hereditary Property, the Joint Embed- ding Property, and the Amalgamation Property, and hence (see, e.g., [11], Theorem 6.1.2) has a Fra¨ıss´elimit, F.

1 ∗ The stated result in [23], Theorem 2.1 is, in fact, that Tfeq does not have SOP1. The proof, however, relies on a faulty result from [7] (Claims 2.11 and 2.14). Those Claims were ﬁxed in [8], a few years after [23] was published. The issue is, in short, that one cannot assume as great a degree of indiscernibility for SOP1 trees as was initially thought. However, given that one can assume that SOP2 trees are highly indiscernible, ∗ Shelah and Usvyatsov’s argument for Theorem 2.1 shows that Tfeq does not have SOP2. It is that argument that we give here, ﬁlling in the details that were less obvious to us than they would have been to the authors ∗ of [23]. It is still conjectured that Tfeq does not have SOP1. CHAPTER 2. PARAMETERIZED EQUIVALENCE RELATIONS 23

Proof. We recall the deﬁnition of each property before proving that K has it.

1. Hereditary Property: If A ∈ K and B is a finitely generated substructure of A, then B ∈ K. (Note that as K consists of finite structures in a relational language, “finitely generated” just means “finite” here.) By universality of T0, any substructure of a (finite) model of T0 is itself a (finite) model of T0. 2. Joint Embedding Property: If A, B ∈ K, then there is C ∈ K such that A and B are embeddable in C. We shall construct C. Suppose

A = {a0, . . . , an−1, r0, . . . , rm−1}, and

B = {b0, . . . , bn0−1, s0, . . . , sm0−1}.

Take the universe of C to be the disjoint union of the universes of A and B, that is,

C = {a0, . . . , an−1, b0, . . . , bn0−1, r0, . . . , rm−1, s0, . . . , sm0−1}

where the ai and bi, and the ri and si are all distinct. (We may replace B by an isomorphic copy, if necessary, to achieve this.) We now deﬁne R(·, ·, ·) on C.

• For i < m, j, k < n, and j0, k0 < n0,

C ri(aj, ak) if and only if A ri(aj, ak); C ¬ri(aj, bj0 ) ∧ ¬ri(bj0 , aj); C ri(bj0 , bk0 ).

• For i0 < m0, j, k < n, and j0, k0 < n0,

C si0 (bj0 , bk0 ) if and only if B si0 (bj0 , bk0 ); C ¬si0 (bj0 , aj) ∧ ¬si0 (aj, bj0 ); C si0 (aj, ak).

In C, each ri has all of the classes it has in A, plus an additional class for the bi’s. Similarly, each si has all of the classes it has in B, plus an additional class for the ai’s. It is clear that each of the ri’s and sj’s is an equivalence relation on the points of C, and that the inclusion maps from A into C and from B into C are embeddings.

3. Amalgamation Property: If A, B, C ∈ K and e : A → B, f : A → C are embeddings, then there are D ∈ K and embeddings g : B → D, h : C → D such that ge = hf. CHAPTER 2. PARAMETERIZED EQUIVALENCE RELATIONS 24

We construct D. Let the universe of D be the disjoint union of B \ e(A), C \ f(A), and A, where

B \ e(A) = {b0, . . . , bnB −1, s0, . . . , smB −1},

C \ f(A) = {c0, . . . , cnC −1, t0, . . . , tmC −1},

A = {a0, . . . , anA−1, r0, . . . , rmA−1}

D C Our definition of R is similar to that of R in the proof of JEP: each si has the same classes as in B, plus an additional class for the elements of C \ f(A), and each ti has the same classes as in C, plus an additional class for the elements of B \ e(A). We must be more careful when extending the definition of the ri’s, however, in order to make sure they satisfy transitivity. Thus, in our formal definition of RD, we consider three separate cases, depending on whether the equivalence relation element is from A, B \ e(A), or C \ f(A).

• For all i < mB, jA, kA < nA, jB, kB < nB, and jC , kC < nC ,

D si(ajA , akA ) if and only if B si(e(ajA ), e(ajA ));

D si(bjB , bkB ) if and only if B si(bjB , bkB );

D si(cjC , ckC ); ( si(ajA , bjB ) ∧ si(bjB , ajA ) if B si(e(ajA ), bjB ), D ¬si(ajA , bjB ) ∧ ¬si(bjB , ajA ) otherwise;

D ¬si(ajA , cjC ) ∧ ¬si(cjC , ajA ) ∧ ¬si(bjB , cjC ) ∧ ¬si(cjC , bjB ).

We can see that for any si, elements of B are si-related in D if and only if they are si-related in B, while elements of C\ f(A) are all si-related to one another, and are not si-related to any element of B.

• For all i < mC , jA, kA < nA, jB, kB < nB, and jC , kC < nC ,

D ti(ajA , akA ) if and only if C ti(f(ajA ), f(akA ));

D ti(cjC , ckC ) if and only if C ti(cjC , ckC );

D ti(bjB , bkB ); ( ti(ajA , cjC ) ∧ ti(cjC , ajA ) if C ti(f(ajA ), cjC ), D ¬ti(ajA , cjC ) ∧ ¬ti(cjC , ajA ) otherwise;

D ¬ti(ajA , bjB ) ∧ ¬ti(bjB , ajA ) ∧ ¬ti(cjC , bjB ) ∧ ¬ti(bjB , cjC ).

As in the deﬁnition of si on D, elements of C are ti-related in D if and only if they are ti-related in C, while the elements of B\ e(A) form a separate ti-class. CHAPTER 2. PARAMETERIZED EQUIVALENCE RELATIONS 25

• For all i < mA, jA, kA < nA, jB, kB < nB, and jC , kC < nC ,

D ri(ajA , akA ) if and only if A ri(ajA , akA )

if and only if B e(ri)(e(ajA ), e(akA ))

if and only if C f(ri)(f(ajA ), f(akA ))

D ri(bjB , bkB ) if and only if B e(ri)(bjB , bkB )

D ri(cjC , ckC ) if and only if C f(ri)(cjC , ckC ) ( ri(ajA , bjB ) ∧ ri(bjB , ajA ) if B e(ri)(e(ajA ), bjB ), D ¬ri(ajA , bjB ) ∧ ¬ri(bjB , ajA ) otherwise; ( ri(ajA , cjC ) ∧ ri(cjC , ajA ) if C f(ri)(f(ajA ), cjC ), D ¬ri(ajA , cjC ) ∧ ¬ri(cjC , ajA ) otherwise;

D ri(bjB , cjC ) ∧ ri(cjC , bjB ) if there is 1 ≤ ` ≤ nA such that

B e(ri)(bjB , e(a`)) and

C f(ri)(f(a`), cjC ),

otherwise, D ¬ri(bjB , cjC ) ∧ ¬ri(cjC , bjB ).

Once again, elements of B are ri-related in D if and only if they are e(ri)-related in B, and elements of C are ri-related in D if and only if they are f(ri)-related in C. We extend ri to (B \ e(A)) × (C \ f(A)) so that bj and ck are ri related just in case there is some element of A that “connects” them. This is well deﬁned: either bj and ck are e(ri)-related (respectively, f(ri)-related) to (images of) all the same elements of A, or to (images of) none of the same elements of A, since e and f preserve the ri-classes of A.

It should be clear that each ri, sj, and tk is an equivalence relation, so D T0. We deﬁne g : B → D and h : C → D in the obvious way: g (B \ e(A)) and h (C \ f(A)) −1 −1 are the identity map, while g e(A) = e and h f(A) = f . The maps g and h are embeddings, and ge = hf.

∗ Let Tfeq be the theory of F. As the theory of the Fra¨ıssélimit of the finite models of T0, ∗ Tfeq is ℵ0-categorical and has elimination of quantifiers (see [11], Theorem 6.4.1), and it is CHAPTER 2. PARAMETERIZED EQUIVALENCE RELATIONS 26

the model completion of T0. For convenience, we spell out the intermediate theory Tfeq: ( ! ) ^ Tfeq := T0 ∪ ∀X∃y0, . . . , yn−1 ¬X(yi, yj) : n < ω i6=j ( ! ) ^ ∪ ∃X0,...,Xn−1 Xi 6= Xj : n < ω i6=j ( ! ) ^ ^ ∪ ∀X0,...,Xn−1∀y0, . . . , yn−1∃y Xi 6= Xj → Xi(y, yi) : n < ω i6=j i

Tfeq says that each equivalence relation has infinitely many classes, that there are infinitely many E-sort elements, and that given any n distinct classes r0, . . . , rn−1, and n (not necessarily distinct) points a0, . . . , an−1, we can find a point that is ri-equivalent to ai for i < n. We shall refer to this as the “cross-cutting axiom.”

∗ Fact 2.1.2. The theory Tfeq is an extension of Tfeq. ∗ Proof. Suppose M Tfeq. We show M Tfeq. ! V 1. For each n < ω, M ∀X∃y0, . . . , yn−1 ¬X(yi, yj) . i6=j M Suppose r ∈ E , and r has m classes for some m < n. We build a model N of T0 extending M: let the universe of N be M ∪ {a0, . . . , an−m−1} (where a0, . . . , an−m−1 are new elements of sort P ). For s ∈ EM \{r}, let the new elements form a single, new s-class. (This is not actually relevant, but we must make some choice with regard to the s-classes of the new elements.) Let each new element form its own r-class. More precisely, we deﬁne RN by: M M N s(b, c) if and only if M s(b, c) for all s ∈ E , b, c ∈ P ; M M N ¬s(b, ai) ∧ ¬s(ai, b) for any s ∈ E , b ∈ P and i < n − m; ^ M N s(ai, aj) for all s ∈ E \{r}; i,j

as desired. ! V 2. For each n < ω, M ∃X0,...,Xn−1 Xi 6= Xj . i6=j

Again, we build a model N of T0 extending M. Let the universe of N be N M ∪ {r0, . . . , rn−1}, and deﬁne R as follows:

M M N s(a, b) if and only if M s(a, b) for all s ∈ E , a, b ∈ P ; M N ri(a, b) for any i < n, any a, b ∈ P .

That is, N adds n equivalence relations to M, each of which has only one class. It is ! V clear that M embeds into N , N T0, and N ∃X0,...,Xn−1 Xi 6= Xj . Since i6=j M is existentially closed for models of T0, it satisﬁes the desired axiom. ! V V 3. For each n < ω, M ∀X0,...,Xn−1∀y0, . . . , yn−1∃y Xi 6= Xj → Xi(y, yi) . i6=j i

M M N s(c, d) if and only if M s(c, d) for all s ∈ E , c, d ∈ P ; M M N ¬s(c, b) ∧ ¬s(b, c) for all s ∈ E , c ∈ P ; ^ N ri(b, ai) ∧ ri(ai, b); i

With this deﬁnition of RN , it is clear that N T , N is an extension of M, and 0 V N ∃y ri(y, ai) . Again, we note that since M is existentially closed for models i

M Since r0, . . . , rn−1 were arbitrary distinct elements of E and a0, . . . , an−1 were arbitrary elements of P M, M satisﬁes the desired axiom.

Fact 2.1.3. If T extends Tfeq, the formula ϕ(x; y, Z) := Z(x, y) has the tree property of the second kind, and in particular, T is not simple.

Proof. Work in any suﬃciently saturated model M of Tfeq. We begin by showing that the ω following type, in variables Xi for i ∈ ω, z(i,j) for i, j ∈ ω, and yα for α ∈ ω is consistent:

ω [ π(Xi, z(j,k), yα : i, j, k ∈ ω, α ∈ ω) = {Xi(yα, z(i,α(i))): i < ω} α∈ωω

∪ {Xi 6= Xj : i < j < ω} [ ∪ {¬Xi(z(i,j), z(i,k)): j < k < ω}. i<ω Given a ﬁnite subset of this type, there are n, m, l ∈ ω such that the subset is contained in ω a type of the following form, in variables Xi for i < n, yα0 , . . . , yαl−1 for αi ∈ ω, and z(i,j) for i < n and j < m, where m > max{αj(i): j < l, i < n}: [ π0(Xi, z(j,k), yαr : i, j < n, k < m, r < l) = {Xi(yαr , z(i,αr(i))): i < n} r

∪ {Xi 6= Xj : i < j < n} [ ∪ {¬Xi(z(i,j), z(i,k)): j < k < m}. i

To realize π0, we begin by choosing any n distinct equivalence relation elements s0, . . . , sn−1 from M. We can do this because EM is infinite. Then, for each i, we choose m elements M a(i,0), . . . , a(i,m−1) of P from distinct si-classes. We can do this because each element of M E has infinitely many classes. Lastly, for each r < l, we find bαr satisfying ^ si(y, a(i,αr(i))). i

Such bαr exist by the cross-cutting axiom of Tfeq. By the choice of these elements,

M π0(si, a(j,k), bαr : i, j < n, k < m, r < l). By compactness, π is consistent and (by saturation) realized in M. Let

ω (si, a(i,j), bα : i, j < ω, α ∈ ω) realize π in M, and for each i, let t(i,j) = si. To recap, we have:

M t(i,α(i))(bα, a(i,α(i))) (2.1) CHAPTER 2. PARAMETERIZED EQUIVALENCE RELATIONS 29 for each α ∈ ωω and i < ω, and

M ¬∃x(t(i,j)(x, a(i,j)) ∧ t(i,k)(x, a(i,k))) (2.2)

for any i < ω and any j 6= k < ω. (This last comes from the fact that t(i,j) = t(i,k) and a(i,j) and a(i,k) are in diﬀerent t(i,j)-classes.) We claim that (a(i,j)t(i,j) : i, j < ω) witnesses that ϕ(x; y, Z) has TP2. • For each i < ω,

{ϕ(x; a(i,j), t(i,j)): j < ω} = {t(i,j)(x, a(i,j)): j < ω}

is inconsistent, by 2.2, above.

• For each α ∈ ωω,

{ϕ(x; a(i,α(i)), t(i,α(i))): i < ω} = {t(i,α(i))(x, a(i,α(i))): i < ω}

is consistent, realized by bα (by 2.1, above).

2.2 Demonstration of subtlety

∗ In this section, we show that Tfeq does not have the tree property of the ﬁrst kind (or, in the terminology of Kim and Kim, that it is subtle). Remark 2.2.1. By Fact 1.2.10 and Corollary 1.5.14, it suﬃces to show that no “complete” formula (that is, a formula that isolates some complete type in its variables) has SOP2.

Remark 2.2.2. Any isolating formula ψ(x0, . . . , xm−1,Y0,...,Yn−1; z0, . . . , zr−1,W0,...,Ws−1) CHAPTER 2. PARAMETERIZED EQUIVALENCE RELATIONS 30 is (equivalent to a formula) of the form ^ ^ ^ ψ(xY ; zW ) := Yk(xi, xj) ∧ ¬Yk(xi, xj) ∧ Yk(xi, zj)

(i,j,k)∈I1 (i,j,k)∈/I1 (i,j,k)∈I2 ^ ^ ^ ∧ ¬Yk(xi, zj) ∧ Yk(zi, zj) ∧ ¬Yk(zi, zj)

(i,j,k)∈/I2 (i,j,k)∈I3 (i,j,k)∈/I3 ^ ^ ^ ∧ Wk(xi, xj) ∧ ¬Wk(xi, xj) ∧ Yk(xi, zj)

(i,j,k)∈J1 (i,j,k)∈/J1 (i,j,k)∈J2 ^ ^ ^ ∧ ¬Yk(xi, zj) ∧ Wk(zi, zj) ∧ ¬Wk(zi, zj)

(i,j,k)∈/J2 (i,j,k)∈J3 (i,j,k)∈/J3 ^ ^ ^ ^ ∧ xi = xj ∧ xi 6= xj ∧ Yi = Yj ∧ Yi 6= Yj

(i,j)∈K1 (i,j)∈/K1 (i,j)∈K2 (i,j)∈/K2 ^ ^ ^ ^ ∧ zi = zj ∧ zi 6= zj ∧ Wi = Wj ∧ Wi 6= Wj

(i,j)∈K3 (i,j)∈/K3 (i,j)∈K4 (i,j)∈/K4 ^ ^ ^ ^ ∧ xi = zj ∧ xi 6= zj ∧ Yi = Wj ∧ Yi 6= Wj

(i,j)∈L1 (i,j)∈/L1 (i,j)∈L2 (i,j)∈/L2

for some I1 ⊆ m × m × n, I2 ⊆ m × r × n, I3 ⊆ r × r × n, J1 ⊆ m × m × s, J2 ⊆ m × r × s, J3 ⊆ r × r × s, K1 ⊆ m × m, K2 ⊆ n × n, K3 ⊆ r × r, K4 ⊆ s × s, L1 ⊆ m × r, and L2 ⊆ n × s. However, we may eliminate the equalities: any equalities among variables of the same ‘type’ (where by ‘type’ we mean both sort and either object or parameter - so xi is the same type as xj, but not the same type as zj, for example) simply make one of the variables redundant. That is, since the formula is consistent in the ﬁrst place, we may replace it by a formula without the redundant variables and the conjuncts containing them. Similarly, suppose there is some equality between an object variable and a parameter variable, say, xi = zj. Then the realization of xi is determined by the parameter at zj. Since the branches of the tree are consistent, the jth point parameter must be the same along any branch, and hence (by Lemma 1.5.15), throughout the tree. If ψ(xY ; zW ) has SOP2, so does the formula 0 0 ψ obtained by removing xi and any conjuncts containing it. (SOP2 for ψ is witnessed by the same parameters as for ψ). CHAPTER 2. PARAMETERIZED EQUIVALENCE RELATIONS 31

Proposition 2.2.3 (Adapted from [23], Theorem 2.1). No formula of the form ^ ^ ^ ψ(xY ; zW ) := Yk(xi, xj) ∧ ¬Yk(xi, xj) ∧ Yk(xi, zj)

(i,j,k)∈I1 (i,j,k)∈/I1 (i,j,k)∈I2 ^ ^ ^ ∧ ¬Yk(xi, zj) ∧ Yk(zi, zj) ∧ ¬Yk(zi, zj)

(i,j,k)∈/I2 (i,j,k)∈I3 (i,j,k)∈/I3 ^ ^ ^ ∧ Wk(xi, xj) ∧ ¬Wk(xi, xj) ∧ Yk(xi, zj)

(i,j,k)∈J1 (i,j,k)∈/J1 (i,j,k)∈J2 ^ ^ ^ ∧ ¬Yk(xi, zj) ∧ Wk(zi, zj) ∧ ¬Wk(zi, zj)

(i,j,k)∈/J2 (i,j,k)∈J3 (i,j,k)∈/J3 ^ ^ ^ ^ ∧ xi 6= xj ∧ Yi 6= Yj ∧ zi 6= zj ∧ Wi 6= Wj i6=j i6=j i6=j i6=j ^ ^ ∧ xi 6= zj ∧ Yi 6= Wj i∈lg(x),j∈lg(z) i∈lg(Y ),j∈lg(W )

(where I1 ⊆ lg(x) × lg(x) × lg(Y ), I2 ⊆ lg(x) × lg(z) × lg(Y ), etc.) has SOP2, and hence, ∗ Tfeq is NTP1.

∗ Proof. We work in a suﬃciently saturated model M of Tfeq. (Recall that in Theorem 1.5.6, we need some degree of saturation to obtain a tree that is 1-fti.) In what follows, we use superscripts to denote coordinates (e.g. ai is the ith coordinate of a), so as to avoid conﬂict with subscripts denoting nodes. Suppose, for a contradiction, that a formula ψ(xY ; zW ) (of the form listed in the state- <ω ment of the proposition) has SOP2. By Theorem 1.5.6, there is a 1-fti tree haαrα : α ∈ 2i witnessing SOP2 for ψ(xY ; zW ) (so lg(aα) = lg(z) and lg(rα) = lg(W ) for each α). Note <ω that by Corollary 1.5.17, since our tree is 1-fti, aηrη ∩ aνrν is the same for any η 6= ν ∈ 2. M M In what follows, let ct := ah0irh0i ∩ ah1irh1i (where c ∈ P and t ∈ E ). By Remark 1.5.18, <ω since the tree is 1-fti, tp(aηrη/ct) = tp(aνrν/ct) for all η, ν ∈ 2. In fact, by the proof of i j k i l j Corollary 1.5.17, for each c ∈ c and t ∈ t there are k, l such that aη = c and rη = t for all η. Take bh0ish0i and bh1ish1i to be realizations of the instances of ψ at the nodes h0i and h1i, respectively:

bh0ish0i ψ(xY ; ah0irh0i); bh1ish1i ψ(xY ; ah1irh1i).

Let N0, N1, and B be the substructures of M with universes

N0 = bh0i ∪ sh0i ∪ ah0i ∪ rh0i,

N1 = bh1i ∪ sh1i ∪ ah1i ∪ rh1i, and

B = ah0i ∪ rh0i ∪ ah1i ∪ rh1i, CHAPTER 2. PARAMETERIZED EQUIVALENCE RELATIONS 32 respectively (regarding each tuple as the set of its coordinates). Note that these are all models of T0, and the diagrams of N0 and N1 are given by ψ. We shall show that N0 and N1 can be amalgamated into a ﬁnite model N of T0 extending B, with bh0ish0i and bh1ish1i being mapped to the same elements of N , b0s0.

Lemma 2.2.4. We can amalgamate N0 and N1 into some N T0 extending B. 0 0 0 0 Proof. We take the universe of N to be {b s , ah0irh0i, ah1irh1i} (where b and s are new, of lengths lg(bh0i) and lg(sh0i), respectively), and deﬁne three maps into N :

fB :B ,→ N is the identity on B ( 0 0 bh0ish0i 7→ b s f0 :N0 ,→ N is given by ah0irh0i 7→ ah0irh0i

( 0 0 bh1ish1i 7→ b s f1 :N1 ,→ N is given by ah1irh1i 7→ ah1irh1i

We claim that we can deﬁne R on N so that each of these maps is an embedding, and N0 N1 B N T0. We begin with R0 := f0(R ) ∪ f1(R ) ∪ R . P Claim 2.2.5. R0 adds no “new relations.” That is, if C ∈ {N0, N1, B} and d, e ∈ C E C and q ∈ C , then fC(d, e, q) ∈ R0 if and only if (d, e, q) ∈ R . Equivalently, if C1 and P P C2 are two distinct structures from the set {N0, N1, B}, and d, e ∈ fC1 (C1 ) ∩ fC2 (C2 ) and E E C1 C2 q ∈ fC1 (C1 ) ∩ fC2 (C2 ), then (d, e, q) ∈ fC1 (R ) if and only if (d, e, q) ∈ fC2 (R ). Proof of claim. There are three cases to consider.

• (d, e, q) ∈ f0(N0) ∩ f1(N1): There are several subcases, depending on whether each of d, e, and q is in B or not, but in all cases the result follows from the fact that N0 and 0i N1 have the same diagram. For example, if there are i, j, and k such that d = b , e = b0j, and q = s0k, then

N0 k i j (d, e, q) ∈ f(R ) ↔ N0 sh0i(bh0i, bh0i) ↔ ψ(xY ; zW ) ` R(xi, xj,Y k) k i j ↔ N1 sh1i(bh1i, bh1i) ↔ (d, e, q) ∈ f(RN1 ).

The arguments in the other subcases are the same, except that some or all of the relevant variables in the second line may be from z and W , rather than x and Y . (We are using the fact that elements of c and t come from the same coordinates in ah0i as in ah1i.) CHAPTER 2. PARAMETERIZED EQUIVALENCE RELATIONS 33

• (d, e, q) ∈ f0(N0) ∩ fB(B): In this case, the result follows from the facts that N0 B f0(N0)∩fB(B) = N0∩B and that N0 and B are both substructures of M, so R and R i i j j k k agree on N0 ∩ B. Suppose d = f(ah0i) = ah0i, e = f(ah0i) = ah0i, and q = f(rh0i) = rh0i. Then we have:

N0 k i j (d, e, q) ∈ f0(R ) ↔ N0 rh0i(ah0i, ah0i) k i j ↔ M rh0i(ah0i, ah0i) k i j ↔ B rh0i(ah0i, ah0i) B ↔ (d, e, q) ∈ fB(R ).

• (d, e, q) ∈ f1(N1) ∩ fB(B): The argument is the same as in the previous case.

N N N We extend R0 to all of P × P × E as follows:

N 0i j k N j 0i k • R (b , ah0i, rh1i) and R (ah0i, b , rh1i) hold if and only if there is l such that:

i l k N1 R(bh1i, ah1i, rh1i) and j l k B R(ah0i, ah1i, rh1i).

N 0i j k N j 0i k • R (b , ah1i, rh0i) and R (ah1i, b , rh0i) hold if and only if there is an l such that:

i l k N0 R(bh0i, ah0i, rh0i) and j l k B R(ah1i, ah0i, rh0i).

N i j 0k N j i 0k • R (ah0i, ah1i, s ) and R (ah1i, ah0i, s ) hold if and only if there is an l such that:

i l k N0 R(ah0i, bh0i, sh0i) and j l k N1 R(ah1i, bh1i, sh1i).

Claim 2.2.6. RN , as defined above, gives an equivalence relation for each i ∈ EN . Proof of claim. The reflexivity of and symmetry of RN come from the definition (and the fact that RN0 ,RN1 , and RB are reflexive and symmetric). We must show that RN (·, ·, q) is transitive for all q ∈ EN .

l N l N l 1. q = rh1i ∈ rh1i. R (u, v, rh1i) and R (v, w, rh1i). CHAPTER 2. PARAMETERIZED EQUIVALENCE RELATIONS 34

N l N1 l a) v ∈ ah1i: then R (u, w, rh1i) follows from one of: transitivity of R (·, ·, rh1i) (if B l u, w ∈ f1(N1)), transitivity of R (·, ·, rh1i) (if u, w ∈ fB(B) = B), or the deﬁnition N 0 of R (if one of u, w comes from b - i.e., f1(N1)\(fB(B)∩f1(N1)) - and the other comes from ah0i - i.e., fB(B) \ (fB(B) ∩ f1(N1)). j b) v = ah0i ∈ ah0i. N l B l • If u, w ∈ fB(B) = B, then R (u, w, rh1i) follows from transitivity of R (·, ·, rh1i). Otherwise, one or both of u, w comes from b0. 0i 0k • Suppose u = b , w = b (i.e., u, b ∈ N \ fB(B)). Then by the deﬁnition of N N 0i j l N j 0k l R , R (b , ah0i, rh1i) and R (ah0i, b , rh1i) imply that there are m and n such that:

N1 i m l B j m l R (bh1i, ah1i, rh1i) and R (ah0i, ah1i, rh1i)

B j n l N1 k n l R (ah0i, ah1i, rh1i) and R (bh1i, ah1i, rh1i) B l B m n l By transitivity of R (·, ·, rh1i), it follows that R (ah1i, ah1i, rh1i). By Claim N1 m n l 2.2.5, R (ah1i, ah1i, rh1i) holds, too. Two applications of the transitivity of N1 l N1 i k l N 0i 0k l R (·, ·, rh1i) give us that R (bh1i, bh1i, rh1i), and thus that R (b , b , rh1i). 0i m B j l • Suppose u = b and w ∈ B. We have ah1i as above. Since R (ah0i, w, rh1i), B l B m l N 0i l transitivity of R (·, ·, rh1i) gives us that R (ah1i, w, rh1i). Then R (b , w, rh1i) N1 l comes from either transitivity of R (·, ·, rh1i) (if w ∈ ah1i, in which case we B N use, once again, the fact that R and R1 agree on elements of ah1irh1i) or N from the deﬁnition of R (if w ∈ ah0i). c) v = b0j ∈ b0. The reasoning here will be the same as in the previous subcase.

l N l N l 2. q = rh0i ∈ rh0i. R (u, v, rh0i) and R (v, w, rh0i). The reasoning here will be the same as in the previous case. 3. q = s0l ∈ s0. RN (u, v, s0l) and RN (v, w, s0l).

a) v ∈ b0. Then RN (u, w, s0l) follows from transitivity in RN1 (respectively, RN0 ) if u and w are both in f1(N1) (respectively, f0(N0)). If u ∈ ah0i and w ∈ ah1i (or vice versa), RN (u, w, s0l) comes from the deﬁnition of RN . j b) v = ah0i ∈ ah0i.

N 0l N0 • If u, w ∈ f0(N0), then R (u, w, s ) comes from transitivity in R . Other- wise, one or both of u, w comes from ah1i. i k N N i j 0l • Suppose u = ah1i, w = ah1i. Then, by deﬁnition of R , R (ah1i, ah0i, s ) and N j k 0l R (ah0i, ah1i, s ) imply that there are m and n such that:

N1 i m l N0 j m l R (ah1i, bh1i, sh1i) and R (ah0i, bh0i, sh0i)

N0 j n l N1 k n l R (ah0i, bh0i, sh0i) and R (ah1i, bh1i, sh1i) CHAPTER 2. PARAMETERIZED EQUIVALENCE RELATIONS 35

N0 N0 m n l N0 By transitivity in R , this gives us that R (bh0i, bh0i, sh0i). Since R and N1 0 0 N1 m n l R agree on their respective preimages of b s , R (bh1i, bh1i, sh1i) holds. N1 N1 i k l Then by transitivity (twice) in R , we have R (ah1i, ah1i, sh1i), and thus N i k 0l R (ah1i, ah1i, s ), as desired. i 0m • Suppose u = ah1i and w ∈ f0(N0). We have b as above. Then, since N0 j l N0 N0 m l R (ah0i, w, sh0i), transitivity in R gives us R (w, bh0i, sh0i). Then, by def- N N1 0 inition of R (if w ∈ ah0i) or by transitivity in R (if w ∈ b ), it follows that N i 0l R (ah1i, w, s ). j c) v = ah1i ∈ ah1i. The reasoning here will be the same as in the previous subcase.

Claim 2.2.6 shows that N T0. We now observe that, by Claim 2.2.5 and the fact that N N N the extension of the deﬁnition of R0 to P × P × E adds no new relations among images of elements from the same structure, f0, f1, and fB are embeddings, and N is the desired amalgam. This ﬁnishes the proof of Lemma 2.2.4.

Since f0 : N0 ,→ N and f1 : N1 ,→ N , and since ψ is quantifier free, 0 0 0 0 N ψ(b s ; ah0irh0i) ∧ ψ(b s ; ah1irh1i). ∗ ∗ ∗ As N T0 and Tfeq is the model completion of T0, N embeds into a model N of Tfeq. Again, since ψ is quantifier free, ∗ 0 0 0 0 N ψ(b s ; ah0irh0i) ∧ ψ(b s ; ah1irh1i). ∗ ∗ Using, once again, the fact that Tfeq is the model completion of T0, we note that Tfeq ∪diag(B) is a complete LB-theory. Since ∗ ∗ (N , ah0i, rh0i, ah1i, rh1i) Tfeq ∪ diag(B), ∗ and since ∃x∃Y (ψ(xY ; ah0irh0i)∧ψ(xY ; ah1irh1i)) is an LB-sentence in Th(N , ah0i, rh0i, ah1i, rh1i), ∗ Tfeq ∪ diag(B) ` ∃x∃Y (ψ(xY ; ah0irh0i) ∧ ψ(xY ; ah1irh1i)). We also know that ∗ (M, ah0i, rh0i, ah1i, rh1i) Tfeq ∪ diag(B), and so M ∃x∃Y (ψ(xY ; ah0irh0i) ∧ ψ(xY ; ah1irh1i)), <ω contradicting the choice of haαrα : α ∈ 2i as an SOP2 tree for ψ(xY ; zW ). Remark 2.2.7. At first glance, one might worry that the proof of Proposition 2.2.3 contradicts ∗ the fact that Tfeq has the tree property, since a tree witnessing TP must contain nodes h0i and h1i such that the instances of the formula in question corresponding to these nodes are inconsistent with each other. However, Proposition 2.2.3 relies on the fact that an SOP2 (or TP1) tree can be chosen to be 1-fti. The same is not always true for formulae with (only) the tree property, as we noted in Remark 1.5.8. 36

Chapter 3

Limit Independence

In this chapter, we begin by examining theories of infinite dimensional vector spaces with a non-degenerate bilinear form, studied by Nicolas Granger in his thesis [9]. The well-behaved non-forking independence in finite dimensional subspaces (which are ω-stable of finite Morley rank) of a model of this theory can be lifted to another independence relation (not quite, but almost, as well-behaved) in the larger structure. We generalize this independence relation construction to any theory whose models can be “approximated” by certain substructures in the same way that Granger’s infinite dimensional vector space is approximated by finite dimensional subspaces.

3.1 Motivation

Here we present the two-sorted theory of an infinite dimensional vector space with a bilinear form, along with Granger’s independence relation for this theory. Most definitions and results in this section are from chapters 9, 10 and 12 of [9]. Granger uses a language with sorts V (for the vector elements) and K (for the field elements). The language contains the following functions and constants:

0V vector additive identity

0K ﬁeld additive identity

1K ﬁeld multiplicative identity

+V : V × V → V vector addition

+K : K × K → K ﬁeld addition

◦K : K × K → K field multiplication γ : K × V → V scalar multiplication [·, ·]: V × V → K bilinear form In the following, we will use the capital letters X, Y , Z, and W for vector variables, the lower case letters x, y, z, and w for field variables, the lower case letters a, b, c, and d for vector CHAPTER 3. LIMIT INDEPENDENCE 37 constants, and lower case greek letters α, β, γ, etc. for field constants. We will denote field and scalar multiplication simply by concatenation of the elements being multiplied, rather than using the symbols given above. We may leave out the subscripts of ‘+V ’ and ‘+K ’ (it should be clear from the context which one we intend) and will make use of ‘P’ (without any subscript). We start with the two-sorted theory of a vector space with a bilinear form. Our base theory, Tu, states that: • K is a field,

• V is a vector space over K, and

• [·, ·] is a bilinear form, that is:

∀X,Y,Z([X + Y,Z] = [X,Z] + [Y,Z]) ∀X,Y,Z([X,Y + Z] = [X,Y ] + [X,Z]) ∀X,Y ∀z([zX, Y ] = z[X,Y ] = [X, zY ]).

(The subscript “u” stands for “universal”: although this theory is not universal, it could be made so by adding symbols for −V , −K , and ÷K . We include only universal axioms for the bilinear form. In particular, Tu does not require the form to be non-degenerate.) We build up to a completion of Tu as follows. • Let T be the two-sorted theory of a vector space with a non-degenerate bilinear form:

T = Tu ∪ {∀X(∀Y ([X,Y ] = 0K ) → X = 0V )}.

• Given a ﬁeld K˜ , T K˜ adds to the above the complete theory of K˜ (for the sort K):

T K˜ := T ∪ T h(K˜ ).

• AT declares that the bilinear form is alternating, while ST declares that it is symmetric:

AT := T ∪ {∀X([X,X] = 0K )};

ST := T ∪ {∀X,Y ([X,Y ] = [Y,X])}.

• Lastly, we specify the dimension. For each n < ω, let Θn be the statement that there are at least n linearly independent vectors:

n−1 n−1 ! X ^ Θn := ∃X0,...,Xn−1∀y0, . . . , yn−1 yiXi = 0V → yi = 0K . i=0 i=0 CHAPTER 3. LIMIT INDEPENDENCE 38

A vector space with an alternating form must be of even dimension, so in that case, we will let the subscript n indicate that the dimension is 2n. If the form is symmetric, a subscript n indicates that the dimension is n:

ATn := AT ∪ {Θ2n ∧ ¬Θ2n+1}, and

STn := ST ∪ {Θn ∧ ¬Θn+1}. The theory of an inﬁnite dimensional vector space with a bilinear form is given by

T∞ := T ∪ {Θn : n < ω}.

˜ K˜ If K is finite and F ∈ {S,A}, F T∞ is ℵ0-categorical, super simple of SU-rank 1, and unstable. (The formula [X,Y ] = 0 has the order property.) See [9], Corollary 6.2.4, Proposition 6.2.5, and Remark 6.2.6. From now on, we assume K˜ is infinite. To obtain a quantifier elimination result when the field is infinite, we expand the language to include predicates for linear independence. For each n ∈ ω, let

n−1 ! n−1 !! X ^ θn(X0,...,Xn−1) := ∀y0, . . . , yn−1 yiXi = 0V → yi = 0K . i=0 i=0

Let Lθ be the language obtained from L by adding each θn as an atomic formula. (More precisely, Lθ is the definitional expansion of L having new n-ary predicate symbols on the K˜ sort V to be interpreted as θn.) Let Lθ be the language obtained from Lθ by adding an elimination set for K˜ . Proposition 3.1.1 ([9], Theorem 9.2.3 and Proposition 9.3.4). Given a field K˜ , F ∈ {S,A}, K˜ K˜ and m ∈ N ∪ {∞}, the theory F Tm has elimination of quantifiers in the language Lθ . In ˜ K˜ particular, if K is an algebraically closed field, then F Tm has elimination of quantifiers in ˜ K˜ the language Lθ. Further, if K is algebraically closed, F Tm is model complete in the language L.

Proposition 3.1.2 ([9], Corollary 9.2.9). Let K˜ be a ﬁeld and m ∈ N ∪ {∞}.

K˜ 1. The theory ATm is complete. ˜ K˜ 2. If K is a field with square roots for each element then the theory STm is complete. Since we find it rather cumbersome to continually refer to a theory with three distinct decorations, we will generally abuse language and notation by speaking of “the theory T∞” K˜ K˜ ˜ (or “the theory Tn”). By this, we mean F T∞ (F Tn ) for some field K and F ∈ {S,A}. We do not mean the incomplete theory which we earlier called T∞ (Tn). Unless otherwise specified, we shall assume that K˜ is algebraically closed.

Proposition 3.1.3 ([9], Corollary 10.1.2). If m is ﬁnite, Tm is ω-stable of ﬁnite Morley rank. CHAPTER 3. LIMIT INDEPENDENCE 39

Fact 3.1.4. T∞ is not simple.

Proof. (See also [9], Proposition 7.4.1.) Consider the formula ϕ(X; Y, y) := p[X,Y ] = yq, <ω and work in any model M = (V,K) T∞. We construct a tree {aηγη : η ∈ ω} witnessing that this formula has the tree property.

• Let aηai be some vector that is linearly independent from {aηj : j ≤ |η|} (the same vector for all i). (This is possible because V is inﬁnite dimensional.)

• Take a countable set of ﬁeld elements {γi : i < ω} distinct from one another and from

{γηj : j ≤ |η|}. Let γηai = γi. (This is possible because K is inﬁnite.) Now we check the tree property conditions:

<ω • For each η ∈ ω, {[X, aηj] = γηj : j ≤ |η|} is consistent, since the aηj’s are linearly independent. (See [9], Lemma 5.2.3.) By compactness, for any β ∈ ωω,

{[X, aβj] = γβj : j < ω} is consistent.

<ω • For any η ∈ ω, {[X, aηai] = γηai : i < ω} is inconsistent, since the aηai’s are all the

same, but the γηai’s are distinct. Hence, the formula [X,Y ] = y (with Y y as the parameter variables) has the tree property in ˜ T∞, and T∞ is not simple. Note that this argument holds whenever K (the field such that ˜ T h(K) ⊂ T∞) is infinite - not just for algebraically closed fields. We need a few more definitions before we can state Granger’s independence relation. Let M T∞ be sufficiently saturated. If A ⊂ M, we call the set of vector elements of A (i.e. M M M A∩V ) AV , and the set of field elements of A (i.e. A∩K ) AK . If char(K ) = 0, Granger M takes KA to be the subfield of K generated by

• AK ,

•{ [a, b]: a, b ∈ AV }, and

• for each n, the sets {α1, . . . , αn} such that there are b1, . . . , bn, b ∈ AV with b1, . . . , bn Pn linearly independent and b = i=1 αibi.

M If char(K ) > 0, he takes KA to be the perfect closure of the above ﬁeld. If A is a set and L is a subﬁeld of KM, we denote the L-linear span of the vector part of M A by spanL(AV ). We denote the K -linear span of AV by hAi.

Remark 3.1.5 ([9], Proposition 9.5.1). For m ∈ N ∪ {∞}, M = (V,K) Tm, and A ⊆ M, 1. dcl(A) = (span (A ),K ), and KA V A CHAPTER 3. LIMIT INDEPENDENCE 40

alg 2. except in the case that m is finite and dim(A) = m − 1, acl(A) = (span alg (AV ),KA ). KA Remark 3.1.6. In the original statement of this Proposition, Granger specifies that M should be “sufficiently saturated.” This is unnecessary, however: since each Tm is model complete with respect to L, dcl(A) and acl(A) are the same in any model of Tm containing A. Granger’s independence relation is as follows:

Deﬁnition 3.1.7 ([9], 12.2.1). Let r ∈ ω, M T∞, C ⊆ B ⊂ M, and a ∈ M. Say that KM tp(a/B) does not Γ-fork (dnΓf) over C if K | K (that is, K and K are algebraically Ca^KC B Ca B M 1 independent over KC in the ﬁeld K ) and one of the following three conditions holds: 1. a ∈ KM

2. a ∈ hCi

3. a∈ / hBi and whenever {b1, . . . , bn} ⊂ BV is linearly independent modulo hCi then the set {[a, b1],..., [a, bn]} is algebraically independent over KB(KCa), the ﬁeld generated over KB by the elements of KCa.

Extend the relation inductively to n-types as follows: tp(a1, . . . , an, an+1/B) dnΓf over C if and only if tp(a1, . . . , an/B) dnΓf over C and tp(an+1/Ba1, . . . , an) dnΓf over Ca1, . . . , an. This gist of this relation is this: dependence either comes from the field or from the existence of an algebraic relation among some of the pairings that was not forced by a linear relation among the vectors being paired. (Or it is trivial dependence: c ∈ hBi \ hCi.) Granger had already defined a very similar relation for finite dimensional vector spaces ([9], Definition 10.2.1), and showed that it was the same as non-forking independence ([9], Proposition 10.2.2) using Theorem 1.4.18, the characterization of simple theories via abstract independence. The finite dimensional relation is almost identical to the infinite dimensional relation, but contains one additional possible condition for c:

4. a ∈ hBi \ hCi and hBi = V and whenever {b1, . . . , bn} ⊂ BV is linearly independent modulo hCi then the set {[a, b1],..., [a, bn]} is algebraically independent over the field KB(KCa). Granger shows that his Γ-non-forking satisfies automorphism invariance, symmetry, transitivity, finite character, extension, and stationarity of types over models. In the finite dimensional case, Γ-non-forking also satisfies local character, but the same is not true in T∞. (In fact, local character does hold in all countable models of T∞, but can fail in uncountable models.)

1Note that independence of ﬁeld elements as computed in M is the same as independence as computed in KM, since KM is stably embedded. CHAPTER 3. LIMIT INDEPENDENCE 41

Remark 3.1.8. Since Γ-non-forking is symmetric for T∞ and T∞ is not simple, Γ-non-forking is not the same as non-forking in T∞ (by Theorem 1.4.16). More concretely, the formula ϕ(X; b) := p[X, b] = 0K ∧ X 6= 0V q (for some b 6= 0V ) divides over ∅ in Tn for each n < ω (and hence, by Proposition 3.1.11 below, Γ-forks over ∅ in T∞), but does not divide over ∅ in T∞. (This is discussed in [9], Example 12.3.3.)

Proof. 1. To see that ϕ(X; b) divides over ∅ in Tn (for n < ω), take a model N Tn containing b, and an indiscernible sequence (bi : i < ω) of elements of N (or some elementary extension of N ) where b0 = b and {b0, . . . , bn−1} (and thus, any set of n distinct elements from the sequence) is linearly independent. Then {ϕ(X; bi): i < ω} is n-inconsistent, by the non-degeneracy of the bilinear form: given distinct elements

bi0 , . . . , bin−1 from the sequence, if [a, bij ] = 0K for j < n, then [a, d] = 0K for all N N d ∈ V , since {bi0 , . . . , bin−1 } spans V . Since the form is non-degenerate, a = 0V , so N 2 ϕ(a; b). M Alternatively, take a ∈ M T∞ such that M ϕ(a; b). Clearly a∈ / K , a∈ / ∅, and {[a, b]} = {0K } is not algebraically independent over Kb(Ka), so tp(a/b) Γ-forks over ∅.

2. If (bi : i < ω) is an indiscernible sequence in M T∞ with b0 = b, then {ϕ(X; bi): i < ω} is consistent. (A finite subset just says that there is some nonzero vector that pairs with finitely many other vectors to give 0K , and this presents no problem in an infinite dimensional space.) So ϕ(X; b) does not divide in T∞.

After establishing that Γ-non-forking in T∞ has many of the nice properties of an independence relation, Granger goes on to show that in a countable dimensional model of T∞, it is the “limit” of Γ-non-forking (and hence, non-forking) in a sequence of ﬁnite dimensional subspaces.

Deﬁnition 3.1.9 ([9], Section 12.1). If M T∞, an approximating sequence for M is a S sequence of homogeneous substructures Nr ⊆ M such that M = r∈ω Nr and for each r, Nr M Nr Tr and K = K .(N is a homogeneous substructure of M if given tuples a, b ∈ N such that there is σ ∈ Aut(M) sending a to b, then there is τ ∈ Aut(M) sending a to b and ﬁxing N setwise.)

Remark 3.1.10. By this definition, only countable dimensional models of T∞ have approximating sequences, since a countable union of finite dimensional subspaces could not cover an uncountable dimensional model. Approximating a model of T∞ by substructures will not be fruitful unless the substructures in question have a nicer theory than T∞, so it would not be helpful to extend this definition by allowing a longer sequence of substructures, the tail end of which consists of infinite dimensional spaces. Admittedly, Granger does not require that an approximating sequence of substructures be a chain of substructures (although he may be assuming it implicitly), so one could imagine CHAPTER 3. LIMIT INDEPENDENCE 42 taking an uncountable sequence of finite dimensional subspaces that simply does not satisfy Nr ⊂ Ns for r < s. This seems unnecessarily messy, however, and there are more reasonable ways to extend the definition. When we generalize this framework in the next section, we shall make use of directed systems rather than sequences, and this will allow us to include uncountable models in our discussion.

Proposition 3.1.11 ([9], Proposition 12.2.3). Let M T∞, C ⊆ B ⊆ M, p(x) ∈ S(B), and (Nr : r ∈ ω) be an approximating sequence for M. Then the following are equivalent. 1. p(x) does not Γ-fork over C.

2. Given any formula ϕ = ϕ(x, b) ∈ p(x), there is Rϕ ∈ ω such that ϕ(x, b) does not fork over C ∩ Nr in the structure Nr for all r ≥ Rϕ. Remark 3.1.12. The original Proposition contained a third statement: “For each ﬁnite b ⊆ B there is Rb ∈ ω such that p(x) Nr∩Cb does not fork over C ∩ Nr in Nr for all r ≥ Rb.” This is not actually equivalent to the other two. The error in the proof lies in Granger’s assumption that p(x) Nr∩Cb has a realization in Nr (that is, an element of Nr that Nr believes satisﬁes the type). If p(x) is a complete type, however, then for every r < ω, p(x) Nr∩Cb is not realized in Nr because it contains the sentence Θr+1. Thus, any complete type (trivially) forks in every Nr.

3.2 Deﬁnitions

We wish to generalize Granger’s “approximation” approach to dealing with the unsimple theory T∞. Given some theory T , we begin with a strongly κ-homogeneous, κ-saturated model M T , and find a directed system, H, of substructures of M. In the case of M = (K,V ) T∞, H is the collection of finite dimensional K-subspaces of (K,V ). Before imposing requirements on H, let us develop some terminology to talk about behaviour in H. Unless otherwise stated, any subset of M we discuss is small (of cardinality less than κ). Definition 3.2.1. 1. For ψ(x) ∈ L, a ∈ M, we say that ψ(a) is eventually true in (or true in the limit of ) H if there is some Na,ψ ∈ H such that a ∈ Na,ψ and for Na,ψ ⊆ N ∈ H, N ψ(a). We define eventually false analogously. 2. Given a formula or finite partial type π(x; y) ∈ L, b ∈ M, and C ⊂ M, we say that π(x, b) eventually forks over C if there is Nπ,b,C ∈ H such that b ∈ Nπ,b,C and for all Nπ,b,C ⊆ N ∈ H, π(x, b) forks over C ∩ N in N . We define eventual non-forking of a finite partial type analogously. 3. Given a (not necessarily complete) type p(x) over B and C ⊆ B, we say that p(x) eventually forks over C if it contains some finite subtype that eventually forks over C, and that it eventually does not fork over C if every finite subtype eventually does not fork over C. CHAPTER 3. LIMIT INDEPENDENCE 43

| lim 4. For subsets A, B, and C of M, we say that A^C B (A is limit independent or eventually independent from B over C) if for each finite tuple a ∈ A, there is Na ∈ H such that a ∈ Na and for all Na ⊆ N ∈ H, | N M aâclM(C)∩N acl (B) ∩ N N where ^| is non-forking independence as computed in N . That is, tpN (a/(aclM(B) ∪ aclM(C)) ∩ N) does not fork over aclM(C) ∩ N in N . Notation: 1. In the following, we shall occasionally use the phrase “for large enough N ” to abbreviate “there is N 0 ∈ H such that for all N 0 ⊆ N ∈ H ...” (For example, the definition of “π(x, b) eventually forks over C” could be rephrased as: “for large enough N , π(x, b) forks over C ∩ N in N .”) 2. Unless otherwise stated, all structures ‘N ’ (possibly with decoration) come from H. 3. We shall refer to aclM(A) as A. Remark 3.2.2. 1. When we say that a formula or type π(x) “forks over A in N ,” we mean that for some formulae ϕ0(x, b0), . . . , ϕn−1(x, bn−1) and some k < ω, the following infinitary type is consistent with T h(N ):

_ [ j {π(x) → ϕi(x, bi)} ∪ {ψ(xi ) ↔ ψ(bi): j < ω, ψ ∈ L(A)} i

| lim 5. In principle, A^C B is not equivalent to the eventual non-forking of any type from M (say, tpM(A/BC)), since limit independence involves the forking of types as computed in the approximating structures, while eventual non-forking of a type involves forking in the approximating structures of a ﬁxed set of formulae. For example,

/| N a^C∩N B ∩ N

might be caused by the forking of some formula from tpN (a/(B ∪ C) ∩ N) that does not appear in tpM(a/B ∪ C).

Deﬁnition 3.2.3. Given M T and H a directed system of substructures N of M, we say that H approximates M just in case conditions 1 through 5 hold. We also identify optional conditions 6 and 7.

1. H covers M: S H = M.

2. Automorphism Invariance: H is closed under automorphism.

3. Convergence of truth value: For any a ∈ M and formula ψ(x) ∈ L, if M ψ(a), then ψ(a) is eventually true in H. (It follows that if M 2 ψ(a), then ψ(a) is eventually false in H: M ¬ψ(a), so for large enough N , N ¬ψ(a), and hence N 2 ψ(a).) 4. Stabilization of (non)forking: For every formula or ﬁnite partial type π(x, y), tuple b, and set C we have one of the following:

• π(x, b) eventually forks over C in H. • π(x, b) eventually does not fork over C in H.

/| lim 5. Strong ﬁnite character of limit dependence: If A^C B, then there are ψ(x, y, z) without parameters, a ∈ A, b ∈ B, and c ∈ C such that M ψ(a, b, c), and ψ(x, b, c) eventually forks over C in H.

6. Stabilization of algebraic closure: If A = A, there is Nalg ∈ H such that for all N N ⊇ Nalg, A ∩ N = acl (A ∩ N). 7. Homogeneity: For each N ∈ H, a, b ∈ M, and A ⊂ M, if tpM(a/A) = tpM(b/A), then tpN (a/A ∩ N) = tpN (b/A ∩ N).

Remark 3.2.4. 1. Condition 6 implies the following:

Given a ∈ M and B ⊂ M, if there is Nalg such that for all N ⊇ Nalg, a ∈ aclN (B ∩ N), then a ∈ B. (That is, if a is eventually in the algebraic closure of B in H, then a is in the algebraic closure of B in M.) CHAPTER 3. LIMIT INDEPENDENCE 45

N N To see this, suppose a ∈ acl (B ∩ N) for N ⊇ Nalg. Then a ∈ acl (B ∩ N) for N ⊇ Nalg, since B ⊆ B. Since B is algebraically closed in M, condition 6 implies that for large enough N , aclN (B ∩ N) = B ∩ N. Hence, for large enough N , a ∈ B ∩ N. Thus a ∈ B, as desired. The converse of this statement follows from condition 3: if a ∈ B, then for large enough N , a ∈ aclN (B ∩ N), since the formula declaring a’s algebraicity over B =n is eventually true in H. (More precisely, if M ϕ(a, b) ∧ (∃ x)ϕ(x, b), then by =n condition 3 there is Nalg such that for all N ⊇ Nalg, N ϕ(a, b) ∧ (∃ x)ϕ(x, b), and hence a ∈ aclN (B ∩ N).) 2. One might consider adding the following uniformity condition: “There is a collection of formulae {ϕi(x, y, z): i ∈ I} such that for any M T , there are parameters {ai : i ∈ I} such that H = {ϕi(x, M, ai): i ∈ I} satisﬁes the requirements of approximation.” If this holds, then approximation is really a property of the theory T , not of the particular model chosen. In the case of Granger’s T∞, the ϕi are of the form

Σ yjZj = X, j

M and ϕi(X, M, a1, . . . , ai) gives rise to the K -vector space generated by {a1, . . . , ai}. | lim Lemma 3.2.5. Suppose H satisfies conditions 1 through 5. Given sets A, B, and C, A^C B if and only if tpM(A/B ∪ C) eventually does not fork over C. /| lim Proof. ←: Suppose A^ C B. By condition 5 on H, there are ψ(x, y, z), a ∈ A, b ∈ B, and c ∈ C such that M ψ(a, b, c) and ψ(x, b, c) eventually forks over C in H. Since ψ(x, b, c) ∈ p = tpM(A/B ∪ C), p eventually forks over C by definition. →: Suppose p = tpM(A/B ∪ C) does not eventually not fork over C. By condition 4 on H, p eventually forks over C, and by definition, there is a finite subtype π(x, b, c) of p that eventually forks over C. Let a ∈ A realize π(x) in M. Since π is finite, by condition 3 on H there is some Ntv ∈ H containing a, b and c such that for N ⊇ Ntv, N π(a, b, c) (that N is, π(x, b, c) ⊂ tp (a/(B ∪ C) ∩ N)). Since π eventually forks over C, there is some Nf such N that for N ⊇ Nf , π(x, b, c) forks over C ∩ N in N . For N ⊇ Ntv ∪ Nf , tp (a/(B ∪ C) ∩ N) forks over C ∩ N in N . That is, /| N a^C∩N B ∩ N. /| lim By definition, A^C B. Remark 3.2.6. In fact, Lemma 3.2.5 does not require condition 4 on H, but the proof is simpler to state with that assumption. lim We have defined ^| without imposing the requirement that the set on the right side must contain the base set. We now show that this does not pose a problem; namely, that if | lim A^C B, we may enlarge the righthand side to contain C, while maintaining the independence. We shall refer to this condition as “normality on the right.” CHAPTER 3. LIMIT INDEPENDENCE 46

Lemma 3.2.7. Suppose H satisﬁes conditions 1 through 5. Given sets A, B, and C, if | lim | lim A^C B, then A^C BC. /| lim Proof. Suppose A^ C BC. By condition 5 on H, there are ψ(x, y, z) ∈ L, a ∈ A, b ∈ BC, and c ∈ C such that M ψ(a, b, c) and ψ(x, b, c) eventually forks over C. Take Ntv, Nf , and Nalg such that for N ⊇ Ntv, N ψ(a, b, c), for N ⊇ Nf , ψ(x, b, c) forks over C ∩N in N , and N N N for N ⊇ Nalg, b ∈ acl (BC∩N). For N ⊇ Ntv ∪Nf ∪Nalg, ψ(x, b, c) ∈ tp (a/ acl (BC∩N)) and forks over C ∩ N in N . That is,

/| N N a^C∩N acl (BC ∩ N). By Corollary 1.4.9, /| N a^C∩N BC ∩ N. By monotonicity of non-forking,

/| N a^C∩N (B ∪ C) ∩ N, and by the deﬁnition of forking independence,

/| N a^C∩N B ∩ N.

/| lim It follows by deﬁnition that A^C B.

3.3 Properties of the relation

We now show that in the event that the structures in H have simple theories, the relation defined in the last section satisfies many of the properties that forking independence sat- lim isfies in a simple theory. In particular, ^| is preserved under taking algebraic closures and has automorphism-invariance, monotonicity, base monotonicity, transitivity, normality, extension, finite character, (a weakened version of) strong finite character, and symmetry. If we impose some additional conditions on H, we also get anti-reflexivity and an independence theorem over algebraically closed sets.

Lemma 3.3.1. Suppose H satisﬁes conditions 1 through 3, and for each N ∈ H, forking and | lim | lim dividing are equivalent in T h(N ). For sets A, B, and C, A^C B if and only if A^C B. Proof. We break the proof up into three parts.

| lim | lim 1. A^C B if and only if A^C B. /| lim ←: Suppose A^ C B. Then there is some ﬁnite a ∈ A such that for arbitrarily large N ∈ H, a /| N B ∩ N. But a ∈ A, too, so by deﬁnition A /| limB. ^C∩N ^C CHAPTER 3. LIMIT INDEPENDENCE 47

/| lim 0 →: Suppose A ^ C B. Then there is some ﬁnite a ∈ A such that for arbitrarily large N ∈ H, a0 /| N B ∩ N. As noted in Remark 3.2.4, by condition 3 there is ^ C∩N 0 N Nalg ∈ H such that for N ⊇ Nalg, a ∈ acl (A ∩ N). Take a ∈ A ﬁnite such that a0 ∈ aclM(a), and for N ⊇ N , a0 ∈ aclN (a). If N ⊇ N and a0 /| N B ∩ N, alg alg ^ C∩N N then aclN (a) /| N B ∩ N by monotonicity of | . Since forking and dividing are the ^ C∩N ^ same in N , it follows by Corollary 1.4.5 that a /| N B ∩ N. Since this is the case for ^ C∩N /| lim arbitrarily large N , A^C B. | lim | lim 2. A^C B if and only if A^C B. /| lim /| N A^C B ⇔ a ∈ A, a^C∩N B ∩ N for arbitrarily large N /| N M ⇔ a ∈ A, a^C∩N acl (B) ∩ N for arbitrarily large N /| lim ⇔ A^C B.

| lim | lim 3. A^C B if and only if A^C B. The proof is the same as in the previous case. That is, the statement follows from the fact that aclM(aclM(C)) = aclM(C).

Combining these three facts, we have:

| lim | lim A^C B if and only if A^C B | lim if and only if A^C B | lim if and only if A^C B.

Theorem 3.3.2. Given a κ-saturated, strongly κ-homogeneous model M of T , approximated by H (that is, satisfying conditions 1 through 5, above), where each N ∈ H has a simple lim theory, ^| has the following properties:

| lim 0 0 0 0 | lim 0 • invariance If A^C B and (A ,B ,C ) ≡ (A, B, C), then A ^C0 B .

| lim 0 0 0 | lim 0 • monotonicity If A^C B, A ⊆ A and B ⊆ B, then A ^C B . | lim | lim • base monotonicity If D ⊆ C ⊆ B and A^D B, then A^C B | lim | lim | lim • transitivity If D ⊆ C ⊆ B then A^D C and A^C B if and only if A^D B. (The ‘if’ - known as partial transitivity - follows from monotonicity and base monotonicity.)

| lim | lim • normality If A^C B, then AC^C B. CHAPTER 3. LIMIT INDEPENDENCE 48

| lim 0 0 0 | lim 0 • extension If A^C B and B ⊂ B , then there is A ≡BC A such that A ^C B . /| lim • finite character If A^ C B, then there are finite tuples a ∈ A, b ∈ B such that /| lim a^C b. ∗ /| lim • strong finite character If A^ C B, then there are finite tuples a ∈ A, b ∈ B, c ∈ C, and a formula ψ(x, y, z) without parameters such that

– ψ(a, b, c), and 0 /| lim 0 – a ^C b for all a ψ(x, b, c). | lim | lim • symmetry A^C B if and only if B^C A.

N Proof. Since T h(N ) is simple for each N ∈ H, ^| is dividing independence in N . In lim the following, we will frequently refer to “Na, as in the deﬁnition of ^| .” Recall that if | lim A^C B, then for each ﬁnite a ∈ A there is Na such that for N ⊇ Na,

| N a^C∩N B ∩ N.

This is the Na to which we are referring. | lim 0 0 0 Invariance. Suppose A^C B and (A, B, C) ≡ (A ,B ,C ). We would like to show that 0 | lim 0 0 0 0 A ^C0 B . Let σ be an automorphism sending A to A , B to B , and C to C . For any a ∈ A, lim let Na be as in the deﬁnition of ^| . By condition 2 on H, σ(N ) ∈ H for any N ∈ H, so we may let Nσ(a) = σ(Na). We claim that Nσ(a) is as required. 0 0 Write a for σ(a), and let N ⊇ Na0 . We must show that

N 0 0 | 0 0 a ^C0∩N 0 B ∩ N .

If not, there are b0 ∈ B0 ∩ N 0, c0 ∈ C0 ∩ N 0 and ψ(x, y, z) such that

ψ(x, b0, c0) ∈ tpN 0 (a0/(B0 ∪ C0) ∩ N 0)

and ψ(x, b0, c0) divides over C0 ∩ N 0 in N 0. Hence, the following type is consistent with T h(N 0) (for some k < ω):

0 0 0 0 ^ {ϕ(yi, zi) ↔ ϕ(b , c ): i < ω, ϕ ∈ L(C ∩ N } ∪ {¬∃x ψ(x, yi, zi): S ⊂ ω, |S| = k}. i∈S

It follows that the image under σ−1 of this type is consistent with T h(N ): ^ {ϕ(yi, zi) ↔ ϕ(b, c): i < ω, ϕ ∈ L(C ∩ N} ∪ {¬∃x ψ(x, yi, zi): S ⊂ ω, |S| = k} i∈S CHAPTER 3. LIMIT INDEPENDENCE 49

(where σ(b) = b0, σ(c) = c0, and σ(N ) = N 0). This implies that ψ(x, b, c) k-divides over C ∩ N in N . Since ψ(x, b0, c0) ∈ tpN 0 (a0/(B0 ∪ C0) ∩ N 0), ψ(x, b, c) ∈ tpN (a/(B ∪ C) ∩ N), and we have /| N a^C∩N B ∩ N. 0 Since N ⊇ Na0 , N ⊇ Na, and the above is a contradiction. | lim Monotonicity. Suppose A0 ⊆ A, B0 ⊆ B, and A^C B. If a ∈ A0, then a ∈ A, and so there is Na such that for N ⊇ Na,

| N a^C∩N B ∩ N.

By monotonicity of non-forking independence in N , and since B0 ∩ N ⊆ B ∩ N,

| N a^C∩N B0 ∩ N.

| lim So the same Na’s witness that A0^C B0. | lim | lim Base monotonicity. Suppose D ⊆ C, and A^D B. We must show that A^C B. If not, then by condition 5 on H, there are a ∈ A, b ∈ B, c ∈ C and ψ such that M ψ(a, b, c) and ψ(x, b, c) eventually forks over C. That is, there is some Nf ∈ H such that for N ⊇ Nf , ψ(x, b, c) forks over C ∩ N in N . Since M ψ(a, b, c), there is some Ntv ∈ H such that for N N ⊇ Ntv, N ψ(a, b, c). So for N ⊇ Nf ∪ Ntv, tp (a/bc) forks over C ∩ N in N . It follows that tpN (a/(B ∪ C) ∩ N) does as well, that is,

/| N a^C∩N B ∩ N.

DC | N for N ⊇ Na , a^D∩N C ∩ N, and CB | N for N ⊇ Na , a^C∩N B ∩ N.

DB DC CB DB Let Na be some element of H containing Na ∪ Na . For N ⊇ Na we have, by N transitivity of ^| , | N a^D∩N B ∩ N. | lim It follows that A^D B, as desired. CHAPTER 3. LIMIT INDEPENDENCE 50

| lim | lim Normality. Suppose A^C B. We must show that AC^C B. If not, then by condition 0 0 5 on H, there are ψ(xy; zw), a ∈ A, c ∈ C, b ∈ B, and c ∈ C such that M ψ(ac; bc ) and 0 0 ψ(xy; bc ) eventually forks over C in H. Take Nf such that for N ⊇ Nf , ψ(xy; bc ) forks over 0 C ∩ N in N . Take Ntv such that for N ⊇ Ntv, N ψ(ac; bc ). For N ⊇ Nf ∪ Ntv, then, tpN (ac/(B ∪ C) ∩ N) forks over C ∩ N in N , that is, /| N ac^C∩N B ∩ N.

N N By monotonicity of | , {a} ∪ (C ∩ N) /| N B ∩ N. By normality of | , a /| N B ∩ N. ^ ^ C∩N ^ ^ C∩N lim For N ⊇ Na (where Na is as in the definition of ^| ), this is a contradiction. Extension. We first show that the extension property for types holds, that is: Claim 3.3.3. Given a partial type π(x) over B that eventually does not fork over C ⊆ B, there is a complete type p ∈ S(B) extending π such that p eventually does not fork over C. Proof of Claim 3.3.3. Subclaim. If π(x) is a partial type that eventually does not fork over C, and ψ(x, b) is a formula, then either π(x) ∪ {ψ(x, b)} eventually does not fork over C, or π(x) ∪ {¬ψ(x, b)} eventually does not fork over C. Proof of subclaim. Suppose both π(x) ∪ {ψ(x, b)} and π(x) ∪ {¬ψ(x, b)} eventually fork over C. Then by the definition of eventual forking for types, there are finite subtypes π0 and π1 of π such that π0(x) ∪ {ψ(x, b)} and π1(x) ∪ {¬ψ(x, b)} eventually fork over C. For i ∈ {0, 1}, let Ni be such that for Ni ⊆ N ∈ H, πi ∪ {ψi(x, b)} forks over C ∩ N in N (where ψ0 = ψ and ψ1 = ¬ψ). For N0 ∪ N1 ⊆ N ∈ H, both of these finite types fork over C ∩ N in N , and we have _ π0(x) ∪ {ψ(x, b)} `N θi(x)

i∈IN and _ π1(x) ∪ {¬ψ(x, b)} `N ηi(x)

i∈IN where each θi and ηi divides over C ∩ N in N (and the particular formulae may depend on N ). It follows that _ π0(x) `N ψ(x, b) → θi(x)

i∈IN _ π1(x) `N ¬ψ(x, b) → ηi(x)

i∈IN

Since π0(x) ∪ π1(x) `N ψ(x, b) ∨ ¬ψ(x, b) (trivially), we have _ _ π0(x) ∪ π1(x) `N θi(x) ∨ ηi(x)

i∈IN i∈IN CHAPTER 3. LIMIT INDEPENDENCE 51

But then π(x) ` W θ (x) ∨ W η (x), and π(x) forks over C ∩ N in N . Since this N i∈IN i i∈IN i happens in all large enough N (with the witnessing disjunctions of dividing formulae possibly changing), π eventually forks over C in H, contrary to our hypothesis. To ﬁnish the proof of the claim, we enumerate the formulae with parameters from B: {ψi(x): ψi(x) ∈ L(B)}. We construct a consistent (in M), eventually non-forking completion of π(x) by induction: • π0(x) = π(x) (This is consistent and eventually non-forking by hypothesis.)

λ S β • For limit cardinals λ < |B|, let π := β<λ π . (Since consistency and eventual non- forking are local properties, and all the πβ’s are consistent and eventually non-forking, so is πλ.)

α • Given a consistent (in M) eventually non-forking type π , consider ψα: by the sub- α α claim, at least one of π ∪ {ψα}, π ∪ {¬ψα} is consistent and eventually non-forking. (The consistency in M follows from the eventual non-forking.) If the former is, let α+1 α α+1 α π := π ∪ {ψα}. Otherwise, let π := π ∪ {¬ψα}.

S α • Let p(x) := α<|B| π . By construction, p is consistent, complete over B, extends π(x), and eventually does not fork over C.

| lim M By Lemma 3.2.5, A^C B implies that p(X) = tp (A/B ∪ C) eventually does not fork over C. Since B ⊂ B0, B ∪ C ⊂ B0 ∪ C, and by Claim 3.3.3, there is an extension q(X) ∈ SM(B0 ∪ C) of p(X) that eventually does not fork over C. 0 M 0 Let A q(X). Since q(X) = tp (A /B0 ∪ C) eventually does not fork over C, we apply Lemma 3.2.5 again to get 0 | lim 0 A ^C B . 0 0 Lastly, since q(X) ⊃ p(X), A ≡B∪C A, and so A ≡BC A. Finite character. We shall show that the following, slightly stronger statement: Claim 3.3.4. If A /| limB, there are a ∈ A, b ∈ B such that for large enough N , a /| N b. ^C ^C∩N (This is slightly stronger than ﬁnite character because we specify that in each N , the forking actually comes from b, rather than b ∩ N.) /| lim 0 0 Proof of Claim. Suppose A^ C B, and let a ∈ A, b ∈ B, c ∈ C, and ψ be as in condition 5 on H. Let ϕ(x, y) ∈ L, n < ω, and b ∈ B be such that:

=n 0 M (∃ x)ϕ(x, b) ∧ ϕ(b , b).

Further, let Nalg be the element of H guaranteed by condition 3 such that

=n 0 for N ⊇ Nalg, N (∃ x)ϕ(x, b) ∧ ϕ(b , b). CHAPTER 3. LIMIT INDEPENDENCE 52

0 N That is, for N ⊇ Nalg, b ∈ acl (b). Next, take Ntv such that for N ⊇ Ntv,

0 0 N ψ(a, b , c ),

0 0 and Nf such that for N ⊇ Nf , ψ(x, b , c ) forks over C ∩ N in N . Then for N ⊇ Ntv ∪ Nf , tpN (a/b0c0) forks over C ∩ N in N , i.e.

/| N 0 a^C∩N b .

If N ⊇ Ntv ∪ Nf ∪ Nalg, monotonicity implies that

/| N N a^C∩N acl (b). By Corollary 1.4.9, /| N a^C∩N b.

∗ /| lim Strong finite character . Suppose A^ C B. Then by condition 5 on H, there are a ∈ A, b ∈ B, c ∈ C, ψ(x, y, z) without parameters, and Nf such that M ψ(a, b, c) and ψ(x, b, c) forks over C ∩ N in N for all N ⊇ Nf . The first part of strong finite character 0 is clearly satisfied by this choice of ψ, a, b, and c. Now suppose that M ψ(a , b, c). By 0 condition 3 on H, there is Ntv such that for N ⊇ Ntv, N ψ(a , b, c). Let N0 be some N 0 element of H containing Nf ∪ Ntv. For N ⊇ N0, tp (a /bc) forks over C ∩ N in N , i.e.

0 /| N a ^C∩N b.

0 /| lim By definition, a ^C b, as desired. /| lim Symmetry. Suppose B^ C A. By Claim 3.3.4, there are b ∈ B, a ∈ A, and Nb such that for all Nb ⊆ N ∈ H, /| N b^C∩N a. N N By symmetry of | , a /| N b, and hence by monotonicity of | , a /| N B ∩ N. By ^ ^ C∩N ^ ^ C∩N /| lim definition, A^C B. Remark 3.3.5. 1. Our ‘strong finite character∗’ is somewhat weaker than standard strong finite character, as we must look in B and C for our parameters, rather than B and C.

2. It is useful to identify where each of the hypotheses on H is used. The covering condition is used throughout. As for the others:

• The proofs of transitivity and symmetry use the full strength of simplicity of each T h(N ). • The automorphism invariance condition is used in the proof of invariance. CHAPTER 3. LIMIT INDEPENDENCE 53

• The proofs of base monotonicity, extension, and strong finite character all depend on Lemma 3.2.5, and hence on the convergence of truth value, stabilization of non-forking, and strong finite character hypotheses. • The proofs of normality and finite character use the convergence of truth value and strong finite character hypotheses. The proof of normality also uses the N N normality of each ^| , and the proof of finite character uses monotonicity of ^| and Corollary 1.4.9. These are all properties of non-forking independence in any theory. • The proof of monotonicity relies only on the monotonicity of non-forking, which is true in any theory. • The extension for types claim (Claim 3.3.3) uses only the forking stabilization hypothesis.

lim Proposition 3.3.6. If M T and H satisfies conditions 1 and 6, then ^| satisfies anti-reflexivity: | lim a^B a implies a ∈ B. | lim | lim Proof. Suppose a^B a. Let Na be as in the definition of ^ . For N ⊇ Na, | N a^B∩N ({a} ∪ B) ∩ N.

| N | N | N N By monotonicity of ^ , a^B∩N a. By anti-reﬂexivity of ^ , a ∈ acl (B ∩ N). By Remark 3.2.4, Condition 6 on H implies that a ∈ aclM(B) = B, as desired. Remark 3.3.7. In Proposition 3.3.6, we do not need to require that the T h(N )’s are simple, since anti-reﬂexivity of forking independence holds in all theories.

Proposition 3.3.8. Suppose M T is κ-saturated and strongly κ-homogeneous, H satisfies conditions 1 through 7 (that is, H approximates M and satisfies the additional conditions of “stabilization of algebraic closure” and “homogeneity”), and the elements of H have theories f that are simple and for which ^| satisfies the Independence Theorem over algebraically closed lim sets. Then ^| satisfies the Independence Theorem over algebraically closed sets. That is, if | lim M M | lim | lim A = A, BÂ C, and d and c are such that tp (d/A) = tp (e/A), dÂ B, and eÂ C, M M | lim then there is a tp (d/AB) ∪ tp (e/AC) with aÂ BC. Proof. Suppose not. Then either tpM(d/AB) ∪ tpM(e/AC) is inconsistent (in which case it is eventually inconsistent, and hence trivially eventually forks, in H), or for each realization /| lim a, aÂ BC. By Lemma 3.2.5 and the fact that A = A, /| lim M aÂ BC ↔ tp (a/A ∪ BC) eventually forks over A ↔ tpM(a/A ∪ BC) eventually forks over A. CHAPTER 3. LIMIT INDEPENDENCE 54

It follows (since this is true for all realizations a), that tpM(d/AB) ∪ tpM(e/AC) does not have an eventually non-forking (over A) extension to SM(A ∪ BC). (Note that

tpM(d/AB) ∪ tpM(e/AC)

is a partial type over A ∪ BC, since (A ∪ B) ∪ (A ∪ C) = A ∪ B ∪ C ⊆ A ∪ BC.) By extension for eventually non-forking types (Claim 3.3.3, above), tpM(d/AB) ∪ tpM(e/AC) itself eventually forks over A. By the definition of eventual non-forking, there is a finite subtype π ⊂ tpM(d/AB)∪tpM(e/AC) that eventually forks over A. The subtype π is of the M M form π1(x, a1, b) ∪ π2(x, a2, c), where π1 ⊂ tp (d/AB), π2 ⊂ tp (e/AC), b ∈ B, and c ∈ C. V V Let ψ1(x, a1, b) = π1 and ψ2(x, a2, c) = π2. Then ψ1(x, a1, b) ∧ ψ2(x, a2, c) eventually forks over A. By the definitions and hypotheses above, we are guaranteed the following elements of H:

•N f , such that for N ⊇ Nf , ψ1 ∧ ψ2 forks over A ∩ N in N ;

•N ψ1,d, such that for N ⊇ Nψ1,d, N ψ1(d, a1, b);

N •N alg, such that for N ⊇ Nalg, acl (A ∩ N) = A ∩ N.

Let N0 be some element of H containing

Nf ∪ Nψ1,d ∪ Nψ2,e ∪ Nb,c,A ∪ Nd,b,A ∪ Ne,c,A ∪ Nalg.

Note also that by Condition 7 on H and the fact that tpM(d/A) = tpM(e/A), we have that for each N ∈ H containing {d, e}, tpN (d/A ∩ N) = tpN (e/A ∩ N). Suppose N ⊇ N0. Then we have: • A ∩ N = aclN (A ∩ N),

| N • bÂ∩N c, | N • dÂ∩N b, | N • eÂ∩N c, and • tpN (d/A ∩ N) = tpN (e/A ∩ N) CHAPTER 3. LIMIT INDEPENDENCE 55

N By our hypothesis that ^| satisﬁes the Independence Theorem over algebraically closed N N sets, tp (d/(A∩N)b)∪tp (e/(A∩N)c) does not fork over A∩N. But since N ⊃ Nψ1,d∪Nψ2,d,

N N tp (d/(A ∩ N)b) ∪ tp (e/(A ∩ N)c) ` ψ1(x, a1, b) ∧ ψ2(x, a2, b),

which, since N ⊇ Nf , forks over A ∩ N in N - a contradiction.

3.4 Example

Granger’s Proposition 3.1.11 shows that Γ-non-forking in M T∞ is equivalent to eventual non-forking in the finite dimensional substructures of M. Above, we showed that under certain assumptions (in particular, the strong finite character assumption), failure of limit independence corresponds to eventual forking of a type. Unfortunately, the strong finite lim character condition does not always hold for ^| in models of T∞ (with H taken to be the collection of finite dimensional substructures). When we restrict to single elements on the lim left and finite dimensional sets on the right and in the base, however, ^| does satisfy our Γ assumptions, and is related to ^| . In the following, let M be a κ-saturated, strongly κ-homogeneous model of T∞ (for some large κ), and H the collection of finite dimensional KM-subspaces of M (that is, N M N the substructures N of M such that K = K and dimK (V ) < ∞). We shall break our convention of referring to tuples with no overline, and so shall revert to writing out “aclM(A)” rather than “A”. We say that a set A is “finite dimensional” if hAi is finite dimensional. We begin by reviewing some of Granger’s results on the relationship between a model of T∞ and its finite dimensional subspaces. Recall that Lθ is the language that results from adding the “linear independence” predicates θn to the two-sorted vector space language, and ˜ K˜ that if K is an algebraically closed field, m ∈ N ∪ {∞}, and F ∈ {S,A}, the theory F Tm has elimination of quantifiers in Lθ. It turns out that there is some local uniformity to this quantifier elimination.

Lemma 3.4.1 ([9], Proposition 9.3.3). 1. For any L-sentence, σ, there is R ∈ ω such that T∞ σ if and only if Tm σ for all m ≥ R.

2. Given any L-formula ψ, there is a quantifier-free Lθ-formula ϕ and some R < ω such that Tm ϕ ↔ ψ for all m ≥ R. (In particular, T∞ ϕ ↔ ψ.) 3. Let F ∈ {S,A} and K be a field with square roots. Every L-formula is equivalent, K modulo F T , to a boolean combination of quantifier-free Lθ-formulae and sentences of the form Θn := ∃X1,...,Xnθn(X1,...,Xn). CHAPTER 3. LIMIT INDEPENDENCE 56

Remark 3.4.2. Given M T∞, the ﬁnite dimensional subspaces of M are in fact Lθ- N substructures of M: for N ∈ H and a1, . . . , an ∈ V ,

N θn(a1, . . . , an) if and only if M θn(a1, . . . , an).

0 Remark 3.4.3. For N 0 ⊆ N ∈ H and A ⊂ N 0, dclN (A) = dclN (A) = dclM(A), and the same quantiﬁer-free Lθ-formulae can be used to deﬁne the elements of dcl(A) in each of these structures. Proof. This follows from the proof of Granger’s Lemma 9.2.6 [9].

Lemma 3.4.4. If A ⊂ N and dim(hAi) < dim(V N ) − 1, aclM(A) ⊆ N. In fact, aclM(A) = aclN (A).

M alg Proof. By Remark 3.1.5, acl (A) = (span alg (AV ),KA ), that is, the ﬁeld-theoretic alge- KA alg N M braic closure of KA and the KA -span of the vector elements of A. Since K = K , alg N N N alg KA ⊆ K . Since AV ⊆ V , span alg (AV ) ⊆ hAi ⊆ V . Further, as noted above, KA is KA alg the same whether computed in M or in N , and so the KA -span of AV is the same in M and in N . Remark 3.4.5. If A is ﬁnite dimensional, then for large enough N , A ⊆ N, and by Lemma 3.4.4, aclN (A) = aclM(A).

Lemma 3.4.6. Suppose N ∈ H, C ⊆ B ⊂ N, a ∈ N, and hBi= 6 V N . Then tpN (a/B) does not Γ-fork over C (in N ) if and only if tpM(a/B) does not Γ-fork over C (in M).

Proof. We begin by considering the case where lg(a) = 1. If tpN (a/B) does not Γ-fork over KN C then, since hBi 6= V N , it follows that K | K and one of the following conditions Ca^KC B holds:

1. a ∈ KN = KM.

2. a ∈ hCi.

3. a∈ / hBi and whenever {b1, . . . , bn} ⊂ BV \ (hCi ∩ B) is linearly independent modulo hCi, {[a, b1],..., [a, bn]} is algebraically independent over KB(KCa).

KN KM First, note that K | K if and only if K | K , since KN = KM. Secondly, note Ca^KC B Ca^KC B that each of the three conditions above is true in N if and only if it is true in M. It follows that tpM(a/B) does not Γ-fork over C. The argument in the other direction is exactly the same. Now suppose that the result holds for tuples of length n, and lg(aa) = n + 1. By the inductive deﬁnition of Γ-forking, tpN (aa/B) does not Γ-fork over C if and only if tpN (a/B) does not Γ-fork over C and tpN (a/Ba) does not Γ-fork over Ca. By the inductive hypothesis CHAPTER 3. LIMIT INDEPENDENCE 57 and the argument above, this is the case if and only if tpM(a/B) does not Γ-fork over C and tpM(a/Ba) does not Γ-fork over Ca, that is, if and only if tpM(aa/B) does not Γ-fork over C. Remark 3.4.7. Working in M or any N ∈ H, suppose C ⊂ B are ﬁnite dimensional sets.

0 0 1. a ∈ hBi \ hCi just in case the following formula is in tp(a/B) (where {b1, . . . , b`} is a 0 0 basis for hBi, and {c1, . . . , ck} is a basis for hCi):

0 0 0 0 0 0 ϕ2(X, b , c ) := p¬θ`+1(b1, . . . , b`,X) ∧ θk+1(c1, . . . , ck,X)q

2. Suppose there are b1, . . . , bn from BV , linearly independent modulo hCi, such that {[a, b1],..., [a, bn]} is algebraically dependent over KB(KCa). Then there is a polyno- 0 ∗ mial q (x1, . . . , xn) with coeﬃcients d from KB(KCa) such that

0 q ([a, b1],..., [a, bn]) = 0.

∗ 0 ∗ ∗ Let q be such that q(x1, . . . , xn, d ) = q (x1, . . . , xn). Since d ∈ KB(KCa), each di fi(z,w) is the result of a rational function (say, ) applied to elements of KB and KCa, gi(z,w) ∗ ∗ ∗ say, b and c . Each bi is defined over B, say by χi(zi, d, c) (where d ∈ B \ C and ∗ c ∈ C), and each ci is defined over Ca, say by ξi(wi, c, a). We may assume χi and ξi 0 0 are quantifier-free. As above, let {c1, . . . , ck} be a basis for hCi. It should be clear, then, that the following formula is in tp(a/B) (where we have abbreviated (b1, . . . , bn) by b).

r s 0 ^ =1 ^ =1 ϕ3(X; c, d, b, c ) := p ∃ zi(χi(zi, d, c)) ∧ ∃ wi(ξi(wi, c, X)) i=1 i=1 r s ^ ^ ∧ ∃y1, . . . , yt∃z1, . . . , zr∃w1, . . . , ws χi(zi, d, c) ∧ ξi(wi, c, X) i=1 i=1 t ^ fi(z, w) ∧ y = ∧ q([X, b ],..., [X, b ], y , . . . , y ) = 0 i g (z, w) 1 n 1 t i=1 i 0 0 ∧ θm+n(c1, . . . , ck, b1 . . . , bn)q.

0 0 Now suppose a ϕ3(X; d, c, b, c ). Witnesses to the zi are field elements defined over B - that is, elements of KB. Witnesses to the wi are field elements defined 0 over Ca - that is, elements of KCa0 . Witnesses to the yi are field elements obtained by applying rational functions to the zi and wi - that is, elements of KB(KCa0 ). It 0 0 follows that ([a , b1],..., [a , bn]) satisfies a polynomial with the yi as coefficients - that 0 0 is, {[a , b1],..., [a , bn]} is algebraic over KB(KCa0 ). Further, by the last part of the 0 /| Γ formula, {b1, . . . , bn} is linearly independent modulo hCi, and so a ^C B. CHAPTER 3. LIMIT INDEPENDENCE 58

The next lemma is a version of Granger’s Proposition 3.1.11. We restrict its scope by considering only 1-types over ﬁnite dimensional sets, and alter it to ﬁt into the “directed system” framework of our approximating substructures.

Lemma 3.4.8. Suppose B is ﬁnite dimensional, C ⊂ B, and p(x) ∈ SM(B). Then the following are equivalent:

1. p(x) does not Γ-fork over C (in M).

2. For all ϕ(x, b) ∈ p there is Nϕ ∈ H such that for N ⊇ Nϕ, ϕ(x, b) does not fork over C ∩ N in N .

3. Let π(x) = {χ(x): χ(x) ∈ p(x) and χ(x) is an Lθ quantiﬁer-free formula}. There is Nπ ∈ H such that for N ⊇ Nπ, π (B ∩ N) does not fork over C ∩ N in N .

Proof. 2 → 1: Suppose that p(x) does Γ-fork over C, and let M p(a). One of the following holds: /| 1. KCa^KC KB, 2. a ∈ hBi \ hCi, or

3. a∈ / hBi and there is {b1, . . . , bn} ⊆ BV linearly independent modulo hCi and {[a, b1],..., [a, bn]} is algebraically independent over KB(KCa).

KM M M N In the first case, tp (KCa/KB) forks over KC , in the structure K . Since K = K for all N ∈ H, this dependence holds in all N containing B ∪{a}. Let N 0 contain B ∪{a}. Suppose 0 KN N 0 ψ(x, d) ∈ tp (KCa/KB) divides over KC . By the stable embeddedness of K , ψ(x, d) 0 N 0 divides over KC in N , as well. It is realized by some α from KCa. Since α ∈ dcl (Ca), there is a quantifier-free Lθ-formula χ(x, c, a) defining α. By Remark 3.4.3, χ(x, c, a) does 0 not depend on N . Similarly, there is ξ(y, b) Lθ quantifier-free defining d. The following 0 formula divides over KC in N :

=1 =1 ϕ1(X; b, c) := p(∃ x)χ(x, c, X) ∧ (∃ y)ξ(y, b) ∧ ∃xy(χ(x, c, X) ∧ ξ(y, b) ∧ ψ(x, y)).q

If (di : i ∈ I) is a KC -indiscernible sequence (with d0 = d) witnessing the division of ψ(x, d), and for each i, σi is a KC -automorphism sending d to di, then the sequence (bi : i ∈ I) = (σi(b): i ∈ I) is a KC -indiscernible sequence witnessing the division of ϕ1(X; b, c). Furthermore, since the di lie outside of KC , we can extract a C-indiscernible sequence from (di : i ∈ I), and hence from (bi : i ∈ I). We see that ϕ1(X; b, c) divides over C 0 0 in N . Since none of χ(x, c, X), ξ(y, b), or ψ(x, y) depended on N , ϕ1(X; b, c) divides over 0 C in all N ⊇ N . It is clear that ϕ1(X; b, c) ∈ p(x). 0 0 In the second case, we recall that B (and thus, C) are finite dimensional. Let {c1, . . . , ck} 0 0 0 0 be a basis for hCi, and {b1, . . . , b`} a basis for hBi. By Remark 3.4.7, ϕ2(X; b , c ) ∈ p(x). 0 0 0 0 0 0 Let N ∈ H contain B, and suppose N ϕ2(a ; b , c ). Then a ∈ hBi \ hCi, and by CHAPTER 3. LIMIT INDEPENDENCE 59 definition tpN 0 (a0/B) Γ-forks over C in N 0. But since Γ-forking is the same as dividing in N 0, tpN 0 (a0/B) divides over C in N 0. We chose a0 to be an arbitrary realization of 0 0 0 0 0 0 ϕ2(X; b , c ), so ϕ2(X; b , c ) divides over C in N (and in any N ⊇ N ). In the third case, we refer again to Remark 3.4.7, and see that for some quantifier- free Lθ-formulae χi(zi, b, c) and ξi(wi, c, X), and some (parameter-free) polynomials fi(z, w), 0 0 0 0 0 gi(z, w), and q(x, y), the formula ϕ3(X; c, d, b, c ) ∈ p(x). If N ⊃ B and N ϕ3(a ; c, b, c ), then (as noted in 3.4.7), tpN 0 (a0/B) Γ-forks (and so divides) over C in N 0. It follows that 0 0 0 ϕ3(X; c, b, c ) divides over C in N , and for all N ⊇ N . This finishes our proof that (2) implies (1), since each possible way p(x) could Γ-fork gives rise to a formula from p(x) that eventually forks in H. 3 → 2: Using the fact that there are Nqf ∈ H and ψ(x, b) a quantifier-free Lθ-formula such that for N ⊇ Nqf , N ∀x(ϕ(x, b) ↔ ψ(x, b)), this implication is clear. 1 → 3: Suppose that p(x) does not Γ-fork over C in M, and let N 0 ∈ H contain B ∪ {a}. N 0 0 By Remark 3.4.2, N π(a). By Lemma 3.4.6, tp (a/B) does not Γ-fork over C (in N ), and hence does not fork over C in N 0. It follows that π(x) does not fork over C (= C ∩ N) in N 0, or in any N ⊇ N 0.

Lemma 3.4.9. If C is ﬁnite dimensional, and ϕ(x, b) is a formula, then either there is Nf such that C ∪ {b} ⊂ Nf and for N ⊇ Nf , ϕ(x, b) forks over C in N , or there is Nnf such that C ∪ {b} ⊂ Nnf and for N ⊇ Nnf , ϕ(x, b) does not fork over C in N . (That is, every formula either eventually forks over C or eventually does not fork over C.)

Proof. Let ϕ(x, b) and C be as in the statement of the lemma. If ϕ(x, b) is not realized in M, then it is eventually not realized in H (for large enough N , N ¬∃xϕ(x, b)), and hence trivially eventually forks over C. Otherwise, we may consider whether or not ϕ(x, b) Γ-forks over C in M: if ϕ(x, b) has a realization whose type over Cb does not Γ-fork over C, then we say ϕ(x, b) does not Γ-fork over C. Otherwise, it does Γ-fork over C. If ϕ(x, b) does not Γ-fork over C, then it belongs to a complete type that does not Γ-fork over C. By Lemma 3.4.8, ϕ(x, b) eventually does not fork over C. It remains to show that if for every a ∈ ϕ(M, b), tpM(a/Cb) Γ-forks over C, then ϕ(x, b) eventually forks over C. Consider N 0 ∈ H such that C ∪ {b} ⊂ N 0, hC ∪ {b}i= 6 V N 0 , and if ψ(x, b) is a quantiﬁer- 0 free Lθ-formula such that M ϕ(x, b) ↔ ψ(x, b), then N ϕ(x, b) ↔ ψ(x, b). (Such an 0 0 N exists by ﬁnite dimensionality of C and Lemma 3.4.1.) Suppose N ϕ(a, b). Then, 0 0 since N is an Lθ-substructure of M (see Remark 3.4.2) and N ψ(a, b), M ψ(a, b), and M N 0 so M ϕ(a, b). By assumption, tp (a/Cb) Γ-forks over C. By Lemma 3.4.6, tp (a/Cb) Γ-forks over C in N 0. It follows that every realization of ϕ(x, b) in N Γ-forks - and hence, forks - over C ∩ N = C in N , so ϕ(x, b) forks over C ∩ N in N . This holds in all N ⊇ N 0, so ϕ(x, b) eventually forks over C, as desired.

Remark 3.4.10. Let M be a κ-saturated, strongly κ-homogeneous model of T∞ for some large κ. We conjecture that one can find an element a and sets B, C with dim(hBC \ Ci) infinite /| lim | Γ M M such that a^C B, but aâclM(C) acl (B) acl (C) (so in particular, by Lemma 3.4.8, there CHAPTER 3. LIMIT INDEPENDENCE 60 is no eventually forking formula in tpM(a/ aclM(B) aclM(C))). The idea would be to find a,

C, and B = {bi : i < ω} such that for arbitrarily large N ∈ H, there are n and bi0 , . . . , bin−1 M with {bij : j < n} linearly independent modulo hacl (C) ∩ Ni and {[a, bi0 ],..., [a, bin−1 ]} algebraically dependent over K(aclM(C)∪aclM(B))∩N (K(aclM(C)∪{a})∩N ), yet there are no n and M bi0 , . . . , bin−1 ∈ B with {bij : j < n} linearly independent modulo hacl (C)i and

{[a, bi0 ],..., [a, bin−1 ]} algebraically dependent over KaclM(C) aclM(B)(KaclM(C)a). In other words, the third condition of Γ-non-forking would hold in arbitrarily large N , but be witnessed by diﬀerent tuples from B each time.

alg Lemma 3.4.11. For any set C and any N ∈ H, KaclM(C)∩N = KC = KaclM(C).

alg Proof. It is clear from the deﬁnitions that KC ⊆ KaclM(C)∩N ⊆ KaclM(C). It remains to alg show that KaclM(C) ⊆ KC , for which it suﬃces to show that the generators of KaclM(C) are alg elements of KC . The generators of KaclM(C) are:

M alg • (acl (C))K = KC M •{ [a, b]: a, b ∈ (acl (C))V = span alg (CV )} KC

M • for each n, the sets {α1, . . . , αn} such that there are b1, . . . , bn, b ∈ (acl (C))V with n b1, . . . , bn linearly independent and b = Σi=1αibi.

M alg As noted above, the set of ﬁeld elements from acl (C) is exactly KC . We must show: alg 1. if a, b ∈ span alg (CV ), [a, b] ∈ KC ; KC n 2. if b1, . . . , bn, b ∈ span alg (CV ) with b1, . . . , bn linearly independent and b = Σi=1αibi, KC alg then α1, . . . , αn ∈ KC . alg Claim 3.4.12. If a, b ∈ span alg (CV ), [a, b] ∈ KC . KC

Proof of Claim. Suppose a, b ∈ span alg (CV ). Then, for some m, there are c1, . . . , cm ∈ CV KC alg and β1, . . . , βm, γ1, . . . , γm ∈ KC such that

m a = Σi=1βici m b = Σi=1γici

Then we have:

[a, b] = [β1c1 + ... + βmcm, γ1c1 + ... + γmcm] m m = Σi=1Σj=1[βici, γjcj] m m = Σi=1Σj=1βiγj[ci, cj]. CHAPTER 3. LIMIT INDEPENDENCE 61

Since {c1, . . . , cm} ⊆ CV , for any 1 ≤ i, j ≤ m,[ci, cj] ∈ KC . Since

alg {β1, . . . , βm, γ1, . . . , γm} ⊆ KC

alg m m alg and KC is a ﬁeld, Σi=1Σj=1βiγj[ci, cj] ∈ KC , as desired.

Claim 3.4.13. If b1, . . . , bn, b ∈ span alg (CV ), {b1, . . . , bn} is linearly independent, and KC n alg b = Σi=1αibi, then α1, . . . , αn ∈ KC .

Proof of Claim. Let b1, . . . , bn, b and α1, . . . , αn be as above. Since b1, . . . , bn, b ∈ span alg (CV ), KC alg there are c1, . . . , cm ∈ CV and {βi,j : 1 ≤ i ≤ n, 1 ≤ j ≤ m} ∪ {γi : 1 ≤ i ≤ m} ⊆ KC such that for 1 ≤ i ≤ n, bi = βi,1c1 + ... + βi,mcm and b = γ1c1 + ... + γmcm.

Remark 3.4.14. We may assume {c1, . . . , cm} is linearly independent.

Proof of Remark. Reordering if necessary, let {c1, . . . , ck} be a maximal linearly independent subset of {c1, . . . , cm}. Suppose k < m. Then for 1 ≤ i ≤ m − k and 1 ≤ j < k, there M are λi,j ∈ K such that ck+i = λi,1c1 + ... + λi,kck. By the deﬁnition of KC (and the fact that c1, . . . , ck are linearly independent vectors from C), each λi,j is an element of KC , and alg alg hence of KC . It follows that each bi (and b) can be written as a KC -linear combination of c1, . . . , ck.

We may now write the original linear equation in terms of the ci’s:

b = α1b1 + ... + αnbn m m m Σi=1γici = α1 (Σi=1β1,ici) + ... + αn (Σi=1βn,ici) m n n Σi=1γici = Σj=1αjβj,1 c1 + ... + Σj=1αjβj,m cm

By the linear independence of {c1, . . . , cm}, we obtain the following system of equations:

γ1 = α1β1,1 + ... + αnβn,1 . .

γm = α1β1,m + ... + αnβn,m

which we may represent with the following matrix equation: CHAPTER 3. LIMIT INDEPENDENCE 62

      β1,1 β2,1 ··· βn,1 α1 γ1  β β ··· β  α   γ   1,2 2,2 n,2   2  2   . . .. .   .  =  .  .  . . . .   .   .  β1,m β2,m ··· βn,m αn γm th We shall refer to the matrix of βi,j’s as “B”. The i column of B is the coordinate vector for bi in terms of the basis {c1, . . . , cm}. (Note: we are not claiming that {c1, . . . , cm} is a basis of hCi or span alg (CV ), but it is a basis of a vector space containing {b1, . . . , bn}.) KC Since {b1, . . . , bn} is linearly independent, the columns of B are linearly independent, and B has rank n. Since B is an m × n matrix of rank n, B has a left-inverse - call it A. From the previous matrix equation, we obtain     α1 γ1 α   γ   2  2   .  = A  .  .  .   .  αn γm alg alg Since each γi and all the entries of A are from KC , each αi is from KC , as desired. Claims 3.4.12 and 3.4.13 complete the proof of the lemma. | lim Lemma 3.4.15. For M T∞ and H as above, ^ satisfies the strong finite character condition with respect to tuples of length one on the left, finite dimensional sets B on the right, and small sets C in the base. (We are not assuming that C ⊆ B. In particular, C might be infinite dimensional.) That is, if a is an element of M, B a finite dimensional /| lim M subset of M, C a small subset of M, and a^C B, then there are ψ(x, y, z) ∈ L, b ∈ acl (B) M M and c ∈ acl (C) such that M ψ(a, b, c) and ψ(x, b, c) eventually forks over acl (C) in H. /| lim /| N M Proof. If a^ C B, then for arbitrarily large N ∈ H, a^ aclM(C)∩N acl (B) ∩ N, that is, tpN (a/(aclM(C) ∪ aclM(B)) ∩ N) forks over aclM(C) ∩ N in N . Recall that Γ-non-forking is the same as non-forking in the finite dimensional substructures, so in arbitrarily large N ∈ H, tpN (a/(aclM(C) ∪ aclM(B)) ∩ N) Γ-forks over aclM(C) ∩ N. In what follows, we shall restrict our attention to N large enough such that the following hold. •{ a} ∪ B ⊂ N. (This is possible because hBi is finite dimensional.) • If a ∈ hCi, then a ∈ hC∩Ni. (If a ∈ hCi, then a is a linear combination of finitely many vectors from C. For large enough N , those vectors are in CV ∩ N, and a ∈ hC ∩ Ni.) •h (aclM(C) ∪ aclM(B)) ∩ Ni= 6 V N . (Since C is small and B is finite dimensional, haclM(C) aclM(B)i = hCBi= 6 V M. Let v ∈ V M \ hCBi. For N containing v, h(aclM(C) ∪ aclM(B)) ∩ Ni= 6 V N .) CHAPTER 3. LIMIT INDEPENDENCE 63

•h C ∩ Ni ∩ hBi = hCi ∩ hBi. As hBi is finite dimensional, hCi ∩ hBi is, as well. It is 0 0 0 0 generated by finitely many elements from C - say, c1, . . . , cn. Once {c1, . . . , cn} ⊂ N, then, hC ∩ Ni ∩ hBi = hCi ∩ hBi. Consider one such N 0. By the definition of Γ-forking, either

K M 0 /| K M M 0 (acl (C)∪{a})∩N ^KaclM(C)∩N0 (acl (C)∪acl (B))∩N or

• a∈ / K, and • a∈ / haclM(C) ∩ N 0i, and • either a ∈ h(aclM(C) ∪ aclM(B)) ∩ N 0i or there is a linearly independent (mod- M 0 M M 0 ulo hacl (C) ∩ N i) set {b1, . . . , bm} from (acl (C) ∪ acl (B))V ∩ N such that {[a, b1],..., [a, bm]} is algebraically dependent over

K(aclM(C)∪aclM(B))∩N 0 (K(aclM(C)∪{a})∩N 0 ), and • either a∈ / h(aclM(C) ∪ aclM(B)) ∩ N 0i \ haclM(C) ∩ N 0i or

h(aclM(C) ∪ aclM(B)) ∩ N 0i= 6 V N 0

M 0 or there is a linearly independent (modulo hacl (C) ∩ N i) set {b1, . . . , bm} from M M 0 (acl (C) ∪ acl (B))V ∩ N such that {[a, b1],..., [a, bm]} is algebraically dependent over K(aclM(C)∪aclM(B))∩N 0 (K(aclM(C)∪{a})∩N 0 ).

By the third assumption on N 0 (that h(aclM(C) ∪ aclM(B)) ∩ N 0i= 6 V N 0 ), the fourth condition holds trivially. Taking that into account,

/| N 0 M M 0 a^aclM(C)∩N 0 acl (C) acl (B) ∩ N if and only if one of the following holds:

0 /| KN 1. K M 0^ K M M 0 ; (acl (C)∪{a})∩N KaclM(C)∩N0 (acl (C)∪acl (B))∩N 2. a ∈ h(aclM(C) ∪ aclM(B)) ∩ N 0i \ haclM(C) ∩ N 0i; 3. a∈ / h(aclM(C) ∪ aclM(B)) ∩ N 0i and there is a linearly independent (modulo M 0 M M 0 hacl (C) ∩ N i) set {b1, . . . , bm} ⊂ ((acl (C) acl (B)) ∩ N )V such that {[a, b1],..., [a, bm]} is algebraically dependent over

K(aclM(C)∪aclM(B))∩N 0 (K(aclM(C)∪{a})∩N 0 ). CHAPTER 3. LIMIT INDEPENDENCE 64

Case 1: K M 0 /| K M M 0 . (We have removed the su- (acl (C)∪{a})∩N ^ KaclM(C)∩N0 (acl (C)∪acl (B))∩N perscript ‘KN 0 ’ because the field, and hence dependence at the level of the field, is the same alg in all the structures in question.) By Lemma 3.4.11, KaclM(C)∩N 0 = KC , so we may rewrite the dependence relation as /| K(aclM(C)∪{a})∩N 0^ alg K(aclM(C)∪aclM(B))∩N 0 . KC We mimic the argument in the first case of the proof of (2) → (1) in Lemma 3.4.8. We know N 0 alg 0 that tp (K(aclM(C)∪{a})∩N 0 /K(aclM(C)∪aclM(B))∩N 0 ) divides over KC in N . Suppose that 0 ψ(x, d) witnesses this (where d ∈ K(aclM(C)∪aclM(B))∩N 0 ), and suppose N ψ(α, d) (where 0 M M 0 0 M 0 α ∈ K(aclM(C)∪{a})∩N 0 ). Then there are b ∈ (acl (C) ∪ acl (B)) ∩ N , c ∈ acl (C) ∩ N , and quantifier-free Lθ-formulae χ, ξ such that

0 =1 0 =1 0 0 0 N (∃ x)χ(x, c , a) ∧ (∃ y)ξ(y, b ) ∧ χ(α, c , a) ∧ ξ(d, b ). It follows that the formula

0 0 =1 0 =1 0 0 0 ϕ1(X; b , c ) := p(∃ x)χ(x, c ,X) ∧ (∃ y)ξ(y, b ) ∧ ∃xy(χ(x, c ,X) ∧ ξ(y, b ) ∧ ψ(x, y))q

N 0 M M 0 alg alg is in tp (a/(acl (B) ∪ acl (C)) ∩ N ) and divides over KC . As above, given a KC - indiscernible sequence (di : i ∈ I) (with d0 = d) (potentially from a suﬃciently saturated elementary extension of N 0) witnessing the division of ψ(x, d), the corresponding alg 0 0 KC -indiscernible sequence (bi : i ∈ I) (with b0 = b) witnesses division of ϕ1(X; b , c ). 0 Also as above, we can extract a C ∩ N -indiscernible sequence from (bi : i ∈ I), wit- 0 0 0 0 0 0 nessing that ϕ1(X; b , c ) divides over C ∩ N in N . Since none of χ(x, c ,X), ξ(y, b ), 0 0 0 0 0 0 ψ(x, y) depended on N , ϕ1(X; b , c ) divides over C ∩ N in all N ⊇ N and ϕ1(X; b , c ) ∈ tpM(a/ aclM(B) aclM(C)). M M 0 M 0 0 0 Case 2: a ∈ h(acl (C) ∪ acl (B)) ∩ N i \ hacl (C) ∩ N i. Let {b1, . . . , bk} be a basis M M 0 0 0 for h(acl (C) ∪ acl (B)) ∩ N i. Note that hb1, . . . , bki ∩ hCi is a ﬁnite dimensional vector 0 0 space. Let {c1, . . . , c`} ⊂ CV be a basis of this intersection. By our second assumption on N 0, a∈ / hCi, and the following formula is in tpM(a/ aclM(B) aclM(C)).

0 0 0 0 0 0 ϕ2(X, b , c ) := p¬θk+1(b1, . . . , bk,X) ∧ θ`+1(c1, . . . , c`,X)q Any realization of this formula is an element of

h(aclM(C) ∪ aclM(B)) ∩ N 0i \ haclM(C) ∩ N 0i,

0 0 M 0 0 0 and so ϕ2(X, b , c ) divides over acl (C) ∩ N in N . The same holds in all N ⊃ N (since 0 0 0 0 0 0 0 0 {c1, . . . , c`} spans hb1, . . . , bki ∩ hCi, realizations of ϕ2(X, b , c ) in any N containing b c do 0 0 M not belong to hCi), and so ϕ2(X, b , c ) eventually forks over acl (C). Case 3: There is a linearly independent (modulo haclM(C) ∩ N 0i = hC ∩ N 0i) set M M 0 {b1, . . . , bm} ⊂ (acl (C) ∪ acl (B))V ∩ N such that {[a, b1],..., [a, bm]} is algebraically CHAPTER 3. LIMIT INDEPENDENCE 65

dependent over K(aclM(C)∪aclM(B))∩N 0 (K(aclM(C)∪{a})∩N 0 ). By Remark 3.4.7, a formula of the N 0 M M 0 0 0 following form is in tp (a/(acl (C) ∪ acl (B)) ∩ N ) (where {c1, . . . , ck} is a basis for the ﬁnite dimensional space hCi ∩ hBi, d are parameters from ((aclM(B) ∪ aclM(C)) ∩ N 0) \ M 0 M 0 (acl (C) ∩ N ), c are parameters from acl (C) ∩ N , we abbreviate (b1, . . . , bm) by b, χi and ξi are quantiﬁer-free Lθ-formulae, q(x1, . . . , xn, y1, . . . , yt) is a polynomial in (x1, . . . , xn), and fi, gi are polynomials without parameters):

r s 0 ^ =1 ^ =1 ϕ3(X; c, b, c , d) := p (∃ zi)χi(zi, d, c) ∧ (∃ wi)ξi(wi, c, X) i=1 i=1 r s ^ ^ ∧ ∃y1, . . . , yt∃z1, . . . , zr∃w1, . . . , ws χi(zi, b, c) ∧ ξi(wi, c, X) i=1 i=1 t ^ fi(z, w) ∧ y = ∧ q([X, b ],..., [X, b ], y , . . . , y ) = 0 i g (z, w) 1 n 1 t i=1 i m k 0 ∧ ∀x1, . . . , xm+k, ∀Y (Σi=1xibi 6= Y ∨ Σi=1xm+ici 6= Y ∨ Y = 0V )q.

By the assumption that hC ∩ Ni ∩ hBi = hCi ∩ hBi, and the fact that χi(zi, d, c) and 0 ξi(wi, c, a) deﬁne the same elements in N as in any larger N or in M,

0 N M M ϕ3(X; c, b, c , d) ∈ tp (a/(acl (B) ∪ acl (C)) ∩ N)

0 0 M M M 0 for N ⊇ N and ϕ3(X; c, b, c , d) ∈ tp (a/ acl (B) acl (C)). Furthermore, ϕ3(X; c, b, c , d) divides over aclM(C) ∩ N in all N ⊇ N 0. As discussed in Remark 3.4.7, realizations a0 of 0 0 this formula are such that {[a , b1],..., [a , bm]} is algebraically dependent over KaclM(B)∩N (K(aclM(C)∪{a0})∩N ). It remains to show that {b1, . . . , bm} is linearly independent modulo haclM(C) ∩ Ni. Claim 3.4.16. A linearly independent set of vectors X is linearly independent modulo the vector space W if and only if hXi ∩ W = {0}.

Proof. → Suppose X = {xi : i ∈ I}, W = hwj : j ∈ Ji (that is, {wj : j ∈ J} is a basis for W ), and v ∈ hXi ∩ W . Then, reordering if necessary, there are x1, . . . , xn, w1, . . . , wm, and scalars λ1, . . . , λn+m such that

v = λ1x1 + ... + λnxn = λn+1w1 + ... + λn+mwm and hence,

λ1x1 + ... + λnxn − (λn+1w1 + ... + λn+mwm) = 0

Since they come from a basis for W , w1, . . . , wm are linearly independent, and so X’s linear independence modulo W implies that {x1, . . . , xn}∪{w1, . . . , wm} is linearly independent. It follows that for all i, λi = 0, and thus that v = 0. CHAPTER 3. LIMIT INDEPENDENCE 66

← Suppose hXi ∩ W = {0}. We must show that if X0 ⊆ X is ﬁnite, there is no (ﬁnite) linearly independent set W0 of vectors from W such that X0 ∪ W0 is linearly dependent. Suppose X0 = {x1, . . . , xn}, and let W0 = {v1, . . . , vm} for some v1, . . . , vm ∈ W linearly independent. Suppose, further, that there are λ1, . . . , λn+m such that

λ1x1 + ... + λnxn + λn+1v1 + ... + λn+mvm = 0 It follows that

n m Σi=1λixi = −Σi=1λn+ivi.

Call this vector v. Since x1, . . . , xn ∈ X and v1, . . . , vm ∈ W , v ∈ hXi ∩ W . It follows that v = 0. Since {x1, . . . , xn} and {v1, . . . , vm} are linearly independent, we must have λi = 0 for each i. It follows that X is linearly independent modulo W .

The last conjunct of ϕ3 states exactly that hb1, . . . , bmi ∩ hCi = {0}, and hence that hb1, . . . , bmi ∩ hC ∩ Ni = {0} for all N under consideration. Remark 3.4.17. We would like to generalize the previous result to tuples a of length greater than one, but unfortunately we conjecture that there are finite tuples a, b, and an infinite /| lim dimensional set C such that a^ C b, and this dependence is not witnessed by any single formula. In particular, we suspect a problem might arise in the following way: there are | Γ M M | Γ M M a1, a2, b, C such that a1a2âclM(C) acl (C) acl (b) (i.e. a1âclM(C) acl (C) acl (b) and Γ M M | M a2âcl (C)a1 acl (C) acl (b)a1), but for arbitrary large N ,

N M M a2 /| M (acl (C) ∪ acl (b) ∪ {a1}) ∩ N, ^(acl (C)∪{a1})∩N

/| lim and hence a1a2^C b. (The fact that in the base, a1 does not lie in the scope of the algebraic closure suggests that one might ﬁnd

M /| M M K(acl (C)∪{a ,a })∩N^K M K(acl (C)∪acl (b)∪{a })∩N , 1 2 (acl (C)∪{a1})∩N 1

M | M M even though Kacl (C)a a ^K M Kacl (C) acl (b)a .) 1 2 acl (C)a1 1

Proposition 3.4.18. Let M T∞, and let H be the collection of finite dimensional sub- lim spaces of M (viewed as L-substructures of M). When ^| is restricted to tuples of length one on the left and finite dimensional sets on the right and in the base, H approximates M and each instance of limit independence corresponds to an instance Γ-independence. More precisely, for an element a and finite dimensional sets B, C,

| lim | Γ M M a^C B if and only if a^aclM(C) acl (B) acl (C). Proof. 1. H covers M: Every vector element generates a one dimensional subspace of V M. CHAPTER 3. LIMIT INDEPENDENCE 67

2. Automorphism invariance: the automorphic image of a ﬁnite dimensional subspace of V M is a ﬁnite dimensional subspace of V M.

3. Convergence of truth value: Suppose a ∈ M, ψ(x) ∈ L, and M ψ(a). We must show that ψ(a) is eventually true in H. By Lemma 3.4.1, there is an Lθ-formula ϕ and an R < ω such that Tm ϕ ↔ ψ for all m ≥ R (and, also by Lemma 3.4.1, 0 T∞ ϕ ↔ ψ). Since T∞ ϕ ↔ ψ, M ϕ(a). Take any N ∈ H containing a and 0 0 of dimension at least R. By Remark 3.4.2, N ϕ(a) and, since N Tm for some 0 0 0 m ≥ R, N ∀x(ψ(x) ↔ ϕ(x)), so N ψ(a). The same holds of all N ⊇ N , so ψ(a) is eventually true in H, as desired.

4. Stabilization of forking: This is Lemma 3.4.9.

5. Strong ﬁnite character: This is Lemma 3.4.15.

6. Stabilization of algebraic closure (for small sets): Suppose that

M alg A = acl (A) = (span alg (AV ),KA ). KA

M Since A is small, there are v1, v2 ∈ V such that {v1, v2} is linearly independent N modulo hAi. In any N containing {v1, v2}, dim(A ∩ N) < dim(V ) − 1. Consider one 0 N alg such N . By Remark 3.1.5, acl (A ∩ N) = (span alg (AV ∩ N),KA∩N ). We show that KA∩N N N acl (A ∩ N)V = AV ∩ N and that acl (A ∩ N)K = AK ∩ N. N Suppose a ∈ acl (A∩N)V = span alg (AV ∩N). That is, there are b1, . . . , bn ∈ AV ∩N KA∩N alg n alg alg and α1, . . . , αn ∈ KA∩N such that a = Σi=1αibi. Since AV ∩N ⊆ AV and KA∩N ⊆ KA , M N a ∈ span alg (AV ) = acl (A)V = AV . Since V is a vector space, a ∈ N. Hence, KA N alg alg alg a ∈ A ∩ N. Now suppose α ∈ acl (A ∩ N)K = KA∩N . Since KA∩N ⊆ KA ,

alg M α ∈ KA = acl (A)K = AK .

As α ∈ N, α ∈ A ∩ N, as desired.

M M 7. Homogeneity: Suppose tp (a/A) = tp (b/A). In particular, the quantiﬁer-free Lθ- type of a over A in M - call it p(x) - is the same as the quantiﬁer free Lθ-type of b over A in M. Let N ∈ H with a, b ∈ N. by Remark 3.4.2,

N p N (a) and N p N (b). N N Since T h(N ) has quantiﬁer elimination in Lθ, tp (a/A ∩ N) = tp (b/A ∩ N). CHAPTER 3. LIMIT INDEPENDENCE 68

| lim M M M By Lemma 3.2.5, a^C B if and only if tp (a/ acl (B) acl (C)) eventually does not fork over aclM(C). By Lemma 3.4.8, this happens if and only if tpM(a/ aclM(B) aclM(C)) does not Γ-fork over aclM(C). It follows that

| lim | Γ M M a^C B if and only if a^aclM(C) acl (C) acl (B). 69

Bibliography

[1] Hans Adler. “Explanation of independence”. PhD thesis. Albert-Ludwigs-Universität Freiburg im Breisgau, 2005. [2] ZoéChatzidakis. “Amalgamation of types in pseudo-algebraically closed fields and applications”. Article in preparation. 2011. [3] ZoéChatzidakis. “Properties of forking in ω-free pseudo-algebraically closed fields”. In: The Journal of Symbolic Logic 67.3 (2002), pp. 957–996. [4] ZoéChatzidakis. “Simplicity and independence for pseudo-algebraically closed fields”. In: Models and computability (Leeds, 1997). Vol. 259. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 1999, pp. 41–61.

[5] Artem Chernikov. “Some remarks on NTP1”. Unpublished note. 2013. [6] Artem Chernikov. “Theories without the tree property of the second kind”. In: Annals of Pure and Applied Logic 165.2 (2014), pp. 695–723. ∗ [7] Mirna Dˇzamonjaand Saharon Shelah. “On C -maximality”. In: Annals of Pure and Applied Logic 125.1-3 (2004), pp. 119–158. ∗ [8] Mirna Dˇzamonjaand Saharon Shelah. “On C -maximality”. Corrected version of the published paper of the same name; available at http://www.uea.ac.uk// h020/692ﬁx4.pdf. 2011. [9] Nicolas Granger. “Stability, simplicity and the model theory of bilinear forms”. PhD thesis. University of Manchester, 1999. [10] Victor Harnik and Leo Harrington. “Fundamentals of forking”. In: Annals of Pure and Applied Logic 26.3 (1984), pp. 245–286. [11] Wilfrid Hodges. A shorter model theory. Cambridge: Cambridge University Press, 1997, pp. x+310. [12] Byunghan Kim. “Forking in simple unstable theories”. In: Journal of the London Math- ematical Society. Second Series 57.2 (1998), pp. 257–267. [13] Byunghan Kim. “Simplicity, and stability in there”. In: Journal of Symbolic Logic 66.2 (2001), pp. 822–836. [14] Byunghan Kim and Hyeung-Joon Kim. “Notions around tree property 1”. In: Annals of Pure and Applied Logic 162.9 (2011), pp. 698–709. BIBLIOGRAPHY 70

[15] Byunghan Kim, Hyeung-Joon Kim, and Lynn Scow. “Tree indiscernibilities, revisited”. Preprint; to appear in ‘Archive for Mathematical Logic’. 2013. [16] Byunghan Kim and Anand Pillay. “Simple theories”. In: Annals of Pure and Applied Logic 88.2-3 (1997). Joint AILA-KGS Model Theory Meeting (Florence, 1995), pp. 149– 164. [17] David Marker. Model theory. Vol. 217. Graduate Texts in Mathematics. An introduction. New York: Springer-Verlag, 2002, pp. viii+342. [18] Lynn Cho Scow. “Characterization Theorems by Generalized Indiscernibles”. PhD thesis. University of California, Berkeley, 2010. [19] Saharon Shelah. Classiﬁcation theory and the number of nonisomorphic models. Vol. 92. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland Publishing Co., 1978, pp. xvi+544. [20] Saharon Shelah. “Simple unstable theories”. In: Annals of Mathematical Logic 19.3 (1980), pp. 177–203. [21] Saharon Shelah. “The universality spectrum: consistency for more classes”. In: Combi- natorics, Paul Erd˝osis eighty, Vol. 1. Bolyai Society Mathematical Studies. Budapest: J´anosBolyai Mathematical Society, 1993, pp. 403–420. [22] Saharon Shelah. “Toward classifying unstable theories”. In: Annals of Pure and Applied Logic 80.3 (1996), pp. 229–255.

[23] Saharon Shelah and Alexander Usvyatsov. “More on SOP1 and SOP2”. In: Annals of Pure and Applied Logic 155.1 (2008), pp. 16–31.