Independence Relations in Theories with the Tree Property · 2018-10-10 · theories are relatively...

Independence Relations in Theories with the Tree Property

by

Gwyneth Fae Harrison-Shermoen

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Logic and the Methodology of Science

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor Thomas Scanlon, ChairProfessor Leo Harrington

Professor John SteelAssociate Professor Antonio Montalban

Professor Martin Olsson

Fall 2013

1

Abstract


by


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 ofindependence in such theories. We discuss the example of the two-sorted theory of an infinitedimensional vector space over an algebraically closed field and with a bilinear form (which werefer to as T∞), examined by Granger in [9]. Granger notes that there is a well-behaved notionof independence, which he calls Γ-non-forking, in this theory, and that it can be viewed asthe 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 inan 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

relation | lim, which is the “limit” of the non-forking independence in the theories of theapproximating substructures. We show that if the approximating substructures have simple

theories, | lim satisfies invariance, monotonicity, base monotonicity, transitivity, normality,

extension, finite character, and symmetry. Under certain additional assumptions, | lim alsosatisfies 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, T ∗feq. We give a proof, explaining in detail the argument of Shelah and Usvyatsov forTheorem 2.1 in [23], that T ∗feq does not have SOP2 (equivalently, TP1). The argument makesuse of a theorem of Kim and Kim, from [14], along with several other lemmas involving treeindiscernibility.

i

To my parents.

ii

Contents

Contents ii

1 Introduction 11.1 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Tree properties, and the SOP hierarchy . . . . . . . . . . . . . . . . . . . . . 41.3 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Forking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Tree indiscernibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Parameterized Equivalence Relations 222.1 The Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Demonstration of subtlety . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Limit Independence 363.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Properties of the relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Bibliography 69

iii

Acknowledgments

First and foremost, I thank my adviser, Tom Scanlon, for his constant encouragement, hisextraordinary generosity with his time and ideas, and his patience, which rivals Penelope’s. Iam also deeply grateful to Zoe Chatzidakis for advising me during my time in France, takingsuch 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 providingme with a wonderful intellectual and social community during my time at Berkeley, and toHipline for providing me with a community outside of grad school, and plenty of opportunitiesto 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 YearFellowships, 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 GraduateDivision, 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 workof Morley in the 1960s and Shelah in the 1970s. Its goal is, roughly, to categorize first-ordertheories 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 consistentwith the theory.

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

|SMn (A)| ≤ κ.

A rich theory has been developed around the notion of stability, and as a result, stabletheories are relatively well understood. There are many mathematical structures with un-stable theories, however, and so there has been a concerted effort to generalize various resultsof stability theory in several different directions (for example, to simple theories, theorieswithout the independence property, and rosy theories).

One of the aspects of stable theories that pops up in many unstable theories is theexistence of a well-behaved independence relation, similar to linear independence in vectorspaces. (In fact, the abstract notion of independence generalizes both linear independence invector spaces and algebraic independence in fields, as well as many other geometric notionsof independence.) Essentially, given sets A, B, and C of elements in whatever structure weare working in, A is “independent” from B over C if B∪C has no more “information” aboutA than C does. (In the case of a vector space over a fixed field, “A is independent from Bover C” boils down to span(A ∪C) ∩ span(B ∪C) = span(C).) Independence relations giveus 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, manyof 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-forkingindependence is a symmetric relation. He and Anand Pillay showed in [16] that if a theoryhas an abstract independence relation satisfying a certain set of axioms, then that theoryis 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 nicenotion 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 simpletheory, but they may satisfy almost all of them. One example of such a theory is the the-ory of ω-free pseudo algebraically closed fields, studied by Chatzidakis in [4], [3], and [2].(Chatzidakis actually identifies several distinct notions of independence, satisfying differentsubsets of the properties of non-forking independence in a simple theory.) Another exampleis the two-sorted theory of an infinite dimensional vector space over an algebraically closedfield 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 independencerelation that Granger singles out (which he refers to as “Γ-non-forking”) can be viewed as thelimit of the non-forking independence in the finite dimensional subspaces, whose theories arestable. 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 approximationby substructures, and define an independence relation that is the limit of the non-forkingindependence 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

substructures N ofM, we define the following notion of independence: A | limC B (A is limitindependent from B over C) if for each finite subset A0 of A, there is NA0 ∈ H such that

A0 ⊆ NA0 and for all NA0 ⊆ N ∈ H, A0| N

aclM(C)∩N aclM(B) ∩N , where | N is non-forkingindependence 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.

• If A |/ limC 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), thentpN (a/A ∩N) = tpN (b/A ∩N).


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

Theorem 1.0.3. Given a κ-saturated, strongly κ-homogeneous modelM of T approximated

by H, where each N ∈ H has a simple theory, |lim

has the following properties: auto-morphism invariance, transitivity, finite character, symmetry, existence, and extension. IfH satisfies the optional conditions and in the theories of its elements, non-forking indepen-

dence satisfies the Independence Theorem over algebraically closed sets, then |lim

satisfiesanti-reflexivity and the Independence Theorem over algebraically closed sets.

This thesis is structured as follows: in the remainder of this chapter, we present ournotational conventions and necessary background from stability and simplicity theory, as wellas some definitions and results of a combinatorial nature on “tree indiscernibility.” Chapter2 examines the two-sorted theory of infinitely many cross-cutting equivalence relations (T ∗feq).Following Shelah and Usvyatsov [23], we give a proof that T ∗feq does not have the tree propertyof 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, definenotions of “approximation” and “limit independence,” and prove Theorem 1.0.3. We then

look at the relationship between Granger’s Γ-non-forking ( | Γ) and limit independence, and

see to what extent | lim generalizes | Γ.

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 (forsome large regular cardinal κ) model C of T . We consider only modelsM of T of cardinalityless than κ and such thatM≺ C. (This is without loss of generality, as every model of T ofcardinality less than κ can be elementarily embedded into C.) We consider only sets A ⊂ Cof 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 denotesets, whileM 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 anequivalence relation, and P a unary predicate. I and J are index sets. K is a field (L andF 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 naturalnumbers (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 arenatural numbers; p and q are types. The letters r, s, and t are occasionally types, but are


more often natural numbers, and u, v, w, x, y, z are variables. In languages with multiplesorts, 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 warnthe reader of exceptions as they arise. We denote the length of the tuple a by lg(a). (Thatis, if a = (a0, . . . , an−1), lg(a) = n.) We abuse notation by writing a ∈ A (where a is a tupleof length n) rather than a ∈ An.

We often denote a union by concatenation, e.g. we may write BC for B ∪ C. Weuse acl(A) to denote the model-theoretic algebraic closure of A, and Kalg to denote thefield-theoretic algebraic closure of K. An automorphism over A or A-automorphism is anautomorphism 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, thatthere is a A-automorphism of C sending B to C. If ∆ ⊂ L, B ≡∆ C means that B and Chave 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 abranch from such a tree, and α ∈ <ωω is an initial segment of such a branch, that is, β � kfor some β ∈ ωω, k < ω. We also work with finitely-branching trees ωq for q < ω, especiallyω2. In all cases, we denote the tree order by E: that is, η E ν if there are k ≤ ` ∈ ω andβ ∈ ωω (respectively, β ∈ ω2) 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(that is, there are a formula ϕ(x; y) and sequences 〈ai : i < ω〉, 〈cσ : σ ∈ 2ω〉 such that� ϕ(ai; cσ) if and only if σ(i) = 0) or the strict order property [19] (that is, there are aformula ϕ(x, y) and a sequence 〈bi : i < ω〉 such that � ∀x(ϕ(x; bi)→ ϕ(x; bj)) if and only ifi < j). In this thesis, we shall focus primarily on unstable theories without the strict orderproperty. In this section, we introduce a number of properties weaker than the strict orderproperty. 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 forany 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 area formula ϕ(x; y), k < ω, and sequences aη ∈ C (η ∈ <ωω) such that:


• for any η ∈ <ωω, {ϕ(x, aη_l) : l < ω} is k-inconsistent, but

• for every β ∈ ωω, {ϕ(x, aβ�n) : 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 theparameter variables have been replaced by parameters.

The tree property is a robust dividing line among unstable theories: the class of simpletheories 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 anindependence relation (to be defined in Section 1.3) satisfying a certain type amalgamationproperty (analogous to stationarity of types in stable theories). The theories of the randomgraph 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 areconsistent, while instances at nodes that are siblings are inconsistent. It leaves open theconsistency of instances at incomparable, non-sibling nodes. It is therefore natural to con-sider 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) ifthere are a formula ϕ(x; y) and tuples {aα : α ∈ <ωω} such that:

• for α0, . . . , αk−1 ∈ <ωω pairwise incomparable, {ϕ(x, aαi) : 1 ≤ i ≤ k} is inconsis-

tent, but

• for β ∈ ωω, {ϕ(x, aβ�n) : 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, T ∗feq (to bediscussed extensively in Chapter 2), is NTP1 [23] but not simple. Recently, Chernikov hasidentified a sufficient condition for a theory to be NTP1, and used it to show that both thetwo-sorted theory of infinite dimensional vector spaces over algebraically closed fields with abilinear form (described in Section 3.1) and the theory of ω-free PAC fields of characteristic0 are NTP1 [5]. This makes use of results from [3], [2], [9], and [6]. Each of these theories isnot 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 aformula ϕ(x; y), tuples {aα : α ∈ <ωω}, and k < ω such that:

• for any α ∈ <ωω, {ϕ(x; aα_i) : i < ω} is k-inconsistent, but


• for any n and any α0, . . . , αn−1 ∈ <ωω, 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 givenas 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 - areNTP2 but not simple [6].

Theorem 1.2.7 ([19], Theorem III.7.11). If a theory T has the tree property, then either Thas 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 orderproperty 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 sequenceϕ = 〈ϕn(xn; yn) : n < ω〉 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);

b) T ∪ {ϕn+1(xn+1, yn+1)} ` ϕn(xn, yn);

c) for m ≤ n, ¬(∃xn,0, . . . , xn,m−1)[∧{ϕn(xn,k, xn,`) : k = ` + 1 mod m}] belongs to

T ;

d) there is a model M of T and anα ∈ M (of length yn, for n < ω, α < λ) such thatanα = an+1

α � lg(yn) and M � ϕn[anβ, anα] 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{ak ∈M lg(x) : k < ω} such that

• M � ϕ(ak, am) for k < m < ω, and

• 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) andtuples aη for η ∈ <ω2 such that

• for every ρ ∈ ω2, the set {ϕ(x, aρ�n) : n < ω} is consistent, while

• if η, ν ∈ <ω2 are incomparable, {ϕ(x, aη), ϕ(x, aν)} is inconsistent.


2. T has SOP1 if there are a formula ϕ(x, y) and tuples aη for η ∈ <ω2 such that

• for ρ ∈ ω2 the set {ϕ(x, aρ�n) : n < ω} is consistent, but

• if ν_〈0〉 E η ∈ <ω2, then {ϕ(x, aη), ϕ(x, aν_〈1〉)} 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 propertyimplies 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. Thereis some disagreement about which properties should be required of a ternary relation in orderfor it to be called an independence relation. Below we give Adler’s definition from [1]. Ingeneral, we shall use the phrase notion of independence as a catch-all term for a relationsatisfying 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:

• invariance: If A | CB and (A′, B′, C ′) ≡ (A,B,C), then A′ | C′B′.

• monotonicity: If A | CB, A′ ⊆ A, and B′ ⊆ B, then A′ | CB′.

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

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

• normality: A | CB implies AC | CB.

• extension: If A | CB and B′ ⊇ B, then there is A′ ≡BC A such that A′ | CB′.

• finite character: If A0|CB for all finite A0 ⊆ A, then A | CB.

• local character: For every A there is a cardinal κ(A) such that for any set B thereis a subset C ⊆ B of cardinality |C| < κ(A) such that A | CB.

An independence relation is strict if it also satisfies anti-reflexivity:

a | Ba implies a ∈ acl(B).


Recalling the slogan from the beginning of this chapter - that A should be independentfrom 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 satisfyare:

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

• symmetry: A | CB if and only if B | CA.

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

– � ϕ(a, b, c), and

– a′ |/ Cb for all a′ such that � ϕ(a′, b, c).

• stationarity: If a | CB and B ⊆ B′, there is a′ ≡BC a such that a′ | CB′, and ifd | CB′ and d ≡BC a, then d ≡B′C a′. (A relation with the property of “stationarityover algebraically closed sets” is one that satisfies the property of stationarity when Cis algebraically closed.)

• independence theorem over a model: If a ≡M b, a | MA, b | MB (for M a modeland A, B ⊇M), and A | MB, then there is c such that c ≡A a, c ≡B b, and c | MAB.(One should read this as a weakening of the previous property. Roughly, if | satisfiesstationarity, independent extensions of types are unique. If | satisfies the indepen-dence theorem (over a model), independent extensions (of types over models) can beamalgamated.)

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 propertiesmentioned 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 nofinite classes), the relation given by

A | CB if and only if A ∩B ⊆ C

is a strict independence relation.

3. The relation given by

A | aCB if and only if acl(AC) ∩ acl(BC) = acl(C)


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

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

A | CB 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 propertiesthey satisfy, paying particular attention to the behaviour of forking and dividing in simpletheories. We give proofs of several easy facts, as this may provide the reader with a bettersense of dividing and forking, which can be difficult to grasp. The reader should assume thatthese facts have already been proven many times, by many people, even where no specificreference is given.

Definition 1.4.1 ([19], Chapter III, Definitions 1.3-1.4). • A formula ϕ(x; b) divides overA 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) andϕ(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

1. p(x) `∨i<n ϕi(x; ai), and

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 indiscernibleover 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 havetaken {bi : i < ω} to be an indiscernible sequence. In that case, the second condition isequivalent to

{ϕ(x; bi) : i < ω} is inconsistent.


Both non-dividing and non-forking can be viewed as notions of independence. We write

a | dCB for “tp(a/BC) does not divide over C” and a | fCB for “tp(a/BC) does not fork overC.”

Fact 1.4.4. For a set C, parameters b /∈ C and c ∈ C, formulae ψ(x, y, z), ϕ(x,w) ∈ L, andn < 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-indiscerniblesequence 〈bici : i ∈ I〉 = 〈bic : i ∈ I〉, with b0c0 = bc (since c ∈ C, ci = c for all i ∈ I) andsuch that

{ψ(x, bi, c) : i ∈ I}is inconsistent. We show that

{(∃=nx)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,

� (∃=nx)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 I ′ ⊆ I such that

� 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,{ψ(x, bi, c) : i ∈ I ′} is k-inconsistent. Contradiction.

Corollary 1.4.5. If A |d

CB, then acl(A) |d

CB.

Proof. We prove the contrapositive. If acl(A) |/ dCB, 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 a′ ∈ acl(A) realizeψ(x; b, c). Since a′ ∈ acl(A), there are n < ω, ϕ(x,w) ∈ L, and a ∈ A such that

ϕ(x, a) ∧ (∃=nx)ϕ(x, a) ∈ tp(a′/A).

It follows that(∃=nx)ϕ(x,w) ∧ ∃x(ϕ(x,w) ∧ ψ(x, b, c)) ∈ tp(a/BC).

By fact 1.4.4 this formula divides over C, so A |/ dCB.


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

Proof. This is clear from the definitions.

Fact 1.4.7. If ψ(x, d) forks over C, and d0, . . . , dn−1 are C-conjugates of d, then∨i<n

ψ(x, di)

forks over C.

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), andso

∃y(ϕ(y, b) ∧ ψ(x, y)) `∨i

ψ(x, di).

By the fact that for i, j < n, tp(di/B) = tp(dj/B), strong κ-homogeneity of C, and Fact1.4.7, ∃y(ϕ(y, b) ∧ ψ(x, y)) forks over C.

Corollary 1.4.9 ([19], Lemma III.6.3(3)). If A |f

CB, then A |f

C acl(B).

Fact 1.4.10. An infinite, A-indiscernible sequence is acl(A)-indiscernible.

Fact 1.4.11. A | CB if and only if A | acl(C)B (for | = | d or | = | f).

Proposition 1.4.12 ([12], Proposition 2.1). If T is simple, ϕ(x, a) divides over A if andonly if ϕ(x, a) forks over A.

Proposition 1.4.13 (Properties of forking independence). Let T be a simple theory. The

independence relation given by A |f

CB if tp(A/BC) forks over C satisfies the axioms ofexistence, anti-reflexivity, monotonicity, base monotonicity, finite character, local character,extension, symmetry, and transitivity.

Proof. Anti-reflexivity, monotonicity, base monotonicity, and finite character follow from thedefinition of forking (and are true in any theory). Existence follows from the equivalenceof 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 provedin [12], Theorems 2.5 and 2.6, respectively.

Remark 1.4.14. If T is simple, we shall write | for forking (equivalently, dividing) indepen-dence.

Remark 1.4.15. In a simple theory, Corollary 1.4.5 can be strengthened to the following:if A | CB, then acl(A) | C acl(B). This is by the equivalence of forking and dividing andsymmetry of | .


The next two theorems give us ways to identify simple theories by looking at the behaviourof 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 provedby 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 |◦CB 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 rela-tion 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 (ofany kind). The problem becomes somewhat more tractable if we may assume certain factsabout the tree. In particular, it is helpful to assume that the tree in question has some levelof indiscernibility. There are several kinds of tree indiscernibility, but the idea behind all ofthem is that given two configurations of nodes on the tree that “look alike,” the parametersdecorating the nodes of those two configurations should have the same type. The differencesamong the various notions of tree indiscernibility come from the different interpretations ofwhat it means for two tree configurations to “look alike.” (For comparison, in the case of anindiscernible sequence, two sets of elements from the sequence “look alike” if they have thesame order type.) For an in-depth discussion of “generalized indiscernibles,” see [18].


Remark 1.5.1 (Notation). 1. In this section, we will make use of overlines to denote tuplesof nodes, and tuples of the tuples decorating those nodes. For example, η denotes, forsome n < ω, a sequence (η0, . . . , ηn−1), and aη denotes (aη0 , . . . , aηn−1). Note that eachaηi 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 anode in a q-branching tree, |η| = k if η ∈ kq.

The following definition is taken from [14], although an equivalent version of parts 1through 3 appeared in [7] several years earlier. As the authors of [14] note, the terminologyof part 4 of this definition is due to Scow [18].

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

2. Given tuples η, ν ∈ <ωq, 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

– η_i 〈t〉 E ηj iff ν_i 〈t〉 E νj.

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

3. We say 〈aη|η ∈ <ωq〉 is 1-fti (or 1-fully tree indiscernible) if for all η, ν ∈ <ωq, η ≈1 νimplies aη ≡ aν .

4. We say a sequence 〈aη|η ∈ <ωq〉 is 1-modeled by a sequence 〈bη|η ∈ <ωq〉 if, for anyd < ω, finite set ∆(x0, . . . , xd−1) of L-formulae, and ∩-closed tupleη = 〈η0, . . . , ηd−1〉 ∈ <ωq, there exists ν ∈ <ωq 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 treethat is 1-fti (see Kim and Kim’s Theorem 1.5.6, below). Importantly, this transformationpreserves 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 ofthe properties defined above.

Example 1.5.4. Let η = (〈1〉, 〈101〉, 〈1001〉). The tuple η is not ∩-closed, because〈101〉 ∩ 〈1001〉 = 〈10〉 /∈ η.


•

η1

•◦

η2•

η3•

•

Example 1.5.5. Let

η = (〈〉, 〈0〉, 〈001〉, 〈01〉, 〈111〉)ν = (〈1〉, 〈101〉, 〈1010〉, 〈10110〉, 〈110〉)

Then η ≈1 ν.

η1

•

•

η5•

•

η2

η4•

η3•

•

ν1

•

•ν5

•

ν2

•

•ν4

ν3

•

•

Theorem 1.5.6 ([14], Proposition 2.3). Any sequence 〈αη|η ∈ <ωq〉 can be 1-modeled (in asufficiently saturated model) by some 1-fti sequence 〈bη|η ∈ <ωq〉.

Remark 1.5.7 (See also [15], Remark 5.2, for the ω-branching case). If 〈aη|η ∈ <ω2〉 witnessesSOP2 for a formula ψ(x, y) and 〈bη|η ∈ <ω2〉 1-models 〈aη|η ∈ <ω2〉 in some model M, then〈bη|η ∈ <ω2〉 is also an SOP2 tree for ψ(x, y).

Proof. 1. First we show that for any β ∈ ω2, {ψ(x, bβ�m)|m ∈ ω} is consistent. Fix sucha β, and consider a finite subset of {ψ(x, bβ�m)|m ∈ ω}. It is contained in a set of theform {ψ(x, bβ�m)|m < n} for some n ∈ ω, so it suffices to show that sets of this formare consistent. Choose n ∈ ω, and let η = 〈β � 0, . . . , β � n − 1〉. This is an ∩-closedtuple. Consider ∆(y) := {∃x(

∧i<n

ψ(x, yi))}. By hypothesis, we can find a ν ∈ <ω2 such

that η ≈1 ν and bη ≡∆ aν . Since ν ≈1 η, we have νi E νj for all i ≤ j. That is, ν lies


on a branch. Since 〈aη|η ∈ <ω2〉 is an SOP2 tree for ψ, {ψ(x, aνi)|i < n} is consistent,and thus M � ∃x(

∧i<n

ψ(x, aνi)). It follows that M � ∃x(∧i<n

ψ(x, bηi)), and thus the set

{ψ(x, bβ�m)|m < n} is consistent, as desired.

2. Now suppose that α, γ ∈ <ω2 are incomparable - we must show that {ψ(x, bα), ψ(x, bγ)}is inconsistent. Let η := 〈α, γ, α ∩ γ〉. This is clearly ∩-closed. Let

∆(y) := {∃x(ψ(x, y0) ∧ ψ(x, y1))}.

We can find ν ∈ <ω2 such that ν ≈1 η and bη ≡∆ aν . Since α ∩ γ is the mostrecent common ancestor of α and γ, it must be that one of α, γ is a descendant of(α∩γ)_〈0〉, while the other is a descendant of (α∩γ)_〈1〉. Without loss of generality,assume (α∩ γ)_〈0〉 E α. It follows that ν_2 〈0〉 E ν0, while ν_2 〈1〉 E ν1, and thus, thatν0 and ν1 are incomparable. Since 〈aη|η ∈ <ω2〉 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 formalargument, and simply observe that in a tree 1-modeled on a tree with TP, nodes that aresiblings might be modeled on nodes that are not siblings in the original tree. Since TP onlyguarantees inconsistency among instances of the formula at sibling nodes, the new tree mightnot satisfy the inconsistency requirement.

The above facts tell us that if a theory T has TP1 (and hence, SOP2), then there area formula ψ(x; y) and a sequence 〈aα|α ∈ <ω2〉 which is a 1-fti tree witnessing SOP2 forψ(x; y). We would also like to be able to narrow down the set of formulae which we mustshow do not have SOP2 in order to show that no formula has SOP2. To that end, we showthat we can “eliminate the disjunction” - that is, if a disjunction of formulae has SOP2, thenone of the disjuncts has SOP2.

Lemma 1.5.9. If 〈aα : α ∈ <ω2〉 is a 1-fti tree witnessing SOP2 for ψ0 ∨ ψ1, then we canfind a tree witnessing SOP2 for either ψ0 or ψ1.

Proof. We may assume that both ψ0(x, a〈〉) and ψ1(x, a〈〉) are consistent - otherwise〈aα : α ∈ <ω2〉 is already a tree for one or the other. (This uses the fact that the tree is1-fti: if, say, ψ0(x, a〈〉) is inconsistent, then so is ψ0(x, aα) for every α ∈ <ω2. It follows thatconsistency of the instances of ψ0∨ψ1 along a branch implies consistency of the correspondinginstances of ψ1. Inconsistency of incomparable instances of the disjunction automaticallyimplies inconsistency of incomparable instances of ψ1, so the original tree witnesses SOP2

for ψ1.) Consider the leftmost branch (i.e. {〈〉, 〈0〉, 〈00〉, 〈000〉, . . .}). Let b realize

{(ψ0 ∨ ψ1)(x; a〈0n〉) : n < ω},


where 〈00〉 = 〈〉, 〈01〉 = 〈0〉, 〈02〉 = 〈00〉, etc. Let f be a function from ω to {0, 1} defined asfollows:

f(j) =

{0 if � ψ0(b; a〈0j〉)

1 otherwise

The function f must take at least one of {0, 1} as its value for arbitrarily large j. We mayassume f(j) = 0 for arbitrarily large j. Now, for any m ∈ ω, consider η := {〈〉, 〈0〉, . . . 〈0m〉}.By our assumption, we can find j0, . . . , jm ∈ ω with j0 < . . . < jm and such that

{ψ0(x, a〈0ji 〉) : i ≤ m}

is consistent. Let ν := {〈0j0〉, 〈0j1〉, . . . , 〈0jm〉}. Clearly, η ≈1 ν. Since our tree is 1-fti, itfollows that {ψ0(x, a〈0i〉) : i ≤ m} is consistent. By compactness, {ψ0(x, a〈0i〉) : i ∈ ω} isconsistent.

We now define a function g : <ω2→ <ω2 by induction, as follows:

g(〈〉) := 〈〉g(α_〈0〉) := g(α)_〈00〉g(α_〈1〉) := g(α)_〈01〉

This is our pruning function: it picks out the nodes from the original tree that will formour new tree. We illustrate the first few steps of the function below. Think of the tree onthe left as the tree we are constructing (the candidate for an SOP2 tree for ψ0(x; y)), andthe 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 nodethat appears in the image of g lies above the left child of the previous node appearing inthe image. This construction gives us a tree all of whose branches “look like” (are 1-similarto) the leftmost branch of the original tree. We shall show that the new tree witnesses thatψ0(x; y) has SOP2.

For α ∈ <ω2, we let bα := ag(α). We claim that 〈bα : α ∈ <ω2〉 is a tree witnessing SOP2

for ψ0.


Claim 1.5.10. Given η, ν ∈ <ω2, η E ν if and only if g(η) E g(ν). (In particular, if η andν 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 distancebetween 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_〈i〉 for some i ∈ {0, 1} and α D η, with the distancebetween η and α equal to n. By the induction hypothesis, g(η) E g(α). By thedefinition of g,

g(ν) = g(α_〈i〉) = g(α)_〈0i〉 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_〈i〉 E η. By the forward direction ofthis claim, g(ν_〈i〉) E g(η), hence g(ν)_〈0i〉 E g(η). It follows that g(η) 5 g(ν), asdesired.

• ν and η are incomparable. We may assume without loss of generality that:

(ν ∩ η)_〈0〉 E ν;

(ν ∩ η)_〈1〉 E η.

Again by the forward direction of the claim, we have:

g(ν) D g((ν ∩ η)_〈0〉) = g(ν ∩ η)_〈00〉 = (g(ν ∩ η)_〈0〉)_〈0〉;g(η) D g((ν ∩ η)_〈1〉) = g(ν ∩ η)_〈01〉 = (g(ν ∩ η)_〈0〉)_〈1〉.

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)_〈0〉 E g(ν).

Proof of remark. If η / ν, then η_〈i〉 E ν, for i = 0 or i = 1. By Claim 1.5.10 and thedefinition of g,

g(η_〈i〉) = g(η)_〈0i〉 = (g(η)_〈0〉)_〈i〉 E ν

Claim 1.5.12. If η, ν ∈ <ω2 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) areincomparable, 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.


Claim 1.5.13. If β ∈ ω2, {ψ0(x, bβ�i) : i ∈ ω} is consistent.

Proof of claim. It suffices to show that for any n ∈ ω, {ψ0(x, bβ�i) : i ≤ n} is consistent. ByRemark 1.5.11, if i < j, then g(β � i)_〈0〉 E g(β � j). Let η := (g(β � i) : i ≤ n). We justobserved that η lies on a branch, and so it is ∩-closed. Now let ν := (〈0i〉 : i ≤ n). This isalso ∩-closed. Furthermore, for i, j ≤ n,

〈0i〉 E 〈0j〉 iff 〈0i〉 = 〈0j〉 or 〈0i〉_〈0〉 E 〈0j〉iff i ≤ j

iff g(β � i) = g(β � j) or g(β � i)_〈0〉 E g(β � j)

iff g(β � i) E g(β � j)

It follows that η ≈1 ν. Since {ψ0(x, a〈0i〉) : i ≤ n} is consistent and 〈aα : α ∈ <ω2〉 is a 1-ftitree, {ψ0(x, ag(β�i)) : i ≤ n} = {ψ0(x, bβ�i) : i ≤ n} is also consistent, as desired.

Claims 1.5.12 and 1.5.13 tell us that 〈bα : α ∈ <ω2〉 is a tree witnessing SOP2 for ψ0, sothe 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 , thereare finitely many complete types in the variables of ψ. Of these, let p0, . . . , pn−1 list theones 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,ψ ≡

∨i<n

ψi. By Lemma 1.5.9 and induction on n,∨i<n

ψi has SOP2 if and only if one of the ψi

does.

We now make an observation about 1-fti trees: if the parameters at two different nodesin such a tree share any elements, then those elements are common to the parameters at allnodes in the tree.

Lemma 1.5.15. Suppose 〈bη : η ∈ <ω2〉 is a 1-fti tree, where the bη are tuples of length n.If for some η, ν ∈ <ω2 and some i ≤ n, biη = biν, then for all ρ ∈ <ω2, biρ = biη.

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.

Claim 1.5.16. If for some ρ ∈ <ω2, biρ = biρ_〈0〉 = biρ_〈1〉, then for all ξ ∈ <ω2, biξ = biρ.

Proof of claim. We split the proof into two cases, depending on whether or not ρ is the rootof the tree.


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

〈〉ξ ≈1 〈〉〈0〉

or〈〉ξ ≈1 〈〉〈1〉

Since both bi〈〉 = bi〈0〉 and bi〈〉 = bi〈1〉, tree-indiscernibility implies that bi〈〉 = biξ.

• Case 2: ρ 6= 〈〉. Then either 〈0〉 E ρ or 〈1〉 E ρ, and hence either

〈〉ρ ≈1 ρρ_〈0〉

or〈〉ρ ≈1 ρρ

_〈1〉.

Since both biρ = biρ_〈0〉 and biρ = biρ_〈1〉, bi〈〉 = biρ by tree-indiscernibility. Then, since

(〈〉, 〈0〉, 〈1〉) ≈1 (ρ, ρ_〈0〉, ρ_〈1〉),

tree-indiscernibility implies that bi〈0〉 = bi〈〉 = bi〈1〉, and we are now in case 1 (giving the

result that for all ξ ∈ <ω2, biξ = bi〈〉 = biρ).

We split the proof of the lemma into two cases, depending on whether or not η and ν arecomparable. In the following, refer to biη = biν as bi.

• Case 1. Suppose η and ν lie on the same branch. Without loss of generality, supposeη_〈0〉 E ν. Then for any ν ′ D η_〈0〉, ην ≈1 ην

′, and so biν′ = biη = bi. In particular,biη_〈0〉 = biη_〈00〉 = biη_〈01〉 = bi. By Claim 1.5.16, we are done (take ρ to be η_〈0〉).

• Case 2. Suppose η and ν are incomparable. Without loss of generality, suppose(η ∩ ν)_〈0〉 E η and (η ∩ ν)_〈1〉 E ν. Then for all η′ D (η ∩ ν)_〈0〉, biη′ = biν = bi

(since (ν, η, η ∩ ν) ≈1 (ν, η′, η ∩ ν)), and for all ν ′ D (η ∩ ν)_〈1〉, biν′ = biη = bi (since(η, ν, η ∩ ν) ≈1 (η, ν ′, η ∩ ν)). In particular,

bi = bi(η∩ν)_〈0〉

= bi(η∩ν)_〈00〉

= bi(η∩ν)_〈01〉

We apply Claim 1.5.16 again, this time taking ρ to be (η ∩ ν)_〈0〉.


Corollary 1.5.17. Suppose 〈bη : η ∈ <ω2〉 is a 1-fti tree, where the bη are tuples of lengthn. If for some η, ν ∈ <ω2 c ∈ bη ∩ bν (that is, c is a coordinate in each tuple, though notnecessarily the same coordinate), then for all ρ ∈ <ω2, c ∈ bρ.

Proof. Take i 6= j < n such that c = biη = bjν . (If i = j, the statement reduces to Lemma1.5.15.) We split the proof into two cases, depending on whether or not η and ν are compa-rable.

• Case 1: Suppose η and ν are comparable. Without loss of generality, we may assumethat η_〈0〉 E ν. Then

(η, ν) ≈1 (η_〈0〉, η_〈00〉)≈1 (η, η_〈00〉)

Since biη = bjν , we have

biη_〈0〉 = bjη_〈00〉

biη = bjη_〈00〉

Hence biη = biη_〈0〉, and by Lemma 1.5.15, c = biη = biρ for all ρ ∈ <ω2.

• Case 2: Suppose η and ν are incomparable. Without loss of generality, we may assumethat (η ∩ ν)_〈0〉 E η and (η ∩ ν)_〈1〉 E ν. It follows that

(η, ν, η ∩ ν) ≈1 ((η ∩ ν)_〈0〉, ν, η ∩ ν).

Since biη = bjν , bi(η∩ν)_〈0〉 = bjν , and we have c = bi(η∩ν)_〈0〉 = biη. By Lemma 1.5.15,

c = biρ for all ρ ∈ <ω2.

Remark 1.5.18. If 〈bη : η ∈ <ω2〉 is a 1-fti tree (where the bη are tuples of length n) and cis an element of the common intersection of 〈bη : η ∈ <ω2〉 (that is, c ∈ bρ for all ρ ∈ <ω2),then for any η, ν ∈ <ω2, tp(bη/c) = tp(bν/c).

Proof. By the proof of Corollary 1.5.17, there is some i < n such that c = biρ, all ρ ∈ <ω2.Since 〈bη : η ∈ <ω2〉 is 1-fti, tp(bη/c) = tp(bη/b

iη) = tp(bν/b

iν) = 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 thedefinable closures of tuples at distinct nodes on a 1-fti tree. We begin by showing thatwe can add elements of the definable closure (in a reasonable way) to a 1-fti tree, and theresulting tree will also be 1-fti.


Lemma 1.5.19. If 〈bη : η ∈ <ω2〉 is a 1-fti tree (where the bη are tuples of length n), ϕ(x, y)is a formula such thatM � (∃=1x)ϕ(x, b〈〉), and for each η ∈ <ω2 cη is the unique realizationof ϕ(x, bη), then the tree 〈bηcη : η ∈ <ω2〉 is 1-fti.

Proof. First note that the statement of the lemma makes sense: ifM � (∃=1x)ϕ(x; b〈〉), thenM � (∃=1x)ϕ(x; bη) for all η ∈ <ω2, so the cη’s exist. We must show that if η and ν aretuples from <ω2 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 thatM � ψ(bη, cη). Then

M � ∃x(ψ(bη, x) ∧∧i<m

ϕ(xi, bηi)).

Since 〈bη : η ∈ <ω2〉 is 1-fti,

M � ∃x(ψ(bν , x) ∧∧i<m

ϕ(xi, bνi)).

Since cνi is the only realization of ϕ(xi, bνi), cν is the only possible realization of ψ(bν , x) ∧∧i<m

ϕ(xi, bνi)). Since we know this formula has a realization, M � ψ(bν , cν). Since ψ was

arbitrary, bηcη ≡ bνcν , and 〈bηcη : η ∈ <ω2〉 is 1-fti.

Corollary 1.5.20. If 〈bη : η ∈ <ω2〉 is a 1-fti tree and for some η, ν (η 6= ν)

c ∈ dcl(bη) ∩ dcl(bν),

then c ∈ dcl(bρ) for all ρ ∈ <ω2.

Proof. Suppose ϕ1 witnesses that c ∈ dcl(bη) and ϕ2 witnesses that c ∈ dcl(bν). For eachρ ∈ <ω2, let cρ be the unique realization of ϕ1(x, bρ), and let dρ be the unique realizationof ϕ2(x, bρ). (So cη = c = dν .) By Lemma 1.5.19, the tree 〈bρcρdρ : ρ ∈ <ω2〉 is 1-fti. Sincec ∈ bηcηdη ∩ bνcνdν , c ∈ bρcρdρ - and hence, c ∈ dcl(bρ) - for all ρ ∈ <ω2, by Corollary 1.5.17.(In fact, by the proof of Corollary 1.5.17, c = cρ = dρ for all ρ ∈ <ω2.)

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 treeproperty 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 T ∗feq. In[23], Shelah and Usvyatsov show that T ∗feq does not have SOP2.1 We give a version of thisproof, 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 shalluse lowercase letters x, y, z to denote variables of sort P , uppercase letters X, Y, Z to denotevariables of sort E, lowercase a, b, c, d to denote parameters of sort P , and lowercase r, s, tto 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 Fraısse limit, F .

1The stated result in [23], Theorem 2.1 is, in fact, that T ∗feq 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 T ∗feq does not have SOP2. It is that argumentthat we give here, filling in the details that were less obvious to us than they would have been to the authorsof [23]. It is still conjectured that T ∗feq 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 asK consists of finite structures in a relational language, “finitely generated”just means “finite” here.) By universality of T0, any substructure of a (finite) modelof 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 Bare embeddable in C.We shall construct C. Suppose

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

B = {b0, . . . , bn′−1, s0, . . . , sm′−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, . . . , bn′−1, r0, . . . , rm−1, s0, . . . , sm′−1}

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

• For i < m, j, k < n, and j′, k′ < n′,

C � ri(aj, ak) if and only if A � ri(aj, ak);C � ¬ri(aj, bj′) ∧ ¬ri(bj′ , aj);C � ri(bj′ , bk′).

• For i′ < m′, j, k < n, and j′, k′ < n′,

C � si′(bj′ , bk′) if and only if B � si′(bj′ , bk′);C � ¬si′(bj′ , aj) ∧ ¬si′(aj, bj′);C � si′(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: IfA,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 .


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}

Our definition of RD is similar to that of RC in the proof of JEP: each si has the sameclasses as in B, plus an additional class for the elements of C \ f(A), and each ti hasthe same classes as in C, plus an additional class for the elements of B \ e(A). Wemust be more careful when extending the definition of the ri’s, however, in order tomake sure they satisfy transitivity. Thus, in our formal definition of RD, we considerthree separate cases, depending on whether the equivalence relation element is fromA, 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 );

D �

{si(ajA , bjB) ∧ si(bjB , ajA) if B � si(e(ajA), bjB),

¬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 theyare 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);

D �

{ti(ajA , cjC ) ∧ ti(cjC , ajA) if C � ti(f(ajA), cjC ),

¬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 ifthey are ti-related in C, while the elements of B \ e(A) form a separate ti-class.


• 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 )

D �

{ri(ajA , bjB) ∧ ri(bjB , ajA) if B � e(ri)(e(ajA), bjB),

¬ri(ajA , bjB) ∧ ¬ri(bjB , ajA) otherwise;

D �

{ri(ajA , cjC ) ∧ ri(cjC , ajA) if C � f(ri)(f(ajA), cjC ),

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

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

Let T ∗feq be the theory of F . As the theory of the Fraısse limit of the finite models of T0,T ∗feq is ℵ0-categorical and has elimination of quantifiers (see [11], Theorem 6.4.1), and it is


the model completion of T0. For convenience, we spell out the intermediate theory Tfeq:

Tfeq := T0 ∪

{∀X∃y0, . . . , yn−1

(∧i 6=j

¬X(yi, yj)

): n < ω

}

∪

{∃X0, . . . , Xn−1

(∧i 6=j

Xi 6= Xj

): n < ω

}

∪

{∀X0, . . . , Xn−1∀y0, . . . , yn−1∃y

(∧i 6=j

Xi 6= Xj →∧i<n

Xi(y, yi)

): n < ω

}Tfeq says that each equivalence relation has infinitely many classes, that there are infinitelymany E-sort elements, and that given any n distinct classes r0, . . . , rn−1, and n (not neces-sarily 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 T ∗feq is an extension of Tfeq.

Proof. Suppose M � T ∗feq. We show M � Tfeq.

1. For each n < ω, M � ∀X∃y0, . . . , yn−1

(∧i 6=j¬X(yi, yj)

).

Suppose r ∈ EM, 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 regardto the s-classes of the new elements.) Let each new element form its own r-class. Moreprecisely, we define RN by:

N � s(b, c) if and only if M � s(b, c) for all s ∈ EM, b, c ∈ PM;

N � ¬s(b, ai) ∧ ¬s(ai, b) for any s ∈ EM, b ∈ PM and i < n−m;

N �∧

i,j<n−m

s(ai, aj) for all s ∈ EM \ {r};

N �∧i 6=j

¬r(ai, aj).

Under this definition ofRN ,N � T0,N extendsM, andN � ∃y0, . . . , yn−1

(∧i 6=j¬r(yi, yj)

).

As M is existentially closed for models of T0, M must already contain at least n r-classes. Since r was arbitrary, we have that

M � ∀X∃y0, . . . , yn−1

(∧i 6=j

¬X(yi, yj)

),


as desired.

2. For each n < ω, M � ∃X0, . . . , Xn−1

(∧i 6=jXi 6= Xj

).

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

N � s(a, b) if and only if M � s(a, b) for all s ∈ EM, a, b ∈ PM;

N � ri(a, b) for any i < n, any a, b ∈ PM.

That is, N adds n equivalence relations to M, each of which has only one class. It is

clear that M embeds into N , N � T0, and N � ∃X0, . . . , Xn−1

(∧i 6=jXi 6= Xj

). Since

M is existentially closed for models of T0, it satisfies the desired axiom.

3. For each n < ω, M � ∀X0, . . . , Xn−1∀y0, . . . , yn−1∃y

(∧i 6=jXi 6= Xj →

∧i<n

Xi(y, yi)

).

Suppose r0, . . . , rn−1 are distinct elements of EM, and a0, . . . , an−1 are (not necessarilydistinct) elements of PM. Again, we build a model N of T0 extending M. Letthe universe of N be the union of M and a single new element, b. For each s ∈EM \ {r0, . . . , rn−1}, let b be in a new s-class of its own. For each ri, put b in the sameri class as ai (and all of the other elements in ai’s ri-class, according to M). Moreprecisely, we define RN as follows:

N � s(c, d) if and only if M � s(c, d) for all s ∈ EM, c, d ∈ PM;

N � ¬s(c, b) ∧ ¬s(b, c) for all s ∈ EM, c ∈ PM;

N �∧i<n

ri(b, ai) ∧ ri(ai, b);

N �

{ri(b, c) ∧ ri(c, b) if M � ri(ai, c)¬ri(b, c) ∧ ¬ri(c, b) otherwise

for any c ∈ PM.

With this definition of RN , it is clear that N � T0, N is an extension of M, and

N � ∃y(∧i<n

ri(y, ai)

). Again, we note that since M is existentially closed for models

of T0, this must already be true in M, that is,

M � ∃y

(∧i<n

ri(y, ai)

).


Since r0, . . . , rn−1 were arbitrary distinct elements of EM and a0, . . . , an−1 were arbi-trary elements of PM, 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 thesecond kind, and in particular, T is not simple.

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

π(Xi, z(j,k), yα : i, j, k ∈ ω, α ∈ ωω) =⋃α∈ωω

{Xi(yα, z(i,α(i))) : i < ω}

∪ {Xi 6= Xj : i < j < ω}

∪⋃i<ω

{¬Xi(z(i,j), z(i,k)) : j < k < ω}.

Given a finite subset of this type, there are n,m, l ∈ ω such that the subset is contained ina type of the following form, in variables Xi for i < n, yα0 , . . . , yαl−1

for αi ∈ ωω, 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) =⋃r<l

{Xi(yαr , z(i,αr(i))) : i < n}

∪ {Xi 6= Xj : i < j < n}

∪⋃i<n

{¬Xi(z(i,j), z(i,k)) : j < k < m}.

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 elementsa(i,0), . . . , a(i,m−1) of PM from distinct si-classes. We can do this because each element ofEM has infinitely many classes. Lastly, for each r < l, we find bαr satisfying∧

i<n

si(y, a(i,αr(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

(si, a(i,j), bα : i, j < ω, α ∈ ωω)

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)


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 thatϕ(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 T ∗feq does not have the tree property of the first kind (or, in theterminology 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)


is (equivalent to a formula) of the form

ψ(xY ; zW ) :=∧

(i,j,k)∈I1

Yk(xi, xj) ∧∧

(i,j,k)/∈I1

¬Yk(xi, xj) ∧∧

(i,j,k)∈I2

Yk(xi, zj)

∧∧

(i,j,k)/∈I2

¬Yk(xi, zj) ∧∧

(i,j,k)∈I3

Yk(zi, zj) ∧∧

(i,j,k)/∈I3

¬Yk(zi, zj)

∧∧

(i,j,k)∈J1

Wk(xi, xj) ∧∧

(i,j,k)/∈J1

¬Wk(xi, xj) ∧∧

(i,j,k)∈J2

Yk(xi, zj)

∧∧

(i,j,k)/∈J2

¬Yk(xi, zj) ∧∧

(i,j,k)∈J3

Wk(zi, zj) ∧∧

(i,j,k)/∈J3

¬Wk(zi, zj)

∧∧

(i,j)∈K1

xi = xj ∧∧

(i,j)/∈K1

xi 6= xj ∧∧

(i,j)∈K2

Yi = Yj ∧∧

(i,j)/∈K2

Yi 6= Yj

∧∧

(i,j)∈K3

zi = zj ∧∧

(i,j)/∈K3

zi 6= zj ∧∧

(i,j)∈K4

Wi = Wj ∧∧

(i,j)/∈K4

Wi 6= Wj

∧∧

(i,j)∈L1

xi = zj ∧∧

(i,j)/∈L1

xi 6= zj ∧∧

(i,j)∈L2

Yi = Wj ∧∧

(i,j)/∈L2

Yi 6= Wj

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, andL2 ⊆ n× s. However, we may eliminate the equalities: any equalities among variables of thesame ‘type’ (where by ‘type’ we mean both sort and either object or parameter - so xi is thesame type as xj, but not the same type as zj, for example) simply make one of the variablesredundant. That is, since the formula is consistent in the first place, we may replace it bya 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 branchesof the tree are consistent, the jth point parameter must be the same along any branch, andhence (by Lemma 1.5.15), throughout the tree. If ψ(xY ; zW ) has SOP2, so does the formulaψ′ obtained by removing xi and any conjuncts containing it. (SOP2 for ψ′ is witnessed bythe same parameters as for ψ).


Proposition 2.2.3 (Adapted from [23], Theorem 2.1). No formula of the form

ψ(xY ; zW ) :=∧

(i,j,k)∈I1

Yk(xi, xj) ∧∧

(i,j,k)/∈I1

¬Yk(xi, xj) ∧∧

(i,j,k)∈I2

Yk(xi, zj)

∧∧

(i,j,k)/∈I2

¬Yk(xi, zj) ∧∧

(i,j,k)∈I3

Yk(zi, zj) ∧∧

(i,j,k)/∈I3

¬Yk(zi, zj)

∧∧

(i,j,k)∈J1

Wk(xi, xj) ∧∧

(i,j,k)/∈J1

¬Wk(xi, xj) ∧∧

(i,j,k)∈J2

Yk(xi, zj)

∧∧

(i,j,k)/∈J2

¬Yk(xi, zj) ∧∧

(i,j,k)∈J3

Wk(zi, zj) ∧∧

(i,j,k)/∈J3

¬Wk(zi, zj)

∧∧i 6=j

xi 6= xj ∧∧i 6=j

Yi 6= Yj ∧∧i 6=j

zi 6= zj ∧∧i 6=j

Wi 6= Wj

∧∧

i∈lg(x),j∈lg(z)

xi 6= zj ∧∧

i∈lg(Y ),j∈lg(W )

Yi 6= Wj

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

Proof. We work in a sufficiently saturated model M of T ∗feq. (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 usesuperscripts to denote coordinates (e.g. ai is the ith coordinate of a), so as to avoid conflictwith subscripts denoting nodes.

Suppose, for a contradiction, that a formula ψ(xY ; zW ) (of the form listed in the state-ment of the proposition) has SOP2. By Theorem 1.5.6, there is a 1-fti tree 〈aαrα : α ∈ <ω2〉witnessing SOP2 for ψ(xY ; zW ) (so lg(aα) = lg(z) and lg(rα) = lg(W ) for each α). Notethat by Corollary 1.5.17, since our tree is 1-fti, aηrη ∩ aνrν is the same for any η 6= ν ∈ <ω2.In what follows, let ct := a〈0〉r〈0〉 ∩ a〈1〉r〈1〉 (where c ∈ PM and t ∈ EM). By Remark 1.5.18,since the tree is 1-fti, tp(aηrη/ct) = tp(aνrν/ct) for all η, ν ∈ <ω2. In fact, by the proof ofCorollary 1.5.17, for each ci ∈ c and tj ∈ t there are k, l such that akη = ci and rlη = tj for allη.

Take b〈0〉s〈0〉 and b〈1〉s〈1〉 to be realizations of the instances of ψ at the nodes 〈0〉 and 〈1〉,respectively:

b〈0〉s〈0〉 � ψ(xY ; a〈0〉r〈0〉);

b〈1〉s〈1〉 � ψ(xY ; a〈1〉r〈1〉).

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

N0 = b〈0〉 ∪ s〈0〉 ∪ a〈0〉 ∪ r〈0〉,N1 = b〈1〉 ∪ s〈1〉 ∪ a〈1〉 ∪ r〈1〉, and

B = a〈0〉 ∪ r〈0〉 ∪ a〈1〉 ∪ r〈1〉,


respectively (regarding each tuple as the set of its coordinates). Note that these are allmodels of T0, and the diagrams of N0 and N1 are given by ψ. We shall show that N0 andN1 can be amalgamated into a finite model N of T0 extending B, with b〈0〉s〈0〉 and b〈1〉s〈1〉being mapped to the same elements of N , b′s′.

Lemma 2.2.4. We can amalgamate N0 and N1 into some N � T0 extending B.

Proof. We take the universe of N to be {b′s′, a〈0〉r〈0〉, a〈1〉r〈1〉} (where b′ and s′ are new, oflengths lg(b〈0〉) and lg(s〈0〉), respectively), and define three maps into N :

fB :B ↪→ N is the identity on B

f0 :N0 ↪→ N is given by

{b〈0〉s〈0〉 7→ b′s′

a〈0〉r〈0〉 7→ a〈0〉r〈0〉

f1 :N1 ↪→ N is given by

{b〈1〉s〈1〉 7→ b′s′

a〈1〉r〈1〉 7→ a〈1〉r〈1〉

We claim that we can define R on N so that each of these maps is an embedding, andN � T0. We begin with R0 := f0(RN0) ∪ f1(RN1) ∪RB.

Claim 2.2.5. R0 adds no “new relations.” That is, if C ∈ {N0,N1,B} and d, e ∈ CPand q ∈ CE, then fC(d, e, q) ∈ R0 if and only if (d, e, q) ∈ RC. Equivalently, if C1 andC2 are two distinct structures from the set {N0,N1,B}, and d, e ∈ fC1(CP1 ) ∩ fC2(CP2 ) andq ∈ fC1(CE1 ) ∩ fC2(CE2 ), then (d, e, q) ∈ fC1(RC1) if and only if (d, e, q) ∈ fC2(RC2).

Proof of claim. There are three cases to consider.

• (d, e, q) ∈ f0(N0) ∩ f1(N1): There are several subcases, depending on whether each ofd, e, and q is in B or not, but in all cases the result follows from the fact that N0 andN1 have the same diagram. For example, if there are i, j, and k such that d = b′i,e = b′j, and q = s′k, then

(d, e, q) ∈ f(RN0)↔ N0 � sk〈0〉(b

i〈0〉, b

j〈0〉)

↔ ψ(xY ; zW ) ` R(xi, xj, Y k)

↔ N1 � sk〈1〉(b

i〈1〉, b

j〈1〉)

↔ (d, e, q) ∈ f(RN1).

The arguments in the other subcases are the same, except that some or all of therelevant variables in the second line may be from z and W , rather than x and Y . (Weare using the fact that elements of c and t come from the same coordinates in a〈0〉 asin a〈1〉.)


• (d, e, q) ∈ f0(N0) ∩ fB(B): In this case, the result follows from the facts thatf0(N0)∩fB(B) = N0∩B and thatN0 and B are both substructures ofM, soRN0 andRB

agree on N0 ∩B. Suppose d = f(ai〈0〉) = ai〈0〉, e = f(aj〈0〉) = aj〈0〉, and q = f(rk〈0〉) = rk〈0〉.Then we have:

(d, e, q) ∈ f0(RN0)↔ N0 � rk〈0〉(a

i〈0〉, a

j〈0〉)

↔M � rk〈0〉(ai〈0〉, aj〈0〉)

↔ B � rk〈0〉(ai〈0〉, aj〈0〉)

↔ (d, e, q) ∈ fB(RB).

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

We extend R0 to all of PN × PN × EN as follows:

• RN (b′i, aj〈0〉, rk〈1〉) and RN (aj〈0〉, b

′i, rk〈1〉) hold if and only if there is l such that:

N1 � R(bi〈1〉, al〈1〉, r

k〈1〉) and

B � R(aj〈0〉, al〈1〉, r

k〈1〉).

• RN (b′i, aj〈1〉, rk〈0〉) and RN (aj〈1〉, b

′i, rk〈0〉) hold if and only if there is an l such that:

N0 � R(bi〈0〉, al〈0〉, r

k〈0〉) and

B � R(aj〈1〉, al〈0〉, r

k〈0〉).

• RN (ai〈0〉, aj〈1〉, s

′k) and RN (aj〈1〉, ai〈0〉, s

′k) hold if and only if there is an l such that:

N0 � R(ai〈0〉, bl〈0〉, s

k〈0〉) and

N1 � R(aj〈1〉, bl〈1〉, s

k〈1〉).

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 thefact that RN0 , RN1 , and RB are reflexive and symmetric). We must show that RN (·, ·, q) istransitive for all q ∈ EN .

1. q = rl〈1〉 ∈ r〈1〉. RN (u, v, rl〈1〉) and RN (v, w, rl〈1〉).


a) v ∈ a〈1〉: then RN (u,w, rl〈1〉) follows from one of: transitivity of RN1(·, ·, rl〈1〉) (if

u,w ∈ f1(N1)), transitivity of RB(·, ·, rl〈1〉) (if u,w ∈ fB(B) = B), or the definition

of RN (if one of u,w comes from b′ - i.e., f1(N1)\ (fB(B)∩f1(N1)) - and the othercomes from a〈0〉 - i.e., fB(B) \ (fB(B) ∩ f1(N1)).

b) v = aj〈0〉 ∈ a〈0〉.• If u,w ∈ fB(B) = B, thenRN (u,w, rl〈1〉) follows from transitivity ofRB(·, ·, rl〈1〉).

Otherwise, one or both of u,w comes from b′.

• Suppose u = b′i, w = b′k (i.e., u, b ∈ N \ fB(B)). Then by the definition ofRN , RN (b′i, aj〈0〉, r

l〈1〉) and RN (aj〈0〉, b

′k, rl〈1〉) imply that there are m and n suchthat:

RN1(bi〈1〉, am〈1〉, r

l〈1〉) and RB(aj〈0〉, a

m〈1〉, r

l〈1〉)

RB(aj〈0〉, an〈1〉, r

l〈1〉) and RN1(bk〈1〉, a

n〈1〉, r

l〈1〉)

By transitivity of RB(·, ·, rl〈1〉), it follows that RB(am〈1〉, an〈1〉, r

l〈1〉). By Claim

2.2.5, RN1(am〈1〉, an〈1〉, r

l〈1〉) holds, too. Two applications of the transitivity of

RN1(·, ·, rl〈1〉) give us that RN1(bi〈1〉, bk〈1〉, r

l〈1〉), and thus that RN (b′i, b′k, rl〈1〉).

• Suppose u = b′i and w ∈ B. We have am〈1〉 as above. Since RB(aj〈0〉, w, rl〈1〉),

transitivity of RB(·, ·, rl〈1〉) gives us that RB(am〈1〉, w, rl〈1〉). Then RN (b′i, w, rl〈1〉)

comes from either transitivity of RN1(·, ·, rl〈1〉) (if w ∈ a〈1〉, in which case we

use, once again, the fact that RB and RN1 agree on elements of a〈1〉r〈1〉) orfrom the definition of RN (if w ∈ a〈0〉).

c) v = b′j ∈ b′. The reasoning here will be the same as in the previous subcase.

2. q = rl〈0〉 ∈ r〈0〉. RN (u, v, rl〈0〉) and RN (v, w, rl〈0〉). The reasoning here will be the sameas in the previous case.

3. q = s′l ∈ s′. RN (u, v, s′l) and RN (v, w, s′l).

a) v ∈ b′. Then RN (u,w, s′l) follows from transitivity in RN1 (respectively, RN0) if uand w are both in f1(N1) (respectively, f0(N0)). If u ∈ a〈0〉 and w ∈ a〈1〉 (or viceversa), RN (u,w, s′l) comes from the definition of RN .

b) v = aj〈0〉 ∈ a〈0〉.• If u,w ∈ f0(N0), then RN (u,w, s′l) comes from transitivity in RN0 . Other-

wise, one or both of u,w comes from a〈1〉.

• Suppose u = ai〈1〉, w = ak〈1〉. Then, by definition of RN , RN (ai〈1〉, aj〈0〉, s

′l) and

RN (aj〈0〉, ak〈1〉, s

′l) imply that there are m and n such that:

RN1(ai〈1〉, bm〈1〉, s

l〈1〉) and RN0(aj〈0〉, b

m〈0〉, s

l〈0〉)

RN0(aj〈0〉, bn〈0〉, s

l〈0〉) and RN1(ak〈1〉, b

n〈1〉, s

l〈1〉)


By transitivity in RN0 , this gives us that RN0(bm〈0〉, bn〈0〉, s

l〈0〉). Since RN0 and

RN1 agree on their respective preimages of b′s′, RN1(bm〈1〉, bn〈1〉, s

l〈1〉) holds.

Then by transitivity (twice) in RN1 , we have RN1(ai〈1〉, ak〈1〉, s

l〈1〉), and thus

RN (ai〈1〉, ak〈1〉, s

′l), as desired.

• Suppose u = ai〈1〉 and w ∈ f0(N0). We have b′m as above. Then, since

RN0(aj〈0〉, w, sl〈0〉), transitivity in RN0 gives us RN0(w, bm〈0〉, s

l〈0〉). Then, by def-

inition of RN (if w ∈ a〈0〉) or by transitivity in RN1 (if w ∈ b′), it follows thatRN (ai〈1〉, w, s

′l).

c) v = aj〈1〉 ∈ a〈1〉. 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 thatthe extension of the definition of R0 to PN ×PN ×EN adds no new relations among imagesof elements from the same structure, f0, f1, and fB are embeddings, and N is the desiredamalgam. This finishes the proof of Lemma 2.2.4.

Since f0 : N0 ↪→ N and f1 : N1 ↪→ N , and since ψ is quantifier free,

N � ψ(b′s′; a〈0〉r〈0〉) ∧ ψ(b′s′; a〈1〉r〈1〉).

As N � T0 and T ∗feq is the model completion of T0, N embeds into a model N ∗ of T ∗feq. Again,since ψ is quantifier free,

N ∗ � ψ(b′s′; a〈0〉r〈0〉) ∧ ψ(b′s′; a〈1〉r〈1〉).

Using, once again, the fact that T ∗feq is the model completion of T0, we note that T ∗feq∪diag(B)is a complete LB-theory. Since

(N ∗, a〈0〉, r〈0〉, a〈1〉, r〈1〉) � T ∗feq ∪ diag(B),

and since ∃x∃Y (ψ(xY ; a〈0〉r〈0〉)∧ψ(xY ; a〈1〉r〈1〉)) is an LB-sentence in Th(N ∗, a〈0〉, r〈0〉, a〈1〉, r〈1〉),T ∗feq ∪ diag(B) ` ∃x∃Y (ψ(xY ; a〈0〉r〈0〉) ∧ ψ(xY ; a〈1〉r〈1〉)). We also know that

(M, a〈0〉, r〈0〉, a〈1〉, r〈1〉) � T∗feq ∪ diag(B),

and soM � ∃x∃Y (ψ(xY ; a〈0〉r〈0〉) ∧ ψ(xY ; a〈1〉r〈1〉)),

contradicting the choice of 〈aαrα : α ∈ <ω2〉 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 contradictsthe fact that T ∗feq has the tree property, since a tree witnessing TP must contain nodes 〈0〉and 〈1〉 such that the instances of the formula in question corresponding to these nodes areinconsistent with each other. However, Proposition 2.2.3 relies on the fact that an SOP2 (orTP1) 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 anon-degenerate bilinear form, studied by Nicolas Granger in his thesis [9]. The well-behavednon-forking independence in finite dimensional subspaces (which are ω-stable of finite Morleyrank) of a model of this theory can be lifted to another independence relation (not quite, butalmost, as well-behaved) in the larger structure. We generalize this independence relationconstruction to any theory whose models can be “approximated” by certain substructuresin the same way that Granger’s infinite dimensional vector space is approximated by finitedimensional subspaces.

3.1 Motivation

Here we present the two-sorted theory of an infinite dimensional vector space with a bilinearform, along with Granger’s independence relation for this theory. Most definitions and resultsin 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 fieldelements). 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 lowercase 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 fieldand scalar multiplication simply by concatenation of the elements being multiplied, ratherthan using the symbols given above. We may leave out the subscripts of ‘+V ’ and ‘+K ’ (itshould be clear from the context which one we intend) and will make use of ‘

∑’ (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 bemade so by adding symbols for −V , −K , and ÷K . We include only universal axioms for thebilinear form. In particular, Tu does not require the form to be non-degenerate.) We buildup 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 ∪ Th(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 thereare at least n linearly independent vectors:

Θn := ∃X0, . . . , Xn−1∀y0, . . . , yn−1

(n−1∑i=0

yiXi = 0V →n−1∧i=0

yi = 0K

).


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}.

If K is finite and F ∈ {S,A}, FT K∞ is ℵ0-categorical, super simple of SU-rank 1, and un-stable. (The formula [X, Y ] = 0 has the order property.) See [9], Corollary 6.2.4, Proposition6.2.5, 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 languageto include predicates for linear independence. For each n ∈ ω, let

θn(X0, . . . , Xn−1) := ∀y0, . . . , yn−1

((n−1∑i=0

yiXi = 0V

)→

(n−1∧i=0

yi = 0K

)).

Let Lθ be the language obtained from L by adding each θn as an atomic formula. (Moreprecisely, Lθ is the definitional expansion of L having new n-ary predicate symbols on thesort V to be interpreted as θn.) Let LKθ be the language obtained from Lθ by adding anelimination set for K.

Proposition 3.1.1 ([9], Theorem 9.2.3 and Proposition 9.3.4). Given a field K, F ∈ {S,A},and m ∈ N ∪ {∞}, the theory FT

Km has elimination of quantifiers in the language LKθ . In

particular, if K is an algebraically closed field, then FTKm has elimination of quantifiers in

the language Lθ. Further, if K is algebraically closed, FTKm is model complete in the language

L.

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

1. The theory ATKm is complete.

2. If K is a field with square roots for each element then the theory STKm is complete.

Since we find it rather cumbersome to continually refer to a theory with three distinctdecorations, we will generally abuse language and notation by speaking of “the theory T∞”(or “the theory Tn”). By this, we mean FT

K∞ (FT

Kn ) 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 Morleyrank.


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,and work in any modelM = (V,K) � T∞. We construct a tree {aηγη : η ∈ <ωω} witnessingthat this formula has the tree property.

• Let aηai be some vector that is linearly independent from {aη�j : j ≤ |η|} (the samevector 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{γη�j : j ≤ |η|}. Let γηai = γi. (This is possible because K is infinite.)

Now we check the tree property conditions:

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

{[X, aβ�j] = γβ�j : j < ω}

is consistent.

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

Hence, the formula [X, Y ] = y (with Y y as the parameter variables) has the tree property inT∞, and T∞ is not simple. Note that this argument holds whenever K (the field such thatTh(K) ⊂ T∞) is infinite - not just for algebraically closed fields.

We need a few more definitions before we can state Granger’s independence relation. LetM � T∞ be sufficiently saturated. If A ⊂ M , we call the set of vector elements of A (i.e.A∩VM) AV , and the set of field elements of A (i.e. A∩KM) AK . If char(KM) = 0, Grangertakes KA to be the subfield of KM 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, . . . , bnlinearly independent and b =

∑ni=1 αibi.

If char(KM) > 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

A by spanL(AV ). We denote the KM-linear span of AV by 〈A〉.Remark 3.1.5 ([9], Proposition 9.5.1). For m ∈ N ∪ {∞}, M = (V,K) � Tm, and A ⊆M ,

1. dcl(A) = (spanKA(AV ), KA), and


2. except in the case that m is finite and dim(A) = m− 1, acl(A) = (spanKalgA

(AV ), KalgA ).

Remark 3.1.6. In the original statement of this Proposition, Granger specifies thatM shouldbe “sufficiently saturated.” This is unnecessary, however: since each Tm is model completewith 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

tp(a/B) does not Γ-fork (dnΓf) over C if KCa| K

M

KCKB (that is, KCa and KB are algebraically

independent over KC in the field KM)1 and one of the following three conditions holds:

1. a ∈ KM

2. a ∈ 〈C〉

3. a /∈ 〈B〉 and whenever {b1, . . . , bn} ⊂ BV is linearly independent modulo 〈C〉 then theset {[a, b1], . . . , [a, bn]} is algebraically independent over KB(KCa), the field generatedover KB by the elements of KCa.

Extend the relation inductively to n-types as follows: tp(a1, . . . , an, an+1/B) dnΓf over C ifand 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 theexistence of an algebraic relation among some of the pairings that was not forced by alinear relation among the vectors being paired. (Or it is trivial dependence: c ∈ 〈B〉 \ 〈C〉.)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 abstractindependence. The finite dimensional relation is almost identical to the infinite dimensionalrelation, but contains one additional possible condition for c:

4. a ∈ 〈B〉 \ 〈C〉 and 〈B〉 = V and whenever {b1, . . . , bn} ⊂ BV is linearly independentmodulo 〈C〉 then the set {[a, b1], . . . , [a, bn]} is algebraically independent over the fieldKB(KCa).

Granger shows that his Γ-non-forking satisfies automorphism invariance, symmetry, tran-sitivity, finite character, extension, and stationarity of types over models. In the finite di-mensional 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 uncountablemodels.)

1Note that independence of field elements as computed in M is the same as independence as computedin KM, since KM is stably embedded.


Remark 3.1.8. Since Γ-non-forking is symmetric for T∞ and T∞ is not simple, Γ-non-forkingis not the same as non-forking in T∞ (by Theorem 1.4.16). More concretely, the formulaϕ(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 � Tncontaining b, and an indiscernible sequence (bi : i < ω) of elements of N (or someelementary extension of N ) where b0 = b and {b0, . . . , bn−1} (and thus, any set of ndistinct 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 elementsbi0 , . . . , bin−1 from the sequence, if [a, bij ] = 0K for j < n, then [a, d] = 0K for alld ∈ V N , since {bi0 , . . . , bin−1} spans V N . Since the form is non-degenerate, a = 0V , soN 2 ϕ(a; b).

Alternatively, take a ∈ M � T∞ such that M � ϕ(a; b). Clearly a /∈ KM, 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 nonzerovector that pairs with finitely many other vectors to give 0K , and this presents noproblem 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 inde-pendence relation, Granger goes on to show that in a countable dimensional model of T∞, itis the “limit” of Γ-non-forking (and hence, non-forking) in a sequence of finite dimensionalsubspaces.

Definition 3.1.9 ([9], Section 12.1). If M � T∞, an approximating sequence for M is asequence of homogeneous substructures Nr ⊆ M such that M =

⋃r∈ωNr and for each r,

Nr � Tr and KNr = KM. (N is a homogeneous substructure of M if given tuples a, b ∈ Nsuch that there is σ ∈ Aut(M) sending a to b, then there is τ ∈ Aut(M) sending a to b andfixing N setwise.)

Remark 3.1.10. By this definition, only countable dimensional models of T∞ have approx-imating sequences, since a countable union of finite dimensional subspaces could not coveran uncountable dimensional model. Approximating a model of T∞ by substructures will notbe fruitful unless the substructures in question have a nicer theory than T∞, so it would notbe helpful to extend this definition by allowing a longer sequence of substructures, the tailend of which consists of infinite dimensional spaces.

Admittedly, Granger does not require that an approximating sequence of substructuresbe a chain of substructures (although he may be assuming it implicitly), so one could imagine


taking an uncountable sequence of finite dimensional subspaces that simply does not satisfyNr ⊂ Ns for r < s. This seems unnecessarily messy, however, and there are more reasonableways to extend the definition. When we generalize this framework in the next section, weshall make use of directed systems rather than sequences, and this will allow us to includeuncountable 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 forkover 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 ⊆ Bthere is Rb ∈ ω such that p(x) �Nr∩Cb does not fork over C ∩Nr in Nr for all r ≥ Rb.” This isnot actually equivalent to the other two. The error in the proof lies in Granger’s assumptionthat p(x) �Nr∩Cb has a realization in Nr (that is, an element of Nr that Nr believes satisfiesthe type). If p(x) is a complete type, however, then for every r < ω, p(x) �Nr∩Cb is notrealized 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 unsimpletheory T∞. Given some theory T , we begin with a strongly κ-homogeneous, κ-saturatedmodel M � T , and find a directed system, H, of substructures of M. In the case ofM = (K,V ) � T∞, H is the collection of finite dimensional K-subspaces of (K,V ). Beforeimposing 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 truein 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 thatπ(x, b) eventually forks over C if there is Nπ,b,C ∈ H such that b ∈ Nπ,b,C and for allNπ,b,C ⊆ N ∈ H, π(x, b) forks over C ∩N in N . We define eventual non-forking of afinite 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 notfork over C.


4. For subsets A,B, and C of M , we say that A | limC B (A is limit independent or even-tually independent from B over C) if for each finite tuple a ∈ A, there is Na ∈ H suchthat a ∈ Na and for all Na ⊆ N ∈ H,

a | NaclM(C)∩N aclM(B) ∩N

where | N 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 enoughN ” to abbreviate“there is N ′ ∈ H such that for all N ′ ⊆ 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 ,” wemean that for some formulae ϕ0(x, b0), . . . , ϕn−1(x, bn−1) and some k < ω, the followinginfinitary type is consistent with Th(N ):

{π(x)→∨i<n

ϕi(x, bi)} ∪⋃i<n

{ψ(xji )↔ ψ(bi) : j < ω, ψ ∈ L(A)}

∪⋃i<n

{¬∃x∧j∈S

ϕi(x, xji ) : S ⊂ ω, |S| = k}.

Notice that under this definition of “forking in N ,” the elements of H need not besaturated.

2. Note that eventual non-forking is not the same as not eventually forking. In principle,a formula or type could both fork in arbitrarily large N ∈ H and not fork in arbitrarilylarge N ∈ H.

3. If p(x) is a complete type that eventually forks over C, then p(x) contains a formulathat eventually forks over C.

4. It is tempting to define eventual forking of an infinite type as the forking of restrictionsof the type in large enough N . This harks back to the problem mentioned in Remark3.1.12: unless the elements of H have the same theory as M (which would make Hrather unhelpful), for any A and any complete type p ∈ SM(A), p �N is false in eachN , and hence trivially forks in N . We do not want every complete type to eventuallyfork, so we must instead define eventual forking of types finitarily.


5. In principle, A | limC B is not equivalent to the eventual non-forking of any type fromM(say, tpM(A/BC)), since limit independence involves the forking of types as computedin the approximating structures, while eventual non-forking of a type involves forkingin the approximating structures of a fixed set of formulae. For example,

a |/ NC∩NB ∩N

might be caused by the forking of some formula from tpN (a/(B ∪ C) ∩ N) that doesnot appear in tpM(a/B ∪ C).

Definition 3.2.3. GivenM � T and H a directed system of substructures N ofM, we saythat H approximates M just in case conditions 1 through 5 hold. We also identify optionalconditions 6 and 7.

1. H covers M:⋃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 ifM 2 ψ(a), then ψ(a) is eventuallyfalse 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), tupleb, 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.

5. Strong finite character of limit dependence: If A |/ limC 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 allN ⊇ Nalg, A ∩N = aclN (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 algebraicclosure of B in H, then a is in the algebraic closure of B in M.)


To see this, suppose a ∈ aclN (B ∩ N) for N ⊇ Nalg. Then a ∈ aclN (B ∩ N) forN ⊇ Nalg, since B ⊆ B. Since B is algebraically closed inM, condition 6 implies thatfor 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 largeenough N , a ∈ aclN (B ∩ N), since the formula declaring a’s algebraicity over Bis eventually true in H. (More precisely, if M � ϕ(a, b) ∧ (∃=nx)ϕ(x, b), then bycondition 3 there is Nalg such that for all N ⊇ Nalg, N � ϕ(a, b) ∧ (∃=nx)ϕ(x, b), andhence a ∈ aclN (B ∩N).)

2. One might consider adding the following uniformity condition: “There is a collection offormulae {ϕi(x, y, z) : i ∈ I} such that for anyM � T , there are parameters {ai : i ∈ I}such that H = {ϕi(x,M, ai) : i ∈ I} satisfies the requirements of approximation.” Ifthis holds, then approximation is really a property of the theory T , not of the particularmodel chosen. In the case of Granger’s T∞, the ϕi are of the form

Σj<iyjZj = X,

and ϕi(X,M, a1, . . . , ai) gives rise to the KM-vector space generated by {a1, . . . , ai}.

Lemma 3.2.5. Suppose H satisfies conditions 1 through 5. Given sets A, B, and C, A |lim

C Bif and only if tpM(A/B ∪ C) eventually does not fork over C.

Proof. ←: Suppose A |/ limC 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. Sinceψ(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 thateventually forks over C. Let a ∈ A realize π(x) in M. Since π is finite, by condition 3 onH there is some Ntv ∈ H containing a, b and c such that for N ⊇ Ntv, N � π(a, b, c) (thatis, π(x, b, c) ⊂ tpN (a/(B ∪C)∩N)). Since π eventually forks over C, there is some Nf suchthat for N ⊇ Nf , π(x, b, c) forks over C ∩N in N . For N ⊇ Ntv ∪Nf , tpN (a/(B ∪C)∩N)forks over C ∩N in N . That is,

a |/ NC∩NB ∩N.

By definition, A |/ limC B.

Remark 3.2.6. In fact, Lemma 3.2.5 does not require condition 4 on H, but the proof issimpler to state with that assumption.

We have defined | lim without imposing the requirement that the set on the right sidemust contain the base set. We now show that this does not pose a problem; namely, that if

A | limC B, we may enlarge the righthand side to contain C, while maintaining the indepen-dence. We shall refer to this condition as “normality on the right.”


Lemma 3.2.7. Suppose H satisfies conditions 1 through 5. Given sets A, B, and C, if

A |lim

C B, then A |lim

C BC.

Proof. Suppose A |/ limC BC. By condition 5 on H, there are ψ(x, y, z) ∈ L, a ∈ A, b ∈ BC,

and c ∈ C such thatM � ψ(a, b, c) and ψ(x, b, c) eventually forks over C. Take Ntv, Nf , andNalg such that for N ⊇ Ntv, N � ψ(a, b, c), for N ⊇ Nf , ψ(x, b, c) forks over C∩N in N , andforN ⊇ Nalg, b ∈ aclN (BC∩N). ForN ⊇ Ntv∪Nf∪Nalg, ψ(x, b, c) ∈ tpN (a/ aclN (BC∩N))and forks over C ∩N in N . That is,

a |/ NC∩N aclN (BC ∩N).

By Corollary 1.4.9,a |/ N

C∩NBC ∩N.

By monotonicity of non-forking,

a |/ NC∩N(B ∪ C) ∩N,

and by the definition of forking independence,

a |/ NC∩NB ∩N.

It follows by definition that A |/ limC B.

3.3 Properties of the relation

We now show that in the event that the structures in H have simple theories, the relationdefined in the last section satisfies many of the properties that forking independence sat-

isfies in a simple theory. In particular, | lim is preserved under taking algebraic closuresand has automorphism-invariance, monotonicity, base monotonicity, transitivity, normality,extension, finite character, (a weakened version of) strong finite character, and symmetry. Ifwe impose some additional conditions on H, we also get anti-reflexivity and an independencetheorem over algebraically closed sets.

Lemma 3.3.1. Suppose H satisfies conditions 1 through 3, and for each N ∈ H, forking and

dividing are equivalent in Th(N ). For sets A,B, and C, A |lim

C B if and only if A |lim

C B.

Proof. We break the proof up into three parts.

1. A | limC B if and only if A | limC B.

←: Suppose A |/ limC B. Then there is some finite a ∈ A such that for arbitrarily large

N ∈ H, a |/ NC∩NB ∩N . But a ∈ A, too, so by definition A |/ lim

C B.


→: Suppose A |/ limC B. Then there is some finite a′ ∈ A such that for arbitrarily

large N ∈ H, a′ |/ NC∩NB ∩ N . As noted in Remark 3.2.4, by condition 3 there is

Nalg ∈ H such that for N ⊇ Nalg, a′ ∈ aclN (A ∩ N). Take a ∈ A finite such that

a′ ∈ aclM(a), and for N ⊇ Nalg, a′ ∈ aclN (a). If N ⊇ Nalg and a′ |/ NC∩NB ∩ N ,

then aclN (a) |/ NC∩NB ∩N by monotonicity of | N . Since forking and dividing are the

same in N , it follows by Corollary 1.4.5 that a |/ NC∩NB ∩N . Since this is the case for

arbitrarily large N , A |/ limC B.

2. A | limC B if and only if A | limC B.

A |/ limC B ⇔ a ∈ A, a |/ N

C∩NB ∩N for arbitrarily large N⇔ a ∈ A, a |/ N

C∩NaclM(B) ∩N for arbitrarily large N

⇔ A |/ limC B.

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

Combining these three facts, we have:

A | limC B if and only if A | limC B

if and only if A | limC B

if and only if A | limC B.

Theorem 3.3.2. Given a κ-saturated, strongly κ-homogeneous modelM of T , approximatedby H (that is, satisfying conditions 1 through 5, above), where each N ∈ H has a simple

theory, |lim

has the following properties:

• invariance If A |lim

C B and (A′, B′, C ′) ≡ (A,B,C), then A′ |lim

C′ B′.

• monotonicity If A |lim

C B, A′ ⊆ A and B′ ⊆ B, then A′ |lim

C B′.

• base monotonicity If D ⊆ C ⊆ B and A |lim

D B, then A |lim

C B

• transitivity If D ⊆ C ⊆ B then A |lim

D C and A |lim

C B if and only if A |lim

D B. (The‘if ’ - known as partial transitivity - follows from monotonicity and base monotonic-ity.)

• normality If A |lim

C B, then AC |lim

C B.


• extension If A |lim

C B and B ⊂ B′, then there is A′ ≡BC A such that A′ |lim

C B′.

• finite character If A |/ limC B, then there are finite tuples a ∈ A, b ∈ B such that

a |/ limC b.

• strong finite character∗ If A |/ limC 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

– a′ |/ limC b for all a′ � ψ(x, b, c).

• symmetry A |lim

C B if and only if B |lim

C A.

Proof. Since Th(N ) is simple for each N ∈ H, | N is dividing independence in N . In

the following, we will frequently refer to “Na, as in the definition of | lim.” Recall that if

A | limC B, then for each finite a ∈ A there is Na such that for N ⊇ Na,

a | NC∩NB ∩N.

This is the Na to which we are referring.

Invariance. Suppose A | limC B and (A,B,C) ≡ (A′, B′, C ′). We would like to show that

A′ | limC′ B′. Let σ be an automorphism sending A to A′, B to B′, and C to C ′. For any a ∈ A,

let Na be as in the definition of | lim. By condition 2 on H, σ(N ) ∈ H for any N ∈ H, sowe may let Nσ(a) = σ(Na). We claim that Nσ(a) is as required.

Write a′ for σ(a), and let N ′ ⊇ Na′ . We must show that

a′ | N′

C′∩N ′B′ ∩N ′.

If not, there are b′ ∈ B′ ∩N ′, c′ ∈ C ′ ∩N ′ and ψ(x, y, z) such that

ψ(x, b′, c′) ∈ tpN′(a′/(B′ ∪ C ′) ∩N ′)

and ψ(x, b′, c′) divides over C ′ ∩ N ′ in N ′. Hence, the following type is consistent withTh(N ′) (for some k < ω):

{ϕ(yi, zi)↔ ϕ(b′, c′) : i < ω, ϕ ∈ L(C ′ ∩N ′} ∪ {¬∃x∧i∈S

ψ(x, yi, zi) : S ⊂ ω, |S| = k}.

It follows that the image under σ−1 of this type is consistent with Th(N ):

{ϕ(yi, zi)↔ ϕ(b, c) : i < ω, ϕ ∈ L(C ∩N} ∪ {¬∃x∧i∈S

ψ(x, yi, zi) : S ⊂ ω, |S| = k}


(where σ(b) = b′, σ(c) = c′, and σ(N ) = N ′). This implies that ψ(x, b, c) k-divides overC ∩ N in N . Since ψ(x, b′, c′) ∈ tpN

′(a′/(B′ ∪ C ′) ∩ N ′), ψ(x, b, c) ∈ tpN (a/(B ∪ C) ∩ N),

and we havea |/ N

C∩NB ∩N.

Since N ′ ⊇ Na′ , N ⊇ Na, and the above is a contradiction.

Monotonicity. Suppose A0 ⊆ A, B0 ⊆ B, and A | limC B. If a ∈ A0, then a ∈ A, and sothere is Na such that for N ⊇ Na,

a | NC∩NB ∩N.

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

a | NC∩NB0 ∩N.

So the same Na’s witness that A0| limC B0.

Base monotonicity. Suppose D ⊆ C, and A | limD B. We must show that A | limC B. Ifnot, then by condition 5 on H, there are a ∈ A, b ∈ B, c ∈ C and ψ such thatM � 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 forN ⊇ Ntv, N � ψ(a, b, c). So for N ⊇ Nf ∪Ntv, tpN (a/bc) forks over C ∩N in N . It followsthat tpN (a/(B ∪ C) ∩N) does as well, that is,

a |/ NC∩NB ∩N.

By base monotonicity for | N ,a |/ N

D∩NB ∩N.

For N ⊇ Na (where Na is as in the definition of | lim), this contradicts our assumption that

A | limD B.

Transitivity. Suppose D ⊆ C ⊆ B, A | limD C, and A | limC B. Then for each a ∈ A thereare NDC

a and NCBa such that:

for N ⊇ NDCa , a | ND∩NC ∩N, and

for N ⊇ NCBa , a | NC∩NB ∩N.

Let NDBa be some element of H containing NDC

a ∪ NCBa . For N ⊇ NDB

a we have, by

transitivity of | N ,

a | ND∩NB ∩N.

It follows that A | limD B, as desired.


Normality. Suppose A | limC B. We must show that AC | limC B. If not, then by condition5 on H, there are ψ(xy; zw), a ∈ A, c ∈ C, b ∈ B, and c′ ∈ C such that M � ψ(ac; bc′) andψ(xy; bc′) eventually forks over C in H. Take Nf such that for N ⊇ Nf , ψ(xy; bc′) forks overC ∩ 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,

ac |/ NC∩NB ∩N.

By monotonicity of | N , {a} ∪ (C ∩N) |/ NC∩NB ∩N . By normality of | N , a |/ N

C∩NB ∩N .

For N ⊇ Na (where Na is as in the definition of | lim), 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 aformula, 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 overC. Then by the definition of eventual forking for types, there are finite subtypes π0 and π1 ofπ 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 = ψand ψ1 = ¬ψ). For N0 ∪N1 ⊆ N ∈ H, both of these finite types fork over C ∩N in N , andwe have

π0(x) ∪ {ψ(x, b)} `N∨i∈IN

θi(x)

andπ1(x) ∪ {¬ψ(x, b)} `N

∨i∈IN

ηi(x)

where each θi and ηi divides over C ∩N in N (and the particular formulae may depend onN ). It follows that

π0(x) `N ψ(x, b)→∨i∈IN

θi(x)

π1(x) `N ¬ψ(x, b)→∨i∈IN

ηi(x)

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

π0(x) ∪ π1(x) `N∨i∈IN

θi(x) ∨∨i∈IN

ηi(x)


But then π(x) `N∨i∈IN θi(x) ∨

∨i∈IN ηi(x), and π(x) forks over C ∩ N in N . Since this

happens in all large enoughN (with the witnessing disjunctions of dividing formulae possiblychanging), 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 comple-tion of π(x) by induction:

• π0(x) = π(x) (This is consistent and eventually non-forking by hypothesis.)

• For limit cardinals λ < |B|, let πλ :=⋃β<λ π

β. (Since consistency and eventual non-

forking are local properties, and all the πβ’s are consistent and eventually non-forking,so is πλ.)

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

• Let p(x) :=⋃α<|B| π

α. By construction, p is consistent, complete over B, extends π(x),and eventually does not fork over C.

By Lemma 3.2.5, A | limC B implies that p(X) = tpM(A/B ∪ C) eventually does notfork over C. Since B ⊂ B′, B ∪ C ⊂ B′ ∪ C, and by Claim 3.3.3, there is an extensionq(X) ∈ SM(B′ ∪ C) of p(X) that eventually does not fork over C.

Let A′ � q(X). Since q(X) = tpM(A′/B′ ∪C) eventually does not fork over C, we applyLemma 3.2.5 again to get

A′ | limC B′.

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 |/ limC B, there are a ∈ A, b ∈ B such that for large enough N , a |/ N

C∩Nb.

(This is slightly stronger than finite character because we specify that in each N , theforking actually comes from b, rather than b ∩N .)

Proof of Claim. Suppose A |/ limC 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:

M � (∃=nx)ϕ(x, b) ∧ ϕ(b′, b).

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

for N ⊇ Nalg, N � (∃=nx)ϕ(x, b) ∧ ϕ(b′, b).


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

N � ψ(a, b′, c′),

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

a |/ NC∩Nb

′.

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

a |/ NC∩N aclN (b).

By Corollary 1.4.9,a |/ N

C∩Nb.

Strong finite character∗. Suppose A |/ limC 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) andψ(x, b, c) forks over C ∩ N in N for all N ⊇ Nf . The first part of strong finite characteris clearly satisfied by this choice of ψ, a, b, and c. Now suppose that M � ψ(a′, b, c). Bycondition 3 on H, there is Ntv such that for N ⊇ Ntv, N � ψ(a′, b, c). Let N0 be someelement of H containing Nf ∪Ntv. For N ⊇ N0, tpN (a′/bc) forks over C ∩N in N , i.e.

a′ |/ NC∩Nb.

By definition, a′ |/ limC b, as desired.

Symmetry. Suppose B |/ limC A. By Claim 3.3.4, there are b ∈ B, a ∈ A, and Nb such

that for all Nb ⊆ N ∈ H,b |/ N

C∩Na.

By symmetry of | N , a |/ NC∩Nb, and hence by monotonicity of | N , a |/ N

C∩NB ∩ N . By

definition, A |/ limC B.

Remark 3.3.5. 1. Our ‘strong finite character∗’ is somewhat weaker than standard strongfinite 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 coveringcondition is used throughout. As for the others:

• The proofs of transitivity and symmetry use the full strength of simplicity ofeach Th(N ).

• The automorphism invariance condition is used in the proof of invariance.


• The proofs of base monotonicity, extension, and strong finite character all dependon Lemma 3.2.5, and hence on the convergence of truth value, stabilizationof non-forking, and strong finite character hypotheses.

• The proofs of normality and finite character use the convergence of truth valueand strong finite character hypotheses. The proof of normality also uses the

normality of each | N , and the proof of finite character uses monotonicity of | N

and Corollary 1.4.9. These are all properties of non-forking independence in anytheory.

• The proof of monotonicity relies only on the monotonicity of non-forking, whichis true in any theory.

• The extension for types claim (Claim 3.3.3) uses only the forking stabilizationhypothesis.

Proposition 3.3.6. If M � T and H satisfies conditions 1 and 6, then | lim satisfiesanti-reflexivity:

a |lim

B a implies a ∈ B.

Proof. Suppose a | limB a. Let Na be as in the definition of | lim. For N ⊇ Na,

a | NB∩N({a} ∪B) ∩N.

By monotonicity of | N , a | NB∩Na. By anti-reflexivity of | N , a ∈ aclN (B∩N). By Remark3.2.4, 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 Th(N )’s are simple,since anti-reflexivity of forking independence holds in all theories.

Proposition 3.3.8. SupposeM � T is κ-saturated and strongly κ-homogeneous, H satisfiesconditions 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

that are simple and for which |f

satisfies the Independence Theorem over algebraically closed

sets. Then |lim

satisfies the Independence Theorem over algebraically closed sets. That is, if

A = A, B |lim

A C, and d and c are such that tpM(d/A) = tpM(e/A), d |lim

A B, and e |lim

A C,

then there is a � tpM(d/AB) ∪ tpM(e/AC) with a |lim

A BC.

Proof. Suppose not. Then either tpM(d/AB) ∪ tpM(e/AC) is inconsistent (in which case itis eventually inconsistent, and hence trivially eventually forks, in H), or for each realizationa, a |/ lim

A BC. By Lemma 3.2.5 and the fact that A = A,

a |/ limA BC ↔ tpM(a/A ∪BC) eventually forks over A

↔ tpM(a/A ∪BC) eventually forks over A.


It follows (since this is true for all realizations a), that tpM(d/AB) ∪ tpM(e/AC) does nothave 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.) Byextension 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 finitesubtype π ⊂ tpM(d/AB)∪ tpM(e/AC) that eventually forks over A. The subtype π is of theform π1(x, a1, b) ∪ π2(x, a2, c), where π1 ⊂ tpM(d/AB), π2 ⊂ tpM(e/AC), b ∈ B, and c ∈ C.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:

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

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

• Nψ2,e, such that for N ⊇ Nψ2,e, N � ψ2(e, a2, c);

• Nb,c,A, such that for N ⊇ Nb,c,A, b | NA∩Nc (since B | limA C, b | limA c by monotonicity);

• Nd,b,A, such that for N ⊇ Nd,b,A, d | NA∩Nb;

• Ne,c,A, such that for N ⊇ Ne,c,A, e | NA∩Nc;

• Nalg, such that for N ⊇ Nalg, aclN (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 thatfor 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),

• b | NA∩Nc,

• d | NA∩Nb,

• e | NA∩Nc, and

• tpN (d/A ∩N) = tpN (e/A ∩N)


By our hypothesis that | N satisfies the Independence Theorem over algebraically closedsets, tpN (d/(A∩N)b)∪tpN (e/(A∩N)c) does not fork over A∩N. But sinceN ⊃ Nψ1,d∪Nψ2,d,

tpN (d/(A ∩N)b) ∪ tpN (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 eventualnon-forking in the finite dimensional substructures of M. Above, we showed that undercertain assumptions (in particular, the strong finite character assumption), failure of limitindependence corresponds to eventual forking of a type. Unfortunately, the strong finite

character condition does not always hold for | lim in models of T∞ (with H taken to be thecollection of finite dimensional substructures). When we restrict to single elements on the

left and finite dimensional sets on the right and in the base, however, | lim does satisfy our

assumptions, and is related to | Γ.

In the following, let M be a κ-saturated, strongly κ-homogeneous model of T∞ (forsome large κ), and H the collection of finite dimensional KM-subspaces of M (that is,the substructures N of M such that KN = KM and dimK(V N ) < ∞). We shall breakour 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 〈A〉 is finitedimensional.

We begin by reviewing some of Granger’s results on the relationship between a model ofT∞ and its finite dimensional subspaces. Recall that Lθ is the language that results fromadding the “linear independence” predicates θn to the two-sorted vector space language, andthat if K is an algebraically closed field, m ∈ N ∪ {∞}, and F ∈ {S,A}, the theory FT

Km

has elimination of quantifiers in Lθ. It turns out that there is some local uniformity to thisquantifier elimination.

Lemma 3.4.1 ([9], Proposition 9.3.3). 1. For any L-sentence, σ, there is R ∈ ω suchthat

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 suchthat 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,modulo FT

K, to a boolean combination of quantifier-free Lθ-formulae and sentences ofthe form

Θn := ∃X1, . . . , Xnθn(X1, . . . , Xn).


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

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

Remark 3.4.3. For N ′ ⊆ N ∈ H and A ⊂ N ′, 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 thesestructures.

Proof. This follows from the proof of Granger’s Lemma 9.2.6 [9].

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

Proof. By Remark 3.1.5, aclM(A) = (spanKalgA

(AV ), KalgA ), that is, the field-theoretic alge-

braic closure of KA and the KalgA -span of the vector elements of A. Since KN = KM,

KalgA ⊆ KN . Since AV ⊆ V N , spanKalg

A(AV ) ⊆ 〈A〉 ⊆ V N . Further, as noted above, Kalg

A is

the same whether computed in M or in N , and so the KalgA -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 Lemma3.4.4, aclN (A) = aclM(A).

Lemma 3.4.6. Suppose N ∈ H, C ⊆ B ⊂ N , a ∈ N , and 〈B〉 6= V N . Then tpN (a/B) doesnot 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

C then, since 〈B〉 6= V N , it follows that KCa| K

N

KCKB and one of the following conditions

holds:

1. a ∈ KN = KM.

2. a ∈ 〈C〉.

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

First, note that KCa| K

N

KCKB if and only if KCa

| KM

KCKB, since KN = KM. Secondly, note

that each of the three conditions above is true in N if and only if it is true inM. It followsthat tpM(a/B) does not Γ-fork over C. The argument in the other direction is exactly thesame.

Now suppose that the result holds for tuples of length n, and lg(aa) = n + 1. By theinductive 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


and the argument above, this is the case if and only if tpM(a/B) does not Γ-fork over C andtpM(a/Ba) does not Γ-fork over Ca, that is, if and only if tpM(aa/B) does not Γ-fork overC.

Remark 3.4.7. Working in M or any N ∈ H, suppose C ⊂ B are finite dimensional sets.

1. a ∈ 〈B〉 \ 〈C〉 just in case the following formula is in tp(a/B) (where {b′1, . . . , b′`} is abasis for 〈B〉, and {c′1, . . . , c′k} is a basis for 〈C〉):

ϕ2(X, b′, c′) := p¬θ`+1(b′1, . . . , b

′`, X) ∧ θk+1(c′1, . . . , c

′k, X)q

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

q′([a, b1], . . . , [a, bn]) = 0.

Let q be such that q(x1, . . . , xn, d∗) = q′(x1, . . . , xn). Since d∗ ∈ KB(KCa), each d∗iis the result of a rational function (say, fi(z,w)

gi(z,w)) applied to elements of KB and KCa,

say, b∗ and c∗. Each b∗i is defined over B, say by χi(zi, d, c) (where d ∈ B \ C andc ∈ C), and each c∗i is defined over Ca, say by ξi(wi, c, a). We may assume χi and ξiare quantifier-free. As above, let {c′1, . . . , c′k} be a basis for 〈C〉. It should be clear,then, that the following formula is in tp(a/B) (where we have abbreviated (b1, . . . , bn)by b).

ϕ3(X; c, d, b, c′) := pr∧i=1

∃=1zi(χi(zi, d, c)) ∧s∧i=1

∃=1wi(ξi(wi, c,X))

∧ ∃y1, . . . , yt∃z1, . . . , zr∃w1, . . . , ws

( r∧i=1

χi(zi, d, c) ∧s∧i=1

ξi(wi, c,X)

∧t∧i=1

yi =fi(z, w)

gi(z, w)∧ q([X, b1], . . . , [X, bn], y1, . . . , yt) = 0

)∧ θm+n(c′1, . . . , c

′k, b1 . . . , bn)q.

Now suppose a′ � ϕ3(X; d, c, b, c′). Witnesses to the zi are field elements definedover B - that is, elements of KB. Witnesses to the wi are field elements definedover Ca′ - that is, elements of KCa′ . Witnesses to the yi are field elements obtainedby applying rational functions to the zi and wi - that is, elements of KB(KCa′). Itfollows that ([a′, b1], . . . , [a′, bn]) satisfies a polynomial with the yi as coefficients - thatis, {[a′, b1], . . . , [a′, bn]} is algebraic over KB(KCa′). Further, by the last part of theformula, {b1, . . . , bn} is linearly independent modulo 〈C〉, and so a′ |/ Γ

CB.


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

Lemma 3.4.8. Suppose B is finite dimensional, C ⊂ B, and p(x) ∈ SM(B). Then thefollowing 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 overC ∩N in N .

3. Let π(x) = {χ(x) : χ(x) ∈ p(x) and χ(x) is an Lθ quantifier-free formula}. There isNπ ∈ 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 letM � p(a). One of the followingholds:

1. KCa|/KCKB,

2. a ∈ 〈B〉 \ 〈C〉, or

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

In the first case, tpKM

(KCa/KB) forks overKC , in the structureKM. SinceKM = KN for allN ∈ H, this dependence holds in all N containing B∪{a}. Let N ′ contain B∪{a}. Suppose

ψ(x, d) ∈ tpKN′

(KCa/KB) divides over KC . By the stable embeddedness of KN′, ψ(x, d)

divides over KC in N ′, as well. It is realized by some α from KCa. Since α ∈ dclN′(Ca),

there is a quantifier-free Lθ-formula χ(x, c, a) defining α. By Remark 3.4.3, χ(x, c, a) doesnot depend on N ′. Similarly, there is ξ(y, b) Lθ quantifier-free defining d. The followingformula divides over KC in N ′:

ϕ1(X; b, c) := p(∃=1x)χ(x, c,X) ∧ (∃=1y)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 divisionof 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 ofϕ1(X; b, c). Furthermore, since the di lie outside of KC , we can extract a C-indiscerniblesequence from (di : i ∈ I), and hence from (bi : i ∈ I). We see that ϕ1(X; b, c) divides over Cin N ′. Since none of χ(x, c,X), ξ(y, b), or ψ(x, y) depended on N ′, ϕ1(X; b, c) divides overC in all N ⊇ N ′. It is clear that ϕ1(X; b, c) ∈ p(x).

In the second case, we recall that B (and thus, C) are finite dimensional. Let {c′1, . . . , c′k}be a basis for 〈C〉, and {b′1, . . . , b′`} a basis for 〈B〉. By Remark 3.4.7, ϕ2(X; b

′, c′) ∈ p(x).

Let N ′ ∈ H contain B, and suppose N ′ � ϕ2(a′; b′, c′). Then a′ ∈ 〈B〉 \ 〈C〉, and by


definition tpN′(a′/B) Γ-forks over C in N ′. But since Γ-forking is the same as dividing

in N ′, tpN′(a′/B) divides over C in N ′. We chose a′ to be an arbitrary realization of

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),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

′(a′/B) Γ-forks (and so divides) over C in N ′. It follows that

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) thateventually forks in H.

3 → 2: Using the fact that there are Nqf ∈ H and ψ(x, b) a quantifier-free Lθ-formulasuch 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 inM, and let N ′ ∈ H contain B∪{a}.By Remark 3.4.2, N � π(a). By Lemma 3.4.6, tpN

′(a/B) does not Γ-fork over C (in N ′),

and hence does not fork over C in N ′. It follows that π(x) does not fork over C (= C ∩N)in N ′, or in any N ⊇ N ′.

Lemma 3.4.9. If C is finite dimensional, and ϕ(x, b) is a formula, then either there is Nfsuch that C ∪ {b} ⊂ Nf and for N ⊇ Nf , ϕ(x, b) forks over C in N , or there is Nnf suchthat C ∪ {b} ⊂ Nnf and for N ⊇ Nnf , ϕ(x, b) does not fork over C in N . (That is, everyformula 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 realizedin M, then it is eventually not realized in H (for large enough N , N � ¬∃xϕ(x, b)), andhence 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 overC, 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. ByLemma 3.4.8, ϕ(x, b) eventually does not fork over C. It remains to show that if for everya ∈ ϕ(M, b), tpM(a/Cb) Γ-forks over C, then ϕ(x, b) eventually forks over C.

Consider N ′ ∈ H such that C ∪{b} ⊂ N ′, 〈C ∪{b}〉 6= V N′, and if ψ(x, b) is a quantifier-

free Lθ-formula such that M � ϕ(x, b) ↔ ψ(x, b), then N ′ � ϕ(x, b) ↔ ψ(x, b). (Such anN ′ exists by finite dimensionality of C and Lemma 3.4.1.) Suppose N ′ � ϕ(a, b). Then,since N ′ is an Lθ-substructure ofM (see Remark 3.4.2) and N ′ � ψ(a, b),M � ψ(a, b), andso M � ϕ(a, b). By assumption, tpM(a/Cb) Γ-forks over C. By Lemma 3.4.6, tpN

′(a/Cb)

Γ-forks over C in N ′. 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 ′,so ϕ(x, b) eventually forks over C, as desired.

Remark 3.4.10. LetM be a κ-saturated, strongly κ-homogeneous model of T∞ for some largeκ. We conjecture that one can find an element a and sets B, C with dim(〈BC \C〉) infinite

such that a |/ limC B, but a | Γ

aclM(C) aclM(B) aclM(C) (so in particular, by Lemma 3.4.8, there


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

with {bij : j < n} linearly independent modulo 〈aclM(C) ∩ N〉 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

bi0 , . . . , bin−1 ∈ B with {bij : j < n} linearly independent modulo 〈aclM(C)〉 and{[a, bi0 ], . . . , [a, bin−1 ]} algebraically dependent overKaclM(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 bydifferent tuples from B each time.

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

Proof. It is clear from the definitions that KalgC ⊆ KaclM(C)∩N ⊆ KaclM(C). It remains to

show that KaclM(C) ⊆ KalgC , for which it suffices to show that the generators of KaclM(C) are

elements of KalgC . The generators of KaclM(C) are:

• (aclM(C))K = KalgC

• {[a, b] : a, b ∈ (aclM(C))V = spanKalgC

(CV )}

• for each n, the sets {α1, . . . , αn} such that there are b1, . . . , bn, b ∈ (aclM(C))V withb1, . . . , bn linearly independent and b = Σn

i=1αibi.

As noted above, the set of field elements from aclM(C) is exactly KalgC . We must show:

1. if a, b ∈ spanKalgC

(CV ), [a, b] ∈ KalgC ;

2. if b1, . . . , bn, b ∈ spanKalgC

(CV ) with b1, . . . , bn linearly independent and b = Σni=1αibi,

then α1, . . . , αn ∈ KalgC .

Claim 3.4.12. If a, b ∈ spanKalgC

(CV ), [a, b] ∈ KalgC .

Proof of Claim. Suppose a, b ∈ spanKalgC

(CV ). Then, for some m, there are c1, . . . , cm ∈ CVand β1, . . . , βm, γ1, . . . , γm ∈ Kalg

C such that

a = Σmi=1βici

b = Σmi=1γici

Then we have:

[a, b] = [β1c1 + . . .+ βmcm, γ1c1 + . . .+ γmcm]

= Σmi=1Σm

j=1[βici, γjcj]

= Σmi=1Σm

j=1βiγj[ci, cj].


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

{β1, . . . , βm, γ1, . . . , γm} ⊆ KalgC

and KalgC is a field, Σm

i=1Σmj=1βiγj[ci, cj] ∈ K

algC , as desired.

Claim 3.4.13. If b1, . . . , bn, b ∈ spanKalgC

(CV ), {b1, . . . , bn} is linearly independent, and

b = Σni=1αibi, then α1, . . . , αn ∈ Kalg

C .

Proof of Claim. Let b1, . . . , bn, b and α1, . . . , αn be as above. Since b1, . . . , bn, b ∈ spanKalgC

(CV ),

there are c1, . . . , cm ∈ CV and {βi,j : 1 ≤ i ≤ n, 1 ≤ j ≤ m} ∪ {γi : 1 ≤ i ≤ m} ⊆ KalgC such

that for 1 ≤ i ≤ n,bi = βi,1c1 + . . .+ βi,mcm

andb = 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 independentsubset of {c1, . . . , cm}. Suppose k < m. Then for 1 ≤ i ≤ m − k and 1 ≤ j < k, thereare λi,j ∈ KM such that ck+i = λi,1c1 + . . . + λi,kck. By the definition of KC (and the factthat c1, . . . , ck are linearly independent vectors from C), each λi,j is an element of KC , and

hence of KalgC . It follows that each bi (and b) can be written as a Kalg

C -linear combination ofc1, . . . , ck.

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

b = α1b1 + . . .+ αnbn

Σmi=1γici = α1 (Σm

i=1β1,ici) + . . .+ αn (Σmi=1βn,ici)

Σmi=1γici =

(Σnj=1αjβj,1

)c1 + . . .+

(Σnj=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:


β1,1 β2,1 · · · βn,1β1,2 β2,2 · · · βn,2

......

. . ....

β1,m β2,m · · · βn,m

α1

α2...αn

=

γ1

γ2...γm

.We shall refer to the matrix of βi,j’s as “B”. The ith column of B is the coordinate vector

for bi in terms of the basis {c1, . . . , cm}. (Note: we are not claiming that {c1, . . . , cm} isa basis of 〈C〉 or spanKalg

C(CV ), but it is a basis of a vector space containing {b1, . . . , bn}.)

Since {b1, . . . , bn} is linearly independent, the columns of B are linearly independent, and Bhas rank n. Since B is an m× n matrix of rank n, B has a left-inverse - call it A. From theprevious matrix equation, we obtain

α1

α2...αn

= A

γ1

γ2...γm

.Since each γi and all the entries of A are from Kalg

C , each αi is from KalgC , as desired.

Claims 3.4.12 and 3.4.13 complete the proof of the lemma.

Lemma 3.4.15. For M � T∞ and H as above, |lim

satisfies the strong finite charactercondition with respect to tuples of length one on the left, finite dimensional sets B on theright, 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 dimensionalsubset of M , C a small subset of M , and a |/ lim

C B, then there are ψ(x, y, z) ∈ L, b ∈ aclM(B)and c ∈ aclM(C) such that M � ψ(a, b, c) and ψ(x, b, c) eventually forks over aclM(C) in H.

Proof. If a |/ limC B, then for arbitrarily large N ∈ H, a |/ N

aclM(C)∩N aclM(B) ∩ N , that is,

tpN (a/(aclM(C) ∪ aclM(B)) ∩N) forks over aclM(C) ∩N in N . Recall that Γ-non-forkingis the same as non-forking in the finite dimensional substructures, so in arbitrarily largeN ∈ 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 followinghold.

• {a} ∪B ⊂ N . (This is possible because 〈B〉 is finite dimensional.)

• If a ∈ 〈C〉, then a ∈ 〈C∩N〉. (If a ∈ 〈C〉, then a is a linear combination of finitely manyvectors from C. For large enough N , those vectors are in CV ∩N , and a ∈ 〈C ∩N〉.)

• 〈(aclM(C) ∪ aclM(B)) ∩N〉 6= V N . (Since C is small and B is finite dimensional,

〈aclM(C) aclM(B)〉 = 〈CB〉 6= VM.

Let v ∈ VM \ 〈CB〉. For N containing v, 〈(aclM(C) ∪ aclM(B)) ∩N〉 6= V N .)


• 〈C ∩ N〉 ∩ 〈B〉 = 〈C〉 ∩ 〈B〉. As 〈B〉 is finite dimensional, 〈C〉 ∩ 〈B〉 is, as well. It isgenerated by finitely many elements from C - say, c′1, . . . , c

′n. Once {c′1, . . . , c′n} ⊂ N ,

then, 〈C ∩N〉 ∩ 〈B〉 = 〈C〉 ∩ 〈B〉.

Consider one such N ′. By the definition of Γ-forking, either

K(aclM(C)∪{a})∩N ′ |/ KaclM(C)∩N′

K(aclM(C)∪aclM(B))∩N ′

or

• a /∈ K, and

• a /∈ 〈aclM(C) ∩N ′〉, and

• either a ∈ 〈(aclM(C) ∪ aclM(B)) ∩ N ′〉 or there is a linearly independent (mod-ulo 〈aclM(C) ∩ N ′〉) set {b1, . . . , bm} from (aclM(C) ∪ aclM(B))V ∩ N ′ such that{[a, b1], . . . , [a, bm]} is algebraically dependent over

K(aclM(C)∪aclM(B))∩N ′(K(aclM(C)∪{a})∩N ′),

and

• either a /∈ 〈(aclM(C) ∪ aclM(B)) ∩N ′〉 \ 〈aclM(C) ∩N ′〉 or

〈(aclM(C) ∪ aclM(B)) ∩N ′〉 6= V N′

or there is a linearly independent (modulo 〈aclM(C) ∩ N ′〉) set {b1, . . . , bm} from(aclM(C) ∪ aclM(B))V ∩ N ′ such that {[a, b1], . . . , [a, bm]} is algebraically dependentover K(aclM(C)∪aclM(B))∩N ′(K(aclM(C)∪{a})∩N ′).

By the third assumption on N ′ (that 〈(aclM(C) ∪ aclM(B)) ∩ N ′〉 6= V N′), the fourth

condition holds trivially. Taking that into account,

a |/ N ′aclM(C)∩N ′ aclM(C) aclM(B) ∩N ′

if and only if one of the following holds:

1. K(aclM(C)∪{a})∩N ′ |/ KN′

KaclM(C)∩N′

K(aclM(C)∪aclM(B))∩N ′ ;

2. a ∈ 〈(aclM(C) ∪ aclM(B)) ∩N ′〉 \ 〈aclM(C) ∩N ′〉;

3. a /∈ 〈(aclM(C) ∪ aclM(B)) ∩N ′〉 and there is a linearly independent (modulo〈aclM(C) ∩N ′〉) set {b1, . . . , bm} ⊂ ((aclM(C) aclM(B)) ∩N ′)V such that{[a, b1], . . . , [a, bm]} is algebraically dependent over

K(aclM(C)∪aclM(B))∩N ′(K(aclM(C)∪{a})∩N ′).


Case 1: K(aclM(C)∪{a})∩N ′ |/ KaclM(C)∩N′

K(aclM(C)∪aclM(B))∩N ′ . (We have removed the su-

perscript ‘KN′’ because the field, and hence dependence at the level of the field, is the same

in all the structures in question.) By Lemma 3.4.11, KaclM(C)∩N ′ = KalgC , so we may rewrite

the dependence relation as

K(aclM(C)∪{a})∩N ′ |/ KalgCK(aclM(C)∪aclM(B))∩N ′ .

We mimic the argument in the first case of the proof of (2)→ (1) in Lemma 3.4.8. We knowthat tpN

′(K(aclM(C)∪{a})∩N ′/K(aclM(C)∪aclM(B))∩N ′) divides over Kalg

C in N ′. Suppose that

ψ(x, d) witnesses this (where d ∈ K(aclM(C)∪aclM(B))∩N ′), and suppose N ′ � ψ(α, d) (where

α ∈ K(aclM(C)∪{a})∩N ′). Then there are b′ ∈ (aclM(C) ∪ aclM(B)) ∩ N ′, c′ ∈ aclM(C) ∩ N ′,

and quantifier-free Lθ-formulae χ, ξ such that

N ′ � (∃=1x)χ(x, c′, a) ∧ (∃=1y)ξ(y, b′) ∧ χ(α, c′, a) ∧ ξ(d, b′).

It follows that the formula

ϕ1(X; b′, c′) := p(∃=1x)χ(x, c′, X) ∧ (∃=1y)ξ(y, b

′) ∧ ∃xy(χ(x, c′, X) ∧ ξ(y, b′) ∧ ψ(x, y))q

is in tpN′(a/(aclM(B) ∪ aclM(C)) ∩ N ′) and divides over Kalg

C . As above, given a KalgC -

indiscernible sequence (di : i ∈ I) (with d0 = d) (potentially from a sufficiently satu-rated elementary extension of N ′) witnessing the division of ψ(x, d), the corresponding

KalgC -indiscernible sequence (bi : i ∈ I) (with b0 = b) witnesses division of ϕ1(X; b

′, c′).

Also as above, we can extract a C ∩ N ′-indiscernible sequence from (bi : i ∈ I), wit-

nessing that ϕ1(X; b′, c′) divides over C ∩ N ′ in N ′. Since none of χ(x, c′, X), ξ(y, b

′),

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)).Case 2: a ∈ 〈(aclM(C) ∪ aclM(B)) ∩ N ′〉 \ 〈aclM(C) ∩ N ′〉. Let {b′1, . . . , b′k} be a basis

for 〈(aclM(C) ∪ aclM(B)) ∩ N ′〉. Note that 〈b′1, . . . , b′k〉 ∩ 〈C〉 is a finite dimensional vectorspace. Let {c′1, . . . , c′`} ⊂ CV be a basis of this intersection. By our second assumption onN ′, a /∈ 〈C〉, and the following formula is in tpM(a/ aclM(B) aclM(C)).

ϕ2(X, b′, c′) := p¬θk+1(b′1, . . . , b

′k, X) ∧ θ`+1(c′1, . . . , c

′`, X)q

Any realization of this formula is an element of

〈(aclM(C) ∪ aclM(B)) ∩N ′〉 \ 〈aclM(C) ∩N ′〉,

and so ϕ2(X, b′, c′) divides over aclM(C) ∩ N ′ in N ′. The same holds in all N ⊃ N ′ (since

{c′1, . . . , c′`} spans 〈b′1, . . . , b′k〉 ∩ 〈C〉, realizations of ϕ2(X, b′, c′) in any N containing b

′c′ do

not belong to 〈C〉), and so ϕ2(X, b′, c′) eventually forks over aclM(C).

Case 3: There is a linearly independent (modulo 〈aclM(C) ∩ N ′〉 = 〈C ∩ N ′〉) set{b1, . . . , bm} ⊂ (aclM(C) ∪ aclM(B))V ∩ N ′ such that {[a, b1], . . . , [a, bm]} is algebraically


dependent over K(aclM(C)∪aclM(B))∩N ′(K(aclM(C)∪{a})∩N ′). By Remark 3.4.7, a formula of the

following form is in tpN′(a/(aclM(C) ∪ aclM(B)) ∩N ′) (where {c′1, . . . , c′k} is a basis for the

finite dimensional space 〈C〉 ∩ 〈B〉, d are parameters from ((aclM(B) ∪ aclM(C)) ∩ N ′) \(aclM(C) ∩ N ′), c are parameters from aclM(C) ∩ N ′, we abbreviate (b1, . . . , bm) by b, χiand ξi are quantifier-free Lθ-formulae, q(x1, . . . , xn, y1, . . . , yt) is a polynomial in (x1, . . . , xn),and fi, gi are polynomials without parameters):

ϕ3(X; c, b, c′, d) := pr∧i=1

(∃=1zi)χi(zi, d, c) ∧s∧i=1

(∃=1wi)ξi(wi, c,X)

∧ ∃y1, . . . , yt∃z1, . . . , zr∃w1, . . . , ws

( r∧i=1

χi(zi, b, c) ∧s∧i=1

ξi(wi, c,X)

∧t∧i=1

yi =fi(z, w)

gi(z, w)∧ q([X, b1], . . . , [X, bn], y1, . . . , yt) = 0

)∧ ∀x1, . . . , xm+k,∀Y (Σm

i=1xibi 6= Y ∨ Σki=1xm+ic

′i 6= Y ∨ Y = 0V )q.

By the assumption that 〈C ∩ N〉 ∩ 〈B〉 = 〈C〉 ∩ 〈B〉, and the fact that χi(zi, d, c) andξi(wi, c, a) define the same elements in N ′ as in any larger N or in M,

ϕ3(X; c, b, c′, d) ∈ tpN (a/(aclM(B) ∪ aclM(C)) ∩N)

for N ⊇ N ′ and ϕ3(X; c, b, c′, d) ∈ tpM(a/ aclM(B) aclM(C)). Furthermore, ϕ3(X; c, b, c′, d)divides over aclM(C) ∩ N in all N ⊇ N ′. As discussed in Remark 3.4.7, realizations a′ ofthis formula are such that {[a′, b1], . . . , [a′, bm]} is algebraically dependent overKaclM(B)∩N(K(aclM(C)∪{a′})∩N). It remains to show that {b1, . . . , bm} is linearly independent

modulo 〈aclM(C) ∩N〉.

Claim 3.4.16. A linearly independent set of vectors X is linearly independent modulo thevector space W if and only if 〈X〉 ∩W = {0}.

Proof. → Suppose X = {xi : i ∈ I}, W = 〈wj : j ∈ J〉 (that is, {wj : j ∈ J} is a basis forW ), and v ∈ 〈X〉 ∩W . Then, reordering if necessary, there are x1, . . . , xn, w1, . . . , wm, andscalars 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 lin-ear 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.


← Suppose 〈X〉 ∩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 linearlyindependent. Suppose, further, that there are λ1, . . . , λn+m such that

λ1x1 + . . .+ λnxn + λn+1v1 + . . .+ λn+mvm = 0

It follows that

Σni=1λixi = −Σm

i=1λn+ivi.

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

The last conjunct of ϕ3 states exactly that 〈b1, . . . , bm〉 ∩ 〈C〉 = {0}, and hence that〈b1, . . . , bm〉 ∩ 〈C ∩N〉 = {0} for all N under consideration.

Remark 3.4.17. We would like to generalize the previous result to tuples a of length greaterthan one, but unfortunately we conjecture that there are finite tuples a, b, and an infinitedimensional set C such that a |/ lim

C b, and this dependence is not witnessed by any singleformula. In particular, we suspect a problem might arise in the following way: there are

a1, a2, b, C such that a1a2| Γ

aclM(C) aclM(C) aclM(b) (i.e. a1| Γ

aclM(C) aclM(C) aclM(b) and

a2| Γ

aclM(C)a1aclM(C) aclM(b)a1), but for arbitrary large N ,

a2|/ N

(aclM(C)∪{a1})∩N(aclM(C) ∪ aclM(b) ∪ {a1}) ∩N,

and hence a1a2|/ limC b. (The fact that in the base, a1 does not lie in the scope of the algebraic

closure suggests that one might find

K(aclM(C)∪{a1,a2})∩N|/K

(aclM(C)∪{a1})∩NK(aclM(C)∪aclM(b)∪{a1})∩N ,

even though KaclM(C)a1a2|K

aclM(C)a1KaclM(C) aclM(b)a1

.)

Proposition 3.4.18. Let M � T∞, and let H be the collection of finite dimensional sub-

spaces of M (viewed as L-substructures of M). When |lim

is restricted to tuples of lengthone on the left and finite dimensional sets on the right and in the base, H approximates Mand each instance of limit independence corresponds to an instance Γ-independence. Moreprecisely, for an element a and finite dimensional sets B, C,

a |lim

C B if and only if a |Γ

aclM(C) aclM(B) aclM(C).

Proof. 1. H covers M: Every vector element generates a one dimensional subspace ofVM.


2. Automorphism invariance: the automorphic image of a finite dimensional subspaceof VM is a finite dimensional subspace of VM.

3. Convergence of truth value: Suppose a ∈ M , ψ(x) ∈ L, and M � ψ(a). We mustshow that ψ(a) is eventually true in H. By Lemma 3.4.1, there is an Lθ-formula ϕand an R < ω such that Tm � ϕ ↔ ψ for all m ≥ R (and, also by Lemma 3.4.1,T∞ � ϕ ↔ ψ). Since T∞ � ϕ ↔ ψ, M � ϕ(a). Take any N ′ ∈ H containing a andof dimension at least R. By Remark 3.4.2, N ′ � ϕ(a) and, since N ′ � Tm for somem ≥ 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

A = aclM(A) = (spanKalgA

(AV ), KalgA ).

Since A is small, there are v1, v2 ∈ VM such that {v1, v2} is linearly independentmodulo 〈A〉. In any N containing {v1, v2}, dim(A ∩N) < dim(V N )− 1. Consider onesuch N ′. By Remark 3.1.5, aclN (A∩N) = (spanKalg

A∩N(AV ∩N), Kalg

A∩N). We show that

aclN (A ∩N)V = AV ∩N and that aclN (A ∩N)K = AK ∩N .

Suppose a ∈ aclN (A∩N)V = spanKalgA∩N

(AV ∩N). That is, there are b1, . . . , bn ∈ AV ∩Nand α1, . . . , αn ∈ Kalg

A∩N such that a = Σni=1αibi. Since AV ∩N ⊆ AV and Kalg

A∩N ⊆ KalgA ,

a ∈ spanKalgA

(AV ) = aclM(A)V = AV . Since V N is a vector space, a ∈ N . Hence,

a ∈ A ∩N . Now suppose α ∈ aclN (A ∩N)K = KalgA∩N . Since Kalg

A∩N ⊆ KalgA ,

α ∈ KalgA = aclM(A)K = AK .

As α ∈ N, α ∈ A ∩N , as desired.

7. Homogeneity: Suppose tpM(a/A) = tpM(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 bover A in M. Let N ∈ H with a, b ∈ N . by Remark 3.4.2,

N � p �N (a)

andN � p �N (b).

Since Th(N ) has quantifier elimination in Lθ, tpN (a/A ∩N) = tpN (b/A ∩N).


By Lemma 3.2.5, a | limC B if and only if tpM(a/ aclM(B) aclM(C)) eventually does notfork 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

a | limC B if and only if a | Γ

aclM(C) aclM(C) aclM(B).

69

Bibliography

[1] Hans Adler. “Explanation of independence”. PhD thesis. Albert-Ludwigs-UniversitatFreiburg im Breisgau, 2005.

[2] Zoe Chatzidakis. “Amalgamation of types in pseudo-algebraically closed fields andapplications”. Article in preparation. 2011.

[3] Zoe Chatzidakis. “Properties of forking in ω-free pseudo-algebraically closed fields”.In: The Journal of Symbolic Logic 67.3 (2002), pp. 957–996.

[4] Zoe Chatzidakis. “Simplicity and independence for pseudo-algebraically closed fields”.In: Models and computability (Leeds, 1997). Vol. 259. London Mathematical SocietyLecture 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: Annalsof Pure and Applied Logic 165.2 (2014), pp. 695–723.

[7] Mirna Dzamonja and Saharon Shelah. “On C∗-maximality”. In: Annals of Pure andApplied Logic 125.1-3 (2004), pp. 119–158.

[8] Mirna Dzamonja and Saharon Shelah. “On C∗-maximality”. Corrected version of thepublished 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”. PhDthesis. University of Manchester, 1999.

[10] Victor Harnik and Leo Harrington. “Fundamentals of forking”. In: Annals of Pure andApplied 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: Annalsof 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 AppliedLogic 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 introduc-tion. New York: Springer-Verlag, 2002, pp. viii+342.

[18] Lynn Cho Scow. “Characterization Theorems by Generalized Indiscernibles”. PhDthesis. 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-HollandPublishing 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 Erdos is eighty, Vol. 1. Bolyai Society Mathematical Studies. Budapest:Janos Bolyai Mathematical Society, 1993, pp. 403–420.

[22] Saharon Shelah. “Toward classifying unstable theories”. In: Annals of Pure and AppliedLogic 80.3 (1996), pp. 229–255.

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

Independence Relations in Theories with the Tree Property · 2018-10-10 · theories are relatively...

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