c 2007 Konstantinos Schoretsanitis FRA¨ISSE´ THEORY FOR METRIC STRUCTURES
BY
KONSTANTINOS SCHORETSANITIS
DISSERTATION
Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate College of the University of Illinois at Urbana-Champaign, 2007
Urbana, Illinois Abstract
In 1954, Roland Fra¨ıss´epublished a paper that answered the following questions: Given a first-order signature
L and a class A of finite L-structures that is closed under isomorphism:
1. find necessary and sufficient conditions on A that guarantee the existence of a “homogeneous” L-
structure M such that the class of L-structures that are isomorphic to finite L-substructures of M is
A;
2. find necessary and sufficient conditions on A that guarantee the existence of an L-structure M such
that Th(M) has QE and is ω-categorical, and such that the class of L-structures that are isomorphic
to finite L-substructures of M is A.
In this thesis we generalize Fra¨ıss´e’sresults to the setting of bounded continuous logic for metric structures.
This logic was presented in 2004 by Ita¨ıBen Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander
Usvyatsov, and it may be considered as a generalization of first-order logic.
We also prove a theorem, in the setting of continuous model theory, that is a generalization of a theorem of H. D. Macpherson about the automorphism groups of ω-categorical structures.
ii To my parents,
Παναγιωτη´ και Aναστασ´ια
iii Acknowledgments
First I am most indebted to my parents for their support over the years.
Also, I would like to thank my advisor C. Ward Henson for his support and helpful comments.
Special thanks to my girlfriend Sasha for her support.
Furthermore, I would like to thank the friends that I made over the years at Urbana-Champaign and with whom I spent some of the most enjoyable years in my life.
A lot of thanks to the Department of Mathematics and the Logic Group for the financial support they provided me over the years.
iv Table of Contents
Chapter 1 Introduction ...... 1
Chapter 2 Introduction to model theory ...... 7 2.1 Metric structures and signatures ...... 7 2.2 Formulas and their interpretations ...... 9 2.3 Model theoretic concepts ...... 14 2.4 Model theoretic theorems ...... 16 2.5 Spaces of types ...... 16
Chapter 3 Ages ...... 19 3.1 Basic definitions ...... 19 3.2 ρ-compact ages ...... 21
Chapter 4 Basic model theory ...... 25 4.1 Quantifier elimination ...... 25 4.2 Inductive theories ...... 27 4.3 Model-complete theories ...... 29 4.4 Model companions ...... 30 4.5 ω-categoricity ...... 31
Chapter 5 Fra¨ıss´eTheorems for complete theories with QE ...... 34 5.1 A Fra¨ıss´eTheorem for complete theories with QE ...... 34 5.2 Axiomatization of complete QE theories ...... 40
Chapter 6 Fra¨ıss´eTheorems for separable structures ...... 45 6.1 Definitions and basic theorems ...... 45 6.2 Near amalgamation property ...... 48 6.3 The dA-metric ...... 50 6.4 Completion of ω-qf-near-homogeneous metric prestructures ...... 56 6.5 Fra¨ıss´etheory for separable structures ...... 59
Chapter 7 Fra¨ıss´eTheorems for complete, ω-categorical theories with QE ...... 69
Chapter 8 Macpherson’s Theorem ...... 74
References ...... 81
Author’s Biography ...... 82
v Chapter 1
Introduction
In 1954, Roland Fra¨ıss´epublished a paper (see [4]) that has become a classic in Model Theory. In this paper
he pointed out that we can think of the class of finite linear orderings as a set of approximations to the
ordering of the rationals, and he described a way of building the rationals out of these finite approximations.
Fra¨ıss´e’sconstruction is important because it works in many other cases too. Starting from a suitable set of
finite structures we can build their “limit” and some of the structures built in this way have turned out to
be remarkably interesting.
In this document we will use the following notation: We say that a set is countable if its cardinality is
equal to the cardinality of a subset of ω. We say that a set is countably infinite if its cardinality is equal to
the cardinality of ω.
For simplicity, for the remainder of the introduction L1 will denote a first-order signature with no function symbols. Some of the theorems that will be stated may be true for more general first-order signatures but
this is not going to be relevant to our later analysis.
The series of questions that Fra¨ıss´eanswered were the following: Given a class A of finite L-structures
that is closed under isomorphisms:
1. find necessary and sufficient conditions on A that guarantee the existence of a “homogeneous” L1-
structure M such that the class of L1-structures that are isomorphic to finite L1-substructures of M is A;
2. find necessary and sufficient conditions on A that guarantee the existence of an L1-structure M such
that Th(M) has QE and is ω-categorical, and such that the class of L1-structures that are isomorphic
to finite L1-substructures of M is A.
In later years several people studied the interplay among A, M, Th(M), and Aut(M).
To state Fra¨ıss´e’sresults precisely we will need some terminology.
Notation 1.1. 1. If A is an L-structure we set age(A) to be the class of L1-structures that are isomorphic
to finite L1-substructures of M.
1 2. If T is a complete L1-theory we set age(T ) to be the class of L1-structures that are isomorphic to finite
L1-substructures of models of T .
We will also need the following definition.
Definition 1.2. Let A be a class of finite L1-structures with the following properties:
• A is closed under isomorphism.
• (Hereditary property, HP for short) If M ∈ A and N is an L1-substructure of M, then N ∈ A.
• (Joint embedding property, JEP for short) If M1 ∈ A, M2 ∈ A, then there exist M ∈ A and L1-
embeddings f1 : M1 → M, f2 : M2 → M.
Then A is called an age in L1. Usually, when L1 is clear from the context, we may just say that A is an age.
Definition 1.3. The cardinality of an age A in L1 is defined to be the number of isomorphism types of elements of A.
Definition 1.4 (Amalgamation property, AP for short). Let A be a class of finite L1-structures. We say
that A has the amalgamation property if for every M, M1, M2 in A and L1-embeddings f1 : M → M1, f2 : M → M2 there exist N in A and L1-embeddings g1 : M1 → N, g2 : M2 → N such that g1 ◦ f1 = g2 ◦ f2.
Note that in general AP does not imply JEP.
Definition 1.5. Let L1 be a first-order signature and let M be an L1-structure. We say that M is ul- trahomogeneous if every isomorphism between finite substructures of M extends to an automorphism of
M.
Here are two theorems of Fra¨ıss´ethat answered the first question.
Theorem 1.6. Let L1 be a countable first-order signature and let M be a countable L1-structure that is ultrahomogeneous. Let A = age(A). Then A is a nonempty, countable age with the AP.
Proof. See [6, Theorem 6.1.7].
Theorem 1.7. Let L1 be a countable first-order signature and let A be a nonempty countable age in L1 with the AP. Then there is a unique L1-structure M such that M is countable, age(M) = A and M is ultrahomogeneous.
Proof. See [6, Theorem 6.1.2].
Here are two theorems of Fra¨ıss´ethat answered the second question.
2 Theorem 1.8. Let L1 be a finite first-order signature and let T be a complete L1-theory that has QE and is ω-categorical. Then age(T ) is an a countably infinite age with the AP.
Proof. See [6, Corollary 6.4.2].
Theorem 1.9. Let L1 be a finite first-order signature and let A be a countably infinite nonempty age with the AP. Then there exists a unique complete theory T that has QE, is ω-categorical, and satisfies age(T ) = A.
Proof. See [6, Theorems 6.4.1 and 7.3.6].
In this thesis we generalize the above theorems to the continuous setting. Let L denote a bounded
continuous signature without function symbols. (See [1], [2].) An age in L is defined in the same way as an
age in L1. We will use the following notation.
A Notation 1.10. Let A be an age. For every n ∈ ω we set Sn to be the set of quantifier-free types in the
variables x1, . . . , xn which are realized in structures in A.
The AP in the continuous setting is defined in exactly in the same way as in the classical first-order
setting. Here is a variation of the AP.
Definition 1.11. Let A be an age. We say that A has the near amalgamation property (near-AP) if for
A every > 0, n, l ∈ ω, M ∈ A, p ∈ Sn+l, anda ¯ = (a1, . . . , an) ⊆ M such thata ¯ |= p n, there exists N in A ¯ ¯ such that M ⊆ N and b = (b1, . . . , bn) ⊆ N such that b |= p and
N max d (ai, bi) < . 1≤i≤n
We note that the AP implies the near-AP. The near-AP is closely related to the notion of strongly
ω-qf-near-homogeneous structures, which is defined next:
Definition 1.12. M is strongly ω-qf-near-homogeneous if for every n ∈ ω, n-tuplesa, ¯ ¯b ⊆ M, such that ¯ qftpM(¯a) = qftpM(b), there exists an automorphism f of M such that
M max d (ai, f(bi)) < . 1≤i≤n
If A is an age with the near-AP, then we can define a natural metric on its quantifier-free type spaces.
A Definition 1.13. Let A be an age. For every n ∈ ω and p, q ∈ Sn , we define
A M ¯ ¯ dn (p, q) = inf{ max d (ai, bi) | M ∈ A, a,¯ b ⊆ M, M |= p(¯a), M |= q(b)}. 1≤i≤n
3 A A Proposition 1.14. Let A be an age with the near-AP. Then for all n ∈ ω, (Sn , dn ) is a metric space.
Proof. See 6.11.
In the classical first-order setting the size of the age is measured by its cardinality. In the continuous
setting the size of an age is measured by its density.
Definition 1.15. Let A be an age with the near-AP.
A A A 1. We say that A is d -compact if for every n ∈ ω, the metric space (Sn , dn ) is compact.
A A A 2. We say that A is d -separable if for every n ∈ ω, the metric space (Sn , dn ) is separable.
A A A 3. We say that A is d -complete if for every n ∈ ω, the metric space (Sn , dn ) is complete.
Definition 1.16. Let A be an age. We say that A is totally bounded if for all > 0 there exists n ≥ 1 such
that for all M ∈ A, M has an -net of size ≤ n.
In this thesis we develop a strong analogy between the classical setting and the continuous setting.
A • A d -compact age in L is the analogue of an age in L1 with finite number of structures (up to isomorphism) of cardinality ≤ n for all n ∈ ω.
A • A d -separable age in L is the analogue of an age in L1 with countable number of structures (up to isomorphism) of cardinality ≤ n for all n ∈ ω.
• An age in L that is totally bounded is the analogue of an age in L1 with a finite number of structures (up to isomorphism).
On the other hand, dA-completeness is a smoothness condition. It is the least kind of regularity that an
age should have so that we can study it from a model theoretic point of view.
The analogues of Theorem 1.6 and Theorem 1.7, respectively, are the following:
Theorem 1.17 (See Theorem 6.30). Let L be a countable bounded continuous signature without function
symbols. Let M be a separable L-structure that is ω-qf-near-homogeneous. Set A = age(M). Then A is a
dA-separable, dA-complete age with the near-AP.
Theorem 1.18 (See Theorem 6.28). Let L be a countable bounded continuous signature without function
symbols. Let A be a dA-separable, dA-complete age with the near-AP. Then there exists a unique separable
L-structure M that is strongly ω-qf-near-homogeneous and satisfies age(M) = A.
The analogues of Theorem 1.8 and Theorem 1.9, respectively are the following:
4 Theorem 1.19 (See Theorem 7.5). Let L be a finite bounded continuous signature without function symbols.
Let T be a complete L-theory that has QE and is ω-categorical. Then age(T ) is a dA-compact age that has
the AP and is not totally bounded.
Theorem 1.20 (See Theorem 7.6). Let L be a finite bounded continuous signature without function symbols.
Let A be an dA-compact age that has the AP and is not totally bounded. Then there exists a unique complete
L-theory T that has QE, is ω-categorical and satisfies age(T ) = A.
In the first order setting, if a theory T in a finite signature has QE, then T is necessarily ω-categorical.
But this is not the case in continuous model theory. (See 7.13) Then, a natural question arises: Can we
characterize QE theories in terms of the properties of their ages in the way we did for ω-categorical theories
with QE? It turns out that it is possible, as we describe next.
The following definition describes another variation of the AP.
Definition 1.21. Let L be a finite bounded continuous signature without function symbols and A an age in
A L. We say that A has the perturbed amalgamation property (PAP) if for every n ∈ ω, > 0, p(¯x) ∈ Sn+1, A A there exists δn (, p) > 0 such that for every M ∈ A,a ¯ = (a1, . . . , an) ⊆ M with ρn(pn, q) < δn (, p), where
q = qftpM(¯a), there exist N ⊇ M in A, and an+1 in N such that ρn+1(p, r) ≤ where r = qftpN(a1, . . . , an+1).
The following two theorems explore the interplay between QE theories and properties of their ages.
Theorem 1.22 (See Theorem 5.4). Let L be a finite bounded continuous signature without function symbols.
Let A be a ρ-compact age with the PAP. Then there exists a unique complete L-theory T that has QE and
satisfies age(T ) = A.
Theorem 1.23 (See Theorem 5.5). Let L be a finite bounded continuous signature without function symbols.
Let T be a complete L-theory that has QE. Then age(T ) is a ρ-compact age with the PAP.
Now we shift our attention to the automorphism groups of ω-categorical structures. The following
theorem provides a link between model theory and the study of permutation groups.
Theorem 1.24. Let L1 be a countable first-order signature and M a countably infinite L1-structure . Set T = Th(M) and G = Aut(M). The following statements are equivalent:
1. T is ω-categorical.
2. For every n ≥ 1, G has only finitely many orbits in its induced action on Mn.
Proof. See [3, p 30].
5 H. D. Macpherson used Theorem 1.24 to prove the following theorem.
Theorem 1.25. Let L1 be a countable first-order signature and M be a countably infinite L1-structure such that Th(M) is ω-categorical. Set G = Aut(M). Then there exists a dense subgroup F ≤ G which is freely generated by countably many elements, where G is equipped with the pointwise convergence topology.
Proof. See [8, Theorem 3.1].
Theorem 1.24 was generalized in the continuous setting by C. Ward Henson as follows.
Theorem 1.26 (See Theorem 4.25). Let M be a noncompact separable L-structure. Set T = Th(M) and
G = Aut(M). The following statements are equivalent:
1. T is an ω-categorical theory.
2. For every > 0, n ≥ 1 there exist n-tuples a¯1 ... a¯l ⊂ M, for some l ∈ ω, such that for every n-tuple ¯b ⊆ M there exist 1 ≤ j ≤ l and F ∈ G such that
M max d (F (aj,i), bi) < . 1≤i≤n
In this thesis we use Theorem 1.26 to prove the following generalization of Macpherson’s Theorem in the continuous setting.
Theorem 1.27 (See Theorem 8.2). Let M be a separable L-structure which is strongly ω-homogeneous, noncompact and such that Th(M) is ω-categorical. Set G = Aut(M). Then there exists a dense subgroup F ≤
G which is freely generated by countably many elements, where G is equipped with the pointwise convergence topology.
6 Chapter 2
Introduction to model theory
In this chapter we give a quick introduction to continuous model theory. Our exposition follows [1] closely.
2.1 Metric structures and signatures
Let (M, d) be a complete bounded metric space. A predicate on M is a uniformly continuous function
from M n into some bounded interval in R, for some n ≥ 1. A function or operation on M is a uniformly continuous function from M n into M, for some n ≥ 1. In each case n is called the arity of the predicate or the function.
A metric structure M based on (M, d) consists of a family (Ri | i ∈ I) of predicates on M, a family
(Fj | j ∈ J) of functions on M, and a family (ak | k ∈ K) of elements of M. When we introduce a metric structure we may denote it as
M = (M,Ri,Fj, ak | i ∈ I, j ∈ J, k ∈ K).
Any of the index sets I, J, K are allowed to be empty. Indeed, they may all be empty, in which case M is a
pure bounded metric space.
The key restrictions on metric structures are: the metric space is complete and bounded, each predicate
takes its values in a bounded interval of reals, and the functions and predicates are uniformly continuous.
All of these restrictions play a role in making the theory work smoothly.
To each metric structure we associate a signature L as follows. To each predicate R of M we associate
a predicate symbol P and an integer a(P ) which is the arity of R; we denote R by P M. To each function
F of M we associate a function symbol f and an integer a(f) which is the arity of F ; we denote F by f M. Finally, to each distinguished element a of M we associate a constant symbol c; we denote a by cM.
So, a signature L gives sets of predicate, function and constant symbols, and associates to each predicate and function symbol its arity. In that respect, L is identical to a signature of first-order model theory. In addition, a signature for metric structures must satisfy more: for each each predicate symbol P , it must
7 provide a closed bounded interval IP of real numbers and a modulus of uniform continuity ∆P . These should
M satisfy the requirements that P takes its values in IP and that ∆P is a modulus of uniform continuity for
M M P . In addition, for each function symbol f, L must provide a modulus of uniform continuity ∆f for f .
Finally, L must provide a non-negative real number DL which is a bound on the diameter of the complete metric space (M, d) on which M is based. We sometimes denote the metric d given by M as dM; this would be consistent with our notation for the interpretation in M of the nonlogical symbols of L. However, we also
find it convenient often to use the same notation d for the logical symbol representing the metric as well as for its interpretation in M; this is consistent with usual mathematical practice and with the handling of the symbol = in first-order logic.
When these requirements are all met and when the predicate, function and constant symbols of L correspond exactly to the predicates, functions and distinguished elements of which M consists, then we say that M is an L-structure.
Basic concepts such as embedding and isomorphism have natural definitions for metric structures:
Definition 2.1. Let L be a signature for metric structures and suppose that M and N are L-structures.
An embedding from M into N is a metric space isometry
T :(M, dM) → (N, dN) that commutes with the interpretations of the predicates, function and constant symbols of L in the following sense:
Whenever P is an n-ary predicate symbol of L and a1, . . . , an ∈ M we have
N M P (T (a1),...,T (an)) = P (a1, . . . , an);
whenever f is an n-ary function symbol of L and a1, . . . , an ∈ M, we have
N M f (T (a1),...,T (an)) = T (f (a1, . . . , an));
and whenever c is a constant symbol of L we have
cN = T (cM).
An isomorphism is a surjective embedding. We say that M and N are isomorphic, and write M =∼ N, if
8 there exists an isomorphism between M and N. An automorphism of M is an isomorphism between M and
itself.
We say M is a substructure of N (and we write M ⊆ N) if M ⊆ N and the inclusion map from M into
N is an embedding of M into N.
2.2 Formulas and their interpretations
Fix a signature L for metric structures, as described above. We assume for simplicity of notation that
DL = 1 and that IP = [0, 1] for every predicate symbol P .
Symbols of L
Among the symbols of L are the predicate, function, and constant symbols; these will be referred to as the nonlogical symbols of L and the remaining ones will be called the logical symbols of L. Among the logical symbols is a symbol d for the metric on the underlying metric space of an L structure; this is treated formally as equivalent to a predicate symbol of arity 2. The logical symbols also include an infinite set VL of variables; usually we take VL to be countable, but there are situations in which it is useful to permit a larger number of variables. The remaining logical symbols consist of a symbol for each continuous function u : [0, 1]n → [0, 1] of finitely many variables n ≥ 1 (these play the roles of connectives) and the symbols sup and inf, which play the role of quantifiers in this logic.
The cardinality of L, denoted card(L), is the smallest infinite cardinal number ≥ the number of nonlogical symbols of L.
Terms of L
Terms are formed inductively, exactly as in the first-order logic. Each variable and constant symbol is an
L-term. If f is an n-ary function symbol and t1, . . . , tn are L-terms, then f(t1, . . . , tn) is an L-term. All L-terms are constructed in this way.
Atomic formulas of L
The atomic formulas of L are the expressions of the form P (t1, . . . , tn) in which P is an n-ary predicate symbol of L and t1, . . . , tn are L-terms, as well as d(t1, t2) in which t1 and t2 are L-terms. Note that the logical symbol d for the metric is treated formally as a binary predicate symbol, exactly analogous to how the equality symbol = is treated in first order logic.
9 Formulas of L
Formulas are also constructed inductively and the basic structure of the induction is similar to the corre-
sponding definition in first-order logic. Continuous functions play the role of connectives and sup and inf
are used formally in the way that quantifiers are used in the first-order logic. The precise definition is as
follows:
Definition 2.2. The class of L-formulas is the smallest class of expressions satisfying the following require-
ments:
1. Atomic formulas of L are L-formulas.
n 2. If u : [0, 1] → [0, 1] is continuous and ϕ1, . . . , ϕn are L-formulas, then u(ϕ1, . . . , ϕn) is an L-formula.
3. If ϕ is an L-structure and x is a variable, then supx ϕ and infx ϕ are L-formulas.
An L formula is quantifier free if it is generated inductively from atomic formulas without using the last clause, namely neither supx nor infx are used. Many syntactic notions from first-order logic can be carried over word for word into this setting. We will assume that this has been done by the reader for many such concepts, including subformula and syntactic substitution of a term for a variable, or formula for a subformula, and so forth.
Free and bound occurrences of variables in L-formulas are defined in a way similar to how this is done in first-order logic. Namely, an occurrence of the variable x is bound if lies within a subformula of the form supx ϕ or infx ϕ, and otherwise it is free. An L-sentence is an L-formula that has no free variables.
When t is a term and the variables occurring in it are among the the variables x1, . . . , xn (which we always take to be distinct in this context), we indicate this by writing t as t(x1, . . . , xn).
Similarly, we write an L-formula as ϕ(x1, . . . , xn) to indicate that its free variables are among x1, . . . , xn.
Prestructures
It is common in mathematics to construct a metric space as the quotient of a pseudometric space or as the completion of such a quotient, and the same is true of metric structures. For that reason we need to consider what we call prestructures and to develop the semantics of continuous logic for them.
As above, we take L to be a fixed signature for metric structures. Let (M0, d0) be a pseudometric space, satisfying that its diameter is ≤ DL. (That is, d0(x, y) ≤ DL for all x, y ∈ M0.) An L-prestructure M0 based on (M0, d0) is a structure consisting of the following data:
10 M0 n 1. for each predicate symbol P of L (of arity n) a function P from M0 into IP that has ∆P as a modulus of uniform continuity;
M0 n 2. for each function symbol f of L (of arity n) a function f from M0 into M0 that has ∆f as a modulus of uniform continuity; and
M0 3. for each constant symbol c of L an element c of M0.
Definition 2.3. Let M0 be an L-prestructure. We say that M0 is a metric L-prestructure if M0 is based on a metric space (M0, d0).
Given an L-prestructure M0, we may form its quotient prestructure as follows. Let (M, d) be the quotient metric space induced by (M0, d) with quotient map π : M0 → M. Then
M n M 1. for each predicate symbol P of L (of arity n) define P from M into IP by setting P (π(x1), . . . , π(xn)) =
M0 P (x1, . . . , xn) for each x1, . . . , xn ∈ M0;
M n M 2. for each function symbol f of L (of arity n) define f form M into M by setting f (π(x1), . . . , π(xn)) =
M0 π(f (x1, . . . , xn)) for each x1, . . . , xn ∈ M0;
3. for each constant symbol c of L define cM = π(cM0 ).
It is obvious that (M, d) has the same diameter as (M0, d0). Also, as noted in [1, page 11], for each
M predicate symbol P and each function symbol f of L, the predicate P is well defined and has ∆P as a
M modulus of uniform continuity and the function f is well defined and has ∆f as a modulus of uniform continuity. In other words, this defines an L-prestructure (which we denote by M) based on the (possibly not complete) metric space (M, d).
Finally we may define an L-structure N by taking a completion of M. This is based on a complete metric space (N, d) that is a completion of (M, d) and its additional structure is defined in the following natural way (made possible by the fact that the predicates and functions given by M are uniformly continuous):
M n 1. for each predicate symbol P of L (of arity n) define P from N into IP to be the unique such function that extends P M and is continuous;
2. for each function symbol f of L (of arity n) define f N from N n into N to be the unique such function
that extends f M and is continuous;
3. for each constant symbol c of L define cN = cM.
11 It is obvious that (N, d) has the same diameter as (M, d). Also, as noted in [1, page 8], for each predicate
N symbol P and each function symbol f of L, the predicate P has ∆P as a modulus of uniform continuity,
N and the function f has ∆f as a modulus of uniform continuity. In other words, N is an L-structure.
Semantics
Let M be any L-prestructure, with (M, d) as its underlying pseudometric space, and let A be any subset of
M. We extend L to a signature L(A) by adding a new constant symbol c(a) to L for each element a ∈ A.
We extend the interpretation given by M in a canonical way, by taking the interpretation of c(a) to be equal
to a itself for each a ∈ A. We call c(a) the name of a in L(A). Indeed, we will often write a instead of c(a)
where no confusion can result from doing so.
Given an L(M)-term t(x1, . . . , xn) we define, exactly as in first-order logic, the interpretation of t in M, which is a function tM : M n → M.
We now come to the key definition in continuous logic for metric structures, in which the semantics of
this logic is defined. For each L(M)-sentence σ, we define the value of σ in M. This value is a real number
in the interval [0, 1] and it is denoted by σM. The definition is by induction on formulas. Note that in the
definition all terms mentioned are L(M)-terms in which no variables occur.
M M M M Definition 2.4. 1. (d(t1, t2)) = d (t1 , t2 ) for any t1, t2;
2. for any n-ary predicate symbol P of L and any t1, . . . , tn,
M M M M (P (t1, . . . , tn)) = P (t1 , . . . , tn );
n 3. for any L(M)-sentences σ1, . . . , σn and any continuous function u : [0, 1] → [0, 1],
M M M (u(σ1, . . . , σn)) = u(σ1 , . . . , σn );
4. for any L(M)-formula ϕ(x),
(sup ϕ(x))M x
is the supremum in [0, 1] of the set {ϕM(a) | a ∈ M};
5. For any L(M)-formula ϕ(x),
(inf ϕ(x))M x
is the infimum in [0, 1] of the set {ϕM(a) | a ∈ M}.
12 M n Definition 2.5. Given an L(M)-formula ϕ(x1, . . . , xn) we let ϕ denote the function from M to [0, 1] defined by
M M ϕ (a1, . . . an) = (ϕ(a1, . . . , an)) .
Definition 2.6. Let ϕ be an L-formula.
1. We say that ϕ is a ∀ L-formula if for every > 0, there exists a formula σ of the form supx¯ ψ, where ψ is a quantifier-free formula, such that for every L-structure M and a ∈ M, |ϕM(a) − σM(a)| ≤ .
2. We say that ϕ is a ∀∃ L-formula if for every > 0, there exists a formula σ of the form supx¯ infy¯ ψ, where ψ is a quantifier-free formula, such that for every L-structure M and a ∈ M, |ϕM(a) − σM(a)| ≤ .
A key fact about formulas in continuous logic is that they define uniformly continuous functions. Indeed, the modulus of uniform continuity for a predicate does not depend on M but only on the data given by the signature L.
Theorem 2.7. Let t(x1, . . . , xn) be an L-term and ϕ(x1, . . . , xn) an L-formula. Then there exist functions
∆t and ∆ϕ from (0, 1] to (0, 1] such that for any L-prestructure M, ∆t is a modulus of uniform continuity for
M n M n the function t : M → M and ∆ϕ is a modulus of uniform continuity for the predicate ϕ : M → [0, 1].
Proof. See [1, Theorem 3.5].
Theorem 2.8. Let M0 be an L-prestructure with underlying pseudometric space (M0, d0); let M be its
quotient L-prestructure with quotient map π : M0 → M and let N be the L-structure that results from
completing M. Let t(x1, . . . , xn) be any L-term and ϕ(x1, . . . , xn) be any L-formula. Then:
M M0 1. t (π(a1), . . . , π(an)) = t (a1, . . . , an) for all a1 . . . , an ∈ M0;
N M 2. t (b1, . . . , bn) = t (b1, . . . , bn) for all b1, . . . , bn ∈ M;
M M0 3. ϕ (π(a1), . . . , π(an)) = ϕ (a1, . . . , an) for all a1, . . . , an ∈ M0;
N M 4. ϕ (b1, . . . , bn) = ϕ (b1, . . . , bn) for all b1, . . . , bn ∈ M.
Proof. See [1, Theorem 3.7].
Conditions of L
An L condition E is a formal expression of the form ϕ = 0, where ϕ is an L-formula. We call E closed if ϕ
is a sentence. If x1, . . . , xn are distinct variables, we write an L-condition as E(x1, . . . , xn) to indicate that
it has the form ϕ(x1, . . . , xn) = 0 (in other words, that the free variables of E are among x1, . . . , xn).
13 Let Ei be the L(M) condition ϕi(x1, . . . , xn) = 0, for i = 1, 2. We say that E1, E2 are logically equivalent if for every L-structure M and every a1, . . . , an we have
M |= E1[a1, . . . , an] iff M |= E2[a1, . . . , an].
If E is the L(M)-condition ϕ(x1, . . . , xn) = 0 and a1, . . . , an are in M, we say E is true of a1, . . . , an in
M M and write M |= E[a1, . . . , an] if ϕ (a1, . . . , an) = 0.
If E is the L(M)-condition ϕ(x1, . . . , xn) = 0 where ϕ is a ∀ L-formula, then we say that E is a ∀ L-condition.
If E is the L(M)-condition ϕ(x1, . . . , xn) = 0 where ϕ is a ∀∃ L-formula, then we say that E is a ∀∃ L-condition.
Definition 2.9. We define a binary function −· : R≥0 × R≥0 → R≥0 by
(x − y) if x ≥ y x −· y = 0 otherwise.
An important type of condition which appears frequently in this thesis is
min(φ −· ψ, σ −· ) = 0 which means if ψ < φ then σ ≤ .
2.3 Model theoretic concepts
Fix a signature L for metric structures. In this section we introduce several of the most fundamental model
theoretic concepts and discuss some of their basic properties.
Definition 2.10. A theory in L is a set of closed L-conditions. If T is a theory in L and M an L-structure,
we say that M is a model of T and write M |= T if M |= E for every condition E in T . We write ModL(T ) for the collection of all L-structures that are models of T . (If L is clear from the context, then we write
simply Mod(T ).)
Let T , Σ be L-theories. We say that T is equivalent to Σ if Mod(T ) = Mod(Σ).
Also, we say that a theory T is satisfiable if there exists an L-structure M such that M |= T .
If T is an L-theory and E is a closed L-condition, we say that E is a logical consequence of T and write
T |= E if M |= E holds for every model M of T . We set Con(T ) to be the set of all closed L-conditions that
are logical consequences of T .
14 Let T be an L-theory and Σ a set of closed L-conditions. We say that Σ axiomatizes T if Con(Σ) =
Con(T ).
If M is an L-structure, the theory of M, denoted Th(M), is the set of closed L-conditions that are true in M. If Con(T ) is a theory of this form, then T will be called complete.
We say that an L-theory T is a ∀ L-theory if there is a set of closed ∀ L-conditions Σ such that
Con(T ) = Con(Σ).
We say that an L-theory T is a ∀∃ L-theory if there exists a set of closed ∀∃ L-conditions Σ such that
Con(T ) = Con(Σ).
Definition 2.11. Suppose that M and N are L-structures.
1. We say that M, N are elementary equivalent, and write M ≡ N, if σM = σN for all L-sentences σ.
Equivalently, this holds if T h(M) = T h(N).
2. If M ⊆ N we say that M is an elementary substructure of N, and write M N, if whenever
ϕ(x1, . . . , xn) is an L-formula and a1, . . . , an are elements of M, we have
M N ϕ (a1, . . . , an) = ϕ (a1, . . . , an).
In this case, we also say that N is an elementary extension of M.
3. A function F from a subset of M into N is an elementary map from M into N if whenever ϕ(x1, . . . , xn)
is an L-formula and a1, . . . , an are elements of the domain of F , we have
M N ϕ (a1, . . . , an) = ϕ (F (a1),...,F (an)).
4. An elementary embedding of M into N is a function from all of M into N that is an elementary map
from M into N.
Remark 2.12. 1. Every elementary map from one metric structure into another is distance preserving.
2. The collection of elementary maps is closed under composition and formation of the inverse.
3. Every isomorphism between metric structures is an elementary embedding.
15 2.4 Model theoretic theorems
Here we state some of the most important theorems of continuous model theory which are generalizations
of the corresponding theorems of first-order model theory.
The following theorem is called the Compactness Theorem for continuous model theory.
Theorem 2.13. Let T be an L-theory. If every finite subset of T has a model then T has a model.
Proof. See [1, Theorem 5.8].
If Λ is a linearly ordered set, a Λ-chain of L-structures is a family of L-structures (Mλ | λ ∈ Λ) such that Mλ ⊆ Mη for λ < η. If this holds, we can define the union of (Mλ | λ ∈ Λ) as an L-prestructure in an obvious way. This union is based on a metric space, but it may not be complete. After taking the completion we get an L-structure that we will refer to as the union of the chain and that we will denote by S λ∈Λ Mλ.
Definition 2.14. A chain of structures (Mλ | λ ∈ λ) is called an elementary chain if Mλ Mη for λ < η.
S Proposition 2.15. If (Mλ | λ ∈ Λ) is an elementary chain and λ ∈ Λ, then Mλ λ∈Λ Mλ.
Proof. See [1, Proposition 7.2].
Recall that the density character of the of a topological space is the smallest cardinality of a dense subset
of the space.
Proposition 2.16 (Downward L¨owenheim-Skolem Theorem). Let κ be an infinite cardinal and assume that
that card(L) ≤ κ. Let M be an L-structure and suppose A ⊆ M has density(A) ≤ κ. Then there exists a
substructure N of M such that
1. N M;
2. A ⊆ N ⊆ M;
3. density(N) ≤ κ.
Proof. See [1, Proposition 7.3].
2.5 Spaces of types
In this section we consider a fixed signature L for metric structures and a fixed L-theory T . Until further
notice we assume that T is a complete theory.
16 Suppose that M is a model of T and A ⊆ M. Denote the L(A)-structure (M, a)a∈A by MA, and set TA to be the L(A)-theory of MA. Note that any model of TA is isomorphic to a structure of the form (N, a)a∈A where N is a model of T .
Definition 2.17. Let TA be as above, β an ordinal, and (xi)i∈β distinct variables.
A set p of L(A)-conditions with all free variables among (xi)i∈β is called a β-type over A if there exists a
model (M, a)a∈A of TA and elements (ei)i∈β of M such that p is the set of all L(A)-conditions E(xi1 , . . . , xin )
for which MA |= E[ei1 , . . . , ein ], where n ∈ min(ω, β) and i1, . . . , in ∈ β.
When this relationship holds, we denote p by tpM((ei)i∈β/A) and we say that (ei)i∈β realizes p in M. (The subscript M will be omitted if doing so causes no confusion; A will be omitted if it is empty.)
The collection of all such β-types over A is denoted by Sβ(TA) or simply by Sβ(A) if the context makes
the theory TA clear.
Now we introduce the logic topology on types. Fix TA as above. If ϕ(xi1 , . . . , xin ) is an L-formula, for
some i1, . . . , in ∈ β, and > 0, we let [ϕ < ] denote the set
{q ∈ Sβ(TA) | for some 0 ≤ δ < the condition ϕ −· δ = 0 is in q}.
Definition 2.18. The logic topology on Sβ(TA) is defined as follows. If p is is Sβ(TA), the basic open neighborhoods of p are the sets of the form [ϕ < ] for which the condition ϕ = 0 is in p and > 0.
Definition 2.19. Let β be an ordinal, and let (xi)i∈β be distinct variables.
A set p of quantifier-free L(A)-conditions with all free variables among (xi)i∈β is called a quantifier-free
β-type over A if there exists a model (M, a)a∈A of TA and elements (ei)i∈β of M such that p is the set of
all quantifier-free L(A)-conditions E(xi1 , . . . , xin ) for which MA |= E[ei1 , . . . , ein ], where n ∈ min(ω, β) and
i1, . . . , in ∈ β.
When this relationship holds, we denote p by qftpM((ei)i∈β/A) and we say that (ei)i∈β realizes p in M. (A will be omitted if it is empty.)
Let p be a quantifier-free β-type over A. If γ ≤ β, then we write p γ to denote the set formed by the
set of all L(A)-conditions whose free variables are among (xi)i∈γ .
If p((xi)i∈β) is a quantifier-free β-type over A and φ(xi1 , . . . , xin ) an L(A)-formula where n ∈ min(ω, β)
p and i1, . . . , in ∈ β. We write φ to denote the unique real number r for which |φ−r| = 0 is in p. Equivalently, φp is defined to be the value φ(¯a) whena ¯ is any realization of p in an L-structure.
Proposition 2.20. For any n ≥ 1, Sn(TA) is compact and Hausdorff with respect to the logic topology.
17 Proof. See [1, Proposition 8.6].
Now we introduce the d-metric on types. Let TA be as above. For each n ≥ 1 we define a natural metric
n on Sn(TA); it is induced as a quotient of the given metric d on M , where (M, a)a∈A is a suitable model of
TA, so we also denote this metric on types by d.
To define the metric, let MA = (M, a)a∈A be any model of TA in which each type in Sn(TA) is realized,
for each n ≥ 1. Let (M, d) be the underlying metric space of M. For p, q ∈ Sn(TA) we define d(p, q) to be
inf{ max d(bj, cj) | MA |= p[b1, . . . , bn], MA |= q[c1, . . . , cn]}. 1≤j≤n
Note that this expression for d(p, q) does not depend on MA, since MA realizes every type of a 2n-tuple
(b1, . . . , bn, c1, . . . , cn) over A. It follows that d is a pseudometric on Sn(TA). Note that if p, q ∈ Sn(A), then
by the Compactness Theorem and our assumption about MA, there exist realizations (b1, . . . , bn) of p and
(c1, . . . , cn) of q in MA such that maxj d(bj, cj) = d(p, q). In particular, if d(p, q) = 0, then p = q; so d is
indeed a metric on Sn(TA).
Proposition 2.21. The d-topology is finer than the logic topology on Sn(TA).
Proof. See [1, Proposition 8.7].
Proposition 2.22. The metric space (Sn(TA), d) is complete.
Proof. See [1, Proposition 8.8].
18 Chapter 3
Ages
In this chapter we fix a bounded continuous signature L without function symbols.
In Chapter 3 we give the definitions of an age and a ρ-compact age and prove some basic propositions that we will need later.
3.1 Basic definitions
Notation 3.1. We denote by ML the class of all L-structures, and by ML the class of all finite L-structures.
If the signature L is clear from the context we simply write M and M respectively.
Definition 3.2. Let A ⊆ M. Then A is called an age in L if it has the following properties:
• A is closed under isomorphism.
• (Hereditary property, HP for short) If M ∈ A and N is an L-substructure of M, then N ∈ A.
• (Joint embedding property, JEP for short) If M1, M2 ∈ A, then there exist M ∈ A and L-embeddings
f1 : M1 → M, f2 : M2 → M.
Usually L will be clear and we will simply say A is an age.
Notation 3.3. 1. Let M an L-structure. We set age(M) to be the class of all finite L-structures which
are isomorphic to a substructure of M.
2. Let T be an L-theory. We set FS(T ) to be the class of all finite L-structures which are isomorphic to
a substructure of a model of T .
3. Let A ⊆ M and α be an ordinal number. We set
A Sα = {qftpM((ai)i∈α) | M ∈ M, age(M) ⊆ A, (ai)i∈α ⊆ M}.
19 4. Let M be an L-structure and A ⊆ M. We set
A Sα (A) = {qftpM((ai)i∈α/A) | M ∈ M, age(M) ⊆ A, (ai)i∈α ⊆ M}.
A 5. We set M to be the class of all L-structures M such that age(M) ⊆ A.
We will also use the following notation.
Notation 3.4. Let M be an L-structure.
1. The diagram of M, denoted by Diag(M), is the set of quantifier-free closed L(M)-conditions that are
true in M.
2. The elementary diagram of M, denoted by EDiag(M), is the set of closed L(M)-conditions that are
true in M.
The following proposition indicates that ages occur very often.
Proposition 3.5. 1. If M is an L-structure, then age(M) is an age.
2. If T is a complete L-theory, then FS(T ) is an age.
Proof. (1): Clearly age(M) has the HP and the JEP.
(2): Clearly FS(T ) has the HP. We show that it has the JEP. Let M, N in FS(T ). Without losing
generality we may assume that M ∩ N = ∅. It is enough to show that the L(M ∪ N)-theory
Diag(M) ∪ Diag(N) ∪ T
has a model. This is an immediate consequence of the compactness theorem and the fact that T is complete.
Notation 3.6. If T is a complete L-theory we write age(T ) instead of FS(T ).
Definition 3.7. (Amalgamation property, AP for short) Let A ⊆ M. We say that A has the amalgamation property if for every M, M1, M2 ∈ A and embeddings f1 : M → M1, f2 : M → M2, there are N in A and embeddings g1 : M1 → N, g2 : M2 → N such that g1 ◦ f1 = g2 ◦ f2.
Proposition 3.8. Let A be an age. The following statements are equivalent:
1. A has the AP.
20 A 2. For every M ∈ A, n, l ∈ ω, p ∈ Sn+l, and a¯ = (a1 . . . , an) ⊆ M such that a¯ |= pn, there exist N in A
such that M ⊆ N and an+1 . . . an+l in N such that (a1, . . . an+l) |= p.
A 3. For every M ∈ A, n ∈ ω, p ∈ Sn+1, and a¯ = (a1 . . . , an) ⊆ M such that a¯ |= p n, there exist N in A
such that M ⊆ N and an+1 in N such that (a1, . . . an+1) |= p.
Proof. Straightforward from the definitions.
3.2 ρ-compact ages
For this section we assume, in addition, that L is finite.
M Definition 3.9. For every n ∈ ω, we define a metric on Sn in the following way. Let (ϕi(¯x))1≤i≤k be an M enumeration of all atomic L-formulas in the variables x1, . . . , xn. If p(¯x), q(¯x) are in Sn then we define
p q ρn(p, q) = max |ϕi(¯x) − ϕi(¯x) |. 1≤i≤k
M Notation 3.10. Let p ∈ Sn for some n ∈ ω, and let (ϕi(¯x))1≤i≤k be an enumeration of all atomic L-
formulas in the free variables x1, . . . , xn. We set
p τp(¯x) = max |ϕi(¯x) − ϕi(¯x) |. 1≤i≤k
M Remark 3.11. Let p ∈ Sn for some n ∈ ω. Then for every metric L-prestructure M and n-tuplea ¯ ⊆ M
M ρn(p, qftpM(¯a)) = τp (¯a).
M Proposition 3.12. For every n ∈ ω, (Sn , ρn) is a metric space.
Proof. Let n ∈ ω, and (ϕi)1≤i≤k be an enumeration of all atomic L-formulas in the free variables x1, . . . , xn.
M k M We define a map Φ : Sn → [0, 1] as follows: for every p ∈ Sn
p p −→ (ϕi | 1 ≤ i ≤ k).
k k M Let ([0, 1] , d) denote [0, 1] equipped with the max metric. We note that for every p, q ∈ Sn , ρn(p, q) = M k k d(Φ(p), Φ(q)). We deduce that we can identify (Sn , ρn) with a metric subspace of ([0, 1] , d). Since ([0, 1] , d) M is a metric space, we conclude that (Sn , ρn) is a metric space.
21 A Notation 3.13. Let A ⊆ M. For every n ∈ ω, we write (Sn , ρn) to denote the the restriction of the metric M A space (Sn , ρn) to Sn .
A Definition 3.14. Let A ⊆ M. We say that A is ρ-compact if for all n ∈ ω,(Sn , ρn) is a complete metric space.
A M Remark 3.15. We note that for all n ∈ ω,(Sn , ρn) is a metric subspace of (Sn , ρn) which is always totally A M bounded. Hence (Sn , ρn) is compact iff it is closed in (Sn , ρn) iff it is a complete metric space.
Proposition 3.16. Let T be a complete L-theory. Then age(T ) is a ρ-compact age.
Proof. By Proposition 3.5 we deduce that age(T ) is an age. The fact that it is ρ-compact is an immediate
consequence of the compactness theorem.
In the following proposition we show that a ρ-compact age is in some sense axiomatizable.
Proposition 3.17. Let A be a ρ-compact age. Then there exists an L-theory Σ such that for every metric
L-prestructure M, M |= Σ iff age(M) ⊆ A.
A Proof. For all n ∈ ω, we have that (Sn , ρn) is totally bounded. Therefore, for all n ∈ ω, > 0, there exists A an -net, {pn,,1, . . . , pn,,mn, } ⊆ Sn . Set Σ to be the set of all closed L-conditions of the form
· sup min (τpn,,i (¯x) − ) = 0, x¯ 1≤i≤mn, for all n ∈ ω, > 0. For every metric L-prestructure M with age(M) ⊆ A, M |= Σ. This proves (⇐). For the direction (⇒), let M be a metric L-prestructure and assume that age(M) * A. Then there exists a finite L-substructure A of M such that A ∈/ A. Leta ¯ be an n-tuple which enumerates A, for some n ∈ ω, and set
p = qftpA(¯a). We will use the following notation.
Notation 3.18. Let (X, d) be a metric space and Y ⊆ X with Y 6= ∅. We define
d(x, Y ) = inf{d(x, y) | y ∈ Y }.
A M We continue the proof of the Proposition 3.17. Since (Sn , ρn) is a compact subspace of (Sn , ρn), we A M A A A deduce that Sn is a closed subset of (Sn , ρn). Since p∈ / Sn , we have ρn(p, Sn ) > 0. Set = ρn(p, Sn )/2. Clearly
min ρn(pn,,i, p) ≥ 2. 1≤i≤mn,
22 Therefore
min τ M (¯a) ≥ 2, pn,,i 1≤i≤mn,
and so
min (τ M (¯a) −· ) ≥ . pn,,i 1≤i≤mn,
Therefore · sup min (τpn,,i (¯x) − ) > 0 x¯ 1≤i≤mn,
is true in M. We conclude that M does not satisfy Σ.
Notation 3.19. Let A be a ρ-compact age. We denote by ΣA the L-theory Σ constructed in the proof of
Proposition 3.17.
Corollary 3.20. Let A be a ρ-compact age. If M is a metric L-prestructure with age(M) ⊆ A, then age(M) ⊆ A.
Proof. Since age(M) ⊆ A, Proposition 3.17 implies that M |= ΣA. From Proposition 2.8, M |= ΣA and so
again from Proposition 3.17, age(M) ⊆ A.
Definition 3.21. Let A ⊆ M. The completion of A is the class B of all finite L-structures M which satisfy the following condition: M is in B, where M = {a1, . . . an} for some n ≥ 1, if qftpM(a1, . . . , an) is in the A M closure of Sn in (Sn , ρn).
Remark 3.22. If B is the completion of A then A ⊆ B.
Proposition 3.23. Let A be an age. If B is the completion of A, then B is a ρ-compact age.
Proof. The fact that B is ρ-compact is immediate. To show that B is an age, it is enough to show that it has the HP and JEP.
For the HP: Let M ∈ B and N ⊆ M. Leta ¯ = (a1, . . . , am) be an enumeration of M, for some m ≥ 1,
B such that for some n ∈ ω,a ¯ n is an enumeration of N. Set p = qftpM(¯a).Since p ∈ Sm , there exists a A sequence (pi | i ∈ ω) in Sm such that limi→∞ ρm(p, pi) = 0. Clearly,
lim ρn(p n, pi n) = 0 (3.1) i→∞
A For all i ∈ ω, pi n ∈ Sn since A has the HP property. By (3.1), and since A ⊆ B and B is ρ-compact,we B conclude that pn ∈ Sn which in turns implies that N ∈ B.
23 ¯ ¯ For the JEP: Let A, B ∈ B. Leta, ¯ b be enumerations of A, B respectively. Set p = qftpA(¯a), q = qftpB(b). There exist sequences (p | i ∈ ω) in SA ,(q | i ∈ ω) in SA , for some n , n ∈ ω such that i n1 i n2 1 2
lim ρn (p, pi) = 0 and lim ρn (q, qi) = 0. i→∞ 1 i→∞ 2
A Since A has the JEP, there exists l ∈ ω such that for every i ∈ ω there exists ri ∈ Sl such that ifc ¯i is a realization of ri then there exist 1 ≤ j1, . . . , jn1 ≤ l and 1 ≤ k1, . . . , kn2 ≤ l such that
(c , . . . , c ) |= p and (c , . . . , c ) |= q i,j1 i,jn1 i i,k1 i,kn2 i
A respectively. Since for all i ∈ ω, ri ∈ Sl , A ⊆ B and B is ρ-compact we deduce that the sequence (ri | i ∈ ω) B has a limit point r in Sl . Without losing generality (we may pass to a subsequence of (ri | i ∈ ω)), we may assume that
lim ρl(r, ri) = 0. i→∞
By applying the pigeon-hole principle, there is a subsequence {rim | m ∈ ω} of {ri | i ∈ ω}, and finite
sequences of natural numbers 1 ≤ j1, . . . , jn1 ≤ l, 1 ≤ k1, . . . , kn2 ≤ l such that for every m ∈ ω
(c , . . . , c ) |= p and (c , . . . , c ) |= q . im,j1 im,jn1 im im,k1 im,kn2 im
B Since r ∈ Sl , there is a finite structure M in B andc ¯ ⊆ M such thatc ¯ |= r. From the above we deduce that
(c , . . . , c ) |= p and (c , . . . , c ) |= q. j1 jn1 k1 kn2
We conclude that we can embed both A and B into M, and therefore B has the JEP.
24 Chapter 4
Basic model theory
In this chapter we fix a bounded continuous signature L without function symbols.
In Chapter 4 we present the basic model theoretic tools that we will use later in this document. In Sections
4.1 and 4.5 we follow [1] closely. In Sections 4.2, 4.3, and 4.4 we prove straightforward generalizations of analogous results in the first-order setting, and we follow [6] closely.
4.1 Quantifier elimination
Definition 4.1. Let T be an L-theory.
1. An L-formula ϕ(x1, . . . , xn) is approximable in T by quantifier-free formulas if for every > 0 there is
a quantifier-free L-formula ψ(x1, . . . , xn) such that for all M |= T and all a1, . . . , an in M, one has
M M |ϕ (a1, . . . , an) − ψ (a1, . . . , an)| ≤ .
2. We say that T has quantifier elimination if every L-formula is approximable in T by quantifier-free
formulas. In this case we also say that T has QE.
Proposition 4.2. Let T be an L-theory and ϕ(x1, . . . , xn) be an L-formula. The following statements are equivalent.
1. ϕ is approximable in T by quantifier-free formulas.
2. Whenever we are given
• models M and N of T ;
• substructures M0 ⊆ M and N0 ⊆ N;
• an isomorphism Φ from M0 onto N0;
• elements a1, . . . , an of M0;
25 we have
M N ϕ (a1, . . . , an) = ϕ (Φ(a1),..., Φ(an)).
Moreover, for the implication (2) ⇒ (1) it suffices to assume (2) only for the cases in which M0, N0 are finitely generated.
Proof. See [5, Proposition 13.12].
Proposition 4.3. Let T be an L-theory. The following statements are equivalent:
1. T admits QE.
FS(T ) 2. For every separable model M of T , n ∈ ω, p ∈ Sn+1 , and a¯ ⊆ M with a¯ |= p n, there exists an
ℵ1-saturated extension N M and an+1 ∈ N such that (¯a, an+1) |= p.
Proof. See [5, Proposition 13.17].
Theorem 4.4. Let T be an L-theory. If T has QE, then FS(T ) has the AP.
Proof. Immediate from Proposition 4.3.
Proposition 4.5. Let T be a satisfiable L-theory that has QE. If there exists an L-structure A which embeds
into every ℵ1-saturated model of T , then T is a complete L-theory.
Proof. We show that for every M, N |= T we have that Th(M) = Th(N). This proves that Con(T ) = Th(M) for some (every) M |= T . This suffices since Th(M) is a complete L-theory.
Fix M, N |= T . Let M1, N1 be ℵ1-saturated elementary extensions of M, N respectively, and A be the common L-structure which embeds into both M1, N1.
M1 Suppose that the closed L-condition σ = 0 is in Th(M1). So we have σ = 0. Set
ψ(x) = max(d(x, x), σ).
M1 Let a ∈ A. We have that ψ (a) = 0. Because T ⊆ Th(M) has QE we have that A M1. So we have
A N1 N1 ψ (a) = 0. As before we have that A N1 and so ψ (a) = 0. This implies that σ = 0, and so
(σ = 0) ∈ Th(N1). So Th(M1) ⊆ Th(N1) and in a similar way we may prove that Th(N1) ⊆ Th(M1).
We deduce that Th(M1) = Th(N1). Since Th(M) = Th(M1) and Th(N) = Th(N1), we conclude that Th(M) = Th(N).
26 4.2 Inductive theories
Definition 4.6. 1. Let A, B be L-structures such that A ⊆ B. We say that A is existentially closed
(e.c) in B (and we write A ⊆ec B) if for every quantifier-free L-formula φ(¯x, y¯), and finite tuplea ¯ ⊆ A
(inf φ(¯x, a¯))A = (inf φ(¯x, a¯))B. x¯ x¯
2. Let K be a class of L-structures. We say that a structure A in K is e.c. in K if for every B in K with
A ⊆ B, A is existentially closed in B.
3. If K is the class of all models of an L-theory T , we refer to e.c. structures in K as e.c models of T .
Definition 4.7. A class of L-structures K is called inductive if the following holds:
1. K is closed under isomorphism.
2. K is closed under unions of chains.
Definition 4.8. Let T be an L-theory. We say that T is inductive if the class K = Mod(T ) is inductive.
Inductive classes are useful because of the following theorem.
Theorem 4.9. Let K be an inductive class of L-structures and A a structure in K. Then there is an e.c. structure B in K such that A ⊆ B.
Proof. Similar to the first-order case. For example, see [6, Theorem 7.2.1].
The following theorem in one direction characterizes syntactically which theories are inductive and in the other direction indicates that there are several examples of inductive theories.
Theorem 4.10. Let T be an L-theory. The following statements are equivalent:
1. T is inductive.
2. T is axiomatized by a set of ∀∃ closed L-conditions.
Proof. (1)⇒(2): See [9, Theorem 3.6].
(2)⇒(1): Similar to the first-order case. For example, see [6, Theorem 2.2.4].
Notation 4.11. Let T be an L-theory. We denote by T∀ the set of all the logical consequences E of T , such that E is a closed ∀ L-condition.
27 The following lemma and proposition will be used later.
Lemma 4.12. Let T be an L-theory. If A |= T∀, then there exists B |= T such that A ⊆ B.
Proof. (of the lemma) Let A |= T∀. Leta ¯ be an enumeration of A. Set p(¯y) = qftpA(¯a) It is enough to show that
p(¯y) ∪ T
has a model. We use compactness. For a contradiction, assume that there exists a finite D ⊆ A and σ(¯x) = 0 ¯ ¯ in qftpA(d), where d is an enumeration of D, such that
{σ(¯x) = 0} ∪ T
has no model. Then,
T |= sup(r −· σ(¯x)) = 0 x¯
for some r > 0. Since A |= T∀ we deduce
(sup(r −· σ(¯x)))A = 0. x¯
This implies
(sup(r −· σ(¯x)))D = 0, x¯
which is a contradiction.
Proposition 4.13. Let T be a ∀∃ L-theory and A a model of T . The following statements are equivalent:
1. A is an e.c model of T .
2. A is an e.c. model of T∀.
Proof. (1)⇒ (2): Let B |= T∀ with A ⊆ B, ψ(¯x, y¯) a quantifier-free L-formula, anda ¯ ⊆ A. It is enough to show that
(inf ψ(¯a, y¯))A = (inf ψ(¯a, y¯))B. y¯ y¯
By Lemma 4.12, there exists C |= T with B ⊆ C. Since A ⊆ B ⊆ C, we deduce that
(inf ψ(¯a, y¯))A ≥ (inf ψ(¯a, y¯))B ≥ (inf ψ(¯a, y¯))C. y¯ y¯ y¯
28 Because A ⊆ C and A is an e.c. model of T we have that
(inf ψ(¯a, y¯))A = (inf ψ(¯a, y¯))C. y¯ y¯
Hence,
(inf ψ(¯a, y¯))A = (inf ψ(¯a, y¯))B. y¯ y¯
(2)⇒(1): Assume (2). First we show that A is a model of T . Since T is a ∀∃ L-theory, a typical L-
condition in T can be written in the form supx¯ infy¯ ψ(¯x, y¯) = 0 with ψ(¯x, y¯) quantifier-free. Leta ¯ be a tuple
A in A. We must show that (infy¯ ψ(¯a, y¯)) = 0. By Lemma 4.12, since A |= T∀ there is C |= T with A ⊆ C.
C A Then, (infy¯ ψ(¯a, y¯)) = 0 and so (infy¯ ψ(¯a, y¯)) = 0 since A is an e.c. model of T∀. Thus A is a model of T .
Clearly A is an e.c. model of T , since every model of T extending A is in fact a model of T∀ too.
4.3 Model-complete theories
Definition 4.14. An L-theory is said to be model-complete if every embedding between its models is
elementary.
Theorem 4.15. Let T be an L-theory. The following statements are equivalent:
1. T is model complete.
2. Every model of T is an e.c. model of T .
3. If A, B are models of T and e : A → B is an embedding, then there are an elementary extension D of
A and an embedding g : B → D such that g ◦ e is the identity on A.
Proof. (1)⇒(2): Let A, B |= T with A ⊆ B. Since T is model complete we deduce that A B. Let φ(¯x, y¯) be a quantifier-free L-formula anda ¯ a finite tuple in A. Since A B, we conclude
(inf φ(¯x, a¯))A = (inf φ(¯x, a¯))B. x¯ x¯
(2)⇒(3): Let A, B |= T such that A ⊆ B. It is enough to show that EDiag(A) ∪ Diag(B) is satisfiable.
This is done by compactness and by using the fact that A ⊆ec B.
(3)⇒(1): Let A0, B0 |= T such that A0 ⊆ B0. Using the hypothesis repeatedly we build a chain
A0 ⊆ B0 ⊆ A1 ⊆ B1 ... , all being models of T , and such that for all i ∈ ω, Ai Ai+1 and Bi Bi+1. Let
29 S S C be the completion of the set-theoretic union of the whole chain. Then C = Ai = Bi. By Proposition
2.15, A0 C and B0 C. It follows that A0 B0.
Proposition 4.16. Let T be an L-theory. If T is model-complete then T is a ∀∃ theory.
Proof. By Proposition 2.15, Mod(T ) is closed under unions of elementary chains and because T is model-
complete we deduce that Mod(T ) is closed under unions of chains. By Theorem 4.10 we conclude that T is
a ∀∃ theory.
4.4 Model companions
Definition 4.17. Let T be an L-theory.
0 0 0 1. We say that an L-theory T is a model companion of T if T is model-complete and T∀ = T∀.
2. We say that T is companionable if it has a model companion.
Theorem 4.18. Let T be a ∀∃ L-theory.
1. If T is companionable, then up to equivalence of theories, its model companion is unique and is the
theory of the class of the e.c. models of T .
2. T is companionable iff the class of e.c. models of T is axiomatized by an L-theory.
Proof. (1): Assume T is companionable with a model companion T 0. Since T 0 is model-complete, by Theorem
4.15 the models of T 0 are precisely the e.c. models of T 0. By Proposition 4.13, the models of T 0 are precisely
0 the e.c. models of T∀ = T∀. Since T is a ∀∃ theory, again by Proposition 4.13, the e.c. models of T∀ are precisely the e.c. models of T . So T 0 axiomatizes the theory of the class of e.c models of T . The uniqueness of T 0 is clear.
(2): The direction(⇒) is immediate from (1).
For the other direction assume that the class of e.c. models of T is axiomatized by an L-theory T 0.
0 First we prove that T∀ = T∀. It is enough to show that that every model of T has an extension which is a model of T 0, and that every model of T 0 has an extension which is a model of T . The first is true by
Corollary 4.9 and the second is obviously true.
Also we note that T 0 is a ∀∃ theory: By Proposition 2.15, Mod(T 0) is closed under unions of elementary chains and because T 0 is model-complete we deduce that Mod(T 0) is closed under unions of chains. By
Theorem 4.10 we conclude that T 0 is a ∀∃ theory.
30 By hypothesis, Mod(T 0) is the class of e.c. models of T . Because T is ∀∃ theory, by Proposition 4.13,
0 0 0 0 0 Mod(T ) is the class of e.c models of T∀. Since T∀ = T∀, Mod(T ) is the class of e.c. models of T∀. Since T is a ∀∃ theory, again by Proposition 4.13, we conclude that Mod(T 0) is precisely the class of e.c. models of
the L-theory T 0. By Theorem 4.15, T 0 is model-complete.
Proposition 4.19. 1. If T , T 0 are model-complete theories and FS(T ) = FS(T 0), then T , T 0 are equiv-
alent.
2. If T , T 0 are theories that have QE and FS(T ) = FS(T 0), then T , T 0 are equivalent.
Proof. We will need the following lemmas:
Lemma 4.20. Let T be an L-theory. Then the models of T∀ are precisely the substructures of models of T .
Proof. Clearly the substructures of models of T are models of T∀. The other direction of the containment is an immediate corollary of Lemma 4.12.
0 0 0 Lemma 4.21. Let T , T be L-theories. Then FS(T ) = FS(T ) iff T∀ is equivalent to T∀.
Proof. (of the lemma) (⇒): Set A = FS(T ) and B = FS(T 0). It is enough to show that for all L-structures
0 0 M, M |= T∀ iff M |= T∀. Let M |= T∀. Because A = B, we have that Diag(M) ∪ T is finitely satisfiable. Compactness implies that there exists an L-structure N such that N |= Diag(M) ∪ T 0. This implies that M
0 0 can embedded in a model of T . By Lemma 4.20 we conclude that that M |= T∀. In the same way we prove 0 M |= T∀ implies M |= T∀. 0 0 (⇐): By Lemma 4.20, the models of T∀, T∀ are precisely the L-substructures of models T , T respectively, 0 0 and so T∀ = T∀ implies that FS(T ) = FS(T ).
Now we prove the proposition.
0 0 (1): Because FS(T ) = FS(T ), by Lemma 4.21, T∀ = T∀. This implies that both theories are model 0 companions of the the L-theory T∀. From Theorem 4.18, the theories T , T are equivalent. (2): It follows from (1).
4.5 ω-categoricity
Definition 4.22. Let κ be a cardinal ≥ card(L). We say that T is κ-categorical if T has noncompact models
and whenever M and N are noncompact models of T having density character equal to κ, one has that M
is isomorphic to N.
31 Theorem 4.23. Suppose that the number of nonlogical symbols in L is countable. Let T be a complete
L-theory that has noncompact models. The following statements are equivalent:
1. T is ω-categorical;
2. For each n ≥ 1, the metric space (Sn(T ), d) is compact.
Proof. See [1, Theorem 12.10].
Theorem 4.24. Suppose that T is ω-categorical and M is the separable model of T . Then M is strongly ω- near-homogeneous in the following sense: if a,¯ ¯b ⊆ M are n-tuples, then for every > 0 there is automorphism
F of M such that
M ¯ max d (F (ai), bi) ≤ d(tp(¯a), tp(b)) + . 1≤i≤n
Proof. See [1, Corollary 12.11].
In Chapter 8 we need the implication (1)⇒(2) in the following theorem. (This theorem is an observation due to C. Ward Henson.)
Theorem 4.25. Let M be a noncompact separable L-structure. Set T = Th(M) and G = Aut(M). The following statements are equivalent:
1. T is an ω-categorical theory.
2. For every > 0, n ≥ 1 there exist n-tuples a¯1 ... a¯l ⊂ M, for some l ∈ ω, such that for every n-tuple ¯b ⊆ M there exist 1 ≤ j ≤ l and F ∈ G such that
M max d (F (aj,i), bi) < . 1≤i≤n
¯ ¯ Proof. (1)⇒(2): Fix > 0, n ≥ 1 and b ⊆ M. Set p = qftpM(b). Since T is ω-categorical, by Theorem
4.23 (Sn(T ), d) is compact. This implies that there exists a finite /2-net {p1, . . . , pl}, for some l ∈ ω, of
(Sn(T ), d). Therefore there exists 1 ≤ j ≤ l such that d(p, pj) < /2. Since M is the unique separable model of T , for every 1 ≤ j ≤ l there exista ¯j ⊆ M such thata ¯j |= pj. Then by Proposition 4.24 there exists an automorphism F of M such that
M max d (F (aj,i), bi) ≤ d(pj, p) + /2 < . 1≤i≤n
(2)⇒(1): By Proposition 4.23 it is enough to show that for all n ≥ 1, (Sn(T ), d) is compact. By
Proposition 2.22 it is enough to show that for all n ≥ 1, (Sn(T ), d) is totally bounded. Fix n ≥ 1. Set D to
32 be the set of n-types realized in M. The hypothesis in (2) implies that D is totally bounded. To show that
(Sn(T ), d) is totally bounded, it is enough to show that D is dense in (Sn(T ), d).
Let p ∈ Sn(T ). Since D is dense in Sn(T ) in the logic topology and L is countable, we deduce that there exists a sequence S in D that converges to p in the logic topology. Since D is totally bounded for the
0 0 d-metric, we deduce that there exists a subsequence S of S such that S is a Cauchy sequence in (Sn(T ), d).
0 Clearly S converges to p in the d-metric. We conclude that D is dense in (Sn(T ), d).
Definition 4.26. Let M be an L-structure. We say that M is strongly ω-homogeneous if for every finite
A ⊆ M if f : A → M is an elementary map with respect to M, then there exists an automorphism g of M
that extends f.
The following easy fact will be used later.
Proposition 4.27. Let M be a separable L-structure such that Th(M) is ω-categorical. The following statements are equivalent:
1. M is strongly ω-homogeneous.
2. For every finite tuple a¯ ⊆ M, Th(M, a¯) is ω-categorical.
Proof. (1)⇒(2): Let Ne be an L(¯a)-structure such that Ne |= Th(M, a¯). We will show that Ne =∼ (M, a¯). Let
N be the reduct of Ne to L. Clearly N |= Th(M) and because Th(M) is ω-categorical we have that M =∼ N. ∼ ¯ ¯ ¯ We deduce that Ne = (M, b) for some b ⊆ M with tpM(¯a) = tpM(b). Since M is strongly ω-homogeneous we conclude that (M, a¯) =∼ (M, ¯b) and hence Ne =∼ (M, a¯). ¯ ¯ ¯ (2)⇒(1): Leta ¯, b be finite tuple in M such that tpM(¯a) = tpM(b). Because tpM(¯a) = tpM(b) we have that Th(M, a¯) = Th(M, ¯b). Because (M, ¯b) |= Th(M, a¯) and Th(M, a¯) is ω-categorical we have that
(M, a¯) =∼ (M, ¯b). We conclude that M is strongly ω-homogeneous.
33 Chapter 5
Fra¨ıss´eTheorems for complete theories with QE
In this chapter we fix a bounded continuous signature L whose only nonological symbols are a finite number of constant and predicate symbols.
In Chapter 5 we prove a generalization to the continuous setting of the Fra¨ıss´eTheorems for complete theories that have QE.
5.1 A Fra¨ıss´eTheorem for complete theories with QE
Definition 5.1. Let A be an age. We say that A has the perturbed amalgamation property (PAP) if for every
A A n ∈ ω, > 0, and p(¯x) ∈ Sn+1, there exists δn (, p) > 0 such that for every M ∈ A,a ¯ = (a1, . . . , an) ⊆ M A with ρn(p n, q) < δn (, p), where q = qftpM(¯a), there exist N ⊇ M in A, and an+1 in N such that
ρn+1(p, r) ≤ , where r = qftpN(a1, . . . , an+1).
Definition 5.2. Let A be an age. We say that A has the uniformly perturbed amalgamation property
A (UPAP) if for every n ∈ ω and > 0, there exists δn () > 0 such that for every M ∈ A whenever we are given
A • p(¯x) ∈ Sn+1
A • a¯ = (a1, . . . , an) ⊆ M such that ρn(pn, q) < δn () where q = qftpM(¯a) then there exist N in A such that M ⊆ N and an+1 in N such that ρn+1(p, r) ≤ , where r = qftpN(a1, . . . , an+1).
Remark 5.3. UPAP is stronger than the PAP, but it turns out that these concepts are equivalent for
ρ-compact ages.
Our main goal in this chapter is to prove the following two theorems, which we call Fra¨ıss´eTheorems for complete QE theories.
Theorem 5.4. Let A be a ρ-compact age with the PAP. Then there exists a unique (up to equivalence of theories) complete L-theory T that has QE and satisfies age(T ) = A.
34 A strong converse of Theorem 5.4 is the following theorem.
Theorem 5.5. Let T be a complete L-theory that has QE. Then age(T ) is a ρ-compact age with the UPAP.
We prove Theorem 5.4 in 5.17 and Theorem 5.5 in 5.23.
We will need the following propositions.
Proposition 5.6. Let A be a ρ-compact age.
1. If A has the PAP, then A has the AP.
2. In particular, if A has the UPAP, then A has the AP.
A Proof. (1): We will use Proposition 3.8. Fix A ∈ A, n ∈ ω, p ∈ Sn+1, anda ¯ = (a1, . . . , an) ⊆ A such that ¯ ¯ ¯ a¯ |= p n. Let b be an enumeration of A \ a¯. Assume that b is an l-tuple for some l ∈ ω. Set q = qftpM(b).
Since A has the PAP, for every m ≥ 1 there exists Am ∈ A such that A ⊆ Am and cm ∈ Am such that ¯ ¯ ρn(p, rm) ≤ 1/m where rm = qftpAm (¯a, cm). Set dm = baa¯ m and sm = qftpAm (dm). Since the sequence A A (sm | m ≥ 1) is in Sl+n+1 and (Sl+n+1, ρl+n+1) is compact, we deduce that the sequence mentioned has a limit point s. Let d¯|= s. Let B be the L-structure generated by d¯ and f : A → B the embedding defined as follows: For every 1 ≤ i ≤ l we set f(bi) = di; for every 1 ≤ i ≤ n we set f(ai) = dl+i. The above embedding shows that without losing generality we may assume that A ⊆ B. Clearly (¯a, dl+n+1) |= p. This suffices. (2): Follows easily from (1) since UPAP implies PAP.
Definition 5.7. Let A be an age.
A A 1. We say that A has the strong JEP if for every family of L-structures (Ai)i∈I in M there exist B ∈ M
and for every i ∈ I an embedding fi : Ai → B.
A 2. We say that A has the strong AP if for every family of L-structures A, (Bi)i∈I in M and embeddings A fi : A → Bi for every i ∈ I, there exist an L-structure C in M and, for every i ∈ I, an embedding
gi : Bi → C such that for every i, j ∈ I, gi ◦ fi = gj ◦ fj.
A A 3. We say that A has the strong PAP if for every n ∈ ω, > 0, and p(¯x) ∈ Sn+1, there exist ∆n (, p) > 0 A A such that for every M ∈ M ,a ¯ = (a1, . . . , an) ⊆ M with ρn(p n, q) < ∆n (, p) where q = qftpM(¯a), A there exist N ⊇ M in M , and an+1 in N such that ρn+1(p, r) ≤ where r = qftpN(a1, . . . , an+1).
Proposition 5.8. Let A be an age. Conditions (1), (2) are equivalent.
1. A has the strong PAP.
35 A A A 2. For every n ∈ ω, > 0, and p ∈ Sn+1, there exists ∆n (, p) > 0 such that for every M ∈ M , A A a¯ = (a1, . . . , an) ⊆ M such that ρn(pn, q) < ∆n (, p), where q = qftpM(¯a), there exist N ⊇ M in M
and an+1 in N such that
min(∆A(, p) −· τ (¯a), τ (¯a, a ) −· ) = 0. n pn p n+1
is true in N.
Proof. Clear from the definition of the strong PAP.
Proposition 5.9. Let A be a ρ-compact age.
1. If A has the JEP, then A has the strong JEP.
2. If A has the AP, then A has the strong AP.
A 3. If A has the PAP, then A has the strong PAP. Moreover, if (δn | n ∈ ω) witness that A has the PAP, A A then taking ∆n = δn , (n ∈ ω) witnesses that A has the strong PAP.
A Proof. (1): Consider L-structures (Ai)i∈I in M , for some index set I. There is no loss of generality to assume that for all i 6= j ∈ I, Ai ∩ Aj = ∅. To show that A has the strong JEP it is enough to show that S the L( i∈I Ai)-theory A [ Σ ∪ Diag(Ai) i∈I is satisfiable, where ΣA is defined in Notation 3.19. This is immediate by compactness, using the fact that
A has the JEP.
A (2): Consider L-structures A, (Bi)i∈I in M and embeddings fi : A → Bi for some index set I. There is no loss of generality to assume that for all i 6= j ∈ I, Bi ∩ Bj = A, and for all i ∈ I, fi is the inclusion map. S To prove that A has the strong AP, it is enough to show that the L( i∈I Bi)-theory
A [ Σ ∪ Diag(Bi) i∈I is satisfiable. This is immediate by compactness, using the fact that A has the AP.
A A (3): It is enough to prove that for all n ∈ ω, > 0, and p ∈ Sn+1, there exists ∆n (, p) > 0 which satisfies A A A the condition in Definition 5.7. We will show that for all n ∈ ω we may choose ∆n such that ∆n = δn . Fix A A A ¯ n ∈ ω, > 0, p ∈ Sn+1, M ∈ M , anda ¯ ⊆ M such that ρn(pn, q) < δn (, p) where q = qftpM(¯a). Let b be ¯ ¯ a tuple which enumerates the elements of M such that bn =a ¯. Set r = qftpM(b). To show that A has the
36 strong PAP it is enough to show that the following set of L-conditions is satisfiable:
A Σ := {τp(x1, . . . , xn, y) −· = 0} ∪ q((xi)i≤n) ∪ r((xi)i∈α) ∪ Σ .
By compactness, it is enough to show that for every l ∈ ω with l ≥ n the following set of formulas is
satisfiable:
A {τp(x1, . . . , xn, y) −· = 0} ∪ q((xi)i≤n) ∪ r((xi)i≤l) ∪ Σ .
Clearly there exists N ∈ A andc ¯ ⊆ N withc ¯ |= r l. Then clearlyc ¯ n |= q. Since A has the PAP and A 0 0 0 ρn(p n, q) < δn (, p), there exist N ∈ A, with N ⊆ N , and d ∈ N such that ρn+1(p, qftp(¯c n, d)) < . Therefore · τp(¯cn, d) − = 0,
and Σ is satisfiable. The fact that for all n ∈ ω we may choose ∆n = δn is immediate from the proof.
Notation 5.10. Let A be an age with the PAP. Set ΨA to be the set of all closed conditions of the form
sup min(δA(, p) −· τ (¯x), inf τ (¯x, x ) −· ) = 0 n pn p n+1 x¯ xn+1
A A for all n ∈ ω, > 0, and p ∈ Sn+1, where (δn | n ∈ ω) witnesses the fact that A has the PAP. Set
T A = ΣA ∪ ΨA.
A A Remark 5.11. Note that at first glance the L-theory T seems to depend on the the functions δn . Never- A theless, in Proposition 5.15 we will show that if A is a ρ-compact age with the PAP and (δn | n ∈ ω) witness the fact that A has the PAP, the theory T A has QE and satisfies FS(T A) = A. By Proposition 4.19, there
A A exists a unique (up to equivalence of theories) such theory and hence T is independent of (δn | n ∈ ω).
Proposition 5.12. Let A be a ρ-compact age with the PAP. Then T A is a satisfiable L-theory and FS(T A) =
A.
We need the following lemmas.
A Lemma 5.13. Let A be a ρ-compact age with the PAP. If A is an L-structure in M , then there exists an A A L-structure B ⊇ A in M such that for all σ ∈ Ψ , and for all finite tuples a¯, if σ is the L-condition
sup min(δA(, p) −· τ (¯x), inf τ (¯x, x ) −· ) = 0 n pn p n+1 x¯ xn+1
37 A for some n ∈ ω, > 0, p ∈ Sn+1, and if a¯ is an n-tuple, then there exists an+1 ∈ B such that
min(δA(, p) −· τ (¯a), τ (¯a, a ) −· ) = 0 n pn p n+1
is true in B.
Proof. Since A has the PAP and is ρ-compact, by Proposition 5.6 we deduce that A has the AP. Then by
Proposition 5.9 we deduce that A has the strong PAP and the strong AP. Let Q be the set of all ordered
pairs (σ, a¯) where σ ∈ ΨA anda ¯ is a tuple as in the statement of the lemma. Since A has the strong PAP,
for every (σ, a¯) ∈ Q there exist an L-structure M(σ,a¯) such that A ⊆ M(σ,a¯), age(M(σ,a¯)) ⊆ A and b ∈ M(σ,a¯) with
min(δA(, p) −· τ (¯a), τ (¯a, b) −· ) = 0. n pn p
From the strong AP we obtain an L-structure M such that for all (σ, a¯) ∈ Q, A ⊆ M(σ,a¯) ⊆ M, and age(M) ⊆ A.
Lemma 5.14. Let A be a ρ-compact age. Then, there is an L-structure A such that age(A) = A.
Proof. (of lemma) Since A has the JEP and is ρ-compact, by Proposition 5.9 we deduce that A has the strong JEP. Let (Ai)i∈I be an enumeration of a set of representatives of A, up to isomorphism; that is, every
A element of A is isomorphic to an element of (Ai)i∈I . By the strong JEP, there exists B ∈ M such that for each i ∈ I, Ai can be embedded into B. Clearly age(B) = A.
Proof. (of Proposition 5.12) By induction on n ∈ ω we define a chain of L-structures (Mn | n ∈ ω) such that
S A n∈ω Mn |= T . For n = 0 we define M0 = A where A is an L-structure obtained by applying Lemma 5.14. S Given the L-structure Mn we define Mn+1 by applying Lemma 5.13 to the structure Mn. Set M = n∈ω Mn and N = M. Clearly M |= ΨA. Also age(M) = A and so by Proposition 3.17 we have that M |= ΣA. Then, from Proposition 3.20 we deduce that N |= ΣA. We conclude that N |= T A.
Since age(M) = A, we have A ⊆ FS(T A) and since ΣA ⊆ T A we have FS(T A) = A.
Proposition 5.15. Let A be a ρ-compact age with the PAP.
1. T A has QE.
2. T A is a complete L-theory.
Proof. (1): By Proposition 5.12 we have FS(T A) = A. Therefore, by Proposition 4.3 it is enough to prove
the following claim.
38 A A Claim 5.16. Let M |= T , where M is separable; consider n ∈ ω, p ∈ Sn+1 and a¯ ⊆ M with a¯ |= p n.
Then there exists an ℵ1-saturated elementary extension N M and an+1 ∈ N such that (¯a, an+1) |= p.
A A Proof. (of claim) Fix a separable L-structure M |= T , n ∈ ω, p ∈ Sn anda ¯ ⊆ M witha ¯ |= pn. For every > 0 we have that
T |= sup min(δA(, p) −· τ (¯x), inf τ (¯x, x ) −· ) = 0. n pn p n+1 x¯ xn+1
So, for every > 0 there exists b ∈ N such that
min(δA(, p) −· τ (¯a), τ (¯a, b ) −· 2) = 0. n pn p
This implies that τp(¯a, b) ≤ 2. Equivalently we have that that ρn+1(p, q) ≤ 2 where q = qftpN(¯a, b). If
N is an ℵ1-saturated extension of M clearly there exists an+1 ∈ N such that (¯a, an+1) |= p.
A A (2): We show that every ℵ1-saturated model of T realizes every element of S1 . Then by Proposition 4.5 we conclude that T A is a complete L-theory.
A A Let M be an ℵ1-saturated model of T . Let q be any 1-type realized in M, say by a, and r ∈ S1 . Since A a ρ-compact age with the PAP, by Proposition 5.6 we deduce that A has the AP. Therefore there exists
A p ∈ S2 such that p1 = q and the restriction of p in the second variable is equal to r. For any > 0,
A · · T |= sup min(δ1 (, p) − τq(x1), inf τp(x1, x2) − ) = 0. x x1 2
M Applying the condition to x1 = a in M and noting that τq (a) = 0, we see that
M |= inf τp(a, x2) ≤ . x2
Since > 0 was arbitrary,
M |= inf τp(a, x2) = 0. x2
Consequently, since M is ℵ1-saturated, there exists b ∈ M such that (a, b) |= p. Hence b |= r. That is, any
A A ℵ1-saturated model of T realizes every element of S1 .
We state again and prove Theorem 5.4.
Theorem 5.17. Let A be a ρ-compact age with the PAP. There exists a unique L-theory T which is complete, has QE and satisfies age(T ) = A.
39 Proof. The existence of an L-theory T with the prescribed properties is immediate from Propositions 5.12
and 5.15. The uniqueness of T comes from Proposition 4.19.
5.2 Axiomatization of complete QE theories
Definition 5.18. Let T be an L-theory. Set A = FS(T ). We say that T has the uniformly perturbed
T A extension property if for all n ∈ ω, > 0 there exists δn () > 0 such that for all p ∈ Sn+1 we have that
T |= sup min(δT () −· τ (¯x), inf τ (¯x, x ) −· ) = 0. n pn p n+1 x¯ xn+1
Proposition 5.19. Let T be an L-theory. The following statements are equivalent:
1. T has QE.
2. T has the uniformly perturbed extension property.
Proof. (2) ⇒ (1): Similar to the proof of the fact that T A has QE in the proof of Theorem 5.15.
(1) ⇒ (2): We will need the following definition and lemma.
Definition 5.20. Let T be an L-theory and A = FS(T ). We say that T has the perturbed extension
A property if for every n ∈ ω, > 0, and p ∈ Sn+1 there exists δ(n, , p) > 0 such that
T |= sup min(δ(n, , p) −· τ (¯x), inf τ (¯x, x ) −· ) = 0. pn p n+1 x¯ xn+1
Lemma 5.21. Let T be an L-theory that has QE. Then T has the perturbed extension property.
Proof. (of lemma) Set A = FS(T ). Assume that T does not have the perturbed extension property. This
A implies that there exists n ∈ ω, > 0, and p ∈ Sn+1 such that for all δ > 0,
T 6|= sup min(δ −· τ (¯x), inf τ (¯x, x ) −· ) = 0. pn p n+1 x¯ xn+1
Taking δ = 1/m for m ≥ 1 we see that
1 T ∪ {τ (¯x) ≤ | m ≥ 1} ∪ { inf τ (¯x, x ) ≥ } pn p n+1 m xn+1 is satisfiable. By model-theoretic compactness, there exist an L-structure M |= T anda ¯ ⊆ M witha ¯ |= pn
40 such that
inf τp(¯a, xn+1) ≥ . xn+1
The above statement implies that there is no an+1 in any elementary extension of M such that
ρ (p, q) < n 2
A where q = qftpM(¯a, an+1). But T has QE and p ∈ Sn+1, so by Proposition 4.3 there exists N M and an+1 ∈ N such that (¯a, an+1) |= p which is a contradiction.
A We continue the proof of Proposition 5.19. From Lemma 5.21, for every n ∈ ω, > 0, p ∈ Sn+1 there exists δ(n, , p) such that
T |= sup min(δ(n, , p) −· τ (¯x), inf τ (¯x, x ) −· ) = 0. pn p n+1 x¯ xn+1
For every n ≥ 0 we define a function
A δn : (0, 1] × Sn+1 → (0, 1] as follows:
δ (, p) = sup{δ ∈ (0, 1] | T |= sup min(δ −· τ (¯x), inf τ (¯x, x ) −· ) = 0}. n pn p n+1 x¯ xn+1
A Claim 5.22. For every n ∈ ω, > 0, δn(, p) is a continuous function of p ∈ (Sn+1, ρn+1).
Proof. (of claim) Fix n ≥ 1, > 0. To show that δn(, p) is continuous it is enough to show that for every
A 0 0 A 0 p ∈ Sn+1, > 0 there exists δ > 0 such that if q ∈ Sn+1 with ρn+1(p, q) < δ , then
0 |δn(, p) − δn(, q)| < .
A 0 0 0 Fix p ∈ Sn+1, > 0. We claim that it suffices to choose δ < . Indeed we have that
0 0 δn(, p) − δ < δn(, q) < δn(, p) + δ ⇔
0 |δn(, p) − δn(, q)| < δ < .
41 For every n ∈ ω, > 0 set
T A δn () = inf{δn(, p) | p ∈ Sn+1}.
A The above infimum is actually a minimum because δn(, p) is a continuous function of p ∈ (Sn+1, ρn+1) A T and (Sn+1, ρn+1) is compact. From Lemma 5.21, for every n ≥ 1, > 0, p ∈ Sn+1() > 0 we have that T T δn(, p) > 0. This implies that δn () > 0. Clearly δn () satisfies the condition in Definition 5.18.
Theorem 5.23. Let T be a complete L-theory that has QE, and let A = age(T ). Then:
1. A is a ρ-compact age with the UPAP.
2. T is axiomatized by T A.
A T 3. For every n ∈ ω let δn , δn be as in Definition 5.2 and Definition 5.18 respectively. Then for every T A A T n ∈ ω, given δn , we may choose δn such that δn = δn .
A T 4. For every n ∈ ω let δn , δn be as in Definition 5.2 and Definition 5.18 respectively. Then, for every A T T A n ∈ ω, given δn , we may choose δn such that δn = δn .
Proof. (1): Since T is a complete L-theory, Proposition 3.16 implies that A is a ρ-compact age. Next we
show that A has the UPAP.
A T A We show that for every n ∈ ω, > 0, it suffices to choose δn = δn . Fix n ∈ ω, > 0, M ∈ A, p ∈ Sn+1 T a¯ ⊆ M such that ρn(p n, q) < δn () where q = qftpM(¯a). Because M ∈ A we have that there exists an L-structure A |= T and M can be embedded in A. Without losing generality we may assume that M is a
substructure of A. Because T has the uniformly perturbed extension property we have that
T |= sup min(δT () −· τ (¯x), inf τ (¯x, x ) −· ) = 0. n pn p n+1 x¯ xn+1
This implies that
(sup min(δT () −· τ (¯x), inf τ (¯x, x ) −· ))A = 0. n pn p n+1 x¯ xn+1
0 0 So we have that for every > there exists b0 ∈ A such that τp(¯a, b0 ) < . Set r0 = qftpA(¯a, b0 ). Because A (Sn+1, ρn+1) is compact we have that
r = lim r0 0→
A exists in Sn+1. Clearly we have that ρn(p, r) ≤ . Because A has the AP we have that there exists N ∈ A
such that M ⊆ N and an+1 ∈ N such that ρn(p, r) ≤ where r = qftpN(¯a, an+1).
42 (2): Since T , T A both have QE and FS(T ) = FS(T A) = A, by Proposition 4.19 we deduce that they
are equivalent, namely T A axiomatizes T .
(3): Immediate from the proof of (1).
(4): Immediate from (2).
Theorem 5.24. Let A be a ρ-compact age. The following statements are equivalent:
1. A has the PAP.
2. A has the UPAP.
Proof. (1)⇒(2): By Theorem 5.4, let T be the unique complete L-theory that has QE and satisfies age(T ) =
A. Then by Theorem 5.23, A has the UPAP.
(2)⇒(1): Trivial.
Remark 5.25. It is natural to ask whether we can strengthen Theorem 5.4 in the following way:
Given an ρ-compact age with the AP, is there a complete L-theory T such that T has QE and satisfies
age(T ) = A?
The following counterexample due to C. Ward Henson answers this question above in the negative.
Let L be the signature of pure metric spaces with d taking values in [0, 1]. Let A be the class of all finite
L-structures M with the following properties:
M 1 1. If x, y ∈ M are distinct, then d (x, y) is in the interval [ 2 , 1].
2. If x, y, z ∈ M are distinct, then at least one of the distances dM(x, y), dM(y, z), dM(x, z), equals 1.
It is clear that the class of L-structures M that satisfy age(M) ⊆ A is axiomatizable by the closed L- conditions,
sup min(d(x, y), d(x, z), d(y, z), 1 −· d(x, y), 1 −· d(x, z), 1 −· d(y, z)) = 0, x,y,z and 1 sup min(d(x, y), −· d(x, y)) = 0. x,y 2
Therefore A is ρ-compact. Also A has the AP: If M1, M2 are in A with intersection M, then for every x ∈ M1 \ M and y ∈ M2 \ M, set d(x, y) = 1. Now we prove that if T is a complete L-theory that satisfies age(T ) = A, then T does not have QE. Set
A S = {p ∈ S3 | p ∈ [max(d(x1, x3), d(x2, x3)) < 1]}.
43 A A Note that S is open in (S3 , ρ3). Map S into S2 by eliminating the x3 variable. The image is {p}, where p
is the unique quantifier free 2-type that contains the condition d(x1, x2) = 1. Note that {p} is not open in
A A 1 (S2 , ρ2), since (S2 , ρ2) is isometric to {0} ∪ [ 2 , 1] with the absolute value metric, with p mapped to 1. But by Theorem 5.5, if T had QE, then A would have the PAP and therefore the image {p} of the open set S should be open, which is a contradiction. We conclude that T does not have QE.
44 Chapter 6
Fra¨ıss´eTheorems for separable structures
In this chapter we fix a countable bounded continuous signature L without function symbols.
The main result of Chapter 6 is Theorem 6.28, which is shown to be an optimal result by its converse in 6.30. In Sections 6.1, 6.2, and 6.3 we provide background concepts and material that are needed in both Chapters 6 and 7. The concepts and results of Sections 6.4 and 6.5 are rather technical and they are introduced solely in order to prove Theorem 6.28.
6.1 Definitions and basic theorems
Definition 6.1. Let M, N be L-structures,a ¯ ⊆ M and ¯b ⊆ N. Let A be the substructure of M generated bya ¯ and B the substructure of N generated by ¯b.
1. f :a ¯ → N is an embedding if it has an extension that is an embedding from A into N.
2. f :a ¯ → ¯b is an isomorphism if it has an extension that is an isomorphism from A onto B.
Definition 6.2. Let M be an L-structure and A = age(M).
1. M is ω-qf-near-homogeneous if for every > 0, all finite tuplesa ¯ ⊆ ¯b ⊆ M, wherea ¯ is an n-tuple for
some n ≥ 1, and every embedding f :a ¯ → M, there exists an embedding g : ¯b → M such that
M max d (f(ai), g(bi)) < . 1≤i≤n
¯ 2. M is strongly ω-qf-near-homogeneous if for every n ∈ ω and n-tuplesa, ¯ b ⊆ M such that qftpM(¯a) = qftpM(¯b), there exists an automorphism f of M such that
M max d (ai, f(bi)) < . 1≤i≤n
Proposition 6.3. Let M be an L-structure and A = age(M). The following statements are equivalent:
45 1. M is ω-qf-near-homogeneous.
A 2. For every > 0, n ∈ ω, l ≥ 1, p ∈ Sn+l, and a¯ = (a1, . . . , an) ⊆ M such that a¯ |= p n, there exists ¯ ¯ b = (b1, . . . , bn+l) in M such that b |= p and
M max d (ai, bi) < . 1≤i≤n
A ¯ 3. For every > 0, n ∈ ω, p ∈ Sn+1, a¯ = (a1, . . . , an) ⊆ M such that a¯ |= p n, there exists b = ¯ (b1, . . . , bn+1) in M such that b |= p and
M max d (ai, bi) < . 1≤i≤n
A ¯ 4. For every > 0, p ∈ Sω , a¯ = (a1, . . . , an) ⊆ M such that a¯ |= p n, there exists b = (bi)1≤i∈ω ⊆ M such that ¯b |= p and
M max d (ai, bi) < . 1≤i≤n
Proof. (1)⇔(2): Immediate.
(2)⇒(3): Immediate.
A (3)⇒(4): Fix > 0, p ∈ Sω ,a ¯ = (a1, . . . , an) ⊆ M such thata ¯ |= pn. We define inductively a chain of ¯ ¯ finite tuples (bi)i∈ω such that for all i ∈ ω, bi is an (n + i)-tuple which realizes p(n + i) and
M max d (bi,j, bi+1,j) < . 1≤j≤n+i 4i+1
¯ ¯ For i = 0 we set b0 =a ¯. Suppose we have defined a chain (bi | i ≤ κ) for some k ∈ ω, such that the conditions ¯ ¯ above are satisfied. By hypothesis, there exists a tuple b such that b |= p(n + k + 1) and
M max d (bk,j, bk+1,j) < . 1≤i≤n+k 4k+1
For all j ≥ 1 {bn,j | n ≥ j} is a Cauchy sequence. Since M is a complete metric space limn→∞ bn+j exists.
Set limn→∞ bn,j = bj.
Clearly (bi)i∈ω |= p.
46 M Now we prove that max1≤j≤n d (aj, bj) < . By the triangle inequality, for all 1 ≤ i ≤ n, j ≥ 1
M M M d (aj, bi,j) ≤ d (b0,j, b1,j) + ··· + d (bi−1,j, bi,j) ≤ + ··· + 4 4i i X ≤ 4k k=1
Then for all 1 ≤ j ≤ n,
i M M X d (aj, bj) = lim d (aj, bi,j) ≤ lim = < . i→∞ i→∞ 4k 3 k=1
(4)⇒(1): Immediate.
Proposition 6.4. 1. Let M be a separable L-structure and N an ω-qf-near-homogeneous L-structure such
that age(M) ⊆ age(N). If a¯ is a finite tuple in M, and f :a ¯ → N is an embedding, then there exists
an embedding g : M → N such that
N max d (f(ai), g(bi)) < . i
2. If M, N are both separable, ω-qf-near-homogeneous L-structures, age(M) = age(N), a¯ ⊆ M, ¯b ⊆ N,
both finite tuples, and f :a ¯ → ¯b is an isomorphism, then there exists an isomorphism g : M → N such
that
N max d (f(ai), g(bi)) < . i
¯ Proof. (1): Let D be a countable dense subset of M,a ¯ be an enumeration of A and d = (di)i∈λ be an ¯ ¯ A enumeration of D for some λ ≤ ω such thata ¯ is an initial segment of d. Set p = qftpM(d). Then p ∈ Sλ for some λ ≤ ω. Since N is ω-qf-near-homogeneous, by Proposition 6.3 there existsc ¯ ⊆ N such thatc ¯ |= p and
N max d (f(ai), ci) < . i
0 0 0 Define a map f : D → N such that for all i ∈ λ, f (di) = ci. Since D is dense in M, f can be extended to an embedding g : M → N. Clearly
N max d (f(ai), g(bi)) < . i
(2): This can be proved by a back-and-forth version of the above argument.
47 Theorem 6.5. 1. Any strongly ω-qf-near-homogeneous structure is ω-qf-near-homogeneous.
2. Any separable ω-qf-near-homogeneous structure is strongly ω-qf-near-homogeneous.
3. If M, N are separable L-structures that are both ω-qf-near-homogeneous and age(M) = age(N) then
M, N are isomorphic.
Proof. (1): Obvious.
(2): Clear by Proposition 6.4(2) if we take M = N.
(3): Immediate from Proposition 6.4.
6.2 Near amalgamation property
Definition 6.6. Let A be an age. We say that A has the near amalgamation property (near-AP) if for
A every > 0, n, l ∈ ω, M ∈ A, p ∈ Sn+l, anda ¯ = (a1, . . . , an) ⊆ M such thata ¯ |= p n, there exists N in A ¯ ¯ such that M ⊆ N and b = (b1, . . . , bn) ⊆ N such that b |= p and
N max d (ai, bi) < . 1≤i≤n
The near-AP will play an important role only in this chapter. The reason is that, in the rest of the document, we will mostly consider ages of complete theories and in that case the following proposition applies.
Proposition 6.7. 1. If A is a ρ-compact age with the near-AP, then A has the AP.
2. Let T be a complete L-theory and A = age(T ). If A has the near-AP, then A has the AP.
Proof. (1): This is immediate consequence of the fact that A is ρ-compact.
(2): By Proposition 3.16 we deduce that A is ρ-compact. Then (1) implies that A has the AP.
The following proposition shows that the near-AP is equivalent to the “1-point near-AP”.
Proposition 6.8. Let A be an age. The following statements are equivalent:
1. A has the near-AP.
48 A 2. For every > 0, n ∈ ω, M ∈ A, p ∈ Sn+1, and a¯ = (a1, . . . , an) ⊆ M such that a¯ |= p n, there exists ¯ ¯ N in A such that M ⊆ N and b = (b1, . . . , bn+1) ⊆ N such that b |= p and
N max d (ai, bi) < . 1≤i≤n
Proof. (1)⇒(2): obvious.
(2)⇒(1): Fix > 0, n ∈ ω, l ≥ 1, M ∈ A, p ∈ Sn+l, anda ¯ ⊆ M witha ¯ |= pn. We will define inductively ¯ a finite chain of finite structures (Nj | 0 ≤ j ≤ l) and finite tuples (bj)0≤j≤l where for all 0 ≤ j ≤ l
¯ bj = (bj,1, . . . , bj,n+j) such that the following properties are true:
¯ • N0 = M and b0 =a ¯.
0 • For all 0 ≤ j ≤ j ≤ l we have that Nj ≤ Nj0 .
¯ • For all 0 ≤ j ≤ l we have that bj ⊆ Nj and
Nj+1 max d (bj+1,i, bj,i) < . 1≤i≤n+j l
¯ For n = 0 we define N0 = M and b0 =a ¯. Suppose that for some k < l we have defined (Nj | 0 ≤ j ≤ k) ¯ 0 and (bj | 0 ≤ j ≤ k) which satisfy the properties stated above for all 0 ≤ j ≤ j ≤ k. Because A has the ¯ ¯ near-AP there exist N ⊇ Nk and b = (b1, . . . , bn+k+1) ⊆ N such that b |= pk + 1 and
N max d (bi, bj,i) < . 1≤i≤n+k l
¯ ¯ Set Nk+1 = N and bk+1 = b. Then clearly for all 1 ≤ i ≤ n
N(l) N(l) N(l) d (ai, bl,i) ≤ d (b0,i, b1,i) + ··· + d (bl−1,i, bl,i).
This implies that
N(l) N(l) N(l) max d (ai, bl,i) ≤ max d (b0,i, b1,i) + ··· + max d (bl−1,i, bl,i) 1≤i≤n 1≤i≤n 1≤i≤n < l = . l
49 This suffices.
6.3 The dA-metric
A Definition 6.9. Let A be an age. For every n ∈ ω, and p, q ∈ Sn we define
A M ¯ ¯ dn (p, q) = inf{ max d (ai, bi) | M ∈ A, a,¯ b ⊆ M, M |= p(¯a), M |= q(b)}. 1≤i≤n
A A Remark 6.10. We note that if A is an age, then for all n ∈ ω, p, q ∈ Sn , we have that dn (p, q) < ∞. This is an immediate consequence of the JEP.
A A Proposition 6.11. Let A be an age with the near-AP. Then for all n ∈ ω, (Sn , dn ) is a metric space.
A A A A Proof. Fix n ∈ ω. Clearly for every p, q ∈ Sn , dn(p, p) = 0 and dn (p, q) = dn (q, p). Clearly also d (p, q) = 0 implies that p = q.
A A Now we show that the triangle inequality holds. Suppose that p, q, r are in Sn . Set dn (p, q) = a, A ¯ ¯ dn (q, r) = b. Fix > 0. There exist M ∈ A anda ¯, b ⊆ M such thata ¯ |= p, b |= q and
M max d (ai, bi) < a + . 1≤i≤n
Similarly there exist N ∈ A and d¯,e ¯ ⊆ N such that d¯|= q,e ¯ |= r and
N max d (di, ei) < b + . 1≤i≤n
¯ ¯ Set s = qftpN(d, e¯). Note that M |= (s n)(b). Because A has the near-AP, for the given > 0 there exist ¯ ¯ M1 in A with M1 ⊇ M and f = (f1, . . . , fn),g ¯ = (g1, . . . , gn) ⊆ M1 such that (f, g¯) |= s and
M1 max d (bi, fi) < . 1≤i≤n
Note that f¯ |= q,g ¯ |= r. Then, from the triangle inequality we have
M1 M1 M1 M1 ∀i ≤ n d (ai, gi) ≤ d (ai, bi) + d (bi, fi) + d (fi, gi).
Combining the above inequalities we get
A A A dn (p, r) < (dn (p, q) + ) + + (dn (q, r) + ).
50 The above is true for all > 0. We deduce that
A A A dn (p, r) ≤ dn (p, q) + dn (q, r).
A A We conclude that (Sn , dn ) is a metric space.
Definition 6.12. Let A be an age with the near-AP.
A A A 1. We say that A is d -separable if for every n ∈ ω, the metric space (Sn , dn ) is separable.
A A A 2. We say that A is d -complete if for every n ∈ ω, the metric space (Sn , dn ) is complete.
Proposition 6.13. Let M be an L-structure which is ω-qf-near-homogeneous. Set A = age(M). Then A is a dA-complete age with the near-AP.
Proof. By Proposition 3.5 we have that A is an age.
A Now we show that A has the near-AP. Fix > 0 n, l ∈ ω, N ∈ A, p ∈ Sn+l, anda ¯ ⊆ N witha ¯ |= p n. ¯ ¯ Because M is ω-qf-near-homogeneous we have that there exists b = (b1, . . . , bn+l) such that b |= p and
M max d (ai, bi) < . 1≤i≤n
Let N0 ⊇ N be a finite L-substructure of M such that ¯b ⊆ N 0.
A A Since A has the near-AP, we deduce that for all n ∈ ω,(Sn , dn ) is a metric space. Now we show that A is dA-complete. We will need the following lemma.
Lemma 6.14. Let M be an L-structure which is ω-qf-near-homogeneous. Put A = age(M). If > 0, p, q
A A ¯ are in Sn , for some n ∈ ω, with dn (p, q) < , and a¯ ⊆ M is such that a¯ |= p, then there exists b ⊆ M such that ¯b |= q and
M max d (ai, bi) < . 1≤i≤n
A A Proof. Fix >, n ∈ ω, p, q ∈ Sn ,a ¯ ⊆ M witha ¯ |= p. Because dn (p, q) < we have that there exists N ∈ A andc, ¯ d¯⊆ N such thatc ¯ |= p, d¯|= q and
N max d (ci, di) < . 1≤i≤n
Set
N 1 = max d (ci, di) < . 1≤i≤n
51 ¯ Because A = age(M) we may assume that N ⊆ M. Set r = qftpN(¯c, d). Choose 2 such that
0 < 2 < − 1.
Becausea ¯ |= r n, Proposition 6.3 implies that there existe ¯ = (e1, . . . , e2n) such thate ¯ |= r and
N max d (ai, ei) < 2. 1≤i≤n
¯ ¯ Set (b1, . . . , bn) = (en+1, . . . , e2n) and b = (b1, . . . , bn). Clearly we have that b |= q. For all 1 ≤ i ≤ n we have that
M M M d (ai, bi) ≤ d (ai, ei) + d (ei, bi).
Therefore
M M M d (ai, bi) ≤ d (ai, ei) + d (ci, di).
We conclude that
M M M max d (ai, bi) ≤ max d (ai, ei) + max d (ci, di) < 2 + 1 < . 1≤i≤n 1≤i≤n 1≤i≤n
A A Now we continue with the proof of Proposition 6.13 and we show that for all n ∈ ω,(Sn , dn ) is complete. A A Fix n ∈ ω. Let (pj | j ∈ ω) be a Cauchy sequence in (Sn , dn ). To prove completeness, we need only to consider sequences that satisfy 1 ∀j0 ≥ j dA(p , p0 ) < . l j j 2j
We define inductively a sequence (¯aj | j ∈ ω) in M such that the following properties are true.
1. For all j ∈ ω,a ¯j |= pj.
2. For all j ≤ j0,
N 1 max d (aj,i, aj0,i) < . 1≤i≤l 2j
For j = 1, picka ¯1 ⊆ M such thata ¯1 |= p1. Assume we have defineda ¯j for all j ≤ k such that the above
0 two conditions are true for all values of j, j ≤ k. We have thata ¯(k) |= pk and that
1 d(p , p ) < . k k+1 2k
52 So from Lemma 6.14 there existsa ¯k+1 ⊆ M such that
N 1 max d (an+1,i, an,i) < . 1≤i≤n 2k
This suffices for the the definition of the sequence (¯aj | j ∈ ω).
Then for every 0 ≤ i ≤ n consider the sequence Si = (aj,i | j ∈ ω). The second condition implies that Si
is a Cauchy sequence in M. Because M is based on a complete metric space we have that limj→∞ Si exists in M. For all 0 ≤ j ≤ n, set
bi = lim Si, j→∞
¯ ¯ A b = (b1, . . . , bn) and p = qftpM(b). Clearly we have that p ∈ Sn and that
A lim dn (p, pj) = 0. j→∞
A A So (Sn , dn ) is complete.
Definition 6.15. Let (X, d1), (X, d2) be metric spaces.
1. We say that d1 is finer than d2 if for every x ∈ X, > 0, there exists δ > 0 such that for all y ∈ X,
d1(x, y) < δ implies d2(x, y) < .
2. We say that d1 is uniformly finer than d2 if for every > 0, there exists δ > 0 such that for every x, y ∈ X
d1(x, y) < δ implies d2(x, y) < .
A A Proposition 6.16. Let A be an age with the near-AP. Then for all n ∈ ω, (Sn , dn ) is uniformly finer than A (Sn , ρn).
A Proof. It is enough to prove that for every n ∈ ω and > 0 there exists δ > 0 such that for every p, q ∈ Sn
A dn (p, q) < δ implies ρn(p, q) < .
Fix n ∈ ω and > 0. Let mP be the number of predicate symbols and mC be the number of constant symbols in L.
For each 1 ≤ i ≤ mP , let ∆Pi be the modulus of uniform continuity that L specifies for the predicate
53 symbol Pi and let a(i) be the arity of the predicate Pi. We will show that it suffices to choose δ > 0 such that δ < min{ , min {∆Pi ()}}. 2 1≤i≤mP
A So assume that p, q ∈ Sn and that A dn (p, q) < δ.
¯ ¯ This implies that there exist M ∈ A anda ¯ = (a1, . . . , an) ⊆ M, b = (b1, . . . , bn) ⊆ M such thata ¯ |= p, b |= q and
M max d (ai, bi) < δ. 1≤i≤n
Lete ¯ = (e1, . . . , emC +n) = (c1, . . . , cmC , x1, . . . , xn). Then there are finitely many atomic L-formulas in x¯ = (x1, . . . , xn) and they have the forms:
• d(ci, cj) where 1 ≤ i, j ≤ mC
• d(ci, xj) and d(xj, ci) where 1 ≤ i ≤ mC , 1 ≤ j ≤ n.
• d(xi, xj) where 1 ≤ i, j ≤ n.
• Pl(ej1 , . . . , eja(l) ) where 1 ≤ l ≤ mP and 1 ≤ j1, . . . , ja(l) ≤ mC + n.
To complete the proof we will need to prove some easy facts.
Claim 6.17. 1. For every i, j with 1 ≤ i ≤ mC , 1 ≤ j ≤ n we have that
p q |d(ci, xj) − d(ci, xj) | <
2. For every i, j with 1 ≤ i, j ≤ n we have that
p q |d(xi, xj) − d(xi, xj) | < .
¯ Proof. (of claim) Fix i, j with 1 ≤ i ≤ mC , 1 ≤ j ≤ n respectively. As above there area, ¯ b ⊆ M, such that a¯ |= p, ¯b |= q and
M max d (ai, bi) < δ ≤ . 1≤i≤n 2
(1): By the triangle inequality, for all 1 ≤ i ≤ mC , 1 ≤ j ≤ n
M M M M M d (ci , aj) ≤ d (ci , bj) + d (bj, aj) ⇔
54 dM(cM, a ) − dM(cM, b ) < . i j i j 2
Similarly, for all 1 ≤ i ≤ n, 1 ≤ j ≤ m,
dM(cM, b ) − dM(cM, a ) < . i j i j 2
Combining the above inequalities we deduce that for all 1 ≤ i ≤ mC , 1 ≤ j ≤ n,
|dM(cM, a ) − dM(cM, b )| < . i j i j 2
We deduce that |d(c , x )p − d(c , x )q| < < . i j i j 2
(2): By the triangle inequality, for all i, j ≤ n
M M M M d (ai, aj) ≤ d (ai, bi) + d (bi, bj) + d (bj, aj) ⇔
dM(a , a ) ≤ + dM(b , b ) + ⇔ i j 2 i j 2
M M d (ai, aj) − d (bi, bj) < , and similarly
M M d (bi, bj) − d (ai, aj) < .
By combining the above two inequalities we deduce that for all 1 ≤ i, j ≤ n,
M M |d (ai, aj) − d (bi, bj)| < .
We conclude that
p q |d(xi, xj) − d(xi, xj) )| < .
Claim 6.18. For every > 0, 1 ≤ l ≤ mP , 1 ≤ j1, . . . , ja(l) ≤ mC + n,
p q |Pl(ej1 , . . . , eja(l) ) − Pl(ej1 , . . . , eja(l) ) | < .
55 ¯ ¯ Proof. (of claim) Fix > 0, 1 ≤ l ≤ mP , 1 ≤ j1, . . . , ja(l) ≤ mC + n. We have thata, ¯ b ⊆ M,a ¯ |= p, b |= q and that
M max d (ai, bi) < δ ≤ ∆P (). 1≤i≤n l
Set ¯ f = (f1, . . . , fmC +n) = (c1, . . . , cmC , a1, . . . , an),
g¯ = (g1, . . . , gmC +n) = (c1, . . . , cmC , b1, . . . , bn).
Clearly
M max d (fi, gi) < ∆Pl (). 1≤i≤mC +n
This implies that
M M |Pl (fj1 , . . . , fja(l) ) − Pl (gj1 , . . . , gja(l) )| < , which in turns implies that
p q |Pl(ej1 , . . . , eja(l) ) − Pl(ej1 , . . . , eja(l) ) | < .
It is easy to see that Claim 6.17 and Claim 6.18 imply that ρn(p, q) < .
6.4 Completion of ω-qf-near-homogeneous metric prestructures
Proposition 6.19. Let M be a metric L-prestructure. Set A = age(M). Suppose that the following condi- tions are true:
• M is ω-qf-near-homogeneous.
A A • For every l ≥ 1, if (pn | n ≥ 1), (qn | n ≥ 1) are Cauchy sequences in (Sl , dl ), then
A lim ρl(pn, qn) = 0 implies lim dl (pn, qn) = 0. n→∞ n→∞
If we set M to be the completion of M, then M is ω-qf-near-homogeneous.
¯ Proof. Fix > 0 and l ∈ ω. Leta ¯ = (a1, . . . , al), b = (b1, . . . , bl+1) in M be such thata ¯ |= q l where ¯ ¯ q = qftpM(b). For all n ∈ ω leta ¯n = (an,1, . . . , an,l), bn = (bn,1, . . . , bn,l+1) be in M such that for all
56 1 ≤ i ≤ l, limn→∞ an,i = ai, and for all 1 ≤ i ≤ l + 1, limn→n bn,i = bi. Set
pn = qftpM(an,1, . . . , an,l), qn = qftpM(bn,1, . . . , bn,l+1).
For all 1 ≤ i ≤ l,(an,i | n ≥ 1), (bn,i | n ≥ 1) are Cauchy sequences in M. So (pn | n ≥ 1) is a
A A A A Cauchy sequence in (Sl , dl ) and (qn | n ≥ 1) is a Cauchy sequence in (Sl+1, dl+1). Clearly we have that A A (qn l | n ≥ 1) is a Cauchy sequence in (Sl , dl ) and that
lim ρl(pn, qn l) = 0. n→∞
By the hypothesis we get
A lim dl (pn, qn l) = 0. n→∞
We define inductively a sequence of (l + 1)-tuples (¯en | n ≥ 1) in M, where for all n ≥ 1e ¯n =
(en,1, . . . , en,l+1), such that if we set sn = qftpM(¯en) then the following conditions are true:
1. (sn | n ≥ 1) is a subsequence of (qn | n ≥ 1); there exists an increasing sequence of natural numbers
(n(k) | k ≥ 1) such that for all k ≥ 1, sk = qn(k).
2. for every i, j ≥ n(k), k ≥ 1, dA (q , q ) < . l+1 i j 2k+2
3.
M max d (ai, e1,i) < + . 1≤i≤l 2 22
4. for all n ≥ 1,
M max d (en,i, en+1,i) < . 1≤i≤l 2n+2
0 0 Definition ofe ¯1: For all i = 1, . . . , l, limn→∞ an,i = ai, so there exists n (1) ≥ 1 such that for all n ≥ n (1)
M max d (ai, an,i) < . (1) 1≤i≤l 2
Also,
A lim dn (pn, qn l) = 0, n→∞
57 so there exists n00(1) ≥ 1 such that for all n ≥ n00(1)
dA(p , q l ) < . l n n 22
A A 000 Further, since (qn | n ≥ 1) is a Cauchy sequence in (Sl+1, dl+1), there exists n (1) ≥ 1 so that for all i, j ≥ n000(1) dA (q , q ) < . l+1 i j 23
0 00 000 Set n(1) = max{n (1), n (1), n (1)}. We have thata ¯n(1) |= pn(1). Because
dA(p , q l) < l n(1) n(1) 22
by Lemma 6.14 there existse ¯1 = (e1,1, . . . , e1,l+1) in M such thate ¯1 |= qn(1) and
M max d (an(1),i, e1,i) < . (2) 1≤i≤l 22
From (1), (2) and the triangle inequality we deduce that
M max d (ai, e1,i) < + . 1≤i≤l 2 22
Inductive step: Assume that we have defined (l+1)-tuplese ¯n in M for all 1 ≤ n ≤ k such that the conditions
A A (1)-(4) are satisfied. We will define tuplee ¯k+1. Since (qn | n ≥ 1) is a Cauchy sequence in (Sl+1, dl+1), there exists n(k + 1) ≥ n(k) such that for all i, j ≥ n(k + 1)
dA (q , q ) < . l+1 i j 2k+3
Then by the induction hypothesis dA (q , q ) < . l+1 n(k+1) n(k) 2k+2
Becausee ¯k |= qn(k), by Lemma 6.14 there exists a tuplee ¯k+1 in M such thate ¯k+1 |= qn(k+1) and
M max d (ek+1,i, ek,i) < . 1≤i≤l+1 2k+2
This suffices for the construction of the sequence (¯en | n ≥ 1).
58 For all 1 ≤ i ≤ l + 1, (en,i | n ≥ 1) is a Cauchy sequence in M, and since M is complete, (en,i | n ≥ 1)
converges in M. For all 1 ≤ i ≤ l, let ei = limn→∞ en,i ande ¯ = (e1, . . . , el+1). To complete the proof that M is ω-qf-near-homogeneous, it is enough to prove the following claim:
Claim 6.20. 1. qftpM(¯e) = q.
M 2. max1≤i≤l d (ai, ei) ≤ .
We prove the claim.
(1): Set r = qftpM(¯e). Then we have that
lim ρl+1(q, qn) = lim ρl+1(r, qn) = 0. n→∞ n→∞
A Therefore ρl+1(q, r) = 0, and because (Sl+1, ρl+1) is a metric space, we conclude that q = r. (2): From conditions (3), (4) and the triangle inequality we have that for all 1 ≤ i ≤ l, n ≥ 1
n−1 n+1 X X dM(a , e ) ≤ dM(a , e ) + dM(e , e ) < . i n,i i 1,i j,i j+1,i 2j j=1 j=1
Therefore, for all 1 ≤ i ≤ l, ∞ M M X d (ai, ei) = lim d (ai, en,i) ≤ = . n→∞ 2j j=1
We conclude that
M max d (ai, ei) ≤ . 1≤i≤l
6.5 Fra¨ıss´etheory for separable structures
In this section our main goal is to prove Theorem 6.28, which is the main theorem of this chapter. First we
prove some propositions.
Proposition 6.21. Let A be an age with the UPAP. If B is the completion of A (as defined in Definition
3.21), then B is a ρ-compact age with the UPAP.
Proof. The fact that B is a ρ-compact age follows from Proposition 3.23.
B A A B We will show that for all n ∈ ω we may choose δn = δn where δn , δn are defined in Definition 5.2. B A Fix n ∈ ω, > 0, p ∈ Sn+1, M ∈ B, anda ¯ ⊆ M such that ρn(p n, q) < δn () where q = qftpM(¯a). Set
59 ¯ ¯ ¯ ρn(p n, q) = δ. Let b be an enumeration of M \ a¯ and l the length of b. Setc ¯ = ba¯ and s = qftpM(¯c). Let m0 ≥ 1 be such that
2 A δ + < δn (). m0
B B A Since B is the completion of A and s ∈ Sl+n, p ∈ Sn+1, for every m ≥ m0 there exist sm ∈ Sl+n and A pm ∈ Sn+1 such that 1 1 ρ (s, s ) < , ρ (p, p ) < . l+n m m n+1 m m
Fix m ≥ m0. Let cm be a realization of sm in an L-structure A and let Mm be a the L-structure generated A by cm in A. Then Mm ∈ . Set qm = qftpMm (cl+1, . . . , cl+n). We have that
ρn(pm n, qm) ≤ ρn(pm n, pn) + ρn(pn, q) + ρn(q, qm) 1 1 < + δ + m m A < δn ().
Since A has the UPAP, there exist Nm ∈ A such that Mm ⊆ Nm and dm ∈ Nm such that
ρn+1(pm, rm) ≤ ,
where rm = qftpMm (cm,l+1, . . . , cm,l+n, dm). Without losing generality we may assume that Nm =c ¯m ∪ A B {dm}. Set tm = qftpMm (¯cm, dm). Then the sequence {tm | m ≥ m0} is in Sl+n+1 ⊆ Sl+n+1 and since B (Sl+n+1, ρl+n+1) is compact we deduce that it has a limit point t. Lete ¯ |= t and N be the L-structure
generated bye ¯. Set r = qftpN(el+1, . . . , el+n+1). We note that
ρn+1(p, rm) ≤ ρn+1(p, pm) + ρn+1(pm, rm) 1 < + m
We deduce that
ρn+1(p, r) = lim ρn+1(p, rm) ≤ . m→∞
¯ Note that M = b ∪ a¯. We define an embedding f : M → N such that for all 1 ≤ i ≤ l, f(bi) = ei; for
all 1 ≤ i ≤ n, f(ai) = el+i. Then without losing generality we may assume that M ⊆ N, and as we noted
above ρn+1(p, r) ≤ where r = qftpN(el+1, . . . , el+n).
60 Proposition 6.22. Let A be an age. The following conditions are equivalent.
1. A has the UPAP.
A 2. For every n ∈ ω, > 0 there exists δn () > 0 such that for every M ∈ A whenever we are given
A • p(x) ∈ Sn+1
• a¯ = (a1, . . . , an) ⊆ M.
then there exist N in A such that M ⊆ N and an+1 in N such that
min(δA() −· τ (¯a), τ (¯a, a ) −· ) = 0. n pn p n+1
is true in N.
Proof. Immediate from the Definition 5.2.
Proposition 6.23. Let A be a dA-separable age with the near-AP. There exists a countable metric L-
prestructure N such that if we set B = age(N), then the following properties are true:
1. B ⊆ A.
2. N is ω-qf-near-homogeneous.
B B A 3. For every n ∈ ω, for every p, q ∈ Sn , dn (p, q) = dn (p, q).
B A A 4. For every n ∈ ω, Sn is dense (Sn , dn ).
5. If, in addition, L is a signature whose nonlogical symbols are a finite number of constant and predicate
symbols, A has the UPAP, and we set D to be the completion of A, then N can be chosen such that
N |= T D.
A A Proof. For every n ∈ ω,(Sn , dn ) is separable. So there exists a countable C ⊆ A such that for every n ∈ ω, C A A Sn is dense in (Sn , dn ). We note that C is not necessarily an age. Let (Mn)n∈ω be an enumeration of a set of representatives of C, up to isomorphism; that is, every element of C is isomorphic to an element of
(Mn)n∈ω. By induction we will define a chain of finite L-structures (Nn | n ∈ ω) such that the following conditions are true:
1. For every n ∈ ω, Mn can be embedded in Nn+1.
61 age(Nn) ¯ ¯ 2. For every n ∈ ω, if p, q ∈ Sl for some l ∈ ω, then there area, ¯ b ⊆ Nn+1 such thata ¯ |= p, b |= q and 1 Nn+1 A max d (ai, bi) < dl (p, q) + . 1≤i≤l n
age(Nn) ¯ 3. For every n ∈ ω, if p ∈ Sl+1 for some l ∈ ω, anda ¯ ⊆ Nn witha ¯ |= pl, then there exists b ⊆ Nn+1 such that ¯b |= p and 1 Nn+1 max d (ai, bi) < . 1≤i≤l n
Now we define the chain of finite L-structures (Nn | n ∈ ω). To do so we need some notation and some lemmas.
Notation 6.24. Let A,(Mn)n∈ω as above. Let M ∈ A and n ∈ ω. Set In(M) to be a structure in A into which M and Mn can both be embedded. Such a structure exists since A has the JEP.
Lemma 6.25. Let A with the near-AP and M ∈ A. For every n ∈ ω there is a structure En(M) in A such age(M) ¯ that M ⊆ En(M) and such that if p, q ∈ Sl for some l ∈ ω, then there exist a,¯ b ⊆ En(M) such that a¯ |= p, ¯b |= q and 1 En(M) A max d (ai, bi) < dl (p, q) + . 1≤i≤l n
Proof. (of lemma) Fix n ∈ ω. Let (pj, qj)j≤k be an enumeration of the set
age(M) {(p, q) | ∃l ∈ ω with p, q ∈ Sl }, for some k ∈ ω. Note that the above set is finite because M is finite. By induction we define a chain of structures (Aj | j ≤ k + 1) in A by
A0 = M
Aj+1 = B ∈ A
¯ ¯ where B ⊇ Aj is such that there existsa ¯, b in B witha ¯ |= pj, b |= qj, and (assuming that pj, qj are l-types)
B A 1 max d (ai, bi) < dl (p, q) + . 1≤i≤l n
A Note that Aj+1 always exists from the definition of the metric dl and the fact that A satisfies the JEP. Set
[ En(M) = Aj = Ak+1. j≤k+1
62 Lemma 6.26. Let A be an age with the near-AP and M ∈ A. For every n ∈ ω there is a structure Hn(M) age(M) in A such that M ⊆ Hn(M), and such that if p ∈ Sl+1 for some l ∈ ω and a¯ ⊆ M with a¯ |= p l, then ¯ ¯ there exists b ⊆ Hn(M) such that b |= p and
1 HN (M) max d (ai, bi) < . i≤i≤l n
Proof. (of lemma) Fix n ∈ ω. Let ((pj, a¯j))j≤k be an enumeration of the set
age(M) {(p, a¯) | ∃l ∈ ω with p ∈ Sl+1 and a¯ |= pl},
for some k ∈ ω. We writea ¯j for (aj,1, . . . , aj,l). Note that the above set is finite because M is finite. By induction we define a finite chain of finite structures (Aj | j ≤ k + 1) in A by
A0 = M
Aj+1 = B ∈ A
¯ ¯ where B ⊇ Aj is such that there exists b ⊆ B with b |= pj, and (assuming pj is an (l + 1)-type)
B 1 max d (aj,i, bi) < . 1≤i≤l n
Note that Aj+1 always exists because of the near-AP of A. Set
[ Hn(M) = Aj = Ak+1. j≤k+1
Now we define by induction on n the chain (Nn)n∈ω announced at the beginning of the proof of Proposition 6.23. Let M ∈ A. Set
N0 = M
Nn+1 = HnEnIn(Nn).
63 Then (Nn)n∈ω clearly satisfy the three conditions stated above. Put
[ N = Nn n∈ω
and B = age(N). Clearly N is countable.
Now we prove that N satisfies the conditions in Proposition 6.23.
(1): Clearly, by construction B ⊆ A.
B (2): Suppose that p ∈ Sl+1 for some l ∈ ω anda ¯ ⊆ N witha ¯ |= pl. Then there exists j1 ∈ ω such that 1 ¯ a¯ ⊆ N(j1). Pick j2 ∈ ω such that < . Set j = max{j1, j2}. Thena ¯ ⊆ N(j) and there exists b ∈ Nj+1 j2 ¯ N such that b |= p and max1≤i≤l d (ai, bi) < . This proves that N is ω-qf-near-homogeneous. (3): We have that (2) implies that B is an age with the near-AP. Because B ⊆ A we have that for every
B l ∈ ω, for every p, q ∈ Sl B A dl (p, q) ≥ dl (p, q).
By condition (3) we get
B A dl (p, q) = dl (p, q).
(4): This is evident from the first condition at the beginning of the proof of Proposition 6.23 that the
chain (Nn | n ∈ ω) satisfies. (5): Let D be the completion of A. From Proposition 6.21 we have that D is a ρ-compact age with the
UPAP. To show that N |= T D it is enough to show that N |= ΣD and N |= ΨD. Since age(N) ⊆ B ⊆ D
and D is a ρ-compact age, by Proposition 3.17 we deduce that N |= ΣD. Therefore it suffices to modify the
construction above to ensure that in addition N |= ΨD.
Let Σ be the set of closed L-conditions
sup min(δD(1/l) −· τ (¯x), inf τ (¯x, x ) −· (1/l)) = 0 m pm p m+1 x¯ xm+1
D D for l, m ∈ ω, and p ∈ Sm, where δm is defined in Definition 5.2. Then Σ is countable and is clearly equivalent D to the theory Ψ . Let (σ)i∈ω be an enumeration of the set Σ and set Σn = {σi | i ≤ n}.
To ensure that N |= Σ we will define, as above, a chain of finite L-structures (Nn | n ∈ ω) which satisfies the three conditions stated at the beginning of the proof of Proposition 6.23 and, in addition, also satisfies
the following condition:
64 • For every n ∈ ω, σ ∈ Σn, anda ¯ ⊆ Nn, if σ is the condition
sup min(δD(1/l) −· τ (¯x), inf τ (¯x, x ) −· (1/l)) = 0 m pm p m+1 x¯ xm+1
A for some l, m ∈ ω, p ∈ Sm, anda ¯ is an m-tuple, then there exists am+1 ∈ Nn+1 such that
D · · min(δm(1/l) − τpm(¯a), inf τp(¯a, am+1) − (1/l)) = 0 xm+1
is true in N.
We will need the following lemma.
Lemma 6.27. Let A be an age which satisfies the near-AP and UPAP. Let Σ be as defined above, and take
M ∈ A. For every n ∈ ω, there is an L-structure Gn(M) ∈ A such that for every σ ∈ Σn, and a¯ ⊆ M, if σ is the sentence
sup min(δA (1/l) −· τ (¯x), inf τ (¯x, x ) −· (1/l)) = 0 m pm p m+1 x¯ xm+1
A for some l, m ∈ ω, p ∈ Sm, and a¯ is an m-tuple, then there exists am+1 ∈ Gn(M) such that
A · · min(δm(1/l) − τpm(¯a), inf τp(¯a, am+1) − (1/l)) = 0 xm+1
is true in Gn(M).
Proof. The lemma is implied by Proposition 6.22 and the near-AP. The details of this proof are similar to the details of the proof of Proposition 5.13.
Now we prove (5). We define by induction the chain (Nn)n∈ω. Let M ∈ A and set
N0 = M
Nn+1 = GnHnEnIn(Nn).
S Then (Nn)n∈ω clearly satisfy conditions (1)-(4) stated the beginning of the proof 6.23. Set N = n∈ω Nn. Then N is a metric L-prestructure and satisfies conditions (1)-(4) of Proposition 6.23.
Here is the main theorem of this chapter.
Theorem 6.28. Let A be a dA-complete, dA-separable age with the near-AP. Then there exists a unique separable L-structure M which satisfies the following properties:
65 1. M is strongly ω-qf-near-homogeneous.
2. age(M) = A.
3. If, in addition, L is a signature whose only nonlogical symbols are a finite number of constant and
predicate symbols, and A has the UPAP, then Th(M) has QE.
Proof. By Proposition 6.23 we have that there exists a metric L-prestructure N such that if we set B =
age(N) then the following properties are true:
• N is countable.
• N is ω-qf-near-homogeneous.
• B ⊆ A.
B B A • For every n ∈ ω, p, q ∈ Sn , dn (p, q) = dn (p, q).
Set M = N. The following obvious fact will be useful.
A Fact 6.29. If A, B are ages with the near-AP and B ⊆ A, then for every n ∈ ω and p, q ∈ Sn we have that B A dn (p, q) ≥ dn (p, q).
(1): We will use Proposition 6.19. By definition N is ω-qf-near-homogeneous. Now we show that N
satisfies the second condition in Proposition 6.19.
0 B B Fix l ∈ ω and let S = (pn | n ≥ 1), S = (qn | n ≥ 1) be Cauchy sequences in (Sl , dl ) such that 0 A A limn→∞ ρl(pn, qn) = 0. By Fact 6.29 we deduce that S and S are Cauchy sequences in (Sl , dl ). Since A A A (Sl , dl ) is complete and limn→∞ ρl(pn, qn) = 0 we deduce that there exists r ∈ Sl such that
A A lim dl (pn, r) = lim dl (qn, r) = 0. n→∞ n→∞
A B A B So we have that limn→∞ dl (pn, qn) = 0. Since for all p, q ∈ Sl dl (p, q) = dl (p, q), we deduce that B limn→∞ dl (pn, qn) = 0. Therefore N satisfies the second condition in Proposition 6.19. We conclude that M is ω-qf-near-homogeneous.
A C (2): Set C = age(M). To prove that A = C, it is enough to prove that for all l ∈ ω, Sl = Sl . C A C Fix l ∈ ω. First we prove that Sl ⊆ Sl . Let p ∈ Sl . Then there existsa ¯ = (a1, . . . , al) ⊆ M such that
a¯ |= p. Because N is dense in M there exists a sequence of l-tuples (¯aj | j ∈ ω) such that for all j ∈ ω,
a¯j = (aj,1, . . . , aj,l) ⊆ N and for all 1 ≤ i ≤ l
M lim d (aj,i, ai) = 0. j→∞
66 B B Set pj = qftpM(¯aj). Clearly (pj | j ∈ ω) is a Cauchy sequence in (Sl , dl ) and
C lim dl (p, pj) = 0. (6.1) j→∞
By Proposition 6.16 and (6.1) we deduce that
lim ρl(p, pj) = 0. (6.2) j→∞
B B Since (pj | j ∈ ω) is a Cauchy sequence in (Sl , dl ), by Fact 6.29 we deduce that it is also Cauchy sequence A A A A A in (Sl , dl ). Because (Sl , dl ) is a complete metric space, there exists r ∈ Sl such that
A lim dj (r, pj) = 0. (6.3) j→∞
By Proposition 6.16 and (6.3) we deduce that
lim ρl(r, pj) = 0. (6.4) j→∞
A By (6.2) and (6.4) we deduce that p = r. We conclude that p is in Sn . A C A B A A Now we prove that Sl ⊆ Sl . Let p ∈ Sl . We have that Sl is dense in (Sl , dl ). So there exists a B sequence (pj | j ∈ ω) in Sl such that A lim dl (p, pj) = 0. j→∞
By Proposition 6.16 we have that
lim ρl(p, pj) = 0. (6.5) j→∞
A A B Clearly (pj | j ∈ ω) is a Cauchy sequence in (Sl , dl ). Since for all p, q ∈ Sl
B A dl (p, q) = dn (p, q),
B B we deduce that (pj | j ∈ ω) is a Cauchy sequence in (Sl , dl ). Since B ⊆ C, by Fact 6.29 we deduce that C C C C (pj | j ∈ ω) is a Cauchy sequence in (Sl , dl ). By Proposition 6.13, (Sl , dl ) is a complete metric space. So C there exists r ∈ Sl such that C lim dl (r, pj) = 0. j→∞
67 By Proposition 6.16
lim ρl(r, pj) = 0 (6.6) j→∞
From (6.5), (6.6) we have that p = r. This implies that p ∈ C.
The uniqueness of the structure M is implied by Theorem 6.5(3).
(3): Let D be the completion of the age A. By Proposition 6.23 we have N |= T D and consequently
M |= T D, where D is a ρ-compact age with UPAP and therefore, by Proposition 5.15, T D has QE. We conclude that Th(M) has QE.
The converse of Theorem 6.28 is the following theorem.
Theorem 6.30. Let M be a separable L-structure which is ω-qf-near-homogeneous. Set A = age(M). Then
A is a dA-complete, dA-separable age with the near-AP.
Proof. By Proposition 6.13, A is a dA-complete age with the near-AP. Since M is separable, we deduce that
A is dA-separable.
68 Chapter 7
Fra¨ıss´eTheorems for complete, ω-categorical theories with QE
In this chapter we fix a continuous signature L whose nonlogical symbols are a finite number of constant
and predicate symbols.
In Chapter 7 we prove a generalization to the continuous setting of the Fra¨ıss´eTheorems for complete,
ω-categorical theories that have QE.
Definition 7.1. Let A be an age. We say that A is totally bounded if for all > 0 there exists n ≥ 1 such
that for all M ∈ A, M has an -net of size ≤ n.
The following remark is clear.
Remark 7.2. Let M be an L-structure and A = age(M). Then, M is compact iff A is totally bounded.
Therefore, if T is a complete theory that is ω-categorical, then age(T ) is not totally bounded.
Definition 7.3. Let A be an age with the near-AP. We say that A is dA-compact if for every n ∈ ω, the
A A metric space (Sn , dn ) is compact.
Remark 7.4. In Proposition 7.10 we will show that if A is a dA-compact age with the near-AP, then A has the AP.
Our main goal in this chapter is to prove the following theorems, which we call Fra¨ıss´eTheorems for complete, QE, ω-categorical theories.
Theorem 7.5. Let T be a complete theory that has QE and is ω-categorical. Then age(T ) is a dA-compact age that has the UPAP and is not totally bounded.
A strong converse of Theorem 7.5 is the following theorem.
Theorem 7.6. Let A be a dA-compact age that has the AP and is not totally bounded. Then there exists a unique complete L-theory T that has QE, is ω-categorical and satisfies age(T ) = A.
First we will prove Theorem 7.5.
69 Proposition 7.7. Let T be a complete L-theory that has QE and let A = age(T ). The following statements
are equivalent:
1. T is ω-categorical.
2. A is dA-compact age that is not totally bounded.
Proof. (1)⇒(2): Because T is ω-categorical, from Theorem 4.23 we conclude that (Sn(T ), d) is compact. Since T is a complete L-theory and has QE, by Theorem 5.5, A is a ρ-compact age with the UPAP, and
A A by Proposition 5.6, A has the AP. So for all n ∈ ω,(Sn , dn ) is a metric space. For all n ∈ ω the space of A types Sn(T ) can be identified with the space of quantifier-free types types Sn , and in addition, (Sn(T ), d) A A A A A can identified with (Sn , d ). Then, for all n ∈ ω,(Sn , dn ) is compact. We conclude that A is d -compact. Since T is ω-categorical, by definition T has a unique separable noncompact model M. Clearly age(M) =
A. Then by Remark 7.2 we conclude that A is not totally bounded.
A A (2)⇒(1): As in the implication (1)⇒(2), for all n ∈ ω,(Sn(T ), d) can identified with (Sn , d ). This
implies that for all n ∈ ω,(Sn(T ), d) is a compact metric space. Since A is not totally bounded, by the compactness theorem we deduce that T has a noncompact model. By Theorem 4.23 we conclude that T is
ω-categorical.
Proof of Theorem 7.5. Since T is a complete L-theory that has QE, by Theorem 5.5 we deduce that A is a ρ-compact age with the UPAP. By the implication (1)⇒(2) in Proposition 7.7 we conclude that A is not totally bounded and dA-compact.
Now we prove Theorem 7.6. We will first need to prove some propositions.
Definition 7.8. Let (Y, d1), (Y, d2) be metric spaces. We say that d1, d2 are uniformly equivalent if for every > 0 there exists δ > 0 such that for every x, y ∈ Y ,
d1(x, y) < δ implies d2(x, y) < ,
and
d2(x, y) < δ implies d1(x, y) < .
u In this case we write (Y, d1) ∼ (Y, d2).
Proposition 7.9. Let (Y, d1), (Y, d2) be metric spaces. Suppose that (Y, d1) is compact and that (Y, d1) is
u finer than (Y, d2). Then (Y, d1) ∼ (Y, d2).
70 Proof. Let IY :(Y, d1) → (Y, d2) be the identity map on Y . Since (Y, d1) is compact and IY is continuous and bijective, we deduce that IY is a homeomorphism. We conclude that IY and its inverse are both uniformly continuous.
Proposition 7.10. Let A be a dA-compact age with the near-AP. Then:
1. A is ρ-compact.
2. A has the AP.
A A A Proof. (1): Fix n ∈ ω. By Proposition 6.16, (Sn , dn ) is finer than (Sn , ρn). By Proposition 7.9 we deduce A A u A A that (Sn , dn )) ∼ (Sn , ρn). We conclude that (Sn , ρn) is compact. (2): Clear.
Proposition 7.11. Let A be an dA-compact age with the AP. Then A has the UPAP.
A A u A A Proof. Since A is d -compact, by Proposition 7.9, for all n ∈ ω,(Sn , ρn) ∼ (Sn , d ). So there exist functions
∆n : (0, 1] → (0, 1]
0 ∆n : (0, 1] → (0, 1]
such that for every ∈ (0, 1]
A ρn(p, q) < ∆n() implies dn (p, q) <
and
A 0 dn (p, q) < ∆n() implies ρn(p, q) < .
A Fix n ∈ ω and > 0. To show that A has the UPAP it is enough to find δn () as in Definition 5.2. We claim A that we may choose δn () such that A 0 δn () = ∆n(∆n+1()).
A To prove the claim, fix M ∈ A, p ∈ Sn+1,a ¯ = (a1, . . . , an) ⊆ M such that if we set q = qftpM(¯a) then
0 ρn(pn, q) < ∆n(∆n+1()).
This implies that
A 0 dn (pn, q) < ∆n+1().
71 ¯ ¯ From Lemma 6.14, there exist A1 ∈ A such that M ⊆ A1 and b = (b1, . . . , bn) ⊆ A1 such that b |= pn, and
0 A1 max d (ai, bi) < ∆n+1(). 1≤i≤n
Because A has the AP, by Proposition 3.8 there exists A2 ∈ A such that A1 ⊆ A2 and c ∈ A2 such that (¯a, c) |= p. Set ¯ d = (d1, . . . , dn+1) = (a1, . . . , an, c),
e¯ = (e1, . . . , en+1) = (b1, . . . , bn, c).
¯ Set r = qftpA2 (d). Clearly,e ¯ |= p and
0 A2 max d (di, ei) < ∆n+1(). 1≤i≤n+1
This implies that
A 0 dn+1(r, p) < ∆n+1(),
which in turn implies that
ρn+1(r, p) < .
Now we restate and prove Theorem 7.6.
Theorem 7.12. Let A be a dA-compact age that has the AP and is not totally bounded. Then there exists
a unique complete L-theory T that has QE, is ω-categorical and satisfies age(T ) = A.
Proof. Since A is dA-compact, by Proposition 7.10 and Proposition 7.11 we deduce that A is ρ-compact and
that it has the UPAP respectively. By Theorem 5.4, there exists a unique complete L-theory T that has
QE. Since A is a dA-compact age that is not totally bounded, by Proposition 7.7 we conclude that T is
ω-categorical.
Remark 7.13. A natural question which arises is the following: If L is a continuous signature whose only
nonlogical symbols are a finite number of constants and predicate symbols, are there any L-theories that
have QE but are not ω-categorical?
The answer is no in the first-order setting: if an L-theory T has QE and L is a finite relational first-order signature, then for every n ∈ ω, Sn(T ) is finite, and therefore T is ω-categorical.
72 The situation is different in the continuous setting. Here is an example of a structure M such that Th(M) has QE but is not ω-categorical. Let L be a signature where the only nonlogical symbol is a constant c. Let
M = {a, b0, b1,... }. We define an L structure M on M as follows:
• We set cM = a.
M • For all i ∈ ω, we set d (a, bi) = 1 − 1/(i + 3).
M • For all i < j ∈ ω, we set d (bi, bj) = 1.
Set T = Th(M).
Claim 7.14. T has QE.
A To prove the claim we will use Proposition 4.3. Set A = age(M). Let M |= T , n ∈ ω, p ∈ Sn+1,a ¯ |= M witha ¯ |= pn. Note that, since M is discrete we have that for all > 0,
sup min( −· τ (¯x), inf τ (¯x, x ) −· ) = 0 pn p n+1 x¯ xn+1
is true in M. Hence for all > 0,
min( −· τpn(¯a), inf τp(¯a, xn+1) −· ) = 0 xn+1
is true in M. We deduce that there exists M0 M and b ∈ M 0 such that (¯a, b) |= p.
Claim 7.15. T is not ω-categorical.
It suffices to find a model of T which realizes a type that is not realized in M. Set A = {a, e, b0, b1 ... }, and let
0 T = T ∪ {d(e, a) = 1} ∪ {d(e, bi) = 1 | i ∈ ω}.
Then T 0 is satisfiable by the compactness theorem, since every finite subset of T 0 is satisfied in M. Every
model of T 0 is a model of T and realizes a type which is not realized in M. We conclude that T is not
ω-categorical.
73 Chapter 8
Macpherson’s Theorem
In this chapter we prove a generalization of the following theorem of H. D. Macpherson.
Theorem 8.1. Let L be a countable first-order signature and M be a countably infinite L-structure such that Th(M) is ω-categorical. Set G = Aut(M). Then there exists a dense subgroup F ≤ G which is freely
generated by countably many elements, where G is equipped with the pointwise convergence topology.
Proof. See [8, Theorem 3.1].
For the remainder of the chapter, L will denote a countable bounded continuous signature. Also, all
the automorphism groups of L-structures will be equipped with the topology of pointwise convergence, and
subgroups of such groups will be equipped with the subspace topology.
The following theorem is a generalization of Theorem 8.1 to the metric setting.
Theorem 8.2. Let M be a separable L-structure which is strongly ω-homogeneous, noncompact and such
that Th(M) is ω-categorical. Set G = Aut(M). Then there exists a dense subgroup F ≤ G which is freely
generated by countably many elements.
The following proposition is a strengthening of the implication (1)⇒(2) in Theorem 4.25.
Proposition 8.3. Let M be a separable L-structure such that Th(M) is ω-categorical and let G be a dense
subgroup of Aut(M). Then for every > 0 and n ≥ 1 there exist n-tuples a¯1 = (a1,1, . . . , a1,n),..., a¯l = ¯ (al,1, . . . , al,n) ⊆ M for some l ≥ 1 such that for every n-tuple b ⊆ M there exist h ∈ G and 1 ≤ j ≤ n such that
M max d (bi, h(aj,i)) < . 1≤i≤n
¯ Proof. Fix > 0 and b = (b1, . . . , bn) ⊆ M. From the implication (1)⇒(2) in Theorem 4.25, there exists
n-tuplesa ¯1 = (a1,1, . . . , a1,n),..., a¯l = (al,1, . . . , al,n) ⊆ M for some l ≥ 1, g ∈ Aut(M) and 1 ≤ j ≤ n such that
M max d (bi, g(aj,i)) < . (8.1) 1≤i≤n 2
74 Because G is dense in Aut(M), there exists h ∈ G such that
M max d (g(aj,i), h(aj,i)) < . (8.2) 1≤i≤n
Equations (8.1), (8.2) and the triangle inequality imply
M max d (bi, h(aj,i)) < . 1≤i≤n
We will need also the following theorem.
Theorem 8.4. Let M be an L-structure such that Th(M) is ω-categorical. Let G ≤ Aut(M) be dense in
Aut(M). Then the union of the totally bounded orbits of G, as it acts on M, is totally bounded.
Proof. Let K be the union of the totally bounded orbits of G. Fix > 0. Then by Proposition 8.3 we have
that there exist a1, . . . , al ∈ M for some l ≥ 1 such that for every b ∈ M there exist h ∈ G and 1 ≤ i ≤ l such that dM(b, h(a )) < . i 3
For each b ∈ K there exists 1 ≤ i ≤ l such that b ∈ G · B(ai, /3). Set
I = {i | G · B(a , ) ∩ K 6= ∅}. i 3
Then we have that
[ K ⊆ G · B(a , ). (8.3) i 3 i∈I
For each i ∈ I we pick bi such that b ∈ G · B(a , ) ∩ K. i i 3
Then G · bi is totally bounded. For each 1 ≤ i ≤ l, let Ni ⊆ G · bi be a finite /3-net for G · bi. Then, for each 1 ≤ i ≤ l we have that [ G · B(b , ) ⊆ B(d, ). (8.4) i 3 d∈Ni
75 Then from (8.3) and (8.4) we have that
[ [ [ K ⊆ G · B(a , ) ⊆ B(d, ). i 3 i∈I i∈I d∈Ni
A consequence of the Theorem 8.4 is the following theorem.
Theorem 8.5. Let M be a separable L-structure such that Th(M) is ω-categorical. Then the set of algebraic elements of M over the empty set is compact.
Proof. Set G = Aut(M). Let K be the union of the totally bounded orbits of G and K0 the union of the algebraic elements of M over the empty set.
First we prove that K is closed. To do so, it is enough to prove that K includes the set of its limit points.
Let a be a limit point of K. we show that G · a is totally bounded. Fix > 0. We show that G · a has a
finite -net. Let b ∈ K such that dM(a, b) < /4. Since b ∈ K, G · b is totally bounded. Let F be a finite
/4-net for G · b.
Claim 8.6. For every x ∈ G · a there exists y ∈ F such that dM(x, y) < /2.
Proof. (of claim) Let g · a ∈ G · a for some g ∈ G. Then dM(g · a, g · b) < /4. Let c ∈ F such that dM(c, g · b) < /4. By the triangle inequality we have dM(g · a, c) < /2.
Claim 8.7. G · a contains a finite subset F 0 that is -dense.
Proof. (of claim) Set
F 00 = {c ∈ F | For some x ∈ G · a dM(c, x) < /2}.
Note that F 00 is finite. For each c ∈ F 00 choose x(c) ∈ G · a such that d(c, x(c)) < /2. Set
F 0 = {x(c) | c ∈ F 00}.
It is easy to see that F 0 is a finite -net for G·a. Indeed, if x ∈ G·a then by Claim 8.6 there exists c ∈ F 00 such
that dM(c, x) < /2. Then dM(c, x(c)) < /2 and by the triangle inequality we get that dM(x, x(c)) < .
Since K is closed and, by Theorem 8.4, totally bounded we deduce that K is compact. To complete the
proof of the Theorem 8.5 it is enough to prove that K = K0. It is enough to prove that a is algebraic over
the empty set iff G · a is totally bounded. If a is algebraic over the empty set then by [1, Exercise 10.8] the
76 set of realizations of p := tpM(a) is compact. By Proposition 4.24 the set of realizations of p is the set G · a. We conclude that G · a is totally bounded. For the other direction, if a is not algebraic over the empty set
then by [1, Exercise 10.8] there exists an ω1-saturated N M such that the set of realization of p in in N
0 has density≥ ω1. By the Downward L¨owenheim-Skolem Theorem there exists a separable L-structure N such that M N0 N, and the set of realizations of p in N0 is noncompact. Since T h(M) is ω-categorical,
we have M =∼ N0. We deduce that the set of realizations of p in M is noncompact. As above, the set of realizations of p in M is G · a. We conclude that G · a is not totally bounded.
Proposition 8.8. Let M be a separable L-structure which is noncompact. If G ≤ Aut(M) is such that for
every a¯ ⊆ M, the union of the totally bounded orbits of Ga¯ is totally bounded, then there is a dense subgroup F ≤ G which is freely generated by countably many elements and F is dense in G.
Proof. We will define by finite approximations a subset {gi | i ∈ ω} of G which will freely generate F . Let A be a countable dense subset of M. Because M is separable, Aut(M) is separable and so G is separable.
Let H be a countable dense subgroup of G. Set B = H · A and
C = {(¯a, ¯b) | a,¯ ¯b ⊂ B ∧ ∃g ∈ H g(¯a) = ¯b}.
¯ Note that card(B) = card(C) = ω and that B is not totally bounded. Let (¯an, bn)n∈ω be an enumeration of
−1 C. Let (wn)n∈ω be an enumeration of the reduced words in {gj, gj | j ∈ ω} so that wn does not involve gj −1 or gj if j > n. Let (xn)n∈ω be an enumeration of B.
Suppose that up to step 3n − 1 we have defined finite partial automorphisms g0,3n−1, . . . , gn−1,3n−1 on B such that each of them is a restriction of an automorphism of M which is in H. ¯ Step 3n: Define a partial automorphism gn,3n by the rule gn,3n(¯an) = bn. For every i < n, we put
gi,3n = gi,3n−1.
Step 3n + 1: Let w = g1 . . . gr where i , . . . , i such that 1 ≤ i , . . . , i ≤ n and , . . . , ∈ {±1}, for n i1 ir 1 r 1 r 1 r some r ≥ 1.
Claim 8.9. There are x ∈ B and extensions gi1,3n+1, . . . , gir ,3n+1, of gi1,3n, . . . , gir ,3n respectively, which are restrictions of automorphisms of M in H, such that:
• g1 ··· gr (x) is defined. i1,3n+1 ir ,3n+1
• Given s with 1 < s ≤ r and the definitions of gr (x), . . . , gs ··· gr (x) ∈ B we have that ir ,3n+1 is,3n+1 ir ,3n+1
O := H ¯ · (g s ··· g r (x)) s a¯s−1,bs−1 is,3n+1 ir ,3n+1
77 is a nonempty, not totally bounded subset of B, where a¯s−1 is a finite tuple which enumerates the
s−1 s−1 elements of B where the partial automorphism g is defined, ¯b = g (¯a ), and H ¯ is−1,3n s−1 is−1,3n s−1 a¯s−1,bs−1 ¯ is the pointwise stabilizer of the tuple (¯as−1, bs−1).
Note that O can be considered as the set of the possible 1-point extensions of gs at the point s is,3n gs ··· gr (x), to partial automorphisms of B, which are restrictions of automorphisms of M in H . is,3n+1 ir ,3n+1
Proof. (of claim) This is proved by induction on s. Note that the induction goes backwards, from r toward
1, and that at the start of the induction we use the fact that B is not totally bounded. Suppose that we have
defined gs ··· gr (x) ∈ B, for some 1 < s ≤ r, such that O is a nonempty, not totally bounded is,3n+1 ir ,3n+1 s subset of B. We claim that for some y ∈ B we have that
Ha¯s−2,¯bs−2 · y
is a nonempty, not totally bounded subset of B, wherea ¯s−2 is a finite tuple which enumerates the elements of B where the partial automorphism gs−2 is defined and ¯b = gs−2 (¯a ). Let K be the union of is−2,3n s−2 is−2,3n s−2 0 the totally bounded orbits of Ha¯s−2,¯bs−2 as it acts on B. Let K be the the union of the totally bounded 0 orbits of Ga¯s−2,¯bs−2 as it acts on M. From hypothesis we have that that K is totally bounded. Because K ⊆ K0, K is totally bounded. Set
P := Ha¯s−2,¯bs−2 · Os.
Clearly,
Os ⊆ P ⊆ B.
Since by induction hypothesis Os is not totally bounded, we deduce that P is not totally bounded. We conclude that P \ K is a nonempty, not totally bounded subset of B. This implies that there exists y ∈ Os such that
Ha¯s−2,¯bs−2 · y
is not totally bounded. If we set
gs−1 ··· gr (x) = y, is−1,3n+1 ir ,3n+1
then
Os−1 = Ha¯s−2,¯bs−2 · y
and Os−1 is not totally bounded.
78 From the claim we conclude that we may find x ∈ B and define gr (x), . . . , w (x) in such a way ir ,3n+1 n that wn(x) 6= x. For every i such that 1 ≤ i ≤ n, if i ∈ {i1, . . . , ir} we extend gi,3n to gi,3n+1 appropriately, otherwise we set gi,3n+1 = gi,3n.
−1 Step 3n + 2: For every i ≤ n, we extend gi,3n+1 to gi,3n+2 to ensure that gi,3n+2 and gi,3n+2 are defined on xn. We can do this in such a way such that for every i ≤ n, gi,3n+2 is a restriction of an automorphism of M in H.
For each i ∈ ω, we set [ gi = gi,n. n∈ω
We have that for all i ∈ ω, gi is a restriction of an element in H. Because for every i ∈ ω, gi is defined on
D, which is dense in M, we have that gi can be extended uniquely to an fi ∈ H. Set
F =< fi | i ∈ ω > .
Claim 8.10. F is dense in H.
It is enough to show that given h ∈ H, > 0, c1, . . . , cn ∈ M for some n ∈ ω, we have that there exists fj ∈ F such that
M max d (h(ci), fj(ci)) < . 1≤i≤n
Fix h ∈ H, > 0, c1, . . . , cn ∈ M. Because D is dense in M there exists d1, . . . , dn ∈ D such that
M max d (ci, di) < . 1≤i≤n 2
¯ ¯ ¯ ¯ ¯ ¯ Set d = (d1, . . . , dn). We have that (d, h(d)) ∈ C. So there exists j ∈ ω such that (d, h(d)) = (¯aj, bj). So we have that
∀i ≤ n fj(di) = h(di).
Because h, fj are isometries we have that for i ≤ n
dM(h(c ), h(d )) = dM(f (c ), f (d )) = dM(c , d ) < . i i j i j i i i 2
From the triangle inequality we have that
79 M M M d (h(ci), fj(ci)) ≤ d (h(ci), fj(di)) + d (fj(di), fj(ci)).
Hence
M M M d (h(ci), fj(ci)) ≤ d (h(ci), h(di)) + d (fj(di), fj(ci)).
Therefore
M M M max d (h(ci), fj(ci)) ≤ max d (h(ci), h(di)) + max d (fj(di), fj(ci)) ≤ + = . 1≤i≤n 1≤i≤n 1≤i≤n 2 2
Theorem 8.11. Let M be a separable L-structure that is strongly ω-homogeneous, noncompact and such that Th(M) is ω-categorical. Let G ≤ Aut(M) be such that for every finite tuple a¯ ⊆ M, Ga¯ is dense in
Auta¯(M). Then there exists a dense subgroup F ≤ G which is freely generated by countably many elements.
Proof. Fix a finite tuplea ¯ in M. Because Th(M) is ω-categorical and M is ω-homogeneous, Proposition 4.27 implies that Th(M, a¯) is ω-categorical. By hypothesis we have that Ga¯ is dense in Aut(M, a¯) = Auta¯(M).
From Theorem 8.4 we have that the union of the totally bounded orbits of Ga¯ is totally bounded. Then by Proposition 8.8 the result follows.
Theorem 8.2 easily follows from Theorem 8.11 if we take G = Aut(M).
80 References
[1] Ita¨ıBen Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov, Model theory for metric structures, to appear in a Newton Institute volume in the Lecture Notes series of the London Math. Society. http://www.math.uiuc.edu/˜henson/ [2] Ita¨ıBen Yaacov and Alexander Usvyatsov, Continuous first order logic and local stability, submitted. http://www.math.univ-lyon1.fr/˜begnac/ [3] Peter J. Cameron, Oligomorphic Permutation Groups, London Mathematical Society Lecture Notes Series, vol 152, 1990. [4] R. Fra¨ıss´e,Sur l’extension aux relations de quelques propriet´esdes ordres, Ann. Sci. Ecole´ Norm. Sup. 71, 1954, 363–388. [5] C. Ward Henson and Jos´eIovino, Ultraproducts in analysis, in Analysis and Logic, London Mathemat- ical Society Lecture Notes Series, vol 262, 2002, 1–113. [6] Wilfrid Hodges, A Shorter Model Theory, Cambridge University Press, 1997.
[7] M. Kat˘etov, On universal metric spaces, Proc. of the 6th Prague Topological Symposium (1986), Frolik (ed). Helderman Verlag Berlin, 1988, pp 323–330.
[8] H. Dugald Macpherson, Groups of automorphisms of ℵ0-categorical structures, Quart. J. Math. Oxford (2), 37, 1986, 449–465. [9] Alexander Usvyatsov, Generalized Vershik’s Theorem and generic metric structures, submitted.
81 Author’s Biography
Konstantinos Schoretsanitis was born in Athens, Greece. After he obtained a Bachelor of Science in Mathe- matics from the University of Patras, Greece, he decided to pursue graduate studies in Mathematics at the
University of Illinois at Urbana-Champaign. He obtained a Master of Science in Mathematics, and he will be granted the Doctor of Philosophy in 2007.
82 Fra¨ıss´eTheory for Metric Structures
Konstantinos Schoretsanitis, Ph.D. Department of Mathematics University of Illinois at Urbana-Champaign, 2007 C. Ward Henson, Advisor
In 1954, Roland Fra¨ıss´epublished a paper that answered the following questions: Given a first-order
signature L and a class A of finite L-structures that is closed under isomorphism:
1. find necessary and sufficient conditions on A that guarantee the existence of a “homogeneous” L-
structure M such that the class of L-structures that are isomorphic to finite L-substructures of M is
A;
2. find necessary and sufficient conditions on A that guarantee the existence of an L-structure M such
that Th(M) has QE and is ω-categorical, and such that the class of L-structures that are isomorphic
to finite L-substructures of M is A.
In this thesis we generalize Fra¨ıss´e’sresults to the setting of bounded continuous logic for metric structures.
This logic was presented in 2004 by Ita¨ıBen Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander
Usvyatsov, and it may be considered as a generalization of first-order logic.
We also prove a theorem, in the setting of continuous model theory, that is a generalization of a theorem of H. D. Macpherson about the automorphism groups of ω-categorical structures.