# Continuous first-order model theory for metric structures Lecture 2 (of 3)

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1 Continuous first-order model theory for metric structures Lecture 2 (of 3) C. Ward Henson University of Illinois Visiting Scholar at UC Berkeley October 21, 2013 Hausdorff Institute for Mathematics, Bonn

2 Some conventions When considering a theory T and its type spaces S n (T ): topological notions always refer to the logic topology (closed set, continuous function, etc); metric notions always refer to the metric induced by distances in models of T. (ball of radius r; uniformly continuous function, etc); except we refer explicitly to the metric topology when we need to.

3 Some notation Consider a theory T and a model M T. For each A M we let A M denote the substructure of M generated by A. The underlying set of A M is the closure in M of the set of all elements t M (a) where t(x) is any term from the signature of T and a is any tuple from A.

4 Suppose M, N T and a M n, b N n. We write a 0 b in M, N if ϕ M (a) = ϕ N (b) for every quantifier-free formula ϕ(x). Note that this is equivalent to the existence of an isomorphism J from a M onto b N satisfying J(a i ) = b i for all i = 1,..., n. Likewise we write a b in M, N if ϕ M (a) = ϕ N (b) for every formula ϕ(x), which is the same as saying (M, a) (N, b)

5 T admits quantifier elimination if for every formula ϕ(x) and every ɛ > 0 there is a quantifier-free formula such that the following condition holds in all models of T : sup x ϕ(x) ψ(x) ɛ Here x = x 1,..., x n is a finite tuple and sup x means sup x1 sup xn.

6 Criterion 1 for QE: types = qf types T admits QE if and only if for all n 1 and all p S n (T ), the type p is determined by the quantifier free formulas it contains. Proof: Consider the restriction map π : S n (T ) S qf n (T ). This map is always continuous and surjective. QE is equivalent to its being a topological homeomorphism. Since both spaces are compact, for QE it suffices that the map is injective.

7 Criterion 2 for QE: back-and-forth property T admits QE if and only if for every ω-saturated models M, N of T and every a M n, b M, a N n, if a 0 a in M, N, then there exists b N such that (a, b) 0 (a, b ) in M, N. Proof: ( ) Use the condition to show that whenever a 0 a in M, N, then a and a are given the same values by formulas of the form inf y ϕ(x, y) in which ϕ is quantifier-free. Use an argument like that for Criterion 1 to show that every formula of this kind is approximated uniformly by quantifier-free formulas. Then use induction on syntactic complexity to show the same is true for all formulas.

8 Criterion 3 for QE: extension of embeddings T admits QE if and only if for every M, N T, every embedding of a substructure of M into N can be extended to an embedding of M into an elementary extension of N. Moreover, if the signature of T has κ many symbols, then it suffices to consider M of density character κ and to consider a fixed elementary extension of N that is κ-saturated.

9 Proof: ( ) Show T verifies Criterion 2. So consider ω-saturated models M, N of T and tuples a M n, a N n and an element b M. Assume a 0 a in M, N. This gives a canonical isomorphism J of a M M onto b N N that maps a exactly to b. The condition above gives an extension of J to an embedding (which we still call J) of M into an elementary extension N of N. This ensures (a, b) 0 (a, J(b)) in M, N. Since N is ω-saturated, we can find b N such that (a, J(b)) 0 (a, b ) in N, N. Therefore (a, b) 0 (a, b ) in M, N as desired.

10 Some examples of theories with QE: APr = theory of atomless probability algebras. H = theory of infinite dimensional Hilbert spaces. ALpL = theory of L p Banach lattices of atomless measure spaces, 1 p < fixed. (Some details given on the blackboard; the first two examples are easy using Criterion 1. The third is more complicated. It is treated using Criterion 3 as Example in the paper Ultraproducts in analysis by WH and José Iovino. See the references at the end of these slides.)

11 M T is an existentially closed (e.c.) model of T if for every M N T, every quantifier-free formula ϕ(x, y), and every a M m (, infy ϕ(a, y) ) M ( = infy ϕ(a, y) ) N. Here y = y 1,..., y n is a finite tuple and inf y means inf y1 inf yn. The condition above is equivalent to requiring the implication ( infy ϕ(a, y) ) N = 0 implies ( infy ϕ(a, y) ) M = 0 for all quantifier-free ϕ.

12 T is model complete if every embedding between models of T is an elementary embedding. (Note: quantifier elimination = model complete.) T is a model companion of T if they have the same signature, T is model complete, and T, T have the same substructures of models.

13 T is inductive if its class of models is closed under unions of arbitrary chains. Theorem Suppose T is an inductive theory and M T. Then there exists an e.c. model N of T such that M N.

14 Theorem Let T be an inductive theory and let T be any theory with the same signature as T. (a) T is a model companion of T if and only if the models of T are exactly the e.c. models of T. (b) In particular, T has a model companion if and only if the class of e.c. models of T is axiomatizable.

15 Theorem Suppose T is a model companion of T. If models of T have the amalgamation property over substructures, then T admits quantifier elimination.

16 Throughout this subsection on separable models, we take T to be a complete theory with a countable signature.

17 isolated types, atomic models A type p S n (T ) is isolated (i.e., principal) if it has the same filter of neighborhoods in the logic topology as in the metric topology.

18 isolated types, atomic models A type p S n (T ) is isolated (i.e., principal) if it has the same filter of neighborhoods in the logic topology as in the metric topology. A structure M is atomic if every type realized in M is isolated (w.r.t. T = Th(M)).

19 Theorem (part of the Omitting Types Theorem) Let p S n (T ). The following are equivalent: (1) p is isolated. (2) p is realized in every model of T.

20 Theorem (Atomic models) (a) T has an atomic model iff for each n N, the isolated types in S n (T ) are dense.

21 Theorem (Atomic models) (a) T has an atomic model iff for each n N, the isolated types in S n (T ) are dense. (b) T has at most one separable atomic model.

22 Theorem (Atomic models) (a) T has an atomic model iff for each n N, the isolated types in S n (T ) are dense. (b) T has at most one separable atomic model. (c) If S n (T ) is separable in the metric topology, for all n N, then T has a separable atomic model.

23 M is ω-near-homogeneous if for any a, b M n realizing the same n-type in M, and any ɛ > 0, there is an automorphism σ of M such that d(σ(a), b) < ɛ. Theorem If M is a separable atomic model of T, then M is ω-near-homogeneous. In particular, this is true of the unique separable model of a separably categorical theory. Caution: in discrete structures (classical model theory), the countable atomic model is always ω-homogeneous. In the setting of metric structures, however, ω-near-homogeneity is often the most one has. We give an example in a few slides.

24 Theorem (Separable categoricity) The following are equivalent: (1) T has exactly one separable model. (2) Every separable model of T is atomic. (3) Every type in S n (T ) is isolated, for all n N. (4) The metric topology is compact on S n (T ), for all n N. (5) The logic topology agrees with the metric topology on S n (T ), for all n N.

25 Corollary Suppose T is separably categorical and T is the restriction of T to a smaller signature. Then T is also separably categorical. Note: this is true in particular if T is the theory of all metric spaces that arise from models of T. A priori T might have separable models that do not come from any model of T. Proof: for each n, consider the restriction map from S n (T ) onto S n (T ); it is continuous, contractive, and surjective. Hence it preserves compactness with respect to the metric topologies.

26 An example of non-homogeneous separably categorical model Let T 0 be the theory of all unit ball structures obtained from the L 1 Banach lattice of an atomless measure space. Note that T 0 is separably categorical, with unique separable model the unit ball B of the L 1 Banach lattice based on [0, 1] with Lebesgue measure. As discussed above, T 0 admits QE. Let T 1 be the theory of all structures (M, f ) where M is is a model of T 0 and f M satisfies f 0 and f 1 = 1. T 1 is a complete theory and admits QE itself. Its separable models are of the form (B, f ).

27 There are two isomorphism classes of separable models of T 1. In one of them, the support of f has full measure as a subset of [0, 1], and in the other, f is supported on a set [0, 1] whose measure is < 1. From this argument it follows that B, the unit ball of the Banach lattice L 1 based on [0, 1] with Lebesgue measure, is not ω-homogeneous, even though it is the unique separable model of T 0. The automorphism group of B has two orbits.

28 In this section we fix T to be the theory of metric spaces of diameter 1. Our objective is to explain the following result, and then to say a few more things about this situation. Theorem T has a model companion T. Moreover: (1) T admits quantifier elimination. (2) T is separably categorical. (3) The unique separable model of T is the Urysohn space U of diameter 1 (sometimes also called the Urysohn sphere). To prove T exists we need to axiomatize the class of e.c. metric spaces of diameter 1. The key to doing this and to proving that T is separably categorical is the following Lemma.

29 Lemma Suppose I is an index set. Consider families (x i i I ) and (y i i I ) from metric spaces X and Y respectively. Let (ɛ i i I ) be any family of real numbers 0. The following are equivalent: 1 There are a metric space Z and isometries f : X Z and g : Y Z satisfying d(f (x i ), g(y i )) ɛ i for all i I. 2 For all i, j I we have d(x i, x j ) d(y i, y j ) ɛ i + ɛ j. Note that if X and Y have diameter 1, then we can easily modify Z in condition 1 so that it also has diameter 1 without changing the statement, simply by truncating its metric at 1.

30 Proof of Lemma: (1 2) is immediate from the triangle inequality. (2 1) Easy amalgamation techniques for metric spaces allow us to assume X = {x i i I } and Y = {y i i I }, and we may assume X Y =. Begin to define a pseudometric d on X Y by taking distances in X and in Y to be the given ones, and to tentatively set the distance between x i and y i to be ɛ i for each i. Then define d(u, v) on X Y to be the infimum of all sums of tentative distances along finite paths joining u to v; allowed edges of the path stay in X or stay in Y or go between x i and y i (points with the same index). Condition 2 is used to show that this definition does not change distances in X or in Y. Finally, Z is taken to be the metric space quotient of (X Y, d).

31 Note: a result with the same essential content as the Lemma above (and essentially the same proof) appears as Proposition 7.1 in the 2008 paper On subgroups of minimal topological groups by Vladimir V. Uspenskij in Topology and its Applications 155 (2008), pp In that paper only finite index sets are considered and the ɛ i are all taken to be the same. We need to allow the ɛ i to vary with i in order to calculate the metric on type spaces over nonempty sets of parameters.

32 It is now easy to axiomatize the class of e.c. metric spaces of diameter 1. Given a finite sequence a = a 1,..., a n in a metric space, let D a (x) be the following quantifier-free formula in the signature of metric spaces: max 1 i,j n d(xi, x j ) d(a i, a j ) Evidently, if b = b 1,..., b n is another n-tuple from some metric space Y, we have that D Y a (b) = 0 if and only if the map taking a i to b i for i = 1,..., n is isometric. Theorem The class of e.c. metric spaces of diameter 1 is axiomatized by the conditions sup x inf y ( Da,b (x, y). D a (x) ) = 0 where a, b = a 1,..., a n, b ranges over all finite metric spaces.

33 Now we know there is a model companion T for the theory T of all metric spaces of diameter 1. Since models of T satisfy the amalgamation property over substructures, we know that T admits QE. In this simple signature, every atomic formula is of the form d(x, y) where x, y are variables. Therefore, for each finite tuple a in a metric space X of diameter 1 we can identify tp X (a) with the matrix ( d(a i, a j ) ). Moreover, the Lemma above yields i,j the following formula for the distance between types: d(tp X (a), tp Y (b)) = 1 2 max d(ai, a j ) d(b i, b j ) i,j This proves that for all n 1, the type space S n (T ) is compact for the metric topology. Hence T is separably categorical.

34 As above, we let U denote the unique separable model of T, which is the Urysohn space of diameter 1 (= the sphere of radius 1 2 in the unbounded Urysohn space). Let M denote a monster model of T (= κ-saturated and κ-homogeneous for some large cardinal κ). We may assume U M. The approach used above can be applied to the type spaces S n (A) for any small subset A M.

35 A 1-type p S 1 (A) is realized by some element b M. Since T admits QE, we may identify p with the function f p : A [0, 1] given by f p (a) = d(a, b). Based on this identification, we see that as sets, S 1 (A) = K 1 (A), the set of Katětov functions with values in [0, 1]. The identification extends to the topological and metric structure, as well. Recall that on K 1 (A) we put the topology of pointwise convergence and the supremum metric. Proposition For each metric space A, the identification above is an isomorphism of topometric spaces between S 1 (A) and K 1 (A). That is, it is a homeomorphism and an isometry.

36 A new example Expand the pure language of metric spaces by adding finitely many predicate symbols P 1,..., P n and for each i = 1,..., n fix a real number C i > 0. Now let T P denote the theory of all structures M for this signature that satisfy Lipschitz axioms of the following form (one for each i): Pi (x) P i (y) j C id(x j, y j )

37 A new example Theorem T P has a model companion TP. Moreover: (1) TP admits quantifier elimination. (2) TP is separably categorical. (3) The underlying metric space of the unique separable model of TP is the Urysohn space U of diameter 1 (i.e., the restriction of TP to the pure metric space signature is T ).

38 Some references Itaï Ben Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov, Model theory for metric structures, in Lecture Notes series of the London Mathematical Society, No. 350, Cambridge University Press, 2008, [Treats QE in chapter 13; separable models in chapter 12.] Itaï Ben Yaacov and Alexander Usvyatsov, On d-finiteness in continuous structures, Fundamenta Mathematicae 194 (2007), [Treats omitting types and separable models in section 1.] C. Ward Henson and José Iovino, Ultraproducts in analysis, in Analysis and Logic, London Mathematical Society Lecture Notes Series, vol. 262, 2002, [Treats atomless L p spaces in Example ]

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