Ultraproducts as a tool in the model theory of metric structures

C. Ward Henson University of Illinois

last revised Apr. 23, 2020

1 Preliminary comments

Ultraproducts play a more significant role in continuous model theory of real-valued structures than in classical model theory of discrete, algebraic structures. In the real-valued setting, ultraproducts do more than proving compactness. They provide an important experimental tool for studying real-valued structures model theoretically. Especially toward verifying axiomatizability of classes of structures and clarifying what is definable.

The material covered here was developed in collaboration with Isaac Goldbring and Bradd Hart.

2 Early uses of ultraproducts

hints in work of Skolem (1934) and von Neumann (1942). a metric ultraproduct occurs in work of Wright, a student of Kaplansky (1954; Ann. of Math.) for discrete structures, the work ofLo´sin 1955; Robinson’s nonstandard analysis in 1961. ultraproducts of some operator algebras in Sakkai (1962), McDuff (1969), Connes (1974), ... . Banach space ultraproducts in Krivine (1967), nonstandard hulls in Luxemburg (1968); used extensively by Lindenstrauss, Pisier, Johnson, ... ; studied by Stern, Heinrich, CWH, Moore, ... . metric geometry: (ultralimits) Gromov (1981), (asymptotic cones) van den Dries–Wilkie (1984).

3 A crash course in model theory for metric structures

Signatures L specify moduli of uniform continuity for their function symbols F and predicate symbols P. L-structures: complete metric spaces equipped with constants, functions, and predicates that interpret the symbols of L. predicates are [0, 1]-valued; predicates and functions must satisfy the given moduli. L-terms: built as usual from constants and function symbols.

Atomic L-formulas: of the form P(t1,..., tn) and d(t1, t2). L-formulas: obtained by closing under continuous connectives u : [0, 1]n → [0, 1] and quantifiers sup and inf. A theory T in L is a set of axioms, which are L-sentences (L-formulas with no free variables). A model of T is an L-structure M such for every sentence σ in T , the value σM is 0. 4 Ultraproducts defined

(Mi | i ∈ I ) are L-structures, U an ultrafilter on I . Q We let N0 = i∈I Mi as a set; its members are (a) = (ai | i ∈ I ), ai ∈ Mi .

We interpret the symbols on N0:

N
M
F 0 (ai | i ∈ I ),... = F i (ai ,...) | i ∈ I ∈ N0 N
M P 0 (ai | i ∈ I ),... = limi→U P i (ai ,...) ∈ [0, 1]

dN0 is only a pseudometric; E((a), (b)) :⇔ dN0 ((a), (b)) = 0 is an equivalence relation and the quotient N0/E has canonical interpretations of the symbols of L, including the metric.

We define the ultraproduct to be N0/E (which is complete) Q and denote it by N := i∈I Mi /U. The image of (a) ∈ N0 in N is denoted (a)U ; note that h i N0
(a)U = (b)U ⇔ 0 = lim d(ai , bi ) = d (a), (b) . i→U 5 Properties of ultraproducts

Lo´s’sTheorem : for every L-formula ϕ(x, y,...) and elements Q (a)U , (b)U ,... ∈ N = Mi /U:

N
Mi ϕ (a)U , (b)U ,... = lim ϕ (ai , bi ,...). i→U Corollary The Compactness Theorem for continuous logic of metric structures.

6 Elementary

Let M, N be L-structures. Let A ⊆ M, B ⊆ N and f : A → B. Definition (a) M and N are elementarily equivalent if σM = σN for all L-sentences σ. We write M ≡ N. (b) The map f is elementary if (M, ha | a ∈ Ai) ≡ (N, hf (a) | a ∈ Ai).

Examples Any isomorphism from M onto N is elementary. The diagonal map DM : M → MU is elementary. Any restriction of an elementary map is elementary The composition of elementary maps is elementary. The inverse of an elementary map is elementary.

7 Let M, N be L-structures. Let A ⊆ M, B ⊆ N and f : A → B. Keisler-Shelah Theorem M and N are elementarily equivalent ∼ if and only if there is an ultrafilter U such that MU = NU . Corollary The map f is elementary iff there is an ultrafilter U and an isomorphism J from MU onto NU such that J ◦ DM = DN ◦ f on A.

8 Definition A class C of L-structures is axiomatizable if there is an L-theory T such that C is the class of all models of T ; that is, C = {M | M is an L-structure and σM = 0 for all σ ∈ T }.

Theorem For any class C of L-structures, the following are equivalent: (a) C is axiomatizable. (b) C is closed under isomorphisms, ultraproducts, and ultraroots.

M is an ultraroot of N if there is an ultrafilter U such that N is isomorphic to MU .

9 T -formulas

Let T be an L-theory and x = x1,..., xn a tuple of distinct variables.

A T -formula ϕ(x) is a sequence (ϕn(x)) of L-formulas such M that ϕn (a) converges as n → ∞, uniformly for all models M of T and all a ∈ Mx . M The interpretation ϕ of a T -formula ϕ(x) = (ϕn(x)) in a model M of T is defined by M M x ϕ (a) := limn ϕn (a) for all a ∈ M . We refer to the interpretations of T -formulas in the models of T (as above) as T -predicates.

Example: for any L-formulas (θn(x)), the weighted sum P −n M n 2 θn (x) is a T -predicate.

10 Assignment of [0, 1]-valued functions to models

Let T be an L-theory; fix a tuple of distinct variables x = x1,..., xn. Denote by F any operation that assigns a function M x F : M → [0, 1] to each model M of T . We will call such an F a T -function.

For example, every T -predicate is a T -function.

Notation: given a T -formula ϕ(x), we let Fϕ denote the M M T -function defined by F := ϕ ; that is: M M Fϕ (a) := ϕ (a).

11 We note two properties of T -predicates F = Fϕ: (iso) If M, N are models of T and J is an isomorphism of M onto N M N, then F ◦ J = F . (diag) If M is a model of T and U is an ultrafilter, with D : M → MU the diagonal embedding, then M M F U ◦ D = F . Fact: a T -function F satisfies both (iso) and (diag) if and only if there is a function∆ F : Sx (T ) → [0, 1] such that for any model M M of T we have: F (a) = ∆F(tpM (a)). (Note that ∆F is unique.) Note: Properties of F that we discuss later correspond to properties of ∆F on the topometric space Sx (T ).

12 Characterization of T -predicates

Let T be an L-theory and let x = x1,..., xn be a tuple of distinct variables. Let F be a T -function. Theorem The following are equivalent:

(1) There exists a T -formula ϕ(x) such that F = Fϕ. (i.e., F is a T -predicate.) (2) F satisfies (iso), and the Los condition N M F ([ai ]) = limi→U F i (ai ) Q holds for every ultraproduct N = Mi /U of models of T and x every [ai ] ∈ N .

(3) F satisfies (iso) and (diag), and ∆F is continuous on Sx (T ). Note: In (2), the ultrapower case of theLos condition is exactly the same as (diag).

13 Assignment of X -valued functions to models

In application settings it can be useful to consider formulas that are allowed to have values in a compact Hausdorff space X . This can be easily implemented in the current framework. Let T be an L-theory; fix a tuple of distinct variables x = x1,..., xn and let X be a compact Hausdorff space. Denote by F any operation that assigns a function M x F : M → X to each model M of T . We will call such an F an X -valued T -function. Note that the conditions (iso) and (diag) make sense for X -valued T -functions F. For any such F, (iso) and (diag) hold iff there exists a function ∆F : Sx (T ) → X such that for every M  T and x M a ∈ M we have F (a) = ∆F(tpM (a)). As before, ∆F is unique, when it exists.

14 Theorem Let F be an X -valued T -function. The following are equivalent: (1) For every continuous function f : X → [0, 1] there exists a (necessarily [0, 1]-valued) T -formula ϕ(x) such that f ◦ F = Fϕ. (2) F satisfies (iso) and (diag), and the Los condition N M F ([ai ]) = limi→U F i (ai ) Q holds for every ultraproduct N = Mi /U of models of T and x every [ai ] ∈ N .

(3) F satisfies (iso) and (diag), and ∆F is continuous on Sx (T ).

In (2), the ultrafilter limit limi→U is taken in the compact space X . The X -valued T -functions satisfying the three equivalent conditions in this theorem will be called X -valued T -predicates. Condition (2) is best suited to testing whether a mathematically natural T -function is in fact a T -predicate; condition (3) is best suited to proving closure properties of the class of T -predicates.

15 Assignment of sets to models

Let T be an L-theory; fix a tuple of distinct variables x = x1,..., xn. We will denote by X any operation that assigns a subset x X(M) ⊆ M to each model M of T . We refer to such an X as a T -assignment of sets.

For example, if ϕ(x) is a T -formula, we consider Xϕ defined by x M Xϕ(M) := {a ∈ M | ϕ (a) = 0}. Such an operation will be referred to as a T -zeroset.

16 Functorial properties of set assignments

We note the following properties of T -zerosets X = Xϕ: (iso) If M, N are models of T and J is an isomorphism of M onto N, then J(X(M)) = X(N). (diag) If M is a model of T and U is an ultrafilter, with D : M → MU the diagonal embedding, then x D(X(M)) = D(M ) ∩ X(MU ). Fact: a T -assignment X satisfies (iso) and (diag) if and only if there is a set τ(X) ⊆ Sx (T ) such that for any model M of T x X(M) = {a ∈ M | tpM (a) ∈ τ(X)}. Note: Properties of X that we discuss later correspond to properties of τ(X) in the topometric space Sx (T ).

17 Proposition Suppose X is a T -assignment of sets and that X satisfies (iso) and (diag). Let M, N be models of T . Then for any elementary embedding J : M → N we have x J(X(M)) = J(M ) ∩ X(N).

Proof. Apply the Keisler-Shelah Theorem to the expansions (M, ha | a ∈ Mi) and (N, hJ(a) | a ∈ Mi). This gives an ultrafilter U and an isomorphism Ψ from MU onto NU which satisfies Ψ ◦ DM = DN ◦ J, where DM , DN are the diagonal embeddings.

18 Further, we note a third basic property of T -zerosets X = Xϕ: x X is closed if X(M) is a closed subset of M for all models M of T . Fact Suppose X is a T -assignment of sets that satisfies (iso) and (diag), so there exists a subset τ(X) ⊆ Sx (T ) that defines X (as on a previous slide). Then

(1) X is closed if and only if τ(X) is a (metrically) closed subset of Sx (T ). (2) define Y by Y(M) = the metric closure of X(M) for all models M of T . Then Y is a closed T -assignment satisfying (iso) and (diag). Moreover, τ(Y) is the metric closure of τ(X) in Sx (T ).

19 Characterization of T -zerosets

Let T be an L-theory and let x = x1,..., xn be a tuple of distinct variables. Let X be a T -assignment of sets. Theorem The following are equivalent: (1) X is a T -zeroset. Q (2) X satisfies (iso) and for every ultraproduct Mi /U of models of T we have Q Q (containment) X(Mi )/U⊆ X( Mi /U) (hence X satisfies (diag) and is closed), and (Gδ) τ(X) is a Gδ in Sx (T ). (3) X satisfies (iso) and (diag), and τ(X) is a closed Gδ in Sx (T ).

Note: If L is countable, then the Gδ conditions in (2) and (3) are automatic.

20 We let Met1 be the category of all complete metric spaces of diameter ≤ 1, with morphisms being isometric embeddings. Also, Mod(T ) is the category of models of T with morphisms being elementary embeddings.

Note: When a T -assignment of sets X is closed and satisfies (iso) and (diag), we may regard X as a functor from Mod(T ) to Met1: for objects M in Mod(T ), this functor assigns X(M) to M. For M, N ∈ Mod(T ) and a morphism f : M → N, we take X(f ) to be the restriction of f to X(M), which maps into X(N).

Therefore, a T -assignment of sets X that is closed and satisfies (iso) and (diag) is called a T -functor.

21 T -definable sets

Let T be an L-theory, and let x = x1,..., xn be a tuple of distinct variables. Let X be a T -functor. Definition We say X is T -definable if the class of T -predicates is closed under the quantifiers supx∈X(M) and infx∈X(M).

So, when X is T -definable, we can relativize quantifiers to X(M) while staying within the expressive power of the signature L applied to models of T . Among other things, this can help giving explicit axioms for T . Note: If X is T -definable, then it is a T -zeroset. Indeed, X(M) is closed in Mx and M dist(a, X(M)) := infy∈X(M) d (a, y).

22 Characterizations of T -definable sets

Let T be an L-theory, and let x = x1,..., xn be a tuple of distinct variables. Let X be a T -functor. Theorem The following are equivalent:

(1) X is T -definable. Q Q (2) (equality) X(Mi )/U = X( Mi /U) Q holds for every ultraproduct Mi /U of models of T .

(3) The function dist(·, τ(X)): Sx (T ) → [0, 1] is continuous. M (4) The T -function P(x) defined by P (a) := dist(a, X(M)) for all models M of T and all a ∈ Mx is a T -predicate.

Note: if X is a T -assignment of sets satisfying (iso) and (equality), then X is automatically a T -functor, and hence it is T -definable.

23 Let T be an L-theory, and let x = x1,..., xn be a tuple of distinct variables. Let X be a T -zeroset. Definition A T -predicate P with zeroset X has the almost-near property if for all  > 0 there exists δ > 0 such that for all models M of T and all x M a ∈ M , we have P (a) < δ implies dist(a, X(M)) ≤ .

Corollary The following are equivalent: (1) X is T -definable. (2) some T -predicate P(x) with X as its zeroset has the almost-near property. (3) every T -predicate P(x) with X as its zeroset has the almost-near property.

24 A framework for applications

Frequently in applications we have a class C of L-structures (closed under isomorphism) that we want to investigate using model theory. We let T = Th(C) and need to study the models of T . Let C∗ ⊆ Mod(T ) be the class of all structures isomorphic to ultraproducts of families from C. We suppose a good understanding of the structures in C∗, but not much knowledge about the full class of models of T .

Facts C∗ is closed under ultraproducts. every model M of T is elementarily equivalent to a member of ∗ ∗ C , and therefore MU is in C for some ultrafilter U.

whether or not (ϕn) defines a T -formula can be verified by considering only members of C.

25 Let X be a compact Hausdorff space. Suppose we have an assignment of X -valued functions F that is only defined for models in C∗. Fact ∗ If F satisfies (iso) and (diag) on C , then there is a function ∆F : Sx (T ) → X such that M F (a) = ∆F(tpM (a)) for all M in C∗ and all a ∈ Mx . Consequently, F has a unique T -function extension to Mod(T ) that satisfies (iso) and (diag) on all models of T . Indeed, we may simply use the preceding equation as a definition.

Fact Suppose F is a T -function satisfying (iso) and (diag) on all models of T . Then, to verify the Loscondition (from a preceding theorem) for F on all models of T , it suffices to verify the condition on all ∗ models in C , and this will imply the continuity of ∆F on Sx (T ).

26 Now suppose we have a uniform assignment of sets X that is only defined for models in C∗. Fact ∗ If X satisfies (iso) and (diag) on C , then there is a set τ(X) ⊆ Sx (T ) such that x X(M) = {a ∈ M | tpM (a) ∈ τ(X)} for all M in C∗. Consequently, X has a unique extension to Mod(T ) that satisfies (iso) and (diag) on all models of T . (Apply the preceding equation as a definition for the other models of T )

Fact Suppose X is a T -assignment of sets satisfying (iso) and (diag) on all models of T . Then, to verify any of the conditions closed or containment, or Gδ, or equality (from preceding theorems) for X on all models of T , it suffices to verify the condition on all models in C∗.

27 Using this framework: Lp(Lq)-Banach lattices

Fix p 6= q in [0, ∞) and let Cpq be the class of Banach lattices of 0 0 0 the form Lp(Lq) := Lp((Σ, B, µ); Lq(Σ , B , µ )), which is the space 0 0 0 of Lq(Σ , B , µ )-valued functions f on Σ such that f is p µ-measurable (in the sense of Bochner) and (kf kq) has finite µ-integral.

In a 1991 monograph, Haydon, Levy, and Raynaud showed that ∗ Cpq is not closed under ultraproducts, and that Cpq is exactly the class of Banach lattices that are bands in members of Cpq.

∗ In 2007, CWH and Raynaud showed that the class Cpq is ∗ axiomatizable. In 2011 they showed that Th(Cpq) has a model companion (with quantifier elimination) and identified its models.

28 Further, let Bpq be the class of Banach space reducts of members ∗ of Cpq. Obviously Bpq is the class of Banach space reducts of ∗ members of Cpq.

∗ In 2018 Raynaud showed that for certain pairs (p, q), the class Bpq is axiomatizable, However, for certain pairs (for example, p = 1 < q < ∞) the question of axiomatizability is open.

And most of the more “advanced” model theoretic questions (stability, etc) about these classes are open.

29 Definability: Example 1

Let C be the class of pointed metric spaces (M, c) of diameter ≤ 1 in which the metric d is an ultrametric (i.e., d satisfies d(x, z) ≤ max(d(x, y), d(y, z))). Let T be the theory of C.

Fix r in the interval 0 < r < 1 and for every (M, c) in C let X(M) = {a ∈ M | d(c, a) ≤ r}.

Obviously X is a T -zeroset. However, X is not T -definable.

Proof: Let (M, c) be a member of C in which there exist elements (an) such that limn d(c, an) = r and r < d(c, an) for all n ∈ N. Let U be a nonprincipal ultrafilter on N. The element a = (an)U of the ultrapower (M, c)U satisfies d(c, a) = r. However, for any element b = (bn)U with d(c, bn) ≤ r for all n, the ultrametric inequality implies d(an, bn) ≥ r for all n, and thus d(a, b) ≥ r.

30 Example 2

Here we consider the theory Pr of probability measure algebras.

Given a model M of Pr we define X(M) = {a ∈ M | a is an atom or 0} and show that X is Pr-definable.

Proof: We need to check that Q Q X(Mi )/U = X( Mi /U) Q holds for every ultraproduct Mi /U of models of Pr.

⊆: It is easy to show by contradiction that if ei is an atom or 0 in Q Mi for all i, then (e)U is an atom or 0 in Mi /U.

⊇: (next slide)

31 Q ⊇: Suppose a = (ai )U ∈ M = Mi /U is an atom and let r = µ(a) > 0. For each i ∈ I , let ei be the atom of largest measure contained in ai , or ei = 0 if ai is atomless. By the first part of the Q proof, either e = (ei )U is an atom in Mi /U or it is 0.

Q We have e ≤ a. Therefore, if e is an atom in Mi /U, we must Q have a = e ∈ X(Mi )/U as needed.

It remains to show that e = 0 is impossible. Otherwise, the set S = {i ∈ I | µ(ei ) ≤ r/2} is in U. For each i ∈ S we may write ai = bi ∪ ci where bi ∩ ci = 0 and µ(ci ) ≤ µ(bi ) ≤ µ(ci ) + r/2. (Put ei ≤ bi and alternate the other atoms ≤ ai between bi and ci and split the atomless part of ai in two equal parts.)

Hence a = (bi )U ∪ (ci )U and (bi )U ∩ (ci )U = 0, while µ((ci )U ) ≤ µ((bi )U ) ≤ µ((ci )U ) + r/2 = µ((ci )U ) + µ(a)/2, which contradicts the assumption that (a)U ) is an atom.

32 Example 3

This example comes from work in progress with Yves Raynaud, in which we are constructing new families of uncountably categorical Banach spaces. These spaces are asymptotically hilbertian in the following strong sense.

Definition A Banach space E is asymptotically 2-hilbertian if every ultrapower of E has the form EU = D(E) ⊕2 H for some Hilbert space H ⊆ EU , where D is the diagonal embedding of E into EU .

Here we only have time to discuss the extent to which the Hilbert part of EU can be definably identified.

33 Let Tb denote a theory that axiomatizes the class of all unit balls of Banach spaces over R. For each such Banach space E, let B(E) denote the unit ball of E, considered as a model of Tb.

For each a ∈ B(E) we let hai denote the linear subspace of E generated by a. We call hai a 2-summand of E if there is a subspace F of E such that E = F ⊕2 hai; note that if this equation holds, then F is uniquely determined by a. Note further that if E = F ⊕2 H and H is a Hilbert space, then hai is a 2-summand of E for every a ∈ H.

For each model B(E) of Tb, we define X(B(E)) := {a ∈ B(E) | hai is a 2-summand of E}.

Proposition

The Tb-assignment X is a Tb-zeroset but it is not Tb-definable.

34 To prove that X is a Tb-zeroset, we verify the conditions in the ultraproduct characterization of zerosets. Condition (iso) is obvious. The ultraproduct containment condition holds because the ⊕2 direct sum operation commutes with the ultraproduct construction. The Gδ condition holds since the signature is countable.

To prove that X is not Tb-definable, consider E = ⊕2(En | n ∈ N), where each En is polyhedral and 2-dimensional, with the unit sphere of En approximating the unit circle more and more closely as n → ∞. Then X(B(E)) = {0}, whereas E is asymptotically 2-hilbertian, so X(B(EU )) will be very large. Thus the ultraproduct equality condition fails to hold.

35 Example 4

Let L be a countable signature and T an L-theory. Recall that we say T is ω-categorical if T has a unique separable model, which is not compact. Proposition The following are equivalent: (1) T is ω-categorical. (2) Every T -zeroset is T -definable.

Proof. x For each model M0 of T and a0 ∈ M0 , let X be defined on models M of T by x X(M) := {a ∈ M | (M, a) ≡ (M0, a0)}. Obviously X is a T -zeroset. The omitting types theorem for continuous logic implies that X is T -definable iff X(M) 6= ∅ for all models M of T . The characterization of separable categoricity then implies the equivalence of (1) and (2). 36