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 j i 2 I ) are L-structures, U an ultrafilter on I . Q We let N0 = i2I Mi as a set; its members are (a) = (ai j i 2 I ), ai 2 Mi . We interpret the symbols on N0: N M F 0 (ai j i 2 I );::: = F i (ai ;:::) j i 2 I 2 N0 N M P 0 (ai j i 2 I );::: = limi!U P i (ai ;:::) 2 [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 := i2I Mi =U. The image of (a) 2 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 '(x; y;:::) and elements Q (a)U ; (b)U ;::: 2 N = Mi =U: N Mi ' (a)U ; (b)U ;::: = lim ' (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 j a 2 Ai) ≡ (N; hf (a) j a 2 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 = fM j M is an L-structure and σM = 0 for all σ 2 T g. 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 '(x) is a sequence ('n(x)) of L-formulas such M that 'n (a) converges as n ! 1, uniformly for all models M of T and all a 2 Mx . M The interpretation ' of a T -formula '(x) = ('n(x)) in a model M of T is defined by M M x ' (a) := limn 'n (a) for all a 2 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 '(x), we let F' denote the M M T -function defined by F := ' ; that is: M M F' (a) := ' (a). 11 We note two properties of T -predicates F = F': (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 '(x) such that F = F'. (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 ] 2 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 2 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 '(x) such that f ◦ F = F'. (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 ] 2 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 '(x) is a T -formula, we consider X' defined by x M X'(M) := fa 2 M j ' (a) = 0g.