A López-Escobar theorem for metric structures, and the topological Vaught conjecture

Abstract

Abstract. We show that a version of López-Escobar's theorem holds in the setting of model theory for metric structures. More precisely, let U denote the Urysohn sphere and let Mod(L, U) be the space of metric L-structures supported on U. Then for any Iso(U)-This answers a question of Ivanov and Majcher-Iwanow. We prove several consequences, for example every orbit equivalence relation of a Polish group action is Borel isomorphic to the isomorphism relation on the set of models of a given Lω 1 ω -sentence that are supported on the Urysohn sphere. This in turn provides a model-theoretic reformulation of the topological Vaught conjecture. §1. Background and statement of main result A well-known theorem of López-Escobar [LE] says roughly that every Borel class of countable structures can be axiomatized by a sentence in the logic where countable conjunctions and disjunctions are allowed. The theorem has been generalized to apply to wider classes of structures, using sentences from a variety of logics (see for example [T, V]).To state López-Escobar's theorem more precisely, let L be a countable first-order language consisting of the relational symbols {R i } where each R i has arity n i . The space Mod(L) of countably infinite L-structures is given byand we note it is compact in the product topology. The space carries a natural S ∞ -action by left-translation on each factor, and the S ∞ -orbits are precisely the isomorphism classes. Next, recall that L ω 1 ω denotes the extension of first-order logic in which countable conjunctions and disjunctions are allowed (formulas are still only allowed to have finitely many free variables). If φ is a sentence of L ω 1 ω then the subset Mod(φ) ⊂ Mod(L) consisting just of the models of φ is clearly S ∞ -invariant (isomorphism invariant), and it is easy to see that 2000 Mathematics Subject Classification. Primary 03C95, 03E15; Secondary 54E50.

Polish G-spaces and continuous logic

“…We may interpret this theorem that U is the continuous counterpart of ω in moving from the case of discrete logic S ∞ -actions to the case of actions on spaces of continuous structures. Some other reasons supporting this intuition can be found in [19] and in Section 2 of [14].…”

“…The proof of Theorem 2.2 of the paper [14] contains a construction which is equivalent to the proof of Theorem 1.2 and is based on Section 2.6 of [4]. Theorem 2.2 of [14] complements it by applications of the universality of the isometry group of the Urysohn sphere U and a version of the López-Escobar theorem. As a result the space Y can be replaced by U and the image of M can be taken to be L ω 1 ω -axiomatisable.…”

“…Note that the classical López-Escobar theorem was one of the ingredients of Corollary 1.13 of [3]. We use the version of the López-Escobar theorem for continuous logic obtained by S.Coskey and M.Lupini in [14] . Very recently I.Ben Yaacov, A.Nies and T.Tsankov have proved in [9] another natural continuous version of the López-Escobar theorem.…”

Expressive power of infinitary [0, 1]-logics

“…In a sequence of papers beginning with his thesis [Ort97], Ortiz develops a logic based on Henson's positive bounded formulas and allows infinitary formulas, but also infinite strings of quantifiers. An early version of [CL16] had infinitary formulas in a logic where the quantifiers sup and inf were replaced by category quantifiers.…”

“…An alternative proof of the existence of Scott sentences in L C ω1,ω goes by first proving a metric version of the López-Escobar Theorem, which characterizes the isomorphism-invariant bounded Borel functions on a space of codes for structures as exactly those functions of the form M → σ M for an L C ω1,ωsentence σ. Using this method Scott sentences in L C ω1,ω were found by Coskey and Lupini [CL16] for structures whose underlying metric space is the Urysohn sphere, and such that all of the distinguished functions and predicates share a common modulus of uniform continuity. Shortly thereafter, Ben Yaacov, Nies, and Tsankov obtained the same result for all complete metric structures.…”