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

Abstract

We show that a version of López-Escobar's theorem holds in the setting of logic for metric structures. More precisely, let $\mathbb{U}$ denote the Urysohn sphere and let $\mathrm{Mod}(\mathcal{L},\mathbb{U})$ be the space of metric $\mathcal{L}$-structures supported on $\mathbb{U}$. Then for any $\mathrm{Iso}(\mathbb{U})$-invariant Borel function $f:\mathrm{Mod}(\mathcal{L},\mathbb{U})\to [0,1]$, there exists a sentence $\varphi$ of ${\mathcal{L}}_{{\omega }_1\omega }$ such that for all $M\in \mathrm{Mod}(\mathcal{L},\mathbb{U})$ we have $f(M)={\varphi }^M$. At the same time we introduce a variant ${\mathcal{L}}_{{\omega }_1\omega }^{\ast }$ of ${\mathcal{L}}_{{\omega }_1\omega }$ in which the usual quantifiers are replaced with category quantifiers, and establish the analogous theorem for ${\mathcal{L}}_{{\omega }_1\omega }^{\ast }$. 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 ${\mathcal{L}}_{{\omega }_1\omega }$-sentence that are supported on the Urysohn sphere. This in turn provides a model-theoretic reformulation of the topological Vaught conjecture.

17 pages

References in corpus (3)

• Model theoretic forcing in analysis
• The conjugacy problem for the automorphism group of the random graph
• The isomorphism relation for separable C*-algebras

Cited by in corpus (2)

• Conjugacy for homogeneous ordered graphs
• The conjugacy problem for automorphism groups of homogeneous digraphs