A López-Escobar theorem for metric structures, and the topological Vaught conjecture
Volume 234 / 2016
Abstract
We show that a version of López-Escobar’s theorem holds in the setting of model theory for metric structures. More precisely, let $\mathbb{U}$ denote the Urysohn sphere and let $\operatorname{Mod}(\mathcal{L},\mathbb{U})$ be the space of metric $\mathcal{L}$-structures supported on $\mathbb{U}$. Then for any $\operatorname{Iso}(\mathbb{U})$-invariant Borel function $f\colon \operatorname{Mod}(\mathcal{L}, \mathbb{U})\rightarrow \lbrack 0,1]$, there exists a sentence $\phi $ of $% \mathcal{L}_{\omega _{1}\omega }$ such that for all $M\in \operatorname{Mod}(\mathcal{L},% \mathbb{U})$ we have $f(M)=\phi ^{M}$. 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.
