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With Martino Lupini. Fundamenta mathematicae, 2016. (lopez-escobar in journal)

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 $\mathop{\mathrm{Mod}}(\mathcal{L},\mathbb{U})$ be the space of metric $\mathcal{L}$-structures supported on $\mathbb{U}$. Then for any $\mathop{\mathrm{Iso}}(\mathbb{U})$-invariant Borel function $f\colon\mathop{\mathrm{Mod}}(\mathcal{L},\mathbb{U})\rightarrow[0,1]$, there exists a sentence $\phi$ of $\mathcal{L}_{\omega_1\omega}$ such that for all $M\in \mathop{\mathrm{Mod}}(\mathcal{L},\mathbb{U})$ 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.