We show that a version of López-Escobar’s theorem holds in the setting of model theory for metric structures. More precisely, let denote the Urysohn sphere and let Mod(,) be the space of metric -structures supported on . Then for any Iso()-invariant Borel function f: Mod(,) → [0,1], there exists a sentence ϕ of such that for all M ∈ Mod(,) we have . 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 -sentence that are supported on the Urysohn sphere. This in turn provides a model-theoretic reformulation of the topological Vaught conjecture.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm135-1-2016,
author = {Samuel Coskey and Martino Lupini},
title = {A L\'opez-Escobar theorem for metric structures, and the topological Vaught conjecture},
journal = {Fundamenta Mathematicae},
volume = {233},
year = {2016},
pages = {55-72},
zbl = {06602781},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm135-1-2016}
}
Samuel Coskey; Martino Lupini. A López-Escobar theorem for metric structures, and the topological Vaught conjecture. Fundamenta Mathematicae, Tome 233 (2016) pp. 55-72. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm135-1-2016/