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

Samuel Coskey, and Martino Lupini. "A López-Escobar theorem for metric structures, and the topological Vaught conjecture." Fundamenta Mathematicae 234.1 (2016): 55-72. <http://eudml.org/doc/286474>.

@article{SamuelCoskey2016,
abstract = {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 $_\{ω₁ω\}$ such that for all M ∈ Mod(,) we have $f(M) = ϕ^\{M\}$. 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 $_\{ω₁ω\}$-sentence that are supported on the Urysohn sphere. This in turn provides a model-theoretic reformulation of the topological Vaught conjecture.},
author = {Samuel Coskey, Martino Lupini},
journal = {Fundamenta Mathematicae},
keywords = {model theory for metric structures; infinitary logic; Polish group action; Urysohn sphere},
language = {eng},
number = {1},
pages = {55-72},
title = {A López-Escobar theorem for metric structures, and the topological Vaught conjecture},
url = {http://eudml.org/doc/286474},
volume = {234},
year = {2016},
}

TY - JOUR
AU - Samuel Coskey
AU - Martino Lupini
TI - A López-Escobar theorem for metric structures, and the topological Vaught conjecture
JO - Fundamenta Mathematicae
PY - 2016
VL - 234
IS - 1
SP - 55
EP - 72
AB - 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 $_{ω₁ω}$ such that for all M ∈ Mod(,) we have $f(M) = ϕ^{M}$. 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 $_{ω₁ω}$-sentence that are supported on the Urysohn sphere. This in turn provides a model-theoretic reformulation of the topological Vaught conjecture.
LA - eng
KW - model theory for metric structures; infinitary logic; Polish group action; Urysohn sphere
UR - http://eudml.org/doc/286474
ER -