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

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Coskey S., Lupini M.

Fundamenta Mathematicae

2016

tom 234

nr 1

55-72

We show that a version of López-Escobar's theorem holds in the setting of model theory for metric structures. More precisely, let 𝕌 denote the Urysohn sphere and let Mod(𝓛,𝕌) be the space of metric 𝓛-structures supported on 𝕌. Then for any Iso(𝕌)-invariant Borel function f: Mod(𝓛,𝕌) → [0,1], there exists a sentence ϕ of $𝓛_{ω₁ω}$ such that for all M ∈ Mod(𝓛,𝕌) 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 $𝓛_{ω₁ω}$-sentence that are supported on the Urysohn sphere. This in turn provides a model-theoretic reformulation of the topological Vaught conjecture.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Coskey S.

1 Lupini M.

1 2016

Wyniki wyszukiwania

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