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

• Autorzy: Samuel Coskey , Martino Lupini
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 03C95: Abstract model theory
• 03E15: Descriptive set theory
• 54E50: Complete metric spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2016
• Tom: 234
• Numer: 1
• Strony: 55-72

Daty

wydano

2016

Twórcy

autor

Samuel Coskey

• Department of Mathematics, Boise State University, 1910 University Dr., Boise, ID 83725-1555, U.S.A.

autor

Martino Lupini

• Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, Room 02.126, 1090 Wien, Austria

Identyfikatory

DOI

10.4064/fm135-1-2016