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

Samuel Coskey¹ ; Martino Lupini²

¹ Department of Mathematics Boise State University 1910 University Dr. Boise, ID 83725-1555, U.S.A.
² Fakultät für Mathematik Universität Wien Oskar-Morgenstern-Platz 1, Room 02.126 1090 Wien, Austria

Fundamenta Mathematicae, Tome 234 (2016) no. 1, pp. 55-72

Résumé

We show that a version of López-Escobar’s theorem holds in the setting of model theory for metric structures. More precisely, let $\mathbb{U}$ denote the Urysohn sphere and let $\operatorname{Mod}(\mathcal{L},\mathbb{U})$ be the space of metric $\mathcal{L}$-structures supported on $\mathbb{U}$. Then for any $\operatorname{Iso}(\mathbb{U})$-invariant Borel function $f\colon \operatorname{Mod}(\mathcal{L}, \mathbb{U})\rightarrow \lbrack 0,1]$, there exists a sentence $\phi $ of $% \mathcal{L}_{\omega _{1}\omega }$ such that for all $M\in \operatorname{Mod}(\mathcal{L},% \mathbb{U})$ 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.

Détail

DOI : 10.4064/fm135-1-2016

Keywords: version pez escobar theorem holds setting model theory metric structures precisely mathbb denote urysohn sphere operatorname mod mathcal mathbb space metric mathcal structures supported mathbb operatorname iso mathbb invariant borel function colon operatorname mod mathcal mathbb rightarrow lbrack there exists sentence phi mathcal omega omega operatorname mod mathcal mathbb have phi answers question ivanov majcher iwanow prove several consequences example every orbit equivalence relation polish group action borel isomorphic isomorphism relation set models given mathcal omega omega sentence supported urysohn sphere turn provides model theoretic reformulation topological vaught conjecture