Definability of groups in $\aleph_0$-stable metric structures

Itaï Ben Yaacov (1)

1 ICJ - Institut Camille Jordan

We prove that in a continuous $\aleph_0$-stable theory every type-definable group is definable. The two main ingredients in the proof are: \begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from \cite{BenYaacov:TopometricSpacesAndPerturbations}, allowing us to prove the theorem in case the metric is invariant under the group action; and \item Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones. \end{enumerate} We conclude this paper with a development of basic stability theory for continuous logic.

Domaines: Logique [math.LO]

Itaï Ben Yaacov. Definability of groups in $\aleph_0$-stable metric structures. 2008. ⟨hal-00259318v1⟩