Randomizations of models as metric structures

Itaï Ben Yaacov (1) , Jérôme Keisler (2)

1 ICJ - Institut Camille Jordan
2 Department of Mathematics [Madison]

The paper \cite{Ke} introduced the notion of a \textbf{randomization} of a first order structure $\CM$. The idea was to form a new structure whose elements are random elements of $\CM$. In this paper we treat randomizations as continuous structures in the sense of the paper \cite{BU}. In this setting, the results of \cite{Ke} show that if $T$ is the complete theory of $\CM$, the theory $T^R$ of randomizations of $\CM$ is a complete theory in continuous logic which admits elimination of quantifiers and has a natural set of axioms. We show that $T^R$ is $\omega$-categorical, $\omega$-stable or stable as a continuous theory if and only if $T$ is $\omega$-categorical, $\omega$-stable or stable as a first order theory.

Mots clés: randomization; continuous logic; metric structures

Domaines: Logique [math.LO]

Itaï Ben Yaacov, Jérôme Keisler. Randomizations of models as metric structures. 2009. ⟨hal-00351833v1⟩