Separable Models of Randomizations

Every complete first order theory has a corresponding complete theory in continuous logic, called the randomization theory. It has two sorts, a sort for random elements of models of the first order theory, and a sort for events. In this paper we establish connections between properties of countable models of a first order theory and corresponding properties of separable models of the randomization theory. We show that the randomization theory has a prime model if and only if the first order theory has a prime model. And the randomization theory has the same number of separable homogeneous models as the first order theory has countable homogeneous models. We also show that when T has at most countably many countable models, each separable model of T is uniquely characterized by a probability density function on the set of isomorphism types of countable models of T . This yields an analogue for randomizations of the results of Baldwin and Lachlan on countable models of ω1-categorical first order theories.