Definable closure in randomizations

TL;DR

This paper characterizes when elements in the randomization of a first-order theory are definable over a set of parameters, providing a precise criterion within the framework of continuous logic.

Contribution

It establishes necessary and sufficient conditions for definability in the randomization of a first-order theory, advancing understanding of definability in continuous model theory.

Findings

Provides a criterion for definability in T^R

Clarifies the structure of definable elements in randomizations

Enhances the theoretical foundation of continuous logic

Abstract

The randomization of a complete first order theory T is the complete continuous theory T^R with two sorts, a sort for random elements of models of T, and a sort for events in an underlying probability space. We give necessary and sufficient conditions for an element to be definable over a set of parameters in a model of T^R.