Randomizations of models as metric structures

TL;DR

This paper explores the properties of randomizations of first order models within continuous logic, demonstrating that key model-theoretic properties are preserved under this operation.

Contribution

It extends Keisler's concept of model randomization to continuous structures, showing preservation of stability, omega-categoricity, and omega-stability.

Findings

Randomization preserves stability properties.

Randomization results in a complete continuous theory.

The theory admits elimination of quantifiers.

Abstract

The notion of a randomization of a first order structure was introduced by Keisler in the paper Randomizing a Model, Advances in Math. 1999. The idea was to form a new structure whose elements are random elements of the original first order structure. In this paper we treat randomizations as continuous structures in the sense of Ben Yaacov and Usvyatsov. In this setting, the earlier results show that the randomization of a complete first order theory is a complete theory in continuous logic that admits elimination of quantifiers and has a natural set of axioms. We show that the randomization operation preserves the properties of being omega-categorical, omega-stable, and stable.