RANDOMIZATIONS OF MODELS AS METRIC STRUCTURES

RANDOMIZATIONS OF MODELS AS METRIC STRUCTURES ITAI BEN YAACOV AND H. JEROME KEISLER Abstract. The notion of a randomization of a first order structure was introduced by Keisler in the paper Randomizations of Models, 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. 1. Introduction In this paper we study randomizations of first order structures in the setting of continuous model theory. Intuitively, a randomization of a first order structure M is a new structure whose elements are random elements of M. In probability theory, one often starts with some structure M and studies the properties of random elements of M. In many cases, the random elements of M have properties analogous to those of the original elements of M. With this idea in mind, the paper Keisler [Kei99] introduced the notion of a randomization of a first order theory T as a new many-sorted first order theory. That approach pre-dated the current development of continuous structures in the paper Ben Yaacov and Usvyatsov [BU].

• signature ofm


• sort kn ?


• complete theory


• ilar model-theoretic


• fullness axioms


• continuous structure


• pre-structure


• no quantifiers