SEPARABLE MODELS OF RANDOMIZATIONS. (22nd December 2015)

• Record Type: Journal Article
• Authors: ANDREWS, URI; KEISLER, H. JEROME
• Is Part Of: Journal of symbolic logic. Volume 80:Number 4(2015)
• Page Start: 1149
• Page End: 1181
• Publication Date: 2015-12-22
• Subjects: Primary 03C15, -- 03B50, -- Secondary 03C20, -- 03C40; continuous model theory, -- metric structure, -- randomization; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/jsl.2015.33
• Languages: English
• ISSNs: 0022-4812

Abstract

Abstract: 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 R 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 ω 1 -categorical first order theories.