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SEPARABLE MODELS OF RANDOMIZATIONS

Published online by Cambridge University Press:  22 December 2015

URI ANDREWS  and
H. JEROME KEISLER
URI ANDREWS
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN-MADISON MADISON, WI, 53706, USAE-mail: andrews@math.wisc.eduURL: http://www.math.wisc.edu/∼andrews
H. JEROME KEISLER
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN-MADISON MADISON, WI, 53706, USAE-mail: keisler@math.wisc.eduURL: http://www.math.wisc.edu/∼keisler
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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 TR 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.

Keywords

Primary 03C1503B50Secondary 03C2003C40continuous model theorymetric structurerandomization
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 4 , December 2015 , pp. 1149 - 1181
Copyright
Copyright © The Association for Symbolic Logic 2015 

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