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Randomizations of models as metric structures. (English) Zbl 1185.03068

Summary: The notion of a randomization of a first-order structure was introduced by H. J. Keisler in the paper [“Randomizing a model”, Adv. Math. 143, No. 1, 124–158 (1999; Zbl 0924.03065)]. 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.

MSC:

03C90 Nonclassical models (Boolean-valued, sheaf, etc.)
03B50 Many-valued logic
03C35 Categoricity and completeness of theories
03C45 Classification theory, stability, and related concepts in model theory

Citations:

Zbl 0924.03065
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