@MISC{Farah_omittingtypes, author = {Ilijas Farah and Menachem Magidor}, title = {OMITTING TYPES IN LOGIC OF METRIC STRUCTURES}, year = {} }

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Abstract

Abstract. The present paper is about omitting types in logic of met-ric structures introduced by Ben Yaacov, Berenstein, Henson and Usvy-atsov. While a complete type is omissible in a model of a complete theory if and only if it is not principal, this is not true for incomplete types by a result of Ben Yaacov. We prove that there is no simple test for determining whether a type is omissible in a model of a theory T in a separable language. More precisely, we find a theory in a separable language such that the set of types omissible in some of its models is a complete Σ12 set and a complete theory in a separable language such that the set of types omissible in some of its models is a complete Π11 set. We also construct (i) a complete theory T and a countable set of types such that each of its finite sets is jointly omissible in a model of T, but the whole set is not and (ii) a complete theory and two types that are separately omissible, but not jointly omissible, in its models. The Omitting Types Theorem is one of the most useful methods for con-structing models of first-order theories with prescribed properties (see [26], [23], or any general text on model theory). It implies, among other facts, the following. (1) If T is a theory in a countable language, then the set of all n-types realized in every model of T is Borel in the logic topology on Sn(T). (2) If T is moreover complete, then any sequence tn, for n ∈ ω, of types each of which can be omitted in a model of T can be simultaneously omitted in a model of T. We note that the types tn in (2) are not required to be complete, but the theory T is. While in classical logic the criterion for a given type to be omissible in a model of a given theory applies regardless of whether the type is com-plete or not, the situation in logic of metric structures is a bit more subtle. The omitting types theorem ([3, §12] or [22, Lecture 4]) has the following straightforward consequences (see Proposition 1.7 for a proof of (3) and Corollary 3.7 for a proof of (4)). (3) If T is a theory in a separable language of logic of metric structures, then the set of all complete n-types realized in every model of T is Borel in the logic topology on Sn(T).