Omitting Types in Logic of Metric Structures

This paper is about omitting types in logic of metric structures introduced by Ben Yaacov, Berenstein, Henson and Usvyatsov. While a complete type is omissible in some model of a countable complete theory if and only if it is not principal, this is not true for the 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 countable language. More precisely, we find a theory in a countable language such that the set of types omissible in some of its models is a complete Σ2 set and a complete theory in a countable language such that the set of types omissible in some of its models is a complete Π1 set. Two more unexpected examples are given: (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 constructing models of first-order theories with prescribed properties (see [28], [25], or any general text in model theory). It implies, among other facts, the following (here Sn(T) denotes the space of complete n-types in theory T). (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 in addition 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. Types tn appearing in (2) are not required to be complete, but the theory T is. While the standard omissibility criterion for a given type in some model of a given theory in classical logic applies regardless of whether the type is complete or not, situation in logic of metric structures is a bit more subtle. The omitting types theorem in logic of metric structures ([2, §12] or [23, Lecture 4]) has following straightforward consequences (see Proposition 1.11 for a proof of (3) and Corollary 4.2 for a proof of (4)). (3) If T is a theory in a countable 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).