Omitting types in logic of metric structures

Abstract

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 $\Sigma^1_2$ set and a complete theory in a countable language such that the set of types omissible in some of its models is a complete $\Pi^1_1$ 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.