Omitting types in logic of metric structures

Ilijas Farah, Menachem Magidor

Published 2014-11-11Version 1

The present paper is about omitting types in logic of metric structures introduced by Ben Yaacov, Berenestein, Henson and Usvyatsov. 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 $\bfT$ 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 $\bSigma^1_2$ set and a complete theory in a separable language such that the set of types omissible in some of its models is a complete $\bPi^1_1$ set. We also construct a complete theory and countably many types that are separately omissible, but not jointly omissible, in its models.