Banach spaces and groups - order properties and universal models

Abstract

We deal with two natural examples of almost-elementary classes: the class of all Banach spaces (over R or C) and the class of all groups. We show both of these classes do not have the strict order property, and find the exact place of each one of them in Shelah's SOP_n (strong order property of order n) hierarchy. Remembering the connection between this hierarchy and the existence of universal models, we conclude, for example, that there are ``few'' universal Banach spaces (under isometry) of regular cardinalities.