Banach spaces and groups — Order properties and universal models

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 that both of these classes do not have the strict order property, and find the exact place of each one of them in Shelah’sSOP _n (strong order property of ordern) 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 density characters. This publication is numbered 789 in the list of publications of Saharon Shelah. The research was supported by The Israel Science Foundation.