Banach spaces and groups - Order properties and universal models

Saharon Shelah, Alex Usvyatsov

Research output: Contribution to journalArticlepeer-review


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's SOPn (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 density characters.

Original languageEnglish
Pages (from-to)245-270
Number of pages26
JournalIsrael Journal of Mathematics
StatePublished - 2006
Externally publishedYes

Bibliographical note

Funding Information:
In this paper we investigate two natural classes--the class of all Banach spaces (real or complex) and the class of all groups, from the point of view of model theory, more precisely, of Shelah's classification theory for classes of logical structures. Classification theory is an attempt to classify theories/classes of models according to their model-theoretical complexity--number of nonisomorphic models, number of nonisomorphic relatively "nice" models, existence of "nice" (saturated, universal) models, etc. In order to analyze structure of classes, certain syntactical properties were defined, some well-known, like the order property (equivalent to non-stability, see [Sh:c], chapter I) and the tree property (equivalent to non-simplicity, see [Sh93], [KimPil]), some less known, e.g., the * This publication is numbered 789 in the list of publications of Saharon Shelah. The research was supported by The Israel Science Foundation. Received January 30, 2003


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