Submission date: 27 August 2016

Abstract:

This paper has three parts. First, we establish some of the basic model theoretic facts about the Tsirelson space of Figiel and Johnson [FJ74], denoted T. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In particular, we show: (1) every strongly separable space, such as T, has the non independence property (NIP); (2) every aleph_0-categorical space is stable if and only if it is strongly separable; consequently T is not aleph_0-categorical; equivalently, its first-order theory (in any countable language) does not characterize T, up to isometric isomorphism; (3) every explicitly de_nable space contains c_0 or ell_p (1 6 p < 1), more precisely, if a Banach space X is aleph_0-categorical in any countable language, then at least one of the following two conditions holds: (a) there exists p in [0,infty) such that for every epsilon>0 there exists a sequence (x_n) which is (1+epsilon)-equivalent to the standard unit basis of ell_p; (b) for every epsilon>0 there exists a sequence (x_n) which is (1 +epsilon)-equivalent to the standard unit basis of c_0.

Mathematics Subject Classification: 46B04, 46B25, 03C45, 03C98

Keywords and phrases: Tsirelson's space, continuous logic, strongly separable, stable space, Rosenthal space, non independence property, aleph_0-categorical, explicitly definable

Full text arXiv 1603.08134: pdf, ps.