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Paper   IPM / M / 16722
School of Mathematics
  Title:   \Aleph0-categorical Banach spaces contain \EllP or C0
  Author(s):  Karim Khanaki
  Status:   To Appear
  Journal: Math. Logic Quart.
  Supported by:  IPM
This paper has three parts. First, we establish some of the basic model theoretic facts about T, the Tsirelson space of Figiel and Johnson [FJ74]. 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 MT , has the non independence property (NIP); (2) every @0-categorical space is stable if and only if it is strongly separable; consequently T is not aleph-categorical; equivalently, its first-order theory (in any countable language) does not characterize T, up to isometric isomorphism; (3) every explicitly definable space contains isometric copies of c_0 or `_p.

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