  We describe a structure over the complex numbers associated with the
non-commutative algebra Aq called quantum 2-tori. These turn out to have
uncountably categorical L_omega1,omega-theory, and are similar to other
pseudo-analytic structures considered by the second author. The first-order
theory of a quantum torus for generic q interprets arithmetic and so is
unstable and undecidable. But certain interesting reduct of the structure, a
quantum line bundle, is superstable.

Logic