  The isomorphism problem in ergodic theory was formulated by von Neumann in
1932 in his pioneering paper Zur Operatorenmethode in der klassischen Mechanik
(Ann. of Math. (2), 33(3):587--642, 1932). The problem has been solved for some
classes of transformations that have special properties, such as the collection
of transformations with discrete spectrum or Bernoulli shifts. This paper shows
that a general classification is impossible (even in concrete settings) by
showing that the collection $E$ of pairs of ergodic, Lebesgue measure
preserving diffeomorphisms $(S,T)$ of the 2-torus that are isomorphic is a
complete analytic set in the $C^\infty$- topology (and hence not Borel).
