  We extend anti-classification results in ergodic theory to the collection of
weakly mixing systems by proving that the isomorphism relation as well as the
Kakutani equivalence relation of weakly mixing invertible measure-preserving
transformations are not Borel sets. This shows in a precise way that
classification of weakly mixing systems up to isomorphism or Kakutani
equivalence is impossible in terms of computable invariants, even with a very
inclusive understanding of ``computability''. We even obtain these
anti-classification results for weakly mixing area-preserving smooth
diffeomorphisms on compact surfaces admitting a non-trivial circle action as
well as real-analytic diffeomorphisms on the $2$-torus.
