  We prove that the relation of bisimilarity between countable labelled
transition systems is $\Sigma_1^1$-complete (hence not Borel), by reducing the
set of non-wellorders over the natural numbers continuously to it.
  This has an impact on the theory of probabilistic and nondeterministic
processes over uncountable spaces, since logical characterizations of
bisimilarity (as, for instance, those based on the unique structure theorem for
analytic spaces) require a countable logic whose formulas have measurable
semantics. Our reduction shows that such a logic does not exist in the case of
image-infinite processes.
