  One of the earliest invariants introduced in the study of finite von Neumann
algebras is the property Gamma of Murray and von Neumann. In this note we prove
that it is not possible to classify separable $\rm{II}_1$ factors satisfying
the property Gamma up to isomorphism by a Borel measurable assignment of
countable structures as invariants. We also show that the same holds true for
the full $\rm{II}_1$ factors.
