  Using Baire category techniques we prove that Araki-Woods factors are not
classifiable by countable structures. As a result, we obtain a far reaching
strengthening as well as a new proof of the well-known theorem of Woods that
the isomorphism problem for ITPFI factors is not smooth. We derive as a
consequence that the odometer actions of Z that preserve the measure class of a
finite non-atomic product measure are not classifiable up to orbit equivalence
by countable structures.
