Using Baire category techniques we prove that Araki-Woods factors are not
classifiable by countable structures. As a result, we obtain a strengthening
and a new proof of the well-known theorem of Woods that the isomorphism problem
for ITPFI factors is not smooth, as well as a new and more direct proof that
the isomorphism relation for injective type III_0 factors is not classifiable
by countable structures.