  In 1976 S. Shelah posed the following problem: for which variety V of
algebras the automorphism group of any free algebra F from V of "large"
infinite rank interprets by means of first-order logic set theory (according to
his results, for every variety V the endomorphism semi-group of F interprets
set theory if rank(F) is an infinite cardinal greater than the power of the
language of V). There are examples of varieties for which the answer is
negative; one such an example, the variety of all algebras in empty language,
is due to Shelah (1973). The author earlier showed that the answer is positive
for any variety of vector spaces over a fixed division ring. In the present
paper it is proved that the same holds for the variety of all groups: the
automorphism group of any infinitely generated free group F interprets set
theory. It follows, in particular, that the group Aut(F) is as undecidable as
possible.
