  We investigate which definable separable metric spaces are countable dense
homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and
give a complete list of zero-dimensional Borel CDH spaces. We also show that
for a Borel X subset of 2^omega the following are equivalent: (1) X is G_delta
in 2^omega, (2) X^omega is CDH and (3) X^omega is homeomorphic to 2^omega or to
omega^omega. Assuming the Axiom of Projective Determinacy the results extend to
all projective sets and under the Axiom of Determinacy to all separable metric
spaces. In particular, modulo large cardinal assumption it is relatively
consistent with ZF that all CDH separable metric spaces are completely
metrizable. We also answer a question of Steprans and Zhou by showing that the
cardinal p = min{kappa: 2^kappa is not CDH}.
