Countable dense homogeneity of definable spaces

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 $X\subseteq 2^{\omega }$ 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 a large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Stepr $\bar {\text{a}}$ns and Zhou, by showing that $\mathfrak{p}= \min \{\kappa : 2^{\kappa }$ is not CDH$\}$.