Hausdorff dimension of metric spaces definable in o-minimal expansions of the real field

Let $R$ be an o-minimal expansion of the real field and $(X,d)$ an $R$-definable metric space. We show that the Hausdorff dimension of $(X,d)$ is an $R$-definable function of its defining parameters, an element of the field of powers of $R$, and is equal to the packing dimension of $(X,d)$. The proof uses a basic topological dichotomy for definable metric spaces due to the second author, and the work of Shiota and the first author on measure theory over nonarchimedean o-minimal structures.