We investigate enumerability properties for classes of random reals which
permit recursive approximations from below. For four classical notions of
randomness (Martin-L\"of randomness, computable randomness, Schnorr randomness,
and Kurtz randomness), as well as for bi-immunity, we detail whether the
left-recursive enumerable members can be enumerated, and similarly for the
complementary left-r.e. classes. We prove a general equivalence between
arithmetic complexity and existence of numberings for classes of left-r.e.
reals and give optimal arithmetic hardness results.