Joel David Hamkins in an answer to my question Countable Dense Sub-Groups of the Reals points out that "one can find an uncountable chain of countable dense additive subgroups of $\mathbb{R}$ whose subset relation has the order type of the continuum $\langle \mathbb{R},<\rangle$."

I would like to know what is the cardinality of the set of countable dense additive subgroups of the reals (up to isomorphism)? Is it undecidable?