What is the cardinality of the set of subgroups of $(\mathbb R, +)$?

First, note that there is a set of size $2^{\aleph_0}$ of real numbers which is linearly independent over $\Bbb Q$, even without the axiom of choice.

Then do the same proof as you did, as it's fine.

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For the first part, Is there any uncountably infinite set that does not generate the reals? is of interest towards a positive answer.