Edit: I think I should rather ask a different question to dismantle my confusion.
So, why is $\mathbb{R}$ not hereditarily countable set? It seems obvious, but I just can't seem to think of any possible proof... does this just rely on the fact that the set cannot be well-ordered in its original order?
Edit: Right. Sorry for asking a stupid question. Just a slight moment of confusion, forgive me.