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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.

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    Do you know what the transitive closure of a set is?2012-09-18
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    It's a set which is countable, all its elements are countable, all the elements of its elements are countable, and so on. An example of a countable set which is not hereditarily countable is $\{\mathbb{R}\}$.2012-09-18
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    You didn’t read Yuval’s comment carefully enough: the set in question is $\{\Bbb R\}$, not $\Bbb R$. $\Bbb R$ is obviously not hereditarily countable, since it’s not even countable. $\{\Bbb R\}$ is countable, since it has only one element, the set $\Bbb R$, but since that element is not itself a countable set, $\{\Bbb R\}$ is not **hereditarily** countable.2012-09-18

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