Why is $\mathbb{R}$ not hereditarily countable set? 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.
 A: First comes the question how do you define the real numbers. In many cases the real numbers are thought of as simply $2^\omega$ or $\omega^\omega$, cardinality-wise there is no difference. However you can also take a set of very large sets, and assume that one is the real numbers.
So there comes a question of natural definition. I will assume that we mean binary $\omega$-sequences. Now what is a sequence? It is a function whose domain is $\omega$ and range included in $\{0,1\}$. It is not too hard to see that this is indeed a hereditarily-countable set.
But what is a hereditarily countable set? It is a set which is countable, all its elements are countable, all their elements are countable, and so on. In short, it is a set whose transitive closure is countable.
However Cantor's theorem tells us that $2^\omega$ has cardinality strictly larger than $\aleph_0$. Therefore the real numbers fail to satisfy this requirement of being a countable set to begin with and thus it is not a hereditarily countable set.
