how to prove that $\#\mathcal{P}(\mathbb{N}) = \#\mathbb{R}$? I'm thinking in associate a subset of $\mathbb{N}$ in a real number by the decimal representation, but i'm failed.
I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these two sets?  I have also learnt that there exists no set whose cardinality is strictly between the natural numbers and the real numbers. Is there a proof of this or at least some intuitiveness behind it?
Let $G = \mathbb{R} / \mathbb{Q}$. Is this an interesting group to study?  Is it isomorphic to any more natural mathematical objects?