Anatomy of $\mathcal P(\mathbb{N})$

Question

How many proper subsets of $\mathcal P(\mathbb{N})$ there is that have cardinality of $2^{|\mathbb{N}|}$ ?

Answer 1

Since $\mathcal P(\mathbb N)$ is in bijective correspondence with $\mathbb R$, it may be somewhat more intuitive to rephrase the question to

How many proper subsets of $\mathbb R$ have the same cardinality as $\mathbb R$ itself?

On one hand there can be at most $2^{|\mathbb R|}=2^{2^{\aleph_0}}$ of them, because that's how many subsets of any cardinality $\mathbb R$ has.

On the other hand, for every $A\in\mathcal P(\mathbb R)$ you can consider the set $$ \{x\le 0\mid x\in A\} \cup (0,1) \cup \{x>1\mid x-1\in A\} $$ which has cardinality $|\mathbb R|$ (and is proper because it doesn't contain $1$).