Cardinality of all subsets of cardinality $\aleph_0$ or $\aleph$ in $\mathbb{R}$

Question

What is the cardinality of all subsets of cardinality $\aleph_0$ in $\mathbb{R}$? And of all subsets of cardinality $\aleph$ in $\mathbb{R}$?

Since both are subsets of $P(\mathbb{R})$ , I conclude both have cardinality less or equal to $2^{\aleph}$. How to proceed from here?

Answer 1

Hint. Let $\mathcal P_\kappa(A)$ mean $\{X\subseteq A\mid |X|=\kappa\}$. Then $$f(X)=\{e^x\mid x\in X\} \cup[-1,0]$$ defines an injection $\mathcal P(\mathbb R)\to\mathcal P_\aleph(\mathbb R)$.

Answer 2

HINT: Assuming the axiom of choice, you can choose for each subset an injective function from $\Bbb N$ or $\Bbb R$ whose range is that set. So you only need to count the functions, namely you need to calculate the cardinals $\aleph^{\aleph_0}$ and $\aleph^\aleph$.

[Without the axiom of choice it is consistent that the set of all countable subsets of $\Bbb R$ has a cardinality which is strictly larger than $\aleph^{\aleph_0}$. Strange, but possible.]