What is aleph null times aleph one?

Question

Could you shed some light on this?

I am guessing it is aleph one, since one cannot pair every element of naturals with its subsets.

Answer 1

If you assume the axiom of choice, the product of any two infinite cardinals is equal to the larger of the two, so yes it is $\aleph_1$.

Answer 2

We can prove that $\aleph_1\cdot\aleph_1=\aleph_1$ by defining a well-order on $\omega_1\times\omega_1$ which is order isomorphic to $\omega_1$. Now we have that: $$\aleph_1\leq\aleph_1\cdot\aleph_0\leq\aleph_1\cdot\aleph_1=\aleph_1$$

Therefore equality is all around.

Note in your argument that $\aleph_1$ need not be the cardinality of $\mathcal P(\Bbb N)$. This assumption is known as The Continuum Hypothesis and was shown to be neither provable nor refutable from the standard axioms of set theory.