Is there a set A s.t. $|\mathcal{P}(A)| = |\mathbb{N}|$?

Question

In other words I am looking for a set that has countably infinite number of subsets.

I have a feeling it doesn't exist, but if so, how do I prove it?

Answer 1

There is not such a set.

If its cardinal is finite then his power set is also finite. And if its cardinal is infinite, therefore at least countable, its power set has at least the continuum for cardinal.