[joshzayin]

Here's perhaps an easier to understand proof.

Let A be a countable set. Now, suppose that there was a surjective function (one that hits every element of P(A)) f: A -> P(A). (Note that f takes elements of A, and produces subsets of A.)

Now, define Y ⊆ A as follows: For all x in A, x is in Y if and only if x is not in f(x).

Thus, Y, which is in P(A), is distinct from every output of f, and so f is not actually surjective.

This means that no surjective function f: A -> P(A) exists.

This is a generalization of Cantor's Diagonal Argument (http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument).