Why should $|2^\mathbb{N}|>|\mathbb{N}^2|$ be true? I've been thinking a bit about infinite things lately, and this question I had wondered about came back to me.
One of the classic expository demonstrations of Cantor's work is the two equally surprising facts that there are as many rationals as natural numbers, but there are more reals than natural numbers. This can be reduced to a statement in cardinal arithmetic, namely that
$$|2^\mathbb{N}|>|\mathbb{N}^2|$$
Now, we can think of these sets as two collections of functions, $2^\mathbb{N}=\{f:\mathbb{N}\to \{0,1\}\}$ and $\mathbb{N}^2=\{f:\{0,1\}\to\mathbb{N}\}$. So it seems that what this statement (and others like it) is saying, is that if you want more functions, you're better off having a large domain than a large codomain.
Is there an intuitive explanation for why this should be true? 

This answer that I wrote explains why $\mathbb N^2$ is countably infinite.
One way to prove that $2^{\mathbb N}$ is uncountably infinite is by Cantor's diagonal argument, which I think has been iterated in stackexchange many times.
