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
$$|2^\mathbb{N}|&gt;|\mathbb{N}^2|$$

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?
The inequality is made somewhat plausible by the observation that $|\mathbb N^2|$ is almost counting the number of $1$-element and $2$-element sets of natural numbers, while $|2^{\mathbb N}|$ counts all the sets of natural numbers.  ("Almost" refers to the facts that each $2$-element set $\{a,b\}$ is counted twice, as $(a,b)$ and as $(b,a)$.)
There is an interesting paper about this theme that claims that the inequality   $|2^\mathbb{N}|&gt;|\mathbb{N}^2|$ need not be strict! So in stead of $|2^\mathbb{N}|&gt;|\mathbb{N}^2|$ we have $|2^\mathbb{N}|\geq|\mathbb{N}^2|$.

Here is the link: http://www.gauge-institute.org/cantorset/CantorSetP.pdf.