[tsimionescu]

> but why does the presence of a mapping imply anything about the "size" of an infinity?

It is used like this because it corresponds to an intuitive property of size. If I say that set X is larger than set Y, it comes naturally to assume that, if I were to lay out their elements one by one in pairs, at some point I would run out of elements from Y but still have more elements in X.

For example, even without knowing how many fingers I have, I can check whether there are more pebbles on a beach than fingers on my hand by putting a pebble on every finger. If there are no more pebbles and I have free fingers, the size of the set of fingers was actually larger than the set of pebbles.

And while of course I would never finish if I started doing this with the naturals and the rationals, I can still prove that it can be done if given infinite time; but that, given infinite time, when comparing the naturals to the reals in the same way, we would run out of naturals and still have more reals left.