What you do to compare sizes of sets A and B is, construct a 1:1 function mapping everything in A to something in B and vice versa.
If such a function exists, they're the same cardinality.
It's a little long-winded, but see:
http://en.wikipedia.org/wiki/Cardinal_number
This idea, which seems obvious only in retrospect, is due to Georg Cantor (the guy with the set, and the paradox).
There is a famous proof about this by Georg Cantor: http://en.wikipedia.org/wiki/Cantors_diagonal_argument
I really love this quote from Wittgenstein:
"Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one".
http://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_the...
That's treating infinite as if it's a limit, when it's not.
If it's uncountable and unmeasurable, than it's non hierarchical.
That's not the assumption, that's the result.
That Naturals are infinite but countable and Reals infinite and uncountable.