To summarize that other thread briefly:
Every rational number between 0 and 1 can be re-expressed using a base number (in the decimal system, 10; in the Cantor argument, 2) to form a series using the digits from 0 to n-1 (0 to 9 in the decimal system; in the Cantor argument, 0 and 1) as the numerators over denominators of the form n^m where m is the sequence of natural numbers (beginning with 1, not 0). Of course, in order to allow for every rational number, the number of columns (decimal or binary "places" or "nths") will have to increase without bounds, generating a number of rows (rational numbers) that increases without bounds. Interestringly, since each row represents a unique rational number, and each row can itself be numbered with the natural numbers, we can see that the sequence of rationals from 0 to 1 is in 1-1 correspondence with the sequence of natural numbers.
What Cantor's argument shows (and what I hope we'll get to in Analysis) is that, once this system for systematically representing all rational numbers is created, it turns out that it also allows the representation of numbers that are not rational, by generating new sequences of digits that by construction differ in some way from every possible rational number. But these too are numbers, and if entered in the table allow the creation of more such numbers, and there is no way to put this set of new numbers in 1-1 correspondence with the sequence of natural numbers.
All this can be said without using "infinity," potential or actual, or the pseudoconcept of "the number of natural numbers." One simply identifies a systematic way of representing rational numbers, and the rest follows.
I think Will's difficulty with infinities is (as I said elsewhere) ontological, not mathematical. The Zeno argument that best highlights the difficulty is not Achilles v Tortoise, but the much balder "stade" argument (from the Greek adverb for "standing" or "motionless", with a suggestion of punning on "stadion", the noun for "measure"). Take an arrow shot from a bow. It appears to move smoothly through the air. But at each instant of time, it has an exact (even, theoretically, calculable) position, and an exact measure from end to end. At each instant, then, the arrow simply IS as it is, in a fixed place at a fixed time. But time is simply a sequence of instants. If in no instant does the arrow move, then how can it move over any sequence of instants?
Unlike the Achilles, this does not involve us in summing smaller and smaller fractions of an interval, a mathematical task to which the concept of "limit" seems wholely adequate. Here we are summing genuine zeroes, which always sum to naught.
Where, then, is the motion? Zeno, of course, was arguing precisely that in reality there is none, and the fact that it appears to us to move shows that our perceptual world is not the real world. But there are other ways to approach the paradox. One (not so easy to defend) is to argue that there cannot be an instant of time. But however you pick it up, this paradox is about how things are in the world, not about mathematical entities.