This re-instated what passes for faith in humanity

[wakim]

Actually, the property that a thing is `larger' than itself is pretty much what characterizes the infinite. But one has to say what one means by `larger' or 'smaller' -- and equal -- before that statement makes any sense.
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Forgive my puckishness: but shouldn't we also know what is meant by "a," "thing," "is," "than," and "itself?"

In his classic 19th century work, Cantor said that two sets have the same cardinality (are `equal in number') if they can be put in one-to-one correspondence with each other.  This agrees with what we think of as `equal' for finite numbers: you have the same number of knives and forks if you can pair them up with none left over.
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Granted.

Any set which can be put in one-to-one correspondence with the natural numbers (that can be `counted') has cardinality equal to `aleph-0', the least infinite cardinal number. (It's a not so trivial result -- the Schroder-Bernstein theorem -- that this one-to-one correspondence definition does order the cardinal numbers.)
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There is a least infinite cardinal number? Doesn't this claim present two problems:
1) If aleph-0 is itself finite, then infinity can now be defined as a finite number plus 1, and thus must be itself finite, and not infinite?
2) If aleph-0 is itself infinite, then infinity is not infinite in that it is larger than any number, as here is an infinite number that has a greater?

Regardless, I see from brief browsing that the idea of trans-finite numbers and aleph-null appears to rest upon the continuum hypothesis that is not just un-proved, but considered un-proveable: Answers.com.

Cantor's truly remarkable discovery was that there are infinite sets that have larger cardinality that that of the natural numbers -- for example the set of all real numbers 0 <= x <= 1. His proof is a classic. Suppose you could put all such numbers (express them as decimals, using infinite 9's in cases of ambiguity such as 0.1 = 0.0999...). Then say you count them as

1 --> 0. a_1 a_2 a_3...
2 --> 0. b_1 b_2 b_3...
3 --> 0. c_1 c_2 c_3...
...

Pick a number 0. x_1 x_2 x_3... where x_1 = 7 if a_1 isn't equal to 7, and a_1 = 6 if a_1=7, x_2 = 7 if b_2 isn't equal to 7 and x_2 = 6 if b_2 = 7, and so on. This number differs from the nth number on the list in the nth decimal place, so it isn't anywhere on the list. So it's not possible put all real numbers in one-to-one correspondence with the integers, meaning that the real numbers have greater cardinality than the integers.
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I grant that any continuous interval is infinitely divisible, so that any finite accounting of it must be incomplete (Hello, Zeno); that a decimal listing of the fractions of that interval may be extended infinitely is corollary. But why can't one make a one-to-one correspondence between these fractions and a set of natural numbers, where for every new fraction "discovered" one more is added to the list of natural numbers? For if the natural numbers are infinite, one can always add one more to the largest number counted to bring it into correspondence with every new fraction "discovered." In other words, for every fraction not on the list isn't there a corresponding larger natural number also not yet listed? And, since both categories are infinite, how can one exhaust the one before the other?

edited for spelling