This re-instated what passes for faith in humanity

[Thecla]

Forgive my puckishness: but shouldn't we also know what is meant by "a," "thing," "is," "than," and "itself?"

Then forgive mine for saying that your original comment ("So what is infinite can be larger than what is infinite? You find no contradiction in stating that a thing may be larger than itself?") struck me as showing some lack of understanding about the mathematics of infinity, and in serious need of defining your terms. ;)

There is a least infinite cardinal number?

Yup: it's aleph_0, the cardinality of the natural numbers.

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?

Actually, we only defined when two sets have cardinal numbers that are equal (in one-to-one correspondence), less than or equal (in one-to-one correspondence with a subset) or greater than or equal. It doesn't make sense in general to add cardinal numbers.

There is another different type of infinite number, called the ordinal numbers, which one can add. They look something like this:

1, 2, 3, 4, ... w, w+1, w+2, ... w+w , ....

Here w (omega) is the first infinite ordinal number, which comes immediately after all the natural numbers. Unlike a finite number, it has no immediate predecessor (i.e. if n < w then there is always another ordinal number m such that n < m < w), which is why your argument in 1) doesn't apply to w.

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?

Here you utterly lost me in philosophical pyrotechnics. ;)

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

The concept of cadinal numbers doesn't rest on the continuum hypothesis at all -- quite the reverse, in fact. Once you've given a precise and meaningful definition of the cardinal numbers, you can ask questions about them. In particular, having shown that c = 2^{aleph_0} -- the cardinality of the continuum (the real numbers) -- is strictly greater than the cardinality aleph_0 of the integers, one can ask if there's another cardinal number between these two. The continuum hypothesis is the statement that there isn't.

What Godel and Cohen proved in the mid-20th century is that this probelm can't be answered within the terms of the standard axioms of set theory (assuming -- as pretty much everyone believes -- that these are not self-contradictory). If the axioms of set theory and the continuum hypothesis are consistent, then the axioms of set theory and the negation of the continuum hypothesis are also consistent. It's one -- and perhaps the most stricking -- example of the dilemma between incompleteness and consistency that enters the foundations of mathematics once one introduces the infinite.

But why can't one make a one-to-one correspondence between these fractions and a set of natural numbers

One can put the fractions in one-to-one correspondence with the natural numbers e.g. put all (positive -- which makes no difference) fractions in a square:

1/1 2/1 3/1 4/1 ...
1/2 2/2 3/2 4/2 ...
1/3 2/3 3/3 4/3 ...
1/4 2/4 3/4 4/4 ...
....

and count them starting at the top left corner down diagonals, neglecting repeats (i.e. 1/1 2/1 1/2 1/3 (2/2) 3/1 4/1 3/2 2/3 1/4...)

So the set of fractions has the same cardinality as the set of integers, and smaller cardinality than the set of real numbers.