This re-instated what passes for faith in humanity

[Thecla]

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?
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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.

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.

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.)

The even numbers, the square numbers etc. can all be put in one-to-one correspondence with the natural numbers, so they all have the same cardinality, even though the even numbers -- say -- are a proper subset of all numbers. (In that sense, one infinite set that is `larger' than another -- in the sense that it strictly contains it -- is the same infinity -- i.e. has the same cardinality -- as the `smaller' set.)

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.