There are several different definitions for comparing infinite quantities, and I've used one of them.
If I defined all the terms, my article would be twice as long (and it's too long as it is.) Whoever wants precise technical terms is welcome to go on Wikipedia.
The standard intuitive explanation seems pretty easy and worthwhile:
Two sets are the same size if and only if you can pair members up.
{1, 2, 3} and {4, 5, 6}, obviously.
{1, 3, 5...} and {2, 4, 6...} are too, you can pair them up (1, 2) (3, 4) (5,6) ...
But some sets aren't the same size; you can't pair them up. It's easy if they're both finite, or one is infinite. If both are infinite, well, if they're not the same size, one has to be larger: there are elements in one that are "left over" after trying to pair them up. That's the bigger one. Cantor proved that this does, in fact, happen. The reals can't be paired up with integers.
Thanks. I don't know if this is the right place to get into this, but I still find that explanation aggravating. Perhaps (probably) if I went and read Cantor's original work, I would be satisfied. But perhaps you can save me the trouble.
Here is my problem. You actually can't pair up the reals with the integers, simply because the reals cannot be enumerated. If you take two sets whose members actually can be enumerated, you can pair them up up long as you want.
To reiterate: I'm claiming that the "pairing" operation is not defined validly here, because it would rely on enumerating the elements of both sets, which you cannot do for the reals. Since the proposed definition of "bigger" rests on the "pairing" operation, that definition doesn't appear to be valid to me.
If I defined all the terms, my article would be twice as long (and it's too long as it is.) Whoever wants precise technical terms is welcome to go on Wikipedia.