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.
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.