The Infinities of Cantor & Dedekind

One way if which you can defend (or explain) the reality of infinite numbers is the way Richard Dedekind did.

It's claimed that an infinite set has the “property” of having a proper subset that is as large as itself. Now how does that generate numerical infinity? Well, if the infinite set itself has a subset which is equal to itself, that in itself seems to imply some kind of infinity. After all, wouldn't you expect an infinite set to include subsets which are themselves infinite in nature?...

I'm not sure if that's the argument or reality though. I suppose it could mean that if the infinite set is truly infinite in nature, then it must have room (as it were) for a subset which is itself infinite. That is, if a set is infinitely large, it has space (as it were) for something which is also infinitely large.

Nonetheless, how can infinities be compared in such a way? How can one infinity be compared to another infinity? More to the point: how can we use such spatial metaphors as “as large as itself” when we compare an infinite set to one of its subsets? Can infinite sets be compared at all – such as the set of natural numbers and the set of even numbers?

Well, in a sense (or even not in a sense) they were compared in this way by George Cantor. He compared one infinite set with another via what's called the “diagonal argument”. That is, he compared the set of real numbers with the set of natural numbers. More specifically, he matched each number from the set of real numbers diagonally (in terms of how he represented it graphically) with one number from the set of natural numbers. At each end of the diagonal line a real number is matched with a natural number.

Actually, Cantor showed that the set of real numbers is larger than the set of natural numbers. Nonetheless, as I said earlier, how can this be the case when we're talking about rival infinite sets? How can one infinity be larger or smaller (or even equal) to another infinite set? More specifically, why did the set of real numbers win in this battle against the set of natural numbers? In other words, why, exactly, are there more real numbers than natural numbers?

Wittgenstein, for one, rejected Cantor's infinite sets. As some kind of constructivist, he thought that Cantor had simply magicked them out of a hat. In other words, Wittgenstein believed that they aren't real or actual.