This is yet another response that has grown long enough to be its own post. Jared, over at Double Edged has been wondering how some infinite sets can be larger than others. For context, you'll have to go over there and read his post, Infinity Plus One? Among other comments, mine is the one dated May28, 5:52 AM. His response is four down from that.
I believe that Jared is begging the question here. That is, his argument against the idea that infinite sets can be different from one another in the number of elements each contains is a simple assertion to the contrary. Infinity equals infinity. If it is true that infinity equals infinity, then the cardinalities of all infinite sets must be equal. However, if it is the case that not all the cardinalities of infinite sets are equal, then it cannot be the case that infinity always equals infinity.
Go back to the cardinalities of the set of counting numbers and the set of even counting numbers. In any given finite set, no matter how large or small, the cardinality of the even set will always be half that of the evens and odds put together (+ or - 1 depending on where you start and end). When you make both of these sets infinite, though, the ratio changes from 1:2 to 1:1. Suppose now that someone didn't think so. You might say that infinity always equals infinity. He would answer that past experience with all those finite sets demonstrates that for any given set of counting numbers the proper subset of even numbers is always half as much; therefore, there are different kinds of infinities. The intuition on either side is just as strong so, to change his position, you're going to need an argument. This will be found in the fact that it is possible to map the proper subset of even numbers onto the set of all counting numbers in a one to one relatinoship. In other words, the mere notion of infinity is not sufficient to establish the comparitive cardinalities of infinite sets. Another step is needed; namely, whether or not the elements of a given infinite set can be put in a one to one correspondence with the set of all counting numbers.
It is the case that any two sets have the same cardinality iff their elements have a one on one connection. The question now is whether it is always the case that any two infinite sets always have this connection. The answer is that it is not. Once all of the counting numbers have been paired with different numbers from the set of all real numbers, there are still real numbers with no place to go.
As to the objection that adding a number to an infinite set does not increase the cardinality, you're right, it doesn't. But that's not what's happening here. The method that both Chuck and I demonstrated reveals what was there all along but was somehow missed. It adds nothing. Consider again the set of even numbers vs. counting numbers. They have an equal cardinality. If I now include the number 7 with the set of even numbers, the cardinality does not change. I have redifined the set and am able to map this set in its entirety onto the set of counting numbers with nothing left over on either side. This is not the case when considering the set of all real numbers. Here, there is no redefinition of the set.
If I had defined the set as "all real numbers except .3582000..." and then later added this number to the set, this would constitute a redefinition of the set. The cardinality thereof would not change. And it would tell me nothing about the comparitive cardinalities between this set and the set of all counting numbers. However, if I define the set as "all real numbers without exception" and then map these onto the set of counting numbers, then the only numbers that I can add to the set are imaginary, such as the square root of negative one. If I did this, the cardinality of the redefined set would not change. On the other hand, if, after pairing each number in the set of all counting numbers with a unique number in the set of all real numbers, I am able to construct a real number that is not already paired to a counting number, I must conclude that the cardinality of the set of all real numbers is larger than the cardinality of the set of all counting numbers. I have added nothing to the set nor have I redefined it. Instead, I have discovered a number that, though in the set all along, managed to escape being mapped onto the counting numbers. It follows, by definition, that this number is left over. "Left over" in this context, has nothing to do with adding anything to an infinite set. Such an addition would have no effect on the cardinality. It simply means that, unlike other infinite sets, not all elements in the set of real numbers can be put in a unique pair with each of the elements in the set of counting numbers.