Originally posted by NicolaiS
I am having a lot or trouble digesting the above quote.
My non-mathematical brain insists in telling me that infinite = infinite and therfore not countable.
By proving by means of diagonalization proof that there are more reals then integers would inmply, in my understanding that the number of integers between 0 and 1 = not infinite.
In other wo ...[text shortened]... soning went wrong, thus I can make a better understanding of "infinite".
Thanks in advance.
Yes, the number of integers between 0 and 1 is not infinite. Depending on whether you mean "strictly between" or "non-strictly" between, it is either 0 or 2.
The point is that the reals are an infinite set (not caring about countable yet), so they can be put into a one-to-one correspondence with any subset of themselves, like the set of reals between 0 and 1 (in other words, for each real, there is exactly one real between 0 and 1).
What the diagonalisation argument shows is that this subset of the reals cannot be put into one-one correspondence with the integers. Therefore, since the reals can be put in one-one correspondence with this subset, the reals cannot be put into one-one correspondence with the integers.
The whole idea is that we've abstracted what "as many as" means for finite sets: the Wolfpack is a finite set of people, which means that there are, say, n people in the Wolfpack. All this means is that we can put the Wolfpack in one-one correspondence with the set {1, 2, 3, ... n}. The extension to countably infinite sets is an extension of this, so it's more sensible to talk in terms of one-one mappings than it is to talk about there being "as many" reals between 0 and 1 as there are reals, for example.
I get the feeling I haven't addressed your confusion.