I've gotten myself in trouble by posting the reaction on a blog with few readers, since I broke the continuity of this thread. I don't want to break the continuity of that one, and I think the exposition is going to descend into discussion soon because I've nearly exhausted my understanding of constructivism, so since you (eist Hardocco) are the only one responding, should we move the discussion into the comments section of the blog?
I've addressed your first point.
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Originally posted by DoctorScribblesI am having a lot or trouble digesting the above quote.
Since there are an infinite number of integers, our list will be infinite.
However, if I can demonstrate that there is a real number that cannot occur in the right-hand side of the list (that is, that for any way you order the reals in correspondence with the integers, there will be a real missing from your list), I have shown that there are in fact more reals between 0 and 1 than there are integers.
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 words: the number of integers between 0 and 1 = infinite minus (at least) 1 and therefor not infinite.
And now I am confused because the number of integers between 0 and 1 = infinite but also not infinite.
I would appreciate it if the good Doctor or Royalchicken could explain to me where my reasoning went wrong, thus I can make a better understanding of "infinite".
Thanks in advance.
Originally posted by NicolaiSYes, 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.
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.
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.