> By infinitely divided, I just mean that on say (0,1), > since we can find infinitely many rationals on that > interval, we actually consider all of them. The > argument that says there are more irrationals than > rationals seems to somehow both allow for and deny > this fact at the same time. I would argue on said > infinitely divided interval, if you were to examine > two adjacent rationals, the distance between them > would be an aleph-naught length decimal expansion of > zeros ending in a one, Here's your basic error. An "aleph-nought length decimal expansion" does NOT have an "ending".
> which I am not sure could be > divided again. This runs counter to popular > intuition: > > "It turns out that, in some sense, the real numbers > would still look like a line under infinite > magnification, but the rational numbers would be dots > separated by spaces. But that is only a vague and > intuitive statement, not anything precise that we can > use in proofs." (excerpted from your first link, end > of second paragraph under "Getting rid of the > pictures") > > This is what I mean by saying I think you can only > argue for that intuition, that picture, if you are > not considering all of the rationals on a given > interval, hence not allowing it to actually be the > case that you can find an infinite number of > rationals between two rationals. In this way, the > list that Cantor finds a contradiction with is > incomplete, because he has stopped the infinite > division of the rationals. No, Cantor's proof has nothing to do with "stopping the infinite division of the rationals". > > Thanks a lot for those links Angela :) I especially > like the first one, and the stuff on infinitesimals > and nonstandard analysis. Thanks a lot.