
Re: RE: How does infinitesimal exist?
Posted:
May 4, 2013 4:30 PM


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 alephnaught length decimal expansion of zeros ending in a one, 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.
Thanks a lot for those links Angela :) I especially like the first one, and the stuff on infinitesimals and nonstandard analysis. Thanks a lot.

