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Topic: How does infinitesimal exist?
Replies: 21   Last Post: Jun 7, 2013 12:13 AM

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Posts: 132
Registered: 11/27/12
Re: RE: How does infinitesimal exist?
Posted: Jun 7, 2013 12:13 AM
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> 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.

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