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NNTP-Posting-Date: Fri, 08 Nov 2024 16:55:34 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (doubling-spaces)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Fri, 8 Nov 2024 08:55:39 -0800
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On 11/08/2024 02:18 AM, WM wrote:
> On 08.11.2024 00:29, Jim Burns wrote:
>  > On 11/7/2024 2:33 PM, WM wrote:
>
>  >> It is impossible however to cover
>  >> the real axis (even many times) by
>  >> the intervals
>  >> J(n) = [n - 1/10, n + 1/10].
>  >
>  > Those are not the cleverly.re.ordered intervals.
> They are the intervals that we start with.
>  >> No boundaries are involved because
>  >> every interval of length 1/5 contains infinitely many rationals and
>  >> therefore is essentially covered by infinitely many intervals of
>  >> length 1/5
>  >> - if Cantor is right.
>  >
>  > I haven't claimed anything at all about
>  > your all.1/5.length intervals.
> Then consider the two only alternatives: Either by reordering (one after
> the other or simultaneously) the measure of these intervals can grow
> from 1/10 of the real axis to infinitely many times the real axis, or not.
>
> My understanding of mathematics and geometry is that reordering cannot
> increase the measure (only reduce it by overlapping). This is a basic
> axiom which will certainly be agreed to by everybody not conditioned by
> matheology. But there is also an analytical proof: Every reordering of
> any finite set of intervals does not increase their measure. The limit
> of a constant sequence is this constant however.
>
> This geometrical consequence of Cantor's theory has, to my knowledge,
> never been discussed. By the way I got the idea after a posting of
> yours: Each of {...,-3,-2,-1,0,1,2,3,...} is the midpoint of an interval.
>
> Regards, WM
>

Perhaps you've never heard of Vitali's doubling-space,
the Vitali and Hausdorff's what became Banach-Tarski
the equi-decomposability, the doubling in signal theory
according to Shannon and Nyquist, and as with regards to
the quasi-invariant measure theory, where: taking a
continuum apart and putting it back together doubles things.

It's part of continuum mechanics and as with regards to infinity.
(Mathematical infinity.)