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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Fri, 8 Nov 2024 11:18:47 +0100
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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