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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: Sat, 9 Nov 2024 22:42:01 +0100
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On 08.11.2024 21:36, Moebius wrote:
> Am 08.11.2024 um 21:35 schrieb Moebius:

>>> What is the measure you are using and what does it give for the real
>>> axis?
>>
>> Ob Du es nochmal schaffst, auf diesen saudummen Scheißdreck NICHT zu 
>> antworten?
> 
> Ich kann es jedenfalls nicht mehr sehen/lesen.

Kein Wunder, weil es Deinen starken Glauben erschüttert.

Use the intervals I(n) = [n - sqrt(2)/2^n, n + sqrt(2)/2^n]. Since n and 
q_n can be in bijection, these intervals are sufficient to cover all 
q_n. That means by clever reordering them you can cover the whole 
positive axis.

The measure of all intervals J(n) = [n - √2/10, n + √2/10] is smaller 
than 3. If no irrationals are outside, then nothing is outside, then the 
measure of the real axis is smaller than 3. That is wrong. Therefore 
there are irrationals outside. That implies that rationals are outside. 
That implies that Cantor's above sequence does not contain all rationals.

And an even more suggestive approximation:
Replace the I(n) by intervals J(n) = [n - 1/10, n + 1/10].
These intervals (without splitting or modifying them) can be reordered, 
to cover the whole positive axis except boundaries.

But note that Cantor's bijection between naturals and rationals does not 
insert any non-natural number into ℕ. It confirms only that both sets 
are very large. Therefore also the above sequence of intervals keeps the 
same intervals and the same reality. And the same density 1/5 for every 
finite interval and therefore also in the limit.

Regards, WM