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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Sat, 9 Nov 2024 13:45:28 -0800
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On 11/9/2024 1:42 PM, WM wrote:
> 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. 

Cantor pairing can handle any natural number and convert into a unique 
pair. Not very large... Infinite indeed. Cantor pairing can handle any 
natural number.



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