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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Sat, 9 Nov 2024 22:30:47 +0100
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On 09.11.2024 15:03, Mikko wrote:
> On 2024-11-08 16:30:23 +0000, WM said:

>>
>> If Cantors enumeration of the rationals is complete, then all rationals
>> are in the sequence 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 
>> 1/5, 2/4, 3/3, 4/2, 5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1,  ... and none 
>> is outside.
> 
> All positive rationals quite obviously are in the sequence. Non-positive
> rationals are not.
> 
>> Therefore also irrational numbers cannot be there.
> 
> That is equally obvious.
> 
>> Of course this is wrong.
> 
> You may call it wrong but that's the way they are.

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 rational are outside. 
That implies that Cantor's above sequence does not contain all rationals.
> 
>> It proves that not all rational numbers are countable and in the 
>> sequence.
> 
> Calling a truth wrong does not prove anything.
> 
Proving that when Cantor is true the real axis has measure 3 proves that 
Cantor is wrong.

Regards, WM