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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)
Date: Sat, 9 Nov 2024 16:03:30 +0200
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On 2024-11-08 16:30:23 +0000, WM said:

> On 08.11.2024 14:09, Mikko wrote:
>> On 2024-11-07 13:21:42 +0000, WM said:
>> 
>>> On 07.11.2024 10:22, Mikko wrote:
>>>> On 2024-11-06 17:55:15 +0000, WM said:
>>>> 
>>>>> On 06.11.2024 16:04, Mikko wrote:
>>>>>> On 2024-11-06 10:01:21 +0000, WM said:
>>>>> 
>>>>>>> I leave ε = 1. No shrinking. Every point outside of the intervals is 
>>>>>>> nearer to an endpoint than to the contents.
>>>>>> 
>>>>>> This discussion started with message that clearly discussed limits when
>>>>>> ε approaches 0. The case ε = 1 was only about a specific unimportant
>>>>>> question.
>>>>> 
>>>>> When ε approaches 0 then the measure of the real axis is, according to 
>>>>> Cantor's results, 0. That shows that his results are wrong.
>>>> 
>>>> It is not the measure of the real axis but the set of rationals. The
>>>> real axis more than just the rationals. The irrationals are also a
>>>> part of the real axis.
>>> 
>>> But not between irrational points.
>> 
>> Real axis contains both real and irrational numbers and nothing else.
>> Between any two points of the real axis there are both rational and
>> irrational points.
> 
> 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.

> It proves that not all rational numbers are countable and in the sequence.

Calling a truth wrong does not prove anything.

-- 
Mikko