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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: Mon, 11 Nov 2024 13:15:56 +0200
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On 2024-11-10 10:54:02 +0000, WM said:

> On 10.11.2024 11:20, Mikko wrote:
>> On 2024-11-09 21:30:47 +0000, WM said:
>> 
>>> 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.
>> 
>> Maybe, maybe not, depending on what is all n.
> 
> It is, as usual, all natural numbers.

The measure of the interval J(n) is √2/5, which is roghly 0,28.
The measure of the set of all those intervals is infinite.
Between the intervals J(n) and (Jn+1) there are infinitely many rational
and irrational numbers but no hatural numbers.

-- 
Mikko