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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: Thu, 21 Nov 2024 11:21:40 +0100
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On 21.11.2024 10:16, Mikko wrote:
> On 2024-11-20 11:42:15 +0000, WM said:

>> The intervals before and after shifting are not different. Only their 
>> positions are.
> 
> The intervals are different. A shifted interval contains a different
> set of numbers.

Consider this simplified argument. Let every unit interval after a 
natural number n which is divisible by 10 be coloured black: (10n, 
10n+1]. All others are white. Is it possible to shift the black 
intervals so that the whole real axis becomes black?

No. Although there are infinitely many black intervals, the white 
intervals will remain in the majority. For every finite distance (0, 
10n)  the relative covering is precisely 1/10, whether or not the 
intervals have been moved or remain at their original sites. That means 
the function decribing this, 1/10, 1/10, 1/10, ... has limit 1/10. That 
is the quotient of the infinity of black intervals and the infinity of 
all intervals.

Regards, WM