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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Tue, 12 Nov 2024 22:38:50 +0100
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On 12.11.2024 20:01, Jim Burns wrote:
> On 11/12/2024 11:43 AM, WM wrote:

>> No, the intervals remain constant
>> in size and multitude.
> 
> Intervals which are constant _only_
> in size and multitude
> are not constant absolutely.

They would suffer to cover all rationals completely if Cantor's 
bijection was complete.
> These intervals
> {[n-⅒,n+⅒]: n∈ℕ⁺}
> cover all naturals ℕ⁺  and
> do not cover all fractions ℕ⁺/ℕ⁺

Right.
> 
>> But the rationals are more in the sense that
>> they include all naturals and 1/2.
> 
> These intervals
> {[i/j-⅒,i/j+⅒]: i/j∈ℕ⁺/ℕ⁺}
> cover all fractions ℕ⁺/ℕ⁺

But these are more intervals.
> 
> These intervals
> {[n-⅒,n+⅒]: n∈ℕ⁺}
> and these intervals
> {[i/j-⅒,i/j+⅒]: i/j∈ℕ⁺/ℕ⁺}
> are different intervals.

In particular the second kind of intervals must be more. And this is the 
solution: The identity of the intervals for the geometric covering is 
irrelevant. I will elaborate o this in the next posting.

Regards, WM