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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: Wed, 13 Nov 2024 17:31:54 +0100
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On 13.11.2024 10:08, Jim Burns wrote:
> On 11/12/2024 4:38 PM, WM wrote:

>>>> 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.
> 
> Are there more, though?
> Or are there fewer?
> i/j ↦ (i+j-1)(i+j-1)+2⋅i
> 
> ⟨ 1/1 1/2 2/1 1/3 2/2 3/1 1/4 2/3 ... ⟩
> ↦
> ⟨ 2   4   6   8   10  12  14  16 ... ⟩

or

 > ⟨ 1/1 1/2 2/1 1/3 2/2 3/1 1/4 2/3 ... ⟩
 > ↦
 > ⟨ 2   3   5   7   11  13  17  19 ... ⟩
> 

or

 > ⟨ 1/1        1/2       2/1  ... ⟩
 > ↦
 > ⟨ 10^10   10^10^10  10^10^10^10  ... ⟩

> Or do infinite sets have different rules
> than finite sets do?

If infinite sets obey the rules sketched above, then set theorists must 
discard geometry
because by shifting intervals the relative covering 1/5 of ℝ+ becomes oo*ℝ,
and analysis
because the constant sequence 1/5, 1/5, 1/5, ... has limit oo,
and logic
because of Bob.

Regards, WM