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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Sun, 10 Nov 2024 12:36:09 -0800
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On 11/10/2024 1:35 AM, WM wrote:
> On 10.11.2024 00:27, Jim Burns wrote:
>> On 11/9/2024 6:45 AM, WM wrote:
> 
>>> Everybody who believes that the intervals
>>> I(n) = [n - 1/10, n + 1/10]
>>> could grow in length or number
>>> to cover the whole real axis
>>> is a fool or worse.
>>
>> Our sets do not change.
>>
>> The set
>>    {[n-⅒,n+⅒]: n∈ℕ⁺}
>> with the midpoints at
>>    ⟨ 1, 2, 3, 4, 5, ... ⟩
>> does not _change_ to the set
>>    {[iₙ/jₙ-⅒,iₙ/jₙ+⅒]: n∈ℕ⁺}
>> with the midpoints at
>>    ⟨ 1/1, 1/2, 2/1, 1/3, 2/2, ... ⟩
> 
> It cannot do so because the reality of the rationals is much larger than 
> the reality of the naturals.[...]

Cantor pairing can create a unique pair of natural numbers from a single 
natural number. Why do think of rationals at all!? Sigh.