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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
 (extra-ordinary)
Date: Sun, 5 Jan 2025 14:18:32 -0800
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On 1/5/2025 9:52 AM, WM wrote:
> On 05.01.2025 18:35, Alan Mackenzie wrote:
>> WM <wolfgang.mueckenheim@tha.de> wrote:
> 
>> Maybe Cantor did say this, as a pioneer early on in the development of
>> set theory.  Things have developed since then, and we see that there is
>> nothing to be gained by construing infinite sets as "fixed quantities";
> 
> It is a precondition of set theory.
> 
>> there is no mathematical proof where such a concept makes the slightest
>> difference.
> 
> Every proof in set theory is based on the invariability of sets.
>>
>>> But all finite initial segments of natural numbers FISONs {1, 2, 3,
>>> ..., n} cover less than 1 % of ℕ.
>>
>> That is a thoroughly unmathematical statement.  To talk about 1% of an
>> infinite set is meaningless.
> 
> It is not meaningless but an abbreviation for the fact that 
> multiplication by 100 does not reach or surpass ℕ.
> 
>> Finally, it is
>> wrong, absurdly wrong.  The union of all FISONs _is_ N.
> 
> That is a a dogma of matheologioy disproved by the fact that every union 
> of FISONs which stay below a certain threshold stays below that 
> threshold. Every FISON stays below 1 % of ℕ.
> 
>> No, not a mathematical proof.
> 
> You mathematical "proofs" contradict the simple theorem stated above. 
> Therefore they are invalid.
>>> {1, 2, 3, ..., 100n} is less than ℕ. That means the set of FISONs will
>>> never cover ℕ, nor will its union reach the invariable quantity.
>>
>> No, it doesn't mean that at all.  The set of FISONs does indeed "cover"
>> N, in the sense that their union is equal to N.  A proof of this is
>> trivial, well within the understanding of a school student studying
>> maths.
> 
> My theorem is better understood by school students 

Oh my!

> not yet stultified by 
> set theory.
> 
> Regards, WM
> 
>