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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Fri, 3 Jan 2025 09:39:01 +0100
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On 02.01.2025 19:19, Jim Burns wrote:
> On 1/2/2025 6:31 AM, WM wrote:
>> On 31.12.2024 21:26, Jim Burns wrote:
>>> On 12/31/2024 1:20 PM, WM wrote:
> 
>>>> Every union of FISONs which
>>>> stay below a certain threshold
>>>> stays below that threshold.
>>>> Note: Every FISON stays below 1 % of ℕ.
>>>
>>> For each finite ⟦0,j⦆, there are
>>> more.than.#⟦0,j⦆.many finite ⟦0,k⦆
>>
>> So it is for all definable natural numbers.
> 
> So it is for each finite.ordinal.

That is impossible, because all finite ordinals can be subtracted from ℕ 
without infinitely many remaining. If you don't understand that, it is 
useless to go on.

>> natural numbers which have only few successors
>> because when they are removed nothing remains
>> ℕ \ {1, 2, 3, ...} = { } .
> 
> ... the non.finite.ordinal natural.numbers.
> 
> We define natural.numbers to be
> only finite.ordinals, of which
> there are more.than.any.finite.many.

Infinitely many can be removed without remainder. But only finitely many 
can be defined by FISONs.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
ℕ \ {1, 2, 3, ...} = { }
Di you understand that?

> If you want to know how Bob disappears,

I know that he never disappears because he is swapped only inside the 
matrix.

Regards, WM