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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Wed, 29 Jan 2025 09:34:35 +0100
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On 28.01.2025 18:18, FromTheRafters wrote:
> WM expressed precisely :
>> On 28.01.2025 11:26, FromTheRafters wrote:
>>> WM expressed precisely :
>>
>>>> An infinite set of FISONs that has the union ℕ needs a first element.
>>>
> The only needed one would be the last one but 
> there is no last one just like there is no last natural number.

The only needed one would be a FISON that does not obey
|ℕ \ {1, 2, 3, ..., n}| = ℵo
but
|ℕ \ {1, 2, 3, ..., n}| = 0

But it is not existing. Therefore you claim that an infinite set is 
necessary. But that is pure matheology.

Assume any set of FISONs with uninon U(F(n)) = ℕ. Then every FISON can 
be dropped as completely useless. Nothing remains. Therefore:
IF U(F(n)) = ℕ THEN ℕ = { }.

FISONs can accomplish only a potentially infinite collection.

Regards, WM