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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Sat, 25 Jan 2025 12:02:39 +0100
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On 24.01.2025 16:44, Jim Burns wrote:
> On 1/24/2025 4:37 AM, WM wrote:
>> On 23.01.2025 16:18, Jim Burns wrote:
> 
> The union of all FISONs covers UF(n)

Simply contradicted by:
∀n ∈ UF(n): |ℕ \ {1, 2, 3, ..., n}| = ℵo
Try to find a counter example. Fail.
> 
> Each FISON is a proper subset of another FISON.
> Each FISON is a proper subset of UF(n)
> No FISON is UF(n)

That is potential infinity.
> 
> Whatever contains each FISON  contains UF(n)

Alas this is not a set but a (potentially in-) finite changing collection.
> 
>> Otherwise Cantor's theorem would require
>> the existence of a first necessary FISON.

Do you know Cantors theorem?
Do you accept Cantors theorem:
"Theorem B: Every embodiment of different numbers of the first and the 
second number class has a smallest number, a minimum."

Do you agree that or every FISON the question whether it is necessary 
can be answered?
>>
>>> Each FISON is a proper subset of ℕ
>>> Each FISON is not ℕ
>>
>> Therefore each FISON can be dropped from
>> the set of candidates.
> 
> Candidates for what office? For UF(n) ?

Candidates for the set of FISONs which are necessary to make UF(n) = ℕ.

>> Up to every FISON
>> |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> 
> For any two FISONs {1,2,...,j} {1,2,...,k}
> their sum {1,2,...,j,j+1,j+2,...,j+k} is a FISON

Therefore the union cannot be larger than a FISON. The infinite union is 
the infinite FISON. But there is no infinite FISON
> ⎛ Consider Bob such that,
> ⎜ before all FISON.end.swaps n⇄n+1
> ⎜ Bob is in the first FISON.end 0
> ⎜
> ⎜ If Bob is in FISON.end n
> ⎜ then
> ⎜ it is after n-1⇄n and before n⇄n+1
> ⎜
> ⎜ If it is after all FISON.end.swaps
> ⎜ then Bob is not.in any FISON.end,
> ⎜ even though
> ⎜ no FISON.end.swap takes Bob
> ⎝ anywhere else.
> 
Swaps cannot eliminate Bob. He remains but i the darkness.

Regards, WM