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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Fri, 21 Feb 2025 10:32:07 +0100
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On 20.02.2025 21:24, Jim Burns wrote:
> On 2/20/2025 2:27 PM, WM wrote:

> Are you denying that {i:j+1≤i≤k+1} exists?
> 
> Are you denying that {i:j+1≤i≤k+1} is
> larger than {i:i≤k} ?

Induction covers all infinitely many elements.
Induction proves that no FISON is changing the assumed union ℕ.
> 
>>>> That contradicts the assumption.
>>>
>>> What is the assumption?
>>
>> The assumption is the existence of S.
>>
>>> What is the contradiction?
>>
>> The contradiction is that
>> induction proves every FISON useless
> 
> Each FISON being uselessᵂᴹ (not.last)

Useless is every FISON smaller than ℕ whether or not being last. (Note 
there is no last FISON.)

> is
> a consequence of ⋃S
> being the only.inductive.subset of ⋃S
> And vice versa.

Whatever
> 
> What are the TWO statements which
> contradict each other?

UF = ℕ and UF = ℕ ==> Ø = ℕ.

Regards, WM