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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Thu, 6 Feb 2025 09:54:04 +0100
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On 06.02.2025 01:46, Jim Burns wrote:
> On 2/5/2025 2:18 PM, WM wrote:

>> The axiom of induction:
>> ∀P( P(1) /\ ∀k(P(k) ==> P(k+1)) ==> ∀n (P(n)))
>>
>> P(1): U(F(n) \ F(1)) = ℕ.
>>
>> P(k): U(F(n) \ {F(1), F(2), ..., F(k)}) = ℕ
>> ==>
>> P(k+1): U(F(n) \ {F(1), F(2), ..., F(k+1)}) = ℕ.
> 
> A description, not a magic spell.
> 
> ...which could also be written...

Why should we use two different writings?
> 
>>>> I can remove all FISONs from U(F(n))
>>>> without changing the claimed union. 
> 
> What you (WM) mean by 'remove all...without changing..."
> is that, by induction,

it is proved that all F(n) can be subtracted like by induction all 
natural numbers can be subtracted from ℕ:

{1} can be subtracted because it is an element of in ℕ.
If {n} has been subtracted, then {n+1} can be subtracted because it is 
an element of in ℕ.

By the axiom of induction the result is the empty set.

Regards, WM