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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Wed, 5 Feb 2025 21:29:38 -0500
Organization: i2pn2 (i2pn.org)
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On 2/5/25 2:18 PM, WM wrote:
> On 05.02.2025 19:43, Jim Burns wrote:
> 
>> It might just matter what a natural number, induction,
>> a FISON, and a union are.
> 
> 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.
> 
> Regards, WM

And thus you can claim that no F(n) is "necessary".

Doesn't mean you can't use a set of them to build the set of Natural 
Numbers.

Unless you want to admit that you logic says we can't factor 36, since 
none of its factors are "necessary" in such a factoring.