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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Sun, 9 Feb 2025 11:59:48 +0100
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On 09.02.2025 01:00, Jim Burns wrote:
> On 2/8/2025 4:54 PM, WM wrote:

>> The axiom of induction says:
>> If any property or predicate P satisfies
>> (P(1) /\ ∀k(P(k) ==> P(k+1)),
>> then it describes all elements of an inductive = infinite set.
> 
> Not for all inductive sets.
> For all minimal.inductive sets.

May be called so.
> 
> minimal.inductive ≠ inductive ≠ infinite

Then you are wrong. Every inductive set is infinite.
> 
>> That is satisfied by the set M of all FISONs which are useless in 
>> U(A(n)) = ℕ.
> 
> Without exception,
> the union of FISONs.after is ℕ

No.

>> Therefore U(F(n)) = ℕ ==>
>> U{ } = { } = ℕ.
> 
> Why that '==>' ?

 From the assumption U(F(n)) = ℕ I have derived that { } = ℕ.
> 
> My best guess at why you claim  U{ } = { } = ℕ
> is that you (WM) are assuming that,
> for some FISON (ie, F(ω-1)) such that
> there are no FISONs.after.

No.
> 
> You (WM) haven't given any other reason.

If U(F(n)) = ℕ, then F(1) can be omitted without changing the result. If 
F(k) can be omitted, then F(k+1) can be omitted too. The set of FISONs 
which can be omitted is an inductive set, i.e., all FISONs.

Regards, WM