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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Mon, 24 Feb 2025 15:59:18 +0100
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On 23.02.2025 23:03, Jim Burns wrote:
> On 2/23/2025 2:32 PM, WM wrote:
>> On 23.02.2025 19:34, Jim Burns wrote:
>>> On 2/23/2025 9:43 AM, WM wrote:
> 
>>>> There is no reason to consider {{F}} at all.
>>>
>>> There is reason, but
>>> only for people wanting to be correct.
>>
>> Peano, Zermelo, or v. Neumann
> 
> ...agree that {{F}} ≠ {F}

and also that 3 ≠ pi.
> 
>> Peano, Zermelo, or v. Neumann create ℕ
> 
> Peano, Zermelo, and v. Neumann assert axioms
> from which the existence of ℕ follows
> in a finite.sequence of not.first.false claims.

These axioms can be applied to show that all FISONs can be removed.
>
>> as well as the set F of all FISONs
>> by induction over the members
>> for use in set theory
>> without being what you erroneously call correct.
> 
> A proof.by.induction shows that
> some set,
> such as the set {x:A(x)} of x such that A(x),
> is inductive.
> 
> The conclusion of a proof.by.induction
> is that {x:A(x)} is the whole set.
> 
> However,
> not just any "whole set" is reliable here.
> It must be a whole set such that
> knowing {x:A(x)} is inductive
> narrows
> which set {x:A(x)} can be
> to one set: that whole set.

The set of finite ordinals after v. Neuman is undoubtedly such a set.

>>>> We omit all F(n) which amounts to remove F.
>>>> Like all natural numbers amount to ℕ (not {ℕ})
> 
> Each natural number is in the domain of ST+F
> ℕ is not in the domain of ST+F

Only all FISONs = natural numbers are the matter of my proof.
According to Zermelo they make up the set ℕ.

Regards, WM