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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Tue, 25 Feb 2025 10:55:35 +0100
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On 25.02.2025 10:17, joes wrote:
> Am Tue, 25 Feb 2025 10:02:47 +0100 schrieb WM:
>> On 24.02.2025 19:01, joes wrote:
>>> Am Mon, 24 Feb 2025 15:59:18 +0100 schrieb WM:
>>
>>>> These axioms can be applied to show that all FISONs can be removed.
>>> Nope. Any number of them can be removed. Indeed, removing all changes
>>> the union, as you have admitted.
>> There are not more than any number.
> There are more FISONs than any natural number: infinitely many.

Yes potential infinity. There are more FISONs than any fixed natural 
number, but the number of FISONs is a natural number because the sequence
{1}
{2, 1}
{3, 2, 1}
....
has no limit. Hint: That would not be a FISON. Further induction 
produces no actual infinity.
> 
>>>> Only all FISONs = natural numbers are the matter of my proof.
>>>> According to Zermelo they make up the set ℕ.
>>> The set N is not a natural number
>> Of course. Nobody said so.
> You pretend to remove the whole set.

I remove by induction all natural numbers. What of ℕ remains in your 
opinion?
>> But Zermelo claims to produce the set ℕ. Is he wrong?
>> Hint: In fact he produces ℕ_def.
> No, you are wrong.

You claimed yourself that induction produces only a finite number of 
elements. Zermelo used induction.

Regards, WM
>