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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Wed, 12 Feb 2025 11:19:55 +0100
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On 12.02.2025 10:46, joes wrote:
> Am Tue, 11 Feb 2025 20:23:28 +0100 schrieb WM:
>> On 11.02.2025 18:42, Jim Burns wrote:
>>> On 2/11/2025 4:31 AM, WM wrote:
>>>> On 10.02.2025 16:16, Jim Burns wrote:
>>
>>>> The set F of FISONs which can be removed without changing the assumed
>>>> result UF = ℕ is the infinite set F of all FISONs.
>>> Yes.
>> Fine.
> Actually, no: you can not remove the whole set, only single elements.

Induction proves all elements and even the set (Zermelo, Z).

> The erroneous step is from „every finite number” to „an infinite number”.

Induction proves an infinite number.

>> Induction covers the whole infinite
>> set.
> No, it only covers the elements.

All elements. None remains. Mithilfe der Definition über induktive 
Mengen lässt sich die Beweismethode der vollständigen Induktion 
rechtfertigen (daher auch der Name induktiv): Soll gezeigt werden, dass 
alle natürlichen Zahlen eine bestimmte Eigenschaft haben, so betrachte 
die Menge E ... [Wikipedia]
> 
>>> How induction works is not well known to you (WM).
>> What is wrong in my application in your opinion?
> You incorrectly include infinity.

No, there is no infinite FISON. All are finite. All can be omitted 
without changing the result.

Regards, WM