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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Mon, 17 Feb 2025 20:36:45 +0100
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On 16.02.2025 23:43, Jim Burns wrote:
> On 2/15/2025 9:51 AM, WM wrote:
>> On 14.02.2025 19:06, Jim Burns wrote:
> 
>>> The set of all natural numbers is not.in
>>> the only inductive subset of the set of
>>> all natural numbers.
>>
>> The set of all natural numbers is constructed
>> by induction.
> 
> By axiom "infinity",
> an inductive set exists.

The axiom applies induction.

> If the set with an only.inductive.subset
> was an element, we'd prove the set has P.
> But the set isn't an element.
> We don't prove the set has P.

Sometimes this is right, sometimes it is wrong.
When all elements of a set are subject to induction then the set is an 
inductive set.
When all elements of a set are removed, then the set is removed.

Example:
If every human has an end, then the human race need not have an end.
If every human has ended, then the human race has ended.

Regards, WM