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From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: The set of necessary FISONs
Date: Sun, 2 Feb 2025 04:55:15 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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Am Sat, 01 Feb 2025 19:23:29 +0100 schrieb WM:
> On 01.02.2025 15:07, joes wrote:
>> Am Sat, 01 Feb 2025 13:46:50 +0100 schrieb WM:
>>> On 31.01.2025 18:51, joes wrote:
>>>> Am Fri, 31 Jan 2025 13:28:35 +0100 schrieb WM:
>>>
>>>>> then all can be discarded,
>>>> No, only finitely consecutive ones.
>>> Induction concerns all natural numbers n as well as all A(n).
>> ω, corresponding to „all”, is not natural.
> ω describes the order type but is not the set ℕ but greater than all
> natnumbers.
Induction proves a sentence for every number, not for the set.
You cannot extend a sentence about numbers to sets.

>>>>> Removing all leaves nothing, in particular no sufficient set for
>>>>> U(F(n)) = ℕ.
>>>> It is obvious that N is not empty.
>>> But the set claimed to have the union ℕ gets empty without changing
>>> its union.
>> Wrong. Finite sets of FISONs do not result in N.
> Induction covers all natural numbers. Otherwise it would not be
> sufficient in the Peano axioms.
It covers the elements of N, not the set itself.

-- 
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.