Re: The set of necessary FISONs

Am Thu, 13 Feb 2025 14:51:44 +0100 schrieb WM:

On 13.02.2025 13:54, joes wrote:

Am Thu, 13 Feb 2025 12:59:56 +0100 schrieb WM:

 

All of us who know induction know that by induction we have obtained
that all FISONs can be removed without changing the result.

The result has certainly changed from a nonempty set, whatever you
think the union of inf. many FISONs is.

The assumption was UF = ℕ and has not changed by omitting any FISON.

Yes, but it does change when you omit *all*. UF is not empty.

 From the assumption UF = ℕ we have obtained U{ } = { } = ℕ.

With P = „You can leave out A(n) and all preceding FISONs” you get an
infinite number of such sentences, where each leaves infinitely many
FISONs in the union.

Induction covers all natural numbers. None is remaining.

Yes, you can remove any natural number of FISONs, but not all,
because their number is not natural.

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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.