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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Thu, 13 Mar 2025 17:43:08 +0100
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On 13.03.2025 17:27, Ben Bacarisse wrote:

> WMaths does (apparently) have one result that is not a theorem of modern
> mathematics.  In WMaths there sets P and E such that
> 
>    E in P   and   P \ {E} = P

That is caused by potential infinity. The sets or better collections are 
not fixed.
> 
> WM himself called this a "surprise" but unfortunately he has never been
> able to offer a proof.

Much more surprising is the idea that all natural numbers can be 
subtracted from ℕ with nothing remaining
ℕ \ {1, 2, 3, ...} = { }.
But when we explicitly subtract only numbers which have ℵo remainders
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
then by magic spell also all remainders vanish.
Otherwise U(FISONs) = ℕ  ==>  Ø = ℕ.

Regards, WM