Re: The existence of dark numbers proved by the thinned out harmonic series.

on 3/17/2025, WM supposed :

On 17.03.2025 11:51, FromTheRafters wrote:

WM presented the following explanation :

On 17.03.2025 02:29, FromTheRafters wrote:

WM brought next idea :

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I informed you several times:
ℕ \ {1, 2, 3, ...} = { }

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When two sets are the same set, the difference set is the empty set.

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When two sets are not the same set, the difference is not empty. This is the case when the set ℕ_def of all numbers which cannot empty ℕ, is subtracted from ℕ
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ℕ \ ℕ_def =/= { }

 That is false if N_def is the same as N

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True.
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, which it is.

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False.
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In fact, every element in a set is "defined" or else you could not find out whether or not it qualifies for set inclusion.

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ℕ is a set which is attained by the natural numbers. "Es ist sogar erlaubt, sich die neugeschaffene Zahl ω als Grenze zu denken, welcher die Zahlen ν zustreben, wenn darunter nichts anderes verstanden wird, als daß ω die erste ganze Zahl sein soll, welche auf alle Zahlen ν folgt, d. h. größer zu nennen ist als jede der Zahlen ν." E. Zermelo (ed.): "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 195. Let Google translate.

 You seem to think that there is N_def followed by a fairy dust of undefined elements which get left behind after some small sequence of finite members (FISON) are 'removed' from N (leaving fairy dust behind as a set) instead of 'constructing' a new set from 'the elements of' the set of naturals.

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So it is! Remember the harmonic series. It is infinite although all defined numbers are separately collected in converging subseries.

I don't know what you expect to achieve along these lines, but it might be interesting how the harmonic series and these contrived subseries based on digits being removed from tailings maps out when compared to other things having been run through the Zeta Function.