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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: The existence of dark numbers proved by the thinned out harmonic
 series.
Date: Mon, 17 Mar 2025 15:17:53 +0100
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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 :
>>
>>>> I informed you several times:
>>>> ℕ \ {1, 2, 3, ...} = { }
>>>
>>> When two sets are the same set, the difference set is the empty set.
>>
>> 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
>>
>> ℕ \ ℕ_def =/= { }
> 
> That is false if N_def is the same as N

True.

>, which it is.

False.

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

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.

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

Regards, WM