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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Fri, 14 Mar 2025 13:09:24 +0100
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On 13.03.2025 18:53, Alan Mackenzie wrote:
> WM <wolfgang.mueckenheim@tha.de> wrote:

> "Definable number" has not been defined by you, except in a sociological
> sense.

Then use numbers defined by induction:

|ℕ \ {1}| = ℵo.
If |ℕ \ {1, 2, 3, ..., n}| = ℵo
then |ℕ \ {1, 2, 3, ..., n+1}| = ℵo.

Here the numbers n belonging to a potentially infinite set are defined. 
This set is called ℕ_def. It strives for ℕ but never reaches it because

>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo infinitely many
>> numbers remain. That is the difference between dark and definable
>> numbers.
> 
> Rubbish!  It's just that the set difference between an infinite set and a
> one of its finite subsets remains infinite.

Yes, just that is the dark part. All definable numbers belong to finite 
sets.

> That doesn't shed any light
> on "dark" or "defi[n]able" numbers.

Du siehst den Wald vor Bäumen nicht.

>> ℕ_def is a subset of ℕ. If ℕ_def had a last
>> element, the successor would be the first dark number.
> 
> If, if, if, ....  "N_def" remains undefined, so it is not sensible to
> make assertions about it.

See above. Every inductive set (Zermelo, Peano, v. Neumann) is definable.

>>> But I can agree with you that there is no first "dark number".  That is
>>> what I have proven.  There is a theorem that every non-empty subset of
>>> the natural numbers has a least member.
> 
>> That theorem is wrong in case of dark numbers.
> 
> That's a very bold claim.  Without further evidence, I think it's fair to
> say you are simply mistaken here.

The potebtially infinite inductive set has no last element. Therefore 
its complement has no first element.
> 
>>>> When |ℕ \ {1, 2, 3, ..., n}| = ℵo, then |ℕ \ {1, 2, 3, ..., n+1}| =
>>>> ℵo. How do the ℵo dark numbers get visible?
> 
> There are no such things as "dark numbers", so talking about their
> visibility is not sensible.

But there are ℵo numbers following upon all numbers of ℕ_def.
> 
>>> There is no such thing as a "dark number".  It's a figment of your
>>> imagination and faulty intuition.
> 
>> Above we have an inductive definition of all elements which have
>> infinitely many dark successors.
> 
> "Dark number" remains undefined, except in a sociological sense.  "Dark
> successor" is likewise undefined.

"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.

Between the striving numbers ν and ω lie the dark numbers.

>> The set ℕ_def defined by induction does not include ℵo undefined numbers.
> 
> The set N doesn't include ANY undefined numbers.

  ℵo

>>>   Quite aside from the fact that there is no
>>> mathematical definition of a "defined" number.  The "definition" you gave
>>> a few posts back was sociological (talking about how people interacted
>>> with eachother) not mathematical.
> 
>> Mathematics is social, even when talking to oneself. Things which cannot
>> be represented in any mind cannot be treated.
> 
> Natural numbers can be "represented in a mind", in fact in any
> mathematician's mind.

Not those which make the set ℕ empty by subtracting them
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
like the dark numbers can do
ℕ \ {1, 2, 3, ...} = { }.

Regards, WM