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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 15:25:22 +0100
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On 14.03.2025 14:35, Alan Mackenzie wrote:
> WM <wolfgang.mueckenheim@tha.de> wrote:
>> 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.
> 
> You're confusing yourself with the outdated notion "potentially
> infinite".  The numbers n in an (?the) inductive set are N, not N_def.
> Why do you denote the natural numbers by "N_def" when everybody else just
> calls them "N"?

Perhaps everybody is unable to see that
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo?

>> It strives for ℕ but never reaches it because .....
> 
> It doesn't "strive" for N.  You appear to be thinking about a process
> taking place in time

Induction and counting are processes. It need not be in time. But it 
fails to complete ℕ.
ℕ \ {1, 2, 3, ...} = { }.
ℕ \ ℕ_def =/= { }.

> "Definable" remains undefined, so there's no point to answer here.  Did
> Zermelo, Peano, or von Neumann use "definable" the way you're trying to
> use it, at all?

Zermelo claimed that without their construction/proof by induction we 
don't know whether infinite sets exist at all.

Um aber die Existenz "unendlicher" Mengen zu sichern, bedürfen wir noch 
des folgenden ... Axioms. [Zermelo: Untersuchungen über die Grundlagen 
der Mengenlehre I, S. 266] The elements are defined by induction in 
order to guarantee the existence of infinite sets.
>> The potentially infinite inductive set has no last element. Therefore
>> its complement has no first element.
> 
> You're letting "potentially infinite" confuse you again.  The inductive
> set indeed has no last element.  So "its complement" (undefined unless we
> assume a base set to take the complement in), if somehow defined, is
> empty.  The empty set has no first element.

The empty set has not ℵo elements.

>> But there are ℵo numbers following upon all numbers of ℕ_def.
> 
> N_def remains undefined,

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

>>> "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.
>> [ "It is even permissible to think of the newly created number as a
>> limit to which the numbers nu tend.  If nothing else is understood,
>> it's held to be the first integer which follows all numbers nu, that
>> is, is bigger than each of the numbers nu." ]
> 
>> Between the striving numbers ν and ω lie the dark numbers.
> 
> That contradicts the long excerpt from Cantor you've just cited.
> According to that, omega is the _first_ number which follows the numbers
> nu.  I.e., there is nothing between nu (which we can identify with N) and
> omega.  There is no place for "dark numbers".

There is place to strive or tend.
>>> 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
> 
> That nonsense has no bearing on the representability of natural numbers
> in a mathematician's mind.  You're just saying that the complement in N
> of a finite subset of N is of infinite size.  Yes, and.... ?
> 
>> like the dark numbers can do
>> ℕ \ {1, 2, 3, ...} = { }.
> 
> Dark numbers remain undefined.

Yes, they cannot be determines as individuals.

> The above identity, more succinctly
> written as N \ N = { } holds trivially, and has nothing to say about the
> mythical "dark numbers".

n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo proves that definable numbers 
are not sufficient.

Regards, WM