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From: Alan Mackenzie <acm@muc.de>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Fri, 14 Mar 2025 15:21:52 -0000 (UTC)
Organization: muc.de e.V.
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WM <wolfgang.mueckenheim@tha.de> wrote:
> 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 sociolog=
ical
>>>> sense.

>>> Then use numbers defined by induction:

>>> |=E2=84=95 \ {1}| =3D =E2=84=B5o.
>>> If |=E2=84=95 \ {1, 2, 3, ..., n}| =3D =E2=84=B5o
>>> then |=E2=84=95 \ {1, 2, 3, ..., n+1}| =3D =E2=84=B5o.

>>> Here the numbers n belonging to a potentially infinite set are define=
d.
>>> This set is called =E2=84=95_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 You're confusing yourself with the outdated notion "potentially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 j=
ust
>> calls them "N"?

> Perhaps everybody is unable to see that
> =E2=88=80n =E2=88=88 =E2=84=95_def: |=E2=84=95 \ {1, 2, 3, ..., n}| =3D=
 =E2=84=B5o?

Everybody can see that, and everybody but you can see it has nothing to
do with the point it purportedly answers.

>>> It strives for =E2=84=95 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=20
> fails to complete =E2=84=95.

Wrong.  It is an "instantaneous" definition which completes N.  There are
not various stages of "N" which are in varying stages of completion.

[ .... ]

>> "Definable" remains undefined, so there's no point to answer here.  Di=
d
>> Zermelo, Peano, or von Neumann use "definable" the way you're trying t=
o
>> use it, at all?

> Zermelo claimed that without their construction/proof by induction we=20
> don't know whether infinite sets exist at all.

Everybody can see that that has nothing to do with the point it
purportedly answers.  "Definable" is not used by mathematicians the way
you attempt to use it.

> Um aber die Existenz "unendlicher" Mengen zu sichern, bed=C3=BCrfen wir=
 noch=20
> des folgenden ... Axioms. [Zermelo: Untersuchungen =C3=BCber die Grundl=
agen=20
> der Mengenlehre I, S. 266] The elements are defined by induction in=20
> 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 inductiv=
e
>> 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 =E2=84=B5o elements.

The empty set has no elements.  What are you trying to say?

>>> But there are =E2=84=B5o numbers following upon all numbers of =E2=84=
=95_def.

>> N_def remains undefined,

>>> |=E2=84=95 \ {1}| =3D =E2=84=B5o.
>>> If |=E2=84=95 \ {1, 2, 3, ..., n}| =3D =E2=84=B5o
>>> then |=E2=84=95 \ {1, 2, 3, ..., n+1}| =3D =E2=84=B5o.

>>>> "Dark number" remains undefined, except in a sociological sense.  "D=
ark
>>>> successor" is likewise undefined.

>>> "Es ist sogar erlaubt, sich die neugeschaffene Zahl =CF=89 als Grenze=
 zu
>>> denken, welcher die Zahlen =CE=BD zustreben, wenn darunter nichts and=
eres
>>> verstanden wird, als da=C3=9F =CF=89 die erste ganze Zahl sein soll, =
welche auf
>>> alle Zahlen =CE=BD folgt, d. h. gr=C3=B6=C3=9Fer zu nennen ist als je=
de der Zahlen =CE=BD."
>>> E. Zermelo (ed.): "Georg Cantor =E2=80=93 Gesammelte Abhandlungen mat=
hematischen
>>> 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 =CE=BD and =CF=89 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 numbe=
rs
>> 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.

The tending takes place, but not in a "place".  That I have to write such
nonsense to answer your point shows the great deterioration which has
taken place in a once vital newsgroup.

>>>> Natural numbers can be "represented in a mind", in fact in any
>>>> mathematician's mind.

>>> Not those which make the set =E2=84=95 empty by subtracting them
>>> =E2=88=80n =E2=88=88 =E2=84=95_def: |=E2=84=95 \ {1, 2, 3, ..., n}| =3D=
 =E2=84=B5o

>> That nonsense has no bearing on the representability of natural number=
s
>> 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
>>> =E2=84=95 \ {1, 2, 3, ...} =3D { }.

>> Dark numbers remain undefined.

> Yes, they cannot be determined as individuals.

They don't exist, as I have proven.  You have not found any flaw in my
proof.

>> The above identity, more succinctly written as N \ N =3D { } holds
>> trivially, and has nothing to say about the mythical "dark numbers".

> n =E2=88=88 =E2=84=95_def: |=E2=84=95 \ {1, 2, 3, ..., n}| =3D =E2=84=B5=
o proves that definable numbers=20
> are not sufficient.

You don't know what the word "prove" means in a mathematical sense.
"Definable numbers" remains undefined, so any "proof" involving them is
meaningless.

> Regards, WM

--=20
Alan Mackenzie (Nuremberg, Germany).