The non-existence of "dark numbers" [was: The existence of dark numbers proven by the thinned out harmonic series]

WM <wolfgang.mueckenheim@tha.de> wrote:

On 12.03.2025 11:22, Alan Mackenzie wrote:

Meaningless.  "Definable number" is itself undefined.

Definition: A natural number is "named" or "addressed" or "identified"
or "(individually) defined" or "instantiated" if it can be communicated,
necessarily by a finite amount of information, in the sense of
Poincaré[1], such that sender and receiver understand the same and can
link it by a finite initial segment (1, 2, 3, ..., n) of natural numbers
to the origin 0. All other natural numbers are called dark natural numbers.

This is bullshit.

 Communication can occur
 - by direct description in the unary system like ||||||| or as many
beeps, raps, or flashes,
 - by a finite initial segment of natural numbers (1, 2, 3, 4, 5, 6, 7),
 - as n-ary representation, for instance binary 111 or decimal 7,
 - by indirect description like "the number of colours of the rainbow",
 - by other words known to sender and receiver like "seven".

Your "dark numbers" have no part in mathematics, don't exist, and can't
exist.  A proof, which I've given to you before, is as follows:

1. Assume that "dark numbers" exist.
2. Every non-empty set of natural numbers contains a least element.
3. The least element of the set of dark numbers, by its very
   definition, has been "named", "addressed", "defined", and
   "instantiated".
4. That least element is thus both a "dark number" and a "light number".
5. This is a contradiction.
6. Therefore the set of dark numbers must be empty.

Jim has supplied at least one other proof.

 [1] "In my opinion a subject is only conceivable if it can be defined
by a finite number of words."

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).