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

On 12.03.2025 13:12, Alan Mackenzie wrote:

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.

Perhaps in your head.

 

  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.

Wrong.

2. Every non-empty set of natural numbers contains a least element.

If the numbers are definable. Learn what potential infinity is.

3. The least element of the set of dark numbers, by its very
    definition, has been "named", "addressed", "defined", and
    "instantiated".

Try to remove all numbers individually from the harmonic series such that none remains. If you can't, find the first one which resists.

Jim has supplied at least one other proof.

He claims that lossless exchange can produce losses. He is in contradiction with logic.
He claims that the following proofs of set existence by induction are different.
{} ∈ Z₀, and for all x: if x ∈ Z₀ then {x} ∈ Z₀.
ℕ \ {1} = ℵo, and if ℕ \ {1, 2, 3, ..., n} = ℵo then ℕ \ {1, 2, 3, ..., n+1} = ℵo.
He is in contradiction with mathematics, at least with Zermelo.
Regards, WM