Re: The non-existence of "dark numbers"

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

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.

Yes, indeed.  But the assumption is for the purposes of a proof by
contradiction.

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

If the numbers are definable.

Meaningless.  Or are you admitting that your "dark numbers" aren't
natural numbers after all?

Learn what potential infinity is.

I know what it is.  It's an outmoded notion of infinity, popular in the
1880s, but which is entirely unneeded in modern mathematics.

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

So you counter my proof by silently snipping elements 4, 5 and 6 of it?
That's not a nice thing to do.

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

Why should I want to do that?

Jim has supplied at least one other proof.

He claims that lossless exchange can produce losses. He is in
contradiction with logic.

Irrelevant to the current discussion.  He has supplied at least one other
proof of the non-existence of "dark numbers".

[ .... ]

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).