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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers" [was: The existence of dark numbers proven by the thinned out harmonic series]
Date: Wed, 12 Mar 2025 11:57:36 -0400
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Alan Mackenzie brought next idea :
> 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 
>> Poincar�[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.

Haha, at least! :D