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From: Alan Mackenzie <acm@muc.de>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Wed, 12 Mar 2025 17:42:19 -0000 (UTC)
Organization: muc.de e.V.
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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 Poincar=C3=A9[1], such that sender and receiver understa=
nd
>>> 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 rainbo=
w",
>>>   - 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=20
> 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=20
> 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

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