Path: news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail
From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Mon, 19 May 2025 20:53:43 +0200
Organization: A noiseless patient Spider
Lines: 60
Message-ID: <100funo$1ous9$1@dont-email.me>
References: <100a8ah$ekoh$1@dont-email.me> <100ccrl$upk6$1@dont-email.me>
 <100cjat$vtec$1@dont-email.me> <100fdbr$1laaq$1@dont-email.me>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Mon, 19 May 2025 20:53:45 +0200 (CEST)
Injection-Info: dont-email.me; posting-host="f03bf24ea76c7f186b2b0e62e98ef69b";
	logging-data="1866633"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX18b8Cv1gFuHW7QePlw6cp3yzytDbpEhAZs="
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:smnwTAquDEFsxj6269bAuwOswLE=
Content-Language: en-US
In-Reply-To: <100fdbr$1laaq$1@dont-email.me>

On 19.05.2025 15:57, Mikko wrote:
> On 2025-05-18 12:20:47 +0000, WM said:
> 
>> On 18.05.2025 12:30, Mikko wrote:
>>> On 2025-05-17 15:00:33 +0000, WM said:
>>>
>>>> Are you aware of the fact that in
>>>>
>>>> {1}
>>>> {1, 2}
>>>> {1, 2, 3}
>>>> ...
>>>> {1, 2, 3, ..., n}
>>>> ...
>>>>
>>>> up to every n infinitely many natural numbers of the whole set
>>>>
>>>> {1, 2, 3, ...}
>>>>
>>>> are missing? Infinitely many of them will never be mentioned 
>>>> individually. They are dark.
>>>
>>> For example, if we pick 5 for n we have
>>>
>>> {1}
>>> {1, 2}
>>> {1, 2, 3}
>>> {1, 2, 3, 4}
>>> {1, 2, 3, 4, 5}
>>>
>>> then 6 and infinitely many other numbers are missing. So numbers
>>> 6, and 7 are dark as are ingfinitely many other numbers.
> 
>> Maybe for a 3-year old child. Doves can count to 7. Earthworms may
>> fail at 1 already.
> 
> Many animals can differentiate quantities up to about 7. As far as
> we know most of them needn't and can't count. They just see the
> difference. Accurate determination of larger quantities may require
> counting.
> 
> None of which is relevant to may observation that if n = 5 then your
> definition makes 6 dark.

If you have no idea of 6, it is dark for you. I you arbitrarily stop at 
5 although you know 6, 5 is not dark for you.
> 
>> It depends on the system. But important is that no system can get over 
>> the infinite gap of dark numbers.
> 
> Why not? Cantor quite obviously gets over quite large infinities.

Yes, but not by counting them one by one. He jumps to ω and can from 
there go on. He could also go back ω-1, ω-2, ... . But there is no 
finite initial segment reaching to ω/n for every n that has a finite 
initial segment. Let alone reaching to ω-n.

Regards, WM
>