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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Sat, 24 May 2025 13:29:53 +0200
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On 24.05.2025 10:13, Mikko wrote:
> On 2025-05-23 08:31:27 +0000, WM said:
> 
>> On 23.05.2025 09:43, Mikko wrote:
>>> Do you mean that every natural number is dark until
>>> someone mentions it but no longer?
>>>
>> Every natural number is dark in a system that cannot represent it in 
>> any form like writing, thinking or whatever. The pocket calculator is 
>> limited to decimal representations below 10^100, the universe is 
>> limited to more or less sophisticated formulas requiring less than 
>> 10^80 bit.
>>
>> In every system almost all natural numbers are and remain dark - if an 
>> actual infinity of them exists.
> 
> That is not a useful concept as it is not possible to know wich numbers are
> presentable in future sysems and which will be actually presented.

But it is fact. Further it need not be deteremined exactly what can be 
presented. It is sufficient, for many purposes, to know that most 
numbers cannot be presented
> 
> At the end of the web page https://mlevanto.github.io/lauseke.html there
> is an arithmetic expression that evaluates to a 65600 digit number. 
> Although
> the value of the expression is not written there I used that digit sequence
> (and several others, some even longer) when I wrote the page.

The numbers that can be used belong to a potentially infinite set. There 
may be much longer sequences. But most natural numbers remain dark - if 
ℕ is actually infinite.
> 
> We don't know whether our universe is finite or infinite. or wheter it
> can be fully described with finite information.

But all that is irrelevant for the fact that all definable numbers make 
up a small minority. Mathematical proof: All numbers defined by finite 
initial segments belong to a (potentially in-) finite set, because an 
actual infinity follows (and two consecutive actual infinities in ℕ are 
impossible):

{1} has infinitely many (ℵo) successors.
If {1, 2, 3, ..., n} has infinitely many (ℵo) successors, then {1, 2, 3, 
...., n, n+1} has infinitely many (ℵo) successors. For every n that can 
be defined.

Regards, WM
>