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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Sun, 25 May 2025 13:38:23 +0200
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On 25.05.2025 12:42, Mikko wrote:
> On 2025-05-24 11:29:53 +0000, WM said:
> 
>> 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.
> 
> But not a mathematical fact.

That depends on the definition of mathematics.
> 
>> Further it need not be determined exactly what can be presented. It 
>> is sufficient, for many purposes, to know that most numbers cannot be 
>> presented
> 
> For many porposes it sufficient to know that most numbers needn't be
> presented.

Here however I am concerned with this question.
> 
>>> 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.
> 
> The set of natural numbers is actually infinite. There is nothing
> potential in mathematics: what is is

Where is it?

> and that is all that can be.

That is clearly wrong. The set of known prime numbers is potentially 
infinite.

The set of numbers which can be chosen as individuals is potentially 
infinite too.

{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.

Regards, WM