Re: Simple enough for every reader?

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