Re: Simple enough for every reader?

On 16.06.2025 09:41, Mikko wrote:

On 2025-06-15 16:15:39 +0000, WM said:
 

On 15.06.2025 12:32, Mikko wrote:

On 2025-06-14 14:00:36 +0000, WM said:

>

Of course the border between visible and dark numbers depends on time and system.

>
And therefore has no mathenatical meaning.

>
That is only true if mathematics is restricted to definable numbers.

 No, it is true anyway.

Please don't claim but argue by arguments:
The complete set of prime numbers is infinite. Only a finite set is known. If each number were known and subtracted from its set, then a last one would be subtracted.

Mathematics is restricted to mathematical things
even when the inspiration comes from something non-mathematical, as it
often comes. Things that depend on time or otherwise on the real world
are not mathematical.

That is a wrong opinion, as best can be seen by geometry. But also the restrictions of reality do limit what can be expressed in mathematics.

But that is not the usual approach. The undefinable prime numbers for instance belong to present mathematics.

 Definability is a mathematical property. Every natural number and in
particular every prime number is definable.

Then define the prime numbers such that none remains undefined. Or define a hitherto undefined prime number.

Then can as they do. The arithmetic order of natural numbers is a known
well-order. But a well-order does not imply that there is a last one.
only that there is a first one.

>
But this does: ℕ \ {1, 2, 3, ...} = { }    .

 No, it does not. There is no last one in { }.

When the natural numbers are subtracted one by one and when finally none remains, then a last one has been subtracted. This is forced by their order.
Regards, WM