Re: Simple enough for every reader?

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. 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.

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. A mathematical property of
real numbers is that not all of them are definable.

But the mathematically interesting fact is that never all numbers can become visible because completeness, well-order and infinity are mutually incompatible.

 What kind of comleteness do you mean?

 The complete set of numbers / prime numbers is infinite. If each number were known and subtracted, a last one would appear.
 

Therefore not all can have a known well-order.

 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 { }.
--
Mikko