Re: Simple enough for every reader?

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

On 14.06.2025 12:58, Mikko wrote:

On 2025-06-13 13:45:20 +0000, WM said:
 

On 13.06.2025 11:53, Mikko wrote:

On 2025-06-12 09:41:51 +0000, WM said:
 

On 12.06.2025 09:56, Mikko wrote:

On 2025-06-11 11:30:26 +0000, WM said:

 

For all natural numbers that can be chosen as individuals:
|ℕ \ {1, 2, 3, ..., n}| = ℵo.
The "..." can be removed by the inductive proof
for all definable n ∈ ℕ:
|ℕ \ {1}| = ℵo.
|ℕ \ {m ∈ ℕ | m < n}| = ℵo
==> |ℕ \ {m ∈ ℕ | m < n+1}| = ℵo.

 That the set difference of an infinite set and a finite set is infinite
is well understood and therefore an uninteresting result.

 That is not the only result.

 That is the result of your proof. Other results can be discussed in
other contexts.

 The following result cannot be circumvented: After mankind will have ceased,

 who will care?

 People who live now and ponder about dark numbers.

At the time they no loger care.

there is a largest natural number ever named.

 That has no mathematical significance. Whether a number is named or not
is a feature of mankind, not of the number.

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

And therefore has no mathenatical meaning.

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?

ℕ \ {1, 2, 3, ...} = { } means that all natural numbers can be subtracted - completely such that none remains. If all were visible with a known well-order, then a last one would be subtracted.

That claim cannot be accepted without a proof. And why is it relevant
wehther a well-order is known? Anyway, there is a last natural number
in the the reverse arithmetic order, and that last one would indeed be
removed. But the reverse arithmetic order is not a well-order.

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.
If you want to argue (rather than just ask) about mathematics it is not
a good idea to reveal that you are not a mathematician.
--
Mikko