Re: Simple enough for every reader?

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.

Interesting is that all defined numbers belong to a finite initial segment.

About natural numbers that is obvious. One interesting thing is that
rational numbers can be ordered so that every one of them belongs
to a finite initial segment. Another intresting thing is that real
numbers cannot be ordered so.
Note that in order to dscuss infinities you must use second or higher
order logic. In first order logic you cannot say that an initial
segment is finite.

Therefore most numbers are dark.

Tha does not follow.
--
Mikko