Re: Simple enough for every reader?

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

On 11.06.2025 09:50, Mikko wrote:

On 2025-06-10 17:06:18 +0000, WM said:
 

 

This expression is represented by
 ((((ℕ \ {1}) \ {2}) \ {3}) ...) = { }

 That may be good enough when you want to prove something that we already
believe anyway. But for a sufficiently rigorous proof of something else
the "..." should be replaced with something more mathematicsl.

 The "..." can be replaced with the singletons of definable natural numbers

 Then do so.

 See above. Three are already given. More can be inserted. But it is impossible to replace them with all natural numbers.

 I can see that you stll have no rigorous proofs,

 I have facts.

In mathematics facts are irrelevant but proofs are essential.

and can't have as you
apparently can't even write your claim without "..." and other informal
expressions.

 Of course I can:
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.

Result: It is impossible to define all natural numbers with none remaining undefined.

You have not derived that "result".
--
Mikko