Re: Simple enough for every reader?

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.

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