Re: The non-existence of "dark numbers"

After serious thinking WM wrote :

On 14.03.2025 14:11, FromTheRafters wrote:

WM wrote on 3/14/2025 :

>

Natural numbers can be "represented in a mind", in fact in any
mathematician's mind.

>
Not those which make the set ℕ empty by subtracting them
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
like the dark numbers can do
ℕ \ {1, 2, 3, ...} = { }.

 The set of natural numbers is never empty. What happens is that you decide which of its elements complies with the definition of elements of the new set, and remove the rest from consideration.

>
The new set defined by individual elements cannot be equal to ℕ.

Why not?  The set which is the improper subset is a set too. Not some ongoing endless unfinished process. Make two proper subsets E and O from the elements of N, make a new set R (for result) by taking the union of E and O. The result of the union operator is a set. How would R be any different than N?