Re: The non-existence of "dark numbers"

on 3/14/2025, WM supposed :

On 14.03.2025 17:09, FromTheRafters wrote:

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?

>
Because |ℕ \ ℕ_def| = ℵo.
ℕ_def contains all numbers the subtraction of which from ℕ does not result in the empty set. Obviously the subtraction of all numbers which cannot empty ℕ cannot empty ℕ. Do you agree?

There you go with the subtraction idea again. The difference set between the set of naturals and your imagined set of 'defined naturals' has cardinality aleph_zero because your 'defined naturals' is a finite set.