Re: The non-existence of "dark numbers" (thread too long, nothing in it)

On 03/14/2025 05:49 PM, FromTheRafters wrote:

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.

Yeah, if it was infinitely-many, there'd be an infinitely-grand.
Of course we all know that's defined away.  It's called restriction
of comprehension, contra expansion of comprehension, and whether
you see that as "true" sort of reflects whether you think about it.
So, when you consider that from at least one perspective
you're looking at it looking at you from the wrong end,
hopefully thusly you can consider your own improvement,
instead of brick-batting and rehabilitating some incorrigeable,
unrecalcitrant, mechanically simulated sock-puppet troll-bot.
That is to say, much your rhetoric about refusing to accept
mathematical inference, applies.
Then, about that there are models of integers _with_ and models
of integers _without_ infinitely grand, or "fragments and extensions:
no standard model", it's a simple consequence of inference.
"Counting" can exist as pebbles in a bad,
infinity _is_ a mental construct, of a
real thing called the infinity of nature.
Natural infinity, ....