Re: The non-existence of "dark numbers"

WM <wolfgang.mueckenheim@tha.de> wrote:

On 14.03.2025 16:21, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

Perhaps everybody is unable to see that
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo?

Everybody can see that, and everybody but you can see it has nothing to
do with the point it purportedly answers.

ℕ_def contains all numbers the subtraction of which from ℕ does not
result in the empty set.

That's gobbledegook.  What does "which" refer to?  To N_def or to a
member of the "all numbers"?

Assuming the former, then if X is any proper subset of N, N \ X is
non-empty.  So by this "definition", N_def is any proper subset of N.

Obviously the subtraction of all numbers which cannot empty ℕ cannot
empty ℕ. Therefore |ℕ \ ℕ_def| = ℵo. Do you agree?

Of course not.  It all depends on the X from which N_def is formed.  If
X is N \ {1}, then N \ X is {1}, and thus |N \ N_def| is 1.

If not, it is useless to discuss with you.

I've been thinking that for quite a few rounds of posts.  However ....

Wrong.  It is an "instantaneous" definition which completes N.

Yes, of course. But ℕ_def is not completed by its definition.

You haven't defined N_def - what appears above is not a coherent
definition.

There are not various stages of "N" which are in varying stages of
completion.

ℕ_def is never complete.

I'll take your word for that.

There is place to strive or tend.

The tending takes place, but not in a "place".

No? Tending means that hitherto undefined natural numbers become
defined. That takes place on the ordinal line.

"Hitherto" ("bis jetzt" in German) is purely a time based adverb.  The
natural numbers are not defined in a time based sequence.  They are
defined all together.

That I have to write such nonsense to answer your point shows the
great deterioration which has taken place in a once vital newsgroup.

Hardly to believe that matheology like tending of ordinals outside of
the ordinal line has ever been useful.

Yes, they cannot be determined as individuals.

They don't exist, as I have proven.

You have proven that you are a matheologian with little ability to
understand arguments contradicting your matheologial belief.

Whatever "matheologian" might mean.  You are the one who is attempting
to establish the existence of so called "dark numbers", so the burden of
proof lies on you.  So far you have failed abjectly.  The definitions
you have attempted to give have been only sociological and the
incoherent attempt from your last post (see above).

"Dark numbers", as far as I am aware, don't appear in any treatment of
the fundamentals of mathematics.  Given the problems with them, it is
only reasonable to conclude they don't exist.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).