Re: The non-existence of "dark numbers"

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

On 15.03.2025 00:11, Alan Mackenzie wrote:

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

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

What does "which" refer to?

It refers to the numbers. An Englishman should comprehend that.

To N_def or to a
member of the "all numbers"?

That is one and the same.

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.

No, ℕ_def contains only definable numbers.

For crying out loud, man!  I'm showing you that your "definition" of
"definable numbers" is no definition at all.  You end up asserting that
the set of definable numbers contains only definable numbers.

Mathematics is not your thing.

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

Of course not.

Then you cannot think logically.

When confronted with your misguided attempts at mathematics, it is very
difficult to follow your "logic", much less agree with it.

It all depends on the X from which N_def is formed.  If
X is N \ {1},

Then its elements are mostly undefined as individuals.

"Undefined as individuals" is an undefined notion, and does not appear in
your "definition" of N_def.  Do you wish to modify your "definition" of
N_def to include this "undefined as individuals" (having stated what it
means)?

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.

It is coherent enough.

It is not.  It remains unclear what N_def is, for example which
particular natural numbers are members of it.  What you proposed as its
"definition" is not even unambiguous, as I've pointed out at some length.

Every element has a finite FISON. ℕ is infinite.  Therefore it cannot
be emptied by the elements of ℕ_def and also not by ℕ_def.

A "finite" FISON?  What other type is there?  What do you mean by
"having" a FISON?  What does it mean to "empty" N by a set or elements of
a set?  What is the significance, if any, of being able to "empty" a set?

None of these notions are standard mathematical ones.  If you want to
communicate clearly with mathematicians, you'd do far better if you used
the standard words with their standard meanings.  But maybe you don't
want to communicate clearly.

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.

Not the defined numbers.

"Defined numbers" remains (still) undefined.  "Defined numbers" appears
not to be a coherent mathematical concept.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).