Re: The non-existence of "dark numbers"

Am Sat, 15 Mar 2025 18:13:57 +0100 schrieb WM:

On 15.03.2025 12:57, Alan Mackenzie wrote:

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

 

I'm showing you that your "definition" of "definable numbers" is no
definition at all.

You are mistaken. Not all numbers have FISONs because ∀n ∈ U(F): |ℕ \
{1, 2, 3, ..., n}| = ℵo.
ℵo nubers have no FISONs.

Those are, by definition, not naturals.

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.

The subtraction of all numbers which cannot empty ℕ cannot empty ℕ.
Simpler logic is hardly possible.

This is so fucked up. If you take out everything, nothing remains.

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,

No. It says simply that no FISON ending with n can be defined.

Then n is infinite and not in fact natural.

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?

Simply try to understand. I have often stated the difference:
∀n ∈ U(F): |ℕ \ {1, 2, 3, ..., n}| = ℵo ℕ \ {1, 2, 3, ...} = { }

It does not say what you think it says.

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.

The set is defined, not its elements. All defined elements

You can't have a set with undefined elements.

Not the defined numbers.

"Defined numbers" remains (still) undefined.

Defined numbers have FISONs ad cannot empty ℕ.

Yes they can. There are no other numbers.

They are placed on the
ordinal line and can tend to ℕ. This cn happen only on the ordinal line.
Your assertion of the contrary is therefore wrong.
 

"Defined numbers" appears not to be a coherent mathematical concept.

The subtraction of all numbers which cannot empty ℕ cannot empty ℕ. The
collection of these numbers is ℕ_def.

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.