Re: The non-existence of "dark numbers"

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

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 numbers have no FISONs.

That's incoherent garbage.  You haven't said what you mean by F.  All
natural numbers "have" a FISON - there is a total logical disconnect
between the clauses around the "because".

If you really think there is a non-empty set of natural numbers which
don't "have" FISONs, then please say what the least natural number in
that set is, or at the very least, how you'd go about finding it.

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.

That wasn't logic; it was incoherent garbage.  You've never said what you
mean by a number "emptying" a set.  It's unclear whether you mean the
subtraction of each number individually, or of all numbers together.
Even "subtraction" is a non-standard word, here.  The opposite of "add"
(hinzufügen) is "remove", not "subtract".

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.

More incoherent garbage.  A FISON is a set.  Sets don't "end" with
anything.

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, ...} = { }

Which doesn't address my question in the sightest.  What do you mean by
"emptying" N by a set or by elements of 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.

You don't want to communicate clearly, do you?  If you did, all your
fantasy constructs like "dark numbers" and "N_def" would collapse to
meaninglessness in the resulting rigorous analysis.

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

Not the defined numbers.

"Defined numbers" remains (still) undefined.

Defined numbers have FISONs and cannot empty ℕ.

Meaningless.  You haven't said what (if anything) you mean by a number
emptying N.  And every natural number "has" a FISON, not just some subset
of them.

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

Of the many assertions I've made, the one you're referring to is unclear.

"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.

Incoherent garbage.  You haven't said what you mean by a number
"emptying" a set.  Even if you did, and it turned out to be a coherent
meaning, it seems likely "all numbers which cannot ...." would not be
uniquely defined.  Hence the definition would collapse in a heap.

The current state of our discussion is that you have failed to give any
coherent definition of "defined numbers"; you have failed to counter my
proof of the non-existence of "dark numbers".

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).