Re: The non-existence of "dark numbers"

On 16.03.2025 13:17, Alan Mackenzie wrote:

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.

 You haven't said what you mean by F.

I did in the discussion with JB: F is the set of FISONs.

All
natural numbers "have" a FISON

Then all natural numbers would be in FISONs. But because of
∀n ∈ U(F): |ℕ \ {1, 2, 3, ..., n}| = ℵo
all FISONs fail to contain all natural numbers.

If you really think there is a non-empty set of natural numbers which
don't "have" FISONs,

Of course there is such a set. It contains almost all natural numbers. This has been proven in the OP: All separated definable natural numbers can be removed from the harmonic series. When only terms containing all definable numbers together remain, then the series diverges. All its terms are dark.

then please say what the least natural number in
that set is, or at the very least, how you'd go about finding it.

The definable numbers are  potentially infinite sequence. With n also n+1 and n^n^n belong to it.

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

 You've never said what you
mean by a number "emptying" a set.

Removing all its elements by subtraction.

It's unclear whether you mean the
subtraction of each number individually, or of all numbers together.

If all natural numbers were individually definable, then there would not be a difference.

Even "subtraction" is a non-standard word, here.  The opposite of "add"
(hinzufügen) is "remove", not "subtract".

The opposite of addition is subtraction. Look for instance: subtraction+of+sets+latex

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.

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

A FISON is a well-ordered set or segment or sequence. It has a largest element.

 

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?

None, but you should pay attention because ℕ is infinite and therefore cannot be emptied by finite sets.

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?

Subtracting a set or elements of a set. See above. Definable elements can be subtracted individually. Undefinable elements can only be subtracted collectively.
 >

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.

You seem unable to learn.

 

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.

You said: The tending takes place, but not in a "place".

 

"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 really have problems to comprehend sentences. Try again.

You haven't said what you mean by a number
"emptying" a set.

Even if I had not, an intelligent reader would know it.

The current state of our discussion is that you have failed to give any
coherent definition of "defined numbers";

A defined number is a number that you can name such that I understand what you mean. In every case you choose almost all numbers will be greater. A child could understand that.
Regards, WM