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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Sun, 16 Mar 2025 17:17:33 +0100
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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