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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Sat, 15 Mar 2025 18:13:57 +0100
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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.

>>>> 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.
> 
>>> 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.

>> 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, ...} = { }

> 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
> 
>> Not the defined numbers.
> 
> "Defined numbers" remains (still) undefined.

Defined numbers have FISONs ad cannot empty ℕ. 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.

Regards, WM