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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 09:56:31 +0100
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On 15.03.2025 00:11, Alan Mackenzie wrote:
> WM <wolfgang.mueckenheim@tha.de> wrote:

>> ℕ_def contains all numbers the subtraction of which from ℕ does not
>> result in the empty set.
> 
> What does "which" refer to?

It refers to the numbers. An Englishman should comprehend that.

> To N_def or to a
> member of the "all numbers"?

That is one and the same.
> 
> Assuming the former, then if X is any proper subset of N, N \ X is
> non-empty.  So by this "definition", N_def is any proper subset of N.

No, ℕ_def contains only definable numbers.
> 
>> 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.

> 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.

>> Yes, of course. But ℕ_def is not completed by its definition.
> 
> You haven't defined N_def - what appears above is not a coherent
> definition.

It is coherent enough. 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.
>>> 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.

Not the defined numbers.

Regards, WM