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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Sat, 15 Mar 2025 18:06:28 -0400
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It happens that Chris M. Thomasson formulated :
> On 3/15/2025 1:56 AM, WM wrote:
>> 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.
>
> When you see { 1, 2, 3 ... }
>
> 4 is dark to you?

There is no four. :)

The elements are one, two, and threedotdotdot.