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From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Fri, 14 Mar 2025 16:21:58 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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Am Fri, 14 Mar 2025 15:25:22 +0100 schrieb WM:
> On 14.03.2025 14:35, Alan Mackenzie wrote:
>> WM <wolfgang.mueckenheim@tha.de> wrote:
>>> On 13.03.2025 18:53, Alan Mackenzie wrote:
>>>> WM <wolfgang.mueckenheim@tha.de> wrote:
>> 
>>>> "Definable number" has not been defined by you, except in a
>>>> sociological sense.
>>> Then use numbers defined by induction: 
>>> |ℕ \ {1}| = ℵo.
>>> If |ℕ \ {1, 2, 3, ..., n}| = ℵo then |ℕ \ {1, 2, 3, ..., n+1}| = ℵo.
>>> Here the numbers n belonging to a potentially infinite set are
>>> defined. This set is called ℕ_def.
>> You're confusing yourself with the outdated notion "potentially
>> infinite".  The numbers n in an (?the) inductive set are N, not N_def.
>> Why do you denote the natural numbers by "N_def" when everybody else
>> just calls them "N"?
> Perhaps everybody is unable to see that ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ...,
> n}| = ℵo?
That's pretty obvious.

>>> It strives for ℕ but never reaches it because .....
>> It doesn't "strive" for N.  You appear to be thinking about a process
>> taking place in time
> Induction and counting are processes. It need not be in time. But it
> fails to complete ℕ.
Whatever. They are infinite in any case, which you continually fail to
comprehend.

> ℕ \ {1, 2, 3, ...} = { }.
Yes, N = {1, 2, 3, ...}.


>>> The potentially infinite inductive set has no last element. Therefore
>>> its complement has no first element.
The complement (wrt what?) is empty.

>> You're letting "potentially infinite" confuse you again.  The inductive
>> set indeed has no last element.  So "its complement" (undefined unless
>> we assume a base set to take the complement in), if somehow defined, is
>> empty.  The empty set has no first element.
> The empty set has not ℵo elements.
Come again?

>>> But there are ℵo numbers following upon all numbers of ℕ_def.
>> N_def remains undefined

>>>> "Dark number" remains undefined, except in a sociological sense. 
>>>> "Dark successor" is likewise undefined.
>>> [ "It is even permissible to think of the newly created number as a
>>> limit to which the numbers nu tend.  If nothing else is understood,
>>> it's held to be the first integer which follows all numbers nu, that
>>> is, is bigger than each of the numbers nu." ]
>>> Between the striving numbers ν and ω lie the dark numbers.
Wrong. There are no other numbers than the "striving" ones.

>> That contradicts the long excerpt from Cantor you've just cited.
>> According to that, omega is the _first_ number which follows the
>> numbers nu.  I.e., there is nothing between nu (which we can identify
>> with N) and omega.  There is no place for "dark numbers".
> There is place to strive or tend.
And it is filled with nu.

>>>> Natural numbers can be "represented in a mind", in fact in any
>>>> mathematician's mind.
>>> Not those which make the set ℕ empty by subtracting them ∀n ∈ ℕ_def:
>>> |ℕ \ {1, 2, 3, ..., n}| = ℵo
There is no number that "makes N empty", wtf.

>> That nonsense has no bearing on the representability of natural numbers
>> in a mathematician's mind.  You're just saying that the complement in N
>> of a finite subset of N is of infinite size.  Yes, and.... ?
And what?

>>> like the dark numbers can do ℕ \ {1, 2, 3, ...} = { }.
I see no dark numbers here.

>> The above identity, more succinctly written as N \ N = { } holds
>> trivially, and has nothing to say about the mythical "dark numbers".
> n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo proves that definable numbers
> are not sufficient.
It doesn't. Why do you think there is a number k such that N = {1, ..., 
k}?

-- 
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.