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From: joes <noreply@example.org>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Thu, 13 Mar 2025 21:10:42 -0000 (UTC)
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Am Thu, 13 Mar 2025 11:35:30 +0100 schrieb WM:
> On 12.03.2025 22:31, Alan Mackenzie wrote:
>> WM <wolfgang.mueckenheim@tha.de> wrote:
>>> On 12.03.2025 18:42, Alan Mackenzie wrote:
>>>> WM <wolfgang.mueckenheim@tha.de> wrote:
>> 
>>>>> If the numbers are definable.
>>>> Meaningless.  Or are you admitting that your "dark numbers" aren't
>>>> natural numbers after all?
>> 
>>>>> Learn what potential infinity is.
>>>> I know what it is.  It's an outmoded notion of infinity, popular in
>>>> the 1880s, but which is entirely unneeded in modern mathematics.
>>> That makes "modern mathematics" worthless.
>> What do you know about modern mathematics?
> I know that it is self-contradictory because it cannot distinguish
> potential and actual infinity.
> When |ℕ| \ |{1, 2, 3, ..., n}| = ℵo, then |ℕ| \ |{1, 2, 3, ..., n+1}| =
> ℵo. This holds for all elements of the inductive set, i.e., all FISONs
> F(n) or numbers n which have more successors than predecessors. Only
> those contribute to the inductive set! Modern mathematics must claim
> that contrary to the definition ℵo vanishes to 0 because ℕ \ {1, 2, 3,
> ...} = { }.
> That is blatantly wrong and shows that modern mathematicians believe in
> miracles. Matheology.
There is no miracle. N is not a FISON.

>> You may recall me challenging
>> others in another recent thread to cite some mathematical result where
>> the notion of potential/actual infinity made a difference.  There came
>> no coherent reply (just one from Ross Finlayson I couldn't make head
>> nor tail of).  Potential infinity isn't helpful and isn't needed
>> anymore.
> 
>>>>>> 3. The least element of the set of dark numbers, by its very
>>>>>>       definition, has been "named", "addressed", "defined", and
>>>>>>       "instantiated".
> It is named but has no FISON. That is the crucial condition.
Then it is larger than omega.

>>>> So you counter my proof by silently snipping elements 4, 5 and 6 of
>>>> it? That's not a nice thing to do.
>>> They were based on the mistaken 3 and therefore useless.
>> You didn't point out any mistake in 3.  I doubt you can.
> I told you that potential infinity has no last element, therefore there
> is no first dark number.
Therefore none.

>>>>> Try to remove all numbers individually from the harmonic series such
>>>>> that none remains. If you can't, find the first one which resists.
>>>> Why should I want to do that?
>>> In order to experience that dark numbers exist and can't be
>>> manipulated.
>> Dark numbers don't exist, as Jim and I have proven.
> When |ℕ| \ |{1, 2, 3, ..., n}| = ℵo, then |ℕ| \ |{1, 2, 3, ..., n+1}| =
> ℵo. How do the ℵo dark numbers get visible?
By induction.

>>> Induction cannot cover all natural numbers but only less than remain
>>> uncovered.
>> The second part of that sentence is gibberish.  Nobody has been talking
>> about "uncovering" numbers, whatever that might mean.  Induction
>> encompasses all natural numbers.  Anything it doesn't cover is not a
>> natural number, by definition.
> Every defined number leaves ℵo undefined numbers.
They are not undefined.

-- 
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.