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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Thu, 13 Mar 2025 11:35:30 +0100
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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?
> 
>> They
> 
> They?
> 
>>>> 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.

   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.
> 
>>> 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.
> 
>>>> 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?

>> 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. Try to find a 
counterexample. Fail.

Regards, WM