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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 17:18:34 +0100
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On 13.03.2025 13:59, Alan Mackenzie wrote:
> WM <wolfgang.mueckenheim@tha.de> wrote:

>> I know that it is self-contradictory because it cannot distinguish
>> potential and actual infinity.
> 
> It can, but doesn't need to.  Potential and actual infinity are needless
> concepts which only serve to confuse and obfuscate.  If you disagree,
> feel free to cite a standard result in standard mathematics which depends
> on these notions.
> 
>> When |ℕ| \ |{1, 2, 3, ..., n}| = ℵo, ....
> 
> The difference operator \
> applies to sets, not to cardinal numbers.

I know, but erroneously I had used the sets. I corrected that but 
without correcting the sign

>> .... 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.
> 
> I.e. all natural numbers.

No. All numbers can be subtracted from ℕ such that none remains:
ℕ \ {1, 2, 3, ...} = { }, let alone ℵo.

>> Only those contribute to the inductive set!
> 
> The inductive set is all natural numbers.  Why must you make such a song
> and dance about it?

Because when only definable numbers are subtracted from ℕ, then
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
infinitely many numbers remain. That is the difference between dark and 
defiable numbers.
> 
>> 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.
> 
> Modern mathematics need not and does not claim such a ridiculous thing.

ℕ \ {1, 2, 3, ...} = { } is wrong?


>>> 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.
> 
> The second part of your sentence does not follow clearly from the first,
> therefore the sentence is false.  And even if it were not false, it has
> no bearing on my item 3.

Try to think better. ℕ_def is a subset of ℕ. If ℕ_def had a last 
element, the successor would be the first dark number.
> 
> But I can agree with you that there is no first "dark number".  That is
> what I have proven.  There is a theorem that every non-empty subset of
> the natural numbers has a least member.

That theorem is wrong in case of dark numbers.
>>>>>> 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?
> 
> There is no such thing as a "dark number".  It's a figment of your
> imagination and faulty intuition.

Above we have an inductive definition of all elements which have 
infinitely many dark successors.

>>>> 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.
> 
> What the heck are you talking about?  What does it even mean for a number
> to "leave" a set of numbers?

The set ℕ_def defined by induction does not include ℵo undefined numbers.

>  Quite aside from the fact that there is no
> mathematical definition of a "defined" number.  The "definition" you gave
> a few posts back was sociological (talking about how people interacted
> with eachother) not mathematical.

Mathematics is social, even when talking to oneself. Things which cannot 
be represented in any mind cannot be treated.

Regards, WM