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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: The non-existence of "dark numbers"
Date: Thu, 13 Mar 2025 17:24:10 -0700
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On 3/13/2025 5:59 AM, Alan Mackenzie wrote:
> WM <wolfgang.mueckenheim@tha.de> wrote:
>> 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:
> 
> [ .... ]
> 
>>>>>> 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.
> 
> 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, ....
> 
> Do you ever bother to check what you write?  The difference operator \
> applies to sets, not to cardinal numbers.  I can guess what you mean, but
> your readers shouldn't have to guess that.
> 
>> .... 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.
> 
>> 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?
> 
>> 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.
> Your understanding of it is what's lacking.
> 
>>> 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.
> 
> What the heck does it mean for a number to "have" a FISON?  Assuming you
> can define that, you need to prove that the least "dark number" "has" no
> FISON.  And assuming you can do that (which I very much doubt), you then
> have to clarify what that condition is crucial to and how.
> 
>>>>> 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.
> 
> 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.
> 
> 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.  On the assumption (yours) that
> "dark numbers" are a subset of the natural numbers, that proves that
> there are no "dark numbers" at all.
> 
>>>>>> 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.
> 
>>>> 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?  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.

v is a number. Let me define it: v = 42

;^)