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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions?
Date: Sat, 5 Oct 2024 23:11:51 -0400
Organization: i2pn2 (i2pn.org)
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On 10/5/24 9:57 AM, Alan Mackenzie wrote:
> Richard Damon <richard@damon-family.org> wrote:
>> On 10/5/24 8:58 AM, Alan Mackenzie wrote:
>>> Richard Damon <richard@damon-family.org> wrote:
> 
>>> [ .... ]
> 
>>>> But actual infinity doesn't exist.
> 
>>> What does it mean for a mathematical concept not to exist?
> 
>> That it doesn't create a usable (non-contradictory) logical system.
> 
> Yes!  At least, sort of.  My understanding of "doesn't exist" is either
> the concept is not (yet?) developed mathematically, or it leads to
> contradictions.  WM's "dark numbers" certainly fall into the first
> category, and possibly the second, too.
> 
> I first came across the terms "potential infinity" and "actual infinity"
> on this newsgroup, not in my degree course a few decades ago.  I'm not
> convinced there is any mathematically valid distinction between them.  If
> there were, I would have heard of it back then.
> 
> Does "actual infinity" create a logical system?  If so, what is unusable
> or contradictory about that system?
> 
> [ .... ]
> 

After a bit of reseach, there does seem to be indications that Aristotle 
did do some reasoning with the terms. I am not sure on the exact 
definitions, but the indications are that "potential" infinity was 
generative, where the numbers are realized as they are needed, and you 
can keep creating more and more of them as you go.

Actual infinity presumed that somehow all the values were created up 
front and none could be added, and he found that logic done on this 
definition was too full of contradictions to be usable, so he concluded 
that "actual infinity" did not really exist.

My guess is that WM doesn't understand this conclusion, or thinks that 
he is somehow smarter than Aristotle and can make it work (when he can't)

or just thinks that since the name given was "actual infinity" that fact 
that it doesn't work just means that infinity can't actually exist.

My guess, from what I have seen from WM, one of the problems with 
"actual infinity" is that it makes it at least seem possible to apply 
the rules of "finite" logic to an infinite logic, and that just breaks 
it. To our finite minds, the rules of infinite logic just don't make 
intuitive sense,