Re: How many different unit fractions are lessorequal than all unit fractions?

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,