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

On 9/6/24 11:00 PM, Ross Finlayson wrote:

On 09/06/2024 07:01 PM, Richard Damon wrote:

On 9/6/24 8:07 AM, WM wrote:

On 06.09.2024 05:08, Richard Damon wrote:

On 9/5/24 10:08 AM, WM wrote:

>

NUF(x) must grow. It cannot grow by more than 1 at any x.

>

This is NOT the "ancient" idea of infinity,

>
NUF(x) must grow. It cannot grow by more than 1 at any x.
Right or wrong in your opinion?
>
Regards, WM
>

>
Only if it exists.
>
If it does, it must be counting some sub-finite values as "unit
fractions" that are not the reciprocal of the Natural Numbers (since
there is no smallest of those unit fractions to count from).
>
So, either it is counting some sub-finite values (actually a lot of them
a countable infinity of them) or it just doesn't exist.
>
Maybe that is your dark numbers, these sub-finite numbers that are
reciprocals of some post-finite values above the infinite set of Natural
Numbers (which have no upper bound) and are below Omega.

 That's a remarkable supposition, I wonder how you'd imagine
both to satisfy to yourself and others that thusly is a
"consistent" form, of course which only requires "internal
consistency" for its own sake, then besides, to suffer the
running of the gauntlet, of those who'd insist it contradicted
theirs. For, their are simple inductive arguments that nothing
ever happens or is, at all.
 I sort of appreciate the sentiment, though, that "infinite"
is big enough to have quite a range.
 

The problem with "consistancy" is that WM's mathematicss isn't consistent with the full Natural Numbers, and unlikely to be helped with the addition of something even more esoteric.
His work doesn't define the set well enough to actually define how it must work, and the best answer is likly to just adopt one of the existing set of sub-finite number, it just needs to have a countably infinite subset of values that can be reasonable defined as "unit fractions".