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

On 11.09.2024 20:52, Jim Burns wrote:

On 9/11/2024 10:35 AM, WM wrote:

Real points on the real line
either are there or are not there.

 Sets of real.line.points and
real.line.points
are not the same things.

But we need not use sets at all if we use the points.

 

In particular
they do not "approach" any other point.

 Define β as
the greatest.lower.bound of visiblesᵂᴹ ⅟ℕᵈᵉᶠ
β = glb.⅟ℕᵈᵉᶠ
 Positive β requires impossibilities.
It requires that ½.β
both IS and IS NOT
undercut by a visibleᵂᴹ unit.fraction.

In some dark cases ½.β does not exist.

⅟ℕᵈᵉᶠ does not change. (Sets do not change.)

Because their elements, here: the points do not change.
But you claim that if β exists, then ½.β exists. And if ½.β exists, then β/4 exists, and so on. That is potential infinity. If all exists, then a smallest exists.
Regards, WM

We use "approach" to say "no gap".