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

On 9/2/2024 4:37 PM, WM wrote:

On 02.09.2024 19:19, Richard Damon wrote:

as any unit fraction you might claim to be
that one has a unit fraction smaller than itself,
so it wasn't the smallest.

>
Your argument stems from visible unit fractions
but becomes invalid in the dark domain.

The darkᵂᴹ domain
  between 0 and visibleᵂᴹ unit.fractions
is empty.
Each positive point is undercut by
visibleᵂᴹ unit.fractions, and thus
each positive point is not darkᵂᴹ.
⎛ Assume otherwise.
⎜ Assume x > 0 is not.undercut by
⎜  visibleᵂᴹ unit.fractions.
⎜
⎜ β ≥ x > 0 is the least.upper.bound of
⎜  points.not.undercut by visibleᵂᴹ unit.fractions.
⎜
⎜ ½⋅β is not.undercut.
⎜
⎜ 2⋅β is undercut.
⎜ visibleᵂᴹ ⅟k < 2⋅β
⎜ visibleᵂᴹ ¼⋅⅟k < ½⋅β
⎜ ½⋅β is undercut.
⎜
⎝ Contradiction.

The problem with your NUF, is that
it is trying to count something from and uncountable end,
one that doesn't actually have an end.

>
The unit fractions end before zero.

The lower.end of unit fractions
is not a visibleᵂᴹ unit.fraction ⅟k > ⅟(k+1)
is not a darkᵂᴹ unit fraction (not.existing)
is not anything not a unit.fraction.
The lower.end of unit fractions
is not.