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

On 9/10/2024 12:16 PM, WM wrote:

On 09.09.2024 21:43, Jim Burns wrote:

The unit fractions
  before x > 0 and before x′ > 0
vary.
For both x and x′, they are ℵ₀.many,
but not the same ℵ₀.many.

>
The same ℵ₀ many are existing in both cases,

No.
They are not the same for ⌊⅟x⌋ < ⌊⅟x′⌋
For ⌊⅟x⌋ < ⌊⅟x′⌋
k ⟼ ⅟⌊k+⅟x⌋: one.to.one
k ⟼ ⅟⌊k+⅟x′⌋: one.to.one
⅟⌊k+⅟x⌋ > ⅟⌊k+⅟x′⌋
  ⅟ℕ∩(0,x]  ⊃≠  ⅟ℕ∩(0,x′]
|⅟ℕ∩(0,x]| = |ℕ| = |⅟ℕ∩(0,x′]|
|ℕ| = ℵ₀
⅟ℕ∩(0,x] is infinite, which is in conflict with
your implicitly.adopted axiom that
there are no infinite sets.
No Grand Council of Mathematicians exists,
in a lair under a dormant mid.Pacific volcano,
to order you to stop using contradictory axioms.
What happens is less cinema.worthy but more inevitable:
If you adopt axioms claiming an infinite set is finite,
there is nothing to which you can be referring.
And there is nothing you or any imaginable
Grand Council of Mathematicians can do to change that.

in addition there are finitely many further unit fractions.
>

and require a minimum length d.

>
No.
There is a greatest.lower.bound 0
but there isn't  a _minimum_

>
There is a minimum larger than
a distance of countably many points.

No.
For each x > 0
ℕ ⟶ (0,½⋅x)
k ⟼ ⅟⌊k+2⋅⅟x⌋
½⋅x is a counter.example to the claim that
🛇 x lower.bounds distances of countably.many.points.
_x > 0  is not minimum distances of countably.many.points_
x ≤ 0 lower.bounds distances of countably.many.points
but
x ≤ 0 is not itself a distance of countably.many.points.
_x ≤ 0  is not minimum distances of countably.many.points_
_Minimum distances of countably.many.points does not exist_

so
each d > 0 is not.minimum.

>
A d of of countably many points is
less than the minimum.

Everything is true of the non.existent minimum.
The minimum is greater than d AND
the minimum is less than d AND
the minimum is in my shirt pocket AND
the minimum tastes like strawberries.
Everything is true: nothing is true.
It is non.existent.
You (WM) don't seem to find the non.existent
something to avoid in what one discusses.
It appears that you prefer the freedom of
claiming whatever pops into your head
about minimumᵂᴹ d > 0 or about darkᵂᴹ numbers
without needing to exert any effort to
figure out how existent things work.
They're non.existent:
You (WM) ask for a (non.existent) counter.example,
and then you're ready to call it a day.