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

On 05.10.2024 01:41, Jim Burns wrote:

On 10/4/2024 4:09 PM, WM wrote:

Otherwise
we could enter the gaps between them.

 You (WM) think that we can't enter gaps because

they are dark.

you (WM) think that a quantifier shift is reliable.
However, a quantifier shift is unreliable.

Don't parrot that nonsense. There is no quantifier shift. Between many unit fractions there are many gaps. But we cannot find them.

>
If ℵ₀ unit fractions do not sit at one point
(and are not dark) then they can be subdivided into smaller parts.

 ∀ᴿx>0:
⎛ uₓ(k) = ⅟⌈k+⅟x⌉
⎜ uₓ: ℕ → (0,x]: one.to.one and not.at.one.point
⎝ NUF(x) = |uₓ(ℕ)| = |ℕ| = ℵ₀

What shall that prove? Between n unit fractions there are n-1 gaps. Each gap can be found for definable unit fractions.
Note also that every pair of unit fractions is a finite set.
Regards, WM