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

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

On 04.10.2024 21:37, Jim Burns wrote:

On 10/4/2024 5:53 AM, WM wrote:

If slightly fewer is not possible,
then ℵ₀ unit fractions sit at one point.

>
No.

>
Otherwise
we could enter the gaps between them.

You (WM) think that we can't enter gaps because
you (WM) think that a quantifier shift is reliable.
However, a quantifier shift is unreliable.
∀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

Slightly fewer is not possible.

>
That proves one point.

...that ω-1 is not possible.

You (WM) think that that's wrong because
you (WM) think that a quantifier shift is reliable.
However, a quantifier shift is unreliable.

>
It is more reliable than your unfounded twaddle.
But it is no quantifier shift but simplest logic:
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
That you (WM) think that you've given an argument
shows
that you (WM) think a quantifier shift is reliable.