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

Am Sat, 05 Oct 2024 11:29:29 +0200 schrieb WM:

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.

You constantly make that incorrect deduction. At one point you seemed
to be recognise that Ax Ey P(x,y) is different from Ey Ax P(x,y).

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.

What about the gap between the last definable and the first dark UF?

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.