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

Am Sat, 05 Oct 2024 20:54:21 +0200 schrieb WM:

On 05.10.2024 15:02, joes wrote:

Am Sat, 05 Oct 2024 11:32:44 +0200 schrieb WM:

On 05.10.2024 10:08, joes wrote:

Am Fri, 04 Oct 2024 22:09:41 +0200 schrieb WM:

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.

Which they can.

Do it. Fail because slightly fewer is not possible.

What do you mean? All the unit fractions have, as you say,
finite distances, which means there are numbers inbetween.

But these numbers and these unit fractions cannot be found. It is
impossible to define a unit fraction having less than ℵo smaller unit
fractions. It is impossible to define a unit fraction being closer to
zero although it is obvious that there are points closer to zero than
ℵo*2^ℵo points.

Yes, the can be found, for example by the arithmetic mean. There simply
are no unit fractions with a finite number of lesser UFs, otherwise those
that are greater couldn’t have an infinite number less than them. You
cannot start from zero, count a finite number of UFs, and reach an
infinity in ever greater steps. It is always possible to increase the
denominator to get a UF closer to 0. Every point has a finite distance
from zero, i.e. an infinite number of points inbetween.

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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.