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

On 9/9/24 6:36 AM, WM wrote:

On 08.09.2024 22:16, joes wrote:

Am Sat, 07 Sep 2024 15:45:08 +0200 schrieb WM:

On 07.09.2024 15:37, joes wrote:

Am Sat, 07 Sep 2024 15:03:35 +0200 schrieb WM:

>

By the way this is independent of the existence of NUF.

Why can they not „fit”?

Because ℵo unit fractions have ℵo gaps. We can use one of the gaps as
x > 0.

What does this mean?

 ℵo unit fractions are claimed to be smaller than every x > 0. If they are existing then I can choose as the x one of the ℵo intervals between two of them. Irrelevant which one, ℵo unit fractions do not fit into it.
 Regards, WM
 

You seem to be conusing "below" and "into".
Is your English that bad?
The gaps (each | is a unit fraction)
  .... | gap | gap | gap | ... gaps | x
the claim is that there are aleph_0 gaps between the unit fractions below x.
Moving x down to one of the gaps doesn't mean we need to compress all those unit fractions into just one of the gaps.
BELOW the x, is still ALeph_0 unit fractions, as by choosing a gap, you only removed a finite number of the Aleph_0 unit fractions, so Aleph_0 still remain.
The fact that you don't understand that Aleph_0 - a finite number = Aleph_0 is your own problem, but *IS* a fact of mathematics.
Note, the interval (0, x) doesn't "begin" at a unit fraction, in part because there is no point it begins at, as our number system is DENSE, and so the lower boundry is 0, so the only point on the boundry there is 0, but that is OUTSIDE the interval. there is no "point" just inside, as it is an unbounded set.