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

On 9/9/24 6:22 AM, WM wrote:

On 08.09.2024 22:29, Moebius wrote:

Am 08.09.2024 um 22:25 schrieb joes:

Am Sun, 08 Sep 2024 21:48:11 +0200 schrieb WM:

On 07.09.2024 16:49, Richard Damon wrote:
>

All those add up to less than x, so they fit.

>

Select any gap between one of the first ℵo unit fractions and its

neighbour. Call its size x. Then ℵo unit fractions cannot fit into the
interval (0, x), independent of the actual size.

>

Why?

>
Weiß niemand.

 ℵo unit fractions cannot fit into one of the ℵo intervals between two of them, because ℵo unit fractions occupy ℵo intervals.
 Regards, WM
 

You seem confused, the Aleph_0 unit fractions don't fit "between" unit fractions, but in the interval (0, x). Since (0 isn't a "unit fraction" you have your meaning messed up.
When we lay out our Aleph_0 unit fractions, when you choose one of the gaps to put the new x into, we loose just a finite number of unit fractions that lied between the old x and the new x, and Aleph_0 minus any finite number is still Aleph_0.
Yes, between two finite unit fractions will only be a finite number of other unit fractions (possibly 0) but the interval (0, x) isn't between two finite unit fractions as its starting point is not a unit fraction, and in fact isn't a finite number in the interval, but is the unbounded edge of the infinite number system which is dense in the area, so there is no point in the set on the boundary. (The point on the boundary is just outside the set).