Re: Replacement of Cardinality

Le 13/08/2024 à 19:32, Moebius a écrit :

Am 13.08.2024 um 19:02 schrieb Jim Burns:

On 8/13/2024 10:21 AM, WM wrote:

Le 12/08/2024 à 19:44, Jim Burns a écrit :

 

There is no ⅟nₓ before the end of the positive axis
without ⅟(nₓ+1) before the end of the positive axis.

>
You cannot see it. It is dark.

 No number can be seen, dark or not.

Numbers can be seen.

 

resulting in a real coordinate x with NUF(x) = 1.

 Assume NUF(x0) = 1 with x0 e IR. This means that there is exactly one unit fraction u such that u < x0. Let's call this unit fraction u0. Then (by definition) there is a (actually exactly one) natural number n such that u0 = 1/n. Let n0 e IN such that u0 = 1/n0. But then (again by definition) 1/(n0 + 1) is a unit fraction which is smaller than u0 and hence smaller than x0. Hence NUF(x0) > 1. Contradiction!

Therefore 1/(n0 + 1) does not exist.
Regards, WM