Re: Replacement of Cardinality

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

On 8/12/2024 9:50 AM, WM wrote:

Le 11/08/2024 à 19:56, Jim Burns a écrit :

What causes an exception: nₓ ∈ ℕ:
⅟nₓ > 0 without ⅟(nₓ+1) > 0 ?

>
The end of the positive axis.

 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. The function NUF(x) is a step-function. It can increase from 0 at x = 0 to greater values, either in a step of size 1 or in a step of size more than 1. But increase by more than 1 is excluded by the gaps between unit fractions. (Note the universal quantifier there, according to which never – in no limit and in no accumulation point – two unit fractions occupy the same point x.) Therefore the step size can only be 1, resulting in a real coordinate x with NUF(x) = 1.
Regards, WM