Re: Replacement of Cardinality

Le 02/08/2024 à 18:24, Richard Damon a écrit :

On 8/2/24 12:05 PM, WM wrote:

The claim is that there is a finite difference that seperates any two specific unit fractions,

Of course.

 Not that there is a single finite difference that seperates any two arbirtry unit fractions.

Between any two arbitrary unit fractions there are ℵ₀*2ℵ₀ points.

 Not understanding the order of the arguement

Between any two arbitrary unit fractions there are ℵ₀*2ℵ₀ points. No order necessary.

 For your example, for any x, there is an eps = 1/(ceil(1/x)+1) that provides a unit fraction smaller than x,

No, eps has to be chosen. x has to be chosen. My claim is that most are dark and cannot be chosen.
Note: For every 1/n there exists a smaller real number. But they cannot be chosen because most 1/n already cannot be chosen. Proof: For every chosen eps you fail to separate infinitely many unit fractions.

and thus NUF(x) can not be 1 for any finite x.

But for x belonging to the first ℵ₀*2ℵ₀ points we have less than ℵ₀ unit fractions.
Regards, WM