Re: Replacement of Cardinality

On 8/19/24 7:29 AM, WM wrote:

Le 17/08/2024 à 16:22, Richard Damon a écrit :

On 8/17/24 9:37 AM, WM wrote:

 

NUF(x) grows from 0 to more, but at no point it grows by more than 1.

 

And there is "no point" that is smaller than all unit fractions but greater than 0, so at that point NUF(x) jumps from 0 to Aleph_0.

 That is superstition.

>
Your problem is NUF(x) may have a clear verbal description, but not a mathematical one, as it is based on a false assumption that there exists a smallest unit fraction.

 That is not an assumption but it is derived from the mathematics:
∀n ∈ ℕ: 1/n - 1/(n+1) > 0
Regards, WM
 

But that doesn't say there is a smallest unit fraction, in fact it says there isn't as 1/(n+1) will ALWAYS be smaller than 1/n, and by your own definitions, since it was just used individually, is defined. In fact, if your system actually has the Natural Numbers, for every n, the number n+1 exists and is well defined.
This means that there is no highest Natural Number, or smallest Unit Fraction, and any system that tries to say otherwise doesn't actually have the Natural Numbers within it.
That describes your system which can not actually support the Natural numbers, and blows up when you assume they must exist in it.