Re: Replacement of Cardinality

On 8/12/24 9:50 AM, WM wrote:

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

On 8/11/2024 8:29 AM, WM wrote:

 

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

>
The end of the positivee axis.

>

∀n ∈ ℕ: 1/n - 1/(n+1) > 0

>
∀n ∈ ℕ: 1/n > 1/(n+1) > 0

 If 1/(n+1) exists.

Which, by the definition of the Natural Numbers, it does.
Of course, if your logic doesn't actually allow the Natural Numbers to exist as a set, then you shouldn't be using it.

>
Each positive unit fraction is not
the first positive unit fraction.
>
What causes an exception: nₓ ∈ ℕ:
⅟nₓ > 0 without ⅟(nₓ+1) > 0 ?

 The end of the positive axis.

Which, by the definition of the Natural Numbers, doesn't exist.
Of course, if your logic doesn't actually allow the Natural Numbers to exist as a set, then you shouldn't be using it.
Since it is clear that you logic can't handle the Natuarl Numbers, you are just al Number, you use of them just shows you don't mind using broken logic.

 Regards, WM