Re: Replacement of Cardinality

On 8/6/2024 9:52 AM, WM wrote:

Le 06/08/2024 à 12:32, Jim Burns a écrit :

On 8/6/2024 4:26 AM, WM wrote:

NUF(x) gives
the number of unit fractions smaller than x.

>

For NUF(x) = 3
⅟ℕᵈᵉᶠ∩(0,x) is finite, namely 3.

>
For NUF(x) = 3.5
⅟ℕᵈᵉᶠ∩(0,x) is fractional, namely 3.5, however,
no such x with NUF(x) = 3.5  exists.

>
That is not of interest.

How sad for you.
You missed an example of something getting named
which does not then pop into existence.
You could have benefited from a little interest.

(We could however subdivide the distance between u_3 and u_4.)

Suppose we subdivided the distance between u_3 and u_4.
Would there be an x with NUF(x) = 3.5 ?

| Assume otherwise.
| Assume NUF(x₃) = 3
|
| u₁ < u₂ < u₃  are all of
| the finite unit fractions in (0,x₃)
|
| However,
| ⅟(1+⅟u₁) < u₁ is also
| a finite unit fraction in (0,x₃)
| 0 < ⅟(1+⅟u₁) < u₁ < u₂ < u₃ < x₃
|
| NUF(x₃) > 3
| Contradiction.
>
Therefore,
no such x with NUF(x) = 3  exists.

>
All that is in vain if you accept mathematics,
in particular ∀n ∈ ℕ: 1/n - 1/(n+1) > 0.

...which is equivalent to
∀n ∈ ℕ:  0 < 1/(n+1) < 1/n