Re: Replacement of Cardinality

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

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

Le 06/08/2024 à 00:19, Jim Burns a écrit :

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

 Also,
no such x with NUF(x) = 3  exists.

At least we have found now a way to express finitely many unit fractions without the accusation of quantifier shift and without the insane result that for all x > 0 NUF(x) = ℵo. That would be wrong even when no gaps between the unit fractions existed.

 | 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.
Regards, WM