Re: Replacement of Cardinality

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

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

NUF(1) = ℵ₀
NUF(x) = ℵ₀

>
NUF(x) gives
the number of unit fractions smaller than x.
>
Following unreadable symbols.
>

For  each x > 0
⅟ℕᵈᵉᶠ∩(0,x) is not finite.

>
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.
Also,
no such x with NUF(x) = 3  exists.
| 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.
Counter.argument:
What about x₃′ = (u₂+u₃)/2 ?
0 < ⅟(1+⅟u₁) < u₁ < u₂ < x₃′
Counter.counter.argument:
0 < ⅟(2+⅟u₁) < ⅟(1+⅟u₁) < u₁ < u₂ < x₃′
NUF(x₃′) > 3
Which unit.fractions changes.
How many unit.fractions stays infinite.
Infinite is not simply big. It's different.
Finite can be big, too, even reallyreally big,
but it's a reallyreally big finite which
shares "common sense" properties with 3 ...
... "common sense" properties which,
upon a more careful look,
may be held in common with reallyreally big finites
but are not held in common with, for example,
how many unit.fractions there are.