Re: Replacement of Cardinality

On 8/19/2024 7:44 AM, WM wrote:

Le 17/08/2024 à 23:44, Jim Burns a écrit :

On 8/17/2024 9:36 AM, WM wrote:

NUF(x) at no point grows by more than 1.

>
At unit.fractions. Not at 0.
>

But NUF(x) grows.

>
At 0. Not at unit fractions.

>
You are not entitled to change the definition of NUF(x).

NUF(x) ≥ NUFᵈᵉᶠ(x) = |⅟ℕᵈᵉᶠ∩(∞,x]|
NUF(0) = NUFᵈᵉᶠ(0) = 0
∀ᴿx > 0: NUF(x) ≥ NUFᵈᵉᶠ(x) > 0
NUF(x) grows at 0
β = glb.{ y∈ℝ: ⅟ℕᵈᵉᶠ∩(∞,y]≠{} }
β = 0
⎛ Assume β > 0
⎜ not.bound 2⋅β > ⅟k ∈ ⅟ℕᵈᵉᶠ
⎜ not.bound ½⋅β > ¼⋅⅟k ∈ ⅟ℕᵈᵉᶠ
⎜ bound ½⋅β < β
⎝ Contradiction.
Therefore, β = 0
∀ᴿx > 0: NUF(x) ≥ NUFᵈᵉᶠ(x) > 0
NUF(x) grows at 0

NUF(x) grows at every unit fractions by 1.

No. ℵ₀+1 = ℵ₀

At 0 there is no unit fractions,
therefore NUF(0) = 0
as for all negative arguments.
The firstincrease happens at the first unit fraction.

No positive lower.bound, darkᵂᴹ or visibleᵂᴹ, exists of
visibleᵂᴹ unit.fractions.