Re: Replacement of Cardinality

On 7/31/2024 9:59 AM, WM wrote:

Le 30/07/2024 à 20:37, Jim Burns a écrit :

NUF(x) = |⅟ℕ∩(0,x]| = ℵ₀
does not imply any unit fraction outside (0,x]

>
But it implies that you can use only
x having ℵ₀ smaller positive points,
in fact even ℵ₀*2ℵ₀.
>
∀ x > 0: NUF(x) = ℵ₀ would be wrong.

Each non.{}.subset of ⅟ℕ∩(0,1] is maximummed.
  ∀S ⊆ ⅟ℕ∩(0,1]: S ≠ {}  ⇒  ∃u = max.S
Each unit.fraction in ⅟ℕ∩(0,1] is down.stepped.
  ∀u ∈ ⅟ℕ∩(0,1] ∃v = ⅟(1+⅟u) = max.⅟ℕ∩(0,u)
Each unit.fraction in ⅟ℕ∩(0,1] is non.max.up.stepped.
  ∀u ∈ ⅟ℕ∩(0,1]: u ≠ max.⅟ℕ∩(0,1]  ⇒
   ∃v = ⅟(-1+⅟u) = min.⅟ℕ∩(u,1]
⎛ Whatever else is
⎜ maximummed and down.stepped and non.max.up.stepped
⎜ has as many as
⎜ maximummed and down.stepped and non.max.up.stepped
⎝ ⅟ℕ∩(0,1]
Each non.{}.subset of ⅟ℕ∩(0,x] is
  a non.{}.subset of ⅟ℕ∩(0,1]
  and is maximummed.
  ∀S ⊆ ⅟ℕ∩(0,x]: S ≠ {}  ⇒  ∃u = max.S
Each unit.fraction in ⅟ℕ∩(0,x] is
  a unit.fraction in ⅟ℕ∩(0,1]
  and is down.stepped.
  ∀u ∈ ⅟ℕ∩(0,x] ∃v = ⅟(1+⅟u) = max.⅟ℕ∩(0,u)
Each unit.fraction in ⅟ℕ∩(0,x] is
  a unit.fraction in ⅟ℕ∩(0,1]
  and is non.max.up.stepped.
  ∀u ∈ ⅟ℕ∩(0,x]: u ≠ max.⅟ℕ∩(0,x]  ⇒
   ∃v = ⅟(-1+⅟u) = min.⅟ℕ∩(u,x]
⅟ℕ∩(0,x] is
maximummed and down.stepped and non.max.up.stepped
⅟ℕ∩(0,x] has as many as ⅟ℕ∩(0,1]
|⅟ℕ∩(0,x]| = |⅟ℕ∩(0,1]| = ℵ₀
∀ᴿx > 0: NUF(x) = ℵ₀
because of
maximumming and down.stepping and non.max.up.stepping.