Re: Replacement of Cardinality

On 8/1/2024 8:13 AM, WM wrote:

Le 31/07/2024 à 19:33, Jim Burns a écrit :

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

>
In the dark domain the maximum cannot be discerned.

⅟ℕᶠⁱⁿ∩(0,1] is the set of finite.unit.fractions.
Each non.{}.subset of ⅟ℕᶠⁱⁿ∩(0,1] is maximummed.
  ∀S ⊆ ⅟ℕᶠⁱⁿ∩(0,1]: S ≠ {}  ⇒  ∃u = max.S
Each finite.unit.fraction in ⅟ℕᶠⁱⁿ∩(0,1] is down.stepped.
  ∀u ∈ ⅟ℕᶠⁱⁿ∩(0,1] ∃v = ⅟(1+⅟u) = max.⅟ℕᶠⁱⁿ∩(0,u)
Each finite.unit.fraction in ⅟ℕᶠⁱⁿ∩(0,1] is non.max.up.stepped.
  ∀v ∈ ⅟ℕᶠⁱⁿ∩(0,1]: v ≠ max.⅟ℕᶠⁱⁿ∩(0,1]  ⇒
   ∃u = ⅟(-1+⅟v) = min.⅟ℕᶠⁱⁿ∩(v,1]
Each non.{}.subset of ⅟ℕᶠⁱⁿ∩(0,1] is maximummed.
Each finite.unit.fraction in ⅟ℕᶠⁱⁿ∩(0,1] is down.stepped.
Each finite.unit.fraction in ⅟ℕᶠⁱⁿ∩(0,1] is non.max.up.stepped.
Therefore,
the finite.unit.fractions in ⅟ℕᶠⁱⁿ∩(0,1] are ℵ₀.many.
∀ᴿx > 0:  ⅟ℕᶠⁱⁿ∩(0,x] ≠ {}
⎛ Assume otherwise.
⎜ Assume ⅟ℕᶠⁱⁿ∩(0,x] = {}
⎜
⎜ β ≥ x > 0 situates the split between
⎜ lower.bounds of ⅟ℕᶠⁱⁿ∩(0,1]  and
⎜ not.lower.bounds of ⅟ℕᶠⁱⁿ∩(0,1]
⎜ 2⋅β > β > ½⋅β > 0
⎜
⎜ ½⋅β < β is a lower.bound of ⅟ℕᶠⁱⁿ∩(0,1]
⎜
⎜ 2⋅β > β is a not.lower.bound of ⅟ℕᶠⁱⁿ∩(0,1]
⎜ finite.unit.fraction ⅟k < 2⋅β exists
⎜ finite.unit.fraction ¼⋅⅟k < ¼⋅2⋅β = ½⋅β exists
⎜ ½⋅β is a not.lower.bound of ⅟ℕᶠⁱⁿ∩(0,1]
⎝ Contradiction.
Therefore,
∀ᴿx > 0:  ⅟ℕᶠⁱⁿ∩(0,x] ≠ {}
(0,x] inherits from its superset (0,1] properties by which
each non.{}.subset of ⅟ℕᶠⁱⁿ∩(0,x] is maximummed,  and
each finite.unit.fraction in ⅟ℕᶠⁱⁿ∩(0,x] is down.stepped,  and
each finite.unit.fraction in ⅟ℕᶠⁱⁿ∩(0,x] is non.max.up.stepped.
Therefore,
the finite.unit.fractions in ⅟ℕᶠⁱⁿ∩(0,x] are ℵ₀.many.
∀ᴿx > 0:  NUFᶠⁱⁿ(x) = ℵ₀
NUFᶠⁱⁿ(x) ≥ NUF(x)
∀ᴿx > 0:  NUF(x) ≥ ℵ₀