Sujet : Re: Replacement of Cardinality
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.logic sci.mathDate : 01. Aug 2024, 19:12:37
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <9822f5da-d61e-44ba-9d70-2850da971b42@att.net>
References : 1 2 3 4 5 6 7 8 9 10 11
User-Agent : Mozilla Thunderbird
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