Sujet : Re: Replacement of Cardinality
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.logic sci.mathDate : 06. Aug 2024, 17:33:55
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <1c55ae2d-09e3-4eb0-b5f1-c64e82fcdcea@att.net>
References : 1 2 3 4 5 6 7 8 9 10 11
User-Agent : Mozilla Thunderbird
On 8/6/2024 9:55 AM, WM wrote:
Le 06/08/2024 à 14:38, Jim Burns a écrit :
NUF(x) ≠ 1
is true everywhere
NUF(x) = 1 ⇒ INVNUF(1) = x
is true everywhere
However,
its truth doesn't imply INVNUF(1) exists.
>
∀n ∈ ℕ: 1/n - 1/(n+1) > 0
implies its existence.
No.
∀n ∈ ℕ: 1/n - 1/(n+1) > 0
implies its NONexistence.
| Assume x₁ exists: NUF(x₁) = 1
| ⅟n is the one unit.fraction in (0,x₁)
| 0 < ⅟n < x₁
|
| However,
| ∀n ∈ ℕ: ⅟n - ⅟(n+1) > 0
| 0 < ⅟(n+1) < ⅟n < x₁
| NUF(x₁) > 1
| Contradiction.
Therefore,
there is no x₁: NUF(x₁) = 1
∀ᴿx: NUF(x) ≠ 1