Sujet : Re: Replacement of Cardinality
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.logic sci.mathDate : 07. Aug 2024, 18:07:09
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <e8a3a66a-7d83-4658-9f4c-23d7dc354fb9@att.net>
References : 1 2 3 4 5 6 7 8 9 10 11
User-Agent : Mozilla Thunderbird
On 8/7/2024 8:38 AM, WM wrote:
Le 06/08/2024 à 19:44, Jim Burns a écrit :
On 8/6/2024 9:52 AM, WM wrote:
Le 06/08/2024 à 12:32, Jim Burns a écrit :
On 8/6/2024 4:26 AM, WM wrote:
Le 06/08/2024 à 00:19, Jim Burns a écrit :
NUF(x) gives
the number of unit fractions smaller than x.
>
For NUF(x) = 3
⅟ℕᵈᵉᶠ∩(0,x) is finite, namely 3.
>
For NUF(x) = 3.5
⅟ℕᵈᵉᶠ∩(0,x) is fractional, namely 3.5, however,
no such x with NUF(x) = 3.5 exists.
(We could however subdivide
the distance between u_3 and u_4.)
>
Suppose we subdivided
the distance between u_3 and u_4.
Would there be an x with NUF(x) = 3.5 ?
>
No, the unit fractions are quantized.
For NUF(x) = 3
⅟ℕᵈᵉᶠ∩(0,x) is finite, namely 3.
For NUF(x) = 3.5
|⅟ℕᵈᵉᶠ∩(0,x)| is fractional, namely 3.5
By definition, INVNUF(3.5) = x
We agree that
saying "INVNUF(3.5)" doesn't prove
INVNUF(3.5) exists
It is equally true that
saying "INVNUF(3)" doesn't prove
INVNUF(3) exists
But we have
proof INVNUF(3) does not exist.
⎛ Assume INVNUF(3) exists.
⎜ NUF(INVNUF(3)) = 3
⎜
⎜ INVNUF(3) >
⎜ ⅟ ⌊⅟INVNUF(3) +1⌋ >
⎜ ⅟ ⌊⅟INVNUF(3) +2⌋ >
⎜ ⅟ ⌊⅟INVNUF(3) +3⌋ >
⎜ ⅟ ⌊⅟INVNUF(3) +4⌋ > 0
⎜
⎜ NUF(INVNUF(3)) > 3
⎝ Contradiction.
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