Re: Replacement of Cardinality

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.