Re: Replacement of Cardinality

On 8/15/2024 9:52 AM, WM wrote:

Le 14/08/2024 à 18:43, Jim Burns a écrit :

On 8/14/2024 8:23 AM, WM wrote:

Therefore there is no step at 0.

>
There is a step at 0.

>
Nonsense.

⎛ Assume otherwise.
⎜ Assume NUF(x) = 0 and x > 0
⎜
⎜ β ≥ x > 0 for
⎜ β greatest.lower.bound of unit.fractionsᵈᵉᶠ
⎜ 2⋅β > β  ⇒  not.lower.bound 2⋅β
⎜ ½⋅β < β  ⇒  lower.bound ½⋅β
⎜
⎜ 2⋅β > ⅟k  ⇐  not.lower.bound 2⋅β
⎜ ¼⋅2⋅β > ¼⋅⅟k
⎜ ¼⋅2⋅β = ½⋅β
⎜ ½⋅β > ¼⋅⅟k  ⇒  not.lower.bound ½⋅β
⎜
⎜ lower.bound ½⋅β and not.lower.bound ½⋅β
⎝ Contradiction.

However,
0 is not a unit.fractionᵈᵉᶠ.

>
Therefore there is no step.
NUF(x) steps only at unit fractions x.

If NUF(x) doesn't step at 0
then lower.bound ½⋅β and not.lower.bound ½⋅β