Re: Replacement of Cardinality

On 8/20/2024 9:07 AM, WM wrote:

Le 19/08/2024 à 20:12, Jim Burns a écrit :

Lemma.
No positive point is a lower.bound (lb) of
all the visibleᵂᴹ unit.fractions.
¬(lb.⅟ℕᵈᵉᶠ > 0)

0 is the greatest lower bound
because
every greater number can be undercut by
a visible unit fraction.

Dark unit fractions cannot be defined.
The GLB that can be defined is 0.

And 0 is glb.⅟ℕᵈᵉᶠ not.in ⅟ℕᵈᵉᶠ
⎛ For set S,
⎜ min.S is defined to be glb.S in S
⎜
⎜ Because two greatest.lb.S cannot exist
⎜ if glb.S is not.in S
⎜ then no glb.S in S exists
⎝ and no min.S exists.
No glb.⅟ℕᵈᵉᶠ in ⅟ℕᵈᵉᶠ exists.
No min.⅟ℕᵈᵉᶠ exists.

It starts from 0.

If ⅟ℕᵈᵉᶠ starts,
⅟ℕᵈᵉᶠ starts from min.⅟ℕᵈᵉᶠ
No min.⅟ℕᵈᵉᶠ exists.
⅟ℕᵈᵉᶠ does not start.
Because 0 is glb.⅟ℕᵈᵉᶠ not.in ⅟ℕᵈᵉᶠ