Re: Replacement of Cardinality

On 8/13/2024 10:27 AM, WM wrote:

Le 12/08/2024 à 20:34, Jim Burns a écrit :

On 8/12/2024 9:47 AM, WM wrote:

Before the bound there is the end,
perhaps dark.

>

0 is
greatest.lower.bound β of visibleᵂᴹ unit.fractions

>
Yes.

Thank you.

But the smallest unit fractions comes before.

No visibleᵂᴹ.or.darkᵂᴹ point ε > 0  is the end of
both visibleᵂᴹ and darkᵂᴹ unit.fractions.
Otherwise,
⎛ ε > 0 is a lower.bound of
⎜ both visibleᵂᴹ and darkᵂᴹ unit.fractions.
⎜
⎜ ε > 0 is a lower.bound of
⎜ visibleᵂᴹ unit.fractions.
⎜
⎜ 0 is the greatest.lower.bound of
⎜ visibleᵂᴹ unit.fractions.
⎜
⎜ ε is a lower.bound greater than the greatest.
⎝ Contradiction.
You (WM) don't seem to want it, but
we could stretch the definition of darkᵂᴹ so that
ε > 0 is the end of darkᵂᴹ but not visibleᵂᴹ
( darkᵂᴹ in the cracks between some visibleᵂᴹ?
However, such an ε > 0 would still leave
the unit.fractions one.ended.
----
0 is the greatest.lower.bound of
both visibleᵂᴹ and darkᵂᴹ unit.fractions.
0 isn't a unit.fraction.