Re: Replacement of Cardinality

On 8/19/2024 1:17 PM, Moebius wrote:

Am 19.08.2024 um 18:58 schrieb Jim Burns:

[...]

>
You got it totally wrong!
>
The dark unit fractions are smaller than
the (all) visible ones.

No positive point is a lower.bound of
all the visibleᵂᴹ unit.fractions. (lemma)
Each darkᵂᴹ unit.fraction is a positive point.
No darkᵂᴹ unit.fraction is a lower.bound of
all the visibleᵂᴹ unit.fractions.
No darkᵂᴹ unit.fraction is smaller than
all the visibleᵂᴹ unit.fractions.
----
Lemma.
No positive point is a lower.bound (lb) of
all the visibleᵂᴹ unit.fractions.
⎛ Assume 0 < lb.⅟ℕᵈᵉᶠ ≤ glb.⅟ℕᵈᵉᶠ = β
⎜ not.bound 2⋅β > ⅟k ∈ ⅟ℕᵈᵉᶠ
⎜ not.bound ½⋅β > ¼⋅⅟k ∈ ⅟ℕᵈᵉᶠ
⎜ bound ½⋅β < β
⎝ Contradiction.
¬(lb.⅟ℕᵈᵉᶠ > 0)

Now:
The visible unit fraction don't have
a smallest one (of course),

More than that.
The visibleᵂᴹ unit fractions don't have
a positive lower bound.
WM has not quite conceded that.
  The last I've seen, he omits 'greatest'.
If he ever does, it is game over.
With or without his concession,
there is no positive lower bound of
visibleᵂᴹ unit fractions,  and
there is no darkᵂᴹ unit.fraction.

but the dark unit fraction do
(at least in mückenmath)!
>
WM: "Dark unit fractions have a smallest element."

WM says a lot of things.
If dark unit.fractions are positive lower bounds of
visible unit fractions,
then they don't exist,
and {} doesn't have a smallest element.