Re: Replacement of Cardinality

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

Le 13/08/2024 à 18:26, Jim Burns a écrit :

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.

But the smallest unit fractions comes before.

 No visibleᵂᴹ.or.darkᵂᴹ point ε > 0  is the end of
both visibleᵂᴹ and darkᵂᴹ unit.fractions.

 Every ε > 0 is visible, because it is chosen.

No visibleᵂᴹ point ε > 0  is the end of
both visibleᵂᴹ and darkᵂᴹ unit.fractions.
No darkᵂᴹ point ε > 0  is the end of
both visibleᵂᴹ and darkᵂᴹ unit.fractions.
There is no lower.end of
both visibleᵂᴹ and darkᵂᴹ unit.fractions.

But ℵo unit fractions occupy an interval y > 0
to settle there.

No.
0.many unit.fractions settle.
⎛ Related:
⎝ The intersection of infinite.end.segments is empty.
For each unit fraction ⅟k
there is an initial sub.segment of (0,1]
which ⅟k is not.in.
There are ℵ₀ unit.fractions in the ⅟k.free segment,
but not ⅟k

Not every point of y has ℵo smaller unit fractions.

No.
∀ᴿy > 0: ∀k ∈ ℕ₁:  0 < ⅟⌊k+⅟y⌋ < y

Every chosen ε > 0 is larger than this interval y.

No.
¬∃ᴿy > 0: ¬∀k ∈ ℕ₁:  0 < ⅟⌊k+⅟y⌋ < y