Re: Replacement of Cardinality

On 8/7/2024 3:05 PM, WM wrote:

Le 07/08/2024 à 20:49, Jim Burns a écrit :

On 8/7/2024 9:05 AM, WM wrote:

Le 07/08/2024 à 04:36, Richard Damon a écrit :

On 8/6/24 4:35 AM, WM wrote:

Right. But with NUF(x) = 1 ==> INVNUF(1) = x we get
∃u ∈ ⅟ℕ, u < x, ∀y > x = INVNUF(1).

>
But INVNUF(1) can't exist, as it will be bigger than
1/ ( ceil(1/INVNUF(1)) +1 ), and 1/ ( ceil(1/INVNUF(1)) +2 )
which are two different unit fractions.

>
Peano is not valid for all dark numbers.

>
For each real x > 0
there are ℵ₀.many visibleᵂᴹ unit.fractions
between x and 0

>
For each visible real.
>

There is no real x > 0 such that
there are fewer than ℵ₀.many visibleᵂᴹ unit fractions
between x and 0

>
They exist but are dark.

If darkᵂᴹ numbers exist,
then there are
visibleᵂᴹ unit.fractions between 0 and the darkᵂᴹ.
There is no positive lower.bound of
visibleᵂᴹ unit.fractions.
Any hypothetically.darkᵂᴹ positive number not so
is a positive lower.bound of visibleᵂᴹ unit.fractions.
It, as with all positive lower.bounds, doesn't exist.
There is no positive lower.bound of
visibleᵂᴹ unit.fractions.
⎛ Assume otherwise.
⎜ Assume β is a positive greatest.lower.bound
⎜
⎜ ½⋅β is a lower.bound
⎜
⎜ 2⋅β is not.a.lower.bound
⎜ ⅟k < 2⋅β a visibleᵂᴹ unit.fraction exists
⎜ ¼⋅⅟k < ¼⋅2⋅β a visibleᵂᴹ unit.fraction exists
⎜ ½⋅β = ¼⋅2⋅β is not.a.lower.bound
⎝ Contradiction.

Darkᵂᴹ numbers do not make _fewer_ visibleᵂᴹ numbers,
do they?

>
No.

∀ᴿx>0: NUFᵛⁱˢ(x) = ℵ₀