Re: Replacement of Cardinality

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.

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

 No.

>
If darkᵂᴹ numbers are what's between [0,1] and (0,1]

 No that is not the case.
 Regards, WM