Re: Replacement of Cardinality

On 7/30/2024 1:30 PM, WM wrote:

Le 29/07/2024 à 20:33, Jim Burns a écrit :

On 7/29/2024 9:23 AM, WM wrote:

NUF(x) cannot grow by more than 1 at any x.

>
NUF(x) = |⅟ℕ∩(0,x]|
NUF(x) cannot grow by more than 1 at any x > 0

>
Correct. At x < 0 or x = 0 NUF(x) = 0 and remains so.
>

¬(0 > 0)

>
Correct.
>

⅟ℕ∩(0,x] has
each non.{}.subset maximummed
each unit.fraction down.stepped
each non.maximum up.stepped
and therefore
ℵ₀.many unit.fractions
>
x > 0  ⇒  NUF(x) = ℵ₀

>
That implies a growth between [0, 1] and (0, 1].

No.
x > 0  ⇒  NUF(x) = ℵ₀
does not imply a growth between [0,1] and (0,1]
(Check your work for a quantifier shift.)
NUF(x) = |⅟ℕ∩(0,x]| = ℵ₀
does not imply any unit fraction outside (0,x]
The unit.fractions in (0,x] are
each non.{}.subset maximummed
each unit.fraction down.stepped
each non.maximum up.stepped.
Things with that order.type are ℵ₀.many.
Finite doesn't need to be small.
Infinite is beyond all big.