Re: Replacement of Cardinality

Le 02/08/2024 à 19:06, Jim Burns a écrit :

(0,x] inherits from its superset (0,1] properties by which,
for ⅟ℕᶠⁱⁿ∩(0,x] finite.unit.fractions in (0,x]
each non.{}.subset is maximummed,  and
each finite.unit.fraction is down.stepped,  and
each finite.unit.fraction in is non.max.up.stepped.
 Therefore,
the finite.unit.fractions in ⅟ℕᶠⁱⁿ∩(0,x] are ℵ₀.many.
 ∀ᴿx > 0:  NUFᶠⁱⁿ(x) = ℵ₀

I recognized lately that you use the wrong definition of NUF.
Here is the correct definition:
There exist NUF(x) unit fractions u, such that for all y >= x: u < y.
Note that the order is ∃ u ∀ y.
NUF(x) = ℵ₀ for all x > 0 is wrong. NUF(x) = 1 for all x > 0 already is wrong since there is no unit fraction smaller than all unit fractions. ℵ₀ unit fractions need ℵ₀*2ℵ₀ points above zero.
Regards, WM