Re: Replacement of Cardinality

On 8/3/2024 10:23 AM, WM wrote:

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.

∀ᴿy ≥ x:  y > u  ⟺  x > u
⎛ Assume otherwise.
⎜ Assume y ≥ x  ∧  ¬(y > u)  ∧  x > u
⎜
⎜ However, '>' is transitive.
⎜ y ≥ x  ∧  x > u  ⇒  y > u
⎝ Contradiction.
Here is an equivalent definition:
There exist NUF(x) unit fractions u, such that
u < x

Note that the order is ∃ u ∀ y.

The order is ∀x ∃u ∀y
∃u ∀x ∀y is an unreliable quantifier shift.

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.

NUF(x) > 1 for all x > 0 is correct since
each unit.fraction is larger than at least two unit.fractions.
and
a positive lower.bound of finite unit.fractions
implies
finite unit.fractions below a lower.bound,
a contradiction.

ℵ₀ unit fractions need ℵ₀*2ℵ₀ points above zero.

The finite unit.fractions in (0,x] are
maximummed and down.stepped and non.max.up.stepped.
The finite unit.fractions in (0,x] are
ℵ₀.many.