Re: Replacement of Cardinality

On 8/4/2024 2:13 PM, WM wrote:

Le 04/08/2024 à 18:39, Jim Burns a écrit :

On 8/4/2024 11:29 AM, WM wrote:

Le 03/08/2024 à 21:54, Jim Burns a écrit :

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

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.

>

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

>
When all x are involved,
the universal quantifier is usually not written.

When a universal quantifier is not written,
it is implicit, and
it can only stand implicitly outside the formula.
∃k≠j:j<k is implicitly ∀j(∃k≠j:j<k)
When a universal quantifier is implicit,
a quantifier shift is impossible to write.
∃k≠j:j<k  always means  ∀j∃k≠j:j<k
∃k≠j:j<k  never means   ∃k∀j≠k:j<k

More of interest are these two claims which are
not both true or both false:
For every x there is u < x.
There is u < x for every x.
The latter is close to my function:
There are NUF(x) u < x.

NUFᵉᵃᶜʰ  :=  |{u∈⅟ℕᵈᵉᶠ: u <ᵉᵃᶜʰ ℝ⁺}|
NUFᵉᵃᶜʰ = 0
NUFᵈᵉᶠ(x)  :=  |{u∈⅟ℕᵈᵉᶠ:u<x}|
∀ᴿx>0: NUFᵈᵉᶠ(x) = ℵ₀
 From ∀x∃U to ∃U∀x is unreliable,
is not.always.correct, is sometimes.incorrect.

Your claim concerns only definable x.

There is no positive lower bound of ⅟ℕᵈᵉᶠ
because
a _positive_ lower.bound of ⅟ℕᵈᵉᶠ implies
there are lower.bounds which don't bound ⅟ℕᵈᵉᶠ
There is no darkᵂᴹ x: 0 < x <ᵉᵃᶜʰ ⅟ℕᵈᵉᶠ
because
darkᵂᴹ x would be a positive lower bound of ⅟ℕᵈᵉᶠ