Re: Replacement of Cardinality (defining numbers as half finite, half infinite)

On 08/05/2024 03:19 PM, Jim Burns wrote:

On 8/5/2024 3:04 PM, WM wrote:

Le 04/08/2024 à 22:16, Jim Burns a écrit :

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

>

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.

>

 From ∀x∃U to ∃U∀x is unreliable,

>
There is no from to.
NUF(x) is so defined.

>
NUFᵉᵃᶜʰ  :=  |{u∈⅟ℕᵈᵉᶠ: u <ᵉᵃᶜʰ ℝ⁺}|
NUFᵈᵉᶠ(x)  :=  |{u∈⅟ℕᵈᵉᶠ: u < x}|
>
∀v ∈ ⅟ℕᵈᵉᶠ:
⎛ ¬(v < u/2 ∈ ℝ⁺)
⎜ ¬(v <ᵉᵃᶜʰ ℝ⁺)
⎝ ¬(v ∈ {u∈⅟ℕᵈᵉᶠ: u <ᵉᵃᶜʰ ℝ⁺})
>
NUFᵉᵃᶜʰ = |{}| = 0
>
⅟ℕᵈᵉᶠ∩(0,1] is maximummed.
⅟ℕᵈᵉᶠ∩(0,x) is maximummed.
>
⅟ℕᵈᵉᶠ∩(0,1] is down.stepped.
⅟ℕᵈᵉᶠ∩(0,x) is down.stepped.
>
⅟ℕᵈᵉᶠ∩(0,1] is non.max.up.stepped.
⅟ℕᵈᵉᶠ∩(0,x) is non.max.up.stepped.
>
⅟ℕᵈᵉᶠ∩(0,1] is ℵ₀.many.
⅟ℕᵈᵉᶠ∩(0,x) is ℵ₀.many.
>
NUF(1) = ℵ₀
NUF(x) = ℵ₀
>

For every number of unit fractions
NUF(x) gives the smallest interval (0, x).

>
For  each real x > 0
for each finite cardinal k > 0
there is a finite.unit.fraction uₖ such that
|⅟ℕᵈᵉᶠ∩[uₖ,x)| = k
uₖ = ⅟⌊k+⅟x⌋
>
For  each x > 0
for each finite cardinal k > 0
there is a finite.unit.fraction uₖ such that
|⅟ℕᵈᵉᶠ∩[uₖ,x)| = k
⅟(1+⅟uₖ) ∈ ⅟ℕᵈᵉᶠ∩(0,uₖ)
k < |⅟ℕᵈᵉᶠ∩(0,x)|
>
For  each x > 0
for each finite number k
k ≠ |⅟ℕᵈᵉᶠ∩(0,x)|
>
For  each x > 0
⅟ℕᵈᵉᶠ∩(0,x) is not finite.
>
>

Not.finite what?
Not finite set, not finite number?
Sometimes it's hard to convey intended inflection
then as with regards to multiple intended and
multiple unintended inflections, here as with
respect to question words or interrogatives,
that the question words so much reflect the
deductive apparatus, vis-a-vis, the inductive
apparatus, intended to convey what it is to be
arrived at, i.e. that to which is to be where
is what is so arrived, at, the syllogism mostly
never has question words, then as with regards
to the syllogism of question words.
That that that that that that, ...,
is a reflection on implicit quantifiers.
So, declaratives, and interrogatives, with
regards to zero and omega w being the two
constants given, of ZF, that, ...,
NU(w)
either isn't or is a thing, as with regards
to whether you think number theory has a point
at infinity, or not.
What I would ask,
is that you surpass,
the inductive impasse,
with the infinite super-task.