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

On 8/5/2024 11:05 PM, Ross Finlayson wrote:

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

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

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?

Not.finite set ⅟ℕᵈᵉᶠ∩(0,x) of
not.infinite.and.not.infinitesimal unit.fractions.
----
Definition.
A finite order is a total order in which
each non.{}.subset is 2.ended ==
each non.{}.subset holds its max and its min.
An infinite order is a total order which
is not finite.
⎛ Lemma:
⎝ No set has both a finite and an infinite order.
A set with a finite order is finite.
A set with an infinite order is infinite.
----
Definition.
Each non.{} S ⊆ ℕᵈᵉᶠ has min.S ∈ S.
Each j ∈ ℕᵈᵉᶠ has successor j+1 ∈ ℕᵈᵉᶠ
0 = min.ℕᵈᵉᶠ
Each non.0 k ∈ ℕᵈᵉᶠ has predecessor k-1 ∈ ℕᵈᵉᶠ
----
Consequences.
Each bounded subset of ℕᵈᵉᶠ is finite.
ℕᵈᵉᶠ and each of its unbounded subsets is infinite.
Each element of infinite ℕᵈᵉᶠ is
finitely.preceded and infinitely.succeeded.
Each is finite, all are infinite.
Similarly,
for each u ∈ ⅟ℕᵈᵉᶠ
⅟ℕᵈᵉᶠ∩[u,1] is finite
⅟ℕᵈᵉᶠ∩(0,u] is infinite

Sometimes it's hard to convey intended inflection

Very true.
I depend upon others to point out those times to me.
Of course 𝐼 know what I mean.
That won't tell me that others know. Or don't.
And, perhaps I merely have room for improvement in
how I express my ideas.
But perhaps I didn't carry the 2.
https://www.gocomics.com/bloomcounty/1988/07/10
It's a significant help to me.

What I would ask,
is that you surpass,
the inductive impasse,
with the infinite super-task.

Induction is a finite task of
reasoning about infinitely.many.