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

On 08/06/2024 04:57 AM, Jim Burns wrote:

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:

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

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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⌋
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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)|
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For  each x > 0
for each finite number k
k ≠ |⅟ℕᵈᵉᶠ∩(0,x)|
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For  each x > 0
⅟ℕᵈᵉᶠ∩(0,x) is not finite.

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Not.finite what?
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Not finite set, not finite number?

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

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Very true.
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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.
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And, perhaps I merely have room for improvement in
how I express my ideas.
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But perhaps I didn't carry the 2.
https://www.gocomics.com/bloomcounty/1988/07/10
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It's a significant help to me.
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What I would ask,
is that you surpass,
the inductive impasse,
with the infinite super-task.

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Induction is a finite task of
reasoning about infinitely.many.
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Thus deductive inference, ..., or, is it?
The super-task or Thomson's Lamp has helped
to establish the infinitary as within reason.