Re: Replacement of Cardinality (infinite middle)

On 08/12/2024 09:25 PM, Jim Burns wrote:

On 8/12/2024 8:28 PM, Ross Finlayson wrote:

On 08/12/2024 04:06 PM, Jim Burns wrote:

On 8/12/2024 4:59 PM, Ross Finlayson wrote:

>

it's
beginning ... ( ... infinitely-many ...) ... end,
where the upper.end of the finites always exists.

>
Note: "always".
>

For ω as I've defined it, no upper.end exists.
>
for each k ∈ ω
𝕆ᶠⁱⁿ(k)
𝕆ᶠⁱⁿ(k+1)
k+1 ∈ ω
k is not the upper end of ω

>

for each k ∉ ω
k is not the upper end of ω

>
ω := {z:𝕆ᶠⁱⁿ(z)}
>

It's like yesterday,
in this thread with the subject of it
talking about
"infinite in the middle and
always with both ends",

>
I have just realized that
I have been overlooking your "always".
>
"ALWAYS with both ends" is finite.
>
⎛ Necessary and sufficient conditions for finiteness
⎜
⎜ [...]
⎜ 3. (Paul Stäckel [1862...1919])
⎜ S can be given a total ordering which is
⎜ well-ordered both forwards and backwards.
⎜ That is, every non-empty subset of S has
⎜ both a least and a greatest element in the subset.
⎝
https://en.wikipedia.org/wiki/Finite_set
>
"Finite" does NOT need to be small.
"Infinite" does NOT mean you can get there, but bigly.
>
"Arguments" without a common understanding of terms
are _at least_ vastly more difficult than they need be.
They might only exist in {}.
>
>

If it's infinite in the middle then, the middle acts
as the fixed-point, thus augmenting automatically yon definition,
and suffering not this.
Or, for example, it's a counter-example.
That introduces two particular terms relevant this discussion
of the vagaries of infinites and infinitesimals: the
counter-example, and particularly the counter-induction,
and the contra-positive, or as of the contra-punctal.
The contra-positive of De Morgan's _laws_ of logic is simply
missing from "material implication". The entire deductive
apparatus and structure resulting is absent it.
The counter-induction applies however you'd like, for
example "next is greatest yet" and "next is not last".
Either way it's deduction to save the day from either
the fallacies of incomplete negations, or fallacies
of incomplete completions, both arrived at from
deductive analysis.