Re: Replacement of Cardinality (infinite middle)

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 {}.