Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary, effectively)

On 12/27/2024 5:24 PM, Ross Finlayson wrote:

On 12/27/2024 01:00 PM, Jim Burns wrote:

[...]

>
The, "almost all", or, "almost everywhere",
does _not_ equate to "all" or "everywhere",

Correct.
⎛ In mathematics, the term "almost all" means
⎜ "all but a negligible quantity".
⎜ More precisely, if X is a set,
⎜ "almost all elements of X" means
⎜ "all elements of X but those in
⎜ a negligible subset of X".
⎜ The meaning of "negligible" depends on
⎜ the mathematical context; for instance,
⎜ it can mean finite, countable, or null.
⎜
⎜ In contrast, "almost no" means
⎜ "a negligible quantity"; that is,
⎜ "almost no elements of X" means
⎜ "a negligible quantity of elements of X".
⎝
https://en.wikipedia.org/wiki/Almost_all

and these days in
sub-fields of mathematics like to do with
topology and the ultrafilter,
it's a usual conceit to
in at least one sense,
not being "actually" correct.

"Almost all" used correctly in a clear context
has a clear meaning, a meaning which isn't "all".
If, despite its being.defined,
one regards "almost all" as not.actual,
one can scratch out "almost all" and
continue, more verbosely, using its definition.
Almost all of ℕ is P
==
(∀)j ∈ ℕ: P(j)
==
∃i ∈ ℕ: ∀j ∈ ℕ:  i<j -> P(j)
That assumes, as background,
certain facts about ℕ
For each j in ℕ,
there are infinitely.many after.j and
there aren't infinitely.many before.j
∀j ∈ ℕ:
∃᳹k ∈ ℕ: j < k  ∧
¬∃᳹i ∈ ℕ: i < j
That assumes, as background,
certain facts about being.finite.
However,
there are some who disagree with those facts.
That's what has my attention, at the moment.
My thought is that
perhaps "almost all" can present
those facts about being.finite
in a less.disagreeable form.