Re: Incompleteness of Cantor's enumeration of the rational numbers

On 14.11.2024 00:16, Jim Burns wrote:

A finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀 in which
each claim is true.or.not.first.false
is
a finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲 of 𝗰𝗹𝗮𝗶𝗺𝘀 in which
each claim is true.
 Some claims are true and we know it
because
they claim that
when we say this, we mean that,
and we, conscious of our own minds, know that
when we say this, we mean that.
 Some 𝗰𝗹𝗮𝗶𝗺𝘀 are not.first.false and we know it
because
we can see that
no assignment of truth.values exists
in which 𝘁𝗵𝗲𝘆 are first.false.
𝗾 is not first.false in ⟨ 𝗽 𝗽⇒𝗾 𝗾 ⟩.
 Some finite 𝘀𝗲𝗾𝘂𝗲𝗻𝗰𝗲𝘀 of 𝗰𝗹𝗮𝗶𝗺𝘀 are
each true.or.not.first.false
and we know it.
 When we know that,
we know each claim is true.
 We know each claim is true, even if
it is a claim physically impossible to check,
like it would be physically impossible
to check each one of infinitely.many.
 

Here is a single claim which is true:
The covering of a geometric figure by a set of similar smaller intervals is independent of the order of the intervals. That holds for every finite figure and, by applying the analytical limit, also for infinite figures like
XOOO...
XOOO...
XOOO...
XOOO...
...
or
0---------_1_--------_2_--------_3_---...
Therefore a geometric representation let alone proof of most of Cantor's bijections is impossible.
Belief in set theory excludes belief in geometry.
All babble about Completely Scattered Space in cases of
I(n) = [n - sqrt(2)/2^n, n + sqrt(2)/2^n]
or
I(n) = [n - 1/2^n, n + 1/2^n]
is to no avail and useless from the outset.
Regards, WM