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

On 11/04/2024 04:18 PM, Moebius wrote:

Am 05.11.2024 um 01:07 schrieb Moebius:
>

For example,
>
        for any two real numbers x, y ... bla bla
>
USUALLY does not exclude x = y. [Might look strange, but is just the
usual math lingo.]

>
Hint: Archimedes Plutonium hat some problems with the convention. Seems
that Jim wants to follow his lead.
>

.
.
.

>
>

It's fair to say that Burns is a great example of dogmatic correctness.
There are others, there's a fuller milieu of "the theory",
as to what constitutes "greater dogmatic correctness".
Burns, or Professor Burns as I sometimes refer to him,
to which he has modestly demurred, is quite great and
here among the best of both foils and straight-men,
even were he a bot, in constancy and a continued
development, and a personal style helping write
first-class logic, in natural language.
Trichotomy, the property for two real numbers that
exactly one of x = y or x < y or x > y, and that
being a transitive property, is about a most fundamental
aspect of "order" or "ordering" theory, where "orderings"
are the fundamental elements of the theory, much like
in "set" theory were sets are fundamental, and elementary.
Then, with regards to ordering and lack thereof, there
is a notion about sampling from trichotomous domains,
what results a "vague fugue", as with regards to that
it's not so much guaranteed that two samples, after
error in precision, close enough together, are ordered.
Then, "sampling, observer, and measurement effects",
generally are seen to include that this makes for
what's called "interval arithmetic" with regards to
the representation of real-valued variables in
fixed or for that matter arbitrary precision machine
numbers.
AP, if you don't recall about 30 years ago, was pretty
great, for a couple notions including "universal atomism"
and "atomic universalism", then though it's largely agreed
that his working or practical theory, if that's not too
generous an appelation, was of insufficient probity
for a candidate in foundations.
So, Burns then is of certainly a high measure of reliability,
then as with regards to a certain perceived inflexibility,
it's not uncommon with regards to the "inductive impasse"
and the role of deductive analysis and the fuller dialectic
in the "analytical bridges (analytische bruecken)", that
the linear curriculum barely has enough time to get one
formalism in place, then that the wider and fuller curriculum,
is largely for more thoroughly logical and widely-mindful
students of _all_ the theory, vis-a-vis, practical working theory.
Even if that's a bot, ..., of which there are plenty.