Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary, not.ultimately.untrue)

On 12/06/2024 10:51 AM, Jim Burns wrote:

On 12/5/2024 9:25 PM, Ross Finlayson wrote:

On 12/05/2024 10:14 AM, Jim Burns wrote:

On 12/4/2024 5:44 PM, Ross Finlayson wrote:

>

About your posited point of detail, or question,
about this yin-yang infinitum,
which is non-inductive, and
a neat also graphical example of the non-inductive,
a counter-example to the naively inductive,
as with regards to whether it's not so
at some finite or not ultimately untrue,
I'd aver that it introduces a notion of "arrival"
at "the trans-finite case",

>

Anyways your point stands that
"not.first.false" is not necessarily
"not.ultimately.untrue",
and so does _not_ decide the outcome.

>
Thank you for what seems to be
a response to my request.
>
You seem to have clarified that
your use of
'not.ultimately.untrue' and 'yin-yang ad infinitum'
is utterly divorced from
my use of
'not.first.false'.

>

A couple thousand years ago,
the Pythagoreans developed a good argument
that √2 is irrational.
>
⎛ The arithmetical case was made that,
⎜ for each rational expression of √2
⎜ that expression is not.first.√2
⎜
⎜ But that can only be true if
⎜ there _aren't any_ rational expressions of √2
⎜
⎜ So, there aren't any,
⎝ and √2 is irrational.
>
Mathematicians,
ever loath to let a good argument go to waste,
took that and applied it (joyously, I imagine)
in a host of other domains.
>
Applied, for example, in the domain of claims.
>
In the domain of claims,
there are claims.
There are claims about rational.numbers,
irrational.numbers, sets, functions, classes, et al.
>
An argument over the domain of claims
makes claims about claims,
claims about claims about rational numbers, et al.
>
We narrow our focus to
claims meeting certain conditions,
that they are in a finite sequence of claims,
each claim of which is true.or.not.first.false.
>
What is NOT a condition on the claims is that
the claims are about only finitely.many, or
are independently verifiable, or,
in some way, leave the infinite unconsidered.
>
We narrow our focus, and then,
for those claims,
we know that none of them are false.
>
We know it by an argument echoing
a thousands.years.old argument.
⎛ There is no first (rational√2, false.claim),
⎝ thus, there is no (rational√2, false.claim).

>
----

You seem to have clarified that
your use of
'not.ultimately.untrue' and 'yin-yang ad infinitum'
is utterly divorced from
my use of
'not.first.false'.

>

No, I say "not.ultimately.untrue" is
_more_ than "not.first.false".

>
Here is how to tell:
>
I have here in my hand a list of claims,
each claim true.or.not.first.false,
considering each point between a split of ℚ
(what I consider ℝ)
>
It is, of course, a finite list, since
I am not a god.like being (trust me on this).
>
If anything here is not.ultimately.untrue
_what_ is not.ultimately.untrue?
The points?
The claims, trustworthily true of the points?
>
>

Clams?
Where are the clams at/from?
If you ask Zeno, he tells you "oh, you want
it that way? Alright then you get nothing."
In a continuous world with continuous motion, ....
The super-classical reasoning and infinitary
reasoning is definitely available since the
ancients and the classical, besides the
classical expositions of the super-classical
of Zeno and Archimedes, for examples, there's
an entire sort of calculus about methods of
exhaustion, which _do_ reach their limit
and _are_ perfect, and simply accessible
to the mind.
You're suffering a great sort of blinders,
and apparently seem switched balk and clam.