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

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

On 12/04/2024 02:12 PM, Jim Burns wrote:

On 12/4/2024 4:39 PM, Ross Finlayson wrote:

On 12/04/2024 11:37 AM, Jim Burns wrote:

On 12/3/2024 8:09 PM, Ross Finlayson wrote:

Yet, I think that I've always been
both forthcoming and forthright
in providing answers, and context,
in this loooong conversation [...]

>
Please continue being forthcoming and forthright
by confirming or correcting my impression that
"yin-yang ad infinitum"
refers to how, up to ω, claim [1] is true,
about immediate [predecessors],
but, from ω onward, it's negation is true.

>
Thank you in advance for confirming or correcting
my impression of what you mean
(something you have not yet done),
in furtherance of your
forthcoming and forthright posting history.

The thing is,
'not.first.false' is not used to describe ordinals,
in the way that 'yin.yang.ad.infinitum'
is used to describe ordinals.
>
'Not.first.false' is used to describe
_claims about ordinals_ of which we are
  here only concerned with finitely.many claims.
There is no 'ad infinitum' for 'not.first.false'.
>
It is in part the absence of 'ad infinitum'
which justifies claims such as [1] and [2]
>
A linearly.ordered _finite_ set must be well.ordered.
If all claims are true.or.not.first.false,
there is no first false claim.
Because well.ordered,
if there is no first false,
then there is no false,
and all those not.first.false claims are justified.
>
The natural numbers are not finitely.many.
But that isn't a problem for this argument,
because it isn't the finiteness of the _numbers_
which it depends upon,
but the finiteness of the claim.sequence.

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'.
⎛ When, inevitably, you and I will have moved on
⎜ to other discussions,
⎜ I (JB) would like to be able to think back on
⎜ at least leaving you (RF) with
⎜ _an awareness of what I am saying_
⎜ even if nothing else was accomplished.
⎜
⎜ Currently, it seems as though
⎜ I have not cleared that low, low bar.
⎜ You seem to be responding to some _other_ 'JB'
⎜
⎜ Upon once more reading what I've said and
⎜ what you've said, I feel
⎜ what.I'm.going.to.call 'intellectual.dizziness':
⎜ something approximating what I felt
⎜ just a couple days ago, when I re.watched
⎜ the Minions movie (2015)
⎜ https://www.imdb.com/title/tt2293640/
⎜ A perpetual dance.and.wave _just beyond_
⎜ the edge of comprehensibility,
⎜
⎜ It is what it is.
⎜
⎝ But, enough about me.
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).