Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-standard)

On 11/19/2024 11:56 AM, Jim Burns wrote:

On 11/19/2024 12:52 PM, Ross Finlayson wrote:

On 11/19/2024 09:50 AM, Ross Finlayson wrote:

On 11/19/2024 09:41 AM, Jim Burns wrote:

On 11/19/2024 8:30 AM, Ross Finlayson wrote:

On 11/19/2024 05:15 AM, Jim Burns wrote:

On 11/18/2024 11:39 PM, Ross Finlayson wrote:

On 11/18/2024 05:45 PM, Ross Finlayson wrote:

On 11/18/2024 04:56 PM, Jim Burns wrote:

>

First things first.
Are there non.well.ordered finite sequences, Ross?

>

Of course I won't address your response.
You're insisting on speaking a language
which sounds misleadingly like mine,
misleadingly like, for example, Cohen's, too.

>

Why would you say, "misleadingly"?

>
Are there non.well.ordered finite sequences, Ross?

>
How about a stoplight?
>
In the temporal, and modal, and finite, thus fixed,
there's a well-ordering of that.
>
In the quasi-modal of whatever variety: there is not.

>
Which I understand as "Yes, there are" and which
you have not corrected.
So, it is "Yes".
>
To answer your question,
that is misleading  because
what is meant by "finite" elsewhere
(by me, by Paul Stäckel, by Wikipedia, ... )
is well.ordered.in.both.directions.
>

What I'm saying is that you can take back
that "not.first.false" guarantees a claim
for inference, when it doesn't.

>
If one changes what words mean,
then claims using them are different, and
have different consequences.
>
Is that the admission you want from me?
Enjoy it in good health.
>

"Broadly" speaking, ....
>
Earlier you disputed that "not.first.false"
and "not.ultimately.untrue" were any different.
>
Now you don't.

>
Earlier I tried to decipher
  what you mean by "not.ultimately.untrue"
>
Now I don't.
>
Now, I strongly suspect that
you have stirred assumptions into "not.ultimately.untrue"
about "not.first.false" and "finite sequences"
that are not true,
and that that's the reason I couldn't catch your meaning.
>
Now, my focus is on
finite (well.ordered.in.both.directions) sequences
of true.or.not.first.false claims.
>

The "bait-and-switch" and "back-slide"
don't go well together.
>
In either order, ....

>
⎛ Necessary and sufficient conditions for finiteness
⎜
⎜ 3. (Paul Stäckel)
⎜ S can be given a total ordering which is
⎜ well-ordered both forwards and backwards.
⎜ That is, every non-empty subset of S has both
⎝ a least and a greatest element in the subset.
>
https://en.wikipedia.org/wiki/Finite_set
>
>

Yeah we looked at that before also,
and I wrote another, different, definition of finite.
It's to be kept in mind,
that for various good
meaning logical and relevant reason,
some have that "initial ordinals _as_ cardinals"
makes an inconsistency with regards to Continuum Hypothesis,
that "Dedekind cuts _as_ reals"
makes an inconsistency with regards to cardinality,
and that "infinity _as_ ordinary"
makes an inconsistency with regards to comprehension.
In discussing "the logic", or here
the logic of sets, ordinals, cardinals, rationals, and reals,
it's altogether a one theory, toutes comprises.
The limit: is the infinite limit.
Calulus: is only _perfect_, else imperfect,
in the infinite limit. And: a reductio arrives
at that elsely, there "merely" unbounded or "merely" inductive,
is incomplete, and one may aver, a "fragment", a sort of
after-the-fact, posterior proposition, that for a bit
of expediency in finger-counting reasoning,
keep the circulus vitiosis, absent the circulus virtus,
the circulus logicus, prior and posterior every-which-way.
The yin-yang ad infinitum, demonstrates geometrically,
an inductive case "not.first.false": that is ultimately.untrue.
There are others usually only more accessible to reason
and reason alone, and there are others.