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

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