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

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

On 11/18/2024 12:59 PM, Ross Finlayson wrote:

On 11/18/2024 07:46 AM, Jim Burns wrote:

>

I plan to turn to your argument
once we have finished with
⎛ A FINITE SEQUENCE OF CLAIMS, each claim of which
⎜ is true.or.not.first.false  is
⎜ a FINITE SEQUENCE OF CLAIMS, each claim of which
⎝ is true.
>
What do you have to say about that, Ross?

>
I already did you keep clipping it.

>

Why don't you look back about the last three posts
and see an example where an inductive argument FAILS
and is nowhere finitely "not.first.false",
that it yet FAILS.

>
My first reaction was that this is not
an inductive argument
⎛ A FINITE SEQUENCE OF CLAIMS, each claim of which
⎜ is true.or.not.first.false  is
⎜ a FINITE SEQUENCE OF CLAIMS, each claim of which
⎝ is true.
>
However, yes,
my argument depends upon the well.ordering of CLAIMS,
and that works out to being an inductive argument.
>
⎛ Assuming transfinite.induction is valid
⎜ in finite sequence P,
⎜ if,
⎜ for each claim,
⎜ its truth is implied by the truth of all prior claims,
⎜ then,
⎜ for each claim,
⎝ that claim is true.
>
That is a transfinite.inductive argument.
For finite sequence P of claims,
( ∀ᴾψ:(⊤ψ⇐∀ᴾξ≺ᴾψ:⊤ξ) ⇒ ∀ᴾφ:⊤φ
>

So, it's a counterexample,
and illustrates why what's not.first.false must
also be not.ultimately.untrue to not FAIL.

>
So, this is why I keep clipping your "counterexample".
>
Your "counterexample" needs
a finite sequence of claims which is NOT well.ordered.
>
You say you have a counter.example.
Congratulations. Your Fields Medal is in the mail.
>
You say Mirimanoff and Finsler and Boffa support you.
Do they have non.well.ordered finite sequences as well?
Let's throw a party!
>
Or, maybe, _kindly_ I should assume you misunderstand.
>

Then about Russell's retro-thesis and

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

Giving an ordinal assignment as "being", cardinals,
results then there are those among CH and not CH,
that would result contradictory models,
and not even necessarily considering Cohen's forcing,
and that he takes an ordinal out of a well-ordering,
since you asked for example of out-of-order ordinals.
Giving an assignment of reals as partitions, or foresplits
as you put it, of rationals, has that each has a distinct
rational, making a case for transfinite induction that
they have the same cardinal, reals and rationals.
Giving an inductive limit that never touches,
means also it never reaches,
thus never crosses,
thus in a reduction, never arrives.
(That, it does, the, "infinite" limit.)
Quantification, comprehension, over the finite
sets, in a theory with expansion of comprehension,
results an extra-ordinary infinite set,
or it's finite.
Now, with regards to your induction about induction,
the example given is a pretty simple sort of geometric
proof involving exhaustion and the limit, yet contrived
to provide a straight line the limit of the construction.
So, the yin-yang recursively has a constant length
the perimeter, yet in the limit, length the diameter.
Induction does not arrive at a correct result,
because it's not complete.
The "finite not-well-ordered" has nothing to do
with it, only, "finite not extra-ordinary".
The extra-ordinary, may still be well-ordered,
about what's called its fixed point.
Then, the counterexample, simply shows a failure
of convergence criterion, that you've claimed
is only possible via induction, a way it does not.
It does not, SUCCEED, thus results,
"not not.ultimately.untrue: FAIL".
"Fixed-point fail"
The counterexample, has set up a fixed point,
it's the destination, and shown that the limit
arrived at by subtracting the areas, does not
match the limit arrived at by bending the circles.
Which is "zero" area. This is courtesy
"pi: the ratio of a circle's perimeter to diameter".
Also there's my oft-repeated,
"Anybody who
buys and/or shills
material implication
is a fool and/or fraud."
Induction: is not complete.
See rule 1.