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

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?