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

On 11/18/2024 08:46 PM, Chris M. Thomasson wrote:

On 11/18/2024 8:39 PM, Ross Finlayson wrote:

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

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

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?

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I already did you keep clipping it.

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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!
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Or, maybe, _kindly_ I should assume you misunderstand.
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Then about Russell's retro-thesis and

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First things first.
Are there non.well.ordered finite sequences, Ross?
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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.
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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.)
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Quantification, comprehension, over the finite
sets, in a theory with expansion of comprehension,
results an extra-ordinary infinite set,
or it's finite.
>
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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.
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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.
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It does not, SUCCEED, thus results,
"not not.ultimately.untrue: FAIL".
>
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"Fixed-point fail"
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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".
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Also there's my oft-repeated,
>
"Anybody who
buys and/or shills
material implication
is a fool and/or fraud."
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Induction: is not complete.
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See rule 1.
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https://www.youtube.com/watch?v=ucPhfzCvKnE&t=3060
>
Michael Dummett has a great account of Frege over on
"Philosophy Overdose" which has posted many recordings
in philosophy.  While, Dummett on Frege's life is not really
reflective as here we read of the "Posthumous Frege", then
it gets into various pragma and requirements of logical theories
about mathematical objects that point to Dummett as a bit
more controversial with regards to standard theory than
usually considered, while as well he can always wrap it up
as the plain analytic, the universals and back-and-forth
and "we must say what we're talking about before we say
what it is", helps establish that any theory of all mathematics
is of course very thorough.
>
Another recording I've much enjoyed is Dirac, various
recordings of Dirac are quite profound.
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https://www.degruyter.com/document/doi/10.1515/agph-2023-0052/pdf?
srsltid=AfmBOooHf3-ryoBRAfRnwj28SyfL_6gPcRVyn_Dj-2Ox3YbdtA_bf5k4
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Here's something about that Chrysippus
is perhaps mis-read, for example by today's
neo-Stoics who are often brainless neck-beards,
looking for an excuse for their misdeeds,
what with regards to that Chrysippus has his
great moods as a defense against Plotinus
and "the less scrupulous then-neo-Stoics" their
funks, anyways that this thesis contends to convey
that ...
>
"My solution finds in Chrysippus a distinction
later made by the medieval logician John Buridan:
that between being possible (there being a state
of affairs that may occur) and being possibly-true
(there being some proposition whose truth-conditions
are that state of affairs). Buridan and Chrysippus
have radically opposing views on the nature of propositions."
>
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Perhaps you'd be better off thinking about it more
thoroughly then always betting on "letting it ride".
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Or, if only in some few "super-classical" cases
that though call into question the sort of
expedient, "certification".
>
Of course you can find a defense of a dialectic
in Aristotle, not merely "see rule 1", or, as so
often is found in those times, "steal rule 1".
(A usual complete flaw in "material implication"
and the "poisoning the tree" of reason.)
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Or, "true Stoics are Ascetics".  And the rest are liars.
It's kind of like "neo-Hegelians":  not Hegelians.
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Buridan is usually enough associated with
indecision paralysis, one imagines he randomly
switch his donkeys.  Yet, there's "the secretary problem",
where, it's usually a good idea to interview at least two.
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https://en.wikipedia.org/wiki/Buridan%27s_donkey
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"One reason why the secretary problem has received
so much attention is that the optimal policy for the
problem (the stopping rule) is simple and selects the
single best candidate about 37% of the time, ...."
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-- https://en.wikipedia.org/wiki/Secretary_problem
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Not to be confused by, "Monty Hall".
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So, perhaps now it's more clear why it takes a
quite fuller and wider dialectic to arrive at
mathematical "infinity" and "continuity".
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Wm says you don't know the power of the dark side?
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https://youtu.be/HEYrRNMr2kg?list=RDMM
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;^)

That one bad apple spoiled the whole damn bunch.
Dummett references a "context" principle, much like
a "domain" principle, that there exists for each text,
a wider con-text, that then given a
modal
relevance
logic
and as well a
geometry
that it's demonstrable that some arguments
by induction: FAIL.
This is where, accordingly, there is a wider
context in the domain, where the truth wins,
where in mathematics there is "right" and "wrong"
and they are not the same, so that without making
aesthetic or subjective judgments, there can be
made judgments of reason, that are right, and
especially, refute those that are wrong.
So, the "Context Principle", and "Domain Principle",
aren't necessarily part of small logics or fragments
of logic, yet, are so in the wider concern with
regards to real notions like the mathematical
"infinity" and "continuity".