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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-standard)
Date: Mon, 18 Nov 2024 20:46:11 -0800
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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?
>>>>
>>>> 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.
>>
>>
> 
> 
> 
> 
> 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.
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