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NNTP-Posting-Date: Wed, 04 Dec 2024 21:58:59 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary, not.ultimately.untrue)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Wed, 4 Dec 2024 13:59:05 -0800
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On 12/04/2024 01:39 PM, Ross Finlayson wrote:
> On 12/04/2024 11:37 AM, Jim Burns wrote:
>> On 12/3/2024 8:09 PM, Ross Finlayson wrote:
>>> On 12/03/2024 04:16 PM, Jim Burns wrote:
>>
>>>> [...]
>>>
>>> it was very brave of you when you admitted that
>>> "not.first.false"
>>> is not so much justifying itself and
>>> not un-justifying itself,
>>
>> Not.first.false claims are justified when
>> they are in finite sequences of such claims,
>> in which each claim is true.or.not.first.false.
>>
>> You (RF) occasionally attribute to me (JB)
>> the most surprising things.
>> Not only do I not recognize them as mine,
>> but they are so distant from mine that
>> I can't guess what you've misunderstood.
>>
>>> with regards to the "yin-yang ad infinitum",
>>> which inductively is a constant
>>> yet in its completion is different,
>>
>> This description of "yin-yang ad infinitum"
>> suggests to me that
>> you are describing what the claims _are about_
>> whereas 'not.first.false' describes
>> the claims _themselves_
>>
>>> with regards to the "yin-yang ad infinitum",
>>> which inductively is a constant
>>> yet in its completion is different,
>>
>> Consider this finite sequence of claims
>> ⎛⎛ By 'ordinals', we mean those which
>> ⎜⎜ have only sets.with.minimums and {}
>> ⎜⎝ ('well.ordered')
>> ⎜
>> ⎜⎛ By 'natural numbers', we mean those which
>> ⎜⎜ have a successor,
>> ⎜⎜ are a successor or 0,  and
>> ⎜⎝ are an ordinal.
>> ⎜
>> ⎜ (not.first.false claim)
>> ⎜
>> ⎜ (not.first.false claim)
>> ⎜
>> ⎜ (not.first.false claim)
>> ⎜
>> ⎜ ...
>> ⎜
>> ⎜ not.first.false claim [1]:
>> ⎜⎛ Each non.zero natural number
>> ⎜⎜ has,
>> ⎜⎜ for it and for each of its non.zero priors,
>> ⎜⎝ an immediate ordinal.predecessor.
>> ⎜
>> ⎜ not.first.false claim [2]:
>> ⎜⎛ The first transfinite ordinal, which we name 'ω',
>> ⎜⎜ and each of its ordinal.followers
>> ⎜⎜ does not have,
>> ⎜⎜ for it and for each of its non.zero priors,
>> ⎜⎜ an immediate ordinal.predecessor.
>> ⎜⎜ That is,
>> ⎜⎜ there is a non.zero prior without
>> ⎝⎝ an immediate ordinal.predecessor.
>>
>> ----
>>> Yet, I think that I've always been
>>> both forthcoming and forthright
>>> in providing answers, and context, in
>>> this loooong conversation [...]
>>
>> Please continue being forthcoming and forthright
>> by confirming or correcting my impression that
>> "yin-yang ad infinitum"
>> refers to how, up to ω, claim [1] is true,
>> about immediate precessors,
>> but, from ω onward, it's negation is true.
>>
>> The thing is,
>> 'not.first.false' is not used to describe ordinals,
>> in the way that 'yin.yang.ad.infinitum'
>> is used to describe ordinals.
>>
>> 'Not.first.false' is used to describe
>> _claims about ordinals_ of which we are
>>   here only concerned with finitely.many claims.
>> There is no 'ad infinitum' for 'not.first.false'.
>>
>> It is in part the absence of 'ad infinitum'
>> which justifies claims such as [1] and [2]
>>
>> A linearly.ordered _finite_ set must be well.ordered.
>> If all claims are true.or.not.first.false,
>> there is no first false claim.
>> Because well.ordered,
>> if there is no first false,
>> then there is no false,
>> and all those not.first.false claims are justified.
>>
>> The natural numbers are not finitely.many.
>> But that isn't a problem for this argument,
>> because it isn't the finiteness of the _numbers_
>> which it depends upon,
>> but the finiteness of the claim.sequence.
>>
>>
>
> Oh, well, tertium non datur or PEM, principle
> of excluded middle, LEM, law of excluded middle,
> TND, no third ground, have that inductive accounts
> may _not_ bring their own completeness, if what
> you'd rather is an inductive account that no
> inductive account is not.ultimately.untrue.
>
> Which is well known since antiquity and also
> as to why the only thing left is for metaphysics
> and foundations to arrive at a theory as with
> regards to not making an inductive account
> after its implied un-founded (and often un-stated)
> assumptions - we should figure out here for whom
> it's called that any theory with finitely many
> axioms is having a neat, and brief, formal counterexample,
> and it's the same for each of them, though you
> can point at Goedel with regards to complete/not-complete
> these what are _partial_, at best, inductive accounts.
> (At best, ....)
>
>
> So, regularity is still a thing of course, and
> monotonic entailment, and for a theory with a modality,
> yet anything mathematical introduces itself to a
> great relevant concern called "the domain of discourse"
> or "the universe of mathematical objects", and then
> there's Zeno again "hey, how about a brief discourse
> on metaphysics?"
>
>
> Also a physics, ..., and "the theory".
>
> So, not.first.false, is only after some pair-wise
> comprehension, because, there are ready example
> that in "super-task comprehension", or what's
> called the illative when it's correct and completes
> and what's usually called "undefined" when it doesn't,
> lazy positivists, it's that not.first.false is either
> not.ultimately.untrue, or it's not.
>
> And sometimes (some times, a modal account), ..., it's not.
>
> Of course there are then _reasons_ _why_
> what is so is so.
>
>
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