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NNTP-Posting-Date: Fri, 06 Dec 2024 23:02:45 +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: Fri, 6 Dec 2024 15:02:53 -0800
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On 12/06/2024 01:32 PM, Ross Finlayson wrote:
> On 12/06/2024 10:51 AM, Jim Burns wrote:
>> On 12/5/2024 9:25 PM, Ross Finlayson wrote:
>>> On 12/05/2024 10:14 AM, Jim Burns wrote:
>>>> On 12/4/2024 5:44 PM, Ross Finlayson wrote:
>>
>>>>> About your posited point of detail, or question,
>>>>> about this yin-yang infinitum,
>>>>> which is non-inductive, and
>>>>> a neat also graphical example of the non-inductive,
>>>>> a counter-example to the naively inductive,
>>>>> as with regards to whether it's not so
>>>>> at some finite or not ultimately untrue,
>>>>> I'd aver that it introduces a notion of "arrival"
>>>>> at "the trans-finite case",
>>>>
>>>>> Anyways your point stands that
>>>>> "not.first.false" is not necessarily
>>>>> "not.ultimately.untrue",
>>>>> and so does _not_ decide the outcome.
>>>>
>>>> Thank you for what seems to be
>>>> a response to my request.
>>>>
>>>> You seem to have clarified that
>>>> your use of
>>>> 'not.ultimately.untrue' and 'yin-yang ad infinitum'
>>>> is utterly divorced from
>>>> my use of
>>>> 'not.first.false'.
>>
>>>> A couple thousand years ago,
>>>> the Pythagoreans developed a good argument
>>>> that √2 is irrational.
>>>>
>>>> ⎛ The arithmetical case was made that,
>>>> ⎜ for each rational expression of √2
>>>> ⎜ that expression is not.first.√2
>>>> ⎜
>>>> ⎜ But that can only be true if
>>>> ⎜ there _aren't any_ rational expressions of √2
>>>> ⎜
>>>> ⎜ So, there aren't any,
>>>> ⎝ and √2 is irrational.
>>>>
>>>> Mathematicians,
>>>> ever loath to let a good argument go to waste,
>>>> took that and applied it (joyously, I imagine)
>>>> in a host of other domains.
>>>>
>>>> Applied, for example, in the domain of claims.
>>>>
>>>> In the domain of claims,
>>>> there are claims.
>>>> There are claims about rational.numbers,
>>>> irrational.numbers, sets, functions, classes, et al.
>>>>
>>>> An argument over the domain of claims
>>>> makes claims about claims,
>>>> claims about claims about rational numbers, et al.
>>>>
>>>> We narrow our focus to
>>>> claims meeting certain conditions,
>>>> that they are in a finite sequence of claims,
>>>> each claim of which is true.or.not.first.false.
>>>>
>>>> What is NOT a condition on the claims is that
>>>> the claims are about only finitely.many, or
>>>> are independently verifiable, or,
>>>> in some way, leave the infinite unconsidered.
>>>>
>>>> We narrow our focus, and then,
>>>> for those claims,
>>>> we know that none of them are false.
>>>>
>>>> We know it by an argument echoing
>>>> a thousands.years.old argument.
>>>> ⎛ There is no first (rational√2, false.claim),
>>>> ⎝ thus, there is no (rational√2, false.claim).
>>
>> ----
>>>> You seem to have clarified that
>>>> your use of
>>>> 'not.ultimately.untrue' and 'yin-yang ad infinitum'
>>>> is utterly divorced from
>>>> my use of
>>>> 'not.first.false'.
>>
>>> No, I say "not.ultimately.untrue" is
>>> _more_ than "not.first.false".
>>
>> Here is how to tell:
>>
>> I have here in my hand a list of claims,
>> each claim true.or.not.first.false,
>> considering each point between a split of ℚ
>> (what I consider ℝ)
>>
>> It is, of course, a finite list, since
>> I am not a god.like being (trust me on this).
>>
>> If anything here is not.ultimately.untrue
>> _what_ is not.ultimately.untrue?
>> The points?
>> The claims, trustworthily true of the points?
>>
>>
>
> Clams?
>
> Where are the clams at/from?
>
> If you ask Zeno, he tells you "oh, you want
> it that way? Alright then you get nothing."
>
> In a continuous world with continuous motion, ....
>
>
> The super-classical reasoning and infinitary
> reasoning is definitely available since the
> ancients and the classical, besides the
> classical expositions of the super-classical
> of Zeno and Archimedes, for examples, there's
> an entire sort of calculus about methods of
> exhaustion, which _do_ reach their limit
> and _are_ perfect, and simply accessible
> to the mind.
>
> You're suffering a great sort of blinders,
> and apparently seem switched balk and clam.
>
>

https://www.youtube.com/watch?v=It2zhClTGfI&t=1020