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NNTP-Posting-Date: Fri, 06 Dec 2024 21:32:33 +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 13:32:26 -0800
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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.