Path: ...!Xl.tags.giganews.com!local-4.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail
NNTP-Posting-Date: Fri, 06 Dec 2024 02:25:31 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary, not.ultimately.untrue)
Newsgroups: sci.math
References: <vg7cp8$9jka$1@dont-email.me> <viag8h$lvep$1@dont-email.me>
 <viaj9q$l91n$1@dont-email.me> <vibvfo$10t7o$1@dont-email.me>
 <vic6m9$11mrq$4@dont-email.me> <vicbp2$1316h$1@dont-email.me>
 <vid4ts$1777k$2@dont-email.me> <vidcv3$18pdu$1@dont-email.me>
 <bdbc0e3d-1db2-4d6a-9f71-368d36d96b40@tha.de> <vier32$1madr$1@dont-email.me>
 <vierv5$1l1ot$2@dont-email.me> <viiqfd$2qq41$5@dont-email.me>
 <vik73d$3a9jm$1@dont-email.me> <vikg6c$3c4tu$1@dont-email.me>
 <9bcc128b-dea8-4397-9963-45c93d1c14c7@att.net>
 <tfGdnSEN5pqwydP6nZ2dnZfqn_cAAAAA@giganews.com>
 <210dfaf2-ad0a-4b39-b7c4-9d5a86198ed9@att.net>
 <rbicnRIWCP4n5NP6nZ2dnZfqnPGdnZ2d@giganews.com>
 <7eded0f4-bd92-49db-925a-4248e823a29b@att.net>
 <D8idnaTTItp6NtL6nZ2dnZfqn_ednZ2d@giganews.com>
 <e604276b-10e0-4919-9665-2c06dae03422@att.net>
 <SKCcnR76RKm3Uc36nZ2dnZfqnPSdnZ2d@giganews.com>
 <0e8fb26a-96f6-4905-800c-57b0d22f1971@att.net>
 <HvGcncRC99LwRs36nZ2dnZfqnPidnZ2d@giganews.com>
 <cd0d1a2b-69b4-49ce-86af-8cc1d73cb454@att.net>
From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Thu, 5 Dec 2024 18:25:28 -0800
User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101
 Thunderbird/38.6.0
MIME-Version: 1.0
In-Reply-To: <cd0d1a2b-69b4-49ce-86af-8cc1d73cb454@att.net>
Content-Type: text/plain; charset=utf-8; format=flowed
Content-Transfer-Encoding: 8bit
Message-ID: <m1Sdnb1u8KcG_c_6nZ2dnZfqn_WdnZ2d@giganews.com>
Lines: 207
X-Usenet-Provider: http://www.giganews.com
X-Trace: sv3-YqW0WHsgh1gjWGgDS96O4NfG3cYvxSQiiWq3oJUQm4jMHYXKE2QZpEkBWwCwKrjK10PIRLj6I8cyd6T!dwKQqq7TxsJOpj/EsqhsgWlcgmHcw43LlvThcnOi8ecb0JYVRiqQtBR0yCzEIBiTBVf8OVMthJB6
X-Complaints-To: abuse@giganews.com
X-DMCA-Notifications: http://www.giganews.com/info/dmca.html
X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers
X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly
X-Postfilter: 1.3.40
Bytes: 9146

On 12/05/2024 10:14 AM, Jim Burns wrote:
> On 12/4/2024 5:44 PM, Ross Finlayson wrote:
>> On 12/04/2024 02:12 PM, Jim Burns wrote:
>>> On 12/4/2024 4:39 PM, Ross Finlayson wrote:
>>>> On 12/04/2024 11:37 AM, Jim Burns wrote:
>>>>> On 12/3/2024 8:09 PM, Ross Finlayson wrote:
>
>>>>>> 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 [predecessors],
>>>>> but, from ω onward, it's negation is true.
>>>
>>> Thank you in advance for confirming or correcting
>>> my impression of what you mean
>>> (something you have not yet done),
>>> in furtherance of your
>>> forthcoming and forthright posting history.
>
>>>>> 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.
>
>> 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'.
>
> ⎛ When, inevitably, you and I will have moved on
> ⎜ to other discussions,
> ⎜ I (JB) would like to be able to think back on
> ⎜ at least leaving you (RF) with
> ⎜ _an awareness of what I am saying_
> ⎜ even if nothing else was accomplished.
> ⎜
> ⎜ Currently, it seems as though
> ⎜ I have not cleared that low, low bar.
> ⎜ You seem to be responding to some _other_ 'JB'
> ⎜
> ⎜ Upon once more reading what I've said and
> ⎜ what you've said, I feel
> ⎜ what.I'm.going.to.call 'intellectual.dizziness':
> ⎜ something approximating what I felt
> ⎜ just a couple days ago, when I re.watched
> ⎜ the Minions movie (2015)
> ⎜ https://www.imdb.com/title/tt2293640/
> ⎜ A perpetual dance.and.wave _just beyond_
> ⎜ the edge of comprehensibility,
> ⎜
> ⎜ It is what it is.
> ⎜
> ⎝ But, enough about me.
>
> 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).
>
>

No, I say "not.ultimately.untrue" is
_more_ than "not.first.false".

The account where you have drawn thinkers
into your shell and closed the door, is
========== REMAINDER OF ARTICLE TRUNCATED ==========