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: <9bcc128b-dea8-4397-9963-45c93d1c14c7@att.net> <210dfaf2-ad0a-4b39-b7c4-9d5a86198ed9@att.net> <7eded0f4-bd92-49db-925a-4248e823a29b@att.net> <0e8fb26a-96f6-4905-800c-57b0d22f1971@att.net> From: Ross Finlayson 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: Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: 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 ==========