Path: ...!Xl.tags.giganews.com!local-3.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail 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 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> <6b1b1e57-fef5-475c-94f8-3e7c9061e70a@att.net> From: Ross Finlayson Date: Fri, 6 Dec 2024 15:02:53 -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: 136 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-VWYebgQhooifGREgeq3Z9QBRcv7IB5P2SuRj8bJv1QNgMlt/FCnhxgeBvn0z9xQzT2rliiBvNp7qZIX!R8dTGAWt7Z+IG0kDlfZuIn9cNyPU2TQnoi1Q1JiyPX6vJyr6QjOJZVT4uu5pL6m8Za91iRPTzUM= 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: 6443 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