Path: ...!Xl.tags.giganews.com!local-3.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Wed, 04 Dec 2024 21:58:59 +0000 Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary, not.ultimately.untrue) Newsgroups: sci.math References: <71758f338eb239b7419418f49dfd8177c59d778b@i2pn2.org> <9bcc128b-dea8-4397-9963-45c93d1c14c7@att.net> <210dfaf2-ad0a-4b39-b7c4-9d5a86198ed9@att.net> <7eded0f4-bd92-49db-925a-4248e823a29b@att.net> From: Ross Finlayson Date: Wed, 4 Dec 2024 13:59:05 -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: 185 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-zhZRA4Zm+4t+M59/vVeQ8nXIbKw3ec9d2AbH76jQiI17rfJrGk1KJqYYSiAmO8629lTEXWHVHkR6905!u3WPcGx8dF3J9jyyORAUY6Nntr/0mmmk3h2VRQXAFnzvFkBoUDThn6JXVTK0Uz5FVn6tKzvlLA== 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: 8416 On 12/04/2024 01:39 PM, Ross Finlayson wrote: > On 12/04/2024 11:37 AM, Jim Burns wrote: >> On 12/3/2024 8:09 PM, Ross Finlayson wrote: >>> On 12/03/2024 04:16 PM, Jim Burns wrote: >> >>>> [...] >>> >>> it was very brave of you when you admitted that >>> "not.first.false" >>> is not so much justifying itself and >>> not un-justifying itself, >> >> Not.first.false claims are justified when >> they are in finite sequences of such claims, >> in which each claim is true.or.not.first.false. >> >> You (RF) occasionally attribute to me (JB) >> the most surprising things. >> Not only do I not recognize them as mine, >> but they are so distant from mine that >> I can't guess what you've misunderstood. >> >>> with regards to the "yin-yang ad infinitum", >>> which inductively is a constant >>> yet in its completion is different, >> >> This description of "yin-yang ad infinitum" >> suggests to me that >> you are describing what the claims _are about_ >> whereas 'not.first.false' describes >> the claims _themselves_ >> >>> with regards to the "yin-yang ad infinitum", >>> which inductively is a constant >>> yet in its completion is different, >> >> Consider this finite sequence of claims >> ⎛⎛ By 'ordinals', we mean those which >> ⎜⎜ have only sets.with.minimums and {} >> ⎜⎝ ('well.ordered') >> ⎜ >> ⎜⎛ By 'natural numbers', we mean those which >> ⎜⎜ have a successor, >> ⎜⎜ are a successor or 0, and >> ⎜⎝ are an ordinal. >> ⎜ >> ⎜ (not.first.false claim) >> ⎜ >> ⎜ (not.first.false claim) >> ⎜ >> ⎜ (not.first.false claim) >> ⎜ >> ⎜ ... >> ⎜ >> ⎜ not.first.false claim [1]: >> ⎜⎛ Each non.zero natural number >> ⎜⎜ has, >> ⎜⎜ for it and for each of its non.zero priors, >> ⎜⎝ an immediate ordinal.predecessor. >> ⎜ >> ⎜ not.first.false claim [2]: >> ⎜⎛ The first transfinite ordinal, which we name 'ω', >> ⎜⎜ and each of its ordinal.followers >> ⎜⎜ does not have, >> ⎜⎜ for it and for each of its non.zero priors, >> ⎜⎜ an immediate ordinal.predecessor. >> ⎜⎜ That is, >> ⎜⎜ there is a non.zero prior without >> ⎝⎝ an immediate ordinal.predecessor. >> >> ---- >>> 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 precessors, >> but, from ω onward, it's negation is true. >> >> 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. >> >> > > Oh, well, tertium non datur or PEM, principle > of excluded middle, LEM, law of excluded middle, > TND, no third ground, have that inductive accounts > may _not_ bring their own completeness, if what > you'd rather is an inductive account that no > inductive account is not.ultimately.untrue. > > Which is well known since antiquity and also > as to why the only thing left is for metaphysics > and foundations to arrive at a theory as with > regards to not making an inductive account > after its implied un-founded (and often un-stated) > assumptions - we should figure out here for whom > it's called that any theory with finitely many > axioms is having a neat, and brief, formal counterexample, > and it's the same for each of them, though you > can point at Goedel with regards to complete/not-complete > these what are _partial_, at best, inductive accounts. > (At best, ....) > > > So, regularity is still a thing of course, and > monotonic entailment, and for a theory with a modality, > yet anything mathematical introduces itself to a > great relevant concern called "the domain of discourse" > or "the universe of mathematical objects", and then > there's Zeno again "hey, how about a brief discourse > on metaphysics?" > > > Also a physics, ..., and "the theory". > > So, not.first.false, is only after some pair-wise > comprehension, because, there are ready example > that in "super-task comprehension", or what's > called the illative when it's correct and completes > and what's usually called "undefined" when it doesn't, > lazy positivists, it's that not.first.false is either > not.ultimately.untrue, or it's not. > > And sometimes (some times, a modal account), ..., it's not. > > Of course there are then _reasons_ _why_ > what is so is so. > > ========== REMAINDER OF ARTICLE TRUNCATED ==========