Path: ...!local-3.nntp.ord.giganews.com!Xl.tags.giganews.com!local-4.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Sat, 21 Dec 2024 23:29:26 +0000 Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary) Newsgroups: sci.math References: <6c74c2e9-17d7-4eb4-afcd-c058309b6c8a@tha.de> <4327ba4a-e810-4565-83e8-b9572018ac35@att.net> <4051acc5-d00a-40d2-8ef7-cf2b91ae75b6@att.net> <8d69d6cd-76bc-4dc1-894e-709d044e68a1@att.net> <7356267c-491b-45c2-b86a-d40c45dfa40c@att.net> <4bf8a77e-4b2a-471f-9075-0b063098153f@att.net> <31180d7e-1c2b-4e2b-b8d6-e3e62f05da43@att.net> From: Ross Finlayson Date: Sat, 21 Dec 2024 15:29:14 -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: <1R-dnTR1L5PL0vr6nZ2dnZfqn_adnZ2d@giganews.com> Lines: 183 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-bdkhwAH9W2tL5cTdaafdTohR4TPbMxjYGSBsnEsgvTGvGR+5rNcpREQd5Um4nbw1fExzUqwqcGaO8be!sxwAbgr6mpPctvMxt9UfxmqKKkfSBvRKGlNGKnRv7/3/4y1y1s4DGvOmGYf7rcucHwzQeA714X4= 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: 7538 On 12/21/2024 11:32 AM, Jim Burns wrote: > On 12/21/2024 6:34 AM, WM wrote: >> On 20.12.2024 19:48, Jim Burns wrote: >>> On 12/19/2024 4:37 PM, WM wrote: > >>>> That means all numbers are lost by loss of >>>> one number per term. >>>> >>>> That implies finite endsegments. >>> >>> Q. What does 'finite' mean? > > Consider end.segments of the finite cardinals. > > Q. What does 'finite' mean, > 'finite', whether darkᵂᴹ or visibleᵂᴹ? > >> Here is a new and better definition of endsegments >> >> E(n) = {n+1, n+2, n+3, ...} with E(0) = ℕ. >> >> ∀n ∈ ℕ : E(n+1) = E(n) \ {n+1} >> means that the sequence of endsegments can decrease only by one >> natnumber per step. > > E(n+1) is larger.than each of > the sets for which there are smaller.by.one sets. > E(n+1) isn't any of > the sets for which there are smaller.by.one sets. > > E(n+1) isn't smaller.by.one than E(n). > E(n+1) is emptier.by.one than E(n) > >> Therefore the sequence of endsegments >> cannot become empty > > Yes, because > the sequence of end.segments > can become emptier.one.by.one, but > it cannot become smaller.one.by.one. > >> (i.e., not all natnumbers can be applied as indices) > > Each finite.cardinal can be applied, > which makes the sequence emptier.by.one > but does not make the sequence smaller.by.one. > >> unless the empty endsegment is reached, > > Each end.segment is emptier.by.one. > No end.segment is smaller.than the first end.segment ℕ > The empty end.segment is not reached. > > No finite.cardinal is in common with > all infinitely.many > infinite.end.segments of > finite.cardinals. > > Nothing other.than a finite.cardinal is in > any end.segment of the finite.cardinals, > or in their intersection. > > The intersection of all infinitely.many > infinite end.segments of finite.cardinals > is not an end.segment > but is empty. > > Q. What does 'finite' mean? > >> unless the empty endsegment is reached, > > The empty end.segment, not.existing, is not.reached. > > The intersection.of.finitely.many is not.empty. > The intersection.of.all is empty. > >> and >> before finite endsegments, >> endsegments containing only 1, 2, 3, or n ∈ ℕ numbers, >> have been passed. > > ⎛ Assume end.segment E(n) of the finite.cardinals > ⎜ holds only finite.cardinal.k.many finite.cardinals. > ⎜ > ⎜ k.sized E(n) holds > ⎜ kᵗʰ.smallest, 1ˢᵗ.largest finite.cardinal E(n)[k] > ⎜ > ⎜ If a finite.cardinal larger.than E(n)[k] exists, > ⎜ it would also be in end.segment E(n) and > ⎜ larger.than 1ˢᵗ.largest E(n)[k]: a contradiction. > ⎜ > ⎜ Thus, > ⎜ a finite.cardinal larger.than E(n)[k] doesn't exist. > ⎜ > ⎜ However, > ⎜⎛ for each finite.cardinal j, > ⎜⎝ larger.than.j finite.cardinal j+1 exists. > ⎜ > ⎜ Larger.than.E(n)[k] finite.cardinal E(n)[k]+1 exists. > ⎝ Contradiction. > > Therefore, > end.segment E(n) of the finite.cardinals does not hold > only finite.cardinal.many finite.cardinals. > > There are no finite end.segments of the finite.cardinals. > > Q. What does 'finite' mean? > >> These however, if existing at all, cannot be seen. >> They are dark. > > Darknessᵂᴹ and visibilityᵂᴹ don't change any of this. > There are no finite end.segments of the finite.cardinals. > > We know it by the method of > assembling finite sequences of claims (proofs), > each claim of which is true.or.not.first.false (valid), > and holding those claims.we.know (theorems), > because > a finite sequence of claims, > each claim of which true.or.not.first.false, > holds only true claims. > > Some claims (definitions) > we know are true because > we know how we have defined things. > > Some claims (valid inferences) > we know are not.first.false because > we can inspect the finite sequence of claims. > > None of _the claims_ are darkᵂᴹ, > whatever the status of _what the claims are about_ > > Darknessᵂᴹ or visibilityᵂᴹ of finite.cardinals > don't change _the claims_ > >>>> That means all numbers are lost by loss of >>>> one number per term. >>>> >>>> That implies finite endsegments. >>> >>> No. >>> Yes, each number is lost by loss of >>> one number per term. >>> However, >>> each end.segment is not finite. >> > >> Then the last endsegment is empty. > > There is no last end.segment of the finite.cardinals. > ⎛ For each finite.cardinal j, > ⎝ larger.than.j finite.cardinal j+1 exists. > contradicts a last end.segment. > > Well, anybody can just build "infinite-middle", is what it is. ========== REMAINDER OF ARTICLE TRUNCATED ==========