Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: FromTheRafters Newsgroups: sci.math Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary) Date: Sat, 30 Nov 2024 06:54:17 -0500 Organization: Peripheral Visions Lines: 69 Message-ID: References: <67d9867b-2614-4475-975c-938bafca5c00@att.net> <4a810760-86a1-44bb-a191-28f70e0b361b@att.net> <23311c1a-1487-4ee4-a822-cd965bd024a0@att.net> <71758f338eb239b7419418f49dfd8177c59d778b@i2pn2.org> Reply-To: erratic.howard@gmail.com MIME-Version: 1.0 Content-Type: text/plain; charset="utf-8"; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Sat, 30 Nov 2024 12:54:28 +0100 (CET) Injection-Info: dont-email.me; posting-host="6f1b240c9bdbb43526c715b31a625539"; logging-data="1807231"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19GLmVXxnjoRuKr8s1gkBZ/UXauhbz/nuQ=" Cancel-Lock: sha1:SYKf+33kU3dQYPxpP5WpUrHdd6c= X-Newsreader: MesNews/1.08.06.00-gb X-ICQ: 1701145376 Bytes: 4450 on 11/30/2024, WM supposed : > On 30.11.2024 11:57, FromTheRafters wrote: >> WM explained : >>> On 29.11.2024 22:50, FromTheRafters wrote: >>>> WM wrote on 11/29/2024 : >>> >>>>> The size of the intersection remains infinite as long as all endsegments >>>>> remain infinite (= as long as only infinite endsegments are considered). >>>> >>>> Endsegments are defined as infinite, >>> >>> Endsegments are defined as endsegments. They have been defined by myself >>> many years ago. >> >> As what is left after not considering a finite initial segment in your new >> set and considering only the tail of the sequence. > > Not quite but roughly. The precise definitions are: > Finite initial segment F(n) = {1, 2, 3, ..., n}. > Endsegment E(n) = {n, n+1, n+2, ...} There it is!! Don't you see that the ellipsis means that endsegments are defined as infinite? > >> Almost all elements are considered in the new set, which means all >> endsegments are infinite. > > Every n that can be chosen has infinitely many successors. Every n that can > be chosen therefore belongs to a collection that is finite but variable. > >>> Try to understand inclusion monotony. The sequence of endsegments >>> decreases. >> >> In what manner are they decreasing? > > They are losing elements, one after the other: > ∀k ∈ ℕ : E(k+1) = E(k) \ {k} > But each endsegment has only one element less than its predecessor. But how is that related to decreasing? What has decreased? >> When you filter out the FISON, the rest, the tail, as a set, stays the same >> size of aleph_zero. > > For all endsegments which are infinite Which they all are, see above. > and therefore have an infinite intersection. The emptyset. >>> As long as it has not decreased below ℵo elements, the intersection has >>> not decreased below ℵo elements. >> >> It doesn't decrease in size at all. > > Then also the size of the intersection does not decrease. Of course not, since it stays at emptyset unless there is a last element -- which there is not since endsegments are infinite. > Look: when endsegments can lose all elements without becoming empty, then > also their intersection can lose all elements without becoming empty. What > would make a difference? Finite sets versus infinite sets. Finite ordered sets have a last element which can be in the intersection of all previously considered finite sets. Infinite ordered sets have no such last element.