Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: "Chris M. Thomasson" Newsgroups: sci.math Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary, infinite-middle) Date: Sun, 1 Dec 2024 15:17:18 -0800 Organization: A noiseless patient Spider Lines: 93 Message-ID: References: <4a810760-86a1-44bb-a191-28f70e0b361b@att.net> <23311c1a-1487-4ee4-a822-cd965bd024a0@att.net> <71758f338eb239b7419418f49dfd8177c59d778b@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Mon, 02 Dec 2024 00:17:19 +0100 (CET) Injection-Info: dont-email.me; posting-host="17232c29395538db7d47c3a356e5c0c8"; logging-data="2975873"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18u6nsQ9a3FPKtvEy84xwKCtJgzUO9zi5w=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:aYJJXsg4iyZ84DCVGJdc8PHzhRc= Content-Language: en-US In-Reply-To: Bytes: 5183 On 11/30/2024 10:27 AM, Ross Finlayson wrote: > On 11/30/2024 03:54 AM, FromTheRafters wrote: >> 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. > > What about the "infinite-middle" models? { 1, ..., 1.5, ... 2, ... 2.5, ..., 3, ... } There are infinite reals in each (...) ? > > This is simply about a symmetric rather than a-symmetric > outset of integers, for example. > > As "sets", with their "ordering", and by that I mean sets > with an ordering, there's first and last, alpha and omega > as it were. > >