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Failed to connect to MySQL: (1203) User howardkn already has more than 'max_user_connections' active connectionsPath: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: WM Newsgroups: sci.logic Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary) Date: Thu, 12 Dec 2024 15:33:20 +0100 Organization: A noiseless patient Spider Lines: 41 Message-ID: References: <3af23566-0dfc-4001-b19b-96e5d4110fee@tha.de> <8a53c5d4-4afd-4f25-b1da-30d57e7fe91c@att.net> <10ebeeea-6712-4544-870b-92803ee1e398@att.net> <1f1a4089-dfeb-45f8-9c48-a36f6a4688fb@att.net> <84818a4f5d3795b746b017ad0861a3d818c5b053@i2pn2.org> <5805ad50ebff3400d1370d8c99790cbc727a340a@i2pn2.org> <1ac93432f1ba567e0f15308b8964bee86b92c706@i2pn2.org> <4e7901e16785581d0d02a2d6474d7d2615c5fac9@i2pn2.org> <2e960b7e409e3af02454365682803fa943f7697a@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Thu, 12 Dec 2024 15:33:20 +0100 (CET) Injection-Info: dont-email.me; posting-host="b813437e7e36d641655e1eddc9d0fc02"; logging-data="2290335"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+0QI3voMiM9tTuY07HWrbDH/XrJVTqSqo=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:+ld1tqA9uq5qDv3F95N4pKQWhAg= Content-Language: en-US In-Reply-To: <2e960b7e409e3af02454365682803fa943f7697a@i2pn2.org> Bytes: 3701 On 12.12.2024 15:23, joes wrote: > Am Thu, 12 Dec 2024 10:12:26 +0100 schrieb WM: >>>> The end of the sequence is defined by ∀k ∈ ℕ : E(k+1) = E(k) \ {k}. > The sequence is endless, has no end, is infinite. If a bijection with ℕ is possible, the sequence can be exhausted so that no natural numbers remains in an endsegment. > >>> None of which are an infinite sets, so trying to take a "limit" of >>> combining them is just improper. >> >> Most endsegments are infinite. But if Cantor can apply all natural >> numbers as indices for his sequences, then all must leave the sequence >> of endsegments. Then the sequence (E(k)) must end up empty. And there >> must be a continuous staircase from E(k) to the empty set. > It makes no sense not being able to „apply” numbers. Clearly Cantor does. He claims it. That means no numbers remain unpaired in endsegments. > The sequence IS continuous. It’s just that you misconceive of the > limit as reachable. Cantor does. If the limit is not reachable, then complete bijections cannot be established. "If we think the numbers p/q in such an order [...] then every number p/q comes at an absolutely fixed position of a simple infinite sequence" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 126] "The infinite sequence thus defined has the peculiar property to contain the positive rational numbers completely, and each of them only once at a determined place." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)] If you accept these claims, then no number must remain in an endsegment. Regards, WM >