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Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: WM <wolfgang.mueckenheim@tha.de> Newsgroups: sci.math Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary) Date: Mon, 20 Jan 2025 13:42:58 +0100 Organization: A noiseless patient Spider Lines: 23 Message-ID: <vmlgci$33ltb$1@dont-email.me> References: <vg7cp8$9jka$1@dont-email.me> <vloule$3eqsr$1@dont-email.me> <ffffed23878945243684de7f2aa9aaaf29564508@i2pn2.org> <vlrej9$2m5k$1@dont-email.me> <d6ed4797-65e8-4004-853c-f07a37af0c11@att.net> <vls4j6$7v2k$3@dont-email.me> <494bfd3b-3c70-4d8d-9c70-ce917c15fc22@att.net> <vm0okb$16cq0$2@dont-email.me> <bff18686-503a-4b7b-9406-b47796f68b47@att.net> <vm15pj$18v7t$1@dont-email.me> <72142d82-0d71-460a-a1be-cadadf78c048@att.net> <vm3hrs$1s9ld$2@dont-email.me> <812e64b1-c85c-48ac-a58c-e8955bc02f8c@att.net> <vm59g4$2b5ib$1@dont-email.me> <b5040865-50e6-4297-a08c-0072e0a2cb0f@att.net> <vm8trc$30hqq$9@dont-email.me> <ec8c4429-b724-4ea6-ad38-80a97cf6b06c@att.net> <vmd6m3$3vamp$1@dont-email.me> <681bacad-63c9-45e1-ba73-4af883caae2d@att.net> <vmebn0$3iir$3@solani.org> <717d95ac-9342-448a-822d-793bdd08abb9@att.net> <vmfpg8$mj54$1@dont-email.me> <f17c3af1-c692-44c9-94f4-bcc57d3541b5@att.net> <vmilhh$252qj$3@dont-email.me> <f3dcd1edaa6ef3cd3083e6f9bdb5a5897d460d84@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Injection-Date: Mon, 20 Jan 2025 13:42:58 +0100 (CET) Injection-Info: dont-email.me; posting-host="ad3059a076be5126998b2dbdaa149b09"; logging-data="3266475"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19Scf23NvN4A5WxsOCA8EJGf3sqgETrgXQ=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:zl8zRLQS4GteEwjG/HvEKafz88s= Content-Language: en-US In-Reply-To: <f3dcd1edaa6ef3cd3083e6f9bdb5a5897d460d84@i2pn2.org> Bytes: 3032 On 19.01.2025 16:05, joes wrote: > Am Sun, 19 Jan 2025 11:52:33 +0100 schrieb WM: >> Cantor claims this also for infinite sets: "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)] > That's not what that means. Some infinite sets are countable, even though > you don't "finish" them. The quote refers to a bijection. But there is no bijection bijection without completeness: to contain the positive rational numbers completely. > >>> There is no step from finite to infinite. >> Not in the visible domain. But there is no loss in lossless exchange - >> even in the dark domain. There lies your fault. > There is only a limit, which does have different properties. A bijection concerns pairing of all elements, no limit. And in particular no loss in explicitly lossless exchange. Regards, WM