Path: ...!weretis.net!feeder9.news.weretis.net!news.quux.org!eternal-september.org!feeder2.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: Wed, 13 Nov 2024 17:14:02 +0100 Organization: A noiseless patient Spider Lines: 20 Message-ID: References: <0e67005f-120e-4b3b-a4d2-ec4bbc1c5662@att.net> <03b90d6c-fff1-411d-9dec-1c5cc7058480@tha.de> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Wed, 13 Nov 2024 17:14:01 +0100 (CET) Injection-Info: dont-email.me; posting-host="61ce7dd57b35d31e814e3d7fc6bc44a6"; logging-data="2408856"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+bkE4DAPNfeISTCcLnwkTg12FgX/Bwqt4=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:i2fWh92+f8U8UlM0lkvtFLFfCrA= Content-Language: en-US In-Reply-To: Bytes: 2662 On 13.11.2024 11:39, Mikko wrote: > On 2024-11-12 13:59:24 +0000, WM said: >> Cantor said that all rationals are within the sequence and hence >> within all intervals. I prove that rationals are in the complement. > > He said that about his sequence and his intervals. Infinitely many of them > are in intervals that do not overlap with any of your J(n). The intervals J(n) = [n - 1/10, n + 1/10] cover the relative measure 1/5 of ℝ+. By translating them to match Cantor's intervals they cover ℝ+ infinitely often. This is impossible. Therefore set theorists must discard geometry. Further all finitely many translations maintain the original relative measure. The sequence 1/5, 1/5, 1/5, ... has limit 1/5 according to analysis. Therefore set theorists must discard analysis. Regards, WM