Path: ...!eternal-september.org!feeder2.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: Mikko Newsgroups: sci.logic Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers Date: Thu, 7 Nov 2024 11:24:32 +0200 Organization: - Lines: 26 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: Thu, 07 Nov 2024 10:24:32 +0100 (CET) Injection-Info: dont-email.me; posting-host="b2129c34fb1c7914224edcfec2ada169"; logging-data="2738258"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/6aN0u7IQh1QnQj05n+tfR" User-Agent: Unison/2.2 Cancel-Lock: sha1:gOAja/lHQNE57kR9H1ZQ0QF5qCE= Bytes: 2208 On 2024-11-06 17:52:01 +0000, WM said: > On 06.11.2024 15:59, Mikko wrote: >> On 2024-11-05 11:26:58 +0000, WM said: >> >>>   Cantor's results are >>>> conclusions of proofs and you have not shown any error in the proofs. >>> >>> I have. This example for instance proves that he did not enumerate all >>> rationals, because the rationals are dense, the intervals are not dense. >> >> You have not proven that. It is fairly easy to prove that there are >> no positive rationals other than those enumerated by Cantor (if I >> recall correctly he enumerated only positive rationals). To prove >> that there are positive rationals that are not included in Cantor's >> enumeration it suffices to show one but you have not shown any. > > I have shown that without rational numbers outside of the intervals > with irrational endpoints covering 3 of infinitely many units the real > axis has measure 3. No, you have not shown it, only said. -- Mikko