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Path: ...!eternal-september.org!feeder2.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: WM <wolfgang.mueckenheim@tha.de> Newsgroups: sci.logic Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary) Date: Fri, 22 Nov 2024 11:53:32 +0100 Organization: A noiseless patient Spider Lines: 28 Message-ID: <vhpnrb$15239$1@dont-email.me> References: <vg7cp8$9jka$1@dont-email.me> <vgl2hj$3794c$1@dont-email.me> <vgleau$bi0i$2@solani.org> <vgnq3i$3qgfe$1@dont-email.me> <vgoka6$3vg2p$1@dont-email.me> <vgq1cm$b5vj$1@dont-email.me> <vgq3ca$beif$1@dont-email.me> <vgsp1c$v1ss$1@dont-email.me> <vgsq2v$v5t1$1@dont-email.me> <vgvm6h$1k8co$1@dont-email.me> <vgvmvr$1kc5f$1@dont-email.me> <vh1vlb$25kic$1@dont-email.me> <vh2j89$29gco$1@dont-email.me> <vh4f7p$2o5hn$1@dont-email.me> <vh4job$2ov2c$1@dont-email.me> <vh78jp$3cbq7$1@dont-email.me> <vh7d5c$3cpaf$1@dont-email.me> <5b8de1bc-9f6c-4dde-a7cd-9e22e8ce19d9@att.net> <vhata3$59e5$2@dont-email.me> <31419fde-62b3-46f3-89f6-a48f1fe82bc0@att.net> <vhc77g$hdd4$1@dont-email.me> <476ae6cb-1116-44b1-843e-4be90d594372@att.net> <vhhr6f$1q0r9$1@dont-email.me> <ffa63cb5-8898-4aa7-80eb-8b2c51c9986d@att.net> <vhkhun$28qt$1@dont-email.me> <vhmtph$j1ek$1@dont-email.me> <vhn1jk$jf6v$1@dont-email.me> <vhn3po$jvo1$1@dont-email.me> <vhn420$jf6v$3@dont-email.me> <vhpg51$13soc$1@dont-email.me> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Injection-Date: Fri, 22 Nov 2024 11:53:31 +0100 (CET) Injection-Info: dont-email.me; posting-host="e66bb22bbe05828732e73e7a13999941"; logging-data="1214569"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+CbDVVYPvK2CF/P6EFU+QnqaK78YGQFp0=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:HVuRaHhFkG7PiIilPOobxGPqkZQ= Content-Language: en-US In-Reply-To: <vhpg51$13soc$1@dont-email.me> Bytes: 3126 On 22.11.2024 09:42, Mikko wrote: > On 2024-11-21 11:03:28 +0000, WM said: >> For every finite (0, n] the relative covering remains f(n) = 1/10, >> independent of shifting. The constant sequence has limit 1/10. > > That is irrelevant to your question whether the whole interval becomes > black if the shifted intervals (n/2, n/2+1) are painted black. It is relevant by three reasons: 1) The limit of the sequence f(n) of relative coverings in (0, n] is 1/10, not 1. Therefore the relative covering 1 would contradict analysis. 2) Since for all intervals (0, n] the relative covering is 1/10, the additional blackies must be taken from the nowhere. 3) Since a shifted blacky leaves a white unit interval where it has left, the white must remain such that the whole real axis can never become black. These facts prevent the Cantor-bijection for different sets of natural numbers. > > And that question is irrelevant to the topic specified on the subject line. > If different sets of natural numbers already cannot be in bijection, then the rationals are also excluded. Regards, WM