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Path: ...!2.eu.feeder.erje.net!feeder.erje.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> Newsgroups: sci.math Subject: Re: Does the number of nines increase? Date: Sun, 7 Jul 2024 13:19:50 -0700 Organization: A noiseless patient Spider Lines: 56 Message-ID: <v6et97$f608$5@dont-email.me> References: <tJf9P9dALSN4l2XH5vdqPbXSA7o@jntp> <v669vp$2pluv$1@dont-email.me> <v66kcm$2rgql$1@dont-email.me> <v66u7k$2t154$1@dont-email.me> <v66v36$2t7em$1@dont-email.me> <v670bh$2tdhr$1@dont-email.me> <v670q1$2tc0j$2@dont-email.me> <v67tgn$35707$2@dont-email.me> <v68s9i$3a5u1$1@dont-email.me> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Injection-Date: Sun, 07 Jul 2024 22:19:52 +0200 (CEST) Injection-Info: dont-email.me; posting-host="0ecdb18ed35c2abf38d5c9c78345642e"; logging-data="497672"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18nunsrMJT/lGTVSXlQ7DCJdJoU2Ozv65E=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:DgqEpiJWgQ38R1oe70xHt3Apua0= Content-Language: en-US In-Reply-To: <v68s9i$3a5u1$1@dont-email.me> Bytes: 3137 On 7/5/2024 6:26 AM, Moebius wrote: > Am 05.07.2024 um 06:40 schrieb Chris M. Thomasson: >> On 7/4/2024 1:30 PM, Moebius wrote: >>> Am 04.07.2024 um 22:23 schrieb Chris M. Thomasson: >>> >>>> I was just thinking that infinite is infinite, >>> >>> No, it isn't. >> >> For some reason I like to think of the density of infinity. The >> Natural numbers are not dense at all when compared to the reals... > > Yeah, but this idea might be rather missleading! > > Hint: The natural numbers are not dense at all when compared to the > rational numbers either, no? > > > But both sets, the set of natural numbers and the set of rational > numbers are _countably infinite_, while the set of real numbers is > _uncountably infinite_ > >>>> there is an infinite number of natural numbers, >>> >>> Right, _countably_ infinitely many. >>> >>>> there are an infinite amount of [real] numbers between say, .0000001 >>>> and .00000001 >>> >>> _Uncountably_ infinitely many. >> >> Okay. I get a little confused by that sometimes. Trying to count the >> reals is not possible because of all those infinite infinities that >> are embedded in them... > > Yeah, a very good metaphor! > >> However The naturals have no infinities between say, 1 and 2. Make any >> sense to you? > > Yeah, somehow. > > Still, the rational numbers are countable! (Not enough "infinite > infinities embedded in them"!) Humm... Not sure about that. How many embedded infinite infinities are "needed" _before_ it can be deemed uncountable? Say between 0 and 1. There seems to be an infinite number of rationals that can fill in the "gap", so to speak. 0 + (1/8 + 1/8 + 1/4 + 1/2) = 1 0 + (1/16 + 1/16 + 1/8 + 1/4 + 1/2) = 1 0 + (1/16 + 1/16 + 1/8 + 2/3 + 1/3 - 1/4) = 1 These are all rational, right?