Deutsch English Français Italiano |
<8734s4r84s.fsf@nosuchdomain.example.com> View for Bookmarking (what is this?) Look up another Usenet article |
Path: ...!news.mixmin.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: Keith Thompson <Keith.S.Thompson+u@gmail.com> Newsgroups: comp.theory Subject: Re: Definition of real number =?utf-8?Q?=E2=84=9D?= --infinitesimal-- Date: Mon, 01 Apr 2024 16:11:15 -0700 Organization: None to speak of Lines: 45 Message-ID: <8734s4r84s.fsf@nosuchdomain.example.com> References: <bebe16f4f02eed7ac4e4d815dc0e1e98f9f0f2a0.camel@gmail.com> <uu6npg$ceq1$1@dont-email.me> <uu79db$gdqk$1@dont-email.me> <875xx4sh0h.fsf@nosuchdomain.example.com> <uu7osb$k31e$1@dont-email.me> <87sf08qzt5.fsf@nosuchdomain.example.com> <uu7q21$k72e$1@dont-email.me> <uu8vf8$vsq2$1@dont-email.me> <uu95mr$114hv$5@dont-email.me> <uu9q43$16c9d$2@dont-email.me> <uu9qqn$16gt9$1@dont-email.me> <uu9s39$16gks$1@dont-email.me> <uu9sj2$16rdo$1@dont-email.me> <uucbe9$1utsv$2@dont-email.me> <uucc0e$1v1p5$1@dont-email.me> <uucdd7$1v8hd$1@dont-email.me> <uucec3$1vh78$1@dont-email.me> <uudnt6$2bun2$1@dont-email.me> <uuegit$2hjc8$1@dont-email.me> <uuev15$2l64e$2@dont-email.me> <uuevt5$2laff$1@dont-email.me> MIME-Version: 1.0 Content-Type: text/plain Injection-Date: Mon, 01 Apr 2024 23:11:19 +0200 (CEST) Injection-Info: dont-email.me; posting-host="3a6d60d87380a0c0f1e42919673cda47"; logging-data="2915168"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19re/FW/48oQckKre1vWlcZ" User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/27.2 (gnu/linux) Cancel-Lock: sha1:uMnAH2xahqAg6gl8pAt/Tdom4qM= sha1:vjfvmtcyOL0adVN2ImxNN6c6+mY= Bytes: 3548 olcott <polcott333@gmail.com> writes: [...] > Since PI is represented by a single geometric point on the number line > then 0.999... would be correctly represented by the geometric point > immediately to the left of 1.0 on the number line or the RHS of this > interval [0,0, 1.0). If there is no Real number at that point then > there is no Real number that exactly represents 0.999... [...] In the following I'm talking about real numbers, and only real numbers -- not hyperreals, or surreals, or any other extension to the real numbers. You assert that there is a geometric point immediately to the left of 1.0 on the number line. (I disagree, but let's go with it for now.) Am I correct in assuming that this means that that point corresponds to a real number that is distinct from, and less than, 1.0? More generally, does each real number correspond to a point on the number line, and does each point on the number line correspond to a real number? (The real numbers can be formally defined without reference to geometry, but let's go with your geometric model for now.) If so, let's call that real number (immediately to the left of 1.0) x. Consider ((x + 1.0)/2.0). Let's call that number y. (The intent is to construct a real number that is exactly halfway between x and 1.0.) Is y a real number? (If not, the real numbers are, unexpectedly, not closed under common arithmetic operations.) Is y less than, equal to, or greater than x? Is y less than, equal to, or greater than 1.0? Again, I am talking *only* about real numbers. Given your past history, I do not expect straight answers to these questions, but I'm prepared to be pleasantly surprised. -- Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com Working, but not speaking, for Medtronic void Void(void) { Void(); } /* The recursive call of the void */