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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: Fri, 29 Mar 2024 17:25:02 -0700 Organization: None to speak of Lines: 53 Message-ID: <875xx4sh0h.fsf@nosuchdomain.example.com> References: <bebe16f4f02eed7ac4e4d815dc0e1e98f9f0f2a0.camel@gmail.com> <uu3qk7$3jc94$1@dont-email.me> <uu444a$3lnuc$1@dont-email.me> <uu44k2$3lrph$1@dont-email.me> <uu50n4$3ca7i$6@i2pn2.org> <uu573n$3tt5t$7@dont-email.me> <uu58nh$3ca7j$2@i2pn2.org> <uu59t9$3ubje$2@dont-email.me> <8734s9u2tl.fsf@nosuchdomain.example.com> <uu5dqp$2tti$2@dont-email.me> <uu6ep9$3dq4u$4@i2pn2.org> <uu6npg$ceq1$1@dont-email.me> <uu79db$gdqk$1@dont-email.me> MIME-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 8bit Injection-Date: Sat, 30 Mar 2024 00:25:06 +0100 (CET) Injection-Info: dont-email.me; posting-host="55d3bec1253354bc3d638b21ffa9fc4c"; logging-data="640901"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19++P6OVlYYCT9JSMAP0T0f" User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/27.2 (gnu/linux) Cancel-Lock: sha1:sFVQiQ2hQLOMa4IU+SgU8A76GNQ= sha1:44ipjf8j4vW2bLKi+2X8/6EN+5w= Bytes: 3715 "Fred. Zwarts" <F.Zwarts@HetNet.nl> writes: > Op 29.mrt.2024 om 16:46 schreef olcott: >> On 3/29/2024 8:13 AM, Richard Damon wrote: >>> On 3/28/24 11:50 PM, olcott wrote: >>>> On 3/28/2024 10:36 PM, Keith Thompson wrote: >>>>> olcott <polcott2@gmail.com> writes: >>>>> [...] >>>>>> It seems dead obvious that 0.999... is infinitesimally less than 1.0. >>>>> >>>>> Yes, it *seems* dead obvious. That doesn't make it true, and in >>>>> fact it >>>>> isn't. >>>>> >>>> >>>> 0.999... means that is never reaches 1.0. >>>> and math simply stipulates that it does even though it does not. >>> >>> >>> 0.999... isn't a "number" in the Real Number system, just an >>> alternate representation for the number 1. >>> >> That is not true. 0.999... means never reaches 1.0 > > Maybe for olcott's unspecified olcott numbers. For real numbers > 0.999... equals 1.0. There are many proofs. See e.g. >> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers olcott almost has a point, in that the sequence of values 0.9, 0.99, 0.999, 0.9999, ... (continuing in the obvious manner) never reaches 1.0. No element of that unending sequence of real numbers is exactly equal to 1.0. What he either doesn't understand, or pretends not to understand, is that the notation "0.999..." does not refer either to any element of that sequence or to the entire sequence. It refers to the *limit* of the sequence. The limit of the sequence happens not to be an element of the sequence, and it's exactly equal to 1.0. This is all stated in terms of the real numbers, which are a well defined set. There are other systems with different properties. If we were talking about the hyperreals, for example, olcott's statement might be correct (though I'm not sure of that). But olcott seems to be insisting, quite incorrectly, that his statements apply to the reals. If he's talking about the reals, he's wrong. If he's talking about something other than the reals, he's boring. Either way, he will not change his mind. Attempts to explain limits and real numbers to him will fail. -- 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 */