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Path: ...!weretis.net!feeder8.news.weretis.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: Andy Walker <anw@cuboid.co.uk> Newsgroups: comp.theory Subject: =?UTF-8?Q?Re=3A_Definition_of_real_number_=E2=84=9D_--infinitesimal?= =?UTF-8?Q?--?= Date: Thu, 28 Mar 2024 20:38:28 +0000 Organization: Not very much Lines: 65 Message-ID: <uu4kg4$3ppio$1@dont-email.me> References: <bebe16f4f02eed7ac4e4d815dc0e1e98f9f0f2a0.camel@gmail.com> <uu3qk7$3jc94$1@dont-email.me> <uu444a$3lnuc$1@dont-email.me> <uu44k2$3lrph$1@dont-email.me> <uu46o5$3lnud$1@dont-email.me> <uu479l$3mm2m$1@dont-email.me> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Thu, 28 Mar 2024 20:38:29 +0100 (CET) Injection-Info: dont-email.me; posting-host="295bd15b0b3e8965cb6722b5b931a25e"; logging-data="3991128"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19aN4FiMdx+DeRe/EMkV3j2" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:MInNtvKzahbchfGzXhmhoI4VSq0= Content-Language: en-GB In-Reply-To: <uu479l$3mm2m$1@dont-email.me> Bytes: 4680 On 28/03/2024 16:53, olcott wrote: >>> Yet it seems that wij is correct that 0.999... would seem to >>> be infinitesimally < 1.0. >> That /cannot/ be correct in the "real" numbers, in which there >> are no infinitesimals [basic axiom of the reals]. In other systems of >> numbers, it could be correct, > Yes. >> but that will depend on what is meant by >> "0.999..", > Approaching yet never reaching 1.0. That is a property of the numbers 0.9, 0.99, 0.999 and so on arranged as a sequence [and of many other sequences], but is not /yet/ a value. Not until you explain what you mean. In conventional mathematics, it is usually taken to mean the limit of that sequence expressed as a real number, where "limit" has a precise meaning as discussed and formalised in the 19thC. That limit is 1. Not a tiny bit less than one, not some new sort of object, but 1, exactly. You and Wij may find that surprising, or even nonsensical, but it is what the mathematics tells us from the axioms of the real numbers and from the definition of "limit". If you want the answer to be different, then that must follow from different axioms and definitions. Until you and/or Wij tell us what those are, there is nothing further useful to be said. >> and note that if you appeal to something that mentions limits >> to define this, then you have to explain how infinite and infinitesimal >> numbers are handled in the definition. Again, there are no infinite or infinitesimal real numbers, so if you want an infinitesimal in your answer, it is incumbent on you to explain what you are using /other than/ conventional maths. >>> One geometric point on the number line. >>> [0.0, 1.0) < [0.0, 1.0] by one geometric point. >> Until you describe the axioms of what you mean by "geometric >> point" and "number line", this is meaningless verbiage. Give your > Of course by geometric point I must mean a box of chocolates and by > number line I mean a pretty pink bow. No one would ever suspect that > these terms have their conventional meanings. I didn't ask what "geometric point" and "number line" are, but what axioms you think they have. In conventional mathematics, those two intervals have /exactly/ the same measure even though they are not exactly the same sets of points. If you get a different answer [and have not simply made a mistake], it /must/ be because you are using different axioms. What are they? >> axioms, and it becomes possible to discuss this. Until then, we are >> entitled to assume that you and Wij are talking about the "traditional" >> "real" numbers [as used in engineering, etc.] in which there are no >> infinitesimals, and so the only interpretation we can make of the size >> of "one geometric point" is the usual "measure", which is zero. > Yet it is never actually zero because it is possible to specify a > line segment that is exactly one geometric point longer than another. > [0.0, 1.0] - [0.0, 1.0) = one geometric point. But "one geometric point" has measure zero. Not "never actually zero", but actually and really zero. Unless, that is, you are using some different and as yet unexplained axioms/definitions. Which are ...? -- Andy Walker, Nottingham. Andy's music pages: www.cuboid.me.uk/andy/Music Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Couperin