Path: ...!weretis.net!feeder6.news.weretis.net!feeder8.news.weretis.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: olcott 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 22:50:49 -0500 Organization: A noiseless patient Spider Lines: 81 Message-ID: References: <8734s9u2tl.fsf@nosuchdomain.example.com> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Fri, 29 Mar 2024 03:50:50 +0100 (CET) Injection-Info: dont-email.me; posting-host="bfd65a280c18a2165003beacad9b3410"; logging-data="96178"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18DWMYAMHM4j5n0NNkcd9P0" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:oWadltWecxWWGCoShzEgwQDXi1o= Content-Language: en-US In-Reply-To: <8734s9u2tl.fsf@nosuchdomain.example.com> Bytes: 4010 On 3/28/2024 10:36 PM, Keith Thompson wrote: > olcott 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... denotes a *limit*. In particular, it's the limit of the value > as the number of 9s increases without bound. That's what the notation That is how it has been misinterpreted yet it has always meant infinitesimally less than 1.0. > "0.999..." *means*. (There are more precise notations for the same > thing, such as "0.9̅" (that's a 9 with an overbar, or "vinculum") or > "0.(9)". > I already know all that. > You have a sequence of numbers: > > 0.9 > 0.99 > 0.999 > 0.9999 > 0.99999 > ... > > Each member of that sequence is strictly less than 1.0, but the *limit* > is exactly 1.0. The limit of a sequence doesn't have to be a member of > the sequence. The limit is, informally, the value that members of the > sequence approach arbitrarily closely. > Yet never reaching. > > >> That we can say this in English yet not say this in conventional >> number systems proves the need for another number system that can >> say this. > > Then I have good news for you. There are several such systems, for > example . > Infinitesimally less than 1.0 means one single geometric point on the number line less than 1.0. > If your point is that you personally like hyperreals better than you > like reals, that's fine, as long as you're clear which number system > you're using. The Infinitesimal number system that I created. > If you talk about things like "0.999..." without > qualification, everyone will assume you're talking about real numbers. > It is already the case that 0.999... specifies Infinitesimally less than 1.0. > And if you're going to play with hyperreal numbers, or surreal numbers, > or any of a number of other extensions to the real numbers, I suggest > that understanding the real numbers is a necessary prerequisite. That > includes understanding that no real number is either infinitesimal or > infinite. > > Disclaimer: I'm not a mathematician. I welcome corrections. > -- Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius hits a target no one else can see." Arthur Schopenhauer