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Failed to connect to MySQL: (1203) User howardkn already has more than 'max_user_connections' active connectionsPath: ...!news.misty.com!weretis.net!feeder6.news.weretis.net!i2pn.org!i2pn2.org!.POSTED!not-for-mail From: Richard Damon Newsgroups: comp.theory Subject: =?UTF-8?Q?Re=3A_Definition_of_real_number_=E2=84=9D_--infinitesimal?= =?UTF-8?Q?--?= Date: Fri, 29 Mar 2024 09:13:13 -0400 Organization: i2pn2 (i2pn.org) 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 13:13:14 -0000 (UTC) Injection-Info: i2pn2.org; logging-data="3598494"; mail-complaints-to="usenet@i2pn2.org"; posting-account="diqKR1lalukngNWEqoq9/uFtbkm5U+w3w6FQ0yesrXg"; User-Agent: Mozilla Thunderbird In-Reply-To: X-Spam-Checker-Version: SpamAssassin 4.0.0 Content-Language: en-US Bytes: 4204 Lines: 93 On 3/28/24 11:50 PM, olcott wrote: > 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... isn't a "number" in the Real Number system, just an alternate representation for the number 1. > >> 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. But "infintesimally" doesn't exist in the Real Number System, it deals onloy with FINITE numbers. > >> "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. Nope. > >> 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. So you are lying about talking about the Reals. > >> 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. >> >