Path: ...!news.mixmin.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: Tue, 2 Apr 2024 16:52:00 -0500 Organization: A noiseless patient Spider Lines: 93 Message-ID: References: <87sf08qzt5.fsf@nosuchdomain.example.com> <8734s4r84s.fsf@nosuchdomain.example.com> <87ttkkpn9y.fsf@nosuchdomain.example.com> <7jOdnYS6Ff5EhJH7nZ2dnZfqnPidnZ2d@brightview.co.uk> <87le5vpqiy.fsf@nosuchdomain.example.com> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Tue, 02 Apr 2024 21:52:01 +0200 (CEST) Injection-Info: dont-email.me; posting-host="6a1facd72526c9019aa65d403fd65586"; logging-data="3634712"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19IfKlTpY6MhaHy4CTIFwtP" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:y5FzSxEd1X2oBUgzZpVn1okakBM= In-Reply-To: Content-Language: en-US Bytes: 6309 On 4/2/2024 4:20 PM, Mike Terry wrote: > On 02/04/2024 19:29, Keith Thompson wrote: >> Mike Terry writes: >>> On 02/04/2024 02:27, Keith Thompson wrote: >>>> olcott writes: >>>>> On 4/1/2024 6:11 PM, Keith Thompson wrote: >>>>>> olcott 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? >>>>> >>>>> IDK, probably not. I am saying that 0.999... exactly equals this >>>>> number. >>>> "IDK, probably not." >>>> Did you even consider taking some time to *think* about this? >>> >>> PO just says things he thinks are true based on his first intuitions >>> when he encountered a topic. He does not "reason" his way to a new >>> carefully thought out theory or even to a single coherent idea. Don't >>> imagine he is thinking of hyperreals or anything - he just "knows" >>> that obviously any number which starts 0.??? is less than one starting >>> 1.??? - because 0 is less than 1 !! Or whatever, it really doesn't >>> matter. >> >> I don't think he's explicitly said that any real number whose decimal >> representation starts with "0." is less than one starting with "1." -- >> but if said that, he'd be right. > >   0.999...  = 1.000...  (so he'd be wrong) > In other words you simply choose to "not believe in" the notion of infinitesimal difference. That doesn't actually make it go away. >> >> What he refuses to understand is that the notation "0.999..." is not a >> decimal representation.  The "..."  notation refers to the limit of a >> sequence, and of course the limit of a sequence does not have to be a >> member of the sequence.  Every member of the sequence (0.9, 0.99, 0.999, >> 0.9999, continuing in the obvious manner) is a real (and rational) >> number that is strictly less than 1.0.  But the limit of the sequence is >> 1.0.  Sequences and their limits can be and are defined rigorously >> without reference to infinitesimals or infinities, > > Ah, I see - you're trying to say that 1.000... is a decimal > representation, but not 0.999...?, which would make sense of why you > think PO would be right above.  That's a new one on me, but I don't go > for that argument at all. > > 0.999... is a decimal representation for the number 1, shortened by ... > which means "continuing in the obvious fashion" or equivalent wording. > I.e. 0.999... is the decimal where every digit after the decimal point > is a 9.  It represents the number 1, as does 1.000....  Yes, there are > two ways to represent the number 1 as an infinite decimal.  Not a problem. > > Anyhow, I have a BA in mathematics, so I understand limits etc..  :)  I > was posting to explain why you're wasting your time trying to explain > abstract ideas to PO, but it's fine with me if people want to do that > for whatever reason. > > Mike. > ps. of course, someone could make a rule that infinitely repeating 9s in > a decimal expansion is outlawed, but that's not normal practice AFAIK. > People just accept there are two representations of certain numbers. > >> >> It can be genuinely difficult to wrap your head around this.  It *is* >> counterintuitive.  And thoughtful challenges to the mathematical >> orthodoxy, like the paper recently discussed in this thread, can be >> useful.  But olcott doesn't offer a coherent alternative. >> >> [...] >> -- Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius hits a target no one else can see." Arthur Schopenhauer