Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: "Fred. Zwarts" Newsgroups: comp.theory Subject: =?UTF-8?Q?Re=3A_Definition_of_real_number_=E2=84=9D_--infinitesimal?= =?UTF-8?Q?--?= Date: Wed, 3 Apr 2024 17:30:58 +0200 Organization: A noiseless patient Spider Lines: 74 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: Wed, 03 Apr 2024 15:31:00 +0200 (CEST) Injection-Info: dont-email.me; posting-host="aa05e2b9d171799075ebf45437bfaa3f"; logging-data="16601"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+1D9bV/NLHZBiQJDofEeSz" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:3cWuAfY69L2VXBOHUbB7L1EO6So= In-Reply-To: Content-Language: en-GB Bytes: 5284 Op 03.apr.2024 om 17:11 schreef olcott: > On 4/3/2024 3:32 AM, Fred. Zwarts wrote: >> Op 02.apr.2024 om 20:51 schreef olcott: >>> On 4/2/2024 1:29 PM, 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. >>>> >>>> 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, >>>> >>> >>> In other words when we pretend that this never ending sequence ends >>> 0.999... ends then we do get to 1.0. >> >> Again fighting windmills. Nobody said the sequence ends. That is >> olcott's own interpretation which he wants to fight. >> > > 0.999... The LFS remains infinitesimally less than 1.0 Fighting windmills again. Fighting his own interpretation of 0.999... Unable to understand the normal interpretation, even when spelled out in detail.