Path: ...!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: Wed, 3 Apr 2024 10:11:24 -0500 Organization: A noiseless patient Spider Lines: 72 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 15:11:25 +0200 (CEST) Injection-Info: dont-email.me; posting-host="bfbe230f41d733c8e8d14e4fa12c421a"; logging-data="6592"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19sLcZIaO8S2CAIlN4Gi9ZQ" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:xSy0p2DQCsJv2uIWNb/YySuv2ok= Content-Language: en-US In-Reply-To: Bytes: 5190 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 >> >> We can also pretend that cats dogs thus cats bark. > -- Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius hits a target no one else can see." Arthur Schopenhauer