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 17:34:28 +0200 Organization: A noiseless patient Spider Lines: 186 Message-ID: References: <875xx4sh0h.fsf@nosuchdomain.example.com> <87sf08qzt5.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 15:34:30 +0200 (CEST) Injection-Info: dont-email.me; posting-host="aa05e2b9d171799075ebf45437bfaa3f"; logging-data="16601"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18R999Z6HS72JTJqWVnxlbi" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:FxdRpbIgfkSr+WkLIC/ARGg+WHs= In-Reply-To: Content-Language: en-GB Bytes: 11145 Op 03.apr.2024 om 17:08 schreef olcott: > On 4/3/2024 3:27 AM, Fred. Zwarts wrote: >> Op 02.apr.2024 om 17:50 schreef olcott: >>> On 4/2/2024 10:38 AM, Fred. Zwarts wrote: >>>> Op 02.apr.2024 om 16:53 schreef olcott: >>>>> On 4/2/2024 4:43 AM, Fred. Zwarts wrote: >>>>>> Op 01.apr.2024 om 22:30 schreef olcott: >>>>>>> On 4/1/2024 2:37 PM, Fred. Zwarts wrote: >>>>>>>> Op 01.apr.2024 om 20:54 schreef olcott: >>>>>>>>> On 4/1/2024 1:39 PM, Fred. Zwarts wrote: >>>>>>>>>> Op 01.apr.2024 om 16:33 schreef olcott: >>>>>>>>>>> On 4/1/2024 2:31 AM, Fred. Zwarts wrote: >>>>>>>>>>>> Op 31.mrt.2024 om 21:42 schreef olcott: >>>>>>>>>>>>> On 3/31/2024 2:26 PM, Fred. Zwarts wrote: >>>>>>>>>>>>>> Op 31.mrt.2024 om 21:02 schreef olcott: >>>>>>>>>>>>>>> On 3/31/2024 1:52 PM, Fred. Zwarts wrote: >>>>>>>>>>>>>>>> Op 30.mrt.2024 om 21:27 schreef olcott: >>>>>>>>>>>>>>>>> On 3/30/2024 3:18 PM, Fred. Zwarts wrote: >>>>>>>>>>>>>>>>>> Op 30.mrt.2024 om 20:57 schreef olcott: >>>>>>>>>>>>>>>>>>> On 3/30/2024 2:45 PM, Fred. Zwarts wrote: >>>>>>>>>>>>>>>>>>>> Op 30.mrt.2024 om 14:56 schreef olcott: >>>>>>>>>>>>>>>>>>>>> On 3/30/2024 7:10 AM, Fred. Zwarts wrote: >>>>>>>>>>>>>>>>>>>>>> Op 30.mrt.2024 om 02:31 schreef olcott: >>>>>>>>>>>>>>>>>>>>>>> On 3/29/2024 8:21 PM, Keith Thompson wrote: >>>>>>>>>>>>>>>>>>>>>>>> olcott writes: >>>>>>>>>>>>>>>>>>>>>>>>> On 3/29/2024 7:25 PM, Keith Thompson wrote: >>>>>>>>>>>>>>>>>>>>>>>> [...] >>>>>>>>>>>>>>>>>>>>>>>>>> What he either doesn't understand, or pretends >>>>>>>>>>>>>>>>>>>>>>>>>> not to understand, is >>>>>>>>>>>>>>>>>>>>>>>>>> that the notation "0.999..." does not refer >>>>>>>>>>>>>>>>>>>>>>>>>> either to any element of >>>>>>>>>>>>>>>>>>>>>>>>>> that sequence or to the entire sequence.  It >>>>>>>>>>>>>>>>>>>>>>>>>> refers to the *limit* of >>>>>>>>>>>>>>>>>>>>>>>>>> the sequence.  The limit of the sequence >>>>>>>>>>>>>>>>>>>>>>>>>> happens not to be an element of >>>>>>>>>>>>>>>>>>>>>>>>>> the sequence, and it's exactly equal to 1.0. >>>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>> In other words when one gets to the end of a >>>>>>>>>>>>>>>>>>>>>>>>> never ending sequence >>>>>>>>>>>>>>>>>>>>>>>>> (a contradiction) thenn (then and only then) >>>>>>>>>>>>>>>>>>>>>>>>> they reach 1.0. >>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>> No. >>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>> You either don't understand, or are pretending >>>>>>>>>>>>>>>>>>>>>>>> not to understand, what >>>>>>>>>>>>>>>>>>>>>>>> the limit of sequence is.  I'm not offering to >>>>>>>>>>>>>>>>>>>>>>>> explain it to you. >>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>> I know (or at least knew) what limits are from my >>>>>>>>>>>>>>>>>>>>>>> college calculus 40 >>>>>>>>>>>>>>>>>>>>>>> years ago. If anyone or anything in any way says >>>>>>>>>>>>>>>>>>>>>>> that 0.999... equals >>>>>>>>>>>>>>>>>>>>>>> 1.0 then they saying what happens at the >>>>>>>>>>>>>>>>>>>>>>> end of a never ending >>>>>>>>>>>>>>>>>>>>>>> sequence and this is a contradiction. >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>> It is clear that olcott does not understand >>>>>>>>>>>>>>>>>>>>>> limits, because he is changing the meaning of the >>>>>>>>>>>>>>>>>>>>>> words and the symbols. Limits are not talking >>>>>>>>>>>>>>>>>>>>>> about what happens at the end of a sequence. It >>>>>>>>>>>>>>>>>>>>>> seems it has to be spelled out for him, otherwise >>>>>>>>>>>>>>>>>>>>>> he will not understand. >>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>> 0.999... Limits basically pretend that we reach the >>>>>>>>>>>>>>>>>>>>> end of this infinite sequence even though that it >>>>>>>>>>>>>>>>>>>>> impossible, and says after we reach this >>>>>>>>>>>>>>>>>>>>> impossible end the value would be 1.0. >>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> No, if olcott had paid attention to the text below, >>>>>>>>>>>>>>>>>>>> or the article I referenced: >>>>>>>>>>>>>>>>>>>>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers >>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> he would have noted that limits do not pretend to >>>>>>>>>>>>>>>>>>>> reach the end. They >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> Other people were saying that math says 0.999... = 1.0 >>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> Indeed and they were right. Olcott's problem seems to >>>>>>>>>>>>>>>>>> be that he thinks that he has to go to the end to >>>>>>>>>>>>>>>>>> prove it, but that is not needed. We only have to go >>>>>>>>>>>>>>>>>> as far as needed for any given ε. Going to the end is >>>>>>>>>>>>>>>>>> his problem, not that of math in the real number system. >>>>>>>>>>>>>>>>>> 0.999... = 1.0 means that with this sequence we can >>>>>>>>>>>>>>>>>> come as close to 1.0 as needed. >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>> That is not what the "=" sign means. It means exactly >>>>>>>>>>>>>>>>> the same as. >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> No, olcott is trying to change the meaning of the symbol >>>>>>>>>>>>>>>> '='. That *is* what the '=' means for real numbers, >>>>>>>>>>>>>>>> because 'exactly the same' is too vague. Is 1.0 exactly >>>>>>>>>>>>>>>> the same as 1/1? It contains different symbols, so why >>>>>>>>>>>>>>>> should they be exactly the same? >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> It never means approximately the same value. >>>>>>>>>>>>>>> It always means exactly the same value. >>>>>>>>>>>>>> >>>>>>>>>>>>>> And what 'exactly the same value' means is explained >>>>>>>>>>>>>> below. It is a definition, not an opinion. >>>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> No matter what you explain below nothing that anyone can >>>>>>>>>>>>> possibly >>>>>>>>>>>>> say can possibly show that 1.000... = 1.0 >>>>>>>>>>>>> >>>>>>>>>>>>> I use categorically exhaustive reasoning thus eliminating the >>>>>>>>>>>>> possibility of correct rebuttals. >>>>>>>>>>>> >>>>>>>>>>>> OK, then it is clear that olcott is not talking about real >>>>>>>>>>>> numbers, because for reals categorically exhaustive >>>>>>>>>>>> reasoning proved that 0.999... = 1 and olcott could not >>>>>>>>>>>> point to an error in the proof. >>>>>>>>>>>> It would have been less confusiong when he had mentioned >>>>>>>>>>>> that explicitly. >>>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> Typo corrected >>>>>>>>>>> No matter what you explain below nothing that anyone can >>>>>>>>>>> possibly >>>>>>>>>>> say can possibly show that 0.999... = 1.0 >>>>>>>>>>> >>>>>>>>>>> 0.999... >>>>>>>>>>> Means an infinite never ending sequence that never reaches 1.0 >>>>>>>>>> >>>>>>>>>> Which nobody denied. >>>>>>>>>> Olcott again changes the question. >>>>>>>>>> The question is not does this sequence end, or does it reach >>>>>>>>>> 1.0, but: which real is represented with this sequence? >>>>>>>>> >>>>>>>>> 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). >>>>>>>> >>>>>>>> In the real number system it is incorrect to talk about a number >>>>>>>> immediately next to another number. So, this is not about real >>>>>>>> numbers. >>>>>>>> >>>>>>> >>>>>>> PI is a real number. >>>>>>> If there is no real number that represents 0.999... >>>>>>> that does not provide a reason to say 0.999... = 1.0. >>>>>> >>>>>> Olcott makes me think of Don Quixote, who was unable to interpret >>>>>> the appearance of a windmill correctly. He interpreted it as >>>>>> nobody else did and therefore he thought he needed to fight it. >>>>>> Similarly, olcott has an incorrect interpretation of 0.999... = >>>>>> 1.0. Nobody has that interpretation, but olcott thinks he has to >>>>>> fight it. >>>>> >>>> >>>>> 0.999... So what do the three dots means to you: Have a dotty day? >>>> >>>> I see olcott does not read (or at least does not understand) what I >>>> write. It has been explained to him so many times in so much detail >>>> what 0.999... = 1 means. His mind seems to be too inflexible to ========== REMAINDER OF ARTICLE TRUNCATED ==========