Path: ...!weretis.net!feeder8.news.weretis.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail
From: olcott <polcott333@gmail.com>
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: <uujris$6e0$2@dont-email.me>
References: <bebe16f4f02eed7ac4e4d815dc0e1e98f9f0f2a0.camel@gmail.com>
 <87sf08qzt5.fsf@nosuchdomain.example.com> <uu7q21$k72e$1@dont-email.me>
 <uu8vf8$vsq2$1@dont-email.me> <uu95mr$114hv$5@dont-email.me>
 <uu9q43$16c9d$2@dont-email.me> <uu9qqn$16gt9$1@dont-email.me>
 <uu9s39$16gks$1@dont-email.me> <uu9sj2$16rdo$1@dont-email.me>
 <uucbe9$1utsv$2@dont-email.me> <uucc0e$1v1p5$1@dont-email.me>
 <uucdd7$1v8hd$1@dont-email.me> <uucec3$1vh78$1@dont-email.me>
 <uudnt6$2bun2$1@dont-email.me> <uuegit$2hjc8$1@dont-email.me>
 <uuev15$2l64e$2@dont-email.me> <uuevt5$2laff$1@dont-email.me>
 <8734s4r84s.fsf@nosuchdomain.example.com> <uufhse$2pgbg$1@dont-email.me>
 <87ttkkpn9y.fsf@nosuchdomain.example.com>
 <7jOdnYS6Ff5EhJH7nZ2dnZfqnPidnZ2d@brightview.co.uk>
 <87le5vpqiy.fsf@nosuchdomain.example.com> <uuhk45$3bqb2$1@dont-email.me>
 <uuj471$3qboe$2@dont-email.me>
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: <uuj471$3qboe$2@dont-email.me>
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 <news.dead.person.stones@darjeeling.plus.com> writes:
>>>> On 02/04/2024 02:27, Keith Thompson wrote:
>>>>> olcott <polcott333@gmail.com> writes:
>>>>>> On 4/1/2024 6:11 PM, Keith Thompson wrote:
>>>>>>> olcott <polcott333@gmail.com> 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 <are> dogs thus cats <do> bark.
> 

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer