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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: =?UTF-8?Q?Re=3A_Definition_of_real_number_=E2=84=9D_--infinitesimal?=
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Date: Wed, 3 Apr 2024 10:40:57 -0500
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On 4/3/2024 10:30 AM, Fred. Zwarts wrote:
> 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 <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
> 
> Fighting windmills again. Fighting his own interpretation of 0.999...
> Unable to understand the normal interpretation, even when spelled out in 
> detail.
> 

When it is conventional to lie, I tell the truth.

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