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From: "Fred. Zwarts" <F.Zwarts@HetNet.nl>
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 17:30:58 +0200
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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.