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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: Tue, 2 Apr 2024 16:52:00 -0500
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On 4/2/2024 4:20 PM, Mike Terry wrote:
> On 02/04/2024 19:29, 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.
> 
>    0.999...  = 1.000...  (so he'd be wrong)
> 

In other words you simply choose to "not believe in"
the notion of infinitesimal difference. That doesn't
actually make it go away.

>>
>> 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,
> 
> Ah, I see - you're trying to say that 1.000... is a decimal 
> representation, but not 0.999...?, which would make sense of why you 
> think PO would be right above.  That's a new one on me, but I don't go 
> for that argument at all.
> 
> 0.999... is a decimal representation for the number 1, shortened by ... 
> which means "continuing in the obvious fashion" or equivalent wording.  
> I.e. 0.999... is the decimal where every digit after the decimal point 
> is a 9.  It represents the number 1, as does 1.000....  Yes, there are 
> two ways to represent the number 1 as an infinite decimal.  Not a problem.
> 
> Anyhow, I have a BA in mathematics, so I understand limits etc..  :)  I 
> was posting to explain why you're wasting your time trying to explain 
> abstract ideas to PO, but it's fine with me if people want to do that 
> for whatever reason.
> 
> Mike.
> ps. of course, someone could make a rule that infinitely repeating 9s in 
> a decimal expansion is outlawed, but that's not normal practice AFAIK.  
> People just accept there are two representations of certain numbers.
> 
>>
>> It can be genuinely difficult to wrap your head around this.  It *is*
>> counterintuitive.  And thoughtful challenges to the mathematical
>> orthodoxy, like the paper recently discussed in this thread, can be
>> useful.  But olcott doesn't offer a coherent alternative.
>>
>> [...]
>>

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