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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 13:51:49 -0500
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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.

We can also pretend that cats <are> dogs thus cats <do> bark.

> 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