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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: Mon, 1 Apr 2024 20:52:25 -0500
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On 4/1/2024 8:27 PM, 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?
> 

Whether it is a real number or not is moot to me.
My key point is that 0.999... = 1.0 is categorically impossible.

>>> More generally, does each real number correspond to a point on the
>>> number line, and does each point on the number line correspond to a real
>>> number?  (The real numbers can be formally defined without reference to
>>> geometry, but let's go with your geometric model for now.)
>>>
>>
>> The line segment [0.0, 1.0] is exactly one geometric point longer than
>> [0.0, 1.10), having all points in common besides the last point.
> 
> Do you believe that to be responsive to my question?  It isn't.  At all.
> 
>>> If so, let's call that real number (immediately to the left of 1.0) x.
>>> Consider ((x + 1.0)/2.0).  Let's call that number y.  (The intent is
>>> to
>>> construct a real number that is exactly halfway between x and 1.0.)
>>> Is y a real number?  (If not, the real numbers are, unexpectedly,
>>> not
>>> closed under common arithmetic operations.)
>>> Is y less than, equal to, or greater than x?
>>> Is y less than, equal to, or greater than 1.0?
>>> Again, I am talking *only* about real numbers.
>>> Given your past history, I do not expect straight answers to these
>>> questions, but I'm prepared to be pleasantly surprised.
> 
> You have answered none of my questions.  I'm mildly disappointed, but
> not really surprised, that you didn't even try to answer any of them.
> 
> My conclusion remains the same: you don't know what you're talking
> about, and your statements about real numbers and limits are wrong,
> incoherent, or both.
> 
> I encourage others to consider this when considering trying to explain
> things to olcott.
> 
> "I can explain it to you, but I can't comprehend it for you."
>      -- Edward I. Koch
> 

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