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From: olcott <polcott2@gmail.com>
Newsgroups: comp.theory
Subject: =?UTF-8?Q?Re=3A_Definition_of_real_number_=E2=84=9D_--infinitesimal?=
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Date: Thu, 28 Mar 2024 21:43:53 -0500
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On 3/28/2024 9:23 PM, Richard Damon wrote:
> On 3/28/24 9:56 PM, olcott wrote:
>> On 3/28/2024 7:07 PM, Richard Damon wrote:
>>> On 3/28/24 12:07 PM, olcott wrote:
>>>> On 3/28/2024 10:59 AM, Andy Walker wrote:
>>>>> On 28/03/2024 13:16, Fred. Zwarts wrote:
>>>>>> It seems that wij wants to define a number type that is different
>>>>>> than the real numbers, but wij uses the same name Real. Very
>>>>>> confusing.
>>>>>
>>>>>      It seems to me to be worse than that.  Wij apparently thinks he
>>>>> /is/ defining the real numbers, and that the traditional 
>>>>> definitions are
>>>>> wrong in some way that he has never managed to explain.  But as he 
>>>>> uses
>>>>> infinity and infinitesimals [in an unexplained way], he is breaking 
>>>>> the
>>>>> Archimedean/Eudoxian axiom, so Wij-reals are not R, and they seem also
>>>>> not to be any of the other usual real-like number systems.  So the 
>>>>> whole
>>>>> of mathematical physics, engineering, ... is left in limbo, with 
>>>>> all the
>>>>> standard theorems inapplicable unless/until Wij tells us much more, 
>>>>> and
>>>>> probably not even then judging by Wij's responses thus far.
>>>>>
>>>>
>>>> Yet it seems that wij is correct that 0.999... would seem to
>>>> be infinitesimally < 1.0. One geometric point on the number line.
>>>> [0.0, 1.0) < [0.0, 1.0] by one geometric point.
>>>
>>> And that depends on WHAT number system you are working in.
>>>
>>> With the classical "Reals", 0.9999.... is 1.00000
>>>
>>
>> Yet that is NOT what 0.999... actually says.
>> It says that it gets infinitely close to 1.0 without every actually 
>> getting there. In other words it is infinitesimally less than 1.0.
> 
> But so close that no number exists between it and 1.0, so they are the 
> same number.
> 

You just admitted that they are not the same number.
It seems dead obvious that 0.999... is infinitesimally less than 1.0.

That we can say this in English yet not say this in conventional
number systems proves the need for another number system that can
say this.


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