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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: Fri, 29 Mar 2024 20:11:38 -0500
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On 3/29/2024 7:25 PM, Keith Thompson wrote:
> "Fred. Zwarts" <F.Zwarts@HetNet.nl> writes:
>> Op 29.mrt.2024 om 16:46 schreef olcott:
>>> On 3/29/2024 8:13 AM, Richard Damon wrote:
>>>> On 3/28/24 11:50 PM, olcott wrote:
>>>>> On 3/28/2024 10:36 PM, Keith Thompson wrote:
>>>>>> olcott <polcott2@gmail.com> writes:
>>>>>> [...]
>>>>>>> It seems dead obvious that 0.999... is infinitesimally less than 1.0.
>>>>>>
>>>>>> Yes, it *seems* dead obvious.  That doesn't make it true, and in
>>>>>> fact it
>>>>>> isn't.
>>>>>>
>>>>>
>>>>> 0.999... means that is never reaches 1.0.
>>>>> and math simply stipulates that it does even though it does not.
>>>>
>>>>
>>>> 0.999... isn't a "number" in the Real Number system, just an
>>>> alternate representation for the number 1.
>>>>
>>> That is not true. 0.999... means never reaches 1.0
>>
>> Maybe for olcott's unspecified olcott numbers. For real numbers
>> 0.999... equals 1.0. There are many proofs. See e.g.
>>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
> 
> olcott almost has a point, in that the sequence of values 0.9, 0.99,
> 0.999, 0.9999, ... (continuing in the obvious manner) never reaches
> 1.0.  No element of that unending sequence of real numbers is exactly
> equal to 1.0.
> 
> What he either doesn't understand, or pretends not to understand, is
> that the notation "0.999..." does not refer either to any element of
> that sequence or to the entire sequence.  It refers to the *limit* of
> the sequence.  The limit of the sequence happens not to be an element of
> the sequence, and it's exactly equal to 1.0.
> 
In other words when one gets to the end of a never ending sequence
(a contradiction) thenn (then and only then) they reach 1.0.

> This is all stated in terms of the real numbers, which are a well
> defined set.  There are other systems with different properties.  If we
> were talking about the hyperreals, for example, olcott's statement might
> be correct (though I'm not sure of that).  But olcott seems to be
> insisting, quite incorrectly, that his statements apply to the reals.
> 

Pi exists at a single geometric point on the number line.

> If he's talking about the reals, he's wrong.  If he's talking about
> something other than the reals, he's boring.  Either way, he will not
> change his mind.  Attempts to explain limits and real numbers to him
> will fail.
> 

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