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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: Sat, 30 Mar 2024 14:57:11 -0500
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On 3/30/2024 2:45 PM, Fred. Zwarts wrote:
> Op 30.mrt.2024 om 14:56 schreef olcott:
>> On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
>>> Op 30.mrt.2024 om 02:31 schreef olcott:
>>>> On 3/29/2024 8:21 PM, Keith Thompson wrote:
>>>>> olcott <polcott2@gmail.com> writes:
>>>>>> On 3/29/2024 7:25 PM, Keith Thompson wrote:
>>>>> [...]
>>>>>>> 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.
>>>>>
>>>>> No.
>>>>>
>>>>> You either don't understand, or are pretending not to understand, what
>>>>> the limit of sequence is.  I'm not offering to explain it to you.
>>>>>
>>>>
>>>> I know (or at least knew) what limits are from my college calculus 40
>>>> years ago. If anyone or anything in any way says that 0.999... equals
>>>> 1.0 then they <are> saying what happens at the end of a never ending
>>>> sequence and this is a contradiction.
>>>>
>>>
>>> It is clear that olcott does not understand limits, because he is 
>>> changing the meaning of the words and the symbols. Limits are not 
>>> talking about what happens at the end of a sequence. It seems it has 
>>> to be spelled out for him, otherwise he will not understand.
>>>
>>
>> 0.999... Limits basically pretend that we reach the end of this 
>> infinite sequence even though that it impossible, and says after we 
>> reach this
>> impossible end the value would be 1.0.
> 
> No, if olcott had paid attention to the text below, or the article I 
> referenced:
>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
> 
> he would have noted that limits do not pretend to reach the end. They 

Other people were saying that math says 0.999... = 1.0

> only tell us that we don't need to go further than needed and that this 
> is reachable for any given rational ε > 0. It is interesting that this 
> is sufficient to construct reals.
> 
>>
>>> 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 = 
>>> 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. 
>>> The = symbol in the context of a limit means in this case:
>>> For each rational ε > 0 (no matter how small) we can find a number N 
>>> {in this case 10log(1/ε)}, such that for all n > N the absolute value 
>>> of the difference between xn and 1.0 is less than ε.
>>> It is not more and not less. Note that it does not speak of what 
>>> happens at the end of the sequence, or about completing the sequence.
>>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number 
>>> system), then he has to specify a rational ε for which no such N can 
>>> be found. If he cannot do that, then he is not speaking about real 
>>> numbers.
>>>
> 
> Good to see that there is no objection against this proof.
> 

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