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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: Sun, 31 Mar 2024 14:02:38 -0500
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On 3/31/2024 1:52 PM, Fred. Zwarts wrote:
> Op 30.mrt.2024 om 21:27 schreef olcott:
>> On 3/30/2024 3:18 PM, Fred. Zwarts wrote:
>>> Op 30.mrt.2024 om 20:57 schreef olcott:
>>>> 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
>>>
>>> Indeed and they were right. Olcott's problem seems to be that he 
>>> thinks that he has to go to the end to prove it, but that is not 
>>> needed. We only have to go as far as needed for any given ε. Going to 
>>> the end is his problem, not that of math in the real number system.
>>> 0.999... = 1.0 means that with this sequence we can come as close to 
>>> 1.0 as needed. 
>>
>> That is not what the "=" sign means. It means exactly the same as.
> 
> No, olcott is trying to change the meaning of the symbol '='. That *is* 
> what the '=' means for real numbers, because 'exactly the same' is too 
> vague. Is 1.0 exactly the same as 1/1? It contains different symbols, so 
> why should they be exactly the same?

It never means approximately the same value.
It always means exactly the same value.

> Therefore, in the construction of reals it is defined how to determine 
> whether two reals are 'exactly' the same. If one real X can be 
> constructed with a sequence of xn and the other real Y with a sequence 
> yn, then we can use X = Y if for every rational ε > 0 we can find an N 
> so that for all n > N |xn - yn| < ε.
> The consequence of this is that for each real we can use an infinite 
> number of Cauchy sequences. E.g. the following sequences
> a: 1/1, 1/1, 1/1, 1/1, etc.
> b: 9/10, 99/100, 999/1000, etc.
> c: 10/9, 100/99, 1000/999, etc.
> d: 1/2, 2/3, 3/4, 4/5, etc.
> e: 1/2, 3/2, 3/4, 5/4, 5/6, 7/6, etc.
> are all sequences that are different representations of the same real 
> which in decimal notation can be written as 1. So, a=b=c=d=e=1.
> Olcott may not like it, but that is how the '=' is defined for reals.
> One may try to create another number system with another meaning for 
> '=', but then we are not talking about reals any more.
> If I do not like that 3+4=7, then I can try to create another system for 
> which 3+4=6 holds, which I like more, but I am no longer speaking of 
> real numbers (and probably nobody is interested in my number system).
> 

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