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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?=
 =?UTF-8?Q?--?=
Date: Sun, 31 Mar 2024 14:42:59 -0500
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On 3/31/2024 2:26 PM, Fred. Zwarts wrote:
> Op 31.mrt.2024 om 21:02 schreef olcott:
>> 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.
> 
> And what 'exactly the same value' means is explained below. It is a 
> definition, not an opinion.
> 

No matter what you explain below nothing that anyone can possibly
say can possibly show that 0.000... = 1.0

I use categorically exhaustive reasoning thus eliminating the
possibility of correct rebuttals.

>>
>>> 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).
> 
> For real numbers, a has exactly the same value as b, c, d, e, f and 1. 
> That is how it is defined. If olcott has another definition of 'exactly 
> the same value', then he is changing the meaning of the words. The 
> meaning of '=' is exactly defined for reals.

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