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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: Mon, 1 Apr 2024 15:30:31 -0500
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On 4/1/2024 2:37 PM, Fred. Zwarts wrote:
> Op 01.apr.2024 om 20:54 schreef olcott:
>> On 4/1/2024 1:39 PM, Fred. Zwarts wrote:
>>> Op 01.apr.2024 om 16:33 schreef olcott:
>>>> On 4/1/2024 2:31 AM, Fred. Zwarts wrote:
>>>>> Op 31.mrt.2024 om 21:42 schreef olcott:
>>>>>> 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 1.000... = 1.0
>>>>>>
>>>>>> I use categorically exhaustive reasoning thus eliminating the
>>>>>> possibility of correct rebuttals.
>>>>>
>>>>> OK, then it is clear that olcott is not talking about real numbers, 
>>>>> because for reals categorically exhaustive reasoning proved that 
>>>>> 0.999... = 1 and olcott could not point to an error in the proof.
>>>>> It would have been less confusiong when he had mentioned that 
>>>>> explicitly.
>>>>>
>>>>
>>>> Typo corrected
>>>> No matter what you explain below nothing that anyone can possibly
>>>> say can possibly show that 0.999... = 1.0
>>>>
>>>> 0.999...
>>>> Means an infinite never ending sequence that never reaches 1.0
>>>
>>> Which nobody denied.
>>> Olcott again changes the question.
>>> The question is not does this sequence end, or does it reach 1.0, 
>>> but: which real is represented with this sequence?
>>
>> Since PI is represented by a single geometric point on the number line
>> then 0.999... would be correctly represented by the geometric point
>> immediately to the left of 1.0 on the number line or the RHS of this
>> interval [0,0, 1.0).
> 
> In the real number system it is incorrect to talk about a number 
> immediately next to another number. So, this is not about real numbers.
> 

PI is a real number.
If there is no real number that represents 0.999...
that does not provide a reason to say 0.999... = 1.0.

>> If there is no Real number at that point then there is no Real number 
>> that exactly represents 0.999...
> 
> Again olcott is changing the meaning of the words and symbols. 0.999... 
> represents a sequence x1 = 0.9, x2 = 0.99, x3 = 0.999, etc. That 
> sequence is not a point. This sequence represents a real number namely 
> exactly 1.0. It has nothing to do with the interval [0, 1). So, bringing 
> up this interval is irrelevant.
> If 0.999... ≠ 1.0, then tell us the value of a rational ε > 0 for which 
> no N can be found such that |xn - 1| < ε for all n > N.
> 
>>
>>> The answer is: This sequence represents one real: 1.
>>> Therefore we can say 0.999... = 1.0. It follows directly from the 
>>> construction of reals.
>>>
>>>>
>>>> If biology "proved" that cats are a kind of dog then no matter
>>>> what this "proof" contains we know in advance that it must be
>>>> incorrect.
>>>
>>> Similarly, if olcott 'proved' that 0.999... ≠ 1 then, no matter what 
>>> this "proof" contains, we know that it must be incorrect. Most 
>>> probably he is changing the question, changing the meaning of the 
>>> words or the symbols, or is talking about olcott numbers instead of 
>>> reals.
>>>
>>>>
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