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From: "Fred. Zwarts" <F.Zwarts@HetNet.nl>
Newsgroups: comp.theory
Subject: =?UTF-8?Q?Re=3A_Definition_of_real_number_=E2=84=9D_--infinitesimal?=
 =?UTF-8?Q?--?=
Date: Wed, 3 Apr 2024 17:34:28 +0200
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Op 03.apr.2024 om 17:08 schreef olcott:
> On 4/3/2024 3:27 AM, Fred. Zwarts wrote:
>> Op 02.apr.2024 om 17:50 schreef olcott:
>>> On 4/2/2024 10:38 AM, Fred. Zwarts wrote:
>>>> Op 02.apr.2024 om 16:53 schreef olcott:
>>>>> On 4/2/2024 4:43 AM, Fred. Zwarts wrote:
>>>>>> Op 01.apr.2024 om 22:30 schreef olcott:
>>>>>>> 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.
>>>>>>
>>>>>> Olcott makes me think of Don Quixote, who was unable to interpret 
>>>>>> the appearance of a windmill correctly. He interpreted it as 
>>>>>> nobody else did and therefore he thought he needed to fight it.
>>>>>> Similarly, olcott has an incorrect interpretation of 0.999... = 
>>>>>> 1.0. Nobody has that interpretation, but olcott thinks he has to 
>>>>>> fight it.
>>>>>
>>>>
>>>>> 0.999... So what do the three dots means to you: Have a dotty day?
>>>>
>>>> I see olcott does not read (or at least does not understand) what I 
>>>> write. It has been explained to him so many times in so much detail 
>>>> what 0.999... = 1 means. His mind seems to be too inflexible to 
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