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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: =?UTF-8?Q?Re=3A_Definition_of_real_number_=E2=84=9D_--infinitesimal?=
 =?UTF-8?Q?--?=
Date: Tue, 2 Apr 2024 22:10:40 -0400
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On 4/2/24 11:50 AM, olcott wrote:
> 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 
>> understand 
> 
> = means exactly the same value.
> You can say that it means something else and you would be wrong.

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