Re: Definition of real number ℝ --infinitesimal--

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.
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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.

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No.
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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.
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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.
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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.
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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.

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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

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he would have noted that limits do not pretend to reach the end. They

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Other people were saying that math says 0.999... = 1.0

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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.

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That is not what the "=" sign means. It means exactly the same as.

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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?

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

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And what 'exactly the same value' means is explained below. It is a definition, not an opinion.
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No matter what you explain below nothing that anyone can possibly
say can possibly show that 1.000... = 1.0
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I use categorically exhaustive reasoning thus eliminating the
possibility of correct rebuttals.

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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.
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Typo corrected
No matter what you explain below nothing that anyone can possibly
say can possibly show that 0.999... = 1.0
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0.999...
Means an infinite never ending sequence that never reaches 1.0

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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?

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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).

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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.
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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.

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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.

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0.999... So what do the three dots means to you: Have a dotty day?

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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

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= means exactly the same value.
You can say that it means something else and you would be wrong.

 Olcott keeps fighting windmills. He keeps interpreting 0.999... = 1 differently from normal the interpretation for real numbers.

I am merely saying what it actually says.
I do not count idiomatic or figure-of-speech meanings as legitimate.
0.999... Specifies an infinite sequence of digits that never end.
0.999... = 1.0
Specifies when an infinite sequence of digits that never ends does end (a contradiction) that value is exactly equal to 1.0.

He keeps fighting his own wrong interpretation. 0.999... = 1 indeed means that 0.999... has exactly the same value as 1, but he keeps interpreting the value of 0.999... in his own way, so that he needs to fight his own interpretation.
 

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it. His seems to be doomed to stick to his own interpretation which he must fight, although nobody agrees with that interpretation. We know how Don Quixote ended.
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In both cases a lot of effort and pain could be saved by adjusting the interpretation to the normal one. However, it seems impossible to help him change his mind such that he will see the correct interpretation.
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--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer