On 3/28/24 12:07 PM, olcott wrote:
On 3/28/2024 10:59 AM, Andy Walker wrote:
On 28/03/2024 13:16, Fred. Zwarts wrote:
It seems that wij wants to define a number type that is different
than the real numbers, but wij uses the same name Real. Very
confusing.
It seems to me to be worse than that. Wij apparently thinks he >>> /is/ defining the real numbers, and that the traditional definitions are >>> wrong in some way that he has never managed to explain. But as he uses >>> infinity and infinitesimals [in an unexplained way], he is breaking the
Archimedean/Eudoxian axiom, so Wij-reals are not R, and they seem also
not to be any of the other usual real-like number systems. So the whole >>> of mathematical physics, engineering, ... is left in limbo, with all the >>> standard theorems inapplicable unless/until Wij tells us much more, and
probably not even then judging by Wij's responses thus far.
Yet it seems that wij is correct that 0.999... would seem to
be infinitesimally < 1.0. One geometric point on the number line.
[0.0, 1.0) < [0.0, 1.0] by one geometric point.
And that depends on WHAT number system you are working in.
With the classical "Reals", 0.9999.... is 1.00000
In some of the hyper real systems, there can be a hyper-finite real
number between them.
The number system that allow for such numbers also define what you can
do with these numbers (and what you can't do).
The problem with poorly defined systems is you can't actually try to do anything with them, because you don't have any tools.
Further, it seems he only defines how these number are written down.
There is no explanation of how to interpret these writings.
Well, quite. It seems that we're supposed to use the standard >>> processes of arithmetic until we get to infinity and similar. But of
course mathematics is concerned with numbers much more than with how
they are notated.
All might become clear if Wij could explain what problem he is >>> really trying to solve. What bridges fall down if "traditional" maths
is used but stay up with Wij-reals? What new puzzles are soluble? Are >>> they somehow more logical, or easier to teach? He seems to think that
"trad" maths is full of holes that he sees but that all the great minds
of the past 2500 years have overlooked. Perhaps it's all or mostly lost >>> in translation, but it's more likely that he is joining the PO Club.
On 3/28/2024 7:07 PM, Richard Damon wrote:
On 3/28/24 12:07 PM, olcott wrote:
On 3/28/2024 10:59 AM, Andy Walker wrote:
On 28/03/2024 13:16, Fred. Zwarts wrote:
It seems that wij wants to define a number type that is different
than the real numbers, but wij uses the same name Real. Very
confusing.
It seems to me to be worse than that. Wij apparently thinks he >>>> /is/ defining the real numbers, and that the traditional definitions
are
wrong in some way that he has never managed to explain. But as he uses >>>> infinity and infinitesimals [in an unexplained way], he is breaking the >>>> Archimedean/Eudoxian axiom, so Wij-reals are not R, and they seem also >>>> not to be any of the other usual real-like number systems. So the
whole
of mathematical physics, engineering, ... is left in limbo, with all
the
standard theorems inapplicable unless/until Wij tells us much more, and >>>> probably not even then judging by Wij's responses thus far.
Yet it seems that wij is correct that 0.999... would seem to
be infinitesimally < 1.0. One geometric point on the number line.
[0.0, 1.0) < [0.0, 1.0] by one geometric point.
And that depends on WHAT number system you are working in.
With the classical "Reals", 0.9999.... is 1.00000
Yet that is NOT what 0.999... actually says.
It says that it gets infinitely close to 1.0 without every actually
getting there. In other words it is infinitesimally less than 1.0.
In some of the hyper real systems, there can be a hyper-finite real
number between them.
The number system that allow for such numbers also define what you can
do with these numbers (and what you can't do).
The problem with poorly defined systems is you can't actually try to
do anything with them, because you don't have any tools.
Further, it seems he only defines how these number are written down. >>>>> There is no explanation of how to interpret these writings.
Well, quite. It seems that we're supposed to use the standard >>>> processes of arithmetic until we get to infinity and similar. But of >>>> course mathematics is concerned with numbers much more than with how
they are notated.
All might become clear if Wij could explain what problem he is >>>> really trying to solve. What bridges fall down if "traditional" maths >>>> is used but stay up with Wij-reals? What new puzzles are soluble? Are >>>> they somehow more logical, or easier to teach? He seems to think that >>>> "trad" maths is full of holes that he sees but that all the great minds >>>> of the past 2500 years have overlooked. Perhaps it's all or mostly
lost
in translation, but it's more likely that he is joining the PO Club.
On 3/28/24 9:56 PM, olcott wrote:
On 3/28/2024 7:07 PM, Richard Damon wrote:
On 3/28/24 12:07 PM, olcott wrote:
On 3/28/2024 10:59 AM, Andy Walker wrote:
On 28/03/2024 13:16, Fred. Zwarts wrote:
It seems that wij wants to define a number type that is different
than the real numbers, but wij uses the same name Real. Very
confusing.
It seems to me to be worse than that. Wij apparently thinks he
/is/ defining the real numbers, and that the traditional
definitions are
wrong in some way that he has never managed to explain. But as he >>>>> uses
infinity and infinitesimals [in an unexplained way], he is breaking >>>>> the
Archimedean/Eudoxian axiom, so Wij-reals are not R, and they seem also >>>>> not to be any of the other usual real-like number systems. So the >>>>> whole
of mathematical physics, engineering, ... is left in limbo, with
all the
standard theorems inapplicable unless/until Wij tells us much more, >>>>> and
probably not even then judging by Wij's responses thus far.
Yet it seems that wij is correct that 0.999... would seem to
be infinitesimally < 1.0. One geometric point on the number line.
[0.0, 1.0) < [0.0, 1.0] by one geometric point.
And that depends on WHAT number system you are working in.
With the classical "Reals", 0.9999.... is 1.00000
Yet that is NOT what 0.999... actually says.
It says that it gets infinitely close to 1.0 without every actually
getting there. In other words it is infinitesimally less than 1.0.
But so close that no number exists between it and 1.0, so they are the
same number.
On 3/28/24 9:56 PM, olcott wrote:
On 3/28/2024 7:07 PM, Richard Damon wrote:
On 3/28/24 12:07 PM, olcott wrote:
On 3/28/2024 10:59 AM, Andy Walker wrote:
On 28/03/2024 13:16, Fred. Zwarts wrote:
It seems that wij wants to define a number type that is different than the real numbers, but wij uses the same name Real. Very confusing.
It seems to me to be worse than that. Wij apparently thinks he
/is/ defining the real numbers, and that the traditional definitions are
wrong in some way that he has never managed to explain. But as he uses
infinity and infinitesimals [in an unexplained way], he is breaking the
Archimedean/Eudoxian axiom, so Wij-reals are not R, and they seem also
not to be any of the other usual real-like number systems. So the whole
of mathematical physics, engineering, ... is left in limbo, with all the
standard theorems inapplicable unless/until Wij tells us much more, and
probably not even then judging by Wij's responses thus far.
Yet it seems that wij is correct that 0.999... would seem to
be infinitesimally < 1.0. One geometric point on the number line.
[0.0, 1.0) < [0.0, 1.0] by one geometric point.
And that depends on WHAT number system you are working in.
With the classical "Reals", 0.9999.... is 1.00000
Yet that is NOT what 0.999... actually says.
It says that it gets infinitely close to 1.0 without every actually getting there. In other words it is infinitesimally less than 1.0.
But so close that no number exists between it and 1.0, so they are the
same number.
That comes out of the ACTUAL definitions of Real Numbers
In some of the hyper real systems, there can be a hyper-finite real number between them.
The number system that allow for such numbers also define what you can do with these numbers (and what you can't do).
The problem with poorly defined systems is you can't actually try to
do anything with them, because you don't have any tools.
Further, it seems he only defines how these number are written down.
There is no explanation of how to interpret these writings.
Well, quite. It seems that we're supposed to use the standard
processes of arithmetic until we get to infinity and similar. But of
course mathematics is concerned with numbers much more than with how they are notated.
All might become clear if Wij could explain what problem he is
really trying to solve. What bridges fall down if "traditional" maths
is used but stay up with Wij-reals? What new puzzles are soluble? Are
they somehow more logical, or easier to teach? He seems to think that
"trad" maths is full of holes that he sees but that all the great minds
of the past 2500 years have overlooked. Perhaps it's all or mostly lost
in translation, but it's more likely that he is joining the PO Club.
On 3/28/2024 9:23 PM, Richard Damon wrote:
On 3/28/24 9:56 PM, olcott wrote:
On 3/28/2024 7:07 PM, Richard Damon wrote:
On 3/28/24 12:07 PM, olcott wrote:
On 3/28/2024 10:59 AM, Andy Walker wrote:
On 28/03/2024 13:16, Fred. Zwarts wrote:
It seems that wij wants to define a number type that is different >>>>>>> than the real numbers, but wij uses the same name Real. Very
confusing.
It seems to me to be worse than that. Wij apparently thinks he
/is/ defining the real numbers, and that the traditional
definitions are
wrong in some way that he has never managed to explain. But as he >>>>>> uses
infinity and infinitesimals [in an unexplained way], he is
breaking the
Archimedean/Eudoxian axiom, so Wij-reals are not R, and they seem >>>>>> also
not to be any of the other usual real-like number systems. So the >>>>>> whole
of mathematical physics, engineering, ... is left in limbo, with
all the
standard theorems inapplicable unless/until Wij tells us much
more, and
probably not even then judging by Wij's responses thus far.
Yet it seems that wij is correct that 0.999... would seem to
be infinitesimally < 1.0. One geometric point on the number line.
[0.0, 1.0) < [0.0, 1.0] by one geometric point.
And that depends on WHAT number system you are working in.
With the classical "Reals", 0.9999.... is 1.00000
Yet that is NOT what 0.999... actually says.
It says that it gets infinitely close to 1.0 without every actually
getting there. In other words it is infinitesimally less than 1.0.
But so close that no number exists between it and 1.0, so they are the
same number.
You just admitted that they are not the same number.
It seems dead obvious that 0.999... is infinitesimally less than 1.0.
That we can say this in English yet not say this in conventional
number systems proves the need for another number system that can
say this.
olcott <polcott2@gmail.com> writes:
[...]
It seems dead obvious that 0.999... is infinitesimally less than 1.0.
Yes, it *seems* dead obvious. That doesn't make it true, and in fact it isn't.
0.999... denotes a *limit*. In particular, it's the limit of the value
as the number of 9s increases without bound. That's what the notation
"0.999..." *means*. (There are more precise notations for the same
thing, such as "0.9̅" (that's a 9 with an overbar, or "vinculum") or "0.(9)".
You have a sequence of numbers:
0.9
0.99
0.999
0.9999
0.99999
...
Each member of that sequence is strictly less than 1.0, but the *limit*
is exactly 1.0. The limit of a sequence doesn't have to be a member of
the sequence. The limit is, informally, the value that members of the sequence approach arbitrarily closely.
<https://en.wikipedia.org/wiki/Limit_of_a_sequence>
That we can say this in English yet not say this in conventional
number systems proves the need for another number system that can
say this.
Then I have good news for you. There are several such systems, for
example <https://en.wikipedia.org/wiki/Hyperreal_number>.
If your point is that you personally like hyperreals better than you
like reals, that's fine, as long as you're clear which number system
you're using.
If you talk about things like "0.999..." without
qualification, everyone will assume you're talking about real numbers.
And if you're going to play with hyperreal numbers, or surreal numbers,
or any of a number of other extensions to the real numbers, I suggest
that understanding the real numbers is a necessary prerequisite. That includes understanding that no real number is either infinitesimal or infinite.
Disclaimer: I'm not a mathematician. I welcome corrections.
olcott <polcott2@gmail.com> writes:
On 3/28/2024 10:36 PM, Keith Thompson wrote:[...]
If your point is that you personally like hyperreals better than you
like reals, that's fine, as long as you're clear which number system
you're using.
The Infinitesimal number system that I created.
Ah, then you're not talking about the conventional real numbers. That's
all I needed to know.
On 29/03/2024 04:29, olcott wrote:
x is said to be infinitesimal
if, and only if, |x| < 1/n for all integers n.
https://en.wikipedia.org/wiki/Hyperreal_number
That's for the hyperreals; there's a clue in the URL.
There are no such "x" in R, by the Archimedean axiom.
0.999... specifies infinitesimally < 1.0
No it doesn't. It specifies different things in different
number systems, which is why mathematicians don't use that notation
in contexts where there could be ambiguity.
and math guys have no way to say that so they
simply round up to 1.0
You've just referred to some "math guys" -- the proponents of
the hyperreals -- who say exactly what they mean by "infinitesimal".
You could equally have referred to the surreals [qv] where similar
statements are made and explained by "math guys". Maths has moved
on over the past few centuries. You and Wij need to move on with
the "math guys".
On 3/29/2024 8:23 AM, Andy Walker wrote:
On 29/03/2024 04:29, olcott wrote:
x is said to be infinitesimal
if, and only if, |x| < 1/n for all integers n.
https://en.wikipedia.org/wiki/Hyperreal_number
That's for the hyperreals; there's a clue in the URL.
There are no such "x" in R, by the Archimedean axiom.
0.999... specifies infinitesimally < 1.0
No it doesn't. It specifies different things in different
number systems, which is why mathematicians don't use that notation
in contexts where there could be ambiguity.
and math guys have no way to say that so they
simply round up to 1.0
You've just referred to some "math guys" -- the proponents of
the hyperreals -- who say exactly what they mean by "infinitesimal".
You could equally have referred to the surreals [qv] where similar
statements are made and explained by "math guys". Maths has moved
on over the past few centuries. You and Wij need to move on with
the "math guys".
Yet my system seems to make more sense.
[0.0, 1.0] - [0.0, 1.0) Only the last point on the number line
of the first interval is not contained in the second interval.
https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
Op 29.mrt.2024 om 15:21 schreef olcott:
On 3/29/2024 8:23 AM, Andy Walker wrote:
On 29/03/2024 04:29, olcott wrote:
x is said to be infinitesimal
if, and only if, |x| < 1/n for all integers n. https://en.wikipedia.org/wiki/Hyperreal_number
That's for the hyperreals; there's a clue in the URL.
There are no such "x" in R, by the Archimedean axiom.
0.999... specifies infinitesimally < 1.0
No it doesn't. It specifies different things in different number systems, which is why mathematicians don't use that notation
in contexts where there could be ambiguity.
and math guys have no way to say that so they
simply round up to 1.0
You've just referred to some "math guys" -- the proponents of the hyperreals -- who say exactly what they mean by "infinitesimal".
You could equally have referred to the surreals [qv] where similar statements are made and explained by "math guys". Maths has moved
on over the past few centuries. You and Wij need to move on with
the "math guys".
Yet my system seems to make more sense.
[0.0, 1.0] - [0.0, 1.0) Only the last point on the number line
of the first interval is not contained in the second interval.
Olcott's system does not (yet) make sense, because olcott has notWhy do you keep quoting something you don't even understand?
defined his system. E.g., what is a point in this context? What is a
number line?
This is how Reals are defined:
https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
If olcott wants to discuss his system, he should define his system at
least as detailed as reals are in this article and indicate where he is deviating from reals, otherwise it is unclear where he is talking about
and discussion do not make sense.
On 3/28/24 11:50 PM, olcott wrote:
On 3/28/2024 10:36 PM, Keith Thompson wrote:
olcott <polcott2@gmail.com> writes:
[...]
It seems dead obvious that 0.999... is infinitesimally less than 1.0.
Yes, it *seems* dead obvious. That doesn't make it true, and in fact it >>> isn't.
0.999... means that is never reaches 1.0.
and math simply stipulates that it does even though it does not.
0.999... isn't a "number" in the Real Number system, just an alternate representation for the number 1.
On 3/29/2024 8:13 AM, Richard Damon wrote:
On 3/28/24 11:50 PM, olcott wrote:
On 3/28/2024 10:36 PM, Keith Thompson wrote:
olcott <polcott2@gmail.com> writes:
[...]
It seems dead obvious that 0.999... is infinitesimally less than 1.0. >>>>Yes, it *seems* dead obvious. That doesn't make it true, and in
fact it
isn't.
0.999... means that is never reaches 1.0.
and math simply stipulates that it does even though it does not.
0.999... isn't a "number" in the Real Number system, just an alternate
representation for the number 1.
That is not true. 0.999... means never reaches 1.0
https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
Op 29.mrt.2024 om 16:46 schreef olcott:
On 3/29/2024 8:13 AM, Richard Damon wrote:
On 3/28/24 11:50 PM, olcott wrote:
On 3/28/2024 10:36 PM, Keith Thompson wrote:
olcott <polcott2@gmail.com> writes:
[...]
It seems dead obvious that 0.999... is infinitesimally less than 1.0. >>>>>Yes, it *seems* dead obvious. That doesn't make it true, and in
fact it
isn't.
0.999... means that is never reaches 1.0.
and math simply stipulates that it does even though it does not.
0.999... isn't a "number" in the Real Number system, just an
alternate representation for the number 1.
That is not true. 0.999... means never reaches 1.0
Maybe for olcott's unspecified olcott numbers. For real numbers 0.999... equals 1.0. There are many proofs. See e.g.
https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
"Fred. Zwarts" <F.Zwarts@HetNet.nl> writes:
Op 29.mrt.2024 om 16:46 schreef olcott:
On 3/29/2024 8:13 AM, Richard Damon wrote:
On 3/28/24 11:50 PM, olcott wrote:That is not true. 0.999... means never reaches 1.0
On 3/28/2024 10:36 PM, Keith Thompson wrote:
olcott <polcott2@gmail.com> writes:
[...]
It seems dead obvious that 0.999... is infinitesimally less than 1.0. >>>>>>Yes, it *seems* dead obvious. That doesn't make it true, and in
fact it
isn't.
0.999... means that is never reaches 1.0.
and math simply stipulates that it does even though it does not.
0.999... isn't a "number" in the Real Number system, just an
alternate representation for the number 1.
Maybe for olcott's unspecified olcott numbers. For real numbers
0.999... equals 1.0. There are many proofs. See e.g.
https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
olcott almost has a point, in that the sequence of values 0.9, 0.99,
0.999, 0.9999, ... (continuing in the obvious manner) never reaches
1.0. No element of that unending sequence of real numbers is exactly
equal to 1.0.
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.
This is all stated in terms of the real numbers, which are a well
defined set. There are other systems with different properties. If we
were talking about the hyperreals, for example, olcott's statement might
be correct (though I'm not sure of that). But olcott seems to be
insisting, quite incorrectly, that his statements apply to the reals.
If he's talking about the reals, he's wrong. If he's talking about
something other than the reals, he's boring. Either way, he will not
change his mind. Attempts to explain limits and real numbers to him
will fail.
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, isIn other words when one gets to the end of a never ending sequence
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.
(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.
This is all stated in terms of the real numbers, which are a well
defined set. There are other systems with different properties. If we
were talking about the hyperreals, for example, olcott's statement might >>> be correct (though I'm not sure of that). But olcott seems to be
insisting, quite incorrectly, that his statements apply to the reals.
Pi exists at a single geometric point on the number line.
Irrelevant.
[...]
olcott <polcott2@gmail.com> writes:
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:[...]
No.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.
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.
Apparently you've forgotten what limits are. I'm still not offering to explain them.
Irrelevant.This is all stated in terms of the real numbers, which are a wellPi exists at a single geometric point on the number line.
defined set. There are other systems with different properties. If we >>>>> were talking about the hyperreals, for example, olcott's statement might >>>>> be correct (though I'm not sure of that). But olcott seems to be
insisting, quite incorrectly, that his statements apply to the reals. >>>>
One geometric point to the left or to the right is incorrect.
You apparently think there's a geometric point immediately to the left
of pi. Real numbers don't work that way.
Say the numeric value corresponding to the geometric point immediately
to the left of pi on the real number line is x. What is the real value
of (pi+x)/2? Is it greater than x? Is it less than pi?
I'm going to drop out of this discussion unless someone says something sufficiently interesting.
Can you quit publishing my email address?
On 2024-03-29 20:18, olcott wrote:
Can you quit publishing my email address?
He's not. You are. His newsreader is just quoting you.
André
On 3/29/2024 9:28 PM, André G. Isaak wrote:
On 2024-03-29 20:18, olcott wrote:
Can you quit publishing my email address?
He's not. You are. His newsreader is just quoting you.
André
Somehow the newsgroup provider started publishing it.
It didn't do this initially.
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, isIn other words when one gets to the end of a never ending sequence
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.
(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.
https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
If olcott had read the article I referenced,
https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
He would have seen a proof for 0.999... = 1. This article also explains
what it means. With a proper interpretation of these words, there is no contradiction. He may not like it, but it has been proven. So, either he continues to talk about his unspecified olcott numbers, or he does not understand the proof for real numbers, or he changes the meaning of the words. Of course, if there are details in the proof he does not
understand, he is free to ask for an explanation.
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, isIn other words when one gets to the end of a never ending sequence
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.
(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.
On 3/30/2024 4:37 AM, Fred. Zwarts wrote:
Op 30.mrt.2024 om 02:31 schreef olcott:
On 3/29/2024 8:21 PM, Keith Thompson wrote:If olcott had read the article I referenced,
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 anIn other words when one gets to the end of a never ending sequence
element of
the sequence, and it's exactly equal to 1.0.
(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.
https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
He would have seen a proof for 0.999... = 1.
Right and another article says that all cats are really a kind of dog.
0.999... means never reaches 1.0.
Anyone saying differently is not telling the truth.
This article also explains what it means. With a proper interpretation
of these words, there is no contradiction. He may not like it, but it
has been proven. So, either he continues to talk about his unspecified
olcott numbers, or he does not understand the proof for real numbers,
or he changes the meaning of the words. Of course, if there are
details in the proof he does not understand, he is free to ask for an
explanation.
On 3/30/2024 5:08 AM, Andy Walker wrote:
On 30/03/2024 01:11, olcott wrote:
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.
Zeno knew better than that some 2400 years ago. He knew that
Achilles caught the tortoise, even though it was an infinite number of
steps; it depends on how fast you complete/encounter the elements of
the sequence. If you travel along the real number line at 1 unit/hour,
starting at 0 at noon, then you reach 0.9 at 12:54, 0.99 at 12:59:24,
0.999 at 12:59:56.4, 0.9999 at 12:59:59.64, 0.99999 at 12:59:59.964,
0.999999 at ..., and this "never ending sequence" is completed by 13:00.
Luckily, none of this matters, as "0.999..." viewed as a real number is
not defined as a "never ending sequence", but as the limit of a sequence
whose terms are defined.
https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes
Zeno essentially "proved" that no one can cross a ten foot wide room
in finite time. He was nuts.
If you switch from the reals to the surreals or hyperreals, then >> you may prefer the way "0.999..." is treated, but you have simply moved
the "never-ending" problem to somewhere else, and the solution is the
same -- you need to sub-divide time appropriately to complete many steps
in a short time. Time is inexorable in the Real World, but messwithable
in theoretical physics and in mathematics.
On 3/30/24 9:56 AM, olcott wrote:
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.
Nope. Shows you don't really understand what limits are.
And are just a pathological liar as you insist that you falsehoods based
on the wrong definitions are the truth.
0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. The
= symbol in the context of a limit means in this case:
For each rational ε > 0 (no matter how small) we can find a number N {in this case 10log(1/ε)}, such that for all n > N the absolute value of the difference between xn and 1.0 is less than ε.
It is not more and not less. Note that it does not speak of what
happens at the end of the sequence, or about completing the sequence.
If olcott wants to prove that 0.999... ≠ 1.0 (in the real number system), then he has to specify a rational ε for which no such N can
be found. If he cannot do that, then he is not speaking about real numbers.
On 3/30/24 10:57 AM, wij wrote:
On Sat, 2024-03-30 at 10:01 -0400, Richard Damon wrote:
On 3/30/24 9:56 AM, olcott wrote:
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.
Nope. Shows you don't really understand what limits are.
And are just a pathological liar as you insist that you falsehoods based on the wrong definitions are the truth.
You are nut who always think he is talking B while reading A. (x!=c) https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-limits.html
Limit is defined on existing numbers, it cannot define the the number it is using.
Things is very simple: "repeating decimal" means the pattern is infinite. (you are worse than olcott in this)
If it does not exit, your math (repeating decimal is ...) is garbage talking
about something does not exist and use it as proof of fact.
What number does the representation 0.abc represent?
it is BY DEFINITION 0 + a * 10^-1 + b * 10^-2 + c * 10^-3
what number does the representation 0.aaa... represent:
The value of lim(n-> inf) Sum(9 * 10^-i) [for i = 1 to n]
If a = 9, what number is that 0.999.... but also the number 1.0 since
they are the same.
For ANY e > 0, there exists an N that for all values of function/series
witn n >= N the difference between the function and 1 is less then e.
BY THE DEFINITION OF LIMIT, that means that 0.999... IS EQUAL TO 1.000
For Reals
Remember n-ary representations are NOT numbers, but representations of
the number.
the value of repeating n-ary representations are defined by limits
Limits are NOT on "a number" but on a function or series (which is a
sort of function of the number of terms being used).
0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 = 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. The
= symbol in the context of a limit means in this case:
For each rational ε > 0 (no matter how small) we can find a number N {in this case 10log(1/ε)}, such that for all n > N the absolute value
of the difference between xn and 1.0 is less than ε.
It is not more and not less. Note that it does not speak of what happens at the end of the sequence, or about completing the sequence. If olcott wants to prove that 0.999... ≠ 1.0 (in the real number system), then he has to specify a rational ε for which no such N can be found. If he cannot do that, then he is not speaking about real numbers.
On Sat, 2024-03-30 at 11:34 -0400, Richard Damon wrote:
On 3/30/24 10:57 AM, wij wrote:
On Sat, 2024-03-30 at 10:01 -0400, Richard Damon wrote:
On 3/30/24 9:56 AM, olcott wrote:
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* ofIn other words when one gets to the end of a never ending sequence >>>>>>>>> (a contradiction) thenn (then and only then) they reach 1.0.
the sequence. The limit of the sequence happens not to be an >>>>>>>>>> element of
the sequence, and it's exactly equal to 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.
Nope. Shows you don't really understand what limits are.
And are just a pathological liar as you insist that you falsehoods based >>>> on the wrong definitions are the truth.
You are nut who always think he is talking B while reading A. (x!=c)
https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-limits.html
Limit is defined on existing numbers, it cannot define the the number it is using.
Things is very simple: "repeating decimal" means the pattern is infinite. >>> (you are worse than olcott in this)
If it does not exit, your math (repeating decimal is ...) is garbage talking
about something does not exist and use it as proof of fact.
What number does the representation 0.abc represent?
it is BY DEFINITION 0 + a * 10^-1 + b * 10^-2 + c * 10^-3
what number does the representation 0.aaa... represent:
The value of lim(n-> inf) Sum(9 * 10^-i) [for i = 1 to n]
If a = 9, what number is that 0.999.... but also the number 1.0 since
they are the same.
I should have provided all that can explain your doubt, but you still
keep insisting 0.999...=1 with no proof, like POOP.
For ANY e > 0, there exists an N that for all values of function/seriesSee the link above. limit says the limit of 0.999... is 1, not 0.999... is 1. You keep talking RD's POOP.
witn n >= N the difference between the function and 1 is less then e.
BY THE DEFINITION OF LIMIT, that means that 0.999... IS EQUAL TO 1.000
For RealsFor your POO Real (not even the obsolete real)
Remember n-ary representations are NOT numbers, but representations ofI already said, limit is defined on existing number system, it cannot define numbers it talks about.
the number.
the value of repeating n-ary representations are defined by limits
Limits are NOT on "a number" but on a function or series (which is aOk, now you changed to another excuse. So, you are not really talking
sort of function of the number of terms being used).
about numbers, right.
0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. The
= symbol in the context of a limit means in this case:
For each rational ε > 0 (no matter how small) we can find a number N >>>>>> {in this case 10log(1/ε)}, such that for all n > N the absolute value >>>>>> of the difference between xn and 1.0 is less than ε.
It is not more and not less. Note that it does not speak of what
happens at the end of the sequence, or about completing the sequence. >>>>>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number >>>>>> system), then he has to specify a rational ε for which no such N can >>>>>> be found. If he cannot do that, then he is not speaking about real >>>>>> numbers.
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 theIn other words when one gets to the end of a never ending sequence >>>>>> (a contradiction) thenn (then and only then) they reach 1.0.
*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.
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
only tell us that we don't need to go further than needed and that this
is reachable for any given rational ε > 0. It is interesting that this
is sufficient to construct reals.
0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞.
The = symbol in the context of a limit means in this case:
For each rational ε > 0 (no matter how small) we can find a number N
{in this case 10log(1/ε)}, such that for all n > N the absolute value
of the difference between xn and 1.0 is less than ε.
It is not more and not less. Note that it does not speak of what
happens at the end of the sequence, or about completing the sequence.
If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
system), then he has to specify a rational ε for which no such N can
be found. If he cannot do that, then he is not speaking about real
numbers.
Good to see that there is no objection against this proof.
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 toIn other words when one gets to the end of a never ending sequence >>>>>>> (a contradiction) thenn (then and only then) they reach 1.0.
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.
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
only tell us that we don't need to go further than needed and that
this is reachable for any given rational ε > 0. It is interesting that
this is sufficient to construct reals.
0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. >>>> The = symbol in the context of a limit means in this case:
For each rational ε > 0 (no matter how small) we can find a number N >>>> {in this case 10log(1/ε)}, such that for all n > N the absolute
value of the difference between xn and 1.0 is less than ε.
It is not more and not less. Note that it does not speak of what
happens at the end of the sequence, or about completing the sequence.
If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
system), then he has to specify a rational ε for which no such N can >>>> be found. If he cannot do that, then he is not speaking about real
numbers.
Good to see that there is no objection against this proof.
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 toIn other words when one gets to the end of a never ending sequence >>>>>>>> (a contradiction) thenn (then and only then) they reach 1.0.
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.
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.
It does not say (nor deny) that 1.0 will be reached. That is
the meaning of the = symbol in the context of limits. It is olcott's
problem that he changes the meaning of the = symbol.
only tell us that we don't need to go further than needed and that
this is reachable for any given rational ε > 0. It is interesting
that this is sufficient to construct reals.
0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. >>>>> The = symbol in the context of a limit means in this case:
For each rational ε > 0 (no matter how small) we can find a number >>>>> N {in this case 10log(1/ε)}, such that for all n > N the absolute
value of the difference between xn and 1.0 is less than ε.
It is not more and not less. Note that it does not speak of what
happens at the end of the sequence, or about completing the sequence. >>>>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
system), then he has to specify a rational ε for which no such N
can be found. If he cannot do that, then he is not speaking about
real numbers.
Good to see that there is no objection against this proof.
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 toIn other words when one gets to the end of a never ending sequence >>>>>>>>> (a contradiction) thenn (then and only then) they reach 1.0.
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.
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.
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:Other people were saying that math says 0.999... = 1.0
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:[...]
No.What he either doesn't understand, or pretends not to >>>>>>>>>>>> understand, isIn other words when one gets to the end of a never ending >>>>>>>>>>> sequence
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.
(a contradiction) thenn (then and only then) they reach 1.0. >>>>>>>>>>
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 >>>>>
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.
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).
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:Other people were saying that math says 0.999... = 1.0
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:[...]
No.What he either doesn't understand, or pretends not to >>>>>>>>>>>> understand, isIn other words when one gets to the end of a never ending >>>>>>>>>>> sequence
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.
(a contradiction) thenn (then and only then) they reach 1.0. >>>>>>>>>>
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 >>>>>
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.
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).
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:[...]
No.What he either doesn't understand, or pretends not to >>>>>>>>>>>>> understand, isIn other words when one gets to the end of a never ending >>>>>>>>>>>> sequence
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.
(a contradiction) thenn (then and only then) they reach 1.0. >>>>>>>>>>>
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.
--
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.
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:[...]
No.What he either doesn't understand, or pretends not to >>>>>>>>>>>>>> understand, isIn other words when one gets to the end of a never ending >>>>>>>>>>>>> sequence
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.
(a contradiction) thenn (then and only then) they reach 1.0. >>>>>>>>>>>>
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.
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:[...]
No.What he either doesn't understand, or pretends not to >>>>>>>>>>>>>>> understand, isIn other words when one gets to the end of a never ending >>>>>>>>>>>>>> sequence
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.
(a contradiction) thenn (then and only then) they reach 1.0. >>>>>>>>>>>>>
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.
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.
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:No, olcott is trying to change the meaning of the symbol '='. That >>>>>> *is* what the '=' means for real numbers, because 'exactly the
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:[...]
No.What he either doesn't understand, or pretends not to >>>>>>>>>>>>>>>> understand, isIn other words when one gets to the end of a never ending >>>>>>>>>>>>>>> sequence
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.
(a contradiction) thenn (then and only then) they reach 1.0. >>>>>>>>>>>>>>
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. >>>>>>
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
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.
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.
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:It never means approximately the same value.
On 3/30/2024 3:18 PM, Fred. Zwarts wrote: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
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, isIn other words when one gets to the end of a never >>>>>>>>>>>>>>>>> ending sequence
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. >>>>>>>>>>>>>>>>>>
(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. >>>>>>>>
contains different symbols, so why should they be exactly the same? >>>>>>>
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).
If there is no Real number at that point then there is no Real number that exactly represents 0.999...
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.
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.
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, isIn other words when one gets to the end of a never >>>>>>>>>>>>>>>>>> ending sequence
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. >>>>>>>>>>>>>>>>>>>
(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.
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.
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.
On 2024-04-01 14:30, olcott wrote:
On 4/1/2024 2:37 PM, Fred. Zwarts wrote:
Op 01.apr.2024 om 20:54 schreef olcott:
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.
I'm a bit unclear why you keep bringing pi into this. pi isn't a
repeating decimal, unlike 0.999... which is.
But if you want to talk about pi, that also can be construed as the
limit of an infinite series:
π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
For any *finite* number of terms, the above series never quite reaches
pi, but the LIMIT of this series is exactly equal to pi, not to some
value one 'geometric point' (which has a length of exactly zero) away
from that limit And for this series your peculiar notion that it is a geometric point away is particularly absurd since it isn't clear whether you'd want it to be one 'geometric point' greater or less than this
limit since the series doesn't converge on its limit from a single direction.
Similarly, the value of the series 9/10 + 99/100 + 999/1000... is
exactly equal to the LIMIT of that series. That's what the notation
0.999... means, by definition.
André
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
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.
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). If there is no Real number at that point then
there is no Real number that exactly represents 0.999...
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.
wij <wyniijj5@gmail.com> writes:
On Tue, 2024-04-02 at 05:46 +0800, wij wrote:
On Mon, 2024-04-01 at 15:03 -0600, André G. Isaak wrote:
On 2024-04-01 14:59, André G. Isaak wrote:
π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
For any *finite* number of terms, the above series never quite reachesObviously, I mean the limit is exactly equal to π/4.
pi, but the LIMIT of this series is exactly equal to pi.
Note that the limit of sequence about is π/4. But none of any number
in the sequence is π/4
To be more correctly:
π/4 = lim 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
No, that's no more correct because the limit is already implied and,
more importantly, the /correct/ limit is implied. The limit is that of
a sequence of partial sums, so to write it without the implied limit one should write either
π/4 = Sigma_{n=0}^oo -1^n * 1/(2n+1)
or
π/4 = lim_{k->oo} Sigma_{n=0}^k -1^n * 1/(2n+1).
Just sticking "lim" in front gains you nothing.
or π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11... +non_zero_remainder
No, because the limit is implied. That's what the informal "..." means.
The only correct equation using the rather informal ... is
π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11... + 0.
You can say what you probably mean using the proper notation as above,
but I doubt you want to do that.
or π/4 ≒ 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
No. The ... implies the limit of the partial sums and, as you know,
that limit is π/4. Exactly π/4.
André already said the no finite partial sum is equal to π/4. What do
you gain by trying to say that again?
Have you got any further in defining the operations on your "numbers as strings or TMs" so that (a+b)/2 is neither a nor b when a=1 and
b=0.999...? That's a fun project, but I don't think you are able to do
it.
Op 02.apr.2024 om 00:21 schreef olcott:
On 4/1/2024 3:59 PM, André G. Isaak wrote:
On 2024-04-01 14:30, olcott wrote:
On 4/1/2024 2:37 PM, Fred. Zwarts wrote:
Op 01.apr.2024 om 20:54 schreef olcott:
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.
I'm a bit unclear why you keep bringing pi into this. pi isn't a
repeating decimal, unlike 0.999... which is.
But if you want to talk about pi, that also can be construed as the
limit of an infinite series:
π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
For any *finite* number of terms, the above series never quite
reaches pi, but the LIMIT of this series is exactly equal to pi, not
to some
When olcott says:
To says that 0.999... = 1.0 means that after the never ending
sequence ends (a contradiction) then we reach exactly 1.0.
he is again changing the meaning of 0.999... = 1.0 from what is defined
for real numbers. It is clear that his interpretation then brings him to--
a contradiction. This is more evidence that this interpretation is
incorrect for the real number system.
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: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.
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, isIn other words when one gets to the end of a never >>>>>>>>>>>>>>>>>>>>> ending sequence
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. >>>>>>>>>>>>>>>>>>>>>>
(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 >>>>>>>>>>>>>>
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?
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.
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: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.
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, isIn other words when one gets to the end of a never >>>>>>>>>>>>>>>>>>>>>> ending sequence
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. >>>>>>>>>>>>>>>>>>>>>>>
(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 >>>>>>>>>>>>>>>
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
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.
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.
On 02/04/2024 19:29, Keith Thompson wrote:
Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
On 02/04/2024 02:27, Keith Thompson wrote:
olcott <polcott333@gmail.com> writes:
On 4/1/2024 6:11 PM, Keith Thompson wrote:"IDK, probably not."
olcott <polcott333@gmail.com> writes:
[...]
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). If there is no Real number at that point then >>>>>>> there is no Real number that exactly represents 0.999...
In the following I'm talking about real numbers, and only real
numbers -- not hyperreals, or surreals, or any other extension to the >>>>>> real numbers.
You assert that there is a geometric point immediately to the left >>>>>> of
1.0 on the number line. (I disagree, but let's go with it for now.) >>>>>> Am I correct in assuming that this means that that point corresponds >>>>>> to
a real number that is distinct from, and less than, 1.0?
IDK, probably not. I am saying that 0.999... exactly equals this
number.
Did you even consider taking some time to *think* about this?
PO just says things he thinks are true based on his first intuitions
when he encountered a topic. He does not "reason" his way to a new
carefully thought out theory or even to a single coherent idea. Don't
imagine he is thinking of hyperreals or anything - he just "knows"
that obviously any number which starts 0.??? is less than one starting
1.??? - because 0 is less than 1 !! Or whatever, it really doesn't
matter.
I don't think he's explicitly said that any real number whose decimal
representation starts with "0." is less than one starting with "1." --
but if said that, he'd be right.
0.999... = 1.000... (so he'd be wrong)
--
What he refuses to understand is that the notation "0.999..." is not a
decimal representation. The "..." notation refers to the limit of a
sequence, and of course the limit of a sequence does not have to be a
member of the sequence. Every member of the sequence (0.9, 0.99, 0.999,
0.9999, continuing in the obvious manner) is a real (and rational)
number that is strictly less than 1.0. But the limit of the sequence is
1.0. Sequences and their limits can be and are defined rigorously
without reference to infinitesimals or infinities,
Ah, I see - you're trying to say that 1.000... is a decimal
representation, but not 0.999...?, which would make sense of why you
think PO would be right above. That's a new one on me, but I don't go
for that argument at all.
0.999... is a decimal representation for the number 1, shortened by ... which means "continuing in the obvious fashion" or equivalent wording.
I.e. 0.999... is the decimal where every digit after the decimal point
is a 9. It represents the number 1, as does 1.000.... Yes, there are
two ways to represent the number 1 as an infinite decimal. Not a problem.
Anyhow, I have a BA in mathematics, so I understand limits etc.. :) I was posting to explain why you're wasting your time trying to explain abstract ideas to PO, but it's fine with me if people want to do that
for whatever reason.
Mike.
ps. of course, someone could make a rule that infinitely repeating 9s in
a decimal expansion is outlawed, but that's not normal practice AFAIK. People just accept there are two representations of certain numbers.
It can be genuinely difficult to wrap your head around this. It *is*
counterintuitive. And thoughtful challenges to the mathematical
orthodoxy, like the paper recently discussed in this thread, can be
useful. But olcott doesn't offer a coherent alternative.
[...]
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: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.
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, isIn other words when one gets to the end of a never >>>>>>>>>>>>>>>>>>>>> ending sequence
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. >>>>>>>>>>>>>>>>>>>>>>
(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 >>>>>>>>>>>>>>
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?
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.
On 4/2/2024 10:27 AM, Fred. Zwarts wrote:
Op 02.apr.2024 om 16:44 schreef olcott:
On 4/2/2024 3:53 AM, Fred. Zwarts wrote:
Op 02.apr.2024 om 00:21 schreef olcott:0.999...
On 4/1/2024 3:59 PM, André G. Isaak wrote:
On 2024-04-01 14:30, olcott wrote:
On 4/1/2024 2:37 PM, Fred. Zwarts wrote:
Op 01.apr.2024 om 20:54 schreef olcott:
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.
I'm a bit unclear why you keep bringing pi into this. pi isn't a
repeating decimal, unlike 0.999... which is.
But if you want to talk about pi, that also can be construed as
the limit of an infinite series:
π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
For any *finite* number of terms, the above series never quite
reaches pi, but the LIMIT of this series is exactly equal to pi,
not to some
When olcott says:
To says that 0.999... = 1.0 means that after the never ending
sequence ends (a contradiction) then we reach exactly 1.0.
he is again changing the meaning of 0.999... = 1.0 from what is defined >>>
is defined to specify an infinite sequence that never reaches 1.0,
when anything else defines it differently this anything else is wrong.
Again he is changing the definition
In other words you believe that 0.999... means have a dotty day, and has nothing to do with infinite sequences of digits.
and is not talking about reals. For reals 0.999... = 1. It has been
explained to him so many times in such
No matter how many times a contradictory statement is explained it never becomes true.
detail. He still does not understand it. It won't help to explain it
again. He sticks to his own illogical world. Not flexible enough to
change his mind when evidence has been provided.
On 4/2/2024 3:53 AM, Fred. Zwarts wrote:
Op 02.apr.2024 om 00:21 schreef olcott:
On 4/1/2024 3:59 PM, André G. Isaak wrote:
On 2024-04-01 14:30, olcott wrote:
On 4/1/2024 2:37 PM, Fred. Zwarts wrote:
Op 01.apr.2024 om 20:54 schreef olcott:
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.
I'm a bit unclear why you keep bringing pi into this. pi isn't a
repeating decimal, unlike 0.999... which is.
But if you want to talk about pi, that also can be construed as the
limit of an infinite series:
π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
For any *finite* number of terms, the above series never quite
reaches pi, but the LIMIT of this series is exactly equal to pi, not
to some
When olcott says:
To says that 0.999... = 1.0 means that after the never ending
sequence ends (a contradiction) then we reach exactly 1.0.
he is again changing the meaning of 0.999... = 1.0 from what is defined
0.999...
is defined to specify an infinite sequence that never reaches 1.0,
when anything else defines it differently this anything else is wrong.
for real numbers. It is clear that his interpretation then brings him
to a contradiction. This is more evidence that this interpretation is
incorrect for the real number system.
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:Which nobody denied.
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: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.
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, isIn other words when one gets to the end of a >>>>>>>>>>>>>>>>>>>>>>> never ending sequence
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. >>>>>>>>>>>>>>>>>>>>>>>>
(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 >>>>>>>>>>>>>>>>
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 >>>>>>>>
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.
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.
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.
Op 02.apr.2024 om 23:52 schreef olcott:
On 4/2/2024 4:20 PM, Mike Terry wrote:
On 02/04/2024 19:29, Keith Thompson wrote:
Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
On 02/04/2024 02:27, Keith Thompson wrote:
olcott <polcott333@gmail.com> writes:
On 4/1/2024 6:11 PM, Keith Thompson wrote:"IDK, probably not."
olcott <polcott333@gmail.com> writes:
[...]
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). If there is no Real number at that point then >>>>>>>>> there is no Real number that exactly represents 0.999...
In the following I'm talking about real numbers, and only real >>>>>>>> numbers -- not hyperreals, or surreals, or any other extension >>>>>>>> to the
real numbers.
You assert that there is a geometric point immediately to the left >>>>>>>> of
1.0 on the number line. (I disagree, but let's go with it for >>>>>>>> now.)
Am I correct in assuming that this means that that point
corresponds
to
a real number that is distinct from, and less than, 1.0?
IDK, probably not. I am saying that 0.999... exactly equals this >>>>>>> number.
Did you even consider taking some time to *think* about this?
PO just says things he thinks are true based on his first intuitions >>>>> when he encountered a topic. He does not "reason" his way to a new
carefully thought out theory or even to a single coherent idea. Don't >>>>> imagine he is thinking of hyperreals or anything - he just "knows"
that obviously any number which starts 0.??? is less than one starting >>>>> 1.??? - because 0 is less than 1 !! Or whatever, it really doesn't
matter.
I don't think he's explicitly said that any real number whose decimal
representation starts with "0." is less than one starting with "1." -- >>>> but if said that, he'd be right.
0.999... = 1.000... (so he'd be wrong)
In other words you simply choose to "not believe in"
the notion of infinitesimal difference. That doesn't
actually make it go away.
It is not a matter of 'believe-in'. In the real number system there are
no infinitesimal differences.
There always a finite ε is used.
Apparently olcott is talking about his undisclosed olcott-numbers, but
he keeps it as a secret what it means.
On 4/3/2024 3:42 AM, Fred. Zwarts wrote:
Op 02.apr.2024 om 23:52 schreef olcott:
On 4/2/2024 4:20 PM, Mike Terry wrote:
On 02/04/2024 19:29, Keith Thompson wrote:
Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
On 02/04/2024 02:27, Keith Thompson wrote:
olcott <polcott333@gmail.com> writes:
On 4/1/2024 6:11 PM, Keith Thompson wrote:"IDK, probably not."
olcott <polcott333@gmail.com> writes:
[...]
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). If there is no Real number at that point >>>>>>>>>> then
there is no Real number that exactly represents 0.999...
In the following I'm talking about real numbers, and only real >>>>>>>>> numbers -- not hyperreals, or surreals, or any other extension >>>>>>>>> to the
real numbers.
You assert that there is a geometric point immediately to the left >>>>>>>>> of
1.0 on the number line. (I disagree, but let's go with it for >>>>>>>>> now.)
Am I correct in assuming that this means that that point
corresponds
to
a real number that is distinct from, and less than, 1.0?
IDK, probably not. I am saying that 0.999... exactly equals this >>>>>>>> number.
Did you even consider taking some time to *think* about this?
PO just says things he thinks are true based on his first intuitions >>>>>> when he encountered a topic. He does not "reason" his way to a new >>>>>> carefully thought out theory or even to a single coherent idea. Don't >>>>>> imagine he is thinking of hyperreals or anything - he just "knows" >>>>>> that obviously any number which starts 0.??? is less than one
starting
1.??? - because 0 is less than 1 !! Or whatever, it really doesn't >>>>>> matter.
I don't think he's explicitly said that any real number whose decimal >>>>> representation starts with "0." is less than one starting with "1." -- >>>>> but if said that, he'd be right.
0.999... = 1.000... (so he'd be wrong)
In other words you simply choose to "not believe in"
the notion of infinitesimal difference. That doesn't
actually make it go away.
It is not a matter of 'believe-in'. In the real number system there
are no infinitesimal differences.
So when they do occur
they cannot be expressed so the convention is to ignore them.
Infinitesimal differences cannot simply be ignored on the
basis the Real number cannot express them.
The Sapir–Whorf hypothesis, also known as the linguistic relativity hypothesis, refers to the proposal that the particular language one
speaks influences the way one thinks about reality. https://www.sciencedirect.com/topics/psychology/sapir-whorf-hypothesis
There always a finite ε is used. Apparently olcott is talking about
his undisclosed olcott-numbers, but he keeps it as a secret what it
means.
Op 03.apr.2024 om 17:11 schreef olcott:
On 4/3/2024 3:32 AM, Fred. Zwarts wrote:
Op 02.apr.2024 om 20:51 schreef olcott:
On 4/2/2024 1:29 PM, Keith Thompson wrote:
Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes:
On 02/04/2024 02:27, Keith Thompson wrote:
olcott <polcott333@gmail.com> writes:
On 4/1/2024 6:11 PM, Keith Thompson wrote:"IDK, probably not."
olcott <polcott333@gmail.com> writes:
[...]
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). If there is no Real number at that point >>>>>>>>>> then
there is no Real number that exactly represents 0.999...
In the following I'm talking about real numbers, and only real >>>>>>>>> numbers -- not hyperreals, or surreals, or any other extension >>>>>>>>> to the
real numbers.
You assert that there is a geometric point immediately to the left >>>>>>>>> of
1.0 on the number line. (I disagree, but let's go with it for >>>>>>>>> now.)
Am I correct in assuming that this means that that point
corresponds
to
a real number that is distinct from, and less than, 1.0?
IDK, probably not. I am saying that 0.999... exactly equals this >>>>>>>> number.
Did you even consider taking some time to *think* about this?
PO just says things he thinks are true based on his first intuitions >>>>>> when he encountered a topic. He does not "reason" his way to a new >>>>>> carefully thought out theory or even to a single coherent idea. Don't >>>>>> imagine he is thinking of hyperreals or anything - he just "knows" >>>>>> that obviously any number which starts 0.??? is less than one
starting
1.??? - because 0 is less than 1 !! Or whatever, it really doesn't >>>>>> matter.
I don't think he's explicitly said that any real number whose decimal >>>>> representation starts with "0." is less than one starting with "1." -- >>>>> but if said that, he'd be right.
What he refuses to understand is that the notation "0.999..." is not a >>>>> decimal representation. The "..." notation refers to the limit of a >>>>> sequence, and of course the limit of a sequence does not have to be a >>>>> member of the sequence. Every member of the sequence (0.9, 0.99,
0.999,
0.9999, continuing in the obvious manner) is a real (and rational)
number that is strictly less than 1.0. But the limit of the
sequence is
1.0. Sequences and their limits can be and are defined rigorously
without reference to infinitesimals or infinities,
In other words when we pretend that this never ending sequence ends
0.999... ends then we do get to 1.0.
Again fighting windmills. Nobody said the sequence ends. That is
olcott's own interpretation which he wants to fight.
0.999... The LFS remains infinitesimally less than 1.0
Fighting windmills again. Fighting his own interpretation of 0.999...
Unable to understand the normal interpretation, even when spelled out in detail.
On 4/3/2024 10:27 AM, Fred. Zwarts wrote:
Op 03.apr.2024 om 17:16 schreef olcott:
On 4/3/2024 3:42 AM, Fred. Zwarts wrote:
Op 02.apr.2024 om 23:52 schreef olcott:
On 4/2/2024 4:20 PM, Mike Terry wrote:
On 02/04/2024 19:29, Keith Thompson wrote:
Mike Terry <news.dead.person.stones@darjeeling.plus.com> writes: >>>>>>>> On 02/04/2024 02:27, Keith Thompson wrote:
olcott <polcott333@gmail.com> writes:PO just says things he thinks are true based on his first
On 4/1/2024 6:11 PM, Keith Thompson wrote:"IDK, probably not."
olcott <polcott333@gmail.com> writes:
[...]
Since PI is represented by a single geometric point on the >>>>>>>>>>>> number lineIn the following I'm talking about real numbers, and only real >>>>>>>>>>> numbers -- not hyperreals, or surreals, or any other
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). If there is no Real number at that >>>>>>>>>>>> point then
there is no Real number that exactly represents 0.999... >>>>>>>>>>> [...]
extension to the
real numbers.
You assert that there is a geometric point immediately to the >>>>>>>>>>> left
of
1.0 on the number line. (I disagree, but let's go with it >>>>>>>>>>> for now.)
Am I correct in assuming that this means that that point >>>>>>>>>>> corresponds
to
a real number that is distinct from, and less than, 1.0?
IDK, probably not. I am saying that 0.999... exactly equals >>>>>>>>>> this number.
Did you even consider taking some time to *think* about this? >>>>>>>>
intuitions
when he encountered a topic. He does not "reason" his way to a new >>>>>>>> carefully thought out theory or even to a single coherent idea. >>>>>>>> Don't
imagine he is thinking of hyperreals or anything - he just "knows" >>>>>>>> that obviously any number which starts 0.??? is less than one >>>>>>>> starting
1.??? - because 0 is less than 1 !! Or whatever, it really doesn't >>>>>>>> matter.
I don't think he's explicitly said that any real number whose
decimal
representation starts with "0." is less than one starting with
"1." --
but if said that, he'd be right.
0.999... = 1.000... (so he'd be wrong)
In other words you simply choose to "not believe in"
the notion of infinitesimal difference. That doesn't
actually make it go away.
It is not a matter of 'believe-in'. In the real number system there
are no infinitesimal differences.
So when they do occur
In the real number system they do not occur. Olcott is fighting
windmills again.
they cannot be expressed so the convention is to ignore them.
Things that do not occur, don't need to be ignored. Olcott is
fighting windmills again.
If there was no word for "murder" in a language yet people are having
their lives taken away against their will murders are still occurring
even if there is no word for it.
Infinitesimal differences cannot simply be ignored on the
basis the Real number cannot express them.
The Sapir–Whorf hypothesis, also known as the linguistic relativity
hypothesis, refers to the proposal that the particular language one
speaks influences the way one thinks about reality.
https://www.sciencedirect.com/topics/psychology/sapir-whorf-hypothesis
There always a finite ε is used. Apparently olcott is talking about
his undisclosed olcott-numbers, but he keeps it as a secret what it
means.
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:Which nobody denied.
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: >>>>>>>>>>>>>>>>>>>>>>>> [...]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.
No, if olcott had paid attention to the text below, >>>>>>>>>>>>>>>>>>>> or the article I referenced:What he either doesn't understand, or pretends >>>>>>>>>>>>>>>>>>>>>>>>>> not to understand, isIn other words when one gets to the end of a >>>>>>>>>>>>>>>>>>>>>>>>> never ending sequence
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. >>>>>>>>>>>>>>>>>>>>>>>>>>
(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. >>>>>>>>>>>>>>>>>>>>
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 >>>>>>>>>>>>>>>>>>
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 >>>>>>>>>>
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.
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.
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.
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.
On 4/3/2024 12:23 PM, Keith Thompson wrote:
"Fred. Zwarts" <F.Zwarts@HetNet.nl> writes:
[...]
Olcott is unable to understand what it says in the context of the
real number system, even when spelled out to him in great
detail. Therefore he sticks to his own (wrong) interpretation and then
starts to fight it. Fighting windmills.
Might I suggest waiting to reply to olcott until he says something
*new*. It could save a lot of time and effort.
0.999... everyone knows that this means infinitely repeating digits
that never reach 1.0 and lies about it. I am not going to start lying
about it.
On 04/03/2024 03:12 PM, Ben Bacarisse wrote:
Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:
olcott <polcott333@gmail.com> writes:
On 4/3/2024 12:23 PM, Keith Thompson wrote:
"Fred. Zwarts" <F.Zwarts@HetNet.nl> writes:
[...]
Olcott is unable to understand what it says in the context of the >>>>>> real number system, even when spelled out to him in greatMight I suggest waiting to reply to olcott until he says something
detail. Therefore he sticks to his own (wrong) interpretation and >>>>>> then
starts to fight it. Fighting windmills.
*new*. It could save a lot of time and effort.
0.999... everyone knows that this means infinitely repeating digits
that never reach 1.0 and lies about it. I am not going to start lying
about it.
(I don't read everything olcott writes, but that *might* be something
new.)
Nobody here is lying. (I'm giving you the benefit of the doubt.)
Some people here are wrong.
You might take a moment to think about *why* so many people would be
motivated to lie about something like this. Is it really plausible
that multiple people (a) know in their hearts that you're right,
but (b) deliberately pretend that you're wrong?
PO is in a genuine bind here. He has almost no ability to understand
other people's mental states, let alone their reasoning. He can't begin
to comprehend what others think, and he struggles to understand what
they write, so he often thinks that people are lying or playing head
games. He's accused me of this numerous times, and (the final straw for
me) that I must be doing this deliberately and sadistically. What other
conclusion can he come to?
Every time PO paraphrases someone's reply to him he gets it wrong. He
simply does not know what people are saying but since they disagree with
something that is obvious to him, they must be stupid, lying or playing
head games.
The classic technique in mediation where each person must reflect back
to the other what it is they believe the other is saying would, were he
capable of it, be useful here. But he would fail at every step.
About the di-aletheic, ....
https://www.youtube.com/watch?v=vbyFehrthIQ&list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY&index=23&t=1305
About statements and fact and retraction, ....
https://www.youtube.com/watch?v=tODnCZvVtLg&list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY&index=15
Iota-values: the word "iota" means "smallest non-zero value".
Real-values: all the values between negative infinity and infinity.
On 04/03/2024 07:35 PM, olcott wrote:
On 4/3/2024 9:11 PM, Ross Finlayson wrote:
On 04/03/2024 03:12 PM, Ben Bacarisse wrote:
Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:
olcott <polcott333@gmail.com> writes:
On 4/3/2024 12:23 PM, Keith Thompson wrote:
"Fred. Zwarts" <F.Zwarts@HetNet.nl> writes:
[...]
Olcott is unable to understand what it says in the context of the >>>>>>>> real number system, even when spelled out to him in greatMight I suggest waiting to reply to olcott until he says something >>>>>>> *new*. It could save a lot of time and effort.
detail. Therefore he sticks to his own (wrong) interpretation and >>>>>>>> then
starts to fight it. Fighting windmills.
0.999... everyone knows that this means infinitely repeating digits >>>>>> that never reach 1.0 and lies about it. I am not going to start lying >>>>>> about it.
(I don't read everything olcott writes, but that *might* be something >>>>> new.)
Nobody here is lying. (I'm giving you the benefit of the doubt.)
Some people here are wrong.
You might take a moment to think about *why* so many people would be >>>>> motivated to lie about something like this. Is it really plausible >>>>> that multiple people (a) know in their hearts that you're right,
but (b) deliberately pretend that you're wrong?
PO is in a genuine bind here. He has almost no ability to understand >>>> other people's mental states, let alone their reasoning. He can't
begin
to comprehend what others think, and he struggles to understand what
they write, so he often thinks that people are lying or playing head
games. He's accused me of this numerous times, and (the final straw >>>> for
me) that I must be doing this deliberately and sadistically. What
other
conclusion can he come to?
Every time PO paraphrases someone's reply to him he gets it wrong. He >>>> simply does not know what people are saying but since they disagree
with
something that is obvious to him, they must be stupid, lying or playing >>>> head games.
The classic technique in mediation where each person must reflect back >>>> to the other what it is they believe the other is saying would, were he >>>> capable of it, be useful here. But he would fail at every step.
About the di-aletheic, ....
https://www.youtube.com/watch?v=vbyFehrthIQ&list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY&index=23&t=1305
About statements and fact and retraction, ....
https://www.youtube.com/watch?v=tODnCZvVtLg&list=PLb7rLSBiE7F4eHy5vT61UYFR7_BIhwcOY&index=15
Iota-values: the word "iota" means "smallest non-zero value".
Real-values: all the values between negative infinity and infinity.
So the geometric point immediately adjacent to 0.0 on the positive
side of the number line would be a real number.
That's kind of the idea where there's a sort of distinction
"real-valued" vis-a-vis just "real numbers", with the idea
that where there are more than one many models of the
linear continuum, a continuous domain, that they all live
in the same space, of real numbers, that they're real-valued.
That is, the linear continuum is complete already,
so any non-standard models live in the same space.
Now, when you say real number, everybody's going to
think that it means the complete ordered field,
or at least everybody with the usual linear curriculum
and formal schooling and the formalism, so when you
say instead "real-valued", it sort of expresses that
if there _is_ a different continuous domain, then
the different models are treated differently and
they're not interchangeable except with regards to
various statements about particularly well-understood
points where they're the same in geometry, here 0 and 1,
these iota-values filling [0,1] empty to full, and
the usual real numbers as real values falling down
after the ordered field of rationals, getting axiomatized
their LUB and thus completion usually negative infinity
to infinity.
There's another Katz been working on some revivals
of studies of infinitesimals, some years ago there
was a paper about the contradistinction of .999... = 1
and .999 < 1, and about for modular and clock arithmetic
that it's very natural that the notations read-out the
same under different meanings.
You can read Ehrlich for a sort of modern survey of
infinitesimals, yet, you might as well just look to
Cavalieri and MacLaurin, or you know, I wrote it up.
Here the point is that "real-valued" then makes for
it sort of suffices that there's a model of real numbers
with "integer part and non-integer part [0,1]" and a
model of real numbers "the ordered field of rationals
closed to least-upper-bound the complete ordered field",
then that it involves book-keeping so they don't get confused.
Which keeps things simple while yet not blind, ....
Just keep in mind that there's an entirely different model
of a continuous domain, zero to one empty to full, than
the usual model called R that is all the rationals plus
filling all the gaps, then there's also to learn about
how the Fourier-style analysis arrives at the signal domain,
so that there are at least three different models altogether,
in the sense of model theory's models, of continous domains,
real values and real-valued.
On 2024-04-04 08:55, olcott wrote:
Different enough to not me equal.
[0.0, 1.0] - [0.0, 1.0) = 0.0...1
0.000...2 - 0.000...1 = 0.000...1
*A good notational convention for infinitesimals*
And that's supposed to mean what exactly? That you take an unending
sequence of zeros and once that unending sequence ends you tack on a 1?
0.999... + 0.000...1 = 1.0
0.999... + 0.000...2 = 1.000...1
0.999... + 0.000...3 = 1.000...2
0.999... + 0.000...n = 1.000...n-1
So what's 0.999... + 0.000...05 ? Is that half an infinitesimal shy of 1?
André
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