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Remembering Prof. Jack Huizenga's retracted critiques of the New Calculus
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John Gabriel
Sep 6, 2017, 2:25:11 AM9/6/17
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Jackie caused a tremendous amount of attention to be focused on my New Calculus with his scorching critique. After I refuted him, he realised he had been impulsive and retracted most of his lies and libel.
Alas, the damage had been done.
9/21/2016 A Harvard alumnus comments! | The New Calculus
Just a few weeks ago, some ignoramus posted a question on that ridiculous site Quora.
The question was:
“What do people think of John Gabriel’s New Calculus, which claims it is the only rigorous formulation of calculus”.
Quora is a question and answer site, that is run by a group of mainstream academics and their sock puppets. They control both the questions and answers. Those who post questions or comments with opposing views, have both their questions and comments mercilessly edited, and deleted if need be. In a sense, it’s Wikipedia-esque, in that the questions and answers are the views of mainstream academics. Nothing that is different or new is tolerated.
One of the fools who sits atop the trash heap, is one Prof. Jack Huizenga, a Harvard alumnus and mathematics teacher at Illinois University. The following link contains a snapshot of Huizenga’s comment: The ridiculous comment at the ridiculous site Quora.
In this very first post, I will address the comments made by this ignoramus (Huizenga) and horrify those, who will realise that this dimwit actually teaches mathematics!
There has been a completely rigorous notion of limit for well over a hundred years. – Huizenga
The great mathematics historian Carl Boyer had this to say:
"Cauchy had stated in his Cours d'analyse that irrational numbers are to be regarded as the limits of sequences of rational numbers. Since a limit is defined as a number to which the terms of the sequence approach in such a way that ultimately the difference between this number and the terms of the sequence can be made less than any given number, the existence of the irrational number depends, in the definition of limit, upon the known existence, and hence the prior definition, of the very quantity whose definition is being attempted.
That is, one cannot define the number 'square root of 2', as the limit of the sequence 1, 1.4, 1.41,1.414, ... because to prove that this sequence has a limit one must assume, in view of the definitions of limits and convergence, the existence of this number as previously demonstrated or defined. Cauchy appears not to have noticed the circularity of the reasoning in this connection, but tacitly assumed that every sequence converging within itself has a limit." - The History of Calculus and its Conceptual Development' (Page. 281) Carl B. Boyer
I doubt that Huizenga ever read this book, and if he did, there is no doubt he did not understand it. Not only has the notion of limit never been rigorous, but it was never challenged by any intelligent mathematician.
Rigorous treatments of infinitesimals are a bit more tricky, but have also been made. – Huizenga
One can only assume the moron is referring to the abortion of non-standard analysis, by a failed Jewish mathematician called Abraham Robinson. Even today, there are many academics (including PhDs) who do not accept non-standard analysis. In my opinion, it is pure rot because infinitesimals
don’t exist.
Reformulations of calculus as attempted in New Calculus, are essentially no interest to mathematicians, as the field of calculus is already extremely well understood and rigorous. – Huizenga
The buffoon wrongly assumes that the New Calculus is only a reformulation, which is evidently false. He goes on to talk about what interests mathematicians, but how could he know? Huizenga is not a mathematician, he is a teacher with no great works behind his name, unless of course one calls his juvenile papers on algebraic geometry a work of any kind.
Needless to say, if calculus were already extremely well understood, we would not have other mathematics PhDs making statements as follows:
Clearly, our calculus course does not prepare scientists in other fields to recognize, understand, and utilize the calculus that many of their fields are based upon. Thus, when it comes to calculus, we don’t get it the first time around, our colleagues don’t get it, and our students are still not getting
it. It’s no wonder that one of the most common occurrences in higher education is that of a non mathematics faculty member discovering that something they were doing is calculus. And at the very least, we feel justified in asserting that there still is a crisis in calculus instruction. – Knisley (Crisis in calculus education)
Knisley goes on to say:
However, in the calculus curriculum, many of the associations are circular. All too often a given concept is associated with a concept that is defined in terms of the original concept. Such connections increase the complexity of a concept without shedding any insight on the concept itself. Not surprisingly, concepts motivated with circular associations are the ones most often memorized with little or no comprehension. – Knisley (Facing the crisis in calculus education)
As for calculus being rigorous, we will hold off on that one until you have read and studied my New Calculus.
There is no point in trying to remove limits or infinitesimals from a discussion of calculus. – Huizenga
It’s easy to see that our moron academic knows nothing about Newton’s calculus and how he (Newton) arrived at the knowledge of it. The author’s main objection to the standard treatment of calculus via real analysis seems to be that he does not understand it. – Huizenga
Well, that’s certainly news to me. I have never made such a claim on the internet or anywhere else. I reject real analysis, not because I don’t understand it (few can ever understand it as well as I), but because it deals with a topic about a non-existent concept – the real number. Real numbers do not exist, because irrational numbers do not exist. In fact, until I arrived on the scene, no one before me and after Euclid, understood what is a number.
In the following comment I debunk the concept of Dedekind cuts and Cauchy sequences:
https://drive.google.com/open?id=0B-mOEooW03iLSTROakNyVXlQUEU
Our buffoon continues:
The errors in New Calculus are too numerous for it, to be worth going through the whole text. – Huizenga
It’s rather funny that he should say this, because he goes on to show that there is not a single error. Unsurprisingly, the baboon dismissed the rest of the text because he could not understand it up till that point. Huizenga mentions that one has to consider an (m,n) pair which is outright false.
In fact, if he had only continued to study and reread the text carefully, he would have soon realised that the values of m and n play no role in that of the gradient. Tsk, tsk.
His definition therefore presupposes that we know the slope of the tangent line (and that we have a notion of a tangent line to a graph of an arbitrary function) and merely computes the slope of a parallel secant line. This is incredibly circular. – Huizenga
Of course we know the slope of the tangent line, provided we know the slope of a parallel secant line. This is grade 8 mathematics! What he states in parenthesis is even more amusing. It demonstrates clearly that he, like many of his colleagues never understood calculus:
If a given function is not continuous and smooth, then any of the methods of calculus are null and void.
One, and only one tangent line exists at every point, provided the function is continuous and smooth, and a given point is not a point of inflection, that is, only half-tangent lines are possible at points of inflection. Our moron then continues to quote the example of the cubic, where there is no
tangent line at x=0 (because an inflection point exists at x=0). Unlike Newton's flawed formulation, the New Calculus handles this correctly.
He further goes to great length to try and convince the reader that he can actually divide the numerator in his difference quotient by m+n. – Huizenga
I don’t go to any great lengths. The proof that every term in the numerator has a factor of m+n can be shown by a high school student. It requires no special knowledge. The first page of the New Calculus website http://thenewcalculus.weebly.com has many examples on this, and includes links
to dynamic Geogebra applets which prove conclusively that it is based on sound analytic geometry.
It is legal to do this in the New Calculus, but illegal in Cauchy's flawed formulation.
That m+n is a factor of every term in the numerator finite difference f(x+n)-f(x-m) can be seen by anyone with a modicum of intelligence, but our chump Huizenga lacks even this. The chump way to see this fact, is to investigate actual functions using the finite difference, and it is soon realised, that this fact is true for all functions. It gets slightly more complicated in the case of terms containing only m and/or n, but this is easy to prove by mathematical induction and constructive proofs, as I have shown in my article
https://drive.google.com/open?id=0B-mOEooW03iLWldTU1ZkTDVQR0E
The easiest proof is given with the equation of a straight line, say f(x)=kx+p. We have from the New Calculus derivative definition: f ' (x) = {k(x+n)+p - [k(x-m)+p] } / (m+n) =k(m+n) / (m+n) = k. Observe that m and n play no role in the value of k which is the gradient. What amuses me, is that so many grade 8 students understand this, and most PhD chumps just don't get it!
Of course the New Calculus is likely to be revised over time to address concerns brought up by people. As this happens, I am confident it will look more and more like standard calculus (or the theory of the symmetric derivative which is closely related). – Huizenga
Another presumptuous claim by our buffoon Huizenga, because the New Calculus has never been revised and there are no plans to revise it whatsoever. There is no need to revise theory that is based on well-defined concepts. It will stand the test of time, just as Euclid’s Elements did.
The statement in parenthesis is quite amusing because it once again demonstrates the lack of understanding displayed by Huizenga. The New Calculus is not just about a new derivative definition, but also about a new integral definition, and much more (*). He might have gotten to that information had he continued studying the text. But ‘open-minded’ academic that he is not, he
ceased to continue, when he could no longer understand what he was reading.
The new calculus derivative definition has nothing in common with the symmetric derivative which Huizenga clearly does not understand. The symmetric derivative requires no special relationship exist between m and n. In fact, the symmetric derivative is used mostly in numeric differentiation.
(*) There are many new methods and theorems in the New Calculus that are not possible using Newton's flawed formulation.
And that covers his comment at Quora. In fact, he proves by all his statements, that there are no errors in the New Calculus, only serious issues in his ability to comprehend.
This same moron thinks that the derivative of sin(x) is not always cos(x). He also fancies that the sine function can take degrees as input, which is outright false. The trigonometric ratios operate only on radian input.
Comments are unwelcome and will be ignored.
Posted on this newsgroup in the interests of public education and to eradicate ignorance and stupidity from mainstream mythmatics.
gils...@gmail.com (MIT)
huiz...@psu.edu (HARVARD)
and...@mit.edu (MIT)
david....@math.okstate.edu (David Ullrich)
djo...@clarku.edu
mar...@gmail.com
Alas, the damage had been done.
9/21/2016 A Harvard alumnus comments! | The New Calculus
Just a few weeks ago, some ignoramus posted a question on that ridiculous site Quora.
The question was:
“What do people think of John Gabriel’s New Calculus, which claims it is the only rigorous formulation of calculus”.
Quora is a question and answer site, that is run by a group of mainstream academics and their sock puppets. They control both the questions and answers. Those who post questions or comments with opposing views, have both their questions and comments mercilessly edited, and deleted if need be. In a sense, it’s Wikipedia-esque, in that the questions and answers are the views of mainstream academics. Nothing that is different or new is tolerated.
One of the fools who sits atop the trash heap, is one Prof. Jack Huizenga, a Harvard alumnus and mathematics teacher at Illinois University. The following link contains a snapshot of Huizenga’s comment: The ridiculous comment at the ridiculous site Quora.
In this very first post, I will address the comments made by this ignoramus (Huizenga) and horrify those, who will realise that this dimwit actually teaches mathematics!
There has been a completely rigorous notion of limit for well over a hundred years. – Huizenga
The great mathematics historian Carl Boyer had this to say:
"Cauchy had stated in his Cours d'analyse that irrational numbers are to be regarded as the limits of sequences of rational numbers. Since a limit is defined as a number to which the terms of the sequence approach in such a way that ultimately the difference between this number and the terms of the sequence can be made less than any given number, the existence of the irrational number depends, in the definition of limit, upon the known existence, and hence the prior definition, of the very quantity whose definition is being attempted.
That is, one cannot define the number 'square root of 2', as the limit of the sequence 1, 1.4, 1.41,1.414, ... because to prove that this sequence has a limit one must assume, in view of the definitions of limits and convergence, the existence of this number as previously demonstrated or defined. Cauchy appears not to have noticed the circularity of the reasoning in this connection, but tacitly assumed that every sequence converging within itself has a limit." - The History of Calculus and its Conceptual Development' (Page. 281) Carl B. Boyer
I doubt that Huizenga ever read this book, and if he did, there is no doubt he did not understand it. Not only has the notion of limit never been rigorous, but it was never challenged by any intelligent mathematician.
Rigorous treatments of infinitesimals are a bit more tricky, but have also been made. – Huizenga
One can only assume the moron is referring to the abortion of non-standard analysis, by a failed Jewish mathematician called Abraham Robinson. Even today, there are many academics (including PhDs) who do not accept non-standard analysis. In my opinion, it is pure rot because infinitesimals
don’t exist.
Reformulations of calculus as attempted in New Calculus, are essentially no interest to mathematicians, as the field of calculus is already extremely well understood and rigorous. – Huizenga
The buffoon wrongly assumes that the New Calculus is only a reformulation, which is evidently false. He goes on to talk about what interests mathematicians, but how could he know? Huizenga is not a mathematician, he is a teacher with no great works behind his name, unless of course one calls his juvenile papers on algebraic geometry a work of any kind.
Needless to say, if calculus were already extremely well understood, we would not have other mathematics PhDs making statements as follows:
Clearly, our calculus course does not prepare scientists in other fields to recognize, understand, and utilize the calculus that many of their fields are based upon. Thus, when it comes to calculus, we don’t get it the first time around, our colleagues don’t get it, and our students are still not getting
it. It’s no wonder that one of the most common occurrences in higher education is that of a non mathematics faculty member discovering that something they were doing is calculus. And at the very least, we feel justified in asserting that there still is a crisis in calculus instruction. – Knisley (Crisis in calculus education)
Knisley goes on to say:
However, in the calculus curriculum, many of the associations are circular. All too often a given concept is associated with a concept that is defined in terms of the original concept. Such connections increase the complexity of a concept without shedding any insight on the concept itself. Not surprisingly, concepts motivated with circular associations are the ones most often memorized with little or no comprehension. – Knisley (Facing the crisis in calculus education)
As for calculus being rigorous, we will hold off on that one until you have read and studied my New Calculus.
There is no point in trying to remove limits or infinitesimals from a discussion of calculus. – Huizenga
It’s easy to see that our moron academic knows nothing about Newton’s calculus and how he (Newton) arrived at the knowledge of it. The author’s main objection to the standard treatment of calculus via real analysis seems to be that he does not understand it. – Huizenga
Well, that’s certainly news to me. I have never made such a claim on the internet or anywhere else. I reject real analysis, not because I don’t understand it (few can ever understand it as well as I), but because it deals with a topic about a non-existent concept – the real number. Real numbers do not exist, because irrational numbers do not exist. In fact, until I arrived on the scene, no one before me and after Euclid, understood what is a number.
In the following comment I debunk the concept of Dedekind cuts and Cauchy sequences:
https://drive.google.com/open?id=0B-mOEooW03iLSTROakNyVXlQUEU
Our buffoon continues:
The errors in New Calculus are too numerous for it, to be worth going through the whole text. – Huizenga
It’s rather funny that he should say this, because he goes on to show that there is not a single error. Unsurprisingly, the baboon dismissed the rest of the text because he could not understand it up till that point. Huizenga mentions that one has to consider an (m,n) pair which is outright false.
In fact, if he had only continued to study and reread the text carefully, he would have soon realised that the values of m and n play no role in that of the gradient. Tsk, tsk.
His definition therefore presupposes that we know the slope of the tangent line (and that we have a notion of a tangent line to a graph of an arbitrary function) and merely computes the slope of a parallel secant line. This is incredibly circular. – Huizenga
Of course we know the slope of the tangent line, provided we know the slope of a parallel secant line. This is grade 8 mathematics! What he states in parenthesis is even more amusing. It demonstrates clearly that he, like many of his colleagues never understood calculus:
If a given function is not continuous and smooth, then any of the methods of calculus are null and void.
One, and only one tangent line exists at every point, provided the function is continuous and smooth, and a given point is not a point of inflection, that is, only half-tangent lines are possible at points of inflection. Our moron then continues to quote the example of the cubic, where there is no
tangent line at x=0 (because an inflection point exists at x=0). Unlike Newton's flawed formulation, the New Calculus handles this correctly.
He further goes to great length to try and convince the reader that he can actually divide the numerator in his difference quotient by m+n. – Huizenga
I don’t go to any great lengths. The proof that every term in the numerator has a factor of m+n can be shown by a high school student. It requires no special knowledge. The first page of the New Calculus website http://thenewcalculus.weebly.com has many examples on this, and includes links
to dynamic Geogebra applets which prove conclusively that it is based on sound analytic geometry.
It is legal to do this in the New Calculus, but illegal in Cauchy's flawed formulation.
That m+n is a factor of every term in the numerator finite difference f(x+n)-f(x-m) can be seen by anyone with a modicum of intelligence, but our chump Huizenga lacks even this. The chump way to see this fact, is to investigate actual functions using the finite difference, and it is soon realised, that this fact is true for all functions. It gets slightly more complicated in the case of terms containing only m and/or n, but this is easy to prove by mathematical induction and constructive proofs, as I have shown in my article
https://drive.google.com/open?id=0B-mOEooW03iLWldTU1ZkTDVQR0E
The easiest proof is given with the equation of a straight line, say f(x)=kx+p. We have from the New Calculus derivative definition: f ' (x) = {k(x+n)+p - [k(x-m)+p] } / (m+n) =k(m+n) / (m+n) = k. Observe that m and n play no role in the value of k which is the gradient. What amuses me, is that so many grade 8 students understand this, and most PhD chumps just don't get it!
Of course the New Calculus is likely to be revised over time to address concerns brought up by people. As this happens, I am confident it will look more and more like standard calculus (or the theory of the symmetric derivative which is closely related). – Huizenga
Another presumptuous claim by our buffoon Huizenga, because the New Calculus has never been revised and there are no plans to revise it whatsoever. There is no need to revise theory that is based on well-defined concepts. It will stand the test of time, just as Euclid’s Elements did.
The statement in parenthesis is quite amusing because it once again demonstrates the lack of understanding displayed by Huizenga. The New Calculus is not just about a new derivative definition, but also about a new integral definition, and much more (*). He might have gotten to that information had he continued studying the text. But ‘open-minded’ academic that he is not, he
ceased to continue, when he could no longer understand what he was reading.
The new calculus derivative definition has nothing in common with the symmetric derivative which Huizenga clearly does not understand. The symmetric derivative requires no special relationship exist between m and n. In fact, the symmetric derivative is used mostly in numeric differentiation.
(*) There are many new methods and theorems in the New Calculus that are not possible using Newton's flawed formulation.
And that covers his comment at Quora. In fact, he proves by all his statements, that there are no errors in the New Calculus, only serious issues in his ability to comprehend.
This same moron thinks that the derivative of sin(x) is not always cos(x). He also fancies that the sine function can take degrees as input, which is outright false. The trigonometric ratios operate only on radian input.
Comments are unwelcome and will be ignored.
Posted on this newsgroup in the interests of public education and to eradicate ignorance and stupidity from mainstream mythmatics.
gils...@gmail.com (MIT)
huiz...@psu.edu (HARVARD)
and...@mit.edu (MIT)
david....@math.okstate.edu (David Ullrich)
djo...@clarku.edu
mar...@gmail.com
Markus Klyver
Sep 6, 2017, 12:47:54 PM9/6/17
to
John Gabriel
Sep 8, 2017, 6:44:24 PM9/8/17
to
https://drive.google.com/open?id=0B-mOEooW03iLSTROakNyVXlQUEU
Markus Klyver
Sep 9, 2017, 6:39:10 AM9/9/17
to
And while I believe your definition is equivalent to Dedekind's with some minor adjustments, it makes definition operations and proving theorems about the real numbers much harder.
Either way, reals are not defined as a limit or an infinite decimal representation.
John Gabriel
Sep 9, 2017, 11:34:05 AM9/9/17
to
> We don't define pi in terms of pi.
A number is the measure of a magnitude.
You can't get away from this. This definition is what it means to be a **number**.
> In the case of the Cauchy model, every real number is an equivalence class of Cauchy sequences.
ALL the sequences in an equivalence class are related by the LIMIT which in the case if incommensurable magnitudes has not been proven to be a **NUMBER**. So it is very circular to state that an equivalence class defines a number, even if one were to accept sets as a valid representation, because the limit that is common to any Cauchy class has not been established as a rational number or anything else. If it is not a rational number, then it is an incommensurable magnitude.
> An an equivalence class is a set. In the case of the Dedekind construction, a real number is a set of rational numbers satisfying certain properties.
>
> And while I believe your definition is equivalent to Dedekind's with some minor adjustments, it makes definition operations and proving theorems about the real numbers much harder.
>
> Either way, reals are not defined as a limit or an infinite decimal representation.
Markus Klyver
Sep 9, 2017, 12:02:18 PM9/9/17
to
I feel I begin to repeat myself, but In the case of the Cauchy model every real number is an equivalence class of Cauchy sequences. It's a set; not a limit or any sort. The *intuitive picture* you can have is of course some form of limit, but formally it is not a limit in any way.
It's not in any way circular either: every real number is a set. A particular set, namely an equivalence class of rational Cauchy sequences. A real number **is** an equivalence class. A particular equivalence class **is** a real number. I can't stress this enough.
All the Cauchy sequences in a certain equivalence class is related by the equivalence relation defining the equivalence class. They are not related by some "limit" of any kind. They are related by the equivalence relation.
I have touched on your definition of number before. You haven't defined what a measure or what a magnitude is. You have only given me synonyms such as "size" and "extent". It's equally vague and nonsensical. And equally, defining pi as the ratio of the circumference to the diameter of a circle requires you to to what lengths of curves are, for example. It also requires a rigours definition of what a circle is.
John Gabriel
Sep 9, 2017, 1:52:28 PM9/9/17
to
<nonsense>
Markus Klyver
Sep 9, 2017, 4:10:06 PM9/9/17
to
konyberg
Sep 9, 2017, 4:20:07 PM9/9/17
to
KON
zelos...@outlook.com
Sep 11, 2017, 2:08:37 AM9/11/17
to
On Saturday, September 9, 2017 at 7:52:28 PM UTC+2, John Gabriel wrote:
> Chuckle. You are a moron therefore you are wrong. I have explained to you the correct concepts and corrected you over and over again. You keep responding with irrelevant crap and "Nah Uh".
>
> <nonsense>
Gabriel, that is a pure example of an ad hominem fallacy.
> Chuckle. You are a moron therefore you are wrong. I have explained to you the correct concepts and corrected you over and over again. You keep responding with irrelevant crap and "Nah Uh".
>
> <nonsense>
We have corrected you over and over. Like your complaint about cauchy sequences, it doesn't matter that there are infinitely many different sequences because, AS WE HAVE ALREADY EXPLAINED TO YOU, it is the equivalence classes we care about! They are the ones we define as the real numbers.
zelos...@outlook.com
Sep 11, 2017, 2:15:16 AM9/11/17
to
On Saturday, September 9, 2017 at 5:34:05 PM UTC+2, John Gabriel wrote:
> They really are! Every sequence in an equivalence class is related by ONE and only ONE attribute - the LIMIT. As I've repeatedly stated, the LIMIT may or may not be a number.
The question is not wether or not the limit is a number, it is wether or not the limit is in Q, but the limit of course exists in R as the sequence itself is part of the equivalence classes that makes up R
> They really are! Every sequence in an equivalence class is related by ONE and only ONE attribute - the LIMIT. As I've repeatedly stated, the LIMIT may or may not be a number.
John Gabriel
Sep 11, 2017, 7:52:16 AM9/11/17
to
You don't fool anyone any longer. I have made sure of this.
In due time you will be cleaning toilets which is more in line with your work aptitude.
zelos...@outlook.com
Sep 11, 2017, 9:28:14 AM9/11/17
to
Dan Christensen
Sep 11, 2017, 12:52:22 PM9/11/17
to
On Wednesday, September 6, 2017 at 2:25:11 AM UTC-4, John Gabriel wrote:
> Jackie caused a tremendous amount of attention to be focused on my New Calculus with his scorching critique. After I refuted him...
Your are delusional, Troll Boy. You have never refuted anyone. Your Wacky New Calclueless is a dead end and a complete waste of time. You cannot even determine the derivative of functions as simple as y=x. No, Troll Boy, it is NOT "undefined." What a moron!
In your goofy number system, you have banned all axioms. So, of course, you cannot derive in the even most elementary results of basic arithmetic, not even 2+2=4.
Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
> Jackie caused a tremendous amount of attention to be focused on my New Calculus with his scorching critique. After I refuted him...
Your are delusional, Troll Boy. You have never refuted anyone. Your Wacky New Calclueless is a dead end and a complete waste of time. You cannot even determine the derivative of functions as simple as y=x. No, Troll Boy, it is NOT "undefined." What a moron!
In your goofy number system, you have banned all axioms. So, of course, you cannot derive in the even most elementary results of basic arithmetic, not even 2+2=4.
Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
Markus Klyver
Sep 16, 2017, 3:54:10 AM9/16/17
to
Markus Klyver
Sep 16, 2017, 3:55:59 AM9/16/17
to
John Gabriel
Sep 16, 2017, 6:38:03 AM9/16/17
to
John Gabriel
Sep 16, 2017, 6:39:05 AM9/16/17
to
On Saturday, 16 September 2017 03:54:10 UTC-4, Markus Klyver wrote:
> Den måndag 11 september 2017 kl. 08:15:16 UTC+2 skrev zelos...@outlook.com:
> > On Saturday, September 9, 2017 at 5:34:05 PM UTC+2, John Gabriel wrote:
> > > They really are! Every sequence in an equivalence class is related by ONE and only ONE attribute - the LIMIT. As I've repeatedly stated, the LIMIT may or may not be a number.
> >
> > The question is not wether or not the limit is a number, it is wether or not the limit is in Q, but the limit of course exists in R as the sequence itself is part of the equivalence classes that makes up R
>
> To prove that Cauchy sequences in ℝ converges in ℝ is not too trivial. You first have to introduce a metric for real numbers
Imbecile. You first have to show that a valid construction of "real" number exists. There is none.
> Den måndag 11 september 2017 kl. 08:15:16 UTC+2 skrev zelos...@outlook.com:
> > On Saturday, September 9, 2017 at 5:34:05 PM UTC+2, John Gabriel wrote:
> > > They really are! Every sequence in an equivalence class is related by ONE and only ONE attribute - the LIMIT. As I've repeatedly stated, the LIMIT may or may not be a number.
> >
> > The question is not wether or not the limit is a number, it is wether or not the limit is in Q, but the limit of course exists in R as the sequence itself is part of the equivalence classes that makes up R
>
> To prove that Cauchy sequences in ℝ converges in ℝ is not too trivial. You first have to introduce a metric for real numbers
<parrot crap>
Markus Klyver
Sep 16, 2017, 3:11:06 PM9/16/17
to
John Gabriel
Sep 16, 2017, 5:16:45 PM9/16/17
to
You would be wise to stop telling me that I don't understand. It is YOU who do not understand.
Markus Klyver
Sep 16, 2017, 7:34:53 PM9/16/17
to
John Gabriel
Sep 16, 2017, 9:47:16 PM9/16/17
to
Markus Klyver
Sep 17, 2017, 3:31:26 AM9/17/17
to
John Gabriel
Sep 17, 2017, 3:44:59 AM9/17/17
to
ALL Cauchy sequences WITHOUT ANY EXCEPTION have a limit. There is not a single Cauchy sequence that does not.
Markus Klyver
Sep 17, 2017, 5:05:15 AM9/17/17
to
John Gabriel
Sep 17, 2017, 9:37:54 AM9/17/17
to
Read what I wrote carefully:
ALL Cauchy sequences WITHOUT ANY EXCEPTION have a limit. There is not a single Cauchy sequence that does not.
> And obviously not Cauchy sequences converge in ℚ.
It is NOT relevant what a Cauchy sequence converges to, only that it converges. If it converges to Q, then we know the limit is a number. If not Q, then the limit is NOT a number. Stop repeating what you read in your silly textbooks. I know all that and have studied it many years ago. I have a better understanding than anyone else.
> That's why we want real numbers;
> we *want* all Cauchy sequences to converge.
Markus Klyver
Sep 17, 2017, 11:24:51 AM9/17/17
to
If you are going to criticize real analysis, make sure you actually understand it first.
John Gabriel
Sep 18, 2017, 8:53:37 AM9/18/17
to
The two series:
1/2 + 1/4 + 1/8 + ... = 1
1/3 + 1/9 + 1/27 + ... = 1/2
generate the sequences:
{1/2; 3/4; 7/8; ...}
{1/3; 4/9; 13/27; ...}
The componentwise difference of these sequences gives:
{ 1/6; 11/36; 85/216; ...}
And the limit of that last sequence is 1/2.
Do you get it now moron?
John Gabriel
Sep 18, 2017, 9:01:52 AM9/18/17
to
zelos...@gmail.com
Sep 18, 2017, 9:28:22 AM9/18/17
to
zelos...@gmail.com
Sep 18, 2017, 9:31:14 AM9/18/17
to
Den lördag 16 september 2017 kl. 12:39:05 UTC+2 skrev John Gabriel:
Dedekind–MacNeille completion is another general construction method that is valid, on rational numbers you get the dedekind completion type.
You got the Eudoxus construction, which is perfectly valid as well.
There are many many more, all work and are generalized such that you can prove that they gives the desired results and for real numbers, are equivalent.
zelos...@gmail.com
Sep 18, 2017, 9:32:39 AM9/18/17
to
And yes, it is Dunning-krüger, very much so, as we can clearly see you do not understand it at all.
zelos...@gmail.com
Sep 18, 2017, 9:35:16 AM9/18/17
to
The sequences are distinct because their componentwise differens is not a null sequence. That is why, not about a limit.
John Gabriel
Sep 18, 2017, 10:22:21 AM9/18/17
to
It is a FACT that:
John Gabriel
Sep 18, 2017, 10:24:29 AM9/18/17
to
The example shows this my stupid one.
John Gabriel
Sep 18, 2017, 11:35:11 AM9/18/17
to
On Monday, 18 September 2017 08:53:37 UTC-4, John Gabriel wrote:
Good luck moron!
Python
Sep 18, 2017, 11:48:01 AM9/18/17
to
John Gabriel wrote:
> To prove me wrong, you would need to find a pair of sequences whose
> componentwise difference sequence is NULL and has NO limit.
x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> To prove me wrong, you would need to find a pair of sequences whose
> componentwise difference sequence is NULL and has NO limit.
y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
Markus Klyver
Sep 18, 2017, 12:41:12 PM9/18/17
to
Also, by your "definition"
1/2 + 1/4 + 1/8 + ... = 1
1/3 + 1/9 + 1/27 + ... = 1/2
Markus Klyver
Sep 18, 2017, 12:43:24 PM9/18/17
to
Markus Klyver
Sep 18, 2017, 12:44:54 PM9/18/17
to
Markus Klyver
Sep 18, 2017, 12:47:59 PM9/18/17
to
burs...@gmail.com
Sep 18, 2017, 2:49:49 PM9/18/17
to
I was first reading the Exodus construction.
But the Eudoxus construction is worth reading.
The Eudoxus Real Numbers
R. D. Arthan
(Submitted on 24 May 2004)
https://arxiv.org/abs/math/0405454
This one is also nice:
chapter 2 of John Harrison’s thesis [5];
John Harrison. Theorem Proving with
the Real Numbers. Technical report,
University of Cambridge Computer Laboratory, 1996.
Basically you can do Cauchy with integers only.
But the Eudoxus construction is worth reading.
The Eudoxus Real Numbers
R. D. Arthan
(Submitted on 24 May 2004)
https://arxiv.org/abs/math/0405454
This one is also nice:
chapter 2 of John Harrison’s thesis [5];
John Harrison. Theorem Proving with
the Real Numbers. Technical report,
University of Cambridge Computer Laboratory, 1996.
Basically you can do Cauchy with integers only.
Markus Klyver
Sep 18, 2017, 3:54:34 PM9/18/17
to
John Gabriel
Sep 18, 2017, 6:49:15 PM9/18/17
to
John Gabriel
Sep 18, 2017, 6:51:31 PM9/18/17
to
ALL Cauchy sequences WITHOUT ANY EXCEPTION have a limit. There is not a single Cauchy sequence that does not.
I did not say what kind of limit, only that for every Cauchy sequence, there is a limit. Nowhere did I say the limit is rational or not rational.
Learn to read dumbo!!!! This is the third time you make the same mistake!!!!
John Gabriel
Sep 18, 2017, 6:53:45 PM9/18/17
to
http://www.maths.qmul.ac.uk/~ig/MAS111/Cauchy%20Criterion.pdf
John Gabriel
Sep 18, 2017, 7:06:22 PM9/18/17
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Python
Sep 19, 2017, 5:00:08 AM9/19/17
to
John Gabriel
Sep 19, 2017, 7:53:17 AM9/19/17
to
On Tuesday, 19 September 2017 05:00:08 UTC-4, Python wrote:
> John Gabriel wrote:
> > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> >> John Gabriel wrote:
> >>> To prove me wrong, you would need to find a pair of sequences whose
> >>> componentwise difference sequence is NULL and has NO limit.
> >>
> >> x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> >>
> >> y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> >
> > They both have the limit sqrt(2). You haven't proven me wrong.
>
> They both have no limit in the field of rationnal numbers Q.
I didn't say they have a limit in the rational numbers. I said they have a "limit".
> John Gabriel wrote:
> > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> >> John Gabriel wrote:
> >>> To prove me wrong, you would need to find a pair of sequences whose
> >>> componentwise difference sequence is NULL and has NO limit.
> >>
> >> x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> >>
> >> y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> >
> > They both have the limit sqrt(2). You haven't proven me wrong.
>
> They both have no limit in the field of rationnal numbers Q.
You should pay attention to detail. You have been so brainwashed by your "schooling", that you automatically see things that aren't there.
Of course there is no limit in the rational numbers which means that the limit is an incommensurable magnitude, NOT a real number.
Python
Sep 19, 2017, 9:44:56 AM9/19/17
to
numbers is abysmal Mr Gabriel. You'd better find another hobby, math
is definitely not your cup of tea.
John Gabriel
Sep 19, 2017, 2:26:41 PM9/19/17
to
On Tuesday, 19 September 2017 09:44:56 UTC-4, Python wrote:
> John Gabriel wrote:
> > On Tuesday, 19 September 2017 05:00:08 UTC-4, Python wrote:
> >> John Gabriel wrote:
> >>> On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> >>>> John Gabriel wrote:
> >>>>> To prove me wrong, you would need to find a pair of sequences whose
> >>>>> componentwise difference sequence is NULL and has NO limit.
> >>>>
> >>>> x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> >>>>
> >>>> y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> >>>
> >>> They both have the limit sqrt(2). You haven't proven me wrong.
> >>
> >> They both have no limit in the field of rationnal numbers Q.
> >
> > I didn't say they have a limit in the rational numbers. I said they have a "limit".
> >
> > You should pay attention to detail. You have been so brainwashed by your "schooling", that you automatically see things that aren't there.
> >
> > Of course there is no limit in the rational numbers which means that the limit is an incommensurable > magnitude, NOT a real number.
>
> Your inability to get the point about Cauchy's construction of real
> numbers is abysmal Mr Gabriel.
So you finally admit you are wrong. That is a first step. An improvement!! Chuckle.
> John Gabriel wrote:
> > On Tuesday, 19 September 2017 05:00:08 UTC-4, Python wrote:
> >> John Gabriel wrote:
> >>> On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> >>>> John Gabriel wrote:
> >>>>> To prove me wrong, you would need to find a pair of sequences whose
> >>>>> componentwise difference sequence is NULL and has NO limit.
> >>>>
> >>>> x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> >>>>
> >>>> y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> >>>
> >>> They both have the limit sqrt(2). You haven't proven me wrong.
> >>
> >> They both have no limit in the field of rationnal numbers Q.
> >
> > I didn't say they have a limit in the rational numbers. I said they have a "limit".
> >
> > You should pay attention to detail. You have been so brainwashed by your "schooling", that you automatically see things that aren't there.
> >
> > Of course there is no limit in the rational numbers which means that the limit is an incommensurable > magnitude, NOT a real number.
>
> Your inability to get the point about Cauchy's construction of real
> numbers is abysmal Mr Gabriel.
Cauchy's poop is abysmal and it swivels in your thinking processes. The only way to survive is to expel the poop so you can think clearly.
> You'd better find another hobby, math is definitely not your cup of tea.
Markus Klyver
Sep 20, 2017, 4:37:45 PM9/20/17
to
Markus Klyver
Sep 20, 2017, 4:42:50 PM9/20/17
to
Den tisdag 19 september 2017 kl. 00:53:45 UTC+2 skrev John Gabriel:
> On Monday, 18 September 2017 18:51:31 UTC-4, John Gabriel wrote:
> On Monday, 18 September 2017 18:51:31 UTC-4, John Gabriel wrote:
> Every Cauchy sequence converges and every convergent sequence is a Cauchy sequence but the imbeciles Klyver and Zelos do not understand this. Chuckle.
>
> http://www.maths.qmul.ac.uk/~ig/MAS111/Cauchy%20Criterion.pdf
You don't seem to understand. You cannot talk about a limit which doesn't exist. A rational sequence cannot converge to anything in ℚ and if an element to converge to doesn't exist, then the sequence is simply divergent.
>
> http://www.maths.qmul.ac.uk/~ig/MAS111/Cauchy%20Criterion.pdf
Bolzano-Weierstrass' is about Cauchy sequences in ℝ, which is a complete field as I have mentioned. Every Cauchy sequence in ℝ converges in ℝ. This (usually) require real numbers to establish though.
burs...@gmail.com
Sep 20, 2017, 5:10:16 PM9/20/17
to
Maybe this could be authorative for what a tangent is:
Am Mittwoch, 20. September 2017 23:02:12 UTC+2 schrieb burs...@gmail.com:
> Ok, a prominent name and geometry:
>
> Hilbert, David; Cohn-Vossen, Stephan (1952).
> Geometry and the Imagination (2nd ed.). Chelsea.
> http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN379425343
Check out: §26 Ebene Kurven,
Abb. 182 and Abb. 183
(Page 155 and Page 156)
Am Mittwoch, 20. September 2017 23:02:12 UTC+2 schrieb burs...@gmail.com:
> Ok, a prominent name and geometry:
>
> Hilbert, David; Cohn-Vossen, Stephan (1952).
> Geometry and the Imagination (2nd ed.). Chelsea.
> http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN379425343
Check out: §26 Ebene Kurven,
Abb. 182 and Abb. 183
(Page 155 and Page 156)
John Gabriel
Sep 20, 2017, 7:30:49 PM9/20/17
to
On Wednesday, 20 September 2017 16:37:45 UTC-4, Markus Klyver wrote:
> Den tisdag 19 september 2017 kl. 00:49:15 UTC+2 skrev John Gabriel:
> > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> > > John Gabriel wrote:
> > > > To prove me wrong, you would need to find a pair of sequences whose
> > > > componentwise difference sequence is NULL and has NO limit.
> > >
> > > x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> > >
> > > y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> >
> > They both have the limit sqrt(2). You haven't proven me wrong.
>
> They don't have a rational limit,
Correct moron. They don't and I never claimed they did.
> Den tisdag 19 september 2017 kl. 00:49:15 UTC+2 skrev John Gabriel:
> > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> > > John Gabriel wrote:
> > > > To prove me wrong, you would need to find a pair of sequences whose
> > > > componentwise difference sequence is NULL and has NO limit.
> > >
> > > x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> > >
> > > y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> >
> > They both have the limit sqrt(2). You haven't proven me wrong.
>
> They don't have a rational limit,
> and it makes no sense to talk about convergence to a number we haven't established yet.
Pathetic. Unbelievably stupid moron you are.
> Without real numbers, a such sequence will diverge.
So, according to you
{3; 3.1; 3.14; ...} diverges? Bwaaa haaaa haaaa.
No my little orangutan, it converges on the incommensurable magnitude called PI. And yes, PI is not a number for otherwise my brilliant ancestors would never have called it an incommensurable magnitude.
Markus Klyver
Sep 21, 2017, 7:53:22 AM9/21/17
to
a ~ b <----> |a_n - b_n| approaches 0 as n approaches infinity
This is the exact construction of real numbers in the Cauchy model. We don't define real numbers as limits of infinite decimal expansions. We define ℝ as a set of equivalence classes.
Yes, the sequence containing the decimal expansion of pi diverges in ℚ. There simply isn't a rational number to converge towards. You need real numbers to claim every rational Cauchy sequence converges.
Unless you have a non-standard definition of a limit, rational sequences "approaching" irrational numbers is simply divergent in ℚ. It makes absolutely no sense under the standard limit definition to talk about "converging" Cauchy sequences that don't converge to an element in ℚ.
John Gabriel
Sep 21, 2017, 7:59:04 AM9/21/17
to
On Wednesday, 20 September 2017 19:30:49 UTC-4, John Gabriel wrote:
> On Wednesday, 20 September 2017 16:37:45 UTC-4, Markus Klyver wrote:
> > Den tisdag 19 september 2017 kl. 00:49:15 UTC+2 skrev John Gabriel:
> > > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> > > > John Gabriel wrote:
> > > > > To prove me wrong, you would need to find a pair of sequences whose
> > > > > componentwise difference sequence is NULL and has NO limit.
> > > >
> > > > x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> > > >
> > > > y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> > >
> > > They both have the limit sqrt(2). You haven't proven me wrong.
> >
> > They don't have a rational limit,
>
> Correct moron. They don't and I never claimed they did.
>
> > and it makes no sense to talk about convergence to a number we haven't established yet.
>
> And yet that is exactly what you do you BIG BABOON!!!!! You define real numbers as the limit of a convergent Cauchy sequence. No need to repeat your bullshit about equivalence classes because the sequences in any equivalence class have one thing in common - it's called a LIMIT. That the componentwise null sequence is a consequence of the LIMIT is NON-REMARKABLE.
>
> Pathetic. Unbelievably stupid moron you are.
>
> > Without real numbers, a such sequence will diverge.
>
> Chuckle.
>
> So, according to you
>
> {3; 3.1; 3.14; ...} diverges? Bwaaa haaaa haaaa.
Can't you see how circular is the Cauchy "definition"? You claim to have real number when you don't have a rational number limit. In other words you assume the prior establishment of the LIMIT as a real number. But being of such low IQ as you are, you didn't notice.
> On Wednesday, 20 September 2017 16:37:45 UTC-4, Markus Klyver wrote:
> > Den tisdag 19 september 2017 kl. 00:49:15 UTC+2 skrev John Gabriel:
> > > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> > > > John Gabriel wrote:
> > > > > To prove me wrong, you would need to find a pair of sequences whose
> > > > > componentwise difference sequence is NULL and has NO limit.
> > > >
> > > > x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> > > >
> > > > y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> > >
> > > They both have the limit sqrt(2). You haven't proven me wrong.
> >
> > They don't have a rational limit,
>
> Correct moron. They don't and I never claimed they did.
>
> > and it makes no sense to talk about convergence to a number we haven't established yet.
>
> And yet that is exactly what you do you BIG BABOON!!!!! You define real numbers as the limit of a convergent Cauchy sequence. No need to repeat your bullshit about equivalence classes because the sequences in any equivalence class have one thing in common - it's called a LIMIT. That the componentwise null sequence is a consequence of the LIMIT is NON-REMARKABLE.
>
> Pathetic. Unbelievably stupid moron you are.
>
> > Without real numbers, a such sequence will diverge.
>
> Chuckle.
>
> So, according to you
>
> {3; 3.1; 3.14; ...} diverges? Bwaaa haaaa haaaa.
Carl Boyer who was much smarter than you will ever be had this to say in his famous book on the history of calculus:
"Cauchy had stated in his Cours d'analyse that irrational numbers are to be regarded as the limits of sequences of rational numbers. Since a limit is defined as a number to which the terms of the sequence approach in such a way that ultimately the difference between this number and the terms of the sequence can be made less than any given number, the existence of the irrational number depends, in the definition of limit, upon the known existence, and hence the prior definition, of the very quantity whose definition is being attempted.
That is, one cannot define the number sqrt(2) as the limit of the sequence 1, 1.4, 1.41, 1.414, ... because to prove that this sequence has a limit one must assume, in view of the definitions of limits and convergence, the existence of this number as previously demonstrated or defined. Cauchy appears not to have noticed the circularity of the reasoning in this connection, but tacitly assumed that every sequence converging within itself has a limit."
The History of Calculus and its Conceptual Development' (Page. 281) Carl B. Boyer
John Gabriel
Sep 21, 2017, 8:07:15 AM9/21/17
to
On Thursday, 21 September 2017 07:53:22 UTC-4, Markus Klyver wrote:
> Den torsdag 21 september 2017 kl. 01:30:49 UTC+2 skrev John Gabriel:
> > On Wednesday, 20 September 2017 16:37:45 UTC-4, Markus Klyver wrote:
> > > Den tisdag 19 september 2017 kl. 00:49:15 UTC+2 skrev John Gabriel:
> > > > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> > > > > John Gabriel wrote:
> > > > > > To prove me wrong, you would need to find a pair of sequences whose
> > > > > > componentwise difference sequence is NULL and has NO limit.
> > > > >
> > > > > x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> > > > >
> > > > > y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> > > >
> > > > They both have the limit sqrt(2). You haven't proven me wrong.
> > >
> > > They don't have a rational limit,
> >
> > Correct moron. They don't and I never claimed they did.
> >
> > > and it makes no sense to talk about convergence to a number we haven't established yet.
> >
> > And yet that is exactly what you do you BIG BABOON!!!!! You define real numbers as the limit of a convergent Cauchy sequence. No need to repeat your bullshit about equivalence classes because the sequences in any equivalence class have one thing in common - it's called a LIMIT. That the componentwise null sequence is a consequence of the LIMIT is NON-REMARKABLE.
> >
> > Pathetic. Unbelievably stupid moron you are.
> >
> > > Without real numbers, a such sequence will diverge.
> >
> > Chuckle.
> >
> > So, according to you
> >
> > {3; 3.1; 3.14; ...} diverges? Bwaaa haaaa haaaa.
> >
> > No my little orangutan, it converges on the incommensurable magnitude called PI. And yes, PI is not a number for otherwise my brilliant ancestors would never have called it an incommensurable magnitude.
>
> Except we don't.
Except you DO!
> Den torsdag 21 september 2017 kl. 01:30:49 UTC+2 skrev John Gabriel:
> > On Wednesday, 20 September 2017 16:37:45 UTC-4, Markus Klyver wrote:
> > > Den tisdag 19 september 2017 kl. 00:49:15 UTC+2 skrev John Gabriel:
> > > > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> > > > > John Gabriel wrote:
> > > > > > To prove me wrong, you would need to find a pair of sequences whose
> > > > > > componentwise difference sequence is NULL and has NO limit.
> > > > >
> > > > > x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> > > > >
> > > > > y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> > > >
> > > > They both have the limit sqrt(2). You haven't proven me wrong.
> > >
> > > They don't have a rational limit,
> >
> > Correct moron. They don't and I never claimed they did.
> >
> > > and it makes no sense to talk about convergence to a number we haven't established yet.
> >
> > And yet that is exactly what you do you BIG BABOON!!!!! You define real numbers as the limit of a convergent Cauchy sequence. No need to repeat your bullshit about equivalence classes because the sequences in any equivalence class have one thing in common - it's called a LIMIT. That the componentwise null sequence is a consequence of the LIMIT is NON-REMARKABLE.
> >
> > Pathetic. Unbelievably stupid moron you are.
> >
> > > Without real numbers, a such sequence will diverge.
> >
> > Chuckle.
> >
> > So, according to you
> >
> > {3; 3.1; 3.14; ...} diverges? Bwaaa haaaa haaaa.
> >
> > No my little orangutan, it converges on the incommensurable magnitude called PI. And yes, PI is not a number for otherwise my brilliant ancestors would never have called it an incommensurable magnitude.
>
> Except we don't.
> We define ℝ as the set of all equivalence classes of rational Cauchy sequences under the equivalence relationship
Your IMAGINARY real number CANNOT be expressed exactly for many reasons. The very first being you need the ellipsis which you claim is the LIMIT that you haven't established as a real number.
A number is NOT a set. In fact a sequence of numbers DOES NOT exist without prior establishment of NUMBER.
>
> a ~ b <----> |a_n - b_n| approaches 0 as n approaches infinity
<irrelevant crap>
Markus Klyver
Sep 21, 2017, 8:20:14 AM9/21/17
to
Just like you, Carl Boyer is wrong. We can axiomatically declare all rational Cauchy sequences to have a limit and axiomatically "construct" real numbers, but the Cauchy construction doesn't do that. We don't axiomatically declare real numbers exist; we define a real number as a particular equivalence class of rational Cauchy sequences.
WE don't define sqrt(2) in terms of limits. Using Cauchy's construction, take any Cauchy sequence {a_k} such that (a_n)^2 approaches 2 as n approaches infinity. Then we define sqrt(2) as the equivalence class [a].
[a] is an equivalence class, and not a limit of a particular Cauchy sequence. You fail to understand this very simple concept over and over again and you keep claiming things that are not true over and over again.
We do not define sqrt(2) as a limit of a rational sequence. A sequence can be constant, and all rational sequences are not convergent in ℚ.
Markus Klyver
Sep 21, 2017, 8:29:34 AM9/21/17
to
a ~ b <----> |a_n - b_n| approaches 0 as n approaches infinity
zelos...@gmail.com
Sep 21, 2017, 8:33:01 AM9/21/17
to
A limit exists only if the element itself exist and most cauchy sequences of rational numbers, this is not the case. There is no rational number it converges to so there is no limit to speak of.
If however working within the real numbers, the limit always exists. Before you start complaining about others maybe you should learn to actually state your positions in clear detailed correct manners, rather than your asinine ways that is juvenile at best?
zelos...@gmail.com
Sep 21, 2017, 8:36:18 AM9/21/17
to
Den måndag 18 september 2017 kl. 16:24:29 UTC+2 skrev John Gabriel:
> On Monday, 18 September 2017 09:35:16 UTC-4, zelos...@gmail.com wrote:
> > > > > Den söndag 17 september 2017 kl. 15:37:54 UTC+2 skrev John Gabriel:
> >
> > The sequences are distinct because their componentwise differens is not a null sequence. That is why, not about a limit.
>
> We are not talking about distinct sequences you ape. That is a GIVEN. If the component difference sequence is a null sequence then the sequences have the same LIMIT.
>
> The example shows this my stupid one.
Again, the limit may not exist within the given structure.
> On Monday, 18 September 2017 09:35:16 UTC-4, zelos...@gmail.com wrote:
> > Den måndag 18 september 2017 kl. 15:01:52 UTC+2 skrev John Gabriel:
> > > On Monday, 18 September 2017 08:53:37 UTC-4, John Gabriel wrote:
> > > > On Sunday, 17 September 2017 11:24:51 UTC-4, Markus Klyver wrote:
> > > On Monday, 18 September 2017 08:53:37 UTC-4, John Gabriel wrote:
> > > > > Den söndag 17 september 2017 kl. 15:37:54 UTC+2 skrev John Gabriel:
> > > > > > On Sunday, 17 September 2017 05:05:15 UTC-4, Markus Klyver wrote:
> > > > > > > Den söndag 17 september 2017 kl. 09:44:59 UTC+2 skrev John Gabriel:
> > > > > > > > On Sunday, 17 September 2017 03:31:26 UTC-4, Markus Klyver wrote:
> > > > > > > > > Den söndag 17 september 2017 kl. 03:47:16 UTC+2 skrev John Gabriel:
> > > > > > > > > > On Saturday, 16 September 2017 19:34:53 UTC-4, Markus Klyver wrote:
> > > > > > > > > > > Den lördag 16 september 2017 kl. 23:16:45 UTC+2 skrev John Gabriel:
> > > > > > > > > > > > On Saturday, 16 September 2017 15:11:06 UTC-4, Markus Klyver wrote:
> > > > > > > > > > > > > Den lördag 16 september 2017 kl. 12:39:05 UTC+2 skrev John Gabriel:
> > > > > > > > > > > > > > On Saturday, 16 September 2017 03:54:10 UTC-4, Markus Klyver wrote:
> > > > > > > > > > > > > > > Den måndag 11 september 2017 kl. 08:15:16 UTC+2 skrev zelos...@outlook.com:
> > > > > > > > > > > > > > > > On Saturday, September 9, 2017 at 5:34:05 PM UTC+2, John Gabriel wrote:
> > > > > > > > > > > > > > > > > They really are! Every sequence in an equivalence class is related by ONE and only ONE attribute - the LIMIT. As I've repeatedly stated, the LIMIT may or may not be a number.
> > > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > > The question is not wether or not the limit is a number, it is wether or not the limit is in Q, but the limit of course exists in R as the sequence itself is part of the equivalence classes that makes up R
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > To prove that Cauchy sequences in ℝ converges in ℝ is not too trivial. You first have to introduce a metric for real numbers
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > Imbecile. You first have to show that a valid construction of "real" number exists. There is none.
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > <parrot crap>
> > > > > > > > > > > > >
> > > > > > > > > > > > > The Cauchy model is a valid one, but you don't understand it.
> > > > > > > > > > > >
> > > > > > > > > > > > Grrr. You are stupid. This is what discourages me from trying to educate you. Listen boy, if I don't understand, then nobody else understands. Did you get this? I am far more intelligent than anyone I have ever known. This is not Dunning Kruger, it is a FACT.
> > > > > > > > > > > >
> > > > > > > > > > > > You would be wise to stop telling me that I don't understand. It is YOU who do not understand.
> > > > > > > > > > >
> > > > > > > > > > > Then I'm quite surprised I have to tell you about how the Cauchy construction works. You obviously didn't know what a Cauchy sequence were, or how real numbers were defined in that model. You wouldn't state something stupid about how irrationals were defined as limits or infinite decimal expansions if you actually knew how the Cauchy construction worked.
> > > > > > > > > >
> > > > > > > > > > Oh Shut up moron. You are an imbecile who knows nothing. I studied real analysis before you were born you dimwit.
> > > > > > > > >
> > > > > > > > > Oh, that's why you say constant rational sequences are not Cauchy sequences?
> > > > > > > >
> > > > > > > > The main attribute of any Cauchy sequence is that it ***converges***. Everything else is irrelevant nonsense. This convergence IMPLIES a LIMIT.
> > > > > > > >
> > > > > > > > ALL Cauchy sequences WITHOUT ANY EXCEPTION have a limit. There is not a single Cauchy sequence that does not.
> > > > > > >
> > > > > > > Den söndag 17 september 2017 kl. 09:44:59 UTC+2 skrev John Gabriel:
> > > > > > > > On Sunday, 17 September 2017 03:31:26 UTC-4, Markus Klyver wrote:
> > > > > > > > > Den söndag 17 september 2017 kl. 03:47:16 UTC+2 skrev John Gabriel:
> > > > > > > > > > On Saturday, 16 September 2017 19:34:53 UTC-4, Markus Klyver wrote:
> > > > > > > > > > > Den lördag 16 september 2017 kl. 23:16:45 UTC+2 skrev John Gabriel:
> > > > > > > > > > > > On Saturday, 16 September 2017 15:11:06 UTC-4, Markus Klyver wrote:
> > > > > > > > > > > > > Den lördag 16 september 2017 kl. 12:39:05 UTC+2 skrev John Gabriel:
> > > > > > > > > > > > > > On Saturday, 16 September 2017 03:54:10 UTC-4, Markus Klyver wrote:
> > > > > > > > > > > > > > > Den måndag 11 september 2017 kl. 08:15:16 UTC+2 skrev zelos...@outlook.com:
> > > > > > > > > > > > > > > > On Saturday, September 9, 2017 at 5:34:05 PM UTC+2, John Gabriel wrote:
> > > > > > > > > > > > > > > > > They really are! Every sequence in an equivalence class is related by ONE and only ONE attribute - the LIMIT. As I've repeatedly stated, the LIMIT may or may not be a number.
> > > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > > The question is not wether or not the limit is a number, it is wether or not the limit is in Q, but the limit of course exists in R as the sequence itself is part of the equivalence classes that makes up R
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > To prove that Cauchy sequences in ℝ converges in ℝ is not too trivial. You first have to introduce a metric for real numbers
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > Imbecile. You first have to show that a valid construction of "real" number exists. There is none.
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > <parrot crap>
> > > > > > > > > > > > >
> > > > > > > > > > > > > The Cauchy model is a valid one, but you don't understand it.
> > > > > > > > > > > >
> > > > > > > > > > > > Grrr. You are stupid. This is what discourages me from trying to educate you. Listen boy, if I don't understand, then nobody else understands. Did you get this? I am far more intelligent than anyone I have ever known. This is not Dunning Kruger, it is a FACT.
> > > > > > > > > > > >
> > > > > > > > > > > > You would be wise to stop telling me that I don't understand. It is YOU who do not understand.
> > > > > > > > > > >
> > > > > > > > > > > Then I'm quite surprised I have to tell you about how the Cauchy construction works. You obviously didn't know what a Cauchy sequence were, or how real numbers were defined in that model. You wouldn't state something stupid about how irrationals were defined as limits or infinite decimal expansions if you actually knew how the Cauchy construction worked.
> > > > > > > > > >
> > > > > > > > > > Oh Shut up moron. You are an imbecile who knows nothing. I studied real analysis before you were born you dimwit.
> > > > > > > > >
> > > > > > > > > Oh, that's why you say constant rational sequences are not Cauchy sequences?
> > > > > > > >
> > > > > > > > The main attribute of any Cauchy sequence is that it ***converges***. Everything else is irrelevant nonsense. This convergence IMPLIES a LIMIT.
> > > > > > > >
> > > > > > > > ALL Cauchy sequences WITHOUT ANY EXCEPTION have a limit. There is not a single Cauchy sequence that does not.
> > > > > > >
> > > > > > > No, that's not the definition of a Cauchy sequence.
> > > > > >
> > > > > > I did not say that was the definition of a Cauchy sequence. I piss and shit on the definitions of mainstream mythmaticians which I know well.
> > > > > >
> > > > > > Read what I wrote carefully:
> > > > > >
> > > > > >
> > > > > > I did not say that was the definition of a Cauchy sequence. I piss and shit on the definitions of mainstream mythmaticians which I know well.
> > > > > >
> > > > > > Read what I wrote carefully:
> > > > > >
> > > > > > ALL Cauchy sequences WITHOUT ANY EXCEPTION have a limit. There is not a single Cauchy sequence that does not.
> > > > > >
> > > > > >
> > > > > > > And obviously not Cauchy sequences converge in ℚ.
> > > > > >
> > > > > > It is NOT relevant what a Cauchy sequence converges to, only that it converges. If it converges to Q, then we know the limit is a number. If not Q, then the limit is NOT a number. Stop repeating what you read in your silly textbooks. I know all that and have studied it many years ago. I have a better understanding than anyone else.
> > > > > >
> > > > > > > That's why we want real numbers;
> > > > > >
> > > > > > That's why you want chocolate? Chuckle. Fool!
> > > > > >
> > > > > >
> > > > > > > we *want* all Cauchy sequences to converge.
> > > > > >
> > > > > > What you want is irrelevant. Only facts matter.
> > > > >
> > > > > But you obviously have no idea what the definition of what a Cauchy sequence even is. No, all Cauchy sequences in ℚ does not converge in ℚ. Hence, we construct real numbers as equivalence classes of Cauchy sequences. The equivalence relation used says nothing about limits; it only requires the component-wise difference to approach zero. And not every Cauchy sequence is equal under a such equivalence relation, as demonstrated by the example I gave you where a_k = 1 for all k and b_k = 2 for all k.
> > > > >
> > > > > If you are going to criticize real analysis, make sure you actually understand it first.
> > > >
> > > > Hey moron, let me give you an example (hope you know what this is). To say the componentwise difference proves the sequences are equivalent is the SAME as saying their LIMIT is the same. The componentwise difference is a result of the LIMIT being the same.
> > > >
> > > > The two series:
> > > >
> > > > 1/2 + 1/4 + 1/8 + ... = 1
> > > >
> > > > 1/3 + 1/9 + 1/27 + ... = 1/2
> > > >
> > > > generate the sequences:
> > > >
> > > > {1/2; 3/4; 7/8; ...}
> > > >
> > > > {1/3; 4/9; 13/27; ...}
> > > >
> > > > The componentwise difference of these sequences gives:
> > > >
> > > > { 1/6; 11/36; 85/216; ...}
> > > >
> > > > And the limit of that last sequence is 1/2.
> > > >
> > > > Do you get it now moron?
> > >
> > > > > >
> > > > > > It is NOT relevant what a Cauchy sequence converges to, only that it converges. If it converges to Q, then we know the limit is a number. If not Q, then the limit is NOT a number. Stop repeating what you read in your silly textbooks. I know all that and have studied it many years ago. I have a better understanding than anyone else.
> > > > > >
> > > > > > > That's why we want real numbers;
> > > > > >
> > > > > > That's why you want chocolate? Chuckle. Fool!
> > > > > >
> > > > > >
> > > > > > > we *want* all Cauchy sequences to converge.
> > > > > >
> > > > > > What you want is irrelevant. Only facts matter.
> > > > >
> > > > > But you obviously have no idea what the definition of what a Cauchy sequence even is. No, all Cauchy sequences in ℚ does not converge in ℚ. Hence, we construct real numbers as equivalence classes of Cauchy sequences. The equivalence relation used says nothing about limits; it only requires the component-wise difference to approach zero. And not every Cauchy sequence is equal under a such equivalence relation, as demonstrated by the example I gave you where a_k = 1 for all k and b_k = 2 for all k.
> > > > >
> > > > > If you are going to criticize real analysis, make sure you actually understand it first.
> > > >
> > > > Hey moron, let me give you an example (hope you know what this is). To say the componentwise difference proves the sequences are equivalent is the SAME as saying their LIMIT is the same. The componentwise difference is a result of the LIMIT being the same.
> > > >
> > > > The two series:
> > > >
> > > > 1/2 + 1/4 + 1/8 + ... = 1
> > > >
> > > > 1/3 + 1/9 + 1/27 + ... = 1/2
> > > >
> > > > generate the sequences:
> > > >
> > > > {1/2; 3/4; 7/8; ...}
> > > >
> > > > {1/3; 4/9; 13/27; ...}
> > > >
> > > > The componentwise difference of these sequences gives:
> > > >
> > > > { 1/6; 11/36; 85/216; ...}
> > > >
> > > > And the limit of that last sequence is 1/2.
> > > >
> > > > Do you get it now moron?
> > >
> > > Your example of the sequences {1; 1; 1; ...} and {2; 2; 2; ...} prove that the LIMIT is the reason sequences are part of the same equivalence class. But you are too stupid to realise this.
> >
> > No it doesn't, this does however demonstrate you have no clue what you are on about.
> >
> >
> > The sequences are distinct because their componentwise differens is not a null sequence. That is why, not about a limit.
>
> We are not talking about distinct sequences you ape. That is a GIVEN. If the component difference sequence is a null sequence then the sequences have the same LIMIT.
>
> The example shows this my stupid one.
Again, the limit may not exist within the given structure.
zelos...@gmail.com
Sep 21, 2017, 8:38:20 AM9/21/17
to
> To prove me wrong, you would need to find a pair of sequences whose componentwise difference sequence is NULL and has NO limit.
>
> Good luck moron!
>
For a sequence with no limit in Q, how about (1/1,2/1, 3/2, 5/3,...)
where each element is the quotient of successive fibbonaci numbers? Tell me what rational number it converges to.
If that rational number doesn't exist, it has no limit in Q.
zelos...@gmail.com
Sep 21, 2017, 8:39:41 AM9/21/17
to
John Gabriel
Sep 21, 2017, 12:16:30 PM9/21/17
to
Markus Klyver
Sep 21, 2017, 1:46:15 PM9/21/17
to
Den torsdag 21 september 2017 kl. 14:39:41 UTC+2 skrev zelos...@gmail.com:
> Den torsdag 21 september 2017 kl. 14:07:15 UTC+2 skrev John Gabriel:
> > On Thursday, 21 September 2017 07:53:22 UTC-4, Markus Klyver wrote:
> > > Den torsdag 21 september 2017 kl. 01:30:49 UTC+2 skrev John Gabriel:
> > > > On Wednesday, 20 September 2017 16:37:45 UTC-4, Markus Klyver wrote:
> > > > > Den tisdag 19 september 2017 kl. 00:49:15 UTC+2 skrev John Gabriel:
> > > > > > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> > > > > > > John Gabriel wrote:
> > > > > > > > To prove me wrong, you would need to find a pair of sequences whose
> > > > > > > > componentwise difference sequence is NULL and has NO limit.
> > > > > > >
> > > > > > > x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> > > > > > >
> > > > > > > y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> > > > > >
> > > > > > They both have the limit sqrt(2). You haven't proven me wrong.
> > > > >
> > > > > They don't have a rational limit,
> > > >
> > > > Correct moron. They don't and I never claimed they did.
> > > >
> > > > > and it makes no sense to talk about convergence to a number we haven't established yet.
> > > >
> > > > And yet that is exactly what you do you BIG BABOON!!!!! You define real numbers as the limit of a convergent Cauchy sequence. No need to repeat your bullshit about equivalence classes because the sequences in any equivalence class have one thing in common - it's called a LIMIT. That the componentwise null sequence is a consequence of the LIMIT is NON-REMARKABLE.
> > > >
> > > > Pathetic. Unbelievably stupid moron you are.
> > > >
> > > > > Without real numbers, a such sequence will diverge.
> > > >
> > > > Chuckle.
> > > >
> > > > So, according to you
> > > >
> > > > {3; 3.1; 3.14; ...} diverges? Bwaaa haaaa haaaa.
> > > >
> > > > No my little orangutan, it converges on the incommensurable magnitude called PI. And yes, PI is not a number for otherwise my brilliant ancestors would never have called it an incommensurable magnitude.
> > >
> > > Except we don't.
> >
> Den torsdag 21 september 2017 kl. 14:07:15 UTC+2 skrev John Gabriel:
> > On Thursday, 21 September 2017 07:53:22 UTC-4, Markus Klyver wrote:
> > > Den torsdag 21 september 2017 kl. 01:30:49 UTC+2 skrev John Gabriel:
> > > > On Wednesday, 20 September 2017 16:37:45 UTC-4, Markus Klyver wrote:
> > > > > Den tisdag 19 september 2017 kl. 00:49:15 UTC+2 skrev John Gabriel:
> > > > > > On Monday, 18 September 2017 11:48:01 UTC-4, Python wrote:
> > > > > > > John Gabriel wrote:
> > > > > > > > To prove me wrong, you would need to find a pair of sequences whose
> > > > > > > > componentwise difference sequence is NULL and has NO limit.
> > > > > > >
> > > > > > > x_0 = 1 ; x_(n+1) = (x_n + 2/x_n)/2
> > > > > > >
> > > > > > > y_0 = 2 ; y_(n+1) = (y_n + 2/y_n)/2
> > > > > >
> > > > > > They both have the limit sqrt(2). You haven't proven me wrong.
> > > > >
> > > > > They don't have a rational limit,
> > > >
> > > > Correct moron. They don't and I never claimed they did.
> > > >
> > > > > and it makes no sense to talk about convergence to a number we haven't established yet.
> > > >
> > > > And yet that is exactly what you do you BIG BABOON!!!!! You define real numbers as the limit of a convergent Cauchy sequence. No need to repeat your bullshit about equivalence classes because the sequences in any equivalence class have one thing in common - it's called a LIMIT. That the componentwise null sequence is a consequence of the LIMIT is NON-REMARKABLE.
> > > >
> > > > Pathetic. Unbelievably stupid moron you are.
> > > >
> > > > > Without real numbers, a such sequence will diverge.
> > > >
> > > > Chuckle.
> > > >
> > > > So, according to you
> > > >
> > > > {3; 3.1; 3.14; ...} diverges? Bwaaa haaaa haaaa.
> > > >
> > > > No my little orangutan, it converges on the incommensurable magnitude called PI. And yes, PI is not a number for otherwise my brilliant ancestors would never have called it an incommensurable magnitude.
> > >
> > > Except we don't.
> >
> > Except you DO!
> >
> > > We define ℝ as the set of all equivalence classes of rational Cauchy sequences under the equivalence relationship
> >
> > Yes, a very circular ill-formed and therefore BAD definition. The rational numbers can be EXPRESSED EXACTLY in terms of the chosen UNIT.
> >
> > Your IMAGINARY real number CANNOT be expressed exactly for many reasons. The very first being you need the ellipsis which you claim is the LIMIT that you haven't established as a real number.
> >
> > A number is NOT a set. In fact a sequence of numbers DOES NOT exist without prior establishment of NUMBER.
> >
> > >
> > > a ~ b <----> |a_n - b_n| approaches 0 as n approaches infinity
> >
> > Non-remarkable. It's a consequence of the fact that a series converges. Also NOTHING to do with infinity which is not even relevant.
> >
> > <irrelevant crap>
>
> >
> > > We define ℝ as the set of all equivalence classes of rational Cauchy sequences under the equivalence relationship
> >
> > Yes, a very circular ill-formed and therefore BAD definition. The rational numbers can be EXPRESSED EXACTLY in terms of the chosen UNIT.
> >
> > Your IMAGINARY real number CANNOT be expressed exactly for many reasons. The very first being you need the ellipsis which you claim is the LIMIT that you haven't established as a real number.
> >
> > A number is NOT a set. In fact a sequence of numbers DOES NOT exist without prior establishment of NUMBER.
> >
> > >
> > > a ~ b <----> |a_n - b_n| approaches 0 as n approaches infinity
> >
> > Non-remarkable. It's a consequence of the fact that a series converges. Also NOTHING to do with infinity which is not even relevant.
> >
> > <irrelevant crap>
>
> A number is whatever we decide it is as a logical construct. As all in mathematics is just logical constructs we can do as we please on that department. We use sets, sequences, pairs of stuff, etc.
THANK YOU! Yes, there isn't really an unified definition of what a number is and have to obey; it depends on the context. Often we mean real or complex numbers, but when constructing these things we use tuples, sets and other already established concepts to create new objects.
John Gabriel
Sep 21, 2017, 2:25:38 PM9/21/17
to
Markus Klyver
Sep 21, 2017, 5:07:20 PM9/21/17
to
John Gabriel
Sep 21, 2017, 8:54:27 PM9/21/17
to
> You don't know what a sequence is, much less what a Cauchy sequence is or how real numbers are defined.
I did not discover the New Calculus because you are smart you moron! I discovered it because I am the smartest. Ask yourself why no one else was able to realise it before me. It won't help actually because you are a confirmed idiot.
<bullshit>
John Gabriel
Sep 21, 2017, 8:57:33 PM9/21/17
to
With {a_n} we associate a sequence {s_n} where s_n = \sum_{k=1}^n a_k.
This means that a sequence is derived from a series. Get it moron?
All this time, both you and Jan Bielawski have been claiming otherwise with Jan recognising you are wrong in the one attempt, but both of you do not understand what is a series or sequence.
You are a pathetic imbecile who knows shit about the very topic you are trying to defend.
John Gabriel
Sep 21, 2017, 9:25:28 PM9/21/17
to
But this is what I have come to expect from nincompoops like you.
zelos...@gmail.com
Sep 22, 2017, 5:05:44 AM9/22/17
to
zelos...@gmail.com
Sep 22, 2017, 5:06:59 AM9/22/17
to
Markus Klyver
Sep 22, 2017, 6:27:54 AM9/22/17
to
Markus Klyver
Sep 22, 2017, 6:30:29 AM9/22/17
to
Again, we do not define real numbers as infinite sums. We define them as equivalence classes. You don't seem to get this very crucial part.
zelos...@gmail.com
Sep 22, 2017, 6:33:26 AM9/22/17
to
Markus Klyver
Sep 22, 2017, 6:42:39 AM9/22/17
to
adrianf...@gmail.com
Sep 22, 2017, 7:00:48 AM9/22/17
to
Hey John Gabriel, your BIG STUPID is showing.
John Gabriel
Sep 22, 2017, 9:06:44 AM9/22/17
to
> Of course you can derive a series from a sequence and a sequence from a series.
That's what is meant by:
----------------------------------------------------
Walter Idiot Rudin (Page 59):
With {a_n} we associate a sequence {s_n} where s_n = \sum_{k=1}^n a_k.
----------------------------------------------------
With {a_n} we associate a sequence {s_n} where s_n = \sum_{k=1}^n a_k.
> But we were talking about sequences of rational numbers and how we can define real numbers as equivalence classes of rational Cauchy sequences.
>
> Again, we do not define real numbers as infinite sums.
You define:
1.000...
and
0.999...
as equivalent Cauchy Sequences. They are equivalent because they hve the same limit!
> We define them as equivalence classes. You don't seem to get this very crucial part.
Markus Klyver
Sep 22, 2017, 4:01:27 PM9/22/17
to
And again, a sequence does not have to be "derived" from a series, even though you always can construct a series from any given sequence {s_n} such that its partial sums form the sequence {s_n}.- The vice versa is also true, you can always construct a sequence from the partial sums of a series.
We are not defining real numbers as infinite decimal expansions; I have repeatedly told you that. 1.000... is a real number, which is not a sequence. In the Cauchy model 1.000... is *an equivalence class* of rational Cauchy sequences. 0.999... is the same equivalence class.
1.000... is a notation for a real number, and in this notation 0.999... = 1.000... This notation is defined using limits.
Neither of them are sequences, and your misuse of mathematical language exposes you actually have no knowledge about this. We define reals as equivalence classes. You don't get this very crucial part, and keep insisting the Cauchy model is circular when it isn't.
John Gabriel
Sep 22, 2017, 8:49:03 PM9/22/17
to
> Rudin defines a sequence in Definition 2.7 in which a sequence is a function f defined from the naturals to some set J.
> And again, a sequence does not have to be "derived" from a series,
> even though you always can construct a series from any given sequence {s_n} such that its partial sums form the sequence {s_n}.- The vice versa is also true, you can always construct a sequence from the partial sums of a series.
>
> We are not defining real numbers as infinite decimal expansions;
1 = 1+0+0+ ... = 1.000...
0.999... = 0.9 + 0.09 + 0.009 + ...
> I have repeatedly told you that. 1.000... is a real number, which is not a sequence. In the Cauchy model 1.000... is *an equivalence class* of rational Cauchy sequences. 0.999... is the same equivalence class.
You are a moron.
>
> 1.000... is a notation for a real number,
>and in this notation 0.999... = 1.000... This notation is defined using limits.
>
> Neither of them are sequences, and your misuse of mathematical language exposes you actually have no knowledge about this. We define reals as equivalence classes. You don't get this very crucial part, and keep insisting the Cauchy model is circular when it isn't.
Markus Klyver
Sep 23, 2017, 8:22:08 AM9/23/17
to
Definition 2.7 reads: "By a sequence, we mean a function f defined on a set J of all the positive integers. If f(n) = x_n [...] it is customary to denote the sequence by {x_n}"
And no, we are not defining real numbers as infinite decimal expansions. This is a lie you keep repeating. We do not define real numbers as infinite decimal expansions. Instead, they can be defined as equivalence classes of rational Cauchy sequences. Equivalence classes are not limits, nor infinite decimal strings. We define decimal expansions in terms of limits, but this require the limit to exist. The limit may or may not exist without reals, but every monotonic and bounded sequence in ℝ is guaranteed to have a limit in ℝ.
And as I said, 1.000... is a real number, which is not a sequence. In the Cauchy model 1.000... is *an equivalence class* of rational Cauchy sequences. 0.999... is the same equivalence class.
In fact, both the decimal expansions 1.000... and 0.999... is well-defined without real numbers, as the limit clearly exist in ℚ. So even without ℝ you would have 0.999... = 1.000... = 1.
You haven't demonstrated how the Cauchy construction is circular, much less you actually understand it at all.
John Gabriel
Sep 23, 2017, 10:31:05 AM9/23/17
to
THERE IS NO DEFINITION 2.7 IN THE THIRD EDITION.
> Definition 2.7 reads: "By a sequence, we mean a function f defined on a set J of all the positive integers. If f(n) = x_n [...] it is customary to denote the sequence by {x_n}"
>
Markus Klyver
Sep 23, 2017, 2:22:59 PM9/23/17
to
John Gabriel
Sep 23, 2017, 2:38:22 PM9/23/17
to
Consider the series: 3/10 + 3/100 + 3/1000 + ...
The general term is given by 3/10^n where n >=1 and n is an +ve integer.
You can take the function f(x) = 1/3 (1 - 10^(-x)) as the sum and from it determine the sequence:
f(0) = 0
f(1) = 0.3
f(2) = 0.33
Therefore,
{0; 0.3; 0.33; 0.333; ...}
And note that the original function f(x) = 3/10^x, is a continuous function.
Conclusion: Definition 2.7 does not contradict anything on page 59, nor is it in a different context at all.
But can a monkey like you understand this? Nah.
Markus Klyver
Sep 23, 2017, 2:48:04 PM9/23/17
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John Gabriel
Sep 23, 2017, 5:37:43 PM9/23/17
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It is sufficient for us to say that sequences are derived from series. Nothing else is needed.
Markus Klyver
Sep 24, 2017, 8:23:12 AM9/24/17
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And all this, we still don't define real numbers as limits of rational Cauchy sequences. We define ℝ as the set of all equivalence classes of rational Cauchy sequences under the equivalence relationship
John Gabriel
Sep 24, 2017, 8:32:19 AM9/24/17
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Markus Klyver
Sep 24, 2017, 8:38:02 AM9/24/17
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John Gabriel
Sep 24, 2017, 8:41:23 AM9/24/17
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Me
Sep 24, 2017, 8:43:38 AM9/24/17
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On Sunday, September 24, 2017 at 2:32:19 PM UTC+2, John Gabriel wrote:
> Rudin makes it clear that sequences are defined from series.
Idiot.
> Rudin makes it clear that sequences are defined from series.
Markus Klyver
Sep 24, 2017, 8:55:44 AM9/24/17
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genm...@gmail.com
Sep 24, 2017, 9:38:05 AM9/24/17
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Markus Klyver
Sep 25, 2017, 11:50:52 AM9/25/17
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In fact, some textbooks defines a series as a sequence.
John Gabriel
Sep 25, 2017, 2:53:20 PM9/25/17
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Your professor is an idiot and that explains why you are an idiot.
Markus Klyver
Sep 25, 2017, 3:53:41 PM9/25/17
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John Gabriel
Sep 25, 2017, 6:37:08 PM9/25/17
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It clearly states that the value (Lim S) of a series (S) is considered to be limit of a particular sequence (Lim S).
Good boy!
Markus Klyver
Sep 26, 2017, 4:06:25 PM9/26/17
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John Gabriel
Sep 26, 2017, 6:36:10 PM9/26/17
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