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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:

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

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
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But you don't understand the Cauchy construction The square root of 2 is not defined as some mystical limit. A such approach is circular. Instead we define a real number as an equivalence class of Cauchy sequences.

### John Gabriel

Sep 8, 2017, 6:44:24 PM9/8/17
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Moron. You need to understand. And it is because of misguided fools like you that I wrote the article

### Markus Klyver

Sep 9, 2017, 6:39:10 AM9/9/17
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But you are wrong. We don't define pi in terms of pi. In the case of the Cauchy model, every real number is an equivalence class of Cauchy sequences. 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.

### John Gabriel

Sep 9, 2017, 11:34:05 AM9/9/17
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I am right.

> We don't define pi in terms of pi.

How you define pi is not a problem. It is simply the ratio of the circumference to the diameter. The problem is that you call this ratio of magnitudes a **number** when you have no means of measuring it.

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.

Yes, but it is still nonsense because numbers are not defined in terms of elements but in terms of a chosen unit and measurement is not partial or approximate, it is COMPLETE. No sequence in an equivalence class is complete.

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.

It debunks any proofs about objects which don't exist, viz. "real" numbers.

>
> Either way, reals are not defined as a limit or an infinite decimal representation.

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.

### Markus Klyver

Sep 9, 2017, 12:02:18 PM9/9/17
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No, you're still wrong. A real number is not defined as a limit or an infinite decimal representation. That is just wrong and false.

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
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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>

### Markus Klyver

Sep 9, 2017, 4:10:06 PM9/9/17
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In what sense am I wrong? You are the one calling people morons and anti-Semitic slur. Glad you don't teach anyone anymore; you can't even be corrected when wrong.

### konyberg

Sep 9, 2017, 4:20:07 PM9/9/17
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This look like a sentence in logic. Can you give where it comes from?

KON

### zelos...@outlook.com

Sep 11, 2017, 2:08:37 AM9/11/17
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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.

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
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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

### John Gabriel

Sep 11, 2017, 7:52:16 AM9/11/17
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In every sense. You are an ignoramus, a dishonest and morally bankrupt piece of shit like most of your Jewish masters. You are one very pathetic lackey.
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
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See? This is what I mean, we show you your idiocy and you go on antisemitic garbage.

### Dan Christensen

Sep 11, 2017, 12:52:22 PM9/11/17
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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

Visit my Math Blog at http://www.dcproof.wordpress.com

### Markus Klyver

Sep 16, 2017, 3:54:10 AM9/16/17
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To prove that Cauchy sequences in ℝ converges in ℝ is not too trivial. You first have to introduce a metric for real numbers and then show that real Cauchy sequences really converge in ℝ. The statement that Cauchy sequences in ℚ also converges in ℝ is trivial since ℚ is trivially embedded into ℝ. You just make equivalence relations between rationals and rational Cauchy sequences, which should be fairly obvious how to do.

### Markus Klyver

Sep 16, 2017, 3:55:59 AM9/16/17
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Maybe you could explain to us why you think we are wrong, instead of adhering to slur and insults.

### John Gabriel

Sep 16, 2017, 6:38:03 AM9/16/17
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Idiot. The explanation has been given to you many times. Now you get only insult because that is all you understand. Mooroooon!!!!!

### John Gabriel

Sep 16, 2017, 6:39:05 AM9/16/17
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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>

### Markus Klyver

Sep 16, 2017, 3:11:06 PM9/16/17
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The Cauchy model is a valid one, but you don't understand it.

### John Gabriel

Sep 16, 2017, 5:16:45 PM9/16/17
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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.

### Markus Klyver

Sep 16, 2017, 7:34:53 PM9/16/17
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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.

### John Gabriel

Sep 16, 2017, 9:47:16 PM9/16/17
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Oh Shut up moron. You are an imbecile who knows nothing. I studied real analysis before you were born you dimwit.

### Markus Klyver

Sep 17, 2017, 3:31:26 AM9/17/17
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Oh, that's why you say constant rational sequences are not Cauchy sequences?

### John Gabriel

Sep 17, 2017, 3:44:59 AM9/17/17
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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.

### Markus Klyver

Sep 17, 2017, 5:05:15 AM9/17/17
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No, that's not the definition of a Cauchy sequence. And obviously not Cauchy sequences converge in ℚ. That's why we want real numbers; we *want* all Cauchy sequences to converge.

### John Gabriel

Sep 17, 2017, 9:37:54 AM9/17/17
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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.

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.

### Markus Klyver

Sep 17, 2017, 11:24:51 AM9/17/17
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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.

### John Gabriel

Sep 18, 2017, 8:53:37 AM9/18/17
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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?

### John Gabriel

Sep 18, 2017, 9:01:52 AM9/18/17
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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.

### zelos...@gmail.com

Sep 18, 2017, 9:28:22 AM9/18/17
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Not at all, convergence does not imply a limit because the limit may not exist. That is why there are ways to construct expanded structures that allows the limit to exist.

### zelos...@gmail.com

Sep 18, 2017, 9:31:14 AM9/18/17
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Den lördag 16 september 2017 kl. 12:39:05 UTC+2 skrev John Gabriel:
There are plenty, you can show that Cauchy sequences, in the general sense that is not dependent on rational numbers, is valid,

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
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There is nothing that says you have to understand it for anyone else to understand it. That logical implication arrow does not exist. In reality most people in mathematics understand all of this vastely better than you. It is just your narcissistic crank ego that doesn't want it to be so.

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
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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.

### John Gabriel

Sep 18, 2017, 10:22:21 AM9/18/17
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Like almost everything else you are ignorant of the very subject you claim to know - real analysis.

It is a FACT that:

### John Gabriel

Sep 18, 2017, 10:24:29 AM9/18/17
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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.

### John Gabriel

Sep 18, 2017, 11:35:11 AM9/18/17
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On Monday, 18 September 2017 08:53:37 UTC-4, 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.

Good luck moron!

### Python

Sep 18, 2017, 11:48:01 AM9/18/17
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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

### Markus Klyver

Sep 18, 2017, 12:41:12 PM9/18/17
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It's not equivalent. If two rational Cauchy sequences have the same rational limit, then their component-wise difference approaches zero as n --> infinity. But the converse does not have to be true since a rational limit may not exist.

1/2 + 1/4 + 1/8 + ... = 1

1/3 + 1/9 + 1/27 + ... = 1/2

are both false because no finite sum will ever equal to the RHS.

### Markus Klyver

Sep 18, 2017, 12:43:24 PM9/18/17
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No, two Cauchy sequences are in the same equivalence class because they are equivalent under a equivalence relation. The equivalence relation we use when defining real numbers mentions nothing about limits.

### Markus Klyver

Sep 18, 2017, 12:44:54 PM9/18/17
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Again, this is false. All rational Cauchy sequences does not have a rational limit. That's we we want to extend the rationals to the reals.

### Markus Klyver

Sep 18, 2017, 12:47:59 PM9/18/17
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Sure. Take any sequences {a_k} and {b_k} which are not the same such that (a_n)^2 and (b_b)^2 both converges to 2 as n ---> infinity.

### burs...@gmail.com

Sep 18, 2017, 2:49:49 PM9/18/17
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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.

### Markus Klyver

Sep 18, 2017, 3:54:34 PM9/18/17
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Never seen a such construction before. Interesting.

### John Gabriel

Sep 18, 2017, 6:49:15 PM9/18/17
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They both have the limit sqrt(2). You haven't proven me wrong.

### John Gabriel

Sep 18, 2017, 6:51:31 PM9/18/17
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Do you suffer from Down Syndrome? The statement:

ALL Cauchy sequences WITHOUT ANY EXCEPTION have a limit. There is not a single Cauchy sequence that does not.

is true you moron!!!!

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
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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

### John Gabriel

Sep 18, 2017, 7:06:22 PM9/18/17
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Has ZERO to do with Eudoxus.

### Python

Sep 19, 2017, 5:00:08 AM9/19/17
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They both have no limit in the field of rationnal numbers Q.

### John Gabriel

Sep 19, 2017, 7:53:17 AM9/19/17
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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