I don't understand which part of this definition is flawed or dubious. The way I look at the Cauchy sequence definition is: the set of real numbers is the set of all numbers that could be used to represent a point on a number line; that seems very elegant. Also, thank you so much for this video - it really helped me understand the definition a lot better.
I don't see any contradiction in the real numbers. The only thing math requires is that there be no contradictions. You're free to construct any object you wish, provided you can get grant money to do your research. If you don't like the Real numbers then you can simply study something else like engineering. Math is not about truth, it is about validity. If A is true then B must be true. That's what math is, it does not make a claim that A is true or not. Only the one follows from the other if it is true.
Perhaps we could define irrational numbers not as completed numbers, but an algorithm that outputs a serious of rationals that go on forever but dont fly off to infinity. For example, the sequence of rationals starting with a/b = 1 --> (a + 2b) / (a+b) generates the digits of root 2. In this way, we describe the fact that rational numbers can be combined to produce something infinite by an asymptotic convergence, without the logical difficulties of irrational numbers. Every instance of (not uncomputable ) irrationals can be presented a limit that approaches an ideal object, but never can be presented fully. For example, we could define pi in terms of the archimedean method involving circumscribed polygons. I dont want to simply say that irrationals are meaningless objects, because there are a lot of very meaningful patterns involving irrationals like phi, pi , e, and root 2.
If this is considered "dubious" then so is the construction of complex numbers via declaring the square root of -1 to arbitrarily exist. Honestly, just because you don't like an idea doesn't imply it's false. Indeed, the complete totally ordered field definition, the Dedekind cut definition, a construction directly from Euclidean geometry, and this one, are all logically equivalent. A definition can't be false; it can only be shown be useful or not by implying useful theorems and/or be shown to be equivalent to other definitions, should those equivalences exist. I think you might just have a different philosophy of mathematics than most people, because you can define your own mathematical universe, and it will be valid, but not more or less true than others'. There are a lot of things this definition explains. For instance, what if a student asks "Why is 1=.999999...?" This answers it beautifully by saying that the reals have no unique representation since there are many representatives one may choose from the given equivalence class, and what's important is that they are closer than any ϵ > 0. It isn't great for every application, such as in abstract set theory, but one thing it certainly is not is false. If that were the case, then I would argue that complex numbers don't exist; but the Clifford algebra generated from a two-dimensional vector space with metric signature (2,0) does, and the unit grade-2 operator that has a square (under the Clifford product) of -1 exists within that structure. It is a different construction of the complex numbers and the imaginary unit "i" but that doesn't mean it is more true or more false than the classical complex number definition; even though I strongly prefer ones that do not take square roots of negative numbers, that is how mathematics grows: by doing things that are indeed logically sound, but at the same time very counterintuitive, to some at least.
I wanted to thank you for this video once again, presenting the idea of these difficult mathematical concepts in a way that even I understand them. (Well, I may have had a "little" experience beforehand but not much.) And I do praise your views and opinions about the logical "weirdnesses" that these concepts have. But at the same time I really also do like Cauchy sequences. Mathematics is weird and feels "buggy" in a way (programmers vocabulary here). It almost seems mathematics is NOT the true language of nature. We as humans have constructed mathematics in a wonderful way but just like a child constructing a Lego castle does not make it a real castle. Well it was a good watch again. Learning is important and your teaching is fantastic in a sense that it's clear and easy to understand.
I was subject to this treatment as an undergraduate, and I did not resist. I'm not sure why you consider the usage of equivalence classes as wrong or dubious. The treatment I was subject to in the construction of the rational numbers from Peano's axioms does the same thing to "sweep under the rug" the difficulties of representing rational numbers as built-up from using sets to construct the integers. In fact, I would go as far to say that the construction of real numbers as equivalence classes of sequences is about as well-defined of a definition as you can get. Nor have I met one undergraduate student who was dissatisfied with this construction while I was an undergraduate in mathematics. To me, this construction feels no more dubious than constructing the complex integers in terms of modding out the set of rational polynomials by the ideal generated by (x^2 + 1). I actually found the construction of real numbers via Cauchy sequences more natural than this monstrosity.
Most undergraduates don't resist the subtle indoctrination they get, because it is not clear to them that this is what it is. They are at a stage in learning when they are presented with such a multitude of new ideas and complicated theories, that they struggle to ingest and absorb them. A true critical examination would require a more careful, slow and thoughtful treatment of the ideas, including discussion of the historical uncertainties that accompanied the developments. It is important to state that professors are not imparting confusions willfully: rather they are just passing on the same flawed theories that they absorbed years before. At least here, now, we have a good chance to square up to the difficulties, examine them carefully and put some real thought into things. Put your prior orientations aside, and try to understand the situations as they really are, and perhaps your perceptions will gradually shift. It is an adventure!
Hi,professor wildberger,I watched this video months ago and now I have something new about the real number system.First,I don't believe the modern real number system completely exists because there are an infinite number of random irrational numbers in the real number set which are based on people's imagination and meaningless definations.What I mean is that a number exists or it is meaningful only if we can represent it geometrically or if we can find it through geometric methonds.For example,the square root of 2 exists because we can draw a right equilateral triangle where the two sides next to its right angle have lengths of 1 then its hypotenuse must be the square root of 2 undoubtably.We can also creat many square roots like this through the pothygarian theorem and we also know pi exists physically.However,I have no idea how to represent a random irrational number geometrically.If you give me a random irrational number 1.12345667448833039........I don't know how it comes from.Real number literally means really existent rumber in the world but from my perspective a lot of irrational numbers are just imagenary,people just assume they are existing but no one can excatly find where these numbers exist in our physical world.So,the whole irrational number set is not only logically problematic as you said but also physically problematic.Additionally,I want to cite a philosophical viewpoint or psychological viewpoint that our imagination is the combination of present elements in the world.For example the unexisting unicorn is the combination of the horn and the horse.Analogously,the random irrational number is the combination of an infinite series of random digital numbers.So,the entire modern mathematical analysis theory is bunch of tricks and flaws and kind of mythology.
I was taught it a little bit differently: I was given the nested interval property as a means of aiding us in our construction of the real numbers. Instead of having one sequence, you have two: say (x_i) and (y_i). We have (x_i) is strictly non-decreasing, (y_i) is strictly non-increasing, x_i 0. We then took as an axiom that there existed a unique number c with x_i
In response to 3:26, the standard definition of a sequence is a function from N into some set (e.g., rationals). Not sure why you declared “we haven’t defined what a sequence means”.
@@njwildberger Sure. A function is a special type of binary relation. More specifically, let X and Y be sets. A binary relation is a subset of X times Y (Cartesian product). A function f from X into Y is a binary relation that satisfies the following two properties: 1) for every x in X there is a y in Y such that (x,y) in f; and 2) if (x, y1) and (x, y2) are both in f, then y1 = y2.
This is an excellent video on the subject. I also thank you for sharing it. It took me 3 reviews to see some of the points, now I am so anxious to see the next one.
The equivalence class of the rational sequence (s_n) is not considered as the limit of the sequence, it is, however, considered to be the limit of the sequence of the following sequence of sequences : ( (s_1,s_1,....), (s_2.s_2,......), ...... ). Defined this way, the arithmetic in the rationals naturally extends to the arithmetic in R. Indeed, given respectively two representatives of the two Cauchy series r_1=(a_1,a_2,....) and r_2=(b_1,b_2,....) you can define a representative of r_1+r_2 by the sequence whose i-th term is defined to be a_i+b_i. This is a perfectly valid definition as, have you chosen any other representative of r_1 and r_2 you will obtain the same sum sequence modulo the equivalence relation. You may reject the idea of "infinity", but it is wrong to say that there is flaws in the definition of the real number. Infinity is not a number but a process, an "infinite" rational sequence (s_1,....) is nothing more than a function s from the natural numbers to the rational, and can be thought as the process of nesting finite sequence one into an other : {s_1} \subset {s_1,s_2} \subset {s_1,s_2,s_3} \subset .... . Although it is impossible to store all the terms physically, we can work with it, as long as we consider a finite number of terms at each time, and that is all we need to do real analysis.
So we define a new number type: algorithmic, where there’s a finitely expressible rule that unambiguously stipulates a non terminating process. This type includes e, pi, gamma, any infinite series, irrational numbers, etc. The cool thing about it is that it’s still countable.
Prof. Wildberger, my complaints about Cauchy sequence are in somewhere different from yours. But before that, i have some questions: 1) Can the Cauchy sequence only be generated over integer steps? Like: s0, s1, s2, s3.... Or can it be generated over or can it be generated over any stepping size? Like: s(1/2), s(2/2), s(3/2).... 2) What i understand by sequences is that, sequences are algorithmic functions taken over integer interval steps instead of the real number line. I know that you think that the definition of functions and real numbers is not logically complete, but please play along. So, from what I understood in your explanation, Cauchy sequences are sequences that when the independent variable goes to the limit of infinity, it approaches to an horizontal asymptote. Is that right? In other words, Cauchy sequences are sequences that as the independent variable goes to the limit of infinity, it converges to a non-infinity value. If so, my first complaint is that they should have not called it a Cauchy sequence. I know giving the name to the contributor is giving tribute and respect to the person. But it confuses the hell out of people. They should call it what it is, "converging sequence" or "sequences with horizontal asymptote". More and more people are going to discover things and we can't just memorize all the names of contributors as time go along. Simple names like irrational trigonometry is perfect. 3) Do Cauchy sequences have to be alternating? From all the example i have seen so far, they always use alternating sequences as examples. But from the definition you showed, it seems to be that as long as the difference of the next generated sequence is smaller than the difference of the previous sequences, they can be considered as Cauchy sequence. Or: |s(k-1) - s(k-2)| < |s(k) - s(k-1)|. If so, then i guess the second part you were talking about Cauchy sequences without limits are of the generating formula like ln(n) because |ln(k-1) - ln(k-2)| < |ln(k) - ln(k-1)|. And as: lim n->inf ln(n) = inf. So it doesn't converge. My next complaint is about how they think that approximating |s(k) - s(k-1)| < eps is enough to prove that the sequence is convergent. This approximates using the squeeze theorem that the top bound will get closer to the the lower bound and closer to the limiting value. But how do we know that it holds true for all sequences? For example, the Collatz Conjecture: how do we know if the upper bound, lower bound and sequence itself will always go to 1? For me, it needs not just an approximation but an actual proof that at |s(inf) + eps| = |s(inf) - eps|. I know that you don't believe in exact values like you mentioned in your previous video. But not being able to obtain an exact value with our current algorithms doesn't mean that the exact value doesn't exist. It just mean that we don't have an algorithm to do that. To prove that there is no exact value, you have to prove that there is no exact value. Saying that we can't find one is not enough. Not having and cannot find it are two different things. And my last complaint is where i completely agree with you. It is about the second part of Cauchy sequences where doesn't have a converging value. If the sequence where to be generated by the function s(n) = ln(n) will satsify |s(k-1) - s(k-2)| < |s(k) - s(k-1)|. But they think of squeezing it with |ln(n) - ln(n-1)| < eps is enough. But the problem is that the value at s(inf) = ln(inf) = inf. It does not converge to proper value.
+John Smith Your sequence S(n) = ln(n) is not a Cauchy sequence. Hence it does not converge to any real number. Ofcourse it does not suffice to prove | s(n ) - s(n-1) | < eps for all natural numbers n if you want to prove that a sequence s(n) is Cauchy Sequence. To prove that a sequence is a Cauchy sequence you have to prove that for every eps > 0 there is a natural number N(eps) such that | s(n) - s(m) | < eps for natural numbers n, m greater than N(eps). Hence your entire argumentation is wrong because you are only talking about | s(n) - s(n-1) | .
I am an undergraduate and I find the idea to be very intuitive and quite simple. Why is it dubious and why is the definition of a cauchy sequence flawed? One could define a sequence as a (partial) injective function from the Natural numbers to the Rational numbers (a partial function would be a finite sequence, otherwise we have an infinite sequence).
Dream World, indeed! How do you define point-set theoretically a classical Platonic line without Real numbers? So we really need that strange continuum, the set of Rationals + "gaps". But if in your dreams lines and planes are ridden with holes, then I can understand your iconoclastic crusade...
Where is your actual objection other than “it seems unnatural to your taste so it must be dubious”? You haven’t shown any logical inconsistency in the theory of reals. The reason of constructing real numbers at all is to show the axioms of a complete ordered field are consistent. This is done by exhibiting such a complete ordered field using only sets which we assume to exist (namely the rationals) as well as rules of constructing new sets (sequences, which are special types of functions, and equivalent classes etc). Equivalent classes of Cauchy sequences is one such exhibition. Once we know complete order field exist (and showing it is unique up to isomorphism) we can unambiguously speak of THE set of real numbers. The axioms of real numbers can now be regarded as proven theorems about rational numbers. From now on, we can work exclusively with those axioms to do arithmetics and calculus, while forgetting the details of construction.
I don't see the point. A sequence _is_ a well defined: Let N be the set of natural numbers and Q be the set of rational numbers than a map f(n) : N → Q is called a sequence of rational number. See for example en.wikipedia.org/wiki/Sequence#Formal_definition or www.encyclopediaofmath.org/index.php/Sequence.
Maral Watanabe How about f(n)=the first number you think about when I shout the number n at you. Does that define a sequence? Why or why not? How about g(n)=the number of atoms in the universe. Same questions. Do you have some good criteria that you could explain to a third party that would allow them to answer all such questions consistently, and the same way you would??
The main problem in your video is that you speak of sequences whithout saying what excatly your understanding of a sequence is. Something like (a) = 2, 4, 6, 8, 10, ... is not a proper definition of a sequence. It may describe the sequence of all even natural number. But "..." does not guarantees that. As long as you do not define (a) exactly a(6) may be 12 but can be 305485038956950389586030000 as well. You may ask me for a natural number as often as you like. I will choose one radomly. But that does not define a sequence. You may even ask me each day or even each hour or even each minute for the rest of life (I cannot garantue that I will answer all these questions) but if you ask that way you will never get sequence in the sense of the above definitions. Say you ask me ten times and I gave you these numbers: 34, 88909, 2, 493948503855900, 4, 49571, 3, 984, 32, 17 Then you have a function S : D -> N where the domain D of S equals the set {1, 2, 3, 4, 5, 6, 7, 9, 10} and N the set of all natural numbers and S(1) = 34 S(2) = 88909 S(3) = 2 S(4) = 493948503855900 S(5) = 4 S(6) = 49571 S(7) = 3 S(8) = 984 S(9) = 32 S(10) = 17 You may call this function a sequence whose domain is D. S is not denfid for 11 or 100. Therefore you cannot ask for S(11) or S(100). You may write down these symbols but they have no mathematical meaning. S is not a sequence that matches the conditions of the links in my previous comment because its domain is D and not N. "g(n)=the number of atoms in the universe" is not sequence but just a sentence. To give it any mathematical meaning you must define what g exactly is. Maybe something like this: Let Ω be the number seconds from the Big Bang till 23:59:59 at 31st of december 2015 and E the set of all natural number smaller or equal Ω then we can define g as function from E to N with g(n) = "number of atoms n seconds after the Big Bang" for all n in E. You should a physists if that defination makes sense. If so g is a well defined function even if we cannot compute g(n) for any n at all.
+Streets, blocks, experimental music You are wrong. Functions are very precisely defined in mathematics, see for example en.wikipedia.org/wiki/Function_(mathematics)#Definition: "The Cartesian product of two sets X and Y is the set of all ordered pairs, written (x, y), where x is an element of X and y is an element of Y. The x and the y are called the components of the ordered pair. The Cartesian product of X and Y is denoted by X × Y. A function f from X to Y is a subset of the Cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset. In other words, for every x in X there is exactly one element y such that the ordered pair (x, y) is contained in the subset defining the function f. This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned."
First of all, I really love most your videos, and thank you a lot for them! However, with all due respect, I think that you need to tone down your rebuttal of practically most of modern mathematics. I've (as a grad. student of Mathematics) spoken with many "finitetists" / constructivists and have a great deal of respect for intuitionistic logic and constructivism in general. I definitely get many of their arguments against classical logic, and I also have great sympathy for people arguing against e.g. the axiom of infinity in ZFC for example. I am actually not sure which side I stand on, but I think I prefer to stand on neither. However, the things you call ""Theorems"" are Theorems accepting the right axioms. It would be much better if you just made clear (which you somewhat have elsewhere) what your problems with e.g. ZFC is, and call it a day. Then build up your foundation for mathematics (e.g. leading to rational trig etc.) the way you do; that's actually great! But quit saying that modern mathematics is logically flawed and confusing and call things "Theorems" etc, this way. Imho this only confuses people even more. For people interested in other mathematicians opinions, see discussions below (I think it is important to get others views on this, and will post these links in other of your videos): 1.) math.stackexchange.com/questions/356264/infinite-sets-dont-exist 2.) math.stackexchange.com/questions/366834/does-mathematics-require-axioms 3.) math.stackexchange.com/questions/617703/what-good-is-infinity 4.) math.stackexchange.com/questions/527248/refuting-the-anti-cantor-cranks
I strongly disagree. If NJ Wildberger had not put forth his arguments as forthrightly as he had, I for one would still be ignorant about the problems with the foundations of mathematics. "But quit saying that modern mathematics is logically flawed and confusing and call things "Theorems" etc, this way. Imho this only confuses people even more." This isn't a good argument against him stating his opinions/arguments as he does. If people are *already* confused about the foundations of mathematics (as you implicitly admit, when you say his rhetoric would only confuse people *'even **_more')_* then it should be the mainstream/establishment mathematics community's responsibility to alleviate that foundational confusion! If Wildberger can make convincing attacks against the standard position, then the standard position should be strengthened to resist such attacks. For, if indeed they *cannot* strengthen the standard position, then in fact Wildberger is *correct* that the standard position is logically dubious! And if it is logically dubious, why should we refrain from pointing this out? Why would we want to tacitly defend the status quo, when the status quo is wrong or misleading or inadequate? Asking him to 'keep quiet' so to speak, or to 'not rock the boat', is, IMHO, very much like a priest asking another priest not to mention his personal doubts about a religious dogma to anyone, for fear of 'losing the flock', and hence losing personal power/security. It doesn't sound like a genuine argument about the truth of the matter, but rather a rationalization to protect one's own entrenched interests at the expense of the greater public and societal good.
That's right. Apparently, because a glorified electronic abacus cannot physically store infinitely many bits of information, infinite series and irrational numbers representing them cannot exist. I suppose that before computers, the biggest possible numbers was related to the number of fingers we have, so perhaps 2^10 or 2^20 if we use our toes. It beggars belief that NJW thinks that has anything to do with math. Computers are just gadgets.
Professor Wildberger does not like the notion of infinite sets. He does not like the idea of the square root of 2. We does not like normal Trigonometry, and opposes the notion of pi.
True, to inadvertently throw the terms sequence, limit and number into the same bag is obviously a very bad idea. This stems from the fact that before I came along, no one understood what Euclid had attempted to write down, that is, the perfect derivation of the number concept from nothing: th-cam.com/video/fT82zT5U37U/w-d-xo.html While Wildberger explains these things using language, this is insufficient. After all, there would be nothing wrong with a new object if it could be used rationally. As Wildberger shows, equivalent Cauchy sequences are a bogus concept because: 1. The definition is circular. Cauchy assumes a number (which is not rational) exists as the limit, which is a *number*. 2. Ignoring the fact that this idea is flawed, defining numbers as "limit" of infinite sequences fails for several reasons: a. In some cases, the limit is the sum. Example: 1/4 = 2/10 + 5/100 = 25/100 = 0.25 b. In other cases, the limit is NOT the sum but is blatantly defined as such. Example: 1/3 = 3/10 + 3/100 + ... = 0.333... c. Finally, when the number cannot be written as p/q (with p and q integers), no one knows what is the limit except that it is given some symbol such as pi, e or sqrt(2). 3. Ignoring all the previous issues, if one has to say that real numbers are sets of rationals, then one must show a unique property, that is, each sequence in an equivalence class has a unique limit, which is the UNIQUE property. Once again, mythmaticians cannot define the limit. In the following article I prove the entire process to be flawed, whether Cauchy sequence or Dedekind cut. drive.google.com/open?id=0B-mOEooW03iLSTROakNyVXlQUEU Wildberger has not done this even though he has the right ideas. For many more hot topics in mathematics: th-cam.com/channels/lBbBVLs3M-d3dNgU4Vop_A.html With regards to improper integrals, there is nothing lost here by not having a definition of real number. For example, an integral that has pi as "limit" is an incommensurable magnitude. Nothing is contradicted and such ill-formed definitions are not required in sound mathematics.
Presumably since sqrt(2) is irrational, but the iterative process of finding it will always produce a rational number, that is another example of why this definition is wrong
No, the iterative procedure to obtain the square root of 2 gives closer and closer approximations to the square root of 2. According to standard analysis, the sequence of closer and closer approximations to the square root of 2, can define the square root of 2. Professor Wildberger does not like this concept.
@@clivegoodman16 but these approximations are all rational aren't they. That means that according to standard analysis the sequences of rationals somehow defines an irrational. I'm glad I studied Physics and not Maths. I have had 30 years of using maths to help design drugs, stuff like that. If I'd studied Maths, I'd have wrapped myself in knots and done nothing with it
@@andywright8803 . Precisely. A real number can be represented by a Cauchy sequence of rational numbers with the proviso that if two such sequences differ by a null sequence they represent the same real number. (A null sequence is a sequence converging to 0 - that is a sequence {s[n]:n a natural number} such that for all epsilon > 0 there exists a natural number N, such that for all natural numbers n such that n>N we have epsilon > s[n] > -epsilon). It is because of this we can represent real numbers by infinite decimal fractions and that we consider 0.9999999 recurring to be the same as 1. Note Professor Wildberger is aware of this definition, but he does not like it. He does not like the idea of infinite sets and sequences. He likes rational numbers - that is numbers which can be represented by a fraction. He does not like numbers such the square root of 2 or pi or functions such as the standard trigonometric ratios such as sine and cosine. I believe his objection is that as we are able to calculate them approximately but not precisely they are somehow illegitimate concepts. Most mathematicians disagree with him.
@@debendragurung3033 Thanks for your comment. What I meant is that all numbers are real numbers. All numbers have infinite decimal places (fore and aft, left and right, up and down, past and future, toward meca normal and otherwise.) Just because all decimal digits are zero dosen't innoculate a number from being Real. All of the rationals are real, just as all of the real numbers are contained in the complex. What I like about Norman J. Wilder is his emphasis on a more concrete style of Math. If you've followed some of his stuff- he's not a big fan of exploring the universe of extremely high numbers. One pertinent question he's asked is (warning, paraphrase) why isn't there more being done on three dimensional calculus and geometery?
@@sallylauper8222 yes indeed. But since you are a scholar of engineer I would love to give a little insight of what this lecture is trying to tell. Please don't mind me. First it talks about sequence ( in particular Cauchy Sequence). And any sequence has one to one relationship with Natural Numbers n1,n2 ,n3 .... and so on .. or simply it is sort of Countable array. But Real numbers is uncountable array. The question arise how would one construct something that is uncountable from good old countable construction. And also much of applied mathematics are more simpler from principles of Countable arrays so called Sequences. So to the Rescue - "Cauchy Sequences". First we construct rationals from integers and we construct all reals from rationals.
I don't understand which part of this definition is flawed or dubious. The way I look at the Cauchy sequence definition is: the set of real numbers is the set of all numbers that could be used to represent a point on a number line; that seems very elegant.
Also, thank you so much for this video - it really helped me understand the definition a lot better.
I don't see any contradiction in the real numbers. The only thing math requires is that there be no contradictions. You're free to construct any object you wish, provided you can get grant money to do your research. If you don't like the Real numbers then you can simply study something else like engineering. Math is not about truth, it is about validity. If A is true then B must be true. That's what math is, it does not make a claim that A is true or not. Only the one follows from the other if it is true.
Perhaps we could define irrational numbers not as completed numbers, but an algorithm that outputs a serious of rationals that go on forever but dont fly off to infinity. For example, the sequence of rationals starting with a/b = 1 --> (a + 2b) / (a+b) generates the digits of root 2. In this way, we describe the fact that rational numbers can be combined to produce something infinite by an asymptotic convergence, without the logical difficulties of irrational numbers. Every instance of (not uncomputable ) irrationals can be presented a limit that approaches an ideal object, but never can be presented fully. For example, we could define pi in terms of the archimedean method involving circumscribed polygons. I dont want to simply say that irrationals are meaningless objects, because there are a lot of very meaningful patterns involving irrationals like phi, pi , e, and root 2.
If this is considered "dubious" then so is the construction of complex numbers via declaring the square root of -1 to arbitrarily exist. Honestly, just because you don't like an idea doesn't imply it's false. Indeed, the complete totally ordered field definition, the Dedekind cut definition, a construction directly from Euclidean geometry, and this one, are all logically equivalent. A definition can't be false; it can only be shown be useful or not by implying useful theorems and/or be shown to be equivalent to other definitions, should those equivalences exist. I think you might just have a different philosophy of mathematics than most people, because you can define your own mathematical universe, and it will be valid, but not more or less true than others'.
There are a lot of things this definition explains. For instance, what if a student asks "Why is 1=.999999...?" This answers it beautifully by saying that the reals have no unique representation since there are many representatives one may choose from the given equivalence class, and what's important is that they are closer than any ϵ > 0. It isn't great for every application, such as in abstract set theory, but one thing it certainly is not is false. If that were the case, then I would argue that complex numbers don't exist; but the Clifford algebra generated from a two-dimensional vector space with metric signature (2,0) does, and the unit grade-2 operator that has a square (under the Clifford product) of -1 exists within that structure. It is a different construction of the complex numbers and the imaginary unit "i" but that doesn't mean it is more true or more false than the classical complex number definition; even though I strongly prefer ones that do not take square roots of negative numbers, that is how mathematics grows: by doing things that are indeed logically sound, but at the same time very counterintuitive, to some at least.
I wanted to thank you for this video once again, presenting the idea of these difficult mathematical concepts in a way that even I understand them. (Well, I may have had a "little" experience beforehand but not much.) And I do praise your views and opinions about the logical "weirdnesses" that these concepts have. But at the same time I really also do like Cauchy sequences. Mathematics is weird and feels "buggy" in a way (programmers vocabulary here). It almost seems mathematics is NOT the true language of nature. We as humans have constructed mathematics in a wonderful way but just like a child constructing a Lego castle does not make it a real castle.
Well it was a good watch again. Learning is important and your teaching is fantastic in a sense that it's clear and easy to understand.
I was subject to this treatment as an undergraduate, and I did not resist. I'm not sure why you consider the usage of equivalence classes as wrong or dubious. The treatment I was subject to in the construction of the rational numbers from Peano's axioms does the same thing to "sweep under the rug" the difficulties of representing rational numbers as built-up from using sets to construct the integers.
In fact, I would go as far to say that the construction of real numbers as equivalence classes of sequences is about as well-defined of a definition as you can get. Nor have I met one undergraduate student who was dissatisfied with this construction while I was an undergraduate in mathematics.
To me, this construction feels no more dubious than constructing the complex integers in terms of modding out the set of rational polynomials by the ideal generated by (x^2 + 1). I actually found the construction of real numbers via Cauchy sequences more natural than this monstrosity.
Most undergraduates don't resist the subtle indoctrination they get, because it is not clear to them that this is what it is. They are at a stage in learning when they are presented with such a multitude of new ideas and complicated theories, that they struggle to ingest and absorb them. A true critical examination would require a more careful, slow and thoughtful treatment of the ideas, including discussion of the historical uncertainties that accompanied the developments.
It is important to state that professors are not imparting confusions willfully: rather they are just passing on the same flawed theories that they absorbed years before.
At least here, now, we have a good chance to square up to the difficulties, examine them carefully and put some real thought into things. Put your prior orientations aside, and try to understand the situations as they really are, and perhaps your perceptions will gradually shift. It is an adventure!
I wish I could meet you ! 💓💓💓💓
Never seen this dedication towards the subject.
Hi,professor wildberger,I watched this video months ago and now I have something new about the real number system.First,I don't believe the modern real number system completely exists because there are an infinite number of random irrational numbers in the real number set which are based on people's imagination and meaningless definations.What I mean is that a number exists or it is meaningful only if we can represent it geometrically or if we can find it through geometric methonds.For example,the square root of 2 exists because we can draw a right equilateral triangle where the two sides next to its right angle have lengths of 1 then its hypotenuse must be the square root of 2 undoubtably.We can also creat many square roots like this through the pothygarian theorem and we also know pi exists physically.However,I have no idea how to represent a random irrational number geometrically.If you give me a random irrational number 1.12345667448833039........I don't know how it comes from.Real number literally means really existent rumber in the world but from my perspective a lot of irrational numbers are just imagenary,people just assume they are existing but no one can excatly find where these numbers exist in our physical world.So,the whole irrational number set is not only logically problematic as you said but also physically problematic.Additionally,I want to cite a philosophical viewpoint or psychological viewpoint that our imagination is the combination of present elements in the world.For example the unexisting unicorn is the combination of the horn and the horse.Analogously,the random irrational number is the combination of an infinite series of random digital numbers.So,the entire modern mathematical analysis theory is bunch of tricks and flaws and kind of mythology.
I was taught it a little bit differently: I was given the nested interval property as a means of aiding us in our construction of the real numbers.
Instead of having one sequence, you have two: say (x_i) and (y_i). We have (x_i) is strictly non-decreasing, (y_i) is strictly non-increasing, x_i 0. We then took as an axiom that there existed a unique number c with x_i
In response to 3:26, the standard definition of a sequence is a function from N into some set (e.g., rationals). Not sure why you declared “we haven’t defined what a sequence means”.
And can you please remind us all of the definition of a “function “ say from natural numbers to rational numbers?
@@njwildberger Sure. A function is a special type of binary relation. More specifically, let X and Y be sets. A binary relation is a subset of X times Y (Cartesian product). A function f from X into Y is a binary relation that satisfies the following two properties: 1) for every x in X there is a y in Y such that (x,y) in f; and 2) if (x, y1) and (x, y2) are both in f, then y1 = y2.
This is an excellent video on the subject. I also thank you for sharing it. It took me 3 reviews to see some of the points, now I am so anxious to see the next one.
Thanks, hope you enjoy the next one too.
The equivalence class of the rational sequence (s_n) is not considered as the limit of the sequence, it is, however, considered to be the limit of the sequence of the following sequence of sequences : ( (s_1,s_1,....), (s_2.s_2,......), ...... ). Defined this way, the arithmetic in the rationals naturally extends to the arithmetic in R. Indeed, given respectively two representatives of the two Cauchy series r_1=(a_1,a_2,....) and r_2=(b_1,b_2,....) you can define a representative of r_1+r_2 by the sequence whose i-th term is defined to be a_i+b_i. This is a perfectly valid definition as, have you chosen any other representative of r_1 and r_2 you will obtain the same sum sequence modulo the equivalence relation. You may reject the idea of "infinity", but it is wrong to say that there is flaws in the definition of the real number.
Infinity is not a number but a process, an "infinite" rational sequence (s_1,....) is nothing more than a function s from the natural numbers to the rational, and can be thought as the process of nesting finite sequence one into an other : {s_1} \subset {s_1,s_2} \subset {s_1,s_2,s_3} \subset .... . Although it is impossible to store all the terms physically, we can work with it, as long as we consider a finite number of terms at each time, and that is all we need to do real analysis.
So we define a new number type: algorithmic, where there’s a finitely expressible rule that unambiguously stipulates a non terminating process. This type includes e, pi, gamma, any infinite series, irrational numbers, etc.
The cool thing about it is that it’s still countable.
That's indeed another excellent video. Brilliantly thought and done!
Slide 4: Shouldn't "conversely" be "equivalently"? And maybe "can't" should be "doesn't"?
thankyou for this amazing explanation
Limits: a simple concept rendered clear in geometry, overly complex in calculus, and senseless by Cauchy sequences.
Prof. Wildberger, my complaints about Cauchy sequence are in somewhere different from yours. But before that, i have some questions:
1) Can the Cauchy sequence only be generated over integer steps? Like: s0, s1, s2, s3.... Or can it be generated over or can it be generated over any stepping size? Like: s(1/2), s(2/2), s(3/2)....
2) What i understand by sequences is that, sequences are algorithmic functions taken over integer interval steps instead of the real number line. I know that you think that the definition of functions and real numbers is not logically complete, but please play along. So, from what I understood in your explanation, Cauchy sequences are sequences that when the independent variable goes to the limit of infinity, it approaches to an horizontal asymptote. Is that right? In other words, Cauchy sequences are sequences that as the independent variable goes to the limit of infinity, it converges to a non-infinity value. If so, my first complaint is that they should have not called it a Cauchy sequence. I know giving the name to the contributor is giving tribute and respect to the person. But it confuses the hell out of people. They should call it what it is, "converging sequence" or "sequences with horizontal asymptote". More and more people are going to discover things and we can't just memorize all the names of contributors as time go along. Simple names like irrational trigonometry is perfect.
3) Do Cauchy sequences have to be alternating? From all the example i have seen so far, they always use alternating sequences as examples. But from the definition you showed, it seems to be that as long as the difference of the next generated sequence is smaller than the difference of the previous sequences, they can be considered as Cauchy sequence. Or: |s(k-1) - s(k-2)| < |s(k) - s(k-1)|. If so, then i guess the second part you were talking about Cauchy sequences without limits are of the generating formula like ln(n) because |ln(k-1) - ln(k-2)| < |ln(k) - ln(k-1)|. And as: lim n->inf ln(n) = inf. So it doesn't converge.
My next complaint is about how they think that approximating |s(k) - s(k-1)| < eps is enough to prove that the sequence is convergent. This approximates using the squeeze theorem that the top bound will get closer to the the lower bound and closer to the limiting value. But how do we know that it holds true for all sequences? For example, the Collatz Conjecture: how do we know if the upper bound, lower bound and sequence itself will always go to 1? For me, it needs not just an approximation but an actual proof that at |s(inf) + eps| = |s(inf) - eps|. I know that you don't believe in exact values like you mentioned in your previous video. But not being able to obtain an exact value with our current algorithms doesn't mean that the exact value doesn't exist. It just mean that we don't have an algorithm to do that. To prove that there is no exact value, you have to prove that there is no exact value. Saying that we can't find one is not enough. Not having and cannot find it are two different things.
And my last complaint is where i completely agree with you. It is about the second part of Cauchy sequences where doesn't have a converging value. If the sequence where to be generated by the function s(n) = ln(n) will satsify |s(k-1) - s(k-2)| < |s(k) - s(k-1)|. But they think of squeezing it with |ln(n) - ln(n-1)| < eps is enough. But the problem is that the value at s(inf) = ln(inf) = inf. It does not converge to proper value.
+John Smith
Your sequence S(n) = ln(n) is not a Cauchy sequence. Hence it does not converge to any real number. Ofcourse it does not suffice to prove | s(n ) - s(n-1) | < eps for all natural numbers n if you want to prove that a sequence s(n) is Cauchy Sequence.
To prove that a sequence is a Cauchy sequence you have to prove that for every eps > 0 there is a natural number N(eps) such that | s(n) - s(m) | < eps for natural numbers n, m greater than N(eps).
Hence your entire argumentation is wrong because you are only talking about | s(n) - s(n-1) | .
I am an undergraduate and I find the idea to be very intuitive and quite simple. Why is it dubious and why is the definition of a cauchy sequence flawed? One could define a sequence as a (partial) injective function from the Natural numbers to the Rational numbers (a partial function would be a finite sequence, otherwise we have an infinite sequence).
Dream World, indeed! How do you define point-set theoretically a classical Platonic line without Real numbers? So we really need that strange continuum, the set of Rationals + "gaps". But if in your dreams lines and planes are ridden with holes, then I can understand your iconoclastic crusade...
I'm pretty sure that he rejects the real line. Only the rational line exists. It just has magic holes in it.
Do you have a playlist of topics related only to real analysis?
Do you think there are really gaps in rational numbers?
Wow... Great lecture sir
If a number is impossible to write, it's "real."
Where is your actual objection other than “it seems unnatural to your taste so it must be dubious”? You haven’t shown any logical inconsistency in the theory of reals. The reason of constructing real numbers at all is to show the axioms of a complete ordered field are consistent. This is done by exhibiting such a complete ordered field using only sets which we assume to exist (namely the rationals) as well as rules of constructing new sets (sequences, which are special types of functions, and equivalent classes etc). Equivalent classes of Cauchy sequences is one such exhibition. Once we know complete order field exist (and showing it is unique up to isomorphism) we can unambiguously speak of THE set of real numbers. The axioms of real numbers can now be regarded as proven theorems about rational numbers. From now on, we can work exclusively with those axioms to do arithmetics and calculus, while forgetting the details of construction.
sir you are great
I don't see the point. A sequence _is_ a well defined:
Let N be the set of natural numbers and Q be the set of rational numbers than a map f(n) : N → Q is called a sequence of rational number.
See for example en.wikipedia.org/wiki/Sequence#Formal_definition or www.encyclopediaofmath.org/index.php/Sequence.
Maral Watanabe How about f(n)=the first number you think about when I shout the number n at you.
Does that define a sequence? Why or why not?
How about g(n)=the number of atoms in the universe. Same questions. Do you have some good criteria that you could explain to a third party that would allow them to answer all such questions consistently, and the same way you would??
The main problem in your video is that you speak of sequences whithout saying what excatly your understanding of a sequence is.
Something like
(a) = 2, 4, 6, 8, 10, ...
is not a proper definition of a sequence. It may describe the sequence of all even natural number. But "..." does not guarantees that. As long as you do not define (a) exactly a(6) may be 12 but can be 305485038956950389586030000 as well.
You may ask me for a natural number as often as you like. I will choose one radomly. But that does not define a sequence. You may even ask me each day or even each hour or even each minute for the rest of life (I cannot garantue that I will answer all these questions) but if you ask that way you will never get sequence in the sense of the above definitions.
Say you ask me ten times and I gave you these numbers:
34, 88909, 2, 493948503855900, 4, 49571, 3, 984, 32, 17
Then you have a function S : D -> N
where the domain D of S equals the set {1, 2, 3, 4, 5, 6, 7, 9, 10} and N the set of all natural numbers
and
S(1) = 34
S(2) = 88909
S(3) = 2
S(4) = 493948503855900
S(5) = 4
S(6) = 49571
S(7) = 3
S(8) = 984
S(9) = 32
S(10) = 17
You may call this function a sequence whose domain is D. S is not denfid for 11 or 100. Therefore you cannot ask for S(11) or S(100). You may write down these symbols but they have no mathematical meaning. S is not a sequence that matches the conditions of the links in my previous comment because its domain is D and not N.
"g(n)=the number of atoms in the universe" is not sequence but just a sentence. To give it any mathematical meaning you must define what g exactly is. Maybe something like this: Let Ω be the number seconds from the Big Bang till 23:59:59 at 31st of december 2015 and E the set of all natural number smaller or equal Ω then we can define g as function from E to N with
g(n) = "number of atoms n seconds after the Big Bang"
for all n in E. You should a physists if that defination makes sense. If so g is a well defined function even if we cannot compute g(n) for any n at all.
+Streets, blocks, experimental music
You are wrong. Functions are very precisely defined in mathematics, see for example en.wikipedia.org/wiki/Function_(mathematics)#Definition:
"The Cartesian product of two sets X and Y is the set of all ordered pairs, written (x, y), where x is an element of X and y is an element of Y. The x and the y are called the components of the ordered pair. The Cartesian product of X and Y is denoted by X × Y.
A function f from X to Y is a subset of the Cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset. In other words, for every x in X there is exactly one element y such that the ordered pair (x, y) is contained in the subset defining the function f. This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned."
First of all, I really love most your videos, and thank you a lot for them! However, with all due respect, I think that you need to tone down your rebuttal of practically most of modern mathematics. I've (as a grad. student of Mathematics) spoken with many "finitetists" / constructivists and have a great deal of respect for intuitionistic logic and constructivism in general. I definitely get many of their arguments against classical logic, and I also have great sympathy for people arguing against e.g. the axiom of infinity in ZFC for example. I am actually not sure which side I stand on, but I think I prefer to stand on neither. However, the things you call ""Theorems"" are Theorems accepting the right axioms. It would be much better if you just made clear (which you somewhat have elsewhere) what your problems with e.g. ZFC is, and call it a day. Then build up your foundation for mathematics (e.g. leading to rational trig etc.) the way you do; that's actually great! But quit saying that modern mathematics is logically flawed and confusing and call things "Theorems" etc, this way. Imho this only confuses people even more. For people interested in other mathematicians opinions, see discussions below (I think it is important to get others views on this, and will post these links in other of your videos):
1.) math.stackexchange.com/questions/356264/infinite-sets-dont-exist
2.) math.stackexchange.com/questions/366834/does-mathematics-require-axioms
3.) math.stackexchange.com/questions/617703/what-good-is-infinity
4.) math.stackexchange.com/questions/527248/refuting-the-anti-cantor-cranks
I strongly disagree. If NJ Wildberger had not put forth his arguments as forthrightly as he had, I for one would still be ignorant about the problems with the foundations of mathematics.
"But quit saying that modern mathematics is logically flawed and confusing and call things "Theorems" etc, this way. Imho this only confuses people even more."
This isn't a good argument against him stating his opinions/arguments as he does. If people are *already* confused about the foundations of mathematics (as you implicitly admit, when you say his rhetoric would only confuse people *'even **_more')_* then it should be the mainstream/establishment mathematics community's responsibility to alleviate that foundational confusion! If Wildberger can make convincing attacks against the standard position, then the standard position should be strengthened to resist such attacks.
For, if indeed they *cannot* strengthen the standard position, then in fact Wildberger is *correct* that the standard position is logically dubious!
And if it is logically dubious, why should we refrain from pointing this out? Why would we want to tacitly defend the status quo, when the status quo is wrong or misleading or inadequate?
Asking him to 'keep quiet' so to speak, or to 'not rock the boat', is, IMHO, very much like a priest asking another priest not to mention his personal doubts about a religious dogma to anyone, for fear of 'losing the flock', and hence losing personal power/security. It doesn't sound like a genuine argument about the truth of the matter, but rather a rationalization to protect one's own entrenched interests at the expense of the greater public and societal good.
oh lord, we want the Limit. Hence all the real numbers. Amen
LOL
good
so you believe that the square root operation is not viable mathematics?
That's right. Apparently, because a glorified electronic abacus cannot physically store infinitely many bits of information, infinite series and irrational numbers representing them cannot exist. I suppose that before computers, the biggest possible numbers was related to the number of fingers we have, so perhaps 2^10 or 2^20 if we use our toes. It beggars belief that NJW thinks that has anything to do with math. Computers are just gadgets.
If math based on rational numbers is flawed that means we can hack it, how do we do that?
Professor Wildberger does not like the notion of infinite sets. He does not like the idea of the square root of 2. We does not like normal Trigonometry, and opposes the notion of pi.
True, to inadvertently throw the terms sequence, limit and number into the same bag is obviously a very bad idea. This stems from the fact that before I came along, no one understood what Euclid had attempted to write down, that is, the perfect derivation of the number concept from nothing:
th-cam.com/video/fT82zT5U37U/w-d-xo.html
While Wildberger explains these things using language, this is insufficient. After all, there would be nothing wrong with a new object if it could be used rationally. As Wildberger shows, equivalent Cauchy sequences are a bogus concept because:
1. The definition is circular. Cauchy assumes a number (which is not rational) exists as the limit, which is a *number*.
2. Ignoring the fact that this idea is flawed, defining numbers as "limit" of infinite sequences fails for several reasons:
a. In some cases, the limit is the sum. Example: 1/4 = 2/10 + 5/100 = 25/100 = 0.25
b. In other cases, the limit is NOT the sum but is blatantly defined as such. Example: 1/3 = 3/10 + 3/100 + ... = 0.333...
c. Finally, when the number cannot be written as p/q (with p and q integers), no one knows what is the limit except that it is given some symbol such as pi, e or sqrt(2).
3. Ignoring all the previous issues, if one has to say that real numbers are sets of rationals, then one must show a unique property, that is, each sequence in an equivalence class has a unique limit, which is the UNIQUE property. Once again, mythmaticians cannot define the limit.
In the following article I prove the entire process to be flawed, whether Cauchy sequence or Dedekind cut.
drive.google.com/open?id=0B-mOEooW03iLSTROakNyVXlQUEU
Wildberger has not done this even though he has the right ideas.
For many more hot topics in mathematics:
th-cam.com/channels/lBbBVLs3M-d3dNgU4Vop_A.html
With regards to improper integrals, there is nothing lost here by not having a definition of real number. For example, an integral that has pi as "limit" is an incommensurable magnitude. Nothing is contradicted and such ill-formed definitions are not required in sound mathematics.
Presumably since sqrt(2) is irrational, but the iterative process of finding it will always produce a rational number, that is another example of why this definition is wrong
No, the iterative procedure to obtain the square root of 2 gives closer and closer approximations to the square root of 2. According to standard analysis, the sequence of closer and closer approximations to the square root of 2, can define the square root of 2. Professor Wildberger does not like this concept.
@@clivegoodman16 but these approximations are all rational aren't they. That means that according to standard analysis the sequences of rationals somehow defines an irrational. I'm glad I studied Physics and not Maths. I have had 30 years of using maths to help design drugs, stuff like that. If I'd studied Maths, I'd have wrapped myself in knots and done nothing with it
@@andywright8803 . Precisely. A real number can be represented by a Cauchy sequence of rational numbers with the proviso that if two such sequences differ by a null sequence they represent the same real number. (A null sequence is a sequence converging to 0 - that is a sequence {s[n]:n a natural number} such that for all epsilon > 0 there exists a natural number N, such that for all natural numbers n such that n>N we have epsilon > s[n] > -epsilon). It is because of this we can represent real numbers by infinite decimal fractions and that we consider 0.9999999 recurring to be the same as 1.
Note Professor Wildberger is aware of this definition, but he does not like it. He does not like the idea of infinite sets and sequences. He likes rational numbers - that is numbers which can be represented by a fraction. He does not like numbers such the square root of 2 or pi or functions such as the standard trigonometric ratios such as sine and cosine. I believe his objection is that as we are able to calculate them approximately but not precisely they are somehow illegitimate concepts. Most mathematicians disagree with him.
🙏🙏🙏🙏🙏
I think of real numbers as numbers with infinite decimal digits, so I'm an engineer. (Guess I need to talk to my guidance councilor...)
What you perhaps meant is irrational numbers. Real numbers includes all including integers and rationals.
@@debendragurung3033 Thanks for your comment. What I meant is that all numbers are real numbers. All numbers have infinite decimal places (fore and aft, left and right, up and down, past and future, toward meca normal and otherwise.) Just because all decimal digits are zero dosen't innoculate a number from being Real. All of the rationals are real, just as all of the real numbers are contained in the complex. What I like about Norman J. Wilder is his emphasis on a more concrete style of Math. If you've followed some of his stuff- he's not a big fan of exploring the universe of extremely high numbers. One pertinent question he's asked is (warning, paraphrase) why isn't there more being done on three dimensional calculus and geometery?
@@sallylauper8222 yes indeed. But since you are a scholar of engineer I would love to give a little insight of what this lecture is trying to tell. Please don't mind me.
First it talks about sequence ( in particular Cauchy Sequence). And any sequence has one to one relationship with Natural Numbers n1,n2 ,n3 .... and so on .. or simply it is sort of Countable array. But Real numbers is uncountable array. The question arise how would one construct something that is uncountable from good old countable construction. And also much of applied mathematics are more simpler from principles of Countable arrays so called Sequences. So to the Rescue - "Cauchy Sequences". First we construct rationals from integers and we construct all reals from rationals.