21 March 2012 The Life of a Great Mathematician - George Cantor: The Father of Set Theory From the School of Mathematics & Statistics University of Newcastle upon Tyne ‘The Postgraduate Magazine’ article by Nasr M. Ahmed George Ferdinand Ludwig Philipp Cantor (1845-1918), a German mathematician who is best known as the founder of set theory. Cantor put forth the modern theory on infinite sets that revolutionized almost every mathematics field. Cantor’s most remarkable achievement was to show, in a mathematically rigorous way, that the concept of infinity is not an undifferentiated one. Not all infinite sets are the same size, and consequently, infinite sets can be compared with one another. However, his new ideas also created many dissenters and made him one of the most assailed mathematicians in history. Hilbert described Cantor’s work as: the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity. Cantor struggled too much to convince people with his golden ideas and died in a mental hospital paying a price for his extraordinary mind!! Cantor was Born in St Petersburg, Russia. He was the eldest of six children. Cantor’s doctoral thesis: In remathema tica ars propendi pluris facienda est quam solvendi “In mathematics the art of asking questions is more valuable than solving problems”, on a question that Carl Friedrich Gauss had left unsettled in his Disquisitiones Arithmeticae (1801). In 1874, he married and eventually had six children. In the same year, Cantor published his first paper on the theory of sets. Cantor’s early interests were in number theory, indeterminate equations and trigonometric series. The subtle theory of trigonometric series seems to have inspired him to look into the foundations of analysis. He produced his beautiful treatment of irrational numbers and commenced in 1874 his revolutionary work on set theory and the theory of the infinite. With this latter work, Cantor created a whole new field of mathematical research. He developed the theory of transfinite numbers- based on a mathematical treatment of the actual infinite and created an arithmetic of transfinite numbers analogous to the arithmetic of finite numbers. Cantor’s earlier work in set theory contained: 1. A proof that the set of real numbers is not denumerable i.e. is not in one-to-one correspondence with the set of natural numbers. [1874] 2. A definition of what it means for two sets M and N to have the same Cardinal number. [1878] 3. A proof that the set of real numbers and the set of points in n-dimensional Euclidean space have the same power. [1878] In 1870 Cantor established a basic uniqueness theorem for trigonometric series: If such a series converges to zero everywhere, then all of its coefficients are zero. To generalize, Cantor [1872] started to allow points at which convergence fails, getting to the following formulation: For a collection P of real numbers, let P’ be the collection of limit points of P , and P(n) the result of n iterations of this operation. If a trigonometric series converges to zero everywhere except on a P where P(n) is empty for some n, then all of its coefficients are zero. Here Cantor was already breaking new ground in considering collections of real numbers defined through an operation. In [1872] Cantor provided his formulation of the real numbers in terms of fundamental sequences of rational numbers. The other well-known formulation of the real numbers is due to Richard Dedekind [1872]. Cantor and Dedekind maintained a fruitful correspondence, especially during the 1870’s. In 1874, Cantor was asking himself whether the unit square could be mapped into a line of unit length with a 1-1 correspondence of points on each. In his letter to Dedekind dated 5 January 1874 he wrote: “Can a surface (say a square that includes the boundary) be uniquely referred to a line (say a straight line segment that includes the end points) so that for every point on the surface there is a corresponding point of the line and, conversely, for every point of the line there is a corresponding point of the surface? I think that answering this question would be no easy job, despite the fact that the answer seems so clearly to be “no” that proof appears almost unnecessary”. In 1877, he wrote to Dedekind proving that there was a 1-1 correspondence of points on the interval [0, 1] and points in p-dimensional space. Cantor was surprised and wrote “I see it, but I don’t believe it!” Cantor submitted his paper to Crelle’s Journal in 1877 which was treated with suspicion by Kronecker (who disliked much of Cantor’s set theory and fundamentally disagreed with Cantor’s work) and only published after Dedekind intervened on Cantor’s behalf. Cantor greatly resented Kronecker’s opposition to his work and never submitted any further papers to Crelle’s Journal (Many of his papers were published in Sweden in the new journal Acta Mathematica, edited and founded by Gösta Mittag-Leffler, one of the first persons to recognize Cantor abilities). Thus, early in his career, Cantor was already having to confront the strong opposition of one of the most eminent mathematicians of his day. Worse yet, transfinite set theory was still in its infancy. It was an untried if not a suspect newcomer to most mathematicians who had been conditioned to reject the absolute infinite in mathematics in favor of arguments involving finite values only. Consequently, Cantor’s new ideas were particularly vulnerable to opposition like Kronecker’s, and Cantor’s resented what he regarded as unfair, premature criticism. In fact, this paper marks the birth of set theory. Previously, all infinite collections had been assumed to be of “the same size”; Cantor was the first to show that there was more than one kind of infinity. In doing so, he became the first to invoke the concept of a 1-to-1 correspondence which became precise in this article. Before in mathematics, infinity had been a forbidden subject. Gauss had stated that infinity should only be used as “a way of speaking” and not as a mathematical value. In 1831 Gauss wrote ‘I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction’ Most mathematicians followed Gauss advice and stayed away. Infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world, for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence. However, in Cantor’s theory, infinity has an actual existence. Cantor considered infinite sets not as merely going on forever but as completed entities, that is having an actual though infinite number of members. He called these actual infinite numbers transfinite numbers. Cantor believed that misuse of the infinite in mathematics had justly inspired a ‘horror of the infinite’ among careful mathematicians of his day, precisely as it did in Gauss. Leibniz and Cantor on the Actual Infinite A well known feature of Leibniz’s philosophy is his espousal of the actual infinite, in defiance of the Aristotelian stricture that the infinite can exist only potentially. Leibniz wrote to Foucher in 1692: ‘I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Thus I believe that there is no part of matter which is not, I do not say divisible, but actually divided; and consequently the least particle ought to be considered as a world full of an infinity of different creatures.’ Cantor indicated his full agreement that there is an “actual infinity of created individuals in the universe as well as on our earth and, in all probability, even in each ever so small extended part of space.” However, Leibniz had refused to countenance infinite number, arguing that its supposition was in conflict with the part-whole axiom, since (as Galileo’s paradox shows) this would lead to an infinite set (the whole) being equal to an infinite proper subset of its elements (the part). Cantor, on the other hand, had followed Dedekind in taking this equality of an infinite set to its proper subset as the defining property of an infinite set. This development, together with the widespread acceptance today of Cantor’s position, has led many commentators to criticize Leibniz’s position as unsound: if there are actually infinitely many creatures, there is an infinite number of them. By considering the infinite sets with a transfinite number of members, Cantor was able to come up with his amazing discoveries and was promoted to full professorship in 1879 for his discoveries. He also defined denumerable sets as sets which can be put into a 1-to-1 correspondence with the natural numbers. Cantor introduces the notion of “power” or “equivalence” of sets; two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor was the first to formulate the continuum hypothesis: There exists no set whose power is greater than that of the naturals and less than that of the reals (in other words, no cardinality bigger than and smaller than C, The real numbers have also been called the continuum, hence the name.). Despite Cantor’s repeated promises to prove the continuum hypothesis, he was never able to do so. his inability to prove the continuum hypothesis caused Cantor considerable anxiety. At that time, Mathematical worries began to trouble Cantor. He began to worry that he could not prove the continuum hypothesis. Early in 1884 he thought he had found a proof, but a few days later he reversed himself completely and thought he could actually disprove the hypothesis. Finally he realized that he had made no progress at all, as he reported in letters to Mittag-Leffler. All the while he had to endure mounting opposition and threats from Kronecker, who said he was preparing an article that would show that “the results of modern function theory and set theory are of no real significance.” Such were circumstance surrounding his hospitalization for what later proved to be recurring cycles of manic depression. The continuum hypothesis is an interesting point in the history of mathematics, and was studied by many mathematicians. In 1963, Paul Cohen obtained a dramatic result by establishing that the continuum hypothesis is independent, that is, neither provable nor refutable from the ordinary set-theoretic axioms. Another mathematical principle which the ordinary set-theoretic axioms fail to settle is the axiom of choice. The independence proofs use the forcing technique for which Cohen won the Field’s medal in 1966. In this sense CH is undecidable, and probably the most famous example of an undecidable statement. In 1891 Cantor published his diagonal argument to show that the integers and the continuum do not have the same cardinality. Cantor proved that there are infinite sets (now known as uncountable sets) which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Cantor’s theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantor’s theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. Cantor then discovered his famous paradox: the set of all sets is its own power set. Therefore, the cardinality of the set of all sets must be bigger than itself! In other words, Cantor had succeeded in proving that for any given transfinite number there’s always a greater transfinite number, so that just as there’s no greatest natural number, there also is no greatest transfinite number. Now consider the set whose members are all possible sets. Surely no set can have more members than this set of all sets. But if this is the case, how can there be a transfinite number greater than the transfinite number of this set? The famous British philosopher and mathematician Bertrand Russell discovered his paradox in 1901 as a result of his work on Cantor’s theorem. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox. Russell’s paradox is the most famous of the logical or set-theoretical paradoxes. The existence of paradoxes in set theory like Cantor’s paradox, Russell’s paradox and others, clearly indicates something is wrong. To solve these paradoxes in set theory, a great deal of literature on the subject has appeared, and numerous attempts at a solution have been offered. So far as mathematics is concerned, there seems to be an easy way out. One has merely to reconstruct set theory on an axiomatic basis sufficiently restrictive to exclude the known antinomies. The first attempt to axiomatise set theory was made by Zermelo in 1908, and subsequent refinement have been made by Fraenkel (1922, 1925), Skolem (1922, 1929), von Newmann (1925, 1928), Bernays (1937-1948), and others. But such a procedure has been criticized as merely avoiding the paradoxes; certainly it doesn’t explain them. Moreover, this procedure carries no guarantee that other kinds of paradoxes will not appear in the future. Gödel showed the limitations of any axiomatic theory and the aims of many mathematicians such as Hilbert could never be achieved. Cantor’s was well aware of the opposition his ideas were encountering, in 1883 he wrote: … I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers. Cantor’s groundbreaking ideas gained him numerous enemies. In 1908 Henri Poincare stated that: Cantor’s set theory would be considered by future generations as “a disease from which one has recovered. However, he was kinder than another critic, Leopold Kronecker who attacked Cantor personally, calling him a “scientific charlatan,” a “renegade” and a “corrupter of youth.” Using his prestige as a professor at the University of Berlin, Kronecker did all he could to suppress Cantor’s ideas. He belittled Cantor’s ideas in front of his students and blocked Cantor’s life ambition of gaining a position at the prestigious University of Berlin. Kronecker regarded Cantor’s theory as a kind of mathematical madness: I don’t know what predominates in Cantor’s theory - philosophy or theology, but I am sure that there is no mathematics there Seeing mathematics headed for the mad-house under Cantor’s leadership, he started to attack the theory and its highly sensitive author with every weapon that came to his hand. Kronecker has been blamed severely for Cantor’s tragedy. Some greats such as Karl Weierstrass and long-time friend Richard Dedekind supported Cantor’s ideas and attacked Kronecker’s actions. However, it was not enough. Under the constant attack by Kronecker, in addition to the absence of welldeserved recognition for his work and being stuck in a third-rate institution Cantor had the first recorded attack of depression in 1884. After he recovered he seemed less confident and wrote: …I don’t know when I shall return to the continuation of my scientific work. At the moment I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness Although he returned to mathematics and attempted to work on the continuum hypothesis again, his attitudes generally had undergone substantial alternation. He began to emphasize other interests. The amount of time he devoted to various literary and historical problems steadily increased, and he read the history and documents of the Elizabethans with great attentiveness in hopes of proving that Francis Bacon was the true author of Shakespear’s plays. When Cantor recovered at the end of June 1884, and entered the depressive phase, he complained that he lacked the energy and interest to return to rigorous mathematical thinking. Cantor made his last major contributions to set theory in 1895 and 1897. On December 16 1899, his youngest son died suddenly. Cantor had been to a lecture on the Bacon-Shakespeare question in Leipzig, and returned home in the evening to find that the child had died in the afternoon. It was a deep disappointment to Cantor as well as a shock. He described his son Rudolf in a letter to Felix Klien a week later, and the tragedy was born in every word he wrote. After his retirement in April 1913, Cantor lived quietly at home. He had been at Halle University for 44 years, 34 of them as ordentlicher Professor; he was remembered as a clear and indeed inspiring teacher. The hostile attitude of many contemporaries severely aggravated Cantor’s emotional ailments and caused several nervous breakdowns. The rest of his life was spent in and out of mental institutions. He died in a mental hospital in Halle on January 6, 1918 at the age of seventy-three. But all that he represented would never die, wrote Edmund Landau to Cantor’s wife. On the 6 January he died suddenly and painlessly after a heart attack, and was buried in Halle next to his son Rudolf. Cantor theory finally began to gain recognition by the turn of the century. Today, Cantor’s set theory has penetrated into almost every branch of mathematics, and it has proved to be of particular importance in topology and in the foundations of real function theory. His theory also has led to many new questions about set theory that should keep mathematicians busy for centuries. Cantor actually described his conviction about the truth of his theory explicitly in quasi-religious terms: My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things. Can we have a Cantorian physics? Even after acknowledging the consistency of Cantor’s theory, it seems that we know nothing about infinity! It appears that in order to avoid all inconsistencies in the temporary physics, physicists need a deeper understanding of the the notion of infinity. Physicists should look seriously to Cantor theory, the only known mathematical theory of infinities, and try to make use of it. In “the Road to Reality” Roger Penrose said “It appears to be a universal feature of the mathematics normally believed to underlie the workings of our physical universe that it has a fundamental dependence on the infinite… we need to take the use of the infinite seriously, particularly in its rule in the mathematical description of the physical continuum. But what kind of infinity is it that we are requiring here?… it is perhaps remarkable, in view of close relationship between mathematics and physics, that issues of such basic importance in mathematics as transfinite set theory and computability have as yet had a very limited impact on our description of the physical world, but only very little use of these ideas has so far been made in mathematical physics.” Cantor showed that there are different sizes of infinity. The smallest one is that of the natural numbers, different infinities continue unendingly to larger and larger scales. Current theoretical physics doesn’t treat infinities like that. This is why I personally believe that the study of real infinities in mathematics may be the first step towards understanding the problem of infinities in physics. There is a class of physical phenomena where infinities occur which may require revolutionary ideas in dealing with the infinities and we shouldn’t forget that Cantor’s theory of infinities has a strong base in physical reality: it is based on arithmetic and set theory. There’s a physical theory called ‘E-infinity’ or ‘Cantorian space-time’ was constructed in 1992. It is the only theory of high energy physics that is based completely on Cantor’s ideas. The theory successfully predicted the accurate values of mass spectrum of elementary particles. I have met the Egyptian author of this theory– Prof. Mohamed.S. El Naschie (who received his education in West Germany (Hamburg and Hannover) and later on in England where he obtained his Ph.D. from University College, London)– several times; the last one was in a library in London. I asked him about what inspired him to make use of Cantor’s ideas in high energy physics, it was surprising to me when he told me: Well, I was inspired by what Wheeler wrote in his famous book ‘Gravitation’ (with Misner and Thorne) about Borel sets: ‘What a line of thought could ever be imagined as leading to four dimensions -or any dimensionality at all- out of more primitive considerations? …… A Borel set is a collection of points (bucket of dust) which have not yet been assembled into a manifold of any particular dimensionality. Recalling the universal sway of the quantum principle, one can imagine probability amplitudes for the points in a Borel set to be assembled into points with this, that, and the other dimensionality……thus one can think of each dimensionality as having a much higher statistical weight than the next higher dimensionality …….. It is too much to imagine that one has yet made enough mistakes in this domain of thought to explore such ideas with any degree of good judgment.’ (Page 1203) In the framework of this physical Cantorian theory, there’s a satisfactory interpretation for the two-slit experiment. One of the startling conclusions of this experiment is that an indivisible quantum particle such as a photon traversing the two-slit screen can be said to have passed through both holes simultaneously!!. Most discussions of the two-slit experiment with particles refer to Feynman’s quote in his lectures: We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. The notion of an object or a point coexisting at two different spatially separated locations simultaneously is, of course, classically impossible and that is the essence of the paradoxical conclusions of the two slit experiment and the origin of the particle-wave duality of quantum physics. Within Cantorian space-time a point could in some sense be said to occupy two different ‘locations’ at the same ‘time’. El Naschie said: ‘Theoretical physicists are conservative by nature and it is important to be that way, but one also has to be open minded about things in maths which may seem at the beginning to be esoteric, such as the Cantor sets that I use… just take Cantor’s ideas into account. Then, theoretical physics will be much easier!’ Absolute Infinite? Cantor was not only a great mathematician, but a very religious man and by some standards a mystic. his mysticism was supported by his mathematics, which to him was at least as strong an argument for the mathematics as for the mysticism. Cantor had a different set of numinous feelings about the infinite. He invented the concept of “Absolute Infinte” and equated it with God. Cantor wrote: “The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto asa mathematical magnitude, number or order type.” NOTE: Before the illustration of some definitions related to this article, a worthy note of attention is that not only Cantor tried to relate the idea of God to his mathematics, but also Kurt Gödeldid in his famous ontological proof. Selected related definitions: 1- Cantor set. (Please write down, this is important!!) Cantor middle third set is probably the best known and studied transfinite set. It is constructed as follows: we take a unit length and remove the middle third except for the end points so that two lines of the length of one third will remain. Then we repeat the same procedure once more with these two remaining lines. That way, we are left with four lines, each of the length one third squared. Continuing this process ad infinitum results in the removal of infinitely many lines which added together will be as long as the unit line which we started with. In other words nothing is left except for a set of points which have no lengths, i.e. measure zero which constitute the so-called Cantor set. The Cantor set contains all points in the interval [0, 1] that are not deleted at any step in this infinite process. That means all points in Cantor set are end points. The total length removed is given by the geometric series equation This means that we removed everything. Nevertheless, this Cantor set possesses a definite quantity that can be used to characterize it mathematically. This quantity is its Hausdorff dimension. For this case, the Hausdorff dimension is given by the ratio of the logarithm of two divided by the logarithm of three. This comes to be approximately 0.63. This is not the only noteworthy property of a Cantor set. A somewhat more surprising fact about this geometrical structure is that it has the cardinality of the continuum. The construction of the Cantor set is illustrated below. Construction of cantor set image So, one could say that, from measure theoretical viewpoint, the classical Cantor set does not exist. However, from the viewpoint of geometrical and topological probability Cantor sets do exist. Moreover, cantor sets possess scale invariance. That means they remain self-similar on all scales regardless of the resolution with which we observe or magnify any part of the set. It is a fractal set. 2- Cantor dust:A multi-dimensional version of the Cantor set. It could be constructed by taking a finite Cartesian product of the Cantor set with itself. Cantor dust has zero measure like the Cantor set. It is illustrated in the following figure. CAntor dust illustration 3- Cantor’s diagonal argument: Published in 1891 as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are known as uncountable sets. 4- Cardinality: A measure of the number of elements of the set. While for finite sets the size is given by a natural number which denotes the number of elements, the Cardinal number is a generalized kind of number used to denote the size of a set. Cardinal numbers can also classify degrees of infinity. Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null (denoted by symbol), the cardinality of the set of natural numbers. 5-Ordinal numbers: Numbers used to denote the position in an ordered sequence. Cantor defined two kinds of infinite numbers, the ordinal numbers and the cardinal numbers. 6- Cantor space: A topological space is a Cantor space if it is homeomorphic to the Cantor set. The canonical example of a Cantor space is the countably infinite topological product of the discrete 2-point space {0, 1}. This is usually written as 2N (where 2 denotes the 2-element set {0,1} with the discrete topology). of course, Cantor set itself is a Cantor space. References: 1- Bell, E. T. Men of Mathematics. New York : Simon & Schuster, 1937 2- Eves, H. An introduction to the history of mathematics. New York, 1953. 3- Penrose, R. The Road to Reality: A Complete Guide to the Laws of the Universe. London, 2004. 4- Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Cambridge, 1979. 5- C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, Freeman, San Francisco, pp 672-678, 795-796, 887 (1973). 6- J.W. Dauben, George Cantor, The Personal matrix of His mathematics, JSTOR Vol. 69, No. 4. 7- I. Grattan -Guinness: towards a biography of George Cantor, Annals of Science, Vol. 27, No. 4 - December 1971. 8- M. S. El Naschie, A review of E infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals, Volume 19, Issue 1, January 2004, Pages 209-236. 9- M. Stern, Memorial Places of Georg Cantor in Halle, The Mathematical Intelligencer, Vol. 10, No.3, 1988.
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