Into this mélange of mathematicians walked a psychologically complicated genius, Georg Cantor (1845–1918), whose pioneering work would lay the foundation for set theory and our modern understanding of infinity.
GEORG CANTOR
Arguably the title of most tormented—both internally and externally—among the greatest mathematicians in history goes to the German mathematician Georg Cantor. We know that Cantor was born in Saint Petersburg, Russia, on March 3, 1845, and that his father, Georg Woldemar Cantor, was born in Copenhagen. We also know that the father’s family moved to Saint Petersburg after the 1807 British bombardment of Copenhagen, in which their home had been destroyed. The father was known to have been a devout Lutheran. Cantor’s mother, Maria Böhm, was born a Roman Catholic. But otherwise, Cantor’s provenance is shrouded in mystery.
Although his parents were married in a Lutheran ceremony in Saint Petersburg in 1842, there are some clues that suggest that Cantor’s grandparents on both sides were Jewish. For one, Georg Woldemar’s father, Jacob Cantor, married a woman whose maiden name was Meier. In addition to the fact that Cantor, Böhm, and Meier are all common Jewish names, an important piece of evidence is found in a letter Cantor wrote to a friend late in life, in which he mentioned that he had “Israelitisch” grandparents. When and how religious conversions took place, if at all, is unknown. Neither do we have much information about the grandparents, including where they were from. A tantalizing detail in the life story of Georg Cantor is the fact that, while he lived as a Lutheran, he chose to use the first letter of the Hebrew alphabet, aleph, to denote the concept he invented for infinity.
Because of the father’s pulmonary illness, the family moved to Frankfurt, Germany, where they enjoyed great economic success with an international wholesale business the father founded and named Cantor & Co. But ultimately the move away from the damp and cold Baltic climate, and the financial success he enjoyed in Frankfurt, didn’t help him much: some years later, Georg Woldemar would die of consumption.
Since Georg was the first of six children, his family had great aspirations for him. His uncle, Joseph Böhm, was the founder and conductor of the Vienna Conservatory. Another uncle was a noted law professor at the University of Kazan in Russia whose writings were used as the legal framework for the Russian Revolution. One of his students had been Leo Tolstoy.
Young Georg Cantor attended private schools in Frankfurt and, at age fifteen, was admitted to a gymnasium, the rigorous German high school in which mathematics and science are taught. While he was boarding at the school, in the city of Darmstadt, his father, Georg Woldemar, wrote him a letter that reveals something about how the family viewed their firstborn son:
I close with these words: Your father, or rather your parents and all other members of the family both in Russia and in Germany and in Denmark, have their eyes on you as the eldest and expect you to be nothing less than a Theodor Schaeffer and, God willing, later perhaps a shining star on the horizon of science.1
Georg Woldemar knew Theodor Schaeffer, his son’s teacher at the gymnasium, and considered him a great scholar. Georg Cantor kept this letter from his father with him throughout his life, perhaps to draw from it the strength he would need to fight against adversity from without and from within, and to surmount the immense hurdles placed in front of him in his quest to understand the nature of infinity.
The year he entered the gymnasium, Cantor confessed to his father that he loved mathematics, and the latter then used this information to push him, in letter after letter, to pursue not only mathematics, but also physics and astronomy. The father told his son about his nighttime dreams. In one of them, he was looking at the stars and marveling at their apparent infinitude. Some historians of mathematics, including E. T. Bell, have identified Georg Woldemar’s extremely high expectations as a possible cause of his son’s mental problems later in life.
Cantor performed well in his school examinations—better in mathematics and the sciences than in geography, history, and the humanities—and the school administration therefore recommended that he pursue a university course of study in the sciences. He was admitted to study mathematics at the renowned Swiss Polytechnic Institute in Zurich, and, after a short stint there, was able to transfer to the University of Berlin, the undisputed world leader in mathematics research at the time. Cantor took courses from many of the great mathematicians in Berlin, but the subject he chose as his concentration was the theory of numbers, and in 1867 he wrote a brilliant dissertation in this area. Weierstrass influenced him greatly, leading him to the study of infinity. Upon obtaining his doctorate, he took the first position he could get—that of a privatdozent (private tutor) at the University of Halle.
Cantor taught at the University of Halle, Germany, from 1869 until the end of his life. The school was founded in 1694; this lithograph depicts one of its “new” buildings under construction in 1836.
Cantor’s life ambition was to become a professor at Berlin, but his views on infinity and irrational numbers quickly came into conflict with those of Kronecker, who was a very influential member of the mathematics faculty in Berlin. The latter continually stood in his way to a professorship at this prestigious university, and Cantor, despite his genius, would remain condemned to teaching at the second-tier University of Halle for the rest of his days.
Cantor was undeterred by attacks against his mathematics, however. Weierstrass, the hulking bigger-than-life presence at Berlin, pioneered approaches to numbers that included quantities that were infinitesimally small and used them to lay a foundation for mathematical analysis. And the German mathematician Dedekind worked in the same direction, defining irrational numbers using the idea of a cut (schnitt in German), which was essentially a sequence of rational numbers that converged to an irrational one. In the work of these two men, Cantor found solace from the mounting attacks of Kronecker, a diminutive man with a bad temper who resembled a small barking dog, ever harassing Cantor for his work on infinity.
Karl Weierstrass, above, Cantor’s doctoral-thesis adviser at the University of Berlin, also served as adviser to Arthur Schoenflies, Cantor’s first biographer.
Cantor started with the work of Weierstrass on the convergence of number sequences. Weierstrass, like Newton, Leibniz, and the ancient Greeks two millennia before them, used the concept of potential infinity—the idea commonly used by any student of calculus today. Saying “let h approach zero,” or “let n go to infinity,” are limit arguments that use potential infinity without ever actually exhibiting an infinite quantity. But at a certain point in his work, Cantor decided to address the question of the actual existence of infinite quantities. It was here, in fact, that Cantor had taken a giant step forward, beyond anything that had passed before him. And it was this act of blasphemy, as it were, that engendered the ire of his detractors—chief among them Kronecker, but also Kronecker’s Berlin colleague Ernst Eduard Kummer.
From work on the convergence of series as some quantity “goes to infinity,” Cantor made the leap to actual infinity—a concept only glimpsed by few of his predecessors, among them Galileo and Bernhard Bolzano (1781–1848). While on house arrest by order of the Inquisition, Galileo turned his attention to pure mathematics and noted that he could draw a one-to-one correspondence between all the integers and all the squared numbers simply by squaring a number, as we’ve seen. Hence, he could “count” all the integers against all the squared integers and find that these two sets were of the same size. But it would seem that there are more integers than squared integers. This is the paradox of infinity: an infinite set can be put into a one-to-one correspondence with a proper subset of itself. (Recall the “Infinite Hotel,” which demonstrates this idea.) Bolzano did the same with continuous numbers. He used the function y = 2x to “count” numbers on an interval and show that an interval twice as large as another could still be viewed as having the same “number of points” as the original interval.
In Halle, Cantor settled down to the mediocre life of an
academic in one of Germany’s lesser institutions of learning and research. While Berlin was teeming with ideas and great mathematicians with whom to discuss them, at Halle there was a dearth of intellectual stimulation. Nevertheless, he found a comfortable life there. In 1875, Cantor married Vally Guttmann, his sister’s friend, who came from a Jewish Berlin family. They began to raise their own family and, with the help of his father’s money, purchased a spacious house with large windows that let in much light. But mathematics is best pursued in a social, intellectual environment where ideas can be shared, discussed, learned, and refined. Because he was in a provincial town and was virtually the only gifted mathematician in his department, Cantor had to work in a vacuum—save for the powerful ideas he had brought with him from the courses he had taken from Weierstrass and others in Berlin.
Cantor, who appears in this undated portrait, suffered his first mental breakdown in 1884.
Weierstrass had further developed a mathematical result using ideas on infinity and irrational numbers pioneered by Bernhard Bolzano, which we know as the Bolzano-Weierstrass theorem. This powerful theorem states that an infinite sequence in a closed and bounded space attains a limit point inside the space itself. Thus, if you define an irrational number as a limit of rational numbers converging to it, then the number is as well defined as any other number, since it is in the original space (consisting of numbers) in which the sequence was defined. The theorem builds on ancient Greek ideas about numbers and is also rooted in Richard Dedekind’s definition of irrational numbers using cuts of rational numbers.
THE CONTINUUM HYPOTHESIS
Cantor came to the idea of actual infinity not by looking at numbers but rather by considering sets. Cantor started from Weierstrass’s idea of defining irrational numbers as limits of sequences of rational numbers. He then decided to look at sets of irrational numbers in intervals of numbers, and then looked for the set of all limit points of any given set of limit points in an interval. Each such set is a “derived set,” as it is derived from the original set of limit points. Thus he created a sequence of sets of limit points. He then asked himself, “When does the derived set of limit points become empty?” That is, he wanted to know if there ever was a case where the set of limit points derived in this way—as limit points of limit points—eventually becomes exhausted.
Cantor was fascinated by the process of deriving infinite sets of numbers from other sets of numbers. Since he used sets in his analysis, he became known not only as the person who explained infinity, but also as the founder of the modern theory of sets. Using the idea of counting, in which we establish a one-to-one correspondence between two sets of numbers—e.g., to count four sheep, we establish an unambiguous one-to-one correspondence that associates the number one with the first sheep, the number two with the second, and so on—Cantor invented an ingenious tool that allowed him to “count” all the rational numbers, even though that set is infinite.
The diagonal array below demonstrates Cantor’s “diagonalization proof” that the infinite number of all the rational numbers is exactly the same as that of the seemingly smaller set of all integers.
Note that the arrows weave through the infinite array, demonstrating how all the integers and the fractions can be listed. The arrows, extending from number to number as shown, produce a one-to-one correspondence between all possible fractions of whole numbers (including integers, such as 2/1 or 4/2, and other ways of representing them) and all the integers by pairing them, an integer and a fraction, all the way to infinity. What we do is order all the fractions (i.e., rational numbers), and count them: 1, 2, 3, … They comprise an infinite set, but they are counted one-to-one against the (infinite) totality of the integers. (The various representations of the same number, such as 2, 4/2, 8/4, etc., do not alter the argument.) So although there are infinitely many rational numbers and infinitely many integers, the number of numbers in each set is identical!
So are all infinite sets equal to each other in their number of elements? In 1874 Cantor proved that the answer to this question is no—there are infinite sets that are “more infinite,” or larger, than the set of integers and rational numbers. Cantor used another incredibly ingenious technique to prove this result: the age-old Greek idea of proof by contradiction. Cantor asked himself a simple question: Can I list all the real numbers as I did with the rational numbers? Lists can be infinite, as long as you can provide a rule that accounts for the progression from one number to the next. In the case of the rational numbers, Cantor had been able to come up with a rule for the list without actually listing every number. Attempting to list all the real numbers—meaning rational and irrational numbers, all together—Cantor began by simply trying to list all the real numbers between zero and one. If such a list existed, then, without any particular order, it would look something like this:
We assume that such a list is exhaustive—i.e., that it includes all the real numbers between zero and one. If this were the case, we would be able to associate each of the infinitely many real numbers with a unique integer, and thus “count” them (in principle; although in practice this would take “forever”). If we can prove that the real numbers between zero and one are indeed “countable,” as mathematicians say, then the argument could perhaps be extended to the set of all real numbers. Its order of infinity would then be the same as that of the integers and rational numbers.
But in fact, this is not the case. Cantor was able to show that, whatever the numbers on this list may be, we can always find a new number that is not on the list! For example, if we isolate the number that makes up the diagonal of all the randomly ordered numbers on the list above, meaning we take the first decimal from the first number on the list, the second decimal from the second number, the third from the third number, and so on, we get: 0.1653 … Now add one to each of these decimals (with the understanding that a nine changes to a zero). We now come up with a brand-new number: 0.2764 … This new number could not possibly be on the infinite list of numbers above because it differs from every number on the list by at least one decimal. Thus real numbers are uncountable, and their order of infinity is higher than the order of infinity of the integers and rational numbers.
Cantor gave a name to the “infinity” of the integers and the rational numbers: aleph-zero. However, he did not know how to classify the order of infinity of the real numbers. It was clearly higher than the order of infinity of the integers and rational numbers, but was there a level of infinity in between these two? Cantor hypothesized that the answer was no, and thought that the order of infinity of the real numbers was indeed the next order of infinity. He called the next-highest order of infinity—the one following that of the integers and rational numbers—aleph-one. Then he spent the rest of his life vainly trying to prove this statement—namely, that the continuum of real numbers is of the next level of infinity after that of the rational numbers. This statement has become known as the continuum hypothesis.
THE RELATIONSHIP BETWEEN Georg Cantor and Leopold Kronecker was one of the most complex and troubling in the history of science. Kronecker came from a wealthy Jewish family of bankers and businessmen, and he never had to worry about his income. Cantor had been Kronecker’s student at the University of Berlin, and at first Kronecker was supportive of his former student and enthusiastic about his first steps in mathematical research in Halle. However, as Cantor began to explore the deep recesses of infinity and discover highly counterintuitive, bizarre properties of infinite sets and numbers, Kronecker became irritated and went on the attack.
In the work of Georg Cantor, we find the bizarre contradictions that lie in the depths of the foundations of mathematics. These contradictions would ultimately hamper Cantor’s attempts to obtain a complete grasp on infinity. We will look at one familiar paradox, Russell’s paradox, which shows that there is no set containing all sets. Put simply, there is no universal “basket,” as it were, containing all baskets (sets) inside it.
Suppose that such a universal set do
es exist. Now consider the set of all sets that do not contain themselves. Does this set contain itself? If it does, then it doesn’t, and if it doesn’t, then it does. Thus what we have is a paradox and it implies that a universal set cannot exist. We can illustrate such impossibility in an easier way with the well-known Barber of Seville paradox. The Barber of Seville shaves all men in Seville who do not shave themselves. Does the barber shave himself? If he does (as the barber), then he doesn’t (as a man of Seville), and if he doesn’t (as the barber), then he does (as a man of Seville). Similar paradoxes to this simplified one plague the very foundations of mathematics.
We know that Kronecker had a philosophical distaste for infinity, irrational numbers, and the continuum, but was his virulence toward Cantor perhaps attributable in part to the fact that he was upset that a former student was making his way in mathematics in spite of him? We don’t know. Either way, Cantor felt the wrath of Kronecker and his Berlin cronies: “It is as if the Berlin mathematics department turns into darkest Africa, with lions roaring and hyenas screaming,” he once wrote a friend to describe the reaction to his name being mentioned at the department.
Weierstrass understood his genius, and Dedekind became his friend, but the disproportionately influential Kronecker continuously stood in his way, taunting Cantor with declarations such as: “God made the integers, all the rest is the work of man.” Irrational numbers (such as π or the square root of 2) are explained using infinite series, and Cantor was a master of these. Today every graduate student takes for granted that there are various levels of infinity—as Cantor has taught us—but to Kronecker this was sheer heresy.
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