2.3 Cantor’s Infinite Numbers 75 however, concluded in Section 21 that this phenomenon was indeed intrinsic to infinite sets (it in fact leads to one possible definition of infinite set) and a more elaborate quantitative comparison of infinite sets is not precluded. In fact, a good part of the book elaborates on the apparently paradoxical relationship of infinite sets with proper subsets, and shows that it must be accorded a central role in the whole theory. Just such a comparison of sizes of infinite sets was undertaken by Georg Cantor, with considerable mathematical rigor. Exercise 2.11: In Example 1 of x20, Bolzano shows that the set of real numbers y between 0 and 12 can be put in one-to-one correspondence with a proper subset of itself, namely the numbers x between 0 and 5, by means of the equation 5y = 12x: Verify that 25y = 12x2 and 5y = 12(5 ¡ x) will also give such one-to-one correspondences. Can you give additional examples? Exercise 2.12: Show that the points of a 1 inch by 1 inch square can be put into one-to-one correspondence with the points of a 2 inch by 2 inch square and with the points of a 2 inch by 1 inch rectangle. Exercise 2.13: Give additional examples of sets that may be put in one- to-one correspondence with proper subsets of themselves. 2.3 Cantor’s Infinite Numbers On the occasion of Georg Cantor’s confirmation, his father wrote a letter to him that Cantor would always remember. It foreshadowed much that was to happen to Cantor in later years. No one knows beforehand into what unbelievably difficult conditions and occupational circumstances he will fall by chance, against what unforeseen and unforeseeable calamities and difficulties he will have to fight in the various situations of life. How often the most promising individuals are defeated after a ten- uous, weak resistance in their first serious struggle following their entry into practical affairs. Their courage broken, they atrophy com- pletely thereafter, and even in the best case they will still be nothing more than a so-called ruined genius!... But they lacked that steady heart, upon which everything depends! Now, my dear son! Believe me, your sincerest, truest and most ex- perienced friend—this sure heart, which must live in us, is: a truly religious spirit.... 76 2. Set Theory: Taming the Infinite PHOTO 2.4. Cantor. But in order to prevent as well all those other hardships and difficul- ties which inevitably rise against us through jealousy and slander of open or secret enemies in our eager aspiration for success in the ac- tivity of our own specialty or business; in order to combat these with success one needs above all to acquire and to appropriate the great- est amount possible of the most basic, diverse technical knowledge and skills. Nowadays these are an absolute necessity if the industri- ous and ambitious man does not want to see himself pushed aside by his enemies and forced to stand in the second or third rank [36, pp. 274 ff.]. By the time Cantor encountered his own great “unforeseen calamities and difficulties” he had acquired that religious spirit and steady heart that his father believed to be so indispensable. Having been irresistably drawn to mathematics, Cantor excelled in it as a student, without neglecting diversity in learning, as his father had admonished him. In 1869, at age 22, three years after he completed his dissertation at the University of Berlin, he joined the University of Halle as a lecturer. Through his colleague Eduard Heine (1821–1881), Cantor became interested in analysis, and soon drew the attention of important 2.3 Cantor’s Infinite Numbers 77 researchers, most notably Leopold Kronecker (1823–1891), a professor at the University of Berlin and a very influential figure in German math- ematics, both mathematically as well as politically. Kronecker held very strong views about the admissibility of certain mathematical techniques and approaches. To him is attributed the statement “God Himself made the whole numbers—everything else is the work of men.” As a consequence, he disliked the style of analysis done by Weierstrass, involving a concept of the real numbers that was not based on whole numbers, and which was not constructive to a high degree. As soon as Cantor began to pursue his set-theoretic research, he lost the support of Kronecker, who quickly be- came one of his strongest critics. Cantor’s desire to obtain a position in the mathematically much more stimulating environment of Berlin never materialized, and Kronecker’s opposition had much to do with it. The series of six articles on infinite linear sets that Cantor published between 1878 and 1884 contained in essence the whole of his set-theoretic discoveries. His creative period was close to an end. Shortly thereafter, in the spring of 1884, he suffered his first mental breakdown. But by the fall he was back working on the Continuum Hypothesis. An important supporter of Cantor in those years was the Swedish mathematician G¨ostaMittag-Leffler (1846–1927). A student of Weierstrass, Mittag-Leffler became professor at the University of Stockholm in 1881. Shortly afterwards he founded the new mathematics journal Acta Mathematica, which quickly became very influential, with articles by the most eminent researchers. His relationship with Cantor was close, and Mittag-Leffler offered to publish Cantor’s pa- pers when other journals were rather reluctant to do so. In 1885, Cantor was about to publish two short articles in the Acta Mathematica contain- ing some new ideas on ordinal numbers. While the first article was being typeset, Cantor received the following letter from Mittag-Leffler, requesting that Cantor withdraw the article: I am convinced that the publication of your new work, before you have been able to explain new positive results, will greatly damage your reputation among mathematicians. I know very well that basi- cally this is all the same to you. But if your theory is once discredited in this way, it will be a long time before it will again command the attention of the mathematical world. It may well be that you and your theory will never be given the justice you deserve in your life- time. Then the theory will be rediscovered in a hundred years or so by someone else, and then it will subsequently be found out that you already had it all. Then, at least, you will be given justice. But in this way [by publishing the article], you will exercise no significant influence, which you naturally desire as does everyone who carries out scientific research [36, p. 138]. The effect on Cantor was devastating. Disillusioned and without friends in the mathematical community, he decided by the end of 1885 to abandon 78 2. Set Theory: Taming the Infinite mathematics. Instead, he turned to philosophy, theology, and history. To his surprise, he found more support for his work among theologians than he had ever received from mathematicians. Soon, however, he was back at work on an extension and generalization of his earlier work. As his last major set-theoretic work, he published a grand summary of his set theory between 1895 and 1897, his Beitr¨agezur Begr¨undungder Transfiniten Mengenlehre (Contributions to the Founding of Transfinite Set Theory) [28, 282–351]. The following excerpt from this work concerns the arithmetic of cardinal numbers. The English translation is taken from [26], pp. 85–97, 103–108. In the wake of this publication, his work was finally recognized for its full significance, and while controversy continued to surround set theory, Cantor was not alone anymore to defend it. Set theory was here to stay. His mathematical career had reached its zenith, and he was able to see that he had indeed created something of lasting significance. Georg Cantor, from Contributions to the Founding of the Theory of Transfinite Numbers x1 The Conception of Power or Cardinal Number By an “aggregate”4 we are to understand any collection into a whole M of definite and separate objects m of our intuition or our thought. These objects are called the “elements” of M. In signs we express this thus: (1) M = fmg: We denote the uniting of many aggregates M; N; P; : : : ; which have no common elements, into a single aggregate by (2) (M; N; P; : : : ): The elements of this aggregate are, therefore, the elements of M, of N, or P; : : : , taken together. We will call by the name “part” or “partial aggregate” of an aggregate M any other aggregate M1 whose elements are also elements of M. If M2 is a part of M1 and M1 is a part of M, then M2 is part of M. Every aggregate M has a definite “power,” which we will also call its “cardinal number.” We will call by the name “power” or “cardinal number” of M the general concept which, by means of our active faculty of thought, arises from the 4The modern terminology is “set.” 2.3 Cantor’s Infinite Numbers 79 aggregate M when we make abstraction of the nature of its various elements m and of the order in which they are given. We denote the result of this double act of abstraction, the cardinal number or power of M, by (3) M: Since every single element m, if we abstract from its nature, becomes a “unit,” the cardinal number M is a definite aggregate composed of units, and this number has existence in our mind as an intellectual image or projection of the given aggregate M. We say that two aggregates M and N are “equivalent,” in signs (4) M » N or N » M; if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other.