The Continuum Hypothesis

Cesare Brazza
History of Mathematics
Rutgers, Spring 2000

"Infinity is up on trial..." (Bob Dylan, Visions of Johanna, cited in In the Light of Logic), pg. 28. These five words suffice to summarize the essence of Cantor's work. Cantor was tormented by opposition throughout his career. After conceiving and then proving his theorems on infinite sets, Cantor struggled against the negative reactions of his peers. It was not until the end of his lifetime that Cantor received the recognition he deserved. Cantor, a devout Christian, always held to his beliefs because to him, they came directly from G-d. "Where G-d was concerned, it was impossible to entertain hypotheses. There were no alternatives to be considered" (p. 238, Georg Cantor). Georg Ferdinand Ludwig Philipp Cantor contributed greatly not only to discrete mathematics, but to every science based in mathematics. "Whatever the disappointments Cantor was to suffer, his transfinite set theory represented a revolution in the history of mathematics. Not a revolution in the sense of returning to ear lier starting points, but more a revolution in the sense of overthrowing older, established prejudices against the infinite in any actual, completed form." (Pg. 118, Georg Cantor). With his theory of sets and his introduction of the concept of infinite nu mbers, Cantor broke through the barriers of previous generations, and has allowed for the further exploration of areas that were previously unattainable.

Georg Cantor was born on March 3, 1845 in St. Petersburg, Russia. He was pushed toward the study of engineering, which his father considered to be a profitable field. However, Georg was always more interested in mathematics. His father, whose approval was very important to Georg, was eventually persuaded to let him study mathematics. "In 1862 he (Georg) had written to his father (who had just consented to his son's pursuing a career in mathematics) in order to explain that 'My soul, my entire being lives in my calling...' ". (p. 239, Georg Cantor). Cantor first attended the University of Zurich before transferring to the University of Berlin, where he received his doctorate. However, the area of mathematics which he tackled, was not immediately accept ed. Curiously, his theories were also used by the Jesuits to "prove" the existence of G-d. "... Cantor's transfinite numbers were to prove no less revolutionary for philosophers and theologians who were concerned with the problem of infinity." (p. 118, Georg Cantor). However, Cantor, who was also a theologian, did not associate himself with any of these proofs. Cantor did, however, consider himself to be an intermediary through whom G-d could communicate "these great, immutable truths" of mathemat ics. "Cantor saw his own role, as mathematician, in terms of a faithful secretary, receiving and describing what had been revealed to him by G-d." (p. 238-239, Georg Cantor). In fact, much of Cantor's beliefs in his work stemmed directly from his beliefs that the Continuum Hypothesis was derived directly from nature. "The principles of mathematics, of set theory, and the transfinite numbers, followed directly from Nature." (p. 238, Georg Cantor). Gauss had previously made a statement that likened the id ea of infinity to a good concept, but not a mathematical value. Because of this statement, many other mathematicians adopted Gauss' notion. One such person was Kronecker, a professor at the University of Berlin, who used his place and prestige to continually discredit Cantor and his theory. Due to this constant frustration and ridicule, Cantor suffered many nervous breakdowns. It was not until much later that he received the recognition he deserved from the London Mathematical Society. However, by this ti me, his nervous breakdowns had gotten the best of him, and he passed away in a mental institution on January 6, 1918.

Cantor's theories are based entirely on the concept of a set. Cantor developed relationships between sets, which would confirm his theory of the infinite. Before looking at the various ways in which Cantor combined and utilized sets, we must first state a few of Cantor's definitions. If two sets, S1 and S2 have elements, which can be placed in a one-to-one correspondence with each other, they were said to be "equinumerous". If a set, S, is equinumerous with some initial segment of the set of natural numbers, then the set is said to be "finite". If a set, S, is finite, or is equinumerous with the set of natural numbers, then it is said to be "denumerable". A number is said to be "transcendental" if it is in the set left after taking away the set of all algebraic numbers from the set of all real numbers. Another way of describing transcendental numbers is as numbers that can never be a solution of an algebraic equation.

From these definitions follow several fundamental results. First, a denumerable union of denumerable sets is denumerable. Second, the set of rational numbers is denumerable. Third, the set of all algebraic numbers is denumerable. Fourth, the set of all real numbe rs is non-denumerable. This fourth statement is the one on which Cantor concentrated most. It says that there is no way to list the set of all real numbers as a "simple sequence of real numbers {x0,x1,x2,...}" (p. 33, In the Light of Logic). Cantor eventually devised a simple proof for this, which made use of his famous diagonal argument. Cantor's diagonal argument shows that the real numbers can not be enumerated. Given any enumeration of the set of real numbers between 0 and 1, it is possible to construct a number that is not in that enumeration. First, we write each member of the subset as an infinite decimal:

Now we can extract a new number composed of the digits on diagonals line, for example the main diagonal, 0.485. We can then alter each of these numbers so that the new number is not equal to the next number on the diagonal line, for example, 0.485 can now be 0.596. With care, it can be shown that this new number is not in the enumeration of the subset of real numbers that we originally considered.

The size of the sets involved clearly plays a major role here. We shall now call the number of elements in a set its "cardinal number". We shall also use the abbreviation "card(S)" to stand for "the cardinal number of the set S". For example, card ({1,2,3}) is 3. To identify the cardinal numbers of infinite sets, Cantor used the Hebrew letter aleph (aleph) with subscripts to represent various infinite cardinal numbers. In this system, the cardinal number for the set of natural numbers is said to be aleph0. However, the cardinal number for the set of real numbers was called c by Cantor. To describe the cardianl number c of the real numbers, we may look at the relationship of the real numbers to the set of rational numbers. For each element in the set of real numbers , there is a set of rational numbers which correspond to it ."One way is to associate with x the subset Qx of Q consisting of all rational r with rIn the Light of Logic). We can then infer that the cardinal number for the set of all real numbers is less than or equal to the cardinal number of the subsets of the set of all rational numbers. The set of all subsets of a set is equinumerous with {0,1} to the power of the set, S. Therefore, the cardinal number of the set of all subsets is said to be 2 to the power of the cardinal number of the set. The cardinal number which Cantor was concerned with, was the cardinal number for the set of subsets of the rational numbers, which could now be considered as 2 aleph0.

So then, c is now less than or equal to 2 aleph0. It is obvious from this that aleph0 is less than 2 aleph0, so we now have aleph0 and 2aleph0 as two transfinite numbers. However, the question of whether or not these were the only two infinite cardinal numbers was now raised. This, in turn, led to the question of there being any cardinal numbers between aleph0 and 2aleph0. We have now come to the core statement of the Continuum Hypothesis. The Continuum Hypothesis states that c is the first cardinal larger than aleph0. But before one can state that, one must first show that there exists such a cardinal number. For this, Cantor created what is known as the Well-Ordering Principle, or WO. This stated that, if for every n onempty subset of the set S, there is a first element, then the set S is said to be well-ordered by a relation < of ordering between its elements. Cantor devoted much of his time and energy to proving that there is not, in fact, any cardinal number betwee n aleph0 and 2aleph0. The Cantorian theory of transfinite cardinal numbers was highly ridiculed. One of the chief critics was actually a former teacher of Cantor's, Leopold Kronecker. This negative publicity was followed by further criticism when paradoxes began to appear in the theory of sets by merely following its ideas to what was believed to be their logical conclusion. However, Cantor did not doubt that his theory was right ,and persevered. "G-d was the source of his inspiration and the ultimate guarantor of the necessary truth of his research. Therefore it could not fail to be absolutely correct, and time would eventually vindicate everything he had done." (p. 239, Georg Cantor). A mathematician named Ernst Zermelo then introduced a way to limit the sizes of sets so as to avoid the paradoxes while still providing an adequate basis for much of Cantor's theory of higher infinities. Zermelo called it the axiomatic theory of sets. As for necessity, Cantor's theory of sets has also been ridiculed. Many have argued that "the necessary use of higher set theory in the mathematics of the finite has yet to be established"(p. 30, In the Light of Logic). They believe that "the actual infinite is not required for the mathematics of the physical world"(p. 30, In the Light of Logic). Cantor, on the other hand, gave an elegant metaphor in a footnote which shows his stance on why the actual infinite is necessary. "Apart from the journey which strives to be carried out in the imagination... or in dreams, I say that a solid ground and base as well as a smooth path are absolutely necessary for secure traveling or wandering, a path which never breaks off, but one whi ch must be and remain passable wherever the journey leads". ---Georg Cantor

There were others, such as David Hilbert, Ernst Zermelo and Leibniz, that would go on to show "that Cantor's higher infinities are in fact necessary for mathematics, even for its most finitary parts" (p. 30, In the Light of Logic). "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 the re is no part of matter which is not --- I do not say divisible --- but actually divisible; and consequently the least particle ought to be considered as a world full of an infinity of different creatures". ---Leibniz

In 1900, to celebrate the onset of a new century, David Hilbert gave a famous lecture in Paris entitled "Mathematische Probleme" (Mathematical Problems), in which he presented 23 unsolved mathematical problems. Of these 23 problems, the one that he put first was Cantor's problem of the cardinal number of the continuum. This prominence showed the importance of Cantor's ideas in Hilbert's view, and thus caught the attention of other mathematicians, notably Ernst Zermelo: "Zermelo had initially worked on topics in analysis and mathematical physics, but in 1899 he became a Privatdozent at Gottingen, where he came under Hilbert's influence and began pursuing set theory" (p. 39, In the Light of Logic). In 1904, Zermelo went on to further support Cantor by proving his Well Ordering Theorem. Zermelo introduced what he called the axiom of choice, or AC. This states that there is a function f on any nonempty set S, which provides for the simultaneous choice from each nonempty subset X of S of a distinguished element of X. By using his axiom of choice to prove Cantor's Well Ordering Theorem, Zermelo was able to get past some of the hurdles that Cantor had had difficulty with. "My main slogan here is that nevertheless, for mathematics, a little bit goes a long way" (p. 124, In the Light of Logic).

Cantor was not the only one interested in the theory of the infinite. Dedekind was another mathematician whose thoughts were similar to Cantor's. He and Cantor had shared their ideas and their discoveries. Richard Dedekind was a teacher at the Polytechni c School in Zurich. It was there that he made several innovations of his own, including his discovery of the "Dedekind cut" and a declaration that he had revealed a "true definition of continuity" (p. 48. Georg Cantor). From this, we see that both Cantor and Dedekind were both attacking the same concept. Although Dedekind did not publish his work for some time, he was eventually persuaded to do so. After reading his papers, Cantor developed a greater interest in the theory of continuity.

Cantor had pushed his way forward, in spite of criticism and his own failing health, to contribute an indispensable and absolutely necessary part of modern day mathematics. Cantor's work contributed greatly towards new studies into the world of the inf inite; much like the way that Charles Hermite and Ferdinand Von Lindemann contributed indispensable mathematical knowledge with their proofs that e and pi are transcendental numbers. "There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our mind, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation." (Charles Hermite, The Mathematical Intelligencer, v. 5, no. 4.) T hroughout his tough career at the University of Berlin, Cantor maintained a steady outlook on his adversaries. "All so-called proofs against the possibility of actually infinite numbers are faulty, as can be demonstrated in every particular case, and as can be concluded on general grounds as well. ...From the outset they expect or even impose all the properties of finite numbers upon the numbers in question, while on the other hand the infinite numbers, if they are to be considere in any form at all, must (in their contrast to the finite numbers) constitute an entirely new kind of number, whose nature is entirely dependent upon the nature of things and is an object of research, but not of our arbitrariness or prejudices". --- Georg Cantor

The Continuum Hypothesis arises in the context of an inevitable evolutionary advance in mathematics. Cantor himself likens his theories to nothing less than divine comprehension of the world around us. " Cantor... was bringing a new understanding, hitherto hidden from mathematicians, into the light." (p. 239, Georg Cantor). The introduction of the concepts that Cantor produced can be compared to the introduction of the number 0 in the ancient Babylonian mathematical system. Originally, the Babylonians did not have a concept for the number 0. During the Seleucid period, however, the Babylonians had a use for such a number, and therefore found a way to, not only discover it, but incorporate it into their mathematical knowledge. In the same way, Cantor saw a use for a new and improved comprehension of the infinite, and therefore went out to discover it. Georg Cantor was continually ridiculed for his stance on infinity, and yet managed to see the need for it . We often take for granted the complex systems we have at our disposal to use in order to find the answers to seemingly easy problems. However, without the contributions of people like Georg Cantor, we would all still be struggling with abstract conc epts of nearly incomprehensible materials. At the very least, our mathematical advances would have been postponed until someone with the same ingenuity as Cantor came alone to give them impetus. "Analysis takes back with one hand what it gives with the ot her. I recoil in fear and loathing from that deplorable evil: continuous functions with no derivatives" (Charles Hermite, Quoted in Comic Sections). With his discovery of transfinite numbers, Cantor inaugurated the development of a new branch of mathematics, which has found application to classical problems.

Bibliography

  1. Joseph Dauben, Georg Cantor, Harvard University Press, Cambridge, Massachusetts and London England, 1979. 404 pp.
  2. Georg Cantor, Transfinite Numbers, The Open Court Publishing Company, Chicago and London, 1915. 211 pp.
  3. Paul Cohen, Set Theory and the Continuum Hypothesis, W.A. Benjamin, Inc., New York, New York, 1966. 154 pp.
    5. Solomon Feferman, In the Light of Logic, Oxford University Press, New York, New York, 1998. 340 pp.
  5. D. MacHale, Comic Sections, Dublin 1993.