Set Theory and the Sizes of Infinity

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Set Theory and the Sizes of Infinity


Set theory, and its transformation of mathematician's ideas of infinity, was mainly the work of one man, the nineteenth-century German mathematician Georg Cantor (1845-1918). Cantor found ways to work with infinite sets, which many believed could not exist. He further alarmed his contemporaries by demonstrating that while all infinite sets are indefinitely large, some are nonetheless larger than others.


One of the earliest philosophers of infinity was Zeno of Elea (495-435 b.c.). His ponderings led him to paradoxes such as one in which Achilles running to overtake a crawling tortoise could never accomplish the feat. First, you see, he must reach the place where the tortoise started, and by that time the tortoise is no longer there. In fact, Zeno "proved" that the entire idea of motion was absurd. Finally, the local authorities lost patience with him and had him executed for treason.

Yet Zeno had been grasping at some ideas that were very profound indeed—but without the mathematical language to handle them. He realized that to frame problems of space and time accurately, they should be broken up into an infinite number of points or instants. Being able to do so awaited understanding of how to work with such infinitesimals using the tools of calculus.

The fourteenth-century scholar Albert of Saxony (c. 1316-1390) was among the first to put forth the idea of comparing infinite quantities. He proved that a beam of infinite length has the same volume as space itself. He did this by conducting a thought experiment in which he sawed the beam into an infinite number of imaginary pieces to build an infinite number of concentric shells in space.

Mathematicians generally regarded infinity as a way of speaking about a limit that could never be reached. They balked at the idea of considering it in more concrete terms, such as an infinitely large group of items. Carl Friedrich Gauss (1777-1855) wrote of his "horror of the actual infinite." Bernhard Bolzano (1781-1848), an Austrian priest, defended the concept of an infinite set in 1847. However, it was the work of Georg Cantor in the 1870s that put the theory of sets, both finite and infinite, in formal mathematical terms.

A set is a grouping of numbers, objects, or ideas of any kind, grouped so that it can be considered as a single entity. For example, a class may be described as a set of students. An individual item within a set (an individual student in our example) is a called an element of the set.

The union of two sets contains all the elements of both sets. If A is the set of all American League baseball teams, and N is the set of all National League baseball teams, then the union of A and N is the set of all major league baseball teams. The intersection of two sets is the set of elements that appear in both sets. Since no team belongs to both the American League and the National League, here the intersection of A and N is a set with no elements, called the null set.

Two sets A and B are equal if all the elements of A are in B, and vice versa. Suppose a clothing store sells jeans. The set of all the jeans it has for sale is called J. The set of blue jeans it sells is called B. If the store only sells blue jeans, then B = J. However, if the store sells several colors, the set B of blue jeans is a subset of the set J of jeans. A proper subset is one that is not the same as the original set, which technically may be considered a subset of itself.

For finite sets, a proper subset is smaller than the original set. That is, suppose we have a set A of integers between 1 and 10 inclusive. Sets are often written in brackets, so here A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Now let's define a set B consisting of only the even numbers between 1 and 10 inclusive. So B = {2, 4, 6, 8, 10}, and B is a subset of A.

In this example, we can see that B is a smaller set than A. If we try to put the elements of A and B into a one-to-one correspondence, we can't do it. We can start out, matching up the 1 from A with the 2 from B, then the 2 from A with the 4 from B, and so on. But clearly we are going to run out of elements in B before we get through with A.

How about one-to-one correspondences for infinite sets? It is easy to see intuitively, for example, that the set of odd integers and the set of even integers can be paired off as far out as you care to count. These infinite sets are the same "size"; in set theory, we say that they have the same cardinality. For finite sets, the cardinality is the number of elements. One-to-one correspondences are used to determine the cardinality of infinite sets.

Bolzano first gave examples of infinite sets that, unlike finite sets, could be paired in a oneto-one correspondence with one of their own proper subsets. For example, the set of all integers can be put into a one-to-one correspondence with the set of odd integers. Since both sets are infinite, we don't have to worry about running out, like we did when we limited the set to integers between 1 and 10. The key issue is that the integers are countable. Once we've matched up a pair and moved on, we don't have to worry about finding more elements between those we've already counted.

Bolzano's examples lent credence to the popular idea that all infinities were the same size. Cantor then came along to prove that this was not the case. He called the cardinality of such countable infinite sets as we have described ℵ0. Then he gave examples of infinite sets that are not countable in the same way.

Consider the real numbers; that is, not only the integers but everything in between, including irrational numbers with decimals that go on forever. You can start matching these up with the set of integers, but at any time you could go back and discover additional irrational numbers between those you'd already counted. The real numbers are uncountable, infinitely more numerous than integers. Therefore, the cardinality of the set of real numbers is greater than that of integers. Some infinities, in other words, are bigger than others; their cardinalities are denoted ℵ1, ℵ2, etc.


Cantor's view of infinites as having an actual existence quickly became controversial. In particular, he was opposed by the influential mathematician Leopold Kronecker (1823-1891), who believed that "God made the integers, and all the rest is the work of man." For the rest of Kronecker's life he bedeviled Cantor, getting his papers rejected from journals and blocking his appointment to teach at the University of Berlin.

Around the turn of the century, mathematicians including Bertrand Russell (1872-1970) discovered a few disturbing paradoxes in Cantor's formulation. Yet at the same time set theory was becoming recognized as fundamental to topology and the analysis of functions. In addition, basic set theory has become a standard part of the modern elementary curriculum introduced in the 1960s as the "new mathematics."


Further Reading

Dauben, Joseph Warren. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1990.

Ferreirós, José. Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Boston: Birkhäuser Verlag, 1999.

Johnson, Phillip E. A History of Set Theory. Boston: Prindle, Weber & Schmidt, 1972.

Zippen, Leo. Uses of Infinity. New York: Random House, 1962.