In mathematics, a set can be thought of as a well defined collection of objects. The "objects" with which we will be concerned are abstract entities such as numbers or other sets. Until late in the nineteenth century, there were few references to sets in the mathematical literature, but this was to change dramatically due to the work of one brilliant German mathematician, Georg Cantor (1845-1918). The official birth of set theory can be traced to an 1874 article by Cantor in Crelle's Journal, one of the most influential mathematics journals of its time. In this article, Cantor suggested that there were different kinds of infinite sets and different orders of infinity. This idea stirred up tremendous controversy among some editors of the journal and among mathematicians in general. Prior to this publication, all infinite sets had been considered alike. No one had ever suggested that there were different orders of infinity. During the period from 1874 through 1897, Cantor continued to develop his set theory, stirring up the mathematical community as it had seldom been stirred before.
If Cantor had restricted his writings to finite sets, nobody would have taken much notice. After all, finite sets, even very large ones, are not mysterious. If you want to know how many elements are in a finite set, you just count them. It might be tedious for very large sets, but, in principle, you can finish the job and report the results of your counting. Determining which of two finite sets is larger is no problem; just count the number of elements in each and declare that the one with the most elements is larger. If they have the same number of elements then they are the same size. Just to make this a little more precise, Cantor introduced the idea of a one-to-one correspondence between the elements of two finite sets: If you can pair each element of set A with exactly one element in set B and vice-versa, then, of course, the sets must have the same number of elements and this pairing of elements is called a one-to-one correspondence. No one objected to this. If every element of a finite set A is also a member of finite set B, then, says Cantor, A is a subset of B. There was no controversy about that. Under this definition, every set is a subset of itself. That was okay with everyone. If a set has no members, then call it the empty set or the "null" set, if you prefer. That was fine. Cantor went on to define the intersection of two sets: The intersection of sets A and B is the set of all elements which are in both set A and set B. He defined union: The union of sets A and B is the set of all elements that are in A or in B or in both. No one had any objections to those definitions. He defined the notion of cardinality for finite sets: The cardinality of a finite set was equal to the number of elements in the set. This ruffled no feathers. None of Cantor's newly minted theory gave anyone pause so long as it applied only to finite sets. Unfortunately, mathematics deals with both the finite and the infinite; and Cantor knew that any set theory that failed to account for infinite sets was virtually useless as a foundation for mathematics. Thus he extended his theory to include the infinite, and the revolution began.
Cantor wanted his theory of finite sets to be logically consistent with whatever he had to say about infinite sets. So, for example, he brought his one-to-one correspondence notion into the world of infinite sets and strange things began to happen. In the finite world, if a set A is a subset of a set B, and if there are elements in B that are not in A; then set B has more elements than set A, and a one-to-one correspondence cannot be set up between the elements of A and the elements of B. In such cases, set A is said to be a "proper" subset of B. Another way of saying this, in Cantor's new language, is to say that the cardinality of set B is greater than the cardinality of set A.
Translate this to the infinite world. The set of natural numbers, {1,2,3,...} is infinite. So is the set of all even natural numbers, {2,4,6,...}, which is clearly a subset of the natural numbers by Cantor's definition. There are obviously elements in {1,2,3...} that are not in {2,4,6,...}; for example 1,3,5, and so on. Therefore, {2,4,6,...} is a proper subset of {1,2,3,...}. So should we not say that the set of natural numbers has a cardinality that is larger than the cardinality of the set of even natural numbers? But wait! It is quite simple to establish a one-to-one correspondence between the natural numbers and the even natural numbers,.e.g., 1 in the natural numbers corresponds to 2 in the even natural numbers, 2 in the naturals corresponds to 4 in the evens, 3 natural to 6 even, and so on. In general, each natural number n corresponds to the even number 2n. Therefore, Cantor says that, since a one-to-one correspondence can be set up between {1,2,3,...} and {2,4,6,...}, these two sets must have the same cardinality. Cantor decided to call this cardinality 0, or "aleph null". ("aleph") is the first letter in the Hebrew alphabet. Thus any set that can be placed in one-to-one correspondence with the set of natural numbers has cardinality 0. Cantor called such sets "denumerably infinite" or "countably infinite." But Cantor was not finished. Next he produced a set, quite familiar to every mathematician, which was infinite but could not be placed in one-to-one correspondence with the set of natural numbers. That set was the interval of real numbers [0,1]. In fact, any interval of real numbers will do, including the entire real number line or "continuum" as Cantor called it. Cantor produced an ingenious proof of this by assuming that a one-to-one correspondence between the natural numbers and the interval [0,1] exists and then constructing a number in that interval which had been left out of the alleged one-to-one correspondence. This proof can be seen in the article "Non-denumerably Infinite Sets." This was the shot heard round the mathematical world. Cantor had exhibited an infinite set that had a greater cardinality than the natural numbers. All infinite sets were not alike. All "infinities" are not the same. Some infinities are greater than other infinities. Cantor went on to show that there are infinities larger than the infinity of real numbers. In fact he showed that there is an infinity of infinities having cardinalities which he called 0, 1, 2, ... . One very influential mathematician, Leopold Kronecker (1823-1891), believed that any such study of infinite sets was absurd because nothing meaningful, in the mathematical sense, could be said about infinity. Kronecker thought that the study of infinity was outside the discipline of mathematics. Because of Kronecker's standing in the mathematical community, Cantor was stung by his criticism. For a time, Cantor even began to doubt himself and the importance of his work. Kronecker died in 1891 and the furor created by his criticism of Cantor's program gradually faded. In 1897, Cantor published the second of a two-volume treatise bringing together all of his ideas about sets and infinity. This treatise looks very much like a present-day textbook on set theory. In fact, it became a cornerstone in the foundation of twentieth century mathematics. Also in 1897, Cantor was honored for his work on the theory of sets at the first International Conference of Mathematicians held in Zurich, Switzerland. With the creation of set theory, Cantor, almost single-handedly, brought forth a revolution in the foundations of mathematics, the impact of which still shapes the structure of the subject at the beginning of the twenty-first century. Today, Cantor's set theory is universally accepted as one of the greatest achievements in the history of mathematics.
This is the complete article, containing 1,341 words
(approx. 4 pages at 300 words per page).
