(1845–1918) German mathematician
The son of a prosperous merchant of St. Petersburg, at that time the capital of Russia, Cantor was educated at the University of Berlin where he completed his PhD in 1868. In 1870 he joined the faculty of the University of Halle and was appointed professor of mathematics in 1879. He spent his entire career at Halle, although it was a career repeatedly interrupted after 1884 by mental illness; he was a manic depressive and was hospitalized first in 1899 and several times thereafter. After 1897 he made no further contribution to mathematics and died of heart failure in 1918 in a mental institution.
Although Cantor's earliest work was concerned with Fourier series, his reputation rests upon his contribution to transfinite set theory. He began with the definition of infinite sets proposed by Dedekind in 1872: a set is infinite when it is similar to a proper part of itself.
In 1874, Cantor published a remarkable paper in Crelle's Journal. Here he first showed that the rational numbers (numbers that can be expressed by dividing one integer by another: 1/2, 1/3, 1/4, 2/3, etc.) are denumerable – they can be put in a one-to-one correspondence with the natural numbers (1, 2, 3, etc.). The usual method of demonstrating this is to set up an array in which the first line contains all the rationals in which the denominator is 1 (1/1, 2/1, 3/1, etc.), the second line has all the rationals with a denominator 2 (1/2, 2/2, 3/2, etc.), and so on. It is then possible to ‘count’ all the fractions in the array by moving diagonally backwards and forwards through the array, and it is clear that every rational number can be put in one-to-one correspondence with an integer. This technique, known as Cantor's diagonal procedure, is not the one Cantor used in the 1874 paper, although he did give the diagonal demonstration later. The set of rational numbers and the set of natural numbers are said to have the same ‘power’.
Cantor then went on to show that the set of all real numbers is not denumerable. He did this by a reductio ad absurdum method. First he assumed that all the real numbers between 0 and 1 are denumerable and expressed as infinite decimal fractions (e.g. ⅓ = 0.333 …). They are arranged in denumerable order:
a 1 = 0. a1 1 a1 2 a1 3 a1 4....
a 2 = 0. a2 1 a2 2 a2 3 a2 4.... etc.
Here, a 1 is the first real number and a 1 1 the first digit, a 1 2 the second, etc. The first digit of the second number a 2 is a 2 1, etc. Cantor then showed that it is possible to construct an infinite decimal that is not in the above set by taking the diagonal containing a 1 1, a 2 2, a 3 3, etc., and changing the digit to 9 if it is 1 and changing the digit to 1 for all other digits. This gives a number that is a real number between 0 and 1, yet is not in the above set. In other words, the real numbers are not denumerable – there is a sense in which there are ‘more’ real numbers than rational numbers or natural numbers. The set of real numbers has a higher power than the set of natural numbers.
Cantor designated the set of natural numbers, the smallest transfinite set, with the symbol ℵ0 and the set of real numbers by the letter c, the number of the continuum. ℵ is the first letter of the Hebrew alphabet, called ‘aleph’. Cantor's symbol ℵ0 is referred to as ‘aleph nul’.
Cantor went on to show that there were in fact an infinite number of transfinite sets. The power set of a set S consists of the subsets of S. Thus let S = (1,2), the power set of S, P(S) = [(1), (2), (1,2), (Ø)], bearing in mind that every set is a member of itself, and that the empty set (Ø) belongs to every set. In general Cantor demonstrated that if a set S has n members then P(S) will have 2n members, that P(S) ≥ S, a result since known as Cantor's theorem.
The theorem applies to all sets, infinite as well as finite. Thus the power set of ℵ0 will be greater than ℵ0 and the process can be continued with the power set of the power set of ℵ0, and so on indefinitely. Cantor had shown that the set of natural numbers had a cardinality of ℵ0, and the real numbers, c, had a cardinality of 2ℵ 0. This enabled him to pose in 1897 the hypothesis that 2ℵ 0 = ℵ1, or, that the continuum (c) is the next highest infinite number after ℵ0. Cantor made little progress with the continuum problem. It remained for Godel and Cohen to illuminate the issue many years later.