diagonal procedure a method, originated by Cantor, for showing that there are infinite sets that cannot be put in one-to-one correspondence with the set of natural numbers (i.e., enumerated). For example, the method can be used to show that the set of real numbers x in the interval 0 ‹ x m 1 is not enumerable. Suppose x0, x1, x2, . . . were such an enumeration (x0 is the real correlated with 0; x1, the real correlated with 1; and so on). Then consider the list formed by replacing each real in the enumeration with the unique non-terminating decimal fraction representing it: (The first decimal fraction represents x0; the second, x1; and so on.) By diagonalization we select the decimal fraction shown by the arrows: and change each digit xnn, taking care to avoid a terminating decimal. This fraction is not on our list. For it differs from the first in the tenths place, from the second in the hundredths place, and from the third in the thousandths place, and so on. Thus the real it represents is not in the supposed enumeration. This contradicts the original assumption. The idea can be put more elegantly. Let f be any function such that, for each natural number n, f(n) is a set of natural numbers. Then there is a set S of natural numbers such that n 1 S S n 2 f(n). It is obvious that, for each n, f(n) & S. See also CANTOR, INFINITY, PHILOSOPHY OF MATHEMATIC. C.S.