infinity

infinity, in mathematics, that which is not finite; it is often indicated by the symbol . A sequence of numbers, a1, a2, a3, … , is said to approach infinity if the numbers eventually become arbitrarily large, i.e., are larger than some number, N, that may be chosen at will to be a million, a billion, or any other large number (see limit). The term infinity is used in a somewhat different sense to refer to a collection of objects that does not contain a finite number of objects. For example, there are infinitely many points on a line, and Euclid demonstrated that there are infinitely many prime numbers. The German mathematician Georg Cantor showed that there are different orders of infinity, the infinity of points on a line being of a greater order than that of prime numbers (see transfinite number). In geometry one may define a point at infinity, or ideal point, as the point of intersection of two parallel lines, and similarly the line at infinity is the locus of all such points; if homogeneous coordinates (x1, x2, x3) are used, the line at infinity is the locus of all points (x1, x2, 0), where x1 and x2 are not both zero. (Homogeneous coordinates are related to Cartesian coordinates by x=x1/x3 and y=x2/x3.)

See A. D. Aczel, The Mystery of the Aleph (2000); D. F. Wallace, Everything and More (2003).