INFINITY, a notion which has assumed the most varied forms, characterized by few common characteristics except the transcendence of our customary notions of boundedness and limit. The chief types of infinitude which come to the attention of the mathematician and philosopher are cardinal infinitude, ordinal in finitude, the infinity of measurement, the co of algebra, the infinite regions of geometry and the infinite of metaphysics.
Cardinal infinitude pertains to assemblages and only to assemblages. An assemblage is finite if it can be exhausted by taking member after member from it. It is what is called re flexive if it can be paired off in a one-one manner with a class derived from it by the removal of one term. Whether all non-finite classes are reflexive is not known, as all sup posed demonstrations of this statement rest in the insecure basis of Zermdo's axiom. (See ASSEMBLAGES, GENERAL THEORY or). The set of all classes which can be paired off in a one-one manner with a given class or assem blage is known as a cardinal number. The cardinal numbers of reflective classes are known as transfinite cardinal numbers. Reflec tive classes are perfectly definite and bounded, though their manner of limitation is not sus ceptible of being based on an enumeration, as is the case with finite classes.
Ordinal infinitude is a property of well-or dered series. A series is an order which estab lishes a definite and unique precedence between any two terms which it concerns, and which is such that if it makes a precede b and b precede c. it makes a precede c. A series is well or dered if every series generated by an operation included in its ordering operation has a first term. A well ordered series is infinite if it contains a cardinally infinite number of terms. Two well ordered series are ordinarily similar if their terms can be put in a one-one corre spondence in such a manner that corresponding terms always bear the same relation of pre cedence or succession to corresponding terms. An ordinal number is the set of all well ordered series similar to a is series. It is trans finite or infinite if it s the number of an infinite series.
Infinitude enters into a system of measure ment in two different ways: as infinitude of subdivision and as infinitude of distance. A
scale of measurement is a system of entities such that every pair has a corresponding real number, which is regarded as the distance separating the members of the i pair. The in finite divisibility of a distance is the fact that the distance can be regarded as the sum of a chain of distances stretching from one point •to the other, and of as small a magnitude as may be desired. This is preconditioned by the fact that between every two real numbers there lies another real number. The infinite extent of a system of measurement consists in the fact that starting from any point it is'possible to find another point removed from it by a dis tance of arbitrarily great magnitude.
Strictly speaking, the symbol ce has no meaning except in some such context as f (m). f (n) is an abbreviation for the limit of f (x) as i x 1 increases without limit, and is only significant when f (x) — f (x') can be made arbitrarily small if. I .r 1 and I x' I are only chosen larger than some number A. f (co) may also be written lim f (x). f (-F cc) or lint f (x) means the limit of f (a + F ib) where a and b are real and a increases through positive values without limit. f (— cc) and lint f (x) have the analogous meaning. f :=3.-00 (+ co) and f (— CO) may exist when f (co) is meaningless.
In ordinary Euclidean geometry parallel pairs of lines and intersecting pairs of lines exhibit many important analogies. The con venient expression of these is rendered pos sible only by the adjunction to the ordinary points, lines and planes of certain entities, called points, lines and planes at infinity. The method of this adjunction differs in the several mathe matical disciplines, for it is .dictated by con siderations rather of convenience than of logi cal necessity. In solid projective geometry (q.v.) the adjoined points form a plane cutting every ordinary line or space in one point. This enables the universal law to be formulated, that every two lines intersect in one point, and only in one point. In the theory of the functions of a complex variable (q.v.), on the other hand, it is much more convenient to regard the en tire infinite region as a single paint, lying on every line.