INFINITY, a term employed in common usage for anything of vast size, but strictly applicable only with the implication of the immeasurably or innumerably great, or of the impossibility of being measured or counted. In the strict sense the term is employed in metaphysics and theology and mathematics. The term is from the Latin in (not) and finis (end) and thus has the etymological significance of "endless." The unlimited straight line has length without limit ; it cannot be measured off by the repeated application of any unit ; it tran scends all possibility of measurement, and in this sense it is said to be infinite. A line is also infinite as regards the number of points on it ; between any two points on a line there is another, and indeed an infinitude of others; on any line segment there is a smaller line segment, and hence an infinitude of line segments such that each of the segments after the first is included in each of those which precede it. Thus a line exhibits both infinitude of measurement and infinitude of subdivision. Space likewise is infinite in extent and it possesses an infinitude of points, of lines, of planes and of bounded portions of space. Endlessness as re gards extent or distance and endlessness as regards subdivision are characteristics of space by which it is seen under the aspect of the infinite. There is a shorter distance than any which we have measured and a longer than any which we can conceive. But boundlessness or endlessness is not equivalent to infinitude. A confusion of the two notions was current in antiquity and has appeared also among modern thinkers, as Hobbes, and Hegel, for instance. A circle, considered in the modern aspect of line is endless, but we would not apply to it the concept of infinitude as regards measurement, though as regards subdivision it exhibits the characteristic quality of infinitude. Space is endless and in finite, the circumference of a circle is endless but finite as regards extent.
The Infinite in Metaphysics and Theology.—Tracing history back to the earliest possible date, we are forced to postulate an earlier period still and then one preceding that, and so on without end. Looking forward we conceive vast stretches of the future ; and beyond any time of which we can think we are forced to postulate a still later time to come. Distances on the earth are small compared with the distance to the sun, and that itself is small compared with the distances of some of the stars; and beyond the furthest reaches of our telescopes we imagine still un ending stretches of space. We have never Seen a thing so small that we cannot imagine a smaller, or measured a thing so great that we cannot conceive a greater. But we have never experi enced the infinitely small or the infinitely great, though we have formed a notion of them. Infinity is a mental concept. In early Greek philosophy (see IONIAN SCHOOL OF PHILOSOPHY) it ap peared under the aspect of the boundless (re, lirraapov ), and as such it was discussed at great length by both Plato and Aristotle. They gave much thought to the question as to which is the more truly real, the finite objects of sense or the universal idea of each thing laid up in the mind of God. They enquired concerning the nature of that unity which underlies the multiplicity of perceived objects; and the same problem, in various forms, has engaged the attention of philosophers throughout the ages.
In Christian theology God is conceived as infinite in power and knowledge and justice and goodness, uncreated and immortal. In some oriental systems, absorption in the infinite is the highest end of man, his perfection resting in the breaking down of human limitations.
The metaphysical and theological conception of the infinite is open to the objection that a finite mind cannot form an adequate or appropriate conception of such an object of thought and there fore cannot form trustworthy judgments concerning it ; and the matter has been extensively debated. Sir William Hamilton's philosophy of the "unconditioned" and Herbert Spencer's doc trine of the infinite "unknowable" give evidence of this contro versy. (See the articles on the thinkers mentioned, and also those On DESCARTES, MALEBRANCHE, SPINOZA and ZENO.) The Mathematical Infinite.—If the law of variation of a magnitude x is such that x becomes and remains greater than any pre-assigned magnitude however large, then x is said to become infinite, and this conception of infinity is denoted by co . If the law of variation of a magnitude x is such that x becomes and remains less than any pre-assigned magnitude, however small, then x is called an infinitesimal and is said to approach the limit zero. These conceptions of the infinite and the infinitesimal are analogous to the metaphysical infinitudes of measurement and of subdivision; but they differ from the latter in a marked way in that the mathematical concepts are defined entirely in terms of that which is finite. The absolutely infinite is in no sense involved. What is described in each case is a type of variation. These conceptions are intimately involved in the articles : FUNCTION, LIMIT, SERIES, NUMBER SEQUENCES, and CALCULUS.
Through the doctrine of geometrical continuity (q.v.) and the application of algebra to geometry arose the important notion of infinity as a localized space-conception, so that mathematicians come to speak of points at infinity, lines at infinity, and planes at infinity. It is said, for instance, that two parallel lines inter sect in a point at infinity; that all circles in a plane pass through two fixed points at infinity (the circular points), and that two spheres intersect in a fixed circle at infinity. (See GEOMETRY.) The most remarkable mathematical doctrine of the infinite is that which is associated with the conception of infinite aggre gates. In the case of any aggregate it is possible to pair the ele ments in such a way that each element in the aggregate is paired with an element of the aggregate. Thus in the set I, 2, 3, we may take I with 2, 2 with 3, 3 with I. In this case the first terms in the pairs exhaust the elements in the given set, and so do the second terms. In any way in which all the elements of a finite aggregate are paired uniquely with elements of the aggregate, the second terms in the pairs (as well as the first) exhaust the aggre gate. This is the distinguishing quality of a finite aggregate as finite. The opposite quality characterizes infinite aggregates as infinite. Thus, in the case of the set of all positive integers, we may pair each integer with its double ; the first terms in the pairs exhaust the aggregate of positive integers, but the second terms do not, since there is no odd number among them. In general an infinite aggregate is an aggregate such that its elements can all be paired uniquely with a part of its elements. This is the positive definition of an infinite aggregate, as opposed to the negative definition which characterizes it as one that is not finite.
If two aggregates are such that the elements of one may be made to correspond with the elements of the other in such a way that each element of either aggregate corresponds to one and just one element of the other aggregate, the relation so established is said to be a one-to-one correspondence between the aggregates. When a one-to-one correspondence is possible we say that the number of elements in one aggregate is the same as the number of elements in the other. In the case of finite aggregates this agrees with the usual conception of the sameness of number of two aggregates. The question arises whether two infinite ag gregates have always the same number of elements; the answer is negative. The number of all integers is the same as the num ber of all rational numbers, but is less (in a suitably defined sense of the term) than the number of all real numbers. The number of points on the segment of a line is equal to the number on the whole line and in fact to the number in the whole of space, but is less than the aggregate of all functions of a real variable.
See E. W. Hobson, Theory of Functions of a Real Variable, vol. I. 3rd ed. (1927), Camb. Univ. Press; B. Bolzano, Paradoxien des Unend lichen (Leipzig, 185o) ; and L. Couturat, De l'infini mathomatique (1896) . (R. D. CA.)