infinity(redirected from Completed infinity)
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See A. D. Aczel, The Mystery of the Aleph (2000); D. F. Wallace, Everything and More (2003).
(in mathematics). “Mathematical infinity is taken from reality, although unconsciously, and therefore it can only be explained from reality and not from itself or from mathematical abstraction” (F. Engels, Anti-Duhring, 1966, p. 396). The material basis of the mathematical infinite can be understood only when it is considered in dialectical harmony with the finite. Every mathematical theory is bound by a compulsory requirement for internal formal consistency. Thus the problem arises of how to unite this requirement with the essentially contradictory character of the reality of infinity. “The annihilation of this inconsistency would be the end of infinity” (ibid., p. 47). The solution to this problem consists of the following: When, in the theory of limits, one considers infinite limits—lim an = ∞—or, in the theory of sets, infinite powers, this does not lead to internal formal inconsistencies in the indicated theories only because these distinct special forms of mathematical infinity are extremely simplified, schematized forms of the different aspects of infinity in the real world.
This article is confined to indicating the different approaches to infinity in mathematics which are discussed in detail in other articles.
(1) The idea of infinitely large and infinitesimal variable quantities in one of the fundamental problems in mathematical analysis. The idea preceding the modern approach to the concept of the infinitesimal, according to which finite quantities were composed of an infinite number of infinitely small “indivisibles” which were not treated as variables but as constants smaller than any finite quantity, can serve as one example of the invalid separation of the infinite from the finite. Only the decomposition of finite quantities into a non-boundedly increasing number of nonbounded decreasing addends has real meaning.
(2) Under entirely different logical circumstances, infinity appears in mathematics in the form of “improper” infinitely distant geometric forms at infinity. Here, for example, a point at infinity on line a is considered as a special fixed object “attached” to the ordinary finite points of the line. However, the continuous connection of the infinite with the finite is detected even here, for example, on projecting from a center lying outside the line, in which to the point at infinity, there turns out to be a corresponding straight line that passes through the center of projection and is parallel to the primary linea.
The extension of the system of real numbers with the two “improper” numbers + ∞ and – ∞, which meets many of the requirements of analysis and the theory of functions of a real variable, is analogous. One can approach the extension of the sequence of natural numbers 1, 2, 3, . . . , with the transfinite numbers ω, ω + 1, . . . , 2ω, 2ω + 1, . . . from the same point of view. In connection with the difference between variable infinitesimal and infinitely large quantities on the one hand and “improper” infinitely large numbers considered as constants on the other, the terms “potential” infinity (for the former) and “actual” infinity (for the latter) have arisen. In this initial sense (for another, modern, sense, see below) the controversy between the advocates of actual and potential infinity can be considered ended. The infinitesimal and the infinitely large quantity, which form the basis of the definition of the derivative as the ratio of two infinitesimals and of the integral as the sum of an infinite number of infinitesimals, and the concepts of mathematical analysis which border on them must be interpreted as potential infinities. In addition to this, under the proper logical circumstances, actual infinitely large “improper” numbers are also rightly included in mathematics (even in many diverse aspects: as quantitative and ordinal transfinite numbers in the theory of sets, as the improper elements + ∞ and – ∞ of the real number system, and so forth).
In mathematics, one has to deal with two methods of adding infinite improper elements to the number system.
(a) From the projective point of view, a straight line has one point at infinity. In the ordinary metric system of coordinates, this point is naturally assigned to the abscissa at ∞. The same addition to the number system of a single infinity without a sign is used in the theory of functions of a complex variable. In elementary analysis, in the study of rational functions f(x) = P(x)lQ(x) where P(x) and Q(x) are polynomials at those points where Q(x) has a zero of a higher order than P(x), it is natural to assume that fix) = ∞.
The following rules of operation are set for the improper element ∞;
∞ + a = ∞ if a is finite
∞ + ∞ has no meaning
∞ · a = ∞ if a ≠ 0
∞ · 0 has no meaning
Inequalities involving ∞ are not considered; it is senseless to ask whether ∞ is greater or smaller than a finite a.
(b) In the study of real functions of a real variable, the system of real numbers is supplemented by the two improper elements + ∞ and – ∞. Then one can suppose that – ∞ < a < + ∞ for any finite a and preserve the fundamental properties of the inequalities in an extended number system. The following rules of operation are established for + ∞ and – ∞;
(+ ∞) + a = + ∞ if a ≠ – ∞
(– ∞) + a = – ∞ if a ≠ + ∞
(+ ∞) + (– ∞) has no meaning
(+ ∞) · a = + ∞ if a > 0
(+ ∞) · a = – ∞ if a < 0
(– ∞) · a = – ∞ if a > 0
(– ∞) · a = + ∞ if a < 0
(+ ∞) · 0 and (∞) · 0 have no meaning
In every mathematical argument it is necessary to be aware of whether we are using the real (not extended) number system or an extended one and in precisely which of the two indicated meanings.
(3) The primary interest as well as the primary difficulty of the mathematical study of infinity is now focused on the problem of the nature of infinite sets of mathematical objects. It is particularly necessary to bear in mind that the total clarity and completeness of the theory of infinitely large and infinitesimal variables attained at present consists only of reducing all the difficulties of this theory to the problem of laying a basis for the study of the number, in which the concept of infinity of systems of numbers is included. The assertion that y is an infinitesimal has meaning only in the designation of the nature of the change of y depending on some other variable x. For example, it is said that y is an infinitesimal as x → a if for any ∊ > 0 there exists a δ > 0 such that ǀx – aǀ < δ implies ǀyǀ < ∊. The assumption that the function y = f(x) is defined for an infinite set of values of x (for example, for all real x sufficiently close to a) enters into this same definition.
In the theory of sets, a profound meaning is attached to the terms “actual” and “potential” infinity which has nothingin common with the designation of every infinite power by an actually infinite number. The point is that infinite systems of mathematical objects—for example, natural or real numbers—are never assigned by simple enumeration as is possible for finite systems of objects. It would be an obvious absurdity to suppose that someone “constructed” a set of natural numbers by enumerating practically “all” of them one after another. Actually, the set of natural numbers is studied by proceeding from the process of constructing its elements by the transition from n to n + 1. In the case of a continuum of real numbers the consideration of one of its elements, a real number, reduces to the study of the process of formation of its successive approximate values, and the consideration of the whole set of real numbers reduces to the study of the general properties of such processes of formation of its elements. In precisely this sense the infinity of a natural series or of the system of all real numbers (a continuum) can be characterized only as potential infinity. One can contrast the point of view of a potential infinity with the view of infinite sets as “actually” assigned, which is independent of the process of constructing them. The solution to the problem to what extent and under what circumstances in the study of infinite sets such abstraction from the process of their construction is valid still cannot be considered completed.
A. N. KOLMOGOROV
in philosophy, a concept used in two different senses: qualitative infinity, which is expressed in the laws of science and which states the universal nature of the connections of phenomena; and quantitative infinity, which is the limitlessness of processes and phenomena.
The problem of qualitative infinity was discussed in ancient philosophy, particularly in connection with cosmogony and problems of the nature of thought. However, it acquired special importance in modern philosophy in connection with the development of natural science and the problems of its logical substantiation (R. Descartes, J. Locke, G. Leibniz). G. Hegel gave a profound philosophical analysis of the problem of infinity. He distinguished between true, (qualitative) and “bad” infinity—a limitless increase in quantity—and connected the category of infinity with the nature of processes of development. Hegel’s ideas were reinterpreted materialistically by Marxism, which emphasized the dialectical interconnection of infinity and the finite and the contradictory nature of infinity. The indication of the connection between infinity and the category of the universal was important. As F. Engels wrote, “the form of universality is the form of self-completeness, hence, of infinity; it is the comprehension of many finites in the infinite” (K. Marx and F. Engels, Soch., 2nd ed., vol. 20, pp. 548–49).
With regard to cosmological problems, quantitative infinity is generally viewed as the infinity of the material world in space and time.
Here, the religious idealistic viewpoint conflicts with the materialist viewpoint. The religious idealistic viewpoint interprets infinity as the infinity of the deity with its timeless-ness, or as the product of consciousness. The materialistic point of view regards infinity as one of the properties of space and time and investigates it, relying on the results of mathematics and cosmology. According to the data of contemporary cosmology, the universe (the material world viewed only as a spatial and temporal distribution of masses) is infinite in space and time. However, its spatial and temporal characteristics taken separately may be both finite and infinite, depending on the frame of reference chosen.
In physics infinity is viewed as infinity “in depth” in connection with the problem of the structure of elementary particles.
REFERENCESFilosofiia estestvoznaniia, issue 1. Moscow, 1966. Pages 28, 191–207.
Naan, G. I. Poniatie beskonechnosti v matematike, fizike i as-tronomii. Moscow, 1965.
Naan, G. I. “Tipy beskonechnogo.” In Einshteinovskii sbornik, 1967. Moscow, 1967.
Zel’manov, A. L. “O beskonechnosti material’nogo mira.” In Dialektika v naukakh o nezhivoi prirode. Moscow, 1964.
I. S. ALEKSEEV
Using the word in the context of sets is sloppy, since different infinite sets aren't necessarily the same size cardinality as each other.
See also aleph 0
See also minus infinity.