Infinity: A term referring to phenomena for which no end exists or can be conceived. Not to be confused with ►unboundedness. It may have been the thought of immensely vast spaces or eons of time that first instilled in humans the idea of infinity. The Italian poet Giacomo Leopardi put it this way:
At first infinity was ascribed simply to very long time intervals or distances, or to very large sets. The Greek word "eon" stands for an age, and in the plural ("eons") also for eternity. At that time, infinitude was not yet regarded as the opposite of finitude, but merely as an enlarged or prolonged version of finitude or as the sum of many finitudes. Starting around 600 B.C., however, the Greek Presocratics began to study infinity more thoroughly. The Greek Concept of Infinity Anaximander (610  546 B.C.) introduced the concept of ►Apeiron, an infinite and infinitely divisible primordial form of matter that is the origin of all finite things: "Infinity is the origin of all things."
Pythagoras (570
 497 B.C.) discovered the relations between the length of the sides of a rectangular triangle (a^{2} +
b^{2} = c^{2}) and the ►irrational numbers as a solution to this equation.
Eudoxus (408  435 B.C.) was the first to postulate infinitely small numbers (anticipating the ►infinitesimal calculus) to determine lengths, surfaces, and volumes. Archimedes (287  212 B.C.) came up with a method of representing indefinitely large numbers to calculate the ►number of sand and also introduced limiting values to calculate the number ►Pi. Aristotle's distinction between actual and potential infinity came to play an important part in the history and mathematics of infinity. After Aristotle there was at first a long intellectual standstill on the topic of infinity. That topic of was eventually taken up by theologians and theologicallyminded philosophers, who brought with them a very different scale of values. While the Greek philosophers typically interpreted infinitude, unlimitedness, in terms of undifferentiatedness and regarded it as an imperfection, it was the finite that struck theologians as imperfect. Perfection lay precisely in the infinite nature of God (and only there). Theological Infinity Augustine (354  430) denied that human beings area able to recognize the infinite on their own. Only ►God is able to do so due to His infinite nature, and only by striving to reach God can humans hope to come closer to infinity and find selffulfillment: "Restless is our heart until it finds rest in Thou." Anselm of
Canterbury (1033  1109) attempted to present a logical justification of faith by employing a concept of perfection in his definition of God: "A being than which nothing greater can be conceived." Enlightenment Infinity Giordano Bruno (1548  1600) described the universe not as potentially but as actually infinite and eternal, with the Earth occupying only a negligible spot within a gigantic space. Anything short of such a universe would be unworthy of the creative power of an infinitely powerful God. "I believe in an infinite universe, the effect of an infinite divine potency, because it has seemed to me unworthy of the divine goodness and power to create a finite world, when able to produce beside it another and others infinite: so that I have declared that there are endless particular worlds similar to this of the Earth. " Bruno paid with his life for this view. Despite his lip service rejection of actual infinity, Leibniz was one of the major figures involved in the mathematical revolution of the 17th century. While mathematics had previously only attempted to describe nature, it now began to deal with abstract objects that did not match anything in nature — objects such as the infinitesimal numbers, which are infinitely small numbers. Paradoxically, this abstract form of mathematics was able to describe the behavior of nature much better than its predecessors. Isaac Newton made use of infinite divisions of mathematical curves in his ►fluxion calculus, defying both Aristotle and the objections of contemporary philosophers and theologians. His efforts were rewarded by his discovery of ►laws of nature concerning gravity and the motions of solid bodies as well as a precise theoretical model of the solar system. The Schism of Infinity Since the time of Newton and Leibniz, infinity has had to tolerate quite diverse treatments in the sciences, mathematics, and philosophy. Mathematicians and natural scientists began to include in their calculations first the infinitely small and then the infinitely large. Consequently, after the 17th century mathematical theories of infinity split off onto a separate route of development, while philosophers and theologians continued theorizing in largely the same way as before: George Berkeley (1685  1753) regarded nature as consisting only of perceptions and thus only of finite things. Only perceptions are real manifestations of God's spirit, outside of which there is nothing in nature. Thus the world exists only insofar as it is perceived either by God our by us. Concepts referring to things that cannot be perceived by the senses, such as 'infinity', have no place in philosophy: "Whether the object of geometry be not the proportions of assignable extensions? And whether there be any need of considering quantities either infinitely great or infinitely small? [...] And whether it be not unnecessary, as well as absurd, to suppose that finite extension is infinitely divisible?" Georg Hegel (1770  1831)
regarded the finite and the infinite as a dialectical unit. The infinite realizes itself only within the finite. But the infinite is not just the negation of the finite; rather, actual, "genuine" infinity is the negation of potential or "bad" infinity.
Potential infinity is like "the straight line, at the two limits of which alone the infinite is". True infinity, by contrast, is like a circle, "a line which has reached itself, which is closed and entirely present, without beginning and end." Mathematical Infinity As has already been mentioned, concepts of infinity in mathematics did not differ noticeably from those in philosophy up to the 17th century. Here, too, infinity was at first accepted only as an approximation, that is, only as potential infinity. The expression _{} approaches infinity if the values a and b approximate each other. However, when a and b are equal, the expression becomes meaningless. Mathematics does not permit division by 0. This resistance to actual infinity first began to crumble in the 17th century with Newton's and Leibniz's ►infinitesimal calculus, the mathematics of infinitely small numbers. Still, although the infinitesimal calculus was obviously successful and accurately described processes in nature (such as planetary motions, for example), in the eyes of many contemporaries actual infinity remained a suspicious, if not specious, notion. S.F.
Lacroix (1765  1843), the author of the standard textbook on infinitesimal calculus in the 18th century, wasted no time in rejecting infinity, writing in the book's introduction, "Infinity as the last element of quantity, is itself an exclusive limit, a limit that quantities can never reach. The associated concept is but a negative concept, for each quantity that I imagine as real and use in my calculus is, for that very reason, not infinite. "
As late as 1831, Carl Friedrich Gauss (1777  1855) complained: " I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely closely." This, however, did not prevent Gauss from developing fundamental methods of differential geometry, which is based on ►infinitesimal calculus and operates with infinitely small numbers. Bernard Bolzano (1781  1841) was the first mathematician to systematically rebel against the anathematizing of infinity. The world, he thought, is full of actual infinites, so that it does not make any sense to exclude them from mathematics. The set of all points in a line is infinite. Every time interval contains infinitely many moments. The number of digits after the decimal point of the square root of 2 is infinite. And the human spirit, according to Bolzano, is capable of imagining an infinity as a whole, for to do so you do not need to imagine every single component of this infinite whole in isolation. Bolzano succeeded in developing his own mathematics, one that enables us to calculate infinite quantities without inconsistency. He also proved that the number of points in a region of the ►number ray does not depend on its length. His results were published posthumously under the title Paradoxes of Infinity. 150 years after its introduction into mathematics, Karl Weierstrass (1815  1897) provided a logical foundation for the ►infinitesimal calculus by deriving the use of infinitely small numbers from limiting values. ► Bernhard Riemann (1826  1866) was the first to prove that a finite surface has the same number of points as an infinite one by projecting the points in the surface of a sphere onto a plane. Richard Dedekind (1831  1916) developed the first mathematical definition of an infinite set. According to Dedekind, a set is infinite if it contains the same number of elements as one of its proper subsets. Conversely, a set is finite if it contains more elements than each of its proper subsets.Last but far from least, ►Georg Cantor (1845  1918) fully rehabilitated the actuality of infinity in mathematics. Against the resistance of conservative mathematicians, he singlehandedly developed ►set theory and the mathematics of ►transfinite, that is, actually infinite numbers. It is not by accident that in the mathematics sections of this dictionary almost every other concept relates to Cantor either directly or indirectly. The mathematician ►David Hilbert summarized Cantor's achievements as follows: "No one shall drive us from the paradise that Cantor created for us." ► Ernst Zermelo (1871  1953) and Abraham Fraenkel together formulated a consistent axiomatic system for Cantor's ►set theory. It contains nine axioms* and constitutes the foundation for almost all branches of mathematics, much like ►Euclid's axiomatic system of geometry. ► Bertrand Russell (1872  1970) discovered ►Russell's paradox, thereby indirectly proving that there is no largest possible infinity. ► Luitzen Brouwer (1881  1966) demonstrated that it is impossible to prove geometrically that the sets of points of planes with different dimensions have the same cardinality. Proving this arithmetically (see ►Dimension), on the other hand, is pretty simple and had already been done much earlier by Cantor. ► Kurt Gödel (1906  1978) proved the incompleteness of all axiomatic theories in 1931; seven years later he proved that the ►continuum hypothesis is not refutable within Zermelo's axiomatic set theory. ► Paul Cohen 1934  2007) proved in 1963 that the opposite of the continuum hypothesis is not refutable either. With that, the continuum hypothesis had been established as one of the undecidable sentences of axiomatic mathematics. * The nine axioms of Zermelo's and Fraenkel's set theory are as follows: I. Axiom of Extensionality: If every element of a set X is also an element of Y and vice versa, then X = Y. II. Axiom of Subsets: If x is any object in the domain then there is a set {x} containing x and only x as element.If a, y are any two objects in the domain then there is always a set {x,y} containing both x and y but no element distinct from x or y. III. Axiom of Specification: If Z is a set, and P is any property that may characterize all elements x of Z, then there is a subset Y of Z containing all and only those x in Z that satisfy the property P. IV. Axiom of Powersets: For each set X there is another set PX containing all and only the subsets of X V. Axiom of Union: For any set X there is a set Y containing all and only those sets that are members of some member of X. VI. Axiom of Choice: If X is a set all of whose elements are distinct from 0 and pairwise disjoint, then their union Sx contains at least one subset that has one and only one element in common with each element of X. VII. Axiom of Infinity: There is at least one set Z that contains the empty set as element and is such that for each of its elements x there is another element of the form {x}. VIII. Axiom of Replacement: If we replace the elements x of a set X definitely with any elements x' of the domain then this domain will contain a set X' that contains all these x' as elements. IX. Axiom of Foundation: Every nonempty set X contains a member y such that X and Y are disjoint sets.
