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Infinity

Infinity


Few concepts in mathematics are more fascinating or confounding than infinity. While mathematicians have a longstanding disagreement over its very definition, one can start with the notion that infinity (denoted by the symbol ) is an unbounded number greater than all real numbers.

Writing about infinity dates back to at least the Greek philosopher Aristotle (384 b.c.e.322 b.c.e.). He stated that infinities come in two varieties; actual infinities (of which he could find no examples) and potential infinities, which he taught were legitimate only as thought. Indeed, the German Karl Gauss (17771855) once scolded a fellow mathematician for using the concept, stating that use of infinity "is never permitted in mathematics."

The French mathematician and philosopher René Descartes (15951650) proposed that because "finite humans" are incapable of producing the concept of infinity, it must come to us by way of an infinite being; that is, Descartes saw the existence of the idea of infinity as an argument for the existence of God. English mathematician John Wallis (16161703) suggested the use of as the symbol for infinity in 1655. Before that time, had sometimes been used in place of M (1000) in Roman numerals.

Defining Infinity

Although students are typically taught that "one cannot divide by 0," it can be argued that = 0 (read as "one divided by infinity"). How is this possible? Observe the following progression.

Note that as the denominator, or the divisor, becomes larger, the value of the fraction (or the "quotient") becomes smaller. What happens if the denominators become very large?

One can see that as the denominator becomes extremely large, the fraction values approach 0. Indeed, if one thinks of infinity as "ultimately large," one can see that the value of the fraction will likewise be "ultimately small," or 0. Hence, one informal (but useful) way to define infinity is "the number that 1 can be divided by to get 0." Actually, there is no need to use the number 1 as the numerator here; any number divided by infinity will produce 0.

Using algebra, one can come up with another definition of infinity. By transforming the following equation we see that infinity is what results if 1 is divided by 0.

If

Then 1 = × 0

And

Notice that this approach to informally defining infinity produces an equation (the middle equation of the three above) in which something times 0 does not give 0! Because of this difficulty, and because the rules of algebra used to write and transform the equations apply to numbers, some mathematicians claim that division by 0 should not be allowed because may not be a defined number. They argue that dividing by 0 does not give infinity, but rather that infinity is undefined.

Another method of attempting to define infinity is to examine sets and their elements. If in counting the elements of a set one-by-one the counting never ends, the set can be said to be infinite.

Infinity as a Slope. Infinity is also sometimes defined as "the slope of a vertical line on the coordinate plane." In coordinate geometry , it is accepted that the slope of any straight line is defined as the change in vertical height divided by the change in horizontal distance between any two points on the line. The slope is often shown as a fraction in lowest terms, and sometimes called "rise over run."

In the figure, the slope of line (a) is ½. If a line is very steep, the rise will be very large compared to the run, giving a very large numerical slope. The slope of line (b) is as . A much steeper line will result in a fraction suchall . Such a line would appear to be vertical, even though it would not be quite vertical if viewed in greater detail. Thus, the slope of extremely steep lines approaches infinity, and the slope of a "completely steep" line, that is, a vertical line, can be thought of as equal to infinity.

Yet on a "completely steep" or vertical line, any two points give a run of 0. This means that one could define the slope of the line as any number over 0. This again allows the conclusion that division by 0 results in infinity, unless one maintains that the slope of a vertical line is undefined.

The Nature of Infinity

Although several definitions of infinity were provided, note that none of them state that infinity is the highest possible number. Consider this: On a number line, how many points are between points 4 and 5? An infinite number, of course, because actual points have no dimension, even though their two-dimensional representations have a very small dimension on the paper, blackboard, or computer screen. But consider further: How many points are between points 4 and 6? Also an infinite number, certainly, but this set appears to be twice as large as the one between points 4 and 5. This use of set theory as an approach to understanding infinity forces one to look at several curious possibilities.

  1. There are different sizes of infinity.
  2. A set with an infinite number of elements is the same size as one of its "smaller" subsets.
  3. Elements can be added to a set that already has an infinite number of elements.

Which of these possibly contradictory statements is true? It may be impossible to answer the question. Galileo (15641642) felt that the second statement was true. The great German mathematician and founder of set theory Georg Cantor (18451918) added to our understanding of infinity by choosing not to see the statements as contradictions at all, but to accept them as simultaneous truths. Cantor defined orders of infinity. An infinite set that can be put into one-to-one correspondence with the counting numbers is the smallest infinite set, called aleph null. Other larger infinite sets are called aleph one, aleph two, and so on. One can see that working with infinity produces various counterintuitive and even paradoxical results; this is why it is such an interesting concept.

There are numerous examples of infinity in pre-college mathematics. One case: it is accepted that 0.999 is exactly equal to 1.0. Yet how can a number which has a 0 in the units place be exactly equal to a number with a one in that place? The idea that there are an infinite number of nines in the first number allows us to make sense of the proposition. The number 0.999 is said to "converge on 1," meaning that 0.999 becomes 1 when the infinite number of nines is considered.

Another example of how infinity comes into play in common mathematics is in the decimal representation of π (pi), or 3.14159. The digits making up π go on forever without any pattern, even though the size of never π gets even as large as 3.15.

No one has ever come across an infinite number of real things. Infinity remains a concept, brought to life only by the imagination.

see also Descartes, and His Coordinate System; Division by Zero; Limit.

Nelson Maylone

Bibliography

Gamow, George. One Two ThreeInfinity: Facts and Speculations of Science. Mineola, NY: Dover Publications, 1998.

Hofstadter, Douglas. Godel, Escher, Bach: An Eternal Braid. New York: Basic Books, 1999.

Morris, Richard. Achilles in the Quantum Universe: The Definitive History of Infinity. New York: Henry Holt and Company, 1997.

Rucker, Rudy. Infinity and the Mind. Princeton, NJ: Princeton University Press. 1995.

Vilenkin, N. In Search of Infinity. New York: Springer Verlag, 1995.

Wilson, Alistair. The Infinite in the Finite. New York: Oxford University Press. 1996.


DEVELOPMENT OF SET THEORY

German mathematician Georg Cantor (18451918) was an active contributor to the development of set theory. He also became known for his definition of irrational numbers.


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Infinity

Infinity


Infinity in a rigorous sense is a mathematical concept, but the notion of boundless entities, such as the number series and time, have since antiquity touched a deep philosophical and religious chord in the human heart.


Ancient and medieval conceptions

To the ancient Greek religious sect known as the Pythagoreans, the notion of limit was valued as conferring intelligibility and definition, while the infinite (apeiron ) was associated with void and primordial matter, imperfection and instability. Plato (c. 428327 b.c.e.) captures this negative sensibility in Philebus when he reports that "the men of old" viewed all beings "as consisting in their nature of Limit and Unlimitedness" (16c). Drawing on this background as well as reacting to it, Aristotle (384322 b.c.e.) adopted the solution of banning anything actually infinite from philosophy. The infinite, he declared, is only "potential," denoting limitless series of successive, finite terms. Time is infinite in this potential sense, without a first beginning or end, but space, which exists all at once, is finite. A similar treatment of infinity is found in Euclidean mathematics, namely in Book 5, definition 4, which allows finite magnitudes as small or as large as desired, but precludes anything actually transfinite.

With the first-century Jewish philosopher Philo and the founder of neoplatonism Plotinus (c. 205270 c.e.), an actual infinite perfection is attributed in a new positive sense to God to mean that divine perfection transcends every finite case and is immense, eternal, incomprehensible, and unsurpassable. The early Christian leader Augustine of Hippo (354430 c.e.) in turn stresses in Confessions Book 7 that God is infinite according to a special immaterial measure of perfection, invisible to the bodily eye. The eighth-century theologian John Damascene speaks of God in De Fide Orthodoxa as "a certain sea of infinite substance" (1, 9). Medieval Jewish mystics such as Isaac the Blind and Azriel of Gerona who were active around the thirteenth century enlist the Hebrew en-sof (infinite) to describe the infinite extension of God's thought. Later cabbalists will use the actual infinite as a proper name and refer to "the En-Sof, Blessed be He."

In the mid-thirteenth century, Latin scholastics became concerned with rationalizing divine infinity by framing a coherent philosophical language to discuss various types of infinity and to explore the properties of the actual infinite, such as its noninductive and reflexive character. Two trends are discernible. Thomas Aquinas (c. 12251274) built on Aristotle to reach God philosophically as infinite (unrestricted) Being, while his Franciscan counterpart, Bonaventure (12211274), drawing more centrally on Augustine, started with a finite degree of ontological perfection and allowed this perfection to be raised to infinity. A new appreciation of the distinction between extension and intensity was thus brought to bear on the infinite, with the notion of intensity serving to mask the paradoxes inherent in the notion of an actual infinite extension. Bonaventure promoted an approach that is introspective rather than cosmological, involving the key premises that the human soul longs for an infinite good (God) and cannot find rest short of reaching it.

Another Franciscan, Peter John Olivi (c. 12481298), clarified the difference that exists between a concept taken unrestrictedly (e.g. being) and the determinate infinite case falling under the concept and denoting God (being of infinite intensity). John Duns Scotus (c. 12651308), also a Franciscan, formulated on this basis a univocal theocentric metaphysics based on adopting the intensive infinite as the "most perfect concept of God naturally available to us in this lifetime." Finally, by stressing the purely semiotic character of the concept and explaining that denoting God by means of the actual infinite does not imply comprehending God, William of Ockham (12881348) helped to secularize the discussion and to give the actual infinite a legitimate place in philosophy. The scientists who introduced ideal elements at infinity in geometry in the seventeenth century, namely Johannes Kepler, René Descartes, and Blaise Pascal, were fully familiar with scholastic mainstreaming of the actual infinite.


Modern conception of infinity

In the seventeenth century, Descartes made infinity a keystone of his metaphysics and philosophy of science. The idea of an actually infinite being is innate in the human mind, he argues, and cannot derive from anything finite, not even by extrapolation. Rather, the human ability to conceptualize the limit of an infinite process proves that the concept of the actual infinite is in us prior to the finite. Descartes also insisted that God alone is actually infinite, so that physical space must be described as merely indefinite rather than infinite. Another seventeenth-century scientist to make creative apologetic use of the actual infinite, based on its mathematical properties, was Blaise Pascal (16231662). In his famous "wager" argument, he invoked the disproportion of an infinite reward to urge human beings to bet their lives on God, no matter how small the odds. Pascal also invoked mathematical incommensurability to argue that charity infinitely exceeds a life devoted to science, just as a life of science infinitely exceeds a life spent on material pleasure.

The taste for images of absolute transcendence has waned among theologians in recent times, prompting renewed interest in the potential infinite. Process theology, in particular, inspired by mathematician and philosopher Alfred North Whitehead (18611947), has explored metaphors connected with the inner unfolding of time and the evolving universe to depict human beings as partners of God's open-ended creativity. Meanwhile, the actual infinite has found rigorous mathematical expression in transfinite set theory, fathered by mathematician Georg Cantor (18451918). Cantor not only extended classical number theory by introducing transfinite numbers but proved that there is a hierarchy of transfinite magnitudes, such that, for instance, the infinite cardinality of the continuum (denoted by c ) is larger than the infinite cardinality of the rational numbers (denoted by aleph-zero ). The religious dimension of transfinite ideation by no means evaporated on account of this new rigor: Cantor actively sought to enlist Catholic theologians in support of his mathematical discoveries, citing as a personal inspiration Augustine's speculation about God's perfect knowledge of numbers. Cantor's fellow mathematician David Hilbert has perhaps best summarized the dual religious and scientific appeal of infinity in the 1925 address designed to herald Cantor's discovery: "the infinite has always stirred the emotions of mankind more deeply than any other questions; the infinite has stimulated and fertilized reason as few other ideas have; but also the infinite, more than any other notion, is in need of clarification."

See also Thomas Aquinas; Aristotle; Plato; Process Thought; Space and Time


Bibliography

davenport, anne. measure of a different greatness: the intensive infinite 1250-1650. leiden, netherlands: brill, 1999.

field, judith. the invention of infinity: mathematics and art in the renaissance. oxford: oxford university press, 1997.

kretzmann, norman, ed. infinity and continuity in antiquity and the middle ages. ithaca, n.y.: cornell university press, 1982.

sweeney, leo. divine infinity in greek and medieval thought. new york: peter lang, 1992.

anne a. davenport

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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).

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infinity

in·fin·i·ty / inˈfinitē/ • n. (pl. -ties) the state or quality of being infinite: the infinity of space. ∎  an infinite or very great number or amount: an infinity of excuses. ∎  Math. a number greater than any assignable quantity or countable number (symbol ∞). ∎  a point in space or time that is or seems infinitely distant: the lawns stretched into infinity.

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infinity

infinity (symbol ) Abstract quantity that represents the magnitude of an object without limit or end. In geometry, the ‘point at infinity’ is where parallel lines can be considered as meeting. In algebra, 1/x approaches infinity as x approaches zero. In set theory, the set of all integers is an example of an infinite set.

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infinity

infinitybanditti, bitty, chitty, city, committee, ditty, gritty, intercity, kitty, nitty-gritty, Pitti, pity, pretty, shitty, slitty, smriti, spitty, titty, vittae, witty •fifty, fifty-fifty, nifty, shifty, swiftie, thrifty •guilty, kiltie, silty •flinty, linty, minty, shinty •ballistae, Christie, Corpus Christi, misty, twisty, wristy •sixty •deity, gaiety (US gayety), laity, simultaneity, spontaneity •contemporaneity, corporeity, femineity, heterogeneity, homogeneity •anxiety, contrariety, dubiety, impiety, impropriety, inebriety, notoriety, piety, satiety, sobriety, ubiety, variety •moiety •acuity, ambiguity, annuity, assiduity, congruity, contiguity, continuity, exiguity, fatuity, fortuity, gratuity, ingenuity, perpetuity, perspicuity, promiscuity, suety, superfluity, tenuity, vacuity •rabbity •improbity, probity •acerbity • witchetty • crotchety •heredity •acidity, acridity, aridity, avidity, cupidity, flaccidity, fluidity, frigidity, humidity, hybridity, insipidity, intrepidity, limpidity, liquidity, lividity, lucidity, morbidity, placidity, putridity, quiddity, rabidity, rancidity, rapidity, rigidity, solidity, stolidity, stupidity, tepidity, timidity, torpidity, torridity, turgidity, validity, vapidity •commodity, oddity •immodesty, modesty •crudity, nudity •fecundity, jocundity, moribundity, profundity, rotundity, rubicundity •absurdity • difficulty • gadgety •majesty • fidgety • rackety •pernickety, rickety •biscuity •banality, duality, fatality, finality, ideality, legality, locality, modality, morality, natality, orality, reality, regality, rurality, tonality, totality, venality, vitality, vocality •fidelity •ability, agility, civility, debility, docility, edibility, facility, fertility, flexility, fragility, futility, gentility, hostility, humility, imbecility, infantility, juvenility, liability, mobility, nihility, nobility, nubility, puerility, senility, servility, stability, sterility, tactility, tranquillity (US tranquility), usability, utility, versatility, viability, virility, volatility •ringlety •equality, frivolity, jollity, polity, quality •credulity, garrulity, sedulity •nullity •amity, calamity •extremity • enmity •anonymity, dimity, equanimity, magnanimity, proximity, pseudonymity, pusillanimity, unanimity •comity •conformity, deformity, enormity, multiformity, uniformity •subcommittee • pepperminty •infirmity •Christianity, humanity, inanity, profanity, sanity, urbanity, vanity •amnesty •lenity, obscenity, serenity •indemnity, solemnity •mundanity • amenity •affinity, asininity, clandestinity, divinity, femininity, infinity, masculinity, salinity, trinity, vicinity, virginity •benignity, dignity, malignity •honesty •community, immunity, importunity, impunity, opportunity, unity •confraternity, eternity, fraternity, maternity, modernity, paternity, taciturnity •serendipity, snippety •uppity •angularity, barbarity, bipolarity, charity, circularity, clarity, complementarity, familiarity, granularity, hilarity, insularity, irregularity, jocularity, linearity, parity, particularity, peculiarity, polarity, popularity, regularity, secularity, similarity, singularity, solidarity, subsidiarity, unitarity, vernacularity, vulgarity •alacrity • sacristy •ambidexterity, asperity, austerity, celerity, dexterity, ferrety, posterity, prosperity, severity, sincerity, temerity, verity •celebrity • integrity • rarity •authority, inferiority, juniority, majority, minority, priority, seniority, sonority, sorority, superiority •mediocrity • sovereignty • salubrity •entirety •futurity, immaturity, impurity, maturity, obscurity, purity, security, surety •touristy •audacity, capacity, fugacity, loquacity, mendacity, opacity, perspicacity, pertinacity, pugnacity, rapacity, sagacity, sequacity, tenacity, veracity, vivacity, voracity •laxity •sparsity, varsity •necessity •complexity, perplexity •density, immensity, propensity, tensity •scarcity • obesity •felicity, toxicity •fixity, prolixity •benedicite, nicety •anfractuosity, animosity, atrocity, bellicosity, curiosity, fabulosity, ferocity, generosity, grandiosity, impecuniosity, impetuosity, jocosity, luminosity, monstrosity, nebulosity, pomposity, ponderosity, porosity, preciosity, precocity, reciprocity, religiosity, scrupulosity, sinuosity, sumptuosity, velocity, verbosity, virtuosity, viscosity •paucity • falsity • caducity • russety •adversity, biodiversity, diversity, perversity, university •sacrosanctity, sanctity •chastity •entity, identity •quantity • certainty •cavity, concavity, depravity, gravity •travesty • suavity •brevity, levity, longevity •velvety • naivety •activity, nativity •equity •antiquity, iniquity, obliquity, ubiquity •propinquity

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