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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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Infinity

Infinity

Resources

As children, we learn to count, and are pleased when first we count to 10, then 100, and then 1,000. By the time we reach 1,000, we may realize that counting to 2,000, or certainly 100,000, is not worth the effort. This is partly because we realize that such projects could take up all our time, and partly because we realize no matter how high we count, it is always possible to count higher. At this point we are introduced to the infinite, and begin to realize what infinity is.

Infinity, written as 1, is not the largest number. It is the term we use to convey the notion that there is no largest number. We say there is an infinite number of numbers.

There are aspects of the infinite that are not altogether intuitive, however. For example, at first glance there would seem to be half as many odd (or even) integers as there are integers all together. Yet it is certainly possible to continue counting by twos forever, just as it is possible to count by ones forever. In fact, we can count by tens, hundreds, or thousands, it does not matter. Once the counting has begun, it never ends.

What of fractions? It seems that just between zero and one there must be as many fractions as there are positive integers. This is easily seen by listing them, 1/1, 1/2, 1/3, 1/ 4, 1/5, 1/6, 1/7, 1/8,. But there are multiples of these fractions as well, for instance, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, and 8/8. Of course many of these multiples are duplicates, 2/8 is the same as 1/4 and so on. It turns out, after all the duplicates are removed, that there is the same number of fractions as there are integers. Not at all an obvious result.

In addition to fractions, or rational numbers, there are irrational numbers, which cannot be expressed as the ratio of whole numbers. Instead, they are recognized by the fact that, when expressed in decimal form, the digits

KEY TERMS

Counting numbers As the name suggests, the counting numbers are 1,2,3, also called the natural numbers. The whole numbers are the counting numbers plus zero.

Transfinite numbers Transfinite numbers were invented by Georg Cantor as a means of expressing the relative size of infinite sets.

to the right of the decimal point never end, and never form a repeating sequence. Terminating decimals, such as 6.125, and repeating decimals, such as 1.333̌ or 6.534m̌ (the bar over the last digits indicates that sequence is to be repeated indefinitely), are rational. Irrational numbers are interesting because they can never be written down. The instant one stops writing down digits to the right of the decimal point, the number becomes rational, though perhaps a good approximation to an irrational number.

It can be proved that there are infinitely more irrational numbers than there are rational numbers, in spite of the fact that every irrational number can be approximated by a rational number. Taken together, the rational and irrational numbers form the set of real numbers.

The word infinite is also used in reference to the very small, or infinitesimal. Consider dividing a line segment in half, then dividing each half, and so on, infinitely many times. This procedure would results in an infinite number of infinitely short line segments. Of course it is not physically possible to carry out such a process; but it is possible to imagine reaching a point beyond which it is not worth the effort to proceed. We understand that the line segments will never have exactly zero length, but after a while no one fully understands what it means to be any shorter. In the language of mathematics, we have approached the limit.

Beginning with the ancient Greeks, and continuing to the turn of the twentieth century, mathematicians either avoided the infinite, or made use of the intuitive concepts of infinitely large or infinitely small. Not until the German mathematician, Georg Cantor (18451918), rigorously defined the transfinite numbers did the notion of infinity finally seem fully understood. Cantor defined the transfinite numbers in terms of the number of elements in an infinite set. The natural numbers have N 0 elements (the first transfinite number). The real numbers have 1 elements (the second transfinite number). Then, any two sets whose elements can be placed in 1-1 correspondence, have the same number of elements. Following this procedure, Cantor showed that the set of integers, the set of odd (or even) integers, and the set of rational numbers all have N 0 elements; and the set of irrational numbers has N 1 elements. He was never able, however, to show that no set of an intermediate size between N 0 and N 1 exists, and this remains unproved today.

Resources

BOOKS

Rucker, Rudy. Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 2004.

Wallace, David Foster. Everything and More: A Compact History of Infinity. New York: W.W. Norton, 2004.

PERIODICALS

Moore, A. W. A Brief History of Infinity. Scientific American 272, No. 4 (1995): 112-16.

Paulos, John Allen. Beyond Numeracy, Ruminations of a Numbers Man New York: Knopf, 1991.

J. R. Maddocks

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Infinity

INFINITY

Infinity is derived etymologically from the Latin, infinitas, which is a combination of in (meaning not) and finis (meaning end, boundary, limit, termination, etc.). In general, the word signifies the state or condition arising from an entity's not having some sort of end, limit, termination, or determining factor. It is predicated both of extramental, actually existing things (such as God, the visible universe, and matter) and of intramental entities (such as logical concepts and mathematical constructs). The condition it signifies in these can connote either perfection or imperfection, depending upon whether the termination, which is thereby stated to be absent, is itself a perfection and whether the entity in question should possess it. For example, when predicated of God, infinity denotes perfection. This is so because it reveals the absence in God of matter and of other intrinsic factors suggesting mere potency, as well as the absence of extrinsic limits such as time, place, or comprehension by a created intellectall of which can be linked with imperfection and none of which is proper to God. On the other hand, when applied to matter, infinity signifies a state of indigence, since it manifests that matter of itself lacks form and act; these are perfective factors that matter must receive if it is to exist within the real universe, and is to be even indirectly intelligible and describable.

This article traces the history of the concept of infinity, which falls into five main sections: (1) that of the ancient Greeks, (2) Neoplatonism, (3) that of the Fathers of the Church, (4) medieval scholasticism, and (5) the period since 1600. The following account treats the first two periods as representative of Greek and Neoplatonic thought, and the last three as representative of Christian and modern thought.

Greek and Neoplatonic Thought

The first period begins in ancient Greece with Anaximander of Miletus (c. 610546 b.c.), who affirmed that "the first principle of existing things is the unlimited" (Fragment 1). It is "eternal and ageless" (Frag. 2), "deathless and indestructible" (Frag. 3), and, most likely, is some basic natural body that is unlimited because quantitatively inexhaustible. The Pythagoreans, as Aristotle reports (Meta. 987a 1519), "thought that finitude and infinity were not attributes of certain other things, e.g., of fire or earth or anything else of this kind, but that infinity itself and unity itself were the substance of the things of which they are predicated." Under their influence plato also made infinity one of the constitutive factors of reality: "All things that are ever said to be consist [so the men of old say] of a one and a many, and have in their nature a conjunction of Limit and Unlimitedness" (Philebus 16C; also see 23C). But Aristotle dominates this first period, as well as most subsequent periods, with respect to quantitative infinity. (see greek philosophy.)

Aristotelian Teaching. According to aristotle (see Phys. 202b 30-208a 27), quantity, motion, and time are all infinite (apeiron or, less frequently, aoriston ), but infinity (apeiria or aoristia ) belongs to the last two only because of their relationship with the former: motion is infinite if the magnitude covered is somehow infinite, and time is so only as a measure of an infinite motion. What, then, is apeiria as found in quantity? Basically it has to do with certain conditions of a line.

Quantitative Infinity. Let AO be an actual line of definite length. Point A terminates it at the beginning and point O at the end; thus, such a line is finite. Because of its definite dimensions, it can be measured, known, and described. Accordingly, its status of finiteness makes it knowable and describable. In it, extension is related to the terminating points in somewhat the same way as matter with respect to form; and it is itself a composite, so to speak, of matter and form (see matter and form). Since perfection and actuality, no less than intelligibility, arise from the presence of form, AO is not only knowable but is perfect and actual as well. Consequently, the condition of finiteness in AO arises from its possession of definite dimensions and is aligned with perfection, actuality, and knowledge.

But how is AO infinite? If AO is finite inasmuch as it has definite dimensions because of its initial and final terminating points, it is infinite inasmuch as it can be conceived without one or other of those terminations. Thus AO is infinite with respect to increase, since no matter what its actual length may be, one can always imagine it as without its final point and thus as extending further. (The same applies to number, which is similarly infinite, since no matter what actual sum is suggested, one can always think of a larger one). AO is also infinite with respect to decrease under certain conditions. Thus, no matter how small it actually becomes by its initial point's receding toward its final one, one can always conceive it as smaller, provided that the recession through subtraction of parts takes place according to a fixed ratio. Thus, let AO be divided at B, C, D, etc., so that AB = ½ AO, BC = ½ BO, CD = ½ CO, and so on. The subtraction of AB, BC, etc., from AO can go on forever, and some of AO will always be left. No matter how small the remaining part becomes, one can conceive of it as still smaller because it, too, is similarly divisible. Consequently, AO is infinite with respect to decrease, when viewed without the initial point it actually has.

Characteristics. What are the characteristics of such infinity? Finitude is, as previously seen, linked with intelligibility, actuality, and perfection, because extension in a finite line is related to its terminal points (and consequent definite dimensions) as matter to form. On the other hand, a line is infinite when its extension is viewed as lacking either its initial or final points and, thus, infinity indicates that a line in such a condition is like matter without form. But form is the source of knowableness, actuality, and perfection. Accordingly, infinity is linked with a state of unintelligibility, mere potentiality, and imperfection. An infinite line, precisely as infinite, is unknowable because it lacks definite dimensions and thus cannot be measured or described. Its infinity, however, is merely a potential condition: every line is actually finite because of its definite length, though it can be considered as subjected to an endless process of addition or of division because of the very nature of quantity, just as primary matter can receive an endless series of substantial forms. An infinite line is imperfect because it is viewed as lacking the determinate dimensions it should and actually does have.

Imperfection of the Infinite. Aristotle frequently stresses this characteristic of imperfection. What is infinite, he explains in an important passage (Phys. 206b 32-207a 2), "turns out to be the contrary of what it is said to be. It is not what has nothing outside it that is infinite, but what always has something outside it." Why so? Because in quantity there is always some part beyond the point one has reached in dividing it, or in building it up by addition. Quantity is infinite because one can always take a part outside what has already been taken. Since, then, that which is infinite always has something outside and beyond, there is always something absent or lacking from it. Thus it is not complete or perfect. In fact, infinity itself is, Aristotle concludes, the very privation of wholeness and perfection, the subject of which is the sensible continuum (ibid., 207b 35-208a 4).

Implications. For Aristotle, then, infinity basically is associated with quantity and is synonymous with imperfection. This synonymity has two important consequences. The Greek philosopher cannot predicate it directly of God Himself (for him, the First Mover and primal Separate Intelligence) but only of His power, and this through an extrinsic predication. That is, His power is so perfect as to be the cause of an infinite effect, viz, the endlessly recurring circular motion of the heavenly bodies through an infinity of time; it is this alone to which infinity directly belongs and through which divine power receives the predication (Meta. 1073a 610). Secondly, the material universe cannot be actually infinite in extent, nor is it merely one of an infinite number of universes, since such sorts of actual infinities are contradictory and impossible. Moreover, it is finite in virtue of the fact that as "uni-verse" it is whole, all-inclusive, complete, and perfect; and whatever is whole, complete, and perfect has an end, which is its limit and termination (Phys. 207a).

Neoplatonism and Plotinus. The second period in the history of infinity, that of neoplatonism, was initiated by plotinus (204270 a.d.). As Aristotle had done before him, Plotinus affirmed that the power of God (the One-Good, the highest hypostasis) is infinite. "He who is capable of making all things, what greatness would He have? He is infinite and, if so, would have no physical magnitude. The Principle would be great in this sense that nothing is more powerful than He or even equally so" (Enn. 6.7.32.14). "The One is the greatest of all things not in physical magnitude but in power, for that which is without extension is great through power. We must also insist that It is not infinite asthough intraversable either in extension or in number but by the unboundedness of Its power" (6.9.6.113; also see2.4.5.1720; 5.5.10.2024). This affirmation is apparently made through extrinsic denomination: the divine power is so great as to be the source of infinite effects material existents that are infinitely numerous because they ceaselessly deploy in endlessly recurring world-cycles (5.7.1.927; 5.7.3.15; 6.2.22.11).

Infinity as Perfection. Unlike Aristotle, though, Plotinus developed a theory of infinity that is synonymous with perfection and that is applicable to God Himself. This theory rests on the insight that form and being are determining and terminating factors wherever found (5.1.7.1926). If something is without form and being, then it is without their determination and, thus, is indeterminate or infinite. If it should possess them but does not, that status of indetermination is coterminous with imperfection. Thus, matter of itself is below form and entity and, accordingly, is indeterminate and simultaneously imperfect (1.8.4.14; 2.4.6.17; 2.4.10.1; 2.4.13.26;2.4.15.16). On the other hand (and of this Aristotle shows no explicit awareness), God rises above the being and form proper only to lower levels of reality, viz, the intelligible, psychical, and sensible universes, and thereby also transcends any formal determination. By this transcen dence He is infinite, and such infinity is aligned with absolute perfection and actual excellence. "Do not remark that [the Good] is in such and such a way because such language would determine It and make It become a particular thing. He who beholds It cannot say that It either is or is not such and such, for thereby he would say that It is one of those beings which can rightly be termed such and such, whereas It really is other than all such beings. Having seen that It is indeterminate, he can enumerate all the beings which come after It and then say that It is nothing of all of them but that It is Total Power which is really master of Itself" (6.8.9.37; see also 6.7.17.1218;5.1.7.1926; 5.5.6.115; 5.5.11.1637).

Infinity and Nonbeing. In thus showing that infinity can be coextensive with perfection and thereby predicable of the divine reality itself, Plotinus made a major contribution to the development of the concept of infinity. But one must remember that this predication is only implicit in Plotinus's text. He explicitly links infinity with nonbeing: the One is stated to be infinite because It transcends being and form. Granted that this infinity of nonbeing can imply that the One Itself is infinite, still this is only an implication.

Christian and Modern Thought

The third period in the history of infinity is occupied by the Fathers of the ChurchSS. Hilary of Poitiers and Augustine in the West; Clement of Alexandria, John Chrysostom, Gregory of Nazianzus, Gregory of Nyssa, Pseudo-Dionysius, and John Damascene in the East. These were all influenced by Sacred Scripture, as well as by Neoplatonism.

Scriptural Teaching. The Bible has only a few and (at best) indirect texts on infinity itself. An example would be the Vulgate's Ps 144.3: "Magnus Dominus et laudabilis nimis et magnitudinis eius non est finis" (the last clause is even less strong in modern translation: " neque explorari potest magnitudo eius"). Still the Scriptures emphasize the awesome power of God (e.g., Gn 17.1; Ps 33.9; Ps 135.6; Jn 1.3), His eternity (Dt 32.40; Gn 21.33), His omnipresence (Dt 4.39; Ps 139.712) coupled with transcendence of any definite place (1 Kgs 8.27; Jb 11.8; Bar 3.25), His otherness from all else (Is 46.9), and the inability of any created intellect to know Him adequately (Rom 11.33; Eph 3.8). The result was that Latin and Greek Fathers of the Church speak of God as infinite in the sense that He is all-powerful, eternal, immense, incomprehensible, and, when under the influence also of Neoplatonism, nonbeing.

Latin and Greek Fathers. St. augustine offers an example of the teaching of the Latin Fathers: "It is evident that the orderly disposition of the universe comes about through a mind [God], and that it can appropriately be called infinite, not in spatial relations, but in power which cannot be understood by human thought.That which is incorporeal can be called both complete and infinite: complete because of its wholeness, infinite because it is not confined by spatial boundaries" (Epist. 118, Fathers of Church ed., 284285). Again: "What is in your mind and heart when you think of a certain substance which is living, perpetual, omnipotent, infinite, everywhere present, everywhere complete, nowhere enclosed? When you think of That, you have a conception of God in your heart" (In evang. Ioh. 1.8 see also Nat. boni 3).

St. john damascene similarly speaks for the Greek Fathers: "[God] is not to be found among beingsnot that He is not but, rather, because He is above all beings and even above being itself. For if knowledge has beings as its objects, then what transcends knowledge also transcends essence and, conversely, what is beyond essence also is beyond knowledge. Therefore, the Divinity is both infinite and incomprehensible, and this alone is comprehensible about HimHis very infinity and incomprehensibility" (De fide orth. 1.4). "We believe in one God: one principle, without beginning, uncreated, unbegotten, indestructible and immortal, eternal, unlimited, uncircumscribed, unbounded, infinite in power, simple, uncompounded, incorporeal" (ibid. 1.8). "Of all the names given to God, the more proper one seems to be that of He Who is . For, like some infinite and indeterminate sea of essence, He has and contains in Himself all beings" (ibid 1.9).

Another author deeply influenced by the Old Testament (in the Septuagint version), who antedates the Fathers and, for that matter, Plotinus too, is philo judaeus of Alexandria, the Jewish theologian and philosopher. For him God is infinite in a threefold way: as incomprehensible, since we can know that He exists but not what He is; as omnipotent, since God freely created the world out of nothing; and as all-good, since He is freely and lovingly provident even over individuals (see H. A. Wolfson, Religious Philosophy [Cambridge, Mass. 1961] 511).

Medieval Scholasticism. The fourth period is that of medieval scholasticism. Although john scotus erigena held a doctrine on divine infinity that seems almost wholly Neoplatonic (see De divisione naturae, 1.56; Patrologia Latina 122; 499D), still from the 10th to the middle of the 13th century Christian authors appear to pay little or no attention to divine infinity. Generally, the term fails even to be listed by theologians among the divine attributes (for example, the Abelardian Ysagoge in theologiam, Robert of Melun's Sententiae, Stephen Langton's Commentarius in sententias, Peter of Poitier's Sententiae ). Occasionally, infinity is applied to divine power or is made synonymous with eternity or with God's incomprehensibility (e.g., Peter of Lombard's Libri IV sententiarum, Hugh of Saint-Victor's De sacramentis christianae fidei, Hugh of Saint-Cher's In sententiarum, Alexander of Hales's Glossa in sent., Albert the Great's In sent. ). But nowhere is it itself discussed at any length.

Bonaventure and Aquinas. After this period of silence, though, the topic is given explicit and detailed attention by Christian authors, two of whom made important contributions to its development and whose position still influences many contemporary scholastics SS. bonaventure (see his In 1 sent. 19.2.3 ad 4; 43.1.1 and ad 3; 43.1.2, written c. 1250) and thomas aquinas (see his In 1 sent., 43.1.1; written c. 1254; ST 1a, 7.12, written c. 1267). Their contribution directly concerns divine infinity, for with their contemporaries they accepted Aristotle's conception of quantitative infinity with reference to lines and numbers and, moreover, agreed that the world is finite in extent and is numerically one. But they broke with Aristotle by predicating infinity of God Himself, as Plotinus also had. Yet their position significantly differs from the Neoplatonist's because it rests upon an obviously different metaphysics.

Aquinaswhose doctrine is considered here for the sake of convenience, although in this matter Bonaventure's position does not differ from Thomas's and, in fact, chronologically anticipates itagreed with Plotinus that forms, and, in general, every sort of act, are determining factors for whatever receives them. Accordingly, a recipient such as matter is indeterminate and infinite (and also imperfect) when considered in itself and as lacking form. But in contrast to Plotinus, Thomas taught also that matter and all other types of potencies are not mere negations, privations, or mental constructs, but are genuinely real and actually existing components within existents, and cause their own sort of determination. Accordingly, a subsistent form or act is without the limiting determination of matter or of potency and, thus, is infinite and infinitely perfect.

Infinity of God. God is such an existent. The divine essence contains no matter or potentiality of any sort and, as such, is totally free from their limitations. Consequently, infinity is a perfection of His very being (see perfec tion, ontological). Whereas each creature is a finite being because it is a composite of act(s) received and determined by potency, God is infinite Being because He is an entirely subsistent, pure act and so without any recipient potency. Perfect Being because He is subsistent existence, God is infinite Being as free from the limiting determination of matter and all potency. (see infinity of god.)

Such a clearly presented and solidly established doctrine was one force that helped focus the attention of Thomas's and Bonaventure's contemporaries and their successors upon infinity as an important topic for discussion. Few if any subsequent scholastic philosophers or theologians failed to investigate "Whether infinity may be attributed as a perfection to the divine being itself," and to give basically the same answer (on duns scotus, see Bettoni, 132159).

Modern Thinkers. Non-scholastic authors in the fifth period, which begins toward the end of the 16th century and continues to the present day, are greatly concerned with infinity too, but the doctrines they elaborate differ greatly from those of preceding eras. With regard to divine infinity they begin with a doctrine that is much the same in content as that of their scholastic predecessors, differing mainly in terminology. For example, R. descartes thought that only God should be called "infinite," whereas quantitative items should be termed "indefinite" (see Reply to Obj. 1, Haldane-Ross transl., 2:17; Principles of Philosophy, 1.24, 26, 27, 1:229230).

God and Infinity. B. spinoza, however, introduced quite a different doctrine, which was demanded by his monism (and, one may add, retained without radical modification by G. W. F. hegel and subsequent monists). Only God is truly real, individual things being mere modes or manifestations of the divine substance. Precisely as individual, as determinate, as this, as finite, they are unreal, since individuality and finitude are mere negations. But God is infinity because He is total reality and sheer affirmation: ("By God I understand Being absolutely infinite. To the essence of that which is absolutely infinite pertains whatever expresses essence and involves no negation" (Ethics, 1.6, ed. J. Wild, 9495).

In an effort to safeguard the reality of individual things while simultaneously retaining God, some subsequent philosophers went to the opposite extreme and made God finiteJ. S. mill, W. james, A. N. white head (see Collins, 285324).

Revolution in Cosmology. The main difference between this period and earlier ones, however, is seen in its cosmology. The Aristotelian notion of the universe as finite in extent and numerically one has been replaced, mainly through the discovery and use of the telescope, by a quite divergent conception. For example, the galaxy seen in the Milky Way, of which the solar system is a tiny part, is simply one among almost innumerable other galaxies. Add this to the fact that the galaxies seem to be receding from one another at enormous speeds, and the inference can easily be drawn that the universe is somehow infinite in extent.

Such a doctrine was anticipated by Giordano bruno and others. Their theorizing has usually been accompanied by a tendency to speak of divine infinity in terms of omnipresence and immensity. God is infinite insofar as He is everywhere present in this infinite universe, which in no way limits, contains, or terminates Him. Unfortunately, in the minds of some, divine infinity is so closely aligned with the infinity of absolute space as almost to seem identical (see pantheism; panentheism).

Modern Mathematics. Coupled with this new cosmology is modern mathematics, including non-Euclidean geometry, which was initiated in the 17th century and came to offer a new approach to mathematical infinity. As developed by Georg cantor, this approach begins with such theorems as, "There are as many negative integers as there are positive integers; there are as many points on a line segment one unit in length as on a line segment two units long , etc." It terminates by defining an infinite set as one that has this property: a proper subset of the set can be put into 1-to-1 reciprocal correspondence with the whole set. In brief, a set is infinite if a part of the set is equal to the whole (see Hausmann, 7689).

Where previously it was assumed that the essence of mathematical infinity lay in quantity and variability, now the concepts of order and multiplicity are regarded as basic. Cantor marked the change by introducing the concept of transfinite numbers. What is a transfinite number? "In general, if we consider any class of sets of elements which are such that they can be put into a one-to-one reciprocal correspondence, we define the property common to every member of the class to be the cardinal number of each set in the class. If we now consider the class of all infinite sets which can be put into a one-to-one reciprocal correspondence, we may define that property of this class to be a transfinite cardinal number" (ibid., 86).

Concluding Summary. Throughout its history infinity has been predicated mainly of quantitative items and of God; in each case it has undergone an evolution of meaning.

With respect to quantity that evolution was postponed until the most recent period. Before that Aristotle's conception held sway. Yet even now his conception is not entirely set aside. True enough, mathematical infinity is newly conceived and interpreted, and the common consensus is that the universe is not the simply bounded system it once was conceived to be. But when one says that the material world is infinite in extent, does he not mean that no matter how large it now actually is, it can still be conceived as (and perhaps will expand to be) of greater extent? If so, this meaning of infinity is still basically Aristotelian.

The evolution with respect to the infinity of God suffered no postponement. Lacking an explicit basis in Sacred Scripture, and linked with imperfection by the ancient Greeks, infinity was first predicated of God only through extrinsic denomination, viz, His power was regarded as infinite because it was capable of producing an infinite number of effects. Plotinus severed this link with imperfection, and infinity became a divine attribute, although explicitly aligned with nonbeing. Utilizing a different metaphysics, Bonaventure and Thomas Aquinas identified infinity with subsistent and all-perfect being. Resorting to a still different metaphysics, Spinoza and other monists made infinity synonymous with reality itself, and tended to reduce all existents other than God to nonentities. In reaction, still others pretended to save individual things by reducing God to their finite and imperfect level.

Bibliography: l. sweeney, "Infinity in Plotinus," Gregorianum 38 (1957) 515535, 713732; "Divine Infinity: 11501250," The Modern Schoolman 35 (195758) 3851; "Another Interpretation of Enneads, VI, 7, 32," ibid. 38 (196061) 289303; "L'Infini quantitatif chez Aristote," Revue philsophique de Louvain 58 (1960) 504528; "Some Mediaeval Opponents of Divine Infinity," Mediaeval Studies 19 (1957) 233245; "Lombard, Augustine and Infinity," Manuscripta 2 (1958) 2440. j. d. collins, God in Modern Philosophy (Chicago 1959) 285324. e. bettoni, Duns Scotus: The Basic Principles of His Philosophy, tr. and ed. b. bonansea (Washington 1961) 132159. c. vollert, "Origin and Age of the Universe Appraised by Science," Theological Studies 18 (1957) 137168. b. a. hausmann, From an Ivory Tower: A Discussion of Philosophical Problems Originating in Modern Mathematics (Milwaukee 1960) 7689. g. gamow, One, Two, Three Infinity (rev. ed. New York 1961). g. cantor, Contributions to the Founding of the Theory of Transfinite Numbers, tr. p. e. b. jourdain (La Salle, Ill. 1941). a. koyrÉ, From the Closed World to the Infinite Universe (Baltimore 1957).

[l. sweeney]

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Infinity

Infinity ★★½ 1996 (PG)

Based on memoirs covering the early years of Nobel Prizewinning physicist Richard Feynman (Broderick) and his romance with aspiring artist Arline Greenbaum (Arquette). They marry despite the fact that Arline is diagnosed with tuberculosis, at this time in the ‘30s a contagious and incurable disease. Richard's recruited to work on the Manhattan Project at Los Alamos, New Mexico, and the narrative travels between his scientific endeavors and Arline's worsening illness in an Albuquerque hospital. Problem is it's neither a character study or a love story but a weak combo. Directing debut of Broderick; screenplay is written by his mother. 119m/C VHS, DVD . Matthew Broderick, Patricia Arquette, James LeGros, Peter Riegert, Dori Brenner, Peter Michael Goetz, Zeljko Ivanek; D: Matthew Broderick; W: Patricia Broderick; C: Toyomichi Kurita; M: Bruce Broughton.

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Infinity

Infinity

The term infinity conveys the mathematical concept of large without bound, and is given the symbol ∞. As children, we learn to count, and are pleased when first we count to 10, then 100, and then 1,000. By the time we reach 1,000, we may realize that counting to 2,000, or certainly 100,000, is not worth the effort. This is partly because we have better things to do, and partly because we realize no matter how high we count, it is always possible to count higher. At this point we are introduced to the infinite, and begin to realize what infinity is and is not.

Infinity is not the largest number. It is the term we use to convey the notion that there is no largest number. We say there is an infinite number of numbers.

There are aspects of the infinite that are not altogether intuitive, however. For example, at first glance there would seem to be half as many odd (or even) integers as there are integers all together. Yet it is certainly possible to continue counting by twos forever, just as it is possible to count by ones forever. In fact, we can count by tens, hundreds, or thousands, it does not matter. Once the counting has begun, it never ends.

What of fractions? It seems that just between zero and one there must be as many fractions as there are positive integers. This is easily seen by listing them, 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8,.... But there are multiples of these fractions as well, for instance, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, and 8/8. Of course many of these multiples are duplicates, 2/8 is the same as 1/4 and so on. It turns out, after all the duplicates are removed, that there is the same number of fractions as there are integers. Not at all an obvious result.

In addition to fractions, or rational numbers, there are irrational numbers, which cannot be expressed as the ratio of whole numbers. Instead, they are recognized by the fact that, when expressed in decimal form, the digits to the right of the decimal point never end, and never form a repeating sequence. Terminating decimals, such as 6.125, and repeating decimals, such as 1.333¯ or 6.534¯ (the bar over the last digits indicates that sequence is to be repeated indefinitely), are rational. Irrational numbers are interesting because they can never be written down. The instant one stops writing down digits to the right of the decimal point, the number becomes rational, though perhaps a good approximation to an irrational number .

It can be proved that there are infinitely more irrational numbers than there are rational numbers, in spite of the fact that every irrational number can be approximated by a rational number . Taken together, the rational and irrational numbers form the set of real numbers .

The word infinite is also used in reference to the very small, or infinitesimal. Consider dividing a line segment in half, then dividing each half, and so on, infinitely many times. This procedure would results in an infinite number of infinitely short line segments. Of course it is not physically possible to carry out such a process; but it is possible to imagine reaching a point beyond which it is not worth the effort to proceed. We understand that the line segments will never have exactly zero length, but after a while no one fully understands what it means to be any shorter. In the language of mathematics , we have approached the limit.

Beginning with the ancient Greeks, and continuing to the turn of the twentieth century, mathematicians either avoided the infinite, or made use of the intuitive concepts of infinitely large or infinitely small. Not until the German mathematician, Georg Cantor (1845-1918), rigorously defined the transfinite numbers did the notion of infinity finally seem fully understood. Cantor defined the transfinite numbers in terms of the number of elements in an infinite set. The natural numbers have u0 elements (the first transfinite number). The real numbers have u1 elements (the second transfinite number). Then, any two sets whose elements can be placed in 1-1 correspondence, have the same number of elements. Following this procedure, Cantor showed that the set of integers, the set of odd (or even) integers, and the set of rational numbers all have u0 elements; and the set of irrational numbers has u1 elements. He was never able, however, to show that no set of an intermediate size between u0 and u1 exists, and this remains unproved today.


Resources

books

Buxton, Laurie. Mathematics for Everyone. New York: Schocken Books, 1985.

Dauben, Joseph Warren. Georg Cantor, His Mathematics and Philosophy of the Infinite. Cambridge: Harvard University Press, 1979.

Paulos, John Allen. Beyond Numeracy, Ruminations of a Numbers Man New York: Knopf, 1991.

periodicals

Moore, A. W. "A Brief History of Infinity." Scientific American 272, no. 4 (1995): 112-16.


J. R. Maddocks

KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Counting numbers

—As the name suggests, the counting numbers are 1,2,3..., also called the natural numbers. The whole numbers are the counting numbers plus zero.

Transfinite numbers

—Transfinite numbers were invented by Georg Cantor as a means of expressing the relative size of infinite sets.

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