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infinity

 
Dictionary: in·fin·i·ty   (ĭn-fĭn'ĭ-tē) pronunciation
 
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n., pl. -ties.
  1. The quality or condition of being infinite.
  2. Unbounded space, time, or quantity.
  3. An indefinitely large number or amount.
  4. Mathematics. The limit that a function f is said to approach at x = a when f(x) is larger than any preassigned number for all x sufficiently near a.
    1. A range in relation to an optical system, such as a camera lens, representing distances great enough that light rays reflected from objects within the range may be regarded as parallel.
    2. A distance setting, as on a camera, beyond which the entire field is in focus.

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Sci-Tech Encyclopedia: Infinity
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The terms infinity and infinite have a variety of related meanings in mathematics. The adjective finite means “having an end,” so infinity may be used to refer to something having no end. In order to give a precise definition, the mathematical domain of discoursemust be specified.

Set theory provides a simple and basic example of an infinite collection—the class of natural numbers, or positive integers. A fundamental property of positive integers is that after each integer there follows a next one, so that there is no last integer. Now it is necessary in mathematics to treat the collection of all positive integers as an entity, and this entity is the simplest infinity, or infinite collection.

The term infinity appears in mathematics in a different sense in connection with limits of functions. For example, consider the function defined by y = 1/x. When x tends to 0, y approaches infinity, and the expression may be written as shown below. \lim_{x\to 0}y=\infty

Precisely, this means that for an arbitrary number a > 0, there exists a number b > 0 such that when 0 < x < b, then y > a, and when −b < x < 0, then y < −a. This example indicates that it is sometimes useful to distinguish +∞ and −∞. The points +∞ and −∞ are pictured at the two ends of the y axis, a line which has no ends in the proper sense of euclidean geometry.

In geometry of two or more dimensions, it is sometimes said that two parallel lines meet at infinity. This leads to the conception of just one point at infinity on each set of parallel lines and of a line at infinity on each set of parallel planes. With such agreements, parts of euclidean geometry can be discussed in the terms of projective geometry. For example, one may speak of the asymptotes of a hyperbola as being tangent to the hyperbola at infinity.

 
Antonyms: infinity
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n

Definition: endlessness
Antonyms: bounds, definiteness, ending, finiteness, limitation, limitedness

 
Hacker Slang: infinity
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1. The largest value that can be represented in a particular type of variable (register, memory location, data type, whatever).

2. minus infinity: The smallest such value, not necessarily or even usually the simple negation of plus infinity. In N-bit twos-complement arithmetic, infinity is 2N-1 - 1 but minus infinity is - (2N-1), not -(2N-1 - 1). Note also that this is different from time T equals minus infinity, which is closer to a mathematician's usage of infinity.

 
Measures and Units: infinity
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mathematics Symbol ∞. A term meaning the boundlessly large, but not necessarily uncountable, actually inclusive of a range of distinct numbers, all being boundlessly large but not mutually equal. The smallest infinity corresponds to the number of integers. It is called aleph-null or aleph-nought, written ℵ0, being the first letter of the Hebrew alphabet with a zero subscript, and, perhaps surprisingly, it can be shown to be identically the number of rational numbers. Despite the general understanding that anything infinite is uncountable, to the mathematician this smallest infinity is regarded as countable; to count an unbounded group one just attaches 1 to the first, 2 to the second, et seq. Such a procedure will not only cover all the natural numbers but also all the integers, indeed all rational numbers. The integers can be so numbered by doubling each unsigned value and adding 1 if originally negative. The rationals m/n between 0 and 1 can be uniquely numbered as m + Σ(n - 2), where Σ means the sum of all positive integers up to that value (numbering ½ as 1, ⅓ as 2, ⅔ as 3, ¼ as 4, ¾ as 6, et seq). The full range of rationals can be shown to be similarly countable. However, rational numbers form only a small minority of numbers. There is no such method for attaching uniquely the integers to all numbers between 0 and 1; they cannot be uniquely numbered in the integer sense, since they are not countable. Their number is symbolized by the letter C or c (for continuum); C is not equal to ℵ0.

Each of these two infinities, individually and separately, is such that most arithmetic on it leaves it unchanged. Multiplication, division, addition, or even subtraction by itself leaves it unchanged. (Hence, for instance, the number of rational numbers between 0 and 1 is the same as between 0 and 100, or any other value; likewise C is the number of numbers in any proper interval.) However, exponentiation changes it; for any distinct infinity ∞, the exponentiated value ∞∞ is not equal to the original ∞. The result of such exponentiation is a new infinite number; starting from aleph-null there is aleph-one; then, if this is self-exponentiated, aleph-two, and so on, i.e. ℵ0, ℵ1, ℵ2,… Since there is no limit to this process, there is no limit to the number of distinct infinities. The number of such would appear to be countable in the mathematician's language, but uncountable in popular terminology. Since any number between 0 and 1 can be represented by a countable string of decimal (else binary else duodecimal else hexadecimal else…) digits, C must equal any finite integer to the power ℵ0. Since this is definitely greater than ℵ0 but not greater than ℵ1, it must equal the latter or some further distinct infinity between those two. So C is not by any means the largest infinity; what is remains a puzzle, as indeed does the meaning of all after ℵ1.

 
Britannica Concise Encyclopedia: infinity
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In mathematics, the useful concept of a process with no end. As represented by the symbol ¥, it is often mistakenly thought to be the largest number or a place on the real number line. Instead, it is the idea of a limit, as in the expression x ® ¥, which suggests that the variable x increases without bound. For example, the function f(x) = 1/x, or the reciprocal of x, tends toward 0 as x approaches infinity as a limit. This process of approaching is crucial to the definition of the derivative and the integral in calculus, as well as to many other concepts of mathematical analysis.

For more information on infinity, visit Britannica.com.

 
Philosophy Dictionary: infinity
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The unlimited; that which goes beyond any fixed bound. Exploration of this notion goes back at least to Zeno of Elea, and extensive mathematical treatment began with Eudoxus of Cnidus (4th c. BC). The mathematical notion was further developed by Cantor and K. T. W. Weierstrass (1815-97) in the 19th century. In the philosophy of space and time problems arise both with the infinitely small (see Bayle's trilemma, Zeno's paradoxes) and with the infinitely large or boundless nature that each seems, intuitively, to possess. Kant argued in the antinomies that consistently regarding space or time as either finite or as infinite, was impossible, and this formed a key element in his idealist theory of them as imposed upon an unknown (noumenal) nature by our own forms of sense. Philosophically one important division is between thinkers such as Cantor who can countenance actual completed infinities, and those such as Aristotle or the mathematician Leopold Krönecker (1823-91) who reject any such notion in favour of merely ‘potential’ infinities.

 
Spotlight: infinity
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From our Archives: Today's Highlights, November 23, 2005

The man who introduced the symbol ∞ for infinity, John Wallis, was born on this date in 1616. The mathematician who made major contributions to trigonometry, calculus, geometry, and the analysis of infinite series was the third Savilian Chair of Geometry. For many years the British Parliament's chief cryptographer, Wallis was part of the initial group of scientists who became the Royal Society.
 
Columbia Encyclopedia: infinity
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infinity, in mathematics, that which is not finite. 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.)

Bibliography

See A. D. Aczel, The Mystery of the Aleph (2000); D. F. Wallace, Everything and More (2003).

 
Quotes About: Infinity
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Quotes:

"There is no more steely barb than that of the Infinite." - Charles Baudelaire

"Offer unto me that which is very dear to thee -- which thou holdest most covetable. Infinite are the results of such an offering." - Bhagavad Gita

"The poetic notion of infinity is far greater than that which is sponsored by any creed." - Joseph Brodsky

"While many people are trying to be in tune with infinite, what they really are is in tune with the indefinite." - Eric Butterworth

"Whenever we encounter the Infinite in man, however imperfectly understood, we treat it with respect. Whether in the synagogue, the mosque, the pagoda, or the wigwam, there is a hideous aspect which we execrate and a sublime aspect which we venerate. So great a subject for spiritual contemplation, such measureless dreaming -- the echo of God on the human wall!" - Victor Hugo

"Seek the Infinite, for that alone is Joy unlimited, imperishable, unfailing, self-sustaining, unconditioned, timeless. When you have this joy, human life becomes a paradise; the light, the grace, the power, the perfections of that which is highest in your inner consciousness, appear in your everyday life." - Swami Omkarananda

See more famous quotes about Infinity

 
Wikipedia: Infinity
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For other uses, see Infinity (disambiguation).
The infinity symbol, ∞, in several typefaces.

Infinity (symbolically represented by ∞) refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology.[1] The word comes from the Latin infinitas or "unboundedness."

In mathematics, "infinity" is often used in contexts where it is treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is a different type of "number" from the real numbers. Infinity is related to limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals,[2] Russell's paradox, non-standard arithmetic, hyperreal numbers, projective geometry, extended real numbers and the absolute Infinite.

Contents

In philosophy

Main article: Infinity (philosophy)

Early Indian views of infinity

The Isha Upanishad of the Yajurveda (c. 4th to 3rd century BC) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".

Pūrṇam adaḥ pūrṇam idam
Pūrṇāt pūrṇam udacyate
Pūrṇasya pūrṇam ādāya
Pūrṇam evāvasiṣyate.
That is whole, this is whole
From the whole, the whole arises[3]
When the whole is taken from the whole
The whole still will remain — Isha Upanishad.

The Indian mathematical text Surya Prajnapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:

  • Enumerable: lowest, intermediate and highest
  • Innumerable: nearly innumerable, truly innumerable and innumerably innumerable
  • Infinite: nearly infinite, truly infinite, infinitely infinite

The Jains were the first to discard the idea that all infinites were the same or equal. They recognized different types of infinities: infinite in length (one dimension), infinite in area (two dimensions), infinite in volume (three dimensions), and infinite perpetually (infinite number of dimensions).

According to Singh (1987), Joseph (2000) and Agrawal (2000), the highest enumerable number N of the Jains corresponds to the modern concept of aleph-null \aleph_0 (the cardinal number of the infinite set of integers 1, 2, ...), the smallest cardinal transfinite number. The Jains also defined a whole system of infinite cardinal numbers, of which the highest enumerable number N is the smallest.

In the Jaina work on the theory of sets, two basic types of infinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asaṃkhyāta ("countless, innumerable") and ananta ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.

Early Greek views of infinity

In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the actual infinite; for example, instead of saying that there are an infinity of primes, Euclid prefers instead to say that there are more prime numbers than contained in any given collection of prime numbers (Elements, Book IX, Proposition 20).

However, recent readings of the Archimedes Palimpsest have hinted that at least Archimedes had an intuition about actual infinite quantities.

In logic

In logic an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."[4]

Mathematical infinity

 This section does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unverifiable material may be challenged and removed. (June 2007)

The notion of infinity has been formalized in various branches of mathematics.

Calculus

Real analysis

Further information: Extended real number line

In real analysis, the symbol \infty, called "infinity", denotes an unbounded limit. x \rightarrow \infty means that x grows without bound, and x \rightarrow -\infty means the value of x is decreasing without bound. If f(t) ≥ 0 for every t, then

  • \int_{a}^{b} \, f(t)\ dt \ = \infty means that f(t) does not bound a finite area from a to b
  • \int_{-\infty}^{\infty} \, f(t)\ dt \ = \infty means that the area under f(t) is infinite.
  • \int_{-\infty}^{\infty} \, f(t)\ dt \ = n means that the total area under f(t) is finite, and equals n

Infinity is also used to describe infinite series:

  • \sum_{i=0}^{\infty} \, f(i) = a means that the sum of the infinite series converges to some real value a.
  • \sum_{i=0}^{\infty} \, f(i) = \infty means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound.

Infinity is often used not only to define a limit but as a value in the affinely extended real number system. Points labeled \infty and -\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat \infty and -\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also introduces a line at infinity in plane geometry, and so forth for higher dimensions.

The extended real number line adds two elements: infinity (\infty), greater than all other extended real numbers, and negative infinity (-\infty), less than all other extended real numbers, for which some arithmetic operations may be performed.

The infinity symbol

 This section may contain original research or unverified claims. Please improve the article by adding references. See the talk page for details. (May 2008)
John Wallis introduced the infinity symbol to mathematical literature.

The precise origin of the infinity symbol, \infty, is unclear. One possibility is suggested by the name it is sometimes called—the lemniscate, from the Latin lemniscus, meaning "ribbon".

John Wallis is usually credited with introducing \infty as a symbol for infinity in 1655 in his De sectionibus conicis. One conjecture about why he chose this symbol is that he derived it from a Roman numeral for 1000 that was in turn derived from the Etruscan numeral for 1000, which looked somewhat like CIƆ and was sometimes used to mean "many." Another conjecture is that he derived it from the Greek letter ω (omega), the last letter in the Greek alphabet.[5]

The infinity symbol is available in standard HTML as &infin; and in LaTeX as \infty. In Unicode, it is the character at code point U+221E, or 8734 in decimal notation.

Complex analysis

As in real analysis, in complex analysis the symbol \infty, called "infinity", denotes an unsigned infinite limit. x \rightarrow \infty means that the magnitude | x | of x grows beyond any assigned value. A point labeled \infty can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold, or Riemann surface, called the extended complex plane or the Riemann sphere. Arithmetic operations similar to those given below for the extended real numbers can also be defined, though there is no distinction in the signs (therefore one exception is that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely z/0 = \infty for any complex number z. In this context is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of \infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformations.

Nonstandard analysis

Main article: Nonstandard analysis

The original formulation of infinitesimal calculus by Newton and Leibniz used infinitesimal quantities. In the twentieth century, it was shown that this treatment could be put on a rigorous footing through various logical systems, including smooth infinitesimal analysis and nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a whole field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to non-standard calculus is fully developed in H. Jerome Keisler's book (see below).

Set theory

Main articles: Cardinality and Ordinal number

A different form of "infinity" are the ordinal and cardinal infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null (\aleph_0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of collections, or sets.

Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. An infinite set can simply be defined as one having the same size as at least one of its "proper" parts; this notion of infinity is called Dedekind infinite.

Cantor defined two kinds of infinite numbers, the ordinal numbers and the cardinal numbers. Ordinal numbers may be identified with well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. If a set is too large to be put in one to one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.

Our intuition gained from finite sets breaks down when dealing with infinite sets. One example of this is Hilbert's paradox of the Grand Hotel.

Cardinality of the continuum

Main article: Cardinality of the continuum

One of Cantor's most important results was that the cardinality of the continuum (\mathbf c) is greater than that of the natural numbers ({\aleph_0}); that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that \mathbf{c} = 2^{\aleph_0} > {\aleph_0} (see Cantor's diagonal argument).

The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, \mathbf{c} = \aleph_1 = \beth_1 (see Beth one). However, this hypothesis can neither be proved nor disproved within the widely accepted Zermelo-Fraenkel set theory, even assuming the Axiom of Choice.

Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space. These results are highly counterintuitive, because they imply that there exist proper subsets of an infinite set S that have the same size as S.

The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval [-0.5π, 0.5π] and R (see also Hilbert's paradox of the Grand Hotel). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points in the side of a square and those in the square.

Cantor also showed that sets with cardinality strictly greater than \mathbf c exist (see his generalized diagonal argument and theorem). They include, for instance:

  • the set of all subsets of R, i.e., the power set of R, written P(R) or 2R
  • the set RR of all functions from R to R

Both have cardinality 2^\mathbf {c} = \beth_2 > \mathbf c (see Beth two).

The cardinal equalities \mathbf{c}^2 = \mathbf{c}, \mathbf c^{\aleph_0} = \mathbf c, and \mathbf c ^{\mathbf c} = 2^{\mathbf c} can be demonstrated using cardinal arithmetic:

\mathbf{c}^2 = (2^{\aleph_0})^2 = 2^{2\times{\aleph_0}} = 2^{\aleph_0} = \mathbf{c},
\mathbf c^{\aleph_0} = (2^{\aleph_0})^{\aleph_0} = 2^{{\aleph_0}\times{\aleph_0}} = 2^{\aleph_0} = \mathbf{c},
\mathbf c ^{\mathbf c} = (2^{\aleph_0})^{\mathbf c} = 2^{\mathbf c\times\aleph_0} = 2^{\mathbf c}.

Geometry and topology

Infinite-dimensional spaces are widely used in geometry and topology. Common examples are the infinite-dimensional complex projective space K(Z,2) and the infinite-dimensional real projective space K(Z/2Z,1).

Fractals

The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have an infinite perimeter resp. an infinite surface area. An example for a fractal curve of infinite length is the Koch snowflake.

Mathematics without infinity

Leopold Kronecker rejected the notion of infinity and began a school of thought, in the philosophy of mathematics called finitism which influenced the philosophical and mathematical school of mathematical constructivism.

Physical infinity

In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting). It is therefore assumed by physicists that no measurable quantity could have an infinite value[citation needed] , for instance by taking an infinite value in an extended real number system (see also: hyperreal number), or by requiring the counting of an infinite number of events. It is for example presumed impossible for any body to have infinite mass or infinite energy. There exists the concept of infinite entities (such as an infinite plane wave) but there are no means to generate such things.

It should be pointed out that this practice of refusing infinite values for measurable quantities does not come from a priori or ideological motivations, but rather from more methodological and pragmatic motivations[citation needed]. One of the needs of any physical and scientific theory is to give usable formulas that correspond to or at least approximate reality. As an example if any object of infinite gravitational mass were to exist, any usage of the formula to calculate the gravitational force would lead to an infinite result, which would be of no benefit since the result would be always the same regardless of the position and the mass of the other object. The formula would be useful neither to compute the force between two objects of finite mass nor to compute their motions. If an infinite mass object were to exist, any object of finite mass would be attracted with infinite force (and hence acceleration) by the infinite mass object, which is not what we can observe in reality. Sometimes infinite result of a physical quantity may mean that the theory being used to compute the result may be approaching the point where it fails. This may help to indicate the limitations of a theory.

This point of view does not mean that infinity cannot be used in physics. For convenience's sake, calculations, equations, theories and approximations often use infinite series, unbounded functions, etc., and may involve infinite quantities. Physicists however require that the end result be physically meaningful. In quantum field theory infinities arise which need to be interpreted in such a way as to lead to a physically meaningful result, a process called renormalization.

However, there are some theoretical circumstances where the end result is infinity. One example is the singularity in the description of black holes. Some solutions of the equations of the general theory of relativity allow for finite mass distributions of zero size, and thus infinite density. This is an example of what is called a mathematical singularity, or a point where a physical theory breaks down. This does not necessarily mean that physical infinities exist; it may mean simply that the theory is incapable of describing the situation properly. Two other examples occur in inverse-square force laws of the gravitational force equation of Newtonian Gravity and Coulomb's Law of electrostatics. At r=0 these equations evaluate to infinities.

Infinity in cosmology

Main article: Physical cosmology

An intriguing question is whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"? This is an important open question of cosmology. Note that the question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology; if one travelled in a straight line through the universe perhaps one would eventually revisit one's starting point.

If, on the other hand, the universe were not curved like a sphere but had a flat topology, it could be both unbounded and infinite. The curvature of the universe can be measured through multipole moments in the spectrum of the Cosmic Background Radiation. As to date, analysis of the radiation patterns recorded by the WMAP satellite hints that the universe has a flat topology. This would be consistent with an infinite physical universe. The Planck satellite scheduled for launch in 2009 is expected to record the Cosmic Background Radiation with ten times higher precision, and will give more insight into the question whether the universe is infinite or not.

Computer representations of infinity

The IEEE floating-point standard specifies positive and negative infinity values; these can be the result of arithmetic overflow, division by zero, or other exceptional operations.

Some programming languages (for example, J and UNITY) specify greatest and least elements, i.e. values that compare (respectively) greater than or less than all other values. These may also be termed top and bottom, or plus infinity and minus infinity; they are useful as sentinel values in algorithms involving sorting, searching or windowing. In languages that do not have greatest and least elements, but do allow overloading of relational operators, it is possible to create greatest and least elements (with some overhead, and the risk of incompatibility between implementations).

Perspective and points at infinity in the arts

Perspective artwork utilizes the concept of imaginary vanishing points, or points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that 'realistically' depict distance and foreshortening of objects. Artist M. C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.

See also

Sister project Wikibooks has a book on the topic of
Sister project Look up infinity in Wiktionary, the free dictionary.

References

  1. ^ About Infinity, by William Kahan.
  2. ^ Large cardinals are quantitative infinities defining the number of things in a collection, which are so large that they cannot be proven to exist in the ordinary mathematics of Zermelo-Fraenkel plus Choice (ZFC).
  3. ^ the whole arises or the whole comes out of (not quite in the sense of subtraction)
  4. ^ Cambridge Dictionary of Philosophy, Second Edition, p. 429
  5. ^ The History of Mathematical Symbols, By Douglas Weaver, Mathematics Coordinator, Taperoo High School with the assistance of Anthony D. Smith, Computing Studies teacher, Taperoo High School.

Other references

External links

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Translations: Infinity
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Dansk (Danish)
n. - uendelighed, uendelig størrelse/antal

idioms:

  • an infinity of    en uendelig størrelse

Nederlands (Dutch)
oneindigheid, eindeloosheid

Français (French)
n. - éternité, infinité

idioms:

  • an infinity of    une infinité de

Deutsch (German)
n. - Unendlichkeit, Grenzenlosigkeit, unendliche Menge

idioms:

  • an infinity of    unendlich viele

Ελληνική (Greek)
n. - απειρία, άπειρο, απεραντοσύνη

idioms:

  • an infinity of    αναρίθμητοι.., απροσμέτρητοι..

Italiano (Italian)
infinito, infinità

idioms:

  • an infinity of    un'infinità di

Português (Portuguese)
n. - infinidade (f)

idioms:

  • an infinity of    uma enorme quantidade de

Русский (Russian)
бесконечность

idioms:

  • an infinity of    множество

Español (Spanish)
n. - eternidad, infinitud, infinidad, infinito

idioms:

  • an infinity of    un sinfín de

Svenska (Swedish)
n. - oändlighet(en), oändlig mängd

中文（简体）(Chinese (Simplified))
无限大, 无限

idioms:

  • an infinity of    无数的..., 无限的..., 无穷的...

中文（繁體）(Chinese (Traditional))
n. - 無限大, 無限

idioms:

  • an infinity of    無數的..., 無限的..., 無窮的...

한국어 (Korean)
n. - 무한, 무한량, (수학의) 무한대

idioms:

  • an infinity of    무수한

日本語 (Japanese)
n. - 無限, 無限のもの, 無限大, 無限遠

idioms:

  • an infinity of    莫大な数

العربيه (Arabic)
‏(الاسم) اللا نهائي, اللا محدوديه‏

עברית (Hebrew)
n. - ‮אינסופיות, אין-סוף, נצח‬

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Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved.  Read more
Sci-Tech Encyclopedia. McGraw-Hill Encyclopedia of Science and Technology. Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.  Read more
Thesaurus. Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved.  Read more
Answers Corporation Antonyms. © 1999-2009 by Answers Corporation. All rights reserved.  Read more
Hacker Slang. The Jargon File. Copyright © 2007.  Read more
Measures and Units. A Dictionary of Weights, Measures, and Units. Copyright © Donald Fenna 2002, 2004. All rights reserved.  Read more
Britannica Concise Encyclopedia. Britannica Concise Encyclopedia. © 2006 Encyclopædia Britannica, Inc. All rights reserved.  Read more
Philosophy Dictionary. The Oxford Dictionary of Philosophy. Copyright © 1994, 1996, 2005 by Oxford University Press. All rights reserved.  Read more
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Columbia Encyclopedia. The Columbia Electronic Encyclopedia, Sixth Edition Copyright © 2003, Columbia University Press. Licensed from Columbia University Press. All rights reserved. www.cc.columbia.edu/cu/cup/  Read more
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Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Infinity".  Read more
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From Today's Highlights
November 23, 2005

To see a world in a grain of sand, And a heaven in a wild flower; Hold infinity in the palm of your hand, And eternity in an hour.
- William Blake

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