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| {{For|other uses of "Infinity" and "Infinite"}}
| | I am Sommer from Bowil. I love to play Guitar. Other hobbies are Woodworking.<br>xunjie といくつかのも開いた10年である、 |
| {{pp-move-indef}}
| | それは明らかに北京オリンピックロンドン五輪相手ではありませんが、 |
| [[File:Infinity symbol.svg|thumb|200px|right|The ∞ symbol in several typefaces]]
| | 太陽の大きな男の子が意味を強調するように、 [http://www.schochauer.ch/_js/r/e/mall/watch/gaga/ gaga �rӋ ���] 高い水準の作品を表示する？加えて、 |
| '''Infinity''' (symbol: '''<big>∞</big>''') is an abstract concept describing something ''without any limit'' and is relevant in a number of fields, predominantly [[mathematics]] and [[physics]]. The English word ''infinity'' derives from [[Latin]] ''infinitas'', which can be translated as "unboundedness", itself [[calque]]d from the Greek word ''apeiros'', meaning "endless".<ref>[http://www.etymonline.com/index.php?allowed_in_frame=0&search=infinity&searchmode=none etymonline] Retrieved 2012-03-06</ref>
| | ローマ世紀のブランドの下着の男性は、 |
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| | 健康的な愛は「ゼロホルムアルデヒド」、 [http://www.steadfast-hawaii.org/img/p/hot/chloe.php ���� �Хå� ���] それは彼のズボンを得ることができる必要があります。 |
| In mathematics, "infinity" is often treated as if it were a [[number]] (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the [[real number]]s. In number systems incorporating [[infinitesimal]]s, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number. [[Georg Cantor]] formalized many ideas related to infinity and [[infinite set]]s during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called [[cardinality|cardinalities]]).<ref>{{cite book
| | 国内ブランドメーカーの何千もの戦略的提携の8主要な学校とディWeina割引の女性のイニシアティブパーソナライズレジャー、 |
| |title=The Princeton Companion to Mathematics
| | そこに第二と第三の段階であるというようになります。[http://www.swiss-equity-fair.ch/pagemedia/js/li/bag/coach/ ���� ؔ�� ���] プリマスの詩に焦点を当て。 |
| |first1=Timothy
| | ファッションで友達Kefanへの完全に身を包んを発表し、 |
| |last1=Gowers
| | それをどのように選択するか？あなたを知らない、 |
| |first2=June
| | 多様なファッション文化、 [http://www.tobler-verlag.ch/flash/ja/li/top/gaga/ �����<br><br>� �٥��] |
| |last2=Barrow-Green
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| |first3=Imre
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| |last3=Leader
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| |publisher=Princeton University Press
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| |year=2008
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| |isbn=0-691-11880-9
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| |page=616
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| |url=http://books.google.com/books?id=LmEZMyinoecC}}, [http://books.google.com/books?id=LmEZMyinoecC&pg=PA616 Extract of page 616]
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| </ref> For example, the set of [[integer]]s is [[Countable set|countably infinite]], while the infinite set of real numbers is [[Uncountable set|uncountable]].<ref>{{harvnb|Maddox|2002|loc=pp. 113 –117}}</ref>
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| ==History==
| | Here is my web site [http://www.schochauer.ch/_js/p/list/jimmychoo/ ジミーチュウ 靴 レディース] |
| {{Main|Infinity (philosophy)}}
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| Ancient cultures had various ideas about the nature of infinity. The [[Maurya Empire|ancient Indians]] and [[ancient Greece|Greeks]] did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
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| ===Early Greek===
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| The earliest recorded idea of infinity comes from [[Anaximander]], a [[Pre-Socratic philosophy|pre-Socratic]] Greek philosopher who lived in [[Miletus]]. He used the word [[apeiron (disambiguation)|apeiron]] which means infinite or limitless.<ref>{{harvnb|Wallace|2004|loc=pg. 44}}</ref> However, the earliest attestable accounts of mathematical infinity come from [[Zeno of Elea]] (c. 490 BCE? – c. 430 BCE?), a [[Pre-Socratic philosophy|pre-Socratic]] Greek philosopher of southern Italy and member of the [[Eleatics|Eleatic]] School founded by [[Parmenides]]. [[Aristotle]] called him the inventor of the [[dialectic]]. He is best known for his [[Zeno's paradoxes|paradoxes]], described by [[Bertrand Russell]] as "immeasurably subtle and profound".
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| In accordance with the traditional view of Aristotle, the [[Hellenistic]] Greeks generally preferred to distinguish the [[potential infinity]] from the [[actual infinity]]; 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 ([[Euclid's Elements|Elements]], Book IX, Proposition 20).
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| However, recent readings of the [[Archimedes Palimpsest]] have hinted that Archimedes at least had an intuition about actual infinite quantities.
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| ===Early Indian===
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| The [[Indian mathematics|Indian mathematical]] text [[Sūryaprajñapti|Surya Prajnapti]] (c. 3rd–4th century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
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| * Enumerable: lowest, intermediate, and highest
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| * Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
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| * Infinite: nearly infinite, truly infinite, infinitely infinite
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| In the Indian work on the theory of sets, two basic types of infinite numbers are distinguished. On both physical and [[Ontology|ontological]] grounds, a distinction was made between [[Asaṃkhyeya|{{IAST|''asaṃkhyāta''}}]] ("countless, innumerable") and ''[[Ananta (infinite)|ananta]]'' ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.
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| <!--=== Buddhism ===
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| In some [[Buddhist]] imagery, including [[Tibetan Buddhism|Tibetan]] Buddhist [[thangka]] and [[vajrayana]] meditation deities such as [[Chenrezig]], the deity is pictured holding a [[Buddhist prayer beads|mala]] twisted in the middle to form a figure 8. This represents the endless (infinite) cycle of existence, of birth, death, and rebirth, i.e., the [infinity of] [[samsara]].{{Citation needed|date=May 2010}}-->
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| ==Mathematics==
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| {{refimprove section|date=December 2013}}
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| ===Infinity symbol===
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| {{main|Infinity symbol}}
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| The infinity symbol <math>\infty</math> (sometimes called the [[lemniscate]]) is a mathematical symbol representing the concept of infinity. The symbol is encoded in [[Unicode]] at {{unichar|221E|infinity|html=|size=200%}} and in [[LaTeX]] as <code>\infty</code>.
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| It was introduced in 1655 by [[John Wallis]],<ref>{{citation
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| | last = Scott | first = Joseph Frederick
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| | edition = 2
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| | isbn = 0-8284-0314-7
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| | page = 24
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| | publisher = [[American Mathematical Society]]
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| | title = The mathematical work of John Wallis, D.D., F.R.S., (1616–1703)
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| | url = http://books.google.com/books?id=XX9PKytw8g8C&pg=PA24
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| | year = 1981}}.</ref><ref>{{citation
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| | last = Martin-Löf | first = Per | author-link = Per Martin-Löf
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| | contribution = Mathematics of infinity
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| | doi = 10.1007/3-540-52335-9_54
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| | location = Berlin
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| | mr = 1064143
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| | pages = 146–197
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| | publisher = Springer
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| | series = Lecture Notes in Computer Science
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| | title = COLOG-88 (Tallinn, 1988)
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| | volume = 417
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| | year = 1990}}.</ref> and, since its introduction, has also been used outside mathematics in modern mysticism<ref>{{citation|title=Dreams, Illusion, and Other Realities|first=Wendy Doniger|last=O'Flaherty|publisher=University of Chicago Press|year=1986|isbn=9780226618555|page=243|url=http://books.google.com/books?id=vhNNrX3bmo4C&pg=PA243}}.</ref> and literary symbology.<ref>{{citation|title=Nabokov: The Mystery of Literary Structures|first=Leona|last=Toker|publisher=Cornell University Press|year=1989|isbn=9780801422119|page=159|url=http://books.google.com/books?id=Jud1q_NrqpcC&pg=PA159}}.</ref>
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| ===Calculus===
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| [[Gottfried Wilhelm Leibniz|Leibniz]], one of the co-inventors of [[infinitesimal calculus]], speculated widely about infinite numbers and their use in mathematics. To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties.<ref>{{SEP|continuity|Continuity and Infinitesimals|[[John Lane Bell]]}}</ref><ref>{{cite journal
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| | last = Jesseph
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| | first = Douglas Michael
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| | year = 1998
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| | title = Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes
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| | journal = [[Perspectives on Science]]
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| | volume = 6
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| | issue = 1&2
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| | pages = 6–40
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| | issn = 1063-6145
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| | oclc = 42413222
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| | url = http://muse.jhu.edu/journals/perspectives_on_science/v006/6.1jesseph.html
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| | accessdate = 16 February 2010
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| | archivedate = 16 February 2010
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| | archiveurl = http://www.webcitation.org/5nZWht6FE
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| }}</ref>
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| ====Real analysis====
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| In [[real analysis]], the symbol <math>\infty</math>, called "infinity", is used to denote an unbounded [[Limit of a function|limit]].<ref>{{harvnb|Taylor|1955|loc=p. 63}}</ref> <math>x \rightarrow \infty</math> means that ''x'' grows without bound, and <math>x \to -\infty</math> means the value of ''x'' is decreasing without bound. If ''f''(''t'') ≥ 0 for every ''t'', then<ref>These uses of infinity for integrals and series can be found in any standard calculus text, such as, {{harvnb|Swokoski|1983|loc=pp. 468-510}}</ref>
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| * <math>\int_{a}^{b} \, f(t)\ dt \ = \infty</math> means that ''f''(''t'') does not bound a finite area from <math>a</math> to <math>b</math>
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| * <math>\int_{-\infty}^{\infty} \, f(t)\ dt \ = \infty</math> means that the area under ''f''(''t'') is infinite.
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| * <math>\int_{-\infty}^{\infty} \, f(t)\ dt \ = a</math> means that the total area under ''f''(''t'') is finite, and equals <math>a</math>
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| Infinity is also used to describe [[infinite series]]:
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| * <math>\sum_{i=0}^{\infty} \, f(i) = a</math> means that the sum of the infinite series [[convergent series|converges]] to some real value <math>a</math>.
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| * <math>\sum_{i=0}^{\infty} \, f(i) = \infty</math> means that the sum of the infinite series [[divergent series|diverges]] in the specific sense that the partial sums grow without bound.
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| Infinity can be used not only to define a limit but as a value in the extended real number system. Points labeled <math>+\infty</math> and <math>-\infty</math> can be added to the [[topological space]] of the real numbers, producing the two-point [[compactification (mathematics)|compactification]] of the real numbers. Adding algebraic properties to this gives us the [[extended real number]]s. We can also treat <math>+\infty</math> and <math>-\infty</math> as the same, leading to the one-point compactification of the real numbers, which is the [[real projective line]].<ref>{{harvnb|Gemignani|1990|loc=p. 177}}</ref> [[Projective geometry]] also refers to a [[line at infinity]] in plane geometry, a [[plane at infinity]] in three dimensional space, and so forth for higher [[dimension]]s.
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| ====Complex analysis====
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| As in real analysis, in [[complex analysis]] the symbol <math>\infty</math>, called "infinity", denotes an unsigned infinite [[Limit (mathematics)|limit]]. <math>x \rightarrow \infty</math> means that the magnitude <math>|x|</math> of ''x'' grows beyond any assigned value. A [[Point at infinity|point labeled <math>\infty</math>]] can be added to the complex plane as a [[topological space]] giving the one-point [[Compactification (mathematics)|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 above 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 <math>z/0 = \infty</math> for any nonzero complex number ''z''. In this context it is often useful to consider [[meromorphic function]]s as maps into the Riemann sphere taking the value of <math>\infty</math> 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 transformation]]s.
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| ===Nonstandard analysis===
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| The original formulation of [[infinitesimal calculus]] by [[Isaac Newton]] and [[Gottfried Leibniz]] used [[infinitesimal]] quantities. In the twentieth century, it was shown that this treatment could be put on a rigorous footing through various [[logical system]]s, 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 [[hyperreal number|hyperreal field]]; there is no equivalence between them as with the Cantorian [[transfinite number|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 {{harvtxt|Keisler|1986}}.
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| ===Set theory===
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| {{Main|Cardinality|Ordinal number}}
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| [[File:Infinity paradoxon - one-to-one correspondence between infinite set and proper subset.gif|thumb|One-to-one correspondence between infinite set and proper subset]]
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| A different form of "infinity" are the [[Ordinal number|ordinal]] and [[cardinal number|cardinal]] infinities of set theory. [[Georg Cantor]] developed a system of [[transfinite number]]s, in which the first transfinite cardinal is [[aleph-null]] <math>(\aleph_0)</math>, the cardinality of the set of [[natural number]]s. 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.
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| 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 subset|proper]] parts; this notion of infinity is called [[Dedekind infinite]]. The diagram gives an example: viewing lines as infinite sets of points, the left half of the lower blue line can be mapped in a one-to-one manner (green correspondences) to the higher blue line, and, in turn, to the whole lower blue line (red correspondences); therefore the whole lower blue line and its left half have the same cardinality, i.e. "size".
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| Cantor defined two kinds of infinite numbers: ordinal numbers and 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 [[sequence]]s which are maps from the positive [[integers]] leads to [[Map (mathematics)|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 [[countable set|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 number]]s, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
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| ====Cardinality of the continuum====
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| {{Main|Cardinality of the continuum}}
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| One of Cantor's most important results was that the cardinality of the continuum <math>\mathbf c</math> is greater than that of the natural numbers <math>{\aleph_0}</math>; that is, there are more real numbers '''R''' than natural numbers '''N'''. Namely, Cantor showed that <math>\mathbf{c} = 2^{\aleph_0} > {\aleph_0}</math> (see [[Cantor's diagonal argument]] or [[Cantor's first uncountability proof]]).
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| 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, <math>\mathbf{c} = \aleph_1 = \beth_1 </math> (see [[Beth number#Beth one|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]].
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| [[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 [[line segment|segment]] of that line, but that this is equal to the number of points on a plane and, indeed, in any [[finite-dimensional]] space.
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| [[File:Peanocurve.svg|thumb|The first three steps of a fractal construction whose limit is a [[space-filling curve]], showing that there are as many points in a one-dimensional line as in a two-dimensional square.]]
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| The first of these results is apparent by considering, for instance, the [[tangent (trigonometric function)|tangent]] function, which provides a [[one-to-one correspondence]] between the [[Interval (mathematics)|interval]] (−π/2, π/2) 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.
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| ===Geometry and topology===
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| {{Main|Dimension (vector space)}}
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| Infinite-[[dimension]]al spaces are widely used in [[geometry]] and [[topology]], particularly as [[classifying space]]s, notably [[Eilenberg−MacLane space]]s. Common examples are the infinite-dimensional [[complex projective space]] [[K(Z,2)]] and the infinite-dimensional [[real projective space]] K(Z/2Z,1).
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| ===Fractals===
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| 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 infinite perimeters—some with infinite, and others with finite surface areas. One such fractal curve with an infinite perimeter and finite surface area is the [[Koch snowflake]].
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| ===Mathematics without infinity===
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| [[Leopold Kronecker]] was skeptical of the notion of infinity and how his fellow mathematicians were using it in 1870s and 1880s. This skepticism was developed in the [[philosophy of mathematics]] called [[finitism]], an extreme form of the philosophical and mathematical schools of [[Mathematical constructivism|constructivism]] and [[intuitionism]].<ref>{{cite book |title=Mathematical Thought from Ancient to Modern Times |last=Kline |first=Morris |authorlink=Morris Kline |year=1972 |publisher=Oxford University Press |location= New York|isbn=0-19-506135-7 |pages=1197–1198 }}</ref>
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| ==Physics==
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| {{Unreferenced section|date=December 2009}}
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| In [[physics]], approximations of [[real number]]s are used for [[Continuum (theory)|continuous]] measurements and [[natural number]]s are used for [[countable|discrete]] measurements (i.e. counting). It is therefore assumed by physicists that no [[observable|measurable quantity]] could have an infinite value,{{Citation needed|date=February 2008}} for instance by taking an infinite value in an [[extended real number line|extended real number]] system, 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. Concepts of infinite things such as an infinite [[plane wave]] exist, but there are no experimental means to generate them.{{Citation needed|date=May 2010}}
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| ===Theoretical applications of physical infinity===
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| The practice of refusing infinite values for measurable quantities does not come from ''[[A priori and a posteriori|a priori]]'' or ideological motivations, but rather from more methodological and pragmatic motivations.{{Citation needed|date=February 2008}} 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.
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| 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 [[function (mathematics)|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]].
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| 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.
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| ===Cosmology===
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| {{Update|Planck|date=March 2013}}
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| In 1584, the Italian philosopher and astronomer [[Giordano Bruno]] proposed an unbounded universe in ''On the Infinite Universe and Worlds'': "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."
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| [[Cosmology|Cosmologists]] have long sought to discover whether infinity exists in our physical [[universe]]: Are there an infinite number of stars? Does the universe have infinite volume? Does space [[Shape of the Universe|"go on forever"]]? This is an open question of [[physical cosmology|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 so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.
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| 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 microwave background radiation|cosmic background radiation]]. As to date, analysis of the radiation patterns recorded by the [[WMAP]] spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe. {{Citation needed|date=July 2013}} The [[Planck (spacecraft)|Planck spacecraft]] launched in 2009 is expected to record the cosmic background radiation with 10 times higher precision, and will give more insight into the question of whether the universe is infinite or not.
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| The concept of infinity also extends to the [[multiverse]] hypothesis, which, when explained by astrophysicists such as [[Michio Kaku]], posits that there are an infinite number and variety of universes.<ref>Kaku, M. (2006). Parallel worlds. Knopf Doubleday Publishing Group.</ref>
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| ==Logic==
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| 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."<ref>''Cambridge Dictionary of Philosophy'', Second Edition, p. 429</ref>
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| ==Computing==
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| The [[IEEE floating-point]] standard (IEEE 754) specifies the positive and negative infinity values. These are defined as the result of [[arithmetic overflow]], [[division by zero]], and other exceptional operations.
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| Some [[programming language]]s, such as [[Java (programming language)|Java]]<ref>{{cite book|last=Gosling|first=James|coauthors=et. al.|title=The Java™ Language Specification|publisher=Oracle America, Inc.|location=California, U.S.A.|date=27 July 2012|edition=Java SE 7|chapter=4.2.3.|accessdate=6 September 2012|url=http://docs.oracle.com/javase/specs/jls/se7/html/jls-4.html#jls-4.2.3}}</ref> and [[J (programming language)|J]],<ref>{{cite book|last=Stokes|first=Roger|title=Learning J|date=July 2012|chapter=19.2.1|url=http://www.rogerstokes.free-online.co.uk/19.htm#10|accessdate=6 September 2012}}</ref> allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as [[Greatest element|greatest and least elements]], as they compare (respectively) greater than or less than all other values. They are useful as [[sentinel value]]s in [[algorithm]]s involving [[sorting]], [[Search algorithm|searching]], or [[window function|windowing]].
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| In languages that do not have greatest and least elements, but do allow [[operator overloading|overloading]] of [[relational operator]]s, it is possible for a programmer to ''create'' the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program, but do implement the floating point [[data type]], the infinity values might still be accessible and usable as the result of certain operations.
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| ==Arts and cognitive sciences==
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| [[Perspective (graphical)|Perspective]] artwork utilizes the concept of imaginary [[vanishing point]]s, or [[point at infinity|points at infinity]], located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms.<ref>{{cite book
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| |title=Mathematics for the nonmathematician
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| |first1=Morris
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| |last1=Kline
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| |publisher=Courier Dover Publications
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| |year=1985
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| |isbn=0-486-24823-2
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| |page=229
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| |url=http://books.google.com/books?id=f-e0bro-0FUC&pg=PA229}}, [http://books.google.com/books?id=f-e0bro-0FUC&pg=PA229 Section 10-7, p. 229]
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| </ref> Artist [[M. C. Escher]] is specifically known for employing the concept of infinity in his work in this and other ways.
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| [[Cognitive science|Cognitive scientist]] [[George Lakoff]] considers the concept of infinity in mathematics and the sciences as a metaphor. This view is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1,2,3,...>.
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| The symbol is often used romantically to represent eternal love. Several types of jewelry are fashioned into the infinity shape for this purpose.
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| ==See also==
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| * [[0.999...]]
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| * [[Aleph number]]
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| * [[Infinite monkey theorem]]
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| * [[Infinite set]]
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| * [[Paradoxes of infinity]]
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| * [[Surreal number]]
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| ==Notes==
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| {{More footnotes|date=June 2009}}
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| {{Reflist|3}}
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| ==References==
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| * {{citation|first=Michael C.|last=Gemignani|title=Elementary Topology|edition=2nd|publisher=Dover|year=1990|isbn=0-486-66522-4}}
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| * {{citation|first=H. Jerome|last=Keisler|title=Elementary Calculus: An Approach Using Infinitesimals|edition=2nd|year=1986|url=http://www.math.wisc.edu/~keisler/calc.html}}
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| <!--
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| * H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
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| -->
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| * {{citation|first=Randall B.|last=Maddox|title=Mathematical Thinking and Writing: A Transition to Abstract Mathematics|publisher=Academic Press|year= 2002|isbn=0-12-464976-9}}
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| * {{citation|first=Earl W.|last=Swokowski|title=Calculus with Analytic Geometry|edition=Alternate|year=1983|publisher=Prindle, Weber & Schmidt|isbn=0-87150-341-7}}
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| * {{citation|first=Angus E.|last=Taylor|title=Advanced Calculus|year=1955|publisher=Blaisdell Publishing Company}}
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| * {{cite book | author=[[David Foster Wallace]] | title=Everything and More: A Compact History of Infinity | publisher=Norton, W. W. & Company, Inc. | year=2004 | isbn=0-393-32629-2}}
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| ==Further reading==
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| * {{cite book | author=Amir D. Aczel | title=The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity | publisher=Pocket Books|place=New York | year=2001 | isbn=0-7434-2299-6}}
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| * [[D. P. Agrawal]] (2000). ''[http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_jaina.htm Ancient Jaina Mathematics: an Introduction]'', [http://infinityfoundation.com Infinity Foundation].
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| * Bell, J. L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.
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| * {{cite book | author=L. C. Jain | title=Exact Sciences from Jaina Sources | year=1982}}
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| * L. C. Jain (1973). "Set theory in the Jaina school of mathematics", ''Indian Journal of History of Science''.
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| * {{cite book | author=George G. Joseph | title=The Crest of the Peacock: Non-European Roots of Mathematics | edition=2nd edition | publisher=[[Penguin Books]] | year=2000 | isbn= 0-14-027778-1}}
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| * {{cite book | author=[[Eli Maor]] | title=To Infinity and Beyond | publisher=Princeton University Press | year=1991 | isbn=0-691-02511-8}}
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| * {{cite book | author=[[Rudy Rucker]] | title=Infinity and the Mind: The Science and Philosophy of the Infinite | publisher=Princeton University Press | year=1995 | isbn=0-691-00172-3}}
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| * {{cite book | author=Navjyoti Singh | title=Jaina Theory of Actual Infinity and Transfinite Numbers | journal=Journal of Asiatic Society | volume=30 | year=1988}}<!-- {{cite book | author=Navjyoti Singh | title=Jaina Theory of Actual Infinity and Transfinite Numbers | journal=Vaishali Institute Research Bulletin| volume=5 | year=1986}}-->
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| ==External links==
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| {{Wiktionary}}
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| {{Wikibooks|Infinity is not a number}}
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| *{{In Our Time|Infinity|p0054927|Infinity}}
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| *''[http://www.earlham.edu/~peters/writing/infapp.htm A Crash Course in the Mathematics of Infinite Sets]'', by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1–59. The stand-alone appendix to ''Infinite Reflections'', below. A concise introduction to Cantor's mathematics of infinite sets.
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| *''[http://www.earlham.edu/~peters/writing/infinity.htm Infinite Reflections]'', by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 (1998) 1–59.
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| *{{cite web|last=Grime|first=James|title=Infinity is bigger than you think|url=http://www.numberphile.com/videos/countable_infinity.html|work=Numberphile|publisher=[[Brady Haran]]}}
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| *[http://pespmc1.vub.ac.be/INFINITY.html ''Infinity'', Principia Cybernetica]
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| *[http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html Hotel Infinity]
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| * John J. O'Connor and Edmund F. Robertson (1998). [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html 'Georg Ferdinand Ludwig Philipp Cantor'], ''[[MacTutor History of Mathematics archive]]''.
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| * John J. O'Connor and Edmund F. Robertson (2000). [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html 'Jaina mathematics'], ''MacTutor History of Mathematics archive''.
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| * Ian Pearce (2002). [http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch5.html 'Jainism'], ''MacTutor History of Mathematics archive''.
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| *[http://www.webcitation.org/query?url=http://uk.geocities.com/frege%40btinternet.com/cantor/Phil-Infinity.htm&date=2009-10-25+04:16:26 Source page on medieval and modern writing on Infinity]
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| *[http://www.washingtonpost.com/wp-srv/style/longterm/books/chap1/mysteryaleph.htm The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity]
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| *[http://dictionary.of-the-infinite.com Dictionary of the Infinite] (compilation of articles about infinity in physics, mathematics, and philosophy)
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| {{Infinity}}
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| [[Category:Infinity| ]]
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| [[Category:Concepts in logic]]
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| [[Category:Philosophy of mathematics]]
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| [[Category:Theology]]
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| [[Category:Philosophical concepts]]
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| {{Link FA|la}}
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