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Line 1: − '''Infinity''' is a designation for that condition which goes on without end. It is not an actual number that can be quantified, but a value which is approached as a limit. It can not be empirically tested as one can never have a sample size of infinity, but logically we know it exists. Infinity has important theoretical applications in mathematics.+ − An example where infinity can be seen as a limit would be in any attempt to divide by zero. The result is an undefined value, but it can be seen with the function <math>F</math><sub>x</sub> = <math>1/x</math> that as x approaches the value of zero, that the resulting answer approaches infinity. − − [[Georg Cantor]]'s diagonal argument is an elegant proof demonstrating that the infinity of real numbers is greater than the infinity of countable integers. The essence of the argument is that in any proposed list of all real numbers, a new real number not in the list can be constructed by taking the digits in a diagonal through the list and changing them to construct a new real number that differs from the nth entry at the nth position right of the decimal point. − + − − However, in some non-standard model of Peano arithematic, ∞ is treated as an actual number. + − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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{{otheruses}}
Infinity is written using the symbol ∞
'''Infinity''' (commonly represented as the symbol '''∞''') comes from the [[Latin]] ''infinitas'' or "unboundedness." It refers to several distinct concepts (usually linked to the idea of "without end") which arise in [[philosophy]], [[mathematics]], and [[theology]].
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" than the [[real numbers]]. Infinity is related to [[limit (mathematics)|limit]]s, [[aleph number]]s, [[class (set theory)|class]]es in [[set theory]], [[Dedekind-infinite set]]s, [[large cardinal]]s,<ref>Large cardinals are quantitative infinities defining the number of things in a [[Set|collection]], which are so large that they cannot be proven to exist in the ordinary mathematics of [[ZFC|Zermelo-Fraenkel plus Choice]] (ZFC).</ref> [[Russell's paradox]], [[non-standard arithmetic]], [[hyperreal number]]s, [[projective geometry]], [[Affinely extended real number system|extended real number]]s and the [[absolute Infinite]].
[[Category:Mathematics]]
=== 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."<ref>''Cambridge Dictionary of Philosophy'', Second Edition, p. 429</ref>
=== Infinity symbol ===
[[Image:John Wallis.jpg|thumb|200px|right|John Wallis introduced the infinity symbol to mathematical literature.]]
The precise origin of the infinity symbol '''∞''' is unclear. One possibility is suggested by the name it is sometimes called—the [[lemniscate]], from the Latin ''lemniscus'', meaning "ribbon." One can imagine walking forever along a simple loop formed from a ribbon.
A popular explanation is that the infinity symbol is derived from the shape of a [[Möbius strip]]. Again, one can imagine walking along its surface forever. However, this explanation is improbable, since the symbol had been in use to represent infinity for over two hundred years before [[August Ferdinand Möbius]] and [[Johann Benedict Listing]] discovered the Möbius strip in [[1858]].
It is also possible that it is inspired by older [[religious]]/[[alchemical]] [[symbolism]]. For instance, it has been found in [[Tibet]]an [[rock carvings]], and the [[ouroboros]], or infinity snake, is often depicted in this shape.
[[John Wallis]] is usually credited with introducing ∞ 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 numerals|Etruscan numeral]] for 1000, which looked somewhat like <font face="Arial Unicode MS, Lucida Sans Unicode">CIƆ</font> 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]].<ref>[http://www.roma.unisa.edu.au/07305/symbols.htm#Infinity 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.</ref>
Another possibility is that the symbol was chosen because it was easy to rotate an "8" character by 90° when [[typesetting]] was done by hand. The symbol is sometimes called a "lazy eight", evoking the image of an "8" lying on its side.
Another popular belief is that the infinity symbol is a clear depiction of the hour glass turned 90°. Obviously, this action would cause the hour glass to take infinite time to empty thus presenting a tangible example of infinity. The invention of the hourglass predates the existence of the infinite symbol allowing this theory to be plausible. <!-- Is this true? -->
The infinity symbol is represented in [[Unicode]] by the character ∞ (U+221E).
== History ==
=== 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".
:'''{{Unicode|Pūrṇam adaḥ pūrṇam idam}}''' (That is full, this is full)
:'''{{Unicode|pūrṇāt pūrṇam udacyate}}''' (From the full, the full is subtracted)
:'''{{Unicode|pūrṇasya pūrṇam ādāya}}''' (When the full is taken from the full)
:'''{{Unicode|pūrṇam evāvasiṣyate'''}} (The full still will remain.) - [[Isha Upanishad]]
The Indian [[Indian mathematics|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 [[Jainism|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 number|aleph-null]] <math>\aleph_0</math> (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 [[Set theory|theory of sets]], two basic types of infinite numbers are distinguished. On both physical and [[Ontology|ontological]] grounds, a distinction was made between {{IAST|''asaṃkhyāta''}} ("countless, innumerable") and ''ananta'' ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.
== Mathematical infinity ==<!-- This section is linked from [[Phase-shift keying]] -->
{{Unreferencedsection|date=June 2007}} Infinity is used in various branches of mathematics.
=== Calculus ===
{{further|[[Limit (mathematics)]], [[Series (mathematics)]], [[Improper integral]]}}
In [[real analysis]], the symbol <math>\infty</math>, called "infinity", denotes an unbounded [[Limit (mathematics)|limit]]. <math>x \rightarrow \infty</math> means that ''x'' grows beyond any assigned value, and <math>x \rightarrow -\infty</math> means x is eventually less than any assigned value. If ''f''(''t'') ≥ 0 for every ''t'', then
* <Math>\int_{a}^{b} \, f(t)\ dt \ = \infty</math> means that ''f''(''t'') does not bound a finite area from a to b
* <Math>\int_{-\infty}^{\infty} \, f(t)\ dt \ = \infty</math> means that the area under ''f''(''t'') is infinite.
* <Math>\int_{-\infty}^{\infty} \, f(t)\ dt \ = 1</math> means that the area under ''f''(''t'') equals 1
Infinity is also used to describe [[infinite series]]:
* <math>\sum_{i=0}^{\infty} \, f(i) = x</math> means that the sum of the infinite series [[convergent series|converges]] to some real value ''x''.
* <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.
==== Algebraic properties ====
{{further|[[Extended real number line]]}}
Infinity is often used not only to define a limit but as a value in the affinely 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 numbers. We can also treat <math>\infty</math> and <math>-\infty</math> as the same, leading to the '''one-point [[compactification (mathematics)|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 called infinity (<math>\infty</math>), greater than all other extended real numbers, and negative infinity (<math>-\infty</math>), less than all other extended real numbers, for which some arithmetic operations may be performed.
==== Complex analysis ====
As in real analysis, in [[complex analysis]] the symbol <math>\infty</math>, called "infinity", denotes an unbounded [[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 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 <math>z/0 = \infty</math> for any complex number ''z''. In this context 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.
=== Nonstandard analysis ===
{{main|Nonstandard analysis}}
The original formulation of the 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 (mathematics)|field]]; there is no equivalence between them as with the Cantorian [[transfinite]]s For example if H is an infinite number, then H + H = 2H, and H + 1 are different infinite numbers.
=== Set theory ===
{{main|Cardinality|Ordinal number}}
A different type of "infinity" are the [[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 [[set]]s.
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]].
Cantor defined two kinds of infinite numbers, the [[ordinal number]]s and the [[aleph number|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.
Our intuition gained from [[finite set]]s breaks down when dealing with [[infinite set]]s. One example of this is [[Hilbert's paradox of the Grand Hotel]].
==== Cardinality of the continuum ====
{{main|Cardinality of the continuum}}
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 whole numbers '''N'''. Namely, Cantor showed that <math>\mathbf{c} = 2^{\aleph_0} > {\aleph_0}</math> (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, <math>\mathbf{c} = \aleph_1</math>. 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 [[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. These results are highly counterintuitive, because they imply that there exist [[proper subset]]s and [[proper superset]]s of an infinite set ''S'' that have the same size as ''S'', although ''S'' contains elements that do not belong to its subsets, and the supersets of ''S'' contain elements that are not included in it.
The first of these results is apparent by considering, for instance, the [[Tangent#Trigonometry|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 curve]]s, 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.
It is also possible to show that sets with cardinality strictly greater than <math>\mathbf c</math> exist. They include, for instance:
* the set of all subsets of '''R''', i.e., the [[power set]] of '''R''', written ''P''('''R''') or 2<sup>'''R'''</sup>
* the set '''R'''<sup>'''R'''</sup> of all functions from '''R''' to '''R'''
Both have cardinality <math>2^\mathbf {c} = \beth_2</math> (see [[Beth number]]).
=== 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 number]]s are used for [[continuum (mathematics)|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, for instance by taking an infinite value in an [[extended real number line|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 and a posteriori (philosophy)|a priori]]'' or ideological motivations, but rather from more methodological and pragmatic motivations. 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.
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)|function]]s, 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]]. One application where infinities arise is the quantification of [[thermodynamic temperature]]s.
However, there are some currently-accepted circumstances where the end result is infinity. One example is [[black holes]]. Physicists have verified that, when a star experiences [[gravitational collapse]], it will eventually shrink down to a point of zero size, and thus have infinite density. This is an example of what is called a [[mathematical singularity]], or a point where the laws of mathematics, and therefore of physics, break down. Physicists have given up hope on the singularity not being real, and have since turned their attention to finding new mathematics where infinities are possible.
=== Infinity in cosmology ===
{{main|Physical cosmology}}
An intriguing question is whether actual infinity exists in our physical [[universe]]: Are there infinitely many stars? Does the universe have infinite volume? [[Shape of the Universe|Does space "go on forever"]]? This is an important 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 walking/sailing/driving straight long enough, you'll return to the exact spot you started from. The universe, at least in principle, might have a similar [[topology]]; if you fly your space ship straight ahead long enough, perhaps you would eventually revisit your starting point. If, however, [[Ultimate fate of the universe|the universe is ever expanding]] then you could never get back to your starting point even on an infinite time scale.
== In computing ==
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 language]]s (for example, [[J programming language|J]] and [[UNITY (programming language)|UNITY]]) specify [[Greatest element|greatest and least elements]], i.e. [[Value (mathematics)|value]]s 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 value]]s in [[algorithm]]s involving [[sorting]], [[searching]] or [[window function|windowing]]. In languages that do not have greatest and least elements, but do allow [[operator overloading|overloading]] of [[relational operator]]s, it is possible to ''create'' greatest and least elements (with some [[Computational overhead|overhead]], and the risk of incompatibility between implementations).
==In the arts==
[[Perspective (graphical)|Perspective]] artwork utilizes the concept of imaginary [[vanishing point]]s located at an infinite distance from the observer. This allows artists to create paintings that realistically depict distance and foreshortening of objects.
A few artists are known specifically for employing the concept of infinity in their works:
* [[M. C. Escher]]
== Notes ==
<div class="references-small">
<references />
</div>
==References==
<div class="references-small">
* {{cite book | author=Amir D. Aczel | title=The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity | publisher=Simon & Schuster Adult Publishing Group | year=2001 | id=ISBN 0-7434-2299-6}}
* [[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].
* {{cite book | author=L. C. Jain | title=Exact Sciences from Jaina Sources | year=1982}}
* L. C. Jain (1973). "Set theory in the Jaina school of mathematics", ''Indian Journal of History of Science''.
* {{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 | id= ISBN 0-14-027778-1}}
* {{cite book | author=Eli Maor | title=To Infinity and Beyond | publisher=Princeton University Press | year=1991 | id=ISBN 0-691-02511-8}}
* 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]]''.
* 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]]''.
* Ian Pearce (2002). [http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch5.html 'Jainism'], ''[[MacTutor History of Mathematics archive]]''.
* {{cite book | author=[[Rudy Rucker]] | title=Infinity and the Mind: The Science and Philosophy of the Infinite | publisher=Princeton University Press | year=1995 | id=ISBN 0-691-00172-3}}
* N. Singh (1988). 'Jaina Theory of Actual Infinity and Transfinite Numbers', ''Journal of Asiatic Society'', Vol. 30.
* {{cite book | author=[[David Foster Wallace]] | title=Everything and More: A Compact History of Infinity | publisher=Norton, W. W. & Company, Inc. | year=2004 | id=ISBN 0-393-32629-2}}
</div>
==External links==
* ''[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.
* ''[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.
* [http://pespmc1.vub.ac.be/INFINITY.html ''Infinity'', Principia Cybernetica]
* [http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html Hotel Infinity]
* [http://samvak.tripod.com/infinite.html The concepts of finiteness and infinity in philosophy]
* [http://uk.geocities.com/frege@btinternet.com/cantor/Phil-Infinity.htm Source page on medieval and modern writing on Infinity]
* [http://www.huge-entity.com/2006/05/perceiving-infinity-schematic-portals.html Perceiving Infinity: A Schematic Toolkit]
* [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]