Infinity Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! ==Mathematics== [[Hermann Weyl]] opened a mathematico-philosophic address given in 1930 with:<ref>{{citation|first=Hermann|last=Weyl|title=Levels of Infinity / Selected Writings on Mathematics and Philosophy|editor=Peter Pesic|year=2012|publisher=Dover|isbn=978-0-486-48903-2|page=17}}</ref> {{blockquote|text=Mathematics is the science of the infinite.}} ===Symbol=== {{Main|Infinity symbol}} 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=}}<ref>{{Cite web|url=https://www.compart.com/en/unicode/U+221E|title=Unicode Character "∞" (U+221E)|last=AG|first=Compart|website=Compart.com|language=en|access-date=2019-11-15}}</ref> and in [[LaTeX]] as <code>\infty</code>.<ref>{{Cite web|url=https://oeis.org/wiki/List_of_LaTeX_mathematical_symbols|title=List of LaTeX mathematical symbols - OeisWiki|website=oeis.org|access-date=2019-11-15}}</ref> It was introduced in 1655 by [[John Wallis]],<ref>{{citation | last = Scott | first = Joseph Frederick | edition = 2 | isbn = 978-0-8284-0314-6 | page = 24 | publisher = [[American Mathematical Society]] | title = The mathematical work of John Wallis, D.D., F.R.S., (1616–1703) | url = https://books.google.com/books?id=XX9PKytw8g8C&pg=PA24 | year = 1981 | url-status=live | archive-url = https://web.archive.org/web/20160509151853/https://books.google.com/books?id=XX9PKytw8g8C&pg=PA24 | archive-date = 2016-05-09 }}</ref><ref>{{citation | last = Martin-Löf | first = Per | author-link = Per Martin-Löf | contribution = Mathematics of infinity | doi = 10.1007/3-540-52335-9_54 | location = Berlin | mr = 1064143 | pages = 146–197 | publisher = Springer | series = Lecture Notes in Computer Science | title = COLOG-88 (Tallinn, 1988) | volume = 417 | year = 1990| isbn = 978-3-540-52335-2 }}</ref> and since its introduction, it 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=978-0-226-61855-5|page=243|url=https://books.google.com/books?id=vhNNrX3bmo4C&pg=PA243|url-status=live|archive-url=https://web.archive.org/web/20160629143323/https://books.google.com/books?id=vhNNrX3bmo4C&pg=PA243|archive-date=2016-06-29}}</ref> and literary [[symbology]].<ref>{{citation|title=Nabokov: The Mystery of Literary Structures|first=Leona|last=Toker|publisher=Cornell University Press|year=1989|isbn=978-0-8014-2211-9|page=159|url=https://books.google.com/books?id=Jud1q_NrqpcC&pg=PA159|url-status=live|archive-url=https://web.archive.org/web/20160509095701/https://books.google.com/books?id=Jud1q_NrqpcC&pg=PA159|archive-date=2016-05-09}}</ref> ===Calculus=== [[Gottfried Wilhelm Leibniz|Gottfried 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 in accordance with the [[Law of continuity]].<ref>{{cite SEP |url-id=continuity |title=Continuity and Infinitesimals |last=Bell |first=John Lane |author-link=John Lane Bell}}</ref><ref name="Jesseph">{{cite journal |last=Jesseph |first=Douglas Michael |date=1998-05-01 |title=Leibniz on the Foundations of the Calculus: The Question of the Reality of Infinitesimal Magnitudes |url=http://muse.jhu.edu/journals/perspectives_on_science/v006/6.1jesseph.html |url-status=dead |journal=[[Perspectives on Science]] |volume=6 |issue=1&2 |pages=6–40 |doi=10.1162/posc_a_00543 |s2cid=118227996 |issn=1063-6145 |oclc=42413222 |archive-url=https://web.archive.org/web/20120111102635/http://muse.jhu.edu/journals/perspectives_on_science/v006/6.1jesseph.html |archive-date=11 January 2012 |access-date=1 November 2019 |via=Project MUSE}}</ref> ====Real analysis==== 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> The notation <math>x \rightarrow \infty</math> means that ''<math>x</math>'' increases without bound, and <math>x \to -\infty</math> means that ''<math>x</math>'' decreases without bound. For example, if <math>f(t)\ge 0</math> for every ''<math>t</math>'', then<ref>These uses of infinity for integrals and series can be found in any standard calculus text, such as, {{harvnb|Swokowski|1983|pp=468–510}}</ref> * <math>\int_{a}^{b} f(t)\, dt = \infty</math> means that <math>f(t)</math> does not bound a finite area from <math>a</math> to <math>b.</math> * <math>\int_{-\infty}^{\infty} f(t)\, dt = \infty</math> means that the area under <math>f(t)</math> is infinite. * <math>\int_{-\infty}^{\infty} f(t)\, dt = a</math> means that the total area under <math>f(t)</math> is finite, and is equal to <math>a.</math> Infinity can also be used to describe [[infinite series]], as follows: * <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> * <math>\sum_{i=0}^{\infty} f(i) = \infty</math> means that the sum of the infinite series properly [[divergent series|diverges]] to infinity, in the sense that the partial sums increase without bound.<ref>{{Cite web|url=http://mathonline.wikidot.com/properly-divergent-sequences|title=Properly Divergent Sequences - Mathonline|website=mathonline.wikidot.com|access-date=2019-11-15}}</ref> In addition to defining a limit, infinity can be also used 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.<ref>{{citation | last1 = Aliprantis | first1 = Charalambos D. | last2 = Burkinshaw | first2 = Owen | edition = 3rd | isbn = 978-0-12-050257-8 | location = San Diego, CA | mr = 1669668 | page = 29 | publisher = Academic Press, Inc. | title = Principles of Real Analysis | url = https://books.google.com/books?id=m40ivUwAonUC&pg=PA29 | year = 1998 | url-status=live | archive-url = https://web.archive.org/web/20150515120230/https://books.google.com/books?id=m40ivUwAonUC&pg=PA29 | archive-date = 2015-05-15 }}</ref> 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 a [[hyperplane at infinity]] for general [[Dimension (mathematics and physics)|dimensions]], each consisting of [[Point at infinity|points at infinity]].<ref>{{citation|first1=Albrecht|last1=Beutelspacher|first2=Ute|last2=Rosenbaum|title=Projective Geometry / from foundations to applications|year=1998|publisher=Cambridge University Press|isbn=978-0-521-48364-3|page=27}}</ref> ====Complex analysis==== [[File:Riemann sphere1.svg|thumb|right|250px|By [[stereographic projection]], the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the [[Riemann sphere]].]] In [[complex analysis]] the symbol <math>\infty</math>, called "infinity", denotes an unsigned infinite [[Limit (mathematics)|limit]]. The expression <math>x \rightarrow \infty</math> means that the magnitude <math>|x|</math> of ''<math>x</math>'' 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]] 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]].<ref>{{Cite book|title=Complex Analysis: An Invitation : a Concise Introduction to Complex Function Theory|first1=Murali|last1=Rao|first2=Henrik|last2=Stetkær|publisher=World Scientific|year=1991|isbn=9789810203757|page=113|url=https://books.google.com/books?id=wdTntZ_N0tYC&pg=PA113}}</ref> Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception 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]] ''<math>z</math>''. 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 (see [[Möbius transformation#Overview|Möbius transformation § Overview]]). ===Nonstandard analysis=== [[File:Números hiperreales.png|450px|thumb|Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1)]] The original formulation of [[infinitesimal calculus]] by [[Isaac Newton]] and Gottfried Leibniz used [[infinitesimal]] quantities. In the second half of the 20th 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 in this sense, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to [[non-standard calculus]] is fully developed in {{harvtxt|Keisler|1986}}. ===Set theory=== {{Main|Cardinality|Ordinal number}} [[File:Infinity paradoxon - one-to-one correspondence between infinite set and proper subset.gif|thumb|One-to-one correspondence between an infinite set and its proper subset]] A different form of "infinity" are the [[Ordinal number|ordinal]] and [[cardinal number|cardinal]] infinities of set theory—a system of [[transfinite number]]s first developed by [[Georg Cantor]]. In this system, the first transfinite cardinal is [[aleph-null]] (<span style="font-family:'Cambria Math';"><big>ℵ</big><sub>0</sub></span>), the cardinality of the set of [[natural number]]s. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, [[Gottlob Frege]], [[Richard Dedekind]] and others—using the idea of collections or sets.<ref name=":1" /> 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 (derived from [[Euclid]]) that the whole cannot be the same size as the part. (However, see [[Galileo's paradox]] where Galileo concludes that positive integers cannot be compared to the subset of positive [[Square number|square integers]] since both are infinite sets.) 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 to the right 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".{{citation needed|date=April 2017}} Cantor defined two kinds of infinite numbers: [[ordinal number]]s and [[cardinal number]]s. Ordinal numbers characterize [[well-ordered]] sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite [[sequence]]s which are maps from the positive [[integers]] leads to [[Function (mathematics)|mappings]] from ordinal numbers to 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 set|uncountable]]''. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory.<ref>{{Cite web|url=https://math.dartmouth.edu/~matc/Readers/HowManyAngels/InfinityMind/IM.html|title=Infinity|website=math.dartmouth.edu|access-date=2019-11-16}}</ref><ref>{{cite book |title=The Infinite |first1=A.W. |last1=Moore |publisher=Routledge |year=1991}} </ref>{{page needed|date=June 2014}} Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.{{citation needed|date=April 2017}} ====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 {{math|'''R'''}} than natural numbers {{math|'''N'''}}. Namely, Cantor showed that <math>\mathbf{c}=2^{\aleph_0}>{\aleph_0}</math>.<ref>{{Cite journal| last = Dauben | first = Joseph | title = Georg Cantor and the Battle for Transfinite Set Theory | url = http://acmsonline.org/home2/wp-content/uploads/2016/05/Dauben-Cantor.pdf | journal = 9th ACMS Conference Proceedings | year = 1993 | page = 4 }}</ref>{{further|Cantor's diagonal argument|Cantor's first set theory article}} 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>.{{further|Beth number#Beth one}}This hypothesis cannot be proved or disproved within the widely accepted [[Zermelo–Fraenkel set theory]], even assuming the [[Axiom of Choice]].<ref>{{harvnb|Cohen|1963|p=1143}}</ref> [[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 also that this is equal to the number of points on a plane and, indeed, in any [[finite-dimensional]] space.{{citation needed|date=April 2017}} [[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]] 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]] ({{math|−{{sfrac|π|2}}, {{sfrac|π|2}}}}) and{{math| '''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 on one side of a square and the points in the square.<ref>{{harvnb|Sagan|1994|pp=10–12}}</ref> ===Geometry=== Until the end of the 19th century, infinity was rarely discussed in [[geometry]], except in the context of processes that could be continued without any limit. For example, a [[line (geometry)|line]] was what is now called a [[line segment]], with the proviso that one can extend it as far as one wants; but extending it ''infinitely'' was out of the question. Similarly, a line was usually not considered to be composed of infinitely many points, but was a location where a point may be placed. Even if there are infinitely many possible positions, only a finite number of points could be placed on a line. A witness of this is the expression "the [[locus (mathematics)|locus]] of ''a point'' that satisfies some property" (singular), where modern mathematicians would generally say "the set of ''the points'' that have the property" (plural). One of the rare exceptions of a mathematical concept involving [[actual infinity]] was [[projective geometry]], where [[points at infinity]] are added to the [[Euclidean space]] for modeling the [[perspective (graphical)|perspective]] effect that shows [[parallel lines]] intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a [[projective plane]], two distinct [[line (geometry)|lines]] intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry. Before the use of [[set theory]] for the [[foundation of mathematics]], points and lines were viewed as distinct entities, and a point could be ''located on a line''. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as ''the set of its points'', and one says that a point ''belongs to a line'' instead of ''is located on a line'' (however, the latter phrase is still used). In particular, in modern mathematics, lines are ''infinite sets''. ===Infinite dimension=== The [[vector space]]s that occur in classical [[geometry]] have always a finite [[dimension (vector space)|dimension]], generally two or three. However, this is not implied by the abstract definition of a vector space, and vector spaces of infinite dimension can be considered. This is typically the case in [[functional analysis]] where [[function space]]s are generally vector spaces of infinite dimension. In topology, some constructions can generate [[topological space]]s of infinite dimension. In particular, this is the case of [[iterated loop space]]s. ===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 infinite perimeters, and can have infinite or finite areas. One such [[fractal curve]] with an infinite perimeter and finite area is the [[Koch snowflake]].{{citation needed|date=April 2017}} ===Mathematics without infinity=== [[Leopold Kronecker]] was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the [[philosophy of mathematics]] called [[finitism]], an extreme form of mathematical philosophy in the general philosophical and mathematical schools of [[Mathematical constructivism|constructivism]] and [[intuitionism]].<ref>{{harvnb|Kline|1972|pp=1197–1198}}</ref> Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see Christianpedia:Copyrights for details). Do not submit copyrighted work without permission! Cancel Editing help (opens in new window) Discuss this page