View source for Infinity

{{Short description|Mathematical concept}}

{{Hatnote group|
{{About||the symbol|Infinity symbol|other uses of "Infinity" and "Infinite"}}
{{Distinguish|Infiniti}}
}}
[[File:SierpinskiTriangle.svg|thumb|The [[Sierpiński triangle]] contains infinitely many (scaled-down) copies of itself.]]
{{Pp-move}} 

'''Infinity''' is something which is boundless, endless, or larger than any [[natural number]]. It is denoted by <math>\infty</math>, called [[the infinity symbol]].

From the time of the [[Ancient Greek mathematics|ancient Greeks]], the [[Infinity (philosophy)|philosophical nature of infinity]] has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol<ref name=":1">{{Cite web |last=Allen |first=Donald |date=2003 |title=The History of Infinity |url=https://www.math.tamu.edu/~dallen/masters/infinity/infinity.pdf |access-date=Nov 15, 2019 |website=Texas A&M Mathematics |archive-date=August 1, 2020 |archive-url=https://web.archive.org/web/20200801202539/https://www.math.tamu.edu/~dallen/masters/infinity/infinity.pdf |url-status=dead}}</ref> and the [[infinitesimal calculus]], mathematicians began to work with [[infinite series]] and what some mathematicians (including [[Guillaume de l'Hôpital|l'Hôpital]] and [[Johann Bernoulli|Bernoulli]])<ref name="Jesseph" /> regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of [[calculus]], it remained unclear whether infinity could be considered as a number or [[Magnitude (mathematics)|magnitude]] and, if so, how this could be done.<ref name=":1" /> At the end of the 19th century, [[Georg Cantor]] enlarged the mathematical study of infinity by studying [[infinite set]]s and [[transfinite number|infinite number]]s, showing that they can be of various sizes.<ref name=":1" /><ref>{{Cite book |last1=Gowers |first1=Timothy |url=https://www.worldcat.org/oclc/659590835 |title=The Princeton companion to mathematics |last2=Barrow-Green |first2=June |publisher=Princeton University Press |others=Imre Leader, Princeton University |year=2008 |isbn=978-1-4008-3039-8 |location=Princeton |language=en |oclc=659590835}}</ref>  For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the [[cardinality]] of the line) is larger than the number of [[integer]]s.<ref>{{harvnb|Maddox|2002|loc=pp. 113–117}}</ref> In this usage, infinity is a mathematical concept, and infinite [[mathematical object]]s can be studied, manipulated, and used just like any other mathematical object.

The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of [[Zermelo–Fraenkel set theory]], on which most of modern mathematics can be developed, is the [[axiom of infinity]], which guarantees the existence of infinite sets.<ref name=":1" /> The mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as [[combinatorics]] that may seem to have nothing to do with them. For example, [[Wiles's proof of Fermat's Last Theorem]] implicitly relies on the existence of [[Grothendieck universe]]s, very large infinite sets,<ref>{{cite journal |last=McLarty |first=Colin |date=15 January 2014|orig-date=September 2010 |title=What Does it Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory |url=https://www.cambridge.org/core/journals/bulletin-of-symbolic-logic/article/abs/what-does-it-take-to-prove-fermats-last-theorem-grothendieck-and-the-logic-of-number-theory/80EDFF3616F8D58590EBA0DCB9FD2E3E |journal=The Bulletin of Symbolic Logic |volume=16 |issue=3 |pages=359–377 |doi=10.2178/bsl/1286284558 |via=Cambridge University Press |s2cid=13475845|url-access=subscription }}</ref> for solving a long-standing problem that is stated in terms of [[elementary arithmetic]].

In [[physics]] and [[cosmology]], it is an open question [[Universe#Size and regions|whether the universe is spatially infinite or not]].

==History==
{{Further|Infinity (philosophy)}}
Ancient cultures had various ideas about the nature of infinity. The [[Vedic period|ancient Indians]] and the [[ancient Greece|Greeks]] did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.

===Early Greek===
The earliest recorded idea of infinity in Greece may be that of [[Anaximander]] (c. 610 – c. 546 BC) a [[Pre-Socratic philosophy|pre-Socratic]] Greek philosopher. He used the word ''[[apeiron]]'', which means "unbounded", "indefinite", and perhaps can be translated as "infinite".<ref name=":1" /><ref>{{harvnb|Wallace|2004|p=44}}</ref>

[[Aristotle]] (350 BC) distinguished ''potential infinity'' from ''[[actual infinity]]'', which he regarded as impossible due to the various [[paradoxes]] it seemed to produce.<ref>{{cite book |author=Aristotle  |url=http://classics.mit.edu/Aristotle/physics.3.iii.html |translator-last1=Hardie|translator-first1=R. P. |translator-last2=Gaye|translator-first2=R. K. |at=Book 3, Chapters 5–8|title=Physics|publisher=The Internet Classics Archive}}</ref> It has been argued that, in line with this view, the [[Hellenistic]] Greeks had a "horror of the infinite"<ref>{{cite book |author=Goodman |first=Nicolas D. |title=Constructive Mathematics |chapter=Reflections on Bishop's philosophy of mathematics |year=1981 |editor1-last=Richman |editor1-first=F. |series=Lecture Notes in Mathematics |publisher=Springer |volume=873|pages=135–145 |doi=10.1007/BFb0090732 |isbn=978-3-540-10850-4}}</ref><ref>Maor, p. 3</ref> which would, for example, explain why [[Euclid]] (c. 300 BC) did not say that there are an infinity of primes but rather "Prime numbers are more than any assigned multitude of prime numbers."<ref>{{Cite journal |last=Sarton |first=George |date=March 1928 |title=''The Thirteen Books of Euclid's Elements''. Thomas L. Heath, Heiberg |url=https://www.journals.uchicago.edu/doi/10.1086/346308 |journal=Isis |volume=10 |issue=1 |pages=60–62 |doi=10.1086/346308 |issn=0021-1753 |via=The University of Chicago Press Journals|url-access=subscription }}</ref> It has also been maintained, that, in proving the [[infinitude of the prime numbers]], Euclid "was the first to overcome the horror of the infinite".<ref>{{Cite book |last=Hutten |first=Ernest Hirschlaff |url=https://archive.org/details/originsofscience0000hutt_n9u7 |title=The origins of science; an inquiry into the foundations of Western thought |date=1962 |publisher=London, Allen and Unwin |others=Internet Archive |isbn=978-0-04-946007-2 |pages=1–241 |language=en |access-date=2020-01-09}}</ref> There is a similar controversy concerning Euclid's [[parallel postulate]], sometimes translated:

{{blockquote|If a straight line falling across two [other] straight lines makes internal angles on the same side [of itself whose sum is] less than two right angles, then the two [other] straight lines, being produced to infinity, meet on that side [of the original straight line] that the [sum of the internal angles] is less than two right angles.<ref>{{cite book|author=Euclid |orig-year=c. 300 BC|translator-last1=Fitzpatrick |translator-first1=Richard |title=Euclid's Elements of Geometry |url=http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf|year=2008 |isbn=978-0-6151-7984-1 |page=6 (Book I, Postulate 5)|publisher=Lulu.com}}</ref>}}

Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...",<ref>{{cite book|last1=Heath|first1=Sir Thomas Little|last2=Heiberg|first2=Johan Ludvig|author-link1=Thomas Heath (classicist)|title=The Thirteen Books of Euclid's Elements|volume=v. 1|publisher=The University Press|year=1908|url=https://books.google.com/books?id=dkk6AQAAMAAJ&q=right+angles+infinite&pg=PR8|page=212}}</ref> thus avoiding the implication that Euclid was comfortable with the notion of infinity.  Finally, it has been maintained that a reflection on infinity, far from eliciting a "horror of the infinite", underlay all of early Greek philosophy and that Aristotle's "potential infinity" is an aberration from the general trend of this period.<ref>{{cite book|last=Drozdek|first=Adam|title=''In the Beginning Was the'' Apeiron'': Infinity in Greek Philosophy''|year=2008|isbn=978-3-515-09258-6|publisher=Franz Steiner Verlag|location=Stuttgart, Germany}}</ref>

===Zeno: Achilles and the tortoise===
{{Main|Zeno's paradoxes#Achilles and the tortoise}}
[[Zeno of Elea]] ({{c.}} 495 – {{c.}} 430&nbsp;BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes,<ref name="Zeno's paradoxes">{{cite web|url=https://plato.stanford.edu/entries/paradox-zeno/ |title=Zeno's Paradoxes |date=October 15, 2010 |website=Stanford University |access-date=April 3, 2017}}</ref> especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described by [[Bertrand Russell]] as "immeasurably subtle and profound".<ref>{{harvnb|Russell|1996|p=347}}</ref>

[[Achilles]] races a tortoise, giving the latter a head start.
*Step #1: Achilles runs to the tortoise's starting point while the tortoise walks forward.
*Step #2: Achilles advances to where the tortoise was at the end of Step #1 while the tortoise goes yet further.
*Step #3: Achilles advances to where the tortoise was at the end of Step #2 while the tortoise goes yet further.
*Step #4: Achilles advances to where the tortoise was at the end of Step #3 while the tortoise goes yet further.
Etc.

Apparently, Achilles never overtakes the tortoise, since however many steps he completes, the tortoise remains ahead of him.

Zeno was not attempting to make a point about infinity. As a member of the [[Eleatic]]s school which regarded motion as an illusion, he saw it as a mistake to suppose that Achilles could run at all. Subsequent thinkers, finding this solution unacceptable, struggled for over two millennia to find other weaknesses in the argument.

Finally, in 1821, [[Augustin-Louis Cauchy]] provided both a satisfactory definition of a limit and a proof that, for {{math|0 < ''x'' < 1}},<ref>{{cite book|last=Cauchy|first=Augustin-Louis|author-link=Augustin-Louis Cauchy|access-date=October 12, 2019|title=Cours d'Analyse de l'École Royale Polytechnique|year=1821|publisher=Libraires du Roi & de la Bibliothèque du Roi|url=https://books.google.com/books?id=UrT0KsbDmDwC&pg=PA1|page=124}}</ref>
<math display="block">a+ax+ax^2+ax^3+ax^4+ax^5+\cdots=\frac{a}{1-x}.</math>

Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with {{math|1=''a'' = 10 seconds}} and {{math|1=''x'' = 0.01}}.  Achilles does overtake the tortoise; it takes him
:<math>10+0.1+0.001+0.00001+\cdots</math><math>=\frac {10}{1-0.01}= \frac {10}{0.99}=10.10101\ldots\text{ seconds}.</math> 

===Early Indian===
The [[Indian mathematics#Jain mathematics (400 BCE – 200 CE)|Jain mathematical]] text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: [[enumerable]], innumerable, and infinite. Each of these was further subdivided into three orders:<ref>{{cite book|author=Ian Stewart|title=Infinity: a Very Short Introduction|url=https://books.google.com/books?id=iewwDgAAQBAJ&pg=PA117|year=2017|publisher=Oxford University Press|isbn=978-0-19-875523-4|page=117|url-status=live|archive-url=https://web.archive.org/web/20170403200429/https://books.google.com/books?id=iewwDgAAQBAJ&pg=PA117|archive-date=April 3, 2017}}</ref>
* Enumerable: lowest, intermediate, and highest
* Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
* Infinite: nearly infinite, truly infinite, infinitely infinite

===17th century===
In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655, [[John Wallis]] first used the notation <math>\infty</math> for such a number in his ''De sectionibus conicis'',<ref>{{Cite book|url=https://books.google.com/books?id=OQZxHpG2y3UC&q=infinity|title=A History of Mathematical Notations|last=Cajori|first=Florian|publisher=Cosimo, Inc.|year=2007|isbn=9781602066854|volume=1|pages=214|language=en}}</ref> and exploited it in area calculations by dividing the region into [[infinitesimal]] strips of width on the order of <math>\tfrac{1}{\infty}.</math><ref>{{harvnb|Cajori|1993|loc=Sec. 421, Vol. II, p. 44}}</ref> But in ''Arithmetica infinitorum'' (1656),<ref>{{Cite web |title=Arithmetica Infinitorum |url=https://archive.org/details/ArithmeticaInfinitorum/page/n5/mode/2up}}</ref> he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c.", as in  "1, 6, 12, 18, 24, &c."<ref>{{harvnb|Cajori|1993|loc=Sec. 435, Vol. II, p. 58}}</ref>

In 1699, [[Isaac Newton]] wrote about equations with an infinite number of terms in his work ''[[De analysi per aequationes numero terminorum infinitas]]''.<ref>{{cite book |title=Landmark Writings in Western Mathematics 1640-1940 |first1=Ivor |last1=Grattan-Guinness |publisher=Elsevier |year=2005 |isbn=978-0-08-045744-4 |page=62 |url=https://books.google.com/books?id=UdGBy8iLpocC |url-status=live |archive-url=https://web.archive.org/web/20160603085825/https://books.google.com/books?id=UdGBy8iLpocC |archive-date=2016-06-03}} [https://books.google.com/books?id=UdGBy8iLpocC&pg=PA62 Extract of p. 62]</ref>

==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&nbsp;''<math>x</math>'' increases without bound, and <math>x \to -\infty</math> means that&nbsp;''<math>x</math>'' decreases without bound. For example, if <math>f(t)\ge 0</math> for every&nbsp;''<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&nbsp;<math>|x|</math> of&nbsp;''<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]]&nbsp;''<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&nbsp;H is an infinite number in this sense, then H&nbsp;+&nbsp;H&nbsp;=&nbsp;2H and H&nbsp;+&nbsp;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" is 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".<ref>{{cite book |title=Classical and Nonclassical Logics: An Introduction to the Mathematics of Propositions |author1=Eric Schechter |edition=illustrated |publisher=Princeton University Press |year=2005 |isbn=978-0-691-12279-3 |page=118 |url=https://books.google.com/books?id=70W4Q-kzdicC}} [https://books.google.com/books?id=70W4Q-kzdicC&pg=PA118 Extract of page 118]</ref>

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 book 
|title=The Infinite 
|author1=A.W. Moore 
|edition=2nd, revised 
|publisher=Routledge 
|year=2012 
|isbn=978-1-134-91213-1 
|page=xiv 
|url=https://books.google.com/books?id=z-UJhZmQnhAC}} [https://books.google.com/books?id=z-UJhZmQnhAC&pg=PR14 Extract of page xiv]</ref> Certain extended number systems, such as the [[Hyperreal number|hyperreal numbers]], incorporate the ordinary (finite) numbers and infinite numbers of different sizes.<ref>{{cite book |title=Infinity and the Mind: The Science and Philosophy of the Infinite |author1=Rudolf V Rucker |edition=illustrated |publisher=Princeton University Press |year=2019 |isbn=978-0-691-19125-6 |page=85 |url=https://books.google.com/books?id=jv2EDwAAQBAJ}} [https://books.google.com/books?id=jv2EDwAAQBAJ&pg=PA85 Extract of page 85]</ref>

====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.<ref>{{cite book |title=Set Theory |author1=Felix Hausdorff |edition= |publisher=American Mathematical Soc. |year=2021 |isbn=978-1-4704-6494-3 |page=44 |url=https://books.google.com/books?id=TFA_EAAAQBAJ}} [https://books.google.com/books?id=TFA_EAAAQBAJ&pg=PA44 Extract of page 44]</ref>

[[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|&minus;{{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&nbsp;1878, but only became intuitively apparent in&nbsp;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 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]].<ref>{{cite book |title=Fractals, Graphics, and Mathematics Education |author1=Michael Frame |author2=Benoit Mandelbrot |edition=illustrated |publisher=Cambridge University Press |year=2002 |isbn=978-0-88385-169-2 |page=36 |url=https://books.google.com/books?id=Wz7iCaiB2C0C}} [https://books.google.com/books?id=Wz7iCaiB2C0C&pg=PA36 Extract of page 36]</ref>

===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>

==Physics==

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). Concepts of infinite things such as an infinite [[plane wave]] exist, but there are no experimental means to generate them.<ref>[http://www.doriclenses.com/administrer/upload/pdf/NOT_AXI_ENG_070212_doricl97_doricle_kvgwQP.pdf Doric Lenses] {{webarchive|url=https://web.archive.org/web/20130124011604/http://www.doriclenses.com/administrer/upload/pdf/NOT_AXI_ENG_070212_doricl97_doricle_kvgwQP.pdf |date=2013-01-24}} – Application Note – Axicons – 2. Intensity Distribution. Retrieved 7 April 2014.</ref>

===Cosmology===
The first published proposal that the universe is infinite came from Thomas Digges in 1576.<ref>John Gribbin (2009), ''In Search of the Multiverse: Parallel Worlds, Hidden Dimensions, and the Ultimate Quest for the Frontiers of Reality'', {{isbn|978-0-470-61352-8}}. p.&nbsp;88</ref> Eight years later, 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."<ref>{{cite book |title=Alien Life Imagined: Communicating the Science and Culture of Astrobiology |edition=illustrated |first1=Mark |last1=Brake |publisher=Cambridge University Press |year=2013 |isbn=978-0-521-49129-7 |page=63 |url=https://books.google.com/books?id=sWGqzfL0snEC&pg=PA63}}</ref>

[[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 still an open question of [[physical cosmology|cosmology]]. 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 with respect to the Earth's curvature, 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.<ref>{{cite book |title=In Quest of the Universe |edition=illustrated |first1=Theo |last1=Koupelis |first2=Karl F. |last2=Kuhn |publisher=Jones & Bartlett Learning |year=2007 |isbn=978-0-7637-4387-1 |page=553 |url=https://books.google.com/books?id=6rTttN4ZdyoC}} [https://books.google.com/books?id=6rTttN4ZdyoC&pg=PA553 Extract of p. 553]</ref>

The curvature of the universe can be measured through [[multipole moments]] in the spectrum of the [[Cosmic microwave background radiation|cosmic background radiation]]. 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.<ref name="NASA_Shape">{{cite web| title=Will the Universe expand forever?| url=http://map.gsfc.nasa.gov/universe/uni_shape.html| publisher=NASA| date=24 January 2014| access-date=16 March 2015| url-status=live| archive-url=https://web.archive.org/web/20120601032707/http://map.gsfc.nasa.gov/universe/uni_shape.html| archive-date=1 June 2012}}</ref><ref name="Fermi_Flat">{{cite web| title=Our universe is Flat| url=http://www.symmetrymagazine.org/article/april-2015/our-flat-universe?email_issue=725| publisher=FermiLab/SLAC| date=7 April 2015| url-status=live| archive-url=https://web.archive.org/web/20150410200411/http://www.symmetrymagazine.org/article/april-2015/our-flat-universe?email_issue=725| archive-date=10 April 2015}}</ref><ref>{{cite journal|title=Unexpected connections|author=Marcus Y. Yoo|journal=Engineering & Science|volume=LXXIV1|date=2011|page=30}}</ref>

However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other. The topology of such games is [[torus|toroidal]] and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space.<ref>{{cite book|last=Weeks|first=Jeffrey|title=The Shape of Space|year=2001|publisher=CRC Press|isbn=978-0-8247-0709-5|url-access=registration|url=https://archive.org/details/shapeofspace0000week}}</ref>

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> Also, [[cyclic model]]s posit an infinite amount of [[Big Bang]]s, resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.<ref name="Nautilus2014">{{cite news |last1=McKee|first1=Maggie |title=Ingenious: Paul J. Steinhardt – The Princeton physicist on what's wrong with inflation theory and his view of the Big Bang |url=http://nautil.us/issue/17/big-bangs/ingenious-paul-j-steinhardt |access-date=31 March 2017 |work=Nautilus |issue=17 |publisher=NautilusThink Inc. |date=25 September 2014 |ref=Chapter 4}}</ref>

==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&nbsp;A) no such series exists or (form&nbsp;B) were it to exist, the thesis would lack the role (e.g.,&nbsp;of justification) that it is supposed to play."<ref>''Cambridge Dictionary of Philosophy'', Second Edition, p.&nbsp;429</ref>

==Computing==
The [[IEEE&nbsp;floating-point]] standard (IEEE 754) specifies a positive and a negative infinity value (and also [[NaN|indefinite]] values). These are defined as the result of [[arithmetic overflow]], [[division by zero]], and other exceptional operations.<ref>{{Cite web|title=Infinity and NaN (The GNU C Library)|url=https://www.gnu.org/software/libc/manual/html_node/Infinity-and-NaN.html|access-date=2021-03-15|website=www.gnu.org}}</ref>

Some [[programming language]]s, such as [[Java (programming language)|Java]]<ref>{{cite book|last=Gosling|first=James |display-authors=etal |title=The Java Language Specification|publisher=Oracle America, Inc.|location=California|date=27 July 2012|edition=Java SE 7|chapter=4.2.3.|access-date=6 September 2012|chapter-url=http://docs.oracle.com/javase/specs/jls/se7/html/jls-4.html#jls-4.2.3|url-status=live|archive-url=https://web.archive.org/web/20120609071157/http://docs.oracle.com/javase/specs/jls/se7/html/jls-4.html#jls-4.2.3|archive-date=9 June 2012}}</ref> and [[J (programming language)|J]],<ref>
{{cite book
|last= Stokes
|first= Roger
|title= Learning J
|date= July 2012
|chapter= 19.2.1
|chapter-url= http://www.rogerstokes.free-online.co.uk/19.htm#10
|access-date= 6 September 2012
|url-status=dead
|archive-url= https://web.archive.org/web/20120325064205/http://www.rogerstokes.free-online.co.uk/19.htm#10
|archive-date= 25 March 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 have uses as [[sentinel value]]s in [[algorithm]]s involving [[sorting]], [[Search algorithm|searching]], or [[window function|windowing]].{{citation needed|date=April 2017}}

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 may still be accessible and usable as the result of certain operations.{{citation needed|date=April 2017}}

In programming, an [[infinite loop]] is a [[loop (computing)|loop]] whose exit condition is never satisfied, thus executing indefinitely.

==Arts, games, and cognitive sciences==
[[Perspective (graphical)|Perspective]] artwork uses the concept of [[vanishing point]]s, roughly corresponding to mathematical [[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 |title=Mathematics for the Nonmathematician |author1=Morris Kline |edition=illustrated, unabridged, reprinted |publisher=Courier Corporation |year=1985 |isbn=978-0-486-24823-3 |page=229 |url=https://books.google.com/books?id=gXSMukf1aFIC}} [https://books.google.com/books?id=gXSMukf1aFIC&pg=PA229 Extract of page 229, Section 10-7]</ref> Artist [[M.C. Escher]] is specifically known for employing the concept of infinity in his work in this and other ways.<ref>{{cite journal |last1=Schattschneider |first1=Doris| volume=57 |issue=6 |pages=706–718 |journal=Notices of the AMS |title=The Mathematical Side of M. C. Escher |url=https://www.ams.org/notices/201006/rtx100600706p.pdf |date=2010}}</ref>

Variations of [[chess]] played on an unbounded board are called [[infinite chess]].<ref>[http://www.chessvariants.com/boardrules.dir/infinite.html Infinite chess at the Chess Variant Pages] {{webarchive|url=https://web.archive.org/web/20170402082426/http://www.chessvariants.com/boardrules.dir/infinite.html |date=2017-04-02}} An infinite chess scheme.</ref><ref>[https://www.youtube.com/watch?v=PN-I6u-AxMg "Infinite Chess, PBS Infinite Series"]  {{webarchive|url=https://web.archive.org/web/20170407211614/https://www.youtube.com/watch?v=PN-I6u-AxMg |date=2017-04-07}} PBS Infinite Series, with academic sources by J. Hamkins (infinite chess: {{cite arXiv |eprint=1302.4377 |last1=Evans |first1=C.D.A |title=Transfinite game values in infinite chess |author2=Joel David Hamkins |class=math.LO |year=2013}} and {{cite arXiv |eprint=1510.08155 |last1=Evans |first1=C.D.A |title=A position in infinite chess with game value $ω^4$ |author2=Joel David Hamkins |author3=Norman Lewis Perlmutter |class=math.LO |year=2015}}).</ref>

[[Cognitive science|Cognitive scientist]] [[George Lakoff]] considers the concept of infinity in mathematics and the sciences as a metaphor. This perspective is based on the basic metaphor of infinity&nbsp;(BMI), defined as the ever-increasing sequence <1, 2, 3, …>.<ref>{{Cite web |url=http://www.se.rit.edu/~yasmine/assets/papers/Embodied%20math.pdf |title=Review of "Where Mathematics comes from: How the Embodied Mind Brings Mathematics Into Being" By George Lakoff and Rafael E. Nunez|first1=Yasmine Nader|last1=Elglaly|first2=Francis |last2=Quek|work=CHI 2009 |access-date=2021-03-25 |archive-date=2020-02-26 |archive-url=https://web.archive.org/web/20200226004335/http://www.se.rit.edu/~yasmine/assets/papers/Embodied%20math.pdf |url-status=dead}}</ref>

==See also==
{{Div col|colwidth=20em}}
* [[0.999...]]
* [[Absolute infinite]]
* [[Aleph number]]
* [[Ananta (infinite)|Ananta]]
* [[Exponentiation]]
* [[Indeterminate form]]
* [[Names of large numbers]]
* [[Infinite monkey theorem]]
* [[Paradoxes of infinity]]
* [[Supertask]]
* [[Surreal number]]
{{Div col end}}

==References==
{{reflist}}

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{{Refend}}

===Sources===
{{Refbegin}}
*{{cite book | first=Amir D. |last=Aczel | title=The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity | publisher=Pocket Books|place=New York | year=2001 | isbn=978-0-7434-2299-4}}
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* Bell, J.L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.
*{{Citation |last=Cohen |first=Paul |title=The Independence of the Continuum Hypothesis |doi=10.1073/pnas.50.6.1143 |journal=[[Proceedings of the National Academy of Sciences of the United States of America]] |volume= 50|issue=6 |pages=1143–1148 |year=1963 |pmid=16578557 |pmc=221287|bibcode=1963PNAS...50.1143C |doi-access=free}}.
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*{{cite book | author=Eli Maor | title=To Infinity and Beyond | publisher=Princeton University Press | year=1991 | isbn=978-0-691-02511-7 | author-link=Eli Maor | url=https://books.google.com/books?id=lXjF7JnHQoIC&q=To+Infinity+and+beyond}}
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{{Refend}}

==External links==
{{Wiktionary}}
{{Wikibooks|Infinity is not a number}}
{{commons category}}
{{wikiquote}}
* {{cite IEP |url-id=infinite |title=The Infinite}}
* {{In Our Time|Infinity|p0054927|Infinity}}
* ''[http://www.earlham.edu/~peters/writing/infapp.htm A Crash Course in the Mathematics of Infinite Sets] {{Webarchive|url=https://web.archive.org/web/20100227033849/http://www.earlham.edu/~peters/writing/infapp.htm |date=2010-02-27}}'', 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] {{Webarchive|url=https://web.archive.org/web/20091105182928/http://www.earlham.edu/~peters/writing/infinity.htm |date=2009-11-05}}'', 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.
* {{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]]|access-date=2013-04-06|archive-url=https://web.archive.org/web/20171022173525/http://www.numberphile.com/videos/countable_infinity.html|archive-date=2017-10-22|url-status=dead}}
* [https://web.archive.org/web/20040910082530/http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html Hotel Infinity]
* 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'] {{Webarchive|url=https://web.archive.org/web/20060916095918/http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html |date=2006-09-16}}, ''[[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'] {{Webarchive|url=https://web.archive.org/web/20081220145242/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html |date=2008-12-20}}, ''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''.
* [https://www.washingtonpost.com/wp-srv/style/longterm/books/chap1/mysteryaleph.htm The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity]
* [http://dictionary.of-the-infinite.com Dictionary of the Infinite] (compilation of articles about infinity in physics, mathematics, and philosophy)

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[[Category:Concepts in logic]]
[[Category:Infinity| ]]
[[Category:Mathematical objects]]
[[Category:Philosophy of mathematics]]